Z. angew. Math. Phys. 53 (2002) 1052–1059 0044-2275/02/061052-8 c 2002 Birkh¨ ° auser Verlag, Basel

Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

Model of diffusion in partially fissured media Horia I. Ene and Dan Poliˇsevski Dedicated to the memory of Eugen So´ os Abstract. We consider an ε -periodic structure formed by two interwoven and connected components which stand for the fissure system and the porous matrix. We assume that on the matrix-fissure interface the pressure has a jump of order ε−1 with respect to the fluid flux which is continuous. We prove that the corresponding homogenized system is exactly that proposed by Barenblatt and al. [1]. Mathematics Subject Classification (2000). 35B27, 76R50, 76S05. Keywords. Partially fissured media, diffusion problem, two-scale convergence.

1. Introduction A fractured porous medium is made up of blocks of an ordinary porous medium with fissures between them. The blocks form the so-called porous matrix (of a certain permeability), while the fissures have a relatively high hidraulic conductivity. In order to describe the flow through a fractured porous medium, Barenblatt and al. [1] (see [2] for a general review) proposed the following system (1.1)

−div (A 5 p1 ) + H(p1 − p2 ) = mg

(1.2)

−div (B 5 p2 ) − H(p1 − p2 ) = (1 − m)g

in Ω in Ω

where p1 and p2 stand for the mean pressure values in the fissures and in the porous matrix, A, B, H, m and g are given coefficients and external forces and Ω denotes the flow domain. It is important to remark that Barenblatt and al. [1] assume that the porous matrix has low permeability and that the fissures have a low bulk volume. If this is the case, most of the flow occurs in the fissures and most of the storage of fluid is in the porous matrix. Due to the fact that at the limit there is no direct flow through the matrix, one can find that B = 0 in (1.2). Such a model of totally fissured media was already studied by a homogenization method in [3].

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In fact the system (1.1)-(1.2) corresponds to the so-called partially fissured medium in which there are substantial paths (directly joining the cells) in addition to the predominant connection with the surrounding fissure system. That means that the cells are not completely isolated from one another by the fissure system and that the matrix is somewhat connected. In the present paper we obtain (1.1)-(1.2) as a model of diffusion in partially fissured media. At the microscale we consider an ε -periodic structure formed by two interwoven and connected components which stand for the fissure and the porous matrix. The only specific assumption is that on the matrix-fissure interface the pressure has a jump of order ε−1 with respect to the fluid flux which is continuous. The homogenization procedure (ε → 0) is a proper one because the characteristic sizes of two regions should be small in comparison with the macroscopic length-scale of the flow domain. In fact the model of diffusion in a partially fractured porous medium is not the only example which leads to problems of the form (1.1)-(1.2). Modelling of heat conduction or convection - diffusion or absorption of a dissolved chemical in a fluid flowing through a porous medium can be treated in a similar manner. Only for historical reasons we present the model of partially fissured media in the framework of a diffusion problem.

2. The partially fissured medium We consider a porous medium composed of two components, the first one being the system of fissures and the second, the solid porous matrix of the structure. It is a convenient reallistic model, introduced in [4], of an ε -periodic structure composed of two phases, both being connected, but only one reaching the boundary of the flow domain. Let Ω be an open connected bounded set in RN (N ≥ 3) , locally located on one side of the boundary ∂Ω , a lipschitz manifold composed of a finite number of connected components. Let Yf be a lipschitz open connected subset of the unit cube Y =]0, 1[N . We assume that Ys = Y \Y¯f has a locally lipschitz boundary and that the intersections of ∂Ys with ∂Y are reproduced identically on opposite faces of the cube, denoted for every i ∈ {1, 2, . . . , N } by (2.1)

Σi = {y ∈ ∂Y, yi = 1} and Σ−i = {y ∈ ∂Y, yi = 0}

We assume that repeating Y by periodicity, the reunion of all the Y¯f parts is a connected domain in RN with a locally C2 boundary; we denote it by RN f N N and consequently RN s = R \Rf . Obviously, the origin of the coordinate system can be set in a fluid ball, that is there exists δ > 0 such that B(0, δ) ⊆ RN f . For any ε ∈]0, 1[ we denote (2.2)

