Electronic Journal of Differential Equations, Vol. 2007(2007), No. 97, pp. 1–29.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
DERIVATION OF MODELS OF COMPRESSIBLE MISCIBLE
DISPLACEMENT IN PARTIALLY FRACTURED RESERVOIRS
CATHERINE CHOQUET
Abstract. We derive rigorously homogenized models for the displacement of
one compressible miscible fluid by another in fractured porous media. We
denote by the characteristic size of the heterogeneity in the medium. A
parameter α ∈ [0, 1] characterizes the cracking degree of the rock. We carefully
define an adapted microscopic model which is scaled by appropriate powers
of . We then study its limit as → 0. Assuming a totally fractured or a
partially fractured medium, we obtain two effective macroscopic limit models.
The first one is a double porosity model. The second one is of single porosity
type but it still contains some effects due to the partial storage in the matrix
part. The convergence is shown using two-scale convergence techniques.
1. Introduction
Most of the natural reservoirs are characterized by the existence of a system of
highly conductive fissures together with a large number of matrix blocks. Because
the fractures can rapidly distribute pollution over vast areas, they are perceived
as controlling the water quality of the entire aquifer, although their storativity
is usually significantly smaller than that of the surrounding matrix. Therefore,
fractured aquifers are considered vulnerable to pollution process. One can assume
that such reservoirs possess two distinct porous structures. The problem of flow
through a fractured environment is thus primarily a problem of flow through a
dual-porosity system. Its two components are hydraulically interconnected and can
not be treated separately. The degree of interconnection between these two flow
systems defines the character of the entire flow domain and is a function of the
hydraulic properties of each of them.
Two main approaches are usually used for the modelling of such reservoirs. The
first one treats the system as a global porous medium with averaged porosity and
permeability. It leads to a single porosity model. The second one uses the concept
of double porosity introduced by Barenblatt et al [5]. Water in the matrix is
considered practically immobile. The less permeable part of the rock contributes
as global sink or source terms for the transported solutes in the fracture. These
two types of model have been rigorously derived using homogenization tools for
2000 Mathematics Subject Classification. 76S05, 35K55, 35B27, 76M50.
Key words and phrases. Miscible compressible displacement; porous medium;
partially fractured reservoir; double porosity; homogenization.
c
2007
Texas State University - San Marcos.
Submitted May 29, 2007. Published July 2, 2007.
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C. CHOQUET
EJDE-2007/97
some displacement problems. Assuming that the reservoir is a porous medium
with highly oscillating porosity and permeability with respect to a parameter and
passing to the limit → 0, one obtains a single porosity model (see for instance
[9]). Modelling the reservoir with a periodic structure controlled by a parameter
> 0 wich represents the size of each block of the matrix, scaling the equations of
the flow in the matrix by appropriate powers of to represent the discontinuities
of the flow, and letting → 0, one gets a double porosity model (see for instance
[4, 10]).
By the way, the former derivation is based on the assumption of a totally fissured
medium, ignoring the heterogeneity of a natural reservoir. The matrix of cells
may also be connected so that some flow occurs directly within the cell matrix.
This is the case of a partially fissured medium considered in [14] and [13]. The
authors introduce two flows in the microscopic model for the matrix. The first
one is the slow scale flow usually introduced for the double porosity model. The
second one is a global flow within the matrix. The homogenization process then
leads to a macroscopic model containing both single porosity and double porosity
characteristics.
The authors in [14] and [13] consider the flow of single phase fluid. The aim
of the present paper is to perform the same work for a miscible and compressible
displacement in a partially fractured medium. In Section 2, we thus derive a new
microscopic model with two flows in the matrix. A special attention is devoted
to the respect of the miscibility assumption. This model is consistent with the
totally fractured case and with the non fractured case. The proportion between the
rapidly varying and the slowly varying part of the flow is specified by a parameter
α ∈ [0, 1]. We derive in Section 3 uniform estimates on the microscopic solutions.
Choosing α = 0 or α > 0, we let the scaling parameter tend to zero in Sections
4 and 5 and we get the associated macroscopic models. We mainly use two-scale
convergence arguments.
Contrary to the case studied in [14, 13], we show that the double porosity part
of the model almost disappears as soon as a direct flow occurs in the matrix. 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.
From a physical viewpoint this contribution aims to give a better understanding
of the transport processes in a more or less fissured medium. From a more mathematical viewpoint, it gives a unified approach for the homogenization in a fractured
and in a non fractured medium.
2. The microscopic system
2.1. Decomposition of the flows. We consider the displacement of two miscible
compressible species in a fractured porous reservoir Ω. The domain Ω thus consists
of two subdomains, the fissures Ωf and the matrix Ωm .
We begin by describing the displacement in the fracture Ωf where we do not
decompose the flow. We follow the lines of [6, 19, 15]. We assume identical compressibilities for the species. We thus consider that the density of the fluid is of the
form
ρ = ρo epf ,
ρo > 0,
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DERIVATION OF MODELS
3
where pf is the pressure in Ωf . Let fi be the concentration of one of the two species
of the mixture and let vi be its velocity in the flow. The conservation of mass of
this component is expressed by the following equation in Ωf ,
∂t (φf ρfi ) + div(ρfi vi ) = ρq fˆi ,
where φf is the porosity of the fracture and q fˆi is a source term such that fˆ1 +fˆ2 = 1.
We then define the average velocity of the flow v f = (f1 v1 + f2 v2 )/(f1 + f2 ) =
f1 v1 + f2 v2 . Due to the Darcy law v f is given by
v f = −(kf /µ(f1 ))∇pf .
We neglect the gravitational terms for sake of clarity in the estimates below. The
permeability of the fracture is kf . The viscosity µ is a nonlinear function depending
on one of the two concentrations in the mixture. We cite for instance the Koval
model [17] where µ is defined for c ∈ (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. Setting ji = ρfi (vi − v f ), the
latter equation becomes
∂t (φf ρfi ) + div(ρfi v f ) + div(ji ) = ρq fˆi .
(2.1)
The tensor ji models the effects of the heterogeneity of the velocities in the mixture. Analogous to Fick’s law this dispersive flux is considered proportional to the
concentration gradient ji = ρD(v f )∇fi , where the dispersion tensor is
D(u) = φf Dm Id + Dp (u) = φf Dm Id + |u| (αl E(u) + αt (Id − E(u))) ,
(2.2)
where E(u)ij = ui uj /|u|2 , αl and αt are the longitudinal and transverse dispersion
constants and Dm is the molecular diffusion. For the usual rates of flow, these reals
are such that αl ≥ αt ≥ Dm > 0. Such a dispersive tensor is characteristic of a
miscible model. Using the definition of the density ρ in (2.1), dividing by ρ > 0,
and using the classical assumption of weak compressibility to neglect the terms
containing v f · ∇pf , we get
φf ∂t fi + φf fi ∂t pf + div(fi v f ) − div(D(v f )∇fi ) = q fˆi .
(2.3)
Bearing in mind f1 + f2 = 1 and summing up the later relation for i = 1, 2, we
model the conservation of the total mass by
φf ∂t pf + div(v f ) = q,
vf = −
kf
∇pf .
µ(f1 )
(2.4)
The flow in Ωf is then completely modelled by (2.4) coupled with
φf ∂t f1 + v f · ∇f1 − div(D(v f )∇f1 ) = q(fˆ1 − f1 ).
(2.5)
We now perform a similar calculation in the matrix part of the domain. But,
following [14], we assume that the flow of each component is made of two parts. The
first component accounts for the global diffusion in the pore system. The second
one corresponds to high frequency spatial variations which lead to local storage in
the matrix. The ith concentration in the matrix mi is then given by
mi = αci + βCi ,
0 ≤ α < 1, α + β = 1.
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C. CHOQUET
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Note that the value of the parameter α may be for instance given by experimental
data on samples of porous media and by stochastic reconstruction (see [7] and the
references therein). The partial concentrations ci and Ci satisfy
∂t (φρci ) + div(ρci vi ) = ρqĉi ,
∂t (φρCi ) + div(ρCi Vi ) = ρq Ĉi ,
(2.6)
where φ is the porosity of the matrix, ρ = ρo ep is the density of the mixture
and p is the pressure in the matrix. We also define two Darcy velocities v =
(c1 v1 + c2 v2 )/(c1 + c2 ) and V = (C1 V1 + C2 V2 )/(C1 + C2 ). We assume that they
both obey a Darcy law depending on the pressure p. The rapidly varying and the
slowly varying components are distinguished by two different permeabilities k and
K satisfying
(2.7)
v = −(k/µ(m1 ))∇p, V = −(K/µ(m1 ))∇p.
Introducing these Darcy velocities in (2.6), we obtain
∂t (φρci ) + div(ρci v) + div(ρci (vi − v)) = ρqĉi ,
∂t (φρCi ) + div(ρCi V ) + div(ρCi (Vi − V )) = ρq Ĉi .
(2.8)
We now use the classical dispersive characteristic of a miscible displacement. To
this aim, we define a global average velocity V = (α(c1 +c2 )v+β(C1 +C2 )V )/(α(c1 +
c2 ) + β(C1 + C2 )), that is
V = α(c1 + c2 )v + β(C1 + C2 )V .
(2.9)
The relative velocities are assumed to satisfy a Fick’s law given by ci (vi −v)+ci (v −
V) = ci (vi − V) = D(V)∇ci and Ci (Vi − V ) + ci (V − V) = Ci (Vi − V) = D(V)∇Ci ,
the tensor D being of the form (2.2). Using the definition of ρ and neglecting the
quadratic velocities terms, we then write (2.8) in the following form for i = 1, 2.
φ∂t di + φdi ∂t p + div(Vdi ) − div(D(V)∇di ) = q dˆi ,
di = ci or Ci .
(2.10)
Finally we use m1 + m2 = 1 to get a relation for the conservation of the total mass
φ∂t p + div(V) = q. The flow in the matrix part is thus governed by the following
set of equations:
φ∂t p + div(V) = q,
V = α(c1 + c2 )v + (1 − α(c1 + c2 ))V ,
k
v=−
∇p,
µ(m1 )
K
V =−
∇p,
µ(m1 )
φ∂t C1 + V · ∇C1 − div(D(V)∇C1 ) = q(Cˆ1 − C1 ),
φ∂t ci + V · ∇ci − div(D(V)∇ci ) = q(ĉi − ci ),
i = 1, 2.
(2.11)
(2.12)
(2.13)
(2.14)
Note that choosing α = 1 and thus β = 0 and k = K, this model is exactly the
same as the one derived in the fracture.
2.2. The scaled microscopic system. Since we neglect the gravitational terms,
we describe the far-field repository by a domain Ω ⊂ R2 with a periodic structure,
controlled by a parameter > 0 which represents the size of each block of the
matrix. Note that all the results of the paper remain true in a domain Ω of R3 (see
Remark 3.3 below for the minor modifications of the proof). The C 1 boundary of
Ω is Γ and ν is the corresponding exterior normal. As in [13], the standard period
( = 1) is a cell Y consisting of a matrix block Ym of external C 1 boundary ∂Ym
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DERIVATION OF MODELS
5
and of a fracture domain Yf . We assume that |Y | = 1. The -reservoir consists of
copies Y covering Ω. The two subdomains of Ω are defined by
Ωf = Ω ∩ ∪ξ∈A (Yf + ξ) , Ωm = Ω ∩ ∪ξ∈A (Ym + ξ) ,
where A is an appropriate infinite lattice. The fracture-matrix interface is denoted
by Γf m = ∂Ωf ∩ ∂Ωm ∩ Ω and νf m is the corresponding unit normal pointing out
Ωf . An example of unit cell in the case of a partially fractured medium is given in
Fig. 1. See [1] for a more complicated admissible 2D-structure.
