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\title{Mathematical models of filtration}
% Thesis title (later): Mathematical models of the filtration process
\author{Arkady Wey}
\college{Mansfield College}
\degree{Doctor of Philosophy}
\degreedate{Hilary Term, 2023} 

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% \begin{abstract}
% Particle filtration using porous membranes is a well-studied industrial process with multiple important applications.
% As filtration occurs, particles become captured in the membrane pores, which causes the filters to clog.
% Clogging poses a problem for filter users since clogged membranes must be cleaned or replaced, which is time consuming and expensive.
% Experimental observations of particle--pore interactions cannot, however, be carried out as filtration proceeds, since these are destructive.
% Meanwhile, mathematical approaches that focus on detailed models of these interactions are often too computationally expensive for filter-scale experiments and parameter sweeps. 
% To solve this problem, in this chapter, we develop a multiscale model of filtration.
% The result is a system of deterministic partial differential equations that govern the changes in the particle concentration and the fluid velocity at any filter depth and time.
% The parameters of this system depend on the evolution of the pore network within the filter. 
% The model will be numerically tractable for experiments on the scale of the whole filter, while still containing details of microscale interactions. 
% In this excerpt, we introduce a network model for the microscale and introduce our multiscale modelling methodology. 
% We then examplify the process by homogenising the network model in a one-dimensional setting, before homogenising the $D$-dimensional system.
% \end{abstract}
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\clearpage
\begin{center}
    \thispagestyle{empty}
    % \vspace*{\fill}
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    \textbf{\emph{To my siblings, Edie and Troy}}
    \\
    \vspace{0.1cm}
    \emph{I love you both so much. I owe you everything. \\This is as much your achievement as it is mine.}
    \\
    \vspace{0.5cm}
    \emph{Per angusta ad augusta, nunc ad meliora}
    \vspace*{\stretch{3}}
    % \vspace*{\fill}
\end{center}
\clearpage

% \begin{acknowledgements}
\section*{Acknowledgements}

\emph{My sincere thanks to the people who helped me think about this thesis...} 
\\
\\
To my supervisors, Prof. Chris Breward, Prof. Jon Chapman, and Prof. Ian Griffiths, for pushing me harder than I have ever been pushed before. Our work is something that I am truly proud of. Ian, you have been the best mentor I could hope for. Thank you for your help with my research, but even more so, for your guidance on personal matters. I am so grateful.
\\
\\
To our industrial collaborators, Dr. Uwe Beuscher and Dr. Vasu Venkateshwaran, at W.L. Gore \& Associates, Inc., for introducing us to the physical problem, and keeping our research relevant outside of mathematics. 
% Collaborative work is always more interesting anyway. 
Also, for the various trips to the USA, which have introduced me to so many friends for life.
\\
\\
To Dr. John Wilkinson and Dr. Edwina Yeo, for proofreading huge portions of this thesis. 
Thank you for helping to turn my drafts into something submittable.
\\
\\
To Tom Babb, Dr. Matthew Shirley, Dr. Kristian Kiradjiev, and Dr. Robert Timms, for their assistance with the ideas underpinning this work on more occasions than I can count. So much of this project has been a collaboration. I am so grateful to be able to call my colleagues my friends. 
\\
\\
To Sophie Hurden and Amanda Guthland-Elmore. I am so sorry that you were unfortunate enough to be my InFoMM administrators. Sophie, I can only imagine how annoying it is dealing with my finances. Thank you. Amanda, thank you so much for taking me under your wing from day one.
\\
\\
To my fellow OCIAM graduates and friends, Dr. Amy K, Andrew A, Anna B, Brady M, Danny B, Dr. Eddy Y, Dr. Ellen L, Dr. Eva A, Georgia B, George B, Dr. Helen S, James H, Joe F, Joe R, Matt C, Dr. Matt S, Markus D, Dr. Michael N, Dr. Meredith E, Nick R, Ollie B, Phil W, Sophie A, Simon F, Tom B, and Torin F, for the mathematical and moral support.
% and the other original members of the SC. 
You will all do amazing things. I am so proud of us. Thanks for all the Najaars trips.
\\
\\
To my cohort, 
Dr. Alex P,
Dr. Christoph H,
Dr. Huining Y,
Dr. Joel D,
Dr. John F,
Dr. Matt S,
Dr. Meredith E,
Dr. Nick B,
Dr. Ollie B,
Dr. Yu T,
for surviving first year.
\\
\\
This research was funded by the Centre for Doctoral Training in Industrially Focused Mathematical Modelling (InFoMM) in the Mathematical Institute at University of Oxford. I would like to thank the Engineering and Physical Sciences Research Council (EPSRC) for my unique opportunity.
% \end{acknowledgements}
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\noindent
\emph{I would also like to thank the people who helped to stop me thinking about this thesis, for which I am at least as grateful...} 
\\
\\
To Edie, Troy, and Ellie, 
% and Adam, Morgan, and Spike, 
for picking me up whenever I'm down. And for the fun and the festivals, to MorgAn and Spike too. I could not have done it without you.
% I will never forget all of our nights... `have you seen the stars tonight?'
\\
\\
To The Cresidents.
To Johnny Corrigan, for the COVID Sainsbury's runs. Thank you for always being right and for doing the bills. 
To George, because you're the only (other) person I know stupid enough to start a Breaking Bad marathon two weeks before submission. There's no one else I'd rather submit with (apart from maybe \emph{our} sibli...). 
To Tom One, for the kebabs, and for being an all-round polite guest. 
To Sam, because you're the only person I know stupid enough to start a marathon. 
To Tom Two, for alerting us all to the dangers of PFAS. 
To Hattie and Liz, but mostly Adam, for finally getting us over the line on Thursday nights at the A and G. 
To Elise, for the extra bike by the washing machine. 
% To Christoph, for the miscellaneous workout equipment. 
And, arguably most importantly, to Naymah, for the towel. 
\\
\\
To The SDM(AI)C, Issy, Joe, Kris, Tom, and our silent participant, Seppe. 
Thanks for the `office chat', Thursday NT-Coffees, and for generally cheering me up.
\\
\\
To the Mansfield lot who made the right decision by leaving in first year, Abi, Conor, and Lottie. Thanks for the wines and the cheeses, for the balls and the bops. Hanks has/will never been/be the same.
\\
\\
To Adam, Conor, Clarence, Lynn, and Tyler, for going to Hel(ston) and back with me.
\\
\\
To Julius and Pav, for inspiring me, teaching me thousands of new things about everything, and for the millions of calls in the COVID months.
\\
\\
To Benson, Conor, and all the other folks down at the CF. We had a few rough quarters, but I'm optimistic about the upcoming aforementioned off-site visit.
\\
\\
To the MtM boys, Aled, Dan, Eli, Nitin, and Tyler, for the holidays, the football, and the Feefs.
% Thanks for backing me from the start.
\\
\\
To The InBendtweeners, 
Mike, Kevin, Vinny, and Paige (and Morty, of course), for the amazing times away. I still root for The Timbers.
% (at least when I root).
\\
\\
Lastly, thanks to my mum and dad. My experiences growing up have taught me to try anything and everything, to expect more, and to never give up.
\\
\\
% I'm writing my acknowledgements at four in the morning, which is why they’re such a mess. Since that’s when a large proportion of this thesis was written... 
\emph{Let's hope the remainder of this thesis is a better read than the last two pages...}

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\begin{abstract}
Filtration is the removal of particles from a fluid suspension 
by forcing it, either using a constant flow rate or pressure drop, 
through a porous material, which causes particles to become captured in the pores. 
Filtration using membranes is a well-studied industrial process with multiple important applications.
% As particle capture and pore clogging occurs, 
% As filtration occurs, particles become captured in the membrane pores, which causes the filters to clog.
% Clogging poses a problem for filter users since clogged membranes must be cleaned or replaced, which is time consuming and expensive.
Experimental observations of particle--pore interactions to gain insight into the filtration process 
are difficult and sometimes destructive.
Some existing mathematical approaches either ignore important microscale information, such as the particle- and pore-size distribution 
or the connectivity of the pores, or are too computationally expensive for filter scale experiments.
In this thesis we motivate, derive, solve, and discuss three novel models for particle filtration: 
(I)
A size-structured model. In this work, the particle and pore sizes are treated as independent variables, 
which gives rise to a system of partial integro--differential equations within which the filter is treated as a macroscale 
continuum. 
We demonstrate that consideration of size structure allows for greater understanding of the filtration process; 
(II) 
A network model. Here, the filter is treated as a network composed of pores, which are modelled as edges, 
and junctions between pores, which are modelled as nodes. 
The connectivity of the network changes as particles deposit in pores. 
This gives rise to a system of ordinary 
algebraic--differential equations for variables defined at each pore and junction of the network. 
We show that this consideration of the structure of the microscale leads to physically intuitive 
behaviour of macroscale quantities, such as the flow rate and pressure;
(III)
A multiscale model. In this work, we extend the method of network homogenisation to 
apply to networks whose connectivity varies in time. 
We use this to derive a system of partial differential equations 
defined on the continuous macroscale geometry that is coupled, via its parameters, with variables that are 
defined on the discrete network that defines the microscale. 
We show that solutions of this multiscale model match those of the network model from (II) in a particular asymptotic limit, 
even though they are obtained at considerably reduced computational cost. 
These three models increase our understanding of the filtration process.
The computational tractability of the models means that they are suitable candidates for filtration process optimisation tasks
that involve repeated solution
% , such as parameter sweeps,
in future work.

% To solve this problem, in this chapter, we develop a multiscale model of filtration.
% The result is a system of deterministic partial differential equations that govern the changes in the particle concentration and the fluid velocity at any filter depth and time.
% The parameters of this system depend on the evolution of the pore network within the filter. 
% The model will be numerically tractable for experiments on the scale of the whole filter, while still containing details of microscale interactions. 
% In this excerpt, we introduce a network model for the microscale and introduce our multiscale modelling methodology. 
% We then examplify the process by homogenising the network model in a one-dimensional setting, before homogenising the $D$-dimensional system.
\end{abstract}

% \newpage{}
% \section*{Acknowledgements}




%% Preview body
\tableofcontents{}
\newpage{}

\setcounter{page}{1}
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% \part{Size-structured model}
% \chapter{Introduction}
% \chapter{Model}
% \chapter{Results}
% \chapter{Conclusions}

% \part{Network model}
% \chapter{Introduction}
% Chapter talking through the network model we settled upon before we realised it was difficult to deal with in network homogenisation.
% \begin{enumerate}
%     \item Derive a physically relevant adhesivity function by considering flow in a single channel.
%     \item Show how to derive a network model, and derive a network model where we track concentration on nodes and edges.
%     \item Reduce this to the model that we will consider in network homogenisation chapter, and explain what adhesivity functions are permitted and which ones are problematic in network homogenisation.
% \end{enumerate}
% \chapter{Model}
% \section{Derivation of network models}
% \section{Pipe model for deposition model}
% \chapter{Results}
% \chapter{Conclusions}

% \part{Multiscale model}

% \chapter{Introduction}
% [Talk through key papers and explain the computational advantages of network homogenisation]
\chapter{Introduction}
\label{chap:Introduction}
We begin, in this chapter, by introducing the physical process of filtration, before 
reviewing the standard approaches to its mathematical modelling. 
We use these to motivate three novel models for filtration that will be derived, solved, and discussed in this thesis, and briefly outline each of these.

\section{Filtration}
First, in this section, we define our use of the term filtration, and 
motivate this thesis by explaining the social and economic importance of the process. 
We briefly summarise the filtration market and industry, and discuss the filtration processes
for which this work is most relevant.
Detailed explanation of filtration is beyond the scope of this study, so we direct the reader to a useful guide to filters and filtration by Sutherland \emph{et al.}, \cite{sutherland2011filters}, within which further explanation to most of the ideas discussed in this section can be found.

\subsection{Definition}
In this thesis, we define filtration to be the separation of solid particles from a fluid in which they are suspended, using a filter, which
is a semi-permeable material 
% membrane 
that consists of a solid substance that is punctured with holes, which we call pores. 
In a paradigmatic filtration process, a diagram of which is shown in Figure \ref{fig:intro_filtration}, the feed is driven into the filter at the inlet, usually either by a constant pressure drop or flow rate. 
Fluid traverses the pores, and, as it does so, particles suspended within it are captured in pores in the filter.
As a result, the filtrate, which is the fluid exiting the filter outlet, contains a lower concentration of particles than the feed.

Uses of filtration are vast and diverse, but can broadly be separated into applications that focus on:
\begin{itemize}
    \item The removal of unwanted particles from a desired fluid. In this case, the particles retained in the filter are a waste product, 
    and the filtrate is an end product;
    % or the waste product of another process that needed cleaning prior to its disposal;
    \item The recovery of desired particles 
    %(often folliwng prioirt crystallisation of precipitation step) 
    from an unwanted fluid. 
    In this case, the particles retained in the filter are the desired product, and the filtrate is waste.
\end{itemize}
\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/introduction/filtration.pdf}
        \centering
        \caption{Diagram showing a paradigmatic 
        filtration process.
        }
        \label{fig:intro_filtration}
\end{figure}


\subsection{Importance}
Filtration is one of the most prevalent and important physical processes, 
and there are scarcely any human activities, domestic or commercial, that are not affected by it \cite{matteson1998filtration}. 
From tea bags \cite{yurtsever2021nonwoven} and coffee papers and presses \cite{eulitz2009effect}, used to remove leaves and granules from water, to the drinking-water purifiers used to separate nitrates and dissolved solids in the kitchen \cite{wu2021critical, ray2021towards}, to
the dust collectors used to keep dirt particles entering vacuum cleaner motors  \cite{wang2001electrostatic}, filters are common in the household.
From the enormous processors in wastewater-treatment plants \cite{hube2020direct, adin1989particle}, 
which use physical processes like sieving for size separation of pebbles on the centimetre scale,
to the delicate nanofibres that use chemical properties like electrostatic adherence for bacteria and virus control in laboratories on the micrometre scale \cite{ikner2012concentration, cashdollar2013methods}, filters are equally common-place within industry.
Filtration is also an important sub-step in many industrial processes where purification is not the primary focus, 
from the removal of debris from hydraulic fluids prior to use in machinery \cite{frith1993control, machal2013design}, as a pre-process step, 
to the decontamination of vehicle and machinery exhaust-fumes \cite{jaworek2007modern, guan2015review}, as a post-process step.

In recent years, concerns over the global environment and the influence of humans upon it have grown. 
Scientists, politicians, and economists, are increasingly worried about three areas. 
\begin{itemize}
    \item Climate change \cite{ipcc2018}: There is more and more evidence \cite{manabe1967thermal,held2006robust} to suggest that 
    long-term shifts in temperature and weather patterns are due to increasingly polluted air \cite{jacobson2009review}, in part 
    caused by growing energy consumption \cite{barker1995global}.
    There is a need for solutions \cite{kumar2015direct, chen2018graphene} that minimise pollutant concentration in air and waste gasses emitted 
    by industrial processes, and that minimise the energy consumed by these processes.
    % Industrial processes and 
    % Anotehr sentence here explaining why pollutants in gas are harming the environment.
    % And energy manufacture are anotehr problem too, so industry needs to use less energy.
    \item Starvation and clean-water shortage \cite{taylor2013ground, van2021global}: Despite the modest growth of the world economy \cite{imf2022}, 
    food and drink starvation has increased in recent years \cite{hinrichsen2002coming}. 
    One reason is that pollutants from industry, such as chemicals and bacteria \cite{malmqvist2002threats, kant2011textile, wu2011dosed}, find their way into waterways, 
    and cause contamination that prevents water from being safe for drinking, cooking, and cleaning. 
    Solutions \cite{macedonio2012efficient} that focus on decreasing the levels of harmful contaminants 
    in water and other liquids are quickly becoming essential.
    % Anotehr sentence here about how contamination of water leaads to less food and drink.
    \item Spread of pathogens \cite{brainard2020facemasks}: 
    The COVID-19 pandemic, which has been responsible for millions of deaths since 2019, has highlighted the importance of the filtration process in medicine, healthcare, and in society more generally. 
    As the pandemic evolved, filtration processes and systems for mechanical ventilation, 
    such as those used in intensive care and anaesthesia settings \cite{gouel2020preliminary}, 
    became some of the most essential mechanisms for the protection of lives. 
    Filters contained in face masks to shield the spread of infectious pathogens 
    have become household items \cite{kwong2021review}.
\end{itemize}
Air and water contamination, energy manufacture and consumption, and bacteria- and virus-spread prevention, 
then, are some of the biggest challenges humanity currently faces.
Problems in these areas are largely due to human activity \cite{ipcc2014ipcc, owa2013water, unicef2022state, lang2015food, morens2020emerging}, and there is an urgent need to change the way life is conducted to help to eradicate these problems.

It is difficult to overstate the role that filtration, as a process and an industry, 
has to play \cite{sutherland2011filters}.
As awareness increases, demand for filters with higher retention, that last longer, and are more energy efficient, increases.
Laws \cite{necr2018} around minimisation of factory waste mean that filters need higher levels of particle retention, 
and to be able to process higher volumes of fluid before their disposal.
Changes to policy \cite{mitchell2004renewable} around energy-usage minimisation mean that filters need to be more energy efficient, 
so that this increased retention and high throughput occur in lower pressure drop or flow rate situations.

\subsection{Utilisation}
The uses of filtration are as vast as they are diverse. 
In 2007, it was estimated that the total market for filtration equipment was worth around $\$38\,\text{bn}$ (USD), 
and was expanding at a rate comfortably in excess of that of the global economy  \cite{sutherland2011filters}.
As we see in Table \ref{tab:market}, 
the market is dominated by solutions for domestic and institutional users, and applications in transport, 
which together account for almost $34\%$ of the market.
Water treatment, the production of bulk chemicals, and 
the manufacture of food and drinks account for another $28\%$, 
and filters for use in laboratories, hospitals, wider healthcare, and the pharmaceutical industry 
constitute a further $11\%$. 
Power generation and other applications make up the remaining $27\%$. 
\begin{table}[!htb]
\centering
\begin{tabular}{|c|c|} 
\hline
 End-user sector & Market share ($\%$)  \\ 
 \hline
 Domestic and institutional & 17.9  \\ 
 Transport & 15.9  \\
 Water treatment & 10.3 \\
 Bulk chemicals & 9.9 \\
 Food and drink & 7.5 \\
 Pharmaceuticals and biochemicals & 6.5 \\
 Power generation & 6.3 \\
 Pulp and paper & 5.1 \\
 Medicine and healthcare & 4.8 \\
 Electricals & 3.5 \\
 Other & 12.3 \\
 \hline
\end{tabular}
    \caption{Table showing the proportions of the filtration market attributed to different end users \cite{sutherland2011filters}.}
    \label{tab:market}
\end{table}

Segregating the market by the phase of fluid that is processed, 
important applications in gas filtration include
the cleaning of air entering living and work spaces \cite{vijayan2015enhancing}, 
protection from exhaust fumes emitted by machinery engines \cite{guan2015review},
and respirators and breathing apparatus \cite{brainard2020facemasks}.

Applications in liquid filtration include
the production of potable water from ground and surface sources \cite{gadgil1998drinking},
refinement of ultrapure water for laboratory use \cite{ma2011ultra},
treatment of waste waters to prepare them for discharge \cite{hube2020direct},
preparation of boiler feed water and the recycling of boiler condensate \cite{koo2011recycling},
processing of transformer oils \cite{salvi2017study},
cleaning of tool coolants and cutting fluids \cite{hudgens1988filtration},
and the cleaning of hydraulic-system fluids \cite{frith1993control}. 
% and purification of all engine liquids, such as fuels, lubricants and coolants [?].
Altogether, filtration is an increasingly important physical process with a diverse set of uses.

\subsection{Particles and the feed}

% Contanimants are presnt in all fluids, that occur naturally or products of proceesses. 
% Where the level of contaminaiton is significant, i,.e too grt fo rhe fluid to be consuiderd, or for it to be subsequently procsses., 
% theere is a need (sometimes obligatory, legally or morally) 
% to get concentration of contaminant below a certain level.
Although some filtration regimes are used to control non-solid particles, such as  
liquid droplets that are well mixed or dissolved in the carrying fluid, 
we will focus our attention on solid particles suspended in a liquid or gas.

The most common way of partitioning the plethora of different solid particles is by their size.
The mean particle size, and the distribution of sizes about this mean, has a major influence on the type of 
filter that is chosen for a retention-, throughput-, and energy-efficient solution.
Particles in feeds range from pebbles that are at least centimetres in diameter 
to tiny dust particles and large molecules on the sub-microscale (see Figure \ref{fig:intro_particle-sizes}).
\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/introduction/particle-sizes.png}
        \centering
        \caption{Diagram showing some types of particles commonly found in air, and their sizes \cite{sutherland2011filters}.
        }
        \label{fig:intro_particle-sizes}
\end{figure}
Particles at the smaller end of this scale include viruses, tobacco smoke, and 
almost all combustion nuclei, 
while those at the larger end include stone particles, such as sand and limestone.
The majority of filtration processes are concerned with the removal of particles 
that are fine enough to have stayed suspended in liquids or atmospheric air for long periods of time,
and are less than $10\mu\text{m}$, say. 
We will focus this study on these, 
since, as we will discuss shortly, we are most interested in filters that contain porous membranes with pore sizes distributed 
over the microscale.


Almost any fluid that contains a particle suspension could be 
considered as a candidate feed.
In this thesis, we focus on fluids, like water, which are well modelled as a continuum with a given viscosity
by traditional rules that govern the macroscopic and microscopic flow in porous media: 
Darcy's law and the Hagen--Poiseuille equation, respectively. 
That is, we are interested in fluids with a Reynolds number, Re, in the region $10^{-1}\le\text{Re}\le10^3$, say, 
which are not sufficiently viscous that edge effects cause substantial changes to the effective flow field, 
nor so inviscid that turbulence causes significant changes on the microscale. 

\subsection{Pores and the filter}
The variety of different filtration devices is enormous, and a full review is far beyond the scope of this introduction.
It suffices to say that their properties, to a large extent, depend on the particles they are used for. 
In general, filters are characterised by the media that they are composed of 
and the arrangement of these media.
Some common filtration media are: absorbent and adsorbent biological materials, and some 
papers and fabrics, for generally smaller particles; and woven wires and plate meshes for larger particles. 
Different configurations lead to different types of filter, 
such as strainers, bags, pleated membranes,
screens, 
tipping pans, 
and
rotary drums, 
% horizontal bed filters, 
% and centrifuges, 
to name just a few (see Figure \ref{fig:intro_filters}).
\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/introduction/filters.pdf}
        \centering
        \caption{Photos showing (left to right)
        strainers, filter bags with their cartridges, and pleated membranes in their cartridges \cite{sutherland2011filters}.
        }
        \label{fig:intro_filters}
\end{figure}

In this thesis, we focus on membrane filters, which are thin, flexible, and semi-permeable
sheets composed of polymers, modified cellulose products,
or occasionally inorganic products, such as ceramics. 
Membrane filters are increasingly used to filter microscale and sub-microscale particles 
in many industrial sectors, including the chemical, petrochemical, and food and beverage processing industries, 
in pharmaceuticals and biotechnology, and especially in the treatment of water \cite{sutherland2011filters}. 
Some examples of membrane technologies and their main applications are shown in Figure \ref{fig:intro_membrane-filters}.
\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/introduction/membrane-filters.png}
        \centering
        \caption{Diagram showing some membranes used for filtration, and their typical pore sizes \cite{sutherland2011filters}.
        }
        \label{fig:intro_membrane-filters}
\end{figure}

% In the 1960s, 
Use of membrane filters began with so-called `non-porous membranes', which are extremely thin films with pores of sizes around $0.1$nm, used for the separation of water from other molecules via reverse osmosis. 
One disadvantage is that these sometimes operate at high transmembrane pressures ($3\times10^6$\,--\,$6\times10^6$ Pa) to obtain appropriate flow rates (at least $1\times10^{-3}$ms${^{-1}}$, say).
In recent years, polymers with slightly larger mean pore sizes have been engineered, 
giving rise to the processes of 
nanofiltration, which needs correspondingly lower transmembrane pressures ($2\times10^6$\,--\,$4\times10^6$ Pa), 
ultrafiltration, which is used for the separation of large organic molecules, and colloidal solids, at pressures in the region of $5\times10^5$\,--\,$1\times10^6$ Pa, and microfiltration, operating at only a few times ambient pressure ($3\times10^5$ Pa), while achieving fluxes around $1\times10^{-3}\text{m}^3\text{s}^{-1}$. 
The lower pressure drops required for nanofiltration, ultrafiltration, and especially microfiltration, enable separation with relatively low energy demands while still providing a filtration capability that is suitable for many tasks.

We are particularly concerned with microfiltration membranes manufactured by W. L. Gore \& Associates, Inc (Gore), which are composed of a material called Expanded Polytetrafluoroethylene (ePTFE) (see \cite{ebnesajjad2016expanded} for a thorough description). 
Properties of ePTFE membranes, such as pore size distribution and average, depend, to a large extent, on the method used to manufacture them \cite{ebnesajjad2017properties}. Several patents exist, and broadly involve the
`expansion' of PTFE, that is, the calendaring, drying, longitudinal stretching, and sintering of an extruded film of the original `un-expanded' fluoropolymer (an organic molecule containing Fluorine atoms, PTFE in this case) \cite{ebnesajjad2017introduction}.
On the macroscale (see Figure \ref{fig:intro_ePTFE} (left)), ePTFE membranes are sheets that have a much larger length and width (around 1m, say) than depth (usually around 1mm, but sometimes as large as 1cm), which refers here to the direction of induced flow in a filtration process).
On the microscale (see Figure \ref{fig:intro_ePTFE} (right)), ePTFE membranes are composed of a mesh of millions of tiny hair-like fibres. This gives rise to a network of hollow tunnels between the fibres. These represent pores in the context of filtration. During filtration, particle-laden fluid is driven through the pore space between fibres, and some particles are captured on `pore walls' (i.e., the fibres between pores). Repeated capture leads to the clogging of individual pores 
and thus, eventually, to the clogging of the filter. 
Typically, ePTFE membranes contain pores of sizes in the range $0.02\mu$m to $40\mu$m and are used in filters designed to filter suspensions containing particles of sizes in a similar range.
In the most extreme case (i.e., where pores are around $0.01\mu$m and the membrane is around $1$cm thick), 
the ePTFE membranes contain around 1,000,000 pores in the depth direction. 
\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/introduction/ePTFE.pdf}
        \centering
        \caption{Diagram showing macroscale and microscale photographs of ePTFE \cite{gore}.
        }
        \label{fig:intro_ePTFE}
\end{figure}
When ePTFE membranes clog, this plastic must be cleaned or replaced, which is time-consuming and expensive, and has negative environmental impacts. 
Industrialists at Gore are interested in understanding the relationship between 
microscale and macroscale phenomena
that lead to clogging, so that they can improve filtration performance in terms of
retention, throughput, and energy consumption. 

% The various processes in which membranes are now used began with reverse
% osmosis, 
% which is a diffusion process in which water molecules move
% through the non-porous membrane, leaving the ions and other impurities behind.
% This is a high pressure process, needing 30–60 bar. 

\subsection{Particle capture and pore clogging}
% We focus on seapartion due to size and physica and chemcial difference (such as charge) betwen the partiles and the proes. 
% This is distinct from other separation methods that fucsoi on other properties such as density. For example, sedinahtiona, whre separation relies on the density difrference beteen te suspended paticle and the suspending fluid. It id the force of gravity wirking oin this density difference or the centrifugal force (if the fitler is a cenrifye) tghat cuases separation via sedimantion.
The mechanisms by which particle capture and pore clogging occur are not completely understood. 
In modelling literature (see \cite{griffiths2014combined}, for example), 
particle capture and pore clogging are assumed to occur by any of four mechanisms, which 
are illustrated in Figure \ref{fig:intro_mechanisms}. 
These are: 
\begin{itemize}
    \item Blocking, where particles pass through larger pores but get stuck
    when they encounter small gaps;
    \item Deposition, where particles pass into larger pores
    and adhere to pore walls, due to electrostatic forces or chemical binding, causing
    pore constriction, for example;
    \item Partial surface blocking, where particles land on the surface
    of the membrane and partially cover a pore entrance;
    \item Caking, where particles
    build up on the surface of the membrane and form a porous layer through which the
    feed must permeate. This layer may evolve to have a distinct permeability from that
    of the actual membrane \cite{ruth1935studies}.
\end{itemize}
\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/introduction/mechanisms.pdf}
        \centering
        \caption{Diagram showing four particle capture and pore clogging mechanisms.
        }
        \label{fig:intro_mechanisms}
\end{figure}
The relative prevalence of these clogging mechanisms
depends on the membrane structure, feed composition, and the process conditions, and plays a part in determining the behaviour of filter properties, such as the flow rate \cite{griffiths2014combined}.
In this thesis, we will focus our attention on blocking and deposition (as in \cite{sanaei2016flow}, for example).

In summary, filtration is a socially and economically important 
process. 
We will focus this study on modelling microfiltration of particles of sizes around $10\mu$m using pores of similar sizes in ePTFE, and pressure drops of around $3\times10^5$ Pa that achieve fluxes of around $1\times10^{-3}\text{m}^3\text{s}^{-1}$, say. 
% nanofiltration, and ultrafiltration.
% In these processes, modest pressure drops and flow rates are used to drive fluid containing 
% microscale and sub-microscale particles through thin semipermeable membranes like ePTFE, which consist of hair-like plastic fibres that are woven randomly to form pores of similar sizes as the particles. 
We will assume that, as fluid traverses the filter, particles are captured in pores via  blocking and deposition, which lead
changes to the effective properties of the filter and eventually clogging. 
% We are interested most in capture that occurs due to the relative sizes of particles and pores, or due to adherence between them.

\section{Existing models for filtration}
Industrialists and researchers have attempted to track and understand particle--pore interactions, and their effects, using many methods. 
One approach is to experimentally 
monitor particle build-up inside membranes. 
This is achieved through a variety 
of techniques, including: microscopy; laser censoring; reflectometry; spectroscopy; and 
3-D imaging, to name just a few \cite{chen2004situ, li1998direct, altmann1997particle, hamachi1999experimental, mairal1999real, cheng2011new, chilcott2002electrical, gaedt2002electrical, abichou2004network, 
bondy2010analysis, 
ferrando2005approach}. 
Experimental methods (see \cite{chan2004characterization} for a review) 
provide useful data on almost every operational aspect of the filtration process.
However, they are often difficult, can take a long time, and are usually expensive, due to the need for repetition. 
Some experiments also involve destruction, and subsequent disposal, of filtration membranes. 
This is problematic, since it leads to financial expense and has negative impacts on the environment
(due to the lack of options for the safe disposal of fluoropolymers, and issues surrounding their recycling) \cite{lohmann2020fluoropolymers}, and can be prohibitive, since industrialists are interested in monitoring particle--pore interactions as filtration proceeds.

As early as 1935 \cite{hermans1935kenntnis}, mathematical modelling of filtration was used as a cheaper and faster alternative to experimental techniques. 
Since then, a vast amount of research has been conducted, which can broadly be segregated into three categories: macroscale models; microscale models; and multiscale models. 

% Experimental methods are expensive because the materials are expensive and because they are destructive blah blah...
% Mathematical and computational methods offer a cheaper and often faster alternative to direct experimental monitoring. 
% Such methods usually fall into one of three categories: 
% effective continuum descriptions defined on the macroscale geometry, which describe the overall properties of a
% filter medium, such as the flow rate given an applied pressure difference and the concentration of particles in the outlet given an inlet concentration; 
% continuum or particle-based
% simulations defined on the microscale geometry, which track individual particles and their interactions with the filter; 
% multiscale models, where variables defined on the macroscale geometry are coupled with those defined on the microscale... blah.

\subsection{Macroscale models}
Macroscale models are descriptions of filter properties that mostly ignore the physical and chemical characteristics of the suspension and the membrane. 
In simple cases, 
the effects of microscale quantities, such as particle and pore size, 
are implicitly included in the value of a `filter coefficient' that is predicted  beforehand, 
as opposed to being calculated as part of the solution. 
One early example is that of Iwasaki \emph{et al.} \cite{iwasaki1937some}, 
who pose a simple model for the exponential decay of particle concentration in the flow 
as a function of filter depth. 
The filter coefficient plays the role of the decay rate, which was later fitted experimentally by Ison and Ives \cite{ison1969removal}.
More complex macroscale models (see \cite{jegatheesan2005deep} for a review) 
usually take the form of systems of differential equations that describe the evolution of the fluid velocity, pressure, and particle concentration, as well as other variables, on a continuous domain. 
For example, Darcy's law, coupled with a continuity equation for the fluid velocity, is often used to describe the proportionality of the flow rate to the pressure gradient by the permeability. 
This is usually coupled to an
advection equation that describes the transport of particles through the filter, often with additional diffusion or reaction terms that account for the global effects of particle--pore interactions. 
The permeability, as well as the diffusivity and reactivity (if they are included), 
are often viewed as other fitting parameters, to be improved by empirical measurement.
These models are generally relatively computationally cheap to solve, but sometimes do not quantitatively agree with observations because they fail to model important microscale mechanisms, such as changes in the pore size distribution due to particle capture.
% They are sometimes improved by using empirical measurementns for the parameters.
Jegatheesan \emph{et al.} provide a useful overview of some prominent research in \cite{jegatheesan2005deep}, and we discuss a few models that are particularly relevant to this thesis in Chapter \ref{chap:ssm_Introduction to the size-structured model}. 

% Mention traditional models: Darcy, Kozeny, O'melia and Ali, Polyakov. 
% Mention more recent advances: Cummings, Sanaei, Kiradjiev,



\subsection{Microscale models}
Microscale models, on the other hand,
seek to resolve the mechanical and chemical interactions 
that cause particles to adhere to pores and decrease their fluid conductance.
Prominent examples include computational fluid dynamics (CFD) simulations, 
which are used to solve highly complex flow equations through exact pore space geometry using finite element methods \cite{rahimi2009cfd}, 
and agent-based models, 
which track the position and behaviour 
of each particle or pore \cite{marrufo2018prediction}. 
More recently, genetic programming techniques \cite{fouladitajar2013membrane} and artificial neural network solvers (ANN) \cite{liu2008evaluation}
have been applied to complicated membrane systems. 
Pore network models (PNMs) \cite{xiong2016review} are another prominent example, in which the microscale pore space is modelled as a network of nodes and edges whose connectivity and other statistical properties are representative of samples of real porous media. 
These usually consist of dynamical systems of algebraic or differential equations that describe the transport of particles through the network, and changes to  the network structure due to particle interactions with the nodes and edges. 
Although microscale models can be highly accurate, they are usually computationally expensive to solve. 
This can render consideration of the entire filter geometry infeasible.
We discuss network models in more detail in Chapter \ref{chap:nm_Introduction to the network model}, and \cite{xiong2016review} provides a good overview of models that employ this framework.
% Discuss non-network approaches:
% Discuss network descriptions: porometry models from Ochateirn, then focus on Griffiths

\subsection{Multiscale models}
Multiscale models occupy the middle-ground  between the two other approaches. 
Like macroscale models, 
these typically consist of differential equations, such as Darcy's law and advection--diffusion--reaction equations, which are defined on a continuum macroscale. 
Unlike macroscale models, 
however, 
the parameters of these equations
are solved as part of the systems, 
and hold information about microscale interactions and geometry. 
Multiscale models provide computational feasibility near that of macroscale models, 
but are generally more accurate, due to explicit microscale dependence. 
Relevant recent work on multiscale models for porous media includes that of Bruna, Chapman, Dalwadi, Griffiths, Please, and Printsypar \cite{bruna2015diffusion, dalwadi2015understanding, dalwadi2016multiscale, 
chapman2017effective, chapman2021homogenization, rooney2021homogenisation, printsypar2019influence, iliev2014multiscale}, among others.
We discuss these at greater length in Chapter \ref{chap:mm_Introduction to the multiscale model}.
% , but [ref] provides 
% a good introduction.
% Mention continuum approaches: Dalwadi, Kiradjiev
% Mention network approaches: Chapman, Shipley, Kropat, Husseini



\section{Three novel models for filtration}
% We now set out a plan for the remainder of this thesis.
In this chapter, we have introduced the physical process of filtration, and summarised the existing approaches to mathematically modelling it.
In the remaining chapters of this thesis, we will motivate, derive, solve, and discuss three novel models 
for filtration.

\subsection{Macroscale model: A size-structured model for filtration}
In Part \ref{part:A size-structured model for filtration}, we build and solve a continuum model for filtration on the macroscale geometry of the filter, which is novel because particle and pore sizes are treated as independent variables upon which parameters of the system depend.

To this end, in Chapter \ref{chap:ssm_Introduction to the size-structured model}, we highlight the importance 
 of the choice and continuous monitoring of the pore-size distribution as a means of controlling the sizes of particles in the filter and filtrate, 
 and thus as a means of filtration-process control and optimisation. 
 We discuss the need for a computationally tractable mathematical model in which particle and pore sizes are fully accounted for. 

In Chapter \ref{chap:ssm_Derivation of the size-structured model}, we derive a size-structured model for filtration 
 by using ideas common in age-structured population models, where age is treated as an independent variable 
 similar to space in more traditional population models.
 We arrive at a system of continuous partial integro--differential equations that govern the temporal and spatial evolution of the size distributions of particles and pores in the filter. 
 
In Chapter \ref{chap:ssm_Solutions of the size-structured model}, we solve this size-structured model
in the cases where particles are mono-dispersed and pores are either mono-dispersed, bi-dispersed, or log-normally poly-dispersed.
 We find that our choice of modelling framework leads to equations that are soluble via explicit and asymptotic 
 methods in certain physically relevant parameter regimes, and are solved via computationally tractable
 classical numerical methods in other regimes of interest.
We find that knowledge of the evolution of the particle- and pore-size distributions leads to increased 
understanding of the behaviour of quantities of interest, such as the flow rate and pressure drop.
 
Finally, in Chapter \ref{chap:ssm_Conclusions for the size-structured model}, we conclude our discussion of the size-structured model 
by noting that our solutions agree with physical intuition, before highlighting the need for analysis of other industrially relevant operating regimes, 
such as poly-dispersed feeds. 
The computational feasibility of this model leads to its potential use in filtration-process optimisation tasks in future work.

\subsection{Microscale model: A network model for filtration}
In Part \ref{part:A network model for filtration}, we build and solve a model for filtration 
in which the microscale structure of the filter is viewed as a network of pores, which we model as edges, and junctions 
at pore ends, which we model as nodes. 

To this end, in Chapter \ref{chap:nm_Introduction to the network model}, 
we note that recent developments in computational image analysis 
have led to algorithms that extract networks from high resolution characterisations of the microscale of porous membranes. 
In the resulting networks, void space is segregated into cylindrical edges, which are viewed as pores that carry flow, 
and spherical nodes, which are junctions at which the direction of particle flow may change.
This gives rise to the potential for a model in which the microscale is represented as a network and the fluid as a continuum 
of particles, which accounts for particle capture that affects network structure. 


In Chapter \ref{chap:nm_Derivation of the network model}, we consider the flow of a fluid containing a continuum of particles 
through a network of pores and junctions. 
The particles can deposit on the inner walls with some adherence probability, which decreases the size of pores, causing a decrease in 
their fluid conductance. 
Continued deposition leads to removal of pores, which alters the connectivity of the network. 
We arrive at a system of ordinary algebraic--differential equations that govern the conductance, particle concentration, and pressure 
on a network whose connectivity is dynamic in time.

In Chapter \ref{chap:nm_Solutions of the network model}, we solve the network model for filters with various pore-size distributions. 
We first focus on the case where the pore-size distribution is uni-modal, which gives rise to a network 
where all edges have the same conductance initially.  
 Later, we explore the case where the pore sizes are distributed log-normally, and find solutions that qualitatively match 
 the solutions for our size-structured model. 
 
Lastly, in Chapter \ref{chap:nm_Conclusions for the network model}, we conclude that our network model allows us to specify an initial microscale structure and monitor temporal changes to this, and 
the effects of these changes on macroscale quantities of interest.
However, although we are able to solve the model on portions of the filter, 
it is prohibitively expensive in cases where the network contains a number of pores 
comparable to that present in entire porous membranes.
We look to our third model as a way of solving this problem.

\subsection{Multiscale model: A multiscale model for filtration}
In Part \ref{part:A multiscale model for filtration}, we build and solve a multiscale model 
for filtration in which the microscale membrane structure is treated as a network of pores and 
junctions, and the macroscale geometry is treated as a continuum of particles and pores.

In particular, in Chapter \ref{chap:mm_Introduction to the multiscale model}, we explore existing models that allow the
simultaneous monitoring of changes to global and local properties of the membrane. 
We find that most models view the microscale as a continuous medium, in which it is not possible to incorporate 
network structure that we have already shown to model the microscale well. 
We conclude that there is a need for a framework for multiscale models, 
in which the microscale is treated as a dynamic network that represents the filter as a whole, while the macroscale is treated as a continuous medium.

In Chapter \ref{chap:mm_Derivation of the multiscale model}, we derive a new multiscale model for filtration, 
which is novel because
 the filter microscale is treated as a periodic network consisting of a repeating smaller network (which we call a cell), while systematic averaging leads to the emergence of a continuum system to represent the filter macroscale.
To derive a network--continuum multiscale system from the discrete microscale problem, we extend an asymptotic method, called network homogenisation, 
to account for networks where the connectivity changes with time. 
The result is a system where partial differential equations defined on the continuous macroscale geometry 
are coupled with ordinary algebraic--differential equations defined on the discrete microscale network. 
Changes to the connectivity of the microscale due to deposition cause changes 
to the parameters of the macroscale system, and thus to macroscale quantities of interest, such as the flow rate and pressure.
% On the microscale, the conductacne and adherecne of pores is controled by a linear equation that whose complexity 
% depends on the numebr of nodes in each cell. 
% On the macroscale, we fnid a system of partial differential equations that goven the pore particle concentration, 
% the flow rate, and the local pressure, given by the parmaeters of the filter, pemreabilirty adn adhesivity (stickiness) of the filter, 
% which are coupled to the micsoalce via the conductance and adherence,

In Chapter \ref{chap:mm_Solutions of the multiscale model}, we solve our multiscale model.
Crucially, we find that the solution of our multiscale model matches that of our network 
model from Part \ref{part:A network model for filtration} well in the asymptotic limit in which many cells constitute the network. 
This is significant as the cost of solution of the multiscale model is a function 
of the number of nodes within a cell, as opposed to the number of nodes in the entire network, 
meaning that simulation of the entire filter is now feasible.
Furthermore, we explore the effect of microscale structure on 
macroscale quantities and show that the pore-size distribution and microscale connectivity are important in determining the permeability and adhesivity. 
% One implication is that it may not be necessary to extract the exact structure of the microscale network to obtain 
% quantitatively accurate solutions, but instead to choose the initial pore conductances correctly. 

In Chapter \ref{chap:mm_Conclusions the multiscale model}, 
we conclude by noting that we have developed a framework for filtration that incorporates
the entire structure of the microscale without the solution becoming prohibitively computationally expensive.
% specified but solutions are much more 
% computationally cheap than solutions of microscale models. 
% Further work will involve the use of 
% actual material data to optimise membrane parameters for filter performance. 
Further work might involve further assessment of the effects of connectivity and pore-size distribution on the effective flow rate and particle concentration, or the inclusion of more complex physical and chemical laws on the microscale, which the framework permits. 
% the importance of exact membrane connectivity, 
%  and further analysis of the extension of this model to non-periodic networks.

Lastly, in Chapter \ref{chap:Conclusion}, we conclude that we have developed three 
novel models for filtration and showed that these increase our understanding of the physical 
process. 
Importantly, solution of all three of our models is more computationally feasible 
than many existing models, which opens them up for the potential for the parameter
sweeps and cost function descents necessary for filtration process optimisation in future work. 


%  \section{Background}
% \comment{
% Filtration is an important industrial process used across a vast
% number of sectors to separate particles from fluids. Applications of filtration include
% water purification and sewage treatment [26, 45, 48], biotechnology [1, 2, 5, 23, 24, 37],
% medicine [29], and food production [10, 11, 14, 17]. Filtration is also used to remove
% environmentally hazardous chemicals from air and other gases [4, 27]. Not sure these
% refs are all sensible. For eg, [27] is about hygroscopy not filtration.
% \\
% A filter membrane is often a porous medium, with an application-dependent and
% manufacturer-determined structure, by which we mean pore size, pore shape, and void
% fraction; an example membrane is shown in Figure 1 (left). The void space between
% the fibres forms a highly complex network of tunnels (commonly referred to as pores)
% on the microscale, as seen in the photos in Figure 1 (right).
% \\
% Our interest is in situations where a fluid containing a suspension of particles,
% known as the feed fluid, is driven through the filter, usually by imposing a constant
% pressure drop or constant flow rate. The particles are forced through the pores and
% are sometimes captured. As a result, the concentration of particles in the fluid passing
% out of the filter is lower than that in the fluid passing in. The number of particles
% \\
% that the filter retains increases until the filter becomes completely clogged [what do
% we mean?], which occurs when the permeability, which is the fluid conductivity of the
% filter, falls below an application-dependent critical value.[Not sure this is true] When
% a filter becomes completely clogged, the filtration process must be halted while the
% filter is cleaned or replaced, which is both expensive and time-consuming.
% \\
% There are four mechanisms that contribute to filter clogging [18], as illustrated in
% Figure 2. These are:
% (i) Blocking, where particles pass through larger pores but get stuck when they
% encounter small gaps;
% (ii) Deposition, where particles pass into larger pores and adhere to pore walls,
% eg due to electrostatic forces or chemical binding, causing pore constriction;
% (iii) Partial surface blocking, where particles land on the surface of the membrane
% and partially cover a pore entrance;
% (iv) Caking, where particles build up on the surface of the membrane and form a
% porous layer through which the feed must permeate. This layer may evolve to
% have a distinct permeability from that of the actual membrane [38].
% The relative prevalence of these clogging mechanisms depends on the membrane struc-
% ture, feed composition, and the process conditions [43].
% \\
% For successful filter operation, a minimum filter efficiency, defined as the propor-
% tion of particles in the feed that are captured, is usually required. A filter performs
% well if the efficiency constraint is met while the lifetime, which is the time until the
% filter is completely clogged [do we mean this?], is maximised. The overarching chal-
% lenge is to control the particle capture in such a way as to increase the filter lifetime
% while satisfying the given filter-efficiency constraint [39].
% \\
% Experimental methods for understanding clogging typically involve direct mon-
% itoring of the particles landing on the membrane surface, which is achieved using
% techniques including microscopy, reflectometry, spectroscopy, and 3D imaging [6, 28].
% \\
% Mathematical and computational methods offer a cheaper and often faster alter-
% native to direct experimental monitoring. Such methods usually fall into one of two
% categories: effective continuum descriptions, which describe the overall properties of a
% filter medium, such as the flow rate given an applied pressure difference and the con-
% centration of particles in the outlet given an inlet concentration, and particle-based
% simulations, which track individual particles and their interactions with the filter. We
% discuss the two branches of study below.
% }


% \section{Aim}
% Our aim in this part of the thesis is to develop a modelling framework in which the polydispersity of pores in a filter and particles in a fluid to be filtered can be easily tracked. 
% We choose to use a size-structured
% methodology, in which the pore and particle sizes are treated as independent variables.
% mono



\part{A size-structured model for filtration}
\label{part:A size-structured model for filtration}

\chapter{Introduction to the size-structured model}
\label{chap:ssm_Introduction to the size-structured model}

\section{Particle and pore size in filtration}
\label{sec:ssm_Particle and pore size in filtration}

In industrial membrane filtration regimes, the size of the particles ranges from the nanoscale to the milliscale \cite{sutherland2011filters}:
\begin{itemize}
    \item Particles on the nanoscale, $10^{-3}$--$10^{-1}\mu \text{m}$, which are only detectable using electron microscopy, include virions (the complete, infective part of a virus) \cite{yang2006nanoporous}, tobacco smoke \cite{keith1982particle}, soot \cite{bensaid2010numerical}, and many products of combustion \cite{chen2016removal};
    \item Microscale particles, of sizes in the range $10^{-1}$--$10^{1}\mu \text{m}$, are only visible through microscopes and
    include bacteria \cite{wand2007removal}, pigments \cite{zhu2017multistage}, aerosols \cite{tien2013granular, davies1983filtration}, and blast-furnace dusts \cite{zeydabadi1997zinc};
    \item 
    On the milliscale, $10^{1}$--$10^{3}\mu \text{m}$, examples of particles, some of which are visible to the naked eye, include pollen \cite{stephen2008new}, and the dust from various types of industrial machinary \cite{wang2008comparison}.
\end{itemize}
%In industrial filtration regimes, particles and pores can range in size from the atomic scale to the milliscale \cite{Yalcinkaya2019}.
% The sizes of the particles in the feed fluid and pores in the filter 
% play a key role in almost every aspect of the filtration process.
% Almost all operational specifics of the filtration regime to be used depend on the particle and pore size. 
% On the sub-microscale, $10^{-4}$--$10^{-2}\mu \text{m}$, for example, reverse-osmosis filtration and nanofiltration methods are often 
% chosen to control the concentration of particles such as metal ions and aqueous salts for applications in waste-water refinement \cite{turhanen2015}. 
% On the microscale, $10^{-2}$--$1\mu\text{m}$, ultrafiltration and microfiltration are the most useful methods for the postprocessing of harmful biproducts in manufacturing plants.  
% The isolation of clay particles, zinc oxide fumes, colloidal silica, asbestos, and atmospheric dust are just a few examples.
% Meanwhile, on the milliscale, $1$--$10^{3}\mu\text{m}$, more conventional filtration methods are used for the separation of larger particles still. Examples include treatment of cement dust in concrete manufacturing processes, and pulvarised-coal removal from air in factories for power generation. 
A common goal in many industrial processes is to segregate particles based on their size \cite{sutherland2011filters}.
Recently, for example, much attention has been paid to the development of filtration materials for protective face masks \cite{ogbuoji2021}.
The aim is to prevent the passing of saliva droplets 
% (containing virions around $10^{-1}\mu\text{m}$) 
of typical sizes $1-10\mu\text{m}$ from an infected person to an uninfected person \cite{chua2020face}, while simultaneously allowing the passage of particles prevalent in air, to prevent unsafe respiration \cite{cabanillas2021effects}.
% In laboratories, depth filers are used to isolate red blood cells from blood [ref: The Mechanisms of Leukocyte Removal by Filtration]. 
% The particles in the feed fluid have a distribution of sizes, and it is necessary to isolate particles in one range from those in another,.
Air purification devices (see \cite{aditya2018review} for a review), which have increased in popularity due to concerns about pollutant levels \cite{katsouyanni2003ambient},
provide another prominent example \cite{vijayan2015enhancing}. 
 Some high-efficiency particulate absorbing (HEPA) filters are designed to remove tire matter, brake dust, and certain vehicle exhaust fumes from air. These particles are $2.5\mu\text{m}$ or less (around 100 times as thin as human hair). 
 % , but do not remove various other non-harmful particles.

The choice of the pore-size distribution, and the filtration process as a whole, depends to a large extent on the particle size \cite{sutherland2011filters}. 
Particles are often retained via blocking, using filters that contain pores in a similar size range, or via deposition, using 
filters containing larger pores that are coated with agents that have some mutual adhesion property with the particles that are to be processed. For example:
\begin{itemize}
    \item Nanoscale particles are often processed using membrane filters, active-carbon filters, and HEPA filters;
    \item Microscale particles are most commonly captured using methods such as electrostatic precipitation, wet scrubbing, and other fabric-dust filtration methods;
    \item 
    Milliscale particles are usually retained using the most conventional filtration methods, which include sieving, venturi scrubbing, cyclonic separation, and filtration via paper and various fabric materials. 
\end{itemize}
Recent research has increasingly focused on controlling the pore-size distribution in the filter as a means of controlling the sizes of particles that are retained \cite{gustafsson2016strategies, kitagawa1992improvement, bowry2002nano}.
For example, in so-called honeycomb-structured
porous membranes, it is possible to control the size of the pores and even their spatial distribution with high accuracy \cite{wong2006water, hernandez2012honeycomb}.
In general, the accuracy with which the pore-size distribution 
can be chosen varies depending on the material used to produce the filter. 
When control of the pore-size distribution is feasible, industrialists are generally interested in designing filters that contain pores with the distribution of sizes that maximises the throughput (total amount of fluid processed before the filter membrane is replaced) while ensuring that some minimum proportion of particles in the desired size is removed by the filter. 
%It can be argued, then, that the size structure of the feed and filter are one of the most important characteristics of the process.
% Recent advances in filtrtion industry mean that we are beginning to be able to maniplate pore sixe distribution. Can control paamteres of the distribution and these can depend on depth, so that pore-size fradients can be introduced.

\section{Existing size dependent models for filtration}
\label{sec:ssm_Existing size-structured models for filtration}

Theoretical work on filtration is vast (see the reviews by Iritani \cite{iritani2013review} and Jegatheesan \cite{jegatheesan2005deep}, for example), and yet,
relative to the subject as a whole, models that explore the relationship between the particle- and pore-size distributions are sparse.
In mathematical studies, filtration processes are frequently idealised by assuming that all particles in the feed fluid are one characteristic size (often the mean of a distribution that is obtained experimentally).
This may be sufficient if the true distribution is tight, 
but seems cavalier in situations where particles are more widely poly-dispersed. 

Indeed, there is experimental evidence to suggest that this simplification has a significant effect on solution behaviour.
In 
% the 2008 paper, 
\cite{chang2008effect}, for example, Chang \emph{et al.} explore mono-dispersed, bi-dispersed, and tri-dispersed feeds, and find that otherwise-equivalent regimes are more efficient when larger particles are present, since more large particles are retained per unit time than small ones.
Thus the solutions exhibit different quantitative behaviour.
One possible reason is that the dominant mechanism of particle capture varies with the particle-size distribution. 
In theory, if particles are larger than pores on average, then blocking 
is more prevalent than deposition, while if the opposite is true, 
then deposition is the more prominent capture mechanism, until pores become sufficiently clogged 
that blocking takes over.


In 
% their 2020 model, 
\cite{griffiths2020role}, Griffiths \emph{et al.} suggest that 
the balance of blocking and deposition affects the behaviour of the flow rate as a function of the total volume of fluid processed. 
As this volume increases, at first the flow rate decreases linearly, as pores are shrunk via deposition, but later it decreases super-linearly, as pores are removed completely via blocking. 
The transition from slower decrease to more rapid decrease occurs more quickly if particles are larger. 
The authors attribute this to the notion that more larger particles lead to pores blocking completely more often, so that a greater proportion of pores are no longer accessible for deposition sooner. 
This interplay between the two clogging mechanisms suggests that the relative sizes of particles and pores are important, 
and that the behaviour of solutions of models based on uni-modal size distributions may be misleading.
%Thus, consideration of the sizes of both particles and pores is important in mathematical models that hope to capture the behaviour of  
%properties such as fluid flow rate and retention-proportion accurately.

One early model that acknowledges the importance of the distribution of pore sizes is that of Kasottis \emph{et al.} \cite{kassotis1985modelling}.
In this work, experiments determine the flux and retention of certain mono-dispersed suspensions in the presence of a poly-dispersed filter, and  
the authors derive a model to explain the results. 
The outcome is an equation for the number of pores present in the filter as a function of the flux and the retention of solid and spherical particles of a given size when the flow down an individual pore is governed by the Hagen--Poiseuille law (theoretically justified by Stokes
% in 1845, 
in \cite{stokes2007theories}).
The immediate utility of the equation is to estimate the number of pores in the filter when the flux and retention are experimentally known.
One could equally see the equation as a means of deducing the flow rate given a known initial pore distribution, 
since the number of pores appears as an integral over the pore-size distribution.
Indeed, the authors go on to predict the proportions of particles that will be retained given particular pore distributions, assuming that 
retention is a result of `geometric considerations only' (akin to blocking in our setup). 
That is, particles that are too large to fit through pores are always captured, but particles that are small enough to 
fit through pores are never retained (so that deposition does not occur), which is one limitation of the work.
%We will expand on several ideas from this work in our model, for example, 
%expressing the total number of pores as an integral over pores of various sizes. 
%We will, however, assume that deposition may also occur in pores that are larger than the particles, 
%and will consider a distribution of particle sizes.

% Kasottis-1984
% An example of an early paper that focues on pore size dustribution. First they do experiments to determine the sieving properoties (flux and retention) of a certain suspension. Then they do modelling to explain it.
% Equation for flux thoruhg filter times the number of aprticles of a particular size X_i that are retaind as a fuctio nof the integra of the distribution of pores of a given size.
% Allows us to estimate the tital number of pores if the flux and retention is measured experimenttally, which is mostly what the paper focuses on. Howevr we could equally imagine that we dont know flux and we do know pore-size distribution.
% Model uses blocking only. Flow thorugh any pore is via Hagen-Posieuille. Then flow of particles thorugh all pores is number of opres that are larger enough to take particles times flow through pore (HP). Particles are spheres.
% Teh shape and sharpness of pore-size distribution were found to be important in determining retention.
% Calculates rejection coefficients as function of pore size – in other workds, preducts what particles will be retained given pores of a given sizem asssuming that rejction is result of geometric considerations only – ie if it is too large then it is blocked if it is smaller than the pore then it does not deposit.

Early models noting that attention should be paid to the entire distribution of particle sizes (as opposed to the mean or maximum) include that of Alon and Adin
% in their 1994 paper, 
\cite{alon1994mathematical}.
The model builds on an earlier one \cite{o1979role} by O'Melia and Ali,
% in 1978, 
which itself builds on the seminal model developed by Kozeny, 
% in 1927, 
\cite{kozeny1927uber}, and by Carman, 
% in 1937, 
\cite{carman1937fluid, carman1956flow}. 
% and in 1956, 
% \cite{.
The result of Kozeny and Carman's original models is a proportionality law between the pressure drop across the filter and the flow rate, via the porosity of the filter and other intrinsic properties of the filter and fluid.
In \cite{o1979role}, O'Melia and Ali extended this model by accounting for particle size by adapting the equation to incorporate the mean diameter of the suspended particles.
Alon and Adin's contribution was to suggest a rule for how the number of particles that have been retained depends on their size. 
The authors compared different models for retention as a function of size, and showed that a power law fitted their experimental data most accurately. 
They concluded that their size-dependent model produced better data fits than simply using the mean particle size, as in O'Melia and Ali's model, for example.

More recently, due to increasing computational resources, 
some attention has been paid to Lagrangian approaches to account for size, where particles and pores are tracked individually.
One such work is that of Marrufo-Hernandez \emph{et al.}, 
% (2018), 
\cite{marrufo2018prediction}. 
Here, the filter is modelled as a box containing a finite number of flat planes, stacked in parallel, which are permeated by circular holes, similarly to in \cite{wei2014understanding}. 
This setting is markedly different from more conventional models in which the filter is seen as a set of quenched spheres, for example, as in \cite{kim2002cake, sung2007permeation, salinas2007studies}.
In Marrufo-Hernandez \emph{et al.} \cite{marrufo2018prediction}, 
particles move subject to Brownian motion accompanied by a constant driving force, and collide with other particles and with pores, due to laws of motion on the microscale.
The authors' presentation of the distributions of particle and pore sizes as functions of time and space mean that
they are able to make detailed conclusions about the filtration process, such as the size structure of the filtrate, and the quantity of pores 
of a given size that remain in the filter at a given time.
One limitation of the model is that the `stacking' of a finite number of planes poses difficult modelling questions, such as how to choose the number of planes correctly, 
since the filters with which we are concerned do not really contain these planes.
More fundamentally, although Lagrangian perspectives such as this offer a comprehensive description of filter behaviour, they are often limited by their significant computational expense. 
Marrufo-Hernandez \emph{et al.} note that their simulations are based on the results obtained by using a batch of 500 particles, 
which is at least two orders of magnitude fewer than the number of particles in a typical filtration process. 

In summary, the sizes of particles and pores are important in determining the key characteristics of filtration processes experimentally. 
Despite this, existing continuum models of filtration capture the large-scale observables of
interest but do not fully account for poly-dispersed feeds and filters, and the simultaneous clogging mechanisms that occur 
due to the relationships between the particle- and pore-size distributions. 
On the other hand, some existing discrete models account for poly-dispersity more completely, but are usually computationally 
expensive.
Therefore, a new, computationally tractable, modelling framework is needed, in which the poly-dispersity of particles and pores, 
and the associated clogging mechanisms, are accounted for. 

\section{A novel size-structured model for filtration}
\label{sec:ssm_A novel size-structured model for filtration}
One candidate framework for describing the evolution of the size distributions of particles and pores is so-called `age-structured' equations, which are common in population and disease models (see the 
% 2008 
book by Brauer \emph{et al.}, \cite{brauer2008mathematical}, for an introduction to these). 
In simple population models, all members of the population are assumed to be interchangeable, and, in particular, behave independently of their age. 
However, in almost all physical populations, age is one of the most important defining characteristics, since different age groups often display different reproduction numbers and survival capacities \cite{brauer2008mathematical}.
The seminal work is that of McKendrick \cite{mckendrick1926applications}, who
% , in 1926, 
was the first to propose an `age-structured' framework to account for populations' age dependency in mathematical models.
% In the McKendrick framework, the 
Here, the key quantity is the density of individuals that are between the ages $a$ and $a +\delta a$ (where $\delta a$ is infinitesimally small), 
$\rho(t,a)$ say.
Assuming that changes to this population density occur due to birth, growth, and death, assigning rates to these phenomena,
and subsequently appealing to the conservation of the total number of individuals, one can derive continuous differential equations that describe the evolution of the population density, $\rho$. 
One such example is the McKendrick--von Foerster equation, which is a first-order linear partial differential equation for the time- and age-dependent population density that was proposed by McKendrick
% in 1926 
and used by von Foerster 
% in 1959 
to describe cell cycles \cite{von1959some}.
In equations such as this, age and time are considered independently, and age is essentially treated as a space-like variable.
One advantage of this approach is that the plethora of solution methods used for 
% general time-space 
time and space dependent partial
differential equations, such as asymptotic analysis, the method of characteristics, and various numerical schemes, are all applicable to find $\rho(t,a)$.
% A variety of these methods are used to find the distribution of individuals at any given time $t$ and age $a$, $\rho(t,a)$.
Once this solution is obtained, secondary quantities of interest can be derived directly from it. 
For example, the total number of individuals in the population at time $t$ is given by $P(t)=\int_0^\infty \rho(t,a) \, \text{d}a$.

To derive a size-structured model for filtration, we will proceed by likening the variable `size' in filtration processes to `age' in population models.  
That is, we will treat size as a space-like variable, and seek distributions of particles and pores that depend on it.

Indeed, in Chapter 3, in Section 3.1 we will begin by assuming that any filter control volume contains a plane like the ones in \cite{marrufo2018prediction}, and propose that
% , at any given time, and at any given depth inside the filter, 
it consists of a continuous distribution of particles and pores of any size 
in the limit where the control volume is infinitesimally small.
Assuming that these distributions are known, we will use them to derive secondary variables of interest, such as the particle and pore concentrations, and the local porosity and permeability of the filter. 
In analogy with the assumption of population conservation in McKendrick's framework, 
we will assume that fluid mass, particles, and pores are conserved, as well as a Hagen--Poiseuille law to link the velocity and pressure, and arrive at a system of partial integro--differential equations 
that describe the evolution of the particle and pore distributions, as well as the flow rate and local pressure.
We will close this system by proposing initial and boundary conditions that model mono-dispersed, or log-normally-dispersed, particles and pores in industrial operational regimes,
before, in Section 3.2, nondimensionalising to prepare the system for analysis and solution. 

In Chapter 4, we will solve the dimensionless model under various assumptions. 
In Section 4.1, we will assume that particles and pores are both mono-dispersed, and explore the cases where particles are larger and smaller than pores. 
We will use these relatively simple scenarios to explore 
the dependence of the model on the various dimensionless parameter groups. 
In particular, we will examine the cases where particles are much smaller than the filter depth, 
and where the concentration of particles in the feed is very small relative to the concentration of pores in the filter.
One advantage of our setup
% (see the definitions of $\Lambda\bloc$ and $\Lambda\depo$ in Section \ref{sec:ssm_Dimensionless model}) 
is that it allows us to circumvent awkward problems that appear in discrete models, such as choosing the `correct number of planes' in \cite{marrufo2018prediction},
by instead choosing dimensionless parameters that are, perhaps, more physically motivated.
In Section 4.2, we will increase the complexity of the problem by supposing that mono-dispersed particles are fed through pores that are log-normally-dispersed. 
We will first assume that the probability of deposition is zero, so that only blocking can occur. 
% Finally, for the remainder of this section, 
We will then explore the case where blocking and deposition occur simultaneously.

In Chapter \ref{chap:ssm_Conclusions for the size-structured model}, we will conclude by summarising and discussing potential avenues for future work.
%
% \paragraph{Strategy that we will use instead: Age-structured models}
% Our solution will be to develop a mdoel of filtration that is continnum and we wil account for sie by doing something similar as age-structuered model in polulation analysis. In the simplest population models for disease, all emebers of the population are assumed to be iterchangeable. More advanced popilational models add some structure to the population such as specificaiton og spatial locaqtion ot age. Age is one of te most important characateristics because individuals with different ages may have different reproduction number of survival capacities [ref]. The first age-structued models were designed by Mckendrick [ref] for studfy of dissaese transimission in populations. In these models, rho is the density of pepole of age a at time at location blah. One can ingtegrate to get the concentration os people etc… Assuming that population changes occur due to death and birth and assigning these rates, one can write down differntial equation that describe the evolution of the population density, where age and time are independent variables, for example, the Mckendric-Lotka equation [ref]. These equations can be solved using the metod of characterestics [ref], or, numerical,y using numerical methods associatged with solution of differentiak equations, such as finite difference or finite elelemets.. thus using this type of theory means that there will be lots of maths that we can readily use.
% We will use a similar theory, apart from for us,. The population density will be the density of pores os size r in the filter at time t at deth z, or ther number of p[articles of size a at position a at time t. The particle and pore size will be independent variables of the system. Using these densities, in the same way as in the population models, we will dervice qwuantities of interest in the filtration industry, suich as the permeability and porosity. Using conservation of mass of fluid and particles and pore we will derive equations fo rhte evolution of these densities. We willl then use the plethora of mekthos available for the solution of these equations, asympttic, and finirte difference. Say something about the fact we will consider deposition and blockinsg.



% In this paper, we will build a size-structured model for filter operation in



% To m do this, we will borrow ideas from afge structured models...


% Hernandez-2018

% 50 G. Viramontes-Gamboa, J. Arauz-Lara, and M. Medina-Noyola, “Tracer diffusion in a Brownian fluid permeating a porous
% medium,” Physical Review Letters 75, 759 (1995).
% 51 G. Viramontes-Gamboa, M. Medina-Noyola, and J. Arauz-Lara, “Brownian motion of colloidal particles in a model porous
% medium,” Physical Review E 52, 4035 (1995).
% 52 M. A. Chávez-Rojo, R. Juárez-Maldonado, and M. Medina-Noyola, “Diffusion of colloidal fluids in random porous media,”
% Physical Review E 77, 040401 (2008). Layer are sort of weird because it means you have to pick the corect number of them but hey dont really exist. Alsoe more expensive the more of them there are, in our contimum setting we will control this with Lambda and same expsne no matter how we kchoose it nad it wil be physically relevant because it will be determined b the particle size.
% )with circular pores in different sizes and random spatial distribution. Explore how the size distribution of the filtrate depends on the incpoing particle size distribution and distribuition of pore hole  sizes. Dynamics are brownina motion and they study normal distributions which pores and particles aren’t. It is Lagrangian approach so they track every particle rather than eulerian approach where feed flui dis a continuum.
% Limiatiotn od this work is that there is no flow, the dynamics are browninan. We expect that particles will fololow the flow in some meaningful way, for example, as we will dervie later, we might expect more partticles to gflow through larger pores because we would expect a larger proportion of the flow to be carried through these. Particles are ent through one by one… particles are randomly distributted. Then theu flow due to browninan motion and some constan force and Klagenvin equation iused to calculate the displacement of the particls.tey do not consdier repulsive or attractgive bahvioru of particles nor will we but mention that we could easil thave efects ike coagulation like screened coulombeic repulsion of van der waals atrtaction. They consdier particle coliiosojm.. particle-wall, particle parti cle, and particle-membrance.. complicated because position of partile shas to be correcteed to acount for walls etc adnd consider relatice momentum of particles due to their mass… we consider these effecsts tot be irrelebant on the macroscale, and seems unncecceesary and easteful of compulation time when all we care about is permeabilirty and flow etc and retnetion etc. Our aim will be to consider eulairian to avod these complications and make theigns compulationall y muich more feasible...they are limiated to 500 particles whi chis a grtoss underexxageration.
% Should e considered as the discretised version of what we do.

% \paragraph{A paper where they do something sikilar ot size structured model}
% Nikanshi-2000
% Closest thing to what we do.
% Consider distribution of particle and pore sizes.
% Consider ‘sedimentation’ of particles on the wall.
% Results are brief but discuss changes to the size-distribution functions. Compare results of poly-dispersed to mono-dispersed. Find that consideration of poly-dispersed leads to significasntly different numierical solutions. Assume that poarticle sixze disribution is known and pore size distribution is known initially. This is doen in two -phase flow – oil and water. Write conservation euquation for fluid and particles and arrive darcy law coupled with advection- reaction equation. The reaction term depends on the distributions of particles and pores and the interaction between them.  Solution is one-dimensional.Numerics is finite elements spatially and finite difference for time. Use upstrteaming scheme. Find that particles are retianed ni the first portion of the filter. Fin that flux dro poff for polydipserse case is greater than mono-dispersed case, and closest thing is when particles are as large as possible, (i.e. not represented by the mean, but ny the largest size). In other words, you cant recreate the poly-dispersed case with one partice size, and using the mean isnt right because blocking matters more than deposition, so using larger ones and assuming blocking only occurs is almost right,l but really drop off is even faster cos deposition also happens.



% Indeed, in section .. we will ...blah



\chapter{Derivation of the size-structured model}
\label{chap:ssm_Derivation of the size-structured model}
In this chapter, we develop a model for filtration in which particle and pore sizes are treated as independent variables. 
We begin by deriving a dimensional system of differential equations for the distributions of these sizes, pose industrially relevant boundary and initial conditions, and then scale the problem to arrive at a dimensionless system to be analysed and solved.
Our focus is on building a model that demonstrates the utility of this size-structured framework for modelling filtration.
Therefore, emphasis is placed on simplicity, to make solutions as intuitive as possible.  
For example, we suppose that the flow is driven only by advection (i.e., we do not consider diffusion, dispersion, or any other transport mechanisms), 
and that the solution depends only on depth (i.e., we ignore possible dependence on spatial variables transverse to the depth direction).

\section{Dimensional model}
\label{sec:ssm_Dimensional model}
We suppose that the filter membrane
(see Figure~\ref{fig:ssm_pores} (Top-Left))
consists of a slab
(see Figure~\ref{fig:ssm_pores} (Bottom-Left))
of solid material (see the grey areas)
containing tunnels of pore space (see the white areas). 
This is located between $z=0$, which we call the inlet, and $z=l$, 
which we call
the outlet.
We assume that a horizontal cross-section 
% through a control volume of the filter, $z+\delta z$ say, where $\delta z \ll 1$, at any depth $0\le z\le l$, 
at a given depth 
(see the red line in Figure~\ref{fig:ssm_pores} (Top-Right))
contains solid material (see the solid parts of the line) perforated by circular pores (see the dashed parts of the red line in Figure~\ref{fig:ssm_pores} (Top-Right)), 
so that, if we look from above (see Figure~\ref{fig:ssm_pores} (Bottom-Right))
then we see a sheet of solid material containing circular pores with a distribution of sizes (i.e., radii).

Fluid, which contains solid spherical particles with various sizes (i.e., radii), 
is driven into the filter at the inlet. 
The particle-laden fluid flows through pores, which capture some of the particles (see Figure \ref{fig:ssm_particles}), causing pores to clog, decreasing their size. 
Fluid, containing any particles that have not been captured, exits the filter at the outlet. 

In the next subsection, we define objects to model the pores in the filter and the particles in the fluid.

\begin{figure}[!htb]
        \includegraphics[width=0.9\textwidth]{./figures/drawings/pores_3.pdf}
        \centering
        \caption{
        (Top--Left) Photograph of an ePTFE membrane on the macroscale.
        The membrane is much thinner than it is wide and long.
        (Bottom--Left) Diagram of an idealised membrane on the microscale.
        The membrane consists of solid space (grey)
        punctured by tunnels of pore space (white).
        (Top--Right) Side-view of the idealised membrane.
        Any depth (e.g., solid red line) consists of pores (dashed red lines) with a distribution of sizes (red arrows).
        (Bottom--Right) Top-view of the idealised membrane.
        Pores vary in radius but are circular, 
        so that, from the top, we see circular pores with a distribution of radii (red arrows).
        }
        \label{fig:ssm_pores}
\end{figure}

\subsection{Particles and pores}
We assume that the membrane is sufficiently homogeneous in the directions transverse to the depth (i.e., $x$ and $y$, see Figure~\ref{fig:ssm_pores}) that the solution does not depend on $x$ or $y$.
We define $\bar{r}(z)$ to be the mean pore-size in the cross-section in the control volume $z+\delta z$ located at $z$, and $\bar{r}^{*}=\bar{r}(l)$ to be the mean pore-size at the outlet. 
We introduce the
pore-size distribution, 
$n(z, t, r)$, 
which is measured in number$/\text{m}^3$ so that $[n]=\text{m}^{-3}$, where we will use 
$[x]$ to mean `dimensions of $x$'.
The distribution $n(z, t, r)\delta r$ is the
number of pores with size in the range $(r, r + \delta r)$ per unit cross-sectional area at depth
$z$ and time $t$. 
It follows that the pore concentration, $N$, which is the number of pores of all sizes per unit filter area at depth $z$ and time $t$ is given by
\begin{align}
    N(z,t) = \int_0^\infty n(z,t,r) \, \text{d}r,
    \label{dim_N_1}
\end{align}
which is measured in $\text{number}/\text{m}^{2}$, so that $[N]=\text{m}^{-2}$.

We relate the local porosity, $\phi(z,t)$, which we define to be the fraction of the total cross-sectional area at $z$ that is pore-space (which is dimensionless), to $n$ using 
\begin{align}
  \phi(z,t) = \int_0^\infty \pi r^2 n(z,t,r) \, \text{d}r.
        \label{eq:ssm_phi_dim_1}
\end{align}


We define $\bar{a}^{*}$ to be the mean particle-size in the feed fluid. 
We introduce the particle-size distribution, $c(z, t, a)$, measured in $\text{number}/\text{m}^4$ with units $[c]=\text{m}^{-4}$, such that $c(z, t, a) \delta a$
is the number of particles with size in the range $(a, a + \delta a)$ per unit filter volume at depth $z$ and time
$t$. 
Thus, the \emph{filter} particle concentration, $C$, which is the number of particles of all sizes in the fluid \emph{per unit filter volume} (i.e., pore space and solid space) at depth $z$ and time $t$ is
\begin{align}
    C(z,t) = \int_0^\infty c(z,t,a) \, \text{d}a,
    \label{dim_C_1}
\end{align}
which is measured in $\text{number}/\text{m}^3$ so that $[C]=\text{m}^{-3}$.

\begin{figure}[!htb]
        \includegraphics[width=0.7\textwidth]{./figures/drawings/size.pdf}
        \centering
        \caption{Diagram of a side-view of the filter as particle-laden fluid moves through it.
        Particle laden fluid enters the filter at the inlet.
        Some particles (green) adhere to pore walls, decreasing the sizes of the pores. 
        Other particles (red) do not adhere to pore walls, and exit the filter at the outlet.
        }
        \label{fig:ssm_particles}
\end{figure}

Lastly, we define the 
\emph{void} particle concentration, 
$C_{\text{V}}$ with $[C_{\text{V}}]=\text{m}^{-3}$, 
to be the number of particles of all sizes in the fluid \emph{per unit void volume} (i.e., pore space and not solid space) at depth $z$ and time $t$.
Since the porosity, $\phi$, represents the ratio of void volume to filter volume, we relate the void particle concentration to the filter particle concentration using 
\begin{align}
     C_{{\text{V}}}(z,t) = \frac{C(z,t)}{\phi(z,t)}.
     \label{eq:def-void-conc}
\end{align}
Hence the void particle concentration, 
$C_{\text{V}}$, 
depends only on the filter particle concentration, $C$, 
and the porosity, $\phi$.
Therefore, in what follows,
 we will focus on writing equations that describe the evolution of $c$ and $n$, 
 use these to determine $C$, $N$, and $\phi$, and then infer $C_{\text{V}}$ from $C$ and $\phi$.
Note that we will usually refer to the filter particle concentration as the particle concentration (dropping the word filter). 
We will always refer to the void particle concentration in full.
% Lastly, we introduce the captured-particle size distribution,
% $g(z,t,a)$, measured in $\text{number}/\text{m}^4$ with units $[g]=\text{m}^{-4}$, such that 
% $g(z, t, a) \delta a$
% is the number of particles with size in the range $(a, a + \delta a)$ per unit filter volume at depth $z$ and time
% $t$. 
% The captured-particle concentration, $G$, is the number of captured particles of all sizes per unit filter volume at depth $z$ and time $t$, and is related to the captured-particle size distribution by
% \begin{align}
%     G(z,t) = \int_0^\infty g(z,t,a) \, \text{d}a,
%     \label{dim_G_1}
% \end{align}
% which has units $[G]=\text{m}^{-3}$.
In the next subsection, we assume that fluid mass is conserved in order to derive equations that govern the pressure and velocity.

\subsection{Conservation of fluid mass and Darcy's law}
For simplicity, we assume that fluid is transported due to advection (and not, for example, diffusion).
Therefore, to derive the equations that govern the fluid flow
through the filter, we first consider a pore of size $r$ and define $q(z, t, r)$ to be the
volume of fluid passing through the pore per unit time, so that $[q]=\text{m}^{3}\text{s}^{-1}$. 
We assume that, at a given depth, the pores act like tubes connected in parallel and thus we relate this flux, $q$, to the fluid pressure, $p(z,t)$, using the Hagen--Poiseuille law \cite{stokes2007theories},
\begin{align}
    q(z,t,r) = - \frac{\pi r^4}{8 \mu} \pp{p(z,t)}{z},
    \label{hagen-poiseuille}
\end{align}
where $\mu$ is the dynamic viscosity of the fluid, with $[\mu]=\text{Pa\hspace{0.2em}s}$.


The volumetric flow rate per unit cross-sectional area through \emph{all} the pores of size $r$ in the cross-section at depth $z$ is $q(z,t,r) n(z,t,r) \delta r$. 
Therefore, multiplying \eqref{hagen-poiseuille} by the pore-size distribution $n(z,t,r)$ and integrating over $r$, we obtain the Darcy equation for the fluid velocity, $u(z,t) = \int_0^{\infty} q n \, \text{d}r$, that is,
\begin{align}
    u(z,t) = - \frac{K(z,t)}{\mu}  \pp{p(z,t)}{z}.
    \label{darcy}
\end{align}
Here, $K(z,t)$ is the local permeability of the filter,
given by 
\begin{align}
    K(z,t) = \ii{\frac{\pi r^4}{8} n(z,t,r)}{r},
\end{align}
so that $[K]=\text{m}^2$. This 
is a measure of the conductivity of the
filter at depth $z$ at time $t$.

We assume that the fluid is incompressible. As a consequence, in our one-dimensional setting, conservation of fluid mass implies that 
\begin{align}
    u = U(t).
    \label{eq:ssm_conservation-of-mass}
\end{align}

Substituting \eqref{eq:ssm_conservation-of-mass} into 
\eqref{darcy}, 
and integrating with respect to the depth over the interval $z$ to $l$, we find that \eqref{darcy} becomes 
\begin{align}
    p(z,t) = p_{\text{out}} + \mu U(t) \int_z^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta,
    \label{eq:ssm_pressure-solution_general}
\end{align}
where $p_{\text{out}}=p(l)$ is assumed to be a known constant outlet pressure.
There are two industrially relevant regimes for the filter:
\begin{itemize}
    \item the constant flow rate regime, in which $U_{\text{c}}$ is a known constant, and
    \eqref{eq:ssm_conservation-of-mass} and \eqref{eq:ssm_pressure-solution_general} become
    \begin{align}
        U(t) = U_{\text{c}}, 
        \quad
        p(z,t) = p_{\text{out}} + \mu U_{\text{c}} \int_z^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta,
    \end{align}
    \item the constant pressure drop regime, where $p(0,t)=p_{\text{in}}$ is a known constant, and
    \eqref{eq:ssm_conservation-of-mass} and \eqref{eq:ssm_pressure-solution_general} become
    \begin{align}
        U(t) = \frac{\Delta p}{\mu \int_0^l \frac{1}{K(z,t)} \, \text{d}z},
        \quad
        p(z,t) = p_{\text{out}} + \Delta p \frac{\int_z^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta}{\int_0^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta},
    \end{align}
    where $\Delta p = p_{\text{in}}-p_{\text{out}}$ is the known pressure drop across the entire filter.
\end{itemize}

We relate $U$ to the void velocity, $U_{\text{V}}$ with $[U_{\text{V}}]=\text{m}\,\text{s}^{-1}$, which 
is the rate at which fluid flows through the pores, using
\begin{align}
    U(t)=\phi(z,t) U_{\text{V}}(z,t).
    \label{eq:ssm_UV_dim_1}
\end{align}

\subsection{Conservation of particles and pores}
Next, we derive partial differential equations that govern the evolution of the particle- and pore-size distributions, $c$ and $n$, by appealing to conservation of particles and pores as particle capture, and the associated pore clogging, occur.

\subsubsection{Particles}
In order to derive the
equation describing the evolution of the particle-size distribution, we consider a small control volume
in the filter and appeal to conservation of particles. 
We assume that the advective flux
of particles entering and leaving the control volume depends on the void velocity $U_{\text{V}}$,
and that the rate of capture of particles of size $a$ (by all pores in the control volume)
is given by $\mathcal{C}(z, t, a)$. 
Shrinking the control volume to zero, the resulting equation for $c(z, t, a)$ reads
\begin{align}
    \pp{c}{t} + U\frac{\partial}{\partial z}\bigg( \frac{c}{\phi}\bigg) = - \mathcal{C},
    \label{eq:ssm_c-eqn-with-C_dim}
\end{align}
where $[\mathcal{C}]=\text{m}^{-4}\text{s}^{-1}$.

% Since particles are conserved and particles that are captured are no longer advected, it follows that 
% \begin{align}
%     \pp{g}{t}  = \mathcal{C}.
%     \label{eq:ssm_c-eqn-with-C_dim}
% \end{align}

\subsubsection{Pores}
To derive the governing equation for the pore-size distribution, we consider a cross-section
through a control volume and assume that the number of pores of size $r$ decreases
when they become clogged, with rate $\mathcal{N}^{-}(z, t, r)$, and increases when larger pores
become clogged and produce pores of size $r$, with rate $\mathcal{N}^{+} (z, t, r)$. Thus,
we write
\begin{align}
    \pp{n}{t} = - \mathcal{N}^{-} + \mathcal{N}^{+},   
    \label{eq:ssm_n-eqn-with-N_dim}
\end{align}
where $[\mathcal{N}^{-}], [\mathcal{N}^{+}]=\text{m}^{-3}\text{s}^{-1}$.

It remains to formulate relationships between the capture and clogging functions, $\mathcal{C}$, $\mathcal{N}^{-}$, and  $\mathcal{N}^{+}$, and the variables $U$, $\phi$, $c$, and $n$.

\subsubsection{Particle capture and pore clogging}
We focus our attention on the
clogging mechanisms of blocking and deposition (see Figure \ref{fig:ssm_lambda}), as defined in Chapter \ref{chap:ssm_Introduction to the size-structured model}, and
neglect partial surface blocking and caking. 
We treat the contribution of these
two mechanisms separately, then combine them together to determine $\mathcal{C}$, $\mathcal{N}^{-}$ ,
and $\mathcal{N}^{+}$.

\begin{figure}[!htbp]
        \includegraphics[width=0.55\textwidth]{./figures/drawings/lambda.pdf}
        \centering
        \caption{Diagram to illustrate blocking and deposition. 
        The blocking and deposition lengths, $\lambda_{\text{B}}$ and 
        $\lambda_{\text{D}}$, are labelled.}
        \label{fig:ssm_lambda}
\end{figure}

We assume that particles are transported into pores by
the \emph{local} fluid flow. 
Thus, we assume that the local speed down pores of size $r$, $v(z, t, r)$ say, with units 
$[v] = \text{s}^{-1}$, is proportional to the void velocity in these pores by the proportion of the local void area occupied by pores of size $r$, 
so that 
\begin{align}
    v(z,t,r) = U_{\text{V}}(z,t) \bigg(\frac{\pi r^2 n(z,t,r)}{\int_{0}^{\infty} \pi r^2 n(z,t,r) \, \text{d}r}  \bigg).
    \label{eq:ssm_v_dim_1}
\end{align}
Rewriting \eqref{eq:ssm_v_dim_1} in terms of the Darcy velocity and the local porosity by recalling \eqref{eq:ssm_phi_dim_1} and \eqref{eq:ssm_UV_dim_1}, it follows that the local-speed distribution is given by 
\begin{align}
    v(z,t,r) = \frac{U(t) \pi r^2 n(z,t,r)}{\big(\phi(z,t)\big)^2}.
    \label{eq:ssm_v_dim_2}
\end{align}

\paragraph{Blocking}

We recall that pores become blocked when particles that
are \emph{larger} than pores are carried into the entrances of the pores by the local fluid flow. 

When particles of size $a$ block pores of size $r\le a$, the number of pores of size $r$ is reduced. This is accompanied by a corresponding reduction in the number of particles of size $a$. We
thus define
\begin{align}
    \mathcal{N}^{-}_{\text{B}}(z,t,r) = v(z,t,r) \int_r^{\infty} c(z,t,a) \, \text{d}a,
    \label{eq:ssm_N-minus_blocking_dim}
\end{align}
and 
\begin{align}
    \mathcal{C}_{\text{B}}(z,t,a) =  \frac{1}{\lambda_{\text{B}}} c(z,t,a) \int_0^{a} v(z,t,r) \, \text{d}r,    
    \label{eq:ssm_C_blocking_dim}
\end{align}
where the subscript B indicates blocking. 
Here, $\lambda_{\text{B}}$ (with $[\lambda_{\text{B}}]=\text{m}$) is the distance over which blocking occurs, which we assume to be
$\mathcal{O}(\bar{a}^{*})$ and call the blocking length (see Figure \ref{fig:ssm_lambda}).


\paragraph{Deposition}

We recall that deposition occurs when particles that are
\emph{smaller} than pores are carried into pores by the local fluid-flow and become
captured.

We define $\varsigma_{\text{D}}$ to be the adherence, which is the probability of capture once a particle enters a pore, where the subscript D indicates deposition.
In general, this might depend on time (e.g., since the probability of adhering to a `clean' pore wall is likely to be different from the probability of adhering to a pore wall that already contains deposited particles), 
position (e.g., since the filter may be manufactured for larger adhesivity at some depths than others),
and the sizes of the particle and pore (e.g., since a larger particle is more likely to move closer to a pore wall and thus more likely to stick), as well as any other variables or parameters of the system, or else an equation 
describing additional physical or chemical relationships between the particle and pore materials. 
To demonstrate a simple case, we assume that the adherence depends only on the size of the particle and pore, so that $\varsigma_{\text{D}}=\varsigma_{\text{D}}(a,r)$.
Note that the adherence is dimensionless, and satisfies $0\le \varsigma_{\text{D}} \le 1$.

When particles of size $a$ deposit in pores of size $r>a$, the number of pores of size $r$ is reduced,  and this is accompanied by a corresponding reduction in the number of particles of size $a$. 
We therefore define
\begin{align}
    \mathcal{N}^{-}_{\text{D}}(z,t,r) = v(z,t,r) \int_0^{r} c(z,t,a) \varsigma_{\text{D}}(a,r) \, \text{d}a,
    \label{eq:ssm_N-minus_deposition_dim}
\end{align}
and 
\begin{align}
    \mathcal{C}_{\text{D}}(z,t,a) =  \frac{1}{\lambda_{\text{D}}} c(z,t,a) \int_a^{\infty} v(z,t,r) \varsigma_{\text{D}}(a,r) \, \text{d}r.
    \label{eq:ssm_C_deposition_dim}
\end{align}
Here, $\lambda_{\text{D}}$, with $[\lambda_{\text{D}}]=\text{m}$, is a measure of the distance over which a particle is removed by sticking to the pore wall, which we assume to be
$\mathcal{O}(\bar{a}^{*})$ and call the deposition length (see Figure \ref{fig:ssm_lambda}).

We suppose that deposition results in the transformation of pores of their original size to pores of a new size. 
We define $\tau_{\text{D}}$ to be the transformation, which is the probability that deposition of particles of size $a$ in pores of size $r>a$ results in pores of size $\rho<r$. 
Much like the adherence, in general this might depend on time, position, and the relative sizes of the particle and pore, or may be given by an equation that describes the physics and chemistry of the deposition mechanism. Again, for simplicity, we assume that this depends only on the sizes of the particle, the pore that the particle enters, and the pore that results from the deposition, so that $\tau_{\text{D}}(a,r,\rho)$.
% \begin{align}
%     \tau_{\text{D}} = \tau_{\text{D}}(a,r,\rho),
% \end{align}
Note that this is a probability density function over $\rho$ for any $a,r$, 
so that $[\tau_{\text{D}}]=\text{m}^{-1}$, and it must be chosen to satisfy 
\begin{align}
    \ii{\tau_{\text{D}}(a,r,\rho)}{\rho} = 1.
    \label{eq:tau-constraint_dim}
\end{align}
Thus, to account for the production of pores of size $r$ due to deposition, we write
\begin{align}
    \mathcal{N}_{\text{D}}^{+}(z,t,r) = \int_{0}^{\infty} v(z,t,\rho) \int_{0}^{\rho} c(z,t,a) \varsigma_{\text{D}}(a,\rho) \tau_{\text{D}}(a,\rho,r) \, \text{d}a \, \text{d}\rho.
    \label{eq:ssm_N-plus_deposition_dim}
\end{align}
Note that we have swapped $r$ and $\rho$ in $\tau_{\text{D}}$ (and $v$) since we are interested in the
production of pores of size $r$ from pores of size $\rho>r$ in \eqref{eq:ssm_N-plus_deposition_dim}.

To combine the effects of blocking and deposition, 
we write 
\begin{align}
    \mathcal{C} = \mathcal{C}_{\text{B}} + \mathcal{C}_{\text{D}},
    \quad
    \mathcal{N}^{+} = \mathcal{N}_{\text{D}}^{+},
    \quad
    \mathcal{N}^{-} = \mathcal{N}_{\text{B}}^{-} + \mathcal{N}_{\text{D}}^{-}.
    \label{eq:ssm_clogging-and-capture_dim}
\end{align}

\subsection{Boundary and initial conditions}

We assume that the concentration of particles in the feed, $C_{\text{in}}^*$, is known, and so we write
\begin{align}
c(0,t,a) = 
C_{\text{in}}^*\phi(0,t) d_{c}(a), \label{c_in_dim_1}
\end{align}
where $\phi$ accounts for the fact that all particles in the feed fluid enter void space when they enter the filter. 
Here, $d_c(a)$ is the distribution of particle sizes in the feed, which is normalised over $a$ so that
\begin{align}
    \ii{d_c(a)}{a} = 1.
    \label{eq:dc-constraint_dim}
\end{align}
Initially, we suppose that there are no particles in the filter, and that the pore distribution is given, but that it is not necessarily constant across the filter.
Thus, we write 
\begin{subequations}
\label{eq:ssm_init-conds_1}
    \begin{align}
c(z,0,a) = 0, 
\quad
n(z,0,r) =  
N_{\text{init}}(z) d_{n}(z,r), 
\tag{\ref{eq:ssm_init-conds_1}a,b}
\end{align}
\end{subequations}
where $N_{\text{init}}(z)$ is the concentration of pores at depth $z$ initially, and $d_n(z,r)$ is the distribution of their sizes, which is normalised over the pore size $r$ so that 
\begin{align}
    \ii{d_n(z,r)}{r} = 1.
    \label{eq:dn-constraint_dim}
\end{align}
Experimentally, it is challenging to quantify $N_{\text{init}}(z)$ directly, since characterisation of
the internal filter structure requires filter destruction. 
Instead, we typically know the
initial porosity of a cross-section, $\phi_{\text{init}}(z)$, and infer $N_{\text{init}}(z)$ using   \eqref{eq:ssm_phi_dim_1}, so that
\begin{align}
N_{\text{init}}(z)
= \frac{\phi_{\text{init}}(z)}{\int_0^\infty \pi r^2 d_n(z,r) \, \text{d}r}.
\label{eq:ssm_Ninit_dim}
\end{align}


\subsection{Operational regimes}
It remains to specify the forms of the functions $\phi_{\text{init}}$, $d_c$, $d_n$, 
$\varsigma_{\text{D}}$, and $\tau_{\text{D}}$, 
which 
depend on the industrial application of the filter.
Here, we specify forms for these functions that represent operational regimes. We use these to demonstrate the solution of the model in the following chapter.

To simplify our analysis, we focus on cases in which the 
local porosity does not vary across the filter initially, so that
\begin{align}
    \phi_{\text{init}}(z) = \phi_{\text{init}}^{*}.
    \label{eq:ssm_phi-choice_dim}
\end{align}

We assume that particles at the inlet are either mono-dispersed, of size $\bar{a}^*$,
or log-normally dispersed, with mean size $\bar{a}^*$. 
It follows that either,
\begin{subequations}
\label{eq:ssm_distr_c_dim}
\begin{align}
    d_c(a)=\updelta(a-\bar{a}^*), 
    \quad \text{or}, \quad
    d_c(a)=\frac{1}{\sqrt{2 \pi} \sigma_c a} 
    e^{- \frac{ 
    \left( 
    \log \left( \frac{a}{\bar{a}^*} \right) + \frac{\sigma_c^2}{2} 
    \right)^2
    }{
    2 \sigma_c^2}},
    \tag{\ref{eq:ssm_distr_c_dim}a,b}    
    \end{align}
\end{subequations}
where $\sigma_c$ is the dimensionless standard deviation of the normal distribution associated with (\ref{eq:ssm_distr_c_dim}b).
We note that (\ref{eq:ssm_distr_c_dim}a) and 
(\ref{eq:ssm_distr_c_dim}b) both satisfy the constraint \eqref{eq:dc-constraint_dim}.
% The condition on the particle-size distribution 
% at the inlet boundary, \eqref{c_in_dim_1}, becomes
% either 
% \begin{subequations}
% \label{eq:ssm_c_in_dim}
% \begin{align}
%     c(0,t,a)=C^*_{\text{in}} 
%     \phi(0,t) \updelta(a-\bar{a}^*), 
%     \quad \text{or} \quad
%     c(0,t,a) = C^*_{\text{in}} 
%     \phi(0,t) \frac{1}{\sqrt{2 \pi} \sigma_c a} 
%     e^{- \frac{ 
%     \left( 
%     \log \left( \frac{a}{\bar{a}^*} \right) + \frac{\sigma_c^2}{2} 
%     \right)^2
%     }{
%     2 \sigma_c^2}},
%     \tag{\ref{eq:ssm_c_in_dim}a,b}    
%     \end{align}
% \end{subequations}
% when either (\ref{eq:ssm_distr_c_dim}a) or 
% (\ref{eq:ssm_distr_c_dim}b) is chosen, respectively.

Similarly, we assume that, initially, pores in the filter are either mono-dispersed, and of size $\bar{r}^{*}$, or log-normally dispersed, with mean size $\bar{r}^{*}$, independently of depth, so that either
\begin{subequations}
\label{eq:ssm_distr_n_dim}
\begin{align}
    d_n(r)=\updelta(r-\bar{r}^*), 
    \quad \text{or} \quad
    d_n(r)=\frac{1}{\sqrt{2 \pi} \sigma_n r} 
    e^{- \frac{ 
    \left( 
    \log \left( \frac{r}{\bar{r}^*} \right) + \frac{\sigma_n^2}{2} 
    \right)^2
    }{
    2 \sigma_n^2}}.
    \tag{\ref{eq:ssm_distr_n_dim}a,b}    
    \end{align}
\end{subequations}
Here $\sigma_n$ is the dimensionless standard deviation of the normal distribution associated with (\ref{eq:ssm_distr_n_dim}b), and we note that (\ref{eq:ssm_distr_c_dim}a) and 
(\ref{eq:ssm_distr_c_dim}b) satisfy the constraint \eqref{eq:dn-constraint_dim}.
% The initial pore concentration, given by \eqref{eq:ssm_Ninit_dim}, 
% becomes a constant, given by either
% \begin{subequations}
% \label{eq:ssm_Ninit-op_dim}
% \begin{align}
%     N^*_{\text{init}} = 
%     \frac{\phi^*_{\text{init}}}{\pi (\bar{r}^*)^2},
%         \quad 
%         \text{or} 
%         \quad
%     N^*_{\text{init}} = 
%     \frac{\phi^*_{\text{init}}}{
%     \frac{\sqrt{\pi}}{\sqrt{2} \sigma_n }
%     \int_0^\infty 
%      r
%     \text{exp}\left(- \frac{ 
%     \left( 
%     \log \left( r/\bar{r}^* \right) + \sigma_n^2/2 
%     \right)^2
%     }{
%     2 \sigma_n^2}\right)
%     \, \text{d}r},
%     \tag{\ref{eq:ssm_Ninit-op_dim}a,b}
% \end{align}
% \end{subequations}
% so that 
% the initial condition on the pore-size distribution, (\ref{eq:ssm_init-conds_dim_summary_1}b), becomes
% either 
% \begin{subequations}
% \label{eq:ssm_c_in_dim}
% \begin{align}
%     n(z,0,r)= \frac{\phi^*_{\text{init}} \updelta(r-\bar{r}^*)}{\pi (\bar{r}^*)^2}, 
%     \quad \text{or} \quad
%     n(z,0,r) = 
%     \frac{ \phi^*_{\text{init}} 
%     \text{exp}\left(- \frac{ 
%     \left( 
%     \log \left( r/\bar{r}^* \right) + \sigma_n^2/2 
%     \right)^2
%     }{
%     2 \sigma_n^2}\right)}{
%     \pi r
%     \int_0^\infty 
%      r
%     \text{exp}\left(- \frac{ 
%     \left( 
%     \log \left( r/\bar{r}^* \right) + \sigma_n^2/2 
%     \right)^2
%     }{
%     2 \sigma_n^2}\right)
%     \, \text{d}r},
%     \tag{\ref{eq:ssm_c_in_dim}a,b}    
%     \end{align}
% \end{subequations}
% when either (\ref{eq:ssm_distr_c_dim}a) or 
% (\ref{eq:ssm_distr_c_dim}b) is chosen, respectively.


Furthermore, we assume that the probability that particles of size $a$ are captured in pores of size $r>a$ is the same regardless of these sizes. As a result, the adherence is constant, so that
\begin{align}
\varsigma_{\text{D}}(a,r)=\varsigma_{\text{D}}.
\label{eq:ssm_sigma-choice_dim}
\end{align}

Lastly, we suppose that when particles of size $a$ are captured in pores of size $\rho>a$, the resulting 
pore is always size $r=\rho-a$, so that 
\begin{align}
    \tau_{\text{D}}(a,\rho,r)= \updelta(r-(\rho-a)), 
    \label{eq:tau-choice_dim}
\end{align}
which 
satisfies \eqref{eq:tau-constraint_dim}.
Note that, even the simplification that $\tau_{\text{D}}=\tau_{\text{D}}(a,r,\rho)$ allows for a huge number of possible forms for $\tau_{\text{D}}$. 
Our choice in \eqref{eq:tau-choice_dim} corresponds to the simple case where deposition of a particle of size $a$ in a pore of size $\rho$ \emph{always} results in a pore of size $r=\rho-a$. 
We might instead consider, for example, the case where $\mu_{\text{D}}=\rho-a$ is the mean resulting pore size and resulting pore sizes are distributed normally about this, so that $\tau(a,\rho,r)_{\text{D}} \sim \text{exp}((\mu-r)/\sqrt{2}\sigma_{\text{D}})^2$. 
The standard deviation here, $\sigma_{\text{D}}$, may model, for example, the extent to which soft particles are deformed when deposition occurs, and the resulting variance in the possible resulting pore size. 
Any probability distribution (over $\rho$) is permitted, allowing for models that account, for example, for the `spreading' of particle mass when deposition occurs.

\subsection{Summary of the dimensional model}

In summary, substituting the specific forms of the blocking and deposition functions, \eqref{eq:ssm_N-minus_blocking_dim}--\eqref{eq:ssm_N-plus_deposition_dim}, into the clogging functions \eqref{eq:ssm_clogging-and-capture_dim}, and using these with the local speed distribution \eqref{eq:ssm_v_dim_2} in the particle and pore equations, \eqref{eq:ssm_c-eqn-with-C_dim}--\eqref{eq:ssm_n-eqn-with-N_dim}, 
the equations that govern the size distributions of particles and pores read
\begin{align}
    \label{eq:ssm_c-eqn_summary_dim}
    \pp{c}{t} + U\frac{\partial}{\partial z}\bigg( \frac{c}{\phi}\bigg) &= 
    -
    \frac{\pi U}{\phi^2} c 
    \Bigg(
    \frac{1}{\lambda_{\text{B}}} \int_0^{a} r^2 n \, \text{d}r
    +
    \frac{1}{\lambda_{\text{D}}} \int_a^{\infty}  \varsigma_{\text{D}} r^2 n\, \text{d}r \Bigg),
    \\
    \pp{n}{t} &=
    - \frac{\pi U}{\phi^2}
    \Bigg(
    r^2 n
    \bigg(
    \int_r^{\infty} c \, \text{d}a
    +
    \int_0^{r} \varsigma_{\text{D}} c  \, \text{d}a
    \bigg)
    \\
    & \quad \quad \quad \quad
    - 
    \int_{0}^{\infty} \rho^2 n(z,t,\rho) \int_{0}^{\rho} \varsigma_{\text{D}} c    \tau_{\text{D}}(a,\rho,r) \, \text{d}a \, \text{d}\rho
    \Bigg).
    \nonumber
\end{align}
Depending on the filtration regime, either
\begin{align}
    U(t) = U_{\text{c}}, 
    \quad
    p(z,t) = p_{\text{out}} + \mu U_{\text{c}} \int_z^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta,
\end{align}
or 
\begin{align}
    U(t) = \frac{\Delta p}{\mu \int_0^l \frac{1}{K(z,t)} \, \text{d}z},
    \quad
    p(z,t) = p_{\text{out}} + \Delta p \frac{\int_z^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta}{\int_0^l \frac{1}{K(\zeta,t)} \, \text{d}\zeta},
\end{align}
are the equations for the flow rate and the pressure, 
where 
\begin{align}
\phi(z,t) = \int_0^\infty \pi r^2 n(z,t,r) \, \text{d}r,
\quad
K(z,t) = \int_0^\infty \frac{\pi r^4}{8} n(z,t,r) \, \text{d}r
\label{eq:ssm_perm_dim_summary} 
\end{align}
define the local porosity and permeability, respectively.
The boundary condition at the inlet at the top of the filter is
\begin{align}
c(0,t,a) = 
C_{\text{in}}^*\phi(0,t) d_{c}(a), \label{c_in_dim_summary_1}
\end{align}
and the initial conditions are 
\begin{align}
c(z,0,a) = 0, 
\quad
n(z,0,r) =  
\frac{\phi_{\text{init}}(z) d_{n}(z,r)}{\int_0^\infty \pi s^2 d_{n}(z,s) \, \text{d}s}.
\label{eq:ssm_init-conds_dim_summary_1}
\end{align}



The equations \eqref{eq:ssm_c-eqn_summary_dim}--\eqref{eq:ssm_init-conds_dim_summary_1} form a closed system for the six solution variables $c$, $n$, $U$, $p$, $\phi$, $K$, 
subject to the specification of the five functions $\phi_{\text{init}}$, $d_{c}$, $d_{n}$, $\varsigma_{\text{D}}$, and $\tau_{\text{D}}$,
 which depend on the choice of operational regime, and any numbers that parametrise these, for example the five parameters $\bar{r}^*, \bar{a}^*$, $\sigma_c$, $\sigma_n$, and $\phi_{\text{init}}$, when these functions are chosen as in \eqref{eq:ssm_distr_c_dim}--\eqref{eq:tau-choice_dim}. 
The six parameters $C_{\text{in}}^{*}$, $\lambda_{\text{B}}$, $\lambda_{\text{D}}$, $\mu$, $l$, and $p_{\text{out}}$, must also be specified, as well as either $U_{\text{c}}$
or 
$p_{\text{in}}$, 
depending on whether a constant flow rate or pressure drop is applied in the filter.



\section{Dimensionless model}
\label{sec:ssm_Dimensionless model}
Next, in this section, we nondimensionalise the size-structured model \eqref{eq:ssm_c-eqn_summary_dim}--\eqref{eq:ssm_init-conds_dim_summary_1}, summarise the resulting dimensionless system, and discuss the associated dimensionless parameter groups.

\subsection{Nondimensionalisation}
We begin by defining $U^{*}$ to be the characteristic Darcy velocity (flux per unit cross-sectional area)
of the fluid, and
$\Delta p^{*}$ to be the characteristic deviation 
of the pressure from the known pressure at the outlet, 
$p_{\text{out}}$.
These depend on the operating regime:
\begin{itemize}
    \item In the constant flow rate regime, we have
    \begin{align}
        U^{*} = U_{\text{c}}, 
        \quad 
        \Delta p^{*} = \frac{8 \mu U_{\text{c}} l}{\bar{r}^{*2}};
    \end{align}
    \item In the constant pressure drop regime, we have
    \begin{align}
        U^{*} = \frac{\Delta p \bar{r}^{*2}}{\mu l},
        \quad
        \Delta p^{*} = \Delta p.
    \end{align}
\end{itemize}
Given these values, we nondimensionalise \eqref{eq:ssm_c-eqn_summary_dim}--\eqref{eq:ssm_init-conds_dim_summary_1} 
using
\begin{align}
    z=l\tilde{z}, 
    \quad
    a=\bar{r}^{*}\tilde{a}, 
    \quad 
    r=\bar{r}^{*}\tilde{r},
    \quad
    U = U^{*} \tilde{U},
    \quad
    p = p_{\text{out}} + \Delta p^{*} \tilde{p},
    \quad
    t = \frac{l}{U^{*}} \tilde{t},
    \label{eq:ssm_nondim_1}
\end{align} 
\begin{align}
    n = \frac{1}{\pi \bar{r}^{*3}} \tilde{n},
    \hspace{0.5em}
    c = \frac{C_{\text{in}}^*}{\bar{r}^*} \tilde{c},
    \hspace{0.5em}
    d_{n} = \frac{1}{\bar{r}^*} \tilde{d}_{n}, 
    \hspace{0.5em}
    d_{c} = \frac{1}{\bar{r}^*} \tilde{d}_{c},
    \hspace{0.5em}
    \tau_{\text{D}} = \frac{1}{\bar{r}^*} \tilde{\tau}_{\text{D}}, 
    \hspace{0.5em}
    K = \frac{\bar{r}^{*2}}{8} \tilde{K},
    \label{eq:ssm_nondim_2}
\end{align}
where variables with tildes are dimensionless.

In \eqref{eq:ssm_nondim_1} we see that the spatial variable is scaled so that the filter has unit length.
The particle and pore sizes are scaled so that 
a pore of mean radius has unit size, and the particle size is measured relative to this.
The timescale, 
\begin{align}
    t^* = \frac{l}{U^*}
\end{align}
is the time that it takes for fluid to move through the filter.

In \eqref{eq:ssm_nondim_2}, we scale the pore-size distribution, $n$, and $d_{n}$, so that the initial distribution, defined 
in \eqref{eq:ssm_init-conds_dim_summary_1}, is a probability density function that integrates to one.
Similarly, we scale the particle-size distributions, $c$ and $d_{c}$, so that
the distribution at the inlet integrates to one, and for unit particle concentration in void space at the inlet. 
Furthermore, the transformation distribution, $\tau_{\text{D}}$, is scaled so that it remains a probability density function that integrates to one, and we scale the local permeability $K$ to balance terms in \eqref{eq:ssm_perm_dim_summary}. 

\subsection{Summary of the dimensionless model}
In dimensionless variables, dropping tildes,
the size-structured model, \eqref{eq:ssm_c-eqn_summary_dim}--\eqref{eq:ssm_init-conds_dim_summary_1},  becomes
\begin{align}
    \label{eq:ssm_c-eqn_summary_nodim}
    \pp{c}{t} + U\frac{\partial}{\partial z}\bigg( \frac{c}{\phi}\bigg) &= 
    -
    \frac{U}{\phi^2} c 
    \Bigg(
    \Lambda_{\text{B}} \int_0^{a} r^2 n \, \text{d}r
    +
    \Lambda_{\text{D}} \int_a^{\infty} \varsigma_{\text{D}} r^2 n  \, \text{d}r \Bigg),
    \\
    \label{eq:ssm_n-eqn_summary_nodim}
    \pp{n}{t} &=
    - \Gamma \frac{U}{\phi^2}
    \Bigg(
    r^2 n
    \bigg(
    \int_r^{\infty} c \, \text{d}a
    +
    \int_0^{r} \varsigma_{\text{D}} c  \, \text{d}a
    \bigg)
    \\
    & \quad \quad \quad \quad
    - 
    \int_{0}^{\infty} \rho^2 n(z,t,\rho) \int_{0}^{\rho}   \varsigma_{\text{D}}  c  \tau_{\text{D}}(a,\rho,r) \, \text{d}a \, \text{d}\rho
    \Bigg),
    \nonumber
\end{align}
where, either
\begin{align}
    U(t) = 1, 
    \quad
    p(z,t) = \int_z^1 \frac{1}{K(\zeta,t)} \, \text{d}\zeta,
        \label{eq:ssm_Up-eqns_const-flow_summary_nodim}
\end{align}
or 
\begin{align}
    U(t) = \frac{1}{\int_0^1 \frac{1}{K(z,t)} \, \text{d}z},
    \quad
    p(z,t) = \frac{\int_z^1 \frac{1}{K(\zeta,t)} \, \text{d}\zeta}{\int_0^1 \frac{1}{K(\zeta,t)} \, \text{d}\zeta},
        \label{eq:ssm_Up-eqns_const-pres_summary_nodim}
\end{align}
define the flow rate and the pressure,
depending on the filtration regime, 
and 
the local porosity and permeability are given by
\begin{align}
\phi(z,t) = \int_0^\infty r^2 n(z,t,r) \, \text{d}r,
\quad
K(z,t) = \int_0^\infty r^4 n(z,t,r) \, \text{d}r
\label{eq:ssm_perm-and-poros_nodim_summary} 
\end{align}
respectively.
The boundary condition at the inlet at the top of the filter is
\begin{align}
c(0,t,a) = \phi(0,t) d_{c}(a), \label{eq:ssm_bound-cond_nodim_summary}
\end{align}
and the initial conditions are 
\begin{subequations}
\label{eq:ssm_init-conds_nodim_summary}
\begin{align}
c(z,0,a) = 0, 
\quad
n(z,0,r) =  
\frac{\phi_{\text{init}}(z) d_{n}(z,r)}{\int_0^\infty s^2 d_{n}(z,s) \, \text{d}s}.
\tag{\ref{eq:ssm_init-conds_nodim_summary}a,b}
\end{align}
\end{subequations}

The dimensionless analogues of 
the functions that specify 
the operational regime, given in 
\eqref{eq:ssm_phi-choice_dim}--\eqref{eq:tau-choice_dim}, 
are 
\begin{subequations}
\label{eq:ssm_distr_c_nodim_summary}
\begin{align}
    d_c(a)=\updelta(a-A), 
    \quad \text{or} \quad
    d_c(a)=\frac{1}{\sqrt{2 \pi} \sigma_c a} 
    e^{- \frac{ 
    \left( 
    \log \left( \frac{a}{A} \right) + \frac{\sigma_c^2}{2} 
    \right)^2
    }{
    2 \sigma_c^2}},
    \tag{\ref{eq:ssm_distr_c_nodim_summary}a,b} 
    \end{align}
\end{subequations}
so that particles are either all size $A$ or this is their mean size, and
\begin{subequations}
\label{eq:ssm_distr_n_nodim_summary}
\begin{align}
    d_n(r)=\updelta(r-1), 
    \quad \text{or} \quad
    d_n(r)=\frac{1}{\sqrt{2 \pi} \sigma_n r} 
    e^{- \frac{ 
    \left( 
    \log \left( r \right) + \frac{\sigma_n^2}{2} 
    \right)^2
    }{
    2 \sigma_n^2}},
    \tag{\ref{eq:ssm_distr_n_nodim_summary}a,b} 
    \end{align}
\end{subequations}
so that, similarly, pores are either all size $1$ or this is their mean size. Also, we have
\begin{subequations}
\label{eq:ssm_phi-init-tau-sig-choice_nodim_summary}
    \begin{align}
    \phi_{\text{init}}(z) = \phi_{\text{init}}^{*},
    \quad
    \varsigma_{\text{D}}(a,r)=\varsigma_{\text{D}},
    \quad
    \tau_{\text{D}}(a,\rho,r) = \updelta(r-(\rho-a)).
        \tag{\ref{eq:ssm_phi-init-tau-sig-choice_nodim_summary}a,b,c}    
\end{align}
\end{subequations}

The equations \eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_init-conds_nodim_summary} form a closed system for the six dimensionless variables $c$, $n$, $U$, $p$, $\phi$, $K$, 
subject to the specification of the operational regime via the five dimensionless functions 
$d_c$, $d_n$,
$\phi_{\text{init}}$, $\varsigma_{\text{D}}$, $\tau_{\text{D}}$. 
Equations \eqref{eq:ssm_distr_c_nodim_summary}--\eqref{eq:ssm_phi-init-tau-sig-choice_nodim_summary} 
are the examples that we will use for these, 
which are parametrised by the four dimensionless numbers:
\begin{itemize}
    \item the size ratio,
    \begin{align}
        A = \frac{\bar{a}^*}{\bar{r}^*},
    \end{align}
    which is the particle size if they are mono-dispersed, as in (\ref{eq:ssm_distr_c_nodim_summary}a), or the mean particle-size if they are log-normally dispersed, as in (\ref{eq:ssm_distr_c_nodim_summary}b) (since $\text{exp}(\mu+\sigma_c^2/2) = A$, where $\mu=\log(A)-\sigma_c^2/2$ is the mean of the normal distribution associated with the log-normal distribution (\ref{eq:ssm_distr_c_nodim_summary}b));
    \item the particle- and pore-size distribution widths, $\sigma_c, \sigma_n$.
    These are the standard deviations of the normal distributions associated with the log-normal distributions (\ref{eq:ssm_distr_c_nodim_summary}b) and (\ref{eq:ssm_distr_n_nodim_summary}b);
    % and thus specify the widths of the particle- and pore-size distributions, respectively, when these are log-normal, as in (\ref{eq:ssm_distr_c_nodim_summary}b) and (\ref{eq:ssm_distr_n_nodim_summary}b),
    \item the initial porosity, $\phi_{\text{init}}^*$,
    which specifies the ratio of pore space to impermeable-material space in the 
    filter at all depths initially.
    \end{itemize}

The system is parametrised by three other dimensionless numbers that result from the specification of the dimensional parameters. These are:
\begin{itemize}
    \item the blocking-length ratio,
    \begin{align}
        \Lambda_{\text{B}} = \frac{l}{\lambda_{\text{B}}},
    \end{align}
    which is the ratio of the filter length to the blocking length;
    \item the deposition-length ratio,
    \begin{align}
        \Lambda_{\text{D}} = \frac{l}{\lambda_{\text{D}}},
    \end{align}
    which is the ratio of the filter length to the deposition length;
    \item the concentration ratio,
    \begin{align}
        \Gamma = \frac{C_{\text{in}}^* l/\bar{r}^*}{1/\pi \bar{r}^{*3}},
        \label{eq:ssm_definition-of-Gamma}
    \end{align}
    which is the ratio of the characteristic concentrations of particles and pores.
\end{itemize}
In practice, we suppose 
that there are at least as many pores as particles per unit area in filters, 
and that filters are much thicker than both the blocking and deposition lengths.  
Thus, we anticipate that 
$\Gamma\ll1$ (or at most that $\Gamma=\mathcal{O}(1)$), 
and that 
$\Lambda_{\text{B}}\gg1$ 
and 
$\Lambda_{\text{D}}\gg1$. 


\chapter{Solutions of the size-structured model}
\label{chap:ssm_Solutions of the size-structured model}
In this chapter, 
we explore solutions of the dimensionless 
size-structured model,
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma},
in various operational regimes.
In Section \ref{sec:ssm_solution-methodology}, we briefly discuss our method of solution.
Then, in Section \ref{sec:ssm_Mono-dispersed particles and mono-dispersed pores}, we consider the case where feed particle sizes are mono-dispersed, and pore sizes are mono-dispersed initially, and remain mono-dispersed or bi-dispersed for all time. 
In Section \ref{sec:ssm_Mono-dispersed particles and poly-dispersed pores}, we focus on the case where the particles are still of a single size, 
but the pore sizes are initially log-normally poly-dispersed.
% Finally, in Section \comment{[]}, 
% we consider the cases where both the particles at the inlet and pores in the filter initially are log-normally poly-dispersed.

\section{Solution methodology}
\label{sec:ssm_solution-methodology}
In general,
the size-structured model, 
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma}, 
is a hyperbolic system of partial integro--differential equations. 
At most, solution variables depend on the single spatial variable, $z$, 
the time variable, $t$, 
and the size variables, $a$ and $r$, 
which behave much like additional spatial variables.
The system is thus a four-dimensional problem.
In the following sections, we will demonstrate that the system can be solved explicitly or asymptotically in various parameter regimes. 
For general parameter regimes, however, we must appeal to computational methods for solution. 

We use the method of lines. That is, we use the method of finite differences to discretise the space and size variables, and then use an initial value problem solver (\texttt{SciPy.solve_ivp()} in \texttt{Python}) to solve the resulting system of ordinary differential equations (in the time variable).
For simplicity, we use a first-order upwind scheme to discretise the spatial variable, and the trapezium rule to discretise the integrals in the size variables.
We note that higher order schemes, such as Lax-Wendroff, are available, and would lead to faster solution convergence.
In our chosen setup, we find that $500$ spatial points, and $100$ particle and pore size points are enough for convergence.

