\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 90, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/90\hfil A double porosity model]
{Homogenization of a double porosity model in deformable media}

\author[A. Ainouz \hfil EJDE-2013/90\hfilneg]
{Abdelhamid Ainouz}  % in alphabetical order

\address{Abdelhamid Ainouz \newline
Laboratoire AMNEDP \\
Mathematics Department, USTHB\\
BP 32 ElAlia, Bab-Ezzouar, Algiers, Algeria}
\email{aainouz@usthb.dz}

\thanks{Submitted November 29, 2012. Published April 5, 2013.}
\subjclass[2000]{35B27, 74Q05, 76M50}
\keywords{Poroelasticity equations; homogenization; two-scale convergence}

\begin{abstract}
 The article addresses the homogenization of a family of micro-models
 for the flow of a slightly compressible fluid in a poroelastic matrix
 containing periodically distributed poroelastic inclusions, with
 low permeabilities and with imperfect contact on the interface.
 The micro-models are based on Biot's system for consolidation processes
 in each phase, with interfacial barrier formulation. Using the two-scale
 convergence technique, it is shown that the derived system is a general
 model of that proposed by Aifantis, plus an extra memory term.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{s1}

The interaction between fluid flow and solid deformation in porous media is
of great importance in petroleum engineering and geomechanics, biosciences,
chemical processes and many industrial applications \cite{cou, cow, wang}.

Some types of porous rocks, like aquifers and petroleum reservoir systems,
may contain fractures. It is known that flows in such media occur mainly in
the fracture region and the dominant fluid storage is in the matrix blocks.
In this situation, the reservoir possesses two porous structures, one related to
the matrix, and the other related to fractures. This notion of double
porosity/permeability has first been introduced by Barenblatt, Zheltov and
Kochina \cite{bzk} to model the flow of a slightly compressible fluid within
naturally fractured porous media. The proposed model is a system of two
partial differential equations in a two-medium description, with Darcy's law
in each phase, plus exchange terms representing the interfacial
coupling that results from the interaction, at the micro-scale, between the
two phases, see \eqref{06}-\eqref{07} below. This was derived under the main
assumption that the fluid pressure is uniformly distributed at the surface
of each phase.

Generally, fractured rock formations present at the micro-scale high degrees
of heterogeneity, and permeability is mainly determined by the size of the
pores and the connectedness of the fracture system. So any mathematical
modeling of fluid flow in such porous media must take into account the rapid
spatial variation of the phenomenological parameters. Furthermore, from the
numerical point of view, modeling of such systems at the local scale yields
a huge number of discretized equations, so computations will be fastidious
and intractable. To deal with such highly heterogeneous domains, the idea is
to replace the medium by an effective one. Homogenization techniques, like the
two-scale convergence method, have been used to rigorously derive an
effective double-porosity model for the Barenblatt, Zheltov and Kochina
(BZK) system, see for instance H. Ene and D. Polisevski \cite{ep}. However,
this model does not take into account the elastic behavior of the solid. In
fact, a rise in pore pressure of the fluid produces a dilation of the solid
mass. On the other hand, compression of the medium will increase pore
pressure. This coupled pressure-deformation was first introduced by Terzaghi
\cite{ter1} in the one-dimensional setting and gave the first soil
consolidation problem for a homogeneous elastic porous medium. Later, M. A.
Biot \cite{biot} has developed in the multidimensional setting a linear
theoretical analysis for the behavior of a fluid saturated poroelastic
medium. The model was based on macroscopic description of the
phenomenological and physical quantities where the representative volume
element is described as the superposition of a particle of fluid and a
particle of solid. Assuming that microstructures are periodically
distributed and that the pore scale is very small compared to the
macroscopic scale, a two-scale asymptotic expansion technique can be used to
rigorously justify the Biot model. The microscopic models are based on
the linear elasticity equations in the skeleton and on the Stokes equations
in the fluid with appropriate transmission conditions. For more details, we
refer the reader to the earlier work by Auriault and Sanchez-Palencia
\cite{as}.

Because of the coupling between the deformation and fluid pressure in double
porosity rocks, which must be understood in order to predict reservoir or
aquifer behavior, the concept of double porosity has been developed by
Aifantis \cite{aif}\ to model oil flow in porous elastic rocks. More
precisely,  Aifantis gave a phenomenological model for flow of a weakly
compressible fluid in a complex and heterogeneous medium where a system of
partial differential equations is given and generalizes Biot's
consolidation model by taking into account the basic physics of flow through
fractured media with interscale couplings. The proposed model reads as
follows:
\begin{gather}
-\mu \Delta \mathbf{u}-(\lambda +\mu )\nabla (\operatorname{div}\mathbf{u})+\alpha
_1\nabla p_1+\alpha _2\nabla p_2 =\mathbf{f},  \label{01}
\\
c_1\partial _{t}p_1+\alpha _1\operatorname{div}(\partial _{t}\mathbf{u})
-K_1\Delta p_1+g(p_1-p_2) = h_1,  \label{02} \\
c_1\partial _{t}p_2+\alpha _2\operatorname{div}(\partial _{t}\mathbf{u})
-K_2\Delta p_2-g(p_1-p_2)  = h_2  \label{03}
\end{gather}
where $u$ is the displacement of the medium; $\lambda $ and $\mu $ are the
dilation and shear moduli of elasticity, respectively; $p_i$ is the
pressure of the fluid in phase $(i) $; $c_i$ the
compressibility, $K_i$ the permeability and $\alpha _i$  the
Biot-Willis parameters \cite{bw2}. We note that if we let the volume of
fissures shrink to zero so that $c_2,\alpha _2,K_2,g$ become
negligible then the system  \eqref{01}-\eqref{03} reduces to the classical
Biot system with single porosity \cite{biot}:
\begin{gather}
-\mu \Delta \mathbf{u}-(\lambda +\mu )\nabla (\operatorname{div}\mathbf{u})+\alpha
_1\nabla p_1 =\mathbf{f},  \label{04} \\
c_1\partial _{t}p_1+\alpha _1\operatorname{div}(\partial _{t}\mathbf{u}
) -K_1\Delta p_1 = h_1.  \label{05}
\end{gather}
On the other hand, by neglecting the deformation effects $\lambda $, $\mu $
and $\alpha _i$ the system \eqref{01}-\eqref{03} reduces to the BZK model
\cite{bzk}:
\begin{gather}
c_1\partial _{t}p_1-K_1\Delta p_1+g(p_1-p_2) = h_1,  \label{06} \\
c_2\partial _{t}p_2-K_2\Delta p_2-g(p_1-p_2) = h_2 \label{07}
\end{gather}

Aifantis' theory of consolidation with the concept of double porosity unify
then the proposed models \eqref{04}-\eqref{05} of Biot for consolidation of
deformable porous media with single porosity and \eqref{06}-\eqref{07} of
BZK model for fluid flow through undeformable porous media with double
porosity. Note also that a mathematical justification of the Aifantis model
has been established in \cite{ain2}. More precisely, it is considered
micro-models with periodically distributed poroelastic inclusions, embedded
in an extra poroelastic matrix, with imperfect contact on the interface. The
micro-model is based on Biot's system for consolidation processes with
interfacial barrier formulation. The macro-model is then derived by means of
the two-scale convergence technique and it reads as follows:
\begin{gather}
-\operatorname{div}\sigma (\mathbf{u}) +\alpha _1\nabla p_1+\alpha
_2\nabla p_2=\mathbf{f},  \label{HP1} \\
\partial _{t}(\widetilde{c_1}p_1+\beta _1:e(\mathbf{u}
) ) -\operatorname{div}(K_1\nabla p_1) +\widetilde{g}
(p_1-p_2) =h_1  \label{HP12} \\
\partial _{t}(\widetilde{c_2}p_2+\beta _2:e(\mathbf{u}
) ) -\operatorname{div}(K_2\nabla p_2) -\widetilde{g}
(p_1-p_2) =h_2  \label{HP2}
\end{gather}
where $\sigma $, $\alpha _i,\beta _i$ and $K_i$ are some effective
tensors, $i=1,2$. See \cite{ain2} for more details. It is then worth
pointing out that the Aifantis model \eqref{01}-\eqref{03} can be seen as a
special case of the homogenized model \eqref{HP1}-\eqref{HP2}
 ($\beta_i=\alpha _i=\gamma _i\mathrm{I}_3$, $\gamma _i$ being a scalar
and $\mathrm{I}_3$ the identity matrix).

In this paper, we consider a family of microscopic models for the fluid flow
in a periodic poroelastic medium made of two constituents : the matrix and
the inclusions, where the material properties change rapidly on a small
scale characterized by a parameter $\varepsilon $ representing the
periodicity of the medium. We shall make the essential assumption that these
inclusions\ have sizes large enough compared with the sizes of pores so that
it makes sense to consider these media as poroelastic materials.

An interesting question is to investigate the limiting behavior of such
media when the flow in the inclusions presents very high frequency spatial
variations as a result of a relatively very low permeability when comparing
to the matrix permeability, since pore flow velocities in the porous matrix
can be high compared to movement through the interconnected pore spaces in
the inclusions. The main difference here from \cite{ain2} is that the coefficients
are scaled analogously to Arbogast et al  \cite{adh1}.
This leads especially to re-scale the flow\ potential in the
inclusions by $\varepsilon ^2$.
The main objective of this paper is to derive a general model from the
point of view of homogenization theory. It will be seen that the macro-model
is in some sense the limit of a family of periodic micro-models in which the
size of the periodicity approach zero. It is shown that the overall behavior
of fluid flow in such heterogeneous media with low permeability at the
micro-scale may present memory terms. It is also shown that in such
anisotropic media, with different coupling interaction properties in
different directions, the Biot-Willis parameters are, as in \cite{ain2},
matrices and no longer scalars, as it is usually considered in the
poroelasticity literature, since it is assumed there that the medium is
homogeneous and isotropic.

The paper is organized as follows. In the next section \ref{s2}, we give the
geometrical setting, the family of the periodic micro-models, and state the
main result of the paper. Section \ref{s3} is devoted to the proof of the
main result with the help of the two-scale convergence technique. 
We conclude this paper with some remarks.

\section{Setting of the micro-model and main result} \label{s2}

The aim of this section is to provide a detailed set up of the studied
microstructure problem, introduce some necessary notations, basic
mathematical tools as well as the notion of two-scale convergence, auxiliary
problems, and then formulate the main result of the paper.