Γmm
Yf
Yf
Ym
Γf m
Yf
Yf
Figure 1. Unit cell of a partially fractured medium
The major difference between a partially fractured structure and a totally fractured one is that the matrix block Ym is not completely surrounded by the fracture
domain Yf . Let us denote by J = (0, T ) the time interval of interest. To homogenize
the reservoir, we shall let tend to zero the size of the cells.
Our starting point consists of the equations derived in the latter subsection.
As we assume a periodic structure in the reservoir, the porosities (φf (x), φ (x)) =
(φf ( x ), φ( x )) and the permeabilities (kf (x), k (x)) = (kf ( x ), k( x )) of the fracture
and of the matrix are periodic of period (Yf , Ym ). These quantities are assumed to
be smooth and bounded, but globally they are discontinuous across Γf m . A slowly
varying flow and a rapidly varying flow occur in the matrix Ωm . The equations for
the rapidly varying flow will be scaled by appropriate powers of to conserve the
flow between the matrix and the fractures as → 0 (cf [4, 14]). Scaling (2.4)-(2.5)
and (2.11)-(2.14), we get
φ ∂t f + v · ∇f − div(D(v )∇f ) = q(fˆ1 − f ) in Ω × J,
(2.15)
f
1
1
f
f
1
1
f
kf
φf ∂t pf + div(v f ) = q, v f = −
∇p
µ(f1 ) f
in Ωf × J,
φ ∂t C1 + V · ∇C1 − div(D (V )∇C1 ) = q(Ĉ1 − C1 )
φ ∂t c1 + V · ∇c1 − div(D (V )∇c1 ) = q(ĉ1 − c1 )
φ
∂t c2
+V
· ∇c2
− div(D (V
φ ∂t p + div(V ) = q,
V s = −α(c1 + c2 )
)∇c2 )
V =
= q(ĉ2 −
V s
+
k
∇p , V h = −(1 − α(c1 +
µ(m1 )
(2.17)
in Ωm × J,
(2.18)
in Ωm × J,
in Ωm × J,
k c2 ))
∇p ,
µ(m1 )
(2.19)
c2 )
V h ,
in Ωm × J,
(2.16)
(2.20)
(2.21)
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C. CHOQUET
EJDE-2007/97
where m1 = αc1 + βC1 , 0 ≤ α < 1, α + β = 1. The flow in the fissures is described
by (2.15)-(2.16). The matrix behavior is described by (2.17)-(2.21). In particular,
(2.17) governs the slowly varying component while (2.18)-(2.19) governs the high
frequency varying ones. The source term q is a nonnegative function of L2 (Ω × J)
and
αĉ1 + β Ĉ1 = fˆ1 , 0 ≤ fˆ1 ≤ 1, ĉ1 + ĉ2 = 1.
We assume that the porosities and the symmetric permeability tensors satisfy
0 < φ− ≤ φf (x), φ(x) ≤ φ−1
− ,
−1
k− |ξ|2 ≤ kf (x)ξ · ξ, k(x)ξ · ξ ≤ k−
|ξ|2 ,
k− > 0, a.e. in Ω, for all ξ ∈ R2 . The viscosity µ ∈ W 1,∞ (0, 1) is such that
0 < µ− ≤ µ(x) ≤ µ+
∀x ∈ (0, 1).
The tensor D is already defined in (2.2). The tensor D has a similar structure
but its diffusive part (α + β2 )Dm Id contains the same proportions of slowly and
rapidly varying flows than the matrix. The main property of these tensors is
D(v f )ξ · ξ ≥ φ− (Dm + αt |v f |)|ξ|2 ,
∀ξ ∈ R2 ,
D (V )ξ · ξ ≥ φ− (Dm (α + β2 ) + αt |V s + 2 V h |)|ξ|2 ,
.
∀ξ ∈ R2 .
(2.22)
The model is completed by the following boundary and initial conditions. We begin
by the continuity relations across the interface Γf m × J.
βD(v f )∇f1 · νf m = D (V )∇C1 · νf m ,
αD(v f )∇f1
αD(v f )∇(1
−
f1 )
·
· νf m = D (V
)∇c1
νf m = −αD(v f )∇f1 · νf m
f1 = αc1 + βC1 ,
v f · νf m = V · νf m , pf
(2.23)
· νf m ,
= D (V
(2.24)
)∇c2
· νf m ,
(2.25)
(2.26)
=p .
(2.27)
We add a zero flux condition out of the full domain Ω.
D(v f )∇f1 · ν = 0 on ∂Ωf ∩ Γ,
D (V
)∇C1
· ν = D (V
v f
)∇c1
· ν = 0 on
· ν = D (V
∂Ωf
)∇c2
(2.28)
· ν = 0 on
∩ Γ, V · ν = 0 on
∂Ωm
∂Ωm
∩ Γ,
∩ Γ.
(2.29)
(2.30)
The initial conditions in Ω are the following.
(f1 (x, 0), C1 (x, 0), c1 (x, 0), c2 (x, 0)) = (χf f1o (x), C1o (x), co1 (x), co2 (x)),
pf (x, 0)
=
χf (x)po (x),
p (x, 0) =
χm (x)po (x).
(2.31)
(2.32)
We assume that po belongs to H 1 (Ω), and that (f1o , C1o , co1 , co2 ) ∈ L∞ (Ω)×(L2 (Ωm ))3
satisfies
0 ≤ f1o (x) ≤ 1 a.e. in Ω,
αco1 (x)
+
βC1o (x)
=
χm f1o (x),
0 ≤ Cα ≤
co1 (x)
+
co2 (x)
(2.33)
≤ 1 a.e. in
Ωm .
(2.34)
The constant 0 ≤ Cα < 1 is introduced to prevent a degeneration in the limit study
of the pressure equation in Ωm (see Section 5).
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DERIVATION OF MODELS
7
2.3. Variational formulation and existence for the microscopic model.
We now state an existence result for the problem (2.15)-(2.21), (2.23)-(2.32). The
usual equations modelling a miscible and compressible displacement in porous media are of the form (2.4)-(2.5). The existence of weak solutions for this problem
is proved in [11]. In the present paper, the decomposition of the flow in the matrix part of the domain induces two additional difficulties. The first one is a new
coupling between concentrations and pressure due to the term α(c1 + c2 ) in (2.21).
But from a mathematical viewpoint this difficulty is similar to the one due to the
concentration dependent viscosity in the Darcy law. The second novelty occurs
at the interface Γf m . One has to link one concentration f1 in the fracture with
three concentrations (C1 , c1 , c2 ) in the matrix. Thus, following [13], we introduce
appropriate concentrations spaces for the problem. Let H be the Hilbert space
H = L2 (Ωf ) × L2 (Ωm ) × L2 (Ωm ) with the inner product
[uf , um , Um ], [ψf , ψm , Ψm ] H Z
Z
Z
=
uf (x) ψf (x) dx +
um (x) ψm (x) dx +
Um (x) Ψm (x) dx.
Ωf
Ωm
Ωm
Let γj : H 1 (Ωj ) → L2 (∂Ωj ) be the usual trace map and χj be the characteristic
function associated with Ωj , j = f, m. Let V be the following Banach space
V = H ∩ (uf , um , Um ) ∈ H 1 (Ωf ) × H 1 (Ωm ) × H 1 (Ωm );
γf uf = αγm
um + βγm
Um on Γf m
endowed with the norm
k(uf , um , Um )kV = kχf uf kL2 (Ω) + kχm um kL2 (Ω) + kχm Um kL2 (Ω)
+ kχf ∇uf k(L2 (Ω))2 + kχm ∇um k(L2 (Ω))2 + kχm ∇Um k(L2 (Ω))2 .
The introduction of similar spaces for the pressure is useless because we only use
one pressure variable in the matrix part of the domain. Note that the pair of spaces
(H , V ) possesses the same “good” properties as (L2 (Ω), H 1 (Ω)). In particular,
we have the compact embedding V ⊂ H . Thus, adapting the proof of [11] to the
present piecewise structure, one can state the following existence result.
Theorem 2.1. Let 0 < < 1. There exists a solution (pf , p , f1 , c1 , C1 , c2 ) of
Problem (2.15)-(2.21), (2.23)-(2.32) in the following sense.
(i) The pressure part (pf , p ) belongs to L2 (J; H 1 (Ωf )) × L2 (J; H 1 (Ωm )) and is
a weak solution of (2.16), (2.20)–(2.21), (2.27), (2.32). Indeed, for any function
ψ ∈ C 1 (J; H 1 (Ω)),
Z
−
(χf φf pf + χm φ p )∂t ψ
Ω×J
kf
k
∇pf + χm (α(c1 + c2 )(1 − 2 ) + 2 )
) ∇p · ∇ψ
µ(f
)
µ(m
Ω×J
1
1
Z
Z
o
= − (χf φf + χm φ )p ψ(x, 0) +
qψ.
Z
χf
+
Ω
(2.35)
Ω×J
(ii) The concentration part (f1 , c1 , C1 , c2 ) is such that (f1 , c1 , C1 ) ∈ L2 (J; V ) ∩
H 1 (J; (V )0 ) and c2 ∈ L2 (J; H 1 (Ωm )) ∩ H 1 (J; (H 1 (Ωm ))0 ). It satisfies for any
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C. CHOQUET
EJDE-2007/97
(df , d1 , D1 ) ∈ L2 (J; V ) and any d2 ∈ L2 (J; H 1 (Ωm )) the following relations.
Z
Ωf ×J
φf ∂t f1 df +
Z
V ·
+
Ωm ×J
Z
Z
φ ∂t c1 d1 +
Ωm ×J
(d1 ∇c1
+
D1 ∇C1 )
D (V )∇c1 · ∇d1 +
+
Ωm ×J
Z
q (fˆ1 − f1 ) df +
=
Ωf ×J
+
Ωf ×J
φ ∂t C1 D1 +
Z
Ωf ×J
(v f · ∇f1 ) df
D(v f )∇f1 · ∇df
D (V )∇C1 · ∇D1
Ωm ×J
Ωm ×J
Ωm ×J
Z
Z
Z
Z
q (ĉ1 − c1 ) d1 +
Z
Ωm ×J
q (Ĉ1 − C1 ) D1 ,
(2.36)
and
Z
φ
Ωm ×J
Z
+
Ωm ×J
Z
=
Ωm ×J
∂t c2
Z
d2 +
Ωm ×J
(V · ∇c2 ) d2
D (V )∇c2 · ∇d2 −
Z
∂Ωm ×J
(D (V )∇c2 · νm ) γm
d2
(2.37)
q (1 − c2 ) d2 .