The advective term in the advection--reaction equation contains the ratio $U/\phi$. 
As a result, the difference between discrete time points needed, $\delta t$ say, depends on the desired final state of the solution. 
To see this, note that we must satisfy the CFL condition, and, therefore, we must ensure that
$\delta t$ is small enough to allow for $U/\phi$ to get large when $\phi$ becomes small near the inlet when the filter is almost completely clogged (i.e., near the end of simulation). 
In practice, we find that $\delta t/\delta z < \phi(0,T)/\phi(0,0)$ is sufficient (since $U$ decreases as a function of time), where $\delta z$ is the difference between spatial points and $T$ is the terminal time.
For example, we simulate the case where the terminal inlet porosity is around $10\%$ of the initial inlet porosity, $\phi(0,T)\approx0.1\phi(0,0)$, and so $\delta t \approx 0.1 \delta z$ is sufficient, and we choose $5000$ temporal points (for $500$ spatial points).

\section[Mono-dispersed particles and mono-dispersed or bi-dispersed pores]{Mono-dispersed particles and mono-dispersed\\or bi-dispersed pores}
\label{sec:ssm_Mono-dispersed particles and mono-dispersed pores}
In this section, we assume that the particles entering the filter at the inlet are all size $A$ (which, from (\ref{eq:ssm_distr_c_nodim_summary}), is constant and not to be confused with the variable $a$), and the pores are all size $1$ initially.
If $A\ge1$ then all particles are larger than all pores, and we expect blocking to occur when a particle enters a pore, 
while if $A<1$ then all particles are smaller than pores, and we expect deposition to occur (with probability $\varsigma_{\text{D}}$) when a particle enters a pore.
We choose (\ref{eq:ssm_distr_c_nodim_summary}a) 
as the distribution of particle-sizes at the inlet, 
so that the boundary condition, \eqref{eq:ssm_bound-cond_nodim_summary}, 
becomes 
\begin{align}
    c(0,t,a) = \phi(0,t) \updelta(a-A).
    \label{eq:ssm_bound-cond_mono-mono}
\end{align}
Further, we choose (\ref{eq:ssm_distr_n_nodim_summary}a) 
as the initial pore-size distribution, so that the initial conditions, 
\eqref{eq:ssm_init-conds_nodim_summary}, 
become
\begin{subequations}
\label{eq:ssm_init-conds_mono-mono}
\begin{align}
c(z,0,a) = 0, 
\quad
n(z,0,r) =  
\phi^*_{\text{init}} \updelta(r-1).
\tag{\ref{eq:ssm_init-conds_mono-mono}a,b}
\end{align}
\end{subequations}
Thus we solve the system 
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
subject to the boundary and initial conditions 
\eqref{eq:ssm_bound-cond_mono-mono}, 
\eqref{eq:ssm_init-conds_mono-mono}. 
We consider first the case $A\ge1$
(when only blocking, and not deposition,
occurs), 
look for explicit asymptotic solutions in the cases $\Gamma\ll1$ (i.e., where the particle concentration is much smaller than the pore concentration) and $\Lambda_{\text{B}}\gg 1$ 
(i.e., where the filter length is much larger than the length over which blocking occurs),
and compare these with a numerical solution. 
We then consider the case $A<1$ in which blocking and deposition both occur, which gives rise to a second pore size, so that the pores become bi-dispersed.

\subsection{Blocking and no deposition}
\label{subsec:ssm_mono-mono_blocking}
% Noting that the case $A<1$ could lead to the emergence of pores or size $0<r<1$, 
We choose $A\ge 1$, since we first want to consider the simple case in which the pore sizes are mono-dispersed for all time. 
This choice allows us to pose the ansatz
\begin{align}
    c(z,t,a) = \updelta(a-A) C(z,t), 
    \quad
    n(z,t,r) = \updelta(r-1) N(z,t).
\end{align}
Using this ansatz, \eqref{eq:ssm_c-eqn_summary_nodim} and \eqref{eq:ssm_n-eqn_summary_nodim}
 become
 \begin{align}
     \pp{C}{t} + U\frac{\partial }{\partial z}\bigg(\frac{C}{N}\bigg)
     & = 
     - \Lambda_{\text{B}} U \frac{C}{N},
     \label{eq:ssm_c-eqn_mono}
     \\
     \pp{N}{t} = - \Gamma U \frac{C}{N}.
     \label{eq:ssm_n-eqn_mono}
 \end{align}
In the constant flow rate case it follows from 
\eqref{eq:ssm_Up-eqns_const-flow_summary_nodim} that the Darcy velocity and pressure are given by 
\begin{align}
    U=1,
    \quad 
    p(z,t) = \int_z^1\frac{1}{N(\zeta,t)}\, \text{d}\zeta,
    \label{eq:ssm_Up-eqns_const-flow_mono}
\end{align}
while in the constant pressure drop case 
it follows from 
\eqref{eq:ssm_Up-eqns_const-pres_summary_nodim}
 that these are given by 
 \begin{align}
     U(t) = \frac{1}{\int_0^1\frac{1}{N(z,t)}\, \text{d}z}, 
     \quad 
     p(z,t) = \frac{\int_z^1\frac{1}{N(\zeta,t)}\, \text{d}\zeta,}{\int_0^1\frac{1}{N(\zeta,t)}\, \text{d}\zeta},
    \label{eq:ssm_Up-eqns_const-pres_mono}
 \end{align}
 since, for mono-dispersed pores, the local porosity and permeability, given by \eqref{eq:ssm_perm-and-poros_nodim_summary}, 
become 
\begin{align}
    \phi(z,t)=N(z,t), 
    \quad 
    K(z,t)=N(z,t),
\end{align}
respectively.
The boundary condition, \eqref{eq:ssm_bound-cond_nodim_summary}, 
becomes 
\begin{align}
    C(0,t) = N(0,t),
        \label{eq:ssm_bound-cond_mono}
\end{align}
and the initial conditions, 
\eqref{eq:ssm_init-conds_nodim_summary}, 
become 
\begin{align}
    C(z,0) = 0, 
    \quad
    N(z,0) = 1.
    \label{eq:ssm_init-conds_mono}
\end{align}
In the following subsections we will consider 
solutions of this system \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono}.

\subsubsection{The case $\Gamma\ll1$ 
(with $\Gamma\Lambda\bloc\ll1$): A characteristic solution}
\label{subsubsec:mono-mono_the-blocking-and-no-deposition-case_the-case-small-Gamma}
In this subsection, we first suppose that the concentration of particles to be removed from the feed fluid is small compared to the concentration of pores in the filter, so that $\Gamma \ll 1$.
% We find an explicit asymptotic solution for the system \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono} in this case.

We seek a solution in the form of an asymptotic expansion in the small parameter, $\Gamma$, given by
\begin{align}
    C = C^{(0)} + \Gamma C^{(1)} + \mathcal{O}(\Gamma^2), 
    \quad
    N = N^{(0)} + \Gamma N^{(1)} + \mathcal{O}(\Gamma^2).
\end{align}
Noting that
\begin{align}
    \frac{C}{N} 
    = 
    \frac{C\order{0}}{N\order{0}} 
    + 
    \Gamma \left( \frac{C\order{1}}{N\order{0}} - \frac{C\order{0}N\order{1}}{(N\order{0})^2}\right)
    + \mathcal{O}(\Gamma^2),
\end{align}
we find that \eqref{eq:ssm_c-eqn_mono} and \eqref{eq:ssm_n-eqn_mono} become
 \begin{align}
     \pp{C^{(0)}}{t} + U^{(0)}\frac{\partial }{\partial z}\bigg(\frac{C^{(0)}}{N^{(0)}}\bigg)
     & = 
     -U^{(0)}\Lambda_{\text{B}}\frac{C^{(0)}}{N^{(0)}}
    +\mathcal{O}(\Gamma \Lambda_{\text{B}}),
     \label{eq:ssm_c-eqn_mono_small-gamma}
     \\
     \pp{N^{(0)}}{t} &= - \Gamma U^{(0)} \frac{C^{(0)}}{N^{(0)}} +\mathcal{O}(\Gamma^2).
     \label{eq:ssm_n-eqn_mono_small-gamma}
 \end{align}
% At leading order in $\Gamma$, 
We find, assuming that $\mathcal{O}(\Gamma)$ terms are small enough to be neglected, that the leading-order pore concentration is independent of time,
 \begin{align}
     N^{(0)}(z,t) = 1,
     \label{eq:N_const_Gamma-small}
 \end{align}
 and that the velocity and pressure are given by
 \begin{align}
     U^{(0)}(t) = 1, 
     \quad
     p^{(0)}(z,t) = 1-z,
 \end{align}
in both the constant flow rate and pressure drop cases, 
since the local porosity and permeability are both constant at leading order when $N\lo$ is constant, 
and given by
\begin{align}
    \phi^{(0)}(z,t) = 1,
    \quad
    K^{(0)}(z,t) = 1.
\end{align}
% It therefore remains to find 
Meanwhile, assuming that $\mathcal{O}(\Gamma \Lambda_{\text{B}})$ terms are small enough to be neglected, the leading-order particle concentration satisfies 
\begin{align}
     \pp{C^{(0)}}{t} + \pp{C^{(0)}}{z}
     & = 
     -\Lambda_{\text{B}}C^{(0)},
     \label{eq:ssm_c-eqn_mono_small-gamma_1}
\end{align}
subject to the boundary and initial conditions
\begin{align}
    C^{(0)}(0,t)= 1, 
    \quad 
    C^{(0)}(z,0)= 0,
\end{align}
which we solve explicitly using the method of characteristics.
Indeed, the characteristic curves of \eqref{eq:ssm_c-eqn_mono_small-gamma_1}, parametrised by the time-like characteristic coordinate, $\tau$, satisfy the characteristic equations
\begin{equation}
\frac{\text{d}t}{\text{d}\tau}=1, 
\quad 
\frac{\text{d}z}{\text{d}\tau}=1, 
\quad 
\frac{\text{d} C^{(0)}}{\text{d}\tau} = - \Lambda_{\text{B}}  C^{(0)},
\label{ch_eq_simp_1}
\end{equation}
which must be solved with initial data, at $\tau=0$, parametrised by $s$ and given by
\begin{align}
t(s) = t_0(s), \quad z(s) = z_0(s), \quad C^{(0)}(s)=C^{(0)}_0(s).
\label{data_gen_simp}
\end{align}
% Here, $s$ parametrises the initial curve along which the initial data at $\tau=0$ is specified. 
The general solutions to \eqref{ch_eq_simp_1} and \eqref{data_gen_simp}
are
\begin{align}
t(s,\tau) = \tau + t_0(s),
 \quad 
z(s,\tau) =  \tau +  z_0(s),
\quad
 C^{(0)}(s,\tau) = C^{(0)}_0(s) e^{- \Lambda_{\text{B}} \tau }.
\label{char_sol_gen}
\end{align}
Eliminating $\tau$ between the first two solutions in \eqref{char_sol_gen},
we find that the characteristic projections are straight lines with unit gradient, given by
\begin{align}
t =  z +  (t_0(s) -z_0(s)). 
\label{ch_proj_simp_1}
\end{align}
We note that the characteristic projection through $t=z=0$, that is, the line $z=t$, splits the domain into characteristic projections emanating from the $t=0$ axis, and those emanating from the $z=0$ axis, as shown in Figure \ref{fig:chars}.
\begin{figure}[!htb]%p]
        \includegraphics[width=0.60\textwidth]{./figures/characteristics/charac.pdf}
        \centering
        \caption{Graph showing the characteristic projections, \eqref{ch_proj_simp_1}, of the equation for the particle concentration at leading order in $\Gamma$, \eqref{eq:ssm_c-eqn_mono_small-gamma_1}. 
        Blue lines propagate information from the initial condition, orange lines propagate information from the boundary condition.
        The black line shows the last time at which the solution depends on information from the initial condition.
        }
        \label{fig:chars}
\end{figure}
After $t=1$, there is no influence of the initial condition on the solution.
We specify the initial data on $t=0$,
\begin{align}
t_0(s)=0, \quad
z_0(s)=s, \quad 
C^{(0)}_0(s)=0, \quad
\text{for} \quad 
0 \le s \le 1, 
\label{data_init_simp}
\end{align}
and find, using \eqref{char_sol_gen}, that the solution is 
\begin{align}
% t = z - s, 
% \quad 
% \text{for} 
\quad
C^{(0)}(z,t) = 0
\quad
\text{for} \quad  
t < z \le 1, 
\quad 
0 \le z \le 1.
\end{align}
Meanwhile, the boundary data on $z=0$ can be parametrised as
\begin{align}
t_0(s)=s, \quad z_0(s)=0, \quad C^{(0)}_0(s)=1, 
\quad 
\text{for}
\quad 
s \ge 0,
\label{data_in_simp}
\end{align}
so that, from \eqref{char_sol_gen}, the solution is
\begin{align}
% t =z+s,
% \quad
C^{(0)}(z,t) = e^{- \Lambda_{\text{B}} z}
\quad
\text{for} \quad  
0 \leq z \le t,  \quad
\ 0 \le z \le 1.
\end{align}
Altogether, we have that
\begin{align}
C^{(0)}(z,t) & 
=
\begin{cases}
e^{-\Lambda_{\text{B}} z} & 
0 \le z \le t,
\\
 0 & 
t < z \le 1,
\label{c_sol_simp_1}
\end{cases} 
\end{align}
for $ 0 \le z \le 1$.

We plot the leading-order particle concentration, $C\lo$ given by \eqref{c_sol_simp_1}, as a function of depth for various times for two values of  
$\Lambda\bloc$, in Figure \ref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1}.
For both values of $\Lambda\bloc$, we see that, 
as time increases, particles travel 
as a wave from the inlet, 
but are almost all captured via blocking in the first $10\%$ of the filter. 
Ahead of the wave front, the concentration of particles is zero because particles have not reached here yet.
Behind the wave front, particles are captured over a layer near the inlet. 
The capture layer has thickness $\mathcal{O}(\Lambda\bloc^{-1})$.
For example, comparing Figure \ref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1_LamB-smaller.pdf} with Figure \ref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1_LamB-larger.pdf}, 
 we see that when the value of $\Lambda\bloc$ is doubled, the thickness of the layer is halved.
 Note that, since the porosity is given by the pore concentration, and this is constant via \eqref{eq:N_const_Gamma-small}, the void concentration, $C_{\mathrm{V}}$, is equal to the concentration, $C$, in this case.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/small-Gamma/part_2__v__posi_1_LamB-smaller.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1_LamB-smaller.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/small-Gamma/part_2__v__posi_1_LamB-larger.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1_LamB-larger.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration at leading-order in $\Gamma\ll1$, $C\lo$, 
    varies with depth, $z$, for various times, $t$.
    The values of $t$ are $0.005$ (blue), $0.010$ (orange), $0.015$ (green), $0.020$ (red), $0.025$ (purple), $0.030$ (brown), $1$ (pink).
    Later $t$ values are identical to the pink curve.
    The arrows show the direction of increasing time.
    The value of $e^{-\Lambda\bloc z}$ is plotted with a black dotted line to aid the eye.
    Note that only the depths $0\le z\le0.1$ are shown.
    \subref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1_LamB-smaller.pdf} 
    The case $\Lambda\bloc=50$.
    \subref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1_LamB-larger.pdf} 
    The case $\Lambda\bloc=100$.
    }
    \label{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1}
\end{figure}

\subsubsection{The case $\Lambda_{\text{B}}\gg1$ (with $\Gamma \Lambda\bloc^{-1}\ll1$)}
An asymptotic solution
In this subsection, we suppose that the filter is much longer than the length over which a particle is captured when a pore is blocked, so that $\Lambda\bloc\gg1$.
% We find an explicit asymptotic solution for the system \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono} in this case.
We seek a solution in the form of an asymptotic expansion in the small parameter, $\Lambda\bloc^{-1}$, given by
\begin{align}
    C = C^{(0)} + \Lambda\bloc^{-1} C^{(1)} + \mathcal{O}(\Lambda\bloc^{-2}), 
    \quad
    N = N^{(0)} + \Lambda\bloc^{-1} N^{(1)} + \mathcal{O}(\Lambda\bloc^{-2}).
\end{align}
Noting that
\begin{align}
    \frac{C}{N} 
    = 
    \frac{C\order{0}}{N\order{0}} 
    + 
    \Lambda\bloc^{-1} \left( \frac{C\order{1}}{N\order{0}} - \frac{C\order{0}N\order{1}}{(N\order{0})^2}\right)
    + \mathcal{O}(\Lambda\bloc^{-2}),
\end{align}
we find that \eqref{eq:ssm_c-eqn_mono} and \eqref{eq:ssm_n-eqn_mono} 
become
 \begin{align}
     \pp{C^{(0)}}{t} + U^{(0)}\frac{\partial }{\partial z}\bigg(\frac{C^{(0)}}{N^{(0)}}\bigg)
     & = 
     -
     \Lambda\bloc U^{(0)}\frac{C^{(0)}}{N^{(0)}}
     -
     U^{(0)}\left( \frac{C\order{1}}{N\order{0}} - \frac{C\order{0}N\order{1}}{(N\order{0})^2}\right) 
     +
     \mathcal{O}\left(\Lambda\bloc^{-1}\right),
     \label{eq:ssm_c-eqn_mono_asymp_1}
     \\
     \pp{N^{(0)}}{t} &= - \Gamma U^{(0)} \frac{C^{(0)}}{N^{(0)}} +\mathcal{O}\left(\Gamma \Lambda\bloc^{-1}\right).
     \label{eq:ssm_n-eqn_mono_asymp_1}
 \end{align}
We find, assuming that $\mathcal{O}(\Lambda\bloc)$ terms are large enough that $\mathcal{O}(1)$ (and thus $\mathcal{O}(\Lambda\bloc^{-1})$) terms can be neglected, 
that the leading order particle concentration is given by
 \begin{align}
    C^{(0)}(z,t) = 0,
    \label{eq:ssm_c-sol_mono_outer}
\end{align}
and so it follows from \eqref{eq:ssm_n-eqn_mono_asymp_1}, 
assuming that $\mathcal{O}(\Gamma\Lambda\bloc^{-1})$ terms are small enough to be neglected, 
that 
\begin{align}
    N^{(0)}(z,t) = 1.
    \label{eq:ssm_n-sol_mono_outer}
 \end{align}
 Solutions \eqref{eq:ssm_c-sol_mono_outer} and \eqref{eq:ssm_n-sol_mono_outer} 
 imply that there are no particles in the filter and that the pore concentration remains at its initial value.
 These solutions 
 % match the initial conditions 
 % \eqref{eq:ssm_init-conds_mono}, 
 % but 
 do not match the boundary condition 
 \eqref{eq:ssm_bound-cond_mono}, however, since particles must enter the filter at the inlet, so the concentration cannot be zero everywhere.

With this in mind, we seek an inner solution that matches
the boundary condition in a layer of thickness 
 $\mathcal{O}(\Lambda\bloc^{-1})$ inside the filter near the inlet.
 We define the short spatial coordinate $\zeta = \Lambda\bloc z$, and find that, in the boundary layer, 
 the leading-order problem becomes 
 \begin{align}
     \frac{\partial }{\partial \zeta}\bigg(\frac{C^{(0)}_{\text{bl}}}{N^{(0)}_{\text{bl}}}\bigg) &= 
     - \frac{C^{(0)}_{\text{bl}}}{N^{(0)}_{\text{bl}}},
     \label{eq:ssm_c-eqn_mono_asymp_2}
     \\
     \pp{N^{(0)}_{\text{bl}}}{t} &= - \Gamma U^{(0)}_{\text{bl}} \frac{C^{(0)}_{\text{bl}}}{N^{(0)}_{\text{bl}}},
    \label{eq:ssm_n-eqn_mono_asymp_2}
 \end{align}
 where the subscript, bl, signifies the solution in the boundary layer. 
 Equation \eqref{eq:ssm_c-eqn_mono_asymp_2} has the solution 
 \begin{align}
     C^{(0)}_{\text{bl}} = N^{(0)}_{\text{bl}}e^{-\zeta},
     \label{eq:ssm_c-sol_inner}
 \end{align}
 where $N^{(0)}_{\text{bl}}$ depends on the form of the velocity, $U^{(0)}_{\text{bl}}$, and therefore the operating regime, through 
 \eqref{eq:ssm_n-eqn_mono_asymp_2}.
Thus, we will consider the constant flow rate and pressure drop regimes separately.

Before proceeding, we note that 
\eqref{eq:ssm_c-sol_inner} does not satisfy the initial conditions, \eqref{eq:ssm_init-conds_mono}, 
and deduce that the boundary-layer particle and pore concentrations 
satisfy a different problem at early times.
Indeed, defining a short temporal coordinate $\tau = \Lambda\bloc t$, 
we find that, at early times, the leading-order problem is 
\begin{align}
    \pp{C^{(0)}_{\text{bl;et}}}{\tau}
    + 
    U^{(0)}_{\text{bl;et}}
    \frac{\partial }{\partial \zeta}\bigg(\frac{C^{(0)}_{\text{bl;et}}}{N^{(0)}_{\text{bl;et}}}\bigg)
     & = 
     -
     U^{(0)}_{\text{bl;et}}\frac{C^{(0)}_{\text{bl;et}}}{N^{(0)}_{\text{bl;et}}},
     \label{eq:et-C}
     \\
     \pp{N^{(0)}_{\text{bl;et}}}{\tau} &= 0,
     \label{eq:et-N}
\end{align}
where the subscript, bl;et, signifies the boundary-layer solution at early times.
Equation \eqref{eq:et-N} is trivially satisfied by
\begin{align}
    N\order{0}_{\text{bl;et}}(z,t)=1,
    \label{eq:sol-N}
\end{align}
and so we find that the particle concentration 
satisfies
\begin{align}
    \pp{C^{(0)}_{\text{bl;et}}}{\tau}
    + 
    \frac{\partial 
    C^{(0)}_{\text{bl;et}}
    }{
    \partial \zeta
    }
     & = 
     -
 C^{(0)}_{\text{bl;et}},
\end{align}
which, returning to the original coordinate system, has the solution
\begin{align}
C^{(0)}_{bl;et}(z,t) & 
=
\begin{cases}
e^{-\Lambda_{\text{B}} z} & 
0 \le z \le t,
\\
 0 & 
t < z \le \Lambda\bloc^{-1},
\end{cases} 
\label{eq:sol-C}
\end{align}
for $ 0 \le z \le \Lambda\bloc^{-1}$.
The early time boundary layer solution, 
given by \eqref{eq:sol-N} and \eqref{eq:sol-C}, 
satisfies the initial conditions, 
\eqref{eq:ssm_init-conds_mono} 
(as well as the boundary condition, 
\eqref{eq:ssm_bound-cond_mono}).
We now return to the boundary-layer problem at 
later times.
Note that, since 
the porosity is equal to the pore concentration which is constant via \eqref{eq:sol-N}, 
the void concentration, $C^{(0)}_{\text{V bl;et}}$, is also equal to the function given by \eqref{eq:sol-C}.

\paragraph{Constant flow rate}
After the early times discussed above, in the constant flow rate regime, we have $U^{(0)}=1$ from \eqref{eq:ssm_Up-eqns_const-flow_mono}, and so 
\eqref{eq:ssm_n-eqn_mono_asymp_2} has the solution
  \begin{align}
     N^{(0)}_{\text{bl}} = 1-\Gamma t e^{-\zeta}. 
 \end{align}
 Therefore, in the original spatial coordinate, 
 the inner solution reads 
  \begin{align}
       C^{(0)}_{\text{bl}}(z,t) = (1-\Gamma t e^{-\Lambda_{\text{B}}z} )e^{-\Lambda_{\text{B}}z},
       \quad
     N^{(0)}_{\text{bl}}(z,t) = 1-\Gamma t e^{-\Lambda_{\text{B}}z}, 
     \label{eq:ssm_sol_mono_inner_const-flow}
 \end{align}
 which matches the outer solution
 \eqref{eq:ssm_c-sol_mono_outer}, 
 \eqref{eq:ssm_n-sol_mono_outer}, 
 and the condition at the inlet boundary, 
 \eqref{eq:ssm_bound-cond_mono}.

 Note, using \eqref {eq:ssm_n-eqn_mono_asymp_2}, that, 
 the boundary-layer void concentration satisfies
\begin{align}
     \frac{\partial }{\partial \zeta}\bigg(C^{(0)}_{\text{V bl}}\bigg) &= 
     - C^{(0)}_{\text{V bl}},
\end{align}
so that 
\begin{align}
    C^{(0)}_{\text{V bl}}(z,t) = e^{-\Lambda\bloc z}.
    \label{eq:cv-sol}
\end{align}
As expected, particles are all removed from the  fluid near the inlet, and so the fluid concentration drops exponentially as a function of depth (see Figure \ref{fig:ssm_mono-mono/large-LamB/const-flow/var_2__v__posi_1.pdf}).

We plot the boundary-layer particle and pore concentrations at leading order in $\Lambda\bloc^{-1}$, 
$C\lo_{\text{bl}}$ and $N\lo_{\text{bl}}$ respectively, given by \eqref{eq:ssm_sol_mono_inner_const-flow}, as functions of depth, $z$, for various times, $t$, in Figure \ref{fig:ssm_mono-mono/large-LamB/const-flow/var_2__v__posi_1.pdf}.
% Comparing Figure 
% \ref{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf} and Figure 
% \ref{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf}, 
As in the $\Gamma\ll1$ case (see Figure \ref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1}), 
% the first thing we note is that 
changes in the solutions occur in a layer 
near the inlet. 
Particles never reach the region outside of this layer, so the pore concentration maintains its 
initial condition here.
% As in the case $\Gamma\ll1$ (see Figure \ref{fig:ssm_mono-mono/small-Gamma/part_2__v__posi_1}), 
The length of this layer is $\mathcal{O}(\Lambda\bloc^{-1})$.

Inside the layer we see that, as particles enter the filter, 
they are captured inside the pores. 
This capture causes pores to become blocked, which in turn decreases the pore concentration 
in the layer (unlike in the $\Gamma\ll1$ case, or in the early time solution). 

Pore decrease is fastest at the inlet, since pores here see the most particles per unit time.
The particle concentration also decreases fastest at the inlet because the concentration of particles driven into the filter decreases at the same rate as the pore concentration, 
to maintain a constant incoming void particle concentration, which is the enforced condition at the boundary. 
At $t=\Gamma^{-1}$ ($t=1$ in this example), all inlet pores are blocked so particles no longer enter the filter.

Note that the filter particle concentration, unlike the void particle concentration (which decreases exponentially in space, see the dashed black line in Figure \ref{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf}), is not spatially monotonic at larger times  (see the red and purple lines in Figure \ref{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf})). 
To see why, note that, as time increases at any particular depth, the void particle concentration remains constant (since new particles enter via the inlet and replace those that have been captured), but the pore concentration (see Figure \ref{fig:ssm_mono-mono/large-LamB/const-flow/pore_2__v__posi_1.pdf})), and therefore the porosity, 
decreases (since particles block pores). 
Since the filter particle concentration is equal to the product of the void particle concentration 
and the porosity (via \eqref{eq:def-void-conc}), the filter particle concentration decreases in time,
fastest near the inlet (where most pores are blocked per unit time). 
Far from the inlet, the filter particle concentration approaches the void particle concentration because there are not enough particles in the fluid to block pores and alter the pore concentration, so that the porosity is constant.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-flow/pore_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/large-LamB/const-flow/pore_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and pore concentration 
    at leading order in $\Lambda\bloc^{-1}$ in the boundary layer near the inlet, 
    $C\lo_{\text{bl}}$ and $N\lo_{\text{bl}}$, respectively, 
    vary with depth, $z$, for various $t$.
    The parameters are $\Lambda\bloc=100$, $\Gamma=1$. 
    The values of $t$ are 
    $0.10$ (blue),
    $0.25$ (orange),
    $0.50$ (green),
    $0.75$ (red),
    $1.00$ (purple).
    The arrows show the direction of increasing time.
    Note that only the depths $0\le z \le 0.1$ are shown.
    \subref{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf} 
    The particle concentration, $C^{(0)}_{\text{bl}}$.
    The void concentration, $C^{(0)}_{\text{V bl}}$ is shown with a dashed black line.
    % The value of $e^{-\Lambda\bloc z}$ is plotted with a dotted black line to aid the eye.
    \subref{fig:ssm_mono-mono/large-LamB/const-flow/pore_2__v__posi_1.pdf} 
    The pore concentration, $N^{(0)}_{\text{bl}}$.
    % The value of $1-\Gamma e^{-\Lambda\bloc z}$ is plotted with a dotted black line to aid the eye.
    }
        \label{fig:ssm_mono-mono/large-LamB/const-flow/var_2__v__posi_1.pdf}
\end{figure}

 \paragraph{Constant pressure drop}
In the constant pressure drop regime, we have
 \begin{align}
     U^{(0)}_{\text{bl}}(t) = \frac{1}{\int_0^1 \frac{1}{N^{(0)}_{\text{bl}}(z,t)} \, \text{d}z}
 \end{align}
  from \eqref{eq:ssm_Up-eqns_const-pres_mono}, 
  and so, 
  in the original spatial coordinate, 
   \eqref{eq:ssm_n-eqn_mono_asymp_2} becomes 
  \begin{align}
        \pp{N^{(0)}_{\text{bl}}}{t} &= - \frac{\Gamma e^{-\Lambda\bloc z}}{\int_0^{1} \frac{1}{N^{(0)}_{\text{bl}}} \, \text{d} z}, 
    \label{eq:ssm_n-eqn_mono_asymp_3}
  \end{align}
where we have used the inner solution, \eqref{eq:ssm_c-sol_inner},  for the particle concentration, $C^{(0)}_{\text{bl}}$.


 Thus (in analogy with the constant flow rate solution
 \eqref{eq:ssm_sol_mono_inner_const-flow}) the pore concentration is given by
 \begin{align}
     N^{(0)}_{\text{bl}}(z,t) = 1-\gamma(t) e^{-\Lambda\bloc z},
        \label{eq:ssm_n-sol_mono_const-pres_inner}
 \end{align}
 where $\gamma$ satisfies
 \begin{align}
     \ff{\gamma}{t} = 
     \frac{\Gamma}{\int_0^1 \frac{1}{1-\gamma(t) e^{-\Lambda\bloc z}} \, \text{d}z},
     \label{eq:ssm_n-eqn_mono_asymp_4}
 \end{align}
and, calculating the integral, we find that \eqref{eq:ssm_n-eqn_mono_asymp_4} simplifies to 
  \begin{align}
     \ff{\gamma}{t} = 
     \frac{\Gamma \Lambda_{\text{B}}}{\log\bigg( \frac{e^{\Lambda_{\text{B}}} -\gamma(t)}{1-\gamma(t)} \bigg)},
     \label{eq:ssm_n-eqn_mono_asymp_5}
 \end{align}
 which is separable. Integrating \eqref{eq:ssm_n-eqn_mono_asymp_5}, we find that
 \begin{align}
     \Lambda_{\text{B}} \gamma(t) + 
     \Big(1-\gamma(t)\Big)
     \log\Big(1-\gamma(t)\Big)
     -e^{\Lambda_{\text{B}}} \Big(1-e^{-\Lambda_{\text{B}}}\gamma(t) \Big) 
     \log\Big(1-e^{-\Lambda_{\text{B}}}\gamma(t)\Big) = \Gamma \Lambda_{\text{B}} t,
     \label{eq:ssm_f_eqn}
 \end{align}
so that 
\begin{align}
    \gamma(t) = \Gamma t + \mathcal{O}(\Lambda\bloc^{-1})
    \quad 
    \text{for}
    \quad 
    0 < t < \frac{1}{\Gamma}.
    \label{eq:small-Gamma}
\end{align}
Thus $\gamma(t)\rightarrow\Gamma t$ in the limit $\Lambda\bloc\rightarrow\infty$ at times until the filtration process ceases (at $t=1/\Gamma$, because this is when pores at the inlet are all blocked).
% , 
% which means that the constant pressure drop solution approximates the constant flow rate solution for $\Lambda\bloc \gg 1$. 
% Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t.pdf} shows 
% % the dependence of $\gamma$ on $t$ in \eqref{eq:ssm_n-eqn_mono_asymp_5}, 
% the behaviour of the solution of the implicit equation \eqref{eq:ssm_n-eqn_mono_asymp_5}
% as $\Lambda\bloc$ increases, 
% for 
% $\Gamma=1$ (Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-1.pdf})
% and 
% $\Gamma=2$ (Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-2.pdf}), 
% and thus provides a visualisation of how 
% $\gamma$ approaches $\Gamma t$.
% In both figures, we see that $\gamma$ increases sub-linearly as a function of $t$ 
%  for any $\Lambda\bloc$. 
 % , and is bounded above by $1$. 
 % We stop when $\gamma=1$ because this is when 
 % $N^{(0)}_{\text{bl}}=0$ at $z=0$.
 % We see that, as $\Lambda\bloc$ increases, 
 % the time at which this occurs increases.
 % We indeed find that $\gamma$ approaches $\Gamma t$ as $\Lambda\bloc$ increases, 
 % so that, 
 % For the values (i.e., $\Lambda\bloc\gg1$) that we are concerned with here, 
 % (e.g., $\Lambda\bloc=100$ (purple line)), 
 % we see that $\gamma$ lies almost exactly on $\Gamma t$ (dotted black line). 
As such, 
as in the constant flow rate regime,
% Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-1.pdf}
% with Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-2.pdf}, 
$\Gamma$ fully determines the gradient
of $\gamma$ (for $\Lambda\bloc\gg1$). 
% of the 
% curve, and that $\gamma'(t)\approx\Gamma$ for large enough $\Lambda\bloc$.
The implication is that, as expected, the relative concentrations of particles and pores dictate the rate of blocking.
% \begin{figure}[!htb]
%      \centering
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-pres/f__v__t_Gamm-1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-1.pdf}
%      \end{subfigure}
%      %\hfill
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-pres/f__v__t_Gamm-2.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-2.pdf}
%      \end{subfigure}
%     \caption{Graphs showing how the function $\gamma$ varies with $t$, for various blocking-parameter values 
%     $\Lambda\bloc$.
%     The values of $\Lambda\bloc$ are 
%     $0.01$ (blue),
%     $0.1$ (orange),
%     $1$ (green),
%     $10$ (red),
%     $100$  (purple).
%     The arrows show the direction of increasing $\Lambda\bloc$.
%     The line $\gamma=\Gamma t$ is shown in a dotted black line.
%     % The value $t=\Gamma^{-1}$ is extrapolated from the $t$ axis 
%     % in the region $0\le \gamma \le 1$ using a dashed black line.
%     % The region of interest is outlined using a dash-dotted black line.
%     \subref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-1.pdf} 
%     The case $\Gamma=1$.
%     \subref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-2.pdf} 
%     The case $\Gamma=2$.
%     }
%     \label{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t.pdf}
% \end{figure}
%  % \correct{}{so that
%  % \begin{align}
%  % \gamma(t) = \Gamma t + \mathcal{O}(\Lambda\bloc^{-1}).     
%  % \end{align} 
%  % One implication is that, since 
%  % $\gamma\rightarrow\Gamma t$ in the limit $\Lambda\bloc\rightarrow\infty$, 
%  % the constant pressure solution approaches 
%  % the constant velocity one.}

Thus, in the constant pressure drop regime, the inner solution reads 
  \begin{align} 
     C^{(0)}_{\text{bl}}(z,t) = (1-\gamma(t) e^{-\Lambda_{\text{B}}z} )e^{-\Lambda_{\text{B}}z},
     \quad
    N^{(0)}_{\text{bl}}(z,t) = 1-\gamma(t) e^{-\Lambda_{\text{B}}z},
     \label{eq:ssm_sol_mono_inner_const-pres}
 \end{align}
 where $\gamma$ satisfies \eqref{eq:small-Gamma}.
 This matches the outer solution
 \eqref{eq:ssm_c-sol_mono_outer}, 
 \eqref{eq:ssm_n-sol_mono_outer}, 
 and the condition at the inlet boundary, 
 \eqref{eq:ssm_bound-cond_mono}.
 Physically,
 the constant pressure drop solution, \eqref{eq:ssm_sol_mono_inner_const-pres},
 matches the constant flow rate solution, 
\eqref{eq:ssm_sol_mono_inner_const-flow}, 
 because, in the former, changes to the flow rate are restricted to the layer over which pores block. 
 This has $\mathcal{O}(\Lambda\bloc^{-1})$ thickness, 
 and we thus expect an $\mathcal{O}(\Lambda\bloc^{-1})$ flow rate change that is negligible for $\Lambda\bloc\gg 1$.
Note that the void particle concentration solution also approaches that in the steady, constant flow rate case, \eqref{eq:cv-sol}.

% To interpret this solution, 
% in Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/var_2__v__time_1.pdf} 
% we plot the values of our leading-order solutions, \eqref{eq:ssm_sol_mono_inner_const-pres}, at the inlet, as functions of time, for various $\Lambda\bloc$. 
% Focusing on Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/pore_2__v__time_1.pdf}, 
% we see that, as time increases, the pore concentration drops from the
% initial value towards zero. 
% We see that, as $\Lambda\bloc$ increases, the gradient of this drop increases.
% For $\Lambda\bloc=100$ (purple line), for example, almost all pores are removed by $t=1$, while for $\Lambda\bloc=10$ (red line), almost $10\%$ of the original pores still remain. 


% From \eqref{eq:ssm_f_eqn}, $\gamma$ and $\Lambda\bloc$ control the time it takes to block all of the pores at the inlet, $T$ say,
% and this is distinct from the constant flow rate case, 
% in which $T=\Gamma^{-1}$.
% As such, returning to Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t.pdf}, 
%  with $\Lambda\bloc=10$,
%  for $\Gamma=1$ we see (red line in
%  Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-1.pdf}) that $T\approx1.1$,
%  while for $\Gamma=2$ we see (red line in Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/f__v__t_Gamm-2.pdf}) 
%  that $T\approx0.55$. 
% \begin{figure}[!htb]
%      \centering
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-pres/part_2__v__time_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/large-LamB/const-pres/part_2__v__time_1.pdf}
%      \end{subfigure}
%      %\hfill
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-pres/pore_2__v__time_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/large-LamB/const-pres/pore_2__v__time_1.pdf}
%      \end{subfigure}
%     \caption{Graphs showing how the inlet leading-order inner particle- and pore-concentration solutions, $C\lo_{\text{bl}}(0,t)$ and $N\lo_{\text{bl}}(0,t)$, respectively, vary with $t$, for various blocking-parameter values 
%     $\Lambda\bloc$.
%     The values of $\Lambda\bloc$ are 
%     $0.01$ (blue),
%     $0.1$ (orange),
%     $1$ (green),
%     $10$ (red),
%     $100$  (purple).
%     The arrows show the direction of increasing $\Lambda\bloc$.
%     % The lines $1-\gamma(t)$ for the corresponding $\Lambda\bloc$ are shown with dotted black lines.
%     \subref{fig:ssm_mono-mono/large-LamB/const-pres/part_2__v__time_1.pdf} 
%     The particle concentration at the inlet, $C\lo_{\text{bl}}(0,t)$.
%     \subref{fig:ssm_mono-mono/large-LamB/const-pres/pore_2__v__time_1.pdf} 
%     The pore concentration at the inlet, $N\lo_{\text{bl}}(0,t)$.
%     }
%     \label{fig:ssm_mono-mono/large-LamB/const-pres/var_2__v__time_1.pdf}
% \end{figure}

% However, 
% % for large enough $\Lambda\bloc$ ($\Lambda\bloc=\mathcal{O}(10^2)$, say), 
% as $\Lambda\bloc\rightarrow\infty$
% we have
% $\gamma\approx\Gamma t$, and $\Lambda\bloc$ has an exponentially small contribution to the blocking time, $T$, 
% which is approximately $T\approx\Gamma^{-1}$ in this limit. 
% Indeed, comparing Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/var_2__v__posi_1.pdf}
% with Figure \ref{fig:ssm_mono-mono/large-LamB/const-flow/var_2__v__posi_1.pdf}, 
% we see that, for $\Lambda\bloc=100$, the constant pressure drop and flow rate solutions are almost identical.
% At $t=1$, the pore concentration in Figure \ref{fig:ssm_mono-mono/large-LamB/const-pres/var_2__v__posi_1.pdf} (purple line) has not quite reached zero at the inlet (because $\gamma$ is not quite $1$ yet), but this is not visible to the naked eye.
% \begin{figure}[!htb]
%      \centering
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-pres/part_2__v__posi_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/large-LamB/const-pres/part_2__v__posi_1.pdf}
%      \end{subfigure}
%      %\hfill
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/large-LamB/const-pres/pore_2__v__posi_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/mono-mono/large-LamB/const-pres/pore_2__v__posi_1.pdf}
%      \end{subfigure}
%     \caption{Graphs showing how the particle concentration and pore concentration 
%     at leading order in $\Lambda\bloc^{-1}$ in the boundary layer near the inlet, 
%     $C\lo_{\text{bl}}$ and $N\lo_{\text{bl}}$, respectively, 
%     vary with depth, $z$, for various $t$.
%     The parameters are $\Lambda\bloc=100$, $\Gamma=1$. 
%     The values of $t$ are 
%     $0.1$ (blue),
%     $0.25$ (orange),
%     $0.5$ (green),
%     $0.75$ (red),
%     $1$ (purple).
%     The arrows show the direction of increasing time.
%     Note that only the depths $0\le z \le 0.1$ are shown.
%     \subref{fig:ssm_mono-mono/large-LamB/const-flow/part_2__v__posi_1.pdf} 
%     The particle concentration, $C^{(0)}_{\text{bl}}$.
%     % The value of $e^{-\Lambda\bloc}$ is plotted with a dotted black line.
%     \subref{fig:ssm_mono-mono/large-LamB/const-flow/pore_2__v__posi_1.pdf} 
%     The pore concentration, $N^{(0)}_{\text{bl}}$.
%     % The value of $1-\Gamma e^{-\Lambda\bloc}$ is plotted with a dotted black line.
%     }
%     \label{fig:ssm_mono-mono/large-LamB/const-pres/var_2__v__posi_1.pdf}
% \end{figure}



\subsubsection{The cases $\Gamma\ll1$ and $\Lambda\bloc\gg1$: A numerical solution}
% $\Gamma=\mathcal{O}(1)$ and $\Lambda_{\text{B}}=\mathcal{O}(1)$}
We use numerical methods to solve the system \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono} for more general values of $\Gamma$ and $\Lambda_{\text{B}}$.
In this subsection, we solve numerically
for $\Gamma\ll1$ or $\Gamma=\mathcal{O}(1)$ with $\Lambda\bloc\gg1$ 
and compare the solutions 
with the asymptotic ones we have already found. 
We use the constant flow rate case since we know the constant pressure drop solution behaves similarly for large enough $\Lambda\bloc$.

For general values of $\Gamma$ and $\Lambda_{\text{B}}$, \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono} is difficult to solve explicitly, since \eqref{eq:ssm_c-eqn_mono} is an 
advection--reaction equation that is 
coupled to \eqref{eq:ssm_n-eqn_mono}. 
We solve the system using a finite-difference scheme that is first order in time and space.
Since the system is hyperbolic, we  
use upwinding \cite{tran2009} in the appropriate direction, so that the known information at the inlet boundary is propagated downstream.
We plot the solutions in Figures \ref{fig:ssm_mono-mono/general/Gamm-0} and \ref{fig:ssm_mono-mono/general/LamB-100}.
% , and compare 
% these with solutions in the two asymptotic regimes, $\Gamma\ll1$ and $\Lambda\bloc\gg1$.

\paragraph{The case $\Gamma\ll1$: Comparison with the characteristic solution in }
We find that the numerical solution agrees well 
with the explicit solution in the case $\Gamma\ll1$. 
To demonstrate this, in Figure \ref{fig:ssm_mono-mono/general/Gamm-0}, we plot the numerical solution for the particle 
concentration, $C$ as a function of depth, $z$, at various times, $t$, and compare it 
to the explicit asymptotic solution at leading order in $\Gamma\ll1$, from 
\eqref{c_sol_simp_1}.

At later times, the explicit solution is an exponentially decaying curve 
over a layer of length $\mathcal{O}(\Lambda\bloc^{-1})$. 
The numerical solution agrees well with the explicit solution, even for 
a relatively small number of spatial points. 
For example, in Figure \ref{fig:ssm_mono-mono/general/Gamm-0/part_3__v__posi_1.pdf} we see that 
$400$ spatial points is enough to match the solutions almost exactly, even though changes occur over 
the first $10\%$ of the filter.

At earlier times, the explicit solution takes the form of a wave that travels through the layer of length 
$\mathcal{O}(\Lambda\bloc^{-1})$.
We must discretise more finely in the spatial coordinate so that the
numerical solution resolves the sharp steps in the explicit solution.
As such, a finer discretisation in the temporal coordinate is also required, to 
ensure that the Courant--Friedrichs--Lewy (CFL) condition \cite{tran2009} is satisfied, for 
convergence of the scheme.
In Figure \ref{fig:ssm_mono-mono/general/Gamm-0/part_3__v__posi_1__fine.pdf}, for example,  
we see that $1000$ spatial points leads to reasonable agreement, and, in particular, correctly captures the wave speed.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/Gamm-0/part_3__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/Gamm-0/part_3__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/Gamm-0/part_3__v__posi_1__fine.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/Gamm-0/part_3__v__posi_1__fine.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration, $C$, varies with depth, $z$, for various
    times, $t$.
    The parameters are $\Lambda_{\text{B}}=100$, 
    % $\Gamma=0$, 
    $\Gamma=0.01$, 
    $A=2$.
    The numerical solution is shown with solid lines. 
    The solution at leading order in $\Gamma\ll1$ is shown in dotted black lines.
    \subref{fig:ssm_mono-mono/general/Gamm-0/part_3__v__posi_1.pdf} 
    The later time solution at time 
    % $t=0.1$ 
    $t=1$ 
    using $400$ spatial points.
    \subref{fig:ssm_mono-mono/general/Gamm-0/part_3__v__posi_1__fine.pdf} 
    The earlier time solution at various times, $t$, using $1000$ spatial points. 
    The values of $t$ are 
    $0.005$ (blue),
    $0.01$ (orange),
    $0.015$ (green),
    $0.02$ (red),
    $0.025$ (purple),
    $0.03$ (brown),
    $0.1$ (pink).
    }
    \label{fig:ssm_mono-mono/general/Gamm-0}
\end{figure}

\paragraph{The case $\Lambda\bloc\gg1$: Comparison with asymptotic solution}
The numerical solution also agrees well with the explicit solution in the case $\Lambda\bloc\gg1$.
To demonstrate this, in Figure \ref{fig:ssm_mono-mono/general/LamB-100}, we plot the numerical solution for the particle and pore concentrations in the case $\Lambda_{\text{B}}=100$ and compare this to the 
boundary layer solution at leading order in $\Lambda\bloc^{-1}\ll1$, $C^\lo_{\text{bl}}$ and $N^\lo_{\text{bl}}$, 
from \eqref{eq:ssm_c-sol_inner}.
% in the case $\Lambda_{\text{B}}\gg1$, given by \eqref{eq:ssm_sol_mono_inner_const-flow}.

We see, as expected, that changes to the numerical solution are restricted to a boundary layer of length $\mathcal{O}(\Lambda\bloc^{-1})$
near the inlet. 
Around $400$ spatial points are enough for good agreement with the asymptotic solution (with $\Lambda\bloc=100$). 
The agreement with $1000$ spatial points (see Figure \ref{fig:ssm_mono-mono/general/LamB-100}) is excellent. 
The number of time points needed depends on the maximum time at which the solution is to be solved. 
As time increases, the local porosity near the inlet decreases, which causes the void velocity to increase. 
Since this velocity drives changes to the particle concentration profile, small time steps (perhaps $\mathcal{O}(10)$ times as small as the spatial steps) are needed at times near the block time, to ensure convergence.

\begin{figure}[!htb]%tb
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/LamB-100/part_3__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/LamB-100/part_3__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/LamB-100/pore_3__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/LamB-100/pore_3__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and pore concentration, 
    $C$ and $N$, vary with depth, $z$, for various times, $t$.
    The parameters are $\Lambda\bloc=100$, $\Gamma=1$, $A=2$. 
    The numerical solutions are shown with solid coloured lines. 
    The spatial domain is discretised using $1000$ points.
    The corresponding solutions at leading order in $\Lambda\bloc^{-1}$ in the boundary layer near the inlet, 
    $C\lo_{\text{bl}}$ and $N\lo_{\text{bl}}$, are shown in black dotted lines.
    The values of $t$ are 
    $0.1$ (blue),
    $0.25$ (orange),
    $0.5$ (green),
    $0.75$ (red),
    $1$ (purple).
    The arrows show the direction of increasing time.
    Note that only the depths $0\le z \le 0.1$ are shown.
    \subref{fig:ssm_mono-mono/general/LamB-100/part_3__v__posi_1.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-mono/general/LamB-100/pore_3__v__posi_1.pdf} 
    The pore concentration, $N^{(0)}_{\text{bl}}$.
    }
    \label{fig:ssm_mono-mono/general/LamB-100}
\end{figure}

% We have a general solution. but we note that teh explcit ones are much cheaper to use. 
% Theyu are useful fo rthings like parameter sweeps...so we use them to se what the parameters do.
\newpage
Thus, in the cases $\Gamma\ll1$ and $\Lambda\bloc\gg1$, we have found explicit solutions that accurately 
approximate the numerical solution but are cheaper computationally. 
These are useful for tasks that involve repeated solution, such as parameter sweeps.
To demonstrate this, we now use the explicit formulae to show how the solution varies as we vary $\Gamma$ (with $\Lambda\bloc\gg1$) 
and $\Lambda\bloc$ (with $\Gamma\ll1$).
% Equipped with a validated numerical solution that holds for all relevant values of $\Gamma$ and $\Lambda_{\text{B}}$, in the following two brief subsections we use this solution to analyse the behaviour of the system as a function of $\Gamma$ and $\Lambda_{\text{B}}$ independently. 

\paragraph{Behaviour as $\Gamma$ varies}
Firstly, we solve the mono-dispersed system, \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono}, for fixed $\Lambda_{\text{B}}$ and various values of $\Gamma$.
We plot the particle and pore concentrations, $C$ and $N$, as functions of depth, $z$, for various values of 
$\Gamma$ at $t=1$ in Figure \ref{fig:ssm_mono-mono/general/sweep-Gamma/var__v__posi_1.pdf}. 
% The behaviour of the solution is similar to that in examples where the solution is shown at 
% increasing times for fixed $\Gamma$ (see Figure \ref{fig:ssm_mono-mono/general/LamB-100}, for example). 
As $t$ increases, $\Gamma$ decreases, and vice versa, from \eqref{eq:ssm_sol_mono_inner_const-flow}.
Therefore, we see that increasing $\Gamma$ accelerates the rate of evolution of the solution, effectively decreasing the timescale, and decreasing the blocking time, $T\approx\Gamma^{-1}$.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-Gamma/part_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/sweep-Gamma/part_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-Gamma/pore_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/sweep-Gamma/pore_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and pore concentration,
    $C$ and $N$, respectively, 
    vary with depth, $z$, for various $\Gamma$.
    The parameters are $\Lambda\bloc=100$, $A=2$, 
    and the solutions are plotted at time $t=1$. 
    The values of $\Gamma$ are
    $0.0$ (blue),
    $0.2$ (orange),
    $0.4$ (green),
    $0.6$ (red),
    $0.8$ (purple).
    $1.0$ (brown).
    The arrows show the direction of increasing $\Gamma$.
    Note that only the depths $0\le z \le 0.1$ are shown.
    \subref{fig:ssm_mono-mono/general/sweep-Gamma/part_2__v__posi_1.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-mono/general/sweep-Gamma/pore_2__v__posi_1.pdf} 
    The pore concentration, $N$.
    }
        \label{fig:ssm_mono-mono/general/sweep-Gamma/var__v__posi_1.pdf}
\end{figure}

% To interpret this, in Figure \ref{fig:ssm_mono-mono/general/sweep-Gamma/part_2__v__posi_1.pdf} we plot the inlet pore concentration, $N(0,t)$, as a function of time, $t$, for various values of $\Gamma$. 
% We note that the concentration of particles at the inlet, $C(0,t)$, is identical (see Figure \ref{fig:ssm_mono-mono/general/sweep-Gamma/var__v__posi_1.pdf}).
% We see that the case $\Gamma=0$ (blue line) corresponds to the case where the pore concentration never changes.
% For non-zero values of $\Gamma$, we see that, as time increases, the inlet pore-concentration decreases linearly towards zero. 
% As $\Gamma$ increases, the rate of this drop increases. 
% That is, as the ratio of particles to pores increases, 
% the time it takes all pores at the inlet to block, $T$, decreases. 
% Checking the times at which $N(0,t)$ reaches zero in Figure \ref{fig:ssm_mono-mono/general/sweep-Gamma/part_2__v__posi_1.pdf}, we see that the block time is always exactly $T=\Gamma^{-1}$. 

% In Figure \ref{fig:ssm_mono-mono/general/sweep-Gamma/T_bloc_1__v__Gamma_coarse_1.pdf}, we plot the block time, $T$, as a function of $\Gamma$. 
% We see that 
As such, the case $\Gamma\gg1$ corresponds to $T\ll1$. 
This represents the case where a huge number of particles are fed into the filter at once 
and block the pores on the surface almost instantaneously.
For example, when $\Gamma=10$ there are $10$ times as many particles as there are pores, and we see that
% we see that 
$T=0.1$, so that the filter is blocked after fluid has passed through the top $10\%$ of its length.
On the other hand, when $\Gamma\ll1$, it follows that $T\gg1$. 
This case models the situation where few particles enter the filter at once, and 
the filter is useful for much longer. 
For example, we see that, when there are $100$ times as many pores as there are particles entering per unit time, 
so that $\Gamma=0.01$, we have $T=100$, and the filter is useful until fluid has traversed its entire length $100$ 
times.
As expected then, the ratio $\Gamma^{-1}$ characterises the useful lifetime of the filter, at least in 
the present case, where all the particles are larger than all the pores.
% \begin{figure}[!htb]
%      \centering
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-Gamma/pore_2__v__time_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/general/sweep-Gamma/pore_2__v__time_1.pdf}
%      \end{subfigure}
%      %\hfill
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-Gamma/T_bloc_1__v__Gamma_coarse_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-mono/general/sweep-Gamma/T_bloc_1__v__Gamma_coarse_1.pdf}
%      \end{subfigure}
%     \caption{Graphs showing how the solution varies with $\Gamma$. 
%     \subref{fig:ssm_mono-mono/general/sweep-Gamma/pore_2__v__time_1.pdf} 
%     Graph showing how the pore concentration at the inlet, $N(0,t)$, varies with time, $t$, 
%     for various values of $\Gamma$.
%     The values of $\Gamma$ are 
%     $0.0$ (blue),
%     $0.01$ (orange),
%     $0.1$ (green),
%     $1$ (red),
%     $10$ (purple).
%     The arrow shows the direction of increasing $\Gamma$.
%     \subref{fig:ssm_mono-mono/general/sweep-Gamma/T_bloc_1__v__Gamma_coarse_1.pdf} 
%     Graph showing how the block time, $T$, varies with $\Gamma$.
%     The smaller graph is to show the region $1\le\Gamma\le10$ without the region 
%     $0.01\le\Gamma < 1$.
%     }
%         \label{fig:ssm_mono-mono/general/sweep-Gamma/var__v__time_1.pdf}
% \end{figure}


\paragraph{Behaviour as $\Lambda_{\text{B}}$ varies}
Secondly, we solve the mono-dispersed system, \eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono}, for fixed $\Gamma$ and various values of $\Lambda_{\text{B}}$.
In Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/var__v__posi_1.pdf} we plot the 
concentrations of particle and pores, $C$ and $N$, as functions of depth, $z$, 
for various values of $\Lambda\bloc$. 
We plot the solutions at $t=1$, and, since $\Gamma=1$, we have that $T=1$, so 
these plots show the solutions when the filtration process ceases. 
In Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1.pdf} we see that 
particles
 have penetrated 
 to a depth within the filter 
 that decreases as $\Lambda\bloc$ increases, 
 as expected. 
 % mass has extended over layer that extends from the inlet into the filter as the particles have been captured.
In Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/pore_2__v__posi_1.pdf} we see that pores have been blocked over a 
layer of corresponding length. 
% In both Figures we see that, as $\Lambda\bloc$ increases, the length of the layer decreases.
We conclude that, 
as the filter length increases compared to the blocking length, 
more of the filter is wasted.
% as the filter length shrinks relative to the particle size, 
% the proportion of the filter used to block particles increases.
% So, as the size of the particles increases (relative to the size of the filter), 
% the proportion of the filter that is used to block particles increases. 
% It follows that, when particles are significantly smaller than the thickness of the filter, 
% the vast majority of the filter is wasted, at least in the present case, in which all the 
% particles are larger than all of the pores.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-LamB/part_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-LamB/pore_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/sweep-LamB/pore_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and pore concentration,
    $C$ and $N$, respectively, 
    vary with depth, $z$, for various $\Lambda\bloc$.
    The parameters are $\Gamma=1$, $A=2$, 
    and the solutions are plotted at time $t=1$. 
    The values of $\Lambda\bloc$ are
    $1$ (blue),
    $10$ (orange),
    $100$ (green),
    $1000$ (red),
    The arrows show the direction of increasing $\Lambda\bloc$.
    \subref{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-mono/general/sweep-LamB/pore_2__v__posi_1.pdf} 
    The pore concentration, $N$.
    }
    \label{fig:ssm_mono-mono/general/sweep-LamB/var__v__posi_1.pdf}
\end{figure}

To calculate the effective length, $l_{\text{eff}}$, the length of the filter that has been used for particle capture,
we 
define a desired threshold
filtrate fluid particle concentration, $C_{\text{eff}}$ say, 
below which the concentration at $z=1$ must remain. 
% Particle concentrations in the fluid that exits the filter that exceed $C_{\text{eff}}$ are unacceptable. 
Given $C_{\text{eff}}$, we say that $l_{\text{eff}}$ is the distance 
from the inlet that is required to reduce the void concentration to $C_{\text{eff}}$. 
In Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1__LamB-10-100.pdf}, 
for example, we plot the void concentration, $C_{\text{V}}$, as a function of depth, $z$, at $t=1$, for various values of $\Lambda\bloc$.
We see how, given an example threshold concentration, $C_{\text{eff}}=0.1$ here, we read off 
values of $l_{\text{eff}}$.
We see that, for a given $C_{\text{eff}}$, 
increasing $\Lambda\bloc$ decreases the value of $l_{\text{eff}}$. 

To understand this quantitatively, we recall that, for large $\Lambda\bloc$, at leading order in $\Lambda\bloc^{-1}$, 
the void concentration is given by \eqref{eq:cv-sol}.
% \eqref{eq:ssm_sol_mono_inner_const-flow}.
% \begin{align}
%     C(z,t) = (1-\Gamma t e^{-\Lambda\bloc z})e^{-\Lambda\bloc z}.
% \end{align}
% So 
% \begin{align}
%     C_{\text{eff}} = (1- e^{-\Lambda\bloc l_{\text{eff}}})e^{-\Lambda\bloc l_{\text{eff}}},
% \end{align}
It follows that 
\begin{align}
    l_{\text{eff}} = -\frac{\text{log}(C_{\text{eff}})}{\Lambda\bloc}.
\end{align}
% where we have used that $T=\Gamma^{-1}$.
% Defining $\zeta =\text{exp}(-\Lambda\bloc l_{\text{eff}})$, 
% we see that $\zeta$ satisfies the quadratic 
% \begin{align}
%     \zeta^2 - \zeta + C_{\text{eff}} = 0,
% \end{align}
% the roots of which are given by
% \begin{align}
%     \zeta_{{\pm}} = \frac{1}{2} \pm \frac{\sqrt{1-4 C_{\text{eff}} }}{2}.
%     \label{eq:ssm_roots}
% \end{align}
% We notice that this has two real and positive solutions provided that $C_{\text{eff}}\le1/4$.
% \begin{align}
%     l_{\text{eff}} = -\frac{1}{\Lambda\bloc}\log\left(\zeta_{\text{eff}}\right), 
% \end{align}
% , as we see in Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1.pdf}.

% We plot $C_{\text{eff}}$ 
% as a function of $\Lambda\bloc l_{\text{eff}}$  in Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/l_effe_1__v__LamB_1.pdf}, and see that 
% $C_{\text{eff}}$ decreases rapidly 
% as $\Lambda\bloc l_{\text{eff}}$ increases. 
% The filter is close to $100\%$ efficient
% by $\Lambda\bloc l_{\text{eff}}=5$. 
% Another intepretation is that 
% $l_{\text{effe}}$ decreases rapidly as 
% $C_{\text{effe}}$ increases, so that, as expected, we find that a less stringent retention threshold means that 
% less of the filter's depth is used.

We plot $l_{\text{eff}}$ as a function of $C_{\text{eff}}$ 
for various values of $\Lambda\bloc$ in Figure \ref{fig:ssm_mono-mono/general/sweep-LamB/l_effe_1__v__LamB_1.pdf}.
As expected, the length, $l_{\text{eff}}$, of filter needed to reduce the concentration to an acceptable level, $C_{\text{eff}}$, 
decreases as $C_{\text{eff}}$ increases.
In other words, a smaller proportion of the filter is needed if the filtration task is less stringent. 
$l_{\text{eff}}$ decreases as $\Lambda\bloc$ increases,
which implies, as expected, that less of the filter is used when particles are smaller in comparison to the 
filter as a whole.
% \correct{
% We see that 
% $l_{{\text{eff}}}=\mathcal{O}(\Lambda\bloc^{-1})$, 
% and thus, as expected, 
% as $\Lambda\bloc$ increases, 
% the filter is larger compared to the particles, 
% and so less of it is used for blocking. 
% As the maximum acceptable filtrate concentration, $C_{\text{eff}}$, increases, 
% the length of the filter used, $l_{\text{eff}}$, decreases.  
% In other words, less of the filter is required if the filtration task is less stringent.}{} 
One implication is that, in this idealistic regime, where there is one particle size and pore size, if the filter is long compared to the particle size, 
% are small compared to the filter,
a very stringent concentration 
constraint (i.e., $C_{\text{eff}}\ll1$) can be obtained at no extra cost, as the filtration process will use more pores but take the same amount of time to run to completion.

% by the time that it is blocked. 
% so that $l_{{\text{eff}}}$ decreases. 
% so that $l_{{\text{eff}}}=\mathcal{O}(\Lambda\bloc^{-1})$. 
% Indeed, we see that $l_{{\text{eff}}}=\mathcal{O}(\Lambda\bloc)$, so that the proportion of the filter that is 
% used is the same as that of the size of the particles as a proportion of the size of the filter. 


\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-mono/general/sweep-LamB/part_2__v__posi_1__LamB-10-100.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1__LamB-10-100.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]
         {./figures/size-structured/mono-mono/general/sweep-LamB/l_effe_1__v__LamB_1.pdf}
         % {./figures/size-structured/mono-mono/general/sweep-LamB/C_effe_1__v__LamB_times_l_effe_1.pdf}
         \caption{}
         \label{fig:ssm_mono-mono/general/sweep-LamB/l_effe_1__v__LamB_1.pdf}
     \end{subfigure}
    \caption{
    % Graphs showing how the solution varies with $\Lambda\bloc$. 
    % Graphs showing the particle concentration and capture-layer length 
    % as functions of depth and the blocking parameter. 
    \subref{fig:ssm_mono-mono/general/sweep-LamB/part_2__v__posi_1__LamB-10-100.pdf}
    Graph showing how the particle concentration, $C$, varies with depth, $z$, at $t=1$ for $\Gamma=1$.
    for two values of $\Lambda\bloc$.
    The values of $\Lambda\bloc$ are 
    $10$ (orange),
    $100$ (green),
    The arrow shows the direction of increasing $\Lambda\bloc$.
    An example efficient concentration threshold, $C_{\text{eff}}$, is plotted 
    with a dotted black line. 
    The lengths of the corresponding capture layers are  
    extrapolated in correspondingly coloured dashed lines.
    \subref{fig:ssm_mono-mono/general/sweep-LamB/l_effe_1__v__LamB_1.pdf} 
    Graph showing how the capture-layer length, $l_{\text{eff}}$, varies with $\Lambda\bloc$, for 
    various efficient concentrations, $C_{\text{eff}}$.
    The values for $C_{\text{eff}}$ are 
    $0.01$ (blue),
    $0.05$ (orange),
    $0.1$ (green),
    $0.2$ (red).
    The arrow shows the direction of increasing $C_{\text{eff}}$.
    }
    \label{fig:ssm_mono-mono/general/sweep-LamB/l_effe}
\end{figure}




\subsection{Blocking and deposition}
\label{subsec:ssm_mono-mono_blocking-and-deposition}
We now consider the case where particles are smaller than pores.
As before, pores are all size $1$ initially, and we choose $A<1$. 

% We find a numerical solution and use this to 
% interpret the deposition-length ratio parameter, $\Lambda_{\text{D}}$ and the adherence parameter, $\varsigma_{\text{D}}$.

We choose the sticking probability $\varsigma\depo=0.1$, 
so that particles deposit in pores that are larger than 
them $10\%$ of the times that they pass through them.
We choose the transformation distribution 
$\tau_{\text{D}}(a,\rho,r) = \updelta(r-(\rho-a))$,
so that pores of size $\rho$ are transformed to 
pores of size $r$ by particles of size $a<\rho$.
It follows that particles of size $A$ can deposit in pores of size $1$, and, when they do so, these pores are transformed to pores of size $R=1-A$. 
We assume that $A\ge 1/2$, so that the new pores of size $R$ are blocked by 
particles of size $A$.  
This special case is the simplest scenario within which to observe blocking and deposition in the same filter, since 
it follows that the pore-size distribution remains (at most) 
bi-dispersed for all time, 
permitting the ansatz 
\begin{align}
    c(z,t,a) = \updelta(a-A)C(z,t), 
    \, \, 
    n(z,t,r) = N_R(z,t) \updelta(r-R) + N_1(z,t) \updelta(r-1),
\end{align}
where $N_R$ and $N_1$ are the concentrations 
of pores of sizes $R$ and $1$ respectively.

Using this ansatz, \eqref{eq:ssm_c-eqn_summary_nodim} and \eqref{eq:ssm_n-eqn_summary_nodim} become
\begin{align}
 \pp{C}{t} + \p{z} \left( \frac{UC}{R^2 N_R+N_1} \right) = & 
 - \frac{UC}{(R^2 N_R+N_1)^2} 
 \left( \Lambda_{\text{B}} R^2 N_R
 +
 \Lambda_{\text{D}} \varsigma_{\text{D}}N_1 
 \right),
 \label{eq:ssm_c-eqn_mono-bi}
\\
\pp{N_R}{t} = & - \frac{\Gamma UC}{(R^2 N_R+N_1)^2}
\left(R^2 N_R
-  \varsigma_{\text{D}} N_1 \right),
 \label{eq:ssm_nR-eqn_mono-bi}
\\
\pp{N_1}{t} = & - \frac{\Gamma UC}{(R^2 N_R+N_1)^2} \varsigma_{\text{D}} N_1 .
 \label{eq:ssm_n1-eqn_mono-bi}
\end{align}
In the constant flow rate case it follows from 
\eqref{eq:ssm_Up-eqns_const-flow_summary_nodim} that the velocity and pressure are 
given by 
\begin{align}
    U=1, 
    \quad
    p(z,t) = \int_{z}^1 \frac{1}{
    R^4 N_R(\zeta,t)+N_1(\zeta,t)}  \, \text{d}\zeta,
\end{align}
while in the constant pressure drop case it follows from \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim}  that these are given by 
\begin{align}
    U(t) = \frac{1}{\int_0^1 \frac{1}{R^4 N_R(z,t)+N_1(z,t)} \, \text{d}z},
    \quad
    p(z,t) = \frac{\int_z^1 \frac{1}{R^4 N_R(\zeta,t)+N_1(\zeta,t)} \, \text{d}\zeta}{\int_0^1 \frac{1}{R^4 N_R(\zeta,t)+N_1(\zeta,t)} \, \text{d}\zeta},
\end{align}
since, for bi-dispersed pores, the local porosity and permeability, defined in 
\eqref{eq:ssm_perm-and-poros_nodim_summary}, 
are given by
\begin{align}
    \phi(z,t) = R^2N_R(z,t) + N_1(z,t),
    \quad
    K(z,t) = R^4N_R(z,t) + N_1(z,t),
    \label{perm-bi}
\end{align}
respectively.
The boundary and initial conditions, \eqref{eq:ssm_bound-cond_nodim_summary} and \eqref{eq:ssm_init-conds_nodim_summary}, become 
\begin{align}
    C(0,t) = R^2N_R(0,t) + N_1(0,t),
\end{align}
and 
\begin{align}
    C(z,0) = 0, 
    \quad
    N_R(z,0) = 0,
    \quad
    N_1(z,0) = 1,
    \label{eq:ssm_init-conds_mono-bi}
\end{align}
since the pore sizes are mono-dispersed initially.
In the remainder of this subsection, we will numerically solve this system, \eqref{eq:ssm_c-eqn_mono-bi}--\eqref{eq:ssm_init-conds_mono-bi}, and analyse the solutions.

\subsubsection{The case $\Gamma=\mathcal{O}(1)$, $\Lambda\bloc\gg1$, and $\Lambda\depo\gg1$: A numerical solution}
% $\Gamma=\mathcal{O}(1)$ and $\Lambda_{\text{B}}=\mathcal{O}(1)$}
For general values of $\Gamma$, $\Lambda_{\text{B}}$, and $\Lambda_{\text{D}}$, we solve the blocking and deposition system, \eqref{eq:ssm_c-eqn_mono-bi}--\eqref{eq:ssm_init-conds_mono-bi}, similarly to the system in the case
where there is no deposition,
\eqref{eq:ssm_c-eqn_mono}--\eqref{eq:ssm_init-conds_mono}.
The particle concentration equation, 
\eqref{eq:ssm_c-eqn_mono-bi}, is now coupled to a reaction equation, \eqref{eq:ssm_nR-eqn_mono-bi} and \eqref{eq:ssm_n1-eqn_mono-bi}, for the concentration of pores of each size, but, from a numerical-methods perspective, this does not pose any new challenges. We now discuss the numerical solutions in the case 
$\Gamma=\mathcal{O}(1)$, $\Lambda\bloc\gg1$, and $\Lambda\depo\gg1$. 

\paragraph{Particle concentration and porosity}
We plot the solutions for the particle concentration, $C$, and the porosity, $\phi=N_1 + R^2 N_R$, 
as functions of depth, $z$,
at early times, in Figure \ref{fig:ssm_mono-bi/general/LamB-200_LamD-100/var_3__v__posi_1__early.pdf}. 
We see that the behaviour is qualitatively similar to 
the early-time behaviour of the system in the blocking only regime, from before. 
Particles travel as a wave through a layer at the top of the filter, and, as they do 
so, 
they are removed so that the concentration 
follows a curve of exponential decay behind the wave front.
% drops exponentially towards zero behind the wave. 
% We see that, 
% for this particular parameter choice, 
% the layer over which 
% particles are captured is longer 
% than before 
% than in the blocking regime. 
We choose $\Lambda\bloc=200$, 
which is twice as large as $\Lambda\bloc=100$ before.
This has the effect of decreasing the 
length of the layer over which particles are captured due to blocking. 
However, 
$\Lambda\depo=100$ and $\varsigma\depo=0.1$, 
so that $\Lambda\depo\varsigma\depo=10$
is ten times as small as $\Lambda\bloc=100$ before.
This has the effect of increasing the length over which particles are captured.
The net effect is that particles are 
captured over a longer length in the present parameter regime. 
% is twice as large as it was before.
% It follows that, if there were no deposition, we would expect particles to be captured over a layer half as long as before. 
% Despite this, capture occurs over a layer that is five times as long.
% The length over which deposition is effective is $\mathcal{O}(\Lambda\depo\varsigma\depo)^{-1}$. 
% Here, $\Lambda\depo=100$ and $\varsigma\depo=0.1$, 
% so that the dual effect of blocking and deposition leads to an effective capture layer of length around five times as long as before.
In general, particles are captured via a mixture of blocking and deposition, 
and the length of the capture layer is determined by $\max \{ \Lambda\bloc^{-1}, (\Lambda\depo\varsigma\depo)^{-1}\}$. 
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/LamB-200_LamD-100/part_3__v__posi_1__early.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/part_3__v__posi_1__early.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1__early.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1__early.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and local porosity, 
    $C$ and $\phi$, vary with depth, $z$, for various times, $t$.
    The parameters are $\Lambda\bloc=200$, $\Lambda\depo=100$, $\Gamma=1$, $A=0.5$, $\varsigma\depo=0.1$. 
    The values of $t$ are 
    $0$ (blue),
    $0.005$ (orange),
    $0.01$ (green),
    $0.02$ (red),
    $0.04$ (purple).
    The arrows show the direction of increasing time.
    Note that only the depths $0\le z \le 0.5$ are shown.
    \subref{fig:ssm_mono-bi/general/LamB-200_LamD-100/part_3__v__posi_1__early.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1__early.pdf} 
    The local porosity, $\phi$.
    }
    \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/var_3__v__posi_1__early.pdf}
\end{figure}

Next, in Figure \ref{fig:ssm_mono-bi/general/LamB-200_LamD-100/var_3__v__posi_1.pdf}, 
we plot the particle concentration and the local porosity as functions of depth at later times. 
We see that the solutions are qualitatively similar to those in the blocking regime.
% ,
% but that changes occur due to blocking and deposition, 
% and thus over a longer layer, for the present parameters.
In Figure \ref{fig:ssm_mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1.pdf}  
we see that the decrease in the porosity is fastest at the inlet, where there are the most particles.  
However, this time, the porosity at the inlet decreases to zero over a longer timescale 
% which is $10$ 
% times as large 
than in the blocking regime. 
This is because, in this example, deposition is ten times less likely to occur than blocking in the previous case (since $\varsigma\depo=0.1$), so that removal of all inlet pores takes around a factor of  $\varsigma\depo^{-1}=10$ times as long so that all inlet pores become clogged around $t=10$ here (as opposed to around $t=1$ before). 
% It follows that $T\approx(\Gamma\varsigma\depo)^{-1}$ when blocking and deposition occur, as opposed to $\Gamma^{-1}$ when only blocking occurs. 

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/LamB-200_LamD-100/part_3__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/part_3__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and local porosity, 
    $C$ and $\phi$, vary with depth, $z$, for various times, $t$.
    The parameters are $\Lambda\bloc=200$, $\Lambda\depo=100$, $\Gamma=1$, $A=0.5$, $\varsigma\depo=0.1$. 
    The values of $t$ are 
    $0$ (blue),
    $2.5$ (orange),
    $5$ (green),
    $7.5$ (red),
    $10$ (purple).
    The arrows show the direction of increasing time.
    Note that only the depths $0\le z \le 0.5$ are shown.
    \subref{fig:ssm_mono-bi/general/LamB-200_LamD-100/part_3__v__posi_1.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-bi/general/LamB-200_LamD-100/poro_2__v__posi_1.pdf} 
    The local porosity, $\phi$.
    }
    \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/var_3__v__posi_1.pdf}
\end{figure}

\paragraph{Pore concentration and permeability}
We plot the local permeability, and the concentrations of large and small pores, as functions of 
depth, for various times, in  
Figure \ref{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1.pdf}. 
We focus on the concentration of large pores first, which we show in 
Figure \ref{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__large.pdf}. 
We see that large pores (solid lines) are removed via deposition (only) 
over the deposition layer.
We see that the permeability (dashed lines) is almost identical to the concentration 
of large pores, since the correction to the permeability due to smaller pores in \eqref{perm-bi} is $R^4N_{R}\approx0.06N_R$.
% These are twice as large as the small pores (which is important because of the weighting by a factor of $r^4$ in the local permeability) and account for all of the void space initially.

We now turn our attention to the concentration of small pores, which is plotted in 
Figure \ref{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__small.pdf}. 
We notice that these are removed due to a complex combination of deposition and blocking. 
To see this, we first note that, 
at earlier times, the concentration of small pores increases over the entire deposition layer. 
This is because, early on, large pores are transformed into small pores as particles (that are smaller than the large pores)
deposit in them. 
% As time increases, we see that the concentration of small pores begins to deviate 
% from the concentration of large pores that have been removed.
% (comparing the orange solid line with the orange dashed line, for example). 
Small pores are being blocked by the particles (which are larger than the small pores) 
at the same time as they are being produced by deposition in the large pores. 
This deviation is more extreme closer to the inlet, because there are the most particles available to 
block the small pores. 
Blocking happens in a shorter layer than deposition because $\Lambda\bloc^{-1}>(\Lambda\depo\varsigma\depo)^{-1}$, 
so particles never reach the small pores further down the filter to remove them. 
At later times, we see that the concentration of small pores begins to decrease near the inlet. 
This is because blocking is $10$ times as likely to occur as deposition (since $\varsigma\depo=0.1$). 
As a result, when 
there are enough small pores, the rate at which they are removed due to blocking exceeds the rate at which they are produced due to deposition.
Note that this is despite the fact that more particles flow through large pores per unit time.
 This example demonstrates the complicated interplay between blocking and deposition in filters with more complicated size distributions.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__large.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__large.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__small.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__small.pdf}
     \end{subfigure}
    \caption{Graphs showing how the local permeability, $K$, and the concentrations of the larger and smaller pores, $N_1$ and $N_R$, 
    vary with depth, $z$, for various times, $t$.
    The parameters are $\Lambda\bloc=200$, $\Lambda\depo=100$, $\Gamma=1$, $A=0.5$, $\varsigma\depo=0.1$. 
    The values of $t$ are 
    $0$ (blue),
    $2.5$ (orange),
    $5$ (green),
    $7.5$ (red),
    $10$ (purple).
    The arrows show the direction of increasing time.
    Note that only the depths $0\le z \le 0.5$ are shown.
    \subref{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__large.pdf} 
    The concentration of the larger pores, $N_1$ (solid lines). 
    The local permeability, $K$ (dashed lines).
    \subref{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1__small.pdf} 
    The concentration of the smaller pores, $N_R$.
    % (solid lines). 
    % The concentration of larger pores that have been lost, $1-N_1$ (dashed lines).
    }
    \label{fig:ssm_mono-bi/general/LamB-200_LamD-100/pore_3__v__posi_1.pdf}
\end{figure}

\paragraph{Behaviour as 
% $\Lambda_{\text{D}}
$\varsigma\depo$ varies}
% Since deposition occurs in this example, 
We now assess 
the effect of varying the sticking probability, 
% $\Lambda\depo$ and 
$\varsigma\depo$.
We solve the system \eqref{eq:ssm_c-eqn_mono-bi}--\eqref{eq:ssm_init-conds_mono-bi} for fixed $\Gamma$, $\Lambda_{\text{B}}$, and $\Lambda_{\text{D}}$, with various values of $\varsigma\depo$.
% To achieve this, and noting that $\Lambda\depo$ and $\varsigma\depo$ 
% only ever appear together, we fix $\Lambda\depo$ and vary $\varsigma\depo$.
We note that parameter sweeps, such as this, are significantly more computationally expensive than in the blocking regime, since before we were equipped with explicit asymptotic solutions.

In Figure \ref{fig:ssm_mono-bi/general/sweep-LamD} we plot the particle concentration and local porosity as functions of depth, $z$, at time $t=10$, for various values of $\varsigma\depo$.
Firstly, we see that as $\varsigma\depo$ increases, the length of the layer over which 
particle capture and pore removal occurs decreases. 
For example, when $\varsigma\depo=0.02$ (blue line),
% for which $\Lambda\depo\varsigma\depo=2$, 
deposition occurs over the entire filter. 
Particles only deposit in pores that are larger (than the particles) $2\%$ of the time,
so the layer over which deposition occurs has length $\mathcal{O}((\Lambda\depo\varsigma\depo)^{-1})=\mathcal{O}(1)$.
Some particles pass all the way to the end of the filter without 
being captured, and so the concentration of particles that remain at the outlet is around $0.025$. 
On the other hand, when the probability of deposition is $10\%$, 
so that $\varsigma\depo=0.1$, 
the deposition layer is length 
$\mathcal{O}((\Lambda\depo\varsigma\depo)^{-1})=\mathcal{O}(10^{-1})$
and % for which $(\Lambda\depo\varsigma\depo)^{-1}=0.1$, 
almost all particles are captured by the time they have traversed the first $10\%$ of the filter. 
The concentration is negligible at the outlet.
Secondly, we see that the rate at which pores are removed at the inlet 
increases as $\varsigma\depo$ increases. 
Thus, for larger $\varsigma\depo$, the filtration process runs for much less time, 
even though the concentration of particles present at the outlet is not substantially different.
One important implication 
% is that, when particles are smaller than pores so that deposition is possible,
% it might be possible to choose $\Lambda\depo$ and $\varsigma\depo$ 
% so to achieve the desired particle retention using the entire filter, 
% rather than just a thin section at the top (as in the blocking regime),
% which is all that is used in a regime in which only blocking is possible (at least when the particles are very small compared to the filter, so that $\Lambda\bloc\ll1$).
% That is,
is that, if it is possible to control the sticking probability, $\varsigma\depo$, 
then it could be chosen so that the capture layer is longer, 
increasing the proportion of pores that are used  
for particle capture, 
thus
extending the useful lifetime of the filter.
% of the filter is extended.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/sweep-LamD/part_3__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/sweep-LamD/part_3__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-bi/general/sweep-LamD/poro_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-bi/general/sweep-LamD/poro_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration and local porosity, 
    $C$ and $\phi$, vary with depth, $z$, for various values of deposition probability,  
    $\varsigma\depo$.
    The parameters are $\Lambda\bloc=200$, $\Lambda\depo=100$, $\Gamma=1$, $A=0.5$, 
    and the solutions are plotted at time $t=10$.
    The values of $\varsigma\depo$ are 
    $0.02$ (blue),
    $0.04$ (orange),
    $0.06$ (green),
    $0.08$ (red),
    $0.1$ (purple).
    The arrows show the direction of increasing $\varsigma\depo$ (and therefore 
    increasing $\Lambda\depo\varsigma\depo$).
    \subref{fig:ssm_mono-bi/general/sweep-LamD/part_3__v__posi_1.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-bi/general/sweep-LamD/poro_2__v__posi_1.pdf} 
    The local porosity, $\phi$.
    }
    \label{fig:ssm_mono-bi/general/sweep-LamD}
\end{figure}

In this section, we have explored the case  
where particles and pores are both one size initially. 
When the particles are larger than the pores, only blocking is possible, 
and we used this case to explore the effect of changing $\Gamma$ and $\Lambda\bloc$. 
When the particles are smaller than the pores, blocking and deposition are both 
possible (when $A$ is chosen appropriately), and we used this case to
explore the effect of changing $\varsigma\depo$.
In the next section, we turn our attention to more complex cases, in which the pores in the filter are poly-dispersed.


\section[Mono-dispersed particles and poly-dispersed pores]{Mono-dispersed particles and poly-dispersed\\pores}
\label{sec:ssm_Mono-dispersed particles and poly-dispersed pores}
In this section, we assume that particles entering the filter at the inlet all have size $A$, so that particles are mono-dispersed everywhere at all times,
\begin{align}
    c(z,t,a) = \updelta(a-A) C(z,t).
\end{align}
% (\ref{eq:ssm_distr_c_nodim_summary}a) as the particle-size distribution at the inlet. 
We suppose that pores in the filter have log-normally dispersed sizes, with mean size 1.
It follows that the boundary condition, 
\eqref{eq:ssm_bound-cond_nodim_summary}, 
becomes 
\begin{align}
    C(0,t) = \phi(0,t),
    \label{eq:ssm_bound-cond_mono-poly}
\end{align}
as in the previous section, while the initial conditions, 
% but that pores in the filter have log-normally dispersed sizes, with mean size $1$.
% (\ref{eq:ssm_distr_n_nodim_summary}b) 
% % as the initial pore-size distribution. 
% The initial conditions, 
\eqref{eq:ssm_init-conds_nodim_summary}, 
become
\begin{subequations}
\label{eq:ssm_init-conds_mono-poly}
\begin{align}
C(z,0) = 0, 
\quad
n(z,0,r) =  n_{\text{init}}(r),
% \frac{\phi^*_{\text{init}} e^{-\sigma_n^2} }{\sqrt{2 \pi} \sigma_n r} 
%     e^{- \frac{ 
%     \left( 
%     \log \left( r \right) + \frac{\sigma_n^2}{2} 
%     \right)^2
%     }{
%     2 \sigma_n^2}},
\tag{\ref{eq:ssm_init-conds_mono-poly}a,b}
\end{align}
\end{subequations}
where we define 
\begin{align}
     n_{\text{init}}(r) = \frac{\phi^*_{\text{init}} e^{-\sigma_n^2} }{\sqrt{2 \pi} \sigma_n r} 
    e^{- \frac{ 
    \left( 
    \log \left( r \right) + \frac{\sigma_n^2}{2} 
    \right)^2
    }{
    2 \sigma_n^2}},
    \label{eq:ssm_definition-of-n-init}
\end{align}
since we calculate that the denominator in (\ref{eq:ssm_init-conds_nodim_summary}b) satisfies
\begin{align}
    \frac{1}{\sqrt{ 2 \pi} \sigma_n} \int_0^\infty r 
    e^{- \frac{ 
    \left( 
    \log \left( r \right) + \frac{\sigma_n^2}{2} 
    \right)^2
    }{
    2 \sigma_n^2}} \, \text{d} r = e^{\sigma_n^2}.
\end{align}
% in (\ref{eq:ssm_init-conds_nodim_summary}a).
In this section, we solve the size-structured model 
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
subject to the boundary and initial conditions 
\eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}.
In Subsection \ref{subsec:ssm_mono-poly_blocking} we consider the case where $\varsigma\depo=0$ 
% so that particles 
% are not captured within pores that are larger than them due to deposition, and
in which only blocking is possible. 
In Subsection \ref{subsec:ssm_mono-poly_blocking-and-deposition}, we then consider the case  $\varsigma\depo=0.001$, so that
% deposition 
% and blocking both occur, but
deposition occurs with $0.1\%$ of the probability 
that blocking does.

% Note that, in this section, 
% we will use the following integral extensively, and so define
% \begin{align}
%     \mathcal{I}(a,b) &= 
%     \int_a^b r^2 d_n(r) \, \text{d}r 
%     \\
%     &= 
%     \frac{1}{\sqrt{ 2 \pi} \sigma_n} \int_a^b r 
%     e^{- \frac{ 
%     \left( 
%     \log \left( r \right) + \frac{\sigma_n^2}{2} 
%     \right)^2
%     }{
%     2 \sigma_n^2}} \, \text{d} r,
%     \\
%      &= 
%      - \frac{1}{2} e^{\sigma_n^2} \text{erf}\left(g(r)\right) \bigg|_a^b,
%     \label{eq:ssm_definition-of-I}
% \end{align}
% where  $a$ and $b$ are any real numbers, $\text{erf}$ is the error function, 
% and $g$ is defined 
% \begin{align}
%     g(r) = \frac{3\sigma_n^2 - \log(r)^2}{2 \sqrt{2} \sigma_n}.
% \end{align}
% We will use that $\mathcal{I}(0, \infty) =  e^{\sigma_n^2}$, 
% $\mathcal{I}(0, A) =  e^{\sigma_n^2}$as in [eq:above].

% We find explicit asymptotic solutions in the cases $\Gamma\ll1$ and $\Lambda_{\text{B}}\gg 1$ when only blocking occurs, 
% and compare these with a numerical solution in 
% the case $\Gamma=\mathcal{O}(1)$ and $\Lambda_{\text{B}}=\mathcal{O}(1)$ when blocking and deposition occur.
% \subsection{The small concentration-ratio case: $\Gamma \ll 1$}

% \subsection{The large blocking-length ratio case: $\Lambda_{\text{B}}\gg1$}

% \subsection{The general case}

\subsection{Blocking and no deposition}
\label{subsec:ssm_mono-poly_blocking}
We first consider 
% the size-structured model 
the system
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
subject to the boundary and initial conditions for mono-dispersed particles and log-normally 
poly-dispersed pores,
\eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}, in the case of zero adherence, $\varsigma_{\text{D}}=0$.
% , 
% so that blocking occurs but deposition does not.
We find a numerical solution for general $\Lambda\bloc$ 
% $\Lambda\bloc=\mathcal{O}(1)$, 
and then use the intuition gained from this solution
to reduce the system to a more simple one when $\Lambda\bloc\gg1$.

% For $\varsigma\depo=0$, t
The equations for the particle- and pore-size distributions, \eqref{eq:ssm_c-eqn_summary_nodim} and \eqref{eq:ssm_c-eqn_summary_nodim}, become
\begin{align}
    \pp{C}{t} + \frac{\partial }{\partial z}\left( \frac{C}{\phi}\right) 
     &= -\frac{\Lambda\bloc C}{\phi^2} \int_0^A r^2 n \, \text{d}r,
     \label{eq:ssm_c-eqn_mono-poly_general}
     \\
     \pp{n}{t} &= \frac{\Gamma C}{\phi^2} r^2 n H(A-r),
     \label{eq:ssm_n-eqn_mono-poly_general}
\end{align}
where $H$ is the Heaviside function.
The velocity and pressure are given by either 
\eqref{eq:ssm_Up-eqns_const-flow_summary_nodim} or \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim}.
In \eqref{eq:ssm_n-eqn_mono-poly_general} we see that the distribution of pores of size $r>A$ is constant and remains at the initial distribution.
We therefore write the porosity and permeability,
\eqref{eq:ssm_perm-and-poros_nodim_summary},
explicitly in terms of pores that are larger and smaller than the particles,
\begin{align}
    \phi = \varphi + \varphi_{[r>A]}, 
    \quad
    K = \kappa+ \kappa_{[r>A]},
    \label{eq:ssm_perm-and-poros_mono-poly_general}
\end{align}
where
\begin{align}
    \varphi(z,t) &= \int_0^A r^2 n(z,t,r) \, \text{d}r, 
    \quad
    \varphi_{[r>A]} = \int_A^{\infty} r^2 n\init \, \text{d}r,
    \\
    \kappa(z,t) &= \int_0^A r^4 n(z,t,r) \, \text{d}r, 
    \quad
    \kappa_{[r>A]} = \int_A^{\infty} r^4 n\init \, \text{d}r.
\end{align}
We call the unknown variables, $\varphi$ and $\kappa$, the small-pore porosity and permeability.
Note that $\varphi_{[r>A]}$ and $\kappa_{[r>A]}$, which we call the large-pore porosity and permeability, are fully specified by the initial conditions, and, indeed, we calculate that
\begin{align}
    \varphi_{[r>A]} &  = \frac{\phi^*\init}{2}(1+\text{erf}(f(A)), 
    \\
    \kappa_{[r>A]} &=  \frac{\phi^*\init e^{5\sigma_n^2}}{2}(1+\text{erf}(g(A)), 
\end{align}
where we define
\begin{align}
    f(A) = \frac{3\sigma_n^2 - 2\log(A)}{2 \sqrt{2} \sigma_n}, 
    \label{eq:ssm_def_of_f}
    \quad
    g(A) = \frac{7\sigma_n^2 - 2\log(A)}{2 \sqrt{2} \sigma_n}.
\end{align}
In the remainder of this subsection we solve \eqref{eq:ssm_c-eqn_mono-poly_general}--\eqref{eq:ssm_perm-and-poros_mono-poly_general}
with  either \eqref{eq:ssm_Up-eqns_const-flow_summary_nodim} or \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim} subject to the boundary and initial conditions
\eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}.

\subsubsection{The case $\Gamma = \mathcal{O}(1)$ and $\Lambda_{\text{B}} \gg 1$: A numerical solution}
For general values of $\Gamma$ and $\Lambda\bloc$, the equation for the particle concentration, 
\eqref{eq:ssm_c-eqn_mono-poly_general}, 
is a partial integro--differential advection--reaction equation that is coupled to a reaction equation, \eqref{eq:ssm_n-eqn_mono-poly_general}, which describes changes to the pore-size distribution.
Note that \eqref{eq:ssm_n-eqn_mono-poly_general} does not contain an integral because there is no deposition and particles are all one size.
These two equations, together with those for the porosity and permeability, \eqref{eq:ssm_perm-and-poros_mono-poly_general},
and either \eqref{eq:ssm_Up-eqns_const-flow_summary_nodim} or \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim}, subject to the boundary and initial conditions, \eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}, are similar to those in the case where the pore sizes are mono-dispersed, 
save for the integral on the right-hand side of \eqref{eq:ssm_c-eqn_mono-poly_general}. 
We therefore solve them using a similar numerical scheme, 
this time using the trapezium rule with 100 pore size points to evaluate the integral. 
That is, we use a first-order upwinding finite-difference scheme and discretise using 400 spatial points, and 
$N_t>400$ temporal points (where the choice of $N_t$ depends on how close to block time, $T$, and therefore how close to a local porosity of zero at the inlet, we will run), noting that we must choose time steps sufficiently small compared to space steps to satisfy the Courant--Friedrichs--Lewy (CFL) condition.
We present the solution for $\Lambda_{\text{B}} \gg 1$ and $\Gamma = \mathcal{O}(1)$ in the constant flow rate regime, and so choose \eqref{eq:ssm_Up-eqns_const-flow_summary_nodim}, rather than \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim}, to specify the velocity and the pressure.

\paragraph{Particle concentration and local porosity}
In Figure \ref{fig:ssm_mono-poly/no-depo/general/conc_part_and_poro}, 
we demonstrate how the particle concentration, $C$, and the local porosity, $\phi$, vary with depth, $z$, at various times, $t$. 
In Figure \ref{fig:ssm_mono-poly/no-depo/general/early-time/conc_part-poro_ttg_2__v__posi_1.pdf},
we see that, at early times, particles block pores at the inlet, which decreases the porosity and thus the concentration.
% , so that the concentration of particles in void space at the 
% inlet remains constant. 
Particles begin to travel through the filter, but are removed over a layer of thickness $\mathcal{O}(\Lambda\bloc^{-1})$.
In Figure \ref{fig:ssm_mono-poly/no-depo/general/conc_part-poro_ttg_2__v__posi_1.pdf} we see that, 
as time continues to increase, this early time behaviour settles into a regime in which particles begin to traverse the filter as a wave.
No particles have reached the region in front of the wave, so the concentration is zero 
and the porosity retains its initial value. 
Behind the wave, all pores that are smaller than the particles have been blocked. Since no pores larger than the particles can be clogged (because there is no deposition), after the wave has passed, the porosity retains some constant value  that is dictated by the proportion of pores that are larger than the particles. 
Behind the wave, the particle concentration matches the inlet concentration, since the fluid carries all 
particles that enter the filter through the larger pores unobstructed.
The concentration and local porosity transition between their two values within a layer of thickness 
$\mathcal{O}(\Lambda\bloc^{-1})$, which is thin when particles are small compared to the depth of the filter.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/early-time/conc_part-poro_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/early-time/conc_part-poro_ttg_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/conc_part-poro_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/conc_part-poro_ttg_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration, $C$  (solid lines), and local porosity, $\phi$ (dashed lines), vary  
    with the depth, $z$, at various times, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, 
    % $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$. 
    % $\varsigma\depo=0$.
    The arrows show the direction of increasing time.
    \subref{fig:ssm_mono-poly/no-depo/general/early-time/conc_part-poro_ttg_2__v__posi_1.pdf} 
    The earlier-time solutions, $C,\phi$.
    The values of $t$ are $0.5$ (blue), $1.0$ (orange),
    $2.0$ (green), $4.0$ (red), $10.0$ (black).
    \subref{fig:ssm_mono-poly/no-depo/general/conc_part-poro_ttg_2__v__posi_1.pdf} 
    The later-time solutions, $C,\phi$.
    The values of $t$ are $10$ (black),
    $20$ (green), $30$ (red), $40$ (purple), $50$ (brown),
    $60$ (pink), $70$ (grey), $80$ (olive), $90$ (cyan),
    $100$ (blue). 
    The dotted black lines show 
    the explicit solution.
    }
    \label{fig:ssm_mono-poly/no-depo/general/conc_part_and_poro}
\end{figure}

\paragraph{Local porosity and local permeability}
In Figure \ref{fig:ssm_mono-poly/no-depo/general/poro-and-perm_local} we show how the local porosity and permeability, $\phi$ and $K$, vary with depth, $z$, at various times. 
In Figure \ref{fig:ssm_mono-poly/no-depo/general/poro_ttg_2__v__posi_1.pdf} we see, as expected, that the value that the porosity is reduced to after the particle wave 
has passed is the large-pore porosity, $\varphi_{[r>A]}<\phi\init^*$, which is the proportion of total pore area that is attributed 
to pores that are larger than the particles. 
% , and can only be decreased via deposition, which does not occur. 
In Figure \ref{fig:ssm_mono-poly/no-depo/general/perm_ttg_2__v__posi_1.pdf}, we see that the local permeability behaves similarly to the local porosity.
It transitions from 
its initial value to the large-pore permeability, $\kappa_{[r>A]}$, over the thin blocking layer. 
The permeability drop is less extreme than the porosity drop, since $K$ is a factor of $r^2$ different from $\phi$, so unaffected larger pores are weighted more heavily. 
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/poro_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/poro_ttg_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/perm_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/perm_ttg_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the local porosity (left) and permeability (right), $\phi$ and $K$, vary with the depth, $z$, at various times, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, 
    % $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$. 
    % $\varsigma\depo=0$.
    The values of $t$ are $10$ (orange),
    $20$ (green), $30$ (red), $40$ (purple), $50$ (brown),
    $60$ (pink), $70$ (grey), $80$ (olive), $90$ (cyan),
    $100$ (blue). 
    The arrows show the direction of increasing time.
    \subref{fig:ssm_mono-poly/no-depo/general/poro_ttg_2__v__posi_1.pdf} 
    The local porosity, $\phi$. 
    The value of the large-pore porosity, $\varphi_{[r>A]}$, 
    is plotted in a black dotted line.
    \subref{fig:ssm_mono-poly/no-depo/general/poro_tota_1__v__time_1.pdf} 
    The local permeability, $K$. 
    The value of the large-pore permeability, $\kappa_{[r>A]}$, 
    is plotted in a black dotted line.
    }
    \label{fig:ssm_mono-poly/no-depo/general/poro-and-perm_local}
\end{figure}


\paragraph{Global porosity and global permeability}
We define the global porosity and global permeability to be 
\begin{align}
    \bar{\phi}(t) = \int_0^{1} \phi(z,t) \, \text{d}z, 
    \quad 
    \bar{K}(t) = \int_0^{1} K(z,t) \, \text{d}z. 
\end{align}
These are distinct from the local porosity and permeability, and we interpret them to be (depth-independent) measures of the `total' porosity 
and permeability (i.e., of the entire filter).
In Figure \ref{fig:ssm_mono-poly/no-depo/general/poro-and-perm_global} we demonstrate 
how these vary with time, $t$. 
We see that these decrease linearly from their initial values, which are dictated by the
initial distribution of pore-sizes, to lower values at some critical time, $t\approx90$. 
After $t\approx90$, these global measures of pore space remain constant for all time. 
We see that the values that the global porosity and permeability are reduced to are $\varphi_{[r>A]}$ 
and $\kappa_{[r>A]}$ respectively. 
The critical time, $t\approx90$ here, is the time at which the particle wave reaches the filter outlet.
Before this, pores are being blocked at the wave-front, and the local porosity and permeability are being reduced, which contributes to the constant reduction of their integrals. 
After this, there are no longer any smaller pores to be blocked at any depth, and the filter is no longer useful, so further changes to the global porosity and permeability do not occur.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/poro_tota_1__v__time_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/poro_tota_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/perm_tota_1__v__time_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/perm_tota_1__v__time_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the global porosity and permeability, $\bar{\phi}$ and $\bar{K}$, vary with time, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, 
    % $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, 
    $\phi^*_{\text{init}}=0.9$. 
    % $\varsigma\depo=0$.
    \subref{fig:ssm_mono-poly/no-depo/general/poro_tota_1__v__time_1.pdf} 
    The global porosity, $\bar{\phi}$.
    The value of the large-pore porosity, $\varphi_{[r>A]}$, 
    is plotted in a black dotted line.
    \subref{fig:ssm_mono-poly/no-depo/general/perm_tota_1__v__time_1.pdf} 
    The global permeability, $\bar{K}$.
    The value of the large-pore permeability, $\kappa_{[r>A]}$, 
    is plotted in a black dotted line.
    }
    \label{fig:ssm_mono-poly/no-depo/general/poro-and-perm_global}
\end{figure}

\paragraph{Pressure and pore-size distribution}
In Figure \ref{fig:ssm_mono-poly/no-depo/general/pres_ttg_2__v__posi_1.pdf} we plot the local pressure as a function of depth at various times. 
We see that, initially, the 
% local 
pressure gradient is uniform. 
% , and 
% that the difference between the inlet and outlet pressure is exactly $\kappa\init^{-1}$. 
At later times, we observe how changes to the local permeability impact the pressure. 
As the local permeability decreases at a given depth, the pressure here increases so that the pressure drop over the entire filter increases. 
At the critical time, $t\approx90$ in this case, the value of $p(0,t)$, 
which is the filter pressure drop (i.e., the pressure drop across the entire filter), 
reaches $1/\kappa_{[r>A]}$, after which there are no 
further changes to the pressure. 
In practice, in constant flow rate regimes, operators monitor the filter pressure drop and ensure that this does not exceed some threshold, at which point the process must be halted due 
to external operational constraints.
We see that, in the present case where there is no deposition, it is relatively simple to construct a pore-size distribution for which the threshold value is not exceeded during the useful lifetime of the filter, by choosing a pore-size distribution for which enough pores are larger than the particles, so that $\kappa_{[r>A]}$ is not too small and thus $p(0,t)$ is not too large.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/pres_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/pres_ttg_2__v__posi_1.pdf}
     \end{subfigure}
     %hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/no-depo/general/pore_ttg_3__v__size_pore_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/no-depo/general/pore_ttg_3__v__size_pore_1.pdf}
     \end{subfigure}
     %\hfill
    \caption{Graphs showing how the pore-size distribution (left) and local pressure (right), $n$ and $p$, vary with the pore size, $r$, and depth, $z$, respectively, at various times $t$.
    The parameters are $\Lambda_{\text{B}}=200$, 
    % $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$. 
    % $\varsigma\depo=0$.
    The arrows show the direction of increasing time.
    \subref{fig:ssm_mono-poly/no-depo/general/pres_ttg_2__v__posi_1.pdf} 
    The local pressure, $p$, as a function of depth, $z$, at various times.
    The values of $t$ are $0$ (blue), $10$ (orange),
    $20$ (green), $30$ (red), $40$ (purple), $50$ (brown),
    $60$ (pink), $70$ (grey), $80$ (olive), $90$ (cyan),
    $100$ (blue). 
    The value of the inverse large-pore permeability, $\kappa_{[r>A]}^{-1}$, 
    is plotted in a black dotted line.
    \subref{fig:ssm_mono-poly/no-depo/general/pore_ttg_3__v__size_pore_1.pdf} 
    The pore-size distribution, $n$, as a function of the pore size, $r$, at various times, $t$. 
    The initial condition is shown in a dashed line.
    The mean particle-size, $A$, is plotted in a solid black line.
    The values of $t$ are $0$ (blue), $41$ (orange),
    $42$ (green), $43$ (red), $44$ (purple), $45$ (brown),
    $46$ (pink), $47$ (grey), $48$ (olive), $49$ (cyan). 
    }
    \label{fig:ssm_mono-poly/no-depo/general/pore-and-pres}
\end{figure}

In Figure \ref{fig:ssm_mono-poly/no-depo/general/pore_ttg_3__v__size_pore_1.pdf} we show the distribution of pore sizes halfway down the filter (i.e., at $z=0.5$), 
for various times. The times are chosen so that we capture the changes 
that occur to the distribution when the particle wave passes through this region of the 
filter. 
We see that the pore distribution is split into two critical regions: pores that are larger than the particles, and those that are smaller. 
The distribution of pores that are larger than the particles never changes, 
because particles pass through these pores without blocking them.
% (since $\varsigma\depo=0$).
However, as time increases, the number of pores of any particular size smaller than the particles 
decreases to zero.
We see that the decrease happens faster for larger pores because there is more flow through these and so more particles are transported
through them per unit time.
However, there are more of these pores to block, so the number of pores of all sizes (smaller than particles) 
are eventually reduced to zero at around the same time, $t\approx 49$ here.
% Before these times, the pore-sizes retain their initial distribution, since particles 
have not reached them. 
After this time, there are no further changes to the distribution, since particles now flow unobstructed through the larger pores. 
% Creation of new small pores is impossible, since particles cannot stick to large pores.

\subsubsection{The case $\Lambda_{\text{B}} \gg 1$ (with $\Gamma=\mathcal{O}(1)$): An earlier time characteristic solution and a later time asymptotic solution}
% We now derive an asymptotic solution in the case where the mono-dispersed particles small compared to the depth of the filter, by considering the system 
% \eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
% subject to the boundary and initial conditions 
% \eqref{eq:ssm_bound-cond_mono-poly} and
% \eqref{eq:ssm_init-conds_mono-poly} with $\Lambda_{\text{B}} \gg 1$.
We use a similar asymptotic 
method to the case where pore sizes were mono-dispersed, 
and so seek an explicit solution at leading order in $\Lambda\bloc^{-1}\ll1$.
Using the numerical solution as a guide, we expect the particle concentration to behave like a travelling wave. 

\paragraph{Earlier times: A characteristic solution}
We begin by considering the solution at early times. 
% We define shorter variables by
We scale the time and depth variables, defining $\hat{t} = \Lambda\bloc t$ and $\hat{z} = \Lambda\bloc z$.
In these new coordinates, \eqref{eq:ssm_c-eqn_mono-poly_general} and \eqref{eq:ssm_n-eqn_mono-poly_general}
become
\begin{align}
    \pp{C}{\hat{t}} + \frac{\partial }{\partial \hat{z}}\left( \frac{C}{\phi}\right) 
     &= -\frac{C}{\phi^2} \varphi
     % _{[r>A]},
     \label{eq:ssm_c-eqn_mono-poly_early}
     \\
     \pp{n}{\hat{t}} &= -\frac{1}{\Lambda\bloc}\frac{\Gamma C}{\phi^2} r^2 n H(A-r),
     \label{eq:ssm_n-eqn_mono-poly_early}
\end{align}
and the boundary and initial conditions, \eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly},
become
\begin{align}
    C(0,\hat{t}) = \phi(0,\hat{t}),
\end{align}
and 
\begin{align}
    C(\hat{z},0) = 0, \quad n(\hat{z},0,r) = n_{\text{init}}(r),
\end{align}
where
\begin{align}
     n_{\text{init}}(r) = \frac{\phi^*_{\text{init}} e^{-\sigma_n^2} }{\sqrt{2 \pi} \sigma_n r} 
    e^{- \frac{ 
    \left( 
    \log \left( r \right) + \frac{\sigma_n^2}{2} 
    \right)^2
    }{
    2 \sigma_n^2}}.
\end{align}
At leading order in $\Lambda\bloc^{-1}\ll1$, \eqref{eq:ssm_c-eqn_mono-poly_early} and \eqref{eq:ssm_n-eqn_mono-poly_early}
become
\begin{align}
    \pp{C\lo}{\hat{t}} + \frac{\partial }{\partial \hat{z}}\left( \frac{C\lo}{\phi\lo}\right) 
     &= -\frac{C\lo}{(\phi\lo)^2} \varphi\lo
     % _{[r>A]},
     \label{eq:ssm_c-eqn_mono-poly_early_lo}
     \\
     \pp{n\lo}{\hat{t}} &= 0.
     % -\frac{1}{\Lambda\bloc}\frac{\Gamma C\lo}{(\phi\lo)^2} r^2 n\lo H(A-r),
     \label{eq:ssm_n-eqn_mono-poly_early_lo}
\end{align}
Using \eqref{eq:ssm_n-eqn_mono-poly_early_lo},  we immediately see that
the leading-order pore density is given by $n\lo = n\init$ for all time.
 Since $n\lo$ is constant, we solve the leading-order particle concentration equation, \eqref{eq:ssm_c-eqn_mono-poly_early_lo}, by the method of characteristics similarly to before, and find, 
 in the original coordinates, that
 \begin{align}
     C\lo(z,t) = 
     \begin{cases}
        0 & z < t/\phi\init^{*}, 
        \\
        \phi\init^{*} e^{-\frac{\Lambda\bloc z}{2}  \left( 1+\text{erf}(f(A)) \right)   }
        & 
        z > t/\phi\init^{*},
     \end{cases}
 \end{align}
 where we recall, from \eqref{eq:ssm_def_of_f}, that
\begin{align}
    f(A) = \frac{3\sigma_n^2 - 2\log(A)}{2 \sqrt{2} \sigma_n}.
 \end{align}
 Similarly to the mono-dispersed case, at early times, particles 
 travel as a steep wave over a thin layer near the inlet, 
 and display behaviour similar to the early-time numerical solution 
 that we showed in Figure \ref{fig:ssm_mono-poly/no-depo/general/early-time/conc_part-poro_ttg_2__v__posi_1.pdf}.
 Behind the wave, the particle concentration exponentially decreases with depth,
 and in front of the wave, there are no particles, since 
 they have not reached these depths at times this early. 
 This time, the gradient of the concentration profile depends on the particle size, $A$, 
 as well as the blocking parameter $\Lambda\bloc$, since $A$ determines the 
 proportion of pores that block particles, and thus the rate at which particles are captured.
 % \comment{[Should probably plot...see agreement with early numerical soln...blah]}
 
\paragraph{Later times: An asymptotic solution}
% Now that we have analysed the early-time behaviour,
At later times, we seek a solution that matches the behaviour of the later-time numerical solution shown in Figure \ref{fig:ssm_mono-poly/no-depo/general/conc_part-poro_ttg_2__v__posi_1.pdf}. 
We seek a travelling wave solution 
and so re-write the problem in a frame that is moving with the wave. 
% switch to the longer coordinates and to the frame where this wave is fixed in time and seek steady (in this frame) solutions.
In front of the wave, 
we assume that $C=0$ 
and that $\phi=\phi\init^*$.
Behind the wave, 
we assume that $C=\varphi_{[r>A]}=\phi$. 

% the porosity will be reduced to the contribution purely from pores with $r>A$, that is, $\varphi_{[r>A]}$, and the particle concentration will also be reduced to $\varphi_{[r>A]}$, to preserve unit void concentration at the inlet. 
% In regions in front of the wave, no particles that can be blocked have arrived. 
% Therefore the porosity is the initial porosity, $\phi\init^*$, and the particle concentration is zero.

Using this intuition, we change to coordinates that are fixed in the frame of the wave, and 
so define $\zeta = \Lambda\bloc z - Vt$, $\tau=t$. 
Equations \eqref{eq:ssm_c-eqn_mono-poly_general} and \eqref{eq:ssm_n-eqn_mono-poly_general}
become 
\begin{align}
    \pp{C}{\tau} - V\pp{C}{\zeta} + \Lambda\bloc\frac{\partial }{\partial \zeta}\left( \frac{C}{\phi}\right) 
     &= -\frac{\Lambda\bloc C}{\phi^2} \varphi,
     \label{eq:ssm_c-eqn_mono-poly_no-depo_large-LamB}
     \\
     \pp{n}{\tau} - V\pp{n}{\zeta} &= \frac{\Gamma C}{\phi^2} r^2 n H(A-r).
     \label{eq:ssm_n-eqn_mono-poly_no-depo_large-LamB}
\end{align}
We seek steady (in this frame) asymptotic solutions in the small parameter, $\Lambda\bloc^{-1}\ll1$, of the form
\begin{align}
    X(\zeta,\tau) &= X^{(0)}(\zeta,\tau) + \frac{1}{\Lambda\bloc}X^{(1)}(\zeta,\tau) + \mathcal{O}\left(\frac{1}{\Lambda\bloc^2}\right). 
\end{align}
At leading order, we see that \eqref{eq:ssm_c-eqn_mono-poly_no-depo_large-LamB} and 
\eqref{eq:ssm_n-eqn_mono-poly_no-depo_large-LamB} imply that 
\begin{align}
    \frac{\partial C\lo}{\partial \zeta}\left( \frac{C\lo}{\phi\lo}\right) 
     &= -\frac{ C\lo}{(\phi\lo)^2} \varphi\lo,
     \label{eq:ssm_c-eqn_mono-poly_no-depo_large-LamB_lo}
     \\
     V\pp{n\lo}{\zeta} &= \frac{\Gamma C\lo}{(\phi\lo)^2} r^2 n\lo H(A-r),
     \label{eq:ssm_n-eqn_mono-poly_no-depo_large-LamB_lo}
\end{align}
which satisfy the conditions 
\begin{align}
    &C\lo \rightarrow \varphi_{[r>A]}, \quad n\lo \rightarrow 0 \quad (\text{for} \quad (r>A) \quad \text{as} \quad \zeta \rightarrow -\infty, 
    \label{eq:ssm_zeta-inf-neg}
    \\
    &C\lo\rightarrow 0, \quad
    \quad \quad \, n\lo \rightarrow n\init \quad \quad \, \, \, \, \quad \quad \quad \quad \text{as} \quad \zeta \rightarrow \infty,
    \label{eq:ssm_zeta-inf-pos}
\end{align}
Note that \eqref{eq:ssm_zeta-inf-neg} implies that
$\varphi\lo \rightarrow 0$ as $\zeta \rightarrow - \infty$ so that $\phi\lo \rightarrow \varphi_{[r>A]}$. 

To solve the leading-order system, \eqref{eq:ssm_c-eqn_mono-poly_no-depo_large-LamB_lo} and \eqref{eq:ssm_n-eqn_mono-poly_no-depo_large-LamB_lo} with \eqref{eq:ssm_zeta-inf-neg} and \eqref{eq:ssm_zeta-inf-pos}, we 
begin by integrating
\eqref{eq:ssm_n-eqn_mono-poly_no-depo_large-LamB_lo} 
with respect to $r$ from $0$ to $A$. 
We find, dropping leading-order notation, that 
\begin{align}
    V \frac{\partial N_{[r<A]}}{\partial \zeta} =\frac{\Gamma C \varphi}{\phi^2} ,
    \label{eq:ssm_N-eqn_mono-poly_no-depo_large-LamB_lo_1}
\end{align}
where we define 
\begin{align}
    N_{[r<A]}(z,t) = \int_0^A n(z,t,r) \, \text{d}r
\end{align}
to be the small-pore concentration, which is the concentration of pores that are smaller than the particles.
% recall that $N$ is the pore concentration, which is the integral of the pore-size distribution over all pore sizes.
Combining \eqref{eq:ssm_N-eqn_mono-poly_no-depo_large-LamB_lo_1} with \eqref{eq:ssm_c-eqn_mono-poly_no-depo_large-LamB_lo} we find that 
\begin{align}
    V \frac{\partial N_{[r<A]}}{\partial \zeta} = -\Gamma
    \frac{\partial}{\partial \zeta} \left( \frac{C}{\phi}\right).
    \label{eq:ssm_N-eqn_mono-poly_no-depo_large-LamB_lo_2}
\end{align}
Integrating \eqref{eq:ssm_N-eqn_mono-poly_no-depo_large-LamB_lo_2} with respect to some temporary variable $\tilde{\zeta}$ from $\zeta$ to $\infty$, and using the boundary
condition as $\zeta\rightarrow\infty$, \eqref{eq:ssm_zeta-inf-pos}, 
we find that
\begin{align}
    V(N_{[r<A]\text{init}}-N_{[r<A]}) = \frac{\Gamma C}{\phi},
    \label{eq:ssm_N-eqn_mono-poly_no-depo_large-LamB_lo_3}
\end{align}
where
\begin{align}
    N_{[r<A]\text{init}} = \int_0^A n\init(r) \, \text{d}r
    \label{four}
\end{align}
is the known initial concentration of pores that are smaller than the particles.
Applying the boundary condition as $\zeta \rightarrow - \infty$,
\eqref{eq:ssm_zeta-inf-neg}, to \eqref{four}, 
and noting that $\phi\rightarrow\varphi_{[r>A]}$ as $\zeta \rightarrow-\infty$,
we find that we must choose
\begin{align}
    V = \frac{\Gamma}{N_{[r<A]\text{init}}}.
    \label{V}
\end{align}
Thus, the speed of the wave is given by 
$\Gamma/\Lambda\bloc N_{[r<A]\text{init}}$.
 Substituting \eqref{V}
 into \eqref{eq:ssm_N-eqn_mono-poly_no-depo_large-LamB_lo_3} and
 returning to leading-order notation,
 we find that the leading-order particle concentration is given by
\begin{align}
    C\lo = \phi\left(1-\frac{N_{[r<A]}}{N_{[r<A]\text{init}}}\right).
    \label{eq:ssm_mono-poly_C-sol_large-LamB}
\end{align}
Furthermore, substituting this solution into the equation for the leading-order pore-size distribution, \eqref{eq:ssm_n-eqn_mono-poly_no-depo_large-LamB_lo}, we conclude that
\begin{align}
    \frac{\partial n\lo}{\partial \zeta} = 
    ( N_{ [r<A] \text{init}}-N_{[r<A]}) \frac{r^2n}{\phi} H(A-r),
    \label{eq:ssm_mono-poly_n-sol_large-LamB}
\end{align}
with  
\begin{align}
    n\lo\rightarrow n\init \quad \text{as} \quad \zeta \rightarrow \infty.
    \label{eq:ssm_zeta-inf-pos_2}
\end{align}
Although \eqref{eq:ssm_mono-poly_n-sol_large-LamB} with \eqref{eq:ssm_zeta-inf-pos_2}
is not straightforward enough to solve explicitly,  
we note that it is a reaction equation that is no longer coupled to the hyperbolic advection--reaction equation,
and is therefore considerably more simple to solve numerically than the original problem.
Thus, \eqref{eq:ssm_mono-poly_n-sol_large-LamB} subject to \eqref{eq:ssm_zeta-inf-pos_2}, 
together with \eqref{eq:ssm_mono-poly_C-sol_large-LamB}, provide a 
leading-order solution in the case $\Lambda\bloc^{-1}\ll1$. 
We plot this solution in Figure \ref{fig:ssm_mono-poly/no-depo/general/conc_part_and_poro}
and observe good agreement with the numerical solution.
% that is more simple 
% than the full problem. 

