Theoretical derivation of Darcy's law for fluid flow in thin porous media

Theoretical derivation of Darcy’s law for fluid flow in thin porous media Francisco Javier SU´ AREZ-GRAU∗ Abstract In this paper we study stationary incompressible Newtonian fluid flow in a thin porous media. The media under consideration is a bounded perforated 3Ddomain confined between two parallel plates. The description of the domain includes two small parameters: εrepresenting the distance between pates and aεconnected to the microstructure of the domain such that aεε. We consider the classical setting of perforated media, i.e. aε-periodically distributed solid (not connected) obstacles of size aε. The goal of this paper is to introduce a version of the unfolding method, depending on both parameters εand aε, and then to derive the corresponding 2DDarcy’s law. AMS classification numbers: 76A20, 76M50, 35B27, 35Q30. Keywords: Homogenization, Stokes equations, Darcy’s law, thin porous media, thin film fluids. 1 Introduction The problem of Stokes fluid flows in periodically perforated domains with obstacles of the same size as the period has been widely treated in the literature. As is well known, such kinds of flows are generally modelled by Darcy’s law, see Darcy [18]. From the mathematical point of view, the transition between Stokes equations to Darcy’s law assuming Dirichlet conditions on the boundary of the obstacles was formally obtained in Sanchez-Palencia [22] and rigorously in Tartar [23]. Since then, several approaches to derive Darcy’s law have been used by classical authors in homogenization theory, such as Allaire [1], Hornung [20] and Lions [21] among others. The goal of this paper is to generalize classical results of perforated media to the case of thin porous media, which by definition includes two small parameters: one called εis connected to the fluid film thickness and the other denoted by aεto the microstructure representing the size of the obstacles and the interspatial distance between them. The case of thin porous media has been recently considered taking as microstructure a periodic array of vertical cylinders of size and period aεconfined between two parallel plates with distance εin Fabricius et al. [19] assuming Dirichlet conditions on the boundary of the cylinders. Thus, depending on the relation between parameters εand aε, different regimes are obtained: proportionally thin porous media (aε≈ε), homogeneously thin porous media (aεε) and very thin porous medium (aεε). For each case, a permeability tensor is obtained by solving local problems defined on a perforated reference cell describing the local geometry of the microstructure. In the critical case, the local problems are 3D, while they are 2D in the other cases, which is a considerable simplification. This result is proved in Fabricius et al. [19] by using the multiscale expansion method, which is a formal but powerful tool to analyze homogenization problems, and later rigorously developed in Anguiano and Su´arez-Grau [11] by using an adaptation of the periodic unfolding method, see Arbogast et al. [2] and Cioranescu et al. [15, 16, 17], which