\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 49, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/49\hfil Mathematical models of a liquid filtration]
{Mathematical models of a liquid filtration \\ from reservoirs}

\author[A. Meirmanov, N. Erygina, S. Mukhambetzhanov \hfil EJDE-2014/49\hfilneg]
{Anvarbek Meirmanov, Nelly Erygina, Saltanbek Mukhambetzhanov}  % in alphabetical order

\address{Anvarbek Meirmanov \newline
Department of higher mathematics and cybernetics,
Kazakh-British Technical University,
Toli Bi 59, 050000, Almaty, Kazakhstan}
\email{anvarbek@list.ru}

\address{Nelly Erygina \newline
Department of mathematics,
Belgorod State University,
ul.Pobedi 85, 308015 Belgorod, Russia}
\email{eryginan@bsu.edu.ru}

\address{Saltanbek Mukhambetzhanov \newline
Department of differential equations,
Al-Farabi Kazakh National University, \newline
Al-Farabi 71, 050000, Almaty, Kazakhstan}
\email{mukhambetzhanov@mail.ru}

\thanks{Submitted January 6, 2014. Published February 19, 2014.}
\subjclass[2000]{35B27, 46E35, 76R99}
\keywords{Lame's equation; Stokes equation; liquid filtration;
 homogenization; \hfill\break\indent  poroelasticity}

\begin{abstract}
 This article studies the filtration from reservoirs into porous
 media under gravity. We start with the exact mathematical model at
 the microscopic level, describing the joint motion of a liquid in reservoir
 and the same liquid and the elastic solid skeleton in the porous medium.
 Then using a homogenization procedure we derive the chain of macroscopic
 models from the poroelasticity equations up to the simplest Darcy's law
 in the porous medium and hydraulics in the reservoir.
\end{abstract}

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

\section{Introduction}

In article we consider a correct description of filtration from reservoirs 
into porous media under gravity. Our approach is based on the way
suggested by  Burridge and  Keller \cite{BK} and Sanchez-Palencia \cite{SP}.
We first describe the problem at the microscopic level using classical 
equations of continuum mechanics and after that derive all possible
homogenized equations, describing the problem at the macroscopic level.

The problem in its simplest setting is modeled by two domains
$\Omega_0$ and $\Omega$ having a common boundary $S^0$.
The domain $\Omega^0$ models a reservoir and is occupied by liquid,
and the domain $\Omega$ models a porous medium.
Throughout this paper we impose the following constraints.

We will use the following assumptions:
\begin{itemize} 
\item[(1)] Let $\chi(\mathbf{y})$ be a 1-periodic function,
$Y_{s}=\{\mathbf{y}\in Y:\chi(\mathbf{y})=0\}$  be the ``solid part" 
of the unit cube $Y=(0,1)^3\subset\mathbb{R}^{3}$, and let the 
``liquid part" $Y_{f}=\{\mathbf{y}\in Y:\chi(\mathbf{y})=1\}$ of  
$Y$ be its open complement.
We write $\gamma =\partial Y_{f} \cap\partial Y_{s}$ and assume that 
$\gamma$ is a Lipschitz continuous surface.

\item[(2)] The domain $E^{\varepsilon}_{f}$ is a periodic repetition in 
$\mathbb{R}^{3}$ of the elementary cell $Y^{\varepsilon}_{f}=\varepsilon Y_{f}$  
and the domain $E^{\varepsilon}_{s}$ is a periodic repetition in 
$\mathbb{R}^{3}$ of the elementary cell $Y^{\varepsilon}_{s}=\varepsilon Y_{s}$.

\item[(3)] The pore space 
$\Omega ^{\varepsilon}_{f}\subset\Omega=\Omega\cap E^{\varepsilon}_{f}$ 
is a periodic repetition in $\Omega$ of the elementary cell $\varepsilon Y_{f}$, 
and the solid skeleton 
$\Omega ^{\varepsilon}_{s}\subset\Omega=\Omega\cap E^{\varepsilon}_{s}$  
is a periodic repetition  in $\Omega$ of  the elementary cell $\varepsilon Y_{s}$. 
The Lipschitz continuous  boundary 
$\Gamma^\varepsilon =\partial \Omega_s^\varepsilon \cap
\partial \Omega_f^\varepsilon$ is a periodic repetition in
$\Omega$ of the boundary $\varepsilon \gamma$.

\item[(4)] $Y_{s}$  and $Y_{f}$ are connected sets.