Furthermore, the following maximum principles hold:
0 ≤ f1 (x, t) ≤ fˆ1
a.e. in Ωf × J,
0 ≤ m1 (x, t) ≤ fˆ1
a.e. in Ωm × J,
0 ≤ Cα ≤
c1 (x, t)
+
c2 (x, t)
≤1
a.e. in
Ωm
(2.38)
× J.
(2.39)
Proof. The proof of this existence result follows the lines of [11] and is based on a
fixed point approach. The necessary a priori estimates are the same as the uniform
ones derived in Section 3 below. We thus do not detail the proof in the present
paper. We only give the details for the crucial maximum principles (2.38)-(2.39).
In view of the method developed in [11], we can assume in what follows that the
Darcy velocities are regularized so that v f and V belong to L∞ (Ω × J). We first
consider the problem of the left-hand sides of estimates (2.38). We note that the
function αc1 + βC1 = m1 satisfies the following system in Ωm × J.
φ ∂t m1 + V · ∇m1 − div(D (V )∇m1 ) = q(fˆ1 − m1 ),
m1
=
f1 ,
D (V )∇m1 · νf m = (α2 + β 2 )D(v f )∇f1 · νf m on Γf m
D (V )∇m1 · ν = 0 on (∂Ωm ∩ Γ) × J,
m1 (x, 0) = αco1 (x) + βC1o (x) = χm (x)f1o (x) in Ωm .
(2.40)
× J,
(2.41)
(2.42)
(2.43)
For any function f , we denote by (f )− the function (f )− = sup(0, −f ). We now
multiply (2.15) by (α2 + β 2 )(f1 )− (respectively (2.40) by (m1 )− ) and we integrate
over Ωf (respectively Ωm ). Noting that (f1 )− = (m1 )− on Γf m , we sum up the
EJDE-2007/97
DERIVATION OF MODELS
9
resulting relations to kill the non zero boundary terms. We get
Z
Z
d
2
2 − 2
− 2
χ (α + β )φf |(f1 ) | + χm φ |(m1 ) | −
(V · ∇m1 ) (m1 )−
2dt Ω f
Ωm
Z
Z
2
2
−
2
2
− (α + β )
(v f · ∇f1 ) (f1 ) − (α + β )
D(v f )∇f1 · ∇(f1 )−
Ωf
Z
−
Ωm
D (V )∇m1 · ∇(m1 )− +
Z
+
Ωf
q
Z
q fˆ1 χf (α2 + β 2 )(f1 )− + χm (m1 )−
(2.44)
Ω
χf (α2
+β
2
)|(f1 )− |2
+
χm |(m1 )− |2
= 0.
Ω
Using −∇f · ∇(f )− = ∇(f )− · ∇(f )− and the basic properties (2.22) of the tensors
(D, D ), we note that
Z
Z
−
φ− (Dm + αt |v f |) |∇(f1 )− |2 dx,
D(v f )∇f1 · ∇(f1 )− dx ≥
Ωf
Ωf
Z
−
Ωm
Z
≥
Ωm
D (V )∇m1 · ∇(m1 )− dx
φ− (Dm (α + β2 ) + αt |V s + 2 V h |) |∇(m1 )− |2 dx.
The convective terms in (2.44) are then estimated as follows using the CauchySchwarz and Young inequalities.
Z
Z
Z α
t −
− 2
(v f · ∇f1 ) (f1 ) ≤
|v f | |∇(f1 ) | + Ckv f k∞
|(f1 )− |2 ,
Ωf
Ωf 2
Ωf
Z
Z
Z
αDm
C
|∇(m1 )− |2 + kV k2∞
|(m1 )− |2 .
(V · ∇m1 ) (m1 )− ≤
2
α
Ωm
Ωm
Ωm
All the terms of (2.44) containing the source q are nonnegative. The previous
estimates then lead to
Z
Z
φ− d
2
2
− 2
− 2
2
2
(α + β )χm |(f1 ) | + χm |(m1 ) | dx + (α + β )φ−
(Dm
2 dt Ω
Ωf
Z
αt
α
(Dm ( + β2 ) + αt |V s + 2 V h |) |∇(m1 )− |2 dx
+ |v f |) |∇(f1 )− |2 dx + φ−
2
2
Ωm
Z
≤C
(α2 + β 2 )χf |(f1 )− |2 + χm |(m1 )− |2 dx.
Ω
Using the Gronwall lemma with Hypotheses (2.33)-(2.34) on the initial data, we
conclude that (f1 )− (x, t) = 0 a.e. in Ωf × J and (m1 )− (x, t) = 0 a.e. in Ωm × J,
that is the first part of (2.38). The second part is proved similarly, multiplying
(2.15) by (α2 + β 2 )(fˆ1 − f1 )− (respectively (2.40) by (fˆ1 − m1 )− ).
We now justify (2.39). To this aim, we consider the problem satisfied by (c1 +c2 )
in Ωm × J:
φ ∂t (c1 + c2 ) + V · ∇(c1 + c2 ) − div(D ∇(c1 + c2 )) = q(1 − c1 − c2 ),
D (V )∇(c1 + c2 ) · ν = 0 on ∂Ωm × J,
(c1 + c2 )(x, 0) = co1 (x) + co2 (x) in Ω.
(2.45)
(2.46)
(2.47)
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C. CHOQUET
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Multiplying (2.45) by (c1 + c2 − Cα )− , integrating over Ωm and using the same
tools as in the previous part of this proof, we state that c1 (x, t) + c2 (x, t) ≥ Cα ≥ 0
almost everywhere in Ωm × J. We now detail the proof of the second part of (2.39).
We multiply (2.45) by (1 − c1 − c2 )− and we integrate over Ωm . Using the basic
properties (2.22) of the tensor D we get
Z
Z
φ− d
|(1 − c1 − c2 )− |2 + φ−
Dm (α + β2 )|∇(1 − c1 − c2 )− |2
2 dt Ωm
Ωm
Z
Z
+
(V · ∇(c1 + c2 )) (1 − c1 − c2 )− −
q(1 − c1 − c2 ) (1 − c1 − c2 )− ≤ 0.
Ωm
Ωm
The fourth term of the left-hand side is nonnegative. The convective term is estimated with the Cauchy-Schwarz and Young inequalities as follows.
Z
(V · ∇(c1 + c2 )) (1 − c1 − c2 )− Ωm
Z
=
Ωm
Z
≤
Ωm
(V · ∇(1 − c1 − c2 )− ) (1 − c1 − c2 )− φ− αDm
|∇(1 − c1 − c2 )− |2 + CkV k2∞
2
Z
Ωm
|(1 − c1 − c2 )− |2 .
Using these estimates in the first relation, we get
Z
Z
α
φ− d
|(1 − c1 − c2 )− |2 dx + φ−
Dm ( + β2 )|∇(1 − c1 − c2 )− |2 dx
2 dt Ωm
2
Ωm
Z
≤C
|(1 − c1 − c2 )− dx|2 dx.
Ωm
The latter relation and the Gronwall lemma combined with Assumption (2.34) let
us conclude that (1 − c1 − c2 )− (x, t) = 0 and thus c1 (x, t) + c2 (x, t) ≤ 1 a.e. in
Ωm × J. This completes the proof of the theorem.
3. Uniform estimates
We begin by stating the following properties of the pressure solutions of the
problem (2.16), (2.20)-(2.21), (2.27), (2.30), (2.32).
Lemma 3.1. The pressure satisfies the following uniform estimates
kpf kL∞ (J;L2 (Ωf )) + kpf kL2 (J;H 1 (Ωf )) ≤ C,
kv f k(L2 (J;L2 (Ωf )))2 ≤ C,
kp kL∞ (J;L2 (Ωm )) ≤ C,
kα1/2 (c1 + c2 )1/2 ∇p k(L2 (J;L2 (Ωm )))2 + k∇p k(L2 (J;L2 (Ωm )))2 ≤ C,
kV s k(L2 (J;L2 (Ωm )))2 ≤ C,
kV h k(L2 (J;L2 (Ωm )))2 ≤ C.
Furthermore the time derivative (χf φf ∂t pf + χm φ ∂t p ) is uniformly bounded in
L2 (J; (H 1 (Ω))0 ).
EJDE-2007/97
DERIVATION OF MODELS
11
Proof. The estimates are derived from integration by parts. We multiply (2.16) by
pf and integrate over Ωf × J. We multiply (2.20) by p and integrate over Ωm × J.
Summing up the resulting relations, we obtain
1
2
Z
Ωf
φf
|pf |2
1
dx +
2
Z
Z
2
φ |p | dx +
Ωf ×J
Ωm
kf
∇p · ∇pf dx dt
µ(f1 ) f
Z
k
∇p · ∇p dx dt
µ(m1 )
Ωm ×J
Z
Z
1
=
(χf φf (x) + χm φ (x)) |po (x)|2 dx +
q (χf pf + χm p ) dx dt.
2 Ω
Ω×J
α(c1 + c2 )(1 − 2 ) + 2
+
Applying the Cauchy-Schwarz and Young inequalities with the properties of φf , φ ,
kf , k and µ in the latter relation, we get
φ−
2
+
Z
|pf |2
Ωf
k−
µ+
φ−
dx +
2
Z
Ωm ×J
Z
k−
|p | dx +
µ
+
Ωm
2
Z
Ωf ×J
|∇pf |2 dx dt
α(c1 + c2 ) |∇p |2 + 2 (1 − α(c1 + c2 )) |∇p |2 dx dt
≤ C kpo kL2 (Ω) , kqkL2 (Ω×J) +
Z
Ωf ×J
|pf |2 dx dt +
Z
|p |2 dx dt.
Ωm ×J
Using the Gronwall lemma, we prove the desired estimates. The result on the time
derivatives then follows straightforward from (2.16), (2.20)-(2.21).
We can now establish the following results concerning the concentrations functions (f1 , C1 , c1 , c2 ).
Lemma 3.2.
(i) The functions (f1 , C1 , c1 , c2 ) are uniformly bounded in the
∞
space L (J; L2 (Ωf )) × (L∞ (J; L2 (Ωf )))3 and are such that
0 ≤ f1 (x, t) ≤ fˆ1 ≤ 1
almost everywhere in Ωf × J,
0 ≤ αc1 (x, t) + βC1 (x, t) ≤ fˆ1 ≤ 1
0 ≤ c1 (x, t) + c2 (x, t) ≤ 1
1/2
almost everywhere in Ωm × J
almost everywhere in Ωm × J;
1/2
(ii) the sequence ((Dm + αt |v f |1/2 )∇f1 ) is uniformly bounded in (L2 (Ωf ×
J))2 ;
(iii) for i = 1, 2, the diffusive terms α1/2 (1 + (c1 + c2 )1/2 |∇p |1/2 )∇ci and (1 +
|∇p |1/2 )∇ci are uniformly bounded in (L2 (Ωm ×J))2 . The same estimates
hold for C1 .