We consider $\Omega $ a bounded and smooth domain in $\mathbb{R}^3$,
$\varepsilon >0$ a sufficiently small parameter ($\varepsilon \ll 1$) and
$Y=]0,1[^3$\ the generic cell of periodicity. We assume that $Y$ is
divided as $Y=Y_1\cup Y_2\cup \Gamma $\ where $Y_1,$\ $Y_2$\ are two
connected open subsets of $Y$ and $\Gamma $ is a smooth surface that
separates them. They are such that
\begin{equation*}
\overline{Y_2}\subset Y,\quad
Y_1\cap Y_2=\emptyset,\quad \Gamma =\overline{Y_1}\cap \overline{Y_2}=\partial
Y_2, \quad 
 \partial Y_1=\Gamma \cup \partial Y.
\end{equation*}
 We denote $\ \mathbf{n}
=(n_i) _{1\leq i\leq 3}$ the unit normal vector on $\partial Y_1$
pointing outward with respect to $Y_1$. Let $\chi _1,\chi _2$ denote
respectively the characteristic function of $Y_1$, $Y_2$ extended by $Y$
-periodicity to $\mathbb{R}^3$. Denote for $x\in \Omega $, $\chi
_i^{\varepsilon }(x)=\chi _i(x/\varepsilon )$ and set
\begin{equation*}
\Omega _i^{\varepsilon }=\{x\in \Omega :\chi _i^{\varepsilon }(x)=1\}\quad
\text{and}\quad
\Gamma ^{\varepsilon }=\overline{\Omega _1^{\varepsilon }}
\cap \overline{\Omega _2^{\varepsilon }}.
\end{equation*}
Let $Z_i=\cup _{e\in \mathbb{Z}^3}(Y_i+e) $, $i=1,2$. As
in \cite{all}, we shall assume that\ the subset $Z_1$ is smooth and
connected open subset of $\mathbb{R}^3$.

With the above assumptions, the material occupying the domain
$\Omega _2^{\varepsilon }$\ is then embedded in the material occupying
$\Omega _1^{\varepsilon }$, and the interface $\Gamma ^{\varepsilon }$ is the
boundary of $\Omega _2^{\varepsilon}$. We observe that the boundary
of $\Omega _1^{\varepsilon }$ consists of two parts the outer boundary
$\partial \Omega $\ and $\Gamma ^{\varepsilon }$. Usually, the
region $\Omega _1^{\varepsilon }$ is referred to as the matrix
while the region $\Omega _2^{\varepsilon }$ is the inclusions. Note
that no connectedness assumption is made on the material part $\Omega
_2^{\varepsilon }$.

Let $T>0$ and $t\in [ 0,T] $ denote the time variable. We set the
space-time domains $Q=(0,T) \times \Omega $,
$\Sigma =(0,T) \times \Gamma $,
$Q_i^{\varepsilon }=(0,T) \times \Omega _i^{\varepsilon }$, and
$\Sigma ^{\varepsilon }=(0,T)\times \Gamma ^{\varepsilon }$.

Let us assume that each phase ($\Omega _1^{\varepsilon }$, $\Omega
_2^{\varepsilon }$) is occupied by a porous and deformable material
through which a slightly compressible and viscous fluid flow diffuses.
Let $\mathbf{u}_i^{\varepsilon }$ denote the displacement of the medium
$\Omega _i^{\varepsilon }$, $i=1,2$. The equation of motion in
$\Omega _1^{\varepsilon }\cup \Omega _2^{\varepsilon }$ is given by
\begin{gather}
-\operatorname{div}\sigma _1^{\varepsilon } = \mathbf{f}_1,\quad \text{in }\Omega
_1^{\varepsilon },  \label{3} \\
-\operatorname{div}\sigma _2^{\varepsilon } = \mathbf{f}_2,\quad \text{in }\Omega
_2^{\varepsilon }  \label{3-0}
\end{gather}
where $\sigma _i^{\varepsilon }$ is the stress tensor which satisfies a
constitutive equation of linear poroelasticity of the form \cite{cou}:
\begin{equation}
\sigma _i^{\varepsilon }=\mathbb{A}_i^{\varepsilon }\mathrm{e}(
\mathbf{u}_1^{\varepsilon }) -\alpha _i^{\varepsilon
}p_i^{\varepsilon }\mathrm{I}_3,\quad \text{in }\Omega _i^{\varepsilon }
\label{3-1}
\end{equation}
and $\mathbf{f}_{i\,}\in L^2(\Omega ) ^3$ is the volume
distributed force in the corresponding medium, $i=1,2$. It is assumed that $
\mathbf{f}_{i\,}$ is independent of $\varepsilon $. In \eqref{3-1},
$\mathbb{A}_1^{\varepsilon }$ and $\mathbb{A}_2^{\varepsilon }$ are fourth rank
elasticity tensors, $\mathrm{e}(\cdot) $ is the
linearized strain tensor, $\mathrm{I}_3$ is the identity matrix, $
p_i^{\varepsilon }$ is the pressure and $\alpha _i^{\varepsilon }$ is
the Biot-Willis parameter in the poroelastic material $\Omega
_i^{\varepsilon }$ \cite{bw2}.

Let $c_1^{\varepsilon }$\ (resp. $c_2^{\varepsilon }$) and $
K_1^{\varepsilon }$ (resp. $K_2^{\varepsilon }$) denote respectively the
porosity and the permeability of the medium $\Omega _1^{\varepsilon }$
(resp. $\Omega _2^{\varepsilon }$). The equation for mass conservation in
each phase reads as follows:
\begin{gather}
\partial _{t}(c_1^{\varepsilon }p_1^{\varepsilon }+\alpha
_1^{\varepsilon }\operatorname{div}\mathbf{u}_1^{\varepsilon }) -\mathrm{
div}(K_1^{\varepsilon }\nabla p_1^{\varepsilon }) =0 \quad
\text{in }\Omega _1^{\varepsilon },  \label{4} \\
\partial _{t}(c_2^{\varepsilon }p_2^{\varepsilon }+\alpha
_2^{\varepsilon }\operatorname{div}\mathbf{u}_2^{\varepsilon }) -\mathrm{
div}(K_2^{\varepsilon }\nabla p_2^{\varepsilon }) =0\quad
\text{in }\Omega _2^{\varepsilon }.  \label{5}
\end{gather}
On the interface $\Gamma ^{\varepsilon }$, we associate to
\eqref{3}-\eqref{3-0} the following transmission conditions:
\begin{equation}
\mathbf{u}_1^{\varepsilon }=\mathbf{u}_2^{\varepsilon },\quad
\sigma_1^{\varepsilon }\cdot \mathbf{n}^{\varepsilon }
=\sigma _2^{\varepsilon}\cdot \mathbf{n}^{\varepsilon }  \label{7}
\end{equation}
and to \eqref{4}-\eqref{5} the well-known open-pore conditions:
\begin{equation}
(K_1^{\varepsilon }\nabla p_1^{\varepsilon }) \cdot \mathbf{n
}^{\varepsilon }=(K_2^{\varepsilon }\nabla p_2^{\varepsilon
}) \cdot \mathbf{n}^{\varepsilon },\ (K_1^{\varepsilon }\nabla
p_1^{\varepsilon }) \cdot \mathbf{n}^{\varepsilon }=-g^{\varepsilon
}(p_1^{\varepsilon }-p_2^{\varepsilon }) .  \label{9}
\end{equation}
where $\mathbf{n}^{\varepsilon }$ stands for the unit normal vector on
$\Gamma ^{\varepsilon }$ pointing outward with respect to
$\Omega _1^{\varepsilon }$, and $g^{\varepsilon }$ is the hydraulic permeability
of the thin layer $\Gamma ^{\varepsilon }$. Taking the limit on the
thickness of the thin layer, one can prove that
these conditions are the only ones that are fully consistent with the
validity of the poroelasticity's equations throughout heterogeneous media
presenting interfaces across which the pressure is discontinuous, see
\cite{gur}. Observe that when $g^{\varepsilon }=\infty $, \eqref{9} reduces to
the standard transmission condition, that is a perfect hydraulic contact on
the interface, and when $g^{\varepsilon }=0$, condition \eqref{9} implies no
flux exchange. Here, in this paper we shall
assume that neither of these conditions is fulfilled. See assumption (H4)
below.

On the exterior boundary $\partial \Omega \backslash \Gamma ^{\varepsilon }$,
we assume the homogeneous Dirichlet boundary conditions:
\begin{equation}
\mathbf{u}_1^{\varepsilon }=\mathbf{0}\quad \text{and}\quad
p_1^{\varepsilon }=0.  \label{10}
\end{equation}

Finally, the system \eqref{4}-\eqref{10} is supplemented by the
initial conditions
\begin{gather}
\mathbf{u}_1^{\varepsilon }(0,\cdot ) =\mathbf{0},\quad
p_1^{\varepsilon }(0,\cdot ) =0\quad \text{in }\Omega
_1^{\varepsilon },  \label{11} \\
\mathbf{u}_2^{\varepsilon }(0,\cdot ) =\mathbf{0},\quad
p_2^{\varepsilon }(0,\cdot ) =0\quad \text{in }\Omega
_2^{\varepsilon }.  \label{12}
\end{gather}

\begin{remark} \rm
The initial conditions \eqref{11}-\eqref{12} are already considered
in the literature, see for e.g. \cite{bms}. Actually, they are stronger
than those studied, for example, by R. E. Showalter \cite{show}.
In fact, we do not need to specify the initial values for the displacements
and the pressures but merely the combinations:
$(c_i^{\varepsilon }p_i^{\varepsilon }+\alpha
_i^{\varepsilon }\operatorname{div}\mathbf{u}_i^{\varepsilon }) $. For
example, we could  impose the following conditions:
\begin{equation}\label{ic}
\lim_{t\to 0^{+}}(c_i^{\varepsilon }p_i^{\varepsilon
}(t)+\alpha _i^{\varepsilon }\operatorname{div}\mathbf{u}_i^{\varepsilon }(t))
=v_i\quad \text{in }L^2(\Omega _i^{\varepsilon }) .
\end{equation}
See \cite{show} for full details. Nevertheless, the choice of the
inhomogeneous initial conditions is rather for technical reasons,
and it is convenient for our purpose. See for e.g. \cite{ain2}.
\end{remark}

