We denote by the characteristic size of the heterogeneity in the medium

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

HOMOGENIZED MODEL FOR FLOW IN PARTIALLY FRACTURED MEDIA

CATHERINE CHOQUET

Abstract. We derive rigorously a homogenized model for the displacement of one compressible miscible fluid by another in a partially fractured porous reservoir. We denote by the characteristic size of the heterogeneity in the medium. A function α characterizes the cracking degree of the rock. Our starting point is an adapted microscopic model which is scaled by appropriate powers of . We then study its limit as → 0. Because of the partially fractured character of the medium, the equation expressing the conservation of total mass in the flow is of degenerate parabolic type. The homogenization process for this equation is thus nonstandard. To overcome this difficulty, we adapt two-scale convergence techniques, convexity arguments and classical compactness tools. The homogenized model contains both single porosity and double porosity characteristics.

1. Introduction and main result

We consider the displacement of a two-component mixture through a highly contrasted porous medium, with fractures and matrix blocks. Assuming that the matrix blocks are disconnected, one usually models this type of setting using the concept of double porosity introduced by Barenblatt et al [5]. The fractured part is responsible for the macro-scale transport and the matrix part can store a concentra- tion longer than is to be expected in a single porous material. The less permeable part of the rock thus contributes as global sink or source terms for the transported solutes in the fracture (see for instance [4, 9]). By the way, the matrix of cells may also be connected so that some flow occurs directly within the cell matrix. We consider here such a partially fissured medium. Most commercial simulators have the added feature of including matrix-matrix connections. But there are very few theoretical derivations of a model for this phenomenon. The uncertainties relating to the size of the physical structure and the fluid content of the reservoir make understanding fluid flow through homogenization a pragmatic approach.

The present paper is an extension of the works [14, 13, 11]. In these latter ref- erences, the degree of interconnection between matrix and fractured part of the

2000Mathematics Subject Classification. 76S05, 35K55, 35B27, 76M50.

Key words and phrases. Miscible compressible displacement; porous medium;

partially fractured reservoir; double porosity; homogenization;

two-scale limit of a degenerate parabolic equation.

c

2009 Texas State University - San Marcos.

Submitted November 20, 2008. Published January 2, 2009.

1

medium, was characterized by a constant averaged parameter. In the present pa- per, a function αdescribes the interconnection. This function may be zero in the totally fractured part Ω0 of the domain Ω, and some technical challenges are en- countered there. The required estimates for the homogenization process involve the degenerating functionαand are technically more challenging than in [11]. The final homogenized equation are different in Ω₀and in Ω\Ω₀. From a more physical viewpoint, this contribution aims to give a model more adapted to the local geom- etry of a natural domain. The interconnection functionα is considered as a first order property of the medium, comparable, for instance, with the fracture intensity function.

We begin by recalling the equations describing the transport of two miscible species in a slightly compressible flow through a homogeneous porous medium, see [6, 20, 15] for details. The unknowns of the problem are the pressure p and the concentrationcof one of the two species of the mixture. Denoting byφthe porosity of the rock and byk its permeability, the mass conservation principles during the displacement are expressed by the equations

φ∂tp+ div(v) =qs, v=− k

µ(c)∇p, (1.1)

φ∂_tc+v· ∇c−div(D(v)∇c) =q_s(ˆc−c). (1.2) The average velocity of the flow v is given by the Darcy law in (1.1). We neglect the gravitational terms. The viscosity µis a nonlinear function depending on the concentration. For instance, in the Koval model [17], µis defined forc∈(0,1) by µ(c) =µ(0)(1 + (M^1/4−1)c)⁻⁴, the constant M =µ(0)/µ(1) being the mobility ratio. Analogous to Fick’s law the dispersive flux is considered proportional to the concentration gradient and the dispersion tensor is

D(v) =φf DmId+Dp(v)

=φf DmId+|v|(αlE(v) +αt(Id− E(v)))

, (1.3) whereE(v)_ij =v_iv_j/|v|², α_l andα_tare the longitudinal and transverse dispersion constants and D_m is the molecular diffusion. For the usual rates of flow, these real numbers are such thatα_l ≥α_t≥D_m >0. The terms containing q_s are the injection and production terms.