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
12
C. CHOQUET
EJDE-2007/97
(2.36) with the test function (df , d1 , D1 ) = (f1 , c1 , C1 ). We get
Z
Z
Z
1
1
φf |f1 |2 dx +
φ (|c1 |2 + |C1 |2 ) dx +
D(v f )∇f1 · ∇f1 dx dt
2 Ωf
2 Ωm
Ωf ×J
Z
+
(D (V )∇c1 · ∇c1 + D (V )∇C1 · ∇C1 ) dx dt
Ωm ×J
Z
+
Ωf ×J
Z
(v f · ∇f1 ) f1 dx dt +
Ω×J
Z
=
V · (c1 ∇c1 + C1 ∇C1 ) dx dt
Ωm ×J
Ω×J
1
2
Ωm ×J
q (χf |f1 |2 + χm (|c1 |2 + |C1 |2 )) dx dt
Z
q fˆ1 f1 dx dt +
q(ĉ1 c1 + Ĉ1 C1 ) dx dt
+
+
Z
Z
φf |f1o |2 + φ (|co1 |2 + |C1o |2 ) dx.
Ω
(3.1)
The convective terms in (3.1) are estimated as follows using the Cauchy-Schwarz
and Young inequalities. In the fractured part, we write
Z
Z
Z
αt |v f | |∇f1 |2 dx + Ckf1 k2∞
|v f | dx,
(v f · ∇f1 ) f1 dx ≤
Ωf ×J 2
Ωf
Ωf ×J
where 0 ≤ f1 (x, t) ≤ 1 a.e. in Ωf × J and v f is uniformly bounded in (L1 (Ωf × J))2
thanks to Lemma 3.1. In the matrix part, the work is more difficult because we do
not have an estimate for c1 and C1 in L∞ (Ωm × J). We thus get firstly
Z
Z
αt V · (c1 ∇c1 + C1 ∇C1 ) ≤
|V s + 2 V h | (|∇c1 |2 + |∇C1 |2 )
Ωm ×J
Ωm ×J 2
Z
+C
(|V s | + |V h |) (|c1 |2 + |C1 |2 ) dx.
Ωf
In such a matrix domain, the Gagliardo-Nirenberg inequality reads
1/2
1/2
kukL4 (Ωm ) ≤ CkukL2 (Ω ) k(α + β)ukH 1 (Ω ) ,
m
m
∀u ∈ H 1 (Ωm ).
The second term of the right-hand side of the latter relation is then treated as
follows using the Gagliardo-Nirenberg inequality and Lemma 3.1.
Z
|V | (|c1 |2 + |C1 |2 )
Ωf
Z
k+
α(c1 + c2 )(1 − ) + |∇p | |c1 |2 + |c2 |2
µ− Ωf
Z
1/2
2
≤C
α2 (c1 + c2 )2 + 2 (1 − α(c1 + c2 ) |∇p |2
≤
Ωf
× kc1 k2L4 (Ωm ) + kC1 k2L4 (Ωm )
≤ C kc1 k2L4 (Ωm ) + kC1 k2L4 (Ωm )
≤ C kc1 kL2 (Ωm ) k(α + β)c1 kH 1 (Ωm ) + kC1 kL2 (Ωm ) k(α + β)C1 kH 1 (Ωm )
Z
C
2
2
kc1 kL2 (Ωm ) + kC1 kL2 (Ωm ) + φ− δ
(α + 2 )Dm |∇c1 |2 + |∇C1 |2 ,
≤
δ
Ωf
EJDE-2007/97
DERIVATION OF MODELS
13
for any δ > 0. The last term in the left-hand side of (3.1) is nonnegative. Using
the latter estimates, the Cauchy-Schwarz and Young inequalities for the right-hand
side source terms and the basic properties (2.22) of the tensors D and D , it follows
from (3.1) that
Z
Z
φ−
αt
2
2
2
(χ |f | + χm (|c1 | + |C1 | )) dx + φ−
(Dm + |v f |) |∇f1 |2 dx dt
2 Ω f 1
2
Ωf ×J
Z
αt
+ φ−
((α + 2 )(1 − δ)Dm + |V s + 2 V h |) |∇c1 |2 + |C1 |2 dx dt
2
Ωm ×J
Z
Z
C
≤C +C
|f1 |2 dx dt +
(|c1 |2 + |C1 |2 ) dx dt.
δ
Ωf ×J
Ωm ×J
We choose 0 < δ < 1. Using the Gronwall lemma yields to the result for f1 , c1 and
C1 . Once we know the estimate for c1 , we obtain similar ones for c2 by multiplying
(2.18) by c1 , (2.19) by c2 , integrating over Ωm and summing up the results to kill
the terms on Γf m . Our claim is proved.
Remark 3.3. This proof fails if we consider a domain Ω ⊂ R3 instead of R2 because
of the use of the Gagliardo-Nirenberg inequality. Nevertheless we can get the same
result using minor modifications. The simplest way is to add a L∞ -estimate for c1
and C1 . To this aim, we note that the function d1 = α
β C1 − c1 is solution of the
following problem in Ωm × J.
α
φ ∂t d1 + V · ∇d1 − div(D (V )∇d1 ) = q( Ĉ1 − ĉ1 − d1 ),
β
D (V )∇d1 · ν = 0 on ∂Ωm × J,
α
d1 (x, 0) = C1o (x) − co1 (x) in Ωm .
β
Similar arguments to the ones used in Theorem 2.1 together with an additional
α o
o
assumption on α
β C1 − c1 lead to a maximum principle for β C1 − c1 . Combining it
with Lemma 3.2 (ii), we get a L∞ -bound for c1 and C1 .
4. The macroscopic model: the case of a totally fractured media
(α = 0)
We assume in this section that α = 0 and then β = 1.
concentrations variables (c1 , c2 ). And we define global
functions θ and ξ by
(
(
p
in
Ω
×
J,
f1
f
f
θ =
ξ
=
p in Ωm × J,
C1
We thus do not consider the
pressure and concentration
in Ωf × J,
in Ωm × J,
(4.1)
and the global porosity Φ , tensor of permeability K and new diffusion tensor D
by Φ = χf φf + χm φ , K = χf kf + χm 2 k and D = χf D + χm D .
4.1. The limit double porosity model. The aim of this section is to derive
rigorously the double porosity model described below. We obtain a macroscopic
fracture system driven by equations in Ω × J, similar to the microscopic ones:
Z
Kf
Yf
φf ∂t Pf − div V f = q −
φ(y) ∂t p dy, V f = −
∇Pf ,
(4.2)
µ(F
1)
Ym
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C. CHOQUET
EJDE-2007/97
Yf
∂t F1 + V f · ∇F1 − div(Df (∇Pf , µ(F1 ))∇F1 ) + q |Yf | F1
Z
Z
Z
k(y)
ˆ
= q f1 − q
C1 dy −
φ(y) ∂t C1 dy +
∇y p · ∇y C1 dy.
Ym
Ym
Ym µ(C1 )
φf
(4.3)
The matrix plays the role of a source and produces the additional right-hand side
source-like terms. The homogenization process gives for this macroscopic level an
Yf
effective porosity φf , an effective rock permeability K f and an homogenized
tensor of diffusion Df defined by
Z
Yf
φf =
φf (y)dy,
(4.4)
Yf
Z
kf (y)(∇y v i (y) + ei ) · (∇y v j (y) + ej )dy
K fij =
1 ≤ i, j ≤ 2,
(4.5)
Yf
Z
Df (∇y wi + ei ) · (∇y wj + ej )dy,
Dfij (∇Pf , µ(F1 )) =
(4.6)
Yf
= kf (y) ∇y v i (y) + ei · ∇y v j (y) + ej . The
functions
(4.7) and (4.8) below.
are respectively solutions of the cell problems
Df = D
Kf (y)
and Kf (y)ij
µ(F1 ) ∇Pf
i
(v )1≤i≤2 and (wi )1≤i≤2
− div(kf (y)(∇y v i (y) + ei )) = 0
y
in Yf ,
kf (y)(∇y v i (y) + ei ) · νy = 0 on Γf m , y 7→ v i (x, y) Y -periodic,
− div(Df (x, y)(∇y wi (x, y) + ei )) = 0
y
Df (x, y)(∇y wi (x, y) + ei ) · νy = 0
(4.7)
in Yf ,
on Γf m , y 7→ wi (x, y) Y -periodic,
(4.8)
where ej is the unit vector in the j-th direction. On the other hand, to each x ∈ Ω
corresponds a matrix block, driven by equations in {x} × Ym × J which give the
new source terms:
k(y)
∇y p,
(4.9)
φ(y) ∂t p + div V h = q, V h = −
y
µ(C1 )
φ(y) ∂t C1 + V · ∇y C1 − div(D(y, V ) ∇y C1 ) + q C1 = q fˆ1 .
(4.10)
h
y
h
The equations (4.2)-(4.3), (4.9)-(4.10) are provided with the following initial and
boundary conditions
K f ∇Pf · ν = 0,
Df ∇F1 · ν = 0
o
Pf (x, 0) = p(x, y, 0) = p (x),
on Γ × J,
p = Pf , C1 = F1
F1 (x, 0) = C1 (x, y, 0) =
f1o (x)
on Γf m , (4.11)
in Ω × Ym . (4.12)
We claim and prove the following convergence result.
Theorem 4.1. As the scaling parameter tends to zero, the microscopic model
(2.15)–(2.17), (2.20)–(2.21), (2.23), (2.26)–(2.32) with α = 0 converges to the double
porosity macroscopic model (4.2)–(4.12).
We have captured the interactions between the local and the global scales. This
model is consistent with the double porosity formulations of the engineering literature, cf [5]. But in [5], the exchanges between fractures and blocks are assumed
quasi-stationary. Without this hypothesis, we obtain additional memory terms.
EJDE-2007/97
DERIVATION OF MODELS
15
The proof of the homogenization process will be carried out by using the twoscale convergence introduced by G.Nguetseng in [18] and developed by Allaire in
[2]. We recall the basic definition and properties of this concept.
Proposition 4.2. A sequence of functions (v ) bounded in L2 (Ω × J) two-scale
2
converges to a limit v o (x, y, t) belonging to L2 (Ω × Y × J), v *v o , if
Z
Z
Z
lim
v (x, t) Ψ(x, x/, t) dx dt =
v o (x, y, t) Ψ(x, y, t) dx dy dt,
→0
Ω×J
Ω×J
Y
for any test function Ψ(x, y, t), Y-periodic in the second variable, satisfying
Z
Z
Z
lim
|Ψ(x, x/, t)|2 dx dt =
|Ψ(x, y, t)|2 dx dy dt.
→0
Ω×J
Ω×J
Y
2
(i) From each bounded sequence (v ) in L (Ω × J) one can extract a subsequence
which two-scale converges.
(ii) Let (v ) be a bounded sequence in L2 (J; H 1 (Ω)) which converges weakly to v
2
1
in L2 (J; H 1 (Ω)). Then v *v and there exists a function v 1 ∈ L2 (Ω × J; Hper
(Y ))
2
such that, up to a subsequence, ∇v *∇v(x, t) + ∇y v 1 (x, y, t).
(iii) Let (v ) be a bounded sequence in L2 (Ω × J) with (∇v ) bounded in (L2 (Ω ×
1
(Y )) such that, up to a
J))2 . Then, there exists a function v o ∈ L2 (Ω × J; Hper
2
2
subsequence, v *v o and ∇v *∇y v o (x, y, t).
