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 Ω_{0}and in Ω\Ω_{0}. 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∂_{t}c+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_{i}v_{j}/|v|^{2}, α_{l} andα_{t}are 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⊂R^{3}with a periodic structure, controlled by a parameter
> 0 which represents the size of each block of the matrix. The C^{1} 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 externalC^{1} 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∈ C^{1}(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∈Ω_{0},

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}∂tf_{1}^{}+v^{}_{f}· ∇f_{1}^{}−div(D(v^{}_{f})∇f_{1}^{}) =qs( ˆf1−f_{1}^{}) in Ω^{}_{f}×J, (1.4)
φ^{}_{f}∂tp^{}_{f}+ div(v^{}_{f}) =qs, v^{}_{f} =− k^{}_{f}

µ(f_{1}^{})∇p^{}_{f} in Ω^{}_{f}×J, (1.5)
φ^{}∂tC_{1}^{}+V^{}· ∇C_{1}^{}−div(D^{}(V^{})∇C_{1}^{}) =qs( ˆC1−C_{1}^{}) in Ω^{}_{m}×J, (1.6)
φ^{}∂_{t}c^{}_{1}+V^{}· ∇c^{}_{1}−div(D^{}(V^{})∇c^{}_{1}) =q_{s}(ˆc_{1}−c^{}_{1}) in Ω^{}_{m}×J, (1.7)
φ^{}∂tc^{}_{2}+V^{}· ∇c^{}_{2}−div(D^{}(V^{})∇c^{}_{2}) =qs(ˆc2−c^{}_{2}) 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(c^{}_{1}+c^{}_{2}) k^{}

µ(m^{}_{1})∇p^{}, V^{}_{h}=−(1−α(c^{}_{1}+c^{}_{2})) k^{}

µ(m^{}_{1})∇p^{}, (1.10)
wherem^{}_{1}=αc^{}_{1}+βC_{1}^{}. 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\Ω_{0} since we can
solely state that (c^{}_{1}+c^{}_{2})(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})∇f_{1}^{}·νf m=D^{}(V^{})∇C_{1}^{}·νf m, (1.11)
αD(v^{}_{f})∇f_{1}^{}·νf m=D^{}(V^{})∇c^{}_{1}·νf m, (1.12)
αD(v^{}_{f})∇(1−f_{1}^{})·ν_{f m}=−αD(v^{}_{f})∇f_{1}^{}·ν_{f m}=D^{}(V^{})∇c^{}_{2}·ν_{f m}, (1.13)
f_{1}^{}=αc^{}_{1}+βC_{1}^{}, α(c^{}_{1}+c^{}_{2}) =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})∇f_{1}^{}·ν = 0 on∂Ω^{}_{f}∩Γ, (1.16)
D^{}(V^{})∇C_{1}^{}·ν=D^{}(V^{})∇c^{}_{1}·ν=D^{}(V^{})∇c^{}_{2}·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

(f_{1}^{}(x,0), C_{1}^{}(x,0), c^{}_{1}(x,0), c^{}_{2}(x,0)) = (f_{1}^{o}(x), C_{1}^{o}(x), c^{o}_{1}(x), c^{o}_{2}(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∈R^{3}. 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 Ω_{0}.

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+β^{2})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∈R^{3},

D^{}(V^{})ξ·ξ≥φ−(Dm(α+β^{2}) +αt|V^{}_{s}+^{2}V^{}_{h}|)|ξ|^{2}, ∀ξ∈R^{3}. (1.21)
We assume thatp^{o}belongs toH^{1}(Ω), and that (f_{1}^{o}, C_{1}^{o}, c^{o}_{1}, c^{o}_{2})∈(L^{∞}(Ω))^{4}satisfies
0≤f_{1}^{o}(x)≤1 a.e. in Ω, (1.22)
γ_{−}≤c^{o}_{1}(x)≤γ_{+}, (γ_{−}, γ_{+})∈R^{2}, a.e. in Ω, (1.23)
αc^{o}_{1}(x) +βC_{1}^{o}(x) =χ^{}_{m}f_{1}^{o}(x), 0≤c^{o}_{1}(x) +c^{o}_{2}(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}^{Y}^{f} +χ_{Ω\Ω}_{0}φ^{Y}^{m}

∂_{t}p_{f}−div K^{H}_{α}
µ(f1)∇pf

=q_{s}−χ_{Ω}_{0}
Z

Y_{m}

φ ∂_{t}p^{0}dy inΩ×J,
φ(y)∂tp^{0}+ div

y (V^{0}) =qs, V^{0}=−k(y)