Zε = {k ∈ ZN , εk + εY ⊆ Ω}

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(2.3)

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Iε = {k ∈ Zε , εk ± εei + εY ⊆ Ω, ∀i ∈ 1, N }

where ei are the unit vectors of the the canonical basis in RN . Finally, we define the solid porous matrix of our structure by ¡ ¢ (2.4) Ωεs = int ∪k∈Iε (εk + εY¯s ) and the system of fissures by ¯ εs Ωεf = Ω\Ω

(2.5)

We remark that Ωεf is connected and has a locally lipschitz boundary, while Ωεs can be in particularly connected also.

3. The diffusion problem For any ε ∈]0, 1[ and α ∈ {s, f } we introduce the transmission factor hε (x) = α h(x/ε) and the components of the tensor of diffusion aεα ij (x) = aij (x/ε) , where α α h(y) and aij (y) = aij (y) are Y -periodic, smooth real functions such that there exists γ > 0 with the property (3.1)

N h(y) ≥ γ and aα ij (y)ξi ξj ≥ γξi ξj ∀y ∈ Y, ∀ξ ∈ R .

At the microscale we search for uεs and uεf which verify the following equations: µ ¶ ∂uεα ∂ aεα = g in Ωεα , ∀α ∈ {s, f } (3.2) − ij ∂xi ∂xj where g ∈ L2 (Ω) . Reminding that the solid matrix is entirely surrounded by fissures, we impose to uεs and uεf the following boundary conditions, the second one being specific to our case: ∂uεs ∂uεf νi = aεf νi on Γε = ∂Ωεs = ∂Ωεs ∩ ∂Ωεf (3.3) aεs ij ij ∂xj ∂xj (3.4)

aεf ij

(3.5)

∂uεf νi = εhε (uεs − uεf ) on Γε ∂xj uεf = 0 on ∂Ω

where ν is the normal on Γε (exterior to Ωεf ). We introduce the Hilbert space (3.6)

Hε = H 1 (Ωεs ) × (H 1 (Ωεf ) ∩ H01 (Ω))

endowed with the scalar product Z Z (3.7) (u, v)Hε = 5us 5 vs + Ωεs

Ωεf

Z 5uf 5 vf + ε

Γε

(us − uf )(vs − vf )

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where the components of any u ∈ Hε are denoted for convenience u = (us , uf ) . For any u, v ∈ Hε we also use the notation Z X Z εα ∂uα ∂vα aij +ε hε (us − uf )(vs − vf ). (3.8) a(u, v) = ∂xi ∂xj Ωεα Γε α=s,f

Now we can present the variational formulation of the problem (3.2)-(3.5): To find uε ∈ Hε such that Z Z (3.9) a(uε , v) = gvs + gvf , ∀ v ∈ Hε . Ωεs

Ωεf

Concerning this problem we first have Theorem 3.1. For any ε ∈]0, 1[ there exists a unique uε ∈ Hε , solution of the problem (3.9). Proof. The coerciveness of the form a(·, ·) follows directly from (3.1). It remains to prove the continuity of the right hand of (3.9). As vs ∈ H 1 (Ωεs ) and vf ∈ H 1 (Ωεf ) ∩ H01 (Ω) , with the techniques introduced by [6] and [7] we have (3.10)

|vs |L2 (Ωεs ) ≤ C(ε| 5 vs |L2 (Ωεs) + ε1/2 |vs |L2 (Γε ) )

(3.11)

ε1/2 |vf |L2 (Γε ) ≤ C(ε| 5 vf |L2 (Ωεf ) + |vf |L2 (Ωεf ) )

(3.12)

|vf |L2 (Ωεf ) ≤ C| 5 vf |L2 (Ωεf )

for some C > 0 , which, moreover, is independent of ε . From (3.10) we obtain (3.13)

|vs |L2 (Ωεs ) ≤ C(ε| 5 vs |L2 (Ωεs ) + ε1/2 |vs − vf |L2 (Γε ) + ε1/2 |vf |L2 (Γε ) ).