To exploit the a priori estimates obtained in the fractured part Ωf , we need to
extend the functions pf and f1 to the whole domain Ω. To this aim, following [1],
we claim that there exists three constants ki = ki (Qf ) > 0, i = 1, 2, 3, and a linear
1
and continuous extension operator Π : H 1 (Ωf ) → Hloc
(Ω) such that Π v = v a.e.
in Ωf and
Z
Z
Z
Z
2
2
2
|Π v| dx ≤ k2
|v| dx,
|∇(Π v)| dx ≤ k3
|∇v|2 dx
Ωf
Ω(k1 )
Ω(k1 )
Ωf
for all v ∈ H 1 (Ωf ), with Ω(k1 ) = {x ∈ Ω | dist(x, Γ) > k1 }. To avoid dealing with
boundary layers, we make the following additional assumption on the structure of
the domain Ω:
Ωm = Ω(k1 ) ∩ ∪k∈Z2 (Ym + k)
and Ωf = Ω \ Ωm .
R
For any subset Ω0 ⊂⊂ Ω, we get with Lemmas 3.1 and 3.2 Ω0 ×J |∇(Π pf )|2 dx dt ≤
R
C and Ω0 ×J |∇(Π f1 )|2 dx dt ≤ C, if < dist(Ω0 , Γ)/k1 . Thus we can state convergence results in any subset Ω0 ⊂⊂ Ω and conclude with a density argument.
But, for sake of simplicity, we prefer assume that the blocks are removed in an
k1 -neighborhood of Γ. We then get
Z
Z
|∇(Π pf )|2 dx dt ≤ C,
|∇(Π f1 )|2 dx dt ≤ C.
Ω×J
Ω×J
2
1
We ensure the existence of functions Pf ∈ L (J; H 1 (Ω)), Pf1 ∈ L2 (Ω × J; Hper
(Y )),
1
1
p ∈ L2 (Ω × J; Hper
(Y )), F1 ∈ L2 (J; H 1 (Ω)), F11 ∈ L2 (Ω × J; Hper
(Y )) and C1 ∈
1
L2 (Ω × J; Hper
(Y )) such that, up to subsequences not relabeled for convenience,
we have the following convergence.
Π pf * Pf weakly in L2 (J; H 1 (Ω)),
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C. CHOQUET
EJDE-2007/97
2
∇(Π pf ) *∇Pf (x, t) + ∇y Pf1 (x, y, t),
2
θ * p(x, y, t),
Π f1 * F1
2
∇θ * ∇y p(x, y, t),
2
weakly in L2 (J; H 1 (Ω)), ∇(Π f1 ) * ∇F1 (x, t) + ∇y F11 (x, y, t),
2
ξ * C1 (x, y, t),
2
∇ξ * ∇y C1 (x, y, t).
2
We also assert that Φ * Φ(y) = χf (y)φf (y) + χm (y)φ(y),
2
K * K(y) = χf (y)kf (y) + χm (y)k(y),
and that we can consider that Φ and K are admissible test functions for the twoscale convergence. We give in the following lemma some additional compactness
result.
Lemma 4.3. The sequence (Π f1 ) is sequentially compact in L2 (Ω × J).
Proof. We begin by proving that the sequence (Φ ∂t ξ ) is uniformly bounded in
L2 (J; (H 2 (Ω))0 ). Let g ∈ L2 (J; H 2 (Ω)). Equations (2.15) and (2.17) give
hΦ ∂t ξ , giL2 (J;(H 2 (Ω))0 )×L2 (J;H 2 (Ω))
Z
Z
Z
K
∇θ
·
∇ξ
)
g
−
D
∇ξ
·
∇g
+
q (fˆ1 − ξ ) g.
=
(
)
µ(ξ
Ω×J
Ω×J
Ω×J
In view of the previous lemmas, we have
Z
K
∇θ · ∇ξ ) g dx dt
(
µ(ξ
)
Ω×J
C
≤
k|K ∇θ |1/2 ∇ξ k(L2 (Ω×J))2 k|K ∇θ |1/2 kL4 (Ω×J) kgkL4 (Ω×J)
µ−
≤ CkgkL2 (J;H 2 (Ω)) ,
Z
D ∇ξ · ∇g dx dt ≤ Ck∇gk(L2 (J;L4 (Ω)))2 ≤ CkgkL2 (J;H 2 (Ω)) ,
Ω×J
Z
q (fˆ1 − ξ ) g dx dt ≤ CkgkL2 (Ω×J) .
Ω×J
Then |hΦ ∂t ξ , gi| ≤ CkgkL2 (J;H 2 (Ω)) and (φf ∂t (Π f1 )) is uniformly bounded in
L2 (J; (H 2 (Ω))0 ). A compactness argument of Aubin’s type ensures that (φf (Π f1 ))
is compact in L2 (J; (H 1 (Ω))0 ). We thus can pass to the limit in the product
hφf (Π f1 ), Π f1 iL2 (J;(H 1 (Ω))0 )×L2 (J;H 1 (Ω)) . Since φf (x) ≥ φ− > 0 almost every
where in Ω, it follows that (Π f1 ) is compact in L2 (Ω × J).
Now, we have the first tools to study the behavior of the microscopic system as
tends to zero. We begin by the pressure equations.
4.2. The Pressure Problem. We want now to pass to the limit in System (2.16),
(2.20)–(2.21), (2.22), (2.30), (2.32) recalled below.
Φ ∂t θ − div((1/µ(ξ ))K ∇θ ) = q
K ∇θ · ν = 0 on Γ × J,
in Ω × J,
θ (x, 0) = po (x)
in Ω.
As in [2], we multiply the first equation by a test function in the form Ψ(x, t) +
∞
Ψ1 (x, x/, t) + ψ(x, x/, t), with Ψ ∈ D(Ω × J), Ψ1 ∈ D(Ω × J; Cper
(Y )) and
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DERIVATION OF MODELS
17
∞
ψ ∈ D(Ω × J; Cper
(Y )) such that ψ(x, y, t) = 0 for y ∈ Yf . Integrating over Ω × J,
we obtain
Z
−
Φ (x) θ ∂t Ψ(x, t) + Ψ1 (x, x/, t) + ψ(x, x/, t) dx dt
ZΩ×J
+
Φ (x) po (x) Ψ(x, 0) + Ψ1 (x, x/, 0) + ψ(x, x/, 0) dx
Ω
Z
K (x) 1
∇θ · ∇Ψ + ∇x Ψ1 + ∇y Ψ1 + ∇x ψ + ∇y ψ dx dt
+
)
µ(ξ
ZΩ×J
=
q Ψ(x, t) + Ψ1 (x, x/, t) + ψ(x, x/, t) dx dt.
Ω×J
Letting → 0, we get
Z
Z
Z
−
φf (y) Pf ∂t Ψ(x, t) −
Ω×J
Yf
Z
φ(y) p ∂t (Ψ(x, t) + ψ(x, y, t))
Ω×J
Ym
Z Z
(χf (y)φf (y) + χm (y)φ(y)) po (Ψ(x, 0) + χm (y)ψ(x, y, 0))
Z
Z
kf (y)
(∇Pf + ∇y Pf1 ) · (∇Ψ + ∇y Ψ1 )
+
µ(F
1)
Ω×J Yf
Z
k x
+ lim
∇p · ∇y ψ(x, , t)
→0 Ω ×J µ(C1 )
m
Z
=
q (Ψ + χm (y)ψ).
+
Ω
Y
Ω×J×Y
By density, this equality holds true for any functions (Ψ, Ψ1 , ψ) ∈ Ho1 (Ω × J) ×
1
1
(Ym )). Another integration by parts shows
(Y )) × L2 (Ω × J; Hper
L2 (Ω × J; Hper
(taking successively the test functions ψ, Ψ1 and Ψ equal to zero) that it is a
variational formulation of the following two-scale homogenized system in Ω × J:
Z
1 Z
Yf
φ(y) ∂t p dy,
φf ∂t Pf − div
kf (y)(∇Pf + ∇y Pf1 )dy = q −
µ(F1 ) Yf
Ym
k (y)
f
− div
(∇Pf + ∇y Pf1 ) = 0 in Yf ,
y
µ(F1 )
k
2
φ(y) ∂t p + div V h = q in Ym , where − ∇p * V h
y
µ(C1 )
kf (y)(∇Pf + ∇y Pf1 ) · νy = 0 on Γf m ,
kf (y)(∇Pf + ∇y Pf1 ) · ν = 0
o
Pf (x, 0) = p(x, y, 0) = p (x)
on Γ,
in Ω × Ym .
Now we eliminate the function Pf1 in the former system. We use the solution
(v i )1≤i≤2 of the cell problem (4.7) and the homogenized permeability tensor K f
P2
i
defined by (4.5). Through the relation Pf1 (x, y, t) =
i=1 ∂xi Pf (x, t) v (y), we
recover the following homogenized system.
Z
K
Yf
f
φf ∂t Pf − div
φ ∂t p dy in Ω × J,
(4.13)
∇Pf = q −
µ(C1 )
Ym
φ ∂t p + div V h = q in Ω × Ym × J,
(4.14)
y
18
C. CHOQUET
EJDE-2007/97
K f ∇Pf · ν = 0 on ∂Ω × J, Pf (x, 0) = p(x, y, 0) = po (x) in Ω × Ym .
(4.15)
We now determine the nonexplicit limit V h using a dilation operator.
4.3. Introduction of an Appropriate Dilation Operator. Due to the nonlinearities and to the strong coupling of the problem, the two-scale convergence does
not provide an explicit form for the source-like terms which model the influence of
the matrix at the macroscopic level. To overcome this difficulty, we follow an idea
of [4], that can also be compared with the unfolding method of Cioranescu et al [12].
Firstly, we recall that in this section we assume no direct flow between the matrix
cells. We thus can assume that the component s of Ωm are strictly separated. For
each > 0, we then define a dilation operator e· mapping measurable functions on
Ωm × J to measurable functions on Ω × Ym × J by
u
e(x, y, t) = u(c (x) + y, t)
for y ∈ Ym , (x, t) ∈ Ω × J,
where c (x) denotes the lattice translation point of the -cell domain containing
x. This dilation allows to reach directly the scale of the standard cell. Since
Ωm = Ω(k1 ) ∩ (∪k∈Z2 (Ym + k)), we recall that c (x) = k for x ∈ (Ym + k). Thus
the function u
e is constant in x on each block (Ym + k), k ∈ Z2 , of Ω. We extend
this operator from Ym to ∪k (Ym + k) periodically. The dilation has the following
properties (cf [4]). Any function u ∈ L2 (J; H 1 (Ωm )) satisfies
ke
ukL2 (Ωm ×J×Ym ) = kukL2 (Ω×J) ,
g
∇y u
e = ∇
x u a.e. in Ω × J × Ym .
Moreover, if (v, w) ∈ (L2 (0, T ; H 1 (Ωm )))2 , then
(e
v , w)
e L2 (Ω×J×Ym ) = (v, w)L2 (Ωm ×J) ,
(e
v , w)L2 (Ω×J×Y ) = (v, w)
e L2 (Ω×J×Y ) ,
g 2 k∇y vek 2
2 = ∇x v
(L (Ω×J×Ym ))
(L (Ωm ×J))2
.
