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Article
Optimising Dead-End Cake Filtration Using Poroelasticity Theory

J. Köry 1

, A. U. Krupp 2, C. P. Please 2

and I. M. Griffiths 2,*

1

School of Mathematics and Statistics, University of Glasgow, 132 University Pl, Glasgow G12 8TA, UK;
Jakub.Koery@glasgow.ac.uk

2 Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford OX2 6GG, UK;

please@maths.ox.ac.uk (C.P.P.)

* Correspondence: ian.griffiths@maths.ox.ac.uk

Abstract: Understanding the operation of filters used to remove particulates from fluids is important
in many practical industries. Typically the particles are larger than the pores in the filter so a cake
layer of particles forms on the filter surface. Here we extend existing models for filter blocking to
account for deformation of the filter material and the cake layer due to the applied pressure that
drives the fluid. These deformations change the permeability of the filter and the cake and hence the
flow. We develop a new theory of compressible-cake filtration based on a simple poroelastic model
in which we assume that the permeability depends linearly on local deformation. This assumption
allows us to derive an explicit filtration law. The model predicts the possible shutdown of the filter
when the imposed pressure difference is sufficiently large to reduce the permeability at some point
to zero. The theory is applied to industrially relevant operating conditions, namely constant flux,
maximising flux and constant pressure drop. Under these conditions, further analytical results are
obtained, which yield predictions for optimal filter design with respect to given properties of the
filter materials and the particles.

Keywords: poroelasticity; filtration; heterogeneous media; caking

Citation: Köry, J.; Krupp, A.U.;

Please, C.P.; Griffiths, I.M. Optimising

1. Introduction

Dead-End Cake Filtration Using

Poroelasticity Theory. Modelling 2021,

2, 18–42. https://doi.org/10.3390/

modelling2010002

Received: 21 November 2020

Accepted: 21 December 2020

Published: 9 January 2021

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Attribution (CC BY) license (https://

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4.0/).

Membranes are an ideal choice for filtering of fluids because they can create highly
selective barriers that can retain particles due to various processes such as size-exclusion
(straining), electro-chemical and other effects [1]. As a result, membrane filtration has
become the method of choice in various applications for separations, not only due to its
economic advantage over other separation techniques due to lower energy costs, but also
because it is better suited to preserve the quality of the product in the food and beverage
industry than other purification processes [2]. Applications in industry range from small-
scale (protein filtration, virus removal) to large-scale (wastewater treatment) processes.
Classical modes of operation include dead-end and cross-flow filtration, depending on
whether the filter membrane surface is, respectively, either perpendicular or parallel to the
flow direction [3].

There are two main different ways in which the particles in the fluid can be retained
by the membrane. In the first, which is characterised by the particles being small compared
to the membrane pores, retention occurs inside the membrane, while in the second, which
is characterised by the particles being bigger than the membrane pores, retention occurs by
particles forming an additional layer on top of the membrane, usually referred to as a cake.
In this work, we will refer to the whole device comprising a filter (the membrane) and a
cake as a filtercake. Modelling of cake formation during both dead-end and cross-flow
filtration has been studied extensively in the literature (see, for example, King & Please [4]
and Sanaei et al. [5]).

Early core work on the build-up of a cake at a membrane surface comprised studies
in compressional rheology. In the pioneering paper by Buscall & White [6] , the authors

Modelling 2021, 2, 18–42. https://doi.org/10.3390/modelling2010002

https://www.mdpi.com/journal/modelling

(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) Modelling 2021, 2

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developed a theory of sedimentation in the form of a two-phase model, which forms the
basis of many subsequent modelling efforts. A notable feature of this paper that permeates
subsequent work is the concept of a yield stress of the cake, which is estimated using
experimental data. The resulting model comprises a one-dimensional nonlinear diffusion
equation for the volume fraction with a moving boundary denoting the free surface of the
cake. The diffusion coefficient depends on both the permeability of the cake and the yield
stress. In Landman et al. (1995) [7], the authors showed how the framework developed by
Buscall & White [6] can be reconciled with simpler rheological relationships in the context
of pressure filtration. In particular, the authors show how the growth of the cake layer can
be predicted for a given applied pressure difference.

Applications of the compressional rheology theory to particular filtration processes
are studied further in Landman et al. (1997) [8], with the objective of using optimal control
to maximise the processing rate. In this case they reduce the nonlinear diffusion equation
derived in Buscall & White [6] to a linearised version to facilitate a more tractable analysis.
Similarly, Kapur et al. [9] consider a simplified version of the model of Buscall & White [6]
to speed up computation.

