Nonlinear Darcy fluid flow with deposition

Transcription

1 Nonlinear Darcy fluid flow wit deposition V.J. Ervin Hyesuk Lee J. Ruiz-Ramírez Marc 23, 2016 Abstract In tis article we consider a model of a filtration process. Te process is modeled using te nonlinear Darcy fluid flow equations wit a varying permeability, coupled wit a deposition equation. Existence and uniqueness of te solution to te modeling equations is establised. A numerical approximation sceme is proposed and analyzed, wit an a priori error estimate derived. Numerical experiments are presented wic support te obtained teoretical results. Key words. Darcy equation, filtration AMS Matematics subject classifications. 65N30 1 Introduction Filtration processes are ubiquitous in our lives. From oil and fuel filters in engines, to filters used in industrial lines to protect sensitive equipment, to ouseold water filtration systems. In tis article we consider te case were te filtration process can be modeled as a fluid flowing troug a porous medium. We make te simplifying assumption tat te rate of particulate deposition in te filter is only dependent on te porosity and te magnitude of te fluid velocity at tat point. Of interest are te modeling equations η t µ eff u + p κ(η) = f, in Ω, (1.1) u = 0, in Ω, (1.2) + dep( u, η) = 0, in Ω, (1.3) subject to suitable boundary and initial conditions. In (1.1)-(1.3) u and p denote te velocity and pressure of te fluid, respectively, µ eff te effective fluid viscosity, and η and κ(η) represent te porosity and permeability trougout te filter (Ω), respectively. (A discussion of te model follows in Section 2.) Department of Matematical Sciences, Clemson University, Clemson, SC 29634, USA. (vjervin@clemson.edu) Department of Matematical Sciences, Clemson University, Clemson, SC , USA (klee@mail.clemson.edu), Partially supported by te NSF under grant no. DMS Department of Matematical Sciences, Clemson University, Clemson, SC 29634, USA. (javier@clemson.edu) 1

2 Te lack of regularity of te fluid velocity, u H div (Ω), leads to an open question of te existence of a solution to (1.1)-(1.3). In order to obtain a modeling system of equations for wic a solution can be sown to exist, we replace η in (1.1) by an smooted porosity, η s. Te approac of regularizing te model wit te introduction of η s is, in part, motivated by te Darcy fluid flow equations, wic can be derived by averaging, e.g. volume averaging [20], omogenization [1], mixture teory [17]. Recently in [9] we considered te case of (steady-state) generalized Newtonian fluid flow troug a porous medium, modeled by equations (1.1), (1.2), wit µ eff κ(η) β( u ). Wit te general assumptions tat β( ) was a positive, bounded, Lipscitz continuous function, bounded away from zero, and wit β( u ) replaced wit β( u s ), existence of a solution was establised. Two smooting operators for u were presented. One was a local averaging operator, wereby u s (x) is obtained by averaging u in a neigborood of x. Te second smooting operator, wic is nonlocal, computes u s (x) using a differential filter applied to u. Tat is, u s is given by te solution to an elliptic differential equation wose rigt and side is u. For establising te existence of a solution te key property tat te smooting operator needs to satisfy is tat it transform a weakly convergent sequence in L 2 (Ω) into a sequence wic converges strongly in L (Ω). A similar model to (1.1)-(1.3) arises in te study of single-pase, miscible displacement of one fluid by anoter in a porous medium. For tis problem η would denote a fluid concentration, and te yperbolic deposition equation (1.3) is replaced by a parabolic transport equation. Existence and uniqueness for tis problem as been investigated and establised by Feng [10] and Cen and Ewing [7]. Because of te connection of tis model to oil extraction, numerical approximation scemes for tis problem ave been well establised. A summary of tese metods is discussed in te recent papers by Bartels, Jensen and Müller [4], and Riviére and Walkington [18]. A steady-state nonlinear Darcy fluid flow problem, wit a permeability dependent on te pressure was investigated by Azaïez, Ben Belgacem, Bernardi, and Corfi [2], and Girault, Murat, and Salgado [13]. For te permeability function Lipscitz continuous, and bounded above and below, existence of a solution (u, p) L 2 (Ω) H 1 (Ω) was establised. Important in andling te nonlinear permeability function, in establising existence of a solution, was te property tat p H 1 (Ω). In [2] te autors also investigated a spectral numerical approximation sceme for te nonlinear Darcy problem, assuming an axisymmetric domain Ω. A convergence analysis for te finite element discretization of tis problem was given in [13]. Following a discussion of te model in Section 2, existence of a solution to te modeling equations is establised in Section 3. An approximation sceme for te filtration model is presented in Section 4, and an a priori error estimate derived. A numerical simulation of te filtration process is presented in Section 5. 2 Discussion of Filtration Model In tis section we discuss te modeling equations we investigate for te filtration process. We assume tat te process can be modeled as fluid flowing troug a porous medium wit a varying permeability. Additionally we assume tat te process as a fixed time orizon, T. (For example, for industrial filters te most practical time to cange filters is during sceduled maintenance periods. Drivers are encouraged to cange te oil filters in teir cars every 3000 miles or every tree monts, wicever comes first.) We use te following parameters/variables to model te process. 2