\item[(5)] The pore space  $\Omega _{f}^{\varepsilon}$ and the solid skeleton  
$\Omega _{s}^\varepsilon$ are connected domains.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1} 
\end{center}
\caption{Filtration from reservoir} \label{fig1}
\end{figure}
Let $Q=\Omega_{0}\cup\Omega\cup S^{0}$, $S=\partial Q$, 
$\bar{S}=\bar{S}^{1}\cup\bar{S}^{2}$.
\end{itemize}



We also assume that $S^1$ is a part of
the plane $\{x_{3}=0\}$, $\mathbf{e}=-\mathbf{e}_{3}$, and
that the domain $Q$ is a subset of the half-space $\{x_{3}<0\}$.
Moreover we suppose that $S^2$ is a $C^2$--smooth surface
and in some small neighborhood of the plane $\{x_{3}=0\}$ it is
represented by the equation $\Phi(x_1,x_2)=0$.

The motion of the liquid in $\Omega^0$ for $t>0$ is governed by the
dimensionless non-stationary Stokes system
\begin{gather} \label{1.1}
\nabla\cdot\mathbf{w}=0,\\
 \label{1.2}
\alpha_{\tau}\varrho_{f}\frac{\partial^2\mathbf{w}}{\partial t^2}=
\nabla\cdot\mathbb{P}_{f}+\varrho_{f}\mathbf{e},\quad
\mathbb{P}_{f}=\alpha_{\mu}
\mathbb{D}(x,\frac{\partial\mathbf{w}}{\partial t})-p\,\mathbb{I},
\end{gather}
and the joint motion of the poroelastic media in $\Omega$ for
$t>0$ is governed by the model \cite{AM1} consisting of the
continuity equation \eqref{1.1} and the dimensionless momentum balance
equation
\begin{equation} \label{1.3}
\alpha_{\tau}\varrho^{\varepsilon}\frac{\partial^2\mathbf{w}}{\partial
t^2}= \nabla\cdot\mathbb P +\varrho^{\varepsilon}\mathbf{e}.
\end{equation}
Here $\nabla\cdot\mathbf{w}$ is the divergence of the vector
$\mathbf{w}$, $\nabla \cdot\mathbb{P}$ is the divergence of the tensor $\mathbb{P}$,
\begin{gather}\label{1.4}
\mathbb{P}=\chi^{\varepsilon}\alpha_{\mu}
\mathbb{D}\big(x,\frac{\partial \mathbf{w}}{\partial t}\big)+
(1-\chi^{\varepsilon})\alpha_{\lambda}\mathbb{D}(x,\mathbf{w})-p\,\mathbb{I},
\\
\mathbb{D}(x,\boldsymbol{v})=\frac{1}{2}\big(\nabla\boldsymbol{v}+
(\nabla\boldsymbol{v})^{*}\big),\quad
\varrho^{\varepsilon}=\varrho_{f}\chi^{\varepsilon}
+\varrho_{s}(1-\chi^{\varepsilon}), \nonumber 
\\
\alpha_{\tau}=\frac{ L}{g\tau^2}, \quad
\alpha_{\mu}=\frac{2\mu}{\tau Lg\,\rho^{\,0}},\quad
\alpha_{\lambda}=\frac{2\lambda}{Lg\,\rho^{\,0}}, \nonumber
\end{gather}
where $\mathbb{I}$ is the unit tensor, and  $L$ is the characteristic 
size of the physical domain in consideration, $\tau$ is the characteristic 
time of the physical process, $\rho^{\,0}$ is the mean density of water,
 $g$ is acceleration due gravity, $\mu$ is the dynamic viscosity, $\lambda$ 
is the elastic constant, $\varrho_{f}$ and $\varrho_{s}$ are the respective
 mean dimensionless densities of the liquid and the solid skeleton, 
correlated with the mean density of water $\rho^{\,0}$.

On the common boundary
$S^0=\partial\Omega\cap\partial\Omega^0$ for $t>0$ the
continuity conditions
\begin{gather}\label{1.5}
\lim_{\mathbf{x}\to \mathbf{x}^0,\, \mathbf{x}\in \Omega^0}
\mathbf{w}(\mathbf{x},t)=
\lim_{\mathbf{x}\to \mathbf{x}^0,\,\mathbf{x}\in\Omega}
\mathbf{w}(\mathbf{x},t),
\\
\label{1.6}
\lim_{\mathbf{x}\to \mathbf{x}^0,\,
\mathbf{x}\in \Omega^0}\mathbb{P}_{f}(\mathbf{x},t)\cdot\mathbf{n}
(\mathbf{x}^0)
=\lim_{\mathbf{x}\to \mathbf{x}^0,\,
\mathbf{x}\in \Omega}\mathbb{P}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x}^0),
\end{gather}
hold  for displacements and for normal tensions. Here
$\mathbf{n}(\mathbf{x}^0)$ is a normal vector to the boundary $S^0$
at $\mathbf{x}^0\in S^0$.