µ(f1) inΩ0×Ym×J,
pf(x, t) =p^{0}(x, y, t) ify∈Γf m, (x, t)∈Ω×J,

K^{H}_{α}∇pf·ν= 0 on∂Ω×J, p_{f}(x,0) =p^{0}(x, y,0) =p^{o}(x) in Ω×Y_{m}.
The homogenized concentrations problem is inΩ×J:

φ_{f}^{Y}^{f}∂_{t}f_{1}+χ_{Ω\Ω}_{0}1

βφ^{Y}^{m}∂_{t}C_{1}+χ_{Ω}_{0}1
β

Z

Y_{m}

φ(y)∂_{t}C_{1}^{0}dy

− K_{Y}^{H}

f

µ(f1)∇p_{f}· ∇f_{1}−χ_{Ω\Ω}_{0} K_{Y}^{H}

m

βµ(f1)∇p_{f}· ∇C_{1}−χ_{Ω}_{0}
β

Z

Y_{m}

k(y)

µ ∇_{y}p^{0}· ∇_{y}C_{1}^{0}dy

−div(D_{f}^{H}(∇pf)∇f1)−χ_{Ω\Ω}_{0}1

βdiv(D_{m}^{H}(∇pf)∇C1)

=qs|Yf|( ˆf1−f1) +1 βqs

Cˆ1−χ_{Ω\Ω}_{0}|Ym|C1−χΩ0

Z

Y_{m}

C_{1}^{0}(·, y,·)dy
,

φ^{Y}^{m}χ_{Ω\Ω}_{0}∂tc1+χΩ_{0}

Z

Y_{m}

φ(y)∂tc^{0}_{1}dy−α

βφ^{Y}^{m}χ_{Ω\Ω}_{0}∂tC1

−α
βχΩ_{0}

Z

Ym

φ(y)∂tC_{1}^{0}dy−χ_{Ω\Ω}_{0}K_{Y}^{H}

m

µ(f1)∇pf· ∇c1−α β∇C1

−χ_{Ω}_{0}
Z

Y_{m}

k(y)

µ ∇_{y}p^{0}· ∇_{y}c^{0}_{1}−α
β∇_{y}C_{1}^{0}

dy

−χ_{Ω\Ω}_{0}div(D^{H}_{m}(∇pf)∇c1) +χ_{Ω\Ω}_{0}α

βdiv(D_{m}^{H}(∇pf)∇C1)

=q_{s}
ˆ

c_{1}−χ_{Ω\Ω}_{0}|Ym|c_{1}−χ_{Ω}_{0}
Z

Y_{m}

c^{0}_{1}(·, y,·)dy

−α βqs

Cˆ1−χ_{Ω\Ω}_{0}|Ym|C1−χΩ_{0}

Z

Ym

C_{1}^{0}(·, y,·)dy
,
and inΩ0×Ym×J

φ(y)∂tC_{1}^{0}−k(y)

µ ∇yp^{0}· ∇yC_{1}^{0}−div

y D(k(y)

µ ∇yp^{0})∇yC_{1}^{0}

=qs( ˆC1−C_{1}^{0}),
φ(y)∂tc^{0}_{1}−k(y)

µ ∇yp^{0}· ∇yc^{0}_{1}−div

y D(k(y)

µ ∇yp^{0})∇yc^{0}_{1}

=qs(ˆc1−c^{0}_{1}),
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}=f_{1}^{o}, c1

_{t=0}=c^{0}_{1}

_{t=0}=c^{o}_{1}, C1

_{t=0}=C_{1}^{0}

_{t=0}=C_{1}^{o},
f1=αc1+βC1 a.e. inΩ×J, f1=βC_{1}^{0} 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]:

H^{}is the Hilbert space H^{}=L^{2}(Ω^{}_{f})×L^{2}(Ω^{}_{m})×L^{2}(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})∈H^{1}(Ω^{}_{f})×H^{1}(Ω^{}_{m})×H^{1}(Ω^{}_{m});

γ_{f}^{}uf =αγ_{m}^{} um+βγ_{m}^{}Um on Γ^{}_{f m}
endowed with the norm

k(uf, um, Um)kV^{}=kχ^{}_{f}ufk_{L}2(Ω)+kχ^{}_{m}umk_{L}2(Ω)+kχ^{}_{m}Umk_{L}2(Ω)