3 Ω R d (d = 2, 3) te region occupied by te filter, u te fluid velocity, p te fluid pressure, η te porosity of te medium, 0 < η < 1, κ te permeability of te medium, 0 < κ <, µ eff te effective fluid viscosity, n te unit outer normal to Ω. We model te fluid flow using te Darcy fluid flow equations: µ eff u + p = 0, in Ω, t (0, T ], (2.1) κ(η) u = 0, in Ω, t (0, T ]. (2.2) Note: We are assuming tat te particulate is sufficiently sparsely distributed in te fluid tat te conservation of mass equation (2.2) is still a valid approximation for te model. Relationsip between permeability κ and porosity η As η 0 te porous medium transitions to a solid medium, in wic case te permeability, κ 0. As η 1 te medium s resistance to te flow goes to zero, i.e., its permeability goes to infinity, and te modeling equations are no longer appropriate to describe te fluid flow. Tere are a number of postulated models for te relationsip between κ and η. Te most commonly cited is te Blake-Kozeny model [5] κ(η) = D 2 p η (1 η) 2, (2.3) were D p represents a material constant te diameter of te particles comprising te porous medium. Remark: Te permeability of granite is millidarcy. In a filtering process te permeability trougout Ω will always be greater tan tat of granite. So, it is reasonable to assume tat κ(η) is bounded from below. At te beginning of te filtering process tere is a prescribed permeability (porosity) trougout Ω. As te filtering process decreases te permeability (porosity) trougout Ω is it also reasonable to assume tat κ(η) is bounded from above. 2.1 Modeling te deposition Te deposition on te particulate in te filter is modeled by an equation describing te cange of porosity. We assume tat η(x, t)/ t is a function of te magnitude of te velocity and te porosity, i.e., η + dep( u, η) = 0, in Ω, t (0, T ]. (2.4) t As a first approximation, we assume tat te deposition function dep(, ) is a separable function of u and η, dep( u, η) = g( u ) (η), in Ω, t (0, T ]. (2.5) Te function g( ) We assume tat if te fluid is flowing too quickly tere is little opportunity for te particles witin 3

4 te fluid to be captured by te filter. Terefore, beyond a critical value for u, say s c, g( u ) is a monotonically decreasing function of u. If u is very small ten, given tat we are modeling a sparsely distributed particulate in te fluid, te rate of deposition will also be very small due to te amount of particulate passing troug te filter. In consideration of te about two situations, we postulate tat g( u ) is a Lipscitz continuous, non-negative, unimodal function, wit maximum value occurring at u = s c. A typical profile for g( ) is given in Figure g(s) s Figure 2.1: Typical profile for g( ). Te function ( ) We assume tat as te porosity decreases te rate at wic deposition occurs also decreases. Tis corresponds to te situation tat as te deposition occurs (i.e., te porosity decreases) tere is less of te filter available for te particulate to adere to. Based on tis, we assume tat (η) is a continuous, non-negative, increasing function. Two simple models for (η) are: (η) = η (η) = η r, r > 1, η(x, t) = η(x, t) = η 0 (x) exp( t 0 g( u(x, s) ) ds), (2.6) η 0 (x) ( 1 + (r 1) η 0 (x) ) r 1 t 1/(r 1), (2.7) 0 g( u(x, s) ) ds were, η 0 (x) denotes te initial porosity distribution trougout te filter. 2.2 Boundary conditions We assume tat te boundary of te filter, Ω, is made up of tree parts: an inflow region, Γ in, an outflow region, Γ out and te walls of te filter, Γ, i.e., Ω = Γ in Γ out Γ. For well-posedness, equations (2.1)-(2.2) require a scalar boundary condition (typically u n or p) be specified on Ω. Inflow boundary condition 4

5 Two pysically reasonable boundary conditions to consider on Γ in are u n = g in and (2.8) p = p in. (2.9) Condition (2.8) specifies te flow profile at te inflow boundary, wereas (2.9) corresponds to a specified pressure ead along te inflow. Outflow boundary condition Te fluid outflow profile will be affected by te deposition occurring in te filter. Terefore specifying an outflow profile is not reasonable for tis problem. Rater, at te outflow boundary we assume a specified pressure p = p out, on Γ out. (2.10) Wall boundary condition Along te walls of te filter we assume a no penetration condition, specifically u n = 0, on Γ. (2.11) For te matematical analysis of tis problem it is convenient to ave omogeneous boundary conditions. Tis is acieved by introducing a suitable cange of variables. For example, in case te specified boundary conditions are u n = g in on Γ in, u n = 0 on Γ, p = p out on Γ out, we introduce functions u b (x, t) (see [11]) and p b (x, t) defined on Ω satisfying u b = 0, in Ω (0, T ], u b n = g in, on Γ in (0, T ], u b n = g in (s) ds/ Γ out, on Γ out (0, T ], Γ in u b t i = 0, on Γ in (0, T ], u b = 0, on Ω\(Γ in Γ out ) (0, T ], were t i, i = 1,..., (d 1) denotes an ortogonal set of tangent vectors on Γ in, and Γ out te measure of Γ out wit respect to ds. p b = 0, in Ω (0, T ], p b = p out, on Γ out (0, T ], p b n = 0, on Ω\Γ out (0, T ]. (In case te pressure is specified on te inflow boundary Γ in ten u b = 0 and te definition of p b is appropriately modified.) Wit te cange of variables: u = u a + u b and p = p a + p b we obtain te following system of 5