 % \comment{[Should probably solve if time...despite this we see good agreement for large enough $\Lambda\bloc$...blah blah] We plot these solution in the above figures and see good agreement with the numerical solution}

\subsection{Blocking and deposition}
\label{subsec:ssm_mono-poly_blocking-and-deposition}
In this subsection, we assume that $\varsigma_{\text{D}}\ne0$ and solve the size-structured model 
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
subject to the boundary and initial conditions for mono dispersed particles and log-normally 
poly-dispersed pores,
\eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}, in the case where blocking and deposition both occur. 
We find an explicit solution at leading order in $\Gamma \ll 1$, and 
a numerical solution for $\Gamma=\mathcal{O}(1)$.

\subsubsection{The case $\Gamma \ll 1$ (with $\Lambda\bloc\gg1$ and $\Lambda\depo\gg1$): A characteristic solution}
We first suppose that the concentration of particles to be removed from the mono-dispersed feed fluid is very small compared to the concentration of log-normally dispersed pores in the filter, and so consider the system 
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
subject to the boundary and initial conditions 
\eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly} in the case $\Gamma \ll 1$. 
% We focus on the constant flow rate case, \eqref{eq:ssm_Up-eqns_const-flow_summary_nodim}.

We seek a solution of the form of an asymptotic expansion in the small parameter, $\Gamma$, given by 
\begin{align}
    X = X^{(0)} + \Gamma X^{(1)} + \mathcal{O}(\Gamma^2), 
\end{align}
where $X$ is any of the solution variables, $C, N, U, p, \phi, K$. 
% Proceeding similarly to in Subsection \ref{subsec:ssm_mono-mono_blocking}, 
At leading order, we find that
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} become
 \begin{align}
     \pp{C^{(0)}}{t} + U^{(0)}\frac{\partial }{\partial z}\bigg(\frac{C^{(0)}}{\phi^{(0)}}\bigg)
     & = 
     -U^{(0)}\frac{F(A)}{\phi^{(0)}} C^{(0)} 
     \label{eq:ssm_c-eqn_mono-poly_small-gamma}
     \\
     \pp{n^{(0)}}{t} &= 0,
     \label{eq:ssm_n-eqn_mono-poly_small-gamma}
 \end{align}
 where we define 
 \begin{align}
     F(A) & = \frac{\Lambda_{\text{B}} \int_0^A r^2 d_n(r) \, \text{d}r + \Lambda_{\text{D}} \varsigma_{\text{D}} \int_A^\infty r^2 d_n(r) \, \text{d}r }{\int_0^\infty r^2 d_n(r) \, \text{d}r},
 \end{align}
and either
 \begin{align}
     U^{(0)}(t) = 1, 
     \quad 
     p^{(0)}(z,t) = \int_z^1 \frac{1}{K^{(0)}(\zeta,t)} \, \text{d}\zeta,
     \label{eq:ssm_Up_const-velo_mono-poly_small-gamma}
 \end{align}
 or 
 \begin{align}
     U^{(0)}(t) = \frac{1}{\int_0^1 \frac{1}{K^{(0)}(z,t)} \, \text{d}z}, 
     \quad 
     p^{(0)}(z,t) = \frac{\int_z^1 \frac{1}{K^{(0)}(\zeta,t)} \, \text{d}\zeta}{  \int_0^1 \frac{1}{K^{(0)}(\zeta,t)} \, \text{d}\zeta},
     \label{eq:ssm_Up_const-pres_mono-poly_small-gamma}
 \end{align}
for
\begin{align}
    \phi^{(0)}(z,t) = \int_0^{ \infty} r^2 n^{(0)}(z,t,r) \, \text{d}r, 
    \quad
    K^{(0)}(z,t) = \int_0^{ \infty} r^4 n^{(0)}(z,t,r) \, \text{d}r. 
     \label{eq:ssm_poro-perm_mono-poly_small-gamma}
\end{align}
The boundary and initial conditions, \eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}, become
\begin{align}
    C^{(0)}(0,t) = \phi^{(0)}(0,t),
    \label{eq:ssm_bound-cond_mono-poly_small-gamma}
\end{align}
and
\begin{subequations}
\label{eq:ssm_init-conds_mono-poly_small-gamma}
\begin{align}
C^{(0)}(z,0) = 0, 
\quad
n^{(0)}(z,0,r) =n_{\text{init}}(r),
\tag{\ref{eq:ssm_init-conds_mono-poly_small-gamma}a,b}
\end{align}
\end{subequations}
for $n_{\text{init}}(r)$ defined in \eqref{eq:ssm_definition-of-n-init}.

We see from \eqref{eq:ssm_n-eqn_mono-poly_small-gamma} that $n^{(0)}$ keeps its initial value, $n_{\text{init}}$, and 
it follows that $\phi^{(0)}$, and $U^{(0)}$, $p^{(0)}$, and $K^{(0)}$ keep their initial values, $\phi^*_{\text{init}}$, and $U_{\text{init}}$, $p_{\text{init}}$ and $K_{\text{init}}$ say, too. 
The latter three solutions are given by substituting the constant solution $n^{(0)}=n_{\text{init}}$ into  either \eqref{eq:ssm_Up_const-velo_mono-poly_small-gamma} or 
\eqref{eq:ssm_Up_const-pres_mono-poly_small-gamma}, 
and \eqref{eq:ssm_poro-perm_mono-poly_small-gamma}.
Thus, it remains to solve 
 \begin{align}
     \pp{C^{(0)}}{t} + 
     \frac{U_{\text{init}}}{\phi^*_{\text{init}}}
     \pp{
     C^{(0)}}{z}
     = 
     -  \frac{U_{\text{init}}}{\phi^*_{\text{init}}} F(A)C^{(0)} ,
     \label{eq:ssm_c-eqn_mono-poly_small-gamma-2}
 \end{align}
 subject to 
 \eqref{eq:ssm_bound-cond_mono-poly_small-gamma} and (\ref{eq:ssm_init-conds_mono-poly_small-gamma}a).

We proceed via the method of characteristics similarly to in Section \ref{subsec:ssm_mono-mono_blocking-and-deposition}, and 
find the explicit solution 
\begin{align}
C^{(0)}(z,t) & 
=
\begin{cases}
(\phi^*_{\text{init}}/U_{\text{init}}) e^{- F(A) z} & 
z \le t (U_{\text{init}}/\phi^*_{\text{init}}),
\\
 0 & 
z > t (U_{\text{init}}/\phi^*_{\text{init}}) ,
\label{eq:ssm_c-sol_mono-poly_small-gamma}
\end{cases} 
\end{align}
for $ 0 \le z \le 1$.
We calculate that
\begin{align}
    F(A) = \frac{1}{2}
    \Bigg( 
    \Lambda_{\text{B}} \bigg( 1-\text{erf} \Big( f(A) \Big) \bigg) + 
    \Lambda_{\text{D}} \varsigma\depo \bigg( 1+\text{erf} \Big( f(A) \Big) \bigg)  
    \Bigg)
\end{align}
where $\text{erf}$ is the error function, and
\begin{align}
    f(A) = \frac{3\sigma_n^2 - 2\log(A)}{2 \sqrt{2} \sigma_n},
\end{align}
from \eqref{eq:ssm_def_of_f}. 
% \paragraph{Functions $\mathrm{erf}(f(A))$ and $F(A)$}
We plot 
% the 
% functions $\text{erf}(f(A))$ and 
$F(A)$ in Figure \ref{fig:ssm_mono-poly/small-Gamma_1}, 
for various $\sigma_n$.
We see that if $A\gg1$ then, 
since $\text{erf}(f(A)) \approx -1$, 
% so that 
$F(A) \approx \Lambda_{\text{B}}$.
Meanwhile, 
if $A\ll1$ then, 
since $\text{erf}(f(A)) \approx 1$, 
$F(A) \approx \Lambda_{\text{D}}$.
As $\sigma_n$ decreases, the transition region of $A$ between these two limiting cases becomes
thinner.
\begin{figure}[!htb]
         \centering
         \includegraphics[width=0.6\textwidth]{./figures/size-structured/mono-poly/yes-depo/small-Gamma/F_1__v__mean_size_part_1.pdf}
    \caption{Graph showing how the function $F$ varies  
    with the particle size, $A$, at various $\sigma_n$.
    The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $U_{\text{init}}=1$, and $\phi^*_{\text{init}}=1$.
    The values of $\sigma_n$ are $0.01$ (blue), $0.05$ (orange),
    $0.1$ (green), $0.2$ (red). 
     The limiting cases $A\gg1$, $F(A)=\Lambda_{\text{B}}$ (dash-dotted),
    and $A\ll1$, $F(A)=\Lambda_{\text{D}}$ (dashed), are plotted in black.
    }
    \label{fig:ssm_mono-poly/small-Gamma_1}
\end{figure}
% \begin{figure}[!htb]
%      \centering
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/small-Gamma/g_1__v__mean_size_part_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-poly/small-Gamma/g_1__v__mean_size_part_1.pdf}
%      \end{subfigure}
%      %\hfill
%      \begin{subfigure}[b]{0.45\textwidth}
%          \centering
%          \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/small-Gamma/F_1__v__mean_size_part_1.pdf}
%          \caption{}
%          \label{fig:ssm_mono-poly/small-Gamma/F_1__v__mean_size_part_1.pdf}
%      \end{subfigure}
%     \caption{Graphs showing how the functions $\text{erf}(f(A))$ and $F(A)$ vary  
%     with the particle size $A$ at various $\sigma_n$.
%     The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
%     $\Gamma=1$, $U_{\text{init}}=1$, and $\phi^*_{\text{init}}=1$.
%     The values of $\sigma_n$ are $0.01$ (blue), $0.05$ (orange),
%     $0.1$ (green), $0.2$ (red). 
%     \subref{fig:ssm_mono-poly/small-Gamma/g_1__v__mean_size_part_1.pdf} 
%     The function $\text{erf}(f(A))$.
%     \subref{fig:ssm_mono-poly/small-Gamma/F_1__v__mean_size_part_1.pdf} 
%     The function $F(A)$.
%      The limiting cases when $A\gg1$, $F(A)=\Lambda_{\text{B}}$ (dash-dotted),
%     and when $A\ll1$, $F(A)=\Lambda_{\text{D}}$ (dashed), are plotted in black.
%     }
%     \label{fig:ssm_mono-poly/small-Gamma_1}
% \end{figure}

\paragraph{Particle concentration and capture-layer length}
We plot the explicit leading-order asymptotic solution, $C^{(0)}(z,t)$, 
given by \eqref{eq:ssm_c-sol_mono-poly_small-gamma}, in Figure \ref{fig:ssm_mono-poly/yes-depo/small-Gamma/part_1__v__posi_1.pdf}.
We see that the particle concentration decreases exponentially over a layer near the inlet of the filter.
We see that the gradient of the concentration profile increases as the particle size increases, so that the length of the  
layer over which particles are captured decreases. 
This is because $\Lambda_{\text{D}}<\Lambda_{\text{B}}$
in this example.
Indeed, if particles are very large, 
and thus larger than all pores, 
then the solution \eqref{eq:ssm_c-sol_mono-poly_small-gamma} matches the solution for 
mono-dispersed particle- and pore-sizes, 
since $F(A)=\Lambda_{\text{B}}$ for $A\gg1$.
In this case, particle capture occurs only via blocking very close to the filter inlet.
As the particle size decreases, 
% some pores are still smaller than the particles, 
% but some are now larger.
particle capture occurs via both blocking and deposition in a layer that is fatter 
than the blocking layer (since $\Lambda_{\text{D}}<\Lambda_{\text{B}}$).
As the particle size decreases more, eventually all pores are larger than the particles. 
Now particle capture occurs only via deposition, which occurs over a still-fatter layer, the thickness of which is dictated by the value of $\Lambda_{\text{D}}$.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/small-Gamma/part_1__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/small-Gamma/part_1__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/small-Gamma/lay_effe_1__v__mean_size_part_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/small-Gamma/lay_effe_1__v__mean_size_part_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the leading-order particle concentration, $C^{(0)}$, 
    varies with depth, $z$, 
    and how the capture-layer length, $l_{\text{eff}}$, varies with the particle size, $A$.
    The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $U_{\text{init}}=1$, $\phi^*_{\text{init}}=1$, and $\varsigma\depo=1$.
    \subref{fig:ssm_mono-poly/yes-depo/small-Gamma/part_1__v__posi_1.pdf} 
    The concentration, $C^{(0)}$, as a function of depth, $z$, at time $t=0.1$ for various $A$.
    The values of $A$ are $0$ (blue), $1$ (orange),
    $2$ (green). 
     The limiting cases when $A\gg1$, $F(A)=\Lambda_{\text{B}}$ (dash-dotted)
    and when $A\ll1$, $F(A)=\Lambda_{\text{D}}$ (dashed), are plotted in black.
    The efficient concentration $C_{\text{eff}}=0.1$ is shown a horizontal dotted purple line, 
    with the capture-layer lengths in the cases $F(A)=\Lambda_{\text{B}}$ (dash-dotted) and $F(A)=\Lambda_{\text{D}}$ (dashed) extrapolated in vertical purple lines.
    \subref{fig:ssm_mono-poly/yes-depo/small-Gamma/lay_effe_1__v__mean_size_part_1.pdf} 
    The capture-layer length, $l_{\text{eff}}$, as a function of the particle size, $A$, for various 
    efficient concentrations, $C_{\text{eff}}$.
    The values of $C_{\text{eff}}$ are  $0.010$ (blue), $0.025$ (orange),
    $0.050$ (green), $0.075$ (red), $0.100$ (purple).
    The cases $F(A)=\Lambda_{\text{B}}$ (dash-dotted) and $F(A)=\Lambda_{\text{D}}$ (dashed) are extrapolated in horizontal purple lines.
    }
    \label{fig:ssm_mono-poly/small-Gamma_2}
\end{figure}

% Similarly to before, 
We calculate the length of the capture layer, $l_{\text{eff}}$, by imposing a required concentration at the inlet, $C_{\text{eff}}$ say, to which the inlet concentration must be reduced by the time particles exit the filter for the filtration regime to be considered effective.
Using \eqref{eq:ssm_c-sol_mono-poly_small-gamma}, 
we see that this length is given by
\begin{align}
    l_{\text{eff}} = \frac{1}{F(A)}\log\Big(\frac{\phi^*_{\text{init}}/U_{\text{init}}}{C_{\text{eff}}}\Big).
\end{align}
We plot the capture-layer length, $l_{\text{eff}}$, as a function of the particle size, $A$, for various 
efficient concentrations, $C_{\text{eff}}$, in Figure \ref{fig:ssm_mono-poly/yes-depo/small-Gamma/lay_effe_1__v__mean_size_part_1.pdf}, 
and demonstrate how we read off a value for the capture-layer length given a required efficient concentration, $0.1$ for example.
% , in Figure \ref{fig:ssm_mono-poly/yes-depo/small-Gamma/lay_effe_1__v__mean_size_part_1.pdf}.
We see that, for any choice of efficient concentration, the capture-layer length 
decreases from some lower limiting-value to an upper limiting-value over a transition region of 
$A$. 
For small enough particles, capture occurs via deposition only, over a layer of length $l_{\text{eff}}=\Lambda_{\text{D}}^{-1} \log((\phi^*_{\text{init}}/U_{\text{init}})/C_{\text{eff}})$. 
For large enough particles, capture occurs via blocking only, over a layer of length $\Lambda_{\text{B}}^{-1} = \log((\phi^*_{\text{init}}/U_{\text{init}})/C_{\text{eff}})$, which is smaller than the deposition layer since $\Lambda_{\text{B}}>\Lambda_{\text{D}}$.
For other particle sizes, 
% in the transition region, 
capture occurs via blocking and deposition 
and the capture-layer length 
lies between these two extremes, 
so that 
$\Lambda_{\text{B}}^{-1}  \log((\phi^*_{\text{init}}/U_{\text{init}})/C_{\text{eff}}) 
\le 
l_{\text{eff}} \le \Lambda_{\text{D}}^{-1} \log((\phi^*_{\text{init}}/U_{\text{init}})/C_{\text{eff}})$. 
The exact value of $l_{\text{eff}}$ depends on the relative densities of pores that are smaller and larger than the particles.


\subsubsection{The case
$\Gamma=\mathcal{O}(1)$, $\Lambda_{\text{B}}\gg1$, 
and $\Lambda_{\text{D}}\gg1$: A numerical solution}
We now suppose that the concentration of particles to be removed from the mono-dispersed feed fluid is comparable
% $\mathcal{O}(1)$ compared 
to the concentration of log-normally dispersed pores in the filter, and so $\Gamma=\mathcal{O}(1)$. 
Thus, we solve the system
\eqref{eq:ssm_c-eqn_summary_nodim}--\eqref{eq:ssm_definition-of-Gamma} 
subject to the boundary and initial conditions 
\eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly} in the case $\Lambda_{\text{B}}\gg1$, 
$\Lambda_{\text{D}}\gg1$, and $\Gamma = \mathcal{O}(1)$. 

The equation for the particle concentration, 
\eqref{eq:ssm_c-eqn_mono-poly_general}, 
is a partial integro--differential advection--reaction equation, 
as it was for the case $\varsigma\depo=0$.
However, since $\varsigma\depo\ne0$, 
% here, this is now coupled to 
\eqref{eq:ssm_n-eqn_mono-poly_general} becomes an integro--differential reaction equation.
The integral appears because now particles can deposit in larger pores 
and transform them to smaller pores.
These equations, with \eqref{eq:ssm_perm-and-poros_mono-poly_general} and either \eqref{eq:ssm_Up-eqns_const-flow_summary_nodim} or \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim} subject to the boundary and initial conditions, \eqref{eq:ssm_bound-cond_mono-poly} and
\eqref{eq:ssm_init-conds_mono-poly}, are of a similar form to that in the case where there was no deposition, despite the extra integral.
We discretise the extra integral using the trapezium rule, and solve via a similar numerical scheme as before.
Again, we present the solution for the constant flow rate regime, and so choose \eqref{eq:ssm_Up-eqns_const-flow_summary_nodim}, rather than \eqref{eq:ssm_Up-eqns_const-pres_summary_nodim}, to specify the velocity and the pressure.

\paragraph{Particle concentration and local porosity}
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/conc_part_and_poro}, 
we demonstrate how the particle concentration, $C$, and the local porosity, $\phi$, vary with depth, $z$, at various times, $t$. 
The first thing we notice is that there are two regions of time, $10\le t\le90$ and $90< t\le300$, within which we observe distinct dynamics.
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/part_ttg_3__v__posi_1.pdf} we see that, in the earlier time region, $10\le t\le90$, particles traverse the filter as a wave. 
As they do so, the amplitude of the wave decreases. 
In the second time region, $90< t\le300$, 
the particle concentration (at any depth) decreases in time with a spatial gradient that is almost uniform. 
Correspondingly, in Figure \ref{fig:ssm_mono-poly/yes-depo/general/poro_ttg_2__v__posi_1.pdf}
we see that, in the first time region, the porosity decreases in a wave across the filter, and decreases linearly with time in the second time-region, maintaining a constant depth-wise gradient.
% , similarly to the particle concentration. 

% To explain this behaviour, 
Firstly, consider the waves in the first time-region.
Similarly to the case of no deposition, 
in front of the wave we see that there are no particles and the porosity retains its initial value.
% neither smaller pores (by which we mean those smaller than the particles) nor larger pores have been reached.
% $C=0$ and $\phi=\phi\init^{*}$.
% Thus there are no particles, and the porosity remains at its initial value. 
Behind the wave, smaller pores have all been blocked by the particles, as before. 
This time, however, since $\varsigma\depo\ne0$, some larger 
pores have decreased in size due to deposition. As a result, 
the porosity drops slightly below the
small-pore porosity, $\varphi_{[r>A]}$, 
which we observed in the no-deposition case.
Correspondingly, the concentration of particles drops below the value observed in the no-deposition scenario, as some particles are also captured in larger pores due to deposition.
The length of the capture-layer, within which the concentration decreases as the waves progresses, 
is still $\mathcal{O}(\Lambda\bloc^{-1})$.

Secondly, consider the linear decrease in the second time region. 
At a critical time similar to that in the blocking regime, $t\approx90$ here, the particle wave reaches the outlet, which means that all original smaller pores have been blocked. After this, pore clogging occurs mostly due to deposition, although blocking still occurs in the smaller pores that have been transformed from larger pores by deposition. 
Although $\Lambda\depo\gg 1$, so that particles are much smaller than the filter, $\varsigma\depo\ll1$, so that the probability that particles are captured via deposition is small.
It follows that the length over which deposition occurs is $\mathcal{O}((\Lambda\depo \varsigma\depo)^{-1})$ ($\mathcal{O}(10)$ here), 
which is much longer than the length of the layer over which blocking occurs, $\mathcal{O}(\Lambda\bloc^{-1})$ ($\mathcal{O}(10^{-2})$ here). 
% In this example, the effective length of deposition is $\mathcal{O}(10^{-1})$ compared to the blocking length which is $\mathcal{O}(100)$. 
Hence, we observe a slow decrease in the particle concentration over the entire length of the filter, 
as particles deposit in the larger pores that were not blocked in the first time region.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/part_ttg_3__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/part_ttg_3__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/poro_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/poro_ttg_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the particle concentration (solid lines) and local porosity (dashed lines), $C$ and $\phi$, vary  
    with the depth, $z$, at various times, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$, $\varsigma\depo=0.001$.
    The times, $t$, at which the wave travels through the filter are shown in blue.
    The values of $t$ shown in blue are $10, 20, \hdots, 70, 80$.
    The blue arrow shows the direction of increasing $t$ for these values.
    The times after the wave has travelled through the filter are shown in orange.
    The values of $t$ shown in orange are $90, 100, 150, 200, 250, 300$.
    The orange arrow shows the direction of increasing $t$ for these values.
    \subref{fig:ssm_mono-poly/yes-depo/general/part_ttg_3__v__posi_1.pdf} 
    The particle concentration, $C$.
    \subref{fig:ssm_mono-poly/yes-depo/general/poro_ttg_2__v__posi_1.pdf} 
    The local porosity, $\phi$.
    }
    \label{fig:ssm_mono-poly/yes-depo/general/conc_part_and_poro}
\end{figure}

\paragraph{Local permeability and local pressure}
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/perm_ttg_2__v__posi_1.pdf} we plot 
the local permeability, $K$, and the local pressure, $p$, as functions of depth, $z$, at various times.
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/perm_ttg_2__v__posi_1.pdf} we see that 
the behaviour of the local permeability is similar to that of the local porosity. 
At earlier times, it decreases in a wave that traverses the filter. 
Unlike in the no-deposition case, the permeability decreases below $\kappa_{[r>A]}$, since particles 
continue to deposit in larger pores after all smaller pores have been blocked.
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/pres_ttg_2__v__posi_1.pdf}, 
we demonstrate how the local pressure, $p$, varies with depth, $z$, at various times.
As in the no-deposition case from before, initially, we see that the local pressure drop is uniform.
% and that the difference between the inlet and outlet pressure is exactly $\kappa\init^{-1}$. 
In the first time region, the decrease to the local permeability at each depth causes an increase to the local pressure at a rate that is determined by the rate at which the permeability is decreased by blocking.
At times near the end of the first time region, $t\approx 90$, we see that the pressure 
at the inlet begins to exceed the value we observed in the no-deposition case, $1/\kappa_{[r>A]}$. 
The local permeability has decreased below $\kappa_{[r>A]}$ at depths behind the wave due to the deposition of particles in larger pores. 
After the critical time, the pressure drop at the inlet required to maintain the constant flow rate 
continues to increase, at a rate that is determined by the rate of 
decrease of the permeability due to deposition. 
When deposition is possible, the filter pressure drop continues to increase until all pores are removed, 
regardless of the choice of the initial pore-size distribution.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/perm_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/perm_ttg_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/pres_ttg_2__v__posi_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/pres_ttg_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the local permeability, $K$, and local pressure, $p$, vary  
    with the depth, $z$, at various times, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$, $\varsigma\depo=0.001$.
    The times, $t$, at which the wave travels through the filter are shown in blue.
    The values of $t$ shown in blue are $10, 20, \hdots, 70, 80$.
    The blue arrow shows the direction of increasing $t$ for these values.
    The times after the wave has travelled through the filter are shown in orange.
    The values of $t$ shown in orange are $90, 100, 150, 200, 250, 300$.
    The orange arrow shows the direction of increasing $t$ for these values.
    \subref{fig:ssm_mono-poly/yes-depo/general/perm_ttg_2__v__posi_1.pdf} 
    The local permeability, $K$.
    \subref{fig:ssm_mono-poly/yes-depo/general/pres_ttg_2__v__posi_1.pdf} 
    The local pressure, $p$.
    }
    \label{fig:ssm_mono-poly/yes-depo/general/perm-and-pres_local}
\end{figure}

\paragraph{Global porosity and global permeability}
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/poro-and-perm_global} we demonstrate 
how the global porosity, $\bar{\phi}$, and global permeability, $\bar{K}$, vary with time, $t$.
These variables are also split into two time regions.
In the earlier one, they decrease linearly until the critical time, $t\approx90$. 
The gradient is similar to that in the no-deposition case, since the majority 
of the clogging is caused by blocking (since this occurs with unit probability), but is steeper, since deposition also occurs behind the waves.
As a result, the variables reach values below the previously observed critical values, 
$\varphi_{[r>A]}$ and $\kappa_{[r>A]}$, by the time that particles have reached the outlet for the first time. 
After this time, $t\approx90$, unlike in the blocking regime, 
the global porosity and permeability continue to decrease. 
This is still due to a combination of blocking and deposition, 
but the rate is slower, because
most smaller pores have already been blocked and the majority of clogging is deposition in larger pores.
The difference in the rates is determined by $\varsigma\depo=0.001$, 
so that 
% Since $\mathcal{O}(\Lambda\depo)=\mathcal{O}(\Lambda\bloc)$, 
deposition only occurs around $0.1\%$ of the time that blocking does.
Filtration processes are usually halted when the ratio of the global permeability as a proportion of the initial global permeability drops below some threshold. 
We use a value of around $0.35$ here. As we have discussed, 
this ratio dictates the inlet pressure required to maintain a constant flux, which must not exceed some maximum value due to external operational constraints such as energy usage or safety precautions. 
However, once the wave has traversed the filter, 
the filtrate is more likely to contain too many particles, because fewer are removed from the fluid due to blocking.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/poro_tota_1__v__time_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/poro_tota_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/perm_tota_1__v__time_1.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/perm_tota_1__v__time_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the global porosity, $\bar{\phi}$, and global permeability, $\bar{K}$, vary with time, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$, $\varsigma\depo=0.001$.
    The values of the large-pore porosity, $\varphi_{[r>A]}$, and large-pore permeability, $\kappa_{[r>A]}$, 
    are plotted in black dotted lines. 
    The times at which corners in the solutions occur are extrapolated in orange dashed lines.
    \subref{fig:ssm_mono-poly/yes-depo/general/poro_tota_1__v__time_1.pdf} 
    The global porosity, $\bar{\phi}$.
    \subref{fig:ssm_mono-poly/yes-depo/general/perm_tota_1__v__time_1.pdf} 
    The global permeability, $\bar{K}$.
    }
    \label{fig:ssm_mono-poly/yes-depo/general/poro-and-perm_global}
\end{figure}

\paragraph{Pore-size distribution}
We plot the inlet pore distribution, $n(0,t,r)$, as a function of the pore size, $r$, 
for various times, $t$, in Figure \ref{fig:ssm_mono-poly/yes-depo/general/pore}. 
Earlier times are displayed in Figure \ref{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__early.pdf}, 
while later times are shown in Figure \ref{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__late.pdf}. 
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly//yes-depo/general/pore_ttg_3__v__size_pore_1__early.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__early.pdf}
     \end{subfigure}
     %hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/size-structured/mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__late.pdf}
         \caption{}
         \label{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__late.pdf}
     \end{subfigure}
     %\hfill
    \caption{Graphs showing how the pore-size distribution at the inlet, 
    $n(0,t,r)$, varies with the pore size, $r$, for various times, $t$.
    The parameters are $\Lambda_{\text{B}}=200$, $\Lambda_{\text{D}}=100$, 
    $\Gamma=1$, $A=1$, $\sigma_n=0.2$, $\phi^*_{\text{init}}=0.9$, $\varsigma\depo=0.001$.
    The arrows show the directions of increasing time.
    The particle size, $A=1$, is highlighted with a solid black line.
    \subref{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__early.pdf} 
    The solution at earlier times.
    The values of $t$ are $0.00$ (blue), $0.25$ (orange),
    $0.50$ (green), $1.00$ (red), $2.00$ (purple), $50$ (brown).
    \subref{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__late.pdf} 
    The solution at later times.
    The values of $t$ are $50$ (blue), $100$ (orange),
    $150$ (green), $200$ (red), $250$ (purple).
    The initial pore-size distribution is plotted in a dashed black line.
    }
    \label{fig:ssm_mono-poly/yes-depo/general/pore}
\end{figure}


Firstly, we focus on the solution at earlier times.
In Figure \ref{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__early.pdf}
we see that, initially (blue line), 
the pore sizes are distributed log-normally, as prescribed. 
At non-zero times, we see that the solution behaviour is split into the two regions, 
$r\le A$ and $r>A$. 
In the first region, we see that the distribution of the smaller pores rapidly drops towards zero, and has almost reached zero 
by $t=2$ (purple line). 
Particles enter the filter at the inlet and immediately block pores that are smaller than the particles. 
This happens 
% with unit probability 
whenever particles flow to smaller pores, and happens over 
a short length, so smaller pores are lost rapidly. 
In the second region, we see that
% the difference between 
the distribution 
evolves much more slowly.
% of 
% larger pores and the initial distribution is negligible for all times other than $t=50$ (brown). 
Particles enter large pores at the inlet and deposit in them. But 
this only happens $0.1 \%$ of the 
time that blocking does, and happens over a longer length scale (since $\Lambda\depo<\Lambda\bloc$), so deposition does not make a visible 
difference to the distribution of larger pores until later times.

Secondly, turning our attention to the solution at later times, in 
Figure \ref{fig:ssm_mono-poly/yes-depo/general/pore_ttg_3__v__size_pore_1__late.pdf} we see that 
the solution is still split into the behaviour of pores that are larger and smaller 
than the particles. 
At later times, all the small pores have been removed at the inlet, due to the blocking that took 
placed at earlier times. 
We see that larger pores are still being removed due to deposition. 
This happens much more slowly than by blocking (noting that the difference in times here is $50$ as opposed to $\mathcal{O}(1)$ in the early time plot). 
We see that deposition in the large pores leads to the production of small pores. 
At times around $100$ (orange line), the rate at which small pores are produced due to 
deposition exceeds than the rate at which small pores are removed due to blocking. 
Even though deposition happens with lower probability, it happens more often, since  
particles are more likely to flow to large pores, and by late times, there are more large pores than small pores to flow to. 
However, by times around $150$ (green line), the number of large pores has decreased. 
The rate of removal of the smallest pores outweighs their production again, 
and the distribution of small pores returns to values near zero.

Thus, we have explored the blocking only, and combined blocking and deposition cases. 
When there is no adherence between particles and pores, so that deposition does not occur, 
particles are removed from the fluid in a wave that traverses the filter, and particle capture terminates when this wave first reaches the outlet, since there are no more small pores to block the particles. 
When deposition can occur,
particles are still removed 
in a wave over roughly the same time interval. 
However, filtration continues after the wave reaches the outlet for the first time, because deposition continues to remove larger pores and create smaller ones that are blocked. 
The filtration process is less efficient in this second time interval, because the majority of particle capture occurs due to deposition, which occurs less frequently than blocking. 

% \section{poly-dispersed particles and poly-dispersed pores}
% \subsection{The blocking and no deposition case}
% \subsection{The blocking and deposition case}
% \comment{
% \begin{itemize}
%     \item Change distributions from normal in transfer of status to log-normal (to deal with negative pore-size issue), and carry out similar analysis as in transfer report. Same solver can be used. 
% \end{itemize}
% }

\chapter{Conclusions for the size-structured model}
\label{chap:ssm_Conclusions for the size-structured model}
In summary, in Part \ref{part:A size-structured model for filtration}, 
in Chapters \ref{chap:ssm_Introduction to the size-structured model}--\ref{chap:ssm_Solutions of the size-structured model} we have motivated, derived, and solved a novel size-structured model 
 for filtration.
 Our model will be useful for increasing our understanding of the filtration process, and for filter optimisation, and there are several extensions 
 to be considered in future work.

\section{Summary}
\subsection{Motivation for the model}
 In Chapter \ref{chap:ssm_Introduction to the size-structured model}, we introduced size-structured modelling in the 
 context of filtration. 
 In particular, in Section \ref{sec:ssm_Particle and pore size in filtration}, we highlighted the importance of the sizes of particles and pores 
 in industrial filtration processes
 and explained that knowledge of the particle size is often key to choosing the distribution of pore sizes in the filter. Tracking both distributions is crucial for evaluation and optimisation of the process.
 
 In Section \ref{sec:ssm_Existing size-structured models for filtration}, we explored existing mathematical models of filtration 
 that account for size.
Broadly, we found that continuum models  
are computationally tractable but do not fully account for 
particle- and pore-size distributions, or the interplay 
between the two. 
On the other hand, discrete models that discuss size structure 
are often computationally expensive, since the states of 
each particle and pore are usually tracked individually.

In Section \ref{sec:ssm_A novel size-structured model for filtration}, we noted that there is a need for a computationally tractable model of filtration that more completely accounts for size structure. We discussed ideas used in age-structured models for population, and concluded that this is a candidate framework.
% from which to build a new model for filtration.

\subsection{Derivation of the model}
In Chapter \ref{chap:ssm_Derivation of the size-structured model}, 
we used ideas from age-structured population models to derive a size-structured model for filtration, 
within which the sizes of particles and pores are treated as independent variables. 
In Section \ref{sec:ssm_Dimensional model}, we modelled the feed fluid and filter as continua of particles and pores, 
and defined distributions of these.
Other quantities of interest, such as the particle and pore concentrations, and the local porosity and permeability of the filter, were derived as variables that depend on these distributions.
We 
included 
 two mechanisms by which particles are captured in pores: blocking (when particles are too large to traverse pores) and deposition (when particles are small enough to traverse pores but stick to pore walls as they do so), 
and built our model around conservation equations for fluid, particles, and pores.
The result of this section is a system of continuous partial integro--differential equations that consist of: 
\begin{itemize}
    \item An advection--reaction equation for the particle distribution that describes the movement of particles through pores due to a constant flow rate or a constant pressure drop, and the reduction of the distribution of particles in the fluid due to blocking and deposition in pores;
    \item A reaction equation for the pore distribution that describes the decrease of the pore distribution due to the capture of particles by either blocking or deposition, and the 
    increase of the distribution of smaller pores 
    due to deposition of particles in larger pores;
    \item A fluid-mass conservation equation and Darcy's equation, which together govern the flow rate and local pressure in the fluid.
\end{itemize}

In Section \ref{sec:ssm_Dimensionless model}, we  
nondimensionalised the model and found that there are four dimensionless parameter groups of interest: $\Lambda\bloc$ (the ratio of the filter length to the blocked-particle length); $\Lambda\depo$ (the ratio of the filter length to the deposited-particle length); 
$\varsigma\depo$ (the probability that deposition of particles in smaller pores occurs); 
and $\Gamma$ (the ratio of the concentrations of particle and pores).
Furthermore, parameters such as the mean particle size, $A$, and the pore size variance parameter, $\sigma_n$, dictate the particle- and pore-size distributions of the specific operational regime.

\subsection{Solution of the model}
In Chapter \ref{chap:ssm_Solutions of the size-structured model}, 
we found explicit, asymptotic, and numerical solutions for 
the dimensionless model in various regimes. 
In Section \ref{sec:ssm_Mono-dispersed particles and mono-dispersed pores}, we focused on the case where the feed and filter are both 
mono-dispersed. 
We considered the scenario where the pores are smaller 
than the particles, so that only blocking occurs. 
We discussed the case in which there is an excess of pores in the filter compared to the number of particles in the feed, $\Gamma\ll1$, 
and the case where the filter is very long compared to the size of a particle, $\Lambda\bloc\gg1$. 
In both cases, we found explicit leading-order asymptotic solutions 
that 
% at leading
% order in the respective small parameters, which agree well 
agree well
with numerical solutions in the appropriate limits. 
Broadly, we found that, when particles are larger than pores, 
particles are captured in a layer very close to the inlet.
The thickness of the capture layer is $\mathcal{O}(\Lambda\bloc^{-1})$.
% That is, as $\Lambda\bloc$ increases we found that the blocking 
% layer becomes thinner, since blocked particles occupy a smaller proportion of the filter a whole. 
One consequence is that, when all particles are larger than the pores, 
only a small proportion of the filter membrane is used to obtain a required particle removal and 
% for an `efficient' reduction in particle concentration, and 
the vast majority of the filter membrane is wasted.
Increasing $\Gamma$, we found, understandably, that as 
the particle concentration increases, the 
filter clogs more quickly.

% Secondly, we considered the scenario where the pores are larger than the particles. 
% Numerical solutions showed that, at earlier times, when there 
% are many large pores, deposition is the dominant capture mechanism and particles are captured over a layer that is of thickness $\mathcal{O}((\Lambda\depo\varsigma\depo)^{-1})$. 
% As deposition occurs, larger pores are transformed to pores that are smaller than the particles, so that blocking and deposition occur simultaneously. 
% The ratio of the number of large and small pores dictates the rate of particle capture, since this dictates 
% the number of particles that are captured frequently over a thin layer due to blocking compared to the number of particles that are captured less frequently (since $\varsigma\depo<1$) over a fatter layer (since $\Lambda\bloc^{-1}<(\Lambda\depo\varsigma\depo)^{-1}$) due to deposition.
% Increasing $\Lambda\depo\varsigma\depo$ led to the deposition layer becoming thinner (similarly to the effect of increasing $\Lambda\bloc$), since this corresponds to less smearing of particle mass when they deposit, or an increase in the probability of particle capture.

In Section \ref{sec:ssm_Mono-dispersed particles and poly-dispersed pores}, we turned our attention to the case where 
particles are all one size, but the distribution of pore sizes 
is log-normal. 
Firstly, we once again considered the case where 
only blocking occurs.
% (this time by setting  $\varsigma\depo=0$).
We found that this situation gives rise to a wave of particles that gradually traverses the entire filter.
Upstream of the wave front, particles have blocked all pores 
that are smaller than the particles, and the concentration is dictated by the number of particles that occupy pores that are larger than the particles.
% , which cannot be blocked. 
Downstream of the wave front, there are no particles, since 
they were all removed further upstream.
In the travelling capture layer, which is of thickness $\mathcal{O}(\Lambda\bloc^{-1})$, 
particles block any smaller pores that still remain, and so the particle concentration transitions from its upstream value to its downstream value.
Pores that are larger than the particles keep their original distribution, since they are unaffected by particles that are smaller than them when deposition occurs with zero probability. 
We found an explicit formula for the speed of the travelling wave that depends on the characteristic ratio of the concentrations of particles in the feed and pores in the filter, $\Gamma$,  the characteristic length over which blocking occurs, $\Lambda\bloc^{-1}$, 
and the initial concentration of pores that are smaller than the particles, $N_{[r<A]\text{init}}$.

Secondly, we considered the case where blocking and deposition both occur. 
In the limit where there are many more pores than particles, $\Gamma\ll1$,
we found an explicit leading-order asymptotic solution and used this 
to show that particles are captured over a layer at the top of the filter, 
the thickness of which decreases as the particle size increases, 
at least when the characteristic 
blocking length is shorter than the 
characteristic deposition length,
so that $\Lambda\bloc^{-1}<(\Lambda\depo\varsigma\depo)^{-1}$ (due to particles spreading when they deposit, or a small sticking probability, $\varsigma\depo\ll1$, for example). 
% because a greater proportion of particles are captured via blocking 
% as opposed to deposition, which occurs over a shorter length (since $\Lambda\bloc^{-1}<(\Lambda\depo\varsigma\depo)^{-1}$).
In the limit  $\Gamma\ll1$, 
we saw that most pores in the filter are not used to capture particles. 
% .
% When particles are large compared to the average pore size, most pores in the filter are wasted, even for the most stringent 
% particle removal constraints.
% This suggests that
% fluids containing comparable concentrations of particles to that of the pores in the filter
% can be processed. 
% an optimal filter would contain more large pores, at least near the inlet, so that some particle removal occurs at greater depth (we outline a simulation to test this hypothesis in the next section).
We then explored the case $\Gamma=\mathcal{O}(1)$. 
We solved the system numerically and showed that, as in the case of no deposition, 
a particle wave travels through the filter. 
This time, however, the particles that are left in larger pores upstream of the wave continue to deposit within them, 
which causes further reduction to the distribution of large 
pores long after the wave front has passed.
The decrease is most severe at the inlet, where particles have had the most chance to deposit. 
Depending on the required retention threshold, 
it could be argued that the useful lifetime of the filter 
is much longer when deposition can occur as opposed to when it cannot (consistent with observations in \cite{sanaei2019membrane}, for example). 
% (that is, when $\varsigma\depo\ne0$ as opposed to when $\varsigma\depo=0$), 
When it can occur, particle capture continues long after the wave has reached the outlet for the first time, until all pores have been removed. 
However, at later times, far fewer particles are captured per unit time, 
so that the fluid exiting the filter is likely to contain many particles. 
% continue to be captured via deposition 
% until all pores are removed.


\section{Further work}
There are several natural extensions to our work, which we will discuss in this section. 

\subsection{Poly-dispersed particles and poly-dispersed pores}
In this thesis, we limited our analysis 
to the cases where either the feed and filter are mono-dispersed, or 
the feed is mono-dispersed and the filter is poly-dispersed. 
However, one of the advantages of this model is that it is a simple framework within which to analyse the case where the feed and filter are both poly-dispersed, a case that industrialists are interested in, to assess the size distribution in the filtrate, for example. 
In Section \ref{sec:ssm_Dimensional model} 
we wrote down the condition at the inlet in the case where 
particles are log-normally distributed, and our future work 
would include solving the model in this case.

\subsection{Depth-dependent initial pore-size distributions}
We focused our analysis on the case where the initial distribution of pore sizes is independent of depth.
It would be interesting to introduce 
depth dependence, via, for example, a 
depth dependent mean pore size parameter, $\bar{r}^*(z)$, 
a depth dependent pore size variance parameter, $\sigma_n(z)$, or a depth dependent initial porosity function, $\phi\init^*(z)$.
In the case where particles are mono-dispersed and pores are poly-dispersed, we found that, once the wave has passed, a gradient 
in the particle concentration and porosity
is established, because deposition, and subsequent blocking, occur more often 
at the top of the filter than at the bottom. 
If we stipulated that, on average, pores increased in size towards the top of the filter (as in \cite{sanaei2018membrane} or \cite{fong2021performance}, for example), then deposition would have 
less effect on the porosity, per occurrence,
at the top of the filter than at the bottom. 
By tailoring the pore size gradient, it might therefore be possible to establish a regime in which the porosity decreases uniformly across the filter. 
This would be favourable as it would mean that the lifetime of the filter would be extended, since it would take longer to reduce the porosity near the inlet (and further down the filter) to zero. Whether this proposed lifetime extension is useful depends on the adherence between particles and pores. 
If this is large enough (close to one), then significant retention continues after the particle wave has traversed the filter.
% It would be simple to explore the effect of an initial porosity gradient using our framework, by introducing a depth dependent initial pore density function, as described in 
% \ref{sec:ssm_Dimensional model}.


% As we have discussed, we have found that, particularly in regimes in which blocking is the dominant mechanism by which 
% particles are captured, particles are caught very close to the inlet 
% and most of the filter is not utilised. 
% This finding supports the idea that it would be useful to introduce a 
% negative porosity gradient across the filter, whereby the mean pore-size is substantially larger than the mean particle-size near the inlet, and decreases with depth. 
% In this case, we would expect deposition to dominate blocking near the inlet, which, for small enough deposition probability, $\varsigma\depo\ll1$, would allow particles to traverse the filter and thus to reach pores much nearer to the outlet before being captured. This situation would utilise 
% a greater proportion of the pores in the filter for particle retention, and probably increase its useful lifetime as a result. 
% Modelling of this scenario would involve choosing a depth-dependent initial pore-size distribution, which, as we showed in Section \ref{sec:ssm_Dimensional model}, is straightforward in our framework.

\subsection{Deposition transformation and sticking probability}
In our simulations, we chose simple forms for the deposition transformation 
distribution and for the sticking probability. 
We supposed that, after deposition, the reduction in pore size was always exactly the particle size, 
and that the adherence between particles and pores is a constant that is independent of their sizes.
Further work would include solving the model using more 
realistic forms for these two functions, motivated by physical intuition, experimental 
evidence, or detailed mathematical modelling of the physics and 
chemistry on the microscale.
Choosing a depth-dependent sticking probability, for example, 
may provide an industrially relevant alternative to 
depth-grading the initial pore-size distribution. 
That is, if the probability of sticking is small (or even zero) near 
the inlet, but increases towards the outlet, 
then for a well-chosen pore-size distribution, we 
would expect a greater number of pores to be used for 
particle retention than in the scenario where the sticking probability is depth independent.

\subsection{Additional particle and pore interactions}
Our model provides a framework for filtration modelling, upon which many additional features could be added. 
We did not discuss particle--particle interactions, for example, but 
it would be relatively simple to include additional terms to describe, say, particle coagulation, by modelling 
this as the sticking together of small particles to form larger particles.
Additionally, our current model does not account for `caking', 
which is the build-up of particles on the surface
of the filter to form a layer through which the feed must permeate.
Extending our model to account for this behaviour would
make it more versatile, and would be another feasible addition, 
since our framework captures the evolution of
the pore distribution, so that the prediction of the onset of caking (when the pores at the surface of the filter become small enough for particle build-up on the surface to begin) would be relatively straightforward.

\subsection{Optimisation and experimental data}

Since repeated solution of our model is computationally feasible, it lends itself to optimisation 
problems, but we did not focus on these in this thesis. 
It would be relatively simple, for example, 
to construct an experiment where a performance metric is returned 
for each value of a given parameter, which is incrementally changed.
The result would be the performance metric, such as the proportion of particles retained, or the useful lifetime of the filter, say, 
as a function of the parameter, which would be useful for improvement of the filtration regime.
Lastly, in future work, it would also be interesting to compare our results 
to experimental data. 
This would allow use to verify our solution, 
and to find exact physically relevant parameter regimes.
\\
\\
\indent In Part \ref{part:A size-structured model for filtration} we have developed a mathematical framework 
for filtration
in which the filter and feed are continua of pores and particles, with their sizes accounted for. 
In future, the model will be useful to increase our understanding of filtration processes and to enhance filter performance.
In Part \ref{part:A network model for filtration}, we will turn our attention to an alternative framework for filtration, 
in which the filter membrane is considered as a network of pores.

% to explore the optimal
% did not focus on optimisation of the filtration process. 
% Our mode
% There are several shortcomings in our current work. Firstly, we have assumed
% very simple functions for the sticking probability and the deposition transformation
% distribution. Selecting more realistic functions, based on physical intuition, experimental evidence, or a detailed description of the microscale chemistry and physics, is
% necessary to improve our model’s predictive power. Nevertheless, we do not expect
% that the choice of these functions will qualitatively alter the results. Secondly, the
% preliminary expose of the power of our framework, as presented in this paper, will
% need to be enhanced by considering poly-dispersed pores and particles and pore distributions that are initially non-uniform in space. These will all be the subject of the
% companion paper [?]. Thirdly, we need to compare our model to experimental data.
% \\
% There are also many additional features that we could build into our framework.
% For example, our model does not account for the build-up of particles on the surface
% of the filter, through which the feed must permeate before it reaches the filter (the
% process of "caking"). Extending our model to account for this caking behaviour would
% make it more versatile. Nevertheless, since our framework captures the evolution of
% the pore distribution, it will be feasible to use such an extended model to predict
% the onset of caking when the pores at the surface of the filter get small enough that
% surface buildup can initiate.
% \\
% The modelling framework presented in this paper provides an alternative to traditional models for filtration in which pore-size evolution is easily accounted for, and
% is thus gives a natural framework for optimising filter operation, enhancing filter performance and increasing our understanding of filtration processes.

% [which suggests that consideration of pore distributions that vary with depth would be useful.] 

% its the start of a framwork since other phenomena can be added... particle coagulation,


% In this paper, we have developed a new size-
% structured framework for describing the evolution of a filter during operation, in which
% we assume that there are continuous distributions of pore sizes in the filter and parti-
% cle sizes in the fluid feed. We assume that pore size and particle size are independent
% variables and, appealing to conservation of mass, we build a set of equations govern-
% ing the evolution of the distribution of particles and pores. We incorporate particle
% capture, and pore evolution, through blocking (that is, a particle being unable to pass
% into a pore which is too small) and deposition (where a particle inside a pore sticks to
% the wall). In this framework, the porosity and permeability of the filter are straight-
% forward to calculation, as functions of space and time, once the pore distribution is
% know. The equations for the evolution of the pore and particle distributions are com-
% plemented by conservation of mass for the fluid and Darcy’s law which determine,
% given the pore distribution, the speed and pressure of the fluid in the filter.
% \\
% As an example, we focus our attention on the case where the pores are initially
% log-normally distributed in size but uniform in space and where the filter is operating
% in a constant flux regime. We nondimensionalise the system and identify the key
% parameters. We explicitly solve the dimensionless equations in the case where there
% are many fewer particles than pores. We see that the clogging mechanism switches
% from blocking to deposition as the size of the particles increases, and the steepness
% of the transition depends on how tight the pore size distribution is. We find that
% the particles are all removed in a small region at the top of the filter, at least for
% \mathcal{O}(1) times. These initial results support the intuitive idea that, to optimise filter
% performance, the filter should be be graded in such a way as to ensure that the
% particles are captured evenly through the depth in order that the whole pore volume
% is efficiently utilised.
% \\
% We then consider the case where there is no deposition. We focus our attention on
% the surface of the filter and solve (numerically) for the distribution of pores there. As
% expected, we find that the particles pass through the large pores without incident but
% get trapped by pores that are smaller than them, reducing the number of available
% pores of that size. We explored the effect of altering the ratio between the size of
% the particles and the average initial size of the pores, and found that the surface
% porosity decreased as the ratio increased (since more of the pores blocked), and that
% this decrease was more marked when the pore-size distribution was tighter.
% \\
% We then considered both blocking and deposition, assuming a simple function for
% the change of pore radius during deposition. Again, we initially focused on the surface
% of the filter. We found that, with the parameters chosen, blocking swiftly removed
% the pores that were smaller than the particles, and then there was a slow removal of
% larger pores by deposition and then subsequent blocking. Add extensions etc
% \\
% There are several shortcomings in our current work. Firstly, we have assumed
% very simple functions for the sticking probability and the deposition transformation
% distribution. Selecting more realistic functions, based on physical intuition, experi-
% mental evidence, or a detailed description of the microscale chemistry and physics, is
% necessary to improve our model’s predictive power. Nevertheless, we do not expect
% that the choice of these functions will qualitatively alter the results. Secondly, the
% preliminary expose of the power of our framework, as presented in this paper, will
% need to be enhanced by considering poly-dispersed pores and particles and pore distri-
% butions that are initially non-uniform in space. These will all be the subject of the
% companion paper [?]. Thirdly, we need to compare our model to experimental data.
% \\
% There are also many additional features that we could build into our framework.
% For example, our model does not account for the build-up of particles on the surface
% of the filter, through which the feed must permeate before it reaches the filter (the
% process of "caking"). Extending our model to account for this caking behaviour would
% make it more versatile. Nevertheless, since our framework captures the evolution of
% the pore distribution, it will be feasible to use such an extended model to predict
% the onset of caking when the pores at the surface of the filter get small enough that
% surface buildup can initiate.
% \\
% The modelling framework presented in this paper provides an alternative to tra-
% ditional models for filtration in which pore-size evolution is easily accounted for, and
% is thus gives a natural framework for optimising filter operation, enhancing filter per-
% formance and increasing our understanding of filtration processes.



\part{A network model for filtration}
\label{part:A network model for filtration}

\chapter{Introduction to the network model}
\label{chap:nm_Introduction to the network model}

\section{Networks in filtration}
\label{sec:nm_Networks in filtration}

In Chapter \ref{chap:Introduction}, we introduced expanded polytetrafluoroethylene (ePTFE) as a filtration membrane material of interest, and explained that the microscale consists of a mesh or fibres. This gives rise to a complex network of pore space between them, through which particle-laden fluid is transported. Intuitively, the exact geometry of the pore space has a significant influence on the effective properties of the filter. 
For example, if the space between fibres is larger, then more fluid is advected through it per unit time, causing a larger effective permeability. 
If the space is smaller, then particles are close to fibres more often, so we might expect larger effective adhesivity (i.e., the overall `stickiness' of the filter that quantifies its ability to capture particles). 

Due to this link between microscale geometry and macroscale behaviour, much attention has been paid to the accurate determination of microscale structure (see \cite{xiong2016review} for a useful review,  and \cite{abichou2004network, aytas2009adsorption, ghassemi2011pore, kohler1996experimental, li1992space, ren2010influence} for some general examples). One significant experimental complication is the evolution of the microscale structure that occurs during filtration. Mechanical, chemical, and electrostatic interactions between particles and fibres (that act as pore walls) cause adherence between them. This leads to deposition of particles on pore walls, and thus changes to the size and shape of pores as filtration occurs. These structural changes are difficult to capture empirically because observations of the process are usually invasive and may even involve destruction of the filter membrane itself.

One alternative approach is to mathematically model these changes to the microscale by representing it as a directed and weighted network of nodes and edges and then posing governing equations on this geometry (see \cite{griffiths2022hybrid, gu2022graphical, gu2022network}, for example). 
In this context, node and edge weights usually represent pore size, and deposition is modelled as a decrease in the concentration of particles in the fluid that is accompanied by changes to these weights, and thus the connectivity of the network.
Unsurprisingly, to a large extent, the success of this strategy depends on the degree to with which the initial network represents the initial microscale of the membrane \cite{xiong2016review}. 

The network representation process, by which a network is obtained, is therefore critical. 
It usually consists of two steps, which are illustrated in Figure \ref{fig:nm_network-representation}. 
First, methods that are broadly referred to as experimental characterisation are used to produce high resolution images of the microscale.
Second, so-called network construction algorithms are used to transform these images into a network of nodes and edges that together represent pore space. 
\begin{figure}[!htb]
        \includegraphics[width=0.7\textwidth]{./figures/network/introduction/multiscale-filters.pdf}
        \centering
        \caption{An illustration of network representation, which is the systematic extraction of a network from the microscale of filtration membranes such as ePTFE. 
        Step 1 is experimental characterisation, in which high resolution images of the microscale are obtained. 
        Step 2 is network extraction, in which the microscale is algorithmically segregated into solid space and pore space, before the pore space is further separated into nodes and edges that form a network.
        }
        \label{fig:nm_network-representation}
\end{figure}
% Before a model can be built, 
% that is constructed by algorithmic analysis of high resolution images that are obtained via experimental characterisation, in which differences in the locations occupied by pore space and solid fibres are clarified. An illustration of the process by which a network is obtained from ePTFE is shown in Figure []. 

% First, various methods are used, sometimes in conjunction, for experimental characterisation, which produces high resolution images of the microscale. 
The choice of characterisation strategy depends heavily on the filtration material. Some common methods include: X-ray computed microtomography (micro-CT) \cite{blunt2013pore, ketcham2001acquisition}; Focused ion beams (FIB) and Scanning electron microscopy (SEM) \cite{keller2011application, abichou2004network, tomutsa2007analysis, curtis2010structural, lemmens2010fib}; Nuclear magnetic resonance (NMR) \cite{blumich2009nmr, callaghan1993principles}; and Mercury intrusion porosimetry (MIP) \cite{giesche2006mercury, y1998new, rouquerol2012characterization}. Characterisation using gas adsorption techniques \cite{thommes2010physical} can also be useful, since the adsorption behaviour of porous materials is a function primarily of their microstructure \cite{xiong2016review}. In most characterisation methods, a trade-off is established between representation and resolution. The portion of the microscale selected must be large enough to permit good representation of the filter as a whole,  whilst still being small enough to permit a resolution that is sufficient for algorithmic post-processing. 

The choice of network construction algorithm depends to a large extent on the characterisation method employed, and on the type of image that this produces. 
% use the outputted characterisation as input for network construction, 
% by segregating areas determined to be pore space into either nodes or edges. O The choice of algorithm depends to a large extent on the type of image  that results from the characterisation step. 
Some examples are: Direct mapping \cite{piri2005three}; Medial axis algorithms \cite{baldwin1996determination, liang1998reconstruction, lindquist1996medial, lindquist2000pore}; and Grain-based models \cite{bryant1992prediction, bryant1993network, bryant1993physically, bakke19973, oren1998extending}. One alternative approach is to use statistical reconstruction methods, which obtain larger networks that have the same statistical properties as characterisations of smaller portions of the membrane \cite{adler1998real, roberts1999chord, yeong1998reconstructing, okabe2004prediction}. For example, images of small membrane samples are analysed for the average coordination number (the number of other nodes that are connected by edges to each node on average) and pore-size distribution, and then a larger network with identical statistical properties is constructed, sometimes with more regular or periodic structure \cite{xiong2016review}.

Over the past few decades, network representation has received considerable attention, and both experimental characterisation and network construction have become substantial subject areas in their own right.
They are, however, far too broad and diverse for complete examination here. It suffices to note that, in general, the output of the these two processes in series is a network of nodes and edges, the connectivity and size distribution of which are known, and preserve or at least statistically represent the topology of the microscale pore space of the filtration medium. 

\section{Existing network models for filtration}
\label{sec:nm_Existing network models for filtration}

Equation systems defined on network geometries are often referred to as pore network models (PNMs). These describe the transport and deposition of particles that occurs in nodes or edges, or both. 
The first PNM was constructed by Fatt in 1956 \cite{fatt1956network}. 
It exploits the analogy between the microscale of porous media and electrical resistor networks. 
Fluid flow plays a similar role as electrical current, and the fluid conductance of edges 
is analogous to the inverse resistance of resistors, 
so that conservative laws for fluid mass can be written as Kirchoff's law \cite{kirchhoff1845about, young1996university}. 
Much of the notation in PNMs is borrowed from percolation theory, as are several of the central ideas. 
For example, in \cite{gine1997percolation}, Grimmett studies the effect of the coordination number of some simple networks with regular structure on the effective properties of one-phase fluid flow.  
PNMs have rapidly grown in sophistication \cite{xiong2016review}, and have since been extended to study the properties of irregular networks and two- and three-phase flow through them, as well as a variety of different physical processes \cite{mehmani2012multiblock, blunt2002detailed, lopez2003predictive, ryazanov2009two, yiotis2006pore}.

In many studies, the notion that the connectivity and size distribution of edges (usually modelled as cylinders with specified initial radius) and nodes (often modelled as spheres of given initial size) affect transport coefficients is central \cite{bryant1992prediction, meyers1999network, jivkov2014network}.
% (Meyers and Liapis, 1999; Raoof
% and Hassanizadeh, 2010a) (Blunt et al., 1992; Knackstedt et al., 1998; Oren, 1994; (Bryntesson, 2002; Jivkov and Xiong, 2014; Laudone et al., 2008; Meyers and Liapis, 1999; Meyers et al., 2001)). 
It has been shown that microscale structure impacts important industrial indicators of filter performance. 
In \cite{griffiths2020role}, for  example, Griffiths \emph{et al.} find that increasing the complexity of random edge connectivity increases the tortuosity of the filter, which
substantially improves its particle removal efficiency. 
Meanwhile, in \cite{sanaei2018membrane}, Sanaei \emph{et al.} find that even incorporating simple symmetric branching structure allows for the study of the effect of pore size gradients on clogging patterns and thus filter performance. 

Generally, PNMs that model the fluid suspension as a continuum and the filter medium as a network are useful for models of the mesoscale of membranes. 
Systems with tens or hundreds of nodes and edges can be used to represent sections of porous media that are too large for simulations that track each particle to be computationally feasible. 
On the other hand, PNMs can be more accurate than PDE systems that usually involve Darcy flow coupled with advection--reaction equations, since these typically ignore microscale geometry that is still important at this scale. 
% PNMs thus provide a useful balance between being numerically cheap enough to get a decent size of network while being expensive because they capture detail.

\section{A novel network model for filtration}
\label{sec:nm_A novel network model for filtration}

Broadly, our ultimate aim in Part \ref{part:A network model for filtration} is to present a relatively simple PNM, with the goal of incorporating this into a multiscale framework in Part \ref{part:A multiscale model for filtration}. 
As such, in Part \ref{part:A network model for filtration}, the objective is not to provide detailed analysis of a complex network model that fully accounts for the intricacies of the phenomena that occur on the microscale of the membrane as filtration occurs. 
Instead, we will derive a relatively straightforward network model and illustrate intuitive solutions in one particular operating regime. 

% In the remainder of Part \ref{part:A network model for filtration}, 
% we will derive and solve a 
To this end, in Chapter \ref{chap:nm_Derivation of the network model}, 
we will derive a novel network model for filtration. 
In Section \ref{sec:nm_Dimensional model}, 
we will begin by supposing that a particle-laden fluid continuum is driven through a network 
of edges, which model pores, and nodes, which model junctions between them. 
We will then consider the conservation of fluid mass, particle mass, and pore volume, to 
derive a governing system. 
We will suppose that edges are cylinders, and will account for deposition by supposing that particles adhere to the inner walls of these cylinders with some adherence probability, which causes changes to their cross-sectional area and thus their conductance.
We will suppose that the initial structure of the network results from the representation methods outlined in Section
\ref{sec:nm_Networks in filtration}, and thus that it is statistically representative of  
a filtration membrane.
Consequently, the pore size distribution, and thus edge conductances and node volumes, as well 
as the connectivity of the network, are all known initially.
The result is a dimensional system of ordinary algebraic--differential equations on a directed and weighted network whose weights and connectivity change with time. 
To demonstrate solution of this system, we will establish suitable initial and boundary conditions for closure that represent a toy operating regime that is simple enough for intuitive interpretation.
In Section \ref{sec:nm_Dimensionless model}, we will nondimensionalise this system, introduce dimensionless parameter groups, and prepare it for solution and analysis. 

In Chapter \ref{chap:nm_Solutions of the network model}, we will briefly discuss solutions of this network model. 
In Section \ref{sec:nm_Solution methodology}, we will begin by discussing our numerical strategy. Emphasis is placed on the computational cost of solution, which, as we have discussed, allows for simulation of tens or hundreds of nodes and edges. 
With this in mind, in Section \ref{sec:nm_Non-random network: Mono-dispersed pores}, 
we describe the numerical solutions of the system on a relatively small network.
We consider a four-regular network containing nodes and edges of equal volume 
and conductance, which corresponds to a membrane with mono-dispersed pore sizes and an average coordination number of four. 
We will show examples of how macroscale properties, such as average particle concentration, effective conductivity, average flow rate, and filter pressure drop, can be derived from the microscale variables that solve the network model.
We will then explore their changes as functions of the system parameters, which correspond to  
the adherence probability in the pores and to the average size of particles in the suspension. 
Consequently, we will find that these parameters affect properties of industrial interest, such as the
efficiency (the proportion of particles that are retained in the filter)  and total throughput (the total quantity of fluid that is processed before termination of the filtration regime).
Solution of the model on random networks with log-normally distributed pore sizes and thus edge conductances is saved for Part \ref{part:A multiscale model for filtration}, Section \ref{sec:mm_Periodic networks}.


% We will suppose that coordination number and pore size disrtivution is known, so that the network is defined with distributions with known parameres for the conductance of edges, which we will show is related to pore sizes. The result is network tha tis weighted y edge condcutances. We will define a model where the fluid is a continuum that carries particles. These particles will deposit in the edges of the network continuously. This will change the sizes, and thus the conductances of the edges. Deposition will thus cause changes to the structure of the network ,so that its connectivity is dynamic. We will then study the simple case of a regular network with uniforml distributed to discuss the qualitiat vve behabioru of the model. We are generally interested in yhow the structure of the network affects the beahvavious of effective paramertres that determine the rate of flow and deposition, but we will save definitions for these, and thorough analysis of this question for Part 3. 

% What will we do differently. We will try to show that in the absence of data about the medium we can use statistics to recreate the same results. We will show tha tthteh network only needs ot be `represenatative’ of  the microscale in terms of coordination number and containing enough nodes.

\chapter{Derivation of the network model}
\label{chap:nm_Derivation of the network model}
% Summary of what we do in chapter.
% Overall aim will be to write down a simple network model and demonstrate why theyre good and what their limitations are.
In this chapter, we develop a model for filtration in which the filter is treated as a network of edges, which model pores, 
and nodes, which model the junctions between them. 
We begin by deriving a dimensional system of algebraic--differential equations for the concentration 
of particles, the conductance of pores, and the pressure. 
We pose industrially relevant boundary and initial conditions, 
and then scale the problem to arrive at a dimensionless system to be analysed and solved.

\section{Dimensional model}
\label{sec:nm_Dimensional model}
We consider a filter membrane that is composed of a network that is embedded in $D$-dimensional space, where $D \in \{ 1,2,3\}$, as illustrated in Figure \ref{fig:nm_network_1}. 
The filter is located between $\bm{x}=\bm{0}$ and $\bm{x}=\bm{l}$,
so that the $m^{\text{th}}$ component of $\bm{l}$, $l^m$, is the length of the network in the $m$ direction.
The network consists of a series of pores, which we model as edges, that are connected to one another at junctions, which we model as nodes.

A particle-laden fluid is driven through the filter in the $x^1$ direction. 
The fluid enters the filter at the inlet, which is located at $x^1=0$, and passes through the pores via the junctions until it 
reaches the outlet, which is located at $x^1=l^1$.
As the fluid traverses the filter, particles in the fluid deposit on the inner walls of the pores due to a mutual adherence between the particles and pores. 
This deposition decreases the cross-sectional area of the pores (see Figure \ref{fig:nm_network_1}), which decreases their fluid conductance.
Eventually, pores are removed completely, which causes a change in the connectivity of the network within the filter. 
In the next subsection, we define objects to track the pores in the filter and the particles in the feed fluid.
\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.90\textwidth]{./figures/network/network_1.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{A diagram showing a two-dimensional filter that is composed of a network of pores, which are visualised as edges, 
        and junctions, which are visualised as nodes.
        Particles in the flow enter through edges crossing the left boundary and exit through those crossing the right boundary after traversing the network.
        Particles deposit in pores, and their volume is spread uniformly across the pore walls, 
        which decreases the cross-sectional area of the pores.}        
        \label{fig:nm_network_1}
    \end{center}
\end{figure}


% [Describe the network. It is any system of nodes and edges that doesn't need to be periodic in the x direction. In filtation, pressure drops are unidirectional as we discussed before, and so we will impose periodict in the y direction. In the next part, we will be interested in periodic networks that repeat over lengthscale. WE will introoduce cells to introduce a difference in length scales. Diagram of an arbitrary network with cells drawn over it. Periodic in the y direction. Particles entering and leaving. Nodes and edges labelled. There are \mathcal{O}(N) nodes in a cell on average. Note that this means we traverse \mathcal{O}($\sqrt{N}$) edges on average. Think of a 4-regular cell, where it's exactly root N. Label $N^1$ to be number of columns of cells, $N^2$ to be number of rows of cellss. Label $L^m$ and $l^m$, and label $\varepsilon$ and $\delta$. Label of a column of nodes, a column of cells, a row of nodes, and a row of cells.]

\subsection{Particles and pores}
We track pores by defining $G_{ij}(T)$ to be the conductance of pore $ij$ at time $T$, so that $[G_{ij}]=\text{m}^D\text{s}^{-1}\text{Pa}^{-1}$ (recall from Chapter \ref{chap:ssm_Derivation of the size-structured model} that we use $[x]$ to mean `the dimensions of $x$'). 
Defining $P_i(T)$ to be the pressure in junction $i$ at time $T$, so that $[P_i]=\text{Pa}$, 
we write 
\begin{align}
    Q_{ij}= G_{ij}\left(P_i-P_j\right).
    \label{eq:nm_flux-definition_dim_1}
\end{align}
Here, $Q_{ij}(T)$ is the volumetric flux through pore $ij$ at time $T$, which is the volume of fluid that 
is transported from junction $i$ to junction $j$ along pore $ij$ per unit time, so that $[Q_{ij}]=\text{m}^{D}\text{s}^{-1}$.

To track particles, we first define $V_i(T)$ to be the volume of junction $i$, so that $[V_i]=\text{m}^D$. 
% We suppose that nodes have a much smaller volume than pores, and assume that deposition 
% does not occur within nodes, so that the node volumes are constant in time.
% Consequently, we write
% \begin{align}
%     V_i = V_i^0,
% \end{align}
% where $V_i^0$ are the constant initial node volumes, which are drawn from some known distribution, with mean $\bar{V}$ say.
Next, we define $C_i(T)$ to be the concentration of particles in junction $i$, which is the number of particles per unit volume of void space (i.e., node volume plus edge volume) in junction $i$ at time $T$, so that $[C_i]=\text{m}^{-D}$. 
For simplicity, we consider the case where the filter and void particle concentrations (discussed in Part \ref{part:A size-structured model for filtration}), are equal: the total node volume, $V_{\text{N}}$, represents junctions that make up the bulk of pore space; 
the total edge volume, $V_{\text{E}}$, represents small conductive pore space regions between junctions; and the total fibre volume, $V_{\text{F}}$, represents tiny fibres that make up the solid space external to the network. 
We choose $V_{\text{E}}, V_{\text{F}} \ll V_{\text{N}}$, and note that, since the porosity (i.e., the ratio of pore space to filter space) is given by $\phi=(V_{\text{N}}+V_{\text{E}})/(V_{\text{N}}+V_{\text{E}}+V_{\text{F}})$, 
it follows that $\phi\approx 1$ (the porosity of ePTFE can exceed $95\%$ \cite{ebnesajjad2017introduction}). 
% Thus we consider $\phi=1$ , and, 
Since void and filter particle concentrations differ only by a factor of $\phi$ (see \eqref{eq:def-void-conc}, for example), it follows that they are approximately equivalent in this simple setting.
% considered in Part \ref{part:A size-structured model for filtration}. 
% However, we assume here that particle capture occurs in pores that are small in comparison to the junctions between them, so that filter space and void space concentrations are identical in this model.
We note that it follows that $V_i(T)C_i(T)$ is the number of particles in node $i$ at time $T$, a property that we will use to derive an equation to describe the evolution of the particle concentration in the following sections.

\subsection{Conservation of fluid}
% We define $Q_{ij}$ to be the volumetric flux of fluid from node $i$ to node $j$, so that $[Q_{ij}]=\text{m}^D\text{s}^{-1}$ 
% (where we recall that by $[x]$ we mean `dimensions of $x$'). 
% Denoting the conductance of the edge from $i$ to $j$, which we call edge $ij$, by $G_{ij}$, and defining $P_i$ to be the pressure in node $i$, with units $[P_i]=\text{Pa}$, it follows that
% \begin{align}
%     Q_{ij} = G_{ij}(P_i-P_j),
%     \label{eq:mm_flux-definition_sum_nodim_1}
% \end{align}
% so that $[G_{ij}]=\text{m}^D\text{s}^{-1}\text{Pa}^{-1}$.

We assume that fluid is conserved in each junction. 
That is, the mass of fluid that enters junction $i$ from pores that are connected to it is equal to the mass that exits junction $i$ to other connected pores. 
It follows that
\begin{align}
    \sum_{j=1}^{N_{\text{f}}} Q_{ij} = 0,
    \label{eq:nm_fluid-conservation_dim_1}
\end{align}
where we define $N_{\text{f}}$ to be the number of junctions in the filter. 
Note that $Q_{ij}=0$ if there is no pore connecting junctions $i$ and $j$.

\subsection{Conservation of particles}
We assume that particles are conserved at each junction. 
We suppose that, as particles move into junction $i$ from junction $j$ (provided that there is a pore $ji$ between these two junctions), some proportion, $0\le A_{ji}(T)\le 1$ say, of the particles are deposited in pore $ji$.
Since the total number of particles is conserved, the number of particles that arrive in junction $i$ from pore $ji$ in the infinitesimal time-step $T+ \delta T$ is equal to the number of particles that leave junction $j$ to the pore $ji$ in this time step minus the number 
of particles that deposit inside the pore during the time step. 
Accounting for the possibility that each junction $i$ is connected to more than one pore $ji$, 
and taking the limit $\delta t \rightarrow 0$, it follows that
\begin{align}
    \pp{(V_i C_i)}{T} &= \sum_{j=1}^{N_{\text{f}}}  \Big(
    (1-A_{ji})Q_{ji} C_j H(P_j-P_i) - Q_{ij} C_i H(P_i-P_j)  \Big).
    \label{eq:nm_particle-conservation_dim_1}
\end{align}
We call $A_{ji}(T)$ (which is equal to $A_{ij}(T)$) the adherence of pore $ji$ at time $T$, with $[A_{ji}]=1$. 
This is the fraction of particles in junction $j$ that deposit in pore $ji$ when they are transported to junction $i$, and therefore represents the `stickiness' between pore $ji$ and the particles.
If $A_{ji}=0$, for example, then no particles stick inside pore $ji$, and all particles that leave junction $j$ arrive in junction $i$, while if $A_{ji}=1$ then all of these particles get stuck in pore $ji$ before they arrive in junction $i$, and contribute to the decrease of the cross-sectional area of the pore.
% We scale adherence to make it so there is $\mathcal{O}(1)$ removal across the entire network. This ensures that networks with different number of cells and nodes within a cell can be compared.
% The adhesivity is an $\mathcal{O}(\varepsilon)$ quantity, which ensures that an $\mathcal{O}(1)$ amount of contaminant is removed after passing through the entire filter, which requires traversing $\varepsilon^{-1}$ cells.
% In \eqref{eq:nm_particle-conservation_dim_1}, 
Meanwhile,
$H$ denotes the Heaviside function,
\begin{align}
    H(P_i-P_j)=
    \begin{cases}
        1 & P_i-P_j>0,
        \\
        0 & P_j-P_i<0.
    \end{cases}
    \label{eq:nm_heaviside-definition_dim_1}
\end{align}
% which we use to indicate the direction of the flow \correct{}{in pores}.
% \correct{If the pressure in junction $i$ is greater than that in junction $j$, then $Q_{ij}$ is positive, and particles flow through pore $ij$ from junction $i$ to junction $j$, 
% while if the pressure in junction $j$ exceeds that in junction $i$, then $Q_{ij}$ is negative, and  particles flow from junction $j$ to junction $i$ (note that $Q_{ji}=-Q_{ij}$).}{
In \eqref{eq:nm_particle-conservation_dim_1},
we consider particle conservation at junction $i$. 
$H$ ensures that only deposition in pores $ji$ upstream of junction $i$
 (i.e., those that connect it to junctions $j$ for which $P_j>P_i$) is accounted for in this conservation law,
to avoid double counting deposition.
Specification of $H(0)$ is not necessary, provided that the imposed filter pressure difference is unidirectional for all time.
% (i.e., does not change direction, in the case of induced backward flow through the filter for cake cleaning, for example).
In this case, for any connected junctions $i$ and $j$, $G_{ij}$ decreases monotonically (due to deposition, via \eqref{eq:nm_conductance-conservation_1_dim}) but never reaches zero, 
and so the inequality $P_i>P_j$ is preserved (via \eqref{eq:nm_fluid-conservation_dim_1}) for all time. 
Therefore, if pore $ij$ conducts fluid from junction $i$ to junction $j$ initially, then it does so for all time. 
We restrict our study to unidirectional filter pressure differences (see (\ref{eq:nm_bc_pres_sum_dim_1})).
In Chapter \ref{chap:mm_Derivation of the multiscale model}, we will see that this permits asymptotic expansion of $H$ as a continuous function, away from the discontinuity at $P_i=P_j$.

% \correct{the first use of this indicator, $H$, ensures that particles only arrive from those junctions $j$, and deposit in those pores $ji$, that are upstream of junction $i$, respectively. 
% The second $H$ is used to ensure that particles are conducted from junction $i$ to pores $ij$ and junctions $j$ that are further downstream.}{}
% In \eqref{eq:nm_particle-conservation_dim_1},
% $H$ ensures that particle deposition occurs in pores upstream of each junction $i$,
% but not in those downstream of it,
% so that the effect of deposition is not `double counted'. 
% To see this, note that, if the pressure in junction $j$ is greater than that in junction $i$,
% so that 
% $P_j>P_i$, 
% then $Q_{ji}>0$,
% and 
% $H(P_j-P_i)=1$ and $H(P_i-P_j)=0$.
% The first term in the right hand side of 
% \eqref{eq:nm_particle-conservation_dim_1}
% is thus non-zero and implies that particles flow from junction $j$ to junction $i$,
% and deposit in pore $ji$ as they do so,
% while the second term
% is zero, which implies that particles do not flow `back' from junction $i$ to junction $j$.
% That is, pore $ji$ conducts fluid \emph{in} to junction $i$, and deposition occurs here.
% Meanwhile,
% if the pressure in junction $i$ is greater than that in junction $j$,
% so that 
% $P_i>P_j$, 
% then $Q_{ij}>0$,
% and $H(P_j-P_i)=0$ and $H(P_i-P_j)=1$.
% The first term in the right hand side of 
% \eqref{eq:nm_particle-conservation_dim_1}
% is thus zero, ensuring that no particles flow from junction $j$ to junction $i$ or deposit in pore $ji$, 
% while the second term
% is non-zero, which implies that particles do flow from junction $i$ to junction $j$.
% That is, pore $ij$ conducts fluid \emph{out} of junction $i$, and deposition does not occur here.
% Note that specification of $H(0)$ is unnecessary 
% terms containing $H(P_i-P_j)$ also contain $G_{ij}$. 
% Therefore, since $P_i=P_j$ if and only if $G_{ij}=0$, terms containing $H(0)$ are equal to zero regardless of the value of $H(0)$. 
% To see this, note that, if $P_i=P_j$ initially, then we set $G_{ij}^{0}=0$, so that pore $ij$ does not exist.
% On the other hand, if $P_i\ne P_j$ initially, 
% then $P_i$ and $P_j$ converge as $G_{ij}$ converges to zero, since the loss of pressure difference in 
% a pore occurs as a result of the loss of its conductance. Despite this, as we will see (from \eqref{eq:nm_conductance-conservation_1_dim}) $G_{ij}$ never reaches zero, and so $P_i\ne P_j$.
% Later, in Chapter \ref{chap:mm_Derivation of the multiscale model}, we will see that this is an important property of the model, since it means that $H$ can expanded as a continuous function away from its discontinuity.

% \comment{[Note that we have assumed that nodes have zero adherence, or else we would see loss terms due to $A_i$, for example.
% The implication is that there is no deposition within them... or are we assuming that nodes are much larger than edges so that porosity is purely a result of node volume... either way, in the end it follows that we can decrease the permeability to zero without altering the porosity. This assumption results in the fact that, later on, we find that deposition does not affect the porosity of the filter (which we define later as ratio of node volume to filter volume), which is one of the things about our model people may find difficult to digest... we should probably write something sensible about this around here?]}.

\subsection{Conservation of pores}
% The system is closed by equations for the node volume and the edge adherence and conductance.  
%  Since there is no deposition within nodes, it follows that the volume of each node does not change, so that
% \begin{align}
%     V_i = V_i^0,
% \end{align}
% where $V_i^0$ are the constant initial node volumes, with mean $\bar{V}$.

% We assume that the adherence of each edge is constant. 
% Moreover, for simplicity, we suppose that these adherences are identical, so that 
% \begin{align}
%     A_{ij}=\bar{A}.
% \end{align}
We assume that pore volume is conserved so that when particles deposit within a pore, the volume of the pore decreases. 
% by the volume that was occupied by those particles in the flow.
%Furthermore, we assume that deposition occurs uniformly, so that particle volume is spread evenly accross the iner wall of the pore. 
As in the particle-conservation equation, \eqref{eq:nm_particle-conservation_dim_1}, 
we suppose that deposition occurs at a rate that is proportional (by a factor of the adherence) to the number of particles that enter 
the pore per unit time.
Assuming that particles deposit uniformly along the length of a pore (see Figure \ref{fig:nm_network_1}), conservation of pore volume implies that 
the decrease in the cross-sectional area of a pore in the infinitesimal time-step $T + \delta T$ is equal to the cross-sectional area occupied by particles that flow into the pore and stick to it in that time step. 
Taking the limit $\delta T \rightarrow 0$, it follows that
% that the cross-sectional area of an edge, which we assume to be constant along its length, decreases proportionally to the flux of particles through it, and to its adherence.
% That is
\begin{align}
    \pp{(\pi R_{ij}^2)}{T} = -S A_{ij}C_{ij}|Q_{ij}|,
    \label{eq:nm_pore-area-conservation_dim_1}
\end{align}
where $R_{ij}$ is the radius of edge $ij$, so that $[R_{ij}]=\text{m}$. Here, $C_{ij}$, which we interpret as the concentration of particles in pore $ij$, is the same as the particle concentration in the junction upstream of the pore, so that
\begin{align}
    C_{ij} = 
    \begin{cases}
        C_i & P_i>P_j,
        \\
        C_j & P_i\le P_j.
    \end{cases}
    \label{eq:nm_pore-conc-definition_dim_1}
\end{align}
The edge-independent constant (for simplicity) of proportionality, $S \ge 0$, with $[S]=\text{m}^{D-1}$, is the particle-size parameter, which represents the cross-sectional area (volume per unit length) that the particles that deposit in the pore occupy. 
The case $S=0$ corresponds to absorption, in which particles deposited in pore $ij$ do not alter its cross-sectional area, 
so that the structure of the network is static. The filter is unaffected by particle retention, and never becomes clogged. We will not discuss this case further.
The case $S<0$ corresponds to pore expansion due to erosion effects, which is also out of scope for the present work.
The case of interest is $S>0$. 
In this scenario, deposition decreases the cross-sectional area of pore $ij$, and the size of $S$ determines the extent of this effect. 

We suppose that flow in pores is laminar, so that it satisfies the Hagen--Poiseuille equation (see \cite{stokes2007theories}, for example), which is given by 
\begin{align}
    P_i-P_j=\frac{8\mu L_{ij} Q_{ij}}{\pi R_{ij}^4},
    \label{eq:nm_hagen-pois_dim_1}
\end{align}
where $L_{ij}$ is the length of pore $ij$, so that $[L_{ij}]=\text{m}$, and $\mu$ is the dynamic viscosity of the fluid, with $[\mu]=\text{Pa}\,\text{s}$.
% We assume that the length of each edge, $L_{ij}$ with $[L_{ij}]=\text{m}$, is constant,
% so that 
% \begin{align}
%     L_{ij}=L^0_{ij},
% \end{align}
% where $L^0_{ij}$ is the initial length of edge $ij$, with mean $\bar{L}$.
Using the definition of the pore flux, \eqref{eq:nm_flux-definition_dim_1}, with the Hagen--Poiseuille equation, \eqref{eq:nm_hagen-pois_dim_1}, 
it follows that the conductance is related to the cross-sectional area via
\begin{align}
    G_{ij}=\frac{\pi R_{ij}^4}{8\mu L_{ij}}.
    \label{eq:nm_conductance-definiton_dim_1}
\end{align}
Using \eqref{eq:nm_conductance-definiton_dim_1} to substitute the cross-sectional area for the conductance in the pore-area conservation equation, \eqref{eq:nm_pore-area-conservation_dim_1}, we arrive at the conductance equation, 
\begin{align}
    \pp{G_{ij}}{T} = - \frac{2S}{(8 \pi \mu )^{\frac{1}{2}} L_{ij}^{\frac{1}{2}}} A_{ij} C_{ij} G_{ij}^{\frac{3}{2}}|P_i-P_j|.
\label{eq:nm_conductance-conservation_1_dim}
\end{align} 
In the next subsection we define boundary conditions for the governing equations, 
\eqref{eq:nm_flux-definition_dim_1}--\eqref{eq:nm_pore-conc-definition_dim_1}.

\subsection{Boundary and initial conditions}
To define the boundary of the filter, we introduce `in' and `out' nodes, which are two nodes that are external to the network in the direction of the flow (see Figure \ref{fig:nm_network_2}).
We refer to these nodes with two further indices: in and out.
We call nodes that are connected to the in and out nodes `inlet' and `outlet' nodes, respectively.
We impose zero adherence on the edges that connect the inlet and outlet nodes to the in and out nodes so that their conductances remain constant. These are not pores, but rather external edges necessary to conduct fluid in and out of the filter.
These represent the inlet and outlet boundaries of the filter.
% That is, $A_{\text{in},i}=0$ and $A_{i,\text{out}}=0$. As a result, the conductances of these edges do not change, using [conductance equation].
We define the external edges to the outermost nodes on the sides of the network that are parallel to the flow as the other boundaries 
of the filter, and suppose that solution variables are periodic across these.
\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.90\textwidth]{./figures/network/network_2.pdf}
        \caption{A diagram showing a network embedded in two dimensions, connected to in and out nodes, which are external. 
        Inlet and outlet nodes 
        are coloured red and blue, respectively.}        
        \label{fig:nm_network_2}
    \end{center}
\end{figure}

We assume that the fluid flow is driven by a known pressure drop from the inlet to the outlet.
We suppose that the pressure at the inlet and outlet are given by $p_{\text{in}}(T)$ and $p_{\text{out}}(T)$ respectively, 
and thus write 
\begin{subequations} \label{eq:bc_pres_dim}
\begin{align}
    P_{\text{in}}(T) = p_{\text{in}}(T), 
    \quad
    P_{\text{out}}(T) = p_{\text{out}}(T),
    \tag{\ref{eq:bc_pres_dim}a,b}
\end{align}
\end{subequations}
for $T\ge0$.
Additionally, the concentration of particles in the feed fluid, $c_{\text{in}}(T)$ say, is known, 
and we write 
\begin{align}
    C_{\text{in}}(T) = c_{\text{in}}(T),
    \label{eq:nm_bc_dim_1}
\end{align}
for $T\ge0$.

Initially, we suppose that there are no particles in the filter, 
so that  
\begin{align}
    C_i(0) = 0, 
    \label{eq:ic_dim}
\end{align}
for $i \in \{1,\hdots,N_{\text{f}}, \text{out} \}$. 
Note here that $i\ne\text{in}$, since $C_i(0) = c_{\text{in}}$ for $i=\text{in}$ (from the boundary condition \eqref{eq:nm_bc_dim_1}).
Furthermore, we suppose that the initial pore-size distribution is given, from which the initial pore conductance values can be deduced.
Thus, we write 
\begin{align}
    G_{ij}(0) = G_{ij}^0,
\end{align}
for $i,j \in \{1,\hdots,N_{\text{f}},\text{in}, \text{out}\}$, where $G_{ij}^0$ is the known initial-conductance of pore $ij$. 

% We assume that there is a unidirectional pressure drop across the network, and a constant supply of particles at the inlet.
% There are no particles in the network initially.
% Using this representation of the network boundary, we write the boundary conditions as
% \begin{subequations} \label{eq:bc_in_dim}
% \begin{align}
%     P_{\text{in}}(T) = p_{\text{in}}, 
%     \quad
%     C_{\text{in}}(T) = c_{\text{in}},
%     \tag{\ref{eq:bc_in_dim}a,b}
% \end{align}
% \end{subequations}
% and 
% \begin{align}
%     P_{\text{out}}(T) = p_{\text{out}} \label{eq:bc_out_dim}
% \end{align}
% for $T\ge0$.
% Here, $p_{\text{in}}$ and $c_{\text{in}}$ are the pressure and concentration at the inlet, and $p_{\text{out}}$ is the pressure at the outlet.
% The initial conditions are 
% \begin{align}
%     C_i(0) = 0, 
%     \label{eq:ic_dim}
% \end{align}
% for $i \in \{1,\hdots,N_{\text{f}}, \text{out} \}$  (note that $C_i(0) = c_{\text{in}}$ from [eq: above] for $i=\text{in}$), and 
% \begin{align}
%     G_{ij}(0) = \hat{G}_{ij}^0
% \end{align}
% for $i,j \in \{1,\hdots,N_{\text{f}},\text{in}, \text{out}\}$.

\subsection{Operational regime}
\label{subsec:nm_Operational regime}
The forms of the functions $L\edge$, $A\edge$, $V_i$, $p_{\text{in}}$, $p_{\text{out}}$, and $c_{\text{in}}$, 
and the values of the parameters $G\edge^0$, 
depend on the industrial application of the filter. 
In this subsection, we specify these in order to demonstrate the solution of the model in the following chapter.

We suppose that deposition in a pore does not affect its length, and so we write
\begin{align}
    L\edge = L\edge^0,
    \label{eq:nm_pore-length_op_dim_1}
\end{align}
where $L\edge^0$ is the constant initial-length of pore $ij$, which is drawn from a known distribution, $\mathcal{L}$ say, with mean $\bar{L}$. 
We write $L\edge^0\sim \mathcal{L}(\bar{L})$ to mean that $L\edge^0$ is drawn from the distribution $\mathcal{L}$ that is specified by the parameter $\bar{L}$, with known probability density function (PDF), $\mathscr{l}$, say.
It remains to specify a particular distribution, $\mathcal{L}$ (and therefore an associated PDF, $\mathscr{l}$), which depends on the structure of the network. 
To simplify, we will consider the case where the network is four-regular, 
which means that each junction is connected to exactly four pores, in a repeating lattice, as in Figure \ref{fig:nm_network_3}. 
In this case, it follows that all pores are the same length, $\bar{L}$, so that
\begin{align}
    \mathscr{l}(L\edge^0) = \updelta(L\edge^0-\bar{L}),
    \label{eq:nm_pore-length-pdf_op_dim_1} 
\end{align}
where $\updelta$ is the PDF for the Dirac-delta distribution.
\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.95\textwidth]{./figures/network/network_3.pdf}
        \caption{A diagram showing a two-dimensional filter represented as a four-regular network.}        
        \label{fig:nm_network_3}
    \end{center}
\end{figure}

For simplicity, 
% we assume that the volume of each node is 
% constant and that all nodes have the same initial volume, so that 
we assume that deposition does not 
occur in junctions
% affect the volumes of the junctions, 
and write
\begin{align}
    V_i = V_i^0,
    \label{eq:nm_node-volume_op_dim_1} 
\end{align}
for some known initial junction-volumes, $V_i^0$ with $[V_i^0]=\text{m}^D$, 
so that
$V_i^0 \sim \mathcal{V}$ with the PDF $\mathscr{v}$.
% \comment{Should I mention here that this is equivalent to assuming that no deposition occurs in junctions. Link this to the assumption that 
% nodes are much larger than edges. Explain that the implication in the multiscale model will be that 
% only pores block, which changes the permeability but does not alter the effective porosity. This is something which people have previously found difficult to digest about our model}
We will focus on the simplest case, in which all junctions have the same volume, $\bar{V}$ say, so that
\begin{align}
    \mathscr{v}(V_i^0) = \updelta(V_i^0-\bar{V}).
    \label{eq:nm_node-volume-pdf_op_dim_1} 
\end{align}

To further simplify our analysis, we assume that the adherence 
% (recall that this is the `stickiness') 
of a pore is 
independent of other variables, such as the pore size, and is therefore constant, 
so that 
\begin{align}
    A\edge = A\edge^0,
    \label{eq:nm_adherence_op_dim_1} 
\end{align}
for $0\le A\edge^0 \le1$ with $[A\edge^0]=1$, so that 
$A\edge^0 \sim \mathcal{A}$ with the PDF $\mathscr{a}$.
Furthermore, we assume that all pores in the filter have the same adherence, and write
\begin{align}
    \mathscr{a}(A\edge^0) = \updelta(A\edge^0-\bar{A}).
    \label{eq:nm_adherence-pdf_op_dim_1} 
\end{align}

We assume that the pressures at the inlet and outlet are known constants, 
and so we write 
\begin{align}
    p_{\text{in}}(T) = p_{\text{in}}^{*}, 
    \quad 
    p_{\text{out}}(T) = p_{\text{out}}^{*}, 
    \label{eq:nm_pressure_op_dim_1} 
\end{align}
where
$[p_{\text{in}}^{*}] = [p_{\text{out}}^{*}] = \text{Pa}\,\text{s}$.

Furthermore, we suppose that the concentration of particles in the feed fluid is also constant 
and known, 
so that 
\begin{align}
    c_{\text{in}}(T) = c_{\text{in}}^{*}, 
    \label{eq:nm_conc_op_dim_1} 
\end{align}
for $[c_{\text{in}}^{*}]=\text{m}^{-D}$.

Finally, we assume that the distribution of pore sizes, $R_{ij}^0$, is known initially. 
Since the initial length distribution is also known (see \eqref{eq:nm_pore-length-pdf_op_dim_1}), it follows (noting, from \eqref{eq:nm_conductance-definiton_dim_1}, that the conductance is completely specified by the pore radius and length) that we are given the initial pore conductance distribution, $\mathcal{G}$, with PDF $\mathscr{g}$, say. 
We consider the cases where, initially, the pore radii, and thus the pore conductances, are either mono-dispersed, with conductance $\bar{G}$, 
or log-normally dispersed, with mean conductance $\bar{G}$. 
That is, either 
\begin{subequations}
\label{eq:nm_cond-pdf_op_dim_1}
\begin{align}
    \mathscr{g}(G\edge^0) = \updelta(G\edge^0-\bar{G}), 
    \quad \text{or} \quad
    \mathscr{g}(G\edge^0)=\frac{1}{\sqrt{2 \pi} \sigma_G G\edge^0} 
    \exp{
    \left(- \frac{ 
    \left( 
    \log \left( \frac{G\edge^0}{\bar{G}} \right) + \frac{\sigma_G^2}{2} 
    \right)^2
    }{
    2 \sigma_G^2}
    \right)}.
    \tag{\ref{eq:nm_cond-pdf_op_dim_1}a,b}    
    \end{align}
\end{subequations}
% In the former case, $G\edge^0 \sim \mathcal{G}(\bar{G})$ where $\mathcal{G}=\text{Dirac}$ is the Dirac-delta distribution.
In the latter case, $G\edge^0 \sim \mathcal{G}(\mu_G,\sigma_G)$ where $\mathcal{G}=\text{Lognormal}$, 
and the mean and standard deviation of the associated normal distribution 
are the dimensionless numbers $\mu_{G}=\log(\bar{G})-\sigma_G^2/2$ and $\sigma_{G}$, so that 
$\log(G\edge^0)\sim \text{Normal}(\mu_G,\sigma_G)$. 
It follows that the expected initial pore conductance is $\mathbb{E}[G\edge^0]=\exp(\mu_G + \sigma_G^2/2) = \bar{G}$, as required. 

\subsection{Summary of the dimensional model}
In summary, using the specific forms of the operational-regime dependent 
functions and parameters, \eqref{eq:nm_pore-length_op_dim_1}--\eqref{eq:nm_cond-pdf_op_dim_1},
the equations for the conservation of fluid, particles, and pores, \eqref{eq:nm_fluid-conservation_dim_1}, \eqref{eq:nm_particle-conservation_dim_1}, and \eqref{eq:nm_conductance-conservation_1_dim}, respectively, read
\begin{align}
    \sum_{j=1}^{N_{\text{f}}} Q_{ij} &= 0,
    \label{eq:nm_fluid-conservation_sum_dim_1}
    \\
    \bar{V}\pp{C_i}{T} &= \sum_{j=1}^{N_{\text{f}}}  \Big(
    (1-\bar{A})Q_{ji} C_j H(P_j-P_i) - Q_{ij} C_i H(P_i-P_j)  \Big),
    \label{eq:nm_particle-conservation_sum_dim_1}
    \\
    \pp{G\edge}{T} &= - \frac{2SA}{(8 \pi \mu \bar{L})^{\frac{1}{2}}} C\edge G\edge^{\frac{3}{2}}|P_i-P_j|,
    \label{eq:nm_conductance-conservation_1_sum_dim}
\end{align} 
respectively,
where the flux down each pore is related to the pressure drop and conductance of the pore via
\begin{align}
    Q_{ij}= G_{ij}\left(P_i-P_j\right).
    \label{eq:nm_flux-definition_sum_dim_1}
\end{align}
% Here, 
% \begin{align}
%     H(P_i-P_j)=
%     \begin{cases}
%         1 & P_i-P_j>0,
%         \\
%         0 & P_j-P_i\le 0,
%     \end{cases}
% \end{align}
% indicates the direction of flow through pores.
The boundary conditions are 
\begin{subequations} \label{eq:nm_bc_pres_sum_dim_1}
\begin{align}
    P_{\text{in}}(T) = p^*_{\text{in}}, 
    \quad
    P_{\text{out}}(T) = p^*_{\text{out}},
    \tag{\ref{eq:nm_bc_pres_sum_dim_1}a,b}
\end{align}
\end{subequations}
and 
\begin{align}
    C_{\text{in}}(T) = c_{\text{in}}^*,
    \label{eq:nm_bc_conc_sum_dim_1}
\end{align}
for $T\ge0$, 
and the initial conditions are  
\begin{align}
    C_i(0) = 0, 
    \label{eq:nm_ic_conc_sum_dim_1}
\end{align}
for $i \in \{1,\hdots,N_{\text{f}}, \text{out} \}$, 
and 
\begin{align}
    G_{ij}(0) = G_{ij}^0,
    \label{eq:nm_ic_cond_sum_dim_1}  
\end{align}
for $i,j \in \{1,\hdots,N_{\text{f}},\text{in}, \text{out}\}$, 
which are drawn from the distribution $\mathcal{G}$ with a PDF of either
\begin{subequations}
\label{eq:nm_cond-pdf_sum_dim_1}
\begin{align}
    \mathscr{g}(G\edge^0) = \updelta(G\edge^0-\bar{G}), 
    \quad \text{or} \quad
    \mathscr{g}(G\edge^0)=\frac{1}{\sqrt{2 \pi} \sigma_G G\edge^0} 
    \exp{
    \left(- \frac{ 
    \left( 
    \log \left( \frac{G\edge^0}{\bar{G}} \right) + \frac{\sigma_G^2}{2} 
    \right)^2
    }{
    2 \sigma_G^2}
    \right)}.
    \tag{\ref{eq:nm_cond-pdf_sum_dim_1}a,b}    
    \end{align}
\end{subequations}

The equations \eqref{eq:nm_fluid-conservation_sum_dim_1}--\eqref{eq:nm_cond-pdf_sum_dim_1} 
form a closed system for the solution variables $Q_{ij}$, $C_{i}$, $P_{i}$, and $G_{ij}$,
subject to the specification of the dimensional parameters $\bar{L}$, $\bar{V}$, $\bar{A}$, 
$S$, $\mu$, $p_{\text{in}}^*$, $p_{\text{out}}^*$, and $c_{\text{in}}^*$, 
along with any other parameters that are needed to specify the probability density functions 
for the initial conductance distribution from which $G\edge^0$ are drawn, $\mathcal{G}$, such as $\sigma_G$ in \eqref{eq:nm_cond-pdf_sum_dim_1}.
Note that our network model is independent of node positions. The structure (in terms of connectivity) of the network enters the system through the pore conductances, $G\edge$.
In the next section, we will nondimensionalise the dimensional network model \eqref{eq:nm_fluid-conservation_sum_dim_1}--\eqref{eq:nm_cond-pdf_sum_dim_1}. 

\section{Dimensionless model}
\label{sec:nm_Dimensionless model}
In this section, we scale the variables of the dimensional network model, \eqref{eq:nm_fluid-conservation_sum_dim_1}--\eqref{eq:nm_cond-pdf_sum_dim_1}, to arrive at a dimensionless model to be analysed and solved in the subsequent chapter.
% \correct{The aim of our nondimensionalisation is to arrive at a system for which it is simple to compare the solutions of systems defined on networks that contain different numbers of nodes}{}.

\subsection{Nondimensionalisation}
We begin by
introducing two dimensionless parameters that make it easy to compare networks that contain different numbers of nodes (and to make our multiscale analysis in Part \ref{part:A multiscale model for filtration} simple). We suppose
 that the network consists of cells, as illustrated in Figure \ref{fig:nm_network_4}.
We define $N^m$ to be the number of cells, which are length $L^m=l^m/N^m$, in the $m$ direction, and define $N$ to be the mean number of nodes within each cell. 
We use these to define the two scaling parameters, 
\begin{align}
    \delta = \frac{1}{\sqrt[D]{N}}, 
    \quad 
    \varepsilon = \frac{1}{N^1}.
    \label{eq:nm_definition-of-delt-and-epsi}
\end{align}
Interpreting these, we see that $\varepsilon^{-1}$ is the number of cells in a row in the flow direction (direction 1) since $\varepsilon$ is the ratio of the cell length to the network length, $\varepsilon=L^1/l^1$ (see the cells boxed in blue in Figure \ref{fig:nm_network_4}). 
Meanwhile, $\delta^{-1}$ is the average number of pores in one cell in a row of nodes in the flow direction (see the pores highlighted and circled in green in Figure \ref{fig:nm_network_4}). 
It follows that $(\delta \varepsilon)^{-1}$ is the average number of pores in a row of nodes that leads from the inlet to the outlet, and is therefore the average number of pores that fluid flows through in order to traverse the filter (see the pores highlighted and circled in orange in Figure \ref{fig:nm_network_4}).
Note that we expect $\delta \varepsilon \ll 1$ to represent a physically relevant filtration regime, since this corresponds to the case where many pores constitute the membrane.
\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.95\textwidth]{./figures/network/network_4.pdf}
        \caption{A diagram showing a two-dimensional filter represented as a four-regular network. 
        Cells are shown with dashed red lines, and one cell is boxed in red.
        A row of cells is boxed in blue.
        A row of pores within a cell are highlighted and circled in green. 
        A row of pores within the network are highlighted and circled in orange.
        In this example:
        $N^1=10$, since each row of cells contains $10$ cells; 
        $N^2=5$, since each column of cells contains $5$ cells;
        $N=4$, since each cell contains $4$ nodes;
        $\delta=1/2$, since $D=2$ and $N=4$; 
        $\varepsilon=1/10$, since $N^1=10$; 
        $(\delta\varepsilon)^{-1}=20$ since each row of pores contains $20$ pores, 
        so that fluid traverses $20$ pores as it traverses the filter.}        
        \label{fig:nm_network_4}
    \end{center}
\end{figure}


Given these scaling parameters, we define dimensionless variables (denoted by tilde) by
\begin{align}
    & 
    L_{ij} = \bar{L} \tilde{L}_{ij},
    \quad
    V_i=\bar{V}\tilde{V}_i,
    \quad
    P_i=(p_{\text{in}}-p_{\text{out}})\tilde{P}_i + p_{\text{out}},
    \quad
    C_i=c_{\text{in}}\tilde{C}_i,
    \label{eq:nm_scaling_1}
    \\
    &
    Q_{ij}=\bar{G}(p_{\text{in}}-p_{\text{out}})\tilde{Q}_{ij}
    \quad
    G_{ij} = \frac{\bar{G}}{{\varepsilon}}\tilde{G}_{ij}, 
    \quad
    T=\frac{\bar{V}}{\delta^2 \varepsilon \bar{G}(p_{\text{in}}-p_{\text{out}})}\tilde{T}.
    \label{eq:nm_scaling_2}
\end{align}
We see that the scalings in \eqref{eq:nm_scaling_1} imply unit pore-lengths and node-volumes, 
\begin{align}
    \tilde{L}\edge^0 = 1,
    \quad
    \tilde{V}_i^0 = 1, 
\end{align}
%     \tilde{G}\edge^0 = \frac{\varepsilon G\edge^0}{\bar{G}}.
and unit concentration in the feed fluid. 
The pressure is scaled for unit pressure drop across the entire network.

In \eqref{eq:nm_scaling_2} we have scaled the remaining variables so that networks with different numbers of nodes in each cell, $N$, and different numbers of cells in the network, $N^m$, can be compared simply.
To see this, we first note that fluid traverses $\mathcal{O}((\delta \varepsilon)^{-1})$ pores in order to traverse the network. 
% since there are $N_{\text{f}}= (N^{1/D}) N^1  = (\delta \varepsilon)^{-1}$ nodes in an average path.
Therefore, the pressure drop in any pore $ij$, $(P_i-P_j)$, is $\mathcal{O}((p_{\text{in}}-p_{\text{out}})\delta \varepsilon))$.
However, in \eqref{eq:nm_scaling_1}, the pore pressure drop is scaled with the network pressure drop, $p_{\text{in}}-p_{\text{out}}$. 
Thus, scaling the pore conductance with $\varepsilon$, as in \eqref{eq:nm_scaling_2},
% , results in a pore flux that is $\mathcal{O}(\delta)$.
% , regardless of the number of nodes in the network, $N_{\text{f}}$.
it follows that the flux through each pore is $\mathcal{O}(\delta \bar{G}(p_{\text{in}}-p_{\text{out}}))$, using \eqref{eq:nm_flux-definition_sum_dim_1}.
As a consequence, the time it takes for fluid to traverse a single pore 
is $\mathcal{O}(T_{\text{p}})$, 
where 
\begin{align}
    T_{\text{p}} = \frac{\bar{V}}{\delta \bar{G} (p_{\text{in}}-p_{\text{out}})}.
\end{align}
Since the fluid traverses $\mathcal{O}((\delta \varepsilon)^{-1})$ pores to traverse the filter (see Figure \ref{fig:nm_network_4}), 
it follows that the time it takes for fluid to travel from the inlet to the outlet is $\mathcal{O}(T_{\text{f}})$, 
where
\begin{align}
    T_{\text{f}} = \frac{\bar{V}}{\delta^2 \varepsilon \bar{G} (p_{\text{in}}-p_{\text{out}})}.
\end{align}
Therefore, in our dimensionless variables, which are given by \eqref{eq:nm_scaling_1} and \eqref{eq:nm_scaling_2}, we expect to have to wait $\mathcal{O}(1)$ time units for fluid to traverse the filter, regardless of the number of nodes it contains (both in terms of the number of cells, 
and the number of nodes within each cell). We summarise our dimensionless network model in the following subsection.

% The adherence is scaled with $\delta$ to impose that the order of $\bar{A}$ is equal to the order of particle removal over the length of any network. 
% That is, under this scaling, $\mathcal{O}(\bar{A})=1$ will cause $\mathcal{O}(1)$ removal of particles by the time they reach the outlet, regardless of the number of edges to be crossed (since $\mathcal{O}(\delta)$ removal occurs on each edge).

\subsection{Summary of the dimensionless model}
In summary, in dimensionless variables, with tildes 
dropped for ease of notation, the equations for the conservation of fluid, particles, and pores, are
\begin{align}
    \sum_{j=1}^{N_{\text{f}}} Q_{ij} &= 0,
    \label{eq:nm_fluid-conservation_sum_nodim_1}
    \\
    \pp{C_i}{T} &= \frac{1}{\delta^2 \varepsilon} \sum_{j=1}^{N_{\text{f}}}  \Big(
    (1-\alpha \delta \varepsilon)Q_{ji} C_j H(P_j-P_i) - Q_{ij} C_i H(P_i-P_j)  \Big),
    \label{eq:nm_particle-conservation_sum_nodim_1} 
    \\
    \pp{G\edge}{T} &= - \frac{ \alpha \beta}{\delta \varepsilon} C\edge G\edge^{\frac{3}{2}} |P_i-P_j|,
    %note that there would be an invisible L^1/2 on the numerator if it wasnt 1.
\label{eq:nm_conductance-conservation_sum_nodim_1}
\end{align} 
where the flux down each pore is related to the pressure drop and conductance of the pore via
\begin{align}
    Q_{ij}= \frac{G_{ij}}{\varepsilon}\left(P_i-P_j\right).
    \label{eq:nm_flux-definition_sum_nodim_1}
\end{align}
% Here, 
% \begin{align}
%     H(P_i-P_j)=
%     \begin{cases}
%         1 & P_i-P_j>0,
%         \\
%         0 & P_j-P_i\le 0,
%     \end{cases}
% \end{align}
% indicates the direction of flow through pores.
The boundary conditions are 
\begin{subequations} \label{eq:nm_bc_pres_sum_nodim_1}
\begin{align}
    P_{\text{in}}(T) = 1, 
    \quad
    P_{\text{out}}(T) = 0,
    \tag{\ref{eq:nm_bc_pres_sum_nodim_1}a,b}
\end{align}
\end{subequations}
and 
\begin{align}
    C_{\text{in}}(T) = 1,
    \label{eq:nm_ic_pres_sum_nodim_1}
\end{align}
for $T\ge0$, 
and the initial conditions are  
\begin{align}
    C_i(0) = 0, 
    \label{eq:nm_ic_conc_sum_nodim_1}
\end{align}
for $i \in \{1,\hdots,N_{\text{f}}, \text{out} \}$, 
and 
\begin{align}
    G_{ij}(0) = G_{ij}^0
    \label{eq:nm_ic_cond_sum_nodim_1}
\end{align}
for $i,j \in \{1,\hdots,N_{\text{f}},\text{in}, \text{out}\}$.

The dimensionless parameters are 
\begin{align}
    \alpha = \frac{\bar{A}}{\delta \varepsilon},
    \label{eq:nm_definition-dimles-parameters_sum_nodim_1}
    \quad
    \beta =  \frac{2S \bar{V} c_{\text{in}} \varepsilon^{\frac{1}{2}}}{(8\pi \mu \bar{L} \bar{G})^{\frac{1}{2}}  }, 
    \quad
    \delta = \frac{1}{\sqrt[D]{N}},
    \quad
    \varepsilon = \frac{1}{N^1}
\end{align}
We see that $\alpha$ is the effective adherence, 
that is, 
the mean adherence scaled by the number of pores in an average path from the inlet to the outlet. 
Note that
the factor $\delta\varepsilon$ in $\alpha$ ensures that when $\alpha=\mathcal{O}(1)$ there is $\mathcal{O}(1)$ removal 
of particles by the time that they reach the outlet (as we will show in Section \ref{sec:nm_Non-random network: Mono-dispersed pores}).
Also, 
$\beta$ is the effective particle size, that is, 
a ratio of the size of the average number of particles entering a pore to the average cross-sectional area of a pore. Note that,
the factor $\varepsilon^{1/2}$ in $\beta$ ensures that when $\alpha=\mathcal{O}(1)$ and $\beta=\mathcal{O}(1)$
there is an $\mathcal{O}(1)$ reduction in the conductance of pores near the inlet by the time that fluid reaches the outlet (again, we will demonstrate this point in Section \ref{sec:nm_Non-random network: Mono-dispersed pores}).
Note that we expect $\alpha=\mathcal{O}(1)$ and
$\beta\ll1$ to represent a physically relevant filtration regime, where the aim is to remove a large proportion of the incoming particles without greatly decreasing the effective conductance of the filter.
The dimensionless analogies of the example PDFs of the initial conductance distribution, $\mathcal{G}$, are
\begin{subequations}
\label{eq:nm_cond-pdf_sum_nodim_1}
\begin{align}
    \mathscr{g}(G\edge^0) = \updelta(G\edge^0-1), 
    \quad \text{or} \quad
    \mathscr{g}(G\edge^0)=\frac{1}{\sqrt{2 \pi} \sigma_G G\edge^0} 
    \exp{
    \left(- \frac{ 
    \left( 
    \log (G\edge^0) + \frac{\sigma_G^2}{2} 
    \right)^2
    }{
    2 \sigma_G^2}
    \right)}.
    \tag{\ref{eq:nm_cond-pdf_sum_nodim_1}a,b}    
    \end{align}
\end{subequations}
Note that, in the dimensionless setting, $\mu_G=-\sigma_G^2/2$ so 
that the expectation of the initial conductances is 
$\mathbb{E}[G\edge^0]=\exp(\mu_G+\sigma_G^2/2)=1$, as we expected, since, in \eqref{eq:nm_scaling_2}, we scaled the conductance 
distribution for unit mean.

The equations \eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_ic_cond_sum_nodim_1} form a closed system for the dimensionless variables $Q_{ij}$, $C_{i}$, $P_{i}$, and $G_{ij}$,
subject to the specification of the dimensionless parameters 
$\alpha$, $\beta$, $\delta$, and $\varepsilon$, 
which are defined in terms of the dimensional parameters in \eqref{eq:nm_definition-dimles-parameters_sum_nodim_1}, 
and any further dimensionless parameters, 
for example $\sigma_G$ in \eqref{eq:nm_cond-pdf_sum_nodim_1}, that 
are needed to specify the initial conductance distribution, $\mathcal{G}$.


% \begin{center}
% \begin{tabular}{|c|c|c|c|} 
%  \hline
%  Parameter & Description & Col2  \\ [0.5ex] 
%  \hline\hline
%  $\alpha$ & 
%  Effective adherence: The `stickiness' between 
%  \newline
%  particles in the flow
%  and pores in the filter, scaled by 
%  \newline
%  the number of pores in an average path
%  \newline
%  from the inlet to the outlet. & 87837 \\ 
%  \hline
%  $\beta$ &  & 78  \\
%  \hline
%  $\delta$ & 545 & 778 \\
%  \hline
%  $\varepsilon$ & 545 & 18744  \\
%  \hline
%  $\sigma_{G}$ & 88 & 788 \\ [1ex] 
%  \hline
% \end{tabular}
% \end{center}

In this chapter, we have derived a novel model for filtration, \eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_cond-pdf_sum_nodim_1}, 
within which the filter is treated as a dynamically structured network of pores and junctions.
The feed is modelled as a continuum of particles, that deposit smoothly on pore walls and so cause changes to the connectivity of the network and thus to the fluid-conductivity of the filter.
% In the next chapter we will solve this model in the cases where the pore sizes are first mono-dispersed and later poly-dispersed.
In the next chapter, we will solve this model in the cases where the pore sizes are mono-dispersed.

% In dimensionless variables, with tildes dropped for ease of notation, 
% the system becomes
% \begin{align}
%     Q_{ij} &= \frac{G_{ij}}{\varepsilon}(P_i-P_j),
%     \\
%     \sum_{j=1}^{N_{\text{f}}} Q_{ij} &= 0,
%     \\
%     V_i\pp{C_i}{T} &= \frac{1}{\delta^2 \varepsilon} \sum_{j=1}^{N_{\text{f}}}  \Big(
%     (1-A_{ji})Q_{ji} C_j H(P_j-P_i) - Q_{ij} C_i H(P_i-P_j)  \Big),
%     \\
%     \pp{G_{ij}}{T} &= - \frac{\beta }{\delta^2 L_{ij}^{\frac{1}{2}}} A_{ij} C_{ij} G_{ij}^{\frac{3}{2}}|P_i-P_j|.
%     \\
%     V_{i} &= \gamma_i,
%     \\
%     A_{ij} &= \alpha,
%     \\
%     L_{ij} &= \lambda_{ij}.
% \end{align}
% where 
% \begin{align}
%     H(P_i-P_j)=
%     \begin{cases}
%         1 & P_i-P_j>0,
%         \\
%         0 & P_j-P_i\le 0,
%     \end{cases}
% \end{align}
% and 
% \begin{align}
%     C_{ij} = 
%     \begin{cases}
%         C_i & P_i>P_j,
%         \\
%         C_j & P_i\le P_j.
%     \end{cases}
% \end{align}
% The boundary conditions become 
% \begin{subequations} \label{eq:bc_in_nondim}
% \begin{align}
%     P_{\text{in}}(T) = 1, 
%     \quad
%     C_{\text{in}}(T) = 1,
%     \tag{\ref{eq:bc_in_nondim}a,b}
% \end{align}
% \end{subequations}
% and 
% \begin{align}
%     P_{\text{out}}(T) = 0
%     \label{eq:bc_out_nondim}
% \end{align}
% for $T\ge0$,
% and the initial conditions are 
% \begin{align}
%     C_i(0) = 0, 
%     \label{eq:ic_nondim}
% \end{align}
% for $i \in \{1,\hdots,N_{\text{f}}, \text{out} \}$  (note that $C_i(0) = 1$ from [eq: above] for $i=\text{in}$), and 
% \begin{align}
%     G_{ij}(0) = G_{ij}^0
% \end{align}
% for $i,j \in \{1,\hdots,N_{\text{f}},\text{in}, \text{out}\}$.
% Note that $G_{ij}^0$ are the $\mathcal{O}(1)$ dimensionless initial-conductances, which are $\varepsilon \bar{G}^{-1}$ times as small as the dimensional ones from [eq: dimensional initial conductance].
% The system [eq:dimensionless system] defines our dimensionless network model for the seven variables $Q_{ij}$, $G_{ij}$, $P_{i}$, $V_{i}$, $C_{i}$, $A_{ij}$, and $L_{ij}$.
% The dimensionless parameters of the model are related to the dimensional parameters via 
% \begin{align}
%     \hat{\alpha} = \bar{A},
%     \quad
%     \hat{\beta} =  \frac{2S \bar{V} c_{\text{in}}}{(8\pi \mu \bar{L} \bar{G})^{\frac{1}{2}}  }, 
%     \quad
%     \gamma_i = \frac{V_i^0}{\bar{V}},
%     \quad
%     \delta = \frac{1}{\sqrt{N}},
%     \quad
%     \varepsilon = \frac{1}{N^1}
%     \quad
%     \lambda_{ij} = \frac{L_{ij}^0}{\bar{L}},
% \end{align}
% and 
% \begin{align}
%     G_{ij}^0 = \varepsilon \frac{\hat{G}_{ij}^0}{\bar{G}}.
% \end{align}


\chapter{Solutions of the network model}
\label{chap:nm_Solutions of the network model}
In this chapter, 
we will describe our method of solution, 
before defining some quantities of interest 
that depend on the solution of the model.
We will then solve 
the dimensionless network model, \eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_cond-pdf_sum_nodim_1}, 
% We will begin with the case where the initial pore conductances are given by
in the case where the initial pore conductances are given by
(\ref{eq:nm_cond-pdf_sum_nodim_1}a), so that edges in the network all have the same conductance initially. 
This corresponds to the scenario where pore sizes are initially mono-dispersed. 
% We will then consider the case where initial pore conductances are given by (\ref{eq:nm_cond-pdf_sum_nodim_1}b), 
% so that edges in the network have conductances drawn from a log-normal distribution.
% This models the scenario where pore sizes are distributed log-normally initially.