+kχ^{}_{f}∇u_{f}k_{(L}2(Ω))^{3}+kχ^{}_{m}∇u_{m}k_{(L}2(Ω))^{3}+kχ^{}_{m}∇U_{m}k_{(L}2(Ω))^{3},
where γ_{j}^{} :H^{1}(Ω^{}_{j})→L^{2}(∂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}^{}=L^{2}(Ω^{}_{m})×L^{2}(Ω^{}_{m})∩

(u_{1}, u_{2})∈H^{1}(Ω^{}_{m})×H^{1}(Ω^{}_{m});

α=αγ_{m}^{} (u_{1}+u_{2}) on Γ^{}_{f m}
endowed with the norm

k(u1, u2)kV_{c}^{} =kχ^{}_{m}u1k_{L}2(Ω)+kχ^{}_{m}u2k_{L}2(Ω)+kχ^{}_{m}∇u1k_{(L}2(Ω))^{3}+kχ^{}_{m}∇u2k_{(L}2(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^{}, f_{1}^{}, c^{}_{1}, C_{1}^{}, c^{}_{2}) of
Problem (1.4)-(1.20)in the following sense.

(i) The pressure part (p^{}_{f}, p^{}) belongs to L^{2}(J;H^{1}(Ω^{}_{f}))×L^{2}(J;H^{1}(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∈ C^{1}(J;H^{1}(Ω)) such thatψ|t=T= 0,

− Z

Ω×J

(χ^{}_{f}φ^{}_{f}p^{}_{f}+χ^{}_{m}φ^{}p^{})∂tψ
+

Z

Ω×J

χ^{}_{f} k^{}_{f}

µ(f_{1}^{})∇p^{}_{f}+χ^{}_{m}(α(c^{}_{1}+c^{}_{2})(1−^{2}) +^{2}) k^{}
µ(m^{}_{1})∇p

· ∇ψ

=− Z

Ω

(χ^{}_{f}φ^{}_{f}+χ^{}_{m}φ^{})p^{o}ψ(x,0) +
Z

Ω×J

qsψ.

(2.1)

(ii) The concentration part (f_{1}^{}, c^{}_{1}, C_{1}^{}, c^{}_{2}) is such that (f_{1}^{}, c^{}_{1}, C_{1}^{}) ∈ L^{2}(J;V^{})∩
H^{1}(J; (V^{})^{0}) and (c^{}_{1}, c^{}_{2}) ∈ L^{2}(J;V c)∩H^{1}(J; (V_{c}^{})^{0}). It satisfies for any test
functions(df, d1, D1)∈L^{2}(J;V^{})andd2∈L^{2}(J;H^{1}(Ω^{}_{m}))the following relations.

Z

Ω^{}_{f}×J

φ^{}_{f}∂_{t}f_{1}^{}d_{f}+
Z

Ω^{}_{m}×J

φ^{}∂_{t}c^{}_{1}d_{1}+
Z

Ω^{}_{m}×J

φ^{}∂_{t}C_{1}^{}D_{1}+
Z

Ω^{}_{f}×J

(v^{}_{f}· ∇f_{1}^{})d_{f}
+

Z

Ω^{}_{m}×J

V^{}·(d_{1}∇c^{}_{1}+D_{1}∇C_{1}^{}) +
Z

Ω^{}_{f}×J

D(v^{}_{f})∇f_{1}^{}· ∇d_{f}
+

Z

Ω^{}_{m}×J

D^{}(V^{})∇c^{}_{1}· ∇d_{1}+
Z

Ω^{}_{m}×J

D^{}(V^{})∇C_{1}^{}· ∇D_{1}

= Z

Ω^{}_{f}×J

qs( ˆf1−f_{1}^{})df+
Z

Ω^{}_{m}×J

qs(ˆc1−c^{}_{1})d1+
Z

Ω^{}_{m}×J

qs( ˆC1−C_{1}^{})D1,
(2.2)
and

Z

Ω^{}_{m}×J

φ^{}∂tc^{}_{2}d2+
Z

Ω^{}_{m}×J

(V^{}· ∇c^{}_{2})d2+
Z

Ω^{}_{m}×J

D^{}(V^{})∇c^{}_{2}· ∇d2

− Z

∂Ω^{}_{m}×J

(D^{}(V^{})∇c^{}_{2}·νm)γ_{m}^{} d2

= Z

Ω^{}_{m}×J

qs(1−c^{}_{2})d2.