6 modeling equations for te filtration process: µ eff κ(η) u a η t + µ eff κ(η) u b + p a = p b, in Ω (0, T ], (2.12) u a = 0, in Ω (0, T ], (2.13) + g( u a + u b ) (η) = 0, in Ω (0, T ], (2.14) u a n = 0, on Γ in Γ (0, T ], (2.15) p a = 0, on Γ out (0, T ], (2.16) η(x, 0) = η 0 (x) in Ω. (2.17) To simplify te notation, we let b = u b, f = p b, β(η) = µ eff /κ(η), and drop te a subscript on u a and p a to obtain β(η)u + β(η)b + p = f, in Ω (0, T ], (2.18) η t u = 0, in Ω (0, T ], (2.19) + g( u + b ) (η) = 0, in Ω (0, T ], (2.20) u n = 0, on Γ in Γ (0, T ], (2.21) p = 0, on Γ out (0, T ], (2.22) η(x, 0) = η 0 (x) in Ω. (2.23) Remark: β(η) is implicitly a function of x troug te dependence of η on x. In te next section we sow tat, under suitable assumptions on β( ), tere exists a solution to (2.18)-(2.23). 3 Existence and Uniqueness In tis section we investigate te existence and uniqueness of solutions to te nonlinear system equations (2.18)-(2.23). We assume tat Ω R d, d = 2 or 3, is a convex polyedral domain and for vectors in R d denotes te Euclidean norm. Weak formulation of (2.18)-(2.23) Let H div (Ω) = { v L 2 (Ω) : v L 2 (Ω) }, and X = {v H div (Ω) : v n = 0 on Γ in Γ}. Define te bilinear form b(, ) and te div-free subspace, Z, of X as b : X L 2 (Ω) R, b(v, q) := q v dω, We use (f, g) := Ω Z := { v X : b(v, q) = 0, q L 2 (Ω) }. Ω f g dω, and f := (f, f) 1/2 to denote te L 2 inner product and te L 2 norm over Ω, respectively, for bot scalar and vector valued functions. 6

7 Additionally, we introduce te norm v X = ( ) 1/2 ( v v + v v) dω. Ω Remark: For v H div (Ω) it follows tat v n H 1/2 ( Ω). For te interpretation of te condition v n = 0 on Γ in Γ see [12, 19]. We make te following assumptions on β( ), g( ) and ( ). Assumptions on β( ) Aβ1 : β( ) : R + R +, Aβ2 : 0 < β min β(s) β max, s R +, Aβ3: β is Lipscitz continuous, β(s 1 ) β(s 2 ) β Lip s 1 s 2. Assumptions on g( ) Ag1 : g( ) : R + {0} R + {0}, Ag2 : g(s) g max, s R + {0}, Ag3: g is Lipscitz continuous, g(s 1 ) g(s 2 ) g Lip s 1 s 2. Assumptions on ( ) A1 : ( ) : R + R + {0}, A2 : (s) max, s R +, A3: is Lipscitz continuous, (s 1 ) (s 2 ) Lip s 1 s 2. Remark: Te assumptions on β, g, and are consistent wit te discussion in Section 2.1. Note tat g max, max sould be given to guarantee positivity of te porosity (0 < η) for some finite time interval. Assumptions on η s Aη s 1. For η(, t) L 2 (Ω), η s (t) L (Ω) C s η(t) L 2 (Ω), Aη s 2. Te mapping η(, t) η s (, t) is linear. Two smooters wic satisfy Aη s 1 and Aη s 2 are discussed in [9]. One is a local averaging operator and te oter a differential smooting operator. We assume tat b and f are continuous wit respect to t on te interval (0, T ) and ave a continuous extension to te interval (0 δ, T ) for some δ > 0, i.e., b, f C 0 (0, T ; L 2 (Ω)). We restate (2.18)-(2.23) as: Given η 0 (x) L 2 (Ω), b, f C 0 (0, T ; L 2 (Ω)) L (0, T ; L 2 (Ω)), find (u, p) L 2 (0, T ; X) L 2 (0, T ; L 2 (Ω)), η H 1 (0, T ; L 2 (Ω)) suc tat for a.e. t, 0 < t < T, (β(η s )u, v) + (β(η s )b, v) b(v, p) = (f, v), v X, (3.1) ( ) η t, ξ b(u, q) = 0, q L 2 (Ω), (3.2) + (g( u + b ) (η), ξ) = 0, ξ L 2 (Ω). (3.3) For te spaces X and L 2 (Ω) we ave te following inf-sup condition inf sup b(v, q) c 0 > 0. (3.4) q L 2 (Ω) v X q v X 7