is introduced in Anguiano and Su´arez-Grau [9]. For other related studies concerning Newtonian fluids in thin porous media such as the derivation of coupled Darcy-Reynolds for fluid flows in thin porous media including a fissure and the modelling of fluid flows in thin porous media with non-homogeneous slip boundary conditions on the cylinders, ∗Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla. 41012-Sevilla (Spain) [email protected] 1 Francisco J. Su´arez-Grau we refer to Anguiano [3, 4] and Anguiano and Su´arez-Grau [10, 13]. In addition, for the case of non-Newtonian power-law fluid flows in thin porous media, we refer to Anguiano [5, 6] and Anguiano and Su´arez-Grau [12] and for the case of non-Newtonian Bingham fluid flows in thin porous media, we refer to Anguiano and Bunoiu [7, 8]. In this paper, the aim is to consider the classical microstructure of perforated domains, i.e. consider aεperiodically distributed solid (not connected) obstacles of the same size and then, to study the influence of the thickness of the domain εto derive the corresponding lower-dimensional Darcy law. In this case, the restriction aεεhas to be naturally imposed. To our knowledge, this problem has only been considered in the case of modelling a thin film passing a thin porous media by Bayada [14], but only the 2D case is considered. In this paper, we consider the 3D case and prove the convergence of the homogenization process when εand aεgo to zero. To to this, the microgeometry of the thin porous media requires a version of the unfolding method depending on both parameters εand aε, which can be applied to other problems and also in the 2D case. As a result, we rigorously derive the corresponding 2D Darcy law, which is different compared to that obtained in the case of homogeneously thin porous media in [11, 19]. The structure of the paper is as follows. In Section 2 we introduce the domain and some useful notation is given in Section 3. In Section 4, we describe the statement of the problem and give the main result (Theorem 4.1), whose proof is provided in Section 5. 2 Definition of the domain Let ωbe a smooth, bounded and connected set in R2. We consider a positive parameter εwhich describes the height of the domain, i.e. we define Qε=ω×(0, ε) as the thin domain without microstructure. To describe the microstructure of the domain Qε, we consider the positive and small parameter aεsuch that lim ε→0 aε ε= 0.(2.1) We denote Y= (−1/2,1/2)3the unitary cube in R3as the reference cell and Tan open connected subset of Ywith a smooth boundary ∂T such that T⊂Y. We denote Yf=Y\T. Thus, for k∈Z3, each cell Yk,aε=aεk+aεYis similar to the unit cell Yrescaled to size aεand Tk,aε=aεk+aεTis similar to Trescaled to size aε. We denote Yfk,aε=Yk,aε\Tk,aε(see Figure 1). We denote by τ(Tk,aε) the set of all translated images of Tk,aε. The set Yf T 1 1 1 a" Yfk,a" Tk,a" a" a" Figure 1: View of the reference cell Y(left) and the rescaled cell Yk,aε(right). τ(Tk,aε) represents the obstacles in R3. The thin porous media Ωεis defined by (see Figure 2) Ωε=Qε\[ k∈Kε Tk,aε,(2.2) where Kε:= k∈Z3:Yk,aε∩Qε6=∅. By construction, Ωεis