We now aim to study a similar flow in a partially fractured porous medium. We thus consider a domain Ω⊂R³with a periodic structure, controlled by a parameter > 0 which represents the size of each block of the matrix. The C¹ boundary of Ω is Γ and ν is the corresponding exterior normal. The standard period (= 1) is a cellY consisting of a matrix block Ym of externalC¹ boundary ∂Ym and of a fracture domainYf. We assume that|Y|= 1. The-reservoir consists of copiesY covering Ω. The two subdomains of Ω are defined by

Ω_f = Ω∩

∪_ξ∈A(Y_f+ξ) , Ω_m= Ω∩

∪_ξ∈A(Y_m+ξ) ,

whereAis an appropriate infinite lattice. The fracture-matrix interface is denoted by Γ_{f m}=∂Ω_f∩∂Ω_m∩Ω andνf mis the corresponding unit normal pointing out Ω_f. See [14] and [1] for some illustrations of admissible structures. To homogenize the reservoir, we shall let tend to zero the sizeof the cells.

Following [14], we assume that the flow is made of two parts. The first component accounts for the global diffusion in the fracture system. The second one corresponds to high frequency spatial variations which lead to local storage in the matrix. A

functionα∈ C¹(Ω) characterizes the interconnection between fractures and matrix.

It is assumed such that

0≤α(x)<1, α(x) +β(x) = 1, α(x) = 0 if and only ifx∈Ω₀,

where Ω0is a bounded open subset of Ω. It may be given by experimental data on samples of porous media and by stochastic reconstruction (see [7] and the references therein). The functionαdescribing theinterconnection intensityis obviously linked with the commonly used concepts of fracture intensity and fracture-size distribution (see for instance [18]). Note that in [14, 13] and [11], the cracking degree was characterized by a constantα∈(0,1).

We thus adapt System (1.1)-(1.2) to such a decomposition of the flow. We also scale the equations for the rapidly varying part by appropriate powers of to conserve the flow between the matrix and the fractures as → 0 (cf [4, 14]).

The complete derivation of the microscopic model is justified in [11]. Denoting by J = (0, T),T >0, the time interval of interest, we consider:

φ_f∂tf₁+v_f· ∇f₁−div(D(v_f)∇f₁) =qs( ˆf1−f₁) in Ω_f×J, (1.4) φ_f∂tp_f+ div(v_f) =qs, v_f =− k_f

µ(f₁)∇p_f in Ω_f×J, (1.5) φ∂tC₁+V· ∇C₁−div(D(V)∇C₁) =qs( ˆC1−C₁) in Ω_m×J, (1.6) φ∂_tc₁+V· ∇c₁−div(D(V)∇c₁) =q_s(ˆc₁−c₁) in Ω_m×J, (1.7) φ∂tc₂+V· ∇c₂−div(D(V)∇c₂) =qs(ˆc2−c₂) in Ω_m×J, (1.8) φ∂tp+ div(V) =qs, V=V_s+V_h, in Ω_m×J, (1.9) V_s=−α(c₁+c₂) k

µ(m₁)∇p, V_h=−(1−α(c₁+c₂)) k

µ(m₁)∇p, (1.10) wherem₁=αc₁+βC₁. The flow in the fractures is described by (1.4)-(1.5). The matrix behavior is described by (1.6)-(1.10). In particular, (1.6) governs the slowly varying component while (1.7)-(1.8) governs the high frequency varying ones. We note that the former system becomes of double degenerate type as→0. Indeed, in the subset Ω0, the parabolic character of (1.6)-(1.9) is only ensured by the term ². Moreover Eq. (1.9) is also of degenerate parabolic type in Ω\Ω₀ since we can solely state that (c₁+c₂)(x, t)≥0 in Ω×J.

The model is completed by the following boundary and initial conditions. We begin by the transmission relations across the interface Γ_{f m}×J.