8 In view of te inf-sup condition we can restate (3.1)-(3.3) as: Given η 0 (x) L 2 (Ω), b, f C 0 (0, T ; L 2 (Ω)) L (0, T ; L 2 (Ω)), find u L 2 (0, T ; Z), η H 1 (0, T ; L 2 (Ω)) suc tat for a.e. t, 0 < t < T, (β(η s )u, v) + (β(η s )b, v) = (f, v), v Z, (3.5) ( ) η t, ξ + (g( u + b ) (η), ξ) = 0, ξ L 2 (Ω). (3.6) Introduce te bounded Darcy projection operator: P η : L 2 (Ω) Z defined by z := P η (r) were, (β(η s )z, v) b(v, λ) = (r, v), v X, b(z, q), = 0, q L 2 (Ω). Tat P η is well defined follows from Aβ1-Aβ3. Note tat u in (3.5) may be written as u = P η (f β(η s )b). (3.7) Additionally, from (3.5) wit te coice v = u, it is straigt forward to see tat u(t) = u(t) X 1 ( f(t) + β max b(t) ) (3.8) β min 1 ( f β L (0,T ; L 2 (Ω)) + β max b L (0,T ; L 2 (Ω))). min Using P η we can restate (3.5), (3.6) as: Given η 0 (x) L 2 (Ω), b, f C 0 (0, T ; L 2 (Ω)) L (0, T ; L 2 (Ω)), find η H 1 (0, T ; L 2 (Ω)) suc tat for a.e. t, 0 < t < T, ( ) η t, ξ + (g( P η (f β(η s )b) + b ) (η), ξ) = 0, ξ L 2 (Ω). (3.9) We recall te Picard-Lindelöf Teorem. (Also know as te Caucy-Lipscitz Teorem). Teorem 3.1 ([14], Teorem I.3.1) Let I denote a domain in R containing te point t 0, Y a Banac space and f : R Y Y. Suppose tat f is continuous in its first variable and locally Lipscitz continuous in its second variable. Ten, tere exists ɛ > 0 suc tat te initial value problem as a unique solution in C 0 (t 0 ɛ, t 0 + ɛ; Y ). u = f(t, u), (3.10) u(t 0 ) = u 0, (3.11) Let F(t, η) = g( P η (f β(η s )b) + b ) (η). Te continuity of F wit respect to t follows from te continuity of f and b wit respect to t, te boundedness of P η, and te continuity of g. To 8

9 investigate te local Lipscitz continuity of F wit respect to η consider te following. F(t, η 1 ) F(t, η 2 ) = g( P η1 (f β(η s 1)b) + b ) (η 1 ) g( P η2 (f β(η s 2)b) + b ) (η 2 ) g( P η1 (f β(η s 1)b) + b ) ((η 1 ) (η 2 )) + (g( P η1 (f β(η s 1)b) + b ) g( P η2 (f β(η s 2)b) + b ) (η 2 ) g max (η 1 ) (η 2 ) + g Lip ( P η1 (f β(η s 1)b) + b P η2 (f β(η s 2)b) + b ) (η 2 ) g max Lip η 1 η 2 + g Lip (P η1 (f β(η s 1)b) P η2 (f β(η s 2)b)) (η 2 ) g max Lip η 1 η 2 + g Lip max P η1 (f β(η s 1)b) P η2 (f β(η s 2)b). (3.12) Now, wit u 1 = P η1 (f β(η1 s)b) Z and u 2 = P η2 (f β(η2 s )b) Z we ave tat (β(η s 1)u 1, v) = (f, v) (β(η s 1)b, v), v Z, (3.13) and (β(η s 2)u 2, v) = (f, v) (β(η s 2)b, v), v Z. (3.14) Subtracting (3.14) from (3.13), and wit te coice v = u 1 u 2, yields (β(η s 1)(u 1 u 2 ), (u 1 u 2 )) = ((β(η s 2) β(η s 1))u 2, (u 1 u 2 )) + ((β(η s 2) β(η s 1))b, (u 1 u 2 )). Hence, wit (3.8), Tus we obtain β min u 1 u 2 β Lip η s 2 η s 1 u 2 + β Lip η s 2 η s 1 b β Lip ( u 2 + b ) η2 s η1 s ( 1 β Lip f + β ) max b + b C s η 2 η 1. β min β min P η1 (f β(η s 1)b) P η2 (f β(η s 2)b) = u 1 u 2 Combining (3.12) and (3.15), we ave β Lip C s βmin 2 ( f + (β max + β min ) b ) η 2 η 1. (3.15) F(t, η 1 ) F(t, η 2 ) F Lip η 2 η 1, were (3.16) F Lip = g max Lip (3.17) β +C Lip ( s g βmin 2 Lip max f L (0,T ; L 2 (Ω)) + (β max + β min ) b L (0,T ; L (Ω))) 2. (3.18) Ten, from Teorem 3.1, we ave tat tere exists ɛ > 0 suc tat tere exists a unique solution η C 0 (0, ɛ; L 2 (Ω)) to (3.9). Regarding te additional regularity of η, formally taking ξ equal to η/ t in (3.9) we ave η t 2 g( P η (f β(η s )b) + b ) (η) η t η t g max max Ω. (3.19) 9