% \section{Solution methodology}
% \subsection{Model reduction}
% In this chapter, we are interested in understanding the behavior of the system, and roles that the parameters play. 
% We therefore seek  to make it as simple as possible.
% To illustrate the behaviour of the solution of the network model, we 
% consider two simplifications. 
% Firstly, we impose that $\gamma_i=1$. 
% In dimensional terms, this is equivalent to assuming that all nodes have the same volume (see [eq: defn of gamma]).
% We do this because we are most interested in 
% how the permeability of the network changes, which 
% we expect to depend on the edge quantities more than the node ones.
% Secondly, we impose that $\lambda_{ij}=1$, which is equivalent to assuming that all edges are the same length (see [eq: defn of lambda]).
% This is not restrictive. 
% We have already decideed that lengths do not change in time. 
% Length only appears in model with conductance. 
% We need to give initial conditions for length and conductance.
% So we may as well set $\gamma_i=1$ and randomise conductance with the distrbution we want instead.
% Since node volumes and edge lengths are scaled on their mean values, these drop out of the system.

% We are interested in comparing network with different numbers of nodes. 
% Therefore, we 
% choose $\alpha$ so that 
% \begin{align}
%     \hat{\alpha} = \delta \epsilon \alpha.
% \end{align}
% Now when $\alpha$ is $\mathcal{O}(1)$, we expect $\mathcal{O}(1)$ removal of particles 
% over the network, since recall that particles traverse 
% $\mathcal{O}((\delta \varepsilon)^{-1})$ edges.
% We are interested in comparing the same amount of removal per unit time, but the conductance equation will be 
% \begin{align}
%     \pp{G_{ij}}{T} &= - \frac{\delta \varepsilon \alpha \hat{\beta} }{\delta^2}  C_{ij} G_{ij}^{\frac{3}{2}}|P_i-P_j|.
% \end{align}
% Therefore, choose $\beta$ so that
% \begin{align}
%     \hat{\beta} = \varepsilon^{-2} \beta.
% \end{align}


% In this case, therefore, we re-write the system
% \begin{align}
%     Q_{ij} &= \frac{G_{ij}}{\varepsilon}(P_i-P_j),
%     \\
%     \sum_{j=1}^{N_{\text{f}}} Q_{ij} &= 0,
%     \\
%     \pp{C_i}{T} &= \frac{1}{\delta^2 \varepsilon} \sum_{j=1}^{N_{\text{f}}}  \Big(
%     (1-\delta \varepsilon \alpha) Q_{ji} C_j H(P_j-P_i) - Q_{ij} C_i H(P_i-P_j)  \Big),
%     \\
%     \pp{G_{ij}}{T} &= - \frac{ \delta \alpha \beta }{\delta^2 \varepsilon}  C_{ij} G_{ij}^{\frac{3}{2}}|P_i-P_j|.
% \end{align}
% The scaling for time is done becuase we understand what the plots mean when we plot on the filter timescale. 
% However, we can see from these equations that there is another natural timesclae that balances the advection--reaction equation.
% Notice that now, when we want to shift to the timescale that balances the advection--deposition equation, we can simply multiple by $\delta^2 \varepsilon$. 
% Call this timescale $T_{\text{b}}$, since it balances that equation but is otherwise fairly meaningless. Mention how it relates to the edge and network timescales. It is a very short timescale, since it takes $\delta^{-1}$ of them to traverse an edge.

% We now have a system for four unknowns, $Q_{ij}$, 
% $G_{ij}$, $P_i$, and $C_i$, that is parametrised by four parameters $\alpha, \beta, \delta, \varepsilon$, and the initial conductances $G_{ij}^0$ (rather than the [some function of $N_{\text{f}}$ that the system was before]).
% Therefore it will be a lot easier to understand the effects of the three important parameters using the reduced model.
% The boundary and initial conditions, 
% and the definitions of the dimensionless parameters are unchanged, and are given by the definitions [equations for dimensionless boundary and iniital conditions and parameters].+

% In the following section we will choose a set of initial conductance, and explore the effect of the main three parameters.

% We will focus on four-regular networks, becausae they are simple and we just want to understand what the network mdoel looks like. We have already established that chaing length is jus tlike chanign conductance. 
% But note that we are choosing asimple connectivity. We will assess later in the thesis whether that matters or not.

\section{Solution methodology}
\label{sec:nm_Solution methodology}
The network model, \eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_cond-pdf_sum_nodim_1}, 
is a system of linear ordinary algebraic--differential equations.
Several computational tools exist for such problems, such as 
\texttt{CasADi}, \cite{Andersson2018},
in \texttt{Python}.
To gain an intuitive understanding of the system, we demonstrate a method to implement it explicitly from first principles (we have verified our results using \texttt{CasADi}).
Later, in Section \ref{sec:mm_Solution methodology}, we will use this to explain why the multiscale model that we discuss in Part \ref{part:A multiscale model for filtration} is computationally advantageous.
Given initial conductances, 
we eliminate the pore fluxes, $Q_{ij}$, by substituting the relationship between the pore flux and the 
pore conductance and junction pressure, \eqref{eq:nm_flux-definition_sum_nodim_1}, 
into 
the fluid-conservation equation, \eqref{eq:nm_fluid-conservation_sum_nodim_1}, so that 
\begin{align}
    \sum_{j=1}^{N_{\text{f}}} \Big(G_{ij}-\updelta_{ij} \sum_{k=1}^{N_{\text{f}}} G_{ik} \Big)P_j = 0,
    \label{eq:nm_fluid-cons_linear-form}
\end{align}
where $\updelta_{ij}$ are elements of the Kronecker-delta matrix, $\bm{\updelta}$.
At $T=0$, we solve \eqref{eq:nm_fluid-cons_linear-form} (given initial conductances) to find the initial pressure, $P_i(0)$, at each junction.
% This is equivalent to finding the initial pressure distribution by running clean water through the filter.
At $T_k>0$ (where $T_k$ is the $k^{\text{th}}$ discrete time-point), 
we use a first-order explicit scheme to discretise the particle-conservation equation, \eqref{eq:nm_particle-conservation_sum_nodim_1}, and
the pore-conductance conservation equation, \eqref{eq:nm_conductance-conservation_sum_nodim_1}. 
That is, we update the particle concentrations, $C_i(T_k)$, and pore conductances, $G_{ij}(T_k)$, by using the particle concentrations, $C_i(T_{k-1})$, pore conductances, $G_{ij}(T_{k-1})$, and junction pressures, $P(T_{k-1})$, at the previous time-point, $T_{k-1}$.
We then update the node pressures, $P(T_{k})$ by solving \eqref{eq:nm_fluid-cons_linear-form} using the pore conductances at the current time step, 
$G_{ij}(T_k)$.

At each time point, then, we solve a linear system that depends on the number of nodes in the network, $N_{\text{f}}$, 
via the number of nodes in each cell, $N$, and the number of cells in the network, $N^1$ and $N^2$, and the connectivity of the network, since this dictates the number of edges. 
For example, a two-dimensional four-regular network contains $\mathcal{O}(N^1N^2(N)^2)$ edges, since there 
are $(2(N)^2-2N)$ in each cell, and there are $N^1N^2$ cells in the network. 
It follows that, in the most expensive case (where matrix sparsity due to local connectivity is not exploited), we 
solve $N_{\text{e}}=\mathcal{O}(N^1N^2(N)^2)$ equations in order to find the pressure and concentration in all nodes, and the conductance of all pores.
We note that this is prohibitively large in cases where there are many nodes in the cell, so that $N\gg1$, and many cells in the network, so that $N^1N^2\gg1$ ($N=\mathcal{O}(10^2)$ and $N^1,N^2=\mathcal{O}(10)$, say, so that $N_{\text{e}}=\mathcal{O}(10^6)$), since solution of a linear system of size $N_{\text{e}}$ requires $\mathcal{O}(N_{\text{e}}^2)$ operations.


\section{Non-random networks: Mono-dispersed pores}
\label{sec:nm_Non-random network: Mono-dispersed pores}
%\section{Non-random networks}
In this section, we define some properties that depend on the solution of the model that we will use to analyse it.
We then consider the solution of the network model, \eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_cond-pdf_sum_nodim_1}, 
in the scenario where pores are mono-dispersed initially,
in which case the initial pore conductances are given by (\ref{eq:nm_cond-pdf_sum_nodim_1}a). 
% so 
% that the initial structure of the network need not be sampled.
We first discuss the simplest case, in which the adherence is zero, $\alpha=0$, so that no deposition can occur.
We then discuss a non-zero adherence case, $\alpha=0.2$ for example, which demonstrates a particular case in which deposition can occur.
After this, 
we explore the effect that changing the parameters $\alpha$, $\beta$, $\delta$, and $\varepsilon$, 
has on the solutions of \eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_cond-pdf_sum_nodim_1}.

\subsection{Macroscale properties}
In filtration applications, industrialists are most interested in certain macroscopic properties of the filter, and how these vary with filter depth and time.
In the present case, where the network is mono-dispersed and four-regular, 
we do not expect variables to vary in the $x^2$ direction, which is transverse to the direction of flow.
As such, we define macroscale quantities to represent the particle concentration, pore conductance, junction pressure,
and fluid flow rate,
which are depth and time dependent, and which are averaged over the transverse direction.
% in the simplest way possible.
Since the present network is two-dimensional and four-regular, this calculation is straightforward, because junctions form `columns' (in the $x^2$ direction) at each depth
(see Figure \ref{fig:nm_network_4}), 
and we simply average the solution variables down these columns. 
That is, calling $x^1=X_j$ the depth of the junctions that are in column $j$ (where $j$ varies between $1$ and $(\delta\varepsilon)^{-1}$),
we define 
\begin{align}
    \bar{C}(X_j,t) &= \frac{\delta}{N^2} \sum_{j=1}^{N^2/\delta} C_j(t),
    \label{eq:nm_definition-of-av-conc}
    \\
    \bar{P}(X_j,t) &= \frac{\delta}{N^2} \sum_{j=1}^{N^2/\delta} P_j(t).
    \label{eq:nm_definition-of-av-pres}
\end{align}
We call these the average particle concentrations and junction pressures, 
which are averaged over the flow-transverse direction, 
since $N^2/\delta$ is the number of nodes in each junction column.
Similarly, defining $x^1=X_{ij}$ to be the depth of the centre of each flow-parallel pore, $ij$, and averaging down each column of pores, 
we obtain the average pore conductances,
\begin{align}
    \bar{G}(X_{ij},t) = \frac{\delta}{N^2} \sum_{j=1}^{N^2/\delta} G_{ij}(t).
    \label{eq:nm_definition-of-av-cond}
\end{align}
This is the conductance in pores that are parallel to the flow, averaged over
the flow-transverse direction.
Lastly, we define the average flow rate through the filter. 
Since fluid mass is conserved in each node, 
this is equal to the rate of flow of fluid into the filter. 
It follows that 
\begin{align}
    U(t) = \frac{\delta}{N^2} \sum_{j=1}^{N^2/\delta} \frac{1}{\delta \varepsilon} G_{\text{in} j} (P_{\text{in}} - P_j),
    \label{eq:nm_definition-of-av-velo}
\end{align}
where we have summed the volumetric flow through all the edges that lead into the filter, and 
divided by $N^2/\delta$, which is the total number of these.
In the following section, we will solve the network model numerically,
and find: 
the average particle concentration, 
$\bar{C}$, given by \eqref{eq:nm_definition-of-av-conc};
the average junction pressure, 
$\bar{P}$, given by \eqref{eq:nm_definition-of-av-pres};
the average pore conductance, 
$\bar{G}$, given by \eqref{eq:nm_definition-of-av-cond};
the average flow rate, $U$, given by \eqref{eq:nm_definition-of-av-velo}. 
We will use these to analyse the effective behaviour of the filter, 
and how this depends upon the dimensionless parameters, 
$\alpha$, $\beta$, $\delta$, and $\varepsilon$.
In what follows, we explore various parameter regimes (rather than the physically relevant filtration regime already discussed: $\alpha=\mathcal{O}(1)$, $\beta \ll1$, $\delta\varepsilon\ll1$) to demonstrate the key properties of the network model.

\subsection{No deposition}
To gain an intuitive understanding of the model, we first consider the simple case
of zero adherence, $\alpha=0$, so that deposition cannot occur.
In Section \ref{sec:change-delt}, we will use this case to analyse the effects of varying $\delta$ and $\varepsilon$.
% In this case, there is no mutual attraction between particles and pores,
% and no deposition will occur. 
% , so particles move through the filter unimpeded 
% by pores.
First, in Figure \ref{fig:nm_alpha-zero}, we display the behaviour of the four average macroscale quantities, 
$\bar{C}$, $\bar{P}$, $\bar{G}$, and $U$, which are defined in \eqref{eq:nm_definition-of-av-conc}--\eqref{eq:nm_definition-of-av-velo}, 
in the no-deposition scenario. 
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-zero/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_alph-zero/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-zero/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_alph-zero/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-zero/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_alph-zero/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-zero/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_alph-zero/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, pore conductance, $\bar{G}$, flow rate, $U$, and junction pressure, $\bar{P}$, vary with depth, $x$ and time, $T$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, 
    $\alpha=0$, $\beta=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    In plots where depth is shown on the independent-variable axis, 
    the values of $T$ are 
    $0$ (blue), 
    $0.25$ (orange), 
    $0.5$ (green), 
    $0.75$ (red), 
    $1$ (purple), 
    and the arrow shows the direction of increasing time.
    \subref{fig:nm_alph-zero/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at various times, $T$.
    \subref{fig:nm_alph-zero/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at various times, $T$.
    Since the solution is the same for all times, 
    only the time $T=1$ is visible.
    \subref{fig:nm_alph-zero/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    \subref{fig:nm_alph-zero/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$, at various times, $T$.
    Since the solution is the same for all times, 
    only the time $T=1$ is visible.
    }
    \label{fig:nm_alpha-zero}
\end{figure}

Observing Figure \ref{fig:nm_alph-zero/conc_2__v__posi_nodes_1.pdf}, 
the first thing we notice is that particles, unimpeded by pores, 
travel in the wave of fluid that traverses the filter, which reaches the outlet in $\mathcal{O}(1)$ time. 
Far upstream of the wave front, at any particular time, the particle concentration matches the concentration 
in the fluid entering the filter at the inlet, since the junctions and pores that make up the void space in the filter here 
are completely filled with particle-laden fluid.
On the other hand, fluid has not had time to reach the area far downstream of the wave front, 
and so the particle concentration is zero here.
Between these two areas, we observe a transition region, 
within which the particle concentration decreases smoothly from the inlet value to zero.
% This transition is gradual because nodes in this region fill with fluid (and therefore particles) gradually.

Since particles cannot stick to pore walls, deposition never occurs, 
and so the cross-sectional area of the pores, and thus their conductance,  
is unchanged as particles traverse them. 
As a result, in Figure \ref{fig:nm_alph-zero/cond_2__v__posi_edges_1.pdf}, 
we observe unit average-conductance within the entire filter, for all time, 
so that the permeability of the filter is unaltered.
It follows that the flow rate of the fluid is unchanged, and thus, in Figure \ref{fig:nm_alph-zero/flux_out_1__v__time_1.pdf}, 
we observe unit average flow rate for all time.

Since the network is four-regular, and pore sizes are mono-dispersed initially, 
the pressure drop in the flow-transverse pores is exactly zero.
This implies that there is no flow in the transverse direction.
As a result, changes in junction pressure occur purely to changes 
to the conductance of flow-parallel pores.
Since these conductances are constant (see Figure \ref{fig:nm_alph-zero/cond_2__v__posi_edges_1.pdf}), in Figure \ref{fig:nm_alph-zero/pres_2__v__posi_nodes_1.pdf}, we see that the junction pressure decreases linearly with depth
in flow-parallel pores, 
so that a unit pressure drop is maintained for all time across the filter. 

In summary, in the no-deposition scenario,
% therefore, 
% particles traverse pores without ever depositing. 
% This means that the conductance of pores is never changed, which means the structure of the filter is 
% never altered. 
% As a result, ]
the fluid travels with unit flow rate for all time, and 
the average junction pressures set up a linearly decreasing profile with depth, 
so that a unit pressure drop across the entire filter depth is preserved for all time.
In the following subsection, we discuss the non-zero adherence case, $\alpha\ne0$, so that deposition can occur. 
This leads to changes to the conductances of pores, and thus the flow rate and pressure.
% We first discuss the case where $\alpha=0$, so that there is no deposition.
% The parameters are $N=9$, $N^1=10$, $N^2=2$ so that $\delta=1/3$ and $\varepsilon=1/10$. The value of 
% $\beta$ is arbitrary since is appears with $\alpha$. 
% This means fluid traverses $(\delta\varepsilon)^{-1}=30$ pores to traverse the filter. 
% So macroscale variables are evaluated at $30$ spatial points.
% See that particles travel as a wave through filter. 
% There is no deposition because pores are not sticky (put somewhere in the intro that only considering the case where particles are smaller than pores 
% and sticking is only mechanisms ie theres no sieving and blokcing etc)
% So conductance doesn't change. 
% Because there is no change to the conductance there is no change to the pressure. 
% And because there is no change to the pressure and conductance there is no change to the flow rate.

\subsection{Deposition}
We now discuss the case of non-zero adherence, $\alpha\ne0$, so that particles do deposit within pores. 
To demonstrate, we consider the case $\alpha=0.2$, which, as we will show, corresponds to a decrease in concentration of around $20\%$ over the length of the filter.
As we will see, 
the behaviour of the solution is markedly different at earlier times, $(T<1)$, and later times, $(T>1)$, 
and we display these cases separately, in Figures 
\ref{fig:nm_alpha-0.1/early} and \ref{fig:nm_alpha-0.1/late}, respectively.

\subsubsection{Earlier times}
% alpha = 0.1 (early time)
We demonstrate the behaviour of the macroscale quantities at earlier times in Figure \ref{fig:nm_alpha-0.1/early}. 
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/early/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/early/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/early/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/early/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/early/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/early/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/early/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/early/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, pore conductance, $\bar{G}$, flow rate, $U$, and junction pressure, $\bar{P}$, vary with depth, $x$, and time, $T$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, 
    $\alpha=0.2$, $\beta=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    In plots where depth is shown on the independent-variable axis, 
    the values of $T$ are 
    $0$ (blue), 
    $0.25$ (orange), 
    $0.5$ (green), 
    $0.75$ (red), 
    $1$ (purple), 
    and the arrow shows the direction of increasing time.
    \subref{fig:nm_alph-0.1/early/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at various times, $T$.
    \subref{fig:nm_alph-0.1/early/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at various times, $T$.
    \subref{fig:nm_alph-0.1/early/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    \subref{fig:nm_alph-0.1/early/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$, at various times, $T$.
    }
    \label{fig:nm_alpha-0.1/early}
\end{figure}
In Figure \ref{fig:nm_alph-0.1/early/conc_2__v__posi_nodes_1.pdf}, 
we see that, as in the case where particles do not deposit in pores (see Figure \ref{fig:nm_alpha-zero}), 
particles travel through the filter in a wave. 
This time, as they do so, particles are captured in pores via deposition, so that their concentration decreases 
smoothly with depth. 

As deposition occurs, particles stick to the inner walls of pores, which reduces their cross-sectional area. 
As we see in Figure \ref{fig:nm_alph-0.1/early/cond_2__v__posi_edges_1.pdf}, 
this causes a decrease in pore conductance as time passes. 
Conductance decrease is fastest in pores nearer the inlet, since there are more particles in the fluid that 
have the chance to adhere to the pores, and particles have been traversing these pores for longer. 
For example, by the time particles reach the outlet for the first time (in $\mathcal{O}(1)$ time), 
the conductance in pores near the inlet has decreased by around $20\%$, 
while in pores near the outlet, the conductance has dropped by less than $5\%$.

As deposition occurs, the drop in the conductance of pores causes a drop in the permeability of the 
filter, and thus a decrease in the average flow rate 
of the fluid traversing the filter.
This drop is slight, since particles have not had time to decrease the conductances of 
pores by much yet, which is why particles still reach the outlet in $\mathcal{O}(1)$ time.
In Figure \ref{fig:nm_alph-0.1/early/flux_out_1__v__time_1.pdf}, we see that the drop in the flow rate is concave. 
At first, for $0\le T<0.25$ say, the rate of decrease of the flow rate is small, because only the pores near the top of the filter 
have been reached by fluid, and so it is only possible for particles to deposit in these. 
The overall conductance of the filter has barely been altered.
Later, for $0.25 \le T\le1$, say, we see that the flow rate decrease is faster. 
In this time interval, fluid has reached more pores in the filter, 
and particles are decreasing the size of many pores at once, 
so that the overall conductance of the filter is being decreased more rapidly.
Despite this, the flow rate has only dropped by around $5\%$ by $T=1$, 
because the cross-sectional area of most pores away from the inlet remain similar 
to their initial values.

As a result, the filter still maintains a pressure--depth profile that is almost linear. 
Indeed, in Figure \ref{fig:nm_alph-0.1/early/pres_2__v__posi_nodes_1.pdf}, we see that 
changes to the pore conductances have not altered the average junction pressure 
substantially by $T=1$ (hence the blue line cannot be detected behind the purple line, similar to that in the graph in the no-deposition case, 
\ref{fig:nm_alph-zero/pres_2__v__posi_nodes_1.pdf}).
At earlier times, then, pores have begun to clog, 
but the decrease in their cross-sectional area has not yet caused 
substantial changes 
to the pressure field that drives the fluid through the filter.

\subsubsection{Later times}
% alpha=0.1 (late time)
To see how this situation develops, 
we plot the macroscale quantities at later times in Figure \ref{fig:nm_alpha-0.1/late}.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/late/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/late/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/late/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/late/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/late/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/late/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/alph-0.1/late/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_alph-0.1/late/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, pore conductance, $\bar{G}$, flow rate, $U$, and junction pressure, $\bar{P}$, vary with depth, $x$, and time, $T$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, 
    $\alpha=0.2$, $\beta=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    In plots where depth is shown on the independent-variable axis, 
    the values of $T$ are 
    $0$ (blue), 
    $5$ (orange), 
    $10$ (green), 
    $15$ (red), 
    $20$ (purple), 
    and the arrow shows the direction of increasing time.
    \subref{fig:nm_alph-0.1/late/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at various times, $T$.
    Since the concentration profile is the same for all times  and $T>0$  that are shown, 
    only the times $T=0$ (blue) and $T=20$ (purple) are visible.
    The dashed black line shows the approximate concentration, $\tilde{C}(x^1)$.
    The dotted black line shows the extrapolation of the outlet concentration, $\tilde{C}(1)$.
    \subref{fig:nm_alph-0.1/late/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at various times, $T$.
    \subref{fig:nm_alph-0.1/late/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    The solution at earlier times, $0\le T\le1$, 
    is shown in an expanded box.
    \subref{fig:nm_alph-0.1/late/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$, at various times, $T$.
    }
    \label{fig:nm_alpha-0.1/late}
\end{figure}
In Figure \ref{fig:nm_alph-0.1/late/conc_2__v__posi_nodes_1.pdf}, 
the first thing we notice is that the average-concentration profile 
is identical at all non-zero times that we display. 
We see that particles have reached the outlet 
% for the first time, 
and that, 
now, the concentration of particles in the flow decreases almost linearly with depth, 
so that, at the outlet, the particle concentration has dropped by around $1/5$ (which is exactly the value of $\alpha$ 
that we use in this example). 

To understand the form of the average particle concentration--depth profile,
we first note that fluid loses a proportion, $\alpha\delta\varepsilon$, of particles as it 
traverses each pore, $ij$. 
% That is, on average, 
%  $C_j=(1-\alpha\delta\varepsilon)C_i$.
 Labelling a row of junctions that lead from the inlet to
 the outlet (see the row of pores highlighted in orange in Figure \ref{fig:nm_network_4}, for example) 
 with indices $1,\hdots,(\delta\varepsilon)^{-1}$ 
 (since there are
 $(\delta\varepsilon)^{-1}$ such junctions), it follows that 
 the concentrations in junction $i$ and junction $i+1$ satisfy $C_{i+1}=(1-\alpha\delta\varepsilon)C_i$. 
 Writing the concentration in junction $i$ as a function of the inlet concentration, 
 we see that 
 $C_i=(1-\alpha\delta\varepsilon)^{i}C_{\text{in}}$, and so we define
\begin{align}
    \tilde{C}(x) = (1-\alpha\delta\varepsilon)^{x/\delta\varepsilon}.
    \label{eq:nm_definition-of-appriximate-conc}
\end{align}
This represents the
expected average-concentration that we observe on the macroscale, 
so that we expect $\tilde{C}\approx C$. As such, 
we call $\tilde{C}$ the approximate concentration. 
In Figure \ref{fig:nm_alph-0.1/late/conc_2__v__posi_nodes_1.pdf} 
we see (comparing the dashed black line with the purple line) that this approximation is exact in a mono-dispersed four-regular network,
since deposition only occurs in pores that are parallel to the direction of the flow (the 1 direction, here).
This is because there is no pressure gradient in the transverse direction to drive particles 
through the flow-transverse pores. 
When the network is mono-dispersed and four-regular, then, 
$\tilde{C}$ provides the average particle concentration at any time that is large enough 
for particles to have traversed the filter, 
% for the first time, 
without the need for numerical 
solution of the network system.

In industrial applications, 
experimentalists are often interested in the efficiency of the filter.
We assume that the efficiency is given by the proportion of the total particles that enter the filter that do not exit it, and, since we have nondimensionalised for unit inlet concentration, we write
\begin{align}
    \eta(T) = 1-C(1,T).
    \label{eq:nm_definition-of-efficiency}
\end{align}
Since \eqref{eq:nm_definition-of-appriximate-conc} provides an approximation for the particle 
concentration, 
we define
\begin{align}
    \tilde{\eta} = 1-\tilde{C}(1) = 1-(1-\alpha\delta\varepsilon)^{1/\delta\varepsilon},
    \label{eq:nm_definition-of-approximate-efficiency}
\end{align}
which we call the approximate efficiency, 
since we expect that $\tilde{\eta}\approx\eta$.
Indeed, since $\tilde{C}=\bar{C}$ for $T$ large enough for particles to have traversed the filter, in the case of a mono-dispersed four-regular network, 
it follows that $\tilde{\eta}=\eta$, so that we have found an explicit 
formula for the efficiency for large enough $T$.
In \eqref{eq:nm_definition-of-approximate-efficiency} we see that this formula depends on the adherence, that is, the proportion of particles 
that are expected to be lost in a particular pore, through $\alpha$, 
and on $(\delta\varepsilon)^{-1}$, which is the approximate number of 
pores that fluid traverses to travel from the inlet to the outlet.
In the present example, we have that $\alpha=0.2$, $\delta=0.5$, and $\varepsilon=0.1$, 
and, using \eqref{eq:nm_definition-of-appriximate-conc}, we calculate that $\tilde{C}(1)\approx0.82$ (see the dotted black line in Figure \ref{fig:nm_alph-0.1/late/conc_2__v__posi_nodes_1.pdf}),
so that, using \eqref{eq:nm_definition-of-approximate-efficiency}, it follows that 
$\eta=\tilde{\eta}\approx0.2$, which means that the filter is around $20\%$ efficient. 
Note that $\alpha=0.2$, which corresponds to $20\%$ adherence in each pore (scaled by $(\delta\varepsilon)^{-1})$. 
Thus we see that the order of $\alpha$ dictates the order of the efficiency.

Returning to our analysis of the solution at later times, shown in Figure \ref{fig:nm_alpha-0.1/late}, 
we next discuss the average pore conductance, 
a plot of which is shown in Figure \ref{fig:nm_alph-0.1/late/cond_2__v__posi_edges_1.pdf}. 
We see (comparing with the solution at earlier times, shown in Figure \ref{fig:nm_alph-0.1/early/cond_2__v__posi_edges_1.pdf}), 
that the conductance of pores continues to decrease as time increases. 
Pores near the inlet still have the smallest conductance, since more particles flow through them 
per unit time. 
However, as time increases, 
the gradient of the average pore conductance--depth profile decreases, 
because the rate of decrease of pore conductance is smaller in pores with less conductance, since they have less cross-sectional area and thus surface area to adhere to, 
and fewer particles are conducted through them per unit time.

As conductance decreases, the average flow rate through the filter continues to decrease, 
as we see in Figure \ref{fig:nm_alph-0.1/late/flux_out_1__v__time_1.pdf}. 
After the earlier times, at which the flow rate decreases rapidly 
(see Figure \ref{fig:nm_alph-0.1/early/flux_out_1__v__time_1.pdf}), 
the flow rate decreases more slowly, 
which corresponds to the decrease in the rate of deposition in pores.
The concave behaviour of the flow rate at earlier times ($0\le T<1$) (see the zoomed region in Figure \ref{fig:nm_alph-0.1/late/flux_out_1__v__time_1.pdf})
is replaced with convex behaviour at later times ($1\le T\le20$), after fluid has reached the outlet for the first time.
By times $T>10$ (green line), for example, 
the pores near the inlet have less than $20\%$ of their original conductance (see Figure \ref{fig:nm_alph-0.1/late/cond_2__v__posi_edges_1.pdf}), 
and so they are almost completely clogged.
As a result, we see that fluid flows through the filter at less than $20\%$ of its initial rate.
This would be problematic from an industrial perspective, since it means that particles are 
being processed $80\%$ more slowly than they were originally. 
In industrial applications, filtration processes are often halted when the flow rate drops below some 
minimum-required threshold, $U_{\text{c}}$ say, and 
industrialists are interested in the amount of fluid that has been processed when 
this first occurs, which is called the total throughput, $\tau$, say. 
We will discuss these two quantities more in the next subsections.

Comparing the average junction pressure--depth profile at later times, 
in Figure \ref{fig:nm_alph-0.1/late/pres_2__v__posi_nodes_1.pdf}, 
with that at earlier times (see Figure \ref{fig:nm_alph-0.1/early/pres_2__v__posi_nodes_1.pdf}), 
 we see that the substantial drop in the average pore conductance (see Figure \ref{fig:nm_alph-0.1/late/cond_2__v__posi_edges_1.pdf}) results
 in much larger changes to the junction pressures than before. 
 As time increases, the pressure in each node drops away from its initial value towards the pressure in the outlet, which is zero. 
As a result, the average junction pressure--depth profile becomes convex, since 
the pressure drops fastest in nodes that are far from the inlet and the outlet, 
because pressures outside of the filter are fixed.

To explain the behaviour in the deposition and no-deposition cases further, 
in the following three subsections we analyse the behaviour of
the solutions as functions of the parameters, $\alpha$, $\beta$, $\delta$ and $\varepsilon$, respectively.

\subsection{Dependence on $\alpha$: Retention efficiency}
We first explore the effects of varying the adherence parameter, $\alpha$.
We will find that the behaviour of the solutions is different 
for smaller values, $1<\alpha<\mathcal{O}(1)$ (we examine the region $0\le\alpha\le1$), 
and larger values, $\alpha=\mathcal{O}((\delta\varepsilon)^{-1})$ (we examine the region $0.25(\delta\varepsilon)^{-1}\le\alpha\le( \delta\varepsilon)^{-1}$), 
and so we will treat these cases separately.

\subsubsection{Smaller $\alpha$ values}
We first plot the four key macroscale quantities in Figure \ref{fig:nm_sweep-alph} for equally spaced values of $\alpha$, 
in $0\le\alpha\le1$.
% sweep alpha 
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, 
    pore conductance, $\bar{G}$, 
    flow rate, $U$, 
    and junction pressure, $\bar{P}$, 
    vary with depth, $x$, and time, $T$, at various values of $\alpha$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\beta=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    The values of $\alpha$ are 
    $0.2$ (blue), 
    $0.4$ (orange), 
    $0.6$ (green), 
    $0.8$ (red), 
    $1$ (purple), 
    and the arrow shows the direction of increasing $\alpha$.
    \subref{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at time $T=2$.
    \subref{fig:nm_sweep-alph/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at time $T=2$.
    \subref{fig:nm_sweep-alph/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    \subref{fig:nm_sweep-alph/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$, at time $T=2$.
    }
    \label{fig:nm_sweep-alph}
\end{figure}

Firstly, in Figure \ref{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf}, 
we see that, as $\alpha$ increases, the 
negative gradient of the average particle concentration--depth profile increases. 
For larger values of $\alpha$, a larger proportion of particles stick to each pore, 
which means that there are fewer particles in the fluid by the time it exits the filter. 

Secondly, we plot the average pore conductance in 
Figure \ref{fig:nm_sweep-alph/cond_2__v__posi_edges_1.pdf}.
We see that, 
for smaller $\alpha$ values, particles are less likely to stick in pores, 
and so the fluid that reaches the region near the outlet still contains a large concentration of 
particles (see Figure \ref{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf}) transported from the inlet. 
Consequently, deposition occurs here, decreasing pore conductance. 
As $\alpha$ increases, particles are increasingly likely to be removed in pores nearer to the inlet, and so fewer particles reach the region near the outlet 
(since 
they have all been removed further up the filter), 
As a result, the pore conductance hardly decreases here. 
Altogether, near the outlet, the pore conductance decreases as $\alpha$ increases for small
$\alpha$, but begins to increase again for larger values.

Thirdly, in Figure \ref{fig:nm_sweep-alph/flux_out_1__v__time_1.pdf}, we see that the $\alpha$ increase that caused the faster pore conductance decrease 
results in a more rapidly decreasing average flow rate. 
That is, the average flow rate decreases as 
$\alpha$ increases. 
Correspondingly, in Figure \ref{fig:nm_sweep-alph/pres_2__v__posi_nodes_1.pdf}, 
we see that the junction pressures are driven towards zero faster for larger values of $\alpha$.

In the previous subsections, we concluded that $\alpha$ controls the proportion of particles that are retained in the filter, 
and, as such, it dictates the filter efficiency. 
To formalise this dependency, we plot the efficiency, $\eta$, and the approximate efficiency, $\tilde{\eta}$, as functions of $\alpha$, 
in Figure \ref{fig:nm_alpha-sweep/effi}.
% efficiency
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/effi_net_1__v__alphas_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/effi_net_1__v__alphas_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/eff_aprx_1__v__alph_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/eff_aprx_1__v__alph_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the efficiency, $\eta$, and approximate efficiency, $\tilde{\eta}$, 
    vary with $\alpha$.
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\beta=1$, $\delta=0.5$, $\varepsilon=0.1$.
    \subref{fig:nm_sweep-alph/effi_net_1__v__alphas_1.pdf} 
    The efficiency, $\eta$, as a function of $\alpha$.
    Blue dots show the numerical solution at values of $\alpha$ equal to 
    $0, 0.2, 0.4, 0.6, 0.8, 1$.
    The blue line shows a spline interpolation between these values.
    The dashed black line shows the approximate efficiency, $\tilde{\eta}$, as a function of $\alpha$.
    \subref{fig:nm_sweep-alph/eff_aprx_1__v__alph_1.pdf} 
    The average efficiency, $\tilde{\eta}$, as a function of $\alpha$.
    The dotted black line shows the asymptote, which is $1$.
    The dashed coloured lines show extrapolations of the approximate 
    efficiency for $\alpha$ values of $1$ (orange), $2$ (green), $3$ (red), $4$ (purple).
    }
    \label{fig:nm_alpha-sweep/effi}
\end{figure}
In Figure \ref{fig:nm_sweep-alph/effi_net_1__v__alphas_1.pdf}, we notice that the efficiency of the filter 
increases as $\alpha$ increases. 
As we saw in Figure \ref{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf}, 
as $\alpha$ increases, more particles are removed in each pore, which means that,
 by the time fluid reaches the outlet, it contains fewer particles when $\alpha$ is larger. 
 As a result, for example, the choice $\alpha=0.2$ leads to an efficiency of less than $20\%$, 
 while $\alpha=1$ leads to an efficiency of around $60\%$.
 We see (comparing the dashed black line with the blue line) that, for the present choice of network, the approximate
 efficiency matches the efficiency exactly.

The accuracy of the approximate efficiency allows us 
to calculate the efficiency for values of $\alpha$ for which 
we have not solved the system numerically. 
Indeed, in Figure \ref{fig:nm_sweep-alph/eff_aprx_1__v__alph_1.pdf}, we extrapolate the approximate-efficiency curve and use it to read off the efficiency for various new values of $\alpha$. 
We see that the efficiency rapidly increases for $0\le\alpha\le3$, so 
that the filter is more than $95\%$ efficient for $\alpha>3$ (dashed red line). 
For $\alpha=4$, the filter is around $99\%$ efficient (dashed purple line), and so almost all particles are retained in the filter. 
The efficiency can never exceed one (dotted black line), since it indicates proportional retention, 
and so the efficiency asymptotically approaches its maximum value for even larger values of $\alpha$.
In an industrial application, Figure \ref{fig:nm_sweep-alph/eff_aprx_1__v__alph_1.pdf} could be used to read off a required 
value of $\alpha$ necessary to achieve some required efficiency value.

\subsubsection{Larger $\alpha$ values}
We analyse the behaviour of the solution 
for larger values of $\alpha$, in the range $0.25(\delta\varepsilon)^{-1}\le \alpha \le (\delta\varepsilon)^{-1}$ say, using Figure \ref{fig:nm_sweep-alph/large-sweep}.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/large-sweep/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/large-sweep/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/large-sweep/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/large-sweep/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/large-sweep/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/large-sweep/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-alph/large-sweep/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-alph/large-sweep/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, pore conductance, $\bar{G}$, 
    flow rate, $U$, 
    and junction pressure, $\bar{P}$, 
    vary with depth, $x$, and time, $T$, 
    at various values of $\alpha$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\beta=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    The values of $\alpha$ are 
    $0.25(\delta \varepsilon^{-1}$ (orange), 
    $0.5(\delta \varepsilon^{-1}$ (green), 
    $0.75(\delta \varepsilon^{-1}$ (red), 
    $(\delta \varepsilon^{-1}$ (purple), 
    and the arrow shows the direction of increasing $\alpha$.
    \subref{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at time $T=1$.
    \subref{fig:nm_sweep-alph/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at time $T=1$.
    \subref{fig:nm_sweep-alph/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    \subref{fig:nm_sweep-alph/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$, at time $T=1$.
    The blue dots and line show the solution at $\alpha=0$, 
    and are included to aid the eye.
    }
    \label{fig:nm_sweep-alph/large-sweep}
\end{figure}
We see that filters with $\alpha$ values that are $\mathcal{O}((\delta\varepsilon)^{-1})$ are $100\%$ efficient: 
all particles that are suspended in the feed fluid are captured in pores within the filter. 
% so that the outlet concentration is zero.

Indeed, in Figure \ref{fig:nm_sweep-alph/conc_2__v__posi_nodes_1.pdf} we see that, for these large values of 
$\alpha$, particles are all captured in a layer that extends from the inlet. 
The thickness of this capture layer is $\mathcal{O}(\alpha^{-1})$, 
so that it becomes thinner for increasing values of $\alpha$. 
When $\alpha=(\delta \varepsilon^{-1}$ (purple line), we see that particles 
are captured over a layer that has length $\delta\varepsilon$. 
This extreme corresponds to the case where pores retain $100\%$ of the particles in the fluid 
that enters them, so that all particles suspended in a given volume of fluid
are captured within the first pore 
that this volume enters. 
As a result, we see that particles are captured within an extremely thin layer at the top of the filter 
(similarly 
to the blocking regime in Part \ref{part:A size-structured model for filtration}).

 Once fluid reaches depths outside the capture layer, it does not contain any particles. 
 It traverses the remainder of the filter without affecting the conductances of the pores that it enters 
 until the conductances of pores at the inlet reach zero, 
 at which time the filter effectively ceases to conduct fluid. 
Correspondingly, in Figure \ref{fig:nm_sweep-alph/cond_2__v__posi_edges_1.pdf}, 
 we see that, for large values of $\alpha$, only the conductances of pores near the inlet 
 are altered.

We observe, in Figure \ref{fig:nm_sweep-alph/flux_out_1__v__time_1.pdf}, that, 
 for large $\alpha$ values, the average flow rate rapidly drops towards zero at times $0\le T<1$, 
 at which fluid has not even reached the outlet for the first time, 
 since the pores near the inlet become clogged 
 very quickly. 
 The average flow rate--time profiles are strictly concave while the flow rate is large enough for reasonable operation 
 (that is, flow rates exceeding $20\%$ of the initial rate, say). 
 By $T=1$, the filter is effectively clogged for all the values of $\alpha$ that we consider here.
Correspondingly, in Figure \ref{fig:nm_sweep-alph/pres_2__v__posi_nodes_1.pdf} we see that the pressure in all internal junctions 
rapidly drops to zero by $T=1$.
% The case $\alpha=\mathcal{O}((\delta\varepsilon)^{-1})$, then, corresponds to the case where the affinity between particles and pores 
% is so great that the filter clogs before fluid has even reached the 
% outlet of the filter for the first time.

Altogether, in this subsection, 
we have shown that 
the parameter $\alpha$ controls the `stickiness' of the pores, and thus 
the proportion of particles that are retained in each pore. 
As a result, the value of $\alpha$ dictates the efficiency of the filter.
As $\alpha$ increases, a greater proportion of particles are captured near the inlet of the filter. 
As a result, pores close to the inlet clog more quickly, 
and the flow rate and pressure drop more rapidly.
In the next subsection, we explore the dependence of the solution on the parameter $\beta$.

\subsection{Dependence on $\beta$: Total throughput}
In this subsection, we discuss the dependence of the solution on the particle size parameter, $\beta$. 
We will find that the behaviour of the solutions is different 
for smaller $\beta$ values, say $\beta=\mathcal{O}(10^{-1})$ (we examine the region $0\le\beta\le0.1$), 
and larger $\beta$ values, say $\beta=\mathcal{O}(1)$ (we examine the region $1\le\beta\le5$, 
and we will treat these two cases separately.
\subsubsection{Smaller $\beta$ values}
We begin by plotting the four key macroscale quantities for equally spaced values of $\beta$ in the range $0\le\beta\le0.1$,
in Figure \ref{fig:nm_sweep-beta/small-sweep}.
% small-sweep
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/small-sweep/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/small-sweep/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/small-sweep/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/small-sweep/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/small-sweep/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/small-sweep/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/small-sweep/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/small-sweep/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, 
    pore conductance, $\bar{G}$, 
    flow rate, $U$, 
    and junction pressure, $\bar{P}$, 
    vary with depth, $x$, and time, $T$, 
    at various values of $\beta$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\alpha=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    The values of $\beta$ are 
    $0.02$ (blue), 
    $0.04$ (orange), 
    $0.06$ (green), 
    $0.08$ (red), 
    $0.1$ (purple), 
    and the arrow shows the direction of increasing $\beta$.
    \subref{fig:nm_sweep-beta/small-sweep/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at time $T=100$.
    The dashed black line shows the approximate concentration, $\tilde{C}$,
    as a
    function of depth.
    Since the average particle concentration is identical for 
    all values of $\beta$ considered, 
    only the solution for $\beta=0.1$ (purple line) is visible.
    \subref{fig:nm_sweep-beta/small-sweep/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at time $T=100$. 
    \subref{fig:nm_sweep-beta/small-sweep/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    The dashed coloured lines show extrapolations of the total throughput, 
    $\tau$, for a given flow rate threshold, $U_{\text{c}}=0.4$ (dashed black line), for the various values of 
    $\beta$.
    \subref{fig:nm_sweep-beta/small-sweep/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$, at time $T=100$.}
    \label{fig:nm_sweep-beta/small-sweep}
\end{figure}

Firstly, in Figure \ref{fig:nm_sweep-beta/small-sweep/conc_2__v__posi_nodes_1.pdf}, 
we see that, once fluid has reached the outlet for the first time, 
the average particle concentration--depth profile is constant, and is given exactly by the approximate 
concentration, $\tilde{C}$ (dashed black line). 
We find that this profile is independent of the choice of $\beta$, 
since pores retain a quantity of particles that is given by $\alpha$, $\delta$, 
and $\varepsilon$, but not $\beta$ (see the definition of $\tilde{C}$, \eqref{eq:nm_definition-of-appriximate-conc}).
As a result, the proportion of the incoming particles 
that are present at the outlet, and thus the efficiency, $\eta$, 
are constants with respect to $\beta$. 

Secondly, we note that, as $\beta$ increases, the average pore conductance decreases, as we show in Figure \ref{fig:nm_sweep-beta/small-sweep/cond_2__v__posi_edges_1.pdf}. 
This is because larger values of $\beta$ correspond to a greater particle size, 
which means that deposition causes a faster decrease in the cross-sectional area of pores, and thus their conductance. 
The gradient of the average pore conductance--depth profile is also independent of $\beta$, since changes in $\beta$ affect all pores equally.

Correspondingly, larger values of $\beta$ result in more rapidly decreasing average flow rates (see Figure \ref{fig:nm_sweep-beta/small-sweep/flux_out_1__v__time_1.pdf}) and average junction pressures (see Figure \ref{fig:nm_sweep-beta/small-sweep/pres_2__v__posi_nodes_1.pdf}).
For $\beta=0.02$ (blue lines), for example, we see that by $T=100$, when fluid has traversed the filter $100$ times, 
fluid still flows at almost $40\%$ of its initial rate. 
For $\beta=0.1$ (purple lines), however, the flow rate drop-off is much more rapid, and by $T=100$ we see that the flow rate is almost zero,
so that the filter is unusable, since pores at the inlet have infinitesimal conductance (see Figure \ref{fig:nm_sweep-beta/small-sweep/cond_2__v__posi_edges_1.pdf}).

As we have discussed, in practice, industrial applications are halted when the average flow rate drops below some threshold, $U_{\text{c}}$, say,
at which the cost associated with the increased process time (due to lower flow rate) outweighs the cost of the cleaning or replacement of the filter membrane. 
For a given flow rate threshold, $U_{\text{c}}$, industrialists are interested in predicting the
total throughput, $\theta$ say, which is the number of times that fluid traverses the filter before the flow rate threshold is subceeded. 
In constant flow rate regimes, 
this is exactly the terminal time, $\tau$ say, 
which is the associated time to reach the threshold, 
so that 
\begin{align}
\theta = \tau.   
\end{align}
In constant pressure drop regimes,
the total throughput is related to the 
terminal time via 
\begin{align}
    \theta = \int_0^\tau U(T) \, \text{d}T.
\end{align}
In either scenario, the terminal time, which is
\begin{align}
    \tau \quad \text{such that} \quad  U(T)<U_{\text{c}} \quad \text{for} \quad T>\tau,
\end{align}
since the average flow rate is always monotonically decreasing in this model, 
is the key quantity.
In Figure \ref{fig:nm_sweep-beta/small-sweep/flux_out_1__v__time_1.pdf}, 
we demonstrate how, given an average flow rate threshold, $U_{\text{c}}$ ($40\%$, for example), 
we read off the terminal time, $\tau$. 
When $\beta=0.02$ (dashed blue lines), for example, $U_{\text{c}}=0.4$ yields $\tau\approx84$. 
As $\beta$ increases (dashed blue lines to dashed purple lines), we see that $\tau$, decreases. 
For $\beta=0.1$ (dashed purple lines), the same flow rate threshold yields a terminal time of just $\tau\approx17$. 
The physical interpretation is that, as the size of the particles increases, 
filters are able to maintain a given efficiency for less time before they become clogged. 

% throughput
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/small-sweep/thro_net_1__v__betas_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/small-sweep/thro_net_1__v__betas_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/small-sweep/thro_aprx_1__v__alph_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/small-sweep/thro_aprx_1__v__alph_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the terminal time, $\tau$, 
    and the approximate terminal time, $\tilde{\tau}$,
    vary with $\beta$, for a given flow rate threshold, $0.4$.
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\alpha=1$, $\delta=0.5$, $\varepsilon=0.1$.
    \subref{fig:nm_sweep-beta/small-sweep/thro_net_1__v__betas_1.pdf} 
    The terminal time, $\tau$, as a function of $\beta$.
    Blue dots show the numerical solution at $\beta$ values of 
    $0.02, 0.04, 0.06, 0.08, 0.1$.
    The blue line shows the function $1.69\beta^{-1}$, found 
    using a log-log plot, which we call the approximate terminal time, $\tilde{\tau}$.
    \subref{fig:nm_sweep-beta/small-sweep/thro_aprx_1__v__alph_1.pdf} 
    The average terminal time, $\tilde{\tau}$, as a function of $\beta$.
    The dashed coloured lines show extrapolations of the approximate 
    terminal time for $\beta$ values of $0.01$ (orange), $0.1$ (green), $1$ (red), $10$ (purple).
    }
    \label{fig:nm_sweep-beta/thro}
\end{figure}

Indeed, in Figure \ref{fig:nm_sweep-beta/small-sweep/thro_net_1__v__betas_1.pdf}, we plot the terminal time 
for the small values of $\beta$ shown in Figure \ref{fig:nm_sweep-beta/small-sweep}.
Using a log-log plot to fit a curve (blue line) to our numerical results (blue dots), 
we find that this decreases like $\beta^{-1}$. 
That is, we define the approximate terminal time to be 
\begin{align}
    \tilde{\tau}=1.69\beta^{-1},
\end{align}
(for $\alpha=1$, note that this is $\alpha$ dependent)
which matches the terminal time almost exactly for $0.02\le \beta\le0.1$, 
so that $\tilde{\tau}\approx\tau$.

This finding means that we are able to approximate the terminal time 
that corresponds to values of $\beta$ for which 
we have not solved the network model numerically. 
As such, in Figure \ref{fig:nm_sweep-beta/small-sweep/thro_aprx_1__v__alph_1.pdf}, we show how, extrapolating the 
approximate terminal time, $\tilde{\tau}$, 
we are able to approximate times at which filtration terminates for large and infinitesimal $\beta$ values, 
without appealing to further numerical simulations.
We see, for example, that for $\beta=0.01$ (dashed orange line), we expect that $\tilde{\tau}\approx170$, 
while for $\beta=10$ (dashed purple line), $\tau\ll1$, and filtration terminates almost immediately.

\subsubsection{Larger $\beta$ values}
To understand why the terminal time is so small for large values of $\beta$, 
we solve the system numerically for $\beta$ values in the region 
$1\le\beta\le5$, 
and plot the solution in Figure \ref{fig:nm_sweep-beta/large-sweep}.
% large-sweep
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/large-sweep/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/large-sweep/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/large-sweep/cond_2__v__posi_edges_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/large-sweep/cond_2__v__posi_edges_1.pdf}
     \end{subfigure}
     %\hfill
          \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/large-sweep/flux_out_1__v__time_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/large-sweep/flux_out_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-beta/large-sweep/pres_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-beta/large-sweep/pres_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, pore conductance, $\bar{G}$, flow rate, $U$, and junction pressure, $\bar{P}$, vary with depth, $x$, and time, $T$, at various values of $\beta$. 
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\alpha=1$, $\delta=0.5$, $\varepsilon=0.1$.
    Coloured dots show the exact value of the quantity at the depths at which it is evaluated.
    Correspondingly coloured lines show a spline interpolation of the quantity between evaluation points.
    The values of $\beta$ are 
    $1$ (blue), 
    $2$ (orange), 
    $3$ (green), 
    $4$ (red), 
    $5$ (purple), 
    and the arrows show the direction of increasing $\beta$.
    \subref{fig:nm_sweep-beta/large-sweep/conc_2__v__posi_nodes_1.pdf} 
    The average particle concentration, $\bar{C}$, 
    as a function of depth, $x$, at time $T=4$.
    \subref{fig:nm_sweep-beta/large-sweep/cond_2__v__posi_edges_1.pdf} 
    The average pore conductance, $\bar{G}$, 
    as a function of depth, $x$, at time $T=4$. 
    \subref{fig:nm_sweep-beta/large-sweep/flux_out_1__v__time_1.pdf} 
    The average flow rate, $U$, 
    as a function of time, $T$.
    The dashed coloured lines show extrapolations of the total throughput, $\tau$, 
    for a given flow rate threshold, $U_{\text{c}}=0.4$, for the various values of 
    $\beta$.
    \subref{fig:nm_sweep-beta/large-sweep/pres_2__v__posi_nodes_1.pdf} 
    The average junction pressure, $\bar{P}$, 
    as a function of depth, $x$.
    }
    \label{fig:nm_sweep-beta/large-sweep}
\end{figure}
Comparing Figure \ref{fig:nm_sweep-beta/large-sweep} with Figure \ref{fig:nm_sweep-beta/small-sweep}, 
we see that the behaviour of the solution is markedly different when $\beta=\mathcal{O}(1)$ as opposed to when 
$\beta=\mathcal{O}(10^{-1})$. 

In particular, we see, in Figure \ref{fig:nm_sweep-beta/large-sweep/conc_2__v__posi_nodes_1.pdf}, 
that particles are all captured within a layer near the inlet, the thickness of which decreases as $\beta$ increases. 
For $\beta=1$, the average particle concentration--depth profile (see blue line in Figure \ref{fig:nm_sweep-beta/large-sweep/conc_2__v__posi_nodes_1.pdf}) matches that in the $\beta=0.1$ case (see purple line in Figure \ref{fig:nm_sweep-beta/small-sweep/conc_2__v__posi_nodes_1.pdf}). 
However, for $\beta=5$ (purple line in Figure \ref{fig:nm_sweep-beta/large-sweep/conc_2__v__posi_nodes_1.pdf}), all particles 
are captured in approximately the first $50\%$ of the filter.

To interpret this, we note that $\beta$ is dictated by the size of particles in the fluid. 
When particles are larger, it takes less time to reduce the flow rate to negligible values, after which only minor changes to 
the average particle concentration can occur. 
If particles are large enough, then pores at the inlet become clogged before fluid 
has traversed the filter and reached the outlet. 
In Figure \ref{fig:nm_sweep-beta/large-sweep/cond_2__v__posi_edges_1.pdf}, we see that, correspondingly to the particle concentration plot,
when $\beta=5$ (purple line), 
only the pores in the first $50\%$ of the filter have been affected by deposition. 
One physical implication is that, when particles are large, clogging occurs near the inlet, and 
much of the filter is unused when the membrane is disposed of, due to a flow rate decrease.

In Figure \ref{fig:nm_sweep-beta/large-sweep/flux_out_1__v__time_1.pdf}, 
we demonstrate how we read off terminal times, $\tau$, for given flow rate thresholds, $U_{\text{c}}$ ($40\%$ here), 
for larger values of $\beta$. 
Comparing the present case, with $\beta$ values that are $\mathcal{O}(1)$ (Figure \ref{fig:nm_sweep-beta/large-sweep/flux_out_1__v__time_1.pdf}), 
with the previous case, in which $\beta$ values are $\mathcal{O}(10^{-1})$ (see Figure \ref{fig:nm_sweep-beta/small-sweep/flux_out_1__v__time_1.pdf}), 
we observe that, as we predicted using the approximate terminal time curve in Figure \ref{fig:nm_sweep-beta/small-sweep/thro_aprx_1__v__alph_1.pdf}, 
the terminal times are much smaller in the present case.

In summary, in this subsection we have shown that 
the parameter $\beta$ controls the size of the particles, and thus 
the change in pore cross-sectional area, and therefore conductance, that occurs due to deposition. 
As a result, the value of $\beta$ dictates the terminal time, $\tau$, and thus the total throughput, $\theta$.
As $\beta$ increases, deposition causes greater decrease in conductance, 
so that pore clogging occurs more rapidly. 
Consequently, the filter clogs over a shorter time, which 
means that the total throughput is smaller.
In the next subsection, we explore the dependence of the solution on the parameters $\delta$ and $\varepsilon$.

\subsection{Dependence on $\delta$ and $\varepsilon$: Discretisation}
\label{sec:change-delt}
Lastly, in this subsection, we assess the effect of changing the scaling parameters, $\delta$ and $\varepsilon$, 
on the solution. 
It is easiest to see the effect of sweeping these two parameters if we set $\alpha=0$, 
which we have shown to correspond to the no deposition scenario.

We plot the average particle concentration as a function of depth in Figure \ref{fig:nm_sweep-delt-and-epsi}, 
and show the solution for various values of $\delta$ (with $\varepsilon$ fixed) and $\varepsilon$ (with $\delta$ fixed) 
in Figures \ref{fig:nm_sweep-delt/conc_2__v__posi_nodes_1.pdf} and \ref{fig:nm_sweep-epsi/conc_2__v__posi_nodes_1.pdf}, 
respectively.
\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-delt/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-delt/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/network/sweep-epsi/conc_2__v__posi_nodes_1.pdf}
         \caption{}
         \label{fig:nm_sweep-epsi/conc_2__v__posi_nodes_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the average particle concentration, $\bar{C}$, 
    varies with depth, $x$, 
    at various values of time, $T$, and $\delta$ (a), and 
    $\varepsilon$ (b).
    The parameters are $N=4$, $N^1=10$, $N^2=2$, $\alpha=0$,
    $\beta=1$. 
    The values of $T$ are 
    $0.00$ (blue), 
    $0.25$ (orange), 
    $0.50$ (green), 
    $0.75$ (red), 
    $1.00$ (purple).
    The black arrows show the direction of increasing time. 
    The coloured arrows show the direction of decreasing 
    $\delta$ (a) and $\varepsilon$ (b) at the 
    correspondingly coloured times.
    \subref{fig:nm_sweep-delt/conc_2__v__posi_nodes_1.pdf} 
    The solution at various values of $\delta$ for $\varepsilon=0.1$. 
    The values of $\delta$ are 
    $1$ (dotted), 
    $0.5$ (dashed), 
    $0.33$ (solid).
    \subref{fig:nm_sweep-epsi/conc_2__v__posi_nodes_1.pdf} 
    The solution at various values of $\varepsilon$ for $\delta=0.5$. 
    The values of $\delta$ are 
    $0.5$ (dotted), 
    $0.1$ (dashed), 
    $0.05$ (solid).
    }
    \label{fig:nm_sweep-delt-and-epsi}
\end{figure}
We recall that particles traverse the filter in a wave.
Comparing Figures \ref{fig:nm_sweep-delt/conc_2__v__posi_nodes_1.pdf} and \ref{fig:nm_sweep-epsi/conc_2__v__posi_nodes_1.pdf}
we see that decreasing the values of $\delta$ and $\varepsilon$ independently has similar effects on 
the solution. 
As $\delta$ and $\varepsilon$ decrease, the transition region of the wave decreases in size. 
Physically, for smaller values of $\delta$ and $\varepsilon$, there are more junctions and pores in the filter, 
and so these are packed more closely to one-another.
As a result, junctions fill with particles more quickly, which increases the sharpness of the wave front,
thus decreasing the dispersion of particles.

% We see in average particle concentration plots for both that decreasing $\delta$ and $\varepsilon$ has the same effect. 
% This effect is the decrease the diffusion in the network. 
% Pores are packed more closely so there is less change in concentration from pore to pore
% Viewingthe microscale network as a discretisation of the macroscale, we see that decreasing these parameters 
% has the effect of decreasing the amoutn of numerical diffision in the behavioru of the effective concentration.
% This is a problem because it means that to accurately represnet macroscale quantities we will 
% need many pores and nodes on the microscale, which we have already noted will be too expensive.
% So we need a method where we can have a big network on the microscale, use this to get macroscale behaviour, 
% but not solve as many equations on the microscale.
% This is what we'll discuss in the next part.

One alternative interpretation is to view the network as a microscale discretisation of the macroscale. 
On the microscale, solution variables are tracked at nodes or pores, which 
are located at discrete points. 
On the macroscale, these solution variables are used to calculate  `average' quantities 
that, when the underlying solution variables are interpolated, vary in continuous space. 
In this setting, $\delta$ and $\varepsilon$ dictate the `closeness' of the points 
at which microscale solution variables are evaluated, and thus the `fineness' 
of the discretisation of the average macroscale quantities.
It follows that decreasing $\delta$ and $\varepsilon$ has the effect of decreasing the numerical 
diffusion in the system, and thus of improving the 
quality of the discretisation, and the accuracy of the average macroscale quantities. 
This corresponds to increasing the number of nodes in a cell, or increasing the number of cells in the 
network.
We showed at the beginning of this chapter, though, that increasing these numbers can be problematic, as it can 
lead to computationally infeasibly large linear systems to solve.
In the following part, we will formalise the view that the network model is a discretisation of the macroscale, 
by developing an asymptotic method. 
We will thus develop a more rigorous method for deriving the average particle concentration, 
pore conductance, junction pressure, and flow rate.
Furthermore, we will show that the quality of these averages increases as $\varepsilon$ and $\delta$ decrease, 
and that this decrease can be achieved without increasing the computational complexity of 
the system.


% We see that $\delta$ and $\varepsilon$ dictate the `closeness' with which pores pack 
% inside the filter, and therefore the fineness of discretisation, when solutions defined at 
% pores and nodes, are viewed as discrete ponts of the solution on the macrcale, within which 
%  the spatial coordinate varies smoothly. 
%  To some extent, then, $\delta$ and $\varepsilon$ control the validity of the `averaging'
%  that takes place to use microcale variables to define macroscale quantities.
%  In the following part, we will formalise this view by using asymptotic methods.

% To sweep $\delta$. 
% We choose $N^1=10$, $N^2=2$ so that $\varepsilon=1/10$, and $\beta$ is irrelevant. 
% Then we choose $N=1$, $N=4$, $N=9$ so that $\delta=1$, $\delta=1/2$, $\delta=1/3$. 
% This means that $\delta^{-1}=1$, $\delta^{-1}=2$, $\delta^{-1}=3$, so equivalent to 
% increasing number of pores in the cell. 
% This means that $(\delta\varepsilon)^{-1}=10$, $(\delta\varepsilon)^{-1}=10$, $(\delta\varepsilon)^{-1}=30$, 
% so pores that fluid traverse to traverse the filter increases.


% To sweep $\varepsilon$. 
% We choose $N=9$ and $N^2=2$ so that $\delta=1/3$ and $\beta$ is irrelevant. 
% Then we choose $N^1=5$, $N^1=10$, $N^1=20$ so that $\varepsilon=1/5$, $\varepsilon=1/10$, $\varepsilon=1/20$. 
% This means that $(\delta\varepsilon)^{-1}=15$, $(\delta\varepsilon)^{-1}=30$, $(\delta\varepsilon)^{-1}=60$,
% so number of pores that fluid traverses to traverse the filter increases.









% In this section we assume that all the conductances in the network are the same. 
% In dimensional terms this is like assuming that all conductances are the mean which means that in dimensionless terms all the initial conductances are one.


% Choose $\beta=\alpha=0$. Show that have travelling waves. Argue that varying $\varepsilon$ and $\delta$ has same effect. So just vary $\varepsilon$ (comment that this will be useful intuition for later) say that this is like changing the length of edges in the network or equivalently how many  nodes there are if the network is the same length. Nothing should change, becasue we have scaled this to  make this the case. Therefore we should just choose a sensible sized network. We'll choose one that has 8 nodes down and 40 across (or something) - say what we're choosing.

% Vary $\alpha$, show that $\bar{A}$ dictates the order of removal, and therefore dictates the efficiency of the filter. Say that $\alpha$ is the `efficiency' parameter, and should be interpreted as the dimensionless stickiness of the filter.

% Vary $\beta$, show that is dictates how kquickly edges are removed, and therefore the lifetime of the filter-- say that it's the `lifetime' parameter which shoudl be interpereted as the dimensionless particle size.
% Show that it's the order of the group $\alpha \beta \delta^{-1/2}$ that dictates the order of shrinkage. If this is order one then we get order one reduction in the effective permeabilit of the network as a whole.


% \section{Random networks}
% Solve the network model in the case where lengths are random. 
% So that the result for concentration is more or less the same. 
% Show that the result for conductance is now noisy.
% But show that its ok, becyause the global permeabliity is the same. 
% Explain that the dimen

\chapter{Conclusions for the network model}
\label{chap:nm_Conclusions for the network model}

In summary, in Part \ref{part:A network model for filtration}, Chapters \ref{chap:nm_Introduction to the network model}--\ref{chap:nm_Solutions of the network model}, we motivated, derived, and solved a novel network model for filtration.
 Our model serves as an introduction to the uses of pore network models (PNMs) in filtration, and there are several extensions to be explored as further work.

\section{Summary}

\subsection{Motivation for the model}
In Chapter \ref{chap:nm_Introduction to the network model}, we began by introducing networks of nodes and edges as useful representations of the microscale of filtration media.
In particular, in Section \ref{sec:nm_Networks in filtration}, we summarised recent research that produces network representations of filtration membranes 
by creating high quality characterisations and then analysing these algorithmically.
The end result is a network whose size  (and thus conductance) distribution and connectivity statistically match 
that of membrane pore space. 

In Section \ref{sec:nm_Existing network models for filtration}, we discussed existing PNMs.
% , that is, the various existing dynamical systems that use pore networks as the underlying geometry.
These are typically systems of differential equations that use the pore space network as an underlying geometry. 
In general, these are computationally cheap enough to model `mesoscopic' (i.e., areas containing hundreds or thousands of pores) properties. 
However, due to the complexity of pore space, and the resulting number of nodes and edges in the network, 
PNMs are too expensive to solve for macroscopic properties across the entire membrane, such as
the effective permeability and adhesivity of the filter. 

With computational constraints in mind, in Section \ref{sec:nm_A novel network model for filtration}, we motivated 
a novel network model for filtration. The aim was to derive and solve a simple system on a network that we supposed to be statistically representative of a membrane, which accounts for the particle deposition that causes dynamic connectivity in the network. 

\subsection{Derivation of the model}

In Chapter \ref{chap:nm_Derivation of the network model}, we derived a network model. 
Indeed, in Section \ref{sec:nm_Dimensional model}, 
we began by assuming that the filter consists of a network of pores and junctions, modelled as cylinders and spheres with known initial size distributions. 
The feed fluid is a continuum of particles that is forced through the network via a constant pressure drop. 
Particles adhere to, and so deposit on, the inner walls of pores, which causes a decrease in their cross-sectional area, and thus their conductance. This causes changes to the network connectivity.
We assumed that fluid and particle mass, and pore volume, are conserved quantities, and arrived at a system of ordinary algebraic--differential equations on a dynamic network that consist of:
\begin{itemize}
    \item A fluid mass conservation equation, which takes the form of a linear algebraic equation for the pressure in junctions as a function of the conductance of pores; 
    \item An ordinary differential equation for the number of particles in each junction that describes 
    the flux of particles in and out of the junction through connected pores, and the loss of particles in the flow as they deposit in pores; 
    \item An ordinary differential equation for the conductance of each pore, which describes the decrease in conductance due to uniform deposition of particle volume along the pore walls caused by an adherence probability between particles and pores.
\end{itemize}

In Section \ref{sec:nm_Dimensionless model}, we nondimensionalised the model, and found four dimensionless parameter groups of interest: $\alpha$ (the adherence parameter); $\beta$ (the particle size parameter); 
$\delta$ (the inverse of the number of pores in a cell); and $\varepsilon$ (the inverse of the number of cells in the network).

\subsection{Solution of the model}
In Chapter \ref{chap:nm_Solutions of the network model}, we found numerical solutions of the dimensionless network model. 
To this end, in Section \ref{sec:nm_Solution methodology}, we began by proposing a suitable numerical scheme, involving solution of a linear problem the size of the number of nodes in the network. This is problematic because networks corresponding to entire membranes can contain $\mathcal{O}(10^4)$ or more pores, and thus nodes and edges, rendering solution computationally challenging. 

Nevertheless, network models are useful for simulating smaller subsections of membranes. 
To demonstrate this, in Section \ref{sec:nm_Non-random network: Mono-dispersed pores}, we 
solved our network model for a four-regular network containing $\mathcal{O}(10^2)$ nodes.
For simplicity, we focused on the case where the network characterises a mono-dispersed filter. 
As a result, pore conductances are all equal initially,
and need not be sampled as part of the solution scheme. 
We derived macroscale quantities: particle concentration; pressure; conductivity; and flow rate, by straightforwardly averaging the microscale solution variables, and explored how these vary with time and with the system parameters. 

We began by discussing the simplest scenario, in which $\alpha=0$ so that 
deposition does not occur.
% there is no mutual adherence between particles and pores and so no deposition occurs. 
In this case, we found, as in Part \ref{part:A size-structured model for filtration}, 
that particles traverse the filter in a wave, with constant unit average conductance across the entire filter. The flow rate and local pressure are unchanged, since pore sizes are unaffected by the particles in the flow. 

We then discussed the case $\alpha\ne 0$, so that deposition does occur. 
We found that the behaviour of the solution splits into two distinct time regions: before and after the fluid has reached the outlet for the first time. 
Before, particles travel as a wave. However, as they traverse the filter, some proportion, which is dictated by $\alpha$, stick to pores, so that the particle concentration decreases in the region behind the wavefront. The deposition of particles in pores causes them to constrict, which decreases their conductance, more rapidly near the inlet, since there are the most particles in the flow here that are available for deposition. 
Because the conductance distribution is no longer uniform across the filter, the flow rate decreases concavely, which decreases the local pressure.  
Once fluid has reached the outlet for the first time, a constant particle concentration profile is maintained, since the deposition probability is constant.
The conductance continues to decrease across the filter, but the rate of decrease slows, since there is less pore surface area available for deposition. As such, changes to velocity occur less rapidly, and decrease becomes convex. 

We analysed this behaviour as functions of the system parameters. 
Firstly, we found that increasing $\alpha$ corresponds to increasing the adherence of each pore.
% , and 
% thus has the effect of increasing the 
% increases the proportion of particles that deposit on each pore. 
Consequently, we found that $\alpha$ dictates the particle retention efficiency: 
If $\alpha$ is larger, then more particles are removed from the flow by each pore, so the 
total removal across the whole filter increases, and thus it is more efficient. 

Secondly, we found that increasing $\beta$ has the effect of increasing the size of particles in the suspension. 
Thus, as $\beta$ increases, the rate at which the volume, and thus the conductance, of each pore decreases is accelerated. Consequently, we found that $\beta$ dictates the total throughout of the filter: 
If $\beta$ is larger, then particles cause a faster decrease in the conductance of each pore, so the flow rate of fluid through the filter drops below some critical threshold more rapidly, 
so that the total throughput decreases. 

Lastly, we explored the effects of decreasing $\delta$ and $\varepsilon$, 
and thus increasing the numbers of nodes and edges in the network. We found that these had a similar effect on the solution.  
As $\delta$ and $\varepsilon$ decrease, nodes and edges pack more tightly, so that the quality of the resolution of the macroscale quantities improves. 
This motivates the interpretation of the network model as a discretisation of the macroscale filter, but presents a numerical challenge: As the number of nodes and edges in the network increases, the approximation of effective properties improves, but solution becomes increasingly computationally expensive.
% until it is infeasible. 

\section{Further work}

% There are several extensions to be considered. 
Further work might generally revolve around more detailed consideration of microscale structure and interactions. 
This could include closer attention to  mechanisms involved in deposition, or consideration of particle size, 
to account for blocking, as in Part \ref{part:A size-structured model for filtration}. 
Porosity gradients could be explored
% Depth-dependent initial pore sizes would be included simply, 
using spatially dependent initial conductance distributions, for example. 
We outline some other extensions in this section.

\subsection{Network connectivity}

We focused on the simplest two-dimensional structure: a four-regular network.
More complex connectivity could be simulated with little additional effort, using, for example, the plethora of existing construction algorithms for networks with random connectivity. 

\subsection{Initial conductance distributions and poly-dispersed pores}

We considered initial edge conductances drawn from a Dirac delta distribution, which corresponds to mono-dispersed pore sizes. 
More accurate capture of microscale structural variation might focus on obtaining a distribution from which edge conductances are sampled, corresponding to poly-dispersed pore sizes
(log-normally, for example, as in Subsection \ref{subsec:nm_Operational regime}). 
Consideration of random initialisations would require repeated simulation, introducing additional computational expense. 

% by supposing that  
% pores are mono-dispersed initially, which resulted in initial conductances that are dramwm from a Dirac delta distribution. Again, with relatively little additional effort, we could randomise initial pore conductances

\subsection{Network representation and experimental data}

Examination of real porous media samples would be another interesting extension. This might involve the coupling of our numerical scheme with an algorithm to derive a network from an actual microscale representation. 
One approach would inspect high resolution images for exact network extraction. 
Another would employ data on the statistical properties of real samples to infer the average coordination number corresponding to a particular material and thus parametrise the algorithm used to generate the network connectivity or else the distribution of initial conductances.
% , such as the log-normal one proposed.

\subsection{Multiscale models}
Although these extensions would be interesting, they emphasise the key problem with our model, and with network models in general. 
More accurate representations of the microscale result in larger networks, which require increased computational resources to solve.
An alternative direction for future study, then, is to focus on the development of models that 
offer the accuracy of network models 
but are more computationally efficient. To this end, in Part \ref{part:A multiscale model for filtration}, we introduce multiscale models as a candidate framework, and explore a particular example. 
In Part \ref{part:A network model for filtration}, we averaged microscale variables in the most straightforward way to obtain macroscale properties.
We will suggest `network homogenisation' 
as a method for the systematic averaging 
 of network quantities to obtain effective ones. 
 We will find that the computational complexity of the effective model depends on the size of a statistically representative cell that contains far fewer nodes than the network as a whole. 
 Yet, solutions of the cheaper effective model still agree with solutions of the more expensive network model in an asymptotic limit. 
 Because of the computational benefit of our multiscale model, we will use it, rather than our network model, to explore the effects of the microscale on the macroscale properties, including some questions concerning connectivity and conductance distribution that we raised earlier in this section. 


% Extensions: 
% - We could spend time deriving a network model that more thouroughly models the microsclae. 
% - If we were to do this, we might focus on: 
% -- Improving the quality of the network representation
% ---- structure: Get real rock samples and actualy run the algorithms to get a real network and then solve the model on a real netowrk with the same connectivity and conductance distribution. 
% --- This seems unfeasible and its not clear that its worth the agro - as we ave disucssed, the network only has to be representative of the actual sample. 
% --- We could therefore find the connectivity and conductance distribution and then choose a netowrk with a regular conectvity with the same average cooridnation number adn same distribution... it's not clear whether we'd get the same results.

% - Even if we did do that though, we're stuck with the same problem.... the network model is too expensive when we have the rigth number of nodes. 
% - So this motivates us to find a better way of approixnuiatng the macroscale quantities of interest given a network model. 
% - I.e.e we are interested in a chepaer splution method where we dont lose accuracy. 
% Once we have done this, we can explpore more complicated questions such as how srtucture affects the solution.

% - To this end, in the next part, we will derive a novel method for systematically averaging th enetwork 
% model. 
% - We will find that solution of this new model depends on the size of a representative cell 
% in the network, rather than the structure of the entire network. 
% - We will find that if the cell represents the network, then the solution of theis afverage model is the same as  that of the network model i nthe asympotic limit where these cells are small and many of them conssitute the filter.
% - If it is statistically representativce, then we will see that inducual runs do not match exactly. 
% - As a result, we will explore the stastics of the multlael model solution  
% and show that these match that of the network.
%  - We will also explore hwat it means for a cell to be preresentaive.. how large does the cel need to be and does a regular structure correctly simulate an iregular structure that we'd find in an actual sample? Ie this wlll allow us to answer some of the furtehr work that we could have done by solving the network model, but it will be more cheap to di it using the multiscale model.l
% possible put some of this in the intro of the next part instead?
 

% \subsection{Theme 1...}

% These scalings are chosen to achieve order 1 pressure drop, order 1 flux, and order 1 removal of particles.

% Note that particles traverse $\mathcal{O}(\delta \varepsilon)$ edges as they traverse the filter, where $\delta=1/\sqrt{N}$. It follows that the pressure drop along edge $ij$ will be $\mathcal{O}(\delta \varepsilon)$. However, we do not scale edge pressure drop with this, since we want pressure drop across the filter to be $1$, not $1/(\delta \varepsilon)$. So instead we scale conductance to make it larger. The scaling ensures that filters with different numbers of cells and nodes within the cell all have $\mathcal{O}(1)$ edge flux and can be compared with each other [note that we want order 1 edge flux because this'll result in order one Darcy velocity, since this is the average edge flux]. Altogether, we scale in this way to make it so $\mathcal{O}(1)$ conductance gives $\mathcal{O}(1)$ edge flux after multiplying with pressure drop. Additionally, we want $\mathcal{O}(1)$ removal of particles over networks with any $\delta$ and $\varepsilon$ that we choose, since we want to compare them, hence we scale edge adherence in this way, so that when $\tilde{A}_{ij}$ is $\mathcal{O}(1)$ we have $\mathcal{O}(1)$ removal over the entire filter. 

% We choose the timescale to remove $\delta$ from the advection--reaction equation (we keep $\varepsilon$ in the equation because, in the next chapter, we will carry out an expansion in this small parameter). To understand the timescale, consider the network time, which is the time taken fill the network with fluid, $T_{\text{n}}$ say. Since edges have infinitesimal volume compared to nodes, this is the total volume of nodes in the network divided by the volume of fluid that enters the network per unit time.
% There are $N^1N^2N$ nodes in the network, with an average volume of $\bar{V}$, so $N^1N^2N\bar{V}$ is the total volume of the (or of nodes in the) network.
% There are $N^2\sqrt{N}$ edges entering the network, each with a flux of $(\bar{G}/(\delta \varepsilon)) (p_{\text{in}}-p_{\text{in}})(\delta \varepsilon)$ through it (since we know the pressure drop on an edge is $\mathcal{O}(\delta \varepsilon)$ smaller than the network pressure drop). Thus the denominator, the flux into the filter, is $N^2\sqrt{N}\bar{G} (p_{\text{in}}-p_{\text{in}})$. 
% It follows that
% \begin{align}
%     T_{\text{n}} = \frac{N^1\sqrt{N} \bar{V}}{\bar{G} (p_{\text{in}}-p_{\text{in}})} = 
%     \frac{\bar{V}}{\delta \varepsilon \bar{G} (p_{\text{in}}-p_{\text{in}})}. 
% \end{align}
% A second timescale of interest is the cell time, 
% \begin{align}
%     T_{\text{c}} = \varepsilon T_{\text{n}} =
%     \frac{\bar{V}}{\delta \bar{G} (p_{\text{in}}-p_{\text{in}})}.
% \end{align}
% This is the amount of time it takes to fill a column of cells with fluid, and we expect to have to wait $\varepsilon^{-1}$ of these to fill the filter.
% A third timescale of interest is the edge time,
% \begin{align}
%     T_{\text{e}} = \delta \varepsilon T_{\text{n}} =
%     \frac{\bar{V}}{ \bar{G} (p_{\text{in}}-p_{\text{in}})}.
% \end{align}
%  This is the amount of time it takes to fill a column of nodes with fluid, and we expect to have to wait $(\delta \varepsilon)^{-1}$ of these to fill the filter.
%  Our chosen timescale is a bit abstract, but we choose it to remove $\delta$ but keep $\varepsilon$ in advection--reaction equation (since we know we'll get rid of $\varepsilon$ using homogenisation.)
%  The timescale is $\delta^2 \varepsilon T_{\text{n}}$, so we expect to wait $\delta^{-2} \varepsilon^{-1}$ to fill the network.

%  Applying these scalings, and dropping tilde on the dimensionless variables, the dimensionless model is
% \begin{align}
%     Q_{ij} &= \frac{G_{ij}}{\delta \varepsilon}(P_i-P_j),
%     \\
%     \sum_{j=1}^{N_{\text{f}}} Q_{ij} &= 0,
%     \\
%     V_i\pp{C_i}{T} &= \sum_{j=1}  \Big(
%     (1-\delta \varepsilon A_{ji} ) \frac{G_{ji}}{ \varepsilon}(P_j-P_i) C_j H(P_j-P_i) - \frac{G_{ij}}{\varepsilon}(P_i-P_j)  \Big)
%     \\
%     \pp{G_{ij}}{T} &= - \varepsilon \beta  A_{ij} C_{ij} G_{ij}^{\frac{3}{2}}|P_i-P_j|,
%     \\
%     V_i &= \upsilon,
%     \\
%     A_{ij} &= A^0
% \end{align}
% where 
% \begin{align}
%     \beta &= \frac{2 \nu \bar{V} c_{\text{in}} }{(8 \pi \mu L_{ij}\bar{G})^{\frac{1}{2}}},
%     \\
%     \upsilon &= \frac{V^0}{\bar{V}}
% \end{align}
% so that $[\beta]=1$ and $[\upsilon]=1$, and $[A^0]=1$ as before.
% Note that if $\mathcal{O}(\beta)=1$ then no removal in limit $\varepsilon \rightarrow 0$. So should choose $\beta=\mathcal{O}(\varepsilon^{-1})$ to compare different networks. Networks with different number of nodes in cell meanwhile are taken care of because timescale changes. 
% \section{Homogenisation of conductance equation [put in correct section]}
% Aim is to homogenise 
% \begin{align}
%     \pp{G_{ij}}{T} &= - \varepsilon \beta  A_{ij} C_{ij} G_{ij}^{\frac{3}{2}}|P_i-P_j|.
% \end{align}
% Using multiple scales we'll get
% \begin{align}
%     \pp{G_{ij}^{\bm{r}}}{T} &= - \varepsilon^2 \beta  A_{ij} c (G_{ij}^{\bm{r}})^{\frac{3}{2}} 
%     \bigg|
%     \sum_{m=1}^D \Delta_{ij}^{r^m} \pp{p^{(0)}}{x^m} \bigg|
%     \\
%     & = 
%     - \varepsilon^2 \beta  A_{ij} c (G_{ij}^{\bm{r}})^{\frac{3}{2}} 
%     \bigg|
%      \Delta_{ij}^{r^1} \pp{p^{(0)}}{x^1} \bigg|,
% \end{align}
% where we've used that pressure gradient is unidirectional.
% Shifting to the effective timescale,  $t=\varepsilon T$, we have
% \begin{align}
%     \pp{c}{t} =     
%     - \varepsilon \beta  A_{ij} c (G_{ij}^{\bm{r}})^{\frac{3}{2}} 
%     \bigg|
%      \Delta_{ij}^{r^1} \pp{p^{(0)}}{x^1} \bigg|.
% \end{align}
% The problem is that this equation can't be processed as it is. The concentration and pressure gradient terms on the right-hand side prevent this, since it's not possible to tabulate conductance with these effective variables.
% To remove them, define the total concentration through point $x$ and time $t$,
% \begin{align}
%     \kappa(x,t) = \int_{0}^t c(x,s) \Big| \pp{p^{(0)}}{x^1} \Big| \, \text{d}s,  
% \end{align}
% which satisfies 
% \begin{align}
%     \pp{\kappa}{t} = c \Big|\pp{p^{(0)}}{x^1}\Big|.
% \end{align}
% It follows that 
% \begin{align}
%     \pp{c}{\kappa} =     
%     - \varepsilon \beta  A_{ij} (G_{ij}^{\bm{r}})^{\frac{3}{2}}
%     \Big| \Delta_{ij}^{r^1} \Big|.
% \end{align}
% This equation can now be preprocessed, since we can
% tabulate values of conductance (and therefore permeability and adhesivity) for values of $\kappa$.



\part{A multiscale model for filtration}
\label{part:A multiscale model for filtration}

\chapter{Introduction to the multiscale model}
\label{chap:mm_Introduction to the multiscale model}

\section{Multiple scales in filtration}
\label{sec:mm_Multiple scales in filration}
Filtration is an inherently multiscale process because particle and pore interactions that occur on the microscale affect macroscale filter properties \cite{iliev2014multiscale}. 
In Part \ref{part:A size-structured model for filtration}, 
we found that the size distributions 
of particles and pores dictate the mechanisms by which particles being captured affects the pore size, which further influences the rate of capture, and thus determines the effective properties of the filter, such as the shape of the flow rate profile. 
In Part \ref{part:A network model for filtration}, 
we established that the geometry of the microscale helps to determine macroscale properties of industrial interest, such as the particle removal efficiency and process termination time.
In special cases, either microscopic or macroscopic effects may dominate, so that operation properties are well predicted by independent microscale or macroscale simulations. 
Generally, however, microscale and macroscale effects are strongly coupled in filtration media, and consideration of multiple scales is required. 
% The macrocale behavour of the filter is clearly a function of interactions between indivuiidual particles and pores that occur on the microscale.

In some ePTFE filters, for example, pore sizes can range from around $0.02\mu$m to $15\mu$m \cite{ebnesajjad2017properties}. 
Although individual sheets of ePTFE are relatively thin, perhaps millimetres deep, it is quite common to stack these or combine them in some other way, so that the effective thickness is significantly larger.
% /of the material can be as large as a centimetre. 
As a consequence, process properties of industrial interest that depend on the entire depth, such as the proportion of the incoming particles that are retained across the membrane, are the result of processes that occur over an area of around a centimetre squared, say, which can contain as many as 1.4 billion 
pores \cite{ebnesajjad2017properties}.
 
This poses significant challenges in models that involve detailed resolution of the microscale or agent-based approaches that track the behaviour of individual particles. 
Non-specialist computers struggle with the memory demands of solvers that seek to solve pore network models (PNMs) like the one in Part \ref{part:A network model for filtration}
when the number of nodes and edges, which represent individual pores, exceeds the order of thousands. 
The problem worsens when industrialists are interested in obtaining optimal filter configurations, requiring repeated solution of models for cost function descents and parameter sweeps.
PNMs alone, then, are insufficient for at least some aspects of filtration modelling, and there is a need for alternative mathematical approaches.
% , becaus emethods that track individual partociels and pores start to struggle computationally with the number of interactions, even when we're only consdiering one dimension. This becomes an 
% even bigger problem when we want ot repeatednly calculate the solution, for things like parameteds sweeps that are necessary for perforamnce optimisation. 
% in a number of pores that is orders of magniutede larger than that feasible for calculation with network models. 

% Luckily, the geometry of ePTFE lends isteld to traditional multiple scales analysis, which is a popula area of mathemtical research.

% Explain what we're going to do in this part, and why.
% The network model is too expensive. 
% We need to make the network smaller. 
% Maybe it would work if we took a portion of the network that's representative and repeated it.
% So first we're going to assume that we've taken a periodic network. 
% We're going to present a formal way of averaging the network model in the case where the network is periodic, and arrive at an effective model. 
% Then we'll show that the effective model matches the 
% periodic network. 
% Then we'll explore under what conditions the effective model will match a non-periodic network... i.e., how to choose a cell that is representative of the non-periodic network.

% Many indusrtial problems involve flow through heterogeneous matreials, often with complex microscales [chapman]. As we have already discussed, this poses a challenge, because solution on the full microscale is often restrictively computationakly expensive, due to the complexity of the micrscale. 
% One common startegfy is to use methods to systematically average (homogenise) the microscale to obtain macroscale (effective; homogenised) properties. 
% This is generally the method of multilple scales,. 

\section{Existing multiscale models for filtration}
\label{sec:mm_Existing multiscale models for filtration}

Multiscale models are a useful tool for deriving the macroscopic behaviour of systems that contain microscale information \cite{hornung1997homogenization}.
 Classical examples include the derivation of: Darcy's law \cite{hornung1997homogenization,sanchez1986homogenization}; 
 the Navier--Stokes equations with various boundary conditions \cite{conca1985numerical};
and advection--diffusion in porous media \cite{auriault1995taylor}. 
 Homogenisation, the method by which effective equations are derived from microscale dynamics, 
 is an active area of research for filtration applications due to the inherent multiscale nature  of many filtration membranes (see Chapter \ref{chap:nm_Introduction to the network model}). 
Indeed, recent work includes \cite{dalwadi2015understanding},
in which Dalwadi \emph{et al.} homogenise the problem of flow through a filter with a near periodic microstructure coupled with particle transport owing to advection, diffusion, and filter adsorption, and use the resulting computationally efficient macroscale equations to investigate why porosity gradients (imposed on the microscale, by varying the pore size) can improve filter efficiency.
In \cite{bruna2015diffusion}, Bruna \emph{et al.} consider the problem of diffusion in a porous medium containing a collection of impenetrable spheres 
and find the corresponding effective diffusion coefficient using two different methods. 
Meanwhile, in \cite{printsypar2019influence}, Printsypar \emph{et al.} investigate how the microstructure of a porous medium influences overall filtration performance, by combining a homogenisation approach with an algorithm that accounts for the agglomeration of fibres when they grow until they touch each other, due to particle deposition.

In these examples,
% as well as many others, 
% the effective system 
% is derived by homogenising a system defined on a
the microscale 
consists of circles of given initial radii that represent fibres in the filter medium and thus obstructions to the suspension.
This is a typical approach, adopted by many authors.
In \cite{dalwadi2016multiscale}, for example, Dalwadi \emph{et al.} propose a similar setup and use the resulting homogenised model to predict the onset of filter clogging. 
This work typifies the homogenisation method, and we use it to outline the process used to average the problem of particle-laden flow through a filter. 

Dalwadi \emph{et al.}, \cite{dalwadi2016multiscale}, begin with the equations for Stoke's flow 
% for a particle-laden fluid through the microscale, 
coupled to an advection--diffusion equation that governs the evolution of the particle concentration. 
Deposition is imposed via a partially absorbing Robin boundary condition that results in growth of the fibres via a volume conservation law. 
The contaminants are small compared to the obstacles, and so particle--particle interactions are unimportant.
The authors proceed with homogenisation by introducing a `microscale' spatial variable that varies more quickly and, crucially, independently, of the `macroscale' one. 
The microscale coordinate is later removed again by introducing periodicity in this variable via repetition of a representative cell, which is a small area of the microscale. 
Intra-cell variation is accounted for with the microscale variable, while inter-cell variation is accounted for with the macroscale variable. 
Then, expanding the velocity, pressure, particle concentration, as well as any other dependent variables, 
in terms of the small parameter that separates the cell size from the filter size, 
the authors obtain effective equations, which 
govern the evolution of homogenised variables that have been averaged over the cell and are thus independent of the microscale variable at the asymptotic order of interest. 
The parameters of these resulting equations depend on the microscale information (the size of the obstacles in this case)
via a cell problem, which, in this example, 
links the effective permeability to the obstacle radius via an integral equation.
It follows that solution of this system allows the authors to obtain the effective permeability, and thus other effective properties of the filter, without losing all the information gained from having knowledge of the microscale, 
unlike purely macroscale models, which ignore microscale structure entirely.


% Explain the process of continuous homogeinsation... use taylor expansion to assume that srtuctsure exusts at any macroscale point and ariables are indpendent and then assume solution is a n asymptotic expansion and find macroscale system where parameteds depend on the microscale via solution of a cell problem which isually takes the form of a integral equation of differentual equation for some variable that dictates the varying structure of the  microscale.
% For example, in Dalwasi, the diffudsisngt of the macroscale system blah blah...

In this example, as well as many others, the microscale geometry is a continuum that is referenced by a continuous spatial variable. 
However, in studies of porous media (as we discussed in Part \ref{part:A network model for filtration}), 
it is common to simplify by approximating the microscale pore space with a 
network, with a discrete set of locations at which system variables, such as the particle concentration and fluid conductance, are measured (nodes and edges, respectively). 
This approximation has led to the derivation of multiscale systems in which the microscale is discrete (i.e., a network). 
The main advantage of homogenising a network over homogenising the exact geometry is the reduction in computational complexity, since in a network there are a finite number of spatial points to consider. 
Firstly, this allows for the study of a larger physical area. 
Secondly, this permits rapid repetition of solution, so that effective parameter \emph{distributions} may be obtained from systems with a stochastic element (like the one considered in Part \ref{part:A network model for filtration}, since the initial conditions are drawn from a distribution).
This Monte Carlo approach is infeasible when the microscale is fully resolved, like in the finite element {method}, for example. We generally refer to the process of obtaining effective equations on the macroscale continuum from network representations of the microscale as `network homogenisation'.

Much of the literature on network homogenisation 
involves the study of the emergence of continuum diffusion from discrete particle jumps on an underlying lattice.
The traditional approach, which is reviewed in \cite{bouchaud1990anomalous} 
in the context of the evolution of Brownian particles in disordered environments, 
uses generating functions and lattice Green functions (see also \cite{havlin1987diffusion, havlin1987diffusion}). 
% Although the key principles are the 
% same, one key difference between network
% and continuum homogenisation is that, 
% in the discrete case, the 
The first step is usually to split the spatial variable into one that gives the position in the cell and a second that references the position of the cell (see \cite{braun1998diffusion, koza1999general}). 
Since the probabilities of particle state transitions do not vary from one cell to another, the traditional process continues by taking a discrete Fourier transform with respect to the second index.

In recent work, by Chapman \emph{et al.} \cite{chapman2017effective, chapman2021homogenization}, however, a slightly different methodology is followed, which more closely 
corresponds to the standard continuous homogenisation method (in \cite{dalwadi2016multiscale}, for example). 
In the first of these, \cite{chapman2017effective}, Chapman \emph{et al.} begin by introducing a node index in the representative cell, which is $\varepsilon\ll 1$ times as small as the network as a whole. 
This index plays the role of the usual continuous microscale spatial variable. 
That is, the dependent variable of interest, the probability of finding a particle at node $i$ at the continuous macroscale position $\bm{x}$, in this case, 
is a function of $i$ and $\bm{x}$ independently. 
Carrying out a Taylor expansion about the point $\bm{x}$, 
and assuming an asymptotic form for the solution as usual, 
the authors obtain a master equation for the probability, which is now explicitly independent of the microscale index (and thus averaged over the cell) at leading order. 
Solutions of the master equation agree with solutions of the underlying dynamics in the asymptotic limit $\varepsilon\rightarrow 0$.
Only the parameters of the homogenised system, the effective diffusion coefficient in this case, depend on the microscale (the transition probabilities, here). 
As in the continuous case, this 
dependence occurs via a cell problem. 
In the discrete scenario, this takes the form of a linear matrix equation, which returns the `cell solution' as a function of the transition probabilities. 

In the second piece, \cite{chapman2021homogenization},
Chapman \emph{et al.}
use a generally similar method. 
This time, though, the microscale contains a continuum fluid rather than particles that transition discretely.
As such, fluid moves through the microscale due to mass conservation laws, 
% identical to 
% those discussed in Part \ref{part:A network model for filtration} (without particles).
as opposed to transition probabilities.
That is, the authors homogenise the problem of
incompressible flow through a network with specified initial edge conductances and node volumes. 
% This is decribed via a conservation equation for fluid mass and for particle mass. 
The microscale setup is thus identical to that in Part \ref{part:A network model for filtration}, 
except for the absence of 
particles in the flow. 
As a result, deposition does not occur, and the network's structure is static. 
Via a similar method as that in \cite{chapman2017effective},
% , because the flow does not contain particles. they emply a similar method as in Shabala, they inrtoduce node index to descirbe the microscla,e and introduce a cell refence that desribes the position of one node relative to another, to account for edges in the network. 
% They rthen taylor expan about a fixewd point, and assusme asymptotic solutions for the pressure and flow rate and particle concetnation and arrive at a continuous system 
Chapman \emph{et al.} \cite{chapman2021homogenization}
arrive at an effective system
defined on the continuum macroscale geometry that 
consists of a continuity equation and 
the Darcy equation, 
which relates the macroscale flow rate to the pressure gradient via the effective permeability. 
The permeability depends on the conductances of the network edges via the cell solution.
% , which is the solution.
% of the linear cell problem in which the matrix elements are made up of the constant edge conductances. 
The computational complexity of the cell problem is a function of the number of nodes in the cell, which is considerably less than that in the entire network.
This allows the authors to obtain a distribution for the effective permeability given only statistical information about the 
network, as opposed to the exact conductance of every edge, via repeated solution of the cell problem. 
In this way, Chapman \emph{et al.} \cite{chapman2021homogenization} examine the influence of the microscale structure, in terms of both the connectivity and the distribution of random conductances, upon the effective properties of the network.

% It is common, in the stufy of porous media, 
% to simplify firtehr more rare to conswider the microsclae as being discontinous, being formed by a network, and the macroscale being considered a continuum. There are some exapmples of this. Usuall this is done by  AGreens fucntions. 

% More recently, peoples have done this via the method that we're goign to use. OUtlin ethe method. Include Jon's work with Rebecca. 

% The most releveant example is arguably Chapman and Wilmoit. Explore single phase flow. Derive effecive permeability. 
% What is problematic in this work is that the network is static. There are no pariles in the flow. It is quialent to setign $\alpha=0$ in our model,m since we have seen that this corresponds to no deposition.
% They get an advection equation withthe permebaility dependnt on the microscale via a linear equation that deepdsn ont eh size of the representatie cell that is beign repeated. 

% In our work we will extend this to consider deposition. This is because our microclae equation contains an extra term that accounts for 
% loos of particles to deposition in the particle conservtation equation. We wil fidn that this adds a reaction term in the advection equation, so that we end up with an advection--reaction equation on the macroscale. As such we will require another effectiv parameter to govern the reactivity of the sysstem. The form of this swill be similar to the peremability and we will find that we can explore it in a simuilar way.

% In that paper, there is a n interest in the how the structure of the network affecrs the effedctgive bhacvioir of the filter. Because the microcale is random, exploration of this topic will invovle the random samplig of these paarameters over many example networks, so that what we end up with is a system where the parametrers are random, sinc ethey depend on the initial srtucture of the network.

\section{A novel multiscale model for filtration}
\label{sec:mm_A novel multiscale model for filtration}

In the remainder of Part \ref{part:A multiscale model for filtration}, 
we will extend the work in \cite{chapman2021homogenization} 
by homogenising the flow \emph{and deposition} of a \emph{particle-laden} fluid through a \emph{dynamic} 
network representation of the filter microscale, that is, the network model discussed in Part \ref{part:A network model for filtration}.

% that rest follows with section, what we expect ot be the key differene between our work and chapmans's, 
% section, 
% difference , 
% etc...

In Chapter \ref{chap:mm_Derivation of the multiscale model}, we will derive the multiscale model corresponding to the network model in Part \ref{part:A network model for filtration}. 
We will begin, in Section \ref{sec:mm_Microscale model}, 
by posing the network model on a periodic network that consists of repeated translation of a representative cell, and introducing the necessary notation, similar to that used in \cite{chapman2021homogenization}. 
The result is a dimensionless periodic network model to be homogenised, with initial conditions corresponding to Dirac-delta distributed of log-normally distributed initial pore conductances. 
In Section \ref{sec:mm_Network homogenisation methodology}, 
we outline the process of network homogenisation. 
In Section \ref{sec:1d_network-homogenisation}, we then carry out homogenisation of our model. 
We first focus on the quasi-one-dimensional case, in which a network with  nodes that are not necessarily connected to their spatially nearest neighbours is embedded in a single macroscopic dimension. 
Then, in Section \ref{sec:Dd_network-homogenisation}, 
we consider the case where a network with arbitrary (though still periodic) connectivity is embedded in a continuum of arbitrary dimension. 
We will see that this case is conceptually similar, although notationally more complex. Proof of the algebraic results that underpin our findings are also more challenging.
Sections \ref{sec:1d_network-homogenisation} and \ref{sec:Dd_network-homogenisation} proceed relatively similarly to the derivation in \cite{chapman2021homogenization}. 
The process is extended, however, by the need to homogenise the equation for the conservation of mass of particles, and 
the corresponding equation for the conservation of the volume of pores. 
As such, we will find that the problem deviates from that in  \cite{chapman2021homogenization} in so much as that the pore conductances, and the cell solutions from which they are derived, are functions of time, so that the structure of the underlying network is dynamic. Darcy's law, which is parametrised by a function of the microscale that represents the effective permeability, is identical to that derived in \cite{chapman2021homogenization}. In our model, 
this is coupled to an advection--reaction equation for the particle concentration, which is parametrised by a second microscale-dependent function, which we call the adhesivity, since it represents the effective ability of the filter to capture particles via adhesion based deposition, and thus the reactivity in the macroscale system.

In Chapter \ref{chap:mm_Solutions of the multiscale model}, we will explore solutions of our multiscale model.
We will begin, in Section \ref{sec:mm_Solution methodology}, by discussing a numerical scheme for solution of a reduction of the model that still contains its interesting properties. 
We will find that even the most straightforward solution method is less computationally complex than solution of the corresponding network model. 
Furthermore, we will suggest a pre-calculation step that further decreases 
computation time substantially.
In Section \ref{sec:mm_Periodic networks}, 
we will exhibit solutions of the multiscale model on various four-regular networks, to demonstrate the case where the underlying structure is indeed exactly periodic, as is assumed in the homogenisation process. 
% \ref{sec:nm_Non-random network: Mono-dispersed pores}, we will next 
% exbihit solutions of the model 
% 
Firstly, we will consider non-random edge conductances, which corresponds to the case where pore sizes are initially mono-dispersed. 
We will find that, despite vast improvements in computational complexity, 
the solution of the multiscale model agrees with that of the network model in the asymptotic limit where many small cells constitute the network.
Secondly,  we will consider log-normally random edge conductances, which corresponds to poly-dispersed initial pore sizes, 
and show that a general run of the multiscale model will agree qualitatively but not quantitatively with the underlying network solution, due to different random initialisations. 
As such, in Section \ref{sec:mm_Non-periodic networks}, we demonstrate the results of repeated calculation of the effective parameters on four regular networks, as well as those with more complex connectivity. 
We will show that the \emph{statistics}
of the multiscale solution do agree with that of the microscale solution in the asymptotic limit, provided that
the repeated cell is statiscally \emph{represantative} of the network 
as a whole. 
In particular, in \cite{chapman2021homogenization}, Chapman \emph{et al.} derive the sampled distribution of the effective permeability and show that its average represents the effective flow dynamics provided that the cell contains enough nodes and has approximately the correct coordination number. 
We find good agreement, and extend these conclusions to the effective adhesivity, thus showing that multiscale models derived from \emph{dynamic} network models are a valid approach for the prediction of the effective properties of filters in which particles deposit due to adhesion between particles and pores on the microscale  (that is perhaps not even periodic).

% In the first chapter wwe will outlin ethe process... 

% In the second section we wil sxplore solutions. blah.


\chapter{Derivation of the multiscale model}
\label{chap:mm_Derivation of the multiscale model}
In this chapter, we derive a multiscale model for filtration in which the microscale and macroscale geometry 
are accounted for simultaneously.
We begin by deriving a system of ordinary algebraic--differential equations on the microscale, which we model as a spatially periodic network of pores and junctions. 
We then develop an asymptotic method, called network homogenisation, to average this system.
Carrying out this method on the periodic-network model, 
we arrive at a system of partial differential equations on the macroscale, which is modelled as a continuum of porous material. 
The parameters of this continuous system, which model the permeability and adhesivity of the membrane, 
depend on the dynamically varying conductances of the pores in the underlying network, 
so that the macroscale dynamics are coupled with the behaviour of microscale.

\section{Microscale model} 
\label{sec:mm_Microscale model}
We begin by modelling the microscale of the filter as a periodic network. 
We first derive a dimensional system of equations that is similar to the non-periodic network model in Part \ref{part:A network model for filtration}, but accounts for simplifications that arise from the periodicity of the network. 
We then nondimensionalise this model, scaling to account for the assumption that the network contains
many cells in the direction of dominant flow.


\subsection{Dimensional model}
As in Part \ref{part:A network model for filtration}, we consider the pressure driven flow of a particle-laden fluid through a filter 
that is modelled as a network of pores and junctions.
Particles in the flow deposit on pore walls due to a mutual adhesivity between the particles and pores, which decreases their cross-sectional area and thus their conductance, so that the structure of the network 
is dynamic. 
Here, unlike in Part \ref{part:A network model for filtration} (see the non-periodic network illustrated in Figure \ref{fig:nm_network_1}), 
the filter is spatially periodic, so that the network is constructed by repeated translation of a cell in $D$-dimensional space, where $D \in \{ 1,2,3\}$, as illustrated in Figure \ref{fig:mm_network_1}. 

\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.90\textwidth]{./figures/multiscale/drawings/network_1.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{A diagram of a two-dimensional filter composed of a periodic network formed from a cell that is repeated vertically and horizontally. Particles in the flow enter through edges crossing the left boundary and exit through those crossing the right boundary, after traversing the network.}        
        \label{fig:mm_network_1}
    \end{center}
\end{figure}

% \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
% \put(60,10){This is text}
% \end{overpic}

% We define $Q_{ij}$ to be the volumetric flux of fluid from node $i$ to node $j$, with units $[Q_{ij}]=\text{m}^D\text{s}^{-1}$. The flux satisfies
% \begin{align}
%     Q_{ij} = 
%     \begin{cases}
%         \phantom{-} |Q_{ij}| & P_i > P_j,
%         \\
%         -|Q_{ij}| & P_i < P_j,
%     \end{cases}
% \end{align}
% where $P_i$ is the pressure in node $i$, with units $[P_i]=\text{Pa}$.
% Denoting the conductance of the edge from $i$ to $j$, which we call edge $ij$, by $G_{ij}$, it follows that
% \begin{align}
%     Q_{ij} = G_{ij}(P_i-P_j),
%     \label{eq:mm_flux-definition_sum_nodim_1}
% \end{align}
% so that $[G_{ij}]=\text{m}^D\text{s}^{-1}\text{Pa}^{-1}$.
% Since mass is conserved at each node, the fluid entering node $i$ from all other nodes equals that exiting node $i$ to all others, and so
% \begin{align}
%     \sum_{j=1} Q_{ij} = 0,
%     \label{eq:conservation-of-mass_without-r_dim}
% \end{align}
% where the sum is over all nodes in the network.

% Conservation of particles implies that 
% \begin{align}
%     V_i\pp{C_i}{\tau} &= \sum_{j=1}  \Big(
%     Q_{ji} C_j H(Q_{ji}) - Q_{ij} C_i H(Q_{ij}) -  A_{ji} Q_{ji} C_j H(Q_{ji})  \Big).
%     \label{eq:mm_particle-conservation_sum_nodim_1_without-r_dim}
% \end{align}
% Here, $V_i$ is the volume of node $i$ so that $[V_i]=\text{m}^D$, while $C_i$ is the concentration of particles in the fluid at this node, and has units $[C_i]=\text{m}^{-D}$.
% Meanwhile, $A_{ij}$, with $[A_{ij}]=1$, is the adhesivity of edge $ij$, which is the fraction by which the particle concentration decreases on edge $ij$ after deposition. 
% The adhesivity is an $\mathcal{O}(\varepsilon)$ quantity, which ensures that an $\mathcal{O}(1)$ amount of contaminant is removed after passing through the entire filter, which requires traversing $\varepsilon^{-1}$ cells.
% Finally, $H$ denotes the Heaviside function, so that 
% \begin{align}
%     H(Q_{ij})=
%     \begin{cases}
%         1 & Q_{ij}>0,
%         \\
%         0 & Q_{ij}\le 0.
%     \end{cases}
% \end{align}
% In the first and second terms on the right-hand side of \eqref{eq:mm_particle-conservation_sum_nodim_1_without-r_dim}, its presence ensures that particles are added to node $i$ from upstream nodes and leave node $i$ to downstream nodes, respectively. These terms describe the advection of particles through the network. 
% In the third term, $H$ ensures that particles are removed in edges upstream of node $i$, and this term thus describes particle deposition.

% The system is closed by an equation that describes the decrease in the conductance of each edge $ij$ due to deposition, say 
% \begin{align}
%     G_{ij} = G(\tau,C_i,P_i,Q_{ij},A_{ij}).
%     \label{eq:conductance_without-r_dim}
% \end{align}
% Here, $G$ is some rule, to be specified, which dictates the evolution of the network.

% A constant pressure drop is enforced across the filter, so that the pressures at the inlet and outlet, $p_{\text{in}}$ and $p_{\text{out}}$, are known. 
% We are also given the concentration of particles in the incoming fluid, $c_{\text{in}}$.
% It follows that 
% \begin{subequations} \label{eq:bc_in_dim}
% \begin{align}
%     P_i = p_{\text{in}}, 
%     \quad
%     C_i = c_{\text{in}}
%     \tag{\ref{eq:bc_in_dim}a,b}
% \end{align}
% \end{subequations}
% at inlet nodes, which are nodes connected to edges that conduct fluid into the network, while
% \begin{align}
%     P_i = p_{\text{out}} \label{eq:bc_out_dim}
% \end{align}
% at nodes connected to edges that conduct fluid out of the network, which we call outlet nodes. 
% There are no particles in the filter initially, 
% and so 
% \begin{align}
%     C_i = 0 \quad \text{at} \quad \tau=0. \label{eq:ic_dim}
% \end{align}
% The system \eqref{eq:mm_flux-definition_sum_nodim_1}--\eqref{eq:ic_dim} defines our dimensional network model.
% Note that it is independent of node positions, and is instead structure-dependent via the conductances. % $G_{ij}$.
The basic setup is the same as that in the dimensional network model from Part \ref{part:A network model for filtration}, 
\eqref{eq:nm_fluid-conservation_sum_dim_1}--\eqref{eq:nm_cond-pdf_sum_dim_1}. 
We call $\varepsilon$ the ratio of the cell length to the filter length, so 
that $\varepsilon^{-1}$ cells constitute the filter depth (see Figure~\ref{fig:mm_network_1}).
When the network is non-periodic, pores and junctions are referenced by indices $i$ and $j$ 
that vary between $1$ and the number of junctions in the filter, $N_{\text{f}}$, say. 
Here, however, we account for the periodicity of the network by introducing one further index called the cell reference, $\bm{r}=(r^1,\hdots,r^D)^{\top}$, which is a vector that describes the position of a cell containing one junction relative to the cell containing another.
We use $i$, $j$, and $\bm{r}$, together, to index a pore, 
so that, for example, $G_{ij}^{\bm{r}}$ is the conductance of the pore between junction $i$ in one cell and junction 
$j$ in a cell at position $\bm{r}$ relative to the first one.
In a two-dimensional network, therefore, $G_{1\,2}^{(1,0)^{\top}}$ denotes the conductance of the pore connecting junction $1$ in one cell to junction $2$ in the cell directly to the right of the first cell.
This convention is illustrated in Figure \ref{fig:mm_network_3-4}.
% [figure -- in a two-dimensional network, $\bm{x} + \varepsilon \bm{X}_i^{\bm{r}=(0,0)}$ is the position of node $i$ in the reference cell, $\bm{x} + \varepsilon \bm{X}_i^{\bm{r}=(1,0)} = \bm{x} + \varepsilon \Big(\bm{X}_i^{\bm{r}=(0,0)} + (1,0) \circ (l^1, l^2) \Big)$ is the position of node $i$ in the cell directly to the right of the reference cell, and $G_{ij}^{\bm{r}=(0,1)}$ denotes the conductance of the edge connecting node $i$ in the reference cell to node $j$ in the neighbouring cell above.]
The indices $i$ and $j$ now vary between $1$ and the number of junctions in the cell, $N$, say.
Junctions in one cell are connected to junctions in another cell up to $K$ cells away, and so the set of possible values of each $r^m$ is $\mathcal{R}=\{-K,\hdots,-1,0,1,\hdots,K\}$, so that the set of possible $\bm{r}$ is $\bm{\mathcal{R}} = \mathcal{R}^D$.
For example, when $K=1$, junctions in a cell are connected to junctions in the same cell or its nearest-neighbouring cells, and $\mathcal{R}=\{-1,0,1\}$ so that $\bm{\mathcal{R}}=\{-1,0,1\}^
D$ (see Figure \ref{fig:mm_network_3-4} (Top)). 
When $K=2$, pores cross up-to two cell boundaries, and $\mathcal{R}=\{-2,-1,0,1,2\}$ so that $\bm{\mathcal{R}}=\{-2,-1,0,1,2\}^D$ (see Figure \ref{fig:mm_network_3-4} (Bottom)).
The symmetric structure of $\mathcal{R}$ follows by noting that if $r^m$ is an element of $\mathcal{R}$, then so too is $-r^m$. 
This is because, for any $i$, $j$, and $\bm{r}$, we have $G_{ij}^{\bm{r}}=G_{ji}^{-\bm{r}}$, by periodicity of the network (see Figure \ref{fig:mm_network_3-4}).
Note that non-planar networks, like the ones in Figure \ref{fig:mm_network_3-4}, are permitted.
\newpage
\begin{figure}[!htb]
    \begin{center}
        \includegraphics[width=0.71\textwidth]{./figures/multiscale/drawings/network_3-4.pdf}
        \caption{Diagrams of sections of two periodic networks that are embedded in two-dimensional space. 
        The conductances of some pores are labelled to illustrate the use of the lower indices, $i$ and $j$, and the upper index, $\bm{r}$, 
        to index the pore between junction $i$ in one cell and junction $j$ in a second cell at position $\bm{r}$ relative to the first.
        (Top)
        A cell of $N=9$ junctions repeated nine times. 
        Junctions in one cell are connected to junctions in cells up to one cell away. 
        It follows that $r^1$ and $r^2$ are drawn from the set $\mathcal{R}=\{-1,0,1\}$, 
        so that $\bm{r}$ is drawn from $\bm{\mathcal{R}}=\{-1,0,1\}^2$.
        (Bottom)
        A cell of $N=2$ junctions repeated four times.
        Junctions in one cell are connected to junctions in cells up to two cells away. 
        It follows that $r^1$ and $r^2$ are drawn from the set $\mathcal{R}=\{-2,-1,0,1,2\}$, 
        so that $\bm{r}$ is drawn from $\bm{\mathcal{R}}=\{-2,-1,0,1,2\}^2$.}
        \label{fig:mm_network_3-4}
    \end{center}
\end{figure}


% \begin{figure}[!htb]
%     \begin{center}
%          \includegraphics[width=0.77\textwidth]{./figures/multiscale/drawings/network_3.pdf}
%         \caption{A diagram of a two-dimensional periodic network composed of a cell of $N=9$ nodes repeated nine times. The conductance on some edges is labelled to illustrate the use of lower indices with $\bm{r}$ to describe which two nodes are connected.
%         In this case, nodes in one cell are connected to nodes in cells up to one cell away, so that $r^1$ and $r^2$ are drawn from the set $\mathcal{R}=\{-1,0,1\}$ and $\bm{r}$ is drawn from $\bm{\mathcal{R}}=\{-1,0,1\}^2$.}         
%         \label{fig:network_3}
%     \end{center}
% \end{figure}
% \begin{figure}[!htb]
%     \begin{center}
%          \includegraphics[width=1.0\textwidth]{./figures/multiscale/drawings/network_4.pdf}
%         \caption{A diagram of a two-dimensional periodic network composed of a cell of $N=2$ nodes repeated four times. The conductance on some edges is labelled to illustrate the use of lower indices combined with $\bm{r}$ to describe which two nodes are connected.
%         For each edge $ij$, we have $G_{ij}^{\bm{r}}=G_{ji}^{-\bm{r}}$, since the network is periodic.
%         In this case, nodes in one cell are connected to nodes in cells up to two cells away, so that $r^1$ and $r^2$ are drawn from the set $\mathcal{R}=\{-2,-1,0,1,2\}$ and $\bm{r}$ is drawn from $\bm{\mathcal{R}}=\{-2,-1,0,1,2\}^2$.}         
%         \label{fig:network_4}
%     \end{center}
% \end{figure}
\newpage
Proceeding similarly to Part \ref{part:A network model for filtration}, we arrive at a dimensional 
network model similar to \eqref{eq:nm_fluid-conservation_sum_dim_1}--\eqref{eq:nm_cond-pdf_sum_dim_1}. 
Accounting for the periodicity of the network, 
the equations for the conservation of fluid, particles, and pores, \eqref{eq:nm_fluid-conservation_sum_dim_1}, \eqref{eq:nm_particle-conservation_sum_dim_1}, and \eqref{eq:nm_conductance-conservation_1_sum_dim}, become
\begin{align}
    \sum_{j=1}^{N} \sum_{\bm{r}\in\bm{\mathcal{R}}} Q\indx &= 0,
    \label{eq:mm_fluid-conservation_sum_dim_1}
    \\
    \bar{V}\pp{C_i}{T} &= \sum_{j=1}^{N} \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big(
    (1-\bar{A})Q_{ji}^{-\bm{r}} C_j H(Q_{ji}^{-\bm{r}}) - Q\indx C_i H(Q_{ij}^{\bm{r}})  \Big),
    \label{eq:mm_particle-conservation_sum_dim_1}
    \\
    \pp{G\indx}{T} &= - \frac{2S\bar{A}}{(8 \pi \mu \bar{L})^{\frac{1}{2}}} C\indx (G\indx)^{\frac{3}{2}}|P_i-P_j|,
    \label{eq:mm_conductance-conservation_1_sum_dim}
\end{align} 
respectively,
where the flux down each pore is related to the pressure drop and conductance of the pore via
\begin{align}
    Q\indx= G\indx\left(P_i-P_j\right).
    \label{eq:mm_flux-definition_sum_dim_1}
\end{align}
Note that, in the summations here, lower indices, $i$ and $j$, vary from $1$ to the number of junctions in the repeating cell, $N$. 
Sums over the upper index, $\bm{r}$, are over all possible cell references in the set $\bm{\mathcal{R}}=\{-K,\hdots-1,0,1\hdots,K\}$, the size of which depends on $K$. 
% When these two sums appear together, as in \eqref{eq:mm_fluid-conservation_sum_dim_1} and \eqref{eq:mm_particle-conservation_sum_dim_1}, they represent a sum over all nodes in the network, $N_{\text{f}}$.
% Here, 
% \begin{align}
%     H(P_i-P_j)=
%     \begin{cases}
%         1 & P_i-P_j>0,
%         \\
%         0 & P_j-P_i\le 0,
%     \end{cases}
% \end{align}
% indicates the direction of flow through pores.