We completement the problem with the Neumann boundary conditions
\begin{equation}\label{1.7}
\mathbb{P}_{f}(\mathbf{x},t)\cdot{\mathbf{n}}=
-p^0(\mathbf{x},t){\mathbf{n}},\quad
\mathbb{P}(\mathbf{x},t)\cdot{\mathbf{n}}=
-p^0(\mathbf{x},t){\mathbf{n}}
\end{equation}
on the part $S_0^1=S^1\cap\overline{\Omega}_0$ of the outer boundary
$S$ of the domain $Q=\Omega^0\cup S^0\cup \Omega$ (which is also the part
of the boundary $\partial\Omega^0$) for $\mathbb{P}_{f}$ and on the part
$S_1^1=S^1\cap\overline{\Omega}$ of the outer boundary $S$ of the domain
$Q$ (which is also the part of the boundary $\partial\Omega$) for $\mathbb{P}$, 
the Dirichlet boundary condition
\begin{equation}\label{1.8}
\mathbf{w}(\mathbf{x},t)=0
\end{equation}
on the part $S^2=S\backslash \overline{S^1}$ of the outer
boundary $S$  for $t>0$, and initial conditions
\begin{equation}\label{1.9}
\mathbf{w}(\mathbf{x},0)=\frac{\partial\mathbf{w}}{\partial
t} (\mathbf{x},0)=0,\quad \mathbf{x}\in Q.
\end{equation}
In \eqref{1.1}--\eqref{1.9} the characteristic function
$\chi^{\varepsilon}(\mathbf{x})$ of the domain
$\Omega^{\varepsilon}_{f}$ is given by the expression
\[
\chi^{\varepsilon}(\mathbf{x})=
\varsigma(\mathbf{x})\chi(\frac{\mathbf{x}}{\varepsilon}),
\]
where $\varsigma(\mathbf{x})$ is the characteristic function
of the domain $\Omega$, $\chi(\mathbf{y})$ is the
characteristic function of the liquid cell $Y_{f}$ in the unit
cube $Y$, and $\mathbf{e}$ is a unit vector in the direction
of gravity. The given function $p^0$ is supposed to be smooth:
\begin{equation}\label{1.10}
\int_{Q_T}\big(|\nabla p^0(\mathbf{x},t)|^2+|\nabla
(\frac{\partial p^0}{\partial t}) (\mathbf
x,t)|^2\big)\,dx\,dt=\mathfrak{P}^2<\infty.
\end{equation}
The main problem in the macroscopic description of the physical problem 
is the boundary conditions on the common boundary for the solutions of 
homogenized equations. There are some particular results obtained by 
 J\"ager and A. Mikeli\'c \cite{Jag-Mic1,Jag-Mic2,Jag-Mic3}
for special geometry of pore space (disconnected solid skeleton) and only 
for domains in $\mathbb{R}^2$.

We study the complete problem in $\mathbb{R}^{3}$ for the arbitrary geometry 
of corresponding pore spaces. Namely, let
\[
\lim_{\varepsilon\searrow 0} \alpha_{\tau}(\varepsilon) =\tau_0,\quad
\lim_{\varepsilon\searrow 0} \alpha_{\mu}(\varepsilon) =\mu_0,\quad
\lim_{\varepsilon\searrow 0} \alpha_{\lambda}(\varepsilon) =\lambda_0,\quad
\lim_{\varepsilon\searrow 0}\frac{\alpha_{\mu}}{\varepsilon^2}=\mu_1.
\]
To derive the desired homogenized problem for the case $\mu_0=0$,
 $\lambda_0=\infty$, and  $0<\tau_0,\,\mu_1<\infty$ we use
 \eqref{1.1}--\eqref{1.9}.
First we fix  $\tau_0>0$, pass to the limit as $\varepsilon\searrow 0$,
and get two different systems
\begin{equation}\label{1.11}
\tau_0\varrho_{f}\frac{\partial\mathbf{v}}{\partial t}+
\nabla p=\varrho_{f}\,\mathbf{e},\quad
\nabla\cdot\mathbf{v}=0
\end{equation}
and
\begin{gather}\label{1.12}
\mathbf{v}^{(f)}=
\int_0^{t}\mathbb{B}^{(f)}(\tau_0;t-\tau)\cdot
\big(-\nabla p^{(f)}\,(\mathbf{x},\tau)+\varrho_{f}\mathbf{e}\big)d\tau,
\\
\label{1.13}
\nabla\cdot\,\mathbf{v}^{(f)}=0
\end{gather}
for the velocity $\mathbf{v}$ and pressure $p$ in $\Omega^0$ and the
velocity $\mathbf{v}^{(f)}$ and pressure $p^{(f)}$ in $\Omega$.