Dewatering is an important application in which cake build-up is a key feature. In
this process, water is removed from solid material or soil with the target product being the
solid material. Stickland et al. (2016) [10] study the dewatering process using the ideas of
compressional rheology in a case where optimisation is the primary objective in forming
the cake layer. Stickland et al. (2018) [11] also show how compressional rheology models
may be used to describe the specific case of wastewater treatment. Hewitt et al. [12] show
how the use of a compressive yield stress is a crucial component in accurately capturing the
dewatering process. They validate their theory using experiments and demonstrate how
the product that is being filtered plays a key role in the behaviour, finding that their model
does not fit the behaviour for the filtration of cellulose. Finally, Eaves et al. [13] study the
dewatering process for paper and slurries. Again the key concept here is the modelling
of water flow through a deforming consolidating material, however the main novelty in
this paper is the more complicated physical configuration of the filtration process that is
considered.

Vacuum filtration is another industrial process in which the build-up of cake is an
important characteristic. A notable differentiating feature of this process to those afore-
mentioned is that desaturation effects must be taken into account to model the air–liquid
capillary imbibition (see for example Stickland et al. (2010) [14]).

While there is an extensive body of literature that focuses on the accurate modelling
of the formation and growth of a cake layer on a membrane, there has been much less
work done to examine how mechanical deformation of both the cake and the underlying
filter material, and the interplay between the two, might affect the behaviour of the entire
system, and it is this question that forms the focus of this study.

We consider an elastic-response model to describe both the membrane and cake
behaviour, which will allow for both a suitable characterisation of the membrane dynamics
while enabling a direct comparison between the behaviour of the two materials. When
a fluid flows through a porous medium, it exerts forces on the porous matrix, inducing
deformations. These deformations in turn affect the local permeability of the material,
which then influences the fluid flow. This coupling between the fluid flow and porous
medium deformation is often modelled using poroelasticity theory, developed in the first
half of the 20th century by Terzaghi and Biot [15,16] and since then applied in a plethora of
industrial as well as biological applications [17–22].

To our knowledge, the first attempt to explicitly model the effects of elastic deforma-
tion on the local permeability of the porous medium was presented in Parker et al. [18].
Authors extended a one-dimensional steady-state poroelastic model—comprising Darcy’s
law for fluid flow, the Navier equation (equipped with a pressure gradient term arising from
Terzaghi’s principle) for deformation and a conservation equation for fluid—by assuming a
variety of constitutive assumptions relating local strain and permeability. In Köry et al. [23],

Modelling 2021, 2

20

the analytic solution under the linear constitutive assumption from Parker et al. [18] was
used to obtain explicit conditions for porous-medium shutdown (deformation leading to
locally zero permeability, which allows no flow through the medium). The analysis was
then extended to the case of non-uniform rest-state permeability, i.e., permeability in the
absence of any flow, with a special emphasis on applications to filtration processes.

Other models introduce more realistic descriptions of flow-deformation coupling
either via power-law constitutive assumptions or using explicitly multi-phase frameworks
(see Lee & Wang [24]), but these are both less intuitive and less amenable to analytical
progress, and become computationally challenging as the number of model parameters
increases. A more complicated model consisting of a poroelastic model (with a more
advanced relationship between the local permeability and deformation) and a fouling
model (modelling the effects of caking and intra-membrane clogging separately) has
recently been employed to explain the observed pressure–time signatures in direct-flow
filtration under constant flux [25].

In this work, we extend the simple poroelastic framework from Köry et al. [23] to
model cake filtration in a dead-end filter. The key objective of this work is to understand
how deformations in both the filter and the cake interact with one another and affect the
behaviour of the overall filtration process. We continue to assume the elastic response is
instantaneous on the time scale of filtration, however the problem is time dependent due
to the cake formation which takes the form of a moving boundary problem. In Section 2,
we formulate the model for a compressible filtercake including the boundary and interface
conditions. We choose to model the system using a linearised theory in order to elucidate
the concept of the interplay between the compression of the cake and the underlying mem-
brane; the modelling ideas we lay out here readily generalise to more complex nonlinear
theories. We nondimensionalise the system and then find the conditions required to both
ensure the validity of small-deformations assumptions and avoid filter shutdown. We
then derive the resulting caking filtration law (Equation (30)). The caking law depends on
the time-dependent pressure drop across the filtercake and two dimensionless material
parameters, reflecting the behaviour of the two porous media (the filter and the cake;
c.f. Köry et al. [23]).