10 Using te establised fact tat η C 0 (0, ɛ; L 2 (Ω)), it ten follows tat η H 1 (0, ɛ; L 2 (Ω)). In order to establis tis result rigorously, we consider a Galerkin approximation of (3.9) in wic te approximation of η/ t does indeed lie in L 2 (0, ɛ; L 2 (Ω)) and ten taking te limit. Next, note tat F(t, η) g max max Ω 1/2. Hence bot F(t, η) and its Lipscitz constant wit respect to η are bounded independent of t and η. Ten, from te proof of Teorem 3.1 (see [14]), ɛ may be cosen suc tat 0 < ɛ < 1/F Lip. As ɛ > 0 can be cosen independent of t and η, te solution can be extended to 0 < t < T. We summarize te above discussion in te following teorem. Teorem 3.2 For β( ) satisfying Aβ1 Aβ3, g( ) satisfying Ag1 Ag3, ( ) satisfying A1 A3,and η( ) satisfying Aη1 Aη2, given η 0 (x) L 2 (Ω), b, f C 0 (0, T ; L 2 (Ω)) L (0, T ; L 2 (Ω)), tere exists a unique solution (u, p) L 2 (0, T ; X) L 2 (0, T ; L 2 (Ω)), η H 1 (0, T ; L 2 (Ω)) satisfying (3.1)-(3.3), for a.e. t, 0 < t < T. Remark: Important in establising te existence and uniqueness of te solution is te assumption tat β(x, t) β min. Under te deposition process eventually (assuming tat te matematical equations correctly model te pysical problem), after some finite time, tis assumption is violated. 4 Finite element approximation In tis section we investigate te finite element approximation to Given η 0 (x) L 2 (Ω), b, f C 0 (0, T ; L 2 (Ω)) L (0, T ; L 2 (Ω)), find (u, p) L 2 (0, T ; X) L 2 (0, T ; L 2 (Ω)), η H 1 (0, T ; L 2 (Ω)) suc tat for a.e. t, 0 < t < T, (β(η s )u, v) + (β(η s )b, v) b(v, p) = (f, v), v X, (4.1) ( ) η t, ξ b(u, q) = 0, q L 2 (Ω), (4.2) + (g( u + b ) η, ξ) = 0, ξ L 2 (Ω). (4.3) Note tat wit regard to te general modeling equations (2.18)-(2.23), ere we ave cosen (η) = η. As before, let Ω R d denote a convex polygonal or polyedral domain and let T be a triangulation of Ω made eiter of triangles or quadrilaterals in R 2 or tetraedra or exaedra in R 3. Tus, te computational domain is defined by Ω = K T K. We assume tat tere exist constants c 1, c 2 suc tat c 1 K c 2 ρ K, were K is te diameter of te cell K, ρ K is te diameter of te biggest neigborood included in K, and = max K T k. For k N, let P T k := span{xα 1 1 xα xα d d : 0 α 1 + α α d k}, and (4.4) P Q k := span{xα 1 1 xα xα d d : 0 α 1, α 2,..., α d k}. (4.5) For K a triangle/tetraedron in R d we let P k (K) = {f : f K P T k }. For K a quadrilateral/exaedron in R d we let P k (K) = {f : f K P Q k }. RT k(t ) is used te denote te Raviart- 10

11 Tomas space of order k [6]. We use te following finite element spaces: X = {RT k (T ) X}, Q = { q L 2 (Ω) : q K P k (K), K T }, R = { r L 2 (Ω) : r K P m (K), K T }, R s = { r C 0 (Ω) : r K P max{1,m} (K), K T }, Z = {v X : (q, v) = 0, q Q }. Note tat as X Q, for v Z we ave tat v = 0, tus v X = v. For N given, let t = T/N, and t n = n t, n = 0, 1,..., N. Additionally, define d t f n := f n f n 1 t Te following norms are used in te analysis, f n := f n + f n 1, f n := f n f n f n 3. ( N 1/2 v := v(t) L (Ω), v := v(t n ) t) 2, v n=0 := sup v(t n ). 0 n N For te a priori error estimates presented below te solution (u, p, η) to (4.1)-(4.3) is required to be sufficiently regular. Te regularity assumptions we assume are, for some δ > 0, u L (0, T ; L (Ω)) L (0, T ; H k+1 (Ω)), p L (0, T ; H k+1 (Ω)), u t L (0, δ; L 2 (Ω)), u tt L 2 (0, T ; L 2 (Ω)),(4.6) η L (0, T ; L (Ω)) L (0, T ; H m+1 (Ω)),(4.7) η t L (0, T ; H m+1 (Ω)), η tt L 2 (0, T ; L 2 (Ω)) L (0, δ; L 2 (Ω)), (4.8) η ttt L 2 (0, T ; L 2 (Ω)). (4.9) Trougout, we use C to denote a generic nonnegative constant, independent of te mes parameter and time step t, wose actual value may cange from line to line in te analysis. Initialization of te Approximation Sceme Te approximation sceme described and analyzed below is a tree-level sceme. To initialize te procedure suitable approximations are required for u n for n = 0, 1, 2, and for ηn for n = 2. Here we state our assumptions on tese initial approximates. (An initialization procedure is presented in te Appendix of [8].) ( u n u n 2 X + η n η n 2 C( t) 4 + C 2k+2 + 2m+2), for n = 0, 1, 2. (4.10) Approximation Sceme Te approximation sceme we investigate is: Given η 0 R, for n = 3,..., N, determine (u n, pn, ηn ) X Q R satisfying were b n 1/2 := b( tn +t n 1 2 ). (β(η n,s )un + β(ηn,s )bn, v) (p n, v) = (f n, v), v X (4.11) (q, u n ) = 0, q Q (4.12) (d t η n, r) + (g( ũn + bn 1/2 )η n, r) = 0, r R, (4.13) 11