βD(v_f)∇f₁·νf m=D(V)∇C₁·νf m, (1.11) αD(v_f)∇f₁·νf m=D(V)∇c₁·νf m, (1.12) αD(v_f)∇(1−f₁)·ν_{f m}=−αD(v_f)∇f₁·ν_{f m}=D(V)∇c₂·ν_{f m}, (1.13) f₁=αc₁+βC₁, α(c₁+c₂) =α, (1.14) v_f·νf m=V·νf m, p_f =p. (1.15) We add a zero flux condition out of the full domain Ω

D(v_f)∇f₁·ν = 0 on∂Ω_f∩Γ, (1.16) D(V)∇C₁·ν=D(V)∇c₁·ν=D(V)∇c₂·ν= 0 on∂Ω_m∩Γ, (1.17) v_f·ν = 0 on∂Ω_f∩Γ, V·ν= 0 on∂Ω_m∩Γ, (1.18)

and the following initial conditions in Ω

(f₁(x,0), C₁(x,0), c₁(x,0), c₂(x,0)) = (f₁^o(x), C₁^o(x), c^o₁(x), c^o₂(x)), (1.19) p_f(x,0) =χ_f(x)p^o(x), p(x,0) =χ_m(x)p^o(x). (1.20) Let us now enumerate the assumptions. The source term qs is a nonnegative function ofL^q(Ω×J),q >2, and

αˆc1+βCˆ1= ˆf1, 0≤fˆ1≤1, ˆc1+ ˆc2= 1.

As we assume a periodic structure in the reservoir, the porosities (φ_f(x), φ(x)) = (φ_f(^x), φ(^x)) and the permeabilities (k_f(x), k(x)) = (k_f(^x), k(^x)) of the fracture and of the matrix are periodic of period (Y_f, Y_m). These quantities are assumed to be smooth and bounded, but globally they are discontinuous across Γ_{f m}. We assume moreover

0< φ₋≤φf(x), φ(x)≤φ⁻¹₋ , k₋|ξ|²≤kf(x)ξ·ξ, k(x)ξ·ξ≤k⁻¹₋ |ξ|², k₋>0, a.e. in Ω, for allξ∈R³. The viscosityµ∈W^1,∞(Ω×(0,1)) is such that

0< µ₋≤µ(x, c)≤µ₊ ∀c∈(0,1), µ(x, c) =µ∈R^∗+ in Ω₀.

For sake of simplicity we have assumed that the viscosity is constant in Ω0. We then can pass to the limit in Ω0without introducing a dilation operator (see [11] Section 4 for the details). The tensor Dis already defined in (1.3). The tensor D has a similar structure but its diffusive part (α+β²)DmIdcontains the proportions of slowly and rapidly varying flows in the matrix. The main property of these tensors is

D(v_f)ξ·ξ≥φ−(Dm+αt|v_f|)|ξ|², ∀ξ∈R³,

D(V)ξ·ξ≥φ−(Dm(α+β²) +αt|V_s+²V_h|)|ξ|², ∀ξ∈R³. (1.21) We assume thatp^obelongs toH¹(Ω), and that (f₁^o, C₁^o, c^o₁, c^o₂)∈(L^∞(Ω))⁴satisfies 0≤f₁^o(x)≤1 a.e. in Ω, (1.22) γ₋≤c^o₁(x)≤γ₊, (γ₋, γ₊)∈R², a.e. in Ω, (1.23) αc^o₁(x) +βC₁^o(x) =χ_mf₁^o(x), 0≤c^o₁(x) +c^o₂(x)≤1 a.e. in Ω_m. (1.24) The main result of the paper is the following.