a periodically perforated domain with obstacles of the same size as the period. We make the assumption that the obstacles τ(¯ Tk,aε) do no intersect the boundary ∂Qε. We denote by Tεthe set of all the obstacles contained in Ωε. Then, Tεis a finite union of obstacles, i.e. Tε=[ k∈Kε Tk,aε. 2 Francisco J. Su´arez-Grau " " Figure 2: View of the domain Ωε(left) and Qε(right). As usual when we deal with thin domains, we will use the dilatation in the variable x3given by z3=x3 ε.(2.3) Then, we define the rescaled porous media e Ωεby (see Figure 3) e Ωε=(x0, z3)∈R3: (x0, εz3)∈Ωε.(2.4) We also introduce the rescaled sets e Yk,aεby (see Figure 3) e Yk,aε=(x0, z3)∈R3: (x0, εz3)∈Yk,aε, and, in the same way, we define the rescaled fluid part e Yfk,aε, the rescaled solid part e Tk,aεof e Yk,aεand the union of rescaled obstacles e Tε. Finally, we denote by Ω the domain with fixed height without microstructure Ω = ω×(0,1). 1 a" e Tk,a" e Yfk,a" a" " a" Figure 3: View of the rescaled domain e Ωε(left) and the rescaled cell e Yk,aε(right). 3 Some notation Along this paper, the points x∈R3will be decomposed as x= (x0, x3) with x0∈R2,x3∈R. We also use the notation x0to denote a generic vector of R2. 3 Francisco J. Su´arez-Grau In order to apply the version of the unfolding method, we need the following notation: for k∈Z3, we define κ:R3→Z3by κ(x) = k⇐⇒ x∈Yk,1.(3.5) Remark that κis well defined up to a set of zero measure in R3, which is given by ∪k∈Z3∂Yk,1. Moreover, for every ε, aε>0, we have κx aε=k⇐⇒ x∈Yk,aεwhich is equivalent to κx0 aε ,εzε aε=k⇐⇒ (x0, z3)∈e Yk,aε. For a vectorial function ϕ= (ϕ0, ϕ3) and a scalar function ψ, we introduce the operators Dε,∇εand divεby (Dεϕ)ij =∂xjϕifor i= 1,2,3, j = 1,2,(Dεϕ)i,3=ε−1∂z3ϕifor i= 1,2,3, ∇εψ= (∇x0ψ, ε−1∂z3ψ)t,divεϕ= divx0ϕ0+ε−1∂z3ϕ3. Let C∞ per(Y) be the space of infinitely differentiable functions in R3that are Y-periodic. By L2 per(Y) (resp. H1 per(Y)) we denote its completion in the norm L2(Y) (resp. H1(Y)). We denote by L2 0the space of functions of L2with null integral and by L2 0,per(Y) the space of functions in L2 per(Y) with zero mean value. We denote by : the full contraction of two matrices, i.e. for A= (aij)1≤i,j≤3and B= (aij)1≤i,j≤3, we have A:B=P3 i,j=1 aijbij. The canonical basis in R3is denoted by {e0, e3}where e0={e1, e2}. Finally, we denote by Oεa generic real sequence, which tends to zero with εand can change from line to line, and by Ca generic positive constant which also can change from line to line. 4 Statement of the problem and main result We consider the following Stokes system in Ωε, with homogeneous boundary conditions on the boundary of the obstacles ∂Tεand on the exterior boundary ∂Qε,        −νdiv(Duε) + ∇pε=fεin Ωε, div uε= 0 in Ωε, uε= 0 on ∂Tε∪∂Qε. (4.6) In the above system, velocity filed uεand (scalar) pressure pεare unknown, while fεrepresents the body force and the viscosity is denoted by νand is assumed strictly positive. To study the asymptotic behavior of the solution we will use the equivalent weak variational formulation of (4.6) which is the following one: find uε∈H1 0(Ωε)3and pε∈L2 0(Ωε) such that νZΩε Duε:Dϕ dx −ZΩε pεdiv ϕ dx =ZΩε fε·ϕ dx, ∀ϕ∈H1 0(Ωε)3.