12 Regarding η n,s, note tat applying a smooter, S, to a function ηn R (typically) does not result in η n,s R s. Terefore, we let S(ηn ) Hm+1 (Ω) C 0 (Ω) denote te result of te smooter applied to η n, and define η n,s (x) = I S(η n )(x), (4.14) were I : C 0 (Ω) R s denotes an interpolation operator. We assume tat te smooted porosity S(η n ) is sufficiently regular suc tat tere exists a constant dependent on S( ), C S(η n ) suc tat S(η n ) I S(η n ) L (Ω) = S(η n ) ηn,s L (Ω) C S(η n ) m+1. (4.15) Te precise dependence of C S(η n ) on S( ) will depend on te particular smooter used. Te computability of te approximation sceme (4.11)-(4.13) is establised in te following lemma. Lemma 4.1 Tere exists a unique solution (u n, pn, ηn ) (X, Q, R ) satisfying (4.11)-(4.13). Proof: For {φ j } N R j=1 a basis for R, and η n = N R j=1 c jφ j, equation (4.13) is equivalent to Ac = d, were c = [c 1, c 2,..., c NR ] T, and for i, j = 1, 2,..., N R, A ij = (( 1 t g( ũn + bn 1/2 ) ) ) φ j, φ i, and d i = Ω ( 1 t 1 ) 2 g( ũn + bn 1/2 ) η n 1 φ i dω. Note tat as Ac = 0 c T Ac = 0 ( 1 t + 1 ) 2 g( ũn + bn 1/2 ) η n ηn dω = 0 ηn = 0 Ω c = 0, it follows tat te (square) linear system (4.13) as a unique solution for η n. Given η n and Aβ2, te existence and uniqueness of (un, pn ) X Q is well known from te approximation teory of Darcy fluid flow equations. Teorem 4.1 For (u, p, η) satisfying (4.1)-(4.3) and (4.6)-(4.9), and (u n, pn, ηn ) satisfying (4.11)- (4.13), and assuming tat C S(η n ) given in (4.15) is bounded by C S η n m+1, we ave tat for t sufficiently small tere exists C > 0 independent of te and t, suc tat for n = 1, 2,..., N, u n u n X + p n p n + ηn η n C (( t) 2 + k+1 + m+1). (4.16) Outline of te proof: Te complete details of te proof are given in [8]. Here we briefly summarize te key steps in te proof. Step 0. Notation. For U n, τ n in Z and R, respectively, let Λ n = u n U n, E n = U n u n ψ n = η n τ n, F n = τ n η n ε u = u n u n, ε η = η n η n. (4.17) 12

13 Step 1. Derive and estimate for E n. Beginning wit (4.11) and (3.1) we obtain β min E n β Lip u n + b n η n,s η n,s + β max Λ n. (4.18) Step 2. Estimate for η n,s η n,s. Wit te triangle inequality, η n,s η n,s η n,s η n,s S(η n ) + S(ηn ) ηn,s η n,s S(η n ) + Ω 1/2 C s ( ψ n + F n ). S(η n ) + Ω 1/2 C s η n ηn (4.19) Step 3. Derive an estimate for F n 2 F n 1 2. Beginning wit (4.13) and (3.3) we obtain F n 2 F n 1 2 t d t ψ n tg 2 Lip η n 2 ( Ẽn 2 + Λ n 2 ) + 2 tg 2 max ψ n 2 + t(6 + 2g 2 max) F n 2 + tr n (u, η), (4.20) were R n (u, η) = d t η n ηn 1/2 2 + g 2 t Lip η n 2 ũ n u n 1/2 2 + gmax η 2 n η n 1/2 2. (4.21) Step 4. Derive a bound for F l 2. Summing (4.20) from n = 3 to n = l, and using te discrete Gronwall s lemma [15, 16], we obtain, wit constants C, C S, w 1, w 2, w 4, w 5, ( F l 2 K w 1 C 2k+2 u 2 k+1 + (w 2C + w 4 CS) 2 2m+2 η 2 m+1 + C 2m+2 tl + ( t) 4 g2 max 48 t 2 η t 2 m+1 dt + ( t) 4 w 5 tl t 2 η tt 2 dt + ( t) tl 2 0 tl t 2 u tt 2 dt + F 2 2 ) η ttt 2 dt. (4.22) Step 5. Derive a bound for E l 2. Combining (4.22) wit (4.19) and (4.18) we obtain a bound for E l 2. Step 6. Error bounds for u l u l 2 and η l η l 2. Te error bounds for u l u l 2 and η l η l 2 ten follow from te bounds for E l 2, F l 2, assumption (4.10), and using u l u l 2 2( E l 2 + Λ l 2 ), η l η l 2 2( F l 2 + ψ l 2 ). Step 7. Error bound for p n p n. Te error bound for p n p n is obtained by firstly using te discrete inf-sup condition 0 < c 0 inf q Q (q, v) sup v X q v X 13