Theorem 1.1. As the scaling parameter tends to zero, the microscopic model (1.4)-(1.20) converges to the following macroscopic model. The homogenized pres- sure problem is

φ_f^Yf +χ_Ω\Ω0φ^Ym

∂_tp_f−div K^H_α µ(f1)∇pf

=q_s−χ_Ω0 Z

Y_m

φ ∂_tp⁰dy inΩ×J, φ(y)∂tp⁰+ div

y (V⁰) =qs, V⁰=−k(y)

µ(f1) inΩ0×Ym×J, pf(x, t) =p⁰(x, y, t) ify∈Γf m, (x, t)∈Ω×J,

K^H_α∇pf·ν= 0 on∂Ω×J, p_f(x,0) =p⁰(x, y,0) =p^o(x) in Ω×Y_m. The homogenized concentrations problem is inΩ×J:

φ_f^Yf∂_tf₁+χ_Ω\Ω01

βφ^Ym∂_tC₁+χ_Ω01 β

Z

Y_m

φ(y)∂_tC₁⁰dy

− K_Y^H

f

µ(f1)∇p_f· ∇f₁−χ_Ω\Ω0 K_Y^H

m

βµ(f1)∇p_f· ∇C₁−χ_Ω0 β

Z

Y_m

k(y)

µ ∇_yp⁰· ∇_yC₁⁰dy

−div(D_f^H(∇pf)∇f1)−χ_Ω\Ω01

βdiv(D_m^H(∇pf)∇C1)

=qs|Yf|( ˆf1−f1) +1 βqs

Cˆ1−χ_Ω\Ω0|Ym|C1−χΩ0

Z

Y_m

C₁⁰(·, y,·)dy ,

φ^Ymχ_Ω\Ω0∂tc1+χΩ₀

Z

Y_m

φ(y)∂tc⁰₁dy−α

βφ^Ymχ_Ω\Ω0∂tC1

−α βχΩ₀

Z

Ym

φ(y)∂tC₁⁰dy−χ_Ω\Ω0K_Y^H

m

µ(f1)∇pf· ∇c1−α β∇C1

−χ_Ω0 Z

Y_m

k(y)

µ ∇_yp⁰· ∇_yc⁰₁−α β∇_yC₁⁰

dy

−χ_Ω\Ω0div(D^H_m(∇pf)∇c1) +χ_Ω\Ω0α

βdiv(D_m^H(∇pf)∇C1)

=q_s ˆ

c₁−χ_Ω\Ω0|Ym|c₁−χ_Ω0 Z

Y_m

c⁰₁(·, y,·)dy

−α βqs

Cˆ1−χ_Ω\Ω0|Ym|C1−χΩ₀

Z

Ym

C₁⁰(·, y,·)dy , and inΩ0×Ym×J

φ(y)∂tC₁⁰−k(y)

µ ∇yp⁰· ∇yC₁⁰−div

y D(k(y)

µ ∇yp⁰)∇yC₁⁰

=qs( ˆC1−C₁⁰), φ(y)∂tc⁰₁−k(y)

µ ∇yp⁰· ∇yc⁰₁−div

y D(k(y)

µ ∇yp⁰)∇yc⁰₁

=qs(ˆc1−c⁰₁), completed by

D^H_f (∇pf)∇f1−1

βD^H_m(∇pf)∇C1

·ν

_Γ×J= 0, D^H_m(∇pf)∇c1−α

βD^H_m(∇pf)∇C1

·ν

_Γ×J = 0, f1

_t=0=f₁^o, c1

_t=0=c⁰₁

_t=0=c^o₁, C1

_t=0=C₁⁰

_t=0=C₁^o, f1=αc1+βC1 a.e. inΩ×J, f1=βC₁⁰ a.e. inΩ0×Γf m×J.

The homogenized quantities K^H_α, D^H_f and D^H_m are defined in (3.3), (3.17) and (3.18) below.

The homogenization process then leads to a macroscopic model containing both single porosity and double porosity characteristics. We show that the double poros- ity part of the model almost disappears as soon as a direct flow occurs in the matrix (see the equations in Ω\Ω0). It emphasizes in particular the role of the dispersion tensor which models all the velocities heterogeneity at the microscopic level. It is characteristic of a miscible flow (see [3] and the references therein). The result is thus quite different of the one obtained in [14, 13]. Nevertheless, even in Ω\Ω0, the model captures the interactions between the matrix and the fractured part.