(4.7) Assuming external body force fε∈L2(Ωε)3, it is well known that (4.6) has a unique weak solution (uε, pε)∈ H1 0(Ωε)3×L2 0(Ωε), see e.g [24]. Our aim is to describe the asymptotic behavior of the velocity uεand the pressure pεof the fluid as εtends to zero and identify an homogenized model coupling the effects of the thickness and the microgeometry of the domain. For this purpose, as usual when we deal with thin domains, we use the dilatation (2.3) in order to have the functions defined in the open set with fixed height e Ωεdefined by (2.4). Namely, we define ˜uε∈H1 0(e Ωε)3and ˜pε∈L2 0(e Ωε) by ˜uε(x0, z3) = uε(x0, εz3),˜pε(x0, z3) = pε(x0, εz3),a.e. (x0, z3)∈e Ωε.(4.8) 4 Francisco J. Su´arez-Grau Using the transformation (2.3) the rescaled system (4.6) can be rewritten as        −νdivε(Dε˜uε) + ∇ε˜pε=˜ fεin e Ωε, divε˜uε= 0 in e Ωε, ˜uε= 0 on ∂e Tε∪Ω, (4.9) where ˜ fεis defined similarly as in (4.8). Moreover, taking in (4.7) as test function ˜ϕ∈H1 0(e Ωε)3with ˜ϕ(x0, z3) = ϕ(x0, εz3) and applying the change of variables (2.3), the variational formulation for the rescaled system (4.9) is then to find ˜uε∈H1 0(e Ωε)3and ˜pε∈L2 0(e Ωε) such that νZe Ωε Dε˜uε:Dε˜ϕ dx0dz3−Ze Ωε ˜pεdivε˜ϕ dx0dz3=Ze Ωε ˜ fε·˜ϕ dx0dz3,∀˜ϕ∈H1 0(e Ωε)3.(4.10) We point out that due to the thickness of the domain, it is usual to assume that the vertical component of the external force can be neglected. Thus, let us apply to the fluid an external body force defined by fε= (f0(x0),0) with f0∈L2(ω)2.(4.11) Our goal then becomes describing the asymptotic behavior of this new sequences ˜uεand ˜pεwhen εtends to zero. However, the sequence of solutions (˜uε,˜pε)∈H1 0(e Ωε)3×L2 0(e Ωε) is not defined in a fixed domain independent of ε and aεbut rather in a varying set e Ωε. Thus, in order to pass to the limit if εtends to zero, convergences in fixed Sovolev spaces (defined in Ω) are used which require first that (˜uε,˜pε) be extended to the whole domain Ω. Then, by definition, an extension ( ˜ Uε,˜ Pε)∈H1 0(Ω)3×L2 0(Ω) of (˜uε,˜pε) is defined on Ω and coincides with (˜uε,˜pε) on e Ωε. Our main result is referred to the asymptotic behavior of the solution of (4.9) and is given by the following theorem. Theorem 4.1. Let (˜uε,˜pε)be the unique solution of problem (4.9). Then the sequence of the extension (a−2 ε˜ Uε,˜ Pε) converges weakly to (u, p)in L2(Ω)3×L2(Ω) with u3= 0. Moreover, defining U(x0) = R1 0u(x0, z3)dz3, it holds U0(x0) = 1 νK(f0(x0)− ∇x0p(x0)) , U3(x0) = 0 in ω, (4.12) and p∈H1(ω)∩L2 0(ω)is the unique solution of the 2DDarcy law      divx01 νK(f0(x0)− ∇x0p(x0)) = 0 in ω, 1 νK(f0(x0)− ∇x0p(x0)) ·n= 0 on ∂ω. (4.13) Here, the permeability tensor K∈R2×2is a symmetric and definite positive matrix with coefficients Kij =ZYf Dywi(y) : Dywj(y)dy, i, j = 1,2, where wi,i= 1,2, is the unique solution of the local Stokes problem              −∆ywi+∇yπi=eiin Yf, divywi= 0 in Yf, wi= 0 in T, wi∈H1 per(Yf)3, πi∈L2 0,per(Yf). (4.14) 5 Francisco J. Su´arez-Grau Remark 4.2. As stated in the Introduction, the derivation of Darcy’s law flows in thin porous media has been studied in [11, 19] where the