In Section 3, we study the implications of the theory to three industrially relevant
operating conditions, namely constant flux, maximising flux, and constant pressure-drop
filtration, placing an emphasis on optimal filter design in each case. We conclude in
Section 4 by summarising the most important outcomes of our analysis and identifying
areas in which the proposed theoretical framework could be improved and extended.

2. Poroelastic Model of Cake Formation

We shall assume that both the cake and filter are compressible and propose a model
for a one-dimensional dead-end filtration device where the increase of resistance of a filter
is due to both the build-up of a cake and the compression of both the filter and the cake.

2.1. Governing Equations

We denote the thickness of the cake in the undeformed configuration in the absence
of fluid flow by ˜Lc and that of the filter in the undeformed configuration by ˜L f . We will
use the assumption of small deformations throughout this work (namely that the applied
pressure difference on the system leads to deformations that are small compared to the
sizes of the two porous media), and so, for example, we may impose the problem on a
region whose boundaries do not depend on the deformations.

2.1.1. Flow and Cake Build-Up

Employing the framework from Köry et al. [23], we introduce a spatial variable ˜x,
impose an input pressure ˜pin at ˜x = ˜L f + ˜Lc and impose an output pressure ˜pout at ˜x = 0
such that ˜pin > ˜pout (see Figure 1). This results in a one-dimensional flow oriented in the
negative ˜x direction. We denote ˜q to be the (positive) fluid flux through the filtercake per

Modelling 2021, 2

21

unit area and allow ˜pin to vary with time but hold ˜pout constant. We assume that all the
particles are bigger than the pores in the filter.

Figure 1. Schematic set-up of the model for compressible filtercake.

Consider the feed solution (the mixture of fluid and particles) with a solid volume
fraction φ assumed to be uniform in space and independent of time approaching the
moving cake front at constant speed ˜v. During (an infinitesimal) time δ˜t, the particles
travel a distance ˜vδ˜t and will cause a growth in the cake size δ ˜Lc. Since the particles
move in the direction opposite to the cake growth, in this time interval a mass of volume
φ ˜A(δ ˜Lc + ˜vδ˜t) has arrived at the cake from the feed where ˜A denotes the cross-sectional
area of the cake. Taking into account particle packing in the cake with φc denoting the solid
volume fraction in the cake in its strain-free state, this mass is transformed into a cake size
increase of φc ˜Aδ ˜Lc. Equating these two quantities, dividing by δ˜t ˜A and taking the limit as
δ˜t

0 gives

→

which may be rearranged to give

Finally, conservation of the fluid across the moving front requires that ˜v = ˜q/(1
upon substitution into (2) yields