To deal with periodic homogenization with microstructures, we shall assume
the following:
\begin{itemize}
\item[(H1) ] There exists $Y$-periodic, fourth rank tensor-valued functions
$\mathbb{A}_i(y) $, $i=1,2$ and continuous on $\mathbb{R}^3$
such that
\begin{gather*}
\mathbb{A}_i^{\varepsilon }(x) =\mathbb{A}_i(\frac{x}{
\varepsilon }) ,\quad x\in \Omega , \\
(\mathbb{A}_i(y) \Xi :\Xi ) \geq C(\Xi :\Xi) .
\end{gather*}
for all $y\in \mathbb{R}^3$ and
$\Xi \in \mathcal{M}_{\mathrm{sym}}^{3\times 3}(\mathbb{R}) $;

\item[(H2)] There exist $Y$-periodic real-valued functions
$c_i(y) $, $i=1,2$ and continuous on $\mathbb{R}^3$ such that
\begin{equation*}
c_i^{\varepsilon }(x) =c_i(\frac{x}{\varepsilon }) ,\quad x\in \Omega
\end{equation*}
and $c_i(y) \geq C>0$ for all $y\in \mathbb{R}^3$;

\item[(H3)] There exist $Y$-periodic matrix-valued functions
$K_i(y) $, $i=1,2$, continuous on $\mathbb{R}^3$ such that
\begin{equation}
K_1^{\varepsilon }(x) =K_1(\frac{x}{\varepsilon }) ,\quad
K_2^{\varepsilon }(x) =\varepsilon ^2K_2(\frac{x}{\varepsilon }) ,\quad
 x\in \Omega \label{sc1}
\end{equation}
and
$\langle K_i\xi ,\xi \rangle \geq C| \xi |^2$, $i=1,2$
for all $y\in \mathbb{R}^3$ and $\xi \in \mathbb{R}^3$;

\item[(H4)] There exists a function
$g\in \mathcal{C}(\mathbb{R}^3)$, $Y$-periodic such that
\begin{equation*}
g^{\varepsilon }(x) =\varepsilon g(x/\varepsilon ),\quad
x\in \mathbb{R}^3\quad\text{and}\quad
\inf_{y\in \mathbb{R}^3}g(y) \geq C>0.
\end{equation*}

\item[(H5)] The Biot-Willis parameter $\alpha _i^{\varepsilon }$ is
defined a.e. in $\Omega $ as follows:
\begin{equation}
\alpha _1^{\varepsilon }(x) =\alpha _1\text{ for }x\in
\Omega _1^{\varepsilon },\quad 
\alpha _2^{\varepsilon }(x) =\varepsilon \alpha _2\quad \text{for }
x\in \Omega _2^{\varepsilon }
\label{sc2}
\end{equation}
where $\alpha _i$ is a positive constant, $i=1,2$.
\end{itemize}

Here and throughout this paper, the quantity $C$ denotes various positive
constants independent of $\varepsilon >0$, of the subscript $i=1,2$ and the
microscopic variable $y\in \mathbb{R}^3$.

\begin{remark}\rm
We have chosen a particular scaling of the permeability coefficients in
\eqref{sc1}. This means that the permeability is much larger in the
network of inclusions than in the porous matrix. This gives that
both terms $\int_{\Omega _1^{\varepsilon }}| \nabla p_1^{\varepsilon }| ^2dx$
and
$\varepsilon ^2\int_{\Omega_2^{\varepsilon }}| \nabla p_2^{\varepsilon }| ^2dx$
have the same order of magnitude and thus leading to a balance in potential
energies. For more details, we refer the reader to Arbogast, Douglas, and
Hornung \cite{adh1} (see also Allaire \cite{all}). In the same way, we also
have taken a special scaling factor of the Biot-Willis parameters in
\eqref{sc2}.
\end{remark}

To set the mathematical framework of our Problem, we  introduce the
following spaces:
\begin{gather*}
\mathbf{H}=H_{0}^{1}(\Omega ) ^3,\quad
L^{\varepsilon }=L^2(\Omega _1^{\varepsilon })
 \times L^2(\Omega _2^{\varepsilon }) , \\
\mathcal{E}_1^{\varepsilon }=\{ q\in H^{1}(\Omega
_1^{\varepsilon }) ;\ q_{|\Gamma }=0\} ,\quad
 \mathcal{E} _2^{\varepsilon }=H^{1}(\Omega _2^{\varepsilon }) ,\quad
\mathcal{E}^{\varepsilon }=\mathcal{E}_1^{\varepsilon }\times \mathcal{E}
_2^{\varepsilon }.
\end{gather*}
The space $\mathbf{H}$ is equipped with the standard norm:
$\|\mathbf{v}\|_{\mathbf{H}}=\|\mathrm{e}(\mathbf{v}) \|_{L^2(\Omega
) ^{3\times 3}}$ and $\mathcal{E}^{\varepsilon }$ with
\begin{equation*}
\| (q_1,q_2) \| _{\mathcal{E}^{\varepsilon}}^2
=\| \nabla q_1\| _{L^2(\Omega_1^{\varepsilon }) }^2
+\varepsilon ^2\| \nabla q_2\| _{L^2(\Omega _2^{\varepsilon })}^2
+\varepsilon \| q_1-q_2\| _{L^2(\Gamma ^{\varepsilon })}^2.
\end{equation*}
See  Monsurr\`{o} \cite{mons}. For a.e. $(t,x) \in Q$, we denote
\begin{gather*}
\mathbf{u}^{\varepsilon }(t,x)
=\chi _1^{\varepsilon }(x) \mathbf{u}_1^{\varepsilon }(t,x) +\chi
_2^{\varepsilon }(x) \mathbf{u}_2^{\varepsilon }(t,x) , \\
 \mathbb{A}^{\varepsilon }(x)
= \chi _1^{\varepsilon }(x) \mathbb{A}_1^{\varepsilon }(x) +\chi
_2^{\varepsilon }(x) \mathbb{A}_2^{\varepsilon }(x) , \\
\mathbf{f}^{\varepsilon }(x)
= \chi _1^{\varepsilon }( x) \mathbf{f}_1(x)
+\chi _2^{\varepsilon }( x) \mathbf{f}_2(x) .
\end{gather*}
Note that, thanks to the transmission condition \eqref{7},
the displacement $\mathbf{u}^{\varepsilon }(t,\cdot ) $ lies in
$\mathbf{H}$ for a.e. $t\in (0,T) $.

Throughout this article, the following notation will be used:
if $\mathcal{F}$ is any Banach space then $L_{T}^{p}(\mathcal{F}) $
denotes the vector-valued functions space defined by
$L_{T}^{p}(\mathcal{F}) =L^{p}(0,T;\mathcal{F}) $

The weak formulation of \eqref{4}-\eqref{12} can be read as follows:
Find $\ (\mathbf{u}^{\varepsilon },p^{\varepsilon }) \in
L_{T}^{\infty }(\mathbf{H}) \times L_{T}^2(\mathcal{E}
^{\varepsilon }) $, such that
$p^{\varepsilon }=(p_1^{\varepsilon },p_2^{\varepsilon }) \in L_{T}^{\infty }(
L^{\varepsilon })$,
\begin{equation*}
\partial _{t}(c_1^{\varepsilon }p_1^{\varepsilon }+\alpha _1
\operatorname{div}\mathbf{u}^{\varepsilon }) \in L_{T}^2(\mathcal{E}
_1^{\varepsilon }{}^{\ast }) ,\partial _{t}(c_2^{\varepsilon
}p_2^{\varepsilon }+\varepsilon \alpha _2\operatorname{div}\mathbf{u}
^{\varepsilon }) \in L_{T}^2(\mathcal{E}_2^{\varepsilon
}{}^{\ast })
\end{equation*}
and for all $\mathbf{v}\in \mathbf{H}$,
 $(q_1,q_2) \in \mathcal{E}^{\varepsilon }$, we have
\begin{gather}
\int_{\Omega }\mathbb{A}^{\varepsilon }\mathrm{e}(\mathbf{u}
^{\varepsilon }) \mathrm{e}(\mathbf{v}) dx+\int_{\Omega
_1^{\varepsilon }}\alpha _1\nabla p_1^{\varepsilon }\mathbf{v}
dx+\int_{\Omega _2^{\varepsilon }}\alpha _2^{\varepsilon }\nabla
p_2^{\varepsilon }\mathbf{v}dx=\int_{\Omega }\mathbf{f}^{\varepsilon }
\mathbf{v}dx,  \label{23}
\\
\begin{aligned}
&\langle \partial _{t}(c_1^{\varepsilon }p_1^{\varepsilon
}+\alpha _1\operatorname{div}\mathbf{u}^{\varepsilon })
,q_1\rangle _{\mathcal{E}_1^{\varepsilon }{}^{\ast },\mathcal{E}
_1^{\varepsilon }}+\int_{\Omega _1^{\varepsilon }}K_1^{\varepsilon
}\nabla p_1^{\varepsilon }\nabla q_1dx \\
&+ \langle \partial _{t}(c_2^{\varepsilon }p_2^{\varepsilon
}+\varepsilon \alpha _2\operatorname{div}\mathbf{u}^{\varepsilon })
,q_2\rangle _{\mathcal{E}_2^{\varepsilon }{}^{\ast },\mathcal{E}
_2^{\varepsilon }}+\int_{\Omega _2^{\varepsilon }}K_2^{\varepsilon
}\nabla p_2^{\varepsilon }\nabla q_2dx\\
&+  \int_{\Gamma ^{\varepsilon }}g^{\varepsilon }(p_1^{\varepsilon
}-p_2^{\varepsilon }) (q_1-q_2) ds^{\varepsilon
}(x) =0,
\end{aligned} \label{24} \\
\mathbf{u}^{\varepsilon }(0,\cdot ) =\mathbf{0}\text{,\ }\chi
_1(\cdot ) p_1^{\varepsilon }(0,\cdot ) +\chi
_2(\cdot ) p_2^{\varepsilon }(0,\cdot ) =0\text{
a.e. in }\Omega .  \label{241}
\end{gather}
Here and throughout this paper $dx$ and $ds^{\varepsilon }(x) $ stand
respectively for the Lebesgue measure on $\mathbb{R}^3$ and the Hausdorff
measure on $\Gamma ^{\varepsilon }$.