These differential equations together with the boundary conditions
\begin{gather}\label{1.14}
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\,\mathbf{x}\in\Omega}
p^{(f)}(\mathbf{x},t)=
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\, \mathbf{x}\in \Omega_0}
p(\mathbf{x},t),
\\
\label{1.15}
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\, \mathbf{x}\in \Omega}
\mathbf{v}^{(f)}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x}^0)=
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\, \mathbf{x}\in\Omega_0}
\mathbf{v}(\mathbf{x},t) \cdot\mathbf{n}(\mathbf{x}^0)
\end{gather}
on the common boundary $S^0$ describe the liquid motion in the domain
$Q$ for $t>0$.

After that we pass to the limit as $\tau_0\searrow 0$ and get the usual
hydraulic equation
\begin{equation}\label{1.16}
\nabla p=\varrho_{f}\,\mathbf{e},\quad
p(\mathbf{x},t)=p^0(t)-\varrho_{f}x_{3}\equiv p_0(\mathbf{x},t)
\end{equation}
in the domain $\Omega^0$ and usual Darcy's system
\begin{equation}\label{1.17}
\mathbf{v}^{(f)}=\frac{1}{\mu_1}\mathbb{B}\cdot
\big(-\nabla p^{(f)}+ \varrho_{f}\mathbf{e}\big),\quad
\nabla\cdot\,\mathbf{v}^{(f)}=0
\end{equation}
in the domain $\Omega$, completed with the continuity condition \eqref{1.14}
on the common boundary $S^0$.

So, if we need to take into account the water flow from reservoir, 
we have to use the first approximation \eqref{1.11}--\eqref{1.15}. 
If we need the simplest model, we use the second approximation 
\eqref{1.16}--\eqref{1.17}.  Other homogenized models of \eqref{1.1}--\eqref{1.9} 
one may find in \cite{AM1}.


The notation of functional spaces and norm there are the same as in \cite{LSU}.

\section{Main results}

\begin{definition} \label{def2.1} \rm
We say that the pair of functions
$\{\mathbf{w}^{\varepsilon},p^{\varepsilon}\}$, such that
\[
p^{\varepsilon}\in
L_2(Q_T),\mathbf{w}^{\varepsilon},
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial
t},\big(\zeta+(1-\zeta)
\chi^{\varepsilon}\big)\nabla\frac{\partial\mathbf{w}^{\varepsilon}}{\partial
t},\nabla\mathbf{w}^{\varepsilon}\in
\mathbf{L}_2(Q_T),
\]
is a weak solution of the problem \eqref{1.1}--\eqref{1.9},
if it satisfies the continuity equation \eqref{1.1} almost
everywhere in $Q_T=Q\times(0,T)$, the boundary condition
\eqref{1.8}, the initial  condition \eqref{1.9} for the
function $\mathbf{w}^{\varepsilon}$, and the integral identity
\begin{equation}\label{2.1}
\begin{aligned}
&\int_{Q_T}\Big(-\tau_0\,\tilde{\varrho}^{\,\varepsilon}
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}\cdot
\frac{\partial\mathbf{\varphi}}{\partial
t}+\big(\zeta\mathbb{P}_{f}+
(1-\zeta)\mathbb{P}\big):\mathbb{D}(x,\mathbf{\varphi})\Big)\,dx\,dt\\
&=\int_{Q_T}\big(\tilde{\varrho}^{\,\varepsilon}\mathbf{e}\cdot
\mathbf{\varphi}-\nabla\cdot(\mathbf{\varphi}\,p^0)\big)\,dx\, dt
\end{aligned}
\end{equation}
for all smooth  functions $\mathbf{\varphi}$, such that
$\mathbf{\varphi}(\mathbf{x},t)=0$ at the boundary
$S^2_T$, and $\mathbf{\varphi}(\mathbf{x},T)=0$, $\mathbf{x}\in Q$.
\end{definition}
In \eqref{2.1} the convolution $\mathbb{A}:\mathbb{B}$ of two tensors 
$\mathbb{A}=(A_{ij})$ and $\mathbb{B}=(B_{ij})$ is defined as 
$\mathbb{A}:\mathbb{B}=\mbox{tr}(\mathbb{A}\cdot\mathbb{B})=
\sum_{i,j=1}^{3}A_{ij}B_{ji}$,
\begin{gather*}
\mathbb{P}_{f}=\alpha_{\mu}
\mathbb{D}(x,\frac{\partial\mathbf{w}^{\,\varepsilon}}{\partial t})
-p^{\,\varepsilon}\,\mathbb{I},\\
\mathbb{P}=\chi^{\varepsilon}\alpha_{\mu}
\mathbb{D}\Big(x,\frac{\partial \mathbf{w}\chi^{\varepsilon}}{\partial t}\Big)+
(1-\chi^{\varepsilon})\alpha_{\lambda}\mathbb{D}(x,\mathbf{w}\chi^{\varepsilon})-
p^{\varepsilon}\chi^{\varepsilon}\,\mathbb{I},
\\
\tilde{\varrho}^{\,\varepsilon}
=\big(\zeta+(1-\zeta)\chi^{\varepsilon}\big)\varrho_{f}+
(1-\zeta)(1-\chi^{\varepsilon})\varrho_{s}
\end{gather*}
and $\zeta=\zeta(\mathbf{x})$ is the characteristic function of
the domain $\Omega^0$ in $Q$.