Indeed, the homogenized permeability and diffusion tensors strongly depend on the transmission functionα. One could compare this effects with some models where

the permeability is concentration dependent: propagation in clays (see [16] and the references therein) or blood flow in micro vessels (see [22] and the references therein) for instance. And in the subset Ω0 where no direct transmission occurs, the model is of double porosity type.

This paper is organized as follows. Section 2 is devoted to the analysis of the microscopic model. We derive uniform estimates for the solutions. Convergence results are stated using two-scale convergence techniques, convexity arguments and classical compactness tools. In Section 3, we pass to the limit→0 and we get the homogenized model described in Theorem 1.1.

2. Analysis of the microscopic model

The existence of weak solutions for the problem (1.4)-(1.20) is proved in [11].

The proof is of course inspired by the statement of the existence of solutions for Problem (1.1)-(1.2) in a homogeneous porous medium (see [10]). But the decom- position of the flow in the matrix part of the domain induces additional difficulties.

Appropriate concentrations spaces for the problem are introduced following [13]:

His the Hilbert space H=L²(Ω_f)×L²(Ω_m)×L²(Ω_m) with the inner product [u_f, u_m, U_m],[ψ_f, ψ_m,Ψ_m]

H

= Z

Ω_f

uf(x)ψf(x)dx+ Z

Ω_m

um(x)ψm(x)dx+ Z

Ω_m

Um(x) Ψm(x)dx, andV is the Banach space

V=H∩

(u_f, u_m, U_m)∈H¹(Ω_f)×H¹(Ω_m)×H¹(Ω_m);

γ_fuf =αγ_m um+βγ_mUm on Γ_{f m} endowed with the norm

k(uf, um, Um)kV=kχ_fufk_L2(Ω)+kχ_mumk_L2(Ω)+kχ_mUmk_L2(Ω)

+kχ_f∇u_fk_(L2(Ω))³+kχ_m∇u_mk_(L2(Ω))³+kχ_m∇U_mk_(L2(Ω))³, where γ_j :H¹(Ω_j)→L²(∂Ω_j) is the usual trace map and χ_j is the characteristic function associated with Ω_j,j=f, m. We also introduce the Banach spaceV_c

V_c=L²(Ω_m)×L²(Ω_m)∩

(u₁, u₂)∈H¹(Ω_m)×H¹(Ω_m);

α=αγ_m (u₁+u₂) on Γ_{f m} endowed with the norm

k(u1, u2)kV_c =kχ_mu1k_L2(Ω)+kχ_mu2k_L2(Ω)+kχ_m∇u1k_(L2(Ω))³+kχ_m∇u2k_(L2(Ω))³. We note that for any fixed >0, the problem is of parabolic type. Then, adapting the proof of [10] to the present piecewise structure, one can state the following existence result (see [11] for a detailed proof).

Theorem 2.1. Let 0 < < 1. There exists a solution (p_f, p, f₁, c₁, C₁, c₂) of Problem (1.4)-(1.20)in the following sense.

(i) The pressure part (p_f, p) belongs to L²(J;H¹(Ω_f))×L²(J;H¹(Ω_m)) and is a weak solution of (1.5), (1.9)-(1.10), (1.15), (1.18) and (1.20). Indeed, for any

function ψ∈ C¹(J;H¹(Ω)) such thatψ|t=T= 0,

− Z

Ω×J

(χ_fφ_fp_f+χ_mφp)∂tψ +

Z

Ω×J

χ_f k_f

µ(f₁)∇p_f+χ_m(α(c₁+c₂)(1−²) +²) k µ(m₁)∇p

· ∇ψ

=− Z

Ω

(χ_fφ_f+χ_mφ)p^oψ(x,0) + Z

Ω×J

qsψ.

(2.1)

(ii) The concentration part (f₁, c₁, C₁, c₂) is such that (f₁, c₁, C₁) ∈ L²(J;V)∩ H¹(J; (V)⁰) and (c₁, c₂) ∈ L²(J;V c)∩H¹(J; (V_c)⁰). It satisfies for any test functions(df, d1, D1)∈L²(J;V)andd2∈L²(J;H¹(Ω_m))the following relations.