microstructure is a periodic array of vertical cylinders of size and period aεand thickness ε. As a result, three different regimes depending on the relation of parameters εand aεare obtained. The relation between the parameters εand aεconsidered in this paper, aεε, can be framed in the regime called homogeneously thin porous media. Although the current study and the previous ones allow to derive the 2D Darcy law (4.13) with permeability tensor K∈R2×2obtained by solving local problems, the main difference between them lies in the local problems (4.14), because the particular structure of the cylinders in the case of homogeneously thin porous media allows local problems to be reduced to problems defined in a 2D reference cell while here they are defined in a 3D reference cell. 5 Proofs In this section we provide the proof of the main result (Theorem 4.1). To to this, we establish some a priori estimates and define the extension of the solution in Subsection 5.1. In Subsection 5.2 we introduce the version of the unfolding method depending on both parameters εand aε. A compactness result, which is the main key when we will pass to the limit later, is addressed in Subsection 5.3. Finally, the proof of the Theorem 4.1 is given in Subsection 5.4. 5.1 A priori estimates In this subsection, we establish sharp a priori estimates of the dilated solution in e Ωε. To do this, we first need the Poincar´e inequality in thin porous media (for the classical version see [23]). Lemma 5.1. There exists a positive constant C, independent of ε, such that kϕkL2(Ωε)3≤CaεkDϕkL2(Ωε)3×3,∀ϕ∈H1 0(Ωε)3.(5.15) Moreover, considering ˜ϕ∈H1 0(e Ωε)3such that ˜ϕ(x0, z3) = ϕ(x0, εz3), then it holds k˜ϕkL2( e Ωε)3≤CaεkDε˜ϕkL2( e Ωε)3×3.(5.16) Proof. We observe that Ωεcan be divided in small cubes of lateral and vertical length aε. We consider the periodic cell Yfand have a Friedrichs inequality ZYf |ϕ|2dz ≤CZYf |Dϕ|2dz , (5.17) for every ϕ∈H1(Yf)3such that ϕ= 0 on ∂T, where the constant Cdepends only on Yf. Then, for every k∈Z3, by the change of variable k+z=x aε , dz =dx a3 ε , ∂z=aε∂x,(5.18) we rescale (5.17) from Yfto Yfk,aε. This yields that, for every function ϕ(x)∈H1(Yfk,aε)3, one has ZYfk,aε |ϕ|2dx ≤Ca2 εZYfk,aε |Dϕ|2dx , with the same constant Cas in (5.17). Summing previous inequality for every k∈ Kε, we get (5.15). In fact, we must consider separately the periods containing a portion of ∂Qε, but they yield at a distance O(aε) of ∂Qε, where ϕis zero, and then the corresponding inequality is immediately obtained. Finally, by taking in (5.15) the change of variables (2.3), we get (5.16). 6 Francisco J. Su´arez-Grau We give a priori estimates for velocity ˜uεin e Ωε. Lemma 5.2. There exists a positive constant C, independent of ε, such that k˜uεkL2( e Ωε)3≤Ca2 ε,kDε˜uεkL2( e Ωε)3×3≤Caε.