The equation \eqref{2.1} obviously contains equations
\eqref{1.2} and \eqref{1.3},  and boundary conditions
\eqref{1.6} and \eqref{1.7}.

The solution of the problem \eqref{1.1}--\eqref{1.9} possesses different 
smoothness in domains $\Omega^{\varepsilon}_{f}$ and $\Omega^{\varepsilon}_{s}$.
 To preserve the best properties - which the solution possesses in the solid 
part - we extend the function $\mathbf{w}^{\varepsilon}$ from the solid part 
$\Omega^{\varepsilon}_{s}$ of the domain $\Omega$ onto the whole  domain  
$\Omega$. To do this we use the extension result 
(see \cite{Conca,NGU1}, and \cite[Lemma B.4.2]{AM1}):
there exists an extension
\begin{equation}\label{2.2}
\mathbf{w}_{s}^{\varepsilon}=\mathbb{E}_{\Omega^{\varepsilon}_{s}}
\big(\mathbf{w}^{\varepsilon}\big),\quad
\mathbb{E}_{\Omega^{\varepsilon}_{s}}:\,
{\mathbf{W}}^1_2(\Omega^{\varepsilon}_{s})\to
{\mathbf{W}}^1_2(\Omega),
\end{equation}
such that
\begin{equation}\label{2.3}
\big(1-\chi^{\varepsilon}(\mathbf{x})\big)
\big(\mathbf{w}^{\varepsilon}(\mathbf{x},t)-
\mathbf{w}_{s}^{\varepsilon}(\mathbf{x},t)\big)=0,\quad
\mathbf{x}\in \Omega,\,\,t\in (0,T),
\end{equation}
and
\begin{equation}\label{2.4}
\begin{gathered}
\int_{\Omega}|\mathbf{w}_{s}^{\varepsilon}(\mathbf{x},t)|^2dx\leqslant
C_0\int_{\Omega^{\varepsilon}_{s}}|\mathbf{w}^{\varepsilon}(\mathbf{x},t)|^2dx,\\
\int_{\Omega}|\mathbb{D}\big(x,\mathbf {w}_{s}^{\varepsilon}(\mathbf{x},t)\big)|^2dx\leqslant
C_0\int_{\Omega^{\varepsilon}_{s}}|\mathbb{D}\big(x,\mathbf{w}^{\varepsilon}
(\mathbf{x},t)\big)|^2dx,\quad t\in (0,T),
\end{gathered}
\end{equation}
where $C_0$ is independent of $\varepsilon$, and $t\in (0,T)$.

\begin{theorem}\label{theorem1}
Let
\begin{equation}\label{2.5}
p^0=p^0(t).
\end{equation}
Then for all $\varepsilon >0$ and for an arbitrary time interval
$[0,T]$ there exists a unique generalized solution of problem
\eqref{1.1}--\eqref{1.9} and
\begin{equation}\label{2.6}
\begin{aligned}
&\max_{0\leq t\leq T}\int_{Q}\Big(\alpha_{\tau}^2
|\frac{\partial^2\mathbf w^{\varepsilon}}{\partial t^2}|^2+
\alpha_{\tau}|\frac{\partial\mathbf w^{\varepsilon}}{\partial t}|^2+
\alpha_{\lambda}(1-\zeta)(1-\chi^{\varepsilon})
|\mathbb{D}(x,\mathbf{w}^{\varepsilon})|^2\Big)dx\\
&+\int_{Q_T}\Big(|p^{\varepsilon}|^2+
\alpha_{\mu}\big(\zeta+(1-\zeta)\chi^{\varepsilon}\big)
|\mathbb{D}(x,\frac{\partial\mathbf{w}^{\varepsilon}}{\partial
t})|^2\Big)\,dx\,dt\leqslant C_0,
\end{aligned}
\end{equation}
here and in what follows, we denote as $C_0$ any constant
independent of $\tau_0$ and $\varepsilon$.
\end{theorem}

\begin{theorem}\label{theorem2}
Under the conditions of Theorem \ref{theorem1} let
\[
\mu_0=0,\quad 0<\mu_1,\quad \tau_0<\infty,\quad
\lambda_0=\infty
\]
$\{\mathbf{w}^{\varepsilon},p^{\varepsilon}\}$ be the weak
solution of the problem \eqref{1.1}--\eqref{1.9} and
$\mathbf{w}_{s}^{\varepsilon}=\mathbb{E}_{\Omega^{\varepsilon}_{s}}
\big(\mathbf{w}^{\varepsilon}\big)$ be an extension
\eqref{2.4} from the domain $\Omega^{\varepsilon}_{s}$ onto the
domain $\Omega$.