14 to sow tat for an arbitrary element Q n Q, c 0 p n (p n Qn sup Qn, v) (p n = sup, v) (Qn, v) v X v X v X v X β max u n un + β Lip u n + b n η n,s η n,s + p n Q n. Te estimate in (4.16) ten follows from interpolation properties of Q and te triangle inequality, p n p n pn Q n + p n Qn, 5 Numerical Computations In tis section we present four numerical examples to illustrate te numerical approximation sceme (4.11)-(4.13). Examples 1 and 2, for wic we ave an exact solution, are cosen to investigate te derived a priori error estimate for te approximation (4.16), and te dependence of te approximation on te smooter. Examples 3 and 4 use te numerical approximation sceme to investigate te performance of several filters. Computations were performed using te deal.ii software package [3]. For te 2-D computations (Examples 1 and 2) te domain was partitioned into quadrilaterals, and for Examples 3 and 4 te domain was partitioned into exaedrons, Ω = K T K. We let discp k = {f : f K P Q k, K T }, and contp k = {f C 0 (Ω) : f K P Q k, K T }, see (4.5). Example 1 and Example 2. We consider Ω = ( 1, 1) (0, 1) and approximate (3.1)-(3.3) for t (0, 0.5]. Te true solution for te velocity and pressure is given by [ u = txy t 2 y 2 tx + t 2 x 2 ty 2 /2 ], p(x, y) = 2t 2 x ty 2. Te function g tat appears in te deposition function is set to g( u ) = u Assuming tat te error in te numerical approximations is of order O( t 2 + k+1 ) (see (4.16)), we cose ( t) 2 k+1. For a function f and its approximations, f n 1, f n 2, computed on partitions on Ω wit mes parameters 1 and 2, we defined te numerical convergence rate r as: r := log( f(n t) f N 1 / f(n t) f N 2 ) log( 1 / 2 ). Te quantity r is defined similarly. Two different smooters were investigated. For te first one, we computed te smooted porosity η s using a local averaging procedure. Specifically, η s (x) = 1 η(x) dω, V (x) V (x) 14

15 were V (x) = δ denotes te area (volume) of te averaging domain V (x). Te second smooter used was te differential smooter δ η s + η s = η in Ω η s = η on Ω. For bot smooters we used for te value of δ, δ = Example 1. For tis example we take η(x, y) = t 2 xy and β(η) = η Computations using te local averaging smooter, wit (u, p, η, η s ) (RT 0, discp 0, discp 0, contp 1 ), and (u, p, η, η s ) (RT 1, discp 1, discp 1, contp 1 ), in te norm are presented in Tables 5.1, and 5.2, respectively. Similar results were also obtained using te norm, and wen using te differential smooter. Te numerical convergence rates are consistent wit tose predicted by Teorem 4.1. t u(t ) u (T ) X r p(t ) p (T ) r η(t ) η (T ) r 1/ E E E /4 2 7/ E E E / E E E /16 2 9/ E E E / E E E / / E E E-05 - Predicted convergence rate (see (4.16)): 1 Table 5.1: Example 1: Convergence rates for (u, p, η, η s ) (RT 0, discp 0, discp 0, contp 1 ). t u(t ) u (T ) X r p(t ) p (T ) r η(t ) η (T ) r E E E / E E E / E E E / E E E / E E E-06 - Predicted convergence rate (see (4.16)): 2 Table 5.2: Example 1: Convergence rates for (u, p, η, η s ) (RT 1, discp 1, discp 1, contp 1 ). Example 2. In tis example we demonstrate te importance of smooting te porosity input into te β( ) function. Using a perturbed porosity field η = t 2 xy sin(169x) cos(169y) (see 15