−

φ) which

Note that it is natural to assume that φ < φc, as the particle packing in the feed fluid should always be less than that in the (strain-free) cake. We assume that once the particles arrive at the cake surface they become bound to the cake and cannot detach and re-enter the feed stream. Thus we do not consider the effects of concentration polarisation here. Because we assume that initially the system is clean so there is no cake we have ˜Lc(0) = 0 as an initial condition. We are often interested in the throughput, ˜ , which is defined as the total T volume of fluid processed per unit area of membrane at a given time, ˜t, d ˜Lc d˜t φ (cid:18) + ˜v = φc (cid:19) d ˜Lc d˜t , d ˜Lc d˜t = φ φc − ˜v. φ d ˜Lc d˜t = φ φ)(φc − (1 − ˜q(˜t). φ) (˜t) = ˜q(ˆt) dˆt. ˜ T ˜t 0 (cid:90) (1) (2) (3) (4) VersionDecember20,2020submittedtoModelling5˜x0˜Lf+˜Lc˜Lf(cid:27)(cid:27)(cid:27)(cid:27)FlowFilterCakePorousgrid6Figure1.Schematicset-upofthemodelforcompressiblefiltercake.FollowingParkeretal.[18]andKöryetal.[23],thepermeabilitiesofthefilter˜kfandofthecake˜kcareassumedtodependlinearlyonthestrain∂˜u(˜x,˜t)/∂˜x,with˜u(˜x,˜t)denotingthedisplacementfieldofthematerialfromitsundeformedstate,andwillvaryspatiallyaswellasintimehere,duetocakebuild-up.Thuswepropose˜ki(cid:18)∂˜ui∂˜x(cid:19)=˜k1,i+˜k2,i∂˜ui∂˜x,(5)wherewehaveintroducedasubscripti,whichhereandhenceforthcanattaintwovalues,f133orc,correspondingrespectivelytothefilterdomain,˜x∈(cid:16)0,˜Lf(cid:17),andthecakedomain,˜x∈134(cid:16)˜Lf,˜Lf+˜Lc(˜t)(cid:17).Notethatweassumethat∂˜ui/∂˜xisagoodapproximationtothestrainandthis135isbecauseweareassumingdeformationsaresmall.Wenotefurtherthat,whileinrealitythe136permeabilitymaypossessamorecomplexrelationshipwiththedeformation,equation(5)isan137appropriateapproximationinthesmall-deformationregimeweareconsidering.138WiththisnotationDarcy’slaw,forslowviscousflowofafluidthroughaporousmaterial,is˜qi=˜ki˜η∂˜pi∂˜x,(6)where˜p(˜x,˜t)isthefluidpressureand˜ηthefluidviscosity.Weassumethefluidtobeincompressible,139whichimpliesthattheflux˜qiisuniforminspace(independentof˜x)forboththecakeandthefilter,140andthecontinuityoffluxatthefilter–cakeinterfaceimpliesthat˜qf(˜x,˜t)=˜qc(˜x,˜t)=˜q(˜t).1412.1.2.Deformation142Weassumesmallone-dimensionaldeformationsandthatthetimescaleofporoelasticresponseismuchsmallerthanthatofcaking(similartoHerterichetal.[25]).Linearporoelasticitytheory[26]theninformsusthatthedisplacementofthefilterandcake˜ui(˜x,˜t)canbemodelledusingasteady-stateNavierequationextendedbytheTerzaghiterm∂˜pi∂˜x=(cid:0)˜λi+2˜µi(cid:1)∂2˜ui∂˜x2,(7)where˜λiand˜µidenotetheeffectiveelasticconstantsofthefilterandthecake(seeKöryetal.[23]for143details).144 Modelling 2021, 2 Following Parker et al. [18] and Köry et al. [23], the permeabilities of the filter ˜k f and of the cake ˜kc are assumed to depend linearly on the strain ∂ ˜u( ˜x, ˜t)/∂ ˜x , with ˜u( ˜x, ˜t) denoting the displacement field of the material from its undeformed state, and will vary spatially as well as in time here, due to cake build-up. Thus we propose ˜ki ∂ ˜ui ∂ ˜x (cid:18) (cid:19) = ˜k1,i + ˜k2,i ∂ ˜ui ∂ ˜x , where we have introduced a subscript i, which here and henceforth can attain two values, f or c, corresponding respectively to the filter domain, ˜x , and the cake domain, 0, ˜L f ˜L f , ˜L f + ˜Lc(˜t) (cid:16) ∈ . Note that we assume that ∂ ˜ui/∂ ˜x is a good approximation to the ˜x strain and this is because we are assuming deformations are small. We note further that, while in reality the permeability may possess a more complex relationship with the deformation, Equation (5) is an appropriate approximation in the small-deformation regime we are considering. (cid:17) (cid:17) With this notation Darcy’s law, for slow viscous flow of a fluid through a porous ∈ (cid:16) material, is ˜qi = ˜ki ˜η ∂ ˜pi ∂ ˜x , where ˜p( ˜x, ˜t) is the fluid pressure and ˜η the fluid viscosity. We assume the fluid to be incompressible, which implies that the flux ˜qi is uniform in space (independent of ˜x) for both the cake and the filter, and the continuity of flux at the filter–cake interface implies that ˜q f ( ˜x, ˜t) = ˜qc( ˜x, ˜t) = ˜q(˜t). 