Then the sequences  $\{\zeta p^{\varepsilon}\}$,
$\{\zeta \mathbf{w}^{\varepsilon}\}$,
$\{\zeta\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}\}$,
$\{\zeta\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}\}$,
$\{(1-\zeta)\chi^{\varepsilon}\,p^{\varepsilon}\}$,
$\{(1-\zeta)\chi^{\varepsilon}\,\mathbf{w}^{\varepsilon}\}$,
$\{(1-\zeta)\chi^{\varepsilon}\,\frac{\partial\mathbf{w}^{\varepsilon}}
{\partial t}\}$ and
$\{(1-\zeta)\chi^{\varepsilon}\,\frac{\partial^2\mathbf{w}^{\varepsilon}}
{\partial t^2}\}$ converge weakly in $L_2(Q_T)$ and
$\mathbf{L}_2(Q_T)$ to the functions $p\in{W}^{1,0}_2(\Omega^0_T)$,
$\mathbf{w}$, $\zeta\mathbf{v}$,
$\zeta\frac{\partial\mathbf{v}} {\partial t}$,
$(1-\zeta)m\,p^{(f)}\in{W}^{1,0}_2(\Omega_T)$,
$(1-\zeta)\mathbf{w}^{(f)}$,
$(1-\zeta)\mathbf{v}^{(f)}$ and
$(1-\zeta)\frac{\partial\mathbf{v}^{(f)}}{\partial t}$ respectively as
 $\varepsilon\to 0$, and
the sequence $\{\mathbf{w}_{s}^{\varepsilon}\}$  converges
strongly in $\mathbf{W}^{1,0}_2(\Omega_T)$ to zero
as $\varepsilon\to 0$.

The limiting pressure $p$ and the limiting velocity
$\mathbf{v}$ of the liquid in the domain $\Omega_0$ satisfy
in $\Omega_0$ for $t>0$ the system
\begin{equation}\label{2.7}
\tau_0\varrho_{f}\frac{\partial\mathbf{v}}{\partial t}+
\nabla p=\varrho_{f}\,\mathbf{e},\quad
\nabla\cdot\mathbf{v}=0
\end{equation}
In the domain $\Omega$ for $t>0$ limiting functions solve the
homogenized system, consisting of the continuity equation
\begin{equation}\label{2.8}
\nabla\cdot\,\mathbf{v}^{(f)}=0,
\end{equation}
and the homogenized momentum balance equation
\begin{equation}\label{2.9}
\mathbf{v}^{(f)}=
\int_0^{t}\mathbb{B}^{(f)}(\tau_0;t-\tau)\cdot
\big(-\nabla p^{(f)}\,(\mathbf{x},\tau)+\varrho_{f}\mathbf{e}\big)d\tau.
\end{equation}
The problem is complemented with the continuity conditions
\begin{gather}\label{2.10}
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\quad \mathbf{x}\in\Omega}
p^{(f)}(\mathbf{x},t)=
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\quad \mathbf{x}\in \Omega_0}
p(\mathbf{x},t), \\
\label{2.11}
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\quad \mathbf{x}\in\Omega}
\mathbf{v}^{(f)}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x}^0)=
\lim_{\mathbf{x}\to \mathbf{x}^0\in S^0,\quad \mathbf{x}\in\Omega_0}
\mathbf{v}(\mathbf{x},t) \cdot\mathbf{n}(\mathbf{x}^0)
\end{gather}
on the common boundary $S^0$, the boundary condition
\begin{equation}\label{2.12}
p(\mathbf{x},t)=p^0(t)
\end{equation}
on the part $S_0^1=S^1\cap\overline{\Omega}_0$ of the
outer boundary $S$, the boundary condition
\begin{equation} \label{2.13}
\mathbf{v}^{(f)}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x})=0
\end{equation}
on the part $S^2$ of the outer boundary $S$, the boundary condition
\begin{equation} \label{2.14}
p^{(f)}(\mathbf{x},t)=p^0(t)
\end{equation}
on the part $S_1^1$ of the outer boundary $S$, and homogeneous initial conditions
\begin{equation} \label{2.15}
\mathbf{v}(\mathbf{x},0)=0,\quad \mathbf{x}\in \Omega_0.
\end{equation}
In \eqref{2.7}--\eqref{2.15},
$\mathbf{n}(\mathbf{x})$ is a unit normal to $S^0$
(or $S^2$) at $\mathbf{x}\in S^0$ (or $S^2$),
\[
\hat{\varrho}=m\,\varrho_{f}+(1-m)\,\varrho_{s},\quad
m=\int_{Y}\chi(\mathbf{y})dy,
\]
and the symmetric matrix $\mathbb{B}^{(f)}(\tau_0;t)$ is given 
by \eqref{4.14} below.