As in Part \ref{part:A network model for filtration}, we enforce a constant pressure drop 
across the filter, and a constant particle concentration 
at the inlet, so 
that the boundary conditions are 
\begin{subequations} \label{eq:mm_bc_pres_sum_dim_1}
\begin{align}
    P_{\text{in}}(T) = p^*_{\text{in}}, 
    \quad
    P_{\text{out}}(T) = p^*_{\text{out}},
    \tag{\ref{eq:mm_bc_pres_sum_dim_1}a,b}
\end{align}
\end{subequations}
and 
\begin{align}
    C_{\text{in}}(T) = c_{\text{in}}^*,
    \label{eq:mm_bc_conc_sum_dim_1}
\end{align}
for $T\ge0$.
There are no particles in the filter initially, and 
initial pore radii, and therefore pore conductances, 
are drawn from the distribution $\mathcal{G}$, which we assume is either a Dirac-delta distribution, or 
a log-normal distribution, to exemplify solutions of the model. 
Thus the initial conditions are  
\begin{align}
    C_i(0) = 0, 
    \label{eq:mm_ic_conc_sum_dim_1}
\end{align}
for $i \in \{1,\hdots,N, \text{out} \}$, 
and 
\begin{align}
    G\indx(0) = G\indxzero,
    \label{eq:mm_ic_cond_sum_dim_1}  
\end{align}
for $i,j \in \{1,\hdots,N,\text{in}, \text{out}\}$, 
where the probability density function (PDF) of $\mathcal{G}$ is either
\begin{subequations}
\label{eq:mm_cond-pdf_sum_dim_1}
\begin{align}
    \mathscr{g}(G\indxzero) = \updelta(G\indxzero-\bar{G}), 
    \quad \text{or} \quad
    \mathscr{g}(G\indxzero)=\frac{1}{\sqrt{2 \pi} \sigma_G G\indxzero} 
    \exp{
    \left(- \frac{ 
    \left( 
    \log \left( \frac{G\indxzero}{\bar{G}} \right) + \frac{\sigma_G^2}{2} 
    \right)^2
    }{
    2 \sigma_G^2}
    \right)}.
    \tag{\ref{eq:mm_cond-pdf_sum_dim_1}a,b}    
    \end{align}
\end{subequations}

The equations \eqref{eq:mm_fluid-conservation_sum_dim_1}--\eqref{eq:mm_cond-pdf_sum_dim_1} 
form a closed system for the solution variables, $Q\indx$, $C_{i}$, $P_{i}$, and $G\indx$,
on a periodic network,
subject to the specification of the dimensional parameters, $\bar{L}$, $\bar{V}$, $\bar{A}$, 
$S$, $\mu$, $p_{\text{in}}^*$, $p_{\text{out}}^*$, and $c_{\text{in}}^*$, 
along with any other parameters that are needed to specify the probability density functions 
for the initial conductance distribution from which $G\indxzero$ are drawn, $\mathcal{G}$, such as $\bar{G}$ and $\sigma_G$ in \eqref{eq:mm_cond-pdf_sum_dim_1}.
As in the non-periodic case, this network model is independent of node positions. The periodic 
connectivity of the network enters the system through the pore conductances, $G\indx$.
In the next section we will nondimensionalise the dimensional network model \eqref{eq:mm_fluid-conservation_sum_dim_1}--\eqref{eq:mm_cond-pdf_sum_dim_1}. 



\subsection{Dimensionless model}
% Unlike in the non-periodic-network model in Part \ref{part:A network model for filtration}, 
In the subsequent sections of this chapter, we will focus on deriving a macroscale system from 
our microscale model, \eqref{eq:mm_fluid-conservation_sum_dim_1}--\eqref{eq:mm_cond-pdf_sum_dim_1}. 
To this end, we will take a network containing a fixed number of junctions in each cell (so that $\delta$ from Part \ref{part:A network model for filtration} is fixed) and consider the case 
in which the network contains many infinitesimally small cells, which corresponds to the asymptotic limit $\varepsilon\rightarrow0$. 
The aim of our nondimensionalisation 
% of the periodic-network microscale model, \eqref{eq:mm_fluid-conservation_sum_dim_1}--\eqref{eq:mm_cond-pdf_sum_dim_1}, 
is to arrive at a scaled system for which this analysis is simple.
% for which it is simple to compare the solutions of systems defined on networks that contain different numbers of pores and junctions.
We therefore scale periodic variables 
% similarly to the non-periodic variables in \eqref{eq:nm_scaling_1} and \eqref{eq:nm_scaling_2}, 
so that 
\begin{align}
    & 
    L\indx = \bar{L} \tilde{L}\indx,
    \quad
    V_i=\bar{V}\tilde{V}_i,
    \quad
    P_i=(p_{\text{in}}-p_{\text{out}})\tilde{P}_i + p_{\text{out}},
    \quad
    C_i=c_{\text{in}}\tilde{C}_i,
    \label{eq:mm_scaling_1}
    \\
    &
    Q\indx=\bar{G}(p_{\text{in}}-p_{\text{out}})\tilde{Q}\indx
    \quad
    G\indx = \frac{\bar{G}}{{\varepsilon}}\tilde{G}\indx, 
    \quad
    T=\frac{ \varepsilon \bar{V} }{\bar{G}(p_{\text{in}}-p_{\text{out}})}\tilde{T}.
    \label{eq:mm_scaling_2}
\end{align}
Similarly to our scaling of the non-periodic variables in \eqref{eq:nm_scaling_1} and \eqref{eq:nm_scaling_2}, 
the scalings in \eqref{eq:mm_scaling_1} mean that
pores and junctions all have unit length and volume initially, respectively,
the feed has unit particle concentration, 
and the filter has a unit pressure drop enforced across it.
We will find (see \eqref{eq:1d_concentration-equation_microscale}), 
that the timescale in \eqref{eq:mm_scaling_2}, 
\begin{align}
    T_{\text{c}} = \frac{\varepsilon \bar{V}}{\bar{G} (p_{\text{in}}-p_{\text{out}})},
    \label{eq:mm_define-T_c}
\end{align}
is the average time it takes fluid to cross a cell.
% is the average time it takes fluid to traverse the filter.
% Fluid reaches the outlet for the first time at $T=\mathcal{O}(1)$, 
% independently of the number of junctions in the cell, $\delta^{-1}$
% and the number of cells in the network $\varepsilon^{-1}$, 
% which are defined in \eqref{eq:nm_definition-of-delt-and-epsi},
% and thus independently of the average number of 
% pores that fluid traverses to travel from the inlet to the outlet the filter, which is
% $\mathcal{O}((\delta\varepsilon)^{-1})$ (see Figure \ref{fig:nm_network_4}).
% It follows that solutions on networks with different numbers of pores and junctions can be compared with ease.

In dimensionless variables, dropping tildes, the equations for the conservation of fluid, particles, and pores, are
\begin{align}
    \sum_{j=1}^{N} \sum_{\bm{r}\in\bm{\mathcal{R}}} Q\indx &= 0,
    \label{eq:mm_fluid-conservation_sum_nodim_1}
    \\
    \pp{C_i}{T} &= \sum_{j=1}^{N} \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Big(
    (1-\alpha \varepsilon)Q_{ji}^{-\bm{r}} C_j H(Q_{ji}^{-\bm{r}}) - Q\indx C_i H(Q\indx )  \Big),
    \label{eq:mm_particle-conservation_sum_nodim_1} 
    \\
    \pp{G\indx}{T} &= - \alpha \beta C\indx (G\indx)^{\frac{3}{2}} |P_i-P_j|,
    %note that there would be an invisible L^1/2 on the numerator if it wasnt 1.
\label{eq:mm_conductance-conservation_sum_nodim_1}
\end{align} 
where the flux down each pore is related to the pressure drop and conductance of the pore via
\begin{align}
    Q\indx= \frac{G\indx}{\varepsilon}\left(P_i-P_j\right).
    \label{eq:mm_flux-definition_sum_nodim_1}
\end{align}
% Here, 
% \begin{align}
%     H(P_i-P_j)=
%     \begin{cases}
%         1 & P_i-P_j>0,
%         \\
%         0 & P_j-P_i\le 0,
%     \end{cases}
% \end{align}
% indicates the direction of flow through pores.
The boundary conditions are 
\begin{subequations} \label{eq:mm_bc_pres_sum_nodim_1}
\begin{align}
    P_{\text{in}}(T) = 1, 
    \quad
    P_{\text{out}}(T) = 0,
    \tag{\ref{eq:mm_bc_pres_sum_nodim_1}a,b}
\end{align}
\end{subequations}
and 
\begin{align}
    C_{\text{in}}(T) = 1,
    \label{eq:mm_ic_pres_sum_nodim_1}
\end{align}
for $T\ge0$, 
and the initial conditions are  
\begin{align}
    C_i(0) = 0, 
    \label{eq:mm_ic_conc_sum_nodim_1}
\end{align}
for $i \in \{1,\hdots,N, \text{out} \}$, 
and 
\begin{align}
    G_{ij}^{\bm{r}}(0) = G_{ij}^{\bm{r}0},
    \label{eq:mm_ic_cond_sum_nodim_1}
\end{align}
for $i,j \in \{1,\hdots,N,\text{in}, \text{out}\}$.
This time, the dimensionless parameters, in terms of the dimensional ones, are 
\begin{align}
    \alpha = \frac{\bar{A}}{\varepsilon},
    \label{eq:mm_definition-dimles-parameters_sum_nodim_1}
    \quad
    \beta =  \frac{2S \bar{V} c_{\text{in}} \varepsilon^{\frac{3}{2}}}{(8\pi \mu \bar{L} \bar{G})^{\frac{1}{2}}  }, 
    \quad
    % \delta = \frac{1}{\sqrt[D]{N}},
    % \quad
    \varepsilon = \frac{1}{N^1}.
\end{align}
We will find (see Section \ref{sec:1d_network-homogenisation}), 
that the factor of $\varepsilon^{-1}$ in $\alpha$ ensures that particle deposition occurs at 
$\mathcal{O}(\varepsilon)$ in each cell, which results in $\mathcal{O}(1)$ removal over the filter for $\alpha=\mathcal{O}(1)$.
Similarly, the extra factor of $\varepsilon$ in $\beta$ ensures $\mathcal{O}(1)$ conductance variation 
for $\alpha=\mathcal{O}(1)$ and $\beta=\mathcal{O}(1)$.
% We showed, in \ref{sec:nm_Non-random network: Mono-dispersed pores}, 
% that the factor of $\delta \varepsilon$ in $\alpha$ ensures that when $\alpha=\mathcal{O}(1)$ then there is $\mathcal{O}(1)$ removal 
% of particles by the time that they reach the outlet, while the factor of $\varepsilon^{1/2}$ in $\beta$ is to ensure that when $\alpha=\mathcal{O}(1)$ and $\beta=\mathcal{O}(1)$
% there is an $\mathcal{O}(1)$ reduction in the conductance of pores near the inlet by the time that fluid reaches the outlet. 
The dimensionless initial pore conductances, $G\indxzero$, are drawn from the initial conductance distribution, $\mathcal{G}$, 
which has a pdf that is either given by 
\begin{subequations}
\label{eq:mm_cond-pdf_sum_nodim_1}
\begin{align}
    \mathscr{g}(G_{ij}^{\bm{r}0}) = \updelta(G_{ij}^{\bm{r}0}-1), 
    \quad \text{or} \quad
    \mathscr{g}(G_{ij}^{\bm{r}0})=\frac{1}{\sqrt{2 \pi} \sigma_G G_{ij}^{\bm{r}0}} 
    \exp{
    \left(- \frac{ 
    \left( 
    \log (G_{ij}^{\bm{r}0}) + \frac{\sigma_G^2}{2} 
    \right)^2
    }{
    2 \sigma_G^2}
    \right)},
    \tag{\ref{eq:mm_cond-pdf_sum_nodim_1}a,b}    
    \end{align}
\end{subequations}
so that, either pores have unit conductance initially, or their conductances 
are distributed log-normally about a unit mean.

The equations \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_ic_cond_sum_nodim_1} form a closed system for the dimensionless periodic-variables $Q\indx$, $C_i$, $P_i$, and $G\indx$,
given the dimensionless parameters 
$\alpha$, $\beta$, and $\varepsilon$. 
These are defined in terms of the dimensional parameters in \eqref{eq:nm_definition-dimles-parameters_sum_nodim_1}, 
and any further dimensionless parameters, 
for example $\sigma_G$ in \eqref{eq:mm_cond-pdf_sum_nodim_1}, that 
are needed to specify the initial conductance distribution, $\mathcal{G}$, from which the initial pore conductances, $G\indxzero$, are drawn.

In this section, we have extended the non-periodic network model that we derived in Part \ref{part:A network model for filtration}, 
\eqref{eq:nm_fluid-conservation_sum_nodim_1}--\eqref{eq:nm_ic_cond_sum_nodim_1}, 
to a periodic-network model, \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_ic_cond_sum_nodim_1}.
In the remainder of this chapter, we will develop an asymptotic method to systematically homogenise the system, 
and use this method to derive an effective system that is defined on the macroscale and microscale geometry of the filter simultaneously.

\section{Network homogenisation methodology}
\label{sec:mm_Network homogenisation methodology}

We are interested in the case where the ratio of the cell length to the network length, $\varepsilon$ defined in \eqref{eq:mm_definition-dimles-parameters_sum_nodim_1}, is small, $\varepsilon\ll1$, so that many cells constitute the network in the dominant direction of the flow.
In this scenario, there are two scales in our model, which are separated by $\varepsilon$. 
The microscale is the scale of each cell, while the macroscale is the scale of the entire network.
In the limit $\varepsilon\rightarrow0$, we will derive a continuous approximation of the discrete periodic-network model,
\eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_ic_cond_sum_nodim_1}, by using the following steps:
\begin{enumerate}[label={Step \arabic*.}, ref={\arabic*}, leftmargin=1.5cm]
    \item  Evaluate all microscale-dependent variables, by which we mean variables indexed by $i,j$, and $\bm{r}$, relative to a common, continuous point, and assume that these variables depend on the macroscopic continuous spatial coordinate and the microscale index \emph{independently}. 
    \label{step:1}
    \vspace{-0.0cm}
    \item Carry out a Taylor-series expansion of each of these variables about the common point in terms of the small parameter, $\varepsilon$.
    \label{step:2}
    \item Assume an asymptotic expansion in terms of the small parameter, $\varepsilon$, as the ansatz for each variable for which the solution is sought.
    \label{step:3}
    \item Solve the resulting master equation at each asymptotic order. Higher-order problems may need to be considered to close lower order problems, to specify the macroscale solution variable. 
    \label{step:4}
\end{enumerate}
We call this process network homogenisation, 
which is a variation of the usual method of multiple scales, in which the microscale consists of discrete spatial points.

To carry out network homogenisation, therefore, we must define discrete spatial points at which the solution variables of the microscale model, 
\eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_ic_cond_sum_nodim_1}, 
$Q\indx$, $C_i$, $P_i$, and $G\indx$, 
are defined.
% Above, and by convention, we use uppercase letters to denote microscale variables. These are quantities defined on a node, which we refer to using a single lower index, or an edge, denoted by two lower indices.
% Lowercase letters, meanwhile, refer to macroscale variables. We will find that these are not indexed by lower indices since they do not vary on the microscale, which means they are node and edge independent.
% Since the network is embedded within $D$-dimensional space,  (where $D \in \{1,2,3\}$, although all of our formulae generalise in an obvious way to an arbitrary number of dimensions), we use upper indices to represent the directional component so that, for example, the spatial variable is $\bm{x}=(x^1,\hdots,x^D)^{\top}$. Following this convention, we define $\bm{l}$ so that $l^m$ (for $m \in \{1,\hdots,D\}$) is the length of the entire network in the $m$ direction and $\varepsilon l^m$ is the length of each cell in the $m$ direction. 
To this end, we define $\bm{X}_i$ so that $\varepsilon X_i^m$ is the component of the position of junction $i$ in the $x^m$ direction relative to some fixed point $\bm{x}$.
It follows that $\bm{x}+\varepsilon\bm{X}_i$ is the position of junction $i$ in the continuous macroscopic spatial variable.
We exemplify this convention in Figure \ref{fig:network_2}.
\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.85\textwidth]{./figures/multiscale/drawings/network_2.pdf}
        \caption{A diagram of a section of a two-dimensional filter with two cells shown. 
        The point $\bm{x}$, the lengths of the cell boundaries, and the distances between nodes $i$ and $j$ and the cell boundaries are labelled.
        Lower indices reference nodes and edges, while upper indices refer to dimensional directions.}        
        \label{fig:network_2}
    \end{center}
\end{figure}
Since the network is periodic, it follows that junction $j$ in the cell at position $\bm{r}$ relative to the cell 
containing $i$ is located at the point $\bm{x}+\varepsilon(\bm{X}_j + \bm{r} \circ\bm{l})$,
where $\circ$ denotes the Hadamard product ($\bm{r} \circ\bm{l} = (r^1l^1, \hdots, r^Dl^D)^{\top}$), 
and $\bm{l}$ is a vector containing the dimensions of the filter, 
so that $l^m$ is the length of the filter in the $x^m$ direction (see Figure \ref{fig:network_2}).
It remains to specify the points, $\varepsilon \bm{X}^{\bm{r}}_{ij}$, say, at which pores are defined (relative to $\bm{x}$).
We choose the midpoint between nodes $i$ and $j$, which we call $\varepsilon \bm{M}_{ij}^{ \bm{r}}$, so that $\varepsilon M_{ij}^{ \bm{r}m}$ is the component of this midpoint in direction $x^m$, as illustrated in Figure \ref{fig:network_5}.
% A simple geometric argument shows that
\begin{align}
    M_{ij}^{\bm{r}m}=\frac{X_j^m+r^ml^m+X_i^m}{2}.
    \label{eq:definition-of-midpoint}
\end{align}

To demonstrate network homogenisation, we define the fixed point $\bm{x}$ to be the left-bottom corner of a particular cell, which we will call the reference cell, and consider Steps \ref{step:1}--\ref{step:4} in the context of three example quantities: 
\begin{enumerate}[label={Quantity \arabic*.}, ref={\arabic*}, leftmargin=2.3cm]
    \item The pressure at junction $i$, $P_i$, in the reference cell (e.g., see $i$ in the left cell in Figure \ref{fig:network_5}).
    \label{quantity:1}
    \item The pressure at junction $j$, $P_j$, which is in a cell at position $\bm{r}$ relative to the reference cell (e.g., see $j$ in the cell at position $\bm{r}=(1,0)^\top$ (i.e., right) relative to the cell that contains $i$, in Figure \ref{fig:network_5}).
    \label{quantity:2}
    \item 
    The conductance of pore  $ij\bm{r}$, $G_{ij}^{\bm{r}}$ (i.e., the pore that connects junction $i$ to junction $j$ in the cell at position $\bm{r}$ relative to the cell containing junction $i$, see Figure \ref{fig:network_5}).
    \label{quantity:3}
\end{enumerate}
Quantities \ref{quantity:1}--\ref{quantity:3}, and the positions at which they are defined in the example case, $D=2$ and $\bm{r}=(1,0)$, are illustrated in Figure \ref{fig:network_5}.
\begin{figure}[!htb]
    \begin{center}
         \includegraphics[width=0.85\textwidth]{./figures/multiscale/drawings/network_5.pdf}
        \caption{A diagram of two adjacent two-dimensional cells. The position $\bm{x}$ in the reference cell, and the positions of the node $i$ in the reference cell, node $j$ in the right-neighbouring cell, and the midpoint between these nodes are labelled above the points.
        Since node $i$ is in the reference cell, the edge to $j$ in the right-neighbouring cell, which is highlighted, is referenced by $\bm{r}=(1,0)^{\top}$.}        
        \label{fig:network_5}
    \end{center}
\end{figure}

\subsection{Step 1: Independent spatial variables}
In Step~\ref{step:1}, we define Quantities~\ref{quantity:1}--\ref{quantity:3} with respect to their microscale and macroscale 
variables independently:
\begin{itemize}
    \item 
For Quantity \ref{quantity:1}, we write 
    \begin{align}
        P_i=P_i(\bm{x}+\varepsilon \bm{X}_i),
    \end{align}
    which depends on both the macroscopic spatial coordinate, $\bm{x}$, and the microscopic index, $i$, independently;
    \item 
For Quantity \ref{quantity:2}, we write 
\begin{align}
    P_j= P_j\Big(\bm{x}+\varepsilon( \bm{X}_j + \bm{r} \circ \bm{l})\Big).
\end{align}
Note (see Figure \ref{fig:network_5}) that $j$ is shifted by $\varepsilon \bm{r} \circ \bm{l}$, since junction $j$ is in a cell at position $\bm{r}$ relative to the reference cell in which junction $i$ is found;
\item
For Quantity \ref{quantity:3}, we write
\begin{align}
    G_{ij}^{\bm{r}}=G_{ij}^{\bm{r}}(\bm{x}+\varepsilon \bm{M}_{ij}^{\bm{r}}).
\end{align}
Note that, although the pore midpoint, $\bm{M}_{ij}^{\bm{r}}$ (defined in \eqref{eq:definition-of-midpoint}), is the natural choice for $\bm{X}_{ij}^{\bm{r}}$, we will find that the process of network homogenisation is independent of this choice provided that $X_{ji}^{ \bm{-r}m}=X_{ij}^{ \bm{r}m}$ for each $m$, which ensures that a quantity defined on an edge, such as the flux, $Q_{ij}^{\bm{r}}$, is defined at the same point regardless of the direction of its flow.
\end{itemize}

\subsection{Step 2: Taylor expansion}
The position at which each variable is defined is important for
Step \ref{step:2}, since it is crucial to carry out Taylor expansion about the common point $\bm{x}$.
For example:
\begin{itemize}
    \item Expanding Quantity \ref{quantity:1} gives  
\begin{align}
    &P_i(\bm{x} + \varepsilon \bm{X}_i) = P_i(\bm{x}) + \varepsilon  \sum_{m=1}^D X_i^m \pp{P_i}{x^m}(\bm{x}) +
    \\
    & \hspace{4.5cm}
    \nonumber
    \frac{\varepsilon^2}{2} \sum_{m=1}^D \sum_{n=1}^D X_i^m X_i^n \pppd{P_i}{x^m}{x^n}(\bm{x});
\end{align}
\item 
Expanding Quantity \ref{quantity:2} gives
\begin{align}
    &P_j\Big(\bm{x} + \varepsilon (\bm{X}_j + \bm{r} \circ \bm{l} )\Big) = P_j(\bm{x}) + \varepsilon  \sum_{m=1}^D (X_j^m+r^ml^m) \pp{P_j}{x^m}(\bm{x}) +
    \\
    & \hspace{3.5cm}
    \nonumber
    \frac{\varepsilon^2}{2} \sum_{m=1}^D \sum_{n=1}^D (X_j^m+r^ml^m) (X_j^n+r^nl^n) \pppd{P_j}{x^m}{x^n}(\bm{x});
\end{align}
\item Expanding 
Quantity \ref{quantity:3} gives 
\begin{align}
    &G_{ij}^{\bm{r}}\Big(
    \bm{x} + \varepsilon \bm{M}_{ij}^{\bm{r}} 
    \Big) = 
    G_{ij}^{\bm{r}}(\bm{x}) + \varepsilon  \sum_{m=1}^D M_{ij}^{ \bm{r}m} \pp{G_{ij}^{\bm{r}}}{x^m}(\bm{x}) +
    \\
    & \hspace{6cm}
    \frac{\varepsilon^2}{2} \sum_{m=1}^D \sum_{n=1}^D M_{ij}^{ \bm{r}m} M_{ij}^{ \bm{r}n} \pppd{G_{ij}^{\bm{r}}}{x^m}{x^n}(\bm{x}).
    \nonumber
\end{align}
\end{itemize}

\subsection{Step 3: Asymptotic expansion}
Step \ref{step:3} involves assuming asymptotic forms for solutions, and, for example:
\begin{itemize}
\item 
    For Quantity~\ref{quantity:1}, we write
    \begin{align}
    P_i(\bm{x}) = P_i^{(0)}(\bm{x}) + \varepsilon P_i^{(1)}(\bm{x}) + \varepsilon^2 P_i^{(2)}(\bm{x}) + \mathcal{O}(\varepsilon^3);
    \label{eq:mm_asymptotic-solution_pressure}
\end{align}
\item 
    For Quantity~\ref{quantity:2}, we write
    \begin{align}
    P_j(\bm{x}) = P_j^{(0)}(\bm{x}) + \varepsilon P_j^{(1)}(\bm{x}) + \varepsilon^2 P_j^{(2)}(\bm{x}) + \mathcal{O}(\varepsilon^3);
    \label{eq:mm_asymptotic-solution_pressure_j}
\end{align}
\item 
For Quantity~\ref{quantity:3}, we write
\begin{align}
    G_{ij}^{\bm{r}}(\bm{x}) = G_{ij}^{\bm{r} (0)}(\bm{x}) + \varepsilon G_{ij}^{\bm{r} (1)}(\bm{x}) + \varepsilon^2 G_{ij}^{\bm{r} (2)}(\bm{x}) + \mathcal{O}(\varepsilon^3).
    \label{eq:mm_asymptotic-solution_conductance}
\end{align}
\end{itemize}


\subsection{Step 4: Asymptotic solution}
In Step 4 we will find some leading-order asymptotic solutions, such as that for the pressure, $P_i^{(0)}$, 
that are independent of the microscale index and are thus given by functions that depend only on the continuous macroscopic spatial variable. 
Other leading-order variables, such as the conductance, $G_{ij}^{\bm{r}}$, 
depend on the microscale and macroscale simultaneously.
That is, for example:
\begin{itemize}
    \item 
    For Quantity~\ref{quantity:1}, we will find that
    $P_i^{(0)}(\bm{x},t)=p^{(0)}(\bm{x},t)$ for any junction in the reference cell, $i \in \{1, \hdots, N\}$;
    \item 
    For Quantity~\ref{quantity:2}, we will find that
    $P_j^{(0)}(\bm{x},t)=p^{(0)}(\bm{x},t)$ for any junction in the neighbouring cell, $j \in \{1, \hdots, N\}$;
    \item
    For Quantity~\ref{quantity:3}, we will find that
    $G_{ij}^{\bm{r}}(\bm{x},t)=G_{ij}^{\bm{r}(0)}(\bm{x},t)$ for any pore in the cells, $i,j \in \{1, \hdots, N\}$ and $\bm{r}\in\bm{\mathcal{R}}$.
\end{itemize}
Analysing higher-order equations, we will obtain a system for the explicitly microscale-independent variables, such as $p^{(0)}$.
The parameters of this system will depend on microscale-dependent variables, such as $G_{ij}^{\bm{r}(0)}$. 
In this way, the evolution of the solution variables will depend on both the continuously varying macroscale spatial-variable, $\bm{x}$, 
and the discretely varying microscale indices, $i,j$, and $\bm{r}$, as we illustrate in Figure \ref{fig:network_6}.
\begin{figure}[!htb]
    \begin{center}
         \includegraphics[width=1.0\textwidth]{./figures/multiscale/drawings/network_6.pdf}
        \caption{A diagram to illustrate a filter in the limit $\varepsilon \rightarrow 0$. 
        Explicitly, macroscale variables
        % , such as $p^{(0)}$, 
        are independent of the microscale structure. 
        However, the parameters of the system that determine these macroscale variables depend on the structure of an infinitesimal cell defined at any continuous point $\bm{x}$.}  
        \label{fig:network_6}
    \end{center}
\end{figure}

In this section, we have summarised the process of network homogenisation.
In the remainder of this chapter, we will carry out Steps~\ref{step:1}--\ref{step:4} of network homogenisation 
to systematically average our periodic-network model, \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_cond-pdf_sum_nodim_1}. 
We will first consider the case where the network is embedded in one-dimensional space, to exemplify a simple version of the process, 
before demonstrating the $D$-dimensional scenario, which is similar, but more notationally complex.

\section{Multiscale model in one dimension} \label{sec:1d_network-homogenisation}
% To demonstrate the network homogenisation methodology, 
In this section, we begin by considering the quasi-one-dimensional case, in which non-spatially-adjacent nodes may be connected in a network that is embedded in one-dimensional space (see Figure \ref{fig:network_7} for an example). This case exemplifies the network homogenisation process while still affording significant notational simplification, since all vectors, such as the cell position, $\bm{x}$, the junction position, $\bm{X}_i$, and the reference, $\bm{r}$, are replaced by scalars, so that we drop the directional upper index, $m$, in the obvious way. In the remainder of this section, we will homogenise the fluid, particle, and pore conservation equations, 
\eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_conductance-conservation_sum_nodim_1}, and systematically find 
a macroscale flow rate.
\begin{figure}[!htb]
    \begin{center}
         \includegraphics[width=0.9\textwidth]{./figures/multiscale/drawings/network_7.pdf}
        \caption{A diagram of a section of a periodic network composed of a cell containing $N=2$ nodes that is repeated three times. 
        The conductance on some edges is labelled.
        The network is quasi-one-dimensional since nodes are organised spatially in one dimension, 
        but non-spatially-adjacent nodes are connected. 
        For example,
        an edge exists between node $1$ in one cell and node $1$ in the cell to the right, but these two nodes are not spatially adjacent, since node $2$ lies between them in the same plane.}  
        \label{fig:network_7}
    \end{center}
\end{figure}


\subsection{Conservation of fluid}
We first homogenise the microscale equation for conservation of fluid, \eqref{eq:mm_fluid-conservation_sum_nodim_1}. Writing this in terms of the junction pressure and pore conductance, using \eqref{eq:mm_flux-definition_sum_nodim_1}, gives
\begin{align}
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r}(P_j-P_i) &= 0.
    \label{eq:mm_1d_fluid-conservation_master_1}
\end{align}
Carrying out Step \ref{step:1} by evaluating $G_{ij}^{r}$, $P_i$, and $P_j$ at $x$, which we choose to be the left-most point in the reference cell, and further Step \ref{step:2}, a Taylor expansion of the pressure and conductance about $x$, we find that \eqref{eq:mm_1d_fluid-conservation_master_1} becomes 
\begin{align}
    & \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    \bigg\{ G_{ij}^{r} \Big(P_j-P_i\Big) \bigg\}
    +
    \varepsilon \bigg\{ 
     \pp{G_{ij}^{r}}{x} M_{ij}^{r} (P_j-P_i)
    +
    G_{ij}^{r} 
    \Big(
     (X_j+ r l) \pp{P_j}{x} -
  X_i 
 \pp{P_i}{x} \Big)
    \bigg\}
        \nonumber
    \\
    &  \hspace{1.5cm}
    +
    \varepsilon^2\bigg\{
    \frac{1}{2}   \ppp{G_{ij}^{r}}{x}
    (M_{ij}^{r} )^2 (P_j-P_i)
    +
     \pp{G_{ij}^{r}}{x} M_{ij}^{r} \Big(  (X_j+ r l)\pp{P_j}{x} -     X_i\pp{P_i}{x} \Big)
    \nonumber
    \\
    &  \hspace{1.5cm}
    +
    \frac{G_{ij}^{r}}{2} \bigg(   (X_j + r l)^2\ppp{P_j}{x} -
      X_i^2 \ppp{P_i}{x} \bigg)
    \bigg\}
    + \mathcal{O}(\varepsilon^3) = 0.
    \label{eq:mm_1d_fluid-conservation_master_2}
\end{align}
Here, and throughout, we use $\{$ and $\}$ to group terms of the same asymptotic order.
We next carry out Step \ref{step:3} by
substituting the asymptotic forms for the pressure and conductance, \eqref{eq:mm_asymptotic-solution_pressure}
and \eqref{eq:mm_asymptotic-solution_conductance}, into \eqref{eq:mm_1d_fluid-conservation_master_2} 
to obtain
\begin{align}
    & 
    \bm{A}^{(0)} \bm{P}^{(0)} + 
    \varepsilon \bigg\{ 
    \bm{A}^{(0)} \bm{P}^{(1)}
    +
    \bm{A}^{(1)} \bm{P}^{(0)} 
    + 
     \bm{D}^{ (0)} \bm{P}^{(0)}
    + 
     \bm{B}^{ (0)} \pp{\bm{P}^{(0)}}{x} 
    \bigg\} +
    \varepsilon^2 \bigg\{
    \bm{A}^{(0)} \bm{P}^{(2)}
    +
    \nonumber
    \\
    & \hspace{1.0cm}
    \bm{A}^{(1)} \bm{P}^{(1)} 
    + 
     \bm{D}^{ (0)} \bm{P}^{(1)}
    + 
     \bm{B}^{ (0)} \pp{\bm{P}^{(1)}}{x} 
    +
    \bm{A}^{(2)} \bm{P}^{(0)} 
    +
     \bm{D}^{ (1)} \bm{P}^{(0)}
    +
     \bm{B}^{ (1)} \pp{\bm{P}^{(0)}}{x}
    +
     \nonumber
    \\
    & \hspace{2.0cm}
      \bm{F}^{ (0)} \bm{P}^{(0)}
    +
      \bm{E}^{ (0)} \pp{\bm{P}^{(0)}}{x}
    +
      \bm{C}^{ (0)} \ppp{\bm{P}^{(0)}}{x}    \bigg\} + \mathcal{O}(\varepsilon^3) = \bm{0},
    \label{eq:mm_1d_fluid-conservation_master_3}
\end{align}
which is the master equation for fluid conservation that we seek to solve.
Here, $\bm{P}^{(s)}$ is a vector of length $N$ in the junction index that contains the $\mathcal{O}(\varepsilon^s)$ pressure solutions,
so that 
\begin{align}
    \bm{P}^{(s)}=
    \left(
    \begin{array}{c}
           P_1^{(s)} \\
           \vdots \\
           P_N^{(s)} 
           \end{array} \right), 
\end{align}
and
$\bm{A}, \bm{B}, \bm{C}, \bm{D}, \bm{E}$, and $\bm{F}$, are matrices of shape ($N,N$) in the junction index, the entries of which are given by
\begin{align}
    A_{ij}^{(s)} &= 
    \sum_{r\in\mathcal{R}} \left(  G_{ij}^{r {(s)}} - \updelta_{ij} \sum_{k=1}^N G_{ik}^{r {(s)}} \right),
    \label{eq:1d_def_A}
    \\
    B_{ij}^{ (s)} &= 
    \sum_{r\in\mathcal{R}} \left( G_{ij}^{r {(s)}}  (X_j + r l
    )- \updelta_{ij} X_i \sum_{k=1}^N G_{ik}^{r {(s)}} \right),
    \label{eq:1d_def_B}
    \\
    C_{ij}^{ (s)} &= 
    \frac{1}{2} \sum_{r\in\mathcal{R}} \left( G_{ij}^{r {(s)}}  (X_j + r l)^2 - \updelta_{ij} X_i^2 \sum_{k=1}^N G_{ik}^{r {(s)}} \right),
    \label{eq:1d_def_C}
    \\
    D_{ij}^{ (s)} &= 
    \sum_{r\in\mathcal{R}} \left( \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} -\updelta_{ij}  \sum_{k=1}^N \pp{G_{ik}^{r {(s)}}}{x} M_{ik}^{r}   \right),
    \label{eq:1d_def_D}
    \\
    E_{ij}^{ (s)} &= 
    \sum_{r\in\mathcal{R}} \left( \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} (X_j + r l)  - \updelta_{ij}  X_i \sum_{k=1}^N  \pp{G_{ik}^{r {(s)}}}{x} M_{ik}^{r}   \right),
    \label{eq:1d_def_E}
    \\
    F_{ij}^{ (s)} &= \frac{1}{2} \sum_{r\in\mathcal{R}} \left( \ppp{G_{ij}^{r {(s)}}}{x}  (M_{ij}^{r})^2  - \updelta_{ij}   \sum_{k=1}^N  \ppp{G_{ik}^{r {(s)}}}{x} (M_{ik}^{r})^2    \right).
    \label{eq:1d_def_F}
\end{align}

To solve the master equation, \eqref{eq:mm_1d_fluid-conservation_master_3}, it is useful to note (see Appendix \ref{sec:1d_proofs_ABCDE} for proofs of (\ref{eq:mm_1d_matrix-lemma_1}a)--(\ref{eq:mm_1d_matrix-lemma_2_D-and-B_E-and-C_E-and-C}b)) that 
\begin{subequations} \label{eq:mm_1d_matrix-lemma_1}
\begin{align}
    \bm{A}^{(s)} \bm{1} = \bm{0},
    \quad 
    \bm{1}^\top \bm{B}^{ (s)} \bm{1} = \bm{0},
    \quad 
    \bm{D}^{ (s)} \bm{1} = \bm{0},
    \quad 
    \bm{F}^{ (s)} \bm{1} = \bm{0},
    \tag{\ref{eq:mm_1d_matrix-lemma_1}a--d}
\end{align}
\end{subequations}
and that
\begin{subequations}
    \label{eq:mm_1d_matrix-lemma_2_D-and-B_E-and-C_E-and-C}
\begin{align}
    \bm{1}^\top \bm{D}^{(s)} = 
    \pp{}{x}\Big(\bm{1}^\top \bm{B}^{(s)} \Big),
    \quad
    \bm{1}^\top \bm{E}^{(s)} \bm{1} =
    \pp{}{x}\Big(\bm{1}^\top \bm{C}^{(s)} \bm{1} \Big),
    \tag{\ref{eq:mm_1d_matrix-lemma_2_D-and-B_E-and-C_E-and-C}a,b}
\end{align}
\end{subequations}
where $\bm{1} = (1,\ldots,1)^\top$ and $\bm{0} = (0,\ldots,0)^\top$ are both vectors of length $N$.
Note additionally that, since $\bm{A}^{(s) \top}$ is symmetric, such that
\begin{align}
    \bm{A}^{(s) \top} = \bm{A}^{(s)},
    \label{eq:1d_matrix-lemma_4}
\end{align}
it follows from (\ref{eq:mm_1d_matrix-lemma_1}a) that
\begin{align}
    \bm{1}^\top \bm{A}^{(s)} = \bm{0}^\top.
    \label{eq:mm_1d_matrix-lemma_3}
\end{align}
We use these results extensively in Step~\ref{step:4}.

Indeed, proceeding with Step \ref{step:4}, at leading order in the master equation, \eqref{eq:mm_1d_fluid-conservation_master_3}, we have that
\begin{align}
    \bm{A}^{(0)} \bm{P}^{(0)} &= \bm{0},
    \label{eq:mm_1d_fluid-conservation_leading-order-problem_1}
\end{align}
and, using the identity (\ref{eq:mm_1d_matrix-lemma_1}a), it follows that
\begin{align}
    \bm{P}^{(0)} = p^{(0)}(x,t) \bm{1},
    \label{eq:mm_1d_fluid-conservation_leading-order-solution_1}
\end{align}
where $p^{(0)}(x,t)$ is the leading-order macroscale pressure solution to be found.
We have discovered that the leading-order pressure at any junction $i$ is independent of $i$,
which means that pressures at junctions in the same cell are identical at leading order.
% Although the network's structure is dynamic due to changes in the conductance, 
% we assume that no region ever becomes fully disconnected. It follows that $\bm{A}^{(0)}$ is irreducible, and further that \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} is unique.
One consequence is that the leading-order intra-cell pressure difference between any two junctions is zero, so that there is no flux through the pore at leading order (using \eqref{eq:mm_flux-definition_sum_nodim_1}),
\begin{align}
    P_j^{(0)}-P_i^{(0)}=0.
    \label{eq:mm_1d_leading-order-pressure-difference}
\end{align}

It remains to find $p^{(0)}$ by
% Pre-multiplying \eqref{eq:mm_1d_fluid-conservation_leading-order-problem_1} by $\bm{1}^{\top}$, 
% for a solvability condition, 
seeking a solvability condition 
% for $p^{(0)}$ is indeterminate from the leading-order problem, and we must proceed to higher-order problems to determine this leading-order solution.
at higher order.
Thus, continuing Step \ref{step:4} by considering first-order terms in the master equation, \eqref{eq:mm_1d_fluid-conservation_master_3}, substituting the leading-order pressure solution, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1}, into \eqref{eq:mm_1d_fluid-conservation_master_3}, and using the identities (\ref{eq:mm_1d_matrix-lemma_1}a) and (\ref{eq:mm_1d_matrix-lemma_1}c), we find that
\begin{align}
    \bm{A}^{(0)} \bm{P}^{(1)} +  \bm{B}^{ (0)} \bm{1} \pp{p^{(0)}}{x} = \bm{0}.
    \label{eq:1d_fluid-conservation_first-order-problem}
\end{align}
Searching for a solvability condition, we pre-multiply both sides of \eqref{eq:1d_fluid-conservation_first-order-problem} by $\bm{1}^{\top}$, but, using (\ref{eq:mm_1d_matrix-lemma_1}b), we see that the resulting equation is satisfied for any $p^{(0)}$. We must, therefore, proceed to even higher order problems to determine the leading-order solution, $p^{(0)}$.
Before this, we note that \eqref{eq:1d_fluid-conservation_first-order-problem} has the solution 
\begin{align}
    \bm{P}^{(1)} = f(x,t) \bm{1} +  \pp{p^{(0)}}{x} \bm{W},
    \label{eq:mm_1d_fluid-conservation_first-order-solution}
\end{align}
where $f(x,t)$ is an arbitrary (see \eqref{eq:1d_master_transport_6}) microscale-independent function, and $\bm{W}$ is a vector of length $N$ that satisfies 
\begin{align}
    \bm{A}^{(0)} \bm{W} = - \bm{B}^{ (0)} \bm{1}.
    \label{eq:mm_1d_cell-problem_1}
\end{align}
Unlike the leading-order pressure, then, the first-order pressure, given by \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, 
is microscale dependent, via $\bm{W}$.
We refer to \eqref{eq:mm_1d_cell-problem_1} as the cell problem, since its solution, $\bm{W}$, which we call the cell solution, determines the dependence of the pressure at 
a continuous spatial point on the structure of the cell located at this point (in the limit $\varepsilon\rightarrow 0$, see Figure~\ref{fig:network_6}) and thus on the microscale network, via the conductances that define $\bm{A}^{(0)}$ and $\bm{B}^{(0)}$ (defined in \eqref{eq:1d_def_A} and \eqref{eq:1d_def_B}).
Since the first term in the right-hand side of \eqref{eq:mm_1d_fluid-conservation_first-order-solution}
is microscale-independent, 
the first-order pressure difference is given by
\begin{align}
    P_j^{(1)}-P_i^{(1)} =  (W_j-W_i) \pp{p^{(0)}}{x},
    \label{eq:mm_1d_first-order-pressure-difference}
\end{align}
which is of interest since this pressure difference determines the pore flux, via 
\eqref{eq:mm_flux-definition_sum_nodim_1}.


% Indeed, since the first-order problem did not yield an equation for the leading-order pressure solution, $p^{(0)}$, to complete Step \ref{step:4} we must proceed to the next order to close the problem.
At second order, by substituting the leading- and first-order solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, into the master equation, \eqref{eq:mm_1d_fluid-conservation_master_3},  
 and using the identities (\ref{eq:mm_1d_matrix-lemma_1}a,c,d), we find that
\begin{align}
&
    \bm{A}^{(0)} \bm{P}^{(2)}
    +
    \bm{A}^{(1)}  \pp{p^{(0)}}{x} \bm{W}
    + 
     \bm{D}^{ (0)} 
     \pp{p^{(0)}}{x} \bm{W}
    + 
     \bm{B}^{ (0)} \pp{}{x}
    \Big(
    f \bm{1} +  \pp{p^{(0)}}{x} \bm{W}
    \Big)
    +
            \nonumber
    \\
    & \hspace{1.0cm}
     \bm{B}^{ (1)} \bm{1} \pp{p }{x}
    +
      \bm{E}^{ (0)} \bm{1} \pp{p^{(0)}}{x}
    +
      \bm{C}^{ (0)} \bm{1} \ppp{p^{(0)}}{x} = \bm{0}.
    \label{eq:mm_1d_fluid-conservation_second-order_problem_1}
\end{align}   
Seeking a solvability condition once more, we pre-multiply both sides of \eqref{eq:mm_1d_fluid-conservation_second-order_problem_1} by
$\bm{1}^{\top}$, and use the identity \eqref{eq:mm_1d_matrix-lemma_3} to eliminate $\bm{P}^{(2)}$. Using the identity (\ref{eq:mm_1d_matrix-lemma_1}b) to simplify, we obtain
\begin{align}
&
    \bm{1}^\top  \bm{D}^{ (0)} 
     \pp{p^{(0)}}{x} \bm{W}
    + 
    \bm{1}^\top  \bm{B}^{ (0)} \pp{}{x}
    \Big( \pp{p^{(0)}}{x} \bm{W}
    \Big)
    +
    \bm{1}^\top   \bm{E}^{ (0)} \bm{1} \pp{p^{(0)}}{x}
    +
    \bm{1}^\top   \bm{C}^{ (0)} \bm{1} \ppp{p^{(0)}}{x} = 0.
    \label{eq:mm_1d_fluid-conservation_second-order_problem_2}
\end{align} 
Furthermore, using the identities in \eqref{eq:mm_1d_matrix-lemma_2_D-and-B_E-and-C_E-and-C} and the product rule to write $\bm{D}^{(0)}$ and $\bm{E}^{(0)}$ in terms of $\bm{B}^{(0)}$ and $\bm{C}^{(0)}$, we simplify
\eqref{eq:mm_1d_fluid-conservation_second-order_problem_2},
and arrive at
\begin{align}
    \pp{}{x} \Big( \sigma^{(0)} \pp{p^{(0)}}{x} \Big) = 0,
    \label{eq:mm_1d_fluid-conservation_second-order_problem_3}
\end{align}
where $\sigma^{(0)}$ is the scalar defined by
\begin{align}
    \sigma^{(0)} = \frac{1}{l}\Big( \bm{1}^\top \bm{B}^{ (0)}  \bm{W} + \bm{1}^\top \bm{C}^{ (0)} \bm{1}
    \Big).
    \label{eq:1d_sigma}
\end{align}
Equation \eqref{eq:mm_1d_fluid-conservation_second-order_problem_3} is the solvability condition for the leading-order pressure solution, $p^{(0)}$, that we seek, 
and thus \eqref{eq:mm_1d_fluid-conservation_second-order_problem_3} is the homogenised version of the equation for conservation of fluid, \eqref{eq:mm_fluid-conservation_sum_nodim_1}. 
We see that we have arrived at an equation, which governs the evolution of the leading-order pressure, $p^{(0)}$, over continuous macroscale space, $x$, which is explicitly independent of the microscale. 
Instead, microscale structure enters the system via the parameter of the equation, $\sigma^{(0)}$, which depends explicitly on the structure of the underlying network via 
$\bm{W}$, $\bm{B}^{(0)}$, $\bm{C}^{(0)}$, which are defined in terms of the pore conductances, $G_{ij}^{r(0)}$, subject 
to the cell problem, \eqref{eq:mm_1d_cell-problem_1}, and the definitions in \eqref{eq:1d_def_B} and \eqref{eq:1d_def_C}, respectively.

\subsection{Darcy's law}
Next, our aim is to relate the leading-order macroscale pressure solution, $p^{(0)}$, to a flow rate.
To this end, we define the Darcy velocity to be
\begin{align}
    u = \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} r Q_{ij}^{r}.
    \label{eq:1d_definition-of-darcy}
\end{align}
To see why \eqref{eq:1d_definition-of-darcy} is indeed the net flux per unit area through a cell, first notice that terms containing $r=0$ correspond to flow through edges with no inter-cell connectivity, and so do not contribute to the sum.
The terms containing $r=1$ correspond to flow out of a cell through edges connected to the cell directly to the right of it, and,
since $Q_{ij}^{r}=-Q_{ji}^{r}$, the terms containing $r=-1$ correspond to flow into a cell through edges connected to the cell directly to the left.
Therefore, averaging these two contributing flows (by dividing by two) gives the flux through the cell.
Now consider terms containing $r=2$, which, we recall, describes the case in which a node in one cell is connected by an edge to a cell two steps to the right. 
In such a case, two copies of the edge will pass through the
right-hand boundary of the reference cell (see Figure \ref{fig:network_8}), and hence $Q_{ij}^{r}$ must be counted twice. 
Similarly, for longer-range connections, multiplying by $r$ ensures that $Q_{ij}^{r}$ is counted the correct number of times (see Appendix \ref{sec:darcy-proof} for proof).
\begin{figure}[H]
    \begin{center}
         \includegraphics[width=0.9\textwidth]{./figures/multiscale/drawings/network_8.pdf}
        \caption{A diagram of a periodic network composed of a cell of $N=2$ nodes repeated three times. 
        The conductance of some pores is labelled.
        The pore with conductance $G_{11}^{(1)}$ connects junction $1$ in one cell junction $1$ in the next cell to the right, since $r=1$. 
        Only one instance of this pore exits the first cell, and so flow through this pore need only be counted once.
        On the other hand, the pore with conductance $G_{22}^{(2)}$ connects junction $2$ in one cell to junction $2$ in the cell 
        two cells to the right, since $r=2$. 
        Two instances of this pore exit the first cell, and so flow through this pore must be counted twice.}  
        \label{fig:network_8}
    \end{center}
\end{figure}

To homogenise the Darcy velocity, \eqref{eq:1d_definition-of-darcy}, we will begin by homogenising the pore flux, $Q_{ij}^{r}$. 
To this end, following Steps \ref{step:1}--\ref{step:3} of network homogenisation, that is, writing the pore flux in terms of the pore pressure drop and conductance using \eqref{eq:mm_flux-definition_sum_nodim_1}, carrying out a Taylor expansion of the pressure and conductance about $x$, 
and substituting in an asymptotic expansion for the conductance, given by \eqref{eq:mm_asymptotic-solution_conductance}, we find that 
\begin{align}
    & Q_{ij}^{r} =
    -\bigg(
    \bigg\{ G_{ij}^{r (0)} \Big( P_j - P_i \Big) \bigg\}  
    + 
    \varepsilon \bigg\{
    G_{ij}^{r (1)} \Big( P_j - P_i \Big) +
    \label{eq:1d_flux_4}
        \\
     & \hspace{1cm}    
    G_{ij}^{r (0)} \Big(
    (X_j + r l) \pp{P_j}{x} 
    - 
     X_i \pp{P_i}{x} \Big)
    +
     \pp{G_{ij}^{r (0)}}{x} M_{ij}^{r} \Big(
    P_j- P_i  \Big)
    \bigg\}
    + \mathcal{O}(\varepsilon^2) 
    \bigg).
    \nonumber
\end{align}
Furthermore, using the leading- and first-order asymptotic pressure solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, and
noting the cancellation of terms by using the leading- and first-order pressure differences, \eqref{eq:mm_1d_leading-order-pressure-difference} and \eqref{eq:mm_1d_first-order-pressure-difference}, we find that 
\begin{align}
    & Q_{ij}^r =
    G_{ij}^{r (0)} 
    \Delta_{ij}^r
    \pp{p^{(0)}}{x} 
    + \mathcal{O}(\varepsilon).
    \label{eq:1d_homogenised-Q}
\end{align}
The leading-order microscale flux is therefore given by
\begin{align}
    & Q_{ij}^{r(0)} =
    G_{ij}^{r (0)} 
    \Delta_{ij}^r
    \pp{p^{(0)}}{x},
    \label{eq:1d_homogenised-Q0}
\end{align}
where
\begin{align}
    \Delta_{ij}^{r} = W_i+X_i - (W_j+X_j+rl), 
    \label{eq:1d_definition-of-delta_temp}
\end{align}
which, we note, satisfies 
\begin{align}
\Delta_{ji}^{-r} = -\Delta_{ij}^{r}.    
\label{eq:1d_delta-identity}
\end{align}

% We call $\Delta_{ij}^r$ the pressure drop
% \correct{}{(per unit pressure gradient)}
% in pore $ijr$.
Equation \eqref{eq:1d_homogenised-Q0} tells us that 
$\Delta_{ij}^{r}$
is the pressure difference (per unit pressure gradient) in pore $ijr$.
% Using \eqref{eq:1d_homogenised-Q0}, we see, 
% in the presence of a unit macroscale pressure-gradient ($\partial p^{(0)}/\partial x=-1$, say, so that fluid flows from left to right), 
% we have $Q_{ij}^{r(0)}=-G_{ij}^{r(0)}\Delta_{ij}^{r}$, 
% so that $\Delta_{ji}^{-r}$ (using \eqref{eq:1d_delta-identity}) is the pressure difference along pore $ijr$ (\correct{}{per unit pressure gradient across the cell}) that causes flow in pore $ijr$.
% Thus the effective length of pore $ijr$ is the contribution that the pressure drop in pore $ijr$ makes to 
% the flux down this pore, relative to the pressure gradient across the whole cell.
% (in contrast to the contribution of the pressure gradient across the cell) 
Equation \eqref{eq:1d_definition-of-darcy} implies that this dictates the effective contribution of the pressure difference in pore $ijr$ to the overall macroscale flow rate, $u$.
% Since the macroscale flow rate, $u$, is related to the pore flux, $Q_{ij}^{r(0)}$ (via \eqref{eq:1d_definition-of-darcy}), 
% $\Delta_{ij}^r$ dictates the \correct{}{effective contribution of pore $ijr$ to the overall macroscale flow rate}. 

% This interpretation of the pressure drop helps us to interpret the cell solution, $\bm{W}$.
% We see that its elements, $W_i$,
It follows that elements of the cell solution $W_i$, 
have dimensions of length (pressure divided by pressure gradient). 
Therefore, we interpret these as the `corrections' to the position of each junction $i$ 
necessary to obtain the imposed `correct' pressure gradient on the macroscale. 
These `corrected' positions (as opposed to the actual positions, $X_i$) are dictated by pore conductances, $G_{ij}^{r(0)}$, via solution of the cell problem, \eqref{eq:mm_1d_cell-problem_1}.
% junction positions, $X_i$, 
% \comment{[These two objects are pretty hard to explain and I'm not sure if it's clear to the reader... should I consider putting in diagrams of a few simple cells and the corresponding deltas and cell solutions?]}
% we interpret $\Delta_{ij}^{r}$ as the leading-order microscale pressure-difference per unit pressure gradient. That is, $\Delta_{ij}^{r}$ dictates the contribution of the edge $(ij,r)$ to the effective pressure drop. For example, in the case of a uniform pressure-gradient from left to right across the cell ($\partial p/\partial x =-1$), $\Delta_{ij}^{r=0}$ is the pressure drop from node $i$ to node $j$, and $G_{ij}^{r=0,(0)} \Delta_{ij}^{r=0}$ is the leading-order flux along this edge.
% From \eqref{eq:1d_definition-of-delta}, we see that $W_i + X_i$ is the pressure per unit pressure gradient at node $i$, so that $(W_i + X_i) - (W_j + X_j)$ is the intra-cell pressure drop per unit pressure gradient. The contribution $rl$ accounts for the possibility that node $j$ is in a cell at position $r \ne 0$ relative to node $i$. 
% One consequence of this definition, which we will use later, is that particles move from node $i$ to node $j$ when $\Delta_{ij}^r$ is negative (or equivalently, from \eqref{eq:1d_delta-identity}, when $\Delta_{ji}^{-r}$ is positive).

% Using \eqref{eq:1d_homogenised-Q0}, we interpret $\Delta_{ij}^{r}$ as the leading-order microscale pressure-difference per unit pressure gradient. That is, $\Delta_{ij}^{r}$ dictates the contribution of the edge $(ij,r)$ to the effective pressure drop. For example, in the case of a uniform pressure-gradient from left to right across the cell ($\partial p/\partial x =-1$), $\Delta_{ij}^{r=0}$ is the pressure drop from node $i$ to node $j$, and $G_{ij}^{r=0,(0)} \Delta_{ij}^{r=0}$ is the leading-order flux along this edge.
% From \eqref{eq:1d_definition-of-delta}, we see that $W_i + X_i$ is the pressure per unit pressure gradient at node $i$, so that $(W_i + X_i) - (W_j + X_j)$ is the intra-cell pressure drop per unit pressure gradient. The contribution $rl$ accounts for the possibility that node $j$ is in a cell at position $r \ne 0$ relative to node $i$. 
% One consequence of this definition, which we will use later, is that particles move from node $i$ to node $j$ when $\Delta_{ij}^r$ is negative (or equivalently, from \eqref{eq:1d_delta-identity}, when $\Delta_{ji}^{-r}$ is positive).

% Following Steps \ref{step:1}--\ref{step:3} of network homogenisation, that is, writing the Darcy velocity in terms of the microscale pressure and conductance by substituting \eqref{eq:mm_flux-definition_sum_nodim_1} into \eqref{eq:1d_definition-of-darcy}, carrying out a Taylor expansion of the pressure and conductance about $x$, and substituting in an asymptotic expansion for the conductance, given by \eqref{eq:mm_asymptotic-solution_conductance}, we arrive at 
% \begin{align}
%     & u =
%     -\frac{1}{ 2 \varepsilon }  \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
%     r \bigg(
%     \bigg\{ G_{ij}^{r (0)} \Big( P_j - P_i \Big) \bigg\}  
%     + 
%     \varepsilon \bigg\{
%     G_{ij}^{r (1)} \Big( P_j - P_i \Big) +
%     \label{eq:1d_flux_4}
%         \\
%      & \hspace{1cm}    
%     G_{ij}^{r (0)} \Big(
%     (X_j + r l) \pp{P_j}{x} 
%     - 
%      X_i \pp{P_i}{x} \Big)
%     +
%      \pp{G_{ij}^{r} (0)}{x} M_{ij}^{r} \Big(
%     P_j- P_i  \Big)
%     \bigg\}
%     + \mathcal{O}(\varepsilon^2) 
%     \bigg).
%     \nonumber
% \end{align}
% Furthermore, using the leading- and first-order asymptotic pressure solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, and
% noting the cancellation of terms by using the leading- and first-order pressure differences, \eqref{eq:mm_1d_leading-order-pressure-difference} and \eqref{eq:mm_1d_first-order-pressure-difference}, we find that 
We find (see Appendix \ref{sec:1d_proofs_k=sig} for proof) that the macroscale solution of the model is independent of the positions of the junctions, $X_i$. 
We expect this, since the microscale model, \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_ic_cond_sum_nodim_1}, is only dependent on the network structure via the conductances of the pores. 
Choosing $X_i=0$ for all $i \in \{1,\hdots,N\}$, 
we see that 
\begin{align}
    \Delta_{ij}^{r} = W_i - (W_j+rl). 
    \label{eq:1d_definition-of-delta}
\end{align}

Substituting the homogenised microscale flux, \eqref{eq:1d_homogenised-Q}, into the Darcy-velocity definition, \eqref{eq:1d_definition-of-darcy}, we find that the homogenised Darcy velocity is
\begin{align}
    & u =
    \frac{1}{ 2 }  \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}
    r G_{ij}^{r (0)} 
    \Delta_{ij}^r
    \pp{p^{(0)}}{x} 
    + \mathcal{O}(\varepsilon),
    \label{eq:1d_homogenised-u}
\end{align}
so that the leading-order Darcy velocity is given by \begin{align}
    & u^{(0)} =
    \frac{1}{ 2 }  \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}
    r G_{ij}^{r (0)} 
    \Delta_{ij}^r
    \pp{p^{(0)}}{x}.
    \label{eq:1d_darcy_full}
\end{align}
% Here 
% \begin{align}
%     \Delta_{ij}^{r} = W_i+X_i - (W_j+X_j+rl), 
%     \label{eq:1d_definition-of-delta}
% \end{align}
% which we interpret, from \eqref{eq:1d_darcy_full}, as the intra-cell pressure drop per unit pressure gradient. 
% \textcolor{blue}{[still not sure quite what to put here, I will show you how this term emerges from homogenising the pressure difference when we meet! (we discussed tortuosity..., but I think it has more to do with permeability since it appears in $k$ and has dimensions length) i.e., since this has dimensions of length, this can be thought of as a measure of the `flux-effective' microscale displacement from node $i$ to node $j$?]}
% Note that the term $G_{ij}^{r (0)} \Delta_{ij}^r \pp{p^{(0)}}{x}$ is the homogenised version of the microscale flux, which will appear extensively in the equation for conservation of particles. 
% \textcolor{blue}{Therefore we interpret $G_{ij}^{r(0)} \Delta_{ij}^{r}$ as the microscale-dependent contribution to the homogenised flux from node $i$ to node $j$?}
Re-writing \eqref{eq:1d_darcy_full}, therefore, we find that the leading-order Darcy velocity satisfies
\begin{align}
    u^{(0)} = - k^{(0)} \pp{p^{(0)}}{x},
    \label{eq:1d_macroscale_darcy_1}
\end{align}
where, using \eqref{eq:1d_delta-identity},
\begin{align}
    k^{(0)} = \frac{1}{2}  \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)} 
    \Delta_{ji}^{-r}.
    \label{eq:1d_definition-of-permeability}
\end{align}
It turns out (see the proof in Appendix \ref{sec:1d_proofs_k=sig}) that $\sigma^{(0)}=k^{(0)}$, and, using this identity in \eqref{eq:mm_1d_fluid-conservation_second-order_problem_3}, we find that the leading-order Darcy velocity is divergence-free, so that
\begin{align}
    \pp{u^{(0)}}{x} = 0.
    \label{eq:1d_macroscale_conservation-of-mass_1}
\end{align}
Carrying out network homogenisation on the microscale equation for conservation of fluid, \eqref{eq:mm_fluid-conservation_sum_nodim_1},  and the pore flux definition, \eqref{eq:mm_flux-definition_sum_nodim_1},
we have shown that the leading-order flow rate and pressure, $u^{(0)}$ and $p^{(0)}$, 
 satisfy the Darcy equation, \eqref{eq:1d_macroscale_darcy_1}, 
 and the expected macroscale equation for fluid conservation, \eqref{eq:1d_macroscale_conservation-of-mass_1}.
 Therefore, \eqref{eq:1d_definition-of-darcy} is indeed the correct definition for the flow rate, $u$. 
Inspecting \eqref{eq:1d_macroscale_darcy_1}, it follows that $k^{(0)}$ is the leading-order permeability of the filter, 
which depends on the microscale via \eqref{eq:1d_definition-of-permeability}.


\subsection{Conservation of particles}
We next homogenise the microscale equation for the conservation of particles, \eqref{eq:mm_particle-conservation_sum_nodim_1}.
Writing this in terms of the junction pressure and pore conductance, using \eqref{eq:mm_flux-definition_sum_nodim_1}, gives
\begin{align}
    \label{eq:1d_master_transport_1}
    \pp{C_i}{T} &= 
    \frac{1}{\varepsilon} 
    \bigg\{ \sum_{j=1}^{N} \sum_{r\in\mathcal{R}} 
    \Big(
    G_{ji}^{-r} (P_j-P_i) C_j H(G_{ji}^{-r}(P_j-P_i)) - 
        \\
    & \hspace{3.5cm} \nonumber
    G_{ij}^r (P_i-P_j) C_i H(G_{ij}^r(P_i-P_j))  \Big)
    \bigg\}
    \\
    & \hspace{4.0cm} \nonumber
    -
     \sum_{j=1}^N \sum_{r\in\mathcal{R}} \alpha G_{ji}^{-r} (P_j-P_i) C_j H(G_{ji}^{-r}(P_j-P_i)).
\end{align}
% , since we expect the first term on right-hand side to correspond to advection, and the second to correspond to particle deposition.
% Using the definition of the microscale flux \eqref{eq:mm_flux-definition_sum_nodim_1} to re-write \eqref{eq:mm_particle-conservation_sum_nodim_1} in terms of the pressure and conductance, and writing the particle  advection and deposition terms separately for convenience, we have
We begin by noting that the Heaviside function, $H$, can be homogenised (since the pressure difference in a pore never reaches zero, as discussed in Section \ref{sec:nm_Dimensional model}) first and carried through the rest of the process.
% without interacting with the other terms of \eqref{eq:1d_master_transport_1}.
Specifically, we carry out Steps \ref{step:1}--\ref{step:3} on the term $H(G_{ij}^r(P_i-P_j))$, and find that  
\begin{align}
    H(Q_{ij}^{r})
    & =
    H\bigg(G_{ij}^{r(0)}(P_i-P_j) + \varepsilon G_{ij}^{r(0)} \bigg\{ X_i  \pp{P_i}{x} -  (X_j+ r l) \pp{P_j}{x} \bigg\} + \mathcal{O}(\varepsilon^2) \bigg) 
    \\
    & =
    H\bigg(\varepsilon \bigg\{  G_{ij}^{r(0)} \Delta_{ij}^r \pp{p^{(0)}}{x} \bigg\} + \mathcal{O}(\varepsilon^2) \bigg),
    \label{eq:1d_homogenised-H}
\end{align}
% \begin{align}
%     H(P_i-P_j)
%     =
%     H\bigg(\varepsilon \bigg\{  \Delta_{ij}^r \pp{p^{(0)}}{x} \bigg\} + \mathcal{O}(\varepsilon^2) \bigg),
%     \label{eq:1d_homogenised-H}
% \end{align}
 where we have used the leading- and first-order pressure solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, respectively.
We therefore define
\begin{align}
    H_{ij}^{r(0)} = H \Big( \Delta_{ij}^r \pp{p^{(0)}}{x} \Big) ,
    \label{eq:1d_definition-of-H}
\end{align}
which satisfies
\begin{align}
    H_{ij}^{r(0)}+H_{ji}^{-r(0)}=1.   
\label{eq:1d_heaviside-identity}
\end{align}
In \eqref{eq:1d_definition-of-H}, note that $H_{ij}^{r(0)}$ is only non-zero when $\Delta_{ij}^r$ is negative, or equivalently, when $\Delta_{ji}^{-r}$ is positive, using \eqref{eq:1d_delta-identity}.
Recalling our interpretation of $\Delta_{ij}^r$, this means that $H_{ij}^{r(0)}$ indicates pores $ijr$ along which particles move from junction $i$ to junction $j$. 
From \eqref{eq:1d_heaviside-identity}, 
it follows that $H_{ji}^{-r(0)}$ indicates the pores $ijr$ along which particles flow from junction $j$ to junction $i$, which are the pores along which deposition occurs. 
Thus $H_{ji}^{-r(0)}$ is the leading-order equivalent of $H$.

We now carry out Steps \ref{step:1}--\ref{step:3} of the network homogenisation process on the remaining terms in the particle conservation equation, \eqref{eq:1d_master_transport_1}.
% Notet that if A is not a constant then where this time we apply a Taylor expansion and an asymptotic expansion to the concentration and the adhesivity function, $A_{ij}^{\bm{r}}$, as well as the conductance and the pressure. 
Using the leading- and first-order pressure solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and
\eqref{eq:mm_1d_fluid-conservation_first-order-solution}, and noting the cancellation of terms that contain the leading-order pressure difference, \eqref{eq:mm_1d_leading-order-pressure-difference}, and the cancellation of terms involving the function $f$
% (which ensures that $f$ is defined arbitrarily in \eqref{eq:mm_1d_fluid-conservation_first-order-solution}) 
via the first-order pressure difference, \eqref{eq:mm_1d_first-order-pressure-difference}, we arrive at
\begin{align}
\label{eq:1d_master_transport_6}
    & \pp{}{T}
    \Big(
    C_i^{(0)} + \varepsilon \Big\{ C_i^{(1)}+ X_i \pp{C_i^{(0)}}{x} \Big\} + 
    \varepsilon^2
    \Big\{
    C_i^{(2)}+ X_i \pp{C_i^{(1)}}{x} 
    +
    \frac{1}{2}   X_i^2
    \pppd{C_i^{(0)}}{x}{x}
    \Big\}
    \Big) = 
    \\
    \nonumber
    & 
    \hspace{1cm}
    \sum_{j=1}^N \sum_{r\in\mathcal{R}}
    \bigg\{
    G_{ij}^{r(0)}
    \Delta_{ji}^{-r} 
    \pp{p^{(0)}}{x}
    (C_j^{(0)}H_{ji}^{-r(0)}+C_i^{(0)}H_{ij}^{r(0)})
    \bigg\}
    +
    \nonumber
    \\
    & \hspace{2cm} \nonumber
    \varepsilon \bigg\{
    G_{ij}^{r(0)}
    \Delta_{ji}^{-r} 
    \pp{p^{(0)}}{x}
    (C_j^{(1)}H_{ji}^{-r(0)}+
        C_i^{(1)}H_{ij}^{r(0)})
    +
        \\
    & \hspace{2cm}
    G_{ij}^{r(0)}
    \Delta_{ji}^{-r} 
    \pp{p^{(0)}}{x}
    \Big( (X_j+r l)\pp{C_j^{(0)}}{x}H_{ji}^{-r(0)}
    +
     X_i\pp{C_i^{(0)}}{x}H_{ij}^{r(0)}\Big)
    +
    \nonumber
    \\
    & \hspace{2cm} 
    \bigg(
    G_{ij}^{r(0)}
    \Big( P_j^{(2)} -
    P_i^{(2)} \Big)
    +
    G_{ij}^{r(1)}
    \Delta_{ji}^{-r} 
    \pp{p^{(0)}}{x}
    +
    \nonumber
    \\
    & \hspace{2cm}
    G_{ij}^{r(0)} \bigg(  (X_j + r l) \pp{}{x} \Big( f+  W_j \pp{p^{(0)}}{x} \Big)
    - 
     X_i \pp{}{x} \Big(f+  W_i \pp{p^{(0)}}{x} \Big) \bigg)
    +
    \nonumber
     \\
     & \hspace{2cm}
    \frac{G_{ij}^{r(0)}}{2} 
    \bigg(   (X_j + r l)^2
    - 
      X_i^2  \bigg)
    \ppp{p^{(0)}}{x}
    +
    \nonumber
     \\
     & \hspace{2cm}
     \pp{G_{ij}^{r(0)}}{x} M_{ij}^{r} 
    \Delta_{ji}^{-r}
    \pp{p^{(0)}}{x}
     \bigg) (C_j^{(0)}H_{ji}^{-r(0)}+C_i^{(0)}H_{ij}^{r(0)}) -
         \nonumber
     \\
     & \hspace{2cm}
     \alpha G_{ij}^{r(0)}
    \Delta_{ji}^{-r} 
    \pp{p^{(0)}}{x}
    C_j^{(0)} H_{ji}^{-r(0)}
     \bigg\} + \mathcal{O}(\varepsilon^2).
    \nonumber
\end{align}
% \begin{align}
% \label{eq:1d_master_transport_6}
%     & V_i \pp{}{\tau}
%     \Big(
%     C_i^{(0)} + \varepsilon \Big\{ C_i^{(1)}+ X_i \pp{C_i^{(0)}}{x} \Big\} + 
%     \varepsilon^2
%     \Big\{
%     C_i^{(2)}+ X_i \pp{C_i^{(1)}}{x} 
%     +
%     \frac{1}{2}   X_i^2
%     \pppd{C_i^{(0)}}{x}{x}
%     \Big\}
%     \Big) = 
%     \\
%     \nonumber
%     & 
%     \hspace{1cm}
%     \sum_{j=1}^N \sum_{r\in\mathcal{R}}
%     \varepsilon 
%     \bigg\{
%     G_{ij}^{r(0)}
%     \Delta_{ji}^{-r} 
%     \pp{p^{(0)}}{x}
%     (C_j^{(0)}H_{ji}^{-r(0)}+C_i^{(0)}H_{ij}^{r(0)})
%     \bigg\}
%     +
%     \nonumber
%     \\
%     & \hspace{2cm} \nonumber
%     \varepsilon^2 \bigg\{
%     G_{ij}^{r(0)}
%     \Delta_{ji}^{-r} 
%     \pp{p^{(0)}}{x}
%     (C_j^{(1)}H_{ji}^{-r(0)}+
%         C_i^{(1)}H_{ij}^{r(0)})
%     +
%         \\
%     & \hspace{2cm}
%     G_{ij}^{r(0)}
%     \Delta_{ji}^{-r} 
%     \pp{p^{(0)}}{x}
%     \Big( (X_j+r l)\pp{C_j^{(0)}}{x}H_{ji}^{-r(0)}
%     +
%      X_i\pp{C_i^{(0)}}{x}H_{ij}^{r(0)}\Big)
%     +
%     \nonumber
%     \\
%     & \hspace{2cm} 
%     \bigg(
%     G_{ij}^{r(0)}
%     \Big( P_j^{(2)} -
%     P_i^{(2)} \Big)
%     +
%     G_{ij}^{r(1)}
%     \Delta_{ji}^{-r} 
%     \pp{p^{(0)}}{x}
%     +
%     \nonumber
%     \\
%     & \hspace{2cm}
%     G_{ij}^{r(0)} \bigg(  (X_j + r l) \pp{}{x} \Big( f+  W_j \pp{p^{(0)}}{x} \Big)
%     - 
%      X_i \pp{}{x} \Big(f+  W_i \pp{p^{(0)}}{x} \Big) \bigg)
%     +
%     \nonumber
%      \\
%      & \hspace{2cm}
%     \frac{G_{ij}^{r(0)}}{2} 
%     \bigg(   (X_j + r l)^2
%     - 
%       X_i^2  \bigg)
%     \ppp{p^{(0)}}{x}
%     +
%     \nonumber
%      \\
%      & \hspace{2cm}
%      \pp{G_{ij}^{r(0)}}{x} M_{ij}^{r} 
%     \Delta_{ji}^{-r}
%     \pp{p^{(0)}}{x}
%      \bigg) (C_j^{(0)}H_{ji}^{-r(0)}+C_i^{(0)}H_{ij}^{r(0)}) -
%          \nonumber
%      \\
%      & \hspace{2cm}
%      G_{ij}^{r(0)}
%     \Delta_{ji}^{-r} 
%     \pp{p^{(0)}}{x}
%     C_j^{(0)} H_{ji}^{-r(0)} A_{ji}^{-r(0)}
%      \bigg\}.
%     \nonumber
% \end{align}
This is the master equation for the conservation of particles that we seek to solve.
% We see that $H_{ji}^{-r(0)}$ disappears from all terms except the one that contains the adherence parameter group, $\alpha\delta$.
% The role of $H_{ji}^{-r(0)}$, therefore, is to ensure that adherence is only active 
% on pores $ijr$ that conduct flow into junction $i$, so that $H_{ji}^{-r(0)}$ is the macroscale version of $H(P_j-P_i)$, 
% as expected.
% on incoming edges relative to node $i$ (that is edges $(ij,r)$ for which $P_j>P_i$ so that $Q_{ij}^r<0$), which was our assumption in the underlying network model. $A_{ji}^{-r(0)}$ is equal to $A_{ij}^{r(0)}$ since the adhesivity function is undirected, and we therefore write $A_{ji}^{-r(0)}$ as a reminder of the direction of flow of particles on edges where this function makes a contribution.
% The final term therefore accounts for deposition, which we have found to be a second-order phenomenon.
% Note that it is not necessary to expand the node volume, $V_i$, since we have assumed that deposition occurs only on edges, and therefore $V_i$ does not vary in time. One important consequence is that the node porosity, that is the ratio of node-space volume to filter volume, is constant throughout filtration. Physically, the assumption is that all particle loss happens in channels, as opposed to in junctions between channels.

We now carry out Step \ref{step:4} to solve the master equation.
% At leading order in the master equation, \eqref{eq:1d_master_transport_6}, we find that
% \begin{align}
%     \bm{1} \circ \pp{\bm{C}^{(0)}}{T} = \bm{0},
% \end{align}
% where 
% $\bm{1} = (1, \hdots, 1)^{\top}$ and 
% $\bm{C} = (C_1, \hdots,C_N)^{\top}$ are the length-$N$ 1-arrays that contain the volumes of and concentrations in the junctions in the cell, respectively.
%  Thus, the leading-order concentration is steady on this timescale. This is because the leading-order pressure difference, \eqref{eq:mm_1d_leading-order-pressure-difference}, is zero, and so the flux that drives particle transport, and therefore concentration variation, is a first-order phenomenon.
% To look for temporal variation in concentration, therefore, we define a longer timescale  $T =\varepsilon \tau $. 
% Although, in this new timescale, a balance of leading-order terms in the master equation, \eqref{eq:1d_master_transport_6}, yields no information,
% a first-order balance gives 
At leading order in \eqref{eq:1d_master_transport_6} we see that 
\begin{align}
    \bm{1} \circ \pp{\bm{C}^{(0)}}{T} =  \bm{S}^{(0)} \bm{C}^{(0)} \pp{p^{(0)}}{x},
    \label{eq:1d_concentration-equation_microscale}
\end{align}
where, recall that $\circ$ denotes the Hadamard product
(and we use it here to remember that the unit sum of the volumes of nodes in the cell appears on the left of this equation, which we use to interpret the parameter $\phi$ in \eqref{eq:1d_definition-of-phi}),
and $\bm{S}^{(0)}$ is a matrix of shape $(N,N)$ with elements
\begin{align}
    S_{ij}^{(0)} = \sum_{r\in\mathcal{R}} \Big( G_{ij}^{r(0)} \Delta_{ji}^{-r} H_{ji}^{-r(0)}
    -
    \updelta_{ij} \sum_{k=1}^N G_{ik}^{r(0)}\Delta_{ik}^{r} H_{ik}^{r(0)}
    \Big).
    \label{eq:1d_definition-of-S}
\end{align}
We identify the terms inside the sums in $\bm{S}^{(0)}$
as the pore fluxes (per unit pressure gradient) into (since $G_{ij}^{r(0)} \Delta_{ji}^{-r}$ is multiplied by $H_{ji}^{-r(0)}$),
and out of (since $G_{ik}^{r(0)} \Delta_{ik}^{r}$ is multiplied by $H_{ik}^{r(0)}$) junction $i$.
% So $\bm{S}^{(0)}$ represents the net intra-cell flux, and \eqref{eq:1d_concentration-equation_microscale} describes the 
% evolution of the leading-order concentration within the cell. 
We note (see the proof in Appendix \ref{sec:1d_proofs_S})
that $\bm{S}^{(0)}$ satisfies
\begin{align}
    \bm{S}^{(0)} \bm{1} = \bm{0} = \bm{1}^{\top} \bm{S}^{(0)},
    \label{eq:1d_lemma_S}
\end{align}
which means that the total mass of particles in the fluid inside the cell is conserved. 
This is to be expected, because there is no intra-cell deposition, since adherence is an $\mathcal{O}(\varepsilon)$ phenomenon. 
We will find that one implication of this conservation is that
$\bm{S}^{(0)}$ 
% is irrelevant in the macroscale-concentration problem, and 
does not appear on longer timescales, on which inter-cell evolution occurs.
Despite this, $\bm{S}^{(0)}$ controls the evolution of the microscale-dependent concentration in \eqref{eq:1d_concentration-equation_microscale}, 
and the timescale, $T_{\text{c}}$ (see \eqref{eq:mm_define-T_c}), is the one over which the concentration settles down to one that is the same for all junctions in the cell. 

Since we are primarily interested in macroscale concentration changes, and thus inter-cell variation, we define a longer time variable, $t = \varepsilon T$. 
Substituting this into the master equation, \eqref{eq:1d_master_transport_6}, a leading-order balance yields
\begin{align}
    \bm{0} =  \bm{S}^{(0)} \bm{C}^{(0)} \pp{p^{(0)}}{x}.
\end{align}
Using the first equation of the identities in \eqref{eq:1d_lemma_S}, it follows that
\begin{align}
    \bm{C}^{(0)} = c^{(0)}(x,t) \bm{1},
    \label{eq:1d_leading-order-concentration}
\end{align}
where $c^{(0)}(x,t)$ is the leading-order concentration solution that we seek.
We see that this is independent of the junction index, $i$, so that it is microscale independent.
The new longer timescale,
\begin{align}
    T_{\text{n}} = \frac{T_{\text{c}}}{\varepsilon} = \frac{\bar{V}}{\bar{G}(p_{\text{in}}^*-p_{\text{in}}^*)},
    \label{eq:mm_define-T_n}
\end{align}
is the one associated with evolution of the particle concentration on the network as a whole.
% We have shown that there is no intra-cell concentration variation over this timescale, and so $t$, which is an order of $\varepsilon$ longer than the timescale $T$ over which we found that intra-cell concentration variation occurs, is the timescale over which we expect to observe macroscale variation.

Proceeding to second order to close the problem by finding an equation that governs $c^{(0)}$, by substituting the leading-order solution for the concentration, given by \eqref{eq:1d_leading-order-concentration}, into the master equation, \eqref{eq:1d_master_transport_6}, and noting that $H_{ij}^{r(0)}$ disappears from terms containing $c^{(0)}$ by using the Heaviside-function identity, \eqref{eq:1d_heaviside-identity},
we find that
% \begin{align}
% \label{eq:1d_second-order-equation_transport_2}
%     \bm{V} \pp{c^{(0)}}{t} 
%     = &
%     \bm{S}^{(0)} \bm{C}^{(1)} \pp{p^{(0)}}{x}
%     + 
%     \bm{K}^{(0)} \bm{1} \pp{p^{(0)}}{x} \pp{c^{(0)}}{x}
%     -
%     \bm{J}^{(0)} \bm{1} \pp{p^{(0)}}{x} c^{(0)}
%     +
%     \\
%     & \hspace{0.5cm} \nonumber
%     \bigg(
%     \bm{A}^{(0)} \bm{P}^{(2)} 
%     +
%     \bm{A}^{(1)}   \bm{W} \pp{p^{(0)}}{x} 
%     +
%      \bm{B}^{(1)} \bm{1} \pp{p^{(0)}}{x}
%     +
%      \bm{B}^{(0)} \pp{}{x} \Big(f \bm{1} 
%     +
%                 \\
%     & \hspace{1cm}
%      \pp{p^{(0)}}{x} \bm{W}\Big) 
%     +
%       \bm{C}^{(0)} \bm{1} \ppp{p^{(0)}}{x}
%     +
%      \bm{D}^{(0)}  \bm{W} \pp{p^{(0)}}{x} 
%     + 
%       \bm{E}^{(0)} \bm{1} \pp{p^{(0)}}{x}
%     \bigg) c^{(0)},
%     \nonumber
% \end{align}
\begin{align}
\label{eq:1d_second-order-equation_transport_2}
    \pp{c^{(0)}}{t}\bm{1}  
    = &
    \bm{S}^{(0)} \bm{C}^{(1)} \pp{p^{(0)}}{x}
    + 
    \bm{K}^{(0)} \bm{1} \pp{p^{(0)}}{x} \pp{c^{(0)}}{x}
    -
    \bm{J}^{(0)} \bm{1} \pp{p^{(0)}}{x} c^{(0)}
    +
    \\
    & \hspace{0.5cm} \nonumber
    \bigg(
    \bm{A}^{(0)} \bm{P}^{(2)} 
    +
    \bm{A}^{(1)}   \bm{W} \pp{p^{(0)}}{x} 
    +
     \bm{B}^{(1)} \bm{1} \pp{p^{(0)}}{x}
    +
     \bm{B}^{(0)} \pp{}{x} \Big(f \bm{1} 
    +
                \\
    & \hspace{1cm}
     \pp{p^{(0)}}{x} \bm{W}\Big) 
    +
      \bm{C}^{(0)} \bm{1} \ppp{p^{(0)}}{x}
    +
     \bm{D}^{(0)}  \bm{W} \pp{p^{(0)}}{x} 
    + 
      \bm{E}^{(0)} \bm{1} \pp{p^{(0)}}{x}
    \bigg) c^{(0)}
    \nonumber
\end{align}
where $\bm{K}^{(0)}$ is the matrix of shape ($N,N$) with elements
\begin{align}
    K_{ij}^{(0)} = \sum_{r\in\mathcal{R}}\bigg( G_{ij}^{r(0)} \Delta_{ji}^{-r} \Big((X_j + r l)H_{ji}^{-r(0)} + X_i H_{ij}^{r(0)} \Big) \bigg),
    \label{eq:1d_definition-of-K}
\end{align}
and 
$\bm{J}^{(0)}$ is another matrix of shape $(N,N)$ with elements
% \begin{align}
%     J_{ij}^{(0)} = \sum_{r\in\mathcal{R}} \Big( G_{ij}^{r(0)} \Delta_{ji}^{-r} A_{ji}^{-r(0)} H_{ji}^{-r(0)} \Big).
%     \label{eq:1d_definition-of-J}
% \end{align}
\begin{align}
    J_{ij}^{(0)} = \sum_{r\in\mathcal{R}} \Big(\alpha  G_{ij}^{r(0)} \Delta_{ji}^{-r} H_{ji}^{-r(0)} \Big).
    \label{eq:1d_definition-of-J}
\end{align}
We identify the term inside the sum in $K_{ij}^{(0)}$ as 
the microscale contribution to the homogenised flux (noting the presence of $G_{ij}^{r(0)} \Delta_{ji}^{-r}$) and so we expect $\bm{K}^{(0)}$ to be a quantity related to macroscale advection.
Meanwhile, we identify the term inside the sum in $J_{ij}^{(0)}$ as the microscale component of the homogenised flux multiplied by the adherence (noting the presence of $\alpha$) in pores conducting particles into junction $i$ (noting the presence of $H_{ji}^{-r(0)}$), and 
so we expect to find that $\bm{J}^{(0)}$ corresponds to deposition in the macroscale equation.

Using the second-order fluid conservation equation, \eqref{eq:mm_1d_fluid-conservation_second-order_problem_1}, we find that the sum of all but the first three terms on the right-hand side of \eqref{eq:1d_second-order-equation_transport_2} is equal to zero, and so, pre-multiplying \eqref{eq:1d_second-order-equation_transport_2} by $\bm{1}^{\top}$ gives
% \begin{align}
% \label{eq:1d_second-order-equation_transport_3}
%     \pp{c^{(0)}}{t} =  
%       \frac{\bm{1}^{\top} \bm{K}^{(0)} \bm{1}}{\bm{1}^{\top} \bm{V}}   \pp{p^{(0)}}{x} \pp{c^{(0)}}{x}
%     - 
%     \frac{\bm{1}^{\top} \bm{J}^{(0)} \bm{1}}{\bm{1}^{\top} \bm{V}}
%      \pp{p^{(0)}}{x} c^{(0)},
% \end{align}
\begin{align}
\label{eq:1d_second-order-equation_transport_3}
    \pp{c^{(0)}}{t} =  
      \frac{\bm{1}^{\top} \bm{K}^{(0)} \bm{1}}{\bm{1}^{\top} \bm{1}}   \pp{p^{(0)}}{x} \pp{c^{(0)}}{x}
    - 
    \frac{\bm{1}^{\top} \bm{J}^{(0)} \bm{1}}{\bm{1}^{\top} \bm{1}}
     \pp{p^{(0)}}{x} c^{(0)},
\end{align}
where we have used the second equation in the identities given by \eqref{eq:1d_lemma_S}.
% We have shown that the long-timescale evolution of the leading-order particle concentration is governed
% by an by an advection--reaction equation. 
This is the governing equation that determines the leading-order particle concentration, $c^{(0)}$, 
and thus closes the problem. 

 To interpret \eqref{eq:1d_second-order-equation_transport_3}, we first note (see Appendix \ref{sec:1d_proofs_K-is-k} for proof) that $\bm{K}^{(0)}$ is proportional to the leading-order permeability, such that 
\begin{align}
    \bm{1}^{\top} \bm{K}^{(0)} \bm{1} 
    = 
    l k^{(0)} 
    = 
    l \sigma^{(0)},
    \label{eq:1d_K-is-k}
\end{align}
so that $\bm{K}^{(0)}$ is indeed an advective term, since it is proportional to the macroscale 
conductivity of the network.
Similarly, we define
\begin{align}
    j^{(0)} = -\frac{\bm{1}^{\top} \bm{J}^{(0)} \bm{1}}{l}
    = 
    -\frac{1}{l}\sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}}
    \alpha  G_{ij}^{r(0)} \Delta_{ji}^{-r} H_{ji}^{-r(0)},
    \label{eq:1d_definition-of-j}
\end{align}
which is a positive scalar quantity (since $\Delta_{ji}^{-r}$ is negative on the relevant pores). 
We call $j^{(0)}$ the adhesivity, since it represents the macroscale 
result of pore adherence on the microscale in the same way that the permeability, $k^{(0)}$, 
represents the macroscale result of pore conductance on the microscale. 
To interpret \eqref{eq:1d_second-order-equation_transport_3} further, we define 
\begin{align}
    v=\bm{1}^{\top} \bm{1} = N,   
    \label{eq:mm_1d_define-v}
\end{align}
which is the total volume of all junctions in the cell.
Using the proportionality of $\bm{K}^{(0)}$ and $k^{(0)}$, \eqref{eq:1d_K-is-k},
and of $\bm{J}^{(0)}$ and $j^{(0)}$, \eqref{eq:1d_definition-of-j}, and the definition of junction volume, $v$ in \eqref{eq:mm_1d_define-v}, we re-write \eqref{eq:1d_second-order-equation_transport_3} in terms of the leading-order permeability and adhesivity, 
which gives
\begin{align}
\label{eq:1d_second-order-equation_transport_4}
    \pp{c^{(0)}}{t} = \frac{l}{v} \Big(k^{(0)} \pp{p^{(0)}}{x}\Big)  \pp{c^{(0)}}{x} 
    +
    \frac{l}{v}
    \Big(j^{(0)} \pp{p^{(0)}}{x}\Big) \, c^{(0)}.
\end{align}
Furthermore, we identify
\begin{align}
    \phi = \frac{v}{l}, 
    \label{eq:1d_definition-of-phi}
\end{align}
as the junction porosity, which is a constant, since it is the scaled 
ratio of the total volume of junctions, within which deposition does not occur, to the cell volume.
Therefore, defining
\begin{align}
    \psi^{(0)} = -j^{(0)} \pp{p^{(0)}}{x},
    \label{eq:1d_definition-of-psi}
\end{align}
which we call the reactivity, 
and using Darcy's law, \eqref{eq:1d_macroscale_darcy_1}, to relate the permeability to the Darcy velocity, $u^{(0)}$, we arrive at
\begin{align}
    \pp{c^{(0)}}{t} + \pp{}{x} \Big(\frac{u^{(0)}}{\phi}c^{(0)}\Big)  = - \frac{\psi^{(0)}}{\phi}  c^{(0)},
    \label{eq:1d_macroscale_transport}
\end{align}
where we have used the macroscale equation for conservation of fluid, \eqref{eq:1d_macroscale_conservation-of-mass_1}, to write the macroscale equation for the conservation of particles, \eqref{eq:1d_second-order-equation_transport_4}, in conservative form.

We have found that the particle concentration, $c^{(0)}$, varies on the continuous macroscale subject to an advection--reaction equation, \eqref{eq:1d_macroscale_transport}.
% Its evolution depends on the parameters of this system, which are the velocity, $u^{(0)}$, and the reactivity, $\psi^{(0)}$. 
% In turn, these are determined by the permeability, $k^{(0)}$, and the adhesivity
% which is a continuous advection--reaction equation with parameters that depend on the microscale properties of the network.
Particles are advected through the filter by the pore velocity, $u^{(0)}\phi^{-1}$, which is related to the pressure, $p^{(0)}$, and the permeability, $k^{(0)}$, which is the effective conductance of the filter, using the Darcy equation \eqref{eq:1d_macroscale_darcy_1}. 
As they do so, particles deposit inside the filter at a rate determined by the reactivity, $\psi^{(0)}\phi^{-1}$, 
which is related to the pressure, $p^{(0)}$, and the adhesivity, $j^{(0)}$, which is the effective adherence of the filter, using the 
reaction equation, \eqref{eq:1d_definition-of-psi}. 
The permeability and the adhesivity are related to the conductance of pores in the network, $G_{ij}^{r(0)}$, via \eqref{eq:1d_definition-of-permeability} 
and \eqref{eq:1d_definition-of-j}, respectively, and thus the macroscale particle concentration depends upon the dynamic structure of the microscale.
% The permeability is defined by \eqref{eq:1d_definition-of-permeability} and is therefore related to the microscale edge conductances.

\subsection{Conservation of pores}
Lastly, we homogenise the equation for the conservation of pores, \eqref{eq:mm_conductance-conservation_sum_nodim_1},
\begin{align}
    \pp{G_{ij}^r}{T} &= - \alpha\beta C_{ij}^{r} (G_{ij}^r)^{\frac{3}{2}} |P_i-P_j|.
    \label{eq:mm_pore-conservation_homog_1}
\end{align}
Carrying out Steps \ref{step:1}--\ref{step:3} of network homogenisation on the remaining terms in \eqref{eq:mm_pore-conservation_homog_1}, 
% Notet that if A is not a constant then where this time we apply a Taylor expansion and an asymptotic expansion to the concentration and the adhesivity function, $A_{ij}^{\bm{r}}$, as well as the conductance and the pressure. 
using the leading- and first-order pressure solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and
\eqref{eq:mm_1d_fluid-conservation_first-order-solution}, and the leading-order particle concentration, \eqref{eq:1d_leading-order-concentration},
and noting the cancellation of the leading-order pressure difference, \eqref{eq:mm_1d_leading-order-pressure-difference}, we find that
\begin{align}
    \pp{G_{ij}^{r(0)}}{T} &= - \varepsilon \bigg\{ \alpha\beta c^{(0)} (G_{ij}^{r(0)})^{\frac{3}{2}} \Big|\Delta_{ij}^r \pp{p^{(0)}}{x} \Big| \bigg\} + \mathcal{O}(\varepsilon^2).
    \label{eq:mm_pore-conservation_homog_2}
\end{align}
A leading-order balance in \eqref{eq:mm_pore-conservation_homog_2} implies that 
\begin{align}
    \pp{G_{ij}^{r(0)}}{T} &= 0.
    \label{eq:mm_pore-conservation_homog_3}
\end{align}
We have found that the conductances of pores do not change over the shorter timescale, $T_{\text{c}}$ (see \eqref{eq:mm_define-T_c}), which is unsurprising, 
since we showed (see \eqref{eq:1d_concentration-equation_microscale}) that deposition occurs over a longer timescale.
Scaling onto the longer timescale over which particle deposition occurs, $T_{\text{n}}$ (see \eqref{eq:mm_define-T_n}), and recalling that $t=\varepsilon T$, 
we find that 
\begin{align}
    \pp{G_{ij}^{r(0)}}{t} &= -  \alpha\beta c^{(0)} (G_{ij}^{r(0)})^{\frac{3}{2}} \Big|\Delta_{ij}^r \pp{p^{(0)}}{x} \Big| + \mathcal{O}(\varepsilon).
    \label{eq:mm_pore-conservation_homog_4}
\end{align}
This time, balancing terms at leading order, we find that
\begin{align}
    % \pp{\bm{G}^{(0)}}{t} &= -  \alpha\beta  \bm{R}^{(0)} \Big| \pp{p^{(0)}}{x} \Big|c^{(0)},
    \pp{\bm{G}^{(0)}}{t} &= -  \alpha\beta  \bm{R}^{(0)} \Big|\pp{p^{(0)}}{x}\Big| c^{(0)},
    \label{eq:mm_pore-conservation_homog_5}
\end{align}
where $\bm{G}^{(0)}$ and $\bm{R}^{(0)}$ are tensors of shape ($N,N,2K+1$) (recalling from Section \ref{sec:mm_Microscale model} that $K$ dictates the size of the set $\mathcal{R}$) with elements 
\begin{align}
    G_{ij}^{r} = G_{ij}^{r(0)}, 
    \quad
    R_{ij}^{r} = (G_{ij}^{r(0)})^{\frac{3}{2}} | \Delta_{ji}^{-r} |.
\end{align}
We have found that the evolution of the conductances of pores within the cell depend on the microscale structure, via the elements of 
$\bm{R}^{(0)}$, and on the evolution of the macroscale concentration and pressure gradient, $c^{(0)}$ and $\partial p^{(0)}/\partial x$, 
respectively.

\subsection{Summary}
In summary, we homogenise the microscale model, \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_cond-pdf_sum_nodim_1}, 
and find a multiscale model. 
In particular, homogenising the microscale equations for the conservation of fluid, particles, and pores,
\eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_conductance-conservation_sum_nodim_1}, 
we find, dropping leading-order notation, that their macroscale equivalents are
\begin{subequations}
\label{eq:mm_1d_conservation_homog_sum}
    \begin{align}
    \pp{u}{x}=0, 
    \quad 
    \pp{c}{t} + \frac{\partial}{\partial x}\Big( \frac{u c}{\phi} \Big) = -\frac{\psi}{\phi}c, 
    \quad 
    % \pp{G_{ij}^r}{t} = -\alpha\beta (G_{ij}^r)^{\frac{3}{2}} \Delta_{ji}^{-r} \Big| \pp{p}{x} \Big| c,
    \pp{G_{ij}^r}{t} = -\alpha\beta (G_{ij}^r)^{\frac{3}{2}} |\Delta_{ji}^{-r}| \Big|\pp{p}{x}\Big| c,
    \tag{\ref{eq:mm_1d_conservation_homog_sum}a,b,c}
\end{align}
\end{subequations}
respectively, where $\phi=N/l$ is the constant junction porosity, which represents the proportion of void 
space in the filter that is attributed to junctions.
Homogenising the microscale flux, \eqref{eq:mm_flux-definition_sum_nodim_1}, 
we find that the Darcy velocity, $u$, is related to the permeability, $k$, 
and the pressure, $p$, by the Darcy equation, 
and that the reactivity, $\psi$, is related to the adhesivity, $j$, and the 
pressure by the reactivity equation, which are respectively given by 
\begin{subequations}
\label{eq:mm_1d_darcy-and-reaction_homog_sum}
    \begin{align}
    u = - k\pp{p}{x}, 
    \quad
    \psi = -j\pp{p}{x}.
    \tag{\ref{eq:mm_1d_darcy-and-reaction_homog_sum}a,b}
\end{align}
\end{subequations}
The permeability and adhesivity, $k$ and $j$, which are the parameters of this system, 
are related to the pore conductances, $G_{ij}^r$,
by the equations 
\begin{subequations}
\label{eq:mm_1d_permeability-and-adhesivity_homog_sum}
    \begin{align}
    k = \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} r G_{ij}^r \Delta_{ji}^{-r}, 
    \quad
    j = -\frac{1}{l} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \alpha G_{ij}^r \Delta_{ji}^{-r} H_{ji}^{-r}. 
    \tag{\ref{eq:mm_1d_permeability-and-adhesivity_homog_sum}a,b}
\end{align}
\end{subequations}
Here, the pore pressure drop is
\begin{align}
    \Delta_{ij}^r = W_i - (W_j+rl),
    \label{eq:mm_1d_pressure-delta_homog_sum}
\end{align}
where the cell solution, $W_i$, is an effective junction position 
that dictates the 
contribution of the pressure drop to the solution evolution on the macroscale, via the permeability 
and adhesivity in \eqref{eq:mm_1d_permeability-and-adhesivity_homog_sum}.
The cell solution is given by solving the linear cell problem, 
\begin{align}
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} \left(  G_{ij}^{r {(s)}} - \updelta_{ij} \sum_{k=1}^N G_{ik}^{r {(s)}} \right) W_j
    = 
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} \left( G_{ij}^{r {(s)}} r l
    \right)
    \label{eq:mm_1d_cell-problem_homog_sum}
\end{align}
where we have chosen the position of junction $i$ to be $X_i=0$ for simplicity, since 
this does not affect the solution.
% Subject to constant adherence, the 
% pore conductances are related via the conductivity equation, 

Homogenising the microscale boundary conditions, \eqref{eq:mm_bc_pres_sum_nodim_1} and 
\eqref{eq:mm_ic_pres_sum_nodim_1}, 
we find that we are given the pressure at the inlet and outlet,
\begin{subequations} \label{eq:mm_1d_bc_pres_homog_sum}
\begin{align}
    p(0,t) = 1, 
    \quad
    p(l,t) = 0,
    \tag{\ref{eq:mm_1d_bc_pres_homog_sum}a,b}
\end{align}
\end{subequations}
and the concentration at the inlet,
\begin{align}
    c(0,t) = 1,
    \label{eq:mm_1d_ic_pres_homog_sum}
\end{align}
for $t\ge0$.
Homogenising the microscale initial conditions, \eqref{eq:mm_ic_conc_sum_nodim_1} and \eqref{eq:mm_ic_cond_sum_nodim_1}, 
we find that there are no particles in the filter initially,
\begin{align}
    c(x,0) = 0, 
    \label{eq:mm_1d_ic_conc_homog_sum}
\end{align}
and that the initial pore conductances are drawn from a known distribution, so that 
\begin{align}
    G_{ij}^{r}(x,0) = G_{ij}^{r0}(x)
    \label{eq:mm_1d_ic_cond_homog_sum}
\end{align}
for $i,j \in \{1,\hdots,N,\text{in}, \text{out}\}$.
Note that the initial conductances, $G_{ij}^{r0}(x)$, may vary with the spatial variable, $x$, 
which permits, for example, enforcement of slowly varying conductance distributions
due to macroscale porosity gradients.

% At leading order we find that nothing changes so we need to wait longer. 
% So rescale onto a longer timescale.
% Redefine time
% \begin{align}
%     \pp{G_{ij}^r}{t} &= - \alpha\gamma\delta c^{(0)} (G_{ij}^{r(0)})^{\frac{3}{2}} \Big|\Delta_{ij}^r \pp{p^{(0)}}{x} \Big| + \mathcal{O}(\varepsilon) .
% \end{align}



% via the matrices $\bm{B}^{(0)}$ and $\bm{C}^{(0)}$ using the definition of $\sigma^{(0)}$ in \eqref{eq:1d_sigma}, since $k^{(0)}=\sigma^{(0)}$.
% Since the permeability also depends on $\bm{W}$ via the definition of $\Delta_{ij}^r$, \eqref{eq:1d_definition-of-delta}, the cell problem, \eqref{eq:mm_1d_cell-problem_1}, must be solved to find $\bm{W}$ as part of the system.

% Note that if $G$ is chosen so that deposition occurs (via a non-zero adhesivity, $A_{ij}^{(0)}$), then the network is dynamic since the edge conductances change in time. 
% Importantly, in this case, the cell problem must be solved at each time instance, since the cell problem, and therefore the value of $\bm{W}$ itself, depends on these microscale conductances via the matrices $\bm{A}^{(0)}$ and $\bm{B}^{(0)}$.
% The macroscale concentration then decreases due to deposition out of the flow, which occurs via the positive parameter $\psi^{(0)}$ defined in \eqref{eq:1d_definition-of-psi}, a reaction term that depends on the edge conductances and the leading-order microscale adhesivity, $A_{ij}^{(0)}$, via $j^{(0)}$ given by \eqref{eq:1d_definition-of-j}, which itself is ultimately determined by the matrix $\bm{J}^{(0)}$ in \eqref{eq:1d_definition-of-J}.

\section{Multiscale model in $D$ dimensions}
\label{sec:Dd_network-homogenisation}
We now present the $D$-dimensional formulation of the homogenisation of the periodic-network model, \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_conductance-conservation_sum_nodim_1}, where $D \in \{1,2,3\}$.
In this setting, the spatial vectors defined in Section \ref{sec:mm_Microscale model}, such as $\bm{x},\bm{X}_i$, and $\bm{r}$, are $D$-dimensional, and we describe dynamics on networks such as that illustrated in Figure \ref{fig:network_9}, as well as Figures \ref{fig:mm_network_1}--\ref{fig:network_5}.
We will find that the $D$-dimensional derivation is similar to the quasi-one-dimensional case presented in Section \ref{sec:1d_network-homogenisation}, and we will therefore find $D$-dimensional analogies to the one-dimensional objects already defined.

\begin{figure}[!htb]
    \begin{center}
         \includegraphics[width=0.7\textwidth]{./figures/multiscale/drawings/network_9.pdf}
        \caption{A diagram of a periodic network embedded in two-dimensional space.
        There are $N=3$ nodes in the cell and this is repeated nine times. 
        The conductance on some edges is labelled.} 
        % to illustrate the nomenclature.} %Node that edges can cross since this is a topological representation of a more complex geometry.}
        %In this case, nodes in one cell are connected to nodes in cells up to two cells away, so that $r^1$ and $r^2$ are drawn from the set $\mathcal{R}=\{-2,-1,0,1,2\}$ and $\bm{r}$ is drawn from $\bm{\mathcal{R}}=\{-2,-1,0,1,2\}^2$.}         
        \label{fig:network_9}
    \end{center}
\end{figure}

\subsection{Conservation of fluid}
As before, we begin by homogenising the microscale equation for the conservation of fluid mass, \eqref{eq:mm_fluid-conservation_sum_nodim_1}. 
As such, using the definition of the pore flux, \eqref{eq:mm_flux-definition_sum_nodim_1}, to write \eqref{eq:mm_fluid-conservation_sum_nodim_1} in terms of the junction pressure, $P_i$, and pore conductance, $G\indx$, and then following Steps \ref{step:1}--\ref{step:3} by evaluating the pressure and conductance at $\bm{x}$, which we define to be the left-bottom corner of the reference cell,  carrying out a Taylor expansion of the pressure and conductance about $\bm{x}$, and assuming asymptotic forms for these quantities, \eqref{eq:mm_asymptotic-solution_pressure} and \eqref{eq:mm_asymptotic-solution_conductance},
we find that
\begin{align}
    & 
    \bm{A}^{(0)} \bm{P}^{(0)} + 
    \varepsilon \bigg\{ 
    \bm{A}^{(0)} \bm{P}^{(1)}
    +
    \bm{A}^{(1)} \bm{P}^{(0)} 
    + 
    \sum_{m=1}^D \bm{D}^{m (0)} \bm{P}^{(0)}
    + 
    \sum_{m=1}^D \bm{B}^{m (0)} \pp{\bm{P}^{(0)}}{x^m} 
    \bigg\} +
    \nonumber
    \\
    & \hspace{0.5cm}
    \varepsilon^2 \bigg\{
    \bm{A}^{(0)} \bm{P}^{(2)}
    +
    \bm{A}^{(1)} \bm{P}^{(1)} 
    + 
    \sum_{m=1}^D \bm{D}^{m (0)} \bm{P}^{(1)}
    + 
    \sum_{m=1}^D \bm{B}^{m (0)} \pp{\bm{P}^{(1)}}{x^m} 
    +
    \bm{A}^{(2)} \bm{P}^{(0)} 
    +
    \nonumber
    \\
    & \hspace{1.0cm}
    \sum_{m=1}^D \bm{D}^{m (1)} \bm{P}^{(0)}
    +
    \sum_{m=1}^D \bm{B}^{m (1)} \pp{\bm{P}^{(0)}}{x^m}
    +
    \sum_{m=1}^D \sum_{n=1}^D \bm{F}^{mn (0)} \bm{P}^{(0)}
    +
    \nonumber
    \\
    & \hspace{1.5cm}
    \sum_{m=1}^D \sum_{n=1}^D \bm{E}^{mn (0)} \pp{\bm{P}^{(0)}}{x^n}
    +
    \sum_{m=1}^D \sum_{n=1}^D \bm{C}^{mn (0)} \pppd{\bm{P}^{(0)}}{x^m}{x^n}    \bigg\} + \mathcal{O}(\varepsilon^3) = \bm{0},
    \label{eq:Dd_master_conservation-of-mass_6}
\end{align}
which is the master equation for the conservation of fluid that we seek.
This is the $D$-dimensional equivalent of \eqref{eq:mm_1d_fluid-conservation_master_3}, where upper indices now denote the spatial direction, so that  $m,n \in \{1, \hdots,D\}$.
As before, $\bm{P}^{(s)}=(P_1^{(s)}, \hdots, P_N^{(s)})^{\top}$ is the vector of $\mathcal{O}(\varepsilon^{(s)})$ pressure solutions in the lower indices, which is length $N$. 
This time, $\bm{A}^{(s)}$, $\bm{B}^{(s)}$, $\bm{C}^{(s)}$, $\bm{D}^{(s)}$, $\bm{E}^{(s)}$, and $\bm{F}^{(s)}$, are defined by the elements
\begin{align}
    A_{ij}^{(s)} &= 
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \left(  G_{ij}^{\bm{r} {(s)}} - \updelta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \right),
    \label{eq:Dd_def_A}
    \\
    B_{ij}^{m (s)} &= 
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m
    )- \updelta_{ij} X_i^m \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \right),
    \label{eq:Dd_def_B}
    \\
    C_{ij}^{mn (s)} &= 
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) - \updelta_{ij} X_i^m X_i^n \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \right),
    \label{eq:Dd_def_C}
    \\
    D_{ij}^{m (s)} &= 
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} -\updelta_{ij}  \sum_{k=1}^N \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} M_{ik}^{\bm{r}m}  \right),
    \label{eq:Dd_def_D}
    \\
    E_{ij}^{mn (s)} &= 
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} (X_j^n + r^nl^n)  - \updelta_{ij}  X_i^n \sum_{k=1}^N  \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} M_{ik}^{\bm{r}m}  \right)
    \label{eq:Dd_def_E}
    \\
    F_{ij}^{mn (s)} &= \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( \pppd{G_{ij}^{\bm{r} {(s)}}}{x^m}{x^n} M_{ij}^{\bm{r}m}  M_{ij}^{\bm{r}n}  - \updelta_{ij}   \sum_{k=1}^N  \pppd{G_{ik}^{\bm{r} {(s)}}}{x^m}{x^n} M_{ik}^{\bm{r}m} M_{ik}^{\bm{r}n}  \right).
    \label{eq:Dd_def_F}
\end{align}
These are 
% $4$-tensors of shape ($N,N,D,D$) 
tensors that are lower indexed by junction indices, $i$ and $j$, and upper spatial indices, $m$ and $n$.
Comparing these tensors with the matrices defined in \eqref{eq:1d_def_A}--\eqref{eq:1d_def_F}, we see that $\bm{A}^{(s)}$ is a matrix of shape 
$(N,N)$ with no spatial indices, and is identical to that defined before, \eqref{eq:1d_def_A}. 
The other objects have gained spatial indices. 
It is simple to verify that when $m=1$, the $3$-tensors of shape $(N,N,D)$ with a single spatial index, $\bm{B}^{(s)}$ and $\bm{D}^{(s)}$, are equal to their one-dimensional versions defined in \eqref{eq:1d_def_B} and \eqref{eq:1d_def_D}, and when $m=1,n=1$, the $4$-tensors of shape $(N,N,D,D)$ with two spatial indices, $\bm{C}^{(s)}$, $\bm{E}^{(s)}$, $\bm{F}^{(s)}$, defined in \eqref{eq:1d_def_C}, \eqref{eq:1d_def_E}, and \eqref{eq:1d_def_F}, are equal to their one-dimensional versions. 
We therefore interpret diagonal elements (in the spatial indices) of these new tensors as their one-dimensional counterparts, and so only off-diagonal elements are new. 
The implication is that the $D$-dimensional master equation, \eqref{eq:Dd_master_conservation-of-mass_6}, contains cross-derivative terms that the one-dimensional equivalent,  \eqref{eq:mm_1d_fluid-conservation_master_3}, does not.
In analogy with \eqref{eq:mm_1d_matrix-lemma_1}--\eqref{eq:mm_1d_matrix-lemma_3}, it can be shown (see Appendix \ref{sec:proofs_ABCDE} for proof) that these new tensors satisfy
\begin{subequations} \label{eq:Dd_matrix-lemma_1}
\begin{align}
    \bm{A}^{(s)} \bm{1} = \bm{0},
    \quad 
    \bm{1}^\top \bm{B}^{m (s)} \bm{1} = \bm{0},
    \quad 
    \bm{D}^{m (s)} \bm{1} = \bm{0},
    \quad 
    \bm{F}^{mn (s)} \bm{1} = \bm{0},
    \tag{\ref{eq:Dd_matrix-lemma_1}a--d}
\end{align}
\end{subequations}
and
\begin{align}
    \bm{1}^\top \bm{D}^{m (s)} &= 
    \pp{}{x^m}\Big(\bm{1}^\top \bm{B}^{m (s)} \Big),
    \label{eq:Dd_matrix-lemma_2_D-and-B}
    \\
    \bm{1}^\top \bm{E}^{mn (s)} \bm{1} &=
    \pp{}{x^m}\Big(\bm{1}^\top \bm{C}^{mn (s)} \bm{1} \Big) 
    +
        \label{eq:Dd_matrix-lemma_2_E-and-C}
    \\
    & \hspace{3cm} \nonumber
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r}(s)}}{x^m}(r^nl^nX_i^m-r^ml^mX_i^n),
\end{align}
for each spatial index $m$ and $n$, where $\bm{1} = (1,\ldots,1)^\top$ and $\bm{0} = (0,\ldots,0)^\top$ are vectors of length $N$ in the junction index, as before. 
Comparing \eqref{eq:Dd_matrix-lemma_2_E-and-C} to
\eqref{eq:mm_1d_matrix-lemma_2_D-and-B_E-and-C_E-and-C}, we see (in Appendix \ref{eq:Dd_proof_matrix-lemma_2_E-and-C}) that off-diagonal elements of these new identities now make non-zero contributions, which, as we will see, is one source of complication in the $D$-dimensional derivation in comparison to the quasi-one-dimensional equivalent.
Note that $\bm{A}^{(s)}$ is still symmetric in the junction indices, and so $\bm{1}^{\top}$ is still a left zero-eigenvector, so that \eqref{eq:1d_matrix-lemma_4} and \eqref{eq:mm_1d_matrix-lemma_3} are unchanged.

We now solve the master equation for fluid conservation, \eqref{eq:Dd_master_conservation-of-mass_6}. 
Balancing leading-order terms, we find that
\begin{align}
    \bm{A}^{(0)} \bm{P}^{(0)} &= \bm{0},
    \label{eq:Dd_leading-order-problem_conservation-of-mass}
\end{align}
as before, which has the solution
\begin{align}
    \bm{P}^{(0)} = p^{(0)}(\bm{x},t) \bm{1}.
    \label{eq:Dd_leading-order-solution_conservation-of-mass}
\end{align}
Here, as usual, $p^{(0)}(\bm{x},t)$ is the leading-order macroscale pressure solution to be found.
% , and we assume that the network remains fully connected, so that \eqref{eq:Dd_leading-order-solution_conservation-of-mass} is unique.
As in the one-dimensional case (see \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1}), the leading-order pressure solution is independent of the microscale, so that the leading-order intra-cell pressure difference is zero,
\begin{align}
    P_j^{(0)}-P_i^{(0)}=0.
    \label{eq:Dd_leading-order-pressure-difference}
\end{align}
We pre-multiply both sides of  \eqref{eq:Dd_leading-order-problem_conservation-of-mass} by $\bm{1}^{\top}$ and see that the resulting equation is satisfied by any $p^{(0)}$, so we must proceed to the next order in $\varepsilon$ for a solvability condition.

At first order, substituting the leading-order pressure solution, \eqref{eq:Dd_leading-order-solution_conservation-of-mass}, into the master equation, \eqref{eq:Dd_master_conservation-of-mass_6},
and using the identities (\ref{eq:Dd_matrix-lemma_1}a) and (\ref{eq:Dd_matrix-lemma_1}c) to simplify, we find that
\begin{align}
    \bm{A}^{(0)} \bm{P}^{(1)} + \sum_{m=1}^D \bm{B}^{m (0)} \bm{1} \pp{p^{(0)}}{x^m} = \bm{0}.
    \label{eq:Dd_first-order-problem_conservation-of-mass}
\end{align}
This has the solution 
\begin{align}
    \bm{P}^{(1)} = f(\bm{x},t) \bm{1} + \sum_{m=1}^D \pp{p^{(0)}}{x^m} \bm{W}^m,
    \label{eq:Dd_first-order-solution_conservation-of-mass}
\end{align}
where $f(\bm{x},t)$ is an arbitrary microscale-independent function, and $\bm{W}^m$ satisfies 
\begin{align}
    \bm{A}^{(0)} \bm{W}^m = - \bm{B}^{m (0)} \bm{1},
    \label{eq:Dd_cell-problem_1}
\end{align}
for each $m \in \{1,\hdots,D\}$.
Comparing the first-order pressure solution, \eqref{eq:Dd_first-order-solution_conservation-of-mass}, to its one-dimensional equivalent, \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, we see that the $D$-dimensional version is a sum of the one-dimensional solutions, as we would expect. 
Meanwhile, comparing the $D$-dimensional cell problem, \eqref{eq:Dd_cell-problem_1}, to the one-dimensional one, \eqref{eq:mm_1d_cell-problem_1}, we see that the cell solution, $\bm{W}$, is now a matrix of shape $(N,D)$. 
As such, $D$ independent linear problems of size $N$ must now be solved to find its elements, $W_i^m$. 
From \eqref{eq:Dd_first-order-solution_conservation-of-mass}, we see that each vector of length $N$ of the cell solution, $\bm{W}^m$, contributes to the first-order pressure in the corresponding direction, $x^m$.
Using \eqref{eq:Dd_def_B}, we see that the first-order pressure difference is non-zero, and is given by
\begin{align}
    P_j^{(1)}-P_i^{(1)} = \sum_{m=1}^D G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m) \pp{p^{(0)}}{x^m},
    \label{eq:Dd_first-order-pressure-difference}
\end{align}
which remains independent of $f$.
As before, the first-order pressure difference, \eqref{eq:Dd_first-order-pressure-difference}, determines the flux down the 
corresponding pore, and this depends on the macroscale pressure gradient, $\partial p^{(0)}/\partial x^m$.
Pre-multiplying both sides of the first order problem, \eqref{eq:Dd_first-order-problem_conservation-of-mass},
by $\bm{1}^{\top}$ and using (\ref{eq:Dd_matrix-lemma_1}b), we see that the resulting equation holds for any $p^{(0)}$, and we must therefore consider higher-order equations to find a solvability condition.