Z

Ω_f×J

φ_f∂_tf₁d_f+ Z

Ω_m×J

φ∂_tc₁d₁+ Z

Ω_m×J

φ∂_tC₁D₁+ Z

Ω_f×J

(v_f· ∇f₁)d_f +

Z

Ω_m×J

V·(d₁∇c₁+D₁∇C₁) + Z

Ω_f×J

D(v_f)∇f₁· ∇d_f +

Z

Ω_m×J

D(V)∇c₁· ∇d₁+ Z

Ω_m×J

D(V)∇C₁· ∇D₁

= Z

Ω_f×J

qs( ˆf1−f₁)df+ Z

Ω_m×J

qs(ˆc1−c₁)d1+ Z

Ω_m×J

qs( ˆC1−C₁)D1, (2.2) and

Z

Ω_m×J

φ∂tc₂d2+ Z

Ω_m×J

(V· ∇c₂)d2+ Z

Ω_m×J

D(V)∇c₂· ∇d2

− Z

∂Ω_m×J

(D(V)∇c₂·νm)γ_m d2

= Z

Ω_m×J

qs(1−c₂)d2.

(2.3)

Furthermore, the following maximum principles hold:

0≤f₁(x, t)≤fˆ1 a.e. inΩ_f ×J, (2.4) 0≤m₁(x, t)≤fˆ1, 0≤c₁(x, t) +c₂(x, t)≤1 a.e. inΩ_m×J, (2.5) γ₋≤c₁(x, t)≤γ₊ a.e. inΩ_m×J. (2.6) We now state some uniform estimates for the solutions of the microscopic sys- tem. We begin by stating the following properties of the pressure solutions of the problem (1.5), (1.9)–(1.10), (1.15), (1.18), (1.20). One of the main difficulties of the homogenization problem appears in the following lemma. Indeed, letting to 0, Equation (1.9) is of degenerate parabolic type because one can only ensure that c₁+c₂≥0. It is a main difference with our former work in [11].

Lemma 2.2. The pressure satisfies the following uniform estimates kp_fkL^∞(J;L^q(Ω_f))+kp_fkL²(J;H¹(Ω_f)) ≤C,

kv_fk_(L2(J;L²(Ω_f)))²≤C, kpk_L∞(J;L^q(Ω_m))≤C,

kα^1/2(c₁+c₂)^1/2∇pk(L²(J;L²(Ω_m)))³+k∇pk(L²(J;L²(Ω_m)))³≤C, kV_sk_(L2(J;L²(Ω_m)))³ ≤C,

kV_hk_(L2(J;L²(Ω_m)))³ ≤C.

Furthermore the time derivative (χ_fφ_f∂_tp_f +χ_mφ∂_tp) is uniformly bounded in L²(J; (H¹(Ω))⁰).

Proof. The estimates are derived from integration by parts. We multiply (1.5) by p_f and integrate over Ω_f×J. We multiply (1.9) byp and integrate over Ω_m×J. Summing up the resulting relations, we obtain

1 2 Z

Ω_f

φ_f|p_f|²dx+1 2 Z

Ω_m

φ|p|²dx+ Z

Ω_f×J

k_f

µ(f₁)∇p_f· ∇p_fdxdt +

Z

Ω_m×J

α(c₁+c₂)(1−²) +² k

µ(m₁)∇p· ∇pdxdt

= 1 2 Z

Ω

(χ_fφ_f(x) +χ_mφ(x))|p^o(x)|²dx+ Z

Ω×J

q_s(χ_fp_f+χ_mp)dxdt.

Applying the Cauchy-Schwarz and Young inequalities with the properties ofφ_f,φ, k_f,k andµin the latter relation, we get

φ₋ 2

Z

Ω_f

|p_f|²dx+φ₋ 2

Z

Ω_m

|p|²dx+k₋ µ₊

Z

Ω_f×J

|∇p_f|²dxdt +k₋

µ₊ Z

Ω_m×J

α(c₁+c₂)|∇p|²+²(1−α(c₁+c₂))|∇p|² dxdt

≤C kp^ok_L2(Ω),kqsk_L2(Ω×J) +

Z

Ω_f×J

|p_f|²dxdt+ Z

Ω_m×J

|p|²dxdt.