Finally, for the solution $\mathbf{v}$, $p$, $\mathbf{v}^{(f)}$, 
and $p^{(f)}$ of the problem \eqref{2.7}--\eqref{2.15} satisfy the estimate
\begin{equation}\label{2.16}
\begin{aligned}
&\int_{\Omega^0_T}\big(\tau_0^2|\frac{\partial\mathbf v}{\partial t}|^2+
\tau_0|\mathbf{v}|^2+|\nabla p|^2\big)\,dx\,dt\\
&+\int_{\Omega_T}\big(\tau_0^2|\frac{\partial\mathbf {v}^{(f)}}{\partial t}|^2+
|\mathbf {v}^{(f)}|^2+|\nabla p^{(f)}|^2\big)\,dx\,dt \leqslant C_0.
\end{aligned}
\end{equation}
\end{theorem}

\begin{theorem}\label{theorem3}
Under the conditions of Theorem \ref{theorem2}, let
$\{\mathbf{v}^{(f,k)},p^{(f,k)},p^{k}\}$
be a solution of  \eqref{2.7}--\eqref{2.15}
with $\tau_0=\frac{1}{k}$.

Then the sequence $\{p^{(f,k)}\}$ converges weakly in $W^{1,0}_2(\Omega_T)$ 
to the function $p^{(f)}$, the sequence  $\{\mathbf{v}^{(f,k)}\}$
converges weakly in $L_2(\Omega_T)$  to the function $\mathbf{v}^{(f)}$, and
the sequence $\{p^{k}\}$  converges  strongly
in $W^{1,0}_2(\Omega^0_T)$ to the function 
$p_0(\mathbf{x},t)=p^0(t)-\varrho_{f}x_{3}$  as $k\to \infty$.

In the domain $\Omega$ for $t>0$ limiting functions solve the
homogenized system, consisting of the continuity equation
\begin{equation}\label{2.17}
\nabla\cdot\,\mathbf{v}^{(f)}=0
\end{equation}
and  Darcy's law
\begin{equation}\label{2.18}
\mathbf{v}^{(f)}=\frac{1}{\mu_1}\mathbb{B}\cdot\big(-\nabla p^{(f)}+
\varrho_{f}\mathbf{e}\big),
\end{equation}
for the liquid component, completed with the boundary conditions
\eqref{2.13}, \eqref{2.14}, and the boundary condition
\begin{equation}\label{2.19}
p^{(f)}=p_0(\mathbf{x},t)
\end{equation}
on the common boundary $S^0$ for $t>0$.

The symmetric constant matrix $\mathbb{B}$ is given by \eqref{5.3}
below.
\end{theorem}