The following result makes the link between two-scale convergence and weak convergence of dilated sequences. We refer to [8] for its detailed proof.
Proposition 4.4. If (v ) is a bounded sequence of L2 (Ωm × J) such that ve converges weakly to v o in L2 (Ω × J; L2per (Ym )) and χm v two-scale converges to v, then
we have v o = v a.e. in Ω × J × Ym .
e and pe . The
We now determine the limit behavior of the dilated solutions C
1
outline of this study is the following. We find the equations satisfied by the dilated
e ) to
solutions in Ω × Ym × J. Some estimates lead to the convergence of (e
p , C
1
2
(p, C1 ) as → 0. On the other hand, for each fixed k ∈ Z , we note that the
e ) is independent of x. Thus we define correspondrestriction in (Ym + k) of (e
p , C
1
e (y, t)). We get enough compactness results to find the
ing functions (e
pk (y, t), C
1k
equations satisfied by their limit (pk , C1k ) for each k ∈ Z2 . Then, by a density
argument, the limit equations for (p, C1 ) are deduced.
e and pe .
We begin with finding the equations satisfied by the dilated solutions C
1
We define a test function ψ̂ by
(
ψ (z − c (x))/, t for z ∈ Ym + c (x)
ψ̂(x, z, t) =
0
for z 6∈ Ym + c (x),
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DERIVATION OF MODELS
19
for any function ψ ∈ L2 (J; Ho1 (Ym )). We multiply (2.20) by ψ̂ and integrate over
Ωm . Since Ωm = ∪x∈Ω (Ym + c (x)) and (Ym + c (x1 )) ∩ (Ym + c (x2 )) = ∅ for
x1 6= x2 , we get for almost every x ∈ Ωm
Z
k (z)
φ (z) ∂t p )(z, t) ψ̂(x, z, t) + 2
) ∇p (z, t) · ∇z ψ̂(x, z, t) dz
µ(C
Ym +c (x)
1
Z
=
q ψ̂(x, z, t) dz.
Ym +c (x)
With the change of variable z 7→ y + c (x) = (y + k), k ∈ Z2 , we recover the
variational formulation of the equation
φ(y) ∂t pe + div qe = qe,
e ))k(y)∇y pe .
qe = −(1/µ(C
1
y
(4.16)
We proceed in the same way for the concentration (2.17). We get
e1 + qe · ∇y C
e1 − div(D(y, qe )∇y C
e1 ) + qe C
e1 = qefˆ1 .
φ(y) ∂t C
y
(4.17)
This system is provided with the following boundary and initial conditions.
f
pe = P
f
e1 = fe
and C
1
pe (x, y, 0) = peo (x, y),
in H 1/2 (∂Ym ) for (x, t) ∈ Ω × J,
e (x, y, 0) = C
fo (x, y)
C
1
1
in Ω × Ym .
(4.18)
(4.19)
Using the basic properties of the dilation operator and the estimates of Section
3, we claim that, for subsequences not relabeled for convenience, the following
convergence take place.
e1 * C1
pe * p, C
weakly in L2 (Ω × J × Ym ),
e * ∇y C1
∇y pe * ∇y p, ∇y C
1
weakly in (L2 (Ω × J × Ym ))2 .
We then choose a fixed k ∈ Z2 . For each fixed > 0, we define the functions pek
e in Ym × J by
and C
1k
(
pe (x, y, t)/x∈(Ym +k) if k is such that (Ym + k) ∩ Ω 6= ∅,
pek (y, t) =
0
otherwise,
(
e (x, y, t)/x∈(Y +k) if k is such that (Ym + k) ∩ Ω 6= ∅,
C
1
m
e1k
C
(y, t) =
0
otherwise.
Roughly speaking, k = (k1 , k2 ) ∈ Z2 is such that (Ym + k) ∩ Ω 6= ∅ if ki < |Ω|i /,
i = 1, 2 (where |Ω|i denotes the value of the measure of Ω in the i-th direction). For
e ) is a solution of Pb. (4.16)-(4.19)
each > 0 such that (Ym + k) ∩ Ω 6= ∅, (e
pk , C
1k
2
in Ym × J. Furthermore, since any f ∈ L (Ω × J) satisfies
kfek kL2 (Ym ×J) =
1
1
kfekL2 ((Ym +k)×Ym ×J) ≤
kfekL2 (Ω×J×Ym ) ,
|Ym |
|Ym |
we have enough regularity properties to get with (4.16)-(4.19) some estimates for
e . They are analogous to those obtained for p and f in the fracture.
pek and C
1
1k
f
We claim that
e (y, t) ≤ 1 a.e. in Ym × J,
1 (Y )) ≤ C,
ke
pk kL∞ (J;L2per (Ym ))∩L2 (J;Hper
0≤C
1k
m
1/2
e1k
e1k
Dm + αt |∇y pek |
2 (Y ))0 ) ≤ C,
∇y C
+ k∂t C
kL2 (J;(Hper
m
(L2 (J×Y ))2
m
20
C. CHOQUET
EJDE-2007/97
where C is a generic constant, independent of and k. Using in particular compacte , we deduce of these latter
ness arguments of Aubin’s type for the concentration C
1k
estimates that, up to subsequences not relabeled for convenience, the following
convergence take place.
pek * pk
e1k
C
* C1k
1
weakly in L2 (J; Hper
(Ym )),
1
weakly in L2 (J; Hper
(Ym )) and a.e. in Ym × J,
1
1
for some limit functions pk ∈ L2 (J; Hper
(Ym )) and Ck ∈ L2 (J; Hper
(Ym )). We can
then easily pass to the limit → 0 in the pressure equation (4.16) satisfied by pek .
We get
φ(y) ∂t pk − div((1/µ(C1k ))k(y)∇y pk ) = q
y
in Ym × J.
(4.20)
This equation is satisfied for each k ∈ Z2 . It reminds to make the link with the limit
e ) by density arguments. On the first hand, for each k ∈ Z2 , there
(p, C1 ) of (e
p , C
1
exists (k) > 0 such that (Ym + k) ∩ Ω 6= ∅ and then pek (y, t) = pe (x, y, t)/x∈(Ym +k)
for any < (k). Since lim→0 |(Ym + k)| = 0, the subset (Ym + k) ∩ Ω tends when
→ 0 to a point {xk } ⊂ Ω. Then pk (y, t) = p(xk , y, t). In the same way, we get
C1k (y, t) = C1 (xk , y, t). On the other hand, lim→0 |Ω\((∪k∈Z2 (Ym +k))∩Ω)| = 0.
Thus the set (∪k∈Z2 {xk }) is dense in Ω. Since the functions (p(xk , y, t), C1 (xk , y, t))
satisfy (4.20) for any k ∈ Z2 , we conclude by density that p is a solution of (4.9).
To end this subsection, we add the following result of strong convergence.
Lemma 4.5. The sequence µ(C1 )−1/2 ∇p satisfies
lim kµ(C1 )−1/2 ∇p k(L2 (Ωm ×J))2 = kµ(C1 )−1/2 ∇y pk(L2 (Ω×J×Ym ))2 .
→0
Using the terminology of [2], µ(C1 )−1/2 ∇p is said strongly two-scale converging
2
to µ(C1 )−1/2 ∇y p. For any bounded sequence v of (L2 (Ω × J))2 such that v * v
and any test admissible function ψ, we can assert that
Z
lim
(µ(C1 )−1/2 ∇p · v ) ψ(x, x/, t) dx dt
→0 Ω×J
Z
Z
=
(µ(C1 )−1/2 ∇y p · v) ψ(x, y, t) dx dy dt.
Ω×J
Ym
Proof. We multiply (4.16) by pe and we integrate over Ω × (0, t) × Ym = Ωt × Ym ,
for t ∈ J. Letting → 0 and using (4.9), we write
Z
Z
d Z
1
k(y)
2
o 2
φ(y) |e
p| −
φ(y) |e
p | +
∇y pe · ∇y pe
lim
e )
→0 2dt Ω×Y
2 Ω×Ym
Ωt ×Ym µ(C
m
1
Z
Z
= lim
qe pe =
qp
→0 Ω ×Y
Ωt ×Ym
t
m
Z
Z
Z
d
1
k(y)
2
o 2
=
φ(y) |p| −
φ(y) |e
p | +
∇y p · ∇y p.
2dt Ω×Ym
2 Ω×Ym
Ωt ×Ym µ(C1 )
Bearing in mind that k is a symmetric definite positive tensor, we conclude in
particular from the letter relation that
1
1
lim ∇y pe 2
=
∇
p
2
1/2 y
e )1/2
→0 µ(C
(L (Ω×J×Ym ))2
(L (Ω×J×Ym ))2
µ(C
)
1
1
EJDE-2007/97
DERIVATION OF MODELS
21
1
1^
] =
lim
∇p
∇p
2
e )1/2
→0 µ(C
→0 µ(C )1/2
(L (Ω×J×Ym ))2
(L2 (Ω×J×Ym ))2
1
1
1
= lim ∇p .
→0 µ(C )1/2
(L2 (Ωm ×J))2
1
= lim Lemma 4.5 is proved.
4.4. Corrector for the flux function and concentration problem. Now we
study the behavior of the concentration problem (2.15)-(2.17) as tends to zero.
We recall it is
Φ ∂t ξ − (1/µ(ξ ))K ∇θ · ∇ξ − div(D ∇ξ ) + q ξ = q fˆ1 ,
D ∇ξ · ν = 0,
ξ (x, 0) = f1o (x).
The starting point of the asymptotic study is similar to the one used for the pressure
problem in Subsection 4.1. We multiply the latter equation by a test function
Ψ(x, t) + Ψ1 (x, x/, t) + ψ(x, x/, t) and we integrate over Ω × J. Letting → 0,
we obtain the limit variational formulation corresponding to (2.15)-(2.17). The
difficulty is due to the nonlinearities involving the Darcy velocity in the convective
terms and in the dispersive ones.
We begin by deriving the matrix equation (4.10). Indeed we have obtained
sufficient compactness results in Subsection 4.2 to pass to the limit via the dilated
e to C1k , we get
equation (4.17). On the one hand, using the a.e. convergence of C
1k
by density
e1 )−1/2 ∇y pe · ∇y C1
φ(y) ∂t C1 − µ(C1 )−1/2 k(y) lim µ(C
→0
e1 )−1/2 ∇y pe ∇y C1
− div µ(C1 )−1/2 D lim µ(C
y
→0
= q (fˆ1 − C1 ).
On the other hand, we note that Lemma 4.5 gives the strong convergence of
e )−1/2 ∇y pe to µ(C1 )−1/2 ∇y p in (L2 (Ω × J × Ym ))2 . It is
µ(C1^
)−1/2 ∇p = µ(C
1
sufficient to explicit all the limits in the former equation. We obtain the limit
equation (4.10).
Passing to the limit in the fractured part is less obvious. We are going to derive
a corrector for the Darcy velocity in the fracture. Our aim is to find a function
V o (x, y, t) ∈ (L2 (Ω × J × Y ))2 such that V o (x, x/, t) is an admissible test function
for the two-scale convergence and such that
lim kχf −(K /µ(ξ ))∇θ − V o (x, x/, t) k(L2 (Ω×J))2 = 0.