Repeated use of (3.11) and (3.12) finally yields (3.14)

|vs |L2 (Ωεs ) + |vf |L2 (Ωεf ) ≤ C|v|Hε

and the proof is obviously completed.

¤

4. The homogenization process At the beginning we give the a priori estimates for the solution of problem (3.9). Setting v = uε in (3.9) we find that (4.1)

{uε }ε is bounded in Hε

It follows immediately that for any α ∈ {s, f } we have (4.2)

|uεα |L2 (Ωεα ) ≤ C and | 5 uεα |L2 (Ωεα ) ≤ C

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where C > 0 is independed of ε . From now on we use the following notations ( ( uεα in Ωεα 5uεα in Ωεα ^ , 5u (4.3) u g εα = εα = 0 in Ω\Ωεα 0 in Ω\Ωεα Using standard methods of the two-scale convergence theory (see [8], [9] ) we prove 1 Theorem 4.1. For any α ∈ {s, f } there exist uα ∈ H01 (Ω) and wα ∈ L2 (Ω; Hper (Yα )/R) such that

(4.4)

u g εα (two-scale) * χα (y)uα

(4.5)

^ 5u εα (two-scale) * χα (y)(5uα + 5y wα (·, y))

where χα is defined by

( χα (y) =

(4.6)

1

in Yα

0

in Y \Yα

Before starting the homogenization process we present the ”local-periodic” problems, which have been already introduced by [10]. For any α ∈ {s, f } we denote the Hilbert space 1 1 (Yα ) = {ϕ ∈ Hloc (RN Hper α ), ϕ is Y − periodic}

(4.7)

1 (Yα )/R which is the unique For any k ∈ {1, 2, . . . , N } there exists wαk ∈ Hper solution of the ”local-periodic” problem Z Z ∂wαk ∂ϕ ∂ϕ 1 aα = − aα , (∀)ϕ ∈ Hper (Yα ) (4.8) ij ik ∂y ∂y ∂y j i i Yα Yα

Lemma 4.2. For any Φα ∈ D(Ω) and ϕα ∈ D(Ω; C∞ per (Y )), α ∈ {s, f } , we have ¶µ ¶ µ X Z ∂wα ∂ϕα ∂Φα ∂uα α + aij (y)χα (y) + + ∂xj ∂yj ∂xi ∂yi α=s,f Ω×Y (4.9) Z Z H(us − uf )(Φs − Φf ) = (χs (y)Φs + χf (y)Φf )g Ω

Ω×Y

where H is given by (4.10)

Z H=

h(y)dσ Γ

Proof. The line of the proof is simple: for Φα ∈ D(Ω) and ϕα ∈ D(Ω; C∞ per (Y )) we set vα = Φα (x) + εϕα (x, x/ε) in (3.9) and then we pass to the limit (ε → 0) . Here we shall present the convergence of the term involving Γε , the only one which seems to introduce some technical novelty.

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We obviously have Z hε (uεs − uεf )[Φs − Φf + ε(ϕs − ϕf )]dσ = ε Γε Z Z (4.11) hε uεs (Φs − Φf )dσ − ε hε uεf (Φs − Φf ) + (terms → o) ε Γε

Γε

2

As Γ is of class C , there exists Ψ ∈ [D(Ω; C1per (Y )]N such that Ψ(·, y) = (Φs − Φf )(·) · ν(y) for y ∈ Γ

(4.12)

Then denoting Ψε (x) = Ψ(x, x/ε) , we have Z Z hε uεs (Φs − Φf )dσ = ε hε uεs Ψε · νdσ = ε ΓZε Z Γε (4.13) divx (hε uεs Ψε ) = − u eεs divy (hε Ψε ) + (terms → o) −ε Ωεs