(2.3)

Furthermore, the following maximum principles hold:

0≤f_{1}^{}(x, t)≤fˆ1 a.e. inΩ^{}_{f} ×J, (2.4)
0≤m^{}_{1}(x, t)≤fˆ1, 0≤c^{}_{1}(x, t) +c^{}_{2}(x, t)≤1 a.e. inΩ^{}_{m}×J, (2.5)
γ_{−}≤c^{}_{1}(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
c^{}_{1}+c^{}_{2}≥0. It is a main difference with our former work in [11].

Lemma 2.2. The pressure satisfies the following uniform estimates
kp^{}_{f}kL^{∞}(J;L^{q}(Ω^{}_{f}))+kp^{}_{f}kL^{2}(J;H^{1}(Ω^{}_{f})) ≤C,

kv^{}_{f}k_{(L}2(J;L^{2}(Ω^{}_{f})))^{2}≤C,
kp^{}k_{L}∞(J;L^{q}(Ω^{}_{m}))≤C,

kα^{1/2}(c^{}_{1}+c^{}_{2})^{1/2}∇p^{}k(L^{2}(J;L^{2}(Ω^{}_{m})))^{3}+k∇p^{}k(L^{2}(J;L^{2}(Ω^{}_{m})))^{3}≤C,
kV^{}_{s}k_{(L}2(J;L^{2}(Ω^{}_{m})))^{3} ≤C,

kV^{}_{h}k_{(L}2(J;L^{2}(Ω^{}_{m})))^{3} ≤C.

Furthermore the time derivative (χ^{}_{f}φ^{}_{f}∂_{t}p^{}_{f} +χ^{}_{m}φ^{}∂_{t}p^{}) is uniformly bounded in
L^{2}(J; (H^{1}(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}|^{2}dx+1
2
Z

Ω^{}_{m}

φ^{}|p^{}|^{2}dx+
Z

Ω^{}_{f}×J

k^{}_{f}

µ(f_{1}^{})∇p^{}_{f}· ∇p^{}_{f}dxdt
+

Z

Ω^{}_{m}×J

α(c^{}_{1}+c^{}_{2})(1−^{2}) +^{2} k^{}

µ(m^{}_{1})∇p^{}· ∇p^{}dxdt

= 1 2 Z

Ω

(χ^{}_{f}φ^{}_{f}(x) +χ^{}_{m}φ^{}(x))|p^{o}(x)|^{2}dx+
Z

Ω×J

q_{s}(χ^{}_{f}p^{}_{f}+χ^{}_{m}p^{})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}|^{2}dx+φ_{−}
2

Z

Ω^{}_{m}

|p^{}|^{2}dx+k_{−}
µ_{+}

Z

Ω^{}_{f}×J

|∇p^{}_{f}|^{2}dxdt
+k_{−}

µ_{+}
Z

Ω^{}_{m}×J

α(c^{}_{1}+c^{}_{2})|∇p^{}|^{2}+^{2}(1−α(c^{}_{1}+c^{}_{2}))|∇p^{}|^{2}
dxdt

≤C kp^{o}k_{L}2(Ω),kqsk_{L}2(Ω×J)
+

Z

Ω^{}_{f}×J

|p^{}_{f}|^{2}dxdt+
Z

Ω^{}_{m}×J

|p^{}|^{2}dxdt.

Using the Gronwall lemma, we prove the desired estimates, but inL^{2}instead 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^{}_{f}^{2}+η)^{q/2−1} (resp.

qp^{}(p^{2}+η)^{q/2−1}) and we integrate by parts over Ω^{}_{f} (resp. Ω^{}_{m}). We obtain
d