→0
Then we will pass to the limit in χf V o (x, x/, t) · ∇ξ instead of χf (K /µ(ξ ))∇θ ·
∇ξ ; respectively we will pass to the limit in χf D(V o (x, x/, t))∇ξ instead of
χf D(K /µ(ξ ))∇θ )∇ξ . We begin with the following result.
Lemma 4.6. The following convergence hold.
Z
K
lim
∇θ · ∇θ dx dt
→0 Ω×J µ(ξ )
Z
Z
kf (y)
=
(∇Pf + ∇y Pf1 ) · (∇Pf
Ω×J Yf µ(F1 )
22
C. CHOQUET
+ ∇y Pf1 ) dx dy dt +
Z
Ω×J
EJDE-2007/97
Z
Ym
k(y)
∇y p · ∇y p dx dy dt.
µ(C1 )
Proof. Let us consider the following energy equation for t ∈ J. We denote Ωt the
set Ω × (0, t).
Z
Z
Z
1
K
Φ (θ (x, t)2 − po (x)2 )dx +
∇θ
·
∇θ
dx
ds
=
q θ dx ds.
)
2 Ω
µ(ξ
Ωt
Ωt
In view of the two-scale convergence of θ we have
Z
Z Z
lim
q θ dx ds =
q (Pf (x, t) + χm (y)p(x, y, t)) dx dy ds,
→0
Ωt
Ωt
Y
and so, in view of the variational formulation obtained in the former subsection,
Z
1 Z
K
2
Φ θ (x, t) dx +
∇θ
·
∇θ
dx
ds
lim
→0 2 Ω
Ωt µ(ξ )
Z Z
Z Z
kf (y)
1
2
Φ(Pf + χm p) (x, y, t) dx dy +
(∇Pf + ∇y Pf1 ) · (∇Pf
=
2 Ω Y
µ(F
1)
Ωt Yf
Z Z
k(y)
+ ∇y Pf1 ) dx dyds +
∇y p · ∇y p dx dyds,
Ωt Ym µ(C1 )
where we recall that Φ(y) = χf (y)φf (y) + χm (y)φ(y). The limit of each term in the
left-hand side of the last relation is larger than the corresponding two-scale limit
in the right-hand side. Thus equality holds for each contribution and Lemma 4.6
is proved.
Now, comparing the results of Lemmas 4.5 and 4.6, we conclude that
Z
K
lim
χf
∇θ · ∇θ dx dt
→0 Ω×J
µ(ξ )
Z
Z
kf (y)
=
(∇Pf + ∇y Pf1 ) · (∇Pf + ∇y Pf1 ) dx dy dt.
µ(F
)
1
Ω×J Yf
We recall that K is a symmetric definite positive tensor and that it is considered
as an admissible test function for the two-scale convergence. Furthermore we have
showed in Lemma 4.3 that χf (Π f1 ) converges almost everywhere in Ω × J to F1 .
The latter relation then leads to
K
kf (y)
1 ∇θ
=
(y)
(∇P
+
∇
P
)
.
lim χf
χ
f
f
y
f 2
→0
µ(ξ )
µ(F1 )
(L2 (Ω×J))2
(L (Ω×J×Y ))2
This relation is sufficient to assert the following result.
Proposition 4.7. Let us define
kf (y)
(∇Pf (x, t) + ∇y Pf1 (x, y, t)).
µ(F1 (x, t))
Assume that
R V o is an admissible test function
R
R for the two-scale convergence, that
is lim→0 Ω×J |V 0 (x, x/, t)|2 dx dt ≤ Ω×J Y |V 0 (x, y, t)|2 dxdy dt. The function
V 0 (x, x/, t) is a corrector for the flux function in the following sense.
K
x lim −χf
∇θ
−
V
, t)
= 0.
(x,
0
→0
µ(ξ )
(L2 (Ω×J))2
V o (x, y, t) = −χf (y)
We then can substitute V 0 (x, x/, t) to χf (K /µ(ξ ))∇θ to pass to the limit in
the different equations. We get (4.3). This completes the proof of Theorem 4.1.
EJDE-2007/97
DERIVATION OF MODELS
23
5. The macroscopic model: the case of a partially fractured medium
(α > 0)
In the present section, we derive rigorously the homogenized model corresponding
to the partially fractured model (2.15)-(2.21), (2.23)-(2.32) when α > 0. The limit
model is described in Theorem 5.5 at the end of the section.
We begin by recalling the estimates derived in Section 3.
kχf pf kL∞ (J;L2 (Ω)) + kχf ∇pf k(L2 (Ω×J))2 ≤ C,
kχm p kL∞ (J;L2 (Ω)) + kχm α(c1 + c2 )∇p k(L2 (Ω×J))2 ≤ C,
kχf f1 kL∞ (Ω×J) + kχf (1 + |∇pf |)∇f1 k(L2 (Ω×J))2 ≤ C,
kχm C1 kL∞ (J;L2 (Ω)) + kχm (1 + |V s |)∇C1 k(L2 (Ω×J))2
kχm ci kL∞ (J;L2 (Ω)) + kχm (1 + |V s |)∇ci k(L2 (Ω×J))2 ≤ C,
≤ C,
i = 1, 2.
For the rest of this article, we assume that the pressure equation (2.20) is not
degenerate, that is the constant Cα defined in (2.34) satisfies
Cα > 0.
The more technical degenerate case when Cα = 0 is postponed to a forthcoming paper. In view of (2.39) in Theorem 2.1, we now have 0 < Cα ≤ c1 (x, t) +
c2 (x, t) ≤ 1 a.e. in Ω × J and kχm ∇p k(L2 (Ω×J))2 ≤ C. Thus there exist func1
tions p ∈ L∞ (J; L2 (Ω)) ∩ L2 (J; H 1 (Ω)), p1 ∈ L2 (Ω × J; Hper
(Y )), (f1 , C1 , c1 , c2 ) ∈
1
(Y )))4 such
(L∞ (J; L2 (Ω)) ∩ L2 (J; H 1 (Ω)))4 and (f11 , C11 , c11 , c12 ) ∈ (L2 (Ω × J; Hper
that, up to extracted subsequences, as ε → 0,
2
θ = χf pf + χm p * p,
2
χf f1 * χf (y)f1 ,
2
χm C1 * χm (y)C1 ,
2
χm ci * χm (y)ci ,
2
∇θ * ∇p(x, t) + ∇y p1 (x, y, t)),
2
χf ∇f1 * χf (y)(∇f1 (x, t) + ∇y f11 (x, y, t)),
2
χm ∇C1 * χm (y)(∇C1 (x, t) + ∇y C11 (x, y, t)),
2
χm ∇ci * χm (y)(∇ci (x, t) + ∇y c1i (x, y, t)), i = 1, 2.
We begin with the following result linking the limit concentrations f1 and m1 =
αc1 + βC1 .
Lemma 5.1. The concentrations f1 (x, t) and m1 (x, t) = αc1 (x, t) + βC1 (x, t) are
equal almost everywhere in Ω × J.
m1 =
Proof. Let c = χf f1 + χm m1 ∈ L2 (J; H 1 (Ω)). It satisfies γf c = γf f1 = γm
2
γm
c and ∇c = χf γf1 + χm m1 ∈ (L2 (Ω × J))2 . We know that c * χf (y)f1 +
2
∞
(Y )))2 we write
χm (y)m1 and ∇c *0. For any Ψ ∈ (Co∞ (Ω; Cper
Z
Z
x
x
x ∇c · Ψ(x, ) dx = −
c div Ψ(x, ) + div Ψ(x, ) dx.
x
y
Ω
Ω
We take the two-scale limits on both sides. We get
Z Z
0=−
(χf (y)f1 (x, t) + χm (y)m1 (x, t)) div Ψ(x, y) dx dy
y
ZΩ ZY
Z Z
=−
f1 (x, t)Ψ(x, s) · νf dx ds −
m1 (x, t)Ψ(x, s) · νm dx ds.
Ω
∂Yf
Ω
∂Ym
24
C. CHOQUET
EJDE-2007/97
This proves that f1 (x, t) = m1 (x, t) for s ∈ ∂Yf ∩ ∂Ym = Γf m and thus f1 (x, t) =
m1 (x, t) a.e. in Ω × J.
We then claim and prove the following compactness result.
Lemma 5.2. The sequences (χf f1 ), (χm m1 ) and (χm (c1 + c2 )) are sequentially
compact in L2 (Ω × J).
Proof. On the one hand, let ψ ∈ L2 (J; H 2 (Ω)). We multiply (2.15) by (α2 +β 2 )χf ψ
and (2.40) by χm ψ. We integrate over Ω × J and sum up the results. Using the
same type of arguments than in the proof of Lemma 4.3, we conclude that the
sequence ∂t (φf (α2 + β 2 )χf f1 + φ χm m1 ) is uniformly bounded in L2 (J; (H 2 (Ω))0 ).
Since (φf (α2 + β 2 )χf f1 + φ χm m1 ) is uniformly bounded in L∞ (Ω × J), a standard
argument of Aubin’s type proves that (φf (α2 +β 2 )χf f1 +φ χm m1 ) lies in a compact
subset of L2 (J; (H 1 (Ω))0 ). Therefore, there is ξ ∈ L2 (J; (H 1 (Ω))0 ), such that, up
to an extracted subsequence,
φf (α2 + β 2 )χf f1 + φ χm m1 → ξ
in L2 (J; (H 1 (Ω))0 ) as → 0.
Two-scale convergence arguments show that
Z
Z
ξ = (α2 + β 2 )(
φf (y)dy)f1 + (
Yf
φ(y)dy)m1 ,
Ym
where m1 = αc1 + βC1 = f1 by Lemma 5.1.
On the other hand, the sequence (χf f1 + χm m1 ) is uniformly bounded in space
2
L (J; H 1 (Ω)). We thus can pass to the limit in the product hφf (α2 + β 2 )χf f1 +
φ χm m1 , χf f1 + χm m1 i(H 1 (Ω))0 ×H 1 (Ω) as follows.
lim χf (α2 + β 2 )φf f1 + χm φ m1 , χf f1 + χf (α2 + β 2 )φf f1
→0
Z
Z
2
2
+ χm φ m1 , χm m1 = (α + β )
φf (y)dy +
φ(y)dy f1 , |Yf |f1
Yf
+
=
(α2 + β 2 )
Z
Z
(α2 + β 2 )
φ(y)dy f1 , |Ym |m1
φf (y)dy +
Yf
Ym
Z
Z
φf (y)dy +
Yf
Ym
φ(y)dy f1 , f1 .
Ym
As a consequence we have
lim (α2 + β 2 )χf φf + χm φ (χf f1 + χm m1 − f1 ), χf f1 + χm m1 − f1
→0
= lim
(α2 + β 2 )χf φf + χm φ (χf f1 + χm m1 ), χf f1 + χm m1
→0
− 2 (α2 + β 2 )χf φf + χm φ (χf f1 + χm m1 ), f1
+ (α2 + β 2 )χf φf + χm φ f1 , f1 = 0.