Using assumptions (H1)--(H5), we  establish the following existence and
uniqueness result whose proof is a slight modification of that given by
Showalter and  Momken \cite{showmomk} and therefore will be omitted.

\begin{theorem}
Assume that {\rm (H1)--(H5)} hold. Then, for any sufficiently small
$\varepsilon >0 $ and
$\mathbf{f}^{\varepsilon }\in \mathbf{L}^2(\Omega ) $, there exists
a unique couple
$(\mathbf{u}^{\varepsilon},p^{\varepsilon }) \in L_{T}^{\infty }(\mathbf{H})
\times L_{T}^2(\mathcal{E}^{\varepsilon }) $, solution of the weak
system \eqref{23}-\eqref{241}, such that
\begin{equation}
\| \mathbf{u}^{\varepsilon }\| _{L_{T}^{\infty }(\mathbf{H}
) }+\| p^{\varepsilon }\| _{L_{T}^2(\mathcal{E}
^{\varepsilon }) }+\| p^{\varepsilon }\| _{L_{T}^{\infty }(
L^{\varepsilon }) }\leq C.  \label{30}
\end{equation}
\end{theorem}

Now, thanks to the a priori estimates \eqref{30}, one is led to study the
limiting behavior of\ the sequence $(\mathbf{u}^{\varepsilon
},p^{\varepsilon }) $ as $\varepsilon $ approaches $0$. To do this, we
shall use the two-scale convergence technique that we shall recall hereafter.

First, we define $\mathcal{C}_{\#}(Y)$ to be the space of all continuous functions on $
\mathbb{R}^3$ which are $Y$-periodic. Let the space $L_{\#}^2(
Y) $ (resp. $L_{\#}^2(Y_i) $, $i=1,2$) to be all
functions belonging to $L_{\mathrm{loc}}^2(\mathbb{R}^3) $
(resp. $L_{\mathrm{loc}}^2(Z_i) $) which are $Y$-periodic,
and $H_{\#}^{1}(Y) $ (resp. $H_{\#}^{1}(Y_i) $) to
be the space of those functions together with their derivatives belonging to
$L_{\#}^2(Y) $ (resp. $L_{\#}^2(Z_i) $).

Now, we recall the definition and main results concerning the method of
two-scale convergence. For more details, we refer the reader to
\cite{all,adh, ngue}.

\begin{definition} \label{def1} \rm
A sequence $(v^{\varepsilon }) \ $in $L^2(\Omega ) $ two-scale converges
to $v\in L^2(\Omega \times Y) $ (we write
$v^{\varepsilon }\overset{2-s}{\rightharpoonup }v$) if,
for any admissible test function
$\varphi \in L^2(\Omega ;\mathcal{C}_{\#}(Y)) $,
\begin{equation*}
\lim_{\varepsilon \to 0}\int_{\Omega }v^{\varepsilon }(
x) \varphi (x,\frac{x}{\varepsilon }) dx
=\int_{\Omega \times Y}v(x,y) \varphi (x,y) \,dx\,dy.
\end{equation*}
\end{definition}

\begin{theorem}\label{t1}
Let $(v^{\varepsilon })$ be a sequence of functions in
$ L^2(\Omega )$ which is uniformly bounded. Then, there exist
$v\in L^2(\Omega \times Y)$ and a subsequence of $(v^{\varepsilon })$ which
two-scale converges to $v$.
\end{theorem}

\begin{theorem}\label{t2}
Let $(v^{\varepsilon })$ be a uniformly bounded sequence in
$H^{1}(\Omega )$ (resp. $H_{0}^{1}(\Omega )$). Then there exist
$v\in H^{1}(\Omega ) $ (resp. $H_{0}^{1}(\Omega )$) and
$\hat{v}\in L^2(\Omega ;H_{\#}^{1}(Y)/\mathbb{R})$ such that, up to a subsequence,
\begin{equation*}
v^{\varepsilon }\overset{2-s}{\rightharpoonup }v;\quad
\nabla v^{\varepsilon }\overset{2-s}{\rightharpoonup }\nabla v
+\nabla _{y}\hat{v}.
\end{equation*}
\end{theorem}

Here and in the sequel the subscript $y$ on a differential operator denotes
the derivative with respect to $y$.

\begin{theorem} \label{t3}
Let $(v^{\varepsilon })$ be a sequence of functions in $ H^{1}(\Omega )$ such that
\begin{equation*}
\| v^{\varepsilon }\| _{L^2(\Omega )
}+\varepsilon \| \nabla v^{\varepsilon }\| _{L^2(
\Omega ) ^3}\leq C.
\end{equation*}
Then, there exist $v\in L^2(\Omega ;H_{\#}^{1}(Y)) $ and a
subsequence of $(v^{\varepsilon }) $, still denoted by $(
v^{\varepsilon }) $ such that
\begin{equation*}
v^{\varepsilon }\overset{2-s}{\rightharpoonup }v,\quad
\varepsilon \nabla v^{\varepsilon }\overset{2-s}{\rightharpoonup }\nabla _{y}v
\end{equation*}
and for every $\varphi \in \mathcal{D}(\Omega ;\mathcal{C}_{\#}(Y)) $,
we have
\begin{equation*}
\lim_{\varepsilon \to 0}\int_{\Gamma ^{\varepsilon }}v^{\varepsilon
}(x) \varphi (x,\frac{x}{\varepsilon })
ds^{\varepsilon }(x) =\int_{\Omega \times \Gamma }v(
x,y) \varphi (x,y) \,dx\,ds(y) .
\end{equation*}
Here and in the sequel $ds(y) $ denotes the Hausdorff measure on
$\Gamma $.
\end{theorem}

The notion of two-scale convergence can easily be generalized to
time-dependent functions without affecting the results stated above.
According to \cite{cs}, we have the following:

\begin{definition}
\label{def2}We say that a sequence $(v^{\varepsilon }) $ in
$L^2(Q) $ two-scale converges to
$v\in L^2(Q\times Y) $ (we always write
$v^{\varepsilon }\overset{2-s}{\rightharpoonup }v $) if, for any
$\varphi \in L^2(Q;\mathcal{C}_{\#}(Y)) $:
\begin{equation*}
\lim_{\varepsilon \to 0}\int_{Q}v^{\varepsilon }(t,x)
\varphi (t,x,\frac{x}{\varepsilon }) \,dt\,dx
=\int_{Q\times Y}v(t,x,y) \varphi (t,x,y) dt\,dx\,dy.
\end{equation*}
\end{definition}

\begin{remark} \label{rem1} \rm
The results stated above still hold for the case of
time-dependent sequences. For if $(v^{\varepsilon }) $ is a
uniformly bounded sequence in $L^2(Q) $, there exists then $
v\in L^2(Q) $ such that, up to a subsequence, $v^{\varepsilon }
\overset{2-s}{\rightharpoonup }v$ in the sense of Definition \ref{def2}.
Moreover, if $(v^{\varepsilon }) $ is uniformly bounded in
$L_{T}^2(H^{1}(\Omega ) ) $, then up to a
subsequence, there exist $v\in L_{T}^2(H^{1}(\Omega )) $ and
$v_{0}\in L^2(Q;H_{\#}^{1}(Y) /\mathbb{R}) $ such that
$v^{\varepsilon }\overset{2-s}{\rightharpoonup }v$ and
$ \nabla v^{\varepsilon }\overset{2-s}{\rightharpoonup }\nabla v+\nabla
_{y}v_{0}$. On the other hand, if a sequence $(v^{\varepsilon })$ is such that
\begin{equation*}
\| v^{\varepsilon }\| _{L^2(Q) }+\varepsilon
\| \nabla v^{\varepsilon }\| _{L^2(Q) }\leq C,
\end{equation*}
then, up to a subsequence, there exists
$v\in L_{T}^2(H_{\#}^{1}(Y) ) $ such that
$v^{\varepsilon }\overset{2-s}{\rightharpoonup }v$ and
$\varepsilon \nabla _{y}v^{\varepsilon }\overset{2-s}{\rightharpoonup }
\nabla _{y}v$.
\end{remark}