16 Figure 5.1), and wit β( ) given by (see Figure 5.2) β(s) = 9.1, 0 s < s , 0.1 s < , 0.3 s < s , 0.4 s < , 0.6 s < s , 0.7 s < , 0.9 s 1.0, we study te convergence rates for two different cases. First, witout smooting te porosity input into te β( ) function (see Table 5.3), and ten wit a smooted porosity input into β( ) (see Table 5.4). For tis example we used te differential smooter wit δ = η y 0 1 x Figure 5.1: Porosity field for Ex. 2 at time t = 0.5. β(s) Figure 5.2: Plot of β(s), Ex.2. s (u, p, η ) (RT 0, discp 0, discp 0 ) t u(t ) u (T ) X r p(t ) p (T ) r η(t ) η (T ) r 1/ E E E /4 2 7/ E E E / E E E /16 2 9/ E E E / E E E-03 - Convergence is not guaranteed by te teory. Table 5.3: Example 2: Convergence rates at T = 0.5, witout smooting te porosity. Te results in Table 5.3 indicate tat witout smooting te convergence rate of te velocity drops drastically and te porosity does not converge. In contrast, using te smooted porosity as input to β( ) te obtained approximations are convergent. 16

17 (u, p, η, η s) (RT 0, discp 0, discp 0, contp 1 ) t u(t ) u (T ) X r p(t ) p (T ) r η(t ) η (T ) r 1/ E E E /4 2 7/ E E E / E E E /16 2 9/ E E E / E E E-04 - Predicted convergence rate: 1 Table 5.4: Example 2: Convergence rates at T = 0.5, using a smooted porosity. Example 3 and Example 4. We consider Ω = ( 1, 1) (0, 1) (0, 1) and approximate (3.1)-(3.3) for t (0, 1]. No flux boundary conditions, u n = 0, were imposed on te walls x = 1, x = 1, y = 1, y = 1, and a zero pressure condition, p = 0, on te outflow boundary z = 1. Four filters, labelled I - IV, wit different initial porosity profiles were investigated, all aving te same initial non void space, ν(0), were ν(t) = (1 η(x, t)) dω. Te computational parameters used were t = 2 5 and = 0.1. Te initial porosity profiles in Filters A - H, see Figures , were I: Te porosity uniform distributed trougout te domain Ω. II: Te porosity increases radially in a continuous fasion. III: Te porosity decreases radially in a continuous fasion. IV: Te porosity decreases continuously in te positive z direction. Ω Figure 5.3: Filter I: Initial porosity field. Figure 5.4: Filter II: Initial porosity field. Example 3. For Example 3 we consider te case of a specified inflow velocity for te fluid, namely u n = f 17

18 Figure 5.5: Filter III: Initial porosity field. Figure 5.6: Filter IV: Initial porosity field. on z = 1, were f (x, t) = (1 x2 )(1 y 2 ) min {1, 4t}, and compare te non void space witin eac filter at time T = 1. Also measure is te maximum pressure witin eac filter at T = 1. Te results are presented in Table 5.5. Plots of te final porosity fields for te filters are presented in Figures Filter I II III IV Nonvoid space ν(1) Max. pressure Table 5.5: Example 3: Non void space ν(t), and maximum pressure witin eac filter at T = 1. In terms of te particulate deposited, all te filters performances were very similar wit a difference between filters of less tan 2.5%. However tere was a significant difference in term of te maximum pressure witin eac filter at T = 1, wit a maximum value of (Filter IV) and a minimum value of 7.03 (Filter III). Figure 5.7: Filter I: Porosity field at T = 1. Figure 5.8: Filter II: Porosity field at T = 1. Example 4. In tis example a inflow pressure, pin, was specified and ten te non void space witin eac filter at T = 1 was calculated, togeter wit te total fluid flow troug te filter. Te inflow pressure 18

19 Figure 5.9: Filter III: Porosity field at T = 1. Figure 5.10: Filter IV: Porosity field at T = 1. used was Te results are given in Table 5.6. p(x, t) = 10 min {1, 4t}. Filter Non void space ν(1) Total flow I II III IV Table 5.6: Example 4: Non void space at T = 1, and total flow from t = 0 to t = 1. Similar to Example 3, te deposition witin te filters was comparable, differing by less tan 6%. Te total flow owever differed by more tan 10% wit te igest total flow of 5.74 occurring for filter I and te lowest total flow of 5.02 occurring for filter IV. References [1] G. Allaire. Homogenization of te Navier-Stokes equations wit a slip boundary condition. Comm. Pure Appl. Mat., 44(6): , [2] M. Azaïez, F. Ben Belgacem, C. Bernardi, and N. Corfi. Spectral discretization of Darcy s equations wit pressure dependent porosity. Appl. Mat. Comput., 217(5): , [3] W. Bangert, T. Heister, L. Heltai, G. Kanscat, M. Kronbicler, M. Maier, B. Turcksin, and T. Young. Te dealii library, version 8.2. Arcive of Numerical Software, 3(1), [4] S. Bartels, M. Jensen, and R. Müller. Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity. SIAM J. Numer. Anal., 47(5): , [5] J. Bear. Dynamics of fluids in porous media. Environmental Science series (New York). American Elsevier,

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