Ω

Using now (4.4) we obtain Z Z hε uεs (Φs − Φf )dσ → − χs (y)us (x)divy [h(y)Ψ(x, y)] = ε Ω×Y Z Z Z ZΓε us ( divy (hΨ)) = us ( h(y)Ψ(x, y)νdσ) = − (4.14) Ys Ω Γ ZΩ us (Φs − Φf ) H Ω

We can prove a similar result for the second term of the right side of (4.11) and thus we obtain Z hε (uεs − uεf )[Φs − Φf + ε(ϕs − ϕf )]dσ → ε ZΓε (4.15) H (us − uf )(Φs − Φf ). Ω

Remark 4.1. For v ∈ D(Ω) and ϕ ∈ C∞ per (Yα ) we set Φα = 0 and ϕα (x, y) = v(x)ϕ(y) in (4.9); we obtain µ ¶ Z ∂ϕ ∂wα ∂uα · aα + =0 (4.16) ij ∂x ∂y ∂y j j i Yα Taking (4.8) in account we find the form of wα with respect to uα : (4.17)

wα (x, y) = wαk (y)

∂uα (x) ∂xk

the solution of (4.8) being unique. Defining the homogenized coefficients by Z α ∂wαj (4.18) Aα = (aα ) ij ij + aik ∂yk Yα

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and introducing (4.17) in (4.9) we can finally sum the homogenization results in the following theorem. ^ eεα and 5u Theorem 4.1. If uε is the solution of (3.9) then u εα defined by (4.3) satisfy for any α ∈ {s, f } . (4.19)

u g εα (two-scale) * χα (y)uα

(4.20)

∂uα ^ 5u 5y wαk (y)) εα (two-scale) * χα (y)(5uα + ∂xk

1 (Yα )/R is the unique solution of the ”local-periodic” problem where wαk ∈ Hper (4.8) and where u = (us , uf ) ∈ H01 (Ω) × H01 (Ω) is the unique solution of the ”homogenized” problem: Z X Z α ∂uα ∂Φα Aij + H(us − uf )(Φs − Φf ) = ∂xj ∂xi Ω α=s,f Ω Z (|Ys |Φs + |Yf |Φf )g, ∀Φ = (Φs , Φf ) ∈ H01 (Ω) × H01 (Ω). Ω

Remark 4.2. Denoting |Ys | = m we notice that (4.21) is the variational formulation of the celebrated (1.1)-(1.2) system.

Acknowlegements This work has been accomplished as part of the European Commision’s Research Program ”HMS-2000”.

References [1] Barenblatt G. I, Zheltov Y. P. and Kochina I. N, On basic conceptions of the theory of homogeneous fluids seepage in fractured rocks (in Russian), Prikl.Mat. i Mekh. 24 (1960), 852-864. [2] Barenblatt G. I., Entov V. M. and Ryzhik V. M., Theory of Fluid Flows Through Natural Rocks, Kluwer Acad. Pub., Dordrecht 1990. [3] Arbogast T., Douglas Jr. J. and Hornung U., Derivation of the double porosity model of single phase flow via homogenization theory, S.I.A.M. J. Math. Anal. 21 (1990), 823-836. [4] Poliˇsevski D., Basic homogenization results for a biconnected ε -periodic structure, submitted for publication, 2002 [5] Sanchez-Palencia E., Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin 1980. [6] Tartar L., Incompressible fluid flow in a porous medium - Convergence of the homogenization process, Appendix of [5]. [7] Allaire G. and Murat F., Homogenization of the Neumann problem with non-isolated holes, Asymptotic Analysis 7 (1993), 81-95.

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[8] Nguetseng G., A general convergence result for a functional related to the theory of homogenization, S.I.A.M. J. Math. Anal. 20 (1989), 608-623. [9] Allaire G., Homogenization and two-scale convergence, S.I.A.M. J. Math. Anal. 23 (1992) 1482-1518. [10] Ene H. I., On the microstructure models of porous media, Rev. Roumaine Math. Pures Appl. 46 (2001), 289-295. Horia I. Ene and Dan Poliˇsevski Romanian Academy Institute of Mathematics P.O.Box 1-764 RO-70700 Bucuresti Romania

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