To state the main result, we introduce the  following three auxiliary problems.
For $j,k\in \{ 1,2,3\} $, let $\mathbf{w}^{jk}\in (
H_{\#}^{1}(Y) /\mathbb{R}) ^3$ be the solution to the
following microscopic system:
\begin{gather*}
-\operatorname{div}_{y}(\mathbb{A}_1\mathrm{e}_{y}(\mathbf{w}^{jk}+
\mathbf{d}^{jk}) ) =0\quad \text{a.e. in }Y_1, \\
-\operatorname{div}_{y}(\mathbb{A}_2\mathrm{e}_{y}(\mathbf{w}^{jk}+
\mathbf{d}^{jk}) ) =0\quad \text{a.e. in }Y_2, \\
\mathbb{A}_1\mathrm{e}_{y}(\mathbf{w}^{jk}+\mathbf{d}^{jk})
\cdot \mathbf{n}=\mathbb{A}_2\mathrm{e}_{y}(\mathbf{w}^{jk}+\mathbf{d
}^{jk}) \cdot \mathbf{n}\quad \text{a.e. on }\Gamma , \\
\mathbb{A}_1\mathrm{e}
_{y}(\mathbf{w}^{jk}+\mathbf{d}^{jk}) \cdot \mathbf{n} \quad
\text{is $Y$-periodic}
\end{gather*}
where $\mathbf{d}^{jk}(y) =(y_{j}\delta _{lk})
_{1\leq l\leq 3}$ and $(\delta _{kj}) $ is the Kr\"{o}necker
symbol. For $j=1,2,3$, let $\pi _{j}\in H^{1}(Y_1) /\mathbb{R}$
be the solution of the following stationary micro-pressure equation:
\begin{gather*}
-\operatorname{div}_{y}(K_1(\nabla \pi _{j}+e_{j}) ) =0
\quad\text{in }Y_1, \\
K_1(\nabla \pi _{j}+e_{j}) \cdot \mathbf{n}=0\quad \text{on }\Gamma
, \\
y\mapsto \pi _{j} \quad\text{is $Y$-periodic}
\end{gather*}
where $e_{j}$ is the $j^{\text{th}}$ vector of the canonical basis of
$\mathbb{R}^3$. Let $\zeta \in L_{T}^{\infty }(H_{\#}^{1}(Y_2) ) $
be the unique solution to the following non
micro-pressure problem of the Robin type:
\begin{gather*}
\partial _{t}(c_2\zeta ) -\operatorname{div}_{y}(K_2\nabla
_{y}\zeta ) = 0 \quad \text{ a.e. in }(0,T) \times Y_2, \\
K_2\nabla _{y}\zeta \cdot \mathbf{n}= -g(y) [ 1-\zeta ] \quad\text{ a.e. on }\Sigma ,
\\
y\mapsto \zeta \quad \text{is $Y$-periodic,} \\
\zeta (0,y) =0, \quad\text{a.e. }y\in Y_2.
\end{gather*}
Now, let us define the homogenized fourth rank tensor
$\widetilde{\mathbb{A}}=(\tilde{a}_{j_1j_2j_3j_{4}}) _{1\leq
j_1,j_2,j_3,j_{4}\leq 3}$, where the coefficients are given by
\begin{equation*}
\tilde{a}_{j_1j_2j_3j_{4}}=\sum_{k_1,k_2=1}^3
\int_{Y}a_{j_1j_2k_1k_2}(y) (\delta
_{j_1k_1}\delta _{j_2k_2}+\mathrm{e}_{k_1k_2,y}(\mathbf{w}
^{j_3j_{4}}) (y) ) dy.
\end{equation*}
Here $(a_{jklm}) $ are the coefficients of the elasticity tensor
$\mathbb{A}$ which are given by
\begin{equation}
\mathbb{A}(y) =\chi _1(y) \mathbb{A}_1(y) +\chi _2(y) \mathbb{A}_2(y)  \label{n0}
\end{equation}
for a.e. $y\in Y$, and $\mathrm{e}_{jk,y}(\cdot) $ is
the linearized elasticity strain tensor where the derivatives are taken with
respect to the microscopic variable $y$. We also define the following
homogenized tensors:
\begin{equation}
\tilde{\sigma}(\mathbf{u})=(\tilde{\sigma}_{jk}(\mathbf{u})) ,\quad
\tilde{K}=(\tilde{K}_{jk}) ,\ B=(b_{jk}) ,\quad
\Lambda =(\lambda _{jk})  \label{n1}
\end{equation}
where for $j,k\in \{ 1,2,3\} $,
\begin{gather}
\tilde{\sigma}_{jk}(\mathbf{u})=\sum_{l,m=1}^3\tilde{a}_{jklm}\mathrm{e}
_{lm}(\mathbf{u}),  \label{n2} \\
\tilde{K}_{jk}=\int_{Y_1}K_1(y) (\nabla _{y}\pi
_{j}+e_{j}) (\nabla \pi _{k}+e_{k}) dy,  \label{n3} \\
b_{jk}=\alpha _1(| Y_1| \delta_{jk}+\int_{\Gamma }\pi _{k}(y) n_{j}ds(y) ) ,
\label{n4} \\
\lambda _{jk}=\alpha _1\int_{Y_1}\sum_{l=1}^3(\delta
_{jl}\delta _{kl}+\frac{\partial w_{l}^{jk}}{\partial y_{l}}) dy.
\label{n5}
\end{gather}
Here $| Y_i| $ denotes the volume of $Y_i$ and $(w_{l}^{ij}) _{1\leq l\leq 3}$
are the components of $\mathbf{w} ^{ij}$. Finally let us define the following
averaging quantities
\begin{gather}
\mathbf{f} = |Y_1|\mathbf{f}_1+|Y_2|\mathbf{f}_2,  \label{n6} \\
\tilde{c}  = \int_{Y_1}c_1(y) dy,  \label{n7} \\
\tilde{g}  = \int_{\Gamma }g(y) ds(y)  \label{n8}
\end{gather}
and the time-dependent functions
\begin{gather}
\theta (t,\tau ) = \alpha _2\int_{\Gamma }\partial _{t}\zeta
(t-\tau ,y) \mathbf{n}ds(y) ,\   \label{n10} \\
\eta (t,\tau ) = -\int_{\Gamma }g(y) \partial
_{t}\zeta (t-\tau ,y) ds(y) .  \label{n11}
\end{gather}

With the above notation, we are now ready to give the main result
 of this article.

\begin{theorem}\label{thp}
Let $(\mathbf{u}^{\varepsilon },p^{\varepsilon }) \in
L^{\infty }(0,T;\mathbf{H}) \times L^2(0,T;\mathcal{E}
^{\varepsilon }) $ be the solution of the weak system \eqref{23}.
Then, up to a subsequence, there exists a unique $(\mathbf{u},p) \in
L^2(0,T;\mathbf{H}_{0}^{1}(\Omega ) \times
H_{0}^{1}(\Omega ) ) $ such that
\begin{gather*}
\mathbf{u}^{\varepsilon } \rightharpoonup \mathbf{u}\text{ in }L^2(
0,T;H_{0}^{1}(\Omega ) ) \quad\text{ weakly,} \\
p_1^{\varepsilon } \rightharpoonup p_1\text{ in }L^2(Q)
\quad \text{ weakly,} \\
p_2^{\varepsilon } \rightharpoonup \int_{Y_2}p_2(y) dy
\text{ in }L^2(Q) \quad \text{ weakly,}
\end{gather*}
where $p=(p_1,\int_{Y_2}p_2(y) dy) $,
\begin{equation*}
p_2(t,x,y) =\int_{0}^{t}p_1(\tau ,x) \partial
_{t}\zeta (t-\tau ,y) d\tau ,\quad \text{a.e. }(t,x,y)
\in Q\times Y_2.
\end{equation*}
and the couple $(\mathbf{u},p_1) $ is a solution to the
homogenized model
\begin{gather*}
-\operatorname{div}\tilde{\sigma}(\mathbf{u})+B\nabla p_1+\int_{0}^{t}\theta
(t,\tau ) p_1(\tau ,x) d\tau = \mathbf{f},\quad \text{ a.e. in }Q, \\
\partial _{t}(\tilde{c}p_1+\Lambda :\mathrm{e}(\mathbf{u}
) ) -\operatorname{div}(\tilde{K}\nabla p_1) +\tilde{g}
p_1-\int_{0}^{t}\eta (t,\tau ) p_1(\tau ,x)
d\tau =0,\quad \text{a.e. in }Q, \\
\mathbf{u}=0, \quad \tilde{K}\nabla p_1\cdot \nu =0 \quad\text{ a.e. on }\Sigma , \\
\mathbf{u}(0,x) = \mathbf{0}\quad \text{a.e. in }\Omega ,p_1(0,x)= 0
\quad \text{a.e. in }\Omega ,
\end{gather*}
Here $\tilde{\sigma}$, $B$, $\theta $, $\mathbf{f}$\textbf{, } $\tilde{c}$, $
\Lambda $, $\tilde{K}$, $\tilde{g}$ and $\eta $ are given in
\eqref{n1}-\eqref{n11}.
\end{theorem}

\section{Proof of main result}\label{s3}

As a direct application of  Theorems \ref{t1}-\ref{t3},
 and of the a priori estimates \eqref{30}, we give without proof the following
two-scale convergence result concerning the solutions
$(\mathbf{u} ^{\varepsilon },p^{\varepsilon }) $ of
 Problem \eqref{23}-\eqref{241}.

\begin{theorem} \label{t4}
There exists a subsequence of $(\mathbf{u}^{\varepsilon},p^{\varepsilon }) $,
solution of \eqref{23}-\eqref{241}, still
denoted $(\mathbf{u}^{\varepsilon },p^{\varepsilon }) $, and
there exist
\begin{gather*}
\mathbf{u}\in L_{T}^{\infty }(\mathbf{H}),\quad
\mathbf{\hat{u}}\in L_{T}^{\infty }(L^2(\Omega
;H_{\#}^{1}(Y) /\mathbb{R}) ) ^3, \\
p_1\in L_{T}^{\infty }(H_{0}^{1}(\Omega ) ),\quad
 \hat{p}_1\in L^2(Q;H_{\#}^{1}(Y) /\mathbb{R}),\\
p_2\in L_{T}^{\infty }(L^2(\Omega ;H_{\#}^{1}(Y)) )
\end{gather*}
such that, for a.e. $t\in (0,T) $,
\begin{gather}
\mathbf{u}^{\varepsilon }(t,\cdot ) \overset{2-s}{
\rightharpoonup }\mathbf{u}(t,\cdot ) \mathbf{,}  \label{31} \\
\chi _1^{\varepsilon }p_1^{\varepsilon }(t,\cdot ) \overset
{2-s}{\rightharpoonup }\chi _1p_{1\,}(t,\cdot ) ,  \label{32}
\\
\chi _1^{\varepsilon }p_2^{\varepsilon }(t,\cdot ) \overset
{2-s}{\rightharpoonup }\chi _2p_2(t,\cdot )  \label{321}
\end{gather}
in the sense of Definition \ref{def1} and
\begin{gather}
\frac{\partial \mathbf{u}^{\varepsilon }}{\partial \mathbf{x}_{j}}\overset{
2-s}{\rightharpoonup }\frac{\partial \mathbf{u}}{\partial \mathbf{x}_{j}}+
\frac{\partial \mathbf{\hat{u}}}{\partial \mathbf{y}_{j}},\quad
j=1,2,3,  \label{33} \\
\chi _1^{\varepsilon }\nabla p_1^{\varepsilon }\overset{2-s}{
\rightharpoonup }\chi _1(\nabla p_1+\nabla _{y}\hat{p}_1) ,
\label{34} \\
\varepsilon \chi _2^{\varepsilon }\nabla p_2^{\varepsilon }\overset{2-s
}{\rightharpoonup }\chi _2\nabla _{y}p_2  \label{35}
\end{gather}
in the sense of Definition \ref{def2}. Moreover, the following convergence holds:
\begin{equation}
\lim_{\varepsilon \to 0}\int_{\Sigma ^{\varepsilon }}\varepsilon
(p_1^{\varepsilon }-p_2^{\varepsilon }) \psi ^{\varepsilon
}\,dt\,ds^{\varepsilon }=\int_{Q\times \Gamma }(p_1-p_2) \psi
\,dt\,dx\,ds,  \label{36}
\end{equation}
for any $\psi \in \mathcal{D}(Q;\mathcal{C}_{\#}(Y)) $ with
 $\psi ^{\varepsilon }(t,x) =\psi (t,x,x/\varepsilon ) $.
\end{theorem}