Proceeding to second order, by substituting the leading- and first-order pressure solutions, \eqref{eq:Dd_leading-order-solution_conservation-of-mass} and \eqref{eq:Dd_first-order-solution_conservation-of-mass}, into the master equation, \eqref{eq:Dd_master_conservation-of-mass_6}, and using (\ref{eq:Dd_matrix-lemma_1}a), (\ref{eq:Dd_matrix-lemma_1}c), and (\ref{eq:Dd_matrix-lemma_1}d) to simplify, we see that
\begin{align}
&
    \bm{A}^{(0)} \bm{P}^{(2)}
    +
    \bm{A}^{(1)} \sum_{m=1}^D \pp{p^{(0)}}{x^m} \bm{W}^m
    + 
    \sum_{m=1}^D \bm{D}^{m (0)} 
    \sum_{n=1}^D \pp{p^{(0)}}{x^n} \bm{W}^n
    + 
                \nonumber
    \\
    & \hspace{1.0cm}
    \sum_{m=1}^D \bm{B}^{m (0)} \pp{}{x^m}
    \Big(
    f \bm{1} + \sum_{n=1}^D \pp{p^{(0)}}{x^n} \bm{W}^n
    \Big)
    +
    \sum_{m=1}^D \bm{B}^{m (1)} \bm{1} \pp{p^{(0)}}{x^m}
    +
                \nonumber
    \\
    & \hspace{2.0cm}
    \sum_{m=1}^D \sum_{n=1}^D \bm{E}^{mn (0)} \bm{1} \pp{p^{(0)}}{x^n}
    +
    \sum_{m=1}^D \sum_{n=1}^D \bm{C}^{mn (0)} \bm{1} \pppd{p^{(0)}}{x^m}{x^n} = \bm{0},
    \label{eq:Dd_second-order-equation_conservation-of-mass_2}
\end{align}
which is the $D$-dimensional equivalent of the one-dimensional second-order conservation of fluid problem, \eqref{eq:mm_1d_fluid-conservation_second-order_problem_1}.
Searching for a solvability condition by pre-multiplying \eqref{eq:Dd_second-order-equation_conservation-of-mass_2} by $\bm{1}^{\top}$ and using the identity \eqref{eq:mm_1d_matrix-lemma_3} to eliminate the second-order pressure, $\bm{P}^{(2)}$, we find that
\begin{align}
&
    \bm{1}^\top \bm{A}^{(1)} \sum_{m=1}^D \pp{p^{(0)}}{x^m} \bm{W}^m
    + 
    \bm{1}^\top \sum_{m=1}^D \bm{D}^{m (0)} 
    \sum_{n=1}^D \pp{p^{(0)}}{x^n} \bm{W}^n
    + 
        \nonumber
    \\
    & \hspace{1.0cm}
    \bm{1}^\top \sum_{m=1}^D \bm{B}^{m (0)} \pp{}{x^m}
    \Big(
    f \bm{1} + \sum_{n=1}^D \pp{p^{(0)}}{x^n} \bm{W}^n
    \Big)
    +
    \bm{1}^\top \sum_{m=1}^D \bm{B}^{m (1)} \bm{1} \pp{p^{(0)}}{x^m}
    +
        \nonumber
    \\
    & \hspace{2.0cm}
    \bm{1}^\top \sum_{m=1}^D \sum_{n=1}^D \bm{E}^{mn (0)} \bm{1} \pp{p^{(0)}}{x^n}
    +
    \bm{1}^\top \sum_{m=1}^D \sum_{n=1}^D \bm{C}^{mn (0)} \bm{1} \pppd{p^{(0)}}{x^m}{x^n} = 0.
\end{align}
Using the identities (\ref{eq:Dd_matrix-lemma_1}a) and (\ref{eq:Dd_matrix-lemma_1}b), and further \eqref{eq:Dd_matrix-lemma_2_D-and-B} and \eqref{eq:Dd_matrix-lemma_2_E-and-C} to simplify, before using the product rule to write the result in terms of $\bm{B}^{m(0)}$ and $\bm{C}^{mn(0)}$, we get 
\begin{align}
&
    \sum_{m=1}^D \pp{}{x^m}\bigg(
    \sum_{n=1}^D 
     \Big(
    \frac{1}{2} \bm{1}^\top \bm{B}^{m (0)} \bm{W}^n
    +
    \frac{1}{2} \bm{1}^\top \bm{B}^{n (0)} \bm{W}^m 
    +
    \bm{1}^\top \bm{C}^{mn (0)} \bm{1} \Big)\bigg) 
    \pp{p^{(0)}}{x^n} 
    + 
    \nonumber
    \\
    & \hspace{1.0cm}
    \frac{1}{2}\sum_{m=1}^D 
    \pp{}{x^m}
    \Bigg(
    \sum_{n=1}^D
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}
    \Big(
    G_{ij}^{\bm{r}(0)}
    (r^nl^nX_i^m-r^ml^mX_i^n)
    +
    \nonumber
    \\
    & \hspace{2.0cm}
    \bm{1}^\top \bm{B}^{m (0)} \bm{W}^n
    -
    \bm{1}^\top \bm{B}^{n (0)} \bm{W}^m 
    \Big) 
    \pp{p^{(0)}}{x^n}
    \bigg) 
    \Bigg)= 0.
    \label{eq:Dd_second-order-equation_conservation-of-mass_5}
\end{align} 
Since the last term in \eqref{eq:Dd_second-order-equation_conservation-of-mass_5} is equal to zero (see Appendix \ref{sec:proofs_extra-terms}), we arrive at
\begin{align}
    \bm{\nabla} \cdot \Big( \bm{\sigma}^{(0)} \bm{\nabla} p^{(0)} \Big) = 0,
    \label{eq:Dd_second-order-equation_conservation-of-mass_6}
\end{align}
where $\bm{\sigma}^{(0)}$ is the symmetric matrix, with shape $(D,D)$, defined by
\begin{align}
    \sigma^{mn(0)} = \frac{1}{\prod_{a=1}^Dl^a}\Big( \frac{1}{2}\bm{1}^{\top} \bm{B}^{m (0)}  \bm{W}^n + \frac{1}{2}\bm{1}^{\top} \bm{B}^{n (0)}  \bm{W}^m + \bm{1}^{\top} \bm{C}^{mn (0)} \bm{1}
    \Big).
    \label{eq:Dd_sigma}
\end{align}
The macroscale fluid conservation equation, \eqref{eq:Dd_second-order-equation_conservation-of-mass_6}, is the $D$-dimensional equivalent of \eqref{eq:mm_1d_fluid-conservation_second-order_problem_3}, which we have shown to be the macroscale Darcy equation.
The parameter of the equation, $\bm{\sigma}^{(0)}$ defined in \eqref{eq:Dd_sigma}, which is a symmetric matrix of shape $(D,D)$, 
is the $D$-dimensional analogy to the scalar $\sigma^{(0)}$ in \eqref{eq:1d_sigma}. 
This dictates the dependence of the macroscale pressure on the microscale, since it depends on the pore conductance, $G_{ij}^{\bm{r}(0)}$, 
via the tensors $\bm{B}^{(0)}$ and $\bm{C}^{(0)}$, which are defined in \eqref{eq:Dd_def_B} and \eqref{eq:Dd_def_D}, respectively, and the cell solution matrix, $\bm{W}$, which is the solution of the cell problem, \eqref{eq:Dd_cell-problem_1}.
We see that the diagonal elements of $\bm{\sigma}^{(0)}$ are equal to $\sigma^{(0)}$. 
% Off-diagonal elements ensure that $\bm{\sigma}^{(0)}$ is symmetric with respect to the spatial indices.
In the one-dimensional setting, we found that $\sigma^{(0)}$ was equal to the scalar permeability, $k^{(0)}$, and similarly, here we will find (see Appendix \ref{sec:proofs_k=sig}) that $\bm{\sigma}^{(0)}$ is the permeability matrix. 
As such, $\sigma^{mn(0)}$ is the permeability in the $x^n$ direction through the surface that is normal to the $x^m$ direction.
% As before, the microscale structure enters the macroscale pressure problem because $\bm{\sigma}^{(0)}$ depends on the conductance of the edges in the microscale network via the matrices $\bm{B}^{m(0)}$ and $\bm{C}^{mn(0)}$, which are given by \eqref{eq:Dd_def_B} and \eqref{eq:Dd_def_C} respectively.

\subsection{Darcy's law}
We now construct a $D$-dimensional macroscale flow rate from microscale fluxes, similar to \eqref{eq:1d_definition-of-darcy}, in order to find the velocity that results from the macroscale pressure, $p^{(0)}$.
% In $D$ dimensions, we relate the macroscale pressure solution, $p^{(0)}$, to a flow rate, by 
To this end, we introduce the $D$-dimensional Darcy velocity vector, $\bm{u}$, where the component in the $x^m$ direction is
\begin{align}
    u^m = \frac{l^m}{2  \prod_n l^n} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} r^m Q_{ij}^{\bm{r}}.
    \label{eq:Dd_definition-of-darcy}
\end{align}
for $m \in \{1,\hdots,D\}$.
Note the division by the cross-sectional area of the surface that is normal to the $x^m$ direction, 
$\prod_{n\ne m}l^{n}$,
% $\varepsilon^{D-1}\prod_{a\ne m}l^{a}$, followed by a scaling with $\varepsilon^{D-1}$, 
so that, analogously with the one-dimensional Darcy velocity, \eqref{eq:1d_definition-of-darcy}, \eqref{eq:Dd_definition-of-darcy} is the $\mathcal{O}(1)$ flux of fluid per unit cross-sectional area through a cell in the $x^m$ direction.

As before, we homogenise the Darcy velocity, \eqref{eq:Dd_definition-of-darcy}, by writing the microscale flux, $Q_{ij}^{\bm{r}}$, in terms of the pressure and conductance, using \eqref{eq:mm_flux-definition_sum_nodim_1}, and following Steps \ref{step:1}--\ref{step:3}. 
Using the $D$-dimensional pressure solutions, \eqref{eq:Dd_leading-order-solution_conservation-of-mass} and \eqref{eq:Dd_first-order-solution_conservation-of-mass}, and noting the cancellations due to the leading- and first-order pressure differences, \eqref{eq:Dd_leading-order-pressure-difference} and \eqref{eq:Dd_first-order-pressure-difference}, we find that the leading-order flux is 
\begin{align}
    Q_{ij}^{\bm{r}(0)} = 
    \sum_{n=1}^D G_{ij}^{\bm{r} (0)} 
    \Delta_{ij}^{r^n}
    \pp{p^{(0)}}{x^n},
    \label{eq:Dd_homogenised-Q0}
\end{align}
where 
\begin{align}
    \Delta_{ij}^{r^m} = W_i^m + X_i^m - (W_j^m + X_j^m + r^ml^m),
    \label{eq:Dd_definition-of-delta}
\end{align}
which satisfies 
\begin{align}
    \Delta_{ji}^{-r^m} = -\Delta_{ij}^{r^m}.
    \label{eq:Dd_delta-identity}
\end{align}
Here, \eqref{eq:Dd_homogenised-Q0}--\eqref{eq:Dd_delta-identity} are analogous is with \eqref{eq:1d_homogenised-Q0}--\eqref{eq:1d_delta-identity}. 
We see that \eqref{eq:Dd_definition-of-delta} defines the elements of a $D$-dimensional pressure drop, $\bm{\Delta}$, 
which is a 4-tensor with shape $(N,N,2K+1,D)$, where we recall that $K$ determines the number of references, $r$, in their set, $\mathcal{R}$.
From \eqref{eq:Dd_homogenised-Q0}, we see that each element, $\Delta_{ij}^{r^m}$, is the leading-order component of the pressure difference across the pore between junction $i$ and the junction $j$ in the cell at reference $r$ in direction $x^m$, per unit pressure gradient.
This dictates the contribution that the $r^m$ component of the pressure drop in pore $ij$ makes to the macroscale pressure gradient across the cell as a whole.
Similarly to before, we interpret the cell solution element, $W_i^m$, as the component of the pressure (per unit pressure gradient) at junction $i$ in the $x^m$ direction that is required to induce the pressure gradient, $\partial p^{(0)}/\partial x^m$. 
It follows (see Appendix \ref{sec:proofs_k=sig}), as in the one-dimensional formulation, that the problem is independent of junction positions, 
so that, choosing $X_i^m=0$, \eqref{eq:Dd_definition-of-delta} may be simplified to
\begin{align}
    \Delta_{ij}^{r^m} = W_i^m - (W_j^m + r^ml^m).
    \label{eq:Dd_definition-of-delta_without-X}
\end{align}
% That is, $r^m$ accounts for that fact that node $j$ is the cell at position $r^m$ in the $m$-direction relative to node $i$, and the matrix $\bm{\Delta}_{ij}$ of size $3 \times D$ (since $r \in \{-1,0,+1 \}$) is the pressure difference per unit pressure gradient between node $i$ and node $j$, with elements $\Delta_{ij}^{r^m}$.

Substituting the homogenised leading-order flux, \eqref{eq:Dd_homogenised-Q0}, into the Darcy velocity, \eqref{eq:Dd_definition-of-darcy}, we find that
\begin{align}
    & u^{m} =
    \frac{l^m}{2 \prod_{a=1}^Dl^a}  \sum_{i=1}^N \sum_{j=1}^N  \sum_{\bm{r}\in\bm{\mathcal{R}}} \sum_{n=1}^D 
    r^m G_{ij}^{\bm{r} (0)} 
    \Delta_{ij}^{r^n}
    \pp{p^{(0)}}{x^n} 
    + \mathcal{O}(\varepsilon),
    \label{eq:Dd_darcy-velocity_2}
\end{align}
in analogy with \eqref{eq:1d_homogenised-u}.
Note that elements of $\Delta_{ij}^{r^n}$
% the upper index in $\Delta_{ij}^{r^n}$, that is, $r \in \mathcal{R}=\{-K,\hdots,0,+1,\hdots,K\}$ with $n \in \{1, \hdots, D \}$, are 
are summed with the corresponding elements of $G_{ij}^{(r^1,\hdots,r^n,\hdots,r^D)^{\top}}$. 
Therefore, writing \eqref{eq:Dd_darcy-velocity_2} in vector form, it follows that the leading-order Darcy velocity satisfies the $D$-dimensional version of Darcy's law. 
That is, we find that
\begin{align}
    \bm{u}^{(0)} = - \bm{k}^{(0)} \bm{\nabla}p^{(0)},
    \label{eq:Dd_macroscale_darcy_1}
\end{align}
where, this time, $\bm{k}^{(0)}$ is a symmetric matrix of shape $(D,D)$, with elements
\begin{align}
    k^{mn(0)} = 
    \frac{l^m}{2 \prod_n l^n}  \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)} 
    \Delta_{ji}^{-r^n}.
    \label{eq:Dd_definition-of-permeability}
\end{align}
This matrix is analogous with the permeability scalar defined in \eqref{eq:1d_definition-of-permeability}.
We find that $\bm{k}^{(0)}=\bm{\sigma}^{(0)}$ (see Appendix \ref{sec:proofs_k=sig}), and, using this identity in the governing equation for the macroscale pressure, \eqref{eq:Dd_second-order-equation_conservation-of-mass_6}, we find that the leading-order Darcy velocity, $\bm{u}^{(0)}$, is divergence-free, so that
\begin{align}
    \bm{\nabla} \cdot \bm{u}^{(0)} = 0,
    \label{eq:Dd_macroscale_conservation-of-mass_1}
\end{align}
in analogy with the one-dimensional fluid conservation equation, \eqref{eq:1d_macroscale_conservation-of-mass_1}.
Inspecting \eqref{eq:Dd_macroscale_darcy_1}, we identify $\bm{k}^{(0)}$ as the leading-order permeability matrix. 
As such, each element, $k^{mn(0)}$, is the permeability in the $x^n$ direction through a cross-section that is normal to the $x^m$ direction.

\subsection{Conservation of particles}
Next, we homogenise the equation for the conservation of particles on the microscale, \eqref{eq:mm_particle-conservation_sum_nodim_1},
which we re-write, substituting the pore flux for the pressure and conductance using \eqref{eq:mm_flux-definition_sum_nodim_1}, as 
\begin{align}
    \label{eq:Dd_master_transport_1}
    \pp{C_i}{T} &= 
    \frac{1}{\varepsilon} 
    \bigg\{ \sum_{j=1}^{N} \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Big(
    G_{ji}^{-\bm{r}} (P_j-P_i) C_j H(G_{ji}^{-\bm{r}}(P_j-P_i)) - 
        \\
    & \hspace{3.25cm} \nonumber
    G_{ij}^{\bm{r}} (P_i-P_j) C_i H(G_{ij}^{\bm{r}}(P_i-P_j))  \Big)
    \bigg\}
    \\
    & \hspace{3.5cm} \nonumber
    -
     \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \alpha G_{ji}^{-\bm{r}} (P_j-P_i) C_j H(G_{ji}^{-\bm{r}}(P_j-P_i)).
\end{align}
As before, we begin by noting that the Heaviside function can be homogenised first and carried through the rest of the process without interacting with the other terms of \eqref{eq:Dd_master_transport_1}.
Specifically, we carry out Steps \ref{step:1}--\ref{step:3} on the term $H(Q\indx)$ and find that  
\begin{align}
    H(Q\indx)
    & =
    H\bigg(G\indx(P_i-P_j) + \varepsilon \bigg\{\sum_{m=1}^D 
    G\indx \bigg(X_i^m  \pp{P_i}{x^m} -  
    \\
    & \hspace{5.5cm} 
    \nonumber 
    (X_j^m +
        r^ml^m) \pp{P_j}{x^m} \bigg) \bigg\} + \mathcal{O}(\varepsilon^2) \bigg) 
    \\
    & =
    H\bigg(\varepsilon \bigg\{ \sum_{m=1}^D G\indx\Delta_{ij}^{r^m} \pp{p^{(0)}}{x^m} \bigg\} + \mathcal{O}(\varepsilon^2) \bigg),
    \label{eq:Dd_homogenised-H}
\end{align}
which is equivalent to \eqref{eq:1d_homogenised-H}.
The first equality is obtained by writing the flux in terms of the conductance and pressure using \eqref{eq:mm_flux-definition_sum_nodim_1}, followed by a Taylor expansion of the pressure and conductance about $\bm{x}$. 
The second equality is a substitution of the asymptotic expansion for pressure, \eqref{eq:mm_asymptotic-solution_pressure}, and use of the leading- and first-order pressure solutions, \eqref{eq:Dd_leading-order-solution_conservation-of-mass} and \eqref{eq:Dd_first-order-solution_conservation-of-mass}.
We therefore define
\begin{align}
    H_{ij}^{\bm{r}(0)} = H\bigg(G_{ij}^{\bm{r}}\sum_{m=1}^D \Delta_{ij}^{r^m} \pp{p^{(0)}}{x^m} \bigg),
    \label{eq:Dd_definition-of-H}
\end{align}
in analogy with \eqref{eq:1d_definition-of-H}, and note that this satisfies
\begin{align}
    H_{ij}^{\bm{r}(0)}+H_{ji}^{-\bm{r}(0)}=1,  
\label{eq:Dd_heaviside-identity}
\end{align}
which is the $D$-dimensional equivalent of \eqref{eq:1d_heaviside-identity}.
As before, we will find that this homogenised Heaviside function, \eqref{eq:Dd_definition-of-H}, disappears from all terms except the one containing the adherence parameter, $\alpha$, which we expect, since we require it to indicate the pores $ij\bm{r}$ that carry fluid into $i$, since it is within these pores that deposition occurs in the underlying network.

We then carry out Steps \ref{step:1}--\ref{step:3} on the rest of the particle conservation equation, \eqref{eq:Dd_master_transport_1}, similarly as in the one-dimensional case. 
We apply a Taylor expansion and an asymptotic expansion to the particle concentration, the conductance, and the pressure, about $\bm{x}$. 
Using the leading- and first-order pressure solutions, \eqref{eq:Dd_leading-order-solution_conservation-of-mass} and
\eqref{eq:Dd_first-order-solution_conservation-of-mass}, and noting the cancellation of terms that contain the leading-order pressure difference, \eqref{eq:Dd_leading-order-pressure-difference}, and the cancellation of terms involving the function $f$ (which ensures that $f$ is arbitrary in \eqref{eq:Dd_first-order-solution_conservation-of-mass}) via the first-order pressure difference, \eqref{eq:Dd_first-order-pressure-difference}, we arrive at
\begin{align}
    & \pp{}{T}
    \Big(
    C_i^{(0)} + \varepsilon \Big\{ C_i^{(1)}+\sum_{m=1}^D X_i^m \pp{C_i^{(0)}}{x} \Big\} +
    \nonumber
    \varepsilon^2
    \Big\{
    C_i^{(2)}
    +
            \\
    &
    \hspace{4cm} \nonumber
    \sum_{m=1}^D X_i^m \pp{C_i^{(1)}}{x} 
    +
    \nonumber
    \frac{1}{2} \sum_{m=1}^D \sum_{n=1}^D X_i^m X_i^n
    \pppd{C_i^{(0)}}{x^m}{x^n}
    \Big\}
    \Big) =
    \\
    \nonumber
    & 
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}
    \bigg\{
    G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Delta_{ji}^{-r^m} 
    \pp{p^{(0)}}{x^m}
    (C_j^{(0)}H_{ji}^{-\bm{r}(0)}+C_i^{(0)}H_{ij}^{\bm{r}(0)})
    \bigg\}
    +
        \nonumber
    \\
    & \hspace{1cm} 
    \varepsilon \bigg\{
    G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Delta_{ji}^{-r^m} 
    \pp{p^{(0)}}{x^m}
    (C_j^{(1)}H_{ji}^{-\bm{r}(0)}+C_i^{(1)}H_{ij}^{\bm{r}(0)})
    +
        \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Delta_{ji}^{-r^m} 
    \pp{p^{(0)}}{x^m}
    \Big(\sum_{n=1}^D (X_j^n+r^nl^n)\pp{C_j^{(0)}}{x^n}H_{ji}^{-\bm{r}(0)}
    +
    \nonumber
    \\
    & \hspace{1cm} \nonumber
    \sum_{n=1}^D X_i^n\pp{C_i^{(0)}}{x^n}H_{ij}^{\bm{r}(0)}\Big)
    +
        \nonumber
    \bigg(
    G_{ij}^{\bm{r}(0)}
    \Big( P_j^{(2)} -P_i^{(2)} \Big)
    +
    G_{ij}^{\bm{r}(1)}
    \sum_{m=1}^D
    \Delta_{ji}^{-r^m}
    \pp{p^{(0)}}{x^m}
    +
            \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{\bm{r}(0)} \bigg( \sum_{m=1}^D (X_j^m + r^ml^m) \pp{}{x^m} \Big( f+ 
    \sum_{n=1}^D W_j^n \pp{p^{(0)}}{x^n} \Big)
    - 
    \sum_{m=1}^D X_i^m \pp{}{x^m} \Big(f+ 
            \nonumber
    \\
    & \hspace{1cm}
    \sum_{n=1}^D W_i^n \pp{p^{(0)}}{x^n} \Big) \bigg)
    +
    \frac{G_{ij}^{\bm{r}(0)}}{2} 
    \bigg( \sum_{m=1}^D \sum_{n=1}^D (X_j^m + r^ml^m)(X_j^n + r^nl^n)
    -
        \nonumber
    \\
    & \hspace{1cm}
    \sum_{m=1}^D \sum_{n=1}^D X_i^m X_i^n  \bigg)
    \pppd{p^{(0)}}{x^m}{x^n}
    +
    \sum_{m=1}^D \pp{G_{ij}^{\bm{r}(0)}}{x^m} M_{ij}^{\bm{r}m} \sum_{n=1}^D 
    \Delta_{ji}^{-r^n}
    \pp{p^{(0)}}{x^n}
     \bigg) (C_j^{(0)}H_{ji}^{-\bm{r}(0)}+
         \nonumber
    \\
    & \hspace{1cm}
    C_i^{(0)}H_{ij}^{\bm{r}(0)})
     -
     G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Delta_{ji}^{-r^m}
    \pp{p^{(0)}}{x^m}
    C_j^{(0)} H_{ji}^{-\bm{r}(0)} A_{ji}^{-\bm{r}(0)}
     \bigg\} + \mathcal{O}(\varepsilon^2),
    %\nonumber
    \label{eq:Dd_master_transport_6}
\end{align}
which is the $D$-dimensional master equation for the conservation of particles. 
This has a similar form as its one-dimensional equivalent, \eqref{eq:1d_master_transport_6}.
For example, the deposition term appears at first order in $\varepsilon$ because there is no pressure difference at leading order.

To simplify the master equation, we continue by carrying out Step \ref{step:4}. 
Similarly to in \eqref{eq:1d_master_transport_6}, a leading-order balance in the $D$-dimensional the master equation, \eqref{eq:Dd_master_transport_6}, yields
% \begin{align}
%     \bm{V} \circ \pp{\bm{C}^{(0)}}{\tau} = \bm{0},
% \end{align}
% where $\bm{V} = (V_1, \hdots,V_N)^{\top}$ and $\bm{C} = (C_1, \hdots,C_N)^{\top}$ are the volume and concentration arrays, as before.
%  Thus, the leading-order concentration is, once again, steady on this timescale. This is because the leading-order pressure difference, \eqref{eq:Dd_leading-order-pressure-difference}, is zero as in the quasi-one-dimensional case, and so the flux that drives particle transport, and therefore concentration variation, remains a first-order phenomenon.

% Rescaling time onto the microscale timescale, $T=\varepsilon \tau$ therefore, we find that a leading-order balance now gives no new information while a first-order balance yields
\begin{align}
    \bm{1} \circ \pp{\bm{C}^{(0)}}{T} = \sum_{m=1}^D \bm{S}^{m(0)} \bm{C}^{(0)} \pp{p^{(0)}}{x^m}.
    \label{eq:Dd_concentration-equation_microscale}
\end{align}
In analogy with \eqref{eq:1d_concentration-equation_microscale}, 
here, $\bm{1} = (1, \hdots,1)^{\top}$ and $\bm{C} = (C_1, \hdots,C_N)^{\top}$ are the vectors of length $N$ 
that contain the junction volumes and concentrations, 
and $\bm{S}^{(0)}$ is a 3-tensor with shape ($N,N,D$) that is the analogy of the matrix defined in \eqref{eq:1d_definition-of-S}, with elements
\begin{align}
    & S_{ij}^{m(0)} = \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big( G_{ij}^{\bm{r}(0)}\Delta_{ji}^{-r^m} H_{ji}^{-\bm{r}(0)}
    -
     \updelta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}(0)}\Delta_{ik}^{r^m} H_{ik}^{\bm{r}(0)}
    \Big).
        \label{eq:Dd_definition-of-S}
\end{align}
% is the $D$-dimensional tensor analogy of the matrix $\bm{S}^{(0)}$ defined in \eqref{eq:1d_definition-of-S}.
In Appendix \ref{sec:proofs_S}, we show that each matrix in this tensor, $\bm{S}^{m(0)}$, has the same form as the equivalent one-dimensional form, \eqref{eq:1d_definition-of-S}, and so satisfies
\begin{align}
    \bm{S}^{m(0)} \bm{1} = \bm{0} = \bm{1}^{\top} \bm{S}^{m(0)}.
    \label{eq:Dd_lemma_S}
\end{align}
% We see that \eqref{eq:Dd_concentration-equation_microscale} is the microscale transport equation equivalent to \eqref{eq:1d_concentration-equation_microscale}.
As before,
on this shorter timescale, $T_{\text{c}}$ defined in \eqref{eq:mm_define-T_c}, we see that \eqref{eq:Dd_concentration-equation_microscale} implies that particles flow in and out of the cell without depositing, so 
that the net particle flux is zero, from \eqref{eq:Dd_lemma_S}. 
Therefore, \eqref{eq:Dd_concentration-equation_microscale} describes the behaviour of the microscale-dependent particle concentration before it settles 
down to the microscale-independent solution that we seek.
% there is no intra-cell deposition since we introduced adhesion as an $\mathcal{O}(\varepsilon)$ phenomenon, and so particle mass is conserved within a given cell, as \eqref{eq:Dd_lemma_S} implies.
% and, as such, we will find that $\bm{S}^{m(0)}$ does not appear over longer timescales, although $\bm{S}^{m(0)}$ controls the evolution of the microscale-dependent concentration in the $m$ direction in \eqref{eq:Dd_concentration-equation_microscale}.

Rescaling time onto a longer timescale, $T_{\text{n}}$ defined in \eqref{eq:mm_define-T_n}, and recalling that $t = \varepsilon T$, 
at leading order we find that
\begin{align}
    \bm{0} = \sum_{m=1}^D \bm{S}^{m(0)} \bm{C}^{(0)} \pp{p^{(0)}}{x^m}.
\end{align}
Therefore, using the first identity in \eqref{eq:Dd_lemma_S}, it follows that
\begin{align}
    \bm{C}^{(0)} = c^{(0)}(\bm{x},t) \bm{1},
    \label{eq:Dd_leading-order-concentration}
\end{align}
which is equivalent to \eqref{eq:1d_leading-order-concentration}.
As usual, $c^{(0)}(\bm{x},t)$ denotes the microscale-independent leading-order concentration solution to be found.

Proceeding to first order to look for an equation that determines $c^{(0)}$, by substituting this leading-order solution, \eqref{eq:Dd_leading-order-concentration}, into the master equation, \eqref{eq:Dd_master_transport_6}, and noting that $H_{ij}^{\bm{r}(0)}$ disappears from terms containing $c^{(0)}$ (using the identity \eqref{eq:Dd_heaviside-identity}),
we find that
\begin{align}
\label{eq:Dd_second-order-equation_transport_2}
    \bm{1} \pp{c^{(0)}}{t} 
    = &
    \sum_{m=1}^D \bm{S}^{m(0)} \bm{C}^{(1)} \pp{p^{(0)}}{x^m}
    + 
    \sum_{m=1}^D \sum_{n=1}^D
    \bm{K}^{mn(0)} \bm{1} \pp{p^{(0)}}{x^m} \pp{c^{(0)}}{x^n}
    -
    \\
    &
    \sum_{m=1}^D
    \bm{J}^{m(0)} \bm{1} \pp{p^{(0)}}{x^m} c^{(0)}
    +
    \bigg(
    \bm{A}^{(0)} \bm{P}^{(2)} 
    +
    \bm{A}^{(1)} \sum_{m=1}^D  \bm{W}^m \pp{p^{(0)}}{x^m} 
    +
    \sum_{m=1}^D \bm{B}^{m(1)} \bm{1} \pp{p^{(0)}}{x^m}
    +
    \nonumber
    \\
    &
    \sum_{m=1}^D \bm{B}^{m(0)} \pp{}{x^m} \Big(f \bm{1} 
    +
    \sum_{n=1}^D \pp{p^{(0)}}{x^n} \bm{W}^n\Big) 
    +
    \sum_{m=1}^D \sum_{n=1}^D \bm{C}^{mn(0)} \bm{1} \pppd{p^{(0)}}{x^m}{x^n}
    +
    \nonumber
    \\
    &
    \sum_{m=1}^D \bm{D}^{m(0)} \sum_{n=1}^D \bm{W}^n \pp{p^{(0)}}{x^n} 
    + 
    \sum_{m=1}^D \sum_{n=1}^D \bm{E}^{mn(0)} \bm{1} \pp{p^{(0)}}{x^n}
    \bigg) c^{(0)},
    \nonumber
\end{align}
where $\bm{K}^{mn(0)}$ is the $mn$ element of a 4-tensor of shape $(N,N,D,D)$, which is equivalent to the  matrix $\bm{K}^{(0)}$ from the one-dimensional case,
\eqref{eq:1d_definition-of-K},
and is defined by
\begin{align}
    K_{ij}^{mn(0)} = \sum_{\bm{r}\in\bm{\mathcal{R}}}\bigg( G_{ij}^{\bm{r}(0)} \Delta_{ji}^{-r^m} \Big((X_j^n + r^nl^n)H_{ji}^{-\bm{r}(0)} + X_i^n H_{ij}^{\bm{r}(0)} \Big) \bigg),
    \label{eq:Dd_definition-of-K}
\end{align}
while $\bm{J}^{m(0)}$ is the $m$ element of a $3$-tensor of shape $(N,N,D)$, which is equivalent to the matrix $\bm{J}^{(0)}$ from before, \eqref{eq:1d_definition-of-J}, and is defined by
\begin{align}
    J_{ij}^{m(0)} = \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big( \alpha G_{ij}^{\bm{r}(0)} \Delta_{ji}^{-r^m}H_{ji}^{-\bm{r}(0)} \Big).
    \label{eq:Dd_definition-of-J}
\end{align}
Using the second-order equation that results from the problem for conservation of mass, \eqref{eq:Dd_second-order-equation_conservation-of-mass_2}, we find that the sum of all but the first three terms on the right-hand side of \eqref{eq:Dd_second-order-equation_transport_2} is equal to zero, and so, pre-multiplying \eqref{eq:Dd_second-order-equation_transport_2} by $\bm{1}^{\top}$ gives
\begin{align}
\label{eq:Dd_second-order-equation_transport_3}
    \pp{c^{(0)}}{t} =  
    \sum_{m=1}^D \sum_{n=1}^D \frac{\bm{1}^{\top} \bm{K}^{mn(0)} \bm{1}}{\bm{1}^{\top} \bm{1}}   \pp{p^{(0)}}{x^m} \pp{c^{(0)}}{x^n}
    - \sum_{m=1}^D
    \frac{\bm{1}^{\top} \bm{J}^{m(0)} \bm{1}}{\bm{1}^{\top} \bm{1}}
     \pp{p^{(0)}}{x^m} c^{(0)},
\end{align}
where we have used the second equality in the identity \eqref{eq:Dd_lemma_S}.
Therefore, as in the one-dimensional case, the macroscale evolution of the leading-order particle concentration is governed by an advection--reaction equation. 

Comparing the definition of $\bm{K}^{mn(0)}$, \eqref{eq:Dd_definition-of-K}, to its one-dimensional counterpart, 
$\bm{K}^{(0)}$ in \eqref{eq:1d_definition-of-K}, we expect $\bm{K}^{mn(0)}$ to be a matrix related to advective terms in the $x^n$ direction on the surface that is normal to the $x^m$ direction, on the macroscale.
Indeed, it follows  (see Appendix \ref{sec:proofs_K-is-k} for proof) 
that  
\begin{align}
    \bm{1}^{\top} \bm{K}^{mn(0)} \bm{1} 
    = 
    \prod_{a=1}^Dl^a k^{mn(0)} 
    = 
    \prod_{a=1}^Dl^a \sigma^{mn(0)},
    \label{eq:Dd_K-is_k}
\end{align}
so that this tensor is proportional to the permeability matrix, $\bm{k}^{(0)}$ of shape $(D,D)$, which corresponds to the effective conductance of the filter.
Similarly, comparing the definition of $\bm{J}^{m(0)}$, \eqref{eq:Dd_definition-of-J}, to its one-dimensional counterpart, 
$\bm{J}^{(0)}$,
\eqref{eq:1d_definition-of-J}, we expect $\bm{J}^{m(0)}$ to be a matrix related to reactive terms in the $x^m$ direction on the macroscale.
Accordingly, we define
\begin{align}
    j^{m(0)} = -\frac{\bm{1}^{\top} \bm{J}^{m(0)} \bm{1}}{\prod_{n=1}^Dl^n} = -\frac{1}{\prod_{n=1}^Dl^n} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \alpha G_{ij}^{\bm{r}(0)} \Delta_{ji}^{-r^m}H_{ji}^{-\bm{r}(0)}.
    \label{eq:Dd_defition-of-j}
\end{align}
We call $\bm{j}^{(0)}$ the adhesivity vector, which is of length $D$, since we expect it to correspond to the effective adherence of the filter.

% so that this tensor is proportional to the permeability tensor $\bm{k}^{(0)}$, which is the effective conductance of the filter.



% Meanwhile, comparing the definition of $\bm{J}^{m(0)}$, \eqref{eq:Dd_definition-of-J}, to the definition of $\bm{J}^{(0)}$, \eqref{eq:1d_definition-of-J}, we expect $\bm{J}^{m(0)}$ to be a matrix related to the reaction terms in the $m$ direction.


To illustrate this, in analogy with the one-dimensional case, we define 
\begin{align}
    v=\bm{1}^{\top} \bm{1} = N,    
\end{align}
which is the total volume of the junctions in a cell. 
% and define $\bm{j}^{(0)}$ by
% \begin{align}
%     j^{m(0)} = -\frac{\bm{1}^{\top} \bm{J}^{m(0)} \bm{1}}{\prod_{a=1}^Dl^a}.
%     \label{eq:Dd_defition-of-j}
% \end{align}
Using the relationships between $\bm{K}^{mn(0)}$ and $k^{mn(0)}$, \eqref{eq:Dd_K-is_k}, and between $\bm{J}^{m(0)}$ and $j^{m(0)}$, \eqref{eq:Dd_defition-of-j}, we re-write
the macroscale concentration equation,
\eqref{eq:Dd_second-order-equation_transport_3}, as
\begin{align}
\label{eq:Dd_second-order-equation_transport_4}
    \pp{c^{(0)}}{t} = \frac{\prod_{m=1}^Dl^m}{v} (\bm{k}^{(0)} \cdot \bm{\nabla}p^{(0)}) \cdot \bm{\nabla}c^{(0)} 
    +
    \frac{\prod_{m=1}^Dl^m}{v}
    (\bm{j}^{(0)} \cdot \bm{\nabla}p^{(0)}) \, c^{(0)}.
\end{align}
Identifying
\begin{align}
    \phi = \frac{v}{\prod_{m=1}^Dl^m} 
\end{align}
 as the $D$-dimensional junction porosity, 
 which is the constant proportion of total filter space occupied
 by junctions, 
 defining the reactivity,
 \begin{align}
    \psi^{(0)} = -\bm{j}^{(0)} \cdot \bm{\nabla }p^{(0)},    
    \label{eq:Dd_definition-of-psi}
 \end{align}
 and using this equation along with Darcy's law, \eqref{eq:Dd_macroscale_darcy_1},
we arrive at
\begin{align}
    \pp{c^{(0)}}{t} + \bm{\nabla} \cdot  \frac{\bm{u}^{(0)}}{\phi} c^{(0)} = - \frac{\psi^{(0)}}{\phi}  c^{(0)}.
    \label{eq:Dd_macroscale_transport}
\end{align}
% which is coupled to the $D$-dimensional equivalent of \eqref{eq:1d_macroscale-conductance}.
In $D$ dimensions, we have found that the particle concentration, $c^{(0)}$, varies on the continuous macroscale subject to an advection--reaction equation, \eqref{eq:Dd_macroscale_transport}. 
That is, particles are advected through the filter by the pore velocity, $\bm{u}^{(0)}\phi^{-1}$, 
which is related to the pressure, $p^{(0)}$, and the permeability, $\bm{k}^{(0)}$, via the Darcy equation \eqref{eq:Dd_macroscale_darcy_1}. 
As they do so, particles deposit in the filter at a rate determined by the reactivity, $\psi^{(0)}\phi^{-1}$, 
which is related to the pressure, $p^{(0)}$, and the adhesivity, via the 
reaction equation, \eqref{eq:Dd_definition-of-psi}. 
The permeability and the adhesivity are related to the conductance of pores in the network, $G_{ij}^{\bm{r}(0)}$, via \eqref{eq:Dd_definition-of-permeability} 
and \eqref{eq:1d_definition-of-j}, respectively, and thus the macroscale particle concentration depends upon the dynamic structure of the microscale.
The analogy is thus with the one-dimensional advection--reaction equation, \eqref{eq:1d_macroscale_transport}, and the interpretation is similar. 
However, we must now discretise macroscale space in $D$ dimensions to find macroscale variables, and solve $D$ linear cell problems, \eqref{eq:Dd_cell-problem_1}, to determine microscale variables. 
This, of course, is more computationally expensive than discretising in one dimension 
and solving one linear problem, to solve in the one-dimensional formulation.
% Its evolution depends on the parameters of this system, which are the velocity, $u^{(0)}$, and the reactivity, $\psi^{(0)}$. 
% In turn, these are determined by the permeability, $k^{(0)}$, and the adhesivity
% which is a continuous advection--reactivity equation with parameters that depend on the microscale properties of the network.


\subsection{Conservation of pores}
Lastly, we homogenise the equation for the conservation of pores, \eqref{eq:mm_conductance-conservation_sum_nodim_1},
\begin{align}
    \pp{G\indx}{T} &= - \alpha\beta C\indx (G\indx)^{\frac{3}{2}} |P_i-P_j|.
    \label{eq:mm_Dd_pore-conservation_homog_1}
\end{align}
Carrying out Steps \ref{step:1}--\ref{step:3} of network homogenisation on \eqref{eq:mm_Dd_pore-conservation_homog_1}, we find that 
% Notet that if A is not a constant then where this time we apply a Taylor expansion and an asymptotic expansion to the concentration and the adhesivity function, $A_{ij}^{\bm{r}}$, as well as the conductance and the pressure. 
% using the leading- and first-order pressure solutions, \eqref{eq:mm_1d_fluid-conservation_leading-order-solution_1} and
% \eqref{eq:mm_1d_fluid-conservation_first-order-solution}, the leading-order particle concentration, \eqref{eq:1d_leading-order-concentration},
% and noting the cancellation of the leading-order pressure difference, \eqref{eq:mm_1d_leading-order-pressure-difference}, we find that
\begin{align}
    \pp{G_{ij}^{\bm{r}(0)}}{T} &= - \varepsilon \bigg\{ \alpha\beta c^{(0)}  (G_{ij}^{\bm{r}(0)})^{\frac{3}{2}} \sum_{n=1}^D 
    \Big| \Delta_{ji}^{-r^n}   
    \pp{p^{(0)}}{x^n} \Big| \bigg\} + \mathcal{O}(\varepsilon^2).
    \label{eq:mm_Dd_pore-conservation_homog_2}
\end{align}
A leading-order balance in \eqref{eq:mm_Dd_pore-conservation_homog_2} implies that 
\begin{align}
    \pp{G_{ij}^{\bm{r}(0)}}{T} &= 0.
    \label{eq:mm_Dd_pore-conservation_homog_3}
\end{align}
We have found that the conductances of pores do not change over this timescale, $T_{\text{c}}$ (see \eqref{eq:mm_define-T_c}), which is unsurprising, 
since we showed (see \eqref{eq:Dd_concentration-equation_microscale}) that deposition does not occur this quickly.
Scaling onto the timescale over which particle deposition does occur, $T_{\text{n}}$ (see \eqref{eq:mm_define-T_n}), and recalling that $t=\varepsilon T$, 
we find that 
\begin{align}
    \pp{G_{ij}^{\bm{r}(0)}}{t} &= -  \alpha\beta c^{(0)}  (G_{ij}^{\bm{r}(0)})^{\frac{3}{2}} \sum_{n=1}^D 
    \Big|  
    \Delta_{ji}^{-r^n}  
    \pp{p^{(0)}}{x^n} \Big| + \mathcal{O}(\varepsilon).
    \label{eq:mm_Dd_pore-conservation_homog_4}
\end{align}
Balancing terms at leading order, we find that
\begin{align}
    \pp{\bm{G}^{(0)}}{t} &= -  \alpha\beta  \bm{R}^{(0)} \Big| \pp{p^{(0)}}{x} \Big|c^{(0)}.
    \label{eq:mm_Dd_pore-conservation_homog_5}
\end{align}
The analogy here is with \eqref{eq:mm_pore-conservation_homog_5}. 
This time, however, $\bm{G}^{(0)}$ and $\bm{R}^{(0)}$ are 4-tensors of shape ($N,N,K,D$) 
(recalling from Section \ref{sec:mm_Microscale model} that $K$$\times$$D$ dictates the size of the set $\bm{\mathcal{R}}$), with elements 
\begin{align}
    G_{ij}^{\bm{r}(0)} = G_{ij}^{\bm{r}(0)}, 
    \quad
    R_{ij}^{\bm{r}(0)} =  (G_{ij}^{\bm{r}(0)})^{\frac{3}{2}} \sum_{n=1}^D |\Delta_{ji}^{-r^n}|.
\end{align}
As in \eqref{eq:mm_pore-conservation_homog_5}, \eqref{eq:mm_Dd_pore-conservation_homog_5} implies that 
the evolution of the conductances of pores within the cell depend on the microscale structure, via the elements of 
$\bm{R}^{(0)}$, and on the evolution of the macroscale concentration and pressure gradient, $c^{(0)}$ and $\partial p^{(0)}/\partial x$, 
respectively.

\subsection{Summary}
In summary, we homogenise the microscale model, \eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_cond-pdf_sum_nodim_1} in $D$ dimensions, 
and find the corresponding multiscale model. 
In particular, this time, homogenising the microscale equations for the conservation of fluid, particles, and pores,
\eqref{eq:mm_fluid-conservation_sum_nodim_1}--\eqref{eq:mm_conductance-conservation_sum_nodim_1}, 
we find, dropping leading-order notation, that their macroscale equivalents are
    \begin{align}
    \bm{\nabla} \cdot \bm{u} &= 0, 
    \label{eq:mm_Dd_fluid-conservation_homog_sum}
    \\ 
    \pp{c}{t} + \bm{\nabla} \cdot  \frac{\bm{u}}{\phi} c &= - \frac{\psi}{\phi}  c, 
    \label{eq:mm_Dd_particle-conservation_homog_sum}
    \\ 
    \pp{G_{ij}^{\bm{r}}}{t} &= -  \alpha\beta c  (G_{ij}^{\bm{r}})^{\frac{3}{2}} \sum_{n=1}^D |\Delta_{ji}^{-r^n}| 
    \Big|   
    \pp{p}{x^n} \Big|,
    \label{eq:mm_Dd_pore-conservation_homog_sum}
    %\tag{\ref{eq:mm_Dd_conservation_homog_sum}a,b,c}
\end{align}
respectively, where $\phi=N/\prod_{m=1}^Dl^m$ is the constant junction porosity, which represents the proportion of void 
space in the filter that is attributed to junctions.
Homogenising the microscale flux, \eqref{eq:mm_flux-definition_sum_nodim_1}, 
we find that the Darcy velocity vector, $\bm{u}$, is related to the permeability matrix, $\bm{k}$, 
and the pressure, $p$, by the Darcy equation, 
and that the reactivity, $\psi$, is related to the adhesivity vector, $\bm{j}$, and the 
pressure by the reactivity equation, which are respectively given by 
\begin{subequations}
\label{eq:mm_Dd_darcy-and-reaction_homog_sum}
    \begin{align}
    \bm{u} = -\bm{k} \bm{\nabla}p, 
    \quad
    \psi = -\bm{j} \cdot \bm{\nabla}p.
    \tag{\ref{eq:mm_Dd_darcy-and-reaction_homog_sum}a,b}
\end{align}
\end{subequations}
The permeability and adhesivity, $\bm{k}$ and $\bm{j}$, which are the parameters of this system, 
are related to the pore conductances, $G_{ij}^{\bm{r}}$,
by the equations 
    \begin{align}
    k^{mn} &= 
    \frac{l^m}{2 \prod_{a=1}^D l^a}  \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r}} 
    \Delta_{ji}^{-r^n},
    \label{eq:mm_Dd_permeability_homog_sum}
    \\
    j^m &= -\frac{1}{\prod_{n=1}^D l^n} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \alpha G_{ij}^{\bm{r}} \Delta_{ji}^{-r^m}H_{ji}^{-\bm{r}}. 
    \label{eq:mm_Dd_adhesivity_homog_sum}
\end{align}
Here, the pore pressure drop is defined by
\begin{align}
    \Delta_{ij}^{r^m} = W_i^m - (W_j^m+r^ml^m),
    \label{eq:mm_Dd_pressure-delta_homog_sum}
\end{align}
where the cell solution, $W_i^m$, is the component of the effective junction position in the $x^m$ direction, 
which dictates the 
contribution of the component of the pressure drop in pore $ij\bm{r}$ in direction $x^m$ to the solution evolution on the macroscale, via the permeability 
and adhesivity, \eqref{eq:mm_Dd_permeability_homog_sum} and \eqref{eq:mm_Dd_adhesivity_homog_sum}.
The cell solution is given by solving the linear cell problem, 
\begin{align}
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Bigg(  G_{ij}^{\bm{r}} - \updelta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}} \Bigg)
    W_j^m
    =
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{\bm{r}}  r^ml^m
    ,
    \label{eq:mm_Dd_cell-problem_homog_sum}
\end{align}
where we have chosen all components of the position of junction $i$ to be $X_i^m=0$ for simplicity, since 
this does not affect the solution.
% Subject to constant adherence, the 
% pore conductances are related via the conductivity equation, 

Homogenising the microscale boundary conditions, \eqref{eq:mm_bc_pres_sum_nodim_1} and 
\eqref{eq:mm_ic_pres_sum_nodim_1}, 
we find that we are given the pressure at the inlet and outlet,
\begin{subequations} \label{eq:mm_Dd_bc_pres_homog_sum}
\begin{align}
    p(0,t) = 1, 
    \quad
    p(l^1,t) = 0,
    \tag{\ref{eq:mm_Dd_bc_pres_homog_sum}a,b}
\end{align}
\end{subequations}
and the concentration at the inlet,
\begin{align}
    c(0,t) = 1,
    \label{eq:mm_Dd_ic_pres_homog_sum}
\end{align}
for $t\ge0$.
Homogenising the microscale initial conditions, \eqref{eq:mm_ic_conc_sum_nodim_1} and \eqref{eq:mm_ic_cond_sum_nodim_1}, 
we find that there are no particles in the filter initially,
\begin{align}
    c(\bm{x},0) = 0, 
    \label{eq:mm_Dd_ic_conc_homog_sum}
\end{align}
and that the initial pore conductances are drawn from a known distribution, so that 
\begin{align}
    G_{ij}^{\bm{r}}(\bm{x},0) = G_{ij}^{\bm{r}0}(\bm{x})
    \label{eq:mm_Dd_ic_cond_homog_sum}
\end{align}
for $i,j \in \{1,\hdots,N,\text{in}, \text{out}\}$.
Note that the initial conductances, $G_{ij}^{\bm{r}0}(\bm{x})$, now vary with the spatial vector, $\bm{x}$, 
so that initial macroscale porosity gradients in any direction are permitted.
We seek solutions of this multiscale model in the next chapter.

% In this chapter, 
% we have ...
% In the next chapter, 
% we will ...

% In summary, dropping leading-order notation, evolution of the concentration, $c$, is governed by 
% \begin{align}
%     \pp{c}{t} + \bm{\nabla} \cdot  \frac{\bm{u}}{\phi} c = - \frac{\psi}{\phi}  c.
%     \label{eq:mm_Dd_particle-conservation_homog_sum}
% \end{align}
% Here, the Darcy velocity vector, $\bm{u}$, and the reaction coefficient, $\psi$, are given by 
% \begin{subequations}
% \label{eq:summary_grad-eqs}
% \begin{align} 
%     \bm{u} =- \bm{k} \bm{\nabla}p, 
%     \quad \quad
%     \psi = -\bm{j} \cdot \bm{\nabla}p,
%     \tag{\ref{eq:summary_grad-eqs}a,b}
% \end{align}
% \end{subequations}
% such that the Darcy velocity satisfies the equation for mass conservation,
% \begin{align}
%     \bm{\nabla} \cdot \bm{u} = 0, \label{eq:summary_cons-of-mass}
% \end{align}
% and the permeability matrix, $\bm{k}$, and the deposition vector, $\bm{j}$, are given by 
% \begin{align}
%     k^{mn} 
%     &= 
%     \frac{l^m}{2 \prod_{a=1}^Dl^a}  \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
%     r^m G_{ij}^{\bm{r}} (W_j^n+X_j^n+r^nl^n - W_i^n - X_i^n), 
%     \label{eq:summary_definition-of-k}
%     \\
%     j^m 
%     & = -
%     \frac{1}{\prod_{a=1}^Dl^a} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}} (W_j^m+X_j^m+r^ml^m - W_i^m - X_i^m)  A_{ji}^{-\bm{r}} H_{ji}^{-\bm{r}}.
%     \label{eq:summary_definition-of-j}
% \end{align}
% The cell solution vector, $\bm{W}$, is given by solving the cell problem, 
% \begin{align}
%     \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big(  G_{ij}^{\bm{r}} - \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}} \Big) W_j^m 
%     =
%     -
%     \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big( G_{ij}^{\bm{r}}  (X_j^m + r^ml^m
%     )- \delta_{ij} X_i^m \sum_{k=1}^N G_{ik}^{\bm{r}} \Big),
%     \label{eq:summary_cell-problem}
% \end{align}
% which depends on the models for the edge conductances, $G_{ij}^{\bm{r}}$,
% and adherences, $A\indx$.
% The boundary conditions are  
% \begin{subequations}
% \label{eq:summary_bc}
% \begin{align}
%     p(t,0) = 1, \quad 
%     p(t,1) = 0, \quad 
%     c(t,0) = 1,
%     \tag{\ref{eq:summary_bc}a--c}
% \end{align}
% \end{subequations}
% and the initial condition is  
% \begin{align}
%     c(0,\bm{x}) = 0.
%     \label{eq:mm_Dd_ic_cond_homog_sum}
% \end{align}

% \subsection{Volume}
% [Show why the most physically meaningful adhesivity functions are too complex to be computationally tractable in network homogenisation - explain that this is because they are flux dependent]
% Since conductance is now varying temporally due to blocking, we need to solve for it.
% We have 
% \begin{align}
%     \pp{V_{ij}}{t} = -\varepsilon \gamma A_{ij} |Q_{ij}^{\bm{r}}| \check{C}_{ij}.
% \end{align}
% We need deal with the modulus sign and the check, which depends in which direction the flow is travelling.
% We will see though that the check doesn't matter because we know that the leading order concentration is independent of the microscale. To deal with the modulus, we note that $A_{ij}$ is always positive, and bring it inside the modulus. Now we can expand. We have
% \begin{align}
%     \pp{V_{ij}}{t} = -\varepsilon \gamma  |A_{ij} G_{ij}^{\bm{r}} (P_j-P_i)| \check{C}_{ij}.
% \end{align}
% Take Taylor series of pressure, and use the asymptotic solution that we have obtained, take Taylor series expansion of conductance and concentration and adhesivity, use asymptotic concentration solution. We have
% \begin{align}
%     &
%     \pp{}{t}
%     \bigg(V_{ij}^{(0)} + \varepsilon \bigg\{ V_{ij}^{(1)} + \frac{d_{ij}^r}{2} \pp{V_{ij}^{(0)}}{x} \bigg\}
%     +
%     + \varepsilon^2 \bigg\{ V_{ij}^{(2)} + \frac{d_{ij}^r}{2} \pp{V_{ij}^{(1)}}{x} + \frac{(d_{ij}^r)^2}{4} \ppp{V_{ij}^{(0)}}{x}\bigg\}\bigg)
%     = 
%     \\
%     & \hspace{5cm}
%     -\varepsilon^2 \gamma  
%     \abs[\Big]{ 
%     A_{ij}^{r(0)} G_{ij}^{r(0)} (W_j-W_i+X_j^m+rL-X_i^m) \pp{p^{(0)}}{x}
%     } c + \mathcal{O}(\varepsilon^3).
%     \nonumber
% \end{align}
% On the usual timescale, at leading order we have 
% \begin{align}
%     \pp{V_{ij}^{(0)}}{t} = 0.
% \end{align}
% We find that the leading order edge volume and therefore conductance does not change on this short timescale, which makes sense since blocking only occurs at $\mathcal{O}(\varepsilon)$.
% Switching to the longer timescale $\hat{t}= \varepsilon t$,
% we also find 
% \begin{align}
%     \pp{V_{ij}^{(0)}}{\hat{t}} = 0,
% \end{align}
% since the pressure difference is first order.
% Switching to a longer timescale $T = \varepsilon^2 t$, comparing second order terms we have 
% \begin{align}
%     \pp{V_{ij}^{(0)}}{T} = - \gamma  
%     \abs[\Big]{ 
%     A_{ij}^{r(0)} G_{ij}^{r(0)} (W_j-W_i+X_j^m+rL-X_i^m) \pp{p^{(0)}}{x}
%     } c.
% \end{align}
% Now, since $V_{ij} = A_{ij}/L_{ij}$, $G_{ij}=R_{ij}^4, A_{ij}=R_{ij}^2$, we have
% \begin{align}
%     \pp{(G_{ij}^{(0)})^{\frac{1}{2}}}{T} = - \frac{\gamma}{L_{ij     }}  
%     \abs[\Big]{ 
%     A_{ij}^{r(0)} G_{ij}^{r(0)} (W_j-W_i+X_j^m+rL-X_i^m) \pp{p^{(0)}}{x}
%     } c.
% \end{align}

% Our chosen function for the adhesivity is
% \begin{align}
%     A_{ij} = 1 - e^{\lambda_{ij} L_{ij}}
% \end{align}
% where 
% \begin{align}
%     \lambda_{ij} &= \sqrt{\left(\frac{\text{Pe}}{2}\right)^2 U_{ij}^2 + \frac{\text{Da}}{\varrho} \frac{1}{R_{ij}}}-\frac{\text{Pe}}{2} U_{ij}
%     \\
%     & = \sqrt{\left(\frac{\text{Pe}}{2}\right)^2 \left( \frac{Q_{ij}}{R_{ij}^2} \right)^2 + \frac{\text{Da}}{\varrho} \frac{1}{R_{ij}}}-\frac{\text{Pe}}{2} \frac{Q_{ij}}{R_{ij}^2}.
% \end{align}
% Here, $U_{ij}=Q_{ij}/A_{ij}$ and $G_{ij}=R_{ij}^4$.
% Expanding, we have
% \begin{align}
%     A_{ij}^{(0)} = 1 - e^{\lambda_{ij}^{(0)} L_{ij}}
% \end{align}
% with
% \begin{align}
%     & 
%     \lambda_{ij}^{(0)} = \sqrt{
%     \varepsilon^2 
%     \left(\frac{\text{Pe}}{2}\right)^2
%     \left( 
%     G_{ij}^{r(0) \frac{1}{2}} (W_i - W_j + X_i^m  -X_j^m- rL)\pp{p^{(0)}}{x}\right)^2 + \frac{\text{Da}}{\varrho} \frac{1}{G_{ij}^{r(0)} }} -
%     \nonumber
%     \\
%     & \hspace{6cm}
%     \varepsilon \frac{\text{Pe}}{2} 
%     G_{ij}^{r(0) \frac{1}{2}} (W_i - W_j + X_i^m  -X_j^m- rL)\pp{p^{(0)}}{x}.
% \end{align}
% So we choose $\text{Pe}=\mathcal{O}(\varepsilon^{-1})$ and $\text{Da}=\mathcal{O}(1)$, so that we have
% \begin{align}
%     & 
%     \lambda_{ij}^{(0)} = \sqrt{
%     \left(\frac{\text{Pe}}{2}\right)^2
%     \left( 
%     G_{ij}^{r(0) \frac{1}{2}} (W_i - W_j + X_i^m  -X_j^m- rL)\pp{p^{(0)}}{x}\right)^2 + \frac{\text{Da}}{\varrho} \frac{1}{G_{ij}^{r(0)} }} -
%     \nonumber
%     \\
%     & \hspace{8cm}
%     \frac{\text{Pe}}{2} 
%     G_{ij}^{r(0) \frac{1}{2}} (W_i - W_j + X_i^m  -X_j^m- rL)\pp{p^{(0)}}{x}.
% \end{align}
% The implication is that diffusivity is small, and particles are blocked due to advection rather than diffusion.

\chapter{Solutions of the multiscale model}
\label{chap:mm_Solutions of the multiscale model}
In this chapter, we solve the dimensionless multiscale model, 
\eqref{eq:mm_Dd_fluid-conservation_homog_sum}--\eqref{eq:mm_Dd_ic_cond_homog_sum}. 
In Section \ref{sec:mm_Solution methodology}, we begin by describing our solution methodology. 
We reduce the problem to a simple and yet physically relevant case, and describe
the resulting numerical system to be solved, highlighting various ways 
in which its computational cost can be decreased.
In Section \ref{sec:mm_Periodic networks}, we then solve the system on periodic networks. 
Firstly, we use the initial conductance distribution (\ref{eq:nm_cond-pdf_sum_nodim_1}a), 
in which the network conductances are non-random, which describes 
the physical scenario of mono-dispersed pore radii and thus conductances. 
We find that solutions of the multiscale model can be made arbitrarily close to 
solutions of the microscale model by increasing the number of repeated cells in the periodic 
network.
Secondly, we use (\ref{eq:nm_cond-pdf_sum_nodim_1}b) for the initial conductance distribution. 
In this case, the conductances of pores are random, which describes 
poly-dispersed pore sizes. 
We find that any single simulation of the multiscale model does not match that of the microscale 
model in general (although their averages do agree), 
due to the randomness of the network. 
This motivates us, in Section \ref{sec:mm_Non-periodic networks}, to discuss the distributions of the solutions 
that arise from repeated Monte Carlo simulations of the multiscale problem, to account for the randomness of the initial conditions. 
We focus on calculation of the parameters of the system, the permeability and adhesivity, 
since these dictate the dependence of the effective solution on the microscale. 
We explain why our system could also be used to approximate the solution of dynamical systems defined 
on non-periodic networks.

\section{Solution methodology}
\label{sec:mm_Solution methodology}
We begin by reducing the multiscale system to a simpler case that we will use to demonstrate the 
solution method. We discuss a numerical scheme, and show that significant reduction in the computational complexity 
of this is possible if microscale dependent parameters are `pre-calculated' 
outside the time-dependent scheme.

\subsection{Model reduction}\label{sec:mm_Model reduction}
To demonstrate the solution of the multiscale model, \eqref{eq:mm_Dd_fluid-conservation_homog_sum}--\eqref{eq:mm_Dd_ic_cond_homog_sum}, 
we simplify by considering the case where the permeability and adhesivity, $\bm{k}$ and $\bm{j}$,
do not vary in the directions that are transverse to the pressure gradient and therefore the flow, $x^2$ and $x^3$, 
so that
\begin{align}
    \pp{\bm{k}}{x^2}=\pp{\bm{k}}{x^3}=\bm{0},
    \quad
    \pp{\bm{j}}{x^2}=\pp{\bm{j}}{x^3}=\bm{0}.
\end{align}
Furthermore, we assume periodicity of all the dependent variables, $\bm{c}$, $G\indx$, $\bm{u}$, $p$, $\psi$, $\bm{k}$, $\bm{j}$, $\bm{W}$, 
across the transverse boundaries, $x^2=0$, $x^2=l^2$ and $x^3=0$, $x^3= l^3$. 
% \begin{align}
%     c(x^1,0,x^3,t)&=c(x^1,l^2,x^3,t),
%     & \quad 
%     c(x^1,x^2,0,t)&=c(x^1,x^2,l^3,t),
%     \label{eq:reduced_periodic-bc_c}
%     \\
%     u(x^1,0,x^3,t)&=u(x^1,l^2,x^3,t),
%     & \quad 
%     u(x^1,x^2,0,t)&=u(x^1,x^2,l^3,t),
%     \label{eq:reduced_periodic-bc_u}
%     \\
%     p(x^1,0,x^3,t)&=p(x^1,l^2,x^3,t),
%     & \quad 
%     p(x^1,x^2,0,t)&=p(x^1,x^2,l^3,t).
%     \label{eq:reduced_periodic-bc_p}
% \end{align}
We then seek spatially univariate solutions that are independent of the transverse directions, so that
\begin{align}
    \pp{c}{x^2}=\pp{c}{x^3}=0,
    \quad
    \pp{G\indx}{x^2}=\pp{G\indx}{x^3}=0,
    \quad
    \pp{\bm{u}}{x^2}=\pp{\bm{u}}{x^3}=\bm{0},
    \quad 
    \pp{p}{x^2}=\pp{p}{x^3}=0.
\end{align}
Under these assumptions, we omit the third dimension, $x^3$, since we expect the dependence of the solution on the two transverse directions to be identical.

Combining the fluid conservation equation, \eqref{eq:mm_Dd_fluid-conservation_homog_sum}, with the particle conservation equation, \eqref{eq:mm_Dd_particle-conservation_homog_sum}, which is the advection--reaction equation that governs the evolution of the particle concentration, $c$, we find that
\begin{align}
    \pp{c}{t} + \frac{u^1}{\phi}\pp{c}{x^1} = -\frac{\psi}{\phi}c.
    \label{eq:mm_particle-conservation_reduced}
\end{align}
The pore conservation equation, \eqref{eq:mm_Dd_pore-conservation_homog_sum}, 
which governs the evolution of the conductances of pores, becomes 
\begin{align}
    \pp{G_{ij}^{\bm{r}}}{t} &= -  \alpha\beta c  (G_{ij}^{\bm{r}})^{\frac{3}{2}} |\Delta_{ji}^{-r^1}| 
    \Big|   
    \pp{p}{x^1} \Big|.
    \label{eq:mm_pore-conservation_reduced}
\end{align}
The Darcy equation, (\ref{eq:mm_Dd_darcy-and-reaction_homog_sum}a), which relates the pressure, $p$, to the Darcy velocity vector, $\bm{u}$, via the permeability matrix, $\bm{k}$, becomes 
\begin{subequations}
\label{eq:mm_darcy_reduced}
\begin{align}
    u^1=-k^{11}\pp{p}{x^1}, 
    \quad 
    u^2=-k^{21}\pp{p}{x^1}.
    \tag{\ref{eq:mm_darcy_reduced}a,b}
\end{align}  
\end{subequations}
% \begin{subequations}
% \label{eq:mm_darcy_reduced}
% \begin{align}
%     u^1=\frac{1}{\int_0^1\frac{1}{k^{11}} \, \text{d}x^1}, 
%     \quad 
%     u^2=\frac{1}{\int_0^1\frac{1}{k^{21}} \, \text{d}x^1}, 
%     \tag{\ref{eq:mm_darcy_reduced}a,b}
% \end{align}  
% \end{subequations}
The reactivity equation, (\ref{eq:mm_Dd_darcy-and-reaction_homog_sum}b), which relates the pressure to the reactivity, $\psi$, via the adhesivity, $\bm{j}$, becomes
\begin{align}
    \psi = - j^1 \pp{p}{x^1}.
    % \psi=\frac{1}{\int_0^1\frac{1}{j^{1}} \, \text{d}x^1}.
    \label{eq:mm_reaction_reduced}
\end{align}
Using the definitions of the permeability and adhesivity, \eqref{eq:mm_Dd_permeability_homog_sum} and \eqref{eq:mm_Dd_adhesivity_homog_sum}
(recalling that the permeability and adhesivity are independent of junction positions, from Appendix \ref{sec:proofs_k=sig}), 
we find that the elements that remain in the system are given by
\begin{align}
    k^{11} 
    &= 
    \frac{l^1}{2 \prod_{m=1}^Dl^m}  \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^1 G_{ij}^{\bm{r}} (W_j^1+r^1l^1 - W_i^1), 
    \label{eq:mm_permeability_reduced}
    \\
    k^{21} 
    &= 
    \frac{l^2}{2 \prod_{m=1}^Dl^m}  \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^2 G_{ij}^{\bm{r}} (W_j^1+r^1l^1 - W_i^1), 
    \\
    j^1 
    & = -
    \frac{1}{\prod_{m=1}^Dl^m} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \alpha G_{ij}^{\bm{r}} (W_j^1+r^1l^1 - W_i^1)  H_{ji}^{-\bm{r}},
    \label{eq:mm_adhesivity_reduced}
\end{align}
where we have used the relevant elements of the pore pressure drop from \eqref{eq:mm_Dd_pressure-delta_homog_sum}, 
\begin{align}
    \Delta_{ij}^{r^1} = W_i^1 - (W_j^1+r^1l^1).
    \label{eq:mm_pressure-delta_reduced}
\end{align}
The cell problem, \eqref{eq:mm_Dd_cell-problem_homog_sum}, simplifies to  
\begin{align}
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big(  G_{ij}^{\bm{r}} - \updelta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}} \Big) W_j^1 
    =
    -
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}}   r^1l^1,
    \label{eq:mm_cell-problem_reduced}
\end{align}
which is one linear problem of size $N$.
Its solution defines the elements of the cell solution matrix, $W_i^1$.
% Notice that the cell problem is a linear equation, $\bm{A} \bm{W}^m = \bm{b}$, where the elements of $\bm{A}$ are defined in \eqref{eq:Dd_def_A}, and we define $\bm{b}$ to be the right-hand side of \eqref{eq:mm_cell-problem_reduced}. 
As in \eqref{eq:mm_Dd_bc_pres_homog_sum}--\eqref{eq:mm_Dd_ic_cond_homog_sum}, we assume a constant pressure drop in the flow direction, $x^1$, 
and a constant concentration of particles at the inlet, 
so that
\begin{subequations} \label{eq:mm_bc_pres_reduced}
\begin{align}
    p(0,t) = 1, 
    \quad
    p(l^1,t) = 0,
    \tag{\ref{eq:mm_bc_pres_reduced}a,b}
\end{align}
\end{subequations}
and 
\begin{align}
    c(0,t) = 1,
    \label{eq:mm_bc_conc_reduced}
\end{align}
for $t\ge0$.
We assume that there are no particles in the filter initially,
\begin{align}
    c(x^1,0) = 0, 
    \label{eq:mm_ic_conc_reduced}
\end{align}
and that the initial pore conductances are drawn from a known distribution, $\mathcal{G}$, so that 
\begin{align}
    G_{ij}^{\bm{r}}(x^1,0) = G_{ij}^{\bm{r}0}(x^1)
    \label{eq:mm_ic_cond_reduced}
\end{align}
for $i,j \in \{1,\hdots,N,\text{in}, \text{out}\}$.
As in Part \ref{part:A network model for filtration} (see \eqref{eq:nm_cond-pdf_sum_nodim_1}), 
we assume that $\mathcal{G}$ is either the Dirac-delta distribution, or the log-normal distribution, 
so that the probability density function, $\mathscr{g}$, of $\mathcal{G}$ is either
\begin{subequations}
\label{eq:mm_cond-pdf_reduced}
\begin{align}
    \mathscr{g}(G_{ij}^{\bm{r}0}) = \updelta(G_{ij}^{\bm{r}0}-1), 
    \quad \text{or} \quad
    \mathscr{g}(G_{ij}^{\bm{r}0})=\frac{1}{\sqrt{2 \pi} \sigma_G G_{ij}^{\bm{r}0}} 
    \exp{
    \left(- \frac{ 
    \left( 
    \log (G_{ij}^{\bm{r}0}) + \frac{\sigma_G^2}{2} 
    \right)^2
    }{
    2 \sigma_G^2}
    \right)},
    \tag{\ref{eq:mm_cond-pdf_reduced}a,b}    
    \end{align}
\end{subequations}
at any position $x^1$.
Comparing the reduced system, \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_cond-pdf_reduced}, with the full system,  \eqref{eq:mm_Dd_fluid-conservation_homog_sum}--\eqref{eq:mm_Dd_ic_cond_homog_sum},
we note that the assumption of spatially univariate effective parameters and solutions affords a significant reduction in mathematical complexity. 
These assumptions are not especially physically restrictive, however: 
although we derived a model that is general enough to model scenarios that involve heterogeneity in the transverse directions, in filtration processes that are driven by unidirectional pressure drops we do not usually expect large transverse variation in filter properties such as permeability. 
In the remainder of this section, we outline the process of solution of the reduced system, \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_cond-pdf_reduced}.
% \textcolor{blue}{[Here, I want to justify this model reduction, because I don't think it would ever be necessary to solve the full system...we'd never be interested in the $y$ variation of permeability, and would only ever be interested in uni-directional pressure-gradients... furthermore, $u^2$ turns out to be $\mathcal{O}(10^{-3})$, so we're finding uni-directional flow solutions without assuming them. The lack of flow in the $y$ direction will mean that there's no interesting variation in that direction anyway]}.


\subsection{Solution without parameter pre-calculation}
To solve the reduced system, \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_cond-pdf_reduced}, 
we begin by noting that 
\begin{subequations}
\label{eq:mm_darcy_reduced_elim-pres}
\begin{align}
    u^1=\frac{1}{\int_0^1\frac{1}{k^{11}} \, \text{d}x^1}, 
    \quad 
    u^2=\frac{1}{\int_0^1\frac{1}{k^{21}} \, \text{d}x^1}, 
    \tag{\ref{eq:mm_darcy_reduced_elim-pres}a,b}
\end{align}  
\end{subequations}
and that
\begin{align}
    % \psi = - j^1 \pp{p}{x^1}.
    \psi=\frac{1}{\int_0^1\frac{1}{j^{1}} \, \text{d}x^1}.
    \label{eq:mm_reaction_reduced_elim-pres}
\end{align}
Here, we have eliminated the pressure gradient from the Darcy equation, \eqref{eq:mm_darcy_reduced}, and from the 
reactivity equation, \eqref{eq:mm_reaction_reduced}, using the conditions on the pressure 
at the boundaries, \eqref{eq:mm_bc_pres_reduced}.

We proceed by using a first-order upwind scheme in space and a first-order scheme in time to discretise the reduced system, using the discrete indices $\xi \in \{1,\hdots,X\}$ and $\tau \in \{1,\hdots,T\}$ to denote discrete space and time.
At each spatial point, $\xi$, and at each temporal point, $\tau$, 
we assume that there is a cell containing $N$ junctions (see Figure).
At $t=0$, 
which corresponds to $\tau=1$, 
we are given the concentration and conductances for all space points, $c[\xi,1]=0$ and $G\indx[\xi,1]=G_{ij}^{\bm{r}0}$,
from \eqref{eq:mm_ic_conc_reduced} and \eqref{eq:mm_ic_cond_reduced}.
At $x=0$, 
which corresponds to $\xi=1$, 
we are given the concentration for all time points, $c[1,\tau]=1$, 
from \eqref{eq:mm_bc_conc_reduced}.

% At $t=0$ (so that $\tau=1$ and $\xi=1$), 
To solve numerically at the time point $\tau$,
we substitute the known conductance on each pore in the cell, $G_{ij}^{\bm{r}}[\xi,\tau]$, into the cell problem, 
\eqref{eq:mm_cell-problem_reduced}, and solve the resulting linear problem 
% of size $N^2\times (2K+1)$ 
to determine the $N$ cell solution elements, 
$W_i[\xi,\tau]$.
Next, we substitute these cell solution elements into the permeability and adhesivity definitions, 
\eqref{eq:mm_permeability_reduced} and \eqref{eq:mm_adhesivity_reduced}, 
to find the current permeability and adhesivity elements, $k^{11}[\xi,\tau]$, $k^{21}[\xi,\tau]$, 
and $j^{1}[\xi,\tau]$.
Next, we substitute these into the modified Darcy equations, \eqref{eq:mm_darcy_reduced_elim-pres}, 
and the reactivity equation, \eqref{eq:mm_reaction_reduced_elim-pres}. 
We use the trapezium rule to discretise the integrals in these equations, and solve these to find 
the velocity and reactivity,
$u^1[\tau]$, $u^2[\tau]$, and $\psi[\tau]$.
We then find the pressure gradient, $(\partial p/\partial x^1)[\xi,\tau]$ by substituting 
the known reactivity, $\psi[\xi,\tau]$, and adhesivity, $j^{1}[\xi,\tau]$, 
into the reactivity equation, 
\eqref{eq:mm_reaction_reduced}.

Next, we use the particle and pore conservation equations, \eqref{eq:mm_particle-conservation_reduced} and \eqref{eq:mm_pore-conservation_reduced}, to calculate the particle concentration and pore conductances at the current spatial point, $\xi$, at the next time point, $\tau+1$.
% In particular, to find the next concentration, $c[\xi,\tau+1]$, we use an upwind discretisation, 
% since, using \eqref{eq:mm_bc_pres_reduced}, we have that $c[1,\tau]=1$ for $\tau \in \{1,\hdots,T\}$.
Substituting the current concentration at the previous spatial point, $c[\xi-1,\tau]$,
the current concentration at the current spatial point, $c[\xi,\tau]$, 
the flow rate and reactivity at the current time point, $u^1[\tau]$ and $\psi[\tau]$, 
and the porosity, $\phi=N/\prod_m^Dl^m$ into the discretised version of \eqref{eq:mm_particle-conservation_reduced}, 
we obtain the new concentration, $c[\xi,\tau+1]$.
Lastly, we substitute $c[\xi,\tau+1]$, $G_{ij}^{\bm{r}}[\xi,\tau]$, $W_i[\xi,\tau]$, and $(\partial p/\partial x^1)[\xi,\tau]$
into the discretised version of \eqref{eq:mm_pore-conservation_reduced} to calculate 
the new conductances, $G_{ij}^{\bm{r}}[\xi,\tau+1]$.
Since the initial concentration and conductances, $c[\xi,1]$ and $G_{ij}^{\bm{r}}[\xi,1]$, 
are given, this method determines the discrete solution variables for all $\xi\in\{1,\hdots,X\}$ and $\tau\in\{1,\hdots,T\}$.

At each spatial point, $\xi$, and time point, $\tau$, we solve a linear problem to 
obtain $(N)^2 \times (2K+1)^D + N + 8$ values, 
so that the overall cost (since $K=\mathcal{O}(1)$ and $D=\mathcal{O}(1)$) 
of solution with no pre-calculation 
is $C_{\text{NPC}}=\mathcal{O}(T(XN^2)^L)$, 
where $\mathcal{O}(S^L)$ for $2<L<3$ is the number of operations involved in the solution of a linear system for $S$ variables. 
In Section \ref{sec:nm_Solution methodology}, we showed that the cost of solving 
the network model is $C_{\text{NM}}=\mathcal{O}((N^1N^2(N)^2)^L)$, 
where we recall that $N^1$ and $N^2$ are the numbers of cells in the network in the $x^1$ and $x^2$ directions. 
In many physical membranes, we might expect to see as many as $10^6$ junctions in 
the flow direction,  
so that $N^1=\mathcal{O}(10^5)$ for $N=\mathcal{O}(10)$, for example. 
This means that the solution of the network model that describes the entire membrane is computationally infeasible.
Crucially, the cost of solution of the multiscale model only depends on the size of one cell within the network.
% which is a problem 
% because there can be at least $\mathcal{O}(10^4)$ junctions in a filter.
% Crucially we note that, unlike the network model, the computational cost of the multiscale solution is independent of the size 
% of the network. 
This is useful, since later in this chapter, we will show that $N=\mathcal{O}(10)$ in the multiscale model is sufficient for representation of the network as a  
whole.
Next, we will show that much of the computational cost of solution is avoidable, by `pre-calculating' 
$G\indx$ and $W_i^m$, and thus the parameters of the macroscale model, $\bm{k}$ and $\bm{j}$, 
before the solution of the time-dependent linear problem that we outlined above.

% Despite this, the main computational cost comes from the size of the cell $N$. 
% This is feasible for moderately sized $N$, and later in this chapter we will discuss how large $N$ needs 
% to be.
% Even so, the expense causes increased computational time.
% This is a result of having to find the conductances of every edge, 
% $G_{ij}^{\bm{r}}$, at every time step. 
% To decrease the cost of solution, we seek a method that avoids this, wich we discusss in the remainder of this section.
% It follows that, at each time point we solve a linear problem of size $\mathcal{O}(N^2\times(2K+1)\times X)$, 
% so that the total computational expense of the solution is $\mathcal{O}(N^2\times(2K+1)\times X\times T)$.

\subsection{Solution with parameter pre-calculation}
As we discussed, the main source of computational cost in the solution of the multiscale model 
emerges from the need to find the conductances of the $\mathcal{O}(N^2)$ pores in the cell at each time and space step, 
and the desire is to derive a numerical scheme for which this is unnecessary. 
To this end, we begin by defining 
\begin{align}
    f(x^1,t) = \int_0^t c(x^1,s) \Big| \frac{\partial p}{\partial x^1}(x^1,s) \Big| \, \text{d}s.
\end{align}
We interpret this as the mass concentration flux of particles, since in dimensional terms we find that 
$[f]=\text{kg}\text{m}^{-3}\text{s}^{-1}$.

Crucially, we note that the mass flux, $f$, satisfies the relation 
\begin{align}
    \pp{f}{t} = c(x^1,t) \Big| \frac{\partial p}{\partial x^1}(x^1,t) \Big|. 
    \label{eq:mm_definition-of-f}
\end{align}
Using \eqref{eq:mm_definition-of-f}, it follows that the pore conductances satisfy 
\begin{align}
    \pp{G\indx}{t} = \pp{G\indx}{f}\pp{f}{t} = \pp{G\indx}{f} c(x^1,t) \Big| \frac{\partial p}{\partial x^1}(x^1,t) \Big|.
    \label{eq:mm_chain-rule}
\end{align}
Substituting the right hand side of \eqref{eq:mm_chain-rule} into the pore conservation equation, 
\eqref{eq:mm_pore-conservation_reduced}, 
we find that
\begin{align}
    \pp{G_{ij}^{\bm{r}}}{f} &= -  \alpha\beta (G_{ij}^{\bm{r}})^{\frac{3}{2}} |W_j^1+r^1l^1-W_i^1|. 
    \label{eq:mm_pore-conservation_modified}
\end{align}
We have removed the explicit time dependence from the pore conservation equation, 
and we now view the pore conductances and the cell solutions as 
explicit functions of the mass flux, $f$, $G_{ij}^{\bm{r}}(f)$ and $W_i^1(f)$.

To see why this is useful, 
we note that the modified pore conservation equation, \eqref{eq:mm_pore-conservation_modified},
with the initial conductance condition, \eqref{eq:mm_ic_cond_reduced}, 
together with the cell problem, \eqref{eq:mm_cell-problem_reduced}, 
now defines an ordinary algebraic--differential initial value problem for the pore conductances and the 
cell solutions that can be solved independently of the rest of the explicitly time-dependent 
macroscale problem. 
We begin by defining points at which the mass flux is discretised, 
$\varphi \in \{1,\hdots,F\}$, say, 
and note that, at $\varphi=1$, 
the conductances $G\indx[1]=G_{ij}^{\bm{r}0}$ are given, 
so that the cell solutions, $W_i^1[1]$, are given by solution of the cell problem, 
\eqref{eq:mm_cell-problem_reduced}. 
We then use a first-order scheme for explicit discretisation of the modified pore conservation 
equation \eqref{eq:mm_pore-conservation_modified}, with $G\indx[1]$ and $W_i^1[1]$, 
to compute the pore conductances at the next mass flux value, $G\indx[2]$. 
In general, for $\varphi>1$,  
we use the current pore conductances, $G\indx[\varphi]$, in the cell problem, \eqref{eq:mm_cell-problem_reduced}, 
to obtain the current cell solutions, $W_i^1[\varphi]$. 
We use the current conductances and cell solutions in \eqref{eq:mm_permeability_reduced} and \eqref{eq:mm_adhesivity_reduced} 
to calculate the current permeability and adhesivity, 
$k^{11}[\xi,\tau]$ and $j^{1}[\xi,\tau]$.
We then use these current conductances and cell solutions to calculate 
the next pore conductances, $G\indx[\varphi+1]$. 
In this way, we pre-calculate a discretisation of the functions 
$G\indx(f)$ and $W_i^1(f)$ and therefore $k^{11}(f)$ and $j^1(f)$ using $C_{\text{PC}}=\mathcal{O}(F(N^2)^L)$ computations.

To solve the multiscale model, we then solve a smaller linear system and interpolate this function. 
In particular, similarly to before, at each space point, $\xi$, 
and at each time point, $\tau$, 
we are given the current concentration, $c[\xi,\tau]$ (due to solution of a forward system at the previous time step), 
and the previous pressure gradient, $(\partial p/\partial x^1)[\xi,\tau-1]$.
We begin by discretising the definition of $f$, \eqref{eq:mm_definition-of-f}, 
and using $c[\xi,\tau]$ and $(\partial p/\partial x^1)[\xi,\tau-1]$ to obtain $f[\xi,\tau]$.
We then interpolate the functions $k^{11}[\varphi]$ and $j^{1}[\varphi]$, using that $\varphi\approx f$ for suitably large $F$ ($\mathcal{O}(10^2)$, say),
which gives $k^{11}[\xi,\tau]$, $k^{21}[\xi,\tau]$, 
and $j^{1}[\xi,\tau]$.
The scheme now proceeds as before. 
That is, we use \eqref{eq:mm_darcy_reduced_elim-pres} and \eqref{eq:mm_reaction_reduced_elim-pres} 
to obtain $u^1[\tau]$, $u^2[\tau]$, and $\psi[\tau]$, 
and then find $(\partial p/\partial x^1)[\xi,\tau]$ by substituting $\psi[\xi,\tau]$ and $j^{1}[\xi,\tau]$ into 
\eqref{eq:mm_reaction_reduced}.
The next concentration value, $c[\xi,\tau+1]$, is then found by substituting these values into 
a first-order discretisation of the particle conservation equation, \eqref{eq:mm_particle-conservation_reduced}, without needing to solve 
the pore conservation equation, \eqref{eq:mm_pore-conservation_reduced}. 

The cost of solution of the multiscale system with pre-calculation, $C_{\text{WPC}}$, 
is the cost of the pre-calculation plus the cost of solution of the time-dependent problem, 
so that $C_{\text{WPC}} = \mathcal{O}(F(N^2)^L)+\mathcal{O}(T(X)^L)<\mathcal{O}(T(XN^2)^L)<\mathcal{O}((N^1N^2(N)^2)^L)$. 
That is, the cost of solution of the multiscale problem with pre-calculation is cheaper than that with no pre-calculation, which is cheaper than 
the infeasible network problem: $C_{\text{WPC}} < C_{\text{NPC}} < C_{\text{NM}}$.
The main cost of the solution has shifted to the pre-calculation step, $C_{\text{PC}}=\mathcal{O}(F(N^2)^L)$, 
and we will show that this need not be large, since $N=\mathcal{O}(10)$ is sufficient for most applications. 
Since the $N^2$ term associated with parameter calculation has been removed from the time-dependent scheme, 
the number of spatial and temporal points, $X$ and $T$, can be chosen, as usual, to be suitably large as to accurately numerically discretise the continuous macroscale system, to satisfy the Courant--Friedrichs--Lewy (CFL) condition \cite{tran2009} for advective systems, for example.
In the next section, we will use the numerical scheme with parameter pre-calculation (WPC) to demonstrate the solution  
of the macroscale model.


% \textcolor{blue}{
% Describe how the system would be solved in general. Explain that this is slow. Explain that if you can precalculate permeablity and adhesivity then this there would be speed up. Derive a constraint on the types of adherence and conductance models for which effective-parameter pre-calculation is possible.}
% Section on what $G_ij$ you can choose
% Talk about the constraint in terms of what you can choose to allow you to do preprocessing - figure this out with Jon.
% Talk about options for functions that satisfy this constraint and explain theat so,merelate to blocking and some relate to deposition 
% Concluide by explaining that we’re going to use simple blocking rule as it’s easiest to explore the relationship between the microscale and effective system. Define G in terms of alpha but don’t say what alpha we’re going to choose.
% \textcolor{blue}{[In this section I will discuss options for what to choose for the model for how the conductance changes, and possibly derive a constraint on the types of rules for which it is possible to  preprocess the effective parameters (i.e. calculate them before the flow calculation}.]

% \section{Model reduction}
% To illustrate the solution of the effective system, \eqref{eq:mm_Dd_particle-conservation_homog_sum}--\eqref{eq:mm_Dd_ic_cond_homog_sum}, and the behaviour of the effective permeability and adhesivity, $\bm{k}$ and $\bm{j}$, we consider the case where these do not vary in the transverse-flow directions, $x^2$ and $x^3$, so that 
% \begin{align}
%     \pp{\bm{k}}{x^2}=\pp{\bm{k}}{x^3}=0,
%     \quad
%     \pp{\bm{j}}{x^2}=\pp{\bm{j}}{x^3}=0.
% \end{align}
% Furthermore, we assume periodicity of all of the dependent variables across the transverse boundaries, $x^2=0$, $x^2 = l^2$ and $x^3=0$, $x^3= l^3$. 
% % \begin{align}
% %     c(x^1,0,x^3,t)&=c(x^1,l^2,x^3,t),
% %     & \quad 
% %     c(x^1,x^2,0,t)&=c(x^1,x^2,l^3,t),
% %     \label{eq:reduced_periodic-bc_c}
% %     \\
% %     u(x^1,0,x^3,t)&=u(x^1,l^2,x^3,t),
% %     & \quad 
% %     u(x^1,x^2,0,t)&=u(x^1,x^2,l^3,t),
% %     \label{eq:reduced_periodic-bc_u}
% %     \\
% %     p(x^1,0,x^3,t)&=p(x^1,l^2,x^3,t),
% %     & \quad 
% %     p(x^1,x^2,0,t)&=p(x^1,x^2,l^3,t).
% %     \label{eq:reduced_periodic-bc_p}
% % \end{align}
% We then seek spatially univariate solutions that are independent of the transverse directions, so that
% \begin{align}
%     \pp{c}{x^2}=\pp{c}{x^3}=0,
%     \quad
%     \pp{u}{x^2}=\pp{u}{x^3}=0,
%     \quad 
%     \pp{p}{x^2}=\pp{p}{x^3}=0.
% \end{align}
% Under these assumptions, we omit the third dimension, $x^3$, since we expect the dependence of the solution on the two transverse directions to be identical.

% The advection--reaction equation, \eqref{eq:mm_Dd_particle-conservation_homog_sum}, which governs the evolution of the particle concentration, $c$, becomes
% \begin{align}
%     \pp{c}{t} + \frac{u^1}{\phi}\pp{c}{x^1} = -\frac{\psi}{\phi}c.
%     \label{eq:mm_particle-conservation_reduced}
% \end{align}
% The Darcy equation, (\ref{eq:summary_grad-eqs}a), which relates the pressure, $p$, to the Darcy velocity, $\bm{u}$, via the permeability, $\bm{k}$, becomes 
% \begin{align}
%     u^1=-k^{11}\pp{p}{x^1}, 
%     \quad 
%     u^2=-k^{21}\pp{p}{x^1}.
%     \label{eq:mm_darcy_reduced}
% \end{align}
% The reactivity equation, (\ref{eq:summary_grad-eqs}b), which relates the pressure to the reactivity, $\psi$, via the adhesivity, $\bm{j}$, becomes
% \begin{align}
%     \psi = - j^1 \pp{p}{x^1}.
%     \label{eq:mm_reaction_reduced}
% \end{align}
% Recall that, in Appendix \ref{sec:proofs_k=sig}, we showed the permeability and adhesivity to be independent of node positions. 
% We may, therefore, arbitrarily set node positions without affecting the solution. We choose $X_i^m=0$ for $i \in \{1,\hdots,N\}$ and $m \in \{1,2\}$, so that the permeability and adhesivity formulae, \eqref{eq:summary_definition-of-k} and \eqref{eq:summary_definition-of-j}, simplify to
% \begin{align}
%     k^{mn} 
%     &= 
%     \frac{l^m}{2 \prod_{a=1}^Dl^a}  \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
%     r^m G_{ij}^{\bm{r}} (W_j^n+r^nl^n - W_i^n), 
%     \label{eq:mm_permeability_reduced}
%     \\
%     j^m 
%     & = -
%     \frac{1}{\prod_{a=1}^Dl^a} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}} (W_j^m+r^ml^m - W_i^m)  A_{ji}^{-\bm{r}} H_{ji}^{-\bm{r}}.
%     \label{eq:mm_adhesivity_reduced}
% \end{align}
% The cell problem, \eqref{eq:summary_cell-problem}, also simplifies to  
% \begin{align}
%     \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big(  G_{ij}^{\bm{r}} - \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}} \Big) W_j^m 
%     =
%     -
%     \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}}   r^ml^m,
%     \label{eq:mm_cell-problem_reduced}
% \end{align}
% which defines the elements of the cell-solution vectors, $W_i^1$ and $W_i^2$ for $i \in \{1,\hdots,N\}$.
% % Notice that the cell problem is a linear equation, $\bm{A} \bm{W}^m = \bm{b}$, where the elements of $\bm{A}$ are defined in \eqref{eq:Dd_def_A}, and we define $\bm{b}$ to be the right-hand side of \eqref{eq:mm_cell-problem_reduced}. 
% We assume a constant pressure drop in the flow direction, $x^1$, and a zero-distribution of particles initially. The corresponding boundary and initial conditions are 
% \begin{subequations}
% \label{eq:reduced_bc}
% \begin{align}
%     p(t,0) = 1, \quad 
%     p(t,1) = 0, \quad 
%     c(t,0) = 1,
%     \tag{\ref{eq:reduced_bc}a--c}
% \end{align}
% \end{subequations}
% and
% \begin{align}
%     c(0,\bm{x}) = 0.
%     \label{eq:mm_ic_cond_reduced}
% \end{align}
% Comparing the reduced system, \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_ic_cond_reduced}, with the full system,  \eqref{eq:mm_Dd_particle-conservation_homog_sum}--\eqref{eq:mm_Dd_ic_cond_homog_sum},
% we note that the assumption of spatially univariate effective parameters and solutions affords a significant reduction in mathematical complexity. 
% These assumptions are not especially physically restrictive, however. Filtration regimes are commonly driven by unidirectional pressure drops, and so we do not expect large transverse variation in filter properties, such as permeability. 
% % \textcolor{blue}{[Here, I want to justify this model reduction, because I don't think it would ever be necessary to solve the full system...we'd never be interested in the $y$ variation of permeability, and would only ever be interested in uni-directional pressure-gradients... furthermore, $u^2$ turns out to be $\mathcal{O}(10^{-3})$, so we're finding uni-directional flow solutions without assuming them. The lack of flow in the $y$ direction will mean that there's no interesting variation in that direction anyway]}.


\section{Periodic networks} \label{sec:mm_Periodic networks}
In this section, to demonstrate the solution of the reduced multiscale model,  \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_cond-pdf_reduced}, 
we consider a microscale given by a four-regular network.
To simplify our analysis, we will consider the case where each cell is composed of four nodes. 
A diagram of the repeating cell is shown in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}. 
We will first assume that the initial conductances of the pores in this cell are given by (\ref{eq:mm_cond-pdf_reduced}a). 
This case models the scenario where the pore-size distribution is mono-dispersed initially, 
which corresponds to a non-random network structure that need not be sampled.
We will then assume that the initial conductances are given by (\ref{eq:mm_cond-pdf_reduced}b), 
which models log-normally poly-dispersed initial pore sizes. 
In this case, the network structure is random, and each initialisation of the problem 
gives rise to a cell with a different set of conductances.
% , so  the solution must be repeated and sampled to 
% find its statistics. 
Throughout, we will consider the case where $\alpha=1$ and $\beta=1$. 
In Part \ref{part:A network model for filtration}, we showed that this corresponds to $\mathcal{O}(1)$ particle removal 
and $\mathcal{O}(1)$ reduction by the time that fluid has traversed the filter once.
We use the numerical solution scheme discussed in Section \ref{sec:mm_Solution methodology}. 
That is, we pre-calculate the permeability and adhesivity parameters of \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_cond-pdf_reduced}, 
$k^{11}$ and $j^1$, as functions of the mass concentration flux, $f$, which acts as a time-like variable. 
We then interpolate these functions to find the parameters at each position and time.

\begin{figure}[!htb]
     \centering
         \includegraphics[width=0.55\textwidth]{./figures/multiscale/drawings/coloured-cell.pdf}
    \caption{A Diagram showing one cell of a four-regular network. 
    Each node is labelled with an index.
    Intra-cell edges that are parallel to the flow direction are coloured blue.
    Intra-cell edges that are perpendicular to the flow direction are coloured orange.
    Inter-cell edges that are parallel to the flow direction are coloured green.
    Inter-cell edges that are perpendicular to the flow direction are coloured red.
    }
\label{fig:mm_periodic/drawings/coloured-cell.pdf}
\end{figure}


\subsection{Non-random networks: Mono-dispersed pores}
We begin by considering the case where the initial pore conductances are given by (\ref{eq:mm_cond-pdf_reduced}a), 
so that all pores have unit conductance initially. 

\subsubsection{Conductance and pressure drop}
In Figure \ref{fig:mm_periodic/non-random/preprocess}, we show the pore conductances, $G\indx$, 
and the pore pressure drops in the direction of the flow, $\Delta_{ij}^{r^1}$, 
as functions of the mass concentration flux, $f$. 
In Figure \ref{fig:mm_periodic/non-random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}, 
we see (dotted black line) that the conductance of  pores that are perpendicular to the flow (see the orange and red edges in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}) does not change with $f$.
Observing Figure \ref{fig:mm_periodic/non-random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}, 
we see that this is because the pressure drop on these pores (orange line) is zero. 
There is no pressure difference across these pores, so that they do not contribute to the macroscale pressure gradient that drives the flow.
On the other hand, returning to Figure \ref{fig:mm_periodic/non-random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}, 
we see (solid coloured lines) that the conductance of the pores that are parallel to the flow (see the green and blue edges in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}) decreases as $f$ increases.
To explain this, we begin by noting, from Figure \ref{fig:mm_periodic/non-random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}, 
that these pores have unit pressure drop, so that they contribute equally to the macroscale pressure gradient 
for any $f$ value.
Since the pore deltas are constant, we find that the pore conservation equation, \eqref{eq:mm_pore-conservation_modified}, 
has an explicit solution, which is given by 
\begin{align}
    G\indx(f) = \frac{4}{(\alpha \beta f+2)^2}, 
    \label{eq:mm_cond-solution_explicit}
\end{align}
which we plot in Figure \ref{fig:mm_periodic/non-random/preprocess/cond_5_v__conc_max_or_tot_1.pdf} (dashed black line).
We observe that, as $F$ increases, so that discretisation of $f$ in our numerical scheme, $\varphi$, becomes finer, 
the numerical solution (coloured lines) approaches this explicit solution, 
and that $F$ need not be too large ($F=\mathcal{O}(10^2)$) before we find good agreement.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the pore conductance, $G\indx$, and the pore pressure drop in the $x^1$ direction, $\Delta_{ij}^{r^1}$, vary with the mass concentration flux, $f$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$.
    \subref{fig:mm_periodic/non-random/preprocess/cond_5_v__conc_max_or_tot_1.pdf} 
    The pore conductance, $G\indx$, 
    as a function of the mass concentration flux, $f$.
    The dotted black line shows the explicit solution on pores that are perpendicular 
    to the $x^1$ direction (see the orange and red pores in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf})
    The dashed black line shows the explicit solution on pores that are parallel 
    to the $x^1$ direction (see the blue and green pores in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}). 
    The solid coloured lines show numerical solutions for various values of $F$.
    The values of $F$ are 
    $50$ (blue), 
    $100$ (orange), 
    $200$ (green), 
    and the arrow shows the direction of increasing $F$.
    \subref{fig:mm_periodic/non-random/preprocess/delt_5_v__conc_max_or_tot_1.pdf} 
    The $x^1$ component of the pore pressure drop, $\Delta_{ij}^{r^1}$, 
    as a function of the mass concentration flux, $f$.
    The blue line shows the solution for pores that are parallel to the $x^1$ direction
    (see the blue and green pores in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}). 
    For example, the pore $i=1, j=2, \bm{r}=(0,0)^{\top}$ (solid blue), for which $\Delta_{12}^{r^1=0}=1$, or the pore $i=2, j=1, \bm{r}=(1,0)^{\top}$ (solid green), for which $\Delta_{21}^{r^1=1}=1$.
    The orange line shows the solution for pores that are perpendicular to the $x^1$ direction
    (see the orange and red pores in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}). 
    For example, the pore $i=1, j=3, \bm{r}=(0,0)^{\top}$ (solid orange), for which $\Delta_{13}^{r^1=0}=0$, or the pore $i=3, j=1, \bm{r}=(0,-1)^{\top}$ (solid red), for which $\Delta_{21}^{r^1=0}=0$.
    }
    \label{fig:mm_periodic/non-random/preprocess}
\end{figure}

\subsubsection{Permeability and adhesivity}
In Figure \ref{fig:mm_periodic/non-random/flow/perm-and-depo}, 
we demonstrate the variation of the permeability and adhesivity, $k^{11}$ and $j^{1}$, 
with the mass concentration flux, $f$, and position, $x^1$. 
In Figure \ref{fig:mm_periodic/non-random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}, 
we see the permeability and adhesivity curves that result from our calculation of the pore conductances (above). 
We see that these are identical. 
Checking the definitions of these parameters, \eqref{eq:mm_permeability_reduced} and \eqref{eq:mm_adhesivity_reduced}, 
we see that this is because $\alpha=1$, and because pore conductances are all the same initially and decrease at the same rate, from \eqref{eq:mm_cond-solution_explicit}.
We demonstrate (dashed black line) how we read off values for the permeability and adhesivity given a discrete approximation of these parameters as functions of $f$ by using an interpolation, which is the strategy that we use in our numerical scheme.
Figure \ref{fig:mm_periodic/non-random/flow/cmax_2__v__posi_1.pdf}, we illustrate (dashed black line) how, given 
a mass concentration flux, our numerical scheme obtains a value for the position at a given time.
Then, in Figures \ref{fig:mm_periodic/non-random/flow/perm_2__v__posi_1.pdf} and \ref{fig:mm_periodic/non-random/flow/depo_2__v__posi_1.pdf}, 
we demonstrate how the permeability and adhesivity are constructed as functions of position and time. 
We see that these decrease as time increases, as we expect (see Part \ref{part:A network model for filtration}). 
This decrease is faster at positions closer to the inlet, since pores here conduct more particles per unit time, 
and thus experience more deposition per unit time.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}
     \end{subfigure}
     %hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/flow/cmax_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/flow/cmax_2__v__posi_1.pdf}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/flow/perm_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/flow/perm_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/flow/depo_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/flow/depo_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the permeability, $k^{11}$, and adhesivity, $j^1$, vary with the mass concentration flux, $f$, and position, $x^1$, at various times, $t$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$.
    The values of $t$ are 
    $0$ (blue), 
    $0.25$ (orange), 
    $0.5$ (green), 
    $0.75$ (red), 
    $1$ (purple).
    The arrows show the direction of increasing time.
    \subref{fig:mm_periodic/non-random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf} 
    The permeability, $k^{11}$, and adhesivity, $j^1$, as functions of $f$. 
    These are the same for a mono-dispersed network.
    The blue dots show the numerical solution at discrete values of $f$, $\varphi$. 
    The blue line shows an interpolation of this numerical solution. 
    The dashed black line shows an extrapolation at 
    $f=0.5$ (which corresponds to position $x^1\approx0.2$ at time $t=0.75$) to determine 
    the permeability and adhesivity, $k^{11}(f\approx0.5)\approx0.625$, $j^{1}(f\approx0.5)\approx0.625$.
    \subref{fig:mm_periodic/non-random/flow/cmax_2__v__posi_1.pdf} 
    The mass concentration flux, $f$, as a function of position, $x^1$, 
    at various times, $t$. 
    The dashed black line shows an extrapolation at 
    position $x^1=0.2$ and time $t=0.75$ to determine the mass concentration flux here, $f(x^1=0.2,t=0.75)\approx0.5$.
    \subref{fig:mm_periodic/non-random/flow/perm_2__v__posi_1.pdf} 
    The permeability, $k^{11}$, as a function of position, $x^1$, 
    at various times, $t$.
    The dashed black line shows an extrapolation at $x^1=0.2$ and $t=0.75$ to confirm 
    that $k^{11}(x^1=0.2,t=0.75)\approx0.625$.
    \subref{fig:mm_periodic/non-random/flow/depo_2__v__posi_1.pdf} 
    The adhesivity, $j^{1}$, as a function of position, $x^1$, 
    at various times, $t$.
    The dashed black line shows an extrapolation at $x^1=0.2$ and $t=0.75$ to confirm 
    that $j^{1}(x^1=0.2,t=0.75)\approx0.625$.
    }
    \label{fig:mm_periodic/non-random/flow/perm-and-depo}
\end{figure}

\subsubsection{Concentration and conductance}
Using Figure \ref{fig:mm_periodic/non-random/flow/comparison}, 
we compare the concentration and permeability (effective conductance) from the multiscale solution
(left plots in the figure)
with the effective concentration and conductance from the microscale solution (see \eqref{eq:nm_definition-of-av-conc}
and \eqref{eq:nm_definition-of-av-cond} from Part \ref{part:A network model for filtration})
(right plots in the figure).
We plot the network solution for decreasing values (reading down the figure) of $\varepsilon$, 
so that we increase the number of cells in the network. 
We see that, when $\varepsilon=\mathcal{O}(10^{-1})$, 
the fit is modest.
The network solution is dispersive because pores and junctions are further apart, and the multiscale solution does not capture this behaviour. 
As $\varepsilon$ decreases, 
the microscale solution becomes less dispersive, since junctions and pores are packed more tightly.
As a result, the accuracy of the fit of the multiscale solution to the microscale solution improves. 
This is impressive since, as discussed in Part \ref{part:A network model for filtration}, 
concentration solutions of the network model are travelling waves that consist of step functions that vary rapidly in space.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.1/conc_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/compare/epsi-0.1/conc_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.1/perm_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/compare/epsi-0.1/perm_2__v__posi_1.pdf}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.05/conc_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/compare/epsi-0.05/conc_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.05/perm_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/compare/epsi-0.05/perm_2__v__posi_1.pdf}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.025/conc_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/compare/epsi-0.025/conc_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.025/perm_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/compare/epsi-0.025/perm_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the homogenised and average network (left) concentrations, $c$ and $\bar{C}$, 
    and (right) conductances,  $k^{11}$ and $\bar{G}$, 
    vary with position, $x^1$, at various times, $t$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$.
    The values of $t$ are 
    $0$ (blue), 
    $0.25$ (blue), 
    $0.5$ (green), 
    $0.75$ (red), 
    $1$ (purple).
    Dots show the average network quantities, interpolated with correspondingly coloured dashed lines. 
    Solid lines of the same colours show the effective quantities. 
    % The arrows show the direction of increasing time.
    \subref{fig:mm_periodic/non-random/compare/epsi-0.1/conc_2__v__posi_1.pdf}
    and
    \subref{fig:mm_periodic/non-random/compare/epsi-0.1/perm_2__v__posi_1.pdf}
    The case $N^1=10$, so that $\varepsilon=0.1$ (see \eqref{eq:nm_definition-of-delt-and-epsi} in Part \ref{part:A network model for filtration}).
    \subref{fig:mm_periodic/non-random/compare/epsi-0.05/conc_2__v__posi_1.pdf}
    and
    \subref{fig:mm_periodic/non-random/compare/epsi-0.05/perm_2__v__posi_1.pdf} 
    $N^1=20$, $\varepsilon=0.05$.
    \subref{fig:mm_periodic/non-random/compare/epsi-0.025/conc_2__v__posi_1.pdf} 
    and
    \subref{fig:mm_periodic/non-random/compare/epsi-0.025/perm_2__v__posi_1.pdf} 
    $N^1=40$, $\varepsilon=0.025$.
    }
    \label{fig:mm_periodic/non-random/flow/comparison}
\end{figure}


\subsubsection{Velocity and reactivity}
We plot the velocity of the flow and reactivity of the filter in 
Figure \ref{fig:mm_periodic/non-random/velocity-and-reactivity}. 
Comparing Figure \ref{./figures/multiscale/periodic/non-random/compare/epsi-0.025/velo_1__v__time_1.pdf}
with Figure \ref{fig:mm_periodic/non-random/flow/psi_2__v__time_t.pdf}, 
the first thing we spot is that these are identical for a mono-dispersed filter (for $\alpha=1$). 
This is because, as we have shown, the permeability and adhesivity are identical in this case. 
As time increases, these functions decrease monotonically. 
As filtration proceeds, the filter becomes less permeable because more particles have deposited in pores, which 
has decreased the conductance of each pore, which decreases the flow rate.
As deposition occurs, pores become smaller, 
which means that pore walls have less surface area for particles to adhere to, which decreases the reactivity (i.e., the ability of the filter to capture particles).
In Figure \ref{./figures/multiscale/periodic/non-random/compare/epsi-0.025/velo_1__v__time_1.pdf}, 
we see that (when 
% $\varepsilon=\mathcal{O}(10^{-2})$, 
$\varepsilon=0.025$) 
the agreement of the macroscale Darcy velocity with the average velocity in the network 
solution (see \eqref{eq:nm_definition-of-av-velo}) is good.
Notice
% , from Figure \ref{fig:mm_periodic/non-random/flow/psi_2__v__time_t.pdf}, 
that when the filter is monodispersed, the reactivity is uniform across the filter, 
which is not the case in poly-dispersed filters.
We will discuss such filters in the second part of this section.
\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/compare/epsi-0.025/velo_1__v__time_1.pdf}
         \caption{}
         \label{./figures/multiscale/periodic/non-random/compare/epsi-0.025/velo_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/non-random/flow/psi_2__v__time_1.pdf}
         \caption{}
         \label{fig:mm_periodic/non-random/flow/psi_2__v__time_t.pdf}
     \end{subfigure}
    \caption{Graphs showing how the velocity, $u$, and reactivity, $\psi$, vary with time $t$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$, $\varepsilon=0.025$.
    \subref{fig:mm_periodic/non-random/preprocess/delt_5_v__conc_max_or_tot_1.pdf} 
    The Darcy velocity, $u$, 
    as a function of time, $t$.
    The solid blue line shows the macroscale solution. 
    The dashed orange line shows the average of the microscale solution (see \eqref{eq:nm_definition-of-av-velo} in Part \ref{part:A network model for filtration}).
    \subref{fig:mm_periodic/non-random/flow/psi_2__v__time_t.pdf} 
    The reactivity, $\psi$, 
    as a function of time, $t$.
    }
    \label{fig:mm_periodic/non-random/velocity-and-reactivity}
\end{figure}


\subsection{Random networks: Poly-dispersed pores}
\label{subsec:mm_Polydispersed pores}
We will now explore the case where the initial pore conductances are given by (\ref{eq:mm_cond-pdf_reduced}b), 
so that they are drawn randomly from a log-normal distribution.
In this case, each initialisation of the underlying network gives rise to a different, independent and identically distributed set of pore conductances. 

\subsubsection{Conductance and pressure drop}
We present the solutions for the pore conductances and pore pressure drops 
as functions of the mass concentration flux in Figure \ref{fig:mm_periodic/random/preprocess} (compare this with the solutions 
for a mono-dispersed filter in Figure \ref{fig:mm_periodic/non-random/preprocess}). 
Observing Figure \ref{fig:mm_periodic/random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}, 
we begin by noting that poly-dispersed solutions (coloured lines) are more complicated than mono-dispersed solutions (black lines). 
We see that the conductances of pores that are parallel (blue and green lines) and perpendicular (orange and red lines) to the direction of the macroscale pressure gradient behave differently from each other. 
The conductance of parallel pores decreases more rapidly than that of perpendicular pores. 
In Figure \ref{fig:mm_periodic/random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}, we see that this 
is because parallel pores have much larger pressure drops than perpendicular pores, 
which means that particles are driven through parallel pores more rapidly so that more particles deposit inside them per unit time.
We see that the parallel and perpendicular pressure deltas are distributed about the corresponding 
pressure drop values in the mono-dispersed case. 
It follows that the expected pressure drops in pores that are parallel and perpendicular to the flow direction are $\mathbb{E}[\Delta_{ij}^{r^1}]=1$ and $\mathbb{E}[\Delta_{ij}^{r^1}]=0$, respectively. 
As filtration proceeds, deposition causes the pressure drops on flow-parallel pores (dashed and solid blue lines) to converge to one another. 
This causes the pressure drops in flow-perpendicular pores to approach zero. 
The result is that, regardless of the initial conductances, as time proceeds, the behaviour of the random network approaches that of the mono-dispersed network, in which there is no microscale flow in pores that are perpendicular to the macroscale pressure gradient.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/preprocess/cond_5_v__conc_max_or_tot_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/preprocess/delt_5_v__conc_max_or_tot_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the pore conductance, $G\indx$, and pore pressure drop in the $x^1$ direction, $\Delta_{ij}^{r^1}$, vary with the mass concentration flux, $f$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$.
    Each coloured line corresponds (with colour and dashing) to an edge in 
    the cell shown in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}.
    Each black line shows the solution in the non-random network case (see Figure \ref{fig:mm_periodic/non-random/preprocess}) for comparison.
    That is, the dashed black line shows the solution on pores that are parallel 
    to the $x^1$ direction for a non-random network (the blue and green edges in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}).
    The dotted black line shows the solution on pores that are perpendicular 
    to the $x^1$ direction for a non-random network (the orange and red pores in Figure \ref{fig:mm_periodic/drawings/coloured-cell.pdf}). \subref{fig:mm_periodic/random/preprocess/cond_5_v__conc_max_or_tot_1.pdf} 
    The pore conductance, $G\indx$, 
    as a function of the mass concentration flux, $f$. \subref{fig:mm_periodic/random/preprocess/delt_5_v__conc_max_or_tot_1.pdf} 
    The $x^1$ component of the pore pressure drop, $\Delta_{ij}^{r^1}$, 
    as a function of the mass concentration flux, $f$.
    }
    \label{fig:mm_periodic/random/preprocess}
\end{figure}

\subsubsection{Permeability and adhesivity}
In Figure \ref{fig:mm_periodic/random/flow/perm-and-depo} we plot the permeability and adhesivity as 
functions of position and the mass concentration flux (compare with the mono-dispersed case in Figure \ref{fig:mm_periodic/non-random/flow/perm-and-depo}).
We illustrate (dashed black lines) how, using the numerical scheme discussed in Section \ref{sec:mm_Solution methodology}, 
we are able to construct permeability and adhesivity functions that depend on position and time from discrete 
representations of these parameters as a function of the mass concentration flux. 
We see that when the filter is poly-dispersed, the permeability and adhesivity are no longer equal, 
as they were in the mono-dispersed case.
For example, in Figure \ref{fig:mm_periodic/random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}, the permeability drops below the mono-dispersed value and the adhesivity consistently lies above it. 
That is, particles are driven through the filter less rapidly (than in the mono-dispersed case), because the filter is more adhesive.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/flow/cmax_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/flow/cmax_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/flow/perm_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/flow/perm_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/flow/depo_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/flow/depo_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the permeability, $k^{11}$, and adhesivity, $j^1$, vary with the mass concentration flux, $f$, 
    and position, $x^1$, at various times, $t$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$.
    The values of $t$ are 
    $0$ (blue), 
    $0.25$ (orange), 
    $0.5$ (green), 
    $0.75$ (red), 
    $1$ (purple).
    The arrows show the direction of increasing time.
    \subref{fig:mm_periodic/non-random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf} 
    The permeability (blue), $k^{11}$, and adhesivity (orange), $j^1$, as functions of $f$. 
    The black dotted line shows the solution in the non-random network case, for comparison. 
    % The dashed black line shows an extrapolation at 
    % $f=0.45$ (which corresponds to position $x^1\approx0.2$ at time $t=0.75$) to determine 
    % the permeability and adhesivity, $k^{11}(f\approx0.5)\approx0.56$, $j^{1}(f\approx0.5)\approx0.66$.
    \subref{fig:mm_periodic/non-random/flow/cmax_2__v__posi_1.pdf} 
    The mass concentration flux, $f$, as a function of position, $x^1$, 
    at various times, $t$. 
    The dashed black line shows an evaluation at 
    position $x^1=0.2$ and time $t=0.75$ to determine the mass concentration flux here, $f(x^1=0.2,t=0.75)\approx0.45$.
    \subref{fig:mm_periodic/non-random/flow/perm_2__v__posi_1.pdf} 
    The permeability, $k^{11}$, as a function of position, $x^1$, 
    at various times, $t$.
    The dashed black line shows an evaluation at $x^1=0.2$ and $t=0.75$ to confirm 
    that $k^{11}(x^1=0.2,t=0.75)\approx0.56$.
    \subref{fig:mm_periodic/non-random/flow/depo_2__v__posi_1.pdf} 
    The adhesivity, $j^{1}$, as a function of position, $x^1$, 
    at various times, $t$.
    The dashed black line shows an evaluation at $x^1=0.2$ and $t=0.75$ to confirm 
    that $j^{1}(x^1=0.2,t=0.75)\approx0.66$.
    }
    \label{fig:mm_periodic/random/flow/perm-and-depo}
\end{figure}

\subsubsection{Concentration and conductance}
Next, in Figure \ref{fig:mm_periodic/random/flow/comparison}, 
we display a comparison between the macroscale and microscale solutions 
for the concentration (left) and conductance (right)  (compare with the mono-dispersed 
case in Figure \ref{fig:mm_periodic/non-random/flow/comparison}) given random initial pore conductances. 
In Figure \ref{fig:mm_periodic/random/compare/epsi-0.05/conc_2__v__posi_1.pdf}, 
we see that the qualitative behaviour of the two concentration solutions agrees. 
Similarly, in Figure \ref{fig:mm_periodic/random/compare/epsi-0.05/perm_2__v__posi_1.pdf}, 
we see that the permeability behaves qualitatively like the average of the 
microscale conductances that result from the log-normally distributed initial values. 
Note that we have used \emph{different} (random) initial pore conductances for the microscale and multiscale simulations, which fully explains differences in the quantitative behaviour of the two solutions (even in the limit $\varepsilon \rightarrow 0$). When the same initial conductances are used, we find, as expected, that the microscale and multiscale solutions agree quantitatively, as in the mono-dispersed case.
Instead, in the poly-dispersed case, we expect the \emph{statistics} (such as the mean and standard deviation) of the distributions of the microscale and multiscale solutions to agree in the limit $\varepsilon\rightarrow 0$ (when the initial pore conductances are chosen randomly). 
We explore this hypothesis in the next section. 


\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/compare/epsi-0.05/conc_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/compare/epsi-0.05/conc_2__v__posi_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/compare/epsi-0.05/perm_2__v__posi_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/compare/epsi-0.05/perm_2__v__posi_1.pdf}
     \end{subfigure}
    \caption{Graphs showing how the macroscale and microscale (left) concentrations, $c$ and $\bar{C}$ (see \eqref{eq:nm_definition-of-av-conc} in Part \ref{part:A network model for filtration}), 
    and (right) conductances (see \eqref{eq:nm_definition-of-av-cond}),  $k^{11}$ and $\bar{G}$, 
    vary with position, $x^1$, at various times, $t$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$, $\varepsilon=0.1$.
    The values of $t$ are 
    $0$ (blue), 
    $0.25$ (blue), 
    $0.5$ (green), 
    $0.75$ (red), 
    $1$ (purple).
    Dots show the average network quantities, interpolated with correspondingly coloured dashed lines. 
    Correspondingly coloured solid lines show the effective quantities. 
    The arrows show the direction of increasing time.
    \subref{fig:mm_periodic/random/flow/cmax_2__v__posi_1.pdf} 
    The macroscale and microscale concentrations, $c$ and $\bar{C}$, as functions of position, $x^1$, 
    at various times, $t$.
    \subref{fig:mm_periodic/random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf} 
    The macroscale and microscale conductances, $G\indx$ and $\bar{G}$, as functions of position, $x^1$, 
    at various times, $t$.
    }
    \label{fig:mm_periodic/random/flow/comparison}
\end{figure}

\subsubsection{Velocity and reactivity}
Before we explore the statistics of the multiscale solution, we present the 
velocity and adhesivity 
in Figure \ref{fig:mm_periodic/random/velocity-and-reactivity} (compare this with the mono-dispersed case 
in Figure \ref{fig:mm_periodic/non-random/velocity-and-reactivity}). 
In Figure \ref{fig:mm_periodic/random/compare/epsi-0.05/velo_1__v__time_1.pdf}, 
we see, as expected, that the poly-dispersed and mono-dispersed solutions exhibit the same qualitative behaviour
(but not quantitative behaviour, even for small $\varepsilon$, due to differences in initial conductances). 
In Figure \ref{fig:mm_periodic/random/flow/psi_2__v__time_t.pdf}, 
we see that, unlike in the mono-dispersed case (see Figure \ref{fig:mm_periodic/non-random/flow/psi_2__v__time_t.pdf}), 
the reactivity is not uniform across the filter. 
Instead, it is largest near the inlet, and decreases with depth. It is proportional to the ratio of the adhesivity and the permeability, and so, at the inlet,
the reactivity increases before decreasing again, 
because the decrease of the permeability is significantly faster than that of the adhesivity here at early times (see Figures \ref{fig:mm_periodic/random/preprocess/perm_prep_3__depo_prep_2__v__conc_max_or_tot_1.pdf}, \ref{fig:mm_periodic/random/flow/perm_2__v__posi_1.pdf}, and \ref{fig:mm_periodic/random/flow/depo_2__v__posi_1.pdf}).
That is, the filter captures more particles nearer the inlet, so that by the time fluid reaches pores further down 
the filter, there are fewer particles in the suspension to be caught. 
As time increases, the reactivity decreases, since smaller pores
% that have been deposited in 
can capture fewer particles, due to their decreased surface area.
\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/compare/epsi-0.05/velo_1__v__time_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/compare/epsi-0.05/velo_1__v__time_1.pdf}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/periodic/random/flow/psi_2__v__time_1.pdf}
         \caption{}
         \label{fig:mm_periodic/random/flow/psi_2__v__time_t.pdf}
     \end{subfigure}
    \caption{Graphs showing how the velocity, $u$, and reactivity, $\psi$, vary with time $t$. 
    The parameters are $N=4$, $\alpha=1$, $\beta=1$.
    \subref{fig:mm_periodic/random/preprocess/delt_5_v__conc_max_or_tot_1.pdf} 
    The Darcy velocity, $u$, 
    as a function of time, $t$.
    The solid blue line shows the macroscale solution. 
    The dashed orange line shows the average of the microscale solution (see \eqref{eq:nm_definition-of-av-velo} in Part \ref{part:A network model for filtration}).
    \subref{fig:mm_periodic/random/flow/psi_2__v__time_t.pdf} 
    The reactivity, $\psi$, 
    as a function of time, $t$.
    }
    \label{fig:mm_periodic/random/velocity-and-reactivity}
\end{figure}


% \subsection{Random periodic networks}
% Match the random periodic network solution to the equivalent effective solution.
In this section, we have demonstrated the solution of distinct initialisations of the multiscale model, \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_cond-pdf_reduced}, and shown that solutions match those of the underlying periodic network in the limit $\varepsilon\rightarrow 0$ (when the initial conductances used are the same between the two models).  
% Agreement is quantitative when the initial conductances are the same, which models mono-dispersed pores, 
% and qualitative when the initial network is random, which models poly-dispersed pores.
In the next section, we will view the solutions of the multiscale model as distributions 
that result from realisations of many distinct realisations of the initial random network.
We will see how sampling the resulting permeability and adhesivity distributions may lead to approximation of microscale solutions 
on networks that are non-periodic.