2.1.2. Deformation We assume small one-dimensional deformations and that the timescale of poroelastic response is much smaller than that of caking (similar to Herterich et al. [25]). Linear poroelasticity theory [26] then informs us that the displacement of the filter and cake ˜ui( ˜x, ˜t) can be modelled using a steady-state Navier equation extended by the Terzaghi term ∂ ˜pi ∂ ˜x = ˜λi + 2 ˜µi (cid:0) (cid:1) ∂2 ˜ui ∂ ˜x2 , where ˜λi and ˜µi denote the effective elastic constants of the filter and the cake (see Köry et al. [23] for details). 2.1.3. Boundary and Interfacial Conditions We fix the end of the filter at ˜x = 0 using a porous grid that offers no resistance to the fluid flow and assume that the open end of the cake at ˜x = ˜L f + ˜Lc(˜t) is free. Then, the boundary conditions read The equations governing ˜p and ˜u, Darcy’s law (6) and the Navier Equation (7), are, respectively, first- and second-order in space. Thus, to close the problem we impose continuity of fluid pressure, displacement and stress: ˜p f (0, ˜t) = ˜pout, ˜u f (0, ˜t) = 0, = ˜pin(˜t), ˜pc (cid:16) ∂ ˜uc ˜L f + ˜Lc(˜t), ˜t (cid:17) ˜L f + ˜Lc(˜t), ˜t (cid:17) (cid:16) ∂ ˜x = 0. ˜p f ( ˜L f , ˜t) = ˜pc( ˜L f , ˜t), ˜u f ( ˜L f , ˜t) = ˜uc( ˜L f , ˜t), ( ˜λ f + 2 ˜µ f ) ( ˜L f , ˜t) = ( ˜λc + 2 ˜µc) ( ˜L f , ˜t). ∂ ˜uc ∂ ˜x ∂ ˜u f ∂ ˜x 22 (5) (6) (7) (8) (9) Modelling 2021, 2 23 Altogether, this forms a moving boundary problem, with the moving boundary located at ˜L f + ˜Lc(t), with three unknown functions ˜p( ˜x, ˜t), ˜u( ˜x, ˜t) and ˜Lc(t). This dimensional model depends on 13 parameters namely: ˜L f , ˜pout, φ, φc, ˜k1,c, ˜k2,c, ˜k1, f , ˜k2, f , ˜η, ˜λ f , ˜µ f , ˜λc, ˜µc, and one input function, ˜pin(˜t). Note that there usually exists a one-to-one correspondence between the permeability and the solid volume fraction of a (strain-free) porous medium. This means that φc in our approach is not an independent parameter and, if needed, could be expressed as function of ˜k1,c. 2.2. Nondimensionalisation Denoting ∆i p := ( ˜pin(0) ˜pout)/( ˜λi + 2 ˜µi), we nondimensionalise as − ˜x = ˜Lc(˜t) = (cid:16) ˜k1,c ˜k1, f (cid:33) ˜L f (cid:32) ˜L f x, (cid:17) Lc(t), ˜ui = ˜L f ∆c p ui, (cid:16) ˜pi = ( ˜pin(0) (cid:17) − ˜pout)pi + ˜pout, (10) ˜q = ( ˜pin(0) ˜pout) q, ˜t = ˜k1, f ˜η ˜L f (cid:32) − = ˜ T (cid:32) (cid:33) (1 − φ)(φc − ˜k1, f φ (1 φ)(φc − − φ˜k2 1, f ( ˜pin(0) (cid:32) φ)˜k1,c ˜L f , (cid:33)T φ) ˜η ˜k1,c ˜L2 f ˜pout) (cid:33) − t, for i = f , c. These scalings are chosen to be natural to the model so that the resulting governing equation for the evolution of the cake thickness possesses the fewest number of parameters, as we shall observe subsequently (see Equation (30)). We introduce the following dimensionless quantities νi = ˜λi + 2 ˜µi ˜λc + 2 ˜µc , ωi = ˜k1,i ˜k1, f , Γi = ∆c p ˜k2,i ˜k1, f , (t) = P ˜pin(˜t) ˜pin(0) ˜pout ˜pout . − − (12) We note that νc = ω f = 1, and so, for convenience, we introduce νf = ν and ωc = ω and use this from hereon whenever there is no ambiguity. Using these scalings and notation the dimensionless filter and the cake regions are rep- (1, 1 + ωc Lc). The dimensionless cake-evolution equation is (0, 1) and x resented by x ∈ ∈ Darcy’s law is where dLc(t) dt = q, q = ki ∂pi ∂x , ki = ωi + Γi ∂ui ∂x , ∂pi ∂x = νi ∂2ui ∂x2 . and the deformation of the two porous media is governed by the Navier equation (11) (13) (14) (15) (16) Modelling 2021, 2 The boundary conditions (8) become p f (0, t) = 0 pc(1 + ωLc(t), t) = (t) P u f (0, t) = 0 ∂uc(1 + ωLc(t), t) ∂x = 0. ν ∂u f ∂x (1, t) = (1, t). ∂uc ∂x Continuity of fluid pressure and displacement at x = 1 become the same conditions for the dimensionless variables, while the dimensionless form of the third interfacial condition from (9) is For this choice of nondimensionalisation, the definition of throughput (4) indicates that = Lc and so the dimensionless throughput and the cake thickness are interchangeable. T The dimensionless model depends only on four model parameters, namely, ν, ω, Γ f and Γc, and one input function . 