dt Z

Ω

χ^{}_{f}(p^{}_{f}^{2}+η)^{q/2}+χ^{}_{m}(p^{2}+η)^{q/2}
+

Z

Ω^{}_{f}

k_{f}^{}

µ(f_{1}^{})(p^{}_{f}^{2}+η)^{q/2−1}∇p^{}_{f}· ∇p^{}_{f}
+

Z

Ω^{}_{f}

k^{}_{f}

µ(f_{1}^{})qp^{}_{f}^{2}(p^{}_{f}^{2}+η)^{q/2−2}∇p^{}_{f}· ∇p^{}_{f}
+

Z

Ω^{}_{m}

(α(c^{}_{1}+c^{}_{2})(1−^{2}) +^{2}) k^{}

µ(m^{}_{1})(p^{2}+η)^{q/2−1}∇p^{}· ∇p^{}
+

Z

Ω^{}_{m}

(α(c^{}_{1}+c^{}_{2})(1−^{2}) +^{2}) k^{}

µ(m^{}_{1})qp^{2}(p^{2}+η)^{q/2−2}∇p^{}· ∇p^{}

= Z

Ω

qsq χ^{}_{f}p^{}_{f}(p^{}_{f}^{2}+η)^{q/2−1}+χ^{}_{m}p^{}(p^{2}+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^{}_{f}p^{}_{f}(p^{}_{f}^{2}+η)^{q/2−1}+χ^{}_{m}p^{}(p^{2}+η)^{q/2−1}

≤C Z

Ω

|qs| χ^{}_{f}(p^{}_{f}^{2}+η)^{(q−1)/2}+χ^{}_{m}(p^{2}+η)^{(q−1)/2}

≤CZ

Ω

χ^{}_{f}(p^{}_{f}^{2}+η)^{q/2}+χ^{}_{m}(p^{2}+η)^{q/2}^{(q−1)/q}Z

Ω

|qs|^{q}^{1/q}

≤CZ

Ω

χ^{}_{f}(p^{}_{f}^{2}+η)^{q/2}+χ^{}_{m}(p^{2}+η)^{q/2}^{(q−1)/q}
.

We conclude with the Gronwall lemma that χ^{}_{f}(p^{}_{f}^{2} +η)^{1/2}+χ^{}_{m}(p^{2}+η)^{1/2} is
uniformly bounded inL^{∞}(J;L^{q}(Ω)). It follows thatχ^{}_{f}p^{}_{f}+χ^{}_{m}p^{} is also uniformly

bounded inL^{∞}(J;L^{q}(Ω)).

We now establish the following results concerning the concentrations functions
(f_{1}^{}, C_{1}^{}, c^{}_{1}, c^{}_{2}).

Lemma 2.3. (i) The functions (f_{1}^{}, C_{1}^{}, c^{}_{1}, c^{}_{2}) are uniformly bounded in the
spaceL^{∞}(J;L^{2}(Ω^{}_{f}))×(L^{∞}(J;L^{2}(Ω^{}_{f})))^{3} and are such that

0≤f_{1}^{}(x, t)≤fˆ1≤1 almost everywhere in Ω^{}_{f}×J,
0≤αc^{}_{1}(x, t) +βC_{1}^{}(x, t)≤fˆ1≤1 almost everywhere in Ω^{}_{m}×J

0≤c^{}_{1}(x, t) +c^{}_{2}(x, t)≤1 almost everywhere inΩ^{}_{m}×J,
γ−≤c^{}_{1}(x, t)≤γ+ almost everywhere in Ω^{}_{m}×J;

(ii) the sequence ((D^{1/2}m +α_{t}^{1/2}|v^{}_{f}|^{1/2})∇f_{1}^{})is uniformly bounded in (L^{2}(Ω^{}_{f}×
J))^{3};

(iii) fori= 1,2, the diffusive termsα^{1/2}(1 + (c^{}_{1}+c^{}_{2})^{1/2}|∇p^{}|^{1/2})∇c^{}_{i} and(1 +

|∇p^{}|^{1/2})∇c^{}_{i}are uniformly bounded in(L^{2}(Ω^{}_{m}×J))^{3}. The same estimates
hold forC_{1}^{}.

Proof. The maximum principles of(i)are a direct consequence of the construction
of the solution (f_{1}^{}, C_{1}^{}, c^{}_{1}, c^{}_{2}) in Theorem 2.1. We write the variational formulation
(2.2) with the test function (d_{f}, d_{1}, D_{1}) = (f_{1}^{}, c^{}_{1}, C_{1}^{}). We get