To determine the limiting equations of the system \eqref{23}-\eqref{241}, we
begin by choosing the adequate admissible test functions.
Let $\mathbf{v}^{\varepsilon }(x) =\mathbf{v}(x) +\varepsilon
\mathbf{\hat{v}}(x,\dfrac{x}{\varepsilon }) $ where
$\mathbf{v} \in \mathcal{D}(\Omega ) ^3$ and
$\mathbf{\hat{v}}\in \mathcal{D}(\Omega ;\mathcal{C}_{\#}^{\infty }(Y) ) ^3$.
Let $q_1^{\varepsilon }(t,x)=\varphi _1(t,x)
+\varepsilon \hat{\varphi}_1(t,x,\dfrac{x}{\varepsilon }) $
and $q_2^{\varepsilon }(t,x)=\varphi _2(t,x,\dfrac{x}{\varepsilon }) $
where $\varphi _1\in \mathcal{D}((0,T) \times
\bar{\Omega}) $ and $\varphi _2$, $\hat{\varphi}_1\in \mathcal{D}
(Q;\mathcal{C}_{\#}^{\infty }(Y) ) $.
Taking $\mathbf{v}=\mathbf{v}^{\varepsilon }$ in \eqref{23}, we have
\begin{equation}
\begin{aligned}
\int_{\Omega }\mathbf{f}^{\varepsilon }\mathbf{v}^{\varepsilon }dx
&= \int_{\Omega }\mathbb{A}^{\varepsilon }(x) \mathrm{e}(
\mathbf{u}^{\varepsilon }) \mathrm{e}(\mathbf{v}^{\varepsilon
}) dx+\int_{\Omega _1^{\varepsilon }}\alpha _1\nabla
p_1^{\varepsilon }\mathbf{v}^{\varepsilon }dx+\varepsilon \int_{\Omega
_2^{\varepsilon }}\alpha _2\nabla p_2^{\varepsilon }\mathbf{v}
^{\varepsilon }dx   \\
&= \int_{\Omega }\mathbb{A}^{\varepsilon }(x) \mathrm{e}(
\mathbf{u}^{\varepsilon }) (\mathrm{e}(\mathbf{v})
(x) +\mathrm{e}_{y}(\mathbf{\hat{v}}) (x,\frac{
x}{\varepsilon }) ) dx\\
&\quad +  \int_{\Omega }(\alpha _1\chi _1^{\varepsilon }(x)
\nabla p_1^{\varepsilon }+\varepsilon \alpha _2\chi _2^{\varepsilon
}(x) \nabla p_2^{\varepsilon }) \mathbf{v}(
x) dx+\varepsilon R_1^{\varepsilon },
\end{aligned}  \label{40}
\end{equation}
where
\begin{align*}
R_1^{\varepsilon }
 &= \int_{\Omega }\mathbb{A}^{\varepsilon }(
x) \mathrm{e}(\mathbf{u}^{\varepsilon }) \mathrm{e}
_{x}(\mathbf{w}) (x,\frac{x}{\varepsilon }) dx
+\alpha _1\int_{\Omega }\chi _1^{\varepsilon }(x) \nabla
p_1^{\varepsilon }\mathbf{w}(x,\frac{x}{\varepsilon }) dx \\
&\quad +\varepsilon \alpha _2\int_{\Omega }\chi _2^{\varepsilon }(
x) \nabla p_2^{\varepsilon }\mathbf{w}(x,\frac{x}{\varepsilon }) dx.
\end{align*}
Observe that $R_1^{\varepsilon }=O(1) $.

Now, we pass to the limit in \eqref{40}. In view of \eqref{33}, and since
$\mathbb{A}^{t}(\mathrm{e}(\mathbf{v}) +\mathrm{e}_{y}(\mathbf{\hat{v}}) ) $
 is an admissible test function, the first integral in the left-hand side
of \eqref{40} converges to
\begin{equation}
\int_{\Omega \times Y}\mathbb{A}(\mathrm{e}(\mathbf{u}) +
\mathrm{e}_{y}(\mathbf{\hat{u}}) ) (\mathrm{e}(
\mathbf{v}) +\mathrm{e}_{y}(\mathbf{\hat{v}}) ) \,dx\,dy
\label{401}
\end{equation}
where the tensor $\mathbb{A}(y) $ is given by \eqref{n0}. In
view of Divergence Lemma and \eqref{34}-\eqref{35}, the second integral of
the left-hand side of \eqref{40} tends to
\begin{equation}
\begin{aligned}
&\alpha _1\int_{\Omega \times Y_1}(\nabla p_1+\nabla _{y}\hat{p}
_1) \mathbf{v}(x) \,dx\,dy+\alpha _2\int_{\Omega \times
Y_2}\nabla _{y}p_2\mathbf{v}(x) \,dx\,dy   \\
&=\alpha _1|Y_1|\int_{\Omega }\nabla p_1\mathbf{v}(x)
dx+\int_{\Omega \times \Gamma }(\alpha _1\hat{p}_1+\alpha
_2p_2) (\mathbf{v\cdot n}) \,dx\,ds,
\end{aligned} \label{37}
\end{equation}
By Theorem \ref{t1}, it follows that
\begin{equation}
\begin{aligned}
\lim_{\varepsilon \to 0}\int_{\Omega }\mathbf{f}^{\varepsilon }
\mathbf{v}^{\varepsilon }(x) dx
&= \lim_{\varepsilon \to0}
\Big(\int_{\Omega }\mathbf{f}^{\varepsilon }(x) \mathbf{v}
(x) dx+\varepsilon \int_{\Omega }\mathbf{f}^{\varepsilon }(
x) \mathbf{\hat{v}}(x,\frac{x}{\varepsilon }) dx\Big)
 \\
&= \int_{\Omega }\mathbf{fv}(x) dx
\end{aligned} \label{403}
\end{equation}
where $\mathbf{f}$ is given by \eqref{n6}. Thus, collecting these
 limits \eqref{401}-\eqref{403}, we obtain the limiting equation of \eqref{40},
\begin{equation}
\begin{aligned}
&\int_{\Omega \times Y}\mathbb{A}[ \mathrm{e}(\mathbf{u}) +
\mathrm{e}_{y}(\mathbf{\hat{u}}) ] [ \mathrm{e}(
\mathbf{v}) +\mathrm{e}_{y}(\mathbf{\hat{v}}) ]
\,dx\,dy+\alpha _1|Y_1|\int_{\Omega }\nabla p_1\mathbf{v}dx   \\
&+\int_{\Omega \times \Gamma }(\alpha _1\hat{p}_1+\alpha
_2p_2) (\mathbf{v\cdot n}) \,dx\,ds=\int_{\Omega }\mathbf{
fv}dx
\end{aligned}\label{45}
\end{equation}
which is valid for a.e. $t\in (0,T) $. Next, we proceed to get
the limiting equation of \eqref{24}. Taking $q_1=q_1^{\varepsilon }$ and
$q_2=q_2^{\varepsilon }$ in \eqref{24}, integrating by parts over
$(0,T) $ and taking into account the initial conditions \eqref{241}, we obtain
\begin{equation}
\begin{aligned}
&-\int_{Q_1^{\varepsilon }}(c_1^{\varepsilon }(x)
p_1^{\varepsilon }+\alpha _1\operatorname{div}\mathbf{u}^{\varepsilon })
\partial _{t}\varphi _1(t,x) \,dt\,dx    \\
&-\int_{Q_2^{\varepsilon
}}c_2^{\varepsilon }(x) p_2^{\varepsilon }\partial
_{t}\varphi _2(t,x,\frac{x}{\varepsilon }) \,dt\,dx\\
&+ \int_{Q_1^{\varepsilon }}K_1(\frac{x}{\varepsilon })
\nabla p_1^{\varepsilon }(\nabla \varphi _1(t,x)
+\nabla _{y}\hat{\varphi}_1(t,x,\frac{x}{\varepsilon })
) \,dt\,dx\\
&+ \int_{Q_2^{\varepsilon }}\varepsilon k_2(\frac{x}{\varepsilon }
) \nabla p_2^{\varepsilon }\nabla _{y}\varphi _2(t,x,\frac{x
}{\varepsilon }) \,dt\,dx\\
&+  \varepsilon \int_{\Sigma ^{\varepsilon }}g(\frac{x}{\varepsilon }
) (p_1^{\varepsilon }-p_2^{\varepsilon }) (
\varphi _1(t,x) -\varphi _2(t,x,\frac{x}{\varepsilon }
) ) \,dt\,ds^{\varepsilon }+\varepsilon R_2^{\varepsilon }=0
\end{aligned}\label{46}
\end{equation}
where
\begin{align*}
R_2^{\varepsilon }
&=\int_{Q_1^{\varepsilon }}-(c_1^{\varepsilon
}(x) p_1^{\varepsilon }+\alpha _1\operatorname{div}\mathbf{u}
^{\varepsilon }) \partial _{t}\hat{\varphi}_1(t,x,\dfrac{x}{
\varepsilon }) \,dt\,dx\\
&\quad + \int_{Q_2^{\varepsilon }}-\alpha _2\operatorname{div}\mathbf{u}^{\varepsilon
}\partial _{t}\varphi _2(t,x,\frac{x}{\varepsilon }) \,dt\,dx\\
&\quad + \int_{Q_1^{\varepsilon }}K_1(\frac{x}{\varepsilon })
\nabla p_1^{\varepsilon }\nabla _{x}\hat{\varphi}_1(t,x,\frac{x}{
\varepsilon }) \,dt\,dx\\
&\quad + \varepsilon \int_{Q_1^{\varepsilon }}K_2(\frac{x}{\varepsilon }
) \nabla p_2^{\varepsilon }\nabla _{x}\varphi _2(t,x,\frac{x
}{\varepsilon }) \,dt\,dx\\
&\quad + \varepsilon \int_{\Sigma ^{\varepsilon }}g(\frac{x}{\varepsilon }
) (p_1^{\varepsilon }-p_2^{\varepsilon }) \hat{\varphi}
_1(t,x) \,dt\,ds^{\varepsilon }.
\end{align*}
The first integral of \eqref{46} is equal to
\begin{equation*}
\int_{\Omega _{T}}-\chi _1(\frac{x}{\varepsilon }) (
c_1(\frac{x}{\varepsilon }) p_1^{\varepsilon }+\alpha _1
\operatorname{div}\mathbf{u}^{\varepsilon }) \partial _{t}\varphi _1(
t,x) \,dt\,dx,
\end{equation*}
and thanks to \eqref{32} and \eqref{33}, it  converges to
\begin{equation*}
\int_{Q\times Y}-\chi _1(y) (c_1(y)
p_1+\alpha _1(\operatorname{div}\mathbf{u}+\operatorname{div}_{y}\mathbf{\hat{u
}}) ) \partial _{t}\varphi _1(t,x) dt\,dx\,dy.
\end{equation*}
In a similar way, by \eqref{321} and \eqref{33}, it follows that
\begin{equation*}
\int_{Q_2^{\varepsilon }}c_2^{\varepsilon }(x)
p_2^{\varepsilon }\partial _{t}\varphi _2(t,x,\frac{x}{\varepsilon
}) \,dt\,dx\to \int_{Q\times Y}\chi _2(y)
c_2(y) p_2\partial _{t}\varphi _2(t,x,y) \,dt\,dx\,dy
\end{equation*}
Now, in view of \eqref{34} one can deduce that
\begin{align*}
&\int_{Q_1^{\varepsilon }}K_1(\frac{x}{\varepsilon })
\nabla p_1^{\varepsilon }(\nabla \varphi _1(t,x)
+\nabla _{y}\hat{\varphi}_1(t,x,\frac{x}{\varepsilon })
) \,dt\,dx\\
&= \int_{Q}\chi _1(\frac{x}{\varepsilon }) K_1(\frac{x}{
\varepsilon }) \nabla p_1^{\varepsilon }(\nabla \varphi
_1(t,x) +\nabla _{y}\hat{\varphi}_1(t,x,\frac{x}{
\varepsilon }) ) \,dt\,dx\\
&\to \int_{Q\times Y}\chi _1(y) K_1(y) (
\nabla p_1+\nabla _{y}\hat{p}_1) (\nabla \varphi (
t,x) +\nabla _{y}\hat{\varphi}_1(t,x,y) ) \,dt\,dx\,dy
\end{align*}
and thanks to \eqref{35}, we also have
\begin{align*}
&\int_{Q_2^{\varepsilon }}\varepsilon k_2(\frac{x}{\varepsilon }
) \nabla p_2^{\varepsilon }\nabla _{y}\varphi _2(t,x,\frac{x
}{\varepsilon }) \,dt\,dx \\
&= \int_{Q}\chi _2(\frac{x}{\varepsilon }) K_2(\frac{x
}{\varepsilon }) \varepsilon \nabla p_2^{\varepsilon }\nabla
_{y}\varphi _2(t,x,\frac{x}{\varepsilon }) \,dt\,dx\\
&\to \int_{Q\times Y}\chi _2(y) K_2(y) \nabla
p_2\nabla _{y}\varphi _2(t,x,y) dt\,dx\,dy.
\end{align*}
By  \eqref{36}, we find that
\begin{align*}
&\varepsilon \int_{\Sigma ^{\varepsilon }}g(\frac{x}{\varepsilon }
) (p_1^{\varepsilon }-p_2^{\varepsilon }) (
\varphi _1(t,x) -\varphi _2(t,x,\frac{x}{\varepsilon }
) ) \,dt\,ds^{\varepsilon }\\
&\to  \int_{Q\times \Gamma }g(y) (p_1-p_2) (
\varphi _1(t,x) -\varphi _2(t,x,y) )
\,dt\,ds\,dy.
\end{align*}