Using the Gronwall lemma, we prove the desired estimates, but inL²instead ofL^q. The result on the time derivatives then follows straightforward from (1.5), (1.9)- (1.10). It remains to show that the pressure is uniformly bounded inL^∞(J;L^q(Ω)).

Letη >0. We multiply Eq. (1.5) (respectively (1.9)) by qp_f(p_f²+η)^q/2−1 (resp.

qp(p²+η)^q/2−1) and we integrate by parts over Ω_f (resp. Ω_m). We obtain d

dt Z

Ω

χ_f(p_f²+η)^q/2+χ_m(p²+η)^q/2 +

Z

Ω_f

k_f

µ(f₁)(p_f²+η)^q/2−1∇p_f· ∇p_f +

Z

Ω_f

k_f

µ(f₁)qp_f²(p_f²+η)^q/2−2∇p_f· ∇p_f +

Z

Ω_m

(α(c₁+c₂)(1−²) +²) k

µ(m₁)(p²+η)^q/2−1∇p· ∇p +

Z

Ω_m

(α(c₁+c₂)(1−²) +²) k

µ(m₁)qp²(p²+η)^q/2−2∇p· ∇p

= Z

Ω

qsq χ_fp_f(p_f²+η)^q/2−1+χ_mp(p²+η)^q/2−1 .

The four last terms of the left hand side of the latter relation are nonnegative. The right hand side term is estimated as follows using the H¨older inequality.

Z

Ω

qsq χ_fp_f(p_f²+η)^q/2−1+χ_mp(p²+η)^q/2−1

≤C Z

Ω

|qs| χ_f(p_f²+η)^(q−1)/2+χ_m(p²+η)^(q−1)/2

≤CZ

Ω

χ_f(p_f²+η)^q/2+χ_m(p²+η)^q/2(q−1)/qZ

Ω

|qs|^q1/q

≤CZ

Ω

χ_f(p_f²+η)^q/2+χ_m(p²+η)^q/2(q−1)/q .

We conclude with the Gronwall lemma that χ_f(p_f² +η)^1/2+χ_m(p²+η)^1/2 is uniformly bounded inL^∞(J;L^q(Ω)). It follows thatχ_fp_f+χ_mp is also uniformly

bounded inL^∞(J;L^q(Ω)).

We now establish the following results concerning the concentrations functions (f₁, C₁, c₁, c₂).

Lemma 2.3. (i) The functions (f₁, C₁, c₁, c₂) are uniformly bounded in the spaceL^∞(J;L²(Ω_f))×(L^∞(J;L²(Ω_f)))³ and are such that

0≤f₁(x, t)≤fˆ1≤1 almost everywhere in Ω_f×J, 0≤αc₁(x, t) +βC₁(x, t)≤fˆ1≤1 almost everywhere in Ω_m×J

0≤c₁(x, t) +c₂(x, t)≤1 almost everywhere inΩ_m×J, γ−≤c₁(x, t)≤γ+ almost everywhere in Ω_m×J;

(ii) the sequence ((D^1/2m +α_t^1/2|v_f|^1/2)∇f₁)is uniformly bounded in (L²(Ω_f× J))³;

(iii) fori= 1,2, the diffusive termsα^1/2(1 + (c₁+c₂)^1/2|∇p|^1/2)∇c_i and(1 +

|∇p|^1/2)∇c_iare uniformly bounded in(L²(Ω_m×J))³. The same estimates hold forC₁.