2.3. Model Restrictions P Using the same methodology as in Köry et al. [23], before solving the problem, we discuss the restrictions on model parameters required so that the small-deformations assumption is valid and so that shutdown is avoided. It can be shown (see Appendix A) = O(1) the small-deformations assumption holds that provided ∆c p 1, ∆ f p in the filtercake for all times. 1 and (cid:28) (cid:28) P The filtercake will shut down if the permeability reaches zero at any point in the depth and this may occur in the filter or in the cake. We now explore the conditions on the dimensionless parameters and the pressure-drop evolution that allow us to avoid this shutdown behaviour. 2.3.1. Avoiding Filter Shutdown We expect the maximum strain in the filter to occur at the porous grid (x = 0). From (A2) and the form of the permeability function (15) this permeability will remain positive provided the applied transmembrane pressure We note that (0) = 1 and so we require P to avoid initial filter shutdown before any cake has deposited. 2.3.2. Avoiding Cake Shutdown To avoid shutdown in the cake, the permeability from (15) needs to be positive everywhere in the cake, which is equivalent to (t) < P ν Γ f . Γ f < ν ∂uc ∂x − (x, t) < ω Γc . 24 (17a) (17b) (17c) (17d) (18) (19) (20) (21) Modelling 2021, 2 Integrating the Navier Equation (16) within the cake and using the boundary condi- tions (17b,d), we conclude for x (1, 1 + ωLc) ∈ which in combination with (21) yields (t) P − pc(x, t) = ∂uc ∂x − (x, t), (t) < pc(x, t) + P ω Γc , and this gives us a restriction to avoid shutdown in the cake. Note that the pressure distribution within the cake is not known a priori and must be determined as part of the solution. 2.4. Solution We now analyse the dimensionless problem (13)–(18) to determine the solution. Sub- stituting the expression for the pressure-gradient term from the Navier Equation (16) into Darcy’s law (14), integrating with respect to x and using the boundary condition (17d), we conclude νΓ f 2 Γc 2 ∂u f ∂x (cid:18) ∂uc ∂x (cid:18) (cid:19) 2 + ν ∂u f ∂x − + ω ∂uc ∂x − (cid:19) 2 q(t)(x f1(t)) = 0 − for x (0, 1), ∈ (24) q(t) x { − [1 + ωLc(t)] = 0 for x (1, 1 + ωLc(t)), } ∈ where f1(t) is an integration constant. This constant can be determined using the continuity of stress at the filter–cake interface (18) (details of the calculations used throughout this section are presented in Appendix B) and we arrive at f1(t) = 1 + ν 2Γ f q(t)   1 1 − (cid:34) − Γ f ω Γcν (cid:32) 1 1 − (cid:114) − 2Γcq(t) ω Lc(t) (cid:33)(cid:35) Equation (24) is subsequently solved to give  2 .    ∂u f ∂x = − 1 Γ f 1 (cid:34) − Γ f ω Γcν (cid:32) 1 1 − (cid:114) − 2Γcq(t) ω Lc(t) (cid:33)(cid:35) 2 + 2Γ f q(t) ν − (cid:118) (cid:117) (cid:117) (cid:116) 1    ∂uc ∂x = − ω Γc (cid:40) 1 − (cid:114) 1 + 2Γcq(t) ω2 (1 + ωLc(t)) x − (cid:104) (cid:41) (cid:105) for x (0, 1), ∈ x 1 − (cid:104)   (cid:105)  for x (1, 1 + ωLc(t)). ∈ Substituting the strain field from (26) into Darcy’s law (14), integrating and using the continuity of pressure across the filter–cake interface as well as appropriate boundary conditions, we arrive at an equation relating q(t), Lc(t) and (t): = ν Γ f P − (cid:118) (cid:117) (cid:117) (cid:116) 1    We further define 1 (cid:34) − ωΓ f νΓc (cid:32) 1 1 − − (cid:114) 2ΓcqLc ω (cid:33)(cid:35) − P 2 2Γ f q . ν    γ f = = Γ f ν ˜pin(0) ˜pout − ˜λ f + 2 ˜µ f ˜k2, f ˜k1, f , γc = = Γc ω ˜pin(0) ˜pout − ˜λc + 2 ˜µc ˜k2,c ˜k1,c , 25 (22) (23) (25) (26) (27) (28)