% We have shown that, when the conductances are random, 
% the macroscale solution of aparticular initialisation doe snot match the microscale oor adifferent initiaisation. 
% This makes sense, as it is the statsitcis of these two soluitons that match. 
% We are interested whether these solutions match `on average'.
% So we will now explore the statistics of the macrosclae solution.
% For this we require to repeat the solution and obsevre the distribution of solutions. 


\section{Non-periodic networks}
\label{sec:mm_Non-periodic networks}
% We will explore the statistics of the solutions distributions in the context of non-periodic networks.
% We use this as an opportunity to illustare a different example of a deposition rule. This loosely corresponds to blcking as opposed to deposition. The analogy is with when the feede is filled with large particles and when the concentration is large enough we assume that they have coagulated in such a  way that this blocks the pore.

% In the this section, we will explore the distributions of the effective parameters as functions of the distribution of the initial conductances. 
% In particular, we will evaluate the effective permeability, $k^{11}$, defined in \eqref{eq:mm_permeability_reduced}, and the effective 
% adhesivity, $j^1$, defined in \eqref{eq:mm_adhesivity_reduced}, as functions of the conductance and adherence values, $G\indx$ and $A\indx$, in \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic}, that result from various distributions of the initial-conductance values, $G_{ij}^{\bm{r}0}$, which also enter the system through \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic}.
% We will use this analysis to help inform our solution of the reduced system, \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:reduced_definition-of-alpha}, which we defined in this section.


In the previous sections, we showed that the behaviour of the solution of the effective system matches the behaviour of the solution of the periodic network system from which it is derived, in the limit $\varepsilon\rightarrow 0$ (see Step 2 in Figure \ref{fig:stats-motivation_1} for an illustration).
% Furthermore, we established that the network structure enters the effective system via the formulae for the
% effective permeability and effective adhesivity, \eqref{eq:mm_permeability_reduced} and \eqref{eq:mm_adhesivity_reduced}, since these depend on the conductances of the edges in the repeating cell and the solutions of the linear cell problem, \eqref{eq:mm_cell-problem_reduced}, which is of size $N$. 
The hope, however, is that the solution of the effective system also matches the solution of a system defined on a non-periodic network,
which we interpret here as a single cell of size $N\rightarrow\infty$.
To achieve this, the repeating cell used for homogenisation must be chosen carefully so that it is \emph{representative} of the non-periodic network, by which we mean that the statistical properties of the conductances, 
and the degree of connectivity of the nodes and edges, which together dictate the structure of the cell, must match those of the entire non-periodic network (see Step 1 in Figure \ref{fig:stats-motivation_1}).
This intuition motivates the question: how should the repeating cell of size $N$ be chosen so that the solution of the resulting effective system matches that of a corresponding system defined on a non-periodic network of size $N\rightarrow \infty$?
\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.65\textwidth]{./figures/multiscale/stats/setup/network-to-effective.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{
        A diagram to illustrate the derivation of a multiscale effective system from one defined on a non-periodic network.
        Step 1 is the construction of a periodic network by repeating a cell that has properties that are representative of the non-periodic network.
        Step 2 is the derivation of an effective system from a system defined on a network, which was detailed in the previous chapter.} 
        \label{fig:stats-motivation_1}
    \end{center}
\end{figure}

To answer this question, we will assess the statistical properties of the distributions of the effective parameters, $k^{11}$ and $j^1$, as functions of the statistical properties of the distributions of conductances, $G_{ij}^{\bm{r}}$, in the cell.
Note, from the formulae \eqref{eq:mm_permeability_reduced} and \eqref{eq:mm_adhesivity_reduced}, that the permeability and adhesivity are both nonlinear with respect to the conductances, since the cell solutions, $W_i^m$, upon which the effective parameters also depend, are solutions of the cell problem, \eqref{eq:mm_cell-problem_reduced}, which is itself a function of the edge conductances. 
This complex dependence of $k^{11}$ and $j^1$ upon $G_{ij}^{\bm{r}}$ via $W_i^m$ 
% (for both intra- and inter-cell $\bm{r}$) 
implies that, even for a network with i.i.d. (independent and
identically distributed) edge conductances, the expectations of the permeability and adhesivity
are not necessarily the same as the permeability and adhesivity of the network with all edge conductances equal to the mean edge conductance, $\bar{G}$, which we will refer to as the mean network.
%[I am trying to explain that to get $W$ one must solve the cell problem, which itself depends on $G$, and so $G$ affects $k$ and $j$ in some nontrivial way...we've found (for example, in our supervision a few weeks ago)  that this fact limits the work that can be done explicitly... does this need further explanation?]. 

We focus our analysis on two properties of the cell: size and structure. 
We vary these independently of each other
(see Figure \ref{fig:stats-motivation_2} for an illustration).
For each structure, and for each size, $N$, we define a probability distribution, $\mathcal{G}$, from which initial conductances, $G_{ij}^{\bm{r}0}$, are drawn. 
We then carry out the process of parameter pre-calculation outlined in Section \eqref{sec:mm_Solution methodology} 
many times to find the resulting distributions for the parameters. 
That is, we sample $\mathcal{G}$ to specify a particular set of $G_{ij}^{\bm{r}0}$ in the cell,
% , and hence a particular set of conductances and adherences.
% , using \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic}. 
and solve the cell problem, \eqref{eq:mm_cell-problem_reduced}, to find the set of corresponding cell solutions, $W_i^m$.
We then use \eqref{eq:mm_permeability_reduced} and \eqref{eq:mm_adhesivity_reduced} 
to calculate $k^{11}$ and $j^1$ for these conductance values.
Since the initial conductances are sampled at random, we repeat this process $S\gg1$ times. 
This process results in discretisations of the probability distributions for the permeability and adhesivity, $\mathcal{K}$ and $\mathcal{J}$, say, which depend on the distribution of the initial conductances, $\mathcal{G}$.
For each structure, we vary $N$ and thus observe how $\mathcal{K}$ and $\mathcal{J}$ vary with the cell size, $N$.
We find the expected values of the parameters, $\mathbb{E}[k^{11}]$ and $\mathbb{E}[j^1]$, and their standard deviations $\mathbb{S}[k^{11}]$ and $\mathbb{S}[j^1]$, by sampling $\mathcal{K}$ and $\mathcal{J}$, 
and compare these four statistics across the different structures.

% Firstly, given a structure, we explore the behaviour of the distributions of the effective parameters as $N$ increases.
% Secondly, we vary the structure (by varying the coordination number, which is the average number of connections to a given node), and compare the statistical properties of the effective parameters for these structures 


\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.8\textwidth]{./figures/multiscale/stats/setup/correct-cell.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{A diagram to illustrate the selection of a repeating cell for a periodic network that is representative of a non-periodic network.
        The left arrow illustrates the choice of the cell size, $N$.
        All cells on the left have the same structure, but have different sizes. 
        The right arrow illustrates the choice of the cell structure. 
        Both cells on the right have the same size and a average degree of six, but different connectivity.} 
        \label{fig:stats-motivation_2}
    \end{center}
\end{figure}


We begin by considering a four-regular structure, which, we recall,
is a cell where each node connects to exactly four other nodes, to form a regular grid in two dimensions.
However, networks in porous media usually have a larger average degree (the average number of other nodes to which each node connects) \cite{xiong2016review}. 
This motivates our study of a \emph{six-regular} structure, which is a cell where each node connects to exactly six other nodes.
The physical networks that motivate this work are irregular in the sense that the size of the neighbourhood of each node varies \cite{xiong2016review}. 
This motivates our analysis of a \emph{six-irregular} cell, which has an average degree of six, although some nodes connect to fewer than six nodes, while others connect to more.

We will keep two properties of all three structures consistent. Firstly, by scaling the cell dimensions, $l^1$ and $l^2$, if necessary, we will ensure that the node density, 
\begin{align}
    \varrho = \frac{N}{l^1l^2},
    \label{eq:results_definition-of-node-density}
\end{align}
which is the number of nodes divided by the area of the cell, is unity. 
% so that 
% \begin{align}
%     \varrho = 1.
%     \label{eq:results_keep-density-constant}
% \end{align}
Secondly, noting that
\begin{align}
    \hat{R}\indx=\frac{1}{\hat{G}\indx},
        \label{eq:results_definition-of-resistance}
\end{align}
where $\hat{G}\indx$ and $\hat{R}\indx$ are the conductance and resistance of pore $ij\bm{r}$ per unit length, 
we will ensure, by scaling $G_{ij}^{\bm{r}0}$ if necessary, that the mean of $\mathcal{G}$ is consistent, so that the mean pore resistance per unit length is the same in all structures.
These two controls ensure that the experiment results are comparable.

% We 
% To fully determine the reduced system, it remains to specify $G\indx$ and $A\indx$ by choosing models for the conductance and adherence,  respectively. 
% [Link to previous section, where we discuss which blocking rules lead to permeability and adhesivity parameters that can be found before time-stepping]
% As a second illustrative example,
\subsection{Blocking}

To demonstrate the generality of our framework, 
we will carry out this analysis using an alternative model for particle--pore interactions, and the resulting changes to the connectivity of the network. 
We assume that each pore becomes clogged instantaneously at the first time at which the concentration within it exceeds some threshold.
At this time, the conductance of the pore is instantaneously reduced to zero and the particles that clogged the pore are removed from the flow.
The analogy is with the `blocking' mechanism discussed in the size-structured model in Part \ref{part:A size-structured model for filtration}, 
where pore radii are instantaneously reduced to zero after interactions with large particles.

\paragraph{Conductance and adherence}
To model blocking in the present framework, we define
\begin{align}
    c^{\text{max}}(\bm{x},t) = \max_{s \le t}(c(\bm{x},s)),
    \label{eq:mm_definition-of-max-conc}
\end{align}
which is the maximum concentration observed at point $\bm{x}$ up to time $t$.
We then assume the conductance model
\begin{align}
    G_{ij}^{\bm{r}}(\bm{x},t) 
    &= 
    G_{ij}^{\bm{r} 0}\Big(1-H\big(t - T_{ij}^{\bm{r},\text{block}}\big)\Big)
    \label{eq:mm_pore-conservation_non-periodic}
    \\
    &=
        \begin{cases}
            G_{ij}^{\bm{r} 0} & c^{\text{max}}(\bm{x},t) \le \alpha G_{ij}^{\bm{r} 0}, 
            \\
            0 & c^{\text{max}}(\bm{x},t) > \alpha G_{ij}^{\bm{r} 0},
        \end{cases}
        \label{eq:mm_pore-conservation_non-periodic_pre-calc}
\end{align}
and the associated adherence model 
\begin{align}
    A_{ij}^{\bm{r}}(\bm{x},t) 
    = 
    % \chi_{ij}^{\bm{r}}(\bm{x},t)
    % = 
        \begin{cases}
            1 & c^{\text{max}}(\bm{x},t) = \alpha G_{ij}^{\bm{r} 0}, 
            \\
            0 & c^{\text{max}}(\bm{x},t) \ne \alpha G_{ij}^{\bm{r} 0}.
        \end{cases}
        \label{eq:mm_adherence_non-periodic}
\end{align}
We recall that $G_{ij}^{\bm{r}0}$ is the initial conductance, and define $T_{ij}^{\bm{r},\text{block}}$ to be the block time, at which the maximum concentration exceeds the threshold in pore $ij\bm{r}$, when the condition $c^{\text{max}} > \alpha G_{ij}^{\bm{r} 0}$ is first satisfied.
We replace the pore conservation equation that corresponds to deposition, \eqref{eq:mm_pore-conservation_reduced},
with \eqref{eq:mm_pore-conservation_non-periodic}. 
This time, we pre-calculate the pore conductances, and thus the permeability and adhesivity, by replacing 
\eqref{eq:mm_pore-conservation_modified} with \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} to find $G\indx(c^{\text{max}})$ (rather than $G\indx(f)$).
The adherence is no longer constant (compare with $\alpha$ in \eqref{eq:mm_definition-dimles-parameters_sum_nodim_1}), and instead depends on the indices of the pore, $ij\bm{r}$, the position of the pore, $\bm{x}$, and time, $t$.

To gain an intuition for this modified multiscale model, in which blocking occurs instead of deposition,
% , \eqref{eq:mm_particle-conservation_reduced}--\eqref{eq:mm_adherence_non-periodic}, 
consider the point $\bm{x}$ at time $t$, which contains a cell in the limit $\varepsilon \rightarrow 0$.
Before $T_{ij}^{\bm{r},\text{block}}$, the concentration of particles in the flow in pore $ij\bm{r}$ has not exceeded the threshold, so the pore conducts all particles through it. 
At the block time, $T_{ij}^{\bm{r},\text{block}}$, the maximum concentration, $c^{\text{max}}$, defined in \eqref{eq:mm_definition-of-max-conc}, first exceeds the threshold, and the adherence of pore $ij\bm{r}$, $A_{ij}^{\bm{r}}$, becomes non-zero according to \eqref{eq:mm_adherence_non-periodic}.
This leads to an instantaneous non-zero adhesivity, $j^1$, according to \eqref{eq:mm_adhesivity_reduced}, which in-turn causes an instantaneous non-zero reactivity, $\psi$, according to \eqref{eq:mm_reaction_reduced}.
This results in the instantaneous removal of free particles from the system, since the reaction term in the advection--reaction equation, \eqref{eq:mm_particle-conservation_reduced}, becomes non-zero. 
At the same time, the conductance of this pore, $G_{ij}^{\bm{r}}$, switches from its initial value to zero, and remains there for the duration of the process, according to \eqref{eq:mm_pore-conservation_non-periodic_pre-calc}.
The linear cell problem, \eqref{eq:mm_cell-problem_reduced}, 
% has the form $\bm{A}\bm{W}^m=\bm{b}$ for each dimension $m \in \{1,2\}$, 
% and elements of the matrix $\bm{A}$ and the vector $\bm{b}$ are 
depends on the pore conductances, $G\indx$. 
Consequently, this instantaneous change in conductance causes an instantaneous change in its solution, the cell solution $W_i^m$, which, in turn, causes changes to the permeability, $k^{11}$, according to \eqref{eq:mm_permeability_reduced}, and to the adhesivity, $j^1$, according to \eqref{eq:mm_adhesivity_reduced}.
Changes to the permeability then affect the velocity of the flow, given by $u^1$ and $u^2$, due to \eqref{eq:mm_darcy_reduced}.
Consequently, the rate of the transport of particles is altered, since the $x^1$-directional velocity, $u^1$, appears in the advective term of the advection--reaction equation, \eqref{eq:mm_particle-conservation_reduced}.
Since, in the limit $\varepsilon \rightarrow 0$, the spatial point $\bm{x}$ contains the entire cell, changes to the conductance and adherence of all edges $ij\bm{r}$ occur here at the same time. 
% As a result, the process outlined above occurs here for each pore $ij\bm{r}$ simultaneously, which affects the solution in a highly complex way.

\paragraph{Threshold parameter}
The threshold concentration above which a pore blocks is dictated by $\alpha$ in \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic}. This parameter contains information about the size or coagulation of particles, for example.
As an illustrative case, we set $\alpha$ so that $\alpha^{-1}$ is the mean, $\bar{G}$, of the distribution from which initial edge conductances are drawn,
\begin{align}
    \alpha = \frac{1}{\bar{G}}.
    \label{eq:reduced_definition-of-alpha}
\end{align}
% and analyse the model in dimensional terms, to demonstrate the solution dependence on $\bar{G}$.
To see why this is a useful test case, we substitute \eqref{eq:reduced_definition-of-alpha} into the conductance and adherence models, \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic}. 
We find that these imply that pore $ij\bm{r}$ at position $\bm{x}$ remains conductive at time $t$ if the initial conductance of the pore,
$G_{ij}^{\bm{r}0}$, satisfies $G_{ij}^{\bm{r}0}\ge c^{\text{max}}(\bm{x},t)\bar{G}$, but blocks if  $G_{ij}^{\bm{r}0}<c^{\text{max}}(\bm{x},t)\bar{G}$.
So the probability of the event $E_{ij}^{\bm{r}}(\bm{x},t)$, which we define to be `pore $ij\bm{r}$ in the cell at position $\bm{x}$ blocks at time $t$',
is the probability that $G_{ij}^{\bm{r}0}$ lies to the left of the scaled mean, $c^{\text{max}}(\bm{x},t)\bar{G}$. 
That is,
\begin{equation}
    \mathbb{P}[E_{ij}^{\bm{r}}(\bm{x},t)] = \text{cdf}(c^{\text{max}}(\bm{x},t)\bar{G}),
    \label{eq:reduced_prob-of-block}
\end{equation}
where $\text{cdf}(G_{ij}^{\bm{r}0}=g)$ is the cumulative density function of the initial conductance distribution, $\mathcal{G}$, at any value $g$, which returns the integral of the probability density function (PDF) (which we recall from \eqref{eq:mm_cond-pdf_reduced}
is denoted by $\mathscr{g}$) to the left of $g$.


\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/reduced/prob_density__v__G_init.pdf}
         \caption{}
         \label{fig:reduced_alpha_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/reduced/cdf__v__G_init.pdf}
         \caption{}
         \label{fig:reduced_alpha_b}
     \end{subfigure}
    \caption{Graphs showing how the PDF and CDF of the initial conductance distribution, $\mathcal{G}$, 
    vary with $G_{ij}^{\bm{r}0}$. 
    The value of the scaled mean, $c^{\text{max}}\bar{G}$, is shown in dashed coloured lines for $c^{\text{max}} \in \{0,1/4,1/2,3/4,1\}$.
    \subref{fig:reduced_alpha_a}
    The pdf of $\mathcal{G}$.
    \subref{fig:reduced_alpha_b}
    The CDF of $\mathcal{G}$.
    %The cumulative density function of the initial conductance distribution is shown in a solid blue line.
    % The value of the scaled mean, $c^{\text{max}}\bar{G}$, is shown in dashed coloured lines for $c^{\text{max}} \in \{0,1/4,1/2,3/4,1\}$.
    The cumulative densities corresponding to the various scaled means are extrapolated using correspondingly coloured dotted lines.}
    \label{fig:reduced_alpha}
\end{figure}


Examples of the PDF and CDF of $\mathcal{G}=\text{Log-normal}$ are shown in Figure \ref{fig:reduced_alpha}, with values of $c^{\text{max}}(\bm{x},t)\bar{G}$ shown with dashed lines.
In Figure \ref{fig:reduced_alpha_a}, we see, for example, that if $G_{ij}^{\bm{r}0}$ lies to the left of the dashed blue line, then pore $ij\bm{r}$ blocks when a maximum concentration value of $c^{\text{max}}=1$ is observed.
% There is less probability density to the left of the other dashed lines, 
% so the pore is less likely to block for these values of $c^{\text{max}}$.
% For $c^{\text{max}}=1$, in \ref{fig:reduced_alpha_a}, we see that the probability that pore $ij\bm{r}$ blocks is the probability that $G_{ij}^{\bm{r}0}$ lies to the left of the dashed blue line, which represents the mean, $\bar{G}$.
Using Figure \ref{fig:reduced_alpha_b}, we find the intersection of $\bar{G}$ with the CDF, shown in a solid blue line, to find that the corresponding probability in the case $c^{\text{max}}$ (dotted blue line), for example, is around $0.56$.
% The cumulative density of $c^{\text{max}}\bar{G}$, and therefore the value $\mathbb{P}[E\indx]=\text{cdf}(c^{\text{max}}\bar{G})$, increases as $c^{\text{max}}$ increases, and is bounded by $0$, when $c^{\text{max}}=0$ (dashed orange line), and $\text{cdf}(\bar{G})=0.55\approx 1/2$, when $c^{\text{max}}=1$ (dashed blue line).
Physically, if the point $\bm{x}$ has received the maximum concentration of particles, any pore with a conductance less than the mean blocks.
On the other hand, for $c^{\text{max}}=0$, in \ref{fig:reduced_alpha_a} we see that the probability that pore $ij\bm{r}$ blocks is the probability that $G_{ij}^{\bm{r}0}$ lies to the left of the dashed orange line, which is at zero.
Using Figure \ref{fig:reduced_alpha_b}, we find the intersection of this line with the CDF and find that the corresponding probability (dotted orange line) is zero.
Physically, if the point $\bm{x}$ has not received any particles, then no pores here block.
Note that the maximum concentration to have reached the point $\bm{x}$ at time $t$ is bounded by these two cases, $0\le c^{\text{max}}(\bm{x},t)\le 1$, and further that we will always consider initial conductance distributions that have $\text{cdf}(\bar{G})\approx 1/2$.
Therefore, since the number of blocked pores in a cell is approximately equal to the number of total pores multiplied by the probability of each pore blocking, it follows that the choice $\alpha^{-1}=\bar{G}$ in \eqref{eq:reduced_definition-of-alpha} ensures that, in a cell with $E$ edges initially, the number of pores that are blocked will range from zero, when no particles have passed through it, to $\text{cdf}(\bar{G})E \approx E/2$, when the concentration of particles is as large as possible.



\subsection{Four-regular structure}
To illustrate our method, we begin by considering a four-regular structure, so that the degree of each node is four. An example of some cells with four-regular structure containing various numbers of nodes, $N$, are shown in Figure \ref{fig:exp-param-dist_4-reg_setup_2}.


\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.9\textwidth]{./figures/multiscale/stats/4reg/setup/cells_4-reg.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{A diagram of cells of various sizes, $N$, with four-regular structure. 
        The cell dimensions, $l^1$ and $l^2$, are 
        set to be $\sqrt{N}$ so that the node density, $\varrho$, satisfies $\varrho=1$ in all cells.
        Consequently, the length of each pore $ij\bm{r}$ is $L\indx=1$.
        We arbitrarily choose the $x^1$ direction to be from left to right. 
        Thus, horizontal edges in this diagram are
        parallel to the pressure gradient, and thus the flow. We call these flow-parallel edges. 
        Vertical edges are transverse to the 
        pressure gradient, and we call these flow-transverse edges.} 
        \label{fig:exp-param-dist_4-reg_setup_2}
    \end{center}
\end{figure}


To satisfy the constraint of unit node-density, we choose $l^1=l^2=\sqrt{N}$ for a given $N$. 
This scales the area of the cell, $l^1l^2$, so that $\varrho=1$ in \eqref{eq:results_definition-of-node-density}, which  ensures spatial invariance between cells with different $N$ (see Figure \ref{fig:exp-param-dist_4-reg_setup_2}).

We draw i.i.d. initial edge conductances from a log-normal distribution, $\mathcal{G}$,
so that $G_{ij}^{\bm{r}0} \sim \text{log}\text{Norm}(\mu,\sigma^2)$.
We choose the parameters of $\mathcal{G}$ to simplify comparison with a related study by Chapman \emph{et al.}, 
\cite{chapman2021homogenization}, and so set the mean of the normal distribution associated with $\mathcal{G}$ to be $\mu=0.5$.
We are interested in the dependence of $\mathcal{K}$ and $\mathcal{J}$ on the variance of $\mathcal{G}$, and thus present scenarios where the standard deviation of the associated normal distribution is either $\sigma=0.03=\sigma_{\text{N}}$ or $\sigma=0.3=\sigma_{\text{W}}$. 
The first value corresponds to the scenario where 
$\mathcal{G}$ is narrow (N) around its mean, and the second corresponds to a wider (W) case.
We plot the PDF and CDF for the cases $G_{ij}^{\bm{r}0} \sim \mathcal{G}(0.5,0.3^2)$
and $G_{ij}^{\bm{r}0} \sim \mathcal{G}(0.5,0.03^2)$ in Figure \ref{fig:exp-param-dist_4-reg_setup_1}.
Here, and throughout, results for the narrower and wider underlying distributions are shown on the left and right respectively.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/setup/prob_density__v__G_init.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_setup_1_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/setup/cdf__v__G_init.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_setup_1_b}
     \end{subfigure}
    \caption{Graphs showing the PDF and CDF of the initial conductance distribution, $\mathcal{G}$. 
    The narrower case is shown in blue. 
    The wider case is shown in orange. 
    The mean initial conductances, $\bar{G}_{\mathrm{W}}$ and $\bar{G}_{\mathrm{N}}$, are shown with correspondingly coloured dashed lines.
    \subref{fig:exp-param-dist_4-reg_setup_1_a}
    The pdf, $\mathscr{g}$, as a function of the initial conductance, $G_{ij}^{\bm{r}0}$.
    % The probability density functions of the narrower and wider initial conductance distributions are shown in solid blue and orange lines.
    % The means of the these distributions,
    % $G_{ij}^{\bm{r}0}=\bar{G}_{\mathrm{N}}$ and $G_{ij}^{\bm{r}0}=\bar{G}_{\mathrm{W}}$, are shown in correspondingly coloured dashed lines.
    \subref{fig:exp-param-dist_4-reg_setup_1_b}
    The CDF as a function of the initial conductance, $G_{ij}^{\bm{r}0}$.
    % The cumulative-density functions of the narrower and wider initial conductance distributions are shown in blue and orange solid lines.
    % The means of the these distributions,
    % $G_{ij}^{\bm{r}0}=\bar{G}_{\mathrm{N}}$ and $G_{ij}^{\bm{r}0}=\bar{G}_{\mathrm{W}}$, are shown in correspondingly coloured dashed lines.
    The cumulative densities of the means,
    $\text{cdf}(\bar{G}_{\mathrm{N}})$ and
    $\text{cdf}(\bar{G}_{\mathrm{W}})$
    are extrapolated in correspondingly coloured dotted lines.}
    \label{fig:exp-param-dist_4-reg_setup_1}
\end{figure}



The mean of the log-normal distribution from which initial conductances are drawn is $\text{exp}(\mu+\sigma^2/2)=\bar{G}$, so that $\bar{G} \approx 1.72461=\bar{G}_{\text{N}}$ in the narrower case, and $\bar{G} \approx 1.64946=\bar{G}_{\text{W}}$ in the wider one (see Figure \ref{fig:exp-param-dist_4-reg_setup_1}).
Note, from Figure \ref{fig:exp-param-dist_4-reg_setup_1_b}, that $\text{cdf}(\bar{G}_{\text{W}})>\text{cdf}(\bar{G}_{\text{N}})\approx 1/2$, which will have consequences for our analysis of the effective adhesivity, $j^1$.
Also note that, since all edges have unit length, the mean resistance per unit length of edges in this structure is $\bar{G}^{-1}$. 
We will preserve this quantity in the other structures that we study.
We present the case $c^{\text{max}}=1$, so that we display 
the permeability and adhesivity distributions, $\mathcal{K}$ and $\mathcal{J}$, 
when the cell contains the largest-feasible concentration.
Note that the qualitative behaviour is similar for any $0\le c^{\text{max}}<1$.
% [NB: I have simulated two cases, either mean or median... consider including both, but only discuss one here]
% As outlined in the previous section, we choose $\alpha$
% in \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic} so that $\alpha^{-1}=\bar{G}$. 


% : the variances are identical and the means are simply scaled by a factor of $c^{\text{max}}$. We expect this, since $c^{\text{max}}$ is the value by which the critical conductance, $c^{\text{max}}\bar{G}$ in \eqref{eq:reduced_prob-of-block}, is scaled.
% In words, for a scaled critical conductance, at which edges block, we expect 
% In the second, edges become blocked if the conductance is less than the median conductance, which is achieved by setting $\alpha^{-1}=\mathbb{M}[G_{ij}^{\bm{r}}]=\text{cdf}(0.5)$, where $\mathbb{M}$ and cdf denote the median and cumulative density function of the distribution from which conductances are drawn. 
% These choices aid consistency between experiments, and therefore comparibility.

\subsubsection{Permeability}
For a cell with $N=1$ (see Figure \ref{fig:exp-param-dist_4-reg_setup_2} for an illustration), the permeability formula, \eqref{eq:mm_permeability_reduced}, simplifies to
\begin{align}
    k^{11}_{N=1}=\frac{1}{2l^2}\sum_{r^1\in \mathcal{R}}\sum_{r^2\in \mathcal{R}}(r^1)^2 l^1 G_{11}^{(r^1,r^2)^{\top}},
    \label{eq:exp-param-dist_4-reg_definition-of-k11}
\end{align}
where $\mathcal{R}=\{-1,0,1\}$. 
Since $r^1=0$ on flow-transverse pores (see Figure \ref{fig:exp-param-dist_4-reg_setup_2}), only the conductances of flow-parallel pores, $G_{11}^{(\pm 1,0)^{\top}}$, contribute to the permeability in \eqref{eq:exp-param-dist_4-reg_definition-of-k11}.
The two flow-parallel pores, $(1,1,(1,0)^{\top})$ and $(1,1,(-1,0)^{\top})$, are identical by periodicity, and so $G_{11}^{(1,0)^{\top}}=G_{11}^{(-1,0)^{\top}}$. 
It follows that there is only one unique value of $G_{ij}^{\bm{r}0}$, and therefore that the distribution, $\mathcal{K}$, of the permeability, $k^{11}$, is identical to the distribution, $\mathcal{G}$, of the initial conductances,
$G_{ij}^{\bm{r}0}$. 
Indeed, in Figure \ref{fig:exp-param-dist_4-reg_perm_N=1}, we plot samples of $\mathcal{K}_{N=1}$ and the PDF of $\mathcal{G}$ 
and see that these match (up to randomness caused by sampling). 
We see that one consequence is that, when $N=1$, the expected permeability, $\mathbb{E}[k^{11}]$, of the four-regular network is exactly equal to the permeability of the mean network, $\bar{k}_4$ say, which is equal to $\bar{G}$ (see the orange and green dashed lines in Figure \ref{fig:exp-param-dist_4-reg_perm_N=1}).
Note, more generally, that regardless of structure and size of the cell, inter-cell pores that are transverse to the flow (those given by $(i,j,\bm{r}=(0,\pm 1)^{\top})$ in the $N=1$ cell) do not contribute to the flow-directional permeability, $k^{11}$ in \eqref{eq:mm_permeability_reduced}, 
since $r^1=0$ for these pores.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__perm_N=1.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_perm_N=1_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__perm_N=1.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_perm_N=1_b}
     \end{subfigure}
    \caption{Graphs showing the pdf, $\mathscr{g}$, of the initial conductance distribution, $\mathcal{G}$, 
    and the resulting permeability distribution, $\mathcal{K}$.
    The parameters are $\mu=0.5$, $S=10,000$, and $N=1$.
    The pdf of $\mathcal{G}$ is shown in a solid orange line. 
    Samples of $\mathcal{K}$ are shown with blue bars.
    The mean of $\mathcal{G}$, $\bar{G}$, which is equal to the permeability of the mean network, $\bar{k}_4$, 
    is shown in a dashed orange line. 
    The expected value of the samples of $\mathcal{K}$, $\mathbb{E}[k^{11}]$, is shown in a dashed green line.
    \subref{fig:exp-param-dist_4-reg_setup_1_a}
    The narrower case, $\sigma=0.03$.
    \subref{fig:exp-param-dist_4-reg_setup_1_b}
    The wider case, $\sigma=0.3$.} 
    \label{fig:exp-param-dist_4-reg_perm_N=1}
\end{figure}



We plot samples of the permeability distribution, and their statistics, for several other $N$, in the cases $\sigma=0.03$ (left) and $\sigma=0.3$ (right), in Figure \ref{fig:exp-param-dist_4-reg_2}.
Since we have used the parameters $\mu=0.5$ and $\sigma=0.3$ to specify $\mathcal{G}$ and thus $\mathcal{K}$ in Figures \ref{fig:exp-param-dist_4-reg_2_b} and \ref{fig:exp-param-dist_4-reg_2_d}, 
it is simple to verify that our results agree well (up to the randomness caused by sampling) 
with those presented in \cite{chapman2021homogenization} (Figure 3).
% , within which the results of an assessment of the permeability in the case $\sigma=0.3$ are presented. 
This helps validate our method and thus the rest of our findings.
We see that it does not follow from our result for $N=1$ that $\mathbb{E}[k^{11}]=\bar{G}$ for $N>1$, since there is now more than one pore for which an initial conductance value is drawn.
Instead, in Figures \ref{fig:exp-param-dist_4-reg_2_a} and \ref{fig:exp-param-dist_4-reg_2_b} we see that as $N$ increases, the spread of the permeability distribution decreases and the expected permeability approaches a constant, which we call $\bar{k}^{11}_{N\rightarrow \infty}$.
We interpret this limit, $\bar{k}^{11}_{N\rightarrow \infty}$, as the approximate (since we do not study cells larger than $N=100$) mean-permeability of an infinitely large non-periodic network. 

\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__perm__with_approx.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_2_a}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__perm__with_approx.pdf}
         \caption{}
     \label{fig:exp-param-dist_4-reg_2_b}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/mean-k_and_std-k__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_2_c}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/mean-k_and_std-k__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_2_d}
    \end{subfigure}
    \caption{Graphs showing how the distribution of samples of the permeability, $k^{11}$, varies with the cell size, $N$, in a four-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    \subref{fig:exp-param-dist_4-reg_2_a}
    The case $\sigma=0.03$.
    Coloured bars show a histogram of the distribution, $\mathcal{K}$, of the permeability, $k^{11}$, for various $N$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate pdf of $\mathcal{K}$.
    \subref{fig:exp-param-dist_4-reg_2_b}
    A similar plot to \subref{fig:exp-param-dist_4-reg_2_a}, but using $\sigma=0.3$. 
    \subref{fig:exp-param-dist_4-reg_2_c}
    The mean (blue dots) and standard deviation (orange dots) of the permeability distribution, $\mathbb{E}[k^{11}]$ and $\mathbb{S}[k^{11}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue curve shows the mean conductance, $\bar{G}$, which is also the permeability of the mean network, $\bar{k}_4$.
    The dashed blue line shows the approximate mean permeability of an infinite non-periodic four-regular network, $\bar{k}^{11}_{N\rightarrow \infty}$, approximated using $N=100$.
    Note that this is almost invisible to the eye, since it lies almost exactly below the solid blue line.
    The solid orange line shows a fit to the standard deviation, achieved by plotting in log-log space.
    \subref{fig:exp-param-dist_4-reg_2_d}
    A similar plot to \subref{fig:exp-param-dist_4-reg_2_c}, but using $\sigma=0.3$.}
    \label{fig:exp-param-dist_4-reg_2}
\end{figure}


\paragraph{Finite cell error}
Firstly, note, from Figures \ref{fig:exp-param-dist_4-reg_2_c} and \ref{fig:exp-param-dist_4-reg_2_d}, that the mean permeability of a cell of size $N$, $\mathbb{E}[k^{11}]$, becomes an increasingly accurate approximation of the mean permeability of the non-periodic network, $\bar{k}^{11}_{N\rightarrow \infty}$, 
as $N$ increases (compare blue dots to dashed blue line).
This is true for all $\sigma$, so that we find that the convergence occurs at approximately the same rate for all initial conductance distributions considered (see Figure \ref{fig:exp-param-dist_4-reg_2_c} for a typical narrower case, and Figure \ref{fig:exp-param-dist_4-reg_2_c} for a wider one).
Indeed, we find that the difference between the mean permeability of the cell of size $N$, $\mathbb{E}[k^{11}]$, and 
the mean permeability of the non-periodic random network, $\bar{k}^{11}_{N\rightarrow \infty}$
(as a fraction of $\bar{k}^{11}_{N\rightarrow \infty}$), satisfies 
\begin{align}
    E_{\text{c}}^{k}=\frac{\left|\mathbb{E}[k^{11}]-\bar{k}^{11}_{N\rightarrow \infty}\right|}{\bar{k}^{11}_{N\rightarrow \infty}} \approx \beta N^{-1},
    \label{eq:error_k_N}
\end{align}
where $\beta$ depends on $\sigma$.
This explains the limit,
$\mathbb{E}[k^{11}] \rightarrow \bar{k}^{11}_{N\rightarrow \infty}$
as
$N \rightarrow \infty$, that we observe.
We interpret this quantity, $E_{\text{c}}^{k}$, which we plot in Figure \ref{fig:exp-param-dist_4-reg_error-k-N}, 
as the error that we make to the mean permeability (note the superscript, $k$) when we replace the random network with a finite cell (note the subscript, c) of size $N$.
The implication of \eqref{eq:error_k_N}
is that, as the size of the cell grows, 
this error decreases like $N^{-1}$.
For example, when $\sigma=0.3$, averaging samples of a size $N=9$ cell leads us to overestimate the mean permeability of the infinitely-large non-periodic random network by approximately $0.5\%$ (see the dashed green line in Figure \ref{fig:exp-param-dist_4-reg_error-k-N_b}).
Even a cell of size just $N=4$ leads to an overestimate of only approximately $1.1\%$ (see the dashed orange line in Figure \ref{fig:exp-param-dist_4-reg_error-k-N_b}).
This result suggests that the multiscale system that we derived from a periodic-network system may also be able to capture the behaviour of the corresponding system defined on a non-periodic network.
Furthermore, note the reduction in the computational complexity of finding the permeability of a small cell as opposed to an entire network (or even a large cell), since, as we have discussed, to find the permeability, \eqref{eq:mm_permeability_reduced}, we solve a linear problem, \eqref{eq:mm_cell-problem_reduced}, of size of the number of nodes in the cell, $N$.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/mean-k_diff__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_error-k-N_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/mean-k_diff__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_error-k-N_b}
     \end{subfigure}
    \caption{Graphs showing how the finite cell error, $E_{\text{c}}^k$, 
    varies with the cell size, $N$.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    Blue dots show the numerical result. 
    Solid blue lines show a fitting.
    The errors that result from using cells of size $N=4$ (orange), $N=9$ (green), 
    and $N=16$ (red) are approximated in coloured dashed lines.
    \subref{fig:exp-param-dist_4-reg_setup_1_a}
    The case $\sigma=0.03$.
\subref{fig:exp-param-dist_4-reg_setup_1_b}
    The case $\sigma=0.3$.}
    \label{fig:exp-param-dist_4-reg_error-k-N}
\end{figure}

\paragraph{Mean network error}

Secondly, note that $\bar{k}^{11}_{N\rightarrow \infty}$ is smaller than the permeability of the mean network, $\bar{k}_4=\bar{G}$, which demonstrates the nonlinear dependence of the permeability on the initial conductances.
This is true for all $\sigma$, although,
for wider initial conductance distributions (see Figure \ref{fig:exp-param-dist_4-reg_2_d}), this difference is more pronounced, 
while for narrower cases (see Figure \ref{fig:exp-param-dist_4-reg_2_c})
the difference is smaller (compare the blue dots with the solid blue line).
Indeed, we find that the difference between the permeability of the mean network, $\bar{k}_4=\bar{G}$, and 
the mean permeability of the non-periodic random network, $k^{11}_{N\rightarrow\infty}$
(as a fraction of $k^{11}_{N\rightarrow\infty}$), satisfies 
\begin{align}
    E_{\text{n}}^k = \frac{\left|\bar{G}-\bar{k}^{11}_{N\rightarrow \infty}\right|}{\bar{k}^{11}_{N\rightarrow \infty}} \approx 0.52\sigma^{2},
    \label{eq:error_k_sigma}
\end{align}
so that
\begin{align}
   \bar{k}^{11}_{N\rightarrow \infty} \rightarrow \bar{G} = e^{\mu + \frac{\sigma^2}{2}}
   \label{eq:results_k-to-infty} 
\end{align}
as
$ \sigma \rightarrow 0$. 
% In \eqref{eq:results_k-to-infty} we have used the definition of $\bar{G}$ in terms of the parameters of the initial conductance distribution, $\mu$ and $\sigma$.
We interpret this quantity, $E_{\text{n}}^k$, which we plot in Figure \ref{fig:exp-param-dist_4-reg_vary-sigma_k}, as the error we make when we replace the permeability (hence the superscript, $k$) of the random network with the permeability of the mean network (hence the subscript, n).
The implication of \eqref{eq:error_k_sigma} is that, as $\sigma$ decreases, so that the initial conductance distribution narrows, the size of this error decreases like $\sigma^2$. 
For example, 
use of $\bar{G}$ in place of $\bar{k}^{11}_{N\rightarrow \infty}$ would lead to an underestimate of the mean permeability of the infinitely large non-periodic random network of approximately $4.4\%$ for $\sigma=0.3$ and approximately $0.04\%$ for $\sigma=0.03$ (see the green and orange dashed lines in Figure \ref{fig:exp-param-dist_4-reg_vary-sigma_k}, respectively).
Note that, if we are willing to accept the error for a given $\sigma$, then replacement leads to a considerable ($C_{\text{PC}}=\mathcal{O}(F(N^2)^L)$, see Section \ref{sec:mm_Solution methodology}) reduction in computational complexity, since the permeability need not be sampled and so the pre-calculation step is redundant. 
Instead, we assume 
that the mean of the permeability distribution is well approximated by the mean of the conductance distribution
% that the statistical properties of the permeability are completely determined by the statistics of the initial conductance distribution, $\mu$ and $\sigma$, 
and solve the multiscale system once using this value. 
% An example of this process was discussed in Subsection \ref{subsec:mm_Polydispersed pores}.

\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.7\textwidth]{./figures/multiscale/stats/4reg/vary-sigma/k_infty_diff__vs__sigma.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{Graph showing how the mean network error, $E_{\text{n}}^k$, 
        varies with the standard deviation of the initial conductance distribution, $\sigma$. 
        Blue dots show the numerical result. 
        Solid blue lines show a fitting. 
        The errors that result from replacing a network consisting of cells 
        containing initial conductances drawn from distributions with standard deviations of $\sigma=0.03$
        and $\sigma=0.3$
        with the mean network are approximated with a dashed orange and green lines, respectively. 
        } 
        \label{fig:exp-param-dist_4-reg_vary-sigma_k}
    \end{center}
\end{figure}



\paragraph{Sampling error}
Thirdly, note that in Figures \ref{fig:exp-param-dist_4-reg_2_c} and \ref{fig:exp-param-dist_4-reg_2_d}
we see that the standard deviation of the permeability, $\mathbb{S}[k^{11}]$, decays with $N$ like $N^{-1/2}$ (compare the orange dots with the solid orange line). This has two important implications. 
Firstly, we note that these standard deviations are larger than the errors caused either by  
replacement by small cells (see Figure \ref{fig:exp-param-dist_4-reg_error-k-N})
or replacement by
the mean network (see Figure \ref{fig:exp-param-dist_4-reg_vary-sigma_k}).
We might consider either replacement acceptable, then, since we do not expect the errors in the solution resulting from either to exceed the error caused by a random sample.
% [Comment on the fact that, since the deviation of the approximate mean from the true value is less than the standard deviation, one could argue that it is acceptable to replace the true mean by the approximate one even in the wider case. Argue that this is the case even at $N=1000$ (by extrapolating the standard deviation curve, which is $\sim N^{-1/2}$), which is approximately the $N$ of interest, since nodes are of size $\sim \mu$m, while filters have a depth $\sim$mm.]
Secondly, the slow decay of the standard deviation with $N$, along with the rapid convergence of the mean with $N$, has further implications when the goal is to evaluate the permeability for a non-periodic network. 
As we have established, the cell problem, \eqref{eq:mm_cell-problem_reduced}, is a linear problem with a computational complexity of at least $\mathcal{O}(N^2)$ \cite{strang1993introduction}.
Since the permeability formula, \eqref{eq:mm_permeability_reduced}, depends on the solution of the cell problem, $W_i^1$, the complexity associated with pre-calculation of the permeability is itself greater than $\mathcal{O}(N^2)$ and in particular is greater than $\mathcal{O}(N^{-1/2})$, which we have shown to be the convergence rate of the standard deviation.
Thus, it is computationally advantageous to average over many realisations of a relatively small cell as opposed to averaging over fewer realisations of a large cell.
As one further remark, note that the permeability distribution for the $N=1$ cell in the narrower case, $\sigma=0.03$, is approximately the distribution for the $N=100$ cell in the wider case, $\sigma=0.3$. One interpretation is that a factor of $10$ increase in the standard deviation of the underlying conductance distribution therefore has a factor of $100$ effect on the size of the cell required for network homogenisation.

\subsubsection{Adhesivity}
Next, in Figure \ref{fig:exp-param-dist_4-reg_3}, we display a similar (see Figure \ref{fig:exp-param-dist_4-reg_2}) assessment of the adhesivity distribution, $\mathcal{J}$, for narrower (left) and wider (right) initial conductance distributions, $\mathcal{G}$. 
We see that the qualitative behaviour of the statistics of the adhesivity is similar to that of the permeability (in Figure \ref{fig:exp-param-dist_4-reg_2}).
That is, as $N$ increases, the variance of the adhesivity also decreases and its expected value approaches a constant, $\bar{j}^1_{N\rightarrow \infty}$ say.
In analogy with $\bar{k}^{11}_{N\rightarrow \infty}$, $\bar{j}^1_{N\rightarrow \infty}$ may be interpreted as the approximate mean-adhesivity of an infinitely large non-periodic random network. 


\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__depo.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_3_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__depo.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_3_b}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/mean-j_and_std-j__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_3_c}
    \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/mean-j_and_std-j__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_3_d}
    \end{subfigure} 
    \caption{Graphs showing how the distribution of samples of the adhesivity, $j^{1}$, varies with the cell size, $N$, in a four-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    \subref{fig:exp-param-dist_4-reg_3_a}
    The case $\sigma=0.03$.
    Coloured bars show a histogram of the distribution, $\mathcal{J}$, of the adhesivity, $j^{1}$, for various $N$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate PDF of $\mathcal{J}$.
    \subref{fig:exp-param-dist_4-reg_3_b}
    A similar plot to \subref{fig:exp-param-dist_4-reg_2_a}, but using $\sigma=0.3$. 
    \subref{fig:exp-param-dist_4-reg_3_c}
    The mean (blue dots) and standard deviation (orange dots) of the adhesivity distribution, $\mathbb{E}[j^{1}]$ and $\mathbb{S}[j^{1}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue curve shows the adhesivity of the mean four-regular network, $\bar{j}_4=\bar{G}\text{cdf}(\bar{G})$ (as we will show).
    The dashed blue line shows the approximate mean adhesivity of an infinite non-periodic four-regular network, $\bar{j}^{1}_{N\rightarrow\infty}$, approximated using $N=100$.
    Note that this is almost invisible to the eye, since it lies almost exactly below the solid blue line.
    The solid orange line shows a fit to the standard deviation, achieved by plotting in log-log space.
    \subref{fig:exp-param-dist_4-reg_3_d}
    A similar plot to \subref{fig:exp-param-dist_4-reg_2_c}, but using $\sigma=0.3$.}
    \label{fig:exp-param-dist_4-reg_3}
\end{figure}




\paragraph{Blocked pore distribution}
Unlike the permeability, however, observing Figure \ref{fig:exp-param-dist_4-reg_3_a} we see that when $\mathcal{G}$ is narrower, the resulting adhesivity distribution is a set of distinct peaks with zero probability-density between them, at any given cell size, $N$. 
To explain this phenomenon, we begin by noting that 
\begin{align}
    j^1 
    & = 
    -\frac{1}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}} (W_j^1 + r^1l^1 - W_i^1)  A_{ji}^{-\bm{r}} H_{ji}^{-r^1} 
    \\
    & = 
    -\frac{1}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}} \Delta_{ji}^{-r^1}  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1}
    \\
    & \approx 
    -\frac{1}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \mathbb{E}[G_{ij}^{\bm{r}} \Delta_{ji}^{-r^1}]  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1}
    \\
    & \approx 
    -\frac{1}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \mathbb{E}[G_{ij}^{\bm{r}}] \mathbb{E}[\Delta_{ji}^{-r^1}]  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1}
    \\
    & =
    -\frac{\bar{G}}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}}  \mathbb{E}[\Delta_{ji}^{-r^1}]  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1}.
    \label{eq:j0-is-sum}
\end{align}
The first equation is the definition of $j^{1}$, \eqref{eq:mm_adhesivity_reduced}, and the second is the definition of
the pore pressure drop, $\Delta_{ij}^{r^1}$, \eqref{eq:Dd_definition-of-delta}. The third uses that, for narrow enough conductance distributions, such that $\sigma \ll 1$, values of the conductance, and therefore the pore pressure drop, lie close to their expectation (compare the values of $G\indx$ and $\Delta_{ij}^{r^1}$ for small $f$ in Figure \ref{fig:mm_periodic/random/preprocess} with those in \ref{fig:mm_periodic/non-random/preprocess}). Note that the fourth equation holds despite the dependence of the directional pressure drop, $\Delta_{ij}^{r^1}$, on the conductance, since $\sigma\ll1$.

Next, we note that
\begin{align}
    \mathbb{E}[\Delta_{ij}^{r^1}] = \cos{\theta_{ij}^{\bm{r}}},
    \label{eq:results_j0-is-cos}
\end{align}
where $\theta_{ij}^{\bm{r}}$ is the angle that the pore $ij\bm{r}$ makes with the direction of macroscale 
pressure gradient ($x^1$ in the present case). 
% [would a figure help? Diagram of an edge with $1$- and $2$-directional components drawn in].
% To see this, recall firstly that $\Delta_{ij}^{r^1}$ is the component of the microscale pressure drop in the $x^1$ direction, which we have chosen arbitrarily to be the direction parallel to the flow.
% This implies that $\Delta_{ij}^{r^1}$ is maximal in the $x^1$ direction.
To see this, note that the pore pressure drop is maximal in pores that 
are parallel to the direction of the pressure gradient, $x^1$, since 
flow down these pores contributes solely to the macroscale behaviour.
It decreases smoothly as the angle the pore makes with the macroscale flow direction increases. 
On pores that are transverse to the flow (see Figure \ref{fig:exp-param-dist_4-reg_setup_2}), 
the pore pressure drop is zero, since flow through these pores does not contribute to flow in the $x^1$ direction.
% Secondly, recall that the cell has been scaled for unit node density, which implies that $0\le\Delta_{ij}^{r^1}\le1$.
% It follows that, for example, in a purely $x^1$-directional pressure gradient, we observe average $x^1$-directional pressure drops per unit pressure gradient, $\mathbb{E}[\Delta_{ij}^{r^1}]$, of one and zero on edges exactly parallel and perpendicular to the flow, respectively.

Choosing edges in the four-regular cell, without loss of generality, to be aligned exactly parallel and perpendicular to the $x^1$ direction (and therefore to the flow, see Figure \ref{fig:exp-param-dist_4-reg_setup_2}), it follows, by substituting \eqref{eq:j0-is-sum} into \eqref{eq:j0-is-sum}, that 

\begin{align}
    j^1
    & \approx 
    -\frac{\bar{G}}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}}  \cos{\theta_{ij}^{\bm{r}}}  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1}.
    \\
    & =
    \bar{G}
    \frac{b^{1\parallel}}{l^1l^2}.
    \label{eq:results_j-is-jtilde}
\end{align}
The second equation, \eqref{eq:results_j-is-jtilde}, is a result of the fact that $\cos \theta\indx\ne0$ only for edges parallel to the flow, 
so that the sum counts the number of edges parallel to the $x^1$ direction that block per unit area in the cell, and $b^{1\parallel}/l^1l^2$ represents this quantity per unit cell-area.
We therefore define 
\begin{align}
    \tilde{j}^1 =
    \bar{G}
    \frac{b^{1\parallel}}{l^1l^2},
    \label{eq:results_definition-of-j0tilde}
\end{align}
which we call the approximate-adhesivity, since $j^1\approx\tilde{j}^1$ by \eqref{eq:results_j-is-jtilde}.
% here
The number of blocked pores, $b^{1\parallel}$, is a random variable drawn from a distribution, $\mathcal{B}^{\parallel}$ say. 
From \eqref{eq:results_definition-of-j0tilde} we see that the behaviour of the approximate adhesivity, $\tilde{j}^1$, 
and thus the structure of the adhesivity, $j^1\sim\mathcal{J}$, 
is given by the structure of the blocked pore distribution, $b^{1\parallel}\sim\mathcal{B}^{\parallel}$, and our aim is to find this.
To this end, we recall that \eqref{eq:mm_pore-conservation_non-periodic_pre-calc} and \eqref{eq:mm_adherence_non-periodic}, together with \eqref{eq:reduced_definition-of-alpha}, imply that when particles with concentration $c^{\text{max}}$ pass through pore $ij\bm{r}$, it remains unblocked if $c^{\text{max}}G\indx<G_{ij}^{\bm{r}0}$, but blocks if $c^{\text{max}}G\indx>G_{ij}^{\bm{r}0}$.
Moreover, note that initial conductances, $G_{ij}^{\bm{r}0}$, are drawn independently and so edges are either blocked or unblocked independently.
It follows that the event that pore $ij\bm{r}$ blocks, $E\indx$, is a two-outcome random variable, and thus a Bernoulli trial with parameter $p=\mathbb{P}[E\indx]=\text{cdf}(c^{\text{max}}\bar{G})$ (see \eqref{eq:reduced_prob-of-block}), 
so that 
\begin{align}
    E\indx \sim \text{Ber}(\text{cdf}(c^{\text{max}}\bar{G})).
    \label{eq:mm_E_distribution}
\end{align}
Since there are $2N$ unique edges in the four-regular cell, and thus $N$ edges parallel to the flow, it follows that the random variable $b^{1\parallel}$, which is the total number of these that are blocked by the concentration $c^{\text{max}}$, follows a Binomial distribution with parameters $n=N$, and $p=\text{cdf}(c^{\text{max}}\bar{G})$, so that 
\begin{align}
    b^{1\parallel} &
    \sim 
    \text{Bin}\left(N,\text{cdf}\left(c^{\text{max}}\bar{G}\right)\right)
    \label{eq:results_b-distribution}
    \\
    &
    \approx 
    \text{Norm}\left(
    N\text{cdf}\left(
    c^{\text{max}}{G}
    \right),
    N\text{cdf}\left(
    c^{\text{max}}\bar{G}
    \right)
    \left(
    1-\text{cdf}\left(c^{\text{max}}\bar{G}
    \right)\right)
    \right).
    \label{eq:results_b-approx}
\end{align}
The approximation \eqref{eq:results_b-approx} follows from the fact that the normal distribution with parameters $\mu=np$ and $\sigma=\sqrt{np(1-p)}$, 
$\text{Norm}(np,np(1-p))$,
provides a good approximation of the binomial distribution with parameters $n$ and $p$, $\text{Bin}(n,p)$, if $n$ is large enough or if $p\approx 1/2$
\cite{schader1989two}.
In our case, $n=N$ and $p=\text{cdf}(c^{\text{max}}\bar{G})$, and we have chosen example initial conductance distributions for which $\text{cdf}(\bar{G})\approx 1/2$, so that this approximation holds.

We plot samples of the distribution of $b^{1\parallel}$ for narrower (left) and wider (right) initial conductance distributions in Figures \ref{fig:exp-param-dist_4-reg_4_a} and \ref{fig:exp-param-dist_4-reg_4_b} (coloured bars) in the case $c^{\text{max}}=1$.
We see that the samples agree well with both the probability-mass function of the binomial distribution (correspondingly coloured dots) in \eqref{eq:results_b-distribution}
and the probability density function of the normal distribution (correspondingly coloured solid lines) in \eqref{eq:results_b-approx}.
In Figure \ref{fig:exp-param-dist_4-reg_4_c}, meanwhile, we illustrate the mean and standard deviation of $b^{1\parallel}$.
Note that these are approximated well by $N/2$ and $\sqrt{N}/2$ for narrow enough conductance distributions, since $\text{cdf}(\bar{G}) \approx 1/2$ for $\sigma \ll 1$, for example $\sigma=\sigma_{N}=0.03$.
In Figure \ref{fig:exp-param-dist_4-reg_4_d}, we see that this proxy for the mean of $b^{1\parallel}$ is less useful for wider conductance distributions, for example $\sigma=\sigma_{\text{W}}=0.3$, for which $\text{cdf}(\bar{G})$ deviates more extremely from $1/2$. The standard-deviation approximation remains useful.


\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__adhe.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_4_a}
     \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__adhe.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_4_b}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/mean-b_and_std-b__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_4_c}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/mean-b_and_std-b__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_4_d}
    \end{subfigure}
    \caption{Graphs showing how the distribution of samples of the number of blocked flow-parallel pores, $b^{1\parallel}$, varies with the cell size, $N$, in a four-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    \subref{fig:exp-param-dist_4-reg_4_a}
    The case $\sigma=0.03$.
    Coloured bars show a histogram of samples of the distribution of blocked flow-parallel pores, $\mathcal{B}^{\parallel}$, for various $N$.
    The binomial distribution that this approximates is plotted with correspondingly coloured dots for each $N$. 
    The normal distribution that this approximates is plotted with correspondingly coloured solid curves for each $N$.
    \subref{fig:exp-param-dist_4-reg_4_b}
    A similar plot to \subref{fig:exp-param-dist_4-reg_2_a}, but using $\sigma=0.3$. 
    \subref{fig:exp-param-dist_4-reg_4_c}
    The mean (blue dots) and standard deviation (orange dots) of the blocked flow-parallel pore distribution, $\mathbb{E}[b^{1\parallel}]$ and $\mathbb{S}[b^{1\parallel}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue line shows the mean of the corresponding binomial distribution, $np=N\text{cdf}(\bar{G})$.
    The dashed line shows an approximation of the mean, $N/2$, which holds since we chose $\bar{G}\approx 1/2$.
    The solid orange line shows the standard deviation of the corresponding binomial distribution, $(np(1-p))^{1/2}=(N\text{cdf}(\bar{G})(1-\text{cdf}(\bar{G})))^{1/2}$.
    The dotted green line shows an approximation of the standard deviation, $\sqrt{N}/2$, which holds since we chose $\bar{G}\approx 1/2$.
    \subref{fig:exp-param-dist_4-reg_4_d}
    A similar plot to \subref{fig:exp-param-dist_4-reg_2_c}, but using $\sigma=0.3$.}
    \label{fig:exp-param-dist_4-reg_4}
\end{figure}





Combining \eqref{eq:results_definition-of-j0tilde} with \eqref{eq:results_b-distribution} 
and
\eqref{eq:results_b-approx}, 
we see that
\begin{align}
    \tilde{j}^{1} 
    & \sim 
    \frac{\bar{G}}{N} \text{Bin}\left(N,\text{cdf}\left(c^{\text{max}}\bar{G}\right)\right)
        \label{eq:results_j0tilde-distribution}
    \\
    &\approx 
    \frac{\bar{G}}{N}
    \text{Norm}\left(
    N\text{cdf}
    \left(
    c^{\text{max}}\bar{G}\right),
    N\text{cdf}
    \left(c^{\text{max}}\bar{G}\right)
    \left(1-\text{cdf}\left(c^{\text{max}}\bar{G}\right)\right)\right),
\label{eq:results_j0tilde-approx}
\end{align}
where we have used that $l^0=l^1=\sqrt{N}$. 
We have found that the random variable, $\tilde{j}^1$, which we call the approximate adhesivity, follows a scaled binomial distribution, as a result of the fact that the number of edges that are blocked in the cell, $b^{1\parallel}$, follows a binomial distribution, \eqref{eq:results_b-distribution}.
The behaviour of $\tilde{j}^1$ is approximately equal to that of the adhesivity, $j^{1}$, due to \eqref{eq:results_j-is-jtilde}, which explains the quasi-discrete behaviour of the adhesivity, illustrated, for example, in Figure \ref{fig:exp-param-dist_4-reg_3_a}. 
% One immediate consequence of \eqref{eq:results_j0tilde-distribution} is that the adhesivity of the mean network, 
% is given by $\bar{j}_4=\bar{G}\text{cdf}(\bar{G})$ (see Figure \ref{fig:exp-param-dist_4-reg_3}).


Some other plots of the adhesivity distribution, $\mathcal{J}$, for particular cell sizes, $N$, are shown in Figure  \ref{fig:exp-param-dist_4-reg_5}, for narrower (left) and wider (right) initial conductance distributions, $\mathcal{G}$. We see that peaks of $j^1$ (solid blue lines) are distributed near to peaks of $\tilde{j}^1$ (dashed orange lines) (see \eqref{eq:results_j0tilde-distribution}).
These are located at values that are $\bar{G}/(l^1 l^2)$ away from each other, due to \eqref{eq:results_definition-of-j0tilde}. 
Each location corresponds to the adhesivity that results from blocking a specific number of pores. 
For example, the first peak corresponds to the case where no pores block, the second to one pore being blocked, and so on. There are $N+1$ such peaks since there are $N$ unique flow-parallel pores in a four-regular cell (plus one peak for the no blocking case).
The height of each peak is dictated by the probability that the corresponding number of pores is blocked, given by the distribution of $b^{1\parallel}$, $\mathcal{B}^{\parallel}$.
 The width of each peak depends on the width of $\mathcal{G}$. For example, in Figures \ref{fig:exp-param-dist_4-reg_5}(a,c,e), we see that when $\mathcal{G}$ is narrower, the peaks of $j^1$ are tight around peaks of the binomial distribution, $\tilde{j}^{1}$, in \eqref{eq:results_j0tilde-distribution}.
 On the other hand, when $\mathcal{G}$ is wider, as is illustrated in Figures \ref{fig:exp-param-dist_4-reg_5}(b,d,f), the binomial structure of $j^1$ is mostly lost, due to the increased spread in pore pressure drops, $\Delta_{ij}^{r^1}$, caused by the increased width of initial conductances, $G_{ij}^{\bm{r}0}$, via the cell problem, \eqref{eq:mm_cell-problem_reduced}. 
Despite the loss of overall binomial structure, the upper and lower limits of the approximate-adhesivity distribution, $\tilde{j}^1$, remain useful to bound the overall width of $\mathcal{J}$.
Indeed, as can be seen in Figure \ref{fig:exp-param-dist_4-reg_5}, values of $j^1$ always lie between $0$, which occurs when no edges block, and the maximal value of $\tilde{j}^1$, $\bar{G}$, which occurs when $N$ edges (the number of edges parallel to the $x^1$ direction) block. 
% Note that the standard deviation of the adhesivity distribution, then, is a result of the standard deviation of the distribution of the number of edges that are blocked, $b^{1\parallel}$, rather than that of the initial conductances.

\begin{figure}
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
    \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__depo__with_approx_N=4.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_5_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__depo__with_approx_N=4.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_5_b}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__depo__with_approx_N=9.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_5_c}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__depo__with_approx_N=9.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_5_d}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/prob_density__v__depo__with_approx_N=16.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_5_e}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/prob_density__v__depo__with_approx_N=16.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_5_f}
    \end{subfigure}
    \caption{Graphs showing the distribution of samples of the adhesivity, $j^{1}$, in a four-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    Coloured bars show a histogram of the distribution, $\mathcal{J}$, of the adhesivity, $j^{1}$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate pdf of $\mathcal{J}$.
    Dashed orange lines show the discrete locations at which probability in the distribution 
    of the approximate adhesivity, $\tilde{j}^1$, is found.
    \subref{fig:exp-param-dist_4-reg_5_a}
    $N=4$ and $\sigma=0.03$.
    \subref{fig:exp-param-dist_4-reg_5_b}
     $N=4$ and $\sigma=0.3$.
    \subref{fig:exp-param-dist_4-reg_5_c}
     $N=9$ and $\sigma=0.03$.
    \subref{fig:exp-param-dist_4-reg_5_d}
     $N=9$ and $\sigma=0.3$.
    \subref{fig:exp-param-dist_4-reg_5_e}
     $N=16$ and $\sigma=0.03$.
    \subref{fig:exp-param-dist_4-reg_5_f}
     $N=16$ and $\sigma=0.3$.}
    \label{fig:exp-param-dist_4-reg_5}
\end{figure}


\paragraph{Finite cell error}
In further analogy with the permeability (see Figures \ref{fig:exp-param-dist_4-reg_2_c} and \ref{fig:exp-param-dist_4-reg_2_d}), in Figures \ref{fig:exp-param-dist_4-reg_3_c} and \ref{fig:exp-param-dist_4-reg_3_d}, we firstly see that the mean adhesivity of the cell of size $N$, $\mathbb{E}[j^{1}]$, becomes an increasingly accurate approximation of the mean adhesivity of the non-periodic network, $\bar{j}^{1}_{N\rightarrow\infty}$, 
as $N$ increases.
As with the permeability, the convergence occurs at approximately the same rate for all initial conductance distributions considered.
Indeed, we find that the difference between the mean adhesivity of the cell of size $N$, $\mathbb{E}[j^{1}]$, and 
the mean adhesivity of the non-periodic random network, $\bar{j}^{1}_{N\rightarrow\infty}$
(as a fraction of $\bar{j}^{1}_{N\rightarrow\infty}$), satisfies 
\begin{align}
    E_{\text{c}}^{j} = \frac{\left|\mathbb{E}[j^{1}]-\bar{j}^{1}_{N\rightarrow \infty}\right|}{\bar{j}^{1}_{N\rightarrow \infty}} \approx \beta N^{-1},
    \label{eq:error_j_N}
\end{align}
where $\beta$ depends on $\sigma$.
Consequently,
$\mathbb{E}[j^{1}] \rightarrow \bar{j}^{1}_{N\rightarrow \infty}$
as
$N \rightarrow \infty$.
We interpret $E_{\text{c}}^{j}$, which we plot in Figure \ref{fig:exp-param-dist_4-reg_error-j-N}, 
as the error that we make in the adhesivity parameter  (note the superscript, $j$) when we replace the random network with a cell  (note the subscript, c) of size $N$.
Comparing \eqref{eq:error_j_N} to \eqref{eq:error_k_N}, we see that, as the size of the cell, $N$, grows, 
this error decreases like $N^{-1}$, just as the corresponding permeability error does.
For the adhesivity, when $\sigma=0.3$, averaging of samples of
a cell of size $N=9$ leads to an underestimate of the mean adhesivity of the infinitely-large non-periodic random network of approximately $2\%$ (see the dashed green line in Figure \ref{fig:exp-param-dist_4-reg_error-j-N_b}), while a cell of size $N=4$ leads to an underestimate of only approximately $4.2\%$ (see the dashed orange line in Figure \ref{fig:exp-param-dist_4-reg_error-j-N_b}).
Note that, although the adhesivity error for a given cell size is larger than the corresponding permeability error (by comparing Figures  \ref{fig:exp-param-dist_4-reg_error-k-N} and  \ref{fig:exp-param-dist_4-reg_error-j-N}), these adhesivity errors are still modest.
This result suggests that the multiscale system may be able to capture the behaviour of the corresponding system defined on a non-periodic network, despite containing a reaction term (via the adhesivity) as well as an advection term (via the permeability) (see \eqref{eq:mm_particle-conservation_reduced}).
The computational complexity of the adhesivity,  \eqref{eq:mm_adhesivity_reduced}, is approximately equal to that of the permeability,  \eqref{eq:mm_permeability_reduced}, since they rely on solution of the same cell problem,  
\eqref{eq:mm_cell-problem_reduced}.
It follows that this result,  \eqref{eq:error_j_N}, offers considerable reduction in the complexity of solving the effective system even when it contains a reaction term, by replacing the network with a smaller cell.


\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.03/alpha-mean/mean-j_diff__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_error-j-N_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/4reg/sigma-0.3/alpha-mean/mean-j_diff__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_4-reg_error-j-N_b}
     \end{subfigure}
    \caption{Graphs showing how the finite cell error, $E_{\text{c}}^j$, 
    varies with the cell size, $N$.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    Blue dots show the numerical result. 
    Solid blue lines show a fitting.
    The errors that result from using cells of size $N=4$ (orange), $N=9$ (green), 
    and $N=16$ (red) are approximated in coloured dashed lines.
    \subref{fig:exp-param-dist_4-reg_setup_1_a}
    The case $\sigma=0.03$.
\subref{fig:exp-param-dist_4-reg_setup_1_b}
    The case $\sigma=0.3$.}
    \label{fig:exp-param-dist_4-reg_error-j-N}
\end{figure}


\paragraph{Mean network error}
Secondly, in Figure \ref{fig:exp-param-dist_4-reg_4}, we note that
$\bar{j}^{1}_{N\rightarrow \infty}$ is close to (but larger than) $\bar{G}\text{cdf}(c^{\text{max}}\bar{G})$ (with $c^{\text{max}}=1$). 
To understand this, note that one important application of the approximate-adhesivity distribution, $\tilde{j}^{1}$, which is distributed as in \eqref{eq:results_j0tilde-distribution}, is that it can be used to approximate the mean of the adhesivity distribution, $\mathbb{E}[j^{1}]$. Since the expected value of a random variable that follows a binomial distribution with parameters $n$ and $p$ is $np$, it follows from  \eqref{eq:results_j0tilde-distribution} that
\begin{align}
    \mathbb{E}[\tilde{j}^{1}] = \bar{G} \text{cdf}(\bar{G}),
\end{align}
and consequently that
\begin{align}
    \mathbb{E}[j^{1}] \approx 
    e^{\mu+\frac{\sigma^2}{2}}
    \text{cdf}\Big(e^{\mu+\frac{\sigma^2}{2}}\Big),
\end{align}
where we have used the approximation \eqref{eq:results_j-is-jtilde}and the definition of $\bar{G}$.
% We interpret this value, $\bar{G} \text{cdf}(\bar{G})$, as the scaled adhesivity of the mean network when $c^{\text{max}}=1$. 
One immediate consequence is that the adhesivity of the mean network
is given by $\bar{j}_4=\bar{G}\text{cdf}(\bar{G})$ (when $c^{\text{max}}=1$).
% (see Figure \ref{fig:exp-param-dist_4-reg_3}).
To see why, note that, in the mean network, the adhesivity is $\bar{G}$, since $\text{cdf}(\bar{G})=1$ because all edges have a conductance equal to the mean.
The value $0\le\text{cdf}(c^{\text{max}}\bar{G})\le 1$ is a scaling that accounts for the fact that, in more general networks, not all edges block because edge conductances are distributed about the mean.
For wider initial conductance distributions (see Figure \ref{fig:exp-param-dist_4-reg_3_d}), the difference between $\jtoinf$ and $\bar{G}\text{cdf}(\bar{G})$ is more pronounced, 
while for narrower cases (see Figure \ref{fig:exp-param-dist_4-reg_3_c})
the difference is smaller.
Indeed, we find that the difference between the scaled adhesivity of the mean network, $\bar{j}_4=\bar{G}\text{cdf}(\bar{G})$, and 
the mean adhesivity of the non-periodic random network, $\bar{j}^{1}_{N\rightarrow\infty}$
(as a fraction of $\bar{j}^{1}_{N\rightarrow\infty}$), satisfies 
\begin{align}
    E_{\text{n}}^j = \frac{\left|\bar{G}\text{cdf}(\bar{G})-\bar{j}^{1}_{N\rightarrow \infty}\right|}{\bar{j}^{1}_{N\rightarrow \infty}} \approx 0.45\sigma^{\frac{3}{2}},
    \label{eq:error_j_sigma}
\end{align}
(in the case $c^{\text{max}}=1$),
so that
\begin{align}
    \bar{j}^{1}_{N\rightarrow \infty} \rightarrow \bar{G}\text{cdf}(\bar{G}) = 
    e^{\mu + \frac{\sigma^2}{2}}
    \left(
    \frac{1}{2}
    \left(1+\text{erf}\left(
    \frac{\log(e^{\mu + \frac{\sigma^2}{2}})- \mu}{\sigma \sqrt{2}}\right)\right)
\right)
\label{eq:results_j-to-infty}
\end{align}
as
$ \sigma \rightarrow 0$, 
where we have used the formula for the cumulative-density function of a log-normal distribution, and the definition of $\bar{G}$.
We interpret this quantity, $E_{\text{n}}^j$, which we plot in Figure \ref{fig:exp-param-dist_4-reg_vary-sigma_j}, as the error we make when we replace the adhesivity (hence the superscript, $j$) of the random network with the scaled  adhesivity associated with the mean network (hence the subscript, n).
The implication of \eqref{eq:error_j_sigma} is that, as $\sigma$ decreases, so that the initial conductance distribution narrows, the size of this error decreases like $\sigma^{3/2}$.
Note that this decay is not as fast as that of the permeability in \eqref{eq:error_k_sigma}, which is $\mathcal{O}(\sigma^2)$.
However, use of $\bar{G}\text{cdf}(\bar{G})$ in place of $\bar{j}^{1}|_{N\rightarrow\infty}$ would lead to an error in the mean adhesivity of the infinitely-large non-periodic random network of approximately $7.4\%$ for $\sigma=0.3$ and approximately $0.25\%$ for $\sigma=0.03$ (see green and orange dashed lines in Figure \ref{fig:exp-param-dist_4-reg_vary-sigma_j}), 
compared to $4.4\%$ and $0.04\%$ for the permeability (compare to the green and orange dashed lines in Figure \ref{fig:exp-param-dist_4-reg_vary-sigma_k}).
As for the permeability (see \eqref{eq:results_k-to-infty}), \eqref{eq:results_j-to-infty}
provides an estimate 
for the mean adhesivity that only depends on the statistics of the initial conductance distribution, $\mu$ and $\sigma$. 
As such, if we accept the associated error, $E_{\text{n}}^j$,
using \eqref{eq:results_j-to-infty} avoids the need for  sampling of the adhesivity, \eqref{eq:mm_adhesivity_reduced}, and consequently the need for solution of the cell problem, \eqref{eq:mm_cell-problem_reduced}. 
The result is a significant reduction of the computational complexity associated with solution of the effective system.

\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.7\textwidth]{./figures/multiscale/stats/4reg/vary-sigma/j_infty_diff__vs__sigma.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{Graph showing how the mean network error, $E_{\text{n}}^j$, 
        varies with the standard deviation of the initial conductance distribution, $\sigma$. 
        Blue dots show the numerical result. 
        Solid blue lines show a fitting. 
        The errors that result from replacing a network consisting of cells 
        containing initial conductances drawn from distributions with standard deviations of $\sigma=0.03$
        and $\sigma=0.3$
        with the mean network are approximated with a dashed orange and green lines, respectively. 
        } 
        \label{fig:exp-param-dist_4-reg_vary-sigma_j}
    \end{center}
\end{figure}
\paragraph{Sampling error}
Thirdly, note that, in
Figures \ref{fig:exp-param-dist_4-reg_3_c} and \ref{fig:exp-param-dist_4-reg_3_d},
the standard deviation of the adhesivity, $\mathbb{S}[j^{1}]$, decays with $N$ like $N^{-1/2}$, 
as did the standard deviation of the permeability, $\mathbb{S}[k^{11}]$ (compare to
Figures \ref{fig:exp-param-dist_4-reg_2_c} and \ref{fig:exp-param-dist_4-reg_2_d}).
Thus, our arguments extend readily to cases of non-zero 
adhesivity. 
Firstly, that is, 
we might accept the errors associated with replacement of $\mathbb{E}[j^1]$ with either $\jtoinf$ or $\bar{G} \text{cdf}(c^{\text{max}} \bar{G})$, in \eqref{eq:error_j_N} and \eqref{eq:error_j_sigma}, since these are smaller than the average sample error.
% [Comment on the fact that, since the deviation of the approximate mean from the true value is less than the standard deviation, one could argue that it is acceptable to replace the true mean by the approximate one even in the wider case. Argue that this is the case even at $N=1000$ (by extrapolating the standard deviation curve, which is $\sim N^{-1/2}$), which is approximately the $N$ of interest, since nodes are of size $\sim \mu$m, while filters have a depth $\sim$mm.]
Secondly, as with the permeability, the $\mathcal{O}(N^{-1/2})$ decay of the standard deviation of the adhesivity indicates that it is better to average over many realisations of a relatively small unit cell than to average over just a few realisations of a large unit cell.

% \newpage

% \textcolor{blue}{\textbf{Plan: Remainder of this chapter}}
% \begin{itemize}
%     \item I'll then analyse a six-regular cell and a six-irregular cell in the same way, and reach similar conclusions.
%     \item We will find that these two cases both have significantly larger parameters, but that these are similar to each other, although six-irregular has much larger standard deviations.
%     \item I will propose that this is because their average connectivity is identical but that the six-irregular case has significantly larger width in the underlying conductance distribution due to the sampling method.  To prove this, I will then compute the six-regular and six-irregular parameters in the case where the six-regular case has the same underlying deviation as the six-irregular case, and I will find that the results are identical on average.
% \end{itemize}

\subsection{Six-regular structure}

Four-regular networks are commonly used, partly because they are easy to visualise and program. 
However, they are not an accurate characterisation of the random networks that represent 
the microscale of porous material,
even when the conductances of their pores are distributed randomly. 
The average degree of porous media networks is often larger than four, and is usually around six.
Even when edges have the same resistance per unit length, 
this difference in average degree has a significant impact on effective properties, 
such as permeability and adhesivity. 
To demonstrate this, we next consider a six-regular network, 
in which each node is connected to exactly six other nodes in a regular lattice.

To construct a six-regular network, 
we first consider the basic cell illustrated in Figure \ref{fig:exp-param-dist_6-reg_setup_2}. 
The lengths of its sides are $L^1=s$ and 
$L^2=\sqrt{3}s$, where $s$ is a scale factor chosen to maintain unit node density, to aid comparison of the effective properties with those of the 
four-regular network. 
There are two nodes in the basic cell, at positions $(0,0)$ and $(s/2, (\sqrt{3}s)/2$, 
and it follows that $s^2=2/\sqrt{3}$. 
We draw initial conductances from the same log-normal distributions as before, but scale the results so that the 
pore resistance per unit length is the same as in the four-regular network ($\approx1/1.72$).
Cells of size $N=2n^2$ for 
$ n \in \{1,2,\hdots 9,10\}$
are then constructed by repeating $n$ copies of the basic cell in the $x^1$ and $x^2$ directions (see Figure \ref{fig:exp-param-dist_6-reg_setup_2}).


\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.9\textwidth]{./figures/multiscale/stats/6reg/setup/cells_6-reg.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{Diagram of the basic construction cell and six-regular cells of sizes $N=2$ and $N=8$ ($N=2n^2$ for $n=1$ and $n=2$) inside larger networks. 
        For each $N$, nodes and edges that are 
        part of the cell are coloured red, and the cell is highlighted with a red box.
        The dimensions of the cell are set to be $L^1=ns$ and $L^2=\sqrt{3}ns$ for $s^2=2/\sqrt{3}$, so that the node density, $\varrho$, satisfies $\varrho=1$ in a cell of any size, $N$.
        Consequently, the length of each pore $ij\bm{r}$ is $L\indx=ns$.
        We arbitrarily choose the $x^1$ direction to be from left to right. 
        Thus, horizontal edges, like the blue edge in the basic cell, are
        parallel to the pressure gradient, and thus the flow. We call these flow-parallel edges. 
        Diagonal edges, like the orange edge in the basic cell, are at an angle 
        $\theta\indx=\pi/3$ to the flow, so that $\cos(\theta\indx)=1/2$. We call these flow-diagonal edges.} 
        \label{fig:exp-param-dist_6-reg_setup_2}
    \end{center}
\end{figure}
\subsubsection{Permeability}

We plot samples of the permeability distribution, $\mathcal{K}$, and their statistics, $\mathbb{E}[k^{11}]$ and
$\mathbb{S}[k^{11}]$ in Figure  \ref{fig:exp-param-dist_6-reg_2} (similarly to the four-regular case in Figure \ref{fig:exp-param-dist_4-reg_2}).
Our plots in Figures \ref{fig:exp-param-dist_6-reg_2_b} and \ref{fig:exp-param-dist_6-reg_2_d} agree well (up to the randomness caused by sampling) 
with those in \cite{chapman2021homogenization} (Figure 4).
Qualitatively, we see that behaviour of $\mathcal{K}$ is similar to that in the four-regular network.
As the cell size increases, the permeability distribution tightens around 
a value that is close to the permeability, $\bar{k}_6$ say, of the mean network, which is a six-regular network in which all pores have the mean conductance per unit length, $\bar{G}$.
However, as expected, we see that the permeability of the six-regular networks is much larger than that of the four-regular networks, owed to the larger average degree.
We find that $\bar{k}_6\approx2.66$, which is substantially larger than $\bar{k}_4\approx1.72$. 
The distribution of the permeability in the smallest cell, $k^{11}_{N=2}$, is no longer the same 
as the initial conductance distribution, $\mathcal{G}$, and has a larger mean, because in a six-regular configuration there are three unique edges 
(rather than one in the four-regular case)
to conduct flow.

\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__perm__with_approx.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_2_a}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__perm__with_approx.pdf}
         \caption{}
     \label{fig:exp-param-dist_6-reg_2_b}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/mean-k_and_std-k__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_2_c}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/mean-k_and_std-k__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_2_d}
    \end{subfigure}
    \caption{Graphs showing how the distribution of samples of the permeability, $k^{11}$, varies with the cell size, $N$, in a six-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    \subref{fig:exp-param-dist_6-reg_2_a}
    The case $\sigma=0.03$.
    Coloured bars show a histogram of the distribution, $\mathcal{K}$, of the permeability, $k^{11}$, for various $N$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate PDF of $\mathcal{K}$.
    \subref{fig:exp-param-dist_6-reg_2_b}
    A similar plot to \subref{fig:exp-param-dist_6-reg_2_a}, but using $\sigma=0.3$. 
    \subref{fig:exp-param-dist_6-reg_2_c}
    The mean (blue dots) and standard deviation (orange dots) of the permeability distribution, $\mathbb{E}[k^{11}]$ and $\mathbb{S}[k^{11}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue curve shows the permeability of the mean six-regular network, $\bar{k}_6$.
    The dashed blue line shows the approximate mean permeability of an infinite non-periodic six-regular network, $\bar{k}^{11}_{N\rightarrow \infty}$, approximated using $N=200$.
    Note that this is almost invisible to the eye, since it lies almost exactly below the solid blue line.
    The solid orange line shows a fit to the standard deviation, achieved by plotting in log-log space.
    The dotted blue line shows the mean conductance $\bar{G}$, which is also the permeability
    of the mean four-regular network, $\bar{k}_4$.
    \subref{fig:exp-param-dist_6-reg_2_d}
    A similar plot to \subref{fig:exp-param-dist_6-reg_2_c}, but using $\sigma=0.3$.}
    \label{fig:exp-param-dist_6-reg_2}
\end{figure}

\paragraph{Finite cell error}
Observing Figures \ref{fig:exp-param-dist_6-reg_2_c} and \ref{fig:exp-param-dist_6-reg_2_d}, 
we see that as $N$ increases, the average permeability of the cell, $\mathbb{E}[k^{11}]$ (blue dots)
approaches the approximate average permeability of the entire network, $\bar{k}^{11}_{N\rightarrow\infty}$ (dashed blue line).
Thus, as before, the error in the permeability that we make when we replace an infinitely large network with a cell of finite size $N$, $E_{\text{c}}^{k}$, decreases with $N$. 
The behaviour of this function is similar to that in the four-regular case, so we do not repeat our analysis here. 
The implication, however, is that this error is always relatively small. 
Even use of the smallest cell, $N=2$, only produces an overestimation of $E_{\text{c}}^{k}\approx 3\%$ when $\sigma=0.3$. This is larger than a similarly sized cell in the four-regular case, but still modest.

\paragraph{Mean network error}
Secondly, comparing the dashed blue line with the solid blue line in Figures \ref{fig:exp-param-dist_6-reg_2_c} and \ref{fig:exp-param-dist_6-reg_2_d}, 
we see that $\bar{k}^{11}_{N\rightarrow\infty}$ is not equal to $\bar{k}_6$. 
That is, because permeability is a function of the conductances, as well as structure, the error we make 
when we replace the random network with the mean network, $E_{\text{n}}^{k}$, is non-zero. 
However, as in the four-regular case, this error is small, and decreases proportionally to the variance of the conductance 
distribution, $\sigma^2$. 
As such, in Figure \ref{fig:exp-param-dist_6-reg_2_c}, for example,
we see that when $\sigma\ll1$, this error is infinitesimal.

\paragraph{Sampling error}
Thirdly, in Figures \ref{fig:exp-param-dist_6-reg_2_c} and \ref{fig:exp-param-dist_6-reg_2_d}, 
we see that the standard deviation of the permeability (see orange dots)
decreases like $N^{-1/2}$ (see orange lines), which is the same rate as in the four-regular scenario. 
It remains computationally advantageous to sample a small cell many times, if the goal is to
obtain the average permeability of a large network.

\subsubsection{Adhesivity}

Next, in  Figure \ref{fig:exp-param-dist_6-reg_3}, we plot samples of the adhesivity distribution, $\mathcal{J}$, for the six-regular network (similarly to the four-regular case in Figure \ref{fig:exp-param-dist_4-reg_3}).
As with the permeability, the adhesivity of the six-regular network is also larger than that of the four-regular network, 
because there are more pores per cell with the potential to become blocked.
Qualitatively, the behaviour of the distribution is similar to that in the four-regular case. 
As the cell size increases, the adhesivity distribution narrows and the value $j^{11}_{N\rightarrow\infty}$ 
becomes more likely.
As before, for each cell size, the adhesivity is distributed between a discrete set of values. 
These are different from those observed in 
the four-regular case.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__depo.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_3_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__depo.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_3_b}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/mean-j_and_std-j__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_3_c}
    \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/mean-j_and_std-j__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_3_d}
    \end{subfigure} 
    \caption{Graphs showing how the distribution of samples of the adhesivity, $j^{1}$, varies with the cell size, $N$, in a six-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    \subref{fig:exp-param-dist_6-reg_3_a}
    The case $\sigma=0.03$.
    Coloured bars show a histogram of the distribution, $\mathcal{J}$, of the adhesivity, $j^{1}$, for various $N$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate pdf of $\mathcal{J}$.
    \subref{fig:exp-param-dist_6-reg_3_b}
    A similar plot to \subref{fig:exp-param-dist_6-reg_2_a}, but using $\sigma=0.3$. 
    \subref{fig:exp-param-dist_6-reg_3_c}
    The mean (blue dots) and standard deviation (orange dots) of the adhesivity distribution, $\mathbb{E}[j^{1}]$ and $\mathbb{S}[j^{1}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue curve shows the adhesivity of the mean six-regular network, $\bar{j}_6=(4/\sqrt{3})\bar{G}\text{cdf}(\bar{G})$ (as we will show).
    The dashed blue line shows the approximate mean adhesivity of an infinite non-periodic six-regular network, $\bar{j}^{1}_{N\rightarrow\infty}$, approximated using $N=200$.
    Note that this is almost invisible to the eye, since it lies almost exactly below the solid blue line.
    The solid orange line shows a fit to the standard deviation, achieved by plotting in log-log space.
    \subref{fig:exp-param-dist_6-reg_3_d}
    A similar plot to \subref{fig:exp-param-dist_6-reg_2_c}, but using $\sigma=0.3$.}
    \label{fig:exp-param-dist_6-reg_3}
\end{figure}

\paragraph{Blocked pore distribution}
To predict the values that the adhesivity takes, we analyse the probability distribution 
for the number of pores that are blocked in the cell. 
The procedure is similar to before. 
We begin by noting that 

\begin{align}
    j^1 
    & = 
    -\frac{1}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}} (W_j^1 + r^1l^1 - W_i^1)  A_{ji}^{-\bm{r}} H_{ji}^{-r^1} 
    \\
    & \approx 
    -\frac{\bar{G}}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}}  \mathbb{E}[\Delta_{ji}^{-r^1}]  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1} 
    \\
    & = 
    -\frac{\bar{G}}{l^1l^2} \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}}  \cos{\theta_{ij}^{\bm{r}}}  
    A_{ji}^{-\bm{r}} 
    H_{ji}^{-r^1}.
    \\
    & =
    \frac{\bar{G}}{l^1l^2}(b^{1\parallel} + \frac{1}{2}b^{1\angle}),
    \label{eq:6-reg_results_j-is-jtilde}
\end{align}
where we have used the relationship between the pore pressure drop and the angle that the pore makes with the flow direction, \eqref{eq:results_j0-is-cos}. 
That is, this time there are two types of pore in the network: flow-parallel pores and 
flow-diagonal pores, see Figure \ref{fig:exp-param-dist_6-reg_setup_2}. 
As before, the expected pressure drop in pores $ij\bm{r}$ that are parallel to the flow
is $\Delta_{ji}^{-r^1}=\cos{(0)}=1$. 
The expected pressure drop in pores $ij\bm{r}$ that are diagonal to the flow 
is $\Delta_{ji}^{-r^1}=\cos{(\pi/3)}=1/2$.
We define $b^{1}$ to be the number of pores that are blocked in the cell, 
drawn from the distribution $\mathcal{B}$. 
Of these, $b^{1\parallel}$ is the number of
blocked flow-parallel pores, with distribution $\mathcal{B}^{\parallel}$, 
while $b^{1\angle}$ is the number of blocked flow-diagonal pores, 
with distribution $\mathcal{B}^{\angle}$.
Therefore, we define 
\begin{align}
    \tilde{j}^1 =
    \frac{\bar{G}}{l^1l^2}(b^{1\parallel}+\frac{1}{2}b^{1\angle}),
    \label{eq:6-reg_results_definition-of-j0tilde}
\end{align}
and call this the approximate adhesivity for the six-regular network, 
since we have shown that $j^1\approx\tilde{j}^1$ in \eqref{eq:6-reg_results_j-is-jtilde}.
To evaluate the approximate adhesivity, $\tilde{j}^1$, 
we must analyse the structure of the distributions for blocked pores, $\mathcal{B}$, $\mathcal{B}^{\parallel}$, 
and $\mathcal{B}^{\angle}$, as before.

Proceeding similarly to the four-regular case, we note that the event that pore $ij\bm{r}$ blocks due to the concentration $c^{\text{max}}$ is 
a Bernoulli trial with probability $p=\text{cdf}(c^{\text{max}}\bar{G})$, as in \eqref{eq:mm_E_distribution}. 
This time, there are $3N$ unique edges in the cell (see Figure \ref{fig:exp-param-dist_6-reg_setup_2}).
% and one in every three edges are flow-parallel, with the other two being flow-diagonal. 
It follows that the number of pores that block due to the concentration 
$c^{\text{max}}$, $b^{1}$,
% , $b^{1\parallel}$, and $b^{1\angle}$, 
should be drawn from a binomial distribution with parameters 
$n=3N$ and $p=\text{cdf}(c^{\text{max}}\bar{G})$,
so that 
\begin{align}
    b^{1} & \sim \text{Bin}(3N,\text{cdf}(c^{\text{max}}\bar{G}))
    \label{eq:mm_6-reg_b1-distribution}
    \\
    & \approx
    \text{Norm}(3N\text{cdf}(c^{\text{max}}\bar{G}),3N\text{cdf}(c^{\text{max}}\bar{G})(1-\text{cdf}(c^{\text{max}}\bar{G}))).
    \label{eq:mm_6-reg_b1-distribution_normal-approx}
\end{align}
We plot samples of $\mathcal{B}$ in Figure \ref{fig:exp-param-dist_6-reg_4},
and compare these to the theoretical distributions that we have derived in 
\eqref{eq:mm_6-reg_b1-distribution} and \eqref{eq:mm_6-reg_b1-distribution_normal-approx}.
We see that the distributions match as expected, and that the normal 
approximation \eqref{eq:mm_6-reg_b1-distribution_normal-approx} is almost exact, 
since $p=\text{cdf}(c^{\text{max}}\bar{G})\approx1/2$, as in the four-regular case.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__adhe.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4_a}
     \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__adhe.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4_b}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/mean-b_and_std-b__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4_c}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/mean-b_and_std-b__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4_d}
    \end{subfigure}
    \caption{Graphs showing how the distribution of samples of the number of blocked pores, $b^{1}$, varies with the cell size, $N$, in a six-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    \subref{fig:exp-param-dist_6-reg_4_a}
    The case $\sigma=0.03$.
    Coloured bars show a histogram of samples of the distribution of blocked pores, $\mathcal{B}$, for various $N$.
    The corresponding binomial distribution is plotted with correspondingly coloured dots for each $N$. 
    The corresponding normal distribution is plotted with correspondingly coloured solid lines for each $N$.
    \subref{fig:exp-param-dist_6-reg_4_b}
    A similar plot to \subref{fig:exp-param-dist_6-reg_2_a}, but using $\sigma=0.3$. 
    \subref{fig:exp-param-dist_6-reg_4_c}
    The mean (blue dots) and standard deviation (orange dots) of the blocked pore distribution, $\mathbb{E}[b^{1}]$ and $\mathbb{S}[b^{1}]$, as functions of $N$, for $\sigma=0.03$. 
    The blue line shows the mean of the corresponding binomial distribution, $np=3N\text{cdf}(\bar{G})$.
    The orange line shows the standard deviation of the corresponding binomial distribution, $(np(1-p))^{1/2}=(3N\text{cdf}(\bar{G})(1-\text{cdf}(\bar{G})))^{1/2}$.
    \subref{fig:exp-param-dist_6-reg_4_d}
    A similar plot to \subref{fig:exp-param-dist_6-reg_2_c}, but using $\sigma=0.3$.}
    \label{fig:exp-param-dist_6-reg_4}
\end{figure}

One of every three unique edges in the six-regular network is flow-parallel, with the other two 
being flow-diagonal (see Figure \ref{fig:exp-param-dist_6-reg_setup_2}). 
It follows that we expect the number of flow-parallel pores that are blocked 
to satisfy
\begin{align}
    b^{1\parallel} & \sim \text{Bin}(N,\text{cdf}(c^{\text{max}}\bar{G}))
    \label{eq:mm_6-reg_b1-hori-distribution}
    \\
    & \approx
    \text{Norm}(N\text{cdf}(c^{\text{max}}\bar{G}),N\text{cdf}(c^{\text{max}}\bar{G})(1-\text{cdf}(c^{\text{max}}\bar{G}))),
    \label{eq:mm_6-reg_b1-hori-distribution_normal-approx}
\end{align}
and the number of flow-diagonal pores that are blocked 
to satisfy
\begin{align}
    b^{1\angle} & \sim \text{Bin}(2N,\text{cdf}(c^{\text{max}}\bar{G}))
    \label{eq:mm_6-reg_b1-diag-distribution}
    \\
    & \approx
    \text{Norm}(2N\text{cdf}(c^{\text{max}}\bar{G}),2N\text{cdf}(c^{\text{max}}\bar{G})(1-\text{cdf}(c^{\text{max}}\bar{G}))).
    \label{eq:mm_6-reg_b1-diag-distribution_normal-approx}
\end{align}
Note that the theoretical distribution of blocked flow-parallel pores, \eqref{eq:mm_6-reg_b1-hori-distribution}, 
is identical to that in the four-regular case, \eqref{eq:results_b-distribution}. 
This helps to explain why we see a substantially larger adhesivity in the six-regular  
network (see the dashed blue lines in Figure \ref{fig:exp-param-dist_6-reg_3}) than in the four-regular network
(compare to the dashed blue lines in Figure \ref{fig:exp-param-dist_4-reg_3}). 
The same average number of flow-parallel pores block, $np=N\text{cdf}(c^{\text{max}}\bar{G})$, but, 
on average, 
$np=2N\text{cdf}(c^{\text{max}}\bar{G})$
flow-diagonal pores 
also block, which contributes to the adhesivity of the network, since 
these have non-zero pore pressure drop in the flow direction, $\Delta_{ij}^{r^1}=1/2$.
We plot samples of $\mathcal{B}^{1\parallel}$ and $\mathcal{B}^{1\angle}$
in Figure \ref{fig:exp-param-dist_6-reg_4-1}. 
As we expect, samples are distributed binomially subject to \eqref{eq:mm_6-reg_b1-hori-distribution} 
and \eqref{eq:mm_6-reg_b1-diag-distribution}, with any small difference due to slight undersampling. 
The accuracy of the normal approximations, \eqref{eq:mm_6-reg_b1-hori-distribution} 
and \eqref{eq:mm_6-reg_b1-diag-distribution}, increases with the cell size, $N$, as we would expect.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__adhe_hori.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4-1_a}
     \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__adhe_hori.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4-1_b}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__adhe_diag.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4-1_c}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__adhe_diag.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_4-1_d}
    \end{subfigure}
    \caption{Graphs showing the distributions of samples of the number of blocked flow-parallel (top) and flow-diagonal (bottom) pores, $b^{1\parallel}$ and $b^{1\angle}$, for various cell sizes, $N$, in a six-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    The cell sizes are $N=2$ and $N=8$.
    Coloured bars show a histogram of samples of the distribution.
    The corresponding binomial distribution is plotted with correspondingly coloured dots. 
    The corresponding normal distribution is plotted with correspondingly coloured solid lines.
    \subref{fig:exp-param-dist_6-reg_4-1_a}
    The case $\sigma=0.03$ for the distribution of $\mathcal{B}^{\parallel}$.
    \subref{fig:exp-param-dist_6-reg_4-1_b}
    The case $\sigma=0.3$ for the distribution of $\mathcal{B}^{\parallel}$.
    \subref{fig:exp-param-dist_6-reg_4-1_c}
    The case $\sigma=0.03$ for the distribution of $\mathcal{B}^{\angle}$.
    \subref{fig:exp-param-dist_6-reg_4-1_d}
    The case $\sigma=0.3$ for the distribution of $\mathcal{B}^{\angle}$.}
    \label{fig:exp-param-dist_6-reg_4-1}
\end{figure}


We plot samples of $j^{1}$ (blue lines),  and the values at which peaks in 
the distribution $\tilde{j}^1$ occur (orange lines) in Figure \ref{fig:exp-param-dist_6-reg_5} (compare with the four-regular case 
in Figure \ref{fig:exp-param-dist_4-reg_5_d}). 
We see that $j^{1}$ is indeed well approximated  by the distribution 
of $\tilde{j}^1$, as we predicted in \eqref{eq:6-reg_results_j-is-jtilde}. 
From \eqref{eq:6-reg_results_definition-of-j0tilde}, we see that $\tilde{j}^1$ is dictated by the distributions of $b^{1\parallel}$ and $b^{1\angle}$, 
\eqref{eq:mm_6-reg_b1-hori-distribution} and \eqref{eq:mm_6-reg_b1-diag-distribution},
and so we see that distinct peaks of $j^{1}$ occur close to discrete values that
correspond to different numbers of edges being blocked. 
This time these values are $\bar{G}/\sqrt{3}N$ apart, 
since this is the adhesivity that corresponds to a single flow-diagonal pore being blocked. 
Here, and throughout, the factor of $\sqrt{3}$ emerges from the scaling of $l^1$ and $l^2$ 
to aid comparison with the four-regular case.
As before, the widths of the spikes in $j^{1}$ are proportional to the standard deviation, $\sigma$, of the conductance 
distribution, $\mathcal{G}$. As a result, when $\sigma$ is larger, for example $\sigma=0.3$ in Figures 
\ref{fig:exp-param-dist_6-reg_5_b} and \ref{fig:exp-param-dist_6-reg_5_d}, discrete structure is 
lost, because the contribution to the adhesivity from one particular number of pores being blocked 
merges with that of another.
The width of the overall distribution is bounded 
% by zero and $(4/\sqrt{3})\bar{G})$, 
% since these values correspond to the 
by the values that correspond to the cases where no pores and all pores block, respectively. 

\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
    \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__depo__with_approx_N=2.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_5_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__depo__with_approx_N=2.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_5_b}
    \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.03/alpha-mean/prob_density__v__depo__with_approx_N=2.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_5_c}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6reg/sigma-0.3/alpha-mean/prob_density__v__depo__with_approx_N=8.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-reg_5_d}
    \end{subfigure}
    \caption{Graphs showing the distribution of samples of the adhesivity, $j^{1}$, in a six-regular network.
    The parameters are $\mu=0.5$, $S=10,000$ and $\sigma=0.03$ (left) or $\sigma=0.3$ (right).
    Coloured bars show a histogram of the distribution, $\mathcal{J}$, of the adhesivity, $j^{1}$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate pdf of $\mathcal{J}$.
    Dashed orange lines show the discrete locations at which probability in the distribution 
    of the approximate adhesivity, $\tilde{j}^1$, is found.
    \subref{fig:exp-param-dist_6-reg_5_a}
    $N=2$ and $\sigma=0.03$.
    \subref{fig:exp-param-dist_6-reg_5_b}
     $N=2$ and $\sigma=0.3$.
    \subref{fig:exp-param-dist_6-reg_5_c}
     $N=8$ and $\sigma=0.03$.
    \subref{fig:exp-param-dist_6-reg_5_d}
     $N=8$ and $\sigma=0.3$.}
    \label{fig:exp-param-dist_6-reg_5}
\end{figure}



\paragraph{Finite cell error}
Returning to the plots of the adhesivity distribution in Figure \ref{fig:exp-param-dist_6-reg_3}, 
in Figures \ref{fig:exp-param-dist_6-reg_3_c} and \ref{fig:exp-param-dist_6-reg_3_d}, 
we see that as $N$ increases, the average adhesivity of the cell, $\mathbb{E}[j^{1}]$ (blue dots)
approaches the approximate average adhesivity of the network, $\bar{j}^{1}_{N\rightarrow\infty}$ (dashed blue line).
Thus, as with the permeability, the adhesivity error that we make when we replace a large network with a smaller cell of size $N$, $E_{\text{c}}^{j}$, decreases with $N$. 
For example, in Figure \ref{fig:exp-param-dist_6-reg_3_d}, we see that the blue dots lie on the 
dashed blue line for cell sizes $N\ge18$, which demonstrates that relatively small cells capture the behaviour of the adhesivity 
of six-regular networks, even when the underlying conductance distribution, $\mathcal{G}$, is relatively wide.


\paragraph{Mean network error}
As usual, the nonlinearity of the adhesivity formula, 
\eqref{eq:mm_adhesivity_reduced}, means that the 
adhesivity of six-regular networks with random conductances is not equal to that 
of the mean six-regular network.
In fact, when the conductance distribution is wider, 
comparing the dashed blue line with the solid blue line in Figure \ref{fig:exp-param-dist_6-reg_3_d}, 
we see that the mean network error, $E_{\text{n}}^{j}$, that we would make if we used 
$\bar{j}_6$ in place of $\bar{j}^{11}_{N\rightarrow\infty}$, is large (compared with the equivalent in a four-regular network, for example). 
For example, this error is around $23\%$ for $\sigma=0.3$, which is considerably larger 
than the mean network error associated with the permeability, $E_{\text{n}}^{k}$. 
On the other hand, when the conductance distribution is more narrow, 
we see that the error is much smaller. 
In Figure \ref{fig:exp-param-dist_6-reg_3_c}, 
we see that when $\sigma=0.03$, $E_{\text{n}}^{j}\approx5\%$.
The behaviour of $E_{\text{n}}^{j}$ with $\sigma$ is similar to that in the four-regular case, and thus we do not 
conduct a more detailed analysis here.

\paragraph{Sampling error}
Lastly, in Figures \ref{fig:exp-param-dist_6-reg_3_c} and \ref{fig:exp-param-dist_6-reg_3_d}, 
we see that the standard deviation of the adhesivity (see orange dots) decreases like that of the permeability, 
proportionally to $N^{-1/2}$ (see orange lines), so that the disadvantages of larger sampling errors associated with the use of smaller cells are favourable over the disadvantages of larger computational costs associated 
with using larger cells.


\subsection{Six-irregular structure}
Next, we assess the impact of random connectivity on the effective properties of the network. Our results so far suggest that networks 
with higher average degree have larger permeability and adhesivity, even when  
the conductivities of their pores 
are drawn from the same distribution. 
We are interested in how well regularly structured networks characterise the effective properties of  
networks with random structure, since regular networks are computationally cheaper to construct. 
We focus on six-irregular networks.
These have an average degree of six, but 
individual nodes may be connected to fewer or more than six other nodes.

To construct a cell of this network, an example of which is shown in Figure \ref{fig:exp-param-dist_6-ireg_setup_2}, we draw the $x^1$ and $x^2$ coordinates of $N$ points from a uniform distribution, and place nodes at these points. 
We then use a Delaunay triangulation to connect these nodes. 
We choose the dimensions of each cell to be $L^1=\sqrt{N}$ and $L^2=\sqrt{N}$, to preserve unit density, and choose a fixed resistance per unit length ($1/1.72$, to make comparison with the other two structures simple), so that pore conductances are inversely proportional to the distance between nodes (see the thicknesses of edges in Figure \ref{fig:exp-param-dist_6-ireg_setup_2}, which are proportional to their conductance. Shorter edges are thicker and thus have larger conductance).
The average degree of the resulting cell is six, which is a property of the Delaunay triangulation method \cite{meijering1953interface}.


\begin{figure}[!htb]
    \begin{center}
    \includegraphics[width=0.5\textwidth]{./figures/multiscale/stats/6ireg/setup/cells_6-ireg.pdf}
    % \begin{overpic}[abs,unit=1mm,width=0.95\textwidth,grid]{./figures/multiscale/drawings/network_3.pdf}
    % %\linethickness{0.1cm}
    % %\put(5,36){\color{red}\vector(1,0){10}}
    % \put(2,38){\parbox{1.2cm}{Particles into filter}}
    % \put(130,38){\parbox{1.2cm}{Particles out of filter}}
    % \end{overpic}
        \caption{Diagram of a cell with size $N=4$
        in a six-irregular network. 
        Nodes and edges that are 
        part of the cell are coloured red, and the cell is highlighted with a red box.
        The dimensions of the cell are set to be $L^1=\sqrt{N}$ and $L^2=\sqrt{N}$ so that the node density, $\varrho$, satisfies $\varrho=1$ in a cell of any size, $N$.
        Node positions are drawn uniformly, and edge lengths are drawn from the distribution that results from node positions and the triangulation.  
        Edge thicknesses are proportional to their 
        inverse lengths, so that shorter edges are thicker.
} 
        \label{fig:exp-param-dist_6-ireg_setup_2}
    \end{center}
\end{figure}


\subsubsection{Permeability}

We plot samples of the permeability distribution for the six-irregular case in 
Figure \ref{fig:exp-param-dist_6-ireg_2} (similarly to Figure \ref{fig:exp-param-dist_6-reg_2}).
These figures agree well (up to the randomness caused by sampling) 
with those presented in \cite{chapman2021homogenization} (Figure 5).
The first thing we note is that the qualitative behaviour of $\mathcal{K}$ is relatively similar to 
that in the four-regular and six-regular cases, despite the fact that the 
underlying conductance distribution, $\mathcal{G}$, is no longer log-normal. 
Despite this, in Figure \ref{fig:exp-param-dist_6-ireg_2_a}, we see that the distribution has multiple peaks 
when the cell size is relatively small. 
This is particularly noticeable in the case $N=4$ (blue line), for example. 
These peaks are real and are due 
to variance in the ratio of inter- and intra-cell pores when $N$ is smaller, as opposed 
to being caused by undersampling. This variance is not present in regular structures (see Figure \ref{fig:exp-param-dist_6-reg_2}, for example), since the number of pores that cross cell boundaries does not vary from sample to sample.

\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6ireg/sigma/alpha-mean/prob_density__v__perm__with_approx.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-ireg_2_a}
     \end{subfigure}
     %\hfill
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6ireg/sigma/alpha-mean/mean-k_and_std-k__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-ireg_2_c}
    \end{subfigure}
    \caption{Graphs showing how the distribution of samples of the permeability, $k^{11}$, varies with the cell size, $N$, in a six-irregular network.
    The number of samples is $S=10,000$.
    \subref{fig:exp-param-dist_6-ireg_2_a}
    Coloured bars show a histogram of the distribution, $\mathcal{K}$, of the permeability, $k^{11}$, for various $N$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate pdf of $\mathcal{K}$.
    \subref{fig:exp-param-dist_6-ireg_2_c}
    The mean (blue dots) and standard deviation (orange dots) of the permeability distribution, $\mathbb{E}[k^{11}]$ and $\mathbb{S}[k^{11}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue curve shows the permeability of the mean six-regular network, $\bar{k}_6$.
    The dashed blue line shows the approximate mean permeability of an infinite non-periodic six-irregular network, $\bar{k}^{11}_{N\rightarrow \infty}$, approximated using $N=100$.
    The solid orange line shows a fit to the standard deviation, achieved by plotting in log-log space.}
    \label{fig:exp-param-dist_6-ireg_2}
\end{figure}

\paragraph{Finite cell error}
Similarly to regular networks, as $N$ increases, in Figure \ref{fig:exp-param-dist_6-ireg_2_a} we see 
that the permeability approaches a uni-modal distribution.
This implies that the error caused by replacing a large network with a finite cell of size $N$, $E_{\text{c}}^k$, decreases 
as $N$ increases. 
We confirm this by comparing the blue dots with the dashed blue line in Figure \ref{fig:exp-param-dist_6-ireg_2_c}. 
We see that the convergence of $\mathbb{E}[k^{11}]$ to $k^{11}_{N\rightarrow\infty}$ is much slower here than in the regular networks, because of the variance in the number of inter-cell edges. 
Even so, the average permeability of a cell with around $N=36$ nodes is similar to that of a much larger network.  

\paragraph{Mean network error}
Importantly, comparing the dashed and solid blue lines in Figure \ref{fig:exp-param-dist_6-ireg_2_c}, 
we see that the average permeability of the six-irregular network is close to that of the mean six-regular network. 
As we hypothesised, the average degree of nodes in the cell plays an important (perhaps more important than the conductance distribution) in determining the permeability of the network.
That the exact number of nodes that each node connects to is different in six-regular and 
six-irregular networks appears to have little effect, so that replacing the six-irregular network with the mean six-regular network leads to small errors, $E_{\text{n}}^k$. 
One implication is the potential for decreased computational cost of simulation, since 
regular structures can be significantly cheaper to program than irregular ones.

\paragraph{Sampling error}
In Figure \ref{fig:exp-param-dist_6-ireg_2_c}, we see that, despite the multiple peaks for smaller values of $N$, 
the standard deviation of the permeability of six-irregular networks decreases like $N^{-1/2}$. 
Thus, even though, in general, much larger cells are required for representation of large irregular than is the case for regular ones, it is still advantageous to average over many samples of small cells as opposed to few samples of large ones. 


\subsubsection{Adhesivity}

Lastly, we analyse the adhesivity of six-irregular networks. 
We plot samples in Figure \ref{fig:exp-param-dist_6-ireg_3}. 
The first thing we note is that the adhesivity distribution, $\mathcal{J}$, does not focus around discrete values, 
as it does for regular networks with small conductance variance. 
This is because of the increase in the width of the initial conductance distribution, $\mathcal{G}$, 
which is caused by the variability in pore lengths that result from the uniform distribution 
of junction locations.
For example, we see that the six-irregular adhesivity distribution resembles the regular case $\sigma=0.3$ in 
Figure \eqref{fig:exp-param-dist_6-reg_3_b} much more 
than the case $\sigma=0.03$ in Figure \eqref{fig:exp-param-dist_6-reg_3_a}.

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6ireg/sigma/alpha-mean/prob_density__v__depo__with_approx.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-ireg_3_a}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6ireg/sigma/alpha-mean/mean-j_and_std-j__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-ireg_3_c}
    \end{subfigure}
    \caption{Graphs showing how the distribution of samples of the adhesivity, $j^{1}$, varies with the cell size, $N$, in a six-irregular network.
    The number of samples is  $S=10,000$.
    \subref{fig:exp-param-dist_6-ireg_3_a}
    Coloured bars show a histogram of the distribution, $\mathcal{J}$, of the adhesivity, $j^{1}$, for various $N$.
    Correspondingly coloured lines interpolate the histogram at bar midpoints, 
    to show an approximate pdf of $\mathcal{J}$.
    \subref{fig:exp-param-dist_6-ireg_3_c}
    The mean (blue dots) and standard deviation (orange dots) of the adhesivity distribution, $\mathbb{E}[j^{1}]$ and $\mathbb{S}[j^{1}]$, as functions of $N$.
    The solid blue curve shows the adhesivity of the mean six-regular network, $\bar{j}_6=(4/\sqrt{3})\bar{G}\text{cdf}(\bar{G})$.
    The dashed blue line shows the approximate mean adhesivity of an infinite non-periodic six-irregular network, $\bar{j}^{1}_{N\rightarrow\infty}$, approximated using $N=100$.
    The solid orange line shows a fit to the standard deviation, achieved by plotting in log-log space.}
    \label{fig:exp-param-dist_6-ireg_3}
\end{figure}

\paragraph{Blocked pore distribution}
The loss of the discrete adhesivity structure observed in regular networks is not related to a change in the behaviour 
of the distribution that governs the number of pores that are blocked, $b^1\sim\mathcal{B}$. 
 To show this, we plot samples (coloured bars) of this distribution in Figure \ref{fig:exp-param-dist_6-ireg_4_a}, 
 and compare these with the distribution of blocked pores in the six regular case (coloured dots and 
 lines). 
 We see that (see Figure \ref{fig:exp-param-dist_6-ireg_4_c}), for any given cell size, $N$, the average number of blocked pores is considerably larger 
 than that in the six-regular case. 
 This suggests a skew in the conductance distribution that encourages 
 more initial conductances to lie below the threshold conductance, due to more long pores. 
 That is, the mean lies further to the left of the median in the six-irregular construction than it does 
 in the log-normal distribution associated with the six-regular construction. 
 This helps to explain why the mean adhesivity of a six-irregular network 
 (see the dashed blue line in Figure \ref{fig:exp-param-dist_6-ireg_3_c} is considerably larger than 
 that of a six-regular network (compare with the dashed blue line in Figure \ref{fig:exp-param-dist_6-reg_3_b}.
 

\begin{figure}[!htb]
     \centering
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6ireg/sigma/alpha-mean/prob_density__v__adhe.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-ireg_4_a}
     \end{subfigure}
     \begin{subfigure}[b]{0.45\textwidth}
         \centering
         \includegraphics[width=\textwidth]{./figures/multiscale/stats/6ireg/sigma/alpha-mean/mean-b_and_std-b__v__N.pdf}
         \caption{}
         \label{fig:exp-param-dist_6-ireg_4_c}
    \end{subfigure}
    \caption{
    Graphs showing how the distribution of samples of the number of blocked pores, $b^{1}$, varies with the cell size, $N$, in a six-irregular network.
    The number of samples is $S=10,000$.
    \subref{fig:exp-param-dist_6-ireg_4_a}
    Coloured bars show a histogram of samples of the distribution of blocked pores, $\mathcal{B}$, for various $N$.
    The corresponding binomial distribution for a six-regular network is plotted with correspondingly coloured dots for each $N$. 
    The corresponding normal distribution for a six-regular network is plotted with correspondingly coloured solid lines for each $N$.
    \subref{fig:exp-param-dist_6-ireg_4_c}
    The mean (blue dots) and standard deviation (orange dots) of the blocked pore distribution, $\mathbb{E}[b^{1}]$ and $\mathbb{S}[b^{1}]$, as functions of $N$, for $\sigma=0.03$. 
    The solid blue line shows the mean of the corresponding binomial distribution for a six-regular network, $np=3N\text{cdf}(\bar{G})$.
    The dashed blue and orange lines show interpolations of the correspondingly coloured dots, to aid the eye.
}
    \label{fig:exp-param-dist_6-ireg_4}
\end{figure}

\paragraph{Finite cell error}
The finite cell error for adhesivity, $E_{\text{c}}^j$, converges to zero relatively slowly 
with $N$ (see the blue dots and dashed blue line in Figure \ref{fig:exp-param-dist_6-ireg_3_c}).
 A cell of size $N\approx49$ is required to capture the adhesivity of larger irregular networks correctly. 
 This is larger than for the six-regular networks.
 This is unsurprising, noting the larger variance in the conductance distribution caused by the variance 
 in pore lengths.

\paragraph{Mean network error}

Comparing the blue solid line to the blue dashed line in Figure \ref{fig:exp-param-dist_6-ireg_3_c}, 
we note that the error in the adhesivity that we would make if we replaced 
the six-irregular network with the mean six-regular network is relatively small ($E_{\text{n}}^j\approx5\%$), especially 
considering that their conductances are drawn from distributions with different statistics. 
% Differences in these statistics are less relevant
 % in the network as a whole than in particular cells. 