As before, we observe that $R_2^{\varepsilon }=O(1) $ and, by
collecting all the preceding limits, we obtain the following limiting equation
of \eqref{24}:
\begin{equation}
\begin{aligned}
&\int_{Q\times Y_1}-(c_1(y) p_1+\alpha _1(
\operatorname{div}\mathbf{u}+\operatorname{div}_{y}\mathbf{\hat{u}}) )
\partial _{t}\varphi _1dt\,dx\,dy\\
&+ \int_{Q\times Y_1}K_1(y) (\nabla p_1+\nabla _{y}
\hat{p}_1) (\nabla \varphi _1+\nabla _{y}\hat{\varphi}
_1) dt\,dx\,dy\\
&+ \int_{Q\times Y_2}(-c_2(y) p_2\partial _{t}\varphi
_2+K_2(y) \nabla _{y}p_2\nabla _{y}\varphi _2)
dt\,dx\,dy\\
&+  \int_{Q\times \Gamma }g(y) (p_1-p_2) (
\varphi _1-\varphi _2) \,dt\,ds\,dy=0.
\end{aligned}  \label{50}
\end{equation}

By a denseness argument,  equations \eqref{45} and \eqref{50} still hold
for any
$$(\mathbf{v},\mathbf{\hat{v}}) \in \mathbf{H}\times
L^2(\Omega ,H^{1}(Y) /\mathbb{R}) ^3
$$
and  any
$$
(\varphi _1,\hat{\varphi}_1,\varphi _2) \in
L_{T}^2(H^{1}(\Omega ) ) \times L^2(
Q;H_{\#}^{1}(Y) /\mathbb{R}) \times L^2(
Q;H_{\#}^{1}(Y) ).
 $$
We can summarize the preceding by
observing that these equations are a weak formulation associated to the
two-scale homogenized system \eqref{471}-\eqref{484}. Indeed, integrating by
parts in \eqref{45} and \eqref{50}, we obtain the system
\begin{gather}
-\operatorname{div}_{y}(\mathbb{A}_1[ \mathrm{e}(\mathbf{u}
) +\mathrm{e}_{y}(\mathbf{\hat{u}}) ] ) = 0\quad \text{ a.e.in }Q\times Y_1,
\label{471} \\
-\operatorname{div}_{y}(\mathbb{A}_2[ \mathrm{e}(\mathbf{u}
) +\mathrm{e}_{y}(\mathbf{\hat{u}}) ] ) =0\quad
\text{a.e.in }Q\times Y_2,  \label{472} \\
\begin{aligned}
&-\operatorname{div}(\int_{Y}\mathbb{A}[ \mathrm{e}(\mathbf{u}
) +\mathrm{e}_{y}(\mathbf{\hat{u}}) ] dy)
+\alpha _1| Y_1| \nabla p_1\\
&+ \int_{\Gamma }(\alpha _1\hat{p}_1+\alpha _2p_2)
\mathbf{n}ds=\mathbf{f}\quad \text{ a.e. in }Q,
\end{aligned}  \label{473}
\end{gather}
and
\begin{gather}
-\operatorname{div}_{y}(K_1(\nabla p_1+\nabla _{y}\hat{p}
_1) ) = 0\quad \text{a.e. in }Q\times Y_1,  \label{474} \\
\partial _{t}(c_2p_2) -\operatorname{div}_{y}(K_2\nabla_{y}p_2)
 = 0 \quad \text{a.e.in }Q\times Y_2,  \label{475} \\
\begin{aligned}
&\partial _{t}(\int_{Y_1}(c_1p_1+\alpha _1(
\operatorname{div}\mathbf{u}+\operatorname{div}_{y}\mathbf{\hat{u}}) )
)
-\operatorname{div}(\int_{Y_1}K_1(\nabla p_1+\nabla _{y} \hat{p}_1) dy)\\
&+\int_{\Gamma }g(y) [ p_1-p_2] ds(y)
=0\quad \text{a.e. in }Q,
\end{aligned} \label{476}
\end{gather}
with the transmission and boundary conditions:
\begin{gather}
\mathbb{A}_1[ \mathrm{e}(\mathbf{u}) +\mathrm{e}
_{y}(\mathbf{\hat{u}}) ] \cdot \mathbf{n}
=\mathbb{A}_2[ \mathrm{e}(\mathbf{u}) +\mathrm{e}
_{y}(\mathbf{\hat{u}}) ] \cdot \mathbf{n}\quad\text{a.e. on }
Q\times \Gamma ,  \label{477} \\
(K_1(\nabla p_1+\nabla _{y}\hat{p}_1) )
\cdot \mathbf{n}=0\quad \text{a.e. on }Q\times \Gamma ,  \label{478} \\
 (K_1(\nabla p_1+\nabla _{y}\hat{p}_1) )
\cdot v=0\quad \text{a.e. on }(0,T) \times \partial \Omega \times Y_1,
\label{4781} \\
K_2\nabla _{y}p_2\cdot \mathbf{n}=-g(y) [ p_1-p_2] \quad
\text{a.e. on }Q\times \Gamma ,  \label{479} \\
\mathbf{u}=0\quad \text{a.e. on }\partial \Omega ,  \label{480} \\
y\mapsto  \mathbf{\hat{u}},\quad \hat{p}_1,p_2
\text{ are $Y$-periodic},  \label{481}
\end{gather}
and the initial conditions:
\begin{gather}
\mathbf{u}(0,x) =\mathbf{0}\quad\text{a.e. in }\Omega ,  \label{482} \\
\mathbf{\hat{u}}(0,x,y) =\mathbf{0}\quad \text{a.e. in }\Omega \times Y,  \label{4821} \\
p_1(0,x) =0\quad \text{ a.e. in }\Omega ,  \label{483} \\
\hat{p}_1(0,x,y) =0\quad \text{a.e. in }\Omega \times Y_1 \\
p_2(0,x,y) =0\quad \text{a.e. in }\Omega \times Y_2.  \label{484}
\end{gather}
Now we decouple the system \eqref{471}-\eqref{484}. In view of the linearity
of the two first equations \eqref{471}-\eqref{472}, we can write that, up to
an additive constant:
\begin{equation}
\mathbf{\hat{u}}(t,x,y)=\sum_{i,j=1}^3\mathrm{e}_{ij}(\mathbf{u}
) (t,x) \mathbf{w}^{ij}(y) +C^{te},\ \text{
a.e. }(t,x,y) \in Q\times Y,  \label{48}
\end{equation}
where, for $i,j\in \{ 1,2,3\} $,
$\mathbf{w}^{ij}\in (H_{\#}^{1}(Y) /\mathbb{R}) ^3$ is the solution to the
 microscopic system
\begin{gather*}
-\operatorname{div}_{y}(\mathbb{A}_1\mathrm{e}_{y}(\mathbf{w}^{ij}+
\mathbf{d}^{ij}) ) =0\quad \text{a.e. in }Y_1,  \\
-\operatorname{div}_{y}(\mathbb{A}_2\mathrm{e}_{y}(\mathbf{w}^{ij}+
\mathbf{d}^{ij}) ) =0\quad \text{a.e. in }Y_2, \\
\mathbb{A}_1\mathrm{e}_{y}(\mathbf{w}^{ij}+\mathbf{d}^{ij})
\cdot \mathbf{n}=\mathbb{A}_2\mathrm{e}_{y}(\mathbf{w}^{ij}+\mathbf{d
}^{ij}) \cdot \mathbf{n}\quad \text{a.e. on }\Gamma , \\
y\mapsto  \mathbf{w}^{ij}\quad Y\text{-periodic}.
\end{gather*}
Here $\mathbf{d}^{kl}=(y_{K}\delta _{il}) _{1\leq i\leq 3}$ and
$(\delta _{ij}) $ is the Kr\"{o}necker symbol.