(5.19) Proof. Considering ˜uεas test function in (4.10) and applying Cauchy-Schwarz’s inequality, we obtain νkDε˜uεk2 L2( e Ωε)3×3≤ k ˜ fεkL2( e Ωε)3k˜uεkL2( e Ωε)3. Taking into account the assumption on fε, which in particular does not depend on x3, and e Ωεhas a fixed height, we have kDε˜uεk2 L2( e Ωε)3×3≤Ck˜uεkL2( e Ωε)3, and from Poincar´e’s inequality (5.16), we get kDε˜uεk2 L2( e Ωε)3×3≤CaεkDε˜uεkL2( e Ωε)3×3, which proves the second estimate in (5.19). This estimate and again Poincar´e’s inequality (5.16) give the first one. The extension of (˜uε,˜pε)to the whole domain Ω.We extend the velocity ˜uεby zero in Ω \e Ωε(this is compatible with the homogeneous boundary condition on ∂Ωε), and denote the extension by ˜ Uε. Obviously, estimates given in Lemma 5.2 remain valid and the extension ˜ Uεis divergence free too. In order to extend the pressure ˜pεto the whole domain Ω, we recall an important result from [23] which is generalizated to thin domains in [9] and is concerned with the extension of the pressure pεto the whole domain Qε. Thus, we first use a restriction operator Rεfrom H1 0(Qε)3into H1 0(Ωε)3which is introduced in [9, Lemma 4.5] as Rε 2, next we extend to Qεthe gradient of the pressure by duality in H−1(Qε)3and finally to Ω by means of the dilatation. Lemma 5.3. There exists a (restriction) operator Rεacting from H1 0(Qε)3into H1 0(Ωε)3such that 1. Rεϕ=ϕ, if ϕ∈H1 0(Ωε)3(elements of H1 0(Ωε)are extended by 0to Qε). 2. divRεϕ= 0 in Ωε, if divϕ= 0 on Qε. 3. For every ϕ∈H1 0(Qε)3, there exists a positive constant C, independent of ϕand ε, such that kRεϕkL2(Ωε)3+aεkDRεϕkL2(Ωε)3×3≤CkϕkL2(Qε)+aεkDx0ϕkL2(Qε)3×2+aεk∂x3ϕkL2(Qε)3.(5.20) Using the restriction operator Rεgiven in Lemma 5.3, we introduce Fεin H−1(Qε)3in the following way hFε, ϕiH−1(Qε)3,H1 0(Qε)3=h∇pε, RεϕiH−1(Ωε)3,H1 0(Ωε)3,for any ϕ∈H1 0(Qε)3,(5.21) and calcule the right hand side of (5.21) by using (4.7), which gives hFε, ϕiH−1(Qε)3,H1 0(Qε)3=−νZΩε Duε:DRεϕ dx +ZΩε f0·(Rεϕ)0dx . (5.22) Using Lemma 5.2 for fixed ε, we see that it is a bounded functional on H1 0(Qε) (se the proof of Lemma 5.4 below), and in fact Fε∈H−1(Qε)3. Moreover, divϕ= 0 implies hFε, ϕi= 0 ,and the DeRham theorem gives the existence of Pεin L2 0(Qε) with Fε=∇Pε. Next, we get for every ˜ϕ∈H1 0(Ω)3where ˜ϕ(x0, z3) = ϕ(x0, εz3), using the change of variables (2.3), that h∇ε˜ Pε,˜ϕiH−1(Ω)3,H1 0(Ω)3=−ZΩ ˜ Pεdivε˜ϕ dx0dy3=−ε−1ZQε Pεdiv ϕ dx =ε−1h∇Pε, ϕiH−1(Qε)3,H1 0(Qε)3. 7 Francisco J. Su´arez-Grau Using the identification (5.22) of Fε, we have h∇ε˜ Pε,˜ϕiH−1(Ω)3,H1 0(Ω)3=ε−1−νZΩε Duε:DRεϕ dx +ZΩε f0·(Rεϕ)0dx, and applying the change of variables (2.3), we obtain h∇ε˜ Pε,˜ϕiH−1(Ω)3,H1 0(Ω)3=−νZe Ωε Dε˜uε:Dε˜ Rε˜ϕ dx0dz3+Ze Ωε f0·(˜ Rε˜ϕ)0dx0dz3,(5.23) where ˜ Rε˜ϕ=Rεϕfor any ˜ϕ∈H1 0(Ω)3where ˜ϕ(x0, z3) = ϕ(x0, εz3). Finally, we estimate the right-hand side of (5.23) and give the estimate of the extended pressure ˜ Pε. Lemma 5.4. There exists a positive constant Cindependent of ε, such that k˜ PεkL2(Ω) ≤C , k∇ε˜ PεkH−1(Ω)3≤C. (5.24) Proof. Applying the dilatation to (5.20), we have that ˜ Rε˜ϕsatisfies the following estimates k˜ Rε( ˜ϕ)kL2( e Ωε)3≤Ck˜ϕkL2(Ω)3+aεkDx0˜ϕkL2(Ω)3×2+aε εk∂z3˜ϕkL2(Ω)3, kDx0˜ Rε˜ϕkL2( e Ωε)3×2≤C1 aε k˜ϕkL2(Ω)3+kDx0˜ϕkL2(Ω)3×2+1 εk∂z3˜ϕkL2(Ω)3, k∂z3˜ Rε˜ϕkL2( e Ωε)3≤Cε aε k˜ϕkL2(Ω)3+εkDx0˜ϕkL2(Ω)3×2+k∂z3˜ϕkL2(Ω)3. (5.25) From the relation (2.1), we have that k˜ Rε˜ϕkL2( e Ωε)3≤Ck˜ϕkH1 0(Ω)3,kDε˜ Rε˜ϕkL2( e Ωε)3×3≤C aε k˜ϕkH1 0(Ω)3.