\paragraph{Sampling error}
Unlike the permeability, the standard deviation of the adhesivity of six-irregular networks is more similar 
to that of six-regular ones, since the relative proportion of intra- and inter-cell pores is irrelevant in 
determining the adhesivity.
Even though a cell of $N\approx49$ is required to capture the adhesivity of a large network 
with low error, it remains computationally advantageous to accept the increased cost incurred by repeated sampling of this relatively small cell.
% , since the decrease in the error associated with choosing an even larger cell is not outweighed 

\subsection{Structure comparison}
In this section we have analysed samples of the permeability and adhesivity of four-regular, six-regular, and six-irregular structures. 
We summarise our findings by comparing the means and standard deviations of 
these samples, $\mathbb{E}[k^{11}]$ and $\mathbb{S}[k^{11}]$, which we plot together in Figure \ref{fig:exp-param-dist_comparison}. 


\begin{figure}[!htb]
     \centering
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
    \includegraphics[width=\textwidth]{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf}
         \caption{}
         \label{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf}
     \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
    \includegraphics[width=\textwidth]{./figures/multiscale/stats/comparison/compare__sd-k__v__N.pdf}
         \caption{}
         \label{./figures/multiscale/stats/comparison/compare__sd-k__v__N.pdf}
     \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
    \includegraphics[width=\textwidth]{./figures/multiscale/stats/comparison/compare__mean-j__v__N.pdf}
         \caption{}
         \label{./figures/multiscale/stats/comparison/compare__mean-j__v__N.pdf}
     \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
         \centering
    \includegraphics[width=\textwidth]{./figures/multiscale/stats/comparison/compare__sd-j__v__N.pdf}
         \caption{}
         \label{./figures/multiscale/stats/comparison/compare__sd-j__v__N.pdf}
     \end{subfigure}
    \caption{Graphs showing how the mean (left) and standard deviation (right) of samples of the permeability (top), $k^{11}$, 
    and adhesivity (bottom), $j^1$, vary with the cell size, $N$.
    The parameters are $S=10,000$ and $\mu=0.3$.
    The four-regular, six-regular, and six-irregular cases are shown with blue, orange, and 
    green correspondingly shaped markers.
    \subref{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf}
    The mean of the permeability distribution, $\mathbb{E}[k^{11}]$,
    as a function of the cell size, $N$.
    The limit $\bar{k}^{11}_{N\rightarrow\infty}$ is shown for each case in a correspondingly coloured dashed line. 
    The permeabilities of the mean four-regular and six-regular networks, $\bar{k}_4$ and $\bar{k}_6$, 
    are shown in blue and orange solid lines, respectively.
    \subref{./figures/multiscale/stats/comparison/compare__sd-k__v__N.pdf}
    The standard deviation of the permeability distribution, $\mathbb{S}[k^{11}]$,
    as a function of the cell size, $N$.
    The solid lines of corresponding colour show fits achieved by plotting in log-log space.
    \subref{./figures/multiscale/stats/comparison/compare__mean-j__v__N.pdf}
    A similar plot as \subref{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf}, 
    but for the mean of the adhesivity distribution, $\mathbb{E}[j^{1}]$.
    \subref{./figures/multiscale/stats/comparison/compare__sd-j__v__N.pdf}
    A similar plot as \subref{./figures/multiscale/stats/comparison/compare__sd-k__v__N.pdf}, 
    but for the standard deviation of the adhesivity distribution, $\mathbb{S}[j^{1}]$.
}
    \label{fig:exp-param-dist_comparison}
\end{figure}

\paragraph{Finite cell error}
In \cite{chapman2021homogenization}, the authors find that, 
as the size of the cell increases, the average permeability of all three networks
approaches a fixed value. 
Our results agree. 
In Figure \ref{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf}, 
we see that as $N$ increases, $\mathbb{E}[k^{11}]$ (see the coloured markers) approaches $k^{11}_{N\rightarrow\infty}$ (see the dashed lines of corresponding colour), 
which can be thought of as the average permeability of a large network with the corresponding structure.
The implication is that, in purely advective macroscale systems, in which particles 
do not interact with pores and cause changes to the structure of the microscale network, 
cells of sizes as small as $N\approx25$ can be used to pre-calculate the permeability parameter of the system, even when the network is irregular. 
In Figure \ref{./figures/multiscale/stats/comparison/compare__mean-j__v__N.pdf}, 
we see that samples of the adhesivity behave similarly, 
so that $\mathbb{E}[j^{1}]$ (see the coloured markers) approaches $j^{1}_{N\rightarrow\infty}$ (see the dashed lines of corresponding colour) as $N$ increases. 
This means that our conclusion can be extended to advection--reaction systems, in which particles do interact with pores in 
the underlying network. 
Convergence of the adhesivity is slightly slower than that of the permeability and so slightly larger cells, say $N\approx49$ (at worst, when the network is irregular), are required for advection--reaction systems than those required for advection only. 
Nevertheless, pre-calculation of the parameters is possible even when the network is dynamic, which decreases the computational expense of solution of multiscale advection--reaction systems by orders of magnitude.



\paragraph{Mean network error}
The relatively small differences in the dashed and solid lines in Figures \ref{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf}
indicates that the permeability of networks is approximated well by a network 
with the same average degree and the mean conductance in every pore. 
Comparing the equivalent lines in \ref{./figures/multiscale/stats/comparison/compare__mean-j__v__N.pdf}, 
we see this approximation is significantly worse for the adhesivity. 
One implication is that, in purely advective systems on static networks, using the statistics of the underlying 
conductance distribution to calculate the permeability will not lead to large errors in the solution. 
Using the conductance statistics to calculate the adhesivity in advection--reaction equations on dynamics networks will lead to much larger errors in the effective solution.

% The permeability is very stronlgy linked to the average degree. 
% This implies that if we know the average degree in a ireegular structure we may as well 
% use the regular structure with the same average degree, since it is cheaper. 
% Furthermore, we might argue that we don't need to sample at all, since the values lie so close 
% to the mean network value that we could use that and accept the error, since this is likely to 
% be smaller than the erro assoacited wth any particular single sample, due to how large the stadrd deviation is. 
% Unofurtinatyrly, this doues not extend to the adhesivity.
%  Irregular value isnt very close to the regular value, and neither of them are particularly well approximated by the 
%   mean network value either.

\paragraph{Sampling error}
Comparing Figures \ref{./figures/multiscale/stats/comparison/compare__mean-k__v__N.pdf} and \ref{./figures/multiscale/stats/comparison/compare__mean-j__v__N.pdf}, 
we see that the standard deviations of the permeability and adhesivity samples, $\mathbb{S}[k^{11}]$ and $\mathbb{S}[j^{1}]$, 
are roughly the same size for a given structure (compare lines of the same colour), and, moreover, they behave in the same way as functions of $N$ (decreasing like $N^{-1/2}$).
Since pre-calculation of the adhesivity, using \eqref{eq:mm_adhesivity_reduced}, 
has the same computational cost as that of the permeability, \eqref{eq:mm_permeability_reduced}, 
the conclusion that is computationally advantageous to sample the smallest cell possible multiple 
times, rather than sampling a larger cell fewer times, extends to systems on networks with dynamic structure. 

\chapter{Conclusions for the multiscale model}
\label{chap:mm_Conclusions the multiscale model}

In Part \ref{part:A multiscale model for filtration}, Chapters \ref{chap:mm_Introduction to the multiscale model}--\ref{chap:mm_Solutions of the multiscale model}, 
we motivated, derived, and solved a novel multiscale model for filtration. 
This strikes a useful compromise 
between computationally expensive network models that accurately capture microscale effects, and macroscale models that  
ignore microscale information but are usually computationally cheaper. 

\section{Summary}

\subsection{Motivation for the model}
In Chapter \ref{chap:mm_Introduction to the multiscale model}, 
we began by introducing multiscale models as a candidate framework for filtration simulations.
In particular, in 
Section \ref{sec:mm_Multiple scales in filration}, we highlighted the multiscale 
nature of the filtration process. 
Particle and pore interactions 
on the microscale dictate flow and particle 
concentration behaviour on the macroscale.
This dependence results in computational feasibility problems for network models in which the entire pore space is resolved. 

Multiscale models were introduced as a potential alternative in Section \ref{sec:mm_Existing multiscale models for filtration}. Examples that consider a continuum
microscale  are common. 
Those that model the microscale as a network are less common, but network homogenisation methods have been used to derive the effective permeability, most notably in \cite{chapman2021homogenization}. 

In Section \ref{sec:mm_A novel multiscale model for filtration}, we motivated a novel multiscale model that accounts for the deposition of particles, as well as their transport. 
This causes changes to the structure of the network, which manifests in a second effective parameter, the adhesivity, in addition to the permeability in \cite{chapman2021homogenization}.  

\subsection{Derivation of the model}
In Chapter \ref{chap:mm_Derivation of the multiscale model}, 
we derived the multiscale model. 
In particular, in Section \ref{sec:mm_Microscale model}, 
we began by adapting the microscale model that we derived and solved in Part \ref{part:A network model for filtration} into a system defined on a repeated cell, representative of the overall microscale, which forms a periodic network with initial conditions corresponding to mono-dispersed and log-normally poly-dispersed pore sizes and thus conductances.
Nondimensionalising, we found that the microscale model depends on three dimensionless parameter groups: 
$\alpha$ (the adherence parameter); 
$\beta$ (the particle size parameter); 
and $\varepsilon$ (the ratio of the cell length to the network length). 

In Section \ref{sec:mm_Network homogenisation methodology}, 
we outlined the process of network homogenisation. Considering periodic node and edge indices as position references that are independent of the macroscale position reference, 
asymptotically expanding quantities defined at nodes and edges, and seeking solutions at the correct asymptotic order,  we explained how to average the network model to obtain an effective model, 
the solution of which approaches that of the network model asymptotically in the small cell limit $\varepsilon\rightarrow 0$.

In Sections \ref{sec:1d_network-homogenisation} and \ref{sec:Dd_network-homogenisation}, we averaged our periodic network model,
firstly in a quasi-one-dimensional setting, 
and secondly in the case where the network is embedded in space of arbitrary dimension. 
We homogenised:
\begin{itemize}
    \item a linear algebraic equation describing fluid mass conservation in nodes. 
    The result is a continuous partial differential equation that describes the spatial independence of the flow rate; 
    \item a definition of the average microscale velocity. 
    The result is the Darcy equation, a partial differential equation describing the proportionality between the effective flow rate and pressure gradient, via the effective permeability;
    \item an ordinary differential equation describing the conservation of the mass of particles as they are transported  and deposited within the microscale geometry. 
    The result is an advection--reaction equation for the effective particle 
    concentration. 
    The advection and reaction rates are dictated by the effective permeability and adhesivity, which average the effects of microscale transport and deposition;
    \item an ordinary differential equation describing pore volume conservation, and thus conductance changes as deposition occurs. 
    The result is a partial differential equation that relates microscale conductance to macroscale variables. 
\end{itemize}
As such, our multiscale model for filtration is a system of continuous partial differential equations that govern the evolution of effective variables through 
 the continuum geometry of the macroscale. 
The parameters of the system, the effective permeability and adhesivity,  depend on the dynamic structure of the discrete network microscale. This is updated via solution of a coupled linear problem that is the size of the number of nodes in the cell. 


\subsection{Solution of the model}

In Chapter \ref{chap:mm_Solutions of the multiscale model}, 
we found numerical solutions of our multiscale model. 
To this end, in Section \ref{sec:mm_Solution methodology}, we started by averaging over flow-transverse directions, to permit quasi-one-dimensional solutions. 
We then outlined a numerical solution scheme, and suggested a simplifying step that involves pre-calculation of potential conductances and resulting permeability and adhesivity values. This decreases the computational expense of the solution considerably. 

We used this method, in Section \ref{sec:mm_Periodic networks}, 
to compute solutions on a four-regular periodic network. 
Firstly, we considered the deterministic initial network case, in which all edge conductances are equal, which corresponds to mono-dispersed pores. 
We found that a single simulation of the multiscale model approaches that of the network model as $\varepsilon\rightarrow 0$, as expected. 
Secondly, we discussed a random network case, in which conductances are drawn from a log-normal distribution, corresponding to poly-dispersed initial pore sizes. 
Individual simulations qualitatively agree with the network model, but not quantitatively, since each initialisation is generally unique. 
We hypothesised that the statistics of  ensemble solutions of the two models should agree. 

To show this, in Section \ref{sec:mm_Non-periodic networks}, we computed sampled distributions of the permeability and adhesivity by repeatedly solving the multiscale model. 
We analysed these distributions for three networks structures: a four-regular network 
with log-normally distributed initial conductances; a six-regular network with initial conductances drawn from an equivalent log-normal distribution; 
a six-irregular network, with initial conductances drawn from a distribution related to the length of edges between nodes with uniformly distributed positions.
% , but with statsitcs to ease comparison. 
For each structure, we computed the sampled mean and standard deviation of the permeability and adhesivity, 
and compared these with the corresponding parameters of: 
a large network with identical structure;
the mean network (a network with the same coordination number with each edge conductance equal to the mean conductance). 
We reached three key conclusions: 
\begin{itemize}
    \item The `finite cell error' (the difference between the permeability or adhesivity of a large network and that of a smaller cell) 
    decreases like $N^{-1}$ 
    as $N$ increases (where $N$ is the number of nodes in the cell). 
    Therefore, the cell becomes increasingly `representative' of the entire network as its size increases, 
    but the cell need not be too large before it represents the network well. 
    % As such behaviour of large networks can be simulated using only a small cell, 
    Since the pre-calculation step is $\mathcal{O}(N^2)$,
    this fact drastically reduces the computational complexity of the multiscale solution, which is already significantly cheaper than solution of the underlying network model. 
    Only slightly larger cells are needed for the adhesivity than for the permeability, which implies that 
    network homogenisation is a useful 
    method for dynamic network models, as well as the static ones discussed in \cite{chapman2021homogenization};
    \item The `mean network error' (the difference between the permeability or adhesivity of the mean network and that of a smaller representative cell) decreases like $\sigma^2$ as $\sigma$ decreases (where $\sigma$ is the standard deviation of the underlying conductance distribution). 
    As such, the random network 
    is increasingly well approximated by the mean network as the conductance distribution narrows. 
    For certain structures and sufficiently narrow conductance distributions, the permeability and adhesivity of the mean network may be used in place of the mean of the distributions from repeated sampling, decreasing computational complexity further;
    % are unneccesary because the mean permeabiltiy and adhesicvity that these would be used ot produice can be repalced with the mean permeabiltiy and adhesivity of the 
    % mean network, which are given by the p[arameters of the underlying conductnace distrivution. 
    % This further decreases the computational expensoce od the multiscale model.
    \item 
    The `sampling error' (the difference between the permeability or adhesivity from a particular run and the mean of a suitably large sample) decreases with $N^{-1/2}$
    as $N$ increases. 
    Since the cost of calculation of the permeability or adhesivity is $\mathcal{O}(N^{2})$, this means that more repeats using a smaller, `less representative' cell is a computationally advantageous strategy compared to fewer repeats using a larger, `more representative' cell, on average.
\end{itemize}
Additionally, we found that, as the coordination number of the underlying network increases, 
its average permeability and adhesivity increases. 
Consequently, users of network and multiscale models should ensure that the coordination number of the chosen network is sufficiently close to that of the porous material of interest. 

 In summary, results for the 
 advective system on the static networks discussed in \cite{chapman2021homogenization} extend to our advection--reaction system on dynamic networks with the same initial structure.
 That is:
 Network homogenisation leads to an effective system that matches the behaviour of the network system in the limit where many cells constitute the network;
 The effective properties of large networks are approximated well by pre-calculating the statistics of small cells, 
 which reduces computational complexity;
Repeatedly sampling the smallest feasible cells is computationally advantageous.
% , which avoids  
%  solving many linear problems on larger cells.
 However, reaction terms lead to the need for a slightly larger repeating cell.
 % , and use of the underlying parameters of the conductance distribution in place of cell sampling is not really an option.

\section{Further work}

There are several potential avenues for further research.

\subsection{Additional particle and pore interactions}
In general, we used a relatively simple
model for the filter microscale, 
to aid simple demonstration of the key aspects of our novel multiscale framework. 
The derivation and inclusion of more complex network dynamics, which more accurately represent the complex particle--pore interactions that occur physically, would be relatively straightforward
and yet useful. Indeed, one advantage of this framework is its generality: we made no assumptions regarding the connectivity; dimension; size distributions; or initial and boundary conditions on the underlying microscale. Decisions regarding these were purely illustrative.

\subsection{Multiscale and microscale solution comparison}

One next step might be to find the temporally varying velocity and particle concentration distributions that follow from the permeability and 
adhesivity distributions established in 
Section \ref{sec:mm_Non-periodic networks}. 
These should show, for example, that the 
solution of the multiscale model on a randomly generated network does indeed agree statistically  
with the equivalent network model in the  
limit $\varepsilon\rightarrow0$, 
as we conjectured in Subsection \ref{subsec:mm_Polydispersed pores}.
% the flow and particle concentration dsitributions that result from the permeability and adhesivity distributions that we caluclated in the last section.  
% It would be interesting to compare the vleocity and concentration dynamics on networks 
% with different connectivity, 
% and show that their statistics agree in the correct asymptotic limit, which would further justify our multiscale method. 
It might be interesting to compare these results across different structures, for example, the three discussed in Section \ref{sec:mm_Non-periodic networks}. 
The permeability and adhesivity
behaviour, which we discussed at length, gives us a good clue that increasing 
the coordination number increases the average velocity and the reaction rate in the advection--reaction equation, so that particles are transported and captured more rapidly, 
resulting in a concentration profile that drops more quickly as the wave of particles traverses the filter.


% - comparae differnt methods of clauclation... ie show that the distributions of the importan properteis is the same int he asymptotic limit. 

\subsection{Industrially relevant properties}
Our model lends itself to the calculation of properties of industrial interest. 
In \cite{chapman2021homogenization}, Chapman \emph{et al.} discuss the ease with which the tortuosity of a filter is calculated 
when its microscale is reduced to a  network. 
This measure of geometric complexity is one of the key parameters for the determination of a material's transport properties. 
The authors go on to show a correlation between tortuosity and permeability. 
 It would be interesting to use our model to extend this analysis to networks with dynamic connectivity, and to establish the relationship between tortuosity and adhesivity, for example. 

% - Use the macroscale model to caluclate interesting filter proeprties... efficiency adn throughput and show that these are identical to the network ones in Part 2. 

\subsection{Optimisation and porosity gradients}
 
 As we demonstrated in Part \ref{part:A network model for filtration}, 
 network models permit simple derivation of secondary filter properties of industrial interest, such as the particle retention efficiency and total throughput. 
 Future work should 
 exploit the relative computational tractability of this model for the calculation of these properties, without loss of accuracy.
 It would be relatively straightforward, for example, to establish the dependence of these properties on the 
 parameters of the underlying conductance distribution, since repeated calculation is feasibly quick. 
 This highlights the potential for optimisation routines that rely on parameter sweeps or cost function descents. 
One experiment might propose a spatially graded mean for the conductance distribution (which is possible, since we asymptotically expanded each pore conductance, to permit macroscale variation), and then 
find the gradient parameter that maximises the total throughput given a threshold requirement for the particle retention efficiency, thus finding the optimal microscale configuration for process performance. Conclusions regarding the desirable porosity gradients could then be compared to those in \cite{dalwadi2015understanding}
and \cite{dalwadi2016multiscale}, for example.

  % or to explor ethe effect of porotist gradients on these. 
  % This is posible since the initial conductance 
  % distriburtion is controllable as a function of position.
  % Future woudl shoudl expooit the relative computtiaonl tractability of our numerical scheme, and 
  % docus on optimisation of controlabel paramets via parmaeter sweeps and simple cost function descenets.
  % For example, we could gradually vary the  mean of the initial microcale conductance distribution, and 
  % find the required value to obtain a given filter efcinecy. Or linearly grade the stadard devitaion, and efind the optimal gradient for maximal total throughput.

\subsection{Network representation and experimental data}
Industrial data would be a useful addition at both the initial and final stages of this framework. 
As we discussed in the context of network models, data on real porous media
samples would lead to the derivation of initial networks corresponding to real membrane samples.
The results of experiments on the same samples could be used for solution comparison, allowing for free parameter tuning, again made possible by the computational tractability of our multiscale method.
Our conclusions could then be used to quantitatively understand real filtration materials, and aid their design.
% - Explore porosity gradients like the literatre... we have homogisned the conductance so distributions that depends on x are permitted. 
% - Use the multilscae mdoel for optimisation.. we have derived somethign thats much much cheaper than network mdoels, this m eans that parameters swee,s and descent algorithsm involved in otimaution are much  more feasible. 
% Optimisation might include... derivation of the corect mean of the lognoram ldsitruviton in the underlying strucure 
% to obtain a given efficiency for as long as possible... or the correct gradint inthe mean to obtain maimum throughput... 


% What we found out in this part:
% When the network is periodic, we can match the result with the effective model and the results agree exactly.
% When the network is non-random, then the average of the conductance down columns is exactly the local permeability. 
% When the network is random, these are not the same, but if we average the permeability of the cell in the right way then these local permeabilities are the same.
% We found that the effective model is a lot cheaper because at worst its cost is the number of time points times the number of spatial points times the size of cell, rather than the number of time points times the size of the network, which is the cost of the microscale model. 
% Additionally, for certain blocking rules, we can pre-process, which means the cost of the effective solution is actually number of time points times number of space points plus number of discretisations of the total concentration, which we control and is small. Note that we we only have to run this pre-process once, which cheapens further.

% Also analysed non-periodic network case. Showed that in theory we can use the mean effective parameters if we're only interested in the mean of the solution. 
% If we're interested in the distribution of the solution, we can use a small cell rather than the entire network. This will be much cheaper, because the cost is number of times times number of spaces times size of cell... We have shown that in practice the size of the cell only needs to be tiny, $N=4$ for example, to get decent agreement in theory.

% What we didn't do is show using simulations that we really can use a small cell for agreement with non-periodic networks in practice... [I should be able to  do this with current solver... Chooose a network that is one large cell and see what the result is. Then choose an effective model with small cells and show that solutions still agree...if not time then further work].



% Conclusions of chapter...
% \begin{itemize}
%     \item The conclusion of the second part of this chapter will be that we only need to care about the coordination number of the network, and then all randomness can be obtained from randomising conductances (meaning that randomising the network structure is irrelevant). This will have further significant computational complexity benefits (on top of what's alraedy been discussed i.e. calculating the parameters before the time dependent problem, and using small cells), since it is much cheaper to construct a regular network as opposed to an irregular one.
%     \item The take home message from this chapter will be that we can massively reduce the computational complexity of solution of the advection-reaction equation by `preprocessing' in three ways:
%     \begin{itemize}
%         \item Finding effective parameters \emph{before} solving the flow problem, and then simply interpolating at temporal and spatial steps (along with a constraint on the rule for blocking for when this is possible).
%         \item Using either a formula that depends only on parameters of the underlying distribution, $\mu$ and $\sigma$, (when it is narrow enough), or using small cells, i.e. those with $N \ll 1000$ (when it is too wide).
%         \item Using regular cell structures even when the physical network  that is being modelled has an irregular (random) structure.
%     \end{itemize}
% \end{itemize}

% \begin{itemize}
%     \item One further take-home message will be that the results that were observed for fixed networks', that is, effective systems that don't contain adhesion terms that affect the network (studied by Jon in his paper), can be extended to effective systems that do contain reaction terms.
%     What we lose is that the rates of convergence with $N$ are slightly smaller for the adhesivity than they are for the permeability, which means that we need to use larger cells for reaction equations that we do for advection equations.



% ----- end of interestign ---------



    
% Furthermore, note that the permeability term accounts for flow through the network, whilst the adhesivity causes changes to the conductynaces of the edges, and therefore causes the network to be dynamic. 
% This result therefpore suggests that our effective system is able to capture the behaviour of the corresponding dynamical system defined on a \emph{dynamic} non-periodic network, as well as that derived on a \emph{dynamic} periodic network. [make this point in the conclusion rather than here]
% Furthermore, even with the addition of adhesion terms it remains possible to reduce the computational complexity of by finding the adhesivity of small cell as opposed to an entire network (or even a large cell), since, to find the adhesivity, \eqref{eq:mm_adhesivity_reduced}, as with the permeabliity before, we must first solve a linear problem, \eqref{eq:mm_cell-problem_reduced}, of size $N$.


% - return to figures and talk abotu tensdigng to  j inf.
% - talk about difference between j and jinf and in same way as k. 
% - Say that an advatange if the approximate distribution is that ti gives the approximation g cdf g.. then talk about how goo dthis approximation is as a function of sigma.
% - talk about the standard deviation and say the same stuff as k.

% From Figure \ref{fig:exp-param-dist_4-reg_3_d}
% we see that when the conductance distribution is wider, the
% mean adhesivity, $\mathbb{E}[j^{0}]$, begins to exceed $\bar{G}_{\text{W}}\text{cdf}(\bar{G}_{\text{W}})$ as the cell size, $N$ , increases.
% This behaviour demonstrates the nonlinear dependence of the effective adhesivity on the underlying conductance. 
% In this case, simply replacing the edge conductances with the mean edge conductance leads to an underestimate of the mean adhesivity by  a small factor [approximately [find exact percentage] $\%$].
% Despite this, we see that sampling of a unit cell with $N=25$, or perhaps even $N=16$, is enough to capture the mean adhesivity well.

% Comparing Figures \ref{fig:exp-param-dist_4-reg_3_c} and  \ref{fig:exp-param-dist_4-reg_3_d}
% with the equivalent figures for permeability, Figures \ref{fig:exp-param-dist_4-reg_2_c} and  \ref{fig:exp-param-dist_4-reg_2_d},
% we see that the convergence of the mean of the adhesivity is fast with $N$, if marginally slower than that of the permeability. 
% One implication is that, if the goal remains to simulate a non-periodic (i.e., $N=\infty$) network, the addition of an adhesion mechanism in the problem is not detrimental to the computational advantage that, as is noted in [ref:Chapman \& Wilmott], is gained in the non-adhesion case. 
% That is, in the adhesion case, suitable approximations of both the required effective parameters are obtained by using cells of size around $N=16$. 
% In other words, the significant additional physical and mathematical complication of a network that is dynamic due to changes in conductance caused by particle--pore adhesion requires minimal additional computational complexity, when compared to the fixed-network case.

% Furthermore, comparing Figures \ref{fig:exp-param-dist_4-reg_3_c} and  \ref{fig:exp-param-dist_4-reg_3_d}
% with Figures \ref{fig:exp-param-dist_4-reg_2_c} and  \ref{fig:exp-param-dist_4-reg_2_d} once more, 
% we see that the behaviour of the standard deviation of the adhesivity, $\mathbb{S}[j^{0}]$, matches that of the permeability, $\mathbb{S}[k^{00}]$. 
% That is, $\mathbb{S}[j^{0}] \sim \mathbb{S}[k^{00}] \sim N^{-1/2}$. 
% Moreover,
% the computational complexity of the adhesivity formula \eqref{eq:summary_definition-of-j} is larger than $\mathcal{O}(N)$, and is similar to that of the permeability formula, \eqref{eq:summary_definition-of-k}.
% Therefore, the computational complexity gained by simulation of a larger cell is not worth the decrease in deviation of the adhesivity from its mean value caused by use of the larger cell.
% As was the case for nonadhesive networks (where only the permeability need be calculated), for adhesive networks it remains computationally advantageous to average over many realisations of a relatively small unit cell than to average over just a few realisations of a large unit cell.
% \end{itemize}

% \newpage

% \textcolor{blue}{\textbf{Plan: Next chapter}}

% I will consider three tests cases 
% \begin{enumerate}
%     \item Periodic network with a $4$-regular cell identical to the cell that is used to calculate the effective parameters.
%     Goal here is to show that network homogenisation works if $\varepsilon$ is small enough.
%     \item Periodic network with a $6$-irregular repeating cell, while using a $6$-regular repeating cell to calculate the effective parameters.
%     Goal here is to show that we can approximate a random periodic network well using a non-random cell.
%     \item Aperiodic network with average coordination number of $6$. Use a $6$-regular cell to calculate the effective parameters. 
%     Goal here is to show that a surprisingly small regular cell can be used to generate the solution of a random large aperiodic network.
% \end{enumerate}

% For each of these tests cases, there are three interesting different ways of finding the solution:
% \begin{enumerate}
%     \item Take a single sample the effective parameters, then find the solution, then resample, and refind, and so on. Then build a statistical representation of the solution.
%     \item Use the large-$N$ means of permeability and adhesivity as the effective parameters in the flow solution. 
%     Goal here is to show that this solution is close to the mean of the statistical solution.
%     \item Use the mean approximations that are functions of the parameters of the underlying conductance distribution as a proxy for the means of the effective parameters. 
%     Goal here is to show that the solution is close to the mean of the statistical solution, and becomes closer if the underlying distribution is narrower.
% \end{enumerate}

% The take-home messages of this chapter should be...
% \begin{itemize}
%     \item Network homogenisation works. This is useful because it is much cheaper to solve the effective problem than the network problem.
%     \item The three preprocessing speed-ups hypothesised in the previosu section also work. I.e., we can find a solution that is reasonable by not sampling the effective parameters at all, and instead use the parameters of the underlying distribution.
% \end{itemize}




\chapter{Conclusion}
\label{chap:Conclusion}

In this chapter, we draw this thesis to a conclusion by summarising our key motivations and ideas, and highlighting three key areas for further research. 
Specific conclusions for Parts \ref{part:A size-structured model for filtration}, \ref{part:A network model for filtration}, and \ref{part:A multiscale model for filtration}
can be found in Chapters \ref{chap:ssm_Conclusions for the size-structured model}, \ref{chap:nm_Conclusions for the network model}, and \ref{chap:mm_Conclusions the multiscale model}, respectively. 


\section{Motivation}
In Chapter \ref{chap:Introduction}, 
we established that filtration is an increasingly important industrial process for which
mathematical models can be broadly separated into three categories:
\begin{itemize}
    \item Macroscale models, 
    which are computationally cheap 
    but sometimes inaccurate because they ignore microscale information; 
    \item Microscale models, which are 
    usually more accurate but more computationally expensive; 
    \item Multiscale models, 
    which provide a useful balance between the other two approaches, although the continuum microscale models that they encompass can still be expensive, or else over-idealise the true geometry. 
\end{itemize}
This motivated the study of three novel models, 
which broadly fit into each of these three categories. 

\section{Summary}
We motivated, derived, and solved these three models in Parts \ref{part:A size-structured model for filtration}, \ref{part:A network model for filtration}, and \ref{part:A multiscale model for filtration}, respectively.

\subsection{Macroscale model: A size structured model for filtration}

In Part \ref{part:A size-structured model for filtration}, 
we explored a novel framework that 
allows for the continuous monitoring of the size distributions of particles in the feed and pores in the filter. 
The parameters of most other macroscale systems are estimated by empirical measurements. 
In our model, the functions that determine the permeability and reactivity of the filter are found dynamically as part of the solution scheme. They depend on the relationship between the particle and pore size distributions, which change subject to models for physical and chemical phenomena. 
As a result, our solutions behave intuitively, and we found various ways of saving computational expense, via asymptotic methods. 

\subsection{Microscale model: A network model for filtration}

In Part \ref{part:A network model for filtration}, 
we derived and analysed a novel microscale model for filtration. 
This serves as a simple example of a framework in which the pore clogging due to particle capture is modelled as changes to the connectivity of a network and decrease in the particle concentration in a continuum fluid. 
Our model is useful to predict important effective properties of filters, 
such as the particle retention efficiency and process termination time. 
Despite this, our network model suffers from the same problem as many others: 
the quality of microscale resolution means that simulation of areas large enough to characterise the filter as a whole is computationally infeasible, 
which motivates the study of an alternative framework. 

\subsection{Multiscale model: A multiscale model for filtration}

In Part \ref{part:A multiscale model for filtration}, we explored a multiscale model for filtration. 
We used network homogenisation to
systematically average the microscale model from Part \ref{part:A network model for filtration} in such a way that the two models agree in a physically relevant asymptotic limit. 
The result is a set of equations that govern 
the macroscale properties of the filter. 
However, 
as in Part \ref{part:A size-structured model for filtration}, 
the parameters of this system are solved as part of the model. 
% (as opposed to traditional macroscale models). 
This time, these depend on the interactions between particles and pores via 
the connectivity of the microscale geometry, as well as the size distribution of pores.
% (as opposed to the ize distrubitons of particles and pores, as in Part \ref{part:A size-structured model for filtration}).  
Unlike in traditional multiscale models, the microscale is a network as opposed to a continuum, which decreases the computational complexity of solution whilst still including connectivity information.
% for less idealised inclusion of microscale geometry, such as the connectivity of pore sapce. 
This allows for the systematic study of the effect of a random microscale on the statistical properties of the parameters of the effective system, via repetition of solution that is infeasible for the more expensive microscale model in Part \ref{part:A network model for filtration}.

\section{Further work}
Avenues for further research can be broadly summarised into three key themes: 
incorporation of additional physical and chemical effects; 
inclusion and calculation of additional industrially relevant filter properties; 
and use of experimental data, especially for optimisation.

\subsection{Additional physical and chemical effects}
In this thesis, in general, we have focused on the use of simple models governing particle--pore interactions to demonstrate the general behaviour of our models. 
There is therefore much scope for the derivation and use of more detailed laws that more accurately take into account physical and chemical microscale phenomena. A more detailed description of the pore size that results from deposition of a particular particle would be simple to incorporate in our size structured framework, for example, via a more complex transformation function. We only discussed blocking and deposition as clogging mechanisms, and ignored partial surface blocking and caking effects introduced in Chapter \ref{chap:Introduction}. 
We ignored diffusive effects in general, 
although we established, in Chapter \ref{chap:Introduction},
that microfiltration generally occurs at low flow rates. In our network model, diffusion would result in terms containing the concentration difference on edges (as opposed to the pressure difference), 
and it would be interesting to analyse the diffusivity parameter that would emerge in the multiscale model, similarly to the permeability and adhesivity. 

% Size structured model: Adding a more realtic way in which the pore distribution changes when particles deposit
% More aparticle and pore interactions. For example we discussed caking and partial blocking in the intrdocution but wehne we only focused on mostly deposition, and some blocking. All three frameworks cold probab;y include others.

\subsection{Industrially relevant properties}
Our focus was on derivation and solution of models for the evolution of primary filter characteristics, such as the flow rate and particle concentration. 
It would be relatively straightforward, and of significant industrial interest, to derive and analyse secondary quantities from these,
such as retention and energy efficiency, and operating time.

% size structured model: 
% Polydispersed particles as well as pores. 
% Porosirt gradients. 
% Networks: More complex initial pore distrivutions. Again ones that depend on space. 
% Deposoiton ruules to accourn for diffusion as well as advectiona dn reaction. Difffusion is clearly important in slower flow, whcih might be important since we eswtablsihed in th eintro that microfiltration occurs at fairly low pressure drops. Diffusion might be included in the netwrok model by edge cocnetnration differemt as opppsoed to edge pressure differemce, for example. Woudl be intersting to see the form of  effective diffusiing paramter. 


\subsection{Experimental data and optimisation}
Empirical data would be a useful addition to any of our models. 
Firstly, initial state information would increase solution accuracy.
In the size structured model, use of the size distributions of actual feeds or membranes would be easily incorporated. Our chosen distributions were for illustrative purposes only. 
In the network and multiscale models, statistical information about the size distribution and coordination number of specific porous media would be incorporated by tuning the parameters of the initial pore conductance distribution and parametrisation of an algorithm for random network structure generation, respectively. 
% and  this might include statstical information about the characterisationo fth enetwork microscale, again size strcutre, but also spefici geometry, such as the coooridnation numebr of the material. Might be able to exxaftly repalicate the networtk, or else choose a statstically correct algorithm for reating a random network with tehe same statistical properties. 
Secondly, solution data would allow for validity checks of our models. Further, systematic analysis of the difference between the mathematical and empirical solutions could allow for parameter tuning, 
made possible by the small numbers of free parameters in our systems, and their computational tractability. 
The combination of correct initial conditions and 
well tuned parameters would render our models suitable for simulation of real filtration experiments, decreasing financial expense and industrial waste.
Future research would also explore the possibility of parameter sweeps for the cost function descents involved in the computational optimisation of filter performance metrics. 
% We would use differences to fit parametres of our models. MNone of our mdoels have a large numebr of oarametes, so this woul dbe possible. 
% Combingin initialisation data nd aolsution data to obtaoin the corect inital conditions adn parameters, we would hten be able to make sconclusions about th  actual operation of actual materials. Note that this is possible because our solves are all omputationally feasible, at least ins omae cases or sowith certain simplifications applied. For example, in Part 1, we were able to find several asymptptoc solutions that could be solved explocity in some cases. We didn't do much of that

% The computational feasibility of our models means 
% that solution repetition is possible. 

% This would hopefuly lead to 
% use for the feasibilty of our models is optimisation. Becaus ethey are computationally feasible we could use repetiiono f solution, while sweeping certain parameters, to optimise mesasures o f filter poerforamnce.  


These avenues for future work would hopefully lead to filters with increased retention efficiency and throughput potential with 
decreased energy demands, 
which were targets that we set out in the introduction of this thesis.


% that can be run in lower pressure drop and flow rate situations, and thus 
% %\part{Appendices}
\appendix
% \chapter{Appendix: Size-structured model}

% \chapter{Appendix: Network model}

% PUT THE FOLLOWING INTO THE APPENDIX


% % \textcolor{blue}{[This isn't necessarily the place for this aside, but note that $j^1$ is equal to $j^0$, but ultimately this doesn't matter because we don't need to calculate it because we're going to be interested in a unidirectional pressure gradient, and we can always re-orient the cell such that this direction is the $0$ direction. We should also make a similar argument to explain why we're only interested in $k^{00}$. I.e., $\partial p/\partial x = 1$ and $\partial p / \partial y = 0$ so that $\bm{\nabla}p = (1,0)^\top$ and therefore in $\bm{j}=(j^0,j^1)^\top$, $j^1$ is not relevant since dotted with pressure gradient. In other words, we have macroscale variation in flow direction but none in transverse direction...]} 
% \textcolor{blue}{Now that I have shown in the previous section that we don't need $j^2$ since we're considering a uni-directional macrosclae pressure gradient, I think the following should go in the appendix...? Note that evaluation of the $x^2$-directional component of the adhesivity, $j^2$, will give the same answer as the $x^1$-directional component, $j^1$.
% To see this, observe that when we evaluate the $x^2$-directional pressure drop component, $\Delta_{ij}^{r^2}$, edges that were previously parallel to the direction of the pressure drop component are now perpendicular, and \emph{vice versa}.
% These edge contributions are now zero, whilst previously perpendicular edges are now parallel and have an average pressure drop contribution of one. In other words, 
% \begin{align}
%     \mathbb{E}[\Delta_{ij}^{r^1}]=\sin{\theta_{ij}^{\bm{r}}}.
%     \label{eq:results_j1-is-sin}
% \end{align} 
% Note, however, that only evaluation of one of either $j^1$ or $j^2$ is necessary in cases where the pressure gradient, $\bm{\nabla}p$, is unidirectional, since the vector $\bm{j}$ gets dotted with the pressure gradient in \eqref{eq:summary_grad-eqs}.
% For a unidirectional pressure gradient diagonal to the edge directions, for example, the pressure gradient dotting with $\bm{j}$ would mean that half the adhesivity contribution would come from $j^1$ and half would come from $j^2$.
% We arbitrarily choose the $x^1$ direction as the direction of the pressure gradient, therefore.[Should probably point out that only need $k^{11}$ for similar reasons]}





\chapter{Appendix: A multiscale model for filtration}
In this appendix, we prove the algebraic results necessary for the network homogenisation of our network model.
We begin with the results we use in the quasi-one-dimensional setting, and then prove the similar (if more notationally complex) results that we use in the $D$-dimensional derivation.

Before we begin, we define the following two operations, which are utilised extensively in the subsequent sections.
\begin{enumerate}[label={Operation \arabic*.}, ref={\arabic*}, leftmargin=2.5cm]
    \item A re-indexing $i\leftrightarrow j$ followed by use of $G_{ji}^{\bm{r}} = G_{ij}^{-\bm{r}}$. We refer to this equality as \emph{the symmetry of conductance}, which holds due to the periodicity of the network, so that these objects describe the conductance of the same link.
    \label{op:1}
    \item A re-indexing
    $\bm{r} \leftrightarrow -\bm{s}$, followed by use of the fact that summing over $-\bm{s}$ is equivalent to summing over $\bm{s}$, since, if $-\bm{s} \in \bm{\mathcal{R}}$, then $\bm{s} \in \bm{\mathcal{R}}$, followed by a re-indexing $\bm{s} \rightarrow \bm{r}$.
    \label{op:2}
\end{enumerate}

\section{Algebraic results in one dimension}
\subsection{Relation of basic matrices} \label{sec:1d_proofs_ABCDE}
In this subsection, we show that 
\begin{align}
    \bm{A}^{(s)} \bm{1} = \bm{0},
    \quad 
    \bm{1}^\top \bm{B}^{ (s)} \bm{1} = \bm{0},
    \quad 
    \bm{D}^{ (s)} \bm{1} = \bm{0},
    \quad 
    \bm{F}^{ (s)} \bm{1} = \bm{0},
    \label{eq:1d_proof_matrix-lemma_1}
\end{align}
and that
\begin{align}
    \bm{1}^\top \bm{D}^{ (s)} &= 
    \pp{}{x}\Big(\bm{1}^\top \bm{B}^{ (s)} \Big),
    \label{eq:1d_proof_matrix-lemma_2_D-and-B}
    \\
    \bm{1}^\top \bm{E}^{ (s)} \bm{1} &=
    \pp{}{x}\Big(\bm{1}^\top \bm{C}^{ (s)} \bm{1} \Big).
    \label{eq:1d_proof_matrix-lemma_2_E-and-C}
\end{align}
To show the results in \eqref{eq:1d_proof_matrix-lemma_1}, firstly, we have
\begin{align}
    \Big(\bm{A}^{(s)} \bm{1}\Big)_i 
    &= 
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} \Big(  G_{ij}^{r {(s)}} - \delta_{ij} \sum_{k=1}^N G_{ik}^{r {(s)}} \Big)
    \\
    &=
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(s)}} - \sum_{j=1}^N \sum_{r\in\mathcal{R}}  G_{ij}^{r {(s)}} 
    \\
    &=
    0.
\end{align}
Secondly, we have
\begin{align}
    \bm{1}^\top \bm{B}^{ (s)} \bm{1}
    &= 
    \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}} \Big( G_{ij}^{r {(s)}}  (X_j + r l
    )- \delta_{ij} X_i \sum_{k=1}^N G_{ik}^{r {(s)}} \Big)
    \\
    &=
    \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}} G_{ij}^{r {(s)}}  (X_j + r l
    )-  \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}}  G_{ij}^{r {(s)}} X_i
    \\
    &=
    \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}} G_{ij}^{r {(s)}}  (X_j + r l
    )-  \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}}  G_{ij}^{-r {(s)}} X_j
    \\
    &=
    \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}} G_{ij}^{r {(s)}}  (X_j + r l
    )-  \sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}}  G_{ij}^{r {(s)}} X_j
    \\
    &=
    l\sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}} G_{ij}^{r {(s)}} r
    \\
    &=
    -l\sum_{i=1}^N\sum_{j=1}^N\sum_{r\in\mathcal{R}} G_{ij}^{r {(s)}} r
    \\
    &=
    0,
\end{align}
where the second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the third uses Operation \ref{op:1} on the second term, the fourth uses Operation \ref{op:2} on the second term, and the sixth uses that if $r \in \mathcal{R}$ then $-r \in \mathcal{R}$.
Thirdly, we have
\begin{align}
    \Big(\bm{D}^{ (s)} \bm{1} \Big)_i
    &= 
    \sum_{j=1}^N \sum_{r\in\mathcal{R}}\Big( \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} -\delta_{ij}  \sum_{k=1}^N \pp{G_{ik}^{r {(s)}}}{x} M_{ik}^{r}   \Big)
    \\
    &=
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} -\sum_{j=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r}
    \\
    &=
    0.
\end{align}
Lastly, we have
\begin{align}
    \Big( \bm{F}^{ (s)} \bm{1} \Big)_i
    &= \frac{1}{2} \sum_{j=1}^N \sum_{r\in\mathcal{R}} \bigg( \ppp{G_{ij}^{r {(s)}}}{x}  (M_{ij}^{r})^2  - \delta_{ij}   \sum_{k=1}^N  \ppp{G_{ik}^{r {(s)}}}{x} (M_{ik}^{r})^2    \bigg)
    \\
    &= 
    \frac{1}{2} \bigg(
    \sum_{j=1}^N \sum_{r\in\mathcal{R}}  \ppp{G_{ij}^{r {(s)}}}{x}  (M_{ij}^{r})^2  - \sum_{j=1}^N \sum_{r\in\mathcal{R}}
    \ppp{G_{ij}^{r {(s)}}}{x}  (M_{ij}^{r})^2  
    \bigg)
    \\
    &=0.
\end{align}

We next show the result given by \eqref{eq:1d_proof_matrix-lemma_2_D-and-B}.
On one hand,
\begin{align}
    \pp{}{x}\Big(\bm{1}^\top \bm{B}^{ (s)} \Big)_j 
    &= \pp{}{x} \Bigg(\sum_{i=1}^N \sum_{r\in\mathcal{R}} \bigg( G_{ij}^{r {(s)}}  (X_j + r l
    )- \delta_{ij} X_i \sum_{k=1}^N G_{ik}^{r {(s)}} \bigg)
    \Bigg)
    \\
    &=
     \sum_{i=1}^N  \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l
    )- \delta_{ij} X_i \sum_{k=1}^N \pp{G_{ik}^{r {(s)}}}{x} \bigg)
    \\
    &=
     \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l
    )-   \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ji}^{r {(s)}}}{x} X_j
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l
    )-   \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{-r {(s)}}}{x} X_j
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l
    )-   \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} X_j
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}  r l,
\end{align}
where the third equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow i$, the fourth uses the symmetry of conductance, and the fifth uses Operation \ref{op:2} on the second term. 
On the other hand, 
\begin{align}
    \Big( \bm{1}^{\top} \bm{D}^{(s)} \Big)_j &= 
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} -\delta_{ij}  \sum_{k=1}^N \pp{G_{ik}^{r {(s)}}}{x} M_{ik}^{r}   \bigg)
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} - \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ji}^{r {(s)}}}{x} M_{ji}^{r} 
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} - \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{\bm{-r} {(s)}}}{x} M_{ji}^{r}  
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} \bigg( M_{ij}^{r} - M_{ji}^{-r} \bigg)  
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}
    r l,
\end{align}
where the second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow i$, the third uses the symmetry of conductance, the fourth uses Operation \ref{op:2} on the second term, and the fifth uses the definition of the midpoint \eqref{eq:definition-of-midpoint}.
We lastly show the result given by \eqref{eq:1d_proof_matrix-lemma_2_E-and-C}.
On one hand, 
\begin{align}
    \pp{}{x}\Big(\bm{1}^\top \bm{C}^{ (s)} \bm{1} \Big) 
    &=
    \pp{}{x} \Bigg( 
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \bigg( G_{ij}^{r {(s)}}  (X_j + r l)^2 -
    \delta_{ij} X_i^2 \sum_{k=1}^N G_{ik}^{r {(s)}} \bigg)
    \Bigg)
    \\
    &=
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l)^2 -
    \delta_{ij} X_i^2 \sum_{k=1}^N \pp{G_{ik}^{r {(s)}}}{x} \bigg)
    \\
    &=
    \frac{1}{2} \Bigg( \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l)^2 -
    \\ & \hspace{6cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  \pp{G_{ij}^{r {(s)}}}{x} X_i^2 \bigg) \Bigg)
    \\
    &=
    \frac{1}{2} \Bigg( \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l)^2 -
    \\ & \hspace{6cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  \pp{G_{ij}^{-r {(s)}}}{x} X_j X_j \bigg) \Bigg)
    \\
    &=
    \frac{1}{2} \Bigg( \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x}  (X_j + r l)^2 -
    \\ & \hspace{6cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  \pp{G_{ij}^{r {(s)}}}{x} X_j X_j \bigg) \Bigg)
    \\
    &=
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}  (X_j r l + X_j r l + r^2 l^2),
\end{align}
where the third equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the fourth uses Operation \ref{op:1} on the second term, and the fifth uses Operation \ref{op:2} on the second term. 
On the other hand,
\begin{align}
    \bm{1}^\top \bm{E}^{ (s)} \bm{1} 
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \bigg( \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} (X_j + r l)  - \delta_{ij}  X_i \sum_{k=1}^N  \pp{G_{ik}^{r {(s)}}}{x} M_{ik}^{r}   \bigg)
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} (X_j + r l)  - 
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}}    \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} X_i
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} (X_j + r l)  - 
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{-r {(s)}}}{x} M_{ji}^{r} X_j
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ij}^{r} (X_j + r l)  - 
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x} M_{ji}^{-r} X_j
    \\
    &=
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}
    \Big( 
    X_j r l + X_j r l + r^2 l^2 + 
    X_j r l +X_i r l
    \Big)
    \\
    &=
    \pp{}{x}\Big(\bm{1}^\top \bm{C}^{ (s)} \bm{1} \Big)  +
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}
    X_j r l 
    +
    \\ & \hspace{6cm} \nonumber
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}
    X_i r l,
    \\
    &=
    \pp{}{x}\Big(\bm{1}^\top \bm{C}^{ (s)} \bm{1} \Big)  +
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{r {(s)}}}{x}
    X_j r l 
    +
    \\ & \hspace{6cm} \nonumber
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
    \sum_{r\in\mathcal{R}} \pp{G_{ij}^{-r {(s)}}}{x}
    X_j r l,
    \\
    &=
    0,
\end{align}
where the second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the third uses Operation \ref{op:1} on the second term, the fourth uses Operation \ref{op:2} on the second term, the fifth uses the definition of the midpoint \eqref{eq:definition-of-midpoint}, the seventh uses Operation \ref{op:1} on the third term, and the eighth uses
Operation \ref{op:2} on the third term.

\subsection{Simplification of permeability terms} \label{sec:1d_proofs_remove-X-from-BC}
In this subsection, we simplify the terms in the permeability scalar $k^{(0)}$.
Firstly, we show that
\begin{align}
    \bm{1}^\top \bm{B}^{ (0)}  \bm{W}  =     -l \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}  W_i.
    \label{eq:1d_remove-X-from-B_mn}
\end{align}
We have
\begin{align}
    \bm{1}^\top \bm{B}^{ (0)}  \bm{W} 
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \left( G_{ij}^{r {(0)}}  (X_j + r l
    )- \delta_{ij} X_i \sum_{k=1}^N G_{ik}^{r {(0)}} \right) W_j
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{r {(0)}} W_j (X_j + r
    l) 
    -
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{r {(0)}} W_i X_i
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{-r {(0)}} W_i (X_i + r
    l) 
    -
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{r {(0)}} W_i X_i
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{r {(0)}} W_i (X_i + (-r
    )l) 
    -
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{r {(0)}} W_i X_i
    \\
    &=
    -l \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}  W_i.  
\end{align}
The first equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the third uses Operation \ref{op:1} on the first sum, and the fourth uses Operation \ref{op:2} on the first term.

Secondly, we prove that 
\begin{align}
    \bm{1}^\top \bm{C}^{ (0)} \bm{1} = 
    - \frac{1}{2}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}
    \Big(
    2 X_i r l
    -
    r^2 l^2
    \Big).
    \label{eq:1d_remove-X-from-C_mn}
\end{align}
We have 
\begin{align}
    \bm{1}^\top \bm{C}^{ (0)} \bm{1} 
    &= 
    \frac{1}{2} \sum_{r\in\mathcal{R}} \left( G_{ij}^{r {(0)}}  (X_j + r l)^2 - \delta_{ij} X_i^2 \sum_{k=1}^N G_{ik}^{r {(0)}} \right)
    \\
    &= 
    \frac{1}{2}
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}  
    X_j^2 
    +
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} 
    2 X_j r l 
    +
    \\
    &
    \nonumber
    \hspace{3.5cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}
    r^2 l^2
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} X_i^2
    \bigg)
    \\
    &= 
    \frac{1}{2}
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{-r {(0)}}  
    X_i^2 
    +
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{-r {(0)}}
    2 X_i r l 
    +
    \\
    &
    \nonumber
    \hspace{3.5cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}
    r^2 l^2
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} X_i^2
    \bigg)
    \\
    &= 
    - \frac{1}{2}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}
    \Big(
    2 X_i r l
    -
    r^2 l^2
    \Big).
\end{align}
The second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$ in the final term, 
the third uses Operation \ref{op:1} on the first and second terms, and the third uses Operation \ref{op:2} on the first and second terms, so that the first and last terms cancel.


\subsection{Independence of node location and permeability equality} \label{sec:1d_proofs_k=sig}
We show that the permeability tensor is independent of the node locations, to show that  
\begin{align}
    k^{(0)} = \sigma^{(0)}.
\end{align}
We begin by simplifying $k^{(0)}$. We have
\begin{align}
    k^{(0)} 
    &= 
    \frac{1}{2}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)} 
    \Big(W_j - 
    W_i
     +
    X_j + r l 
    - 
    X_i \Big) 
    \\
    &=
    \frac{1}{2}  
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)} 
    \Big(W_j 
     +
    X_j + r l    
    \Big)
    -
    \\
    \nonumber
    &
    \hspace{6cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)}
    \Big(
    W_i
    + 
    X_i \Big)
    \bigg)
    \\
    &=
    \frac{1}{2}  
    \bigg(
    \sum_{i=1}^N\sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{-r (0)} 
    \Big(W_i 
     +
    X_i + r l    
    \Big)
    -
    \\
    \nonumber
    &
    \hspace{6cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)}
    \Big(
    W_i
    + 
    X_i \Big)
    \bigg)
    \\
    &=
    \frac{1}{2}  
    \bigg(
    \sum_{i=1}^N\sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    (-r) G_{ij}^{r (0)} 
    \Big(W_i 
     +
    X_i + (-r)l    
    \Big)
    -
    \\
    \nonumber
    &
    \hspace{6cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)}
    \Big(
    W_i
    + 
    X_i \Big)
    \bigg)
    \\
    &=
    -   
    \sum_{i=1}^N\sum_{j=1}^N \sum_{r\in\mathcal{R}}  
    r G_{ij}^{r (0)} 
    \Big(W_i
     +
    X_i - \frac{r l}{2} \Big),
    \label{eq:1d_simplified-k}
\end{align}
where the first equation uses the definition of the permeability, \eqref{eq:1d_definition-of-permeability}, the third uses Operation \ref{op:1} on the first term, and the fourth uses Operation \ref{op:2} on the first term.
We simplify $\sigma^{(0)}$ similarly. 
Indeed, using the simplified forms of the permeability terms, $\bm{1}^\top \bm{B}^{ (0)}  \bm{W}$ and $\bm{1}^\top \bm{C}^{ (0)}  \bm{1}$,  \eqref{eq:1d_remove-X-from-B_mn} and \eqref{eq:1d_remove-X-from-C_mn}, we have
\begin{align}
    \sigma^{(0)} 
    &= 
    \frac{1}{l}\Big( \bm{1}^\top \bm{B}^{ (0)}  \bm{W} + \bm{1}^\top \bm{C}^{ (0)} \bm{1}
    \Big)
    \\
    &=
    - \frac{1}{2 l}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    G_{ij}^{r {(0)}}
    \Big(
    2 l r W_i
    +
    2 X_i r l
    -
    (r l)^2
    \Big)
    \\
    &=
    -
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}
    \Big(
    W_i
    +
    X_i
    -
    \frac{r l}{2}
    \Big)
    \\
    &= k^{(0)},
\end{align}
where the last equation uses the simplified definition of $k^{(0)}$, \eqref{eq:1d_simplified-k}.
 
We will show that $k^{(0)}$, and therefore $\sigma^{(0)}$, is independent of $X_a^b$.
Using the simplified definition of $k^{(0)}$, \eqref{eq:1d_simplified-k}, we have 
\begin{align}
    \pp{k^{(0)}}{X_a^b} 
    &= 
    \pp{}{X_a^b}
    \bigg(
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}
    \Big(
    W_i
    +
    X_i
    -
    \frac{r l}{2}
    \Big)
    \bigg)
    \\
    &=
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}
    \Big(
    \pp{W_i}{X_a^b}
    +
    \delta_{ia}\delta^{nb}
    \Big).
    \label{eq:1d_dkdx}
\end{align}
Now, we note that
\begin{align}
    \Big(\bm{A}^{(0)} \pp{\bm{W}}{X_a^b} \Big)_i &=
    \Big(\pp{}{X_a^b}
     (\bm{A}^{(0)}  \bm{W}) \Big)_i
    \\
    &=
    \Big(\pp{}{X_a^b}
    (- \bm{B}^{ (0)}  \bm{1}) \Big)_i
    \\
    &=
    -\pp{}{X_a^b}
    \Bigg(
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} \Big( G_{ij}^{r {(0)}}  (X_j + r l
    )- \delta_{ij} X_i \sum_{k=1}^N G_{ik}^{r {(0)}} \Big)
    \Bigg)
    \\
    &=
    -\pp{}{X_a^b}
    \Bigg(
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}  (X_j + r l
    )- \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} X_i
    \Bigg)
    \\
    &=
    -
    \Bigg(
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} \delta_{ja}\delta^{nb} 
    - 
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} \delta_{ia} \delta^{nb}
    \Bigg)
    \\
    &=
    -
    \Bigg(
    \sum_{r\in\mathcal{R}} 
    \Big(
    G_{ia}^{r {(0)}}  
    - 
    \delta_{ia}
    \sum_{k=1}^N  G_{ik}^{r {(0)}} 
    \Big)
    \Bigg) \delta^{nb}
    \\
    &=
    -A_{ia}^{(0)} \delta^{nb}
    \\
    &=
    -\Big(\bm{A}^{(0)} \bm{e}_a \delta^{nb} \Big)_i,
\end{align}
where the first equation uses the cell problem, \eqref{eq:mm_1d_cell-problem_1}.
Therefore
\begin{align}
    \bm{A}^{(0)} \Big( 
    \pp{\bm{W}}{X_a^b}+
    \bm{e}_a \delta^{nb}
    \Big)=\bm{0},
\end{align}
and so, using that $\bm{A}^{(0)} \bm{1} = \bm{0}$ from \eqref{eq:mm_1d_matrix-lemma_1}, we arrive at
\begin{align}
    \pp{\bm{W}}{X_a^b}
     = \alpha \bm{1} - \bm{e}_a \delta^{nb},
\end{align}
so that
\begin{align}
    \pp{W_i}{X_a^b} = \alpha - \delta_{ia} \delta^{nb},
    \label{eq:1d_dwdx}
\end{align}
for some function $\alpha(x,t)$ that is independent of the microscale.
Substituting this result, \eqref{eq:1d_dwdx}, into the equation for the derivative of $k^{(0)}$, \eqref{eq:1d_dkdx}, we find that
\begin{align}
    \pp{k^{(0)}}{X_a^b} 
    &=
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}
    \Big(
    \pp{W_i}{X_a^b}
    +
    \delta_{ia}\delta^{nb}
    \Big)
    \\
    &=
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}
    \alpha
    \\
    &=
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    (-r) G_{ij}^{r {(0)}}
    \alpha
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
    r G_{ij}^{r {(0)}}
    \alpha
    \\
    &=
    -\pp{k^{(0)}}{X_a^b},
\end{align}
where the third equation uses Operation \ref{op:1} followed by Operation \ref{op:2}.
We have therefore shown that
\begin{align}
    \pp{k^{(0)}}{X_a^b}  = 0,
\end{align}
and therefore that 
\begin{align}
    \pp{}{X_a^b} \bigg( \frac{1}{l}\Big( \bm{1}^\top \bm{B}^{ (0)}  \bm{W} + \bm{1}^\top \bm{C}^{ (0)} \bm{1}
    \Big) \bigg) 
    &=
    \pp{\sigma^{(0)}}{X_a^b}
    \\
    &=
    \pp{k^{(0)}}{X_a^b}
    \\
    &= 0,
\end{align}
where the first equation uses the definition of $\sigma^{(0)}$.
So $k^{(0)}$ and $\sigma^{(0)}$, and therefore $\bm{B}^{ (0)}$ and $\bm{C}^{ (0)}$, are independent of $X_a^b$, which permits the choice $X_a^b=0$ in the original definitions of the latter two, \eqref{eq:1d_def_B} and \eqref{eq:1d_def_C}, allowing us to re-write these as
\begin{align}
    B_{ij}^{ (0)} &= 
    \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} r l,
    \\
    C_{ij}^{ (0)} &= 
    \frac{1}{2} \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}  r^2 l^2.
\end{align}
This completes our proof.

Using these more simple definitions, we also note that $\bm{B}^{ (0)}$ is skew-symmetric so that
\begin{align}
    \Big(\bm{B}^{ (0)}\Big)^\top = -\bm{B}^{ (0)},
\end{align}
and $\bm{C}^{ (0)}$ is symmetric so that
\begin{align}
    \Big(\bm{C}^{ (0)}\Big)^\top = \bm{C}^{ (0)},
\end{align}
since, using Operation \ref{op:2},
\begin{align}
    B_{ji}^{ (0)} &= 
    \sum_{r\in\mathcal{R}} G_{ji}^{r {(0)}} r l 
    \\ 
    &=
    \sum_{r\in\mathcal{R}} G_{ij}^{-r {(0)}} r l 
    \\
    &=
    \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} (-r)l 
    \\
    &=
    -\sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} r l
    \\
    &=
    -B_{ij}^{ (0)},
\end{align}
and 
\begin{align}
    C_{ji}^{ (0)} &= 
    \frac{1}{2} \sum_{r\in\mathcal{R}} G_{ji}^{r {(0)}}  r^2 l^2
    \\
    &=
    \frac{1}{2} \sum_{r\in\mathcal{R}} G_{ij}^{-r {(0)}}  r^2 l^2
    \\
    &=
    \frac{1}{2} \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}}  (-r)l (-r)l
    \\
    &=
    \frac{1}{2} \sum_{r\in\mathcal{R}} G_{ij}^{r {(0)}} r^2 l^2
    \\
    &= C_{ij}^{ (0)}.
\end{align}

\subsection{Intra-cell particle conservation} \label{sec:1d_proofs_S}
In this subsection, we show that $\bm{1}^{\top} \bm{S}^{(0)} = \bm{0}^{\top}$ and $\bm{S}^{(0)} \bm{1} = \bm{0}$.
We first show that $\bm{S}^{(0)} \bm{1} = \bm{0}$. 
We have 
\begin{align}
    \Big(\bm{S}^{(0)} \bm{1}\Big)_i 
    & = 
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
   \Big( G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i) H_{ji}^{-r(0)}
    +
    \\ & \hspace{3cm} \nonumber 
    \delta_{ij} \sum_{k=1}^N G_{ik}^{r(0)}(W_k-W_i+X_k+r l-X_i) H_{ik}^{r(0)}
    \Big)
    \\
    &=
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} 
   \Big( G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i)
    \\
    &=
    \sum_{j=1}^N \sum_{r\in\mathcal{R}} \Big( G_{ij}^{r(0)} - \delta_{ij} \sum_{k=1}^N G_{ik}^{r(0)} \Big) W_j +
    \\ & \hspace{3cm} \nonumber
    \sum_{j=1}^N \sum_{r\in\mathcal{R}}  \Big( 
    G_{ij}^{r(0)} (X_j + r l) - \delta_{ij} \sum_{k=1}^N G_{ik}^{r(0)} X_i \Big)
    \\
    &=
    (\bm{A}^{(0)} \bm{W})_i + (\bm{B}^{(0)} \bm{1})_i 
    \\
    &= 
    0.
\end{align}
The first equation uses the definition of $\bm{S}^{(0)}$, \eqref{eq:1d_definition-of-S}, the second uses $\delta_{ij}$ followed by the Heaviside identity, \eqref{eq:1d_heaviside-identity}, 
and the fifth uses the cell problem,

Next, we show that $\bm{1}^{\top} \bm{S}^{(0)} = \bm{0}^{\top}$.
We have 
\begin{align}
    \Big(\bm{1}^{\top} \bm{S}^{(0)}\Big)_j 
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} 
   \Big( G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i) H_{ji}^{-r(0)}
    +
    \\ 
    & \hspace{2.5cm}
    \nonumber \delta_{ij} \sum_{k=1}^N G_{ik}^{r(0)}(W_k-W_i+X_k+r l-X_i) H_{ik}^{r(0)}
    \Big)
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} 
    \Big( G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i) H_{ji}^{-r(0)}
    +
    \\ 
    & \hspace{2.5cm}
    \nonumber 
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} G_{ji}^{r(0)}(W_i-W_j+X_i+r l-X_j) H_{ji}^{r(0)}
    \Big)
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} 
    \Big( G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i) H_{ji}^{-r(0)}
    -
    \\ 
    & \hspace{2.5cm}
    \nonumber 
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} G_{ji}^{r(0)}(W_j-W_i+X_j-r l-X_i) H_{ji}^{r(0)}
    \Big)
    \\
    &=
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} 
    \Big( G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i) H_{ji}^{-r(0)}
    -
    \\ 
    & \hspace{2.5cm}
    \nonumber 
    \sum_{i=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)}(W_j-W_i+X_j+r l-X_i) H_{ji}^{-r(0)}
    \Big)
    \\
    &=
    0.
\end{align}
The second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow i$ in the second term, the third uses a factorisation of $-1$ in the second term to achieve like-terms,
and the fourth uses the symmetry of conductance in the second term followed by Operation \ref{op:2} on the second term.

\subsection{Interpretation of advective terms}\label{sec:1d_proofs_K-is-k}
In this subsection we show that the coefficient in the advective term of equation for conservation of particles, \eqref{eq:1d_macroscale_transport}, is related to the permeability by
\begin{align}
    \bm{1}^{\top} \bm{K}^{(0)} \bm{1} 
    = 
    l k^{(0)} 
    = 
    l \sigma^{(0)}.
\end{align}
We have
\begin{align}
    \bm{1}^{\top} \bm{K}^{(0)} \bm{1} 
    &= 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i ) (X_j + 
    r l)H_{ji}^{-r(0)}
    +
    \\
    & \hspace{2.5cm}
    \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i )
     X_i H_{ij}^{r(0)} 
    \\
    &= 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i ) (X_j + 
    r l)H_{ji}^{-r(0)}
    +
    \\
    & \hspace{2.5cm}
    \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{\bm{-r}(0)} (W_i- W_j + X_i + r l - X_j )
     X_j H_{ji}^{r(0)} 
    \\
    &= 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i ) (X_j + 
    r l)H_{ji}^{-r(0)}
    +
    \\
    & \hspace{2.5cm}
    \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_i- W_j + X_i - r l - X_j )
     X_j H_{ji}^{-r(0)} 
    \\
    &= 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i )  
    r l H_{ji}^{-r(0)},
\end{align}
where the first equation uses an expansion of the definition of $K_{ij}^{(0)}$, \eqref{eq:1d_definition-of-K}, the second uses Operation \ref{op:1} on the second term, and the third uses Operation \ref{op:2} on the second term.
Defining
\begin{align}
    \Lambda_{ij}^{r(0)} =
    G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i )  r l,
    \label{eq:1d_definition-of-Lambda}
\end{align}
and noting that $\Lambda_{ji}^{-r(0)}=\Lambda_{ij}^{r(0)}$, it follows that
\begin{align}
    \bm{1}^{\top} \bm{K}^{(0)} \bm{1} 
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \Lambda_{ij}^{r(0)} H_{ji}^{-r(0)}
    +
    \frac{1}{2} \sum_{j=1}^N \sum_{i=1}^N \sum_{r\in\mathcal{R}} \Lambda_{ji}^{r(0)} H_{ij}^{-r(0)}
    \\
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{-r} \Lambda_{ij}^{r(0)} H_{ji}^{-r(0)}
    +
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \Lambda_{ij}^{-r(0)} H_{ij}^{-r(0)}
    \\
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} \Lambda_{ij}^{r(0)}
    \\
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{r\in\mathcal{R}} G_{ij}^{r(0)} (W_j- W_i + X_j + r l - X_i )  
    r l
    \\
    &= l k^{(0)},
\end{align}
where the first equation uses a splitting of terms followed by a re-labelling $i\leftrightarrow j$ in the second term, the second uses the symmetry of $\Lambda_{ij}^{r(0)}$, the third uses Operation \ref{op:2} on the second term followed by  the Heaviside function identity, \eqref{eq:1d_heaviside-identity}, the fourth uses the definition of $\Lambda_{ij}^{r(0)}$, \eqref{eq:1d_definition-of-Lambda}, and the fifth uses the definition of the permeability, \eqref{eq:1d_definition-of-permeability}.



\section{Algebraic results in $D$ dimensions}
In this section, we prove the algebraic results necessary for network homogenisation in $D$ dimensions.
\subsection{Relation of basic matrices} \label{sec:proofs_ABCDE}
In this subsection, we show that 
\begin{align}
    \bm{A}^{(s)} \bm{1} = \bm{0},
    \quad 
    \bm{1}^\top \bm{B}^{m (s)} \bm{1} = \bm{0},
    \quad 
    \bm{D}^{m (s)} \bm{1} = \bm{0},
    \quad 
    \bm{F}^{mn (s)} \bm{1} = \bm{0},
    \label{eq:Dd_proof_matrix-lemma_1}
\end{align}
and that
\begin{align}
    \bm{1}^\top \bm{D}^{m (s)} &= 
    \pp{}{x^m}\Big(\bm{1}^\top \bm{B}^{m (s)} \Big),
    \label{eq:Dd_proof_matrix-lemma_2_D-and-B}
    \\
    \bm{1}^\top \bm{E}^{mn (s)} \bm{1} &=
    \pp{}{x^m}\Big(\bm{1}^\top \bm{C}^{mn (s)} \bm{1} \Big) 
    +
    \\ & \hspace{3cm} \nonumber
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r}(s)}}{x^m}(r^nl^nX_i^m-r^ml^mX_i^n).
    \label{eq:Dd_proof_matrix-lemma_2_E-and-C}
\end{align}
To show the results in \eqref{eq:Dd_proof_matrix-lemma_1}, firstly, we have
\begin{align}
    \Big(\bm{A}^{(s)} \bm{1}\Big)_i 
    &= 
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big(  G_{ij}^{\bm{r} {(s)}} - \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \Big)
    \\
    &=
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(s)}} - \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{\bm{r} {(s)}} 
    \\
    &=
    0.
\end{align}
Secondly, we have
\begin{align}
    \bm{1}^\top \bm{B}^{m (s)} \bm{1}
    &= 
    \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}} \Big( G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m
    )- \delta_{ij} X_i^m \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \Big)
    \\
    &=
    \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m
    )-  \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{\bm{r} {(s)}} X_i^m
    \\
    &=
    \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m
    )-  \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{-\bm{r} {(s)}} X_j^m
    \\
    &=
    \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m
    )-  \sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{\bm{r} {(s)}} X_j^m
    \\
    &=
    l^m\sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(s)}} r^m
    \\
    &=
    -l^m\sum_{i=1}^N\sum_{j=1}^N\sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(s)}} r^m
    \\
    &=
    0,
\end{align}
where the second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the third uses Operation \ref{op:1} on the second term, the fourth uses Operation \ref{op:2} on the second term, and the sixth uses that if $r^m \in \mathcal{R}$ then $-r^m \in \mathcal{R}$.
Thirdly, we have
\begin{align}
    \Big(\bm{D}^{m (s)} \bm{1} \Big)_i
    &= 
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}\Big( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} -\delta_{ij}  \sum_{k=1}^N \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} M_{ik}^{\bm{r}m}  \Big)
    \\
    &=
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} -\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m}
    \\
    &=
    0.
\end{align}
Lastly, we have
\begin{align}
    \Big( \bm{F}^{mn (s)} \bm{1} \Big)_i
    &= \frac{1}{2} \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pppd{G_{ij}^{\bm{r} {(s)}}}{x^m}{x^n} M_{ij}^{\bm{r}m}  M_{ij}^{\bm{r}n}  - \delta_{ij}   \sum_{k=1}^N  \pppd{G_{ik}^{\bm{r} {(s)}}}{x^m}{x^n} M_{ik}^{\bm{r}m} M_{ik}^{\bm{r}n}  \bigg)
    \\
    &= 
    \frac{1}{2} \bigg(
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  \pppd{G_{ij}^{\bm{r} {(s)}}}{x^m}{x^n} M_{ij}^{\bm{r}m}  M_{ij}^{\bm{r}n}  - \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}
    \pppd{G_{ij}^{\bm{r} {(s)}}}{x^m}{x^n} M_{ij}^{\bm{r}m} M_{ij}^{\bm{r}n}  
    \bigg)
    \\
    &=0.
\end{align}

We next show the result given by \eqref{eq:Dd_proof_matrix-lemma_2_D-and-B}.
On one hand,
\begin{align}
    \pp{}{x^m}\Big(\bm{1}^\top \bm{B}^{m (s)} \Big)_j 
    &= \pp{}{x^m} \Bigg(\sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m
    )- 
    \\ & \hspace{6.5cm} \nonumber 
    \delta_{ij} X_i^m \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \bigg)
    \Bigg)
    \\
    &=
     \sum_{i=1}^N  \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m
    )- \delta_{ij} X_i^m \sum_{k=1}^N \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} \bigg)
    \\
    &=
     \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m
    )-   \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ji}^{\bm{r} {(s)}}}{x^m} X_j^m
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m
    )-   \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{-\bm{r} {(s)}}}{x^m} X_j^m
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m
    )-   \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} X_j^m
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  r^ml^m,
\end{align}
where the third equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow i$, the fourth the symmetry of conductance, and the fifth uses Operation \ref{op:2} on the second term. 
On the other hand, 
\begin{align}
    \Big( \bm{1}^{\top} \bm{D}^{m(s)} \Big)_j &= 
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} -\delta_{ij}  \sum_{k=1}^N \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} M_{ik}^{\bm{r}m}  \bigg)
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} - \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ji}^{\bm{r} {(s)}}}{x^m} M_{ji}^{\bm{r}m} 
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} - \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{-r} {(s)}}}{x^m} M_{ji}^{\bm{r}m}  
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} \bigg( M_{ij}^{\bm{r}m} - M_{ji}^{-\bm{r}m} \bigg)  
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}
    r^ml^m,
\end{align}
where the second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow i$, the third uses the symmetry of conductance, the fourth uses Operation \ref{op:2} on the second term, and the fifth uses the definition of the midpoint \eqref{eq:definition-of-midpoint}.
We lastly show the result given by \eqref{eq:Dd_proof_matrix-lemma_2_E-and-C}.
On one hand, 
\begin{align}
    \pp{}{x^m}\Big(\bm{1}^\top \bm{C}^{mn (s)} \bm{1} \Big) 
    &=
    \pp{}{x^m} \Bigg( 
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( G_{ij}^{\bm{r} {(s)}}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) -
    \\
    & \hspace{6cm} \nonumber
    \delta_{ij} X_i^m X_i^n \sum_{k=1}^N G_{ik}^{\bm{r} {(s)}} \bigg)
    \Bigg)
    \\
    &=
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) -
    \\
    & \hspace{6cm} \nonumber
    \delta_{ij} X_i^m X_i^n \sum_{k=1}^N \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} \bigg)
    \\
    &=
    \frac{1}{2} \Bigg( \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) -
    \\
    & \hspace{5cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} X_i^m X_i^n \bigg) \Bigg)
    \\
    &=
    \frac{1}{2} \Bigg( \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) -
    \\
    & \hspace{5cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  \pp{G_{ij}^{-\bm{r} {(s)}}}{x^m} X_j^m X_j^n \bigg) \Bigg)
    \\
    &=
    \frac{1}{2} \Bigg( \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) -
    \\
    & \hspace{5cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} X_j^m X_j^n \bigg) \Bigg)
    \\
    &=
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}  (X_j^m r^nl^n + X_j^n r^ml^m + r^ml^mr^nl^n),
\end{align}
where the third equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the fourth uses Operation \ref{op:1} on the second term, and the fifth uses Operation \ref{op:2} on the second term. 
On the other hand,
\begin{align}
    \bm{1}^\top \bm{E}^{mn (s)} \bm{1} 
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} (X_j^n + r^nl^n)  - 
    \\ & \hspace{7cm} \nonumber 
    \delta_{ij}  X_i^n \sum_{k=1}^N  \pp{G_{ik}^{\bm{r} {(s)}}}{x^m} M_{ik}^{\bm{r}m}  \bigg)
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} (X_j^n + r^nl^n)  - 
    \\ & \hspace{6cm} \nonumber 
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}}    \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} X_i^n
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} (X_j^n + r^nl^n)  - 
    \\ & \hspace{6cm} \nonumber 
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{-\bm{r} {(s)}}}{x^m} M_{ji}^{\bm{r}m} X_j^n
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ij}^{\bm{r}m} (X_j^n + r^nl^n)  - 
    \\ & \hspace{6cm} \nonumber 
    \sum_{i=1}^N \sum_{j=1}^N
        \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m} M_{ji}^{-\bm{r}m} X_j^n
        \\
        &=
        \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
        \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}
        \Big( 
        X_j^m r^nl^n + X_j^n r^ml^m + r^ml^m r^nl^n + 
        \\
        & \hspace{7.5cm} \nonumber
        X_j^n r^ml^m +X_i^m r^nl^n
        \Big)
    \\
    &=
    \pp{}{x^m}\Big(\bm{1}^\top \bm{C}^{mn (s)} \bm{1} \Big)  +
    \\ & \hspace{3.5cm} \nonumber
    \frac{1}{2}\sum_{i=1}^N \sum_{j=1}^N
    \sum_{\bm{r}\in\bm{\mathcal{R}}} \pp{G_{ij}^{\bm{r} {(s)}}}{x^m}
    \Big( 
    X_j^n r^ml^m +X_i^m r^nl^n
    \Big),
\end{align}
where the second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the third uses Operation \ref{op:1} on the second term, the fourth uses Operation \ref{op:2} on the second term, and the fifth uses the definition of the midpoint, \eqref{eq:definition-of-midpoint}.

\subsection{Simplification of permeability terms} \label{sec:proofs_remove-X-from-BC}
In this subsection, we simplify the terms in the permeability tensor, $\bm{k}^{(0)}$.
Firstly, we show that
\begin{align}
    \bm{1}^\top \bm{B}^{m (0)}  \bm{W}^n  = -l^m \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}  W_i^n.
    \label{eq:Dd_remove-X-from-B_mn}
\end{align}
We have
\begin{align}
    \bm{1}^\top \bm{B}^{m (0)}  \bm{W}^n 
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( G_{ij}^{\bm{r} {(0)}}  (X_j^m + r^ml^m
    )- \delta_{ij} X_i^m \sum_{k=1}^N G_{ik}^{\bm{r} {(0)}} \right) W_j^n
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}} W_j^n (X_j^m + r
    ^ml^m) 
    -
    \\ & \hspace{6cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}} W_i^n X_i^m
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{-\bm{r} {(0)}} W_i^n (X_i^m + r
    ^ml^m) 
    -
    \\ & \hspace{6cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}} W_i^n X_i^m
    \\
    &=
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}} W_i^n (X_i^m + (-r
    ^m)l^m) 
    -
    \\ & \hspace{6cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}} W_i^n X_i^m
    \\
    &=
    -l^m \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}  W_i^n.  
\end{align}
The first equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$, the third uses Operation \ref{op:1} on the first sum, and the fourth uses Operation \ref{op:2} on the first term.
Since the above was calculated without loss of generality in $m$ and $n$, it trivially follows that
\begin{align}
    \bm{1}^\top \bm{B}^{n (0)}  \bm{W}^m 
    =  
    -l^n \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^n G_{ij}^{\bm{r} {(0)}}  W_i^m.
    \label{eq:Dd_remove-X-from-B_nm}
\end{align}

Secondly, we prove that 
\begin{align}
    \bm{1}^\top \bm{C}^{mn (0)} \bm{1} = 
    - \frac{1}{2}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}
    \Big(
    X_i^mr^nl^n + X_i^nr^ml^m
    -
    r^ml^mr^nl^n
    \Big).
    \label{eq:Dd_remove-X-from-C_nm}
\end{align}
We have 
\begin{align}
    \bm{1}^\top \bm{C}^{mn (0)} \bm{1} 
    &= 
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} \left( G_{ij}^{\bm{r} {(0)}}  (X_j^m + r^ml^m)(X_j^n + r^nl^n) - \delta_{ij} X_i^m X_i^n \sum_{k=1}^N G_{ik}^{\bm{r} {(0)}} \right)
    \\
    &= 
    \frac{1}{2}
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}  
    X_j^mX_j^n 
    +
    \\ & \hspace{1.2cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}
    \Big(X_j^m r^nl^n + X_j^n r^ml^m\Big)
    +
    \\ & \hspace{1.2cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}
    r^ml^mr^nl^n
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} X_i^m X_i^n
    \bigg)
    \\
    &= 
    \frac{1}{2}
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{-\bm{r} {(0)}}  
    X_i^mX_i^n 
    +
    \\ & \hspace{1.2cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{-\bm{r} {(0)}}
    \Big(X_i^mr^nl^n + X_i^nr^ml^m\Big)
    +
    \\ & \hspace{1.2cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}
    r^ml^mr^nl^n
    - 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} X_i^m X_i^n
    \bigg)
    \\
    &= 
    - \frac{1}{2}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}
    \Big(
    X_i^mr^nl^n + X_i^nr^ml^m
    -
    r^ml^mr^nl^n
    \Big).
\end{align}
The second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow j$ in the final term, 
the third uses Operation \ref{op:1} on the first and second terms, and the third uses Operation \ref{op:2} on the first and second terms, so that the first and last terms cancel.

\subsection{Cancellation of fluid conservation terms} \label{sec:proofs_extra-terms}
In this subsection, we show that
\begin{align}
G_{ij}^{\bm{r}(0)}(r^nl^nX_i^m-r^ml^mX_i^n) 
+
\bm{1}^{\top} \bm{B}^{m(0)} \bm{W}^n 
-
\bm{1}^{\top} \bm{B}^{n(0)} \bm{W}^m  
&= 0.
\label{eq:Dd_lemma_extra-terms}
\end{align}
First note, using the cell problem, \eqref{eq:Dd_cell-problem_1}, that
\begin{align}
    0 &= (\bm{A}^{(0)} \bm{W}^m)_i + (\bm{B}^{m(0)} \bm{1})_i
    \\
    &=
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \bigg( G_{ij}^{\bm{r} {(0)}}  (W_j^m+ X_j^m + r^ml^m
    )- \delta_{ij} (W_i^m+X_i^m) \sum_{k=1}^N G_{ik}^{\bm{r} {(0)}} \bigg)
    \\
    &=
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{\bm{r} {(0)}}  \Big( y_j^m - y_i^m + r^ml^m  \Big),
\end{align}
where we have defined $\bm{y}^m$ by $y_i^m=X_i^m+W_i^m$, so that
\begin{align}
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} r^ml^m
        =
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  G_{ij}^{\bm{r} {(0)}}  \Big( y_i^m -y_j^m \Big). \label{eq:Dd_cell-problem_2}
\end{align}
Since we have formulated the cell problem independently of $X_a^b$ in \eqref{eq:Dd_cell-problem_2}, the fluid conservation problem itself is independent of $X_a^b$, and it follows that
\begin{align}
\sum_{i=1}^N \sum_{j=1}^N  \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (r^nl^nX_i^m- r^ml^m X_i^n)
+ 
\bm{1}^{\top} \bm{B}^{m(0)} \bm{W}^n 
- \bm{1}^{\top} \bm{B}^{n(0)} \bm{W}^m  
& =
\\
\sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
G_{ij}^{\bm{r}(0)}
\Big(
r^nl^ny_i^m-r^ml^my_i^n\Big) 
& =
\\
\sum_{i=1}^N 
\bigg( y_i^m \Big(
\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
G_{ij}^{\bm{r}(0)}
r^nl^n
\Big)
-
y_i^n
\Big(
\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} r^ml^m\Big) \bigg)
& =
\\
\sum_{i=1}^N 
\bigg( y_i^m \Big(
\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
G_{ij}^{\bm{r}(0)}
(y_i^n-y_j^n)
\Big)
-
y_i^n
\Big(
\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (y_i^m-y_j^m)\Big) \bigg)
& =
\\
\sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
G_{ij}^{\bm{r}(0)}
\Big(
y_i^ny_j^m
-
y_i^my_j^n
\Big) 
&= 0.
\end{align}
% \begin{alignat}{2}
% \sum_{i=1}^N \sum_{j=1}^N & \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)}  && \hspace{-0.4cm} (r^nl^nX_i^m- r^ml^m X_i^n)
% + 
% \bm{1}^{\top} \bm{B}^{m(0)} \bm{W}^n 
% - 
% \nonumber
% \\
% & \bm{1}^{\top} \bm{B}^{n(0)} \bm{W}^m  
% &&=
% \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
% G_{ij}^{\bm{r}(0)}
% \Big(
% r^nl^ny_i^m-r^ml^my_i^n\Big)
% \\
% & &&=
% \sum_{i=1}^N 
% \bigg( y_i^m \Big(
% \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
% G_{ij}^{\bm{r}(0)}
% r^nl^n
% \Big)
% -
% y_i^n
% \Big(
% \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} r^ml^m\Big) \bigg)
% \\
% & &&=
% \sum_{i=1}^N 
% \bigg( y_i^m \Big(
% \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
% G_{ij}^{\bm{r}(0)}
% (y_i^n-y_j^n)
% \Big)
% -
% y_i^n
% \Big(
% \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (y_i^m-y_j^m)\Big) \bigg)
% \\
% & &&=
% \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
% G_{ij}^{\bm{r}(0)}
% \Big(
% y_i^ny_j^m
% -
% y_i^my_j^n
% \Big) 
% \\
% & &&= 0.
% \end{alignat}
The first equation uses the simplification of the permeability terms, that is, that $\bm{1}^{\top} \bm{B}^{m(0)} \bm{W}^n$ and $\bm{1}^{\top} \bm{B}^{n(0)} \bm{W}^m$ can also be written independently of $X_a^b$, which we showed in Appendix \eqref{sec:proofs_remove-X-from-BC} through \eqref{eq:Dd_remove-X-from-B_mn} and \eqref{eq:Dd_remove-X-from-B_nm}, the third uses the re-formulation of the cell problem, \eqref{eq:Dd_cell-problem_2}, and the fourth uses a splitting of the sum, followed by Operation \ref{op:1} on the second term, followed by Operation \ref{op:2} on the second term.


\subsection{Independence of node location and permeability equality} \label{sec:proofs_k=sig}
We show that the permeability matrix is independent of the node locations, and, consequently, that  
\begin{align}
    k^{mn(0)} = \sigma^{mn(0)}.
\end{align}
We begin by simplifying $k^{mn(0)}$. We have
\begin{align}
    k^{mn(0)} 
    &= 
    \frac{l^m}{2 \prod_{a=1}^Dl^a}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)} 
    \Big(W_j^n - 
    W_i^n
     +
    X_j^n + r^nl^n 
    - 
    X_i^n \Big) 
    \\
    &=
    \frac{l^m}{2 \prod_{a=1}^Dl^a}  
    \bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)} 
    \Big(W_j^n 
     +
    X_j^n + r^nl^n    
    \Big)
    -
    \\
    \nonumber
    &
    \hspace{6cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)}
    \Big(
    W_i^n
    + 
    X_i^n \Big)
    \bigg)
    \\
    &=
    \frac{l^m}{2 \prod_{a=1}^Dl^a}  
    \bigg(
    \sum_{i=1}^N\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{-\bm{r} (0)} 
    \Big(W_i^n 
     +
    X_i^n + r^nl^n    
    \Big)
    -
    \\
    \nonumber
    &
    \hspace{6cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)}
    \Big(
    W_i^n
    + 
    X_i^n \Big)
    \bigg)
    \\
    &=
    \frac{l^m}{2 \prod_{a=1}^Dl^a}  
    \bigg(
    \sum_{i=1}^N\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    (-r^m) G_{ij}^{\bm{r} (0)} 
    \Big(W_i^n 
     +
    X_i^n + (-r^n)l^n    
    \Big)
    -
    \\
    \nonumber
    &
    \hspace{6cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)}
    \Big(
    W_i^n
    + 
    X_i^n \Big)
    \bigg)
    \\
    &=
    - \frac{l^m}{2 \prod_{a=1}^Dl^a}  
    \sum_{i=1}^N\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)} 
    \Big(W_i^n
     +
    X_i^n - \frac{r^n l^n}{2} \Big),
    \label{eq:Dd_simplified-k}
\end{align}
where the first equation uses the definition of the permeability, \eqref{eq:Dd_definition-of-permeability}, the third uses Operation \ref{op:1} on the first term, and the fourth uses Operation \ref{op:2} on the first term.
We simplify $\sigma^{mn(0)}$ similarly. 
Indeed, using the simplified forms of the permeability terms $\bm{1}^\top \bm{B}^{m (0)}  \bm{W}^n$ and $\bm{1}^\top \bm{B}^{n (0)}  \bm{W}^m$, \eqref{eq:Dd_remove-X-from-B_mn} and \eqref{eq:Dd_remove-X-from-B_nm}, we have
\begin{align}
    \sigma^{mn(0)} 
    &= 
    \frac{1}{\prod_m l^m}\Big( \frac{1}{2}\bm{1}^\top \bm{B}^{m (0)}  \bm{W}^n + \frac{1}{2}\bm{1}^\top \bm{B}^{n (0)}  \bm{W}^m + \bm{1}^\top \bm{C}^{mn (0)} \bm{1}
    \Big)
    \\
    &=
    - \frac{1}{2 \prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}}
    \Big(
    l^m 
    r^m   W_i^n
    +
    l^n 
    r^n   W_i^m
    \label{eq:Dd_simplified-sigma}
    \\
    & \hspace{5cm}
    \nonumber
    +
    X_i^mr^nl^n + X_i^nr^ml^m
    -
    r^ml^mr^nl^n
    \Big).
\end{align}

We note that for diagonal elements, that is, the case $n=m$, the result follows simply, since
\begin{align}
    \sigma^{mm(0)} 
    &= 
    \frac{1}{\prod_m l^m}\Big( \bm{1}^\top \bm{B}^{m (0)}  \bm{W}^m + \bm{1}^\top \bm{C}^{mm (0)} \bm{1}
    \Big)
    \\
    &=
    - \frac{1}{2 \prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}}
    \Big(
    2 l^m r^m W_i^m
    +
    2 X_i^mr^ml^m
    -
    (r^ml^m)^2
    \Big)
    \\
    &=
    - \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}
    \Big(
    W_i^m
    +
    X_i^m
    -
    \frac{r^ml^m}{2}
    \Big)
    \\
    &= k^{mm(0)},
\end{align}
where the first equation uses the simplified definition of $k^{mm(0)}$, \eqref{eq:Dd_simplified-sigma}, and the second uses the simplified definition of $k^{mm(0)}$, \eqref{eq:Dd_simplified-k}.
For off-diagonal elements, and thus in general, we have the case $n \neq m$, and from \eqref{eq:Dd_simplified-sigma} it instead follows that
\begin{align}
    \sigma^{mn(0)} 
    &=
    - \frac{1}{2 \prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} {(0)}}
    \Big(
    l^m 
    r^m   W_i^n
    +
    l^n 
    r^n   W_i^m
    +
    \\ & \hspace{6cm} \nonumber 
    X_i^mr^nl^n + X_i^nr^ml^m
    -
    r^ml^mr^nl^n
    \Big)
    \\
    &=
    - \frac{1}{2 \prod_m l^m}
    \bigg(
    l^m \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} r^m
    G_{ij}^{\bm{r} {(0)}}
    \Big(
    W_i^n
    + X_i^n
    - \frac{r^nl^n}{2}
    \Big)
    +
    \\
    \nonumber
    & \hspace{4cm}
    l^n \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} r^n
    G_{ij}^{\bm{r} {(0)}}
    \Big( W_i^m
    +
    X_i^m 
    -
    \frac{r^ml^m}{2}
    \Big)
    \\
    &= \frac{1}{2}(k^{mn(0)} + k^{nm(0)}),
    \label{eq:Dd_sigma-is-half-ks}
\end{align}
where the last equation uses the simplification of $k^{mn(0)}$, \eqref{eq:Dd_simplified-k}, once more.
It remains, therefore, to show that $k^{nm(0)}=k^{mn(0)}$ for $n \neq m$. 

To this end, we will show that $k^{mn(0)}$, and therefore $\sigma^{mn(0)}$, is independent of $X_a^b$.
Using the simplified definition of $k^{mn(0)}$, \eqref{eq:Dd_simplified-k}, we have 
\begin{align}
    \pp{k^{mn(0)}}{X_a^b} 
    &= 
    \pp{}{X_a^b}
    \bigg(
    - \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}
    \Big(
    W_i^n
    +
    X_i^n
    -
    \frac{r^nl^n}{2}
    \Big)
    \bigg)
    \\
    &=
    - \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}
    \Big(
    \pp{W_i^n}{X_a^b}
    +
    \delta_{ia}\delta^{nb}
    \Big).
    \label{eq:Dd_dkdx}
\end{align}
Now, we note that
\begin{align}
    \Big(\bm{A}^{(0)} \pp{\bm{W}^n}{X_a^b} \Big)_i &=
    \Big(\pp{}{X_a^b}
     (\bm{A}^{(0)}  \bm{W}^n) \Big)_i
    \\
    &=
    \Big(\pp{}{X_a^b}
    (- \bm{B}^{n (0)}  \bm{1}) \Big)_i
    \\
    &=
    -\pp{}{X_a^b}
    \Bigg(
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big( G_{ij}^{\bm{r} {(0)}}  (X_j^n + r^nl^n
    )- \delta_{ij} X_i^n \sum_{k=1}^N G_{ik}^{\bm{r} {(0)}} \Big)
    \Bigg)
    \\
    &=
    -\pp{}{X_a^b}
    \Bigg(
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}  (X_j^n + r^nl^n
    )- \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} X_i^n
    \Bigg)
    \\
    &=
    -
    \Bigg(
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} \delta_{ja}\delta^{nb} 
    - 
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} \delta_{ia} \delta^{nb}
    \Bigg)
    \\
    &=
    -
    \Bigg(
    \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Big(
    G_{ia}^{\bm{r} {(0)}}  
    - 
    \delta_{ia}
    \sum_{k=1}^N  G_{ik}^{\bm{r} {(0)}} 
    \Big)
    \Bigg) \delta^{nb}
    \\
    &=
    -A_{ia}^{(0)} \delta^{nb}
    \\
    &=
    -\Big(\bm{A}^{(0)} \bm{e}_a \delta^{nb} \Big)_i,
\end{align}
where the first equation uses the cell problem, \eqref{eq:Dd_cell-problem_1}.
Therefore
\begin{align}
    \bm{A}^{(0)} \Big( 
    \pp{\bm{W}^n}{X_a^b}+
    \bm{e}_a \delta^{nb}
    \Big)=\bm{0},
\end{align}
and so, using that $\bm{A}^{(0)} \bm{1} = \bm{0}$ from \eqref{eq:Dd_matrix-lemma_1}, we arrive at
\begin{align}
    \pp{\bm{W}^n}{X_a^b}
     = \alpha \bm{1} - \bm{e}_a \delta^{nb},
\end{align}
so that
\begin{align}
    \pp{W_i^n}{X_a^b} = \alpha - \delta_{ia} \delta^{nb},
    \label{eq:Dd_dwdx}
\end{align}
for some function $\alpha(x,t)$ that is independent of the microscale.
Substituting this result, \eqref{eq:Dd_dwdx}, into the equation for the derivative of $k^{mn(0)}$, \eqref{eq:Dd_dkdx}, we find that
\begin{align}
    \pp{k^{mn(0)}}{X_a^b} 
    &=
    - \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}
    \Big(
    \pp{W_i^n}{X_a^b}
    +
    \delta_{ia}\delta^{nb}
    \Big)
    \\
    &=
    - \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}
    \alpha
    \\
    &=
    - \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    (-r^m) G_{ij}^{\bm{r} {(0)}}
    \alpha
    \\
    &=
    \frac{l^m}{\prod_m l^m}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    r^m G_{ij}^{\bm{r} {(0)}}
    \alpha
    \\
    &=
    -\pp{k^{mn(0)}}{X_a^b},
\end{align}
where the third equation uses Operation \ref{op:1} followed by Operation \ref{op:2}.
We have therefore shown that
\begin{align}
    \pp{k^{mn(0)}}{X_a^b}  = 0.
\end{align}
Since this was proved for general $m$ and $n$, it trivially follows that
\begin{align}
    \pp{k^{nm(0)}}{X_a^b}  = 0,
\end{align}
and therefore that 
\begin{align}
    \pp{}{X_a^b} \bigg( \frac{1}{\prod_m l^m}\Big( \frac{1}{2}\bm{1}^\top \bm{B}^{m (0)}  \bm{W}^n + \frac{1}{2}\bm{1}^\top \bm{B}^{n (0)}  \bm{W}^m + \bm{1}^\top \bm{C}^{mn (0)} \bm{1}
    \Big) \bigg) 
    &=
    \\
    \pp{\sigma^{mn(0)}}{X_a^b} 
    &=
    \\
    \frac{1}{2} \pp{}{X_a^b} (k^{mn(0)} + k^{nm(0)}) 
    &= 0,
\end{align}
where the first equation uses the definition of $\sigma^{mn(0)}$, \eqref{eq:Dd_sigma}, and the second uses the relationship between $\sigma^{mn(0)}$ and $k^{mn(0)}$,
\eqref{eq:Dd_sigma-is-half-ks}.
So $k^{mn(0)}$ and $\sigma^{mn(0)}$, and therefore $\bm{B}^{m (0)}$ and $\bm{C}^{mn (0)}$, are independent of $X_a^b$, which permits the choice $X_a^b=0$ in the original definitions of the latter two, \eqref{eq:Dd_def_B} and \eqref{eq:Dd_def_C}, allowing us to re-write these as
\begin{align}
    B_{ij}^{m (0)} &= 
    \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} r^ml^m,
    \\
    C_{ij}^{mn (0)} &= 
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}  r^ml^m r^nl^n.
\end{align}
Using these more simple definitions, we note that $\bm{B}^{m (0)}$ is skew-symmetric in the lower indices, so that
\begin{align}
    \Big(\bm{B}^{m (0)}\Big)^\top = -\bm{B}^{m (0)}
\end{align}
and $\bm{C}^{mn (0)}$ is symmetric in the lower indices, so that
\begin{align}
    \Big(\bm{C}^{mn (0)}\Big)^\top = \bm{C}^{mn (0)},
\end{align}
since, using Operation \ref{op:2}, we have that
\begin{align}
    B_{ji}^{m (0)} &= 
    \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ji}^{\bm{r} {(0)}} r^ml^m 
    \\ 
    &=
    \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{-\bm{r} {(0)}} r^ml^m 
    \\
    &=
    \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} (-r^m)l^m 
    \\
    &=
    -\sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} r^m l^m
    \\
    &=
    -B_{ij}^{m (0)},
\end{align}
and that
\begin{align}
    C_{ji}^{mn (0)} &= 
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ji}^{\bm{r} {(0)}}  r^ml^m r^nl^n
    \\
    &=
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{-\bm{r} {(0)}}  r^ml^m r^nl^n
    \\
    &=
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}}  (-r^m)l^m (-r^n)l^n
    \\
    &=
    \frac{1}{2} \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r} {(0)}} r^ml^m r^nl^n
    \\
    &= C_{ij}^{mn (0)}.
\end{align}
As a result, we have symmetry in the upper indices of the permeability terms, such that
\begin{align}
    (\bm{W}^n)^\top \bm{B}^{m (0)} \bm{1} 
    &=
     \Big( (\bm{B}^{m (0)} \bm{1})^\top \bm{W}^n \Big)^\top 
     \\
     &=
     (\bm{B}^{m (0)} \bm{1})^\top \bm{W}^n
     \\
     &=
     (-\bm{A}^{(0)} \bm{W}^m)^\top \bm{W}^n
     \\
     &=
     -( \bm{W}^m)^\top \bm{A}^{(0)} \bm{W}^n
     \\
     &=
     ( \bm{W}^m)^\top \bm{B}^{n(0)} \bm{1}.
     \label{eq:Dd_wB1_mn-is-wB1_nm}
\end{align}
Now, choosing $X_a^b=0$ in the simplified definition of $k^{mn(0)}$, \eqref{eq:Dd_simplified-k}, we have
\begin{align}
    k^{mn(0)}-k^{nm(0)} &= 
    - \frac{l^m}{\prod_m l^m}  
    \sum_{i=1}^N\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^m G_{ij}^{\bm{r} (0)} 
    \Big(W_i^n
    - \frac{r^n l^n}{2} \Big)
    +
    \\
    &
    \nonumber
    \hspace{3cm}
    \frac{l^n}{\prod_n l^n}  
    \sum_{i=1}^N\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  
    r^n G_{ij}^{\bm{r} (0)} 
    \Big(W_i^m
     - \frac{r^m l^m}{2} \Big)
     \\
     &=
    - \frac{1}{\prod_m l^m}
    \bigg(
    \sum_{i=1}^N\sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    G_{ij}^{\bm{r} (0)}
    \Big(
    l^m  
    r^m  
    W_i^n
    -
    l^n
    r^n 
    W_i^m \Big)
     \bigg)
     \\
     &=
    - \frac{1}{\prod_m l^m}
    \Big( (\bm{W}^n)^\top \bm{B}^{m (0)} \bm{1} -
    (\bm{W}^m)^\top \bm{B}^{n (0)} \bm{1} \Big)
    \\
    &=
    0,
    \label{eq:Dd_symmetry-of-k}
\end{align}
where we have used this the symmetry, \eqref{eq:Dd_wB1_mn-is-wB1_nm}, in the last equation.
Therefore, using \eqref{eq:Dd_sigma-is-half-ks}, we arrive at
\begin{align}
    \sigma^{mn(0)}
    &= \frac{1}{2}(k^{mn(0)} + k^{nm(0)}) = k^{mn(0)}. 
\end{align}

\subsection{Intra-cell particle conservation} \label{sec:proofs_S}
In this subsection, we show that $\bm{1}^{\top} \bm{S}^{m(0)} = \bm{0}^{\top}$ and $\bm{S}^{m(0)} \bm{1} = \bm{0}$.

We first show that $\bm{S}^{m(0)} \bm{1} = \bm{0}$. 
We have 
\begin{align}
    \Big(\bm{S}^{m(0)} \bm{1}\Big)_i 
    & = 
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
   \Big( G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m) H_{ji}^{-\bm{r}(0)}
    +
    \\ 
    & \hspace{2cm}
    \nonumber \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}(0)}(W_k^m-W_i^m+X_k^m+r^ml^m-X_i^m) H_{ik}^{\bm{r}(0)}
    \Big)
    \\
    &=
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
   \Big( G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m)
    \\
    &=
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Big( G_{ij}^{\bm{r}(0)} - \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}(0)} \Big) W_j^m +
    \\
    & \hspace{2.5cm} \nonumber
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}  \Big( 
    G_{ij}^{\bm{r}(0)} (X_j^m + r^ml^m) - \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}(0)} X_i^m \Big)
    \\
    &=
    (\bm{A}^{(0)} \bm{W}^m)_i + (\bm{B}^{m(0)} \bm{1})_i 
    \\
    &= 
    0.
\end{align}
The first equation uses the definition of $\bm{S}^{m(0)}$, \eqref{eq:Dd_definition-of-S}, the second uses $\delta_{ij}$ followed by the Heaviside function identity, \eqref{eq:Dd_heaviside-identity}, and the fifth uses the cell problem \eqref{eq:Dd_cell-problem_1}.

Next, we show that $\bm{1}^{\top} \bm{S}^{m(0)} = \bm{0}^{\top}$. 
We have 
\begin{align}
    \Big(\bm{1}^{\top} \bm{S}^{m(0)}\Big)_j 
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
   \Big( G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m) H_{ji}^{-\bm{r}(0)}
    +
    \\ 
    & \hspace{2cm}
    \nonumber \delta_{ij} \sum_{k=1}^N G_{ik}^{\bm{r}(0)}(w_k^m-W_i^m+X_k^m+r^ml^m-X_i^m) H_{ik}^{\bm{r}(0)}
    \Big)
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Big( G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m) H_{ji}^{-\bm{r}(0)}
    +
    \\ 
    & \hspace{1.5cm}
    \nonumber 
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ji}^{\bm{r}(0)}(W_i^m-W_j^m+X_i^m+r^ml^m-X_j^m) H_{ji}^{\bm{r}(0)}
    \Big)
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Big( G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m) H_{ji}^{-\bm{r}(0)}
    -
    \\ 
    & \hspace{1.5cm}
    \nonumber 
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ji}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m-r^ml^m-X_i^m) H_{ji}^{\bm{r}(0)}
    \Big)
    \\
    &=
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} 
    \Big( G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m) H_{ji}^{-\bm{r}(0)}
    -
    \\ 
    & \hspace{1.5cm}
    \nonumber 
    \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)}(W_j^m-W_i^m+X_j^m+r^ml^m-X_i^m) H_{ji}^{-\bm{r}(0)}
    \Big)
    \\
    &=
    0.
\end{align}
The second equation uses $\delta_{ij}$ followed by a re-indexing $k \rightarrow i$ in the second term, the third uses a factorisation of $-1$ in the second term to achieve like-terms,
and the fourth uses the symmetry of conductance in the second term followed by Operation \ref{op:2} on the second term.

\subsection{Interpretation of advective terms} \label{sec:proofs_K-is-k}
In this subsection we show that the coefficient in the advective term of the particle-conservation equation, \eqref{eq:Dd_macroscale_transport}, is related to the permeability by
\begin{align}
    \bm{1}^{\top} \bm{K}^{mn(0)} \bm{1} 
    = 
    \prod_{a=1}^Dl^a k^{mn(0)} 
    = 
    \prod_{a=1}^Dl^a \sigma^{mn(0)}.
\end{align}
We have
\begin{align}
    \bm{1}^{\top} \bm{K}^{mn(0)} \bm{1} 
    &= 
    \\ & \hspace{-1cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m ) (X_j^n + 
    r^nl^n)H_{ji}^{-\bm{r}(0)}
    +
    \\ & \hspace{0cm} 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m )
     X_i^n H_{ij}^{\bm{r}(0)} =
    \\ & \hspace{-1cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m ) (X_j^n + 
    r^nl^n)H_{ji}^{-\bm{r}(0)}
    +
    \\ & \hspace{0cm} 
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{-r}(0)} (W_i^m- W_j^m + X_i^m + r^ml^m - X_j^m )
     X_j^n H_{ji}^{\bm{r}(0)} = 
    \\ & \hspace{-1cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m ) (X_j^n + 
    r^nl^n)H_{ji}^{-\bm{r}(0)}
    +
    \\ & \hspace{0cm}
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_i^m- W_j^m + X_i^m - r^ml^m - X_j^m )
     X_j^n H_{ji}^{-\bm{r}(0)} = 
    \\ & \hspace{-1cm} \nonumber
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m )  
    r^nl^n H_{ji}^{-\bm{r}(0)},
\end{align}
where the first equation uses an expansion of the definition of $K_{ij}^{mn(0)}$, \eqref{eq:Dd_definition-of-K}, the second uses Operation \ref{op:1} on the second term, and the third uses Operation \ref{op:2} on the second term.
Defining
\begin{align}
    \Lambda_{ij}^{\bm{r}mn(0)} =
    G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m )  r^nl^n,
    \label{eq:Dd_definition-of-Lambda}
\end{align}
and noting that $\Lambda_{ji}^{-\bm{r}mn(0)}=\Lambda_{ij}^{\bm{r}mn(0)}$, it follows that
\begin{align}
    \bm{1}^{\top} \bm{K}^{mn(0)} \bm{1} 
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Lambda_{ij}^{\bm{r}mn(0)} H_{ji}^{-\bm{r}(0)}
    +
    \frac{1}{2} \sum_{j=1}^N \sum_{i=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Lambda_{ji}^{\bm{r}mn(0)} H_{ij}^{-\bm{r}(0)}
    \\
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{-\bm{r}} \Lambda_{ij}^{\bm{r}mn(0)} H_{ji}^{-\bm{r}(0)}
    +
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Lambda_{ij}^{-\bm{r}mn(0)} H_{ij}^{-\bm{r}(0)}
    \\
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} \Lambda_{ij}^{\bm{r}mn(0)}
    \\
    &=  
    \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}} G_{ij}^{\bm{r}(0)} (W_j^m- W_i^m + X_j^m + r^ml^m - X_i^m )  
    r^nl^n
    \\
    &= \prod_{a=1}^Dl^a k^{nm(0)}
    \\
    &= \prod_{a=1}^Dl^a k^{mn(0)},
\end{align}
where the first equation uses a splitting of terms followed by a re-labelling $i\leftrightarrow j$ in the second term, the second uses the symmetry of $\Lambda_{ij}^{\bm{r}mn(0)}$, the third uses Operation \ref{op:2} on the second term followed by the Heaviside function identity, \eqref{eq:Dd_heaviside-identity}, the fourth uses the definition of $\Lambda_{ij}^{\bm{r}mn(0)}$, \eqref{eq:Dd_definition-of-Lambda}, the fifth uses the definition of the permeability tensor, \eqref{eq:Dd_definition-of-permeability}, and the sixth uses the symmetry of the permeability matrix, \eqref{eq:Dd_symmetry-of-k}.

\section{Darcy velocity}
\label{sec:darcy-proof}
In this appendix, we show that our definitions of the Darcy velocity, \eqref{eq:1d_definition-of-darcy} and \eqref{eq:Dd_definition-of-darcy}, are the correct choice, using an alternative but equivalent formulation. 
As an illustrative example, consider the $m=1$ case in two dimensions. 
Intuitively, the net flux per unit area out of a reference cell into cells with a right-facing component is
\begin{align}
    \hat{u}^1 = \frac{1}{l^2} \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_+^1} Q_{ij}^{\bm{r}}, 
\end{align}
where $\bm{\mathcal{R}}_+^1 = \{ \bm{r} \in \bm{\mathcal{R}} \, | \, r^1 = 1 \}$ is the 
set of references for cell positions immediately to the right of a cell, and $l^2$ is the vertical length, so that $l^1l^2$ represents the cross-sectional area in the two-dimensional case.
Now, we observe that 
\begin{align}
    \hat{u}^1
    &=
    \frac{1}{2 l^2}
    \Bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_+^1}  r^1  Q_{ij}^{\bm{r}} +
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_+^1}  r^1  Q_{ji}^{\bm{r}}
    \Bigg)
    \\
    &=
    \frac{1}{2 l^2}
    \Bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_+^1} r^1  Q_{ij}^{\bm{r}} -
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_+^1}  r^1  Q_{ij}^{-\bm{r}}
    \Bigg)
    \\
    &=
    \frac{1}{2 l^2}
    \Bigg(
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_+^1}  r^1  Q_{ij}^{\bm{r}} +
    \sum_{i=1}^N \sum_{j=1}^N \sum_{\bm{r} \in \bm{\mathcal{R}}_-^1} r^1 Q_{ij}^{\bm{r}}
    \Bigg)
    \\
    &= u^1,
\end{align}
where $\bm{\mathcal{R}}_-^m = \{ \bm{r} \in \bm{\mathcal{R}} \, | \, r^m = -1 \}$ denotes the set of positions of cells immediately to the left of the reference cell.
The first equation is a relabelling $i \leftrightarrow j$ in the second term, and the second equation uses $Q_{ij}^{\bm{r}}=Q_{ji}^{-\bm{r}}$ in the second term.
The final equation is our original definition of the Darcy velocity, 
\eqref{eq:1d_definition-of-darcy} and \eqref{eq:Dd_definition-of-darcy}.
So the flux through the cell is equal to half of the flux out of the cell to all other cells, due to the periodicity of the network.


\printbibliography %Prints bibliography
%\bibliography{thesis.bib}

\end{document}

\begin{align}
\label{eq:1d_master_transport_6}
    & V_i \pp{}{\tau}
    \Big(
    C_i^{(0)} + \varepsilon \Big\{ C_i^{(1)}+ X_i \pp{C_i^{(0)}}{x} \Big\} + 
    \varepsilon^2
    \Big\{
    C_i^{(2)}+ X_i \pp{C_i^{(1)}}{x} 
    +
    \frac{1}{2}   X_i^2
    \pppd{C_i^{(0)}}{x}{x}
    \Big\}
    \Big) = 
    \\
    \nonumber
    & 
    \sum_{j=1}^N \sum_{r\in\mathcal{R}}
    \varepsilon 
    \bigg\{
    G_{ij}^{r(0)}
    \Big(
    W_j - W_i + X_j + r l - X_i 
    \Big)
    \pp{p^{(0)}}{x}
    (C_j^{(0)}H_{ji}^{-r(0)}+C_i^{(0)}H_{ij}^{r(0)})
    \bigg\}
    +
    \\
    & \hspace{1cm}
    \varepsilon^2 \bigg\{
    G_{ij}^{r(0)}
    \Big(
    W_j - W_i + X_j + r l - X_i 
    \Big)
    \pp{p^{(0)}}{x}
    (C_j^{(1)}H_{ji}^{-r(0)}+C_i^{(1)}H_{ij}^{r(0)})
    +
    \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{r(0)}
    \Big(
    W_j - W_i + X_j + r l - X_i 
    \Big)
    \pp{p^{(0)}}{x}
    \Big( (X_j+r l)\pp{C_j^{(0)}}{x}H_{ji}^{-r(0)}
    +
     X_i\pp{C_i^{(0)}}{x}H_{ij}^{r(0)}\Big)
    +
        \nonumber
    \\
    & \hspace{1cm} 
    \bigg(
    G_{ij}^{r(0)}
    \Big( P_j^{(2)} -P_i^{(2)} \Big)
    +
    G_{ij}^{r(1)}
    \Big(
    W_j - W_i + X_j + r l - X_i \Big)
    \pp{p^{(0)}}{x}
    +
    \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{r(0)} \bigg(  (X_j + r l) \pp{}{x} \Big( f+  W_j \pp{p^{(0)}}{x} \Big)
    - 
     X_i \pp{}{x} \Big(f+  W_i \pp{p^{(0)}}{x} \Big) \bigg)
    +
    \nonumber
    \\
    & \hspace{1cm}
    \frac{G_{ij}^{r(0)}}{2} 
    \bigg(   (X_j + r l)^2
    - 
      X_i^2  \bigg)
    \ppp{p^{(0)}}{x}
    +
    \nonumber
    \\
    & \hspace{1cm}
     \pp{G_{ij}^{r(0)}}{x} M_{ij}^{r}  \Big(
    W_j - W_i + X_j + r l - X_i \Big)
    \pp{p^{(0)}}{x}
     \bigg) (C_j^{(0)}H_{ji}^{-r(0)}+C_i^{(0)}H_{ij}^{r(0)}) -
     \nonumber
     \\
     & \hspace{1cm}
     G_{ij}^{r(0)}
     \Big(
    W_j - W_i + X_j + r l - X_i \Big)
    \pp{p^{(0)}}{x}
    C_j^{(0)} H_{ji}^{-r(0)} A_{ji}^{-r(0)}
     \bigg\}.
    \nonumber
\end{align}



\begin{align}
\label{eq:Dd_master_transport_6}
    & V_i \pp{}{\tau}
    \Big(
    C_i^{(0)} + \varepsilon \Big\{ C_i^{(1)}+\sum_{m=1}^D X_i^m \pp{C_i^{(0)}}{x} \Big\} +
    \nonumber
    \\
    &
    \hspace{3cm}
    \varepsilon^2
    \Big\{
    C_i^{(2)}+\sum_{m=1}^D X_i^m \pp{C_i^{(1)}}{x} 
    +
    \frac{1}{2} \sum_{m=1}^D \sum_{n=1}^D X_i^m X_i^n
    \pppd{C_i^{(0)}}{x^m}{x^n}
    \Big\}
    \Big) =
    \\
    \nonumber
    & 
    \sum_{j=1}^N \sum_{\bm{r}\in\bm{\mathcal{R}}}
    \varepsilon 
    \bigg\{
    G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Big(
    W_j^m - W_i^m + X_j^m + r^ml^m - X_i^m 
    \Big)
    \pp{p^{(0)}}{x^m}
    (C_j^{(0)}H_{ji}^{-\bm{r}(0)}+C_i^{(0)}H_{ij}^{\bm{r}(0)})
    \bigg\}
    +
    \\
    & \hspace{1cm}
    \varepsilon^2 \bigg\{
    G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Big(
    W_j^m - W_i^m + X_j^m + r^ml^m - X_i^m 
    \Big)
    \pp{p^{(0)}}{x^m}
    (C_j^{(1)}H_{ji}^{-\bm{r}(0)}+C_i^{(1)}H_{ij}^{\bm{r}(0)})
    +
    \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D
    \Big(
    W_j^m - W_i^m + X_j^m + r^ml^m - X_i^m 
    \Big)
    \pp{p^{(0)}}{x^m}
    \Big(\sum_{n=1}^D (X_j^n+r^nl^n)\pp{C_j^{(0)}}{x^n}H_{ji}^{-\bm{r}(0)}
    +
    \nonumber
    \\
    & \hspace{1cm} 
    \sum_{n=1}^D X_i^n\pp{C_i^{(0)}}{x^n}H_{ij}^{\bm{r}(0)}\Big)
    +
    \bigg(
    G_{ij}^{\bm{r}(0)}
    \Big( P_j^{(2)} -P_i^{(2)} \Big)
    +
    \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{\bm{r}(1)}
    \sum_{m=1}^D\Big(
    W_j^m - W_i^m + X_j^m + r^ml^m - X_i^m \Big)
    \pp{p^{(0)}}{x^m}
    +
    \nonumber
    \\
    & \hspace{1cm}
    G_{ij}^{\bm{r}(0)} \bigg( \sum_{m=1}^D (X_j^m + r^ml^m) \pp{}{x^m} \Big( f+ \sum_{n=1}^D W_j^n \pp{p^{(0)}}{x^n} \Big)
    - 
    \sum_{m=1}^D X_i^m \pp{}{x^m} \Big(f+ \sum_{n=1}^D W_i^n \pp{p^{(0)}}{x^n} \Big) \bigg)
    +
    \nonumber
    \\
    & \hspace{1cm}
    \frac{G_{ij}^{\bm{r}(0)}}{2} 
    \bigg( \sum_{m=1}^D \sum_{n=1}^D (X_j^m + r^ml^m)(X_j^n + r^nl^n)
    - 
    \sum_{m=1}^D \sum_{n=1}^D X_i^m X_i^n  \bigg)
    \pppd{p^{(0)}}{x^m}{x^n}
    +
    \nonumber
    \\
    & \hspace{1cm}
    \sum_{m=1}^D \pp{G_{ij}^{\bm{r}(0)}}{x^m} M_{ij}^{\bm{r}m} \sum_{n=1}^D \Big(
    W_j^n - W_i^n + X_j^n + r^nl^n - X_i^n \Big)
    \pp{p^{(0)}}{x^n}
     \bigg) (C_j^{(0)}H_{ji}^{-\bm{r}(0)}+C_i^{(0)}H_{ij}^{\bm{r}(0)}) 
     -
     \nonumber
     \\
     & \hspace{1cm}
     G_{ij}^{\bm{r}(0)}
    \sum_{m=1}^D \Big(
    W_j^m - W_i^m + X_j^m + r^ml^m - X_i^m \Big)
    \pp{p^{(0)}}{x^m}
    C_j^{(0)} H_{ji}^{-\bm{r}(0)} A_{ji}^{-\bm{r}(0)}
     \bigg\},
    \nonumber
\end{align}