Since α2 + β 2 > 0, φf , φ ≥ φ− > 0, this shows that (χf f1 + χm m1 ) strongly
converges to f1 in L2 (Ω × J). A similar calculation using (2.45) gives the result for
χm (c1 + c2 ). The proof of the Lemma is complete.
We now have the tools to pass to the limit in the pressure equation. The structure
of the problem satisfied by θ is similar with the one of the pressure problem in the
fractured part in the totally fractured setting (Section 4). We thus do not detail the
EJDE-2007/97
DERIVATION OF MODELS
25
details of the convergence. We claim that the limit homogenized pressure problem
in Ω × J is the following.
Z k(y) kf (y)
(∇p + ∇y p1 ) dy = q,
φ ∂t p − div
+ χm (y)α(c1 + c2 )
χf (y)
µ(f1 )
µ(m1 )
Y
kf (y)
k(y) − div χf (y)
(∇p + ∇y p1 ) = 0 in Y,
+ χm (y)α(c1 + c2 )
y
µ(f1 )
µ(m1 )
kf (y)
k(y) (∇p + ∇y p1 ) · ν = 0 on Γ × J,
χf (y)
+ χm (y)α(c1 + c2 )
µ(f1 )
µ(m1 )
where φ is defined by
Z
Z
φ=
φ(y) dy = φf
φf (y) dy +
Yf
Yf
Ym
+φ
.
(5.1)
Ym
Let v j (x, y, t) be the Y -periodic solution of the cell-problem
− div (χf (y)kf (y) + χm (y)α(c1 + c2 )k(y))(∇y v j + ej ) = 0 in Y,
y
Z
v j dy = 0.
(5.2)
Y
Defining an homogenized tensor of permeability K = (K ij )1≤i,j≤2 in Ω by
Z
K ij (x, t) =
χf (y)kf (y) + χm (y)α(c1 + c2 )k(y)
Y
(5.3)
× (∇y v i (x, y, t) + ei )(∇y v j (x, y, t) + ej ) dy,
and setting
p1 (x, t, y) =
2
X
v j (x, y, t) ∂j p(x, t),
j=1
we claim the following result.
Proposition 5.3. The homogenized pressure problem is
1
φ ∂t p + div v = q, v = −
K∇p in Ω × J,
µ(f1 )
v · ν Γ×J = 0, pt=0 = po ,
(5.4)
(5.5)
where φ (respectively K) is the homogenized porosity (respectively permeability tensor) given in (5.1) (respectively in (5.3)).
As in Section 4, we have to introduce a corrector for the Darcy velocity in order
to pass to the limit in the concentration equation. We denote
kf (y)
k(y)
+ χm (y)α(c1 + c2 )
)(∇p + ∇y p1 ),
µ(f1 )
µ(f1 )
v εo (x, t) = v o (x, t, x/ε), (x, t) ∈ Ω × J.
v o (x, t, y) = −(χf (y)
Following the lines of Subsection 4.4, we can prove the following corrector result.
Lemma 5.4. We have, for a subsequence,
Z
|χf v f + χm V − v εo |2 dx dt → 0
Ω×J
as ε → 0.
26
C. CHOQUET
EJDE-2007/97
Let us now turn to the concentration problem. Once again, the technical difficulties are not greater than the ones of Subsection 4.4 in the fractured part. We thus
only give the outline of the convergence study. At the limit, we need to characterize
the behavior of two global concentrations, f1 = m1 and (c1 + c2 ).
We begin with f1 . We multiply (α2 + β 2 )(2.15) + (2.40) by a test function
∞
ψ(x, t) + ψ1 (x, x/, t), with ψ ∈ D(Ω × J), ψ1 ∈ D(Ω × J; Cper
(Y )). The terms on
Γf m are equal to zero when we integrate by parts. Noting that χf (α2 +β 2 )kf ∇pf +
χm k ∇p = (α2 + β 2 )(χf kf ∇pf + χm k ∇p ) + (1 − α2 − β 2 )χm k ∇p and passing
to the limit, we get the following homogenized problem in Ω × J.
(α2 + β 2 )
Ym +φ
∂t f1 −
K∇p · ∇f1
µ(f1 )
Z
χm (y)v o (x, t, y) · (∇f1 + ∇y f11 ) dy − div(D(∇p, µ(f1 ))∇f1 )
+ (1 − α2 − β 2 )
(α2 + β 2 )φf
Yf
Ym
= (α + β )|Yf | + |Ym | q (fˆ1 − f1 ),
2
2
(5.6)
D(∇p, µ(f1 ))∇f1 · ν Γ×J = 0,
f1 t=0 = f1o .
(5.7)
The function f11 and the tensor of diffusion D = (Dij )1≤i,j≤2 are defined by
f11 (x, t, y) =
2
X
wj (x, t, y) ∂j f1 ,
(5.8)
j=1
Dij (∇p, µ(f1 ))
Z
=
(α2 + β 2 )χf (y) + χm (y) D(v o ) ∇y wi (x, y) + ei · ∇y wj (x, y) + ej dy,
Y
(5.9)
where wj (x, t, y) is the Y -periodic solution of the following cell-problem for (x, t) ∈
Ω × J:
− div (α2 + β 2 )χf (y) + χm (y) D(v o )(∇y wj + ej ) = 0 in Y,
y
Z
(5.10)
wj (x, t, y) dy = 0, j = 1, 2.
Y
For the asymptotic study of (c1 + c2 ) we exploit Problem (2.45)-(2.47). It leads to
the following homogenized problem
Z
Ym
φ ∂t (c1 + c2 ) +
χm (y)v o (x, t, y) · (∇(c1 + c2 ) + ∇y (c11 + c12 )) dy
Ym
(5.11)
− div(Dm (∇p, µ(f1 ))∇(c1 + c2 ))
= |Ym |q (1 − c1 − c2 )
in Ω × J,
Dm ∇(c1 + c2 ) · ν Γ×J = 0,
(c1 + c2 )t=0 = co1 + co2 .
(5.12)
The functions c11 , c12 and the homogenized tensor of diffusion Dm = (Dmij )1≤i,j≤2
are defined by
(c11 + c12 )(x, t, y) =
2
X
j=1
j
wm
(x, t, y) ∂j (c1 + c2 ),
(5.13)
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DERIVATION OF MODELS
Z
i
j
D(v o )(∇y wm
+ ei ) · (∇y wm
+ ej ) dy,
Dmij (∇p, µ(f1 )) =
27
(5.14)
Ym
j
where wm
(x, t, y) is the Y -periodic solution of the following cell-problem for (x, t) ∈
Ω × J:
j
(x, t, y) + ej )) = 0 in Ym ,
− div(D(v o )(x, t, y)(∇y wm
y
(5.15)
j
D(v o )(∇y wm
+ ej )) · νy = 0 on Γf m , j = 1, 2.
We summarize the results of the present section in the following theorem.
Theorem 5.5. The homogenized partially fractured model in Ω × J is:
(φf
Yf
1
) ∂t p + div v = q, v = −
K∇p,
µ(f1 )
v · ν Γ×J = 0, pt=0 = po ,
Ym
+φ
p1 (x, t, y) =
2
X
v j (x, y, t) ∂j p(x, t),
j=1
v o (x, t, y) = −(χf (y)
kf (y)
k(y)
+ χm (y)α(c1 + c2 )
)(∇p + ∇y p1 ),
µ(f1 )
µ(f1 )
Ym ∂t f1 + (α2 + β 2 )v · ∇f1
+φ
Z
+ (1 − α2 − β 2 )
χm (y)v o (x, t, y) · (∇f1 + ∇y f11 ) dy − div(D(∇p, µ(f1 ))∇f1 )
(α2 + β 2 )φf
Yf
Ym
= (α + β )|Yf | + |Ym | q (fˆ1 − f1 ),
2
2
f11 (x, t, y) =
2
X
wj (x, t, y) ∂j f1 ,
j=1
D(∇p, µ(f1 ))∇f1 · ν Γ×J = 0, f1 t=0 = f1o ,
Z
Ym
φ ∂t (c1 + c2 ) +
χm (y)v o (x, t, y) · (∇(c1 + c2 ) + ∇y (c11 + c12 )) dy
Ym
− div(Dm (∇p, µ(f1 ))∇(c1 + c2 ))
= |Ym |q (1 − c1 − c2 ),
(c11 + c12 )(x, t, y) =
2
X
j
wm
(x, t, y) ∂j (c1 + c2 ),
j=1
Dm ∇(c1 + c2 ) · ν Γ×J = 0,
(c1 + c2 )t=0 = co1 + co2 .
j
The auxiliary functions v j , wj , wm
are defined by the cell problems (5.2), (5.10),
(5.15). The homogenized permeability is given in (5.3), while the diffusion tensors
D and Dm are given in (5.9) and (5.14).
Remark 5.6. The latter model is consistent with the one corresponding to a nonfractured porous medium. Indeed, letting α → 1 and |Ym | → 0, one gets
1
Y
φ ∂t p + div v = q, v = −
K∇p,
µ(f1 )
v · ν Γ×J = 0, pt=0 = po ,
28
C. CHOQUET
p1 (x, t, y) =
2
X
v j (x, y, t) ∂j p(x, t),
EJDE-2007/97
v o (x, t, y) = −
j=1
k(y)
(∇p + ∇y p1 ),
µ(f1 )
Y
φ ∂t f1 + v · ∇f1 − div(D(∇p, µ(f1 ))∇f1 ) = q (fˆ1 − f1 ),
D(∇p, µ(f1 ))∇f1 · ν = 0, f1 = f1o .
Γ×J
t=0
Note that this model is the homogenized form corresponding to the following microscopic model in a non-fractured porous medium (see [9] for a rigorous derivation):
k
∇p in Ω × J,
µ(f1 )
φ ∂t f1 + v · ∇f1 − div(D(v )∇f1 ) = q(fˆ1 − f1 ) in Ω × J,
v · ν Γ×J = 0, D(v )∇f1 · ν Γ×J = 0, pf t=0 = po , f1 t=0 = f1o .
φ ∂t p + div(v ) = q,
v = −
Remark 5.7. Let us add some “physical” comments on the homogenized model
of miscible displacement in partially fractured media described in Theorem 5.5.
Contrary to the one-component flow considered in [13], it seems that the double
porosity characteristics (that is a system of equations in the homogenized fracture
coupled with one in a microscopic matrix block) disappear as soon as a direct flow
occurs in the matrix part, that is as soon as α > 0. This corresponds to some
experimental data ([3] and references therein). However, the model of Theorem
5.5 is also not a single porosity model as the one described in Remark 5.6. It
really contains some matrix effects. We recall in particular that the smaller α is,
the more important is the storage in the matrix. And in the model of Theorem
5.5, the homogenized permeability is concentration (α(c1 + c2 )) dependent, and
the convective effects in the homogenized fracture depend on α. One can 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 [20] and the references therein) for instance.
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Catherine Choquet
Laboratoire Analyse Topologie Probabilités, CNRS UMR 6632, Université P. Cézanne,
Faculté des Sciences et Techniques de Saint-Jérôme, Case Cour A, 13397 Marseille
Cedex 20, France
E-mail address: c.choquet@univ-cezanne.fr