Proof. The maximum principles of(i)are a direct consequence of the construction of the solution (f₁, C₁, c₁, c₂) in Theorem 2.1. We write the variational formulation (2.2) with the test function (d_f, d₁, D₁) = (f₁, c₁, C₁). We get

1 2

Z

Ω_f

φ_f|f₁|²dx+1 2 Z

Ω_m

φ(|c₁|²+|C₁|²)dx+ Z

Ω_f×J

D(v_f)∇f₁· ∇f₁dx dt +

Z

Ω_m×J

(D(V)∇c₁· ∇c₁+D(V)∇C₁· ∇C₁)dx dt +

Z

Ω_f×J

(v_f· ∇f₁)f₁dx dt+ Z

Ω_m×J

V·(c₁∇c₁+C₁∇C₁)dx dt +

Z

Ω×J

qs(χ_f|f₁|²+χ_m(|c₁|²+|C₁|²))dx dt (2.7)

= Z

Ω×J

qsfˆ1f₁dx dt+ Z

Ω_m×J

qs(ˆc1c₁+ ˆC1C₁)dx dt +1

2 Z

Ω

φ_f|f₁^o|²+φ(|c^o₁|²+|C₁^o|²) dx.

The convective terms in (2.7) are estimated as follows using the Cauchy-Schwarz and Young inequalities. In the fractured part, we write

Z

Ω_f×J

(v_f· ∇f₁)f₁dx ≤

Z

Ω_f×J

αt

2 |v_f| |∇f₁|²dx+Ckf₁k²_∞ Z

Ω_f

|v_f|dx, where 0≤f₁(x, t)≤1 a.e. in Ω_f×J andv_f is uniformly bounded in (L¹(Ω_f×J))³ thanks to Lemma 2.2. In the matrix part, we get firstly

Z

Ω_m×J

V·(c₁∇c₁+C₁∇C₁) ≤

Z

Ω_m×J

αt

2 |V_s+²V_h|(|∇c₁|²+|∇C₁|²) +C

Z

Ω_f

(|V_s|+|V_h|) (|c₁|²+|C₁|²)dx.

The second term of the right-hand side of the latter relation is treated as follows using Lemma 2.2.

Z

Ω_f

|V_s+V_h|(|c₁|²+|C₁|²)

≤ k₊ µ−

Z

Ω_f

α(c₁+c₂)(1−) +

|∇p| |c₁|²+|C₁|²

≤CZ

Ω_f

α²(c₁+c₂)²+²(1−α(c₁+c₂)²

|∇p|²1/2

× kc₁k²_L∞(Ω_m)+kC₁k²_L∞(Ω_m)

≤ C

δ kc₁k²_L∞(Ω_m)+kC₁k²_L∞(Ω_m)

+φ₋δ Z

Ω_f

(α+²)D_m |∇c₁|²+|∇C₁|² , for any δ >0. The last term in the left-hand side of (2.7) is nonnegative. Using the latter estimates, the Cauchy-Schwarz and Young inequalities for the right-hand side source terms and the basic properties (1.21) of the tensorsDandD, it follows from (2.7) that

φ₋ 2

Z

Ω

(χ_f|f₁|²+χ_m(|c₁|²+|C₁|²))dx+φ₋ Z

Ω_f×J

(D_m+αt

2 |v_f|)|∇f₁|²dxdt +φ₋

Z

Ω_m×J

((α+²)(1−δ)D_m+αt

2 |V_s+²V_h|) |∇c₁|²+|∇C₁|² dxdt

≤C δ +C

Z

Ω_f×J

|f₁|²dxdt.

We choose 0< δ <1. We use the Gronwall lemma to infer from the latter relation that p

α+β²∇c₁ and |α(c₁+c₂) +³(1−α(c₁+c₂))∇p|^1/2∇c₁ are uniformly bounded in (L²(Ω×J))³. The estimates for f₁, c₁ andC₁ follow. Once we know the estimate for c₁, we obtain similar ones forc₂ by multiplying (1.7) by c₁, (1.8) byc₂, integrating over Ω_m and summing up the results to kill the terms on Γ_{f m}.

Our claim is proved.

We now have sufficient estimates to state the first convergence result. The proof of the homogenization process will be carried out by using the two-scale convergence introduced by G.Nguetseng in [19] and developed by Allaire in [2]. The basic definition and properties of this concept follow.