Similarly, in view of \eqref{474}, \eqref{478} and \eqref{481} one can write
that
\begin{equation}
\hat{p}_1(t,x,y) =\sum_{i=1}^3\frac{\partial p_1}{\partial
x_i}(t,x) \pi _i(y) +C^{te},\quad \text{a.e. }(t,x,y) \in
 Q\times Y_1,  \label{49}
\end{equation}
where, for $i=1,2,3$, the micro-pressure
$\pi _i\in H^{1}(Y_1) /\mathbb{R}$ is the solution of the  stationary
equation
\begin{gather*}
-\operatorname{div}_{y}(K_1(\nabla \pi _i+e_i) ) =0\quad
\text{in }Y_1, \\
K_1(\nabla \pi _i+e_i) \cdot \mathbf{n}=0\quad \text{on }\Gamma
, \\
y\mapsto \pi _i\quad Y\text{-periodic.}
\end{gather*}
Here $e_i$ is the $i^{\text{th}}$ vector of the canonical basis of
$\mathbb{R}^3$. Let us denote
\begin{gather*}
\widetilde{\mathbb{A}} = (\tilde{a}_{i_1i_2i_3i_{4}})
_{1\leq i_1,i_2,i_3,i_{4}\leq 3},\ \  \\
\tilde{a}_{i_1i_2i_3i_{4}}
= \sum_{j_1,j_2=1}^3\int_{Y}a_{i_1i_2j_1j_2}(y)
(\delta _{i_1j_1}\delta _{i_2j_2}+\mathrm{e}
_{j_1j_2,y}(\mathbf{w}^{i_3i_{4}}) (y) )dy,
\end{gather*}
where $(a_{ijlm}) $ are the coefficients of the elasticity
tensor $\mathbb{A}$ and
\begin{equation*}
\mathrm{e}_{ij,y}(\mathbf{w})
=\frac{1}{2}\Big(\frac{\partial
w_i}{\partial y_{j}}+\frac{\partial w_{j}}{\partial y_i}\Big) ,\quad
\mathbf{w}=(w_{j}) _{1\leq j\leq 3}.
\end{equation*}
Also define the effective stress tensor
\begin{equation*}
\tilde{\sigma}(\mathbf{u})=(\tilde{\sigma}_{ij}(\mathbf{u}))
_{1\leq i,j\leq 3},\quad
 \tilde{\sigma}_{ij}(\mathbf{u})=\sum_{l,m=1}^3
\tilde{a}_{ijlm}\mathrm{e}_{lm}(\mathbf{u})\text{,\ }
\end{equation*}
the effective permeability tensor
\begin{equation*}
\tilde{K}=(\tilde{K}_{ij}) _{1\leq i,j\leq 3},\quad
 \tilde{K} _{ij}=\int_{Y_1}K_1(y) (\nabla _{y}\pi
_i+e_i) (\nabla \pi _{j}+e_{j}) dy,
\end{equation*}
the effective Biot-Willis matrices:
\begin{gather*}
B = (b_{ij}) ,\quad b_{ij}=\alpha _1(|
Y_1| \delta _{ij}+\int_{\Gamma }\pi _{j}(y)
n_ids(y) ) ,\quad \mathbf{n}=(n_i) _{1\leq
i\leq 3} \\
\Lambda =(\lambda _{ij}) _{1\leq i,j\leq 3},\quad
\lambda_{ij}=\alpha _1\int_{Y_1}\sum_{m=1}^3\Big(\delta _{im}\delta _{jm}+
\frac{\partial w_{m}^{ij}}{\partial y_{m}}\Big) dy, \\
\mathbf{w}^{ij} = (w_{m}^{ij}) _{1\leq m\leq 3}
\end{gather*}
and finally the  averaging quantities
\begin{equation*}
\tilde{c}=\int_{Y_1}c_1(y) dy,\quad
\tilde{g}=\int_{\Gamma }g(y) ds(y) .
\end{equation*}
Then from \eqref{48}-\eqref{49} we deduce the homogenized system
\begin{gather}
-\operatorname{div}\tilde{\sigma}(\mathbf{u})+B\nabla p_1+\alpha
_2\int_{\Gamma }p_2\mathbf{n}ds(y) =\mathbf{f}\quad
\text{a.e. in }Q,  \label{d1} \\
\partial _{t}(\tilde{c}p_1+\Lambda :\mathrm{e}(\mathbf{u}
) ) -\operatorname{div}(\tilde{K}\nabla p_1) +\tilde{g}p_1
-\int_{\Gamma }g(y) p_2ds(y) =0,\quad \text{a.e. in }Q,
\label{d2} \\
\partial _{t}(c_2p_2) -\operatorname{div}_{y}(K_2\nabla
_{y}p_2) =0\quad \text{a.e. in }Q\times Y_2,  \label{d3} \\
c_2\nabla _{y}p_2\cdot \mathbf{n}=-g(y) [ p_1-p_2] \quad
\text{a.e. on }Q\times \Gamma ,  \label{d31} \\
\mathbf{u}=0, \quad \tilde{K}\nabla p_1\cdot \nu =0\quad
\text{a.e. on }(0,T) \times \Sigma ,  \label{d4} \\
y\mapsto  p_2\quad Y\text{-periodic,}  \label{d5} \\
\mathbf{u}(0,x) =\mathbf{0}\quad \text{a.e. in }\Omega ,\quad
p_1(0,x) =0\quad \text{a.e. in }\Omega ,  \label{d6} \\
p_2(0,x,y) =0\quad \text{a.e. in }\Omega \times Y_2.  \label{d9}
\end{gather}

Now, we  establish a relation between the two pressures $p_1$
and $p_2$. To this aim, let $\zeta \in L^{\infty }(
0,T;H_{\#}^{1}(Y_2) ) $ be the unique solution to the
following microscopic and non homogeneous Robin problem
\begin{gather*}
\partial _{t}(c_2\zeta ) -\operatorname{div}_{y}(K_2\nabla
_{y}\zeta ) =0\quad \text{a.e.in }(0,T) \times Y_2, \\
K_2\nabla _{y}\zeta \cdot \mathbf{n}\quad =-g(y) [ 1-\zeta ] \quad
\text{a.e. on }\Sigma , \\
y\mapsto \zeta \quad Y\text{-periodic,} \\
\zeta (0,y) =0\quad \text{a.e. }y\in Y_2.
\end{gather*}
Since $c_2,K_2,g$ are time-independent and $p_1$ is independent of 
$y$, using the Laplace transform method, one can then easily see that
\begin{equation}
p_2(t,x,y) =\int_{0}^{t}p_1(\tau ,x) \partial
_{t}\zeta (t-\tau ,y) d\tau ,\ \text{a.e. }(t,x,y)
\in Q\times Y_2.  \label{rel1}
\end{equation}
Therefore, the homogenized system \eqref{d1}-\eqref{d9} can be rewritten as
\begin{gather*}
-\operatorname{div}\tilde{\sigma}(\mathbf{u})+B\nabla p_1+\int_{0}^{t}\theta
(t,\tau ) p_1(\tau ,x) d\tau =\mathbf{f}\quad \text{ a.e. in }Q, \\
\partial _{t}(\tilde{c}p_1+\Lambda :\mathrm{e}(\mathbf{u}
) ) -\operatorname{div}(\tilde{K}\nabla p_1) +\tilde{g}p_1 
-\int_{0}^{t}\eta (t,\tau ) p_1(\tau ,x) d\tau
=0,\quad \text{a.e. in }Q, \\
\mathbf{u}=0, \quad \tilde{K}\nabla p_1\cdot \nu =0\quad
\text{a.e. on }(0,T) \times \partial \Omega , \\
\mathbf{u}(0,x)=\mathbf{0},\quad p_1(0,x) =0 \quad\text{a.e. in }\Omega ,
\end{gather*}
where we have denoted
\begin{gather*}
\theta (t,\tau ) = \alpha _2\int_{\Gamma }\partial _{t}\zeta
(t-\tau ,y) \mathbf{n}ds(y) ,  \\
\eta (t,\tau )  = \int_{\Gamma }g(y) \partial
_{t}\zeta (t-\tau ,y) ds(y) .
\end{gather*}
Finally, let us observe that the overall pressure of the fluid flow in the
microstructure model which is 
\begin{equation*}
P^{\varepsilon }(t,x) =\chi _1^{\varepsilon
}(x)p_1^{\varepsilon }(t,x) +\chi _2^{\varepsilon
}(x)p_2^{\varepsilon }(t,x)
\end{equation*}
for a.e. $(t,x) \in Q$. The two-scale converges to $\chi
_1(y)p_1(t,x) +\chi _2(y)p_2(t,x,y) $, and
thanks to \eqref{rel1}, converges then weakly in $L^2(Q) $ to
\begin{equation*}
|Y_1|p_1(t,x) +\int_{0}^{t}\int_{Y_2}p_1(\tau
,x) \partial _{t}\zeta (t-\tau ,y) dyd\tau .
\end{equation*}
This concludes the proof of Theorem \ref{thp}.

\subsection*{Conclusion}

We have used the homogenization theory to derive a macro-model for fluid
flow in composite poroelastic with microstructures, in which inclusions are
fully embedded and with very low permeabilities. We have shown that the
overall behavior of fluid flow in such heterogeneous media with low
permeability at the micro-scale may present memory terms. We also have shown
that in such cases, the Biot-Willis parameters are, as in \cite{ain2},
matrices and no longer scalars, as it is usually considered in the
poroelasticity literature, since it is assumed there that the medium is
homogeneous and isotropic. Nevertheless, anisotropic media may present
different coupling interaction properties in different directions at the
micro-scale, and which lead at the macro-scale to such anisotropic
Biot-Willis parameters. Finally, let us mention that the result of the paper
remains valid if one considers non homogeneous initial conditions or with
any volume distributed source densities in each phases.

\subsection*{Acknowledgments}
This work was achieved during the stay of the author at 
the Laboratoire de math\'{e}matiques de Versailles, Universit\'{e} 
de Versailles-Saint-Quentin-en-Yvelines, France in October, 2012. 
This stay was supported by EGIDE program through the  TASSILI 
project Analyse des \'{e}quations aux d\'{e}riv\'{e}es partielles 
en domaines non born\'{e}s  No.\ (C.M.E.P.) 11 MDU 835 and
No.\ (EGIDE) 24471NA. The author also acknowledges the support
 of the  Algerian ministry of higher education and scientific 
research through the  C.N.E.P.R.U. project 
Techniques de mod\'{e}lisation en milieux h\'{e}t\'{e}rog\`{e}nes 
et couches minces No.\ B00220090078. 
The author also thanks the anonymous referees for their careful 
reading of the paper and for raising questions, valuable 
suggestions  and comments on an earlier version of this work  
that allowed to improve it.

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\end{document}