(5.26) Thus, from Cauchy-Schwarz’s inequality and using estimates for Dε˜uεin (5.19), assumption of f0given in (4.11) and estimates of the dilated restricted operator (5.26), we obtain Ze Ωε Dε˜uε:Dε˜ Rε˜ϕ dx0dz3≤CaεkDε˜ Rε˜ϕkL2( e Ωε)3×3≤Ck˜ϕkH1 0(Ω)3, Ze Ωε f0·(˜ Rε˜ϕ)0dx0dz3≤Ck˜ Rε˜ϕkL2( e Ωε)3≤Ck˜ϕkH1 0(Ω)3, which together with (5.23) gives h∇ε˜ Pε,˜ϕiH−1(Ω)3,H1 0(Ω)3≤Ck˜ϕkH1 0(Ω)3. This implies the second estimate given in (5.24) and using the Ne˘cas inequality, there exists a representative ˜ Pε∈L2 0(Ω) such that k˜ PεkL2(Ω) ≤Ck∇ ˜ PεkH−1(Ω)3≤Ck∇ε˜ PεkH−1(Ω)3, which implies the first estimate given in (5.24). 5.2 Adaptation of the unfolding method The change of variables (2.3) does not provide the information we need about the behavior of of the solution in the microstructure associated to e Ωε. To solve this difficulty, we use an adaptation of the unfolding method (for classical versions see [2, 15, 16, 17]) which is related with the change of variables applied applied in [14] to study the porous part in the case of modelling of a thin film passing a thin porous media. In a simple way, it consists of dividing the domain e Ωεinto cubes of lateral length aεand vertical length aε/ε. 8 Francisco J. Su´arez-Grau Definition 5.5. Let ˜ϕbe in L2(e Ωε)and ˜ ψbe in L2(Ω). We define the functions ˆϕε∈L2(R3×Yf)3and ˆ ψε∈ L2(R3×Y)3by ˆϕε(x0, z3, y) = ˜ϕaεκx0 aε ,εz3 aεe0+aεy0,aε εκx0 aε ,εz3 aεe3+aε εy3,a.e. (x0, z3, y)∈R3×Yf,(5.27) ˆ ψε(x0, z3, y) = ˜ ψaεκx0 aε ,εz3 aεe0+aεy0,aε εκx0 aε ,εz3 aεe3+aε εy3,a.e. (x0, z3, y)∈R3×Y, (5.28) assuming ˜ϕ(resp. ˜ ψ) is extended by zero outside e Ωε(resp. Ω), where the function κis defined by (3.5). Remark 5.6. The restrictions of ˆϕεto e Yk,aε×Yf(resp. ˆ ψεto e Yk,aε×Y) does not depend on (x0, z3), while as a function of yit is obtained from ( ˜ϕ, ˜ ψ)by using the change of variables y0=x0−aεk0 aε , y3=εz3−aεk3 aε ,(5.29) which transforms e Yfk,aεinto Yf(resp. e Yk,aεinto Y). Proposition 5.7. Let ˜ϕbe in H1(e Ωε)and ˜ ψbe in H1(Ω). Then, we have kˆϕεkL2(R3×Yf)3=k˜ϕkL2( e Ωε)3,kDy0ˆϕεkL2(R3×Yf)3×2=aεkDx0˜ϕkL2( e Ωε)3×2,k∂y3ˆϕεkL2(R3×Yf)3=aε εk∂z3˜ϕkL2( e Ωε)3, kˆ ψεkL2(R3×Y)3=k˜ ψkL2(Ω)3,kDy0ˆ ψεkL2(R3×Y)3×2=aεkDx0˜ ψkL2(Ω)3×2,k∂y3ˆ ψεkL2(R3×Y)3=aε εk∂z3˜ ψkL2(Ω)3. Proof. We will only make the proof for ˆϕε. The procedure for ˆ ψεis similar, so we omit it. Taking into account the definition (5.27) of ˆϕε, we obtain ZR3×Yf |Dy0ˆϕε(x0, z3, y)|2dx0dz3dy =X k∈Z3Ze Yk,aεZYf |Dy0ˆϕε(x0, z3, y)|2dydx0dz3 =X k∈Z3Ze Yk,aεZYfDy0˜ϕ(aεk0+aεy0, aεε−1k3+aεε−1y3)2dydx0dz3. We observe that ˜ϕdoes not depend on (x0, z3), then we can deduce ZR3×Yf |Dy0ˆϕε(x0, z3, y)|2dx0dz3dy =a3 ε εX k∈Z3ZYfDy0˜ϕ(aεk0+aεy0, aεε−1k3+aεε−1y3)2dy. By the change of variables (5.29), we obtain ZR3×Yf |Dy0ˆϕε(x0, z3, y)|2dx0dz3dy =a2 εX k∈Z3Ze Yfk,aε |Dx0˜ϕ(x0, z3)|2dx0dz3 =a2 εZe Ωε |Dx0˜ϕ(x0, z3)|2dx0dz3. Thus, we get the property for Dy0ˆϕε. Similarly, we have ZR3×Yf |∂y3ˆϕε(x0, z3, y)|2dx0dz3dy =a3 ε εX k∈Z3ZYf∂y3˜ϕ(aεk0+aεy0, aεε−1k3+aεε−1y3)2dy. 9 Francisco J. Su´arez-Grau [2] T. Arbogast, J. Douglas J.R., and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990), no. 4, 823-836. [3] M. 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