1 2

Z

Ω^{}_{f}

φ^{}_{f}|f_{1}^{}|^{2}dx+1
2
Z

Ω^{}_{m}

φ^{}(|c^{}_{1}|^{2}+|C_{1}^{}|^{2})dx+
Z

Ω^{}_{f}×J

D(v^{}_{f})∇f_{1}^{}· ∇f_{1}^{}dx dt
+

Z

Ω^{}_{m}×J

(D^{}(V^{})∇c^{}_{1}· ∇c^{}_{1}+D^{}(V^{})∇C_{1}^{}· ∇C_{1}^{})dx dt
+

Z

Ω^{}_{f}×J

(v^{}_{f}· ∇f_{1}^{})f_{1}^{}dx dt+
Z

Ω^{}_{m}×J

V^{}·(c^{}_{1}∇c^{}_{1}+C_{1}^{}∇C_{1}^{})dx dt
+

Z

Ω×J

qs(χ^{}_{f}|f_{1}^{}|^{2}+χ^{}_{m}(|c^{}_{1}|^{2}+|C_{1}^{}|^{2}))dx dt (2.7)

= Z

Ω×J

qsfˆ1f_{1}^{}dx dt+
Z

Ω^{}_{m}×J

qs(ˆc1c^{}_{1}+ ˆC1C_{1}^{})dx dt
+1

2 Z

Ω

φ^{}_{f}|f_{1}^{o}|^{2}+φ^{}(|c^{o}_{1}|^{2}+|C_{1}^{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}· ∇f_{1}^{})f_{1}^{}dx
≤

Z

Ω^{}_{f}×J

αt

2 |v^{}_{f}| |∇f_{1}^{}|^{2}dx+Ckf_{1}^{}k^{2}_{∞}
Z

Ω^{}_{f}

|v^{}_{f}|dx,
where 0≤f_{1}^{}(x, t)≤1 a.e. in Ω^{}_{f}×J andv^{}_{f} is uniformly bounded in (L^{1}(Ω^{}_{f}×J))^{3}
thanks to Lemma 2.2. In the matrix part, we get firstly

Z

Ω^{}_{m}×J

V^{}·(c^{}_{1}∇c^{}_{1}+C_{1}^{}∇C_{1}^{})
≤

Z

Ω^{}_{m}×J

αt

2 |V^{}_{s}+^{2}V^{}_{h}|(|∇c^{}_{1}|^{2}+|∇C_{1}^{}|^{2})
+C

Z

Ω^{}_{f}

(|V^{}_{s}|+|V^{}_{h}|) (|c^{}_{1}|^{2}+|C_{1}^{}|^{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}|(|c^{}_{1}|^{2}+|C_{1}^{}|^{2})

≤ k_{+}
µ−

Z

Ω^{}_{f}

α(c^{}_{1}+c^{}_{2})(1−) +

|∇p^{}| |c^{}_{1}|^{2}+|C_{1}^{}|^{2}

≤CZ

Ω^{}_{f}

α^{2}(c^{}_{1}+c^{}_{2})^{2}+^{2}(1−α(c^{}_{1}+c^{}_{2})^{2}

|∇p^{}|^{2}1/2

× kc^{}_{1}k^{2}_{L}∞(Ω^{}_{m})+kC_{1}^{}k^{2}_{L}∞(Ω^{}_{m})

≤ C

δ kc^{}_{1}k^{2}_{L}∞(Ω^{}_{m})+kC_{1}^{}k^{2}_{L}∞(Ω^{}_{m})

+φ_{−}δ
Z

Ω^{}_{f}

(α+^{2})D_{m} |∇c^{}_{1}|^{2}+|∇C_{1}^{}|^{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}|f_{1}^{}|^{2}+χ^{}_{m}(|c^{}_{1}|^{2}+|C_{1}^{}|^{2}))dx+φ_{−}
Z

Ω^{}_{f}×J

(D_{m}+αt

2 |v^{}_{f}|)|∇f_{1}^{}|^{2}dxdt
+φ_{−}

Z

Ω^{}_{m}×J

((α+^{2})(1−δ)D_{m}+αt

2 |V^{}_{s}+^{2}V^{}_{h}|) |∇c^{}_{1}|^{2}+|∇C_{1}^{}|^{2}
dxdt

≤C δ +C

Z

Ω^{}_{f}×J

|f_{1}^{}|^{2}dxdt.

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

α+β^{2}∇c^{}_{1} and |α(c^{}_{1}+c^{}_{2}) +^{3}(1−α(c^{}_{1}+c^{}_{2}))∇p^{}|^{1/2}∇c^{}_{1} are uniformly
bounded in (L^{2}(Ω×J))^{3}. The estimates for f_{1}^{}, c^{}_{1} andC_{1}^{} follow. Once we know
the estimate for c^{}_{1}, we obtain similar ones forc^{}_{2} by multiplying (1.7) by c^{}_{1}, (1.8)
byc^{}_{2}, 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.