\section{Proof of Theorem \ref{theorem1}}

The proof of this theorem is straightforward and is based on the energy equalities
\begin{equation}\label{3.1}
\begin{aligned}
&\frac{1}{2}\int_{Q}\Big(\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}
|\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}(\mathbf{x},t)|^2+
\alpha_{\lambda}(1-\zeta)(1-\chi^{\varepsilon})
|\mathbb{D}\big(x,\mathbf{w}^{\varepsilon}(\mathbf{x},t)\big)|^2\Big)dx\\
&+\alpha_{\mu}\int_0^{t}\int_{Q}\big(\zeta+(1-\zeta)\chi^{\varepsilon}\big)
\mathbb{D}\big(x,\frac{\partial\mathbf{w}^{\varepsilon}}
{\partial\tau}(\mathbf{x},\tau)\big)|^2dxd\tau\\
&=\int_0^{t}\int_{Q}\tilde{\varrho}^{\,\varepsilon}\mathbf{e}\cdot
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial
t}(\mathbf{x},\tau)dxd\tau,
\end{aligned} \end{equation}
and
\begin{equation}\label{3.2}
\begin{aligned}
&\frac{1}{2}\int_{Q}\Big(\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}
|\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}(\mathbf{x},t)|^2+
\alpha_{\lambda}(1-\zeta)(1-\chi^{\varepsilon})|\mathbb{D}\big(x,
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}
(\mathbf{x},t)\big)|^2\Big) dx\\
&+\alpha_{\mu}\int_0^{t}\int_{Q}\big(\zeta+(1-\zeta)\chi^{\varepsilon}\big)
|\mathbb{D}\big(x,\frac{\partial^2\mathbf{w}^{\varepsilon}}
{\partial \tau^2}(\mathbf{x},\tau)\big)|^2 dxd\tau\\
&=\frac{1}{2}\int_{Q}\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}
|\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}|^2
(\mathbf{x},0)dx=I_0.
\end{aligned}
\end{equation}
We may use, for example, Galerkin's method. This method shows that
for any $t\geqslant 0$ and any divergent free function
$\mathbf{\varphi}\in W^1_2(Q)$, vanishing at
$\mathbf{x}\in S^2$, the equality
\begin{align*}
&\int_{Q}\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}
\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}(\mathbf{x},t)
\cdot\mathbf{\varphi}(\mathbf{x})dx+
\int_{Q}\big(\zeta\mathbb{P}_{f}+(1-\zeta)\mathbb{P}\big)(\mathbf{x},t):
\mathbb{D}\big(x,\mathbf{\varphi}(\mathbf{x})\big)dx\\
&=\int_{Q}\tilde{\varrho}^{\,\varepsilon}\mathbf{e}\cdot
\mathbf{\varphi}(\mathbf{x})dx
\end{align*}
holds.
For $t=0$ $\mathbb{P}_{f}+(1-\zeta)\mathbb{P}=0$, we have
\[
\int_{Q}\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}
\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}(\mathbf{x},0)
\cdot\mathbf{\varphi}(\mathbf{x}) dx=
\int_{Q}\tilde{\varrho}^{\,\varepsilon}\mathbf{e}\cdot
\mathbf{\varphi}(\mathbf{x})dx.
\]
In particular, Galerkin's method states that
$\frac{\partial^2\mathbf{w}^{\varepsilon}}
{\partial t^2}(\mathbf{x},0)$ is a divergent free function
in $Q$. Therefore, for
\begin{gather*}
\mathbf{\varphi}(\mathbf{x})=
\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}(\mathbf{x},0),
\\
\int_{Q}\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}
|\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial
t^2}(\mathbf{x},0)|^2dx=
\int_{Q}\tilde{\varrho}^{\,\varepsilon}\mathbf{e}\cdot
\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}(\mathbf{x},0)dx,
\end{gather*}
which implies
\[
\int_{Q}\alpha_{\tau}\tilde{\varrho}^{\,\varepsilon}|\frac{\partial^2
\mathbf{w}^{\varepsilon}}{\partial
t^2}(\mathbf{x},0)|^2dx \leqslant\frac{C_0}{\alpha_{\tau}}.
\]
The above relation and \eqref{3.2} provide an estimate of the
time derivative
$\frac{\partial^2\mathbf{w}^{\varepsilon}}
{\partial t^2}$ in \eqref{2.6}.

To estimate the right-hand side of \eqref{3.1} we use
representations
\[
\tilde{\varrho}^{\,\varepsilon}=\varrho_{f}+(1-\zeta)(1-\chi^{\varepsilon})
(\varrho_{s}-\varrho_{f}),\quad\mathbf{e}=-\nabla x_{3},
\]
integration by parts and the continuity equation \eqref{1.1},
\[
\varrho_{f}\int_{Q}\mathbf{e}\cdot\mathbf{w}^{\varepsilon}dx=-\varrho_{f}
\int_{Q}(\nabla x_{3})\cdot\mathbf{w}^{\varepsilon}dx=0.
\]
So,
\begin{align*}
I&=\int_0^{t}\int_{Q}\tilde{\varrho}^{\,\varepsilon}\mathbf{e}
\cdot\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}dxd\tau\\
&=-\varrho_{f} \int_{Q}(\nabla x_{3})\cdot\mathbf{w}^{\varepsilon}dx+
(\varrho_{s}-\varrho_{f})\int_{\Omega}(1-\chi^{\varepsilon})\mathbf{e}\cdot
\mathbf{w}^{\varepsilon}dx\\
&=(\varrho_{s}-\varrho_{f})\int_{\Omega}(1-\chi^{\varepsilon})\mathbf{e}\cdot
\mathbf{w}^{\varepsilon}dx.
\end{align*}
Next we apply the  H\"{o}lder inequality,
\begin{align*}
I&\leqslant(\varrho_{s}-\varrho_{f})\big(\int_{\Omega}dx\big)^{1/2}
\big(\int_{\Omega}(1-\chi^{\varepsilon})|\mathbf{w}^{\varepsilon}|^2dx
\big)^{1/2}\\
&\leqslant \frac{(\varrho_{s}-\varrho_{f})^2}{2\delta}|\Omega|+\frac{\delta}{2}
\int_{\Omega}(1-\chi^{\varepsilon})|\mathbf{w}^{\varepsilon}|^2dx
\end{align*}
and the extension
$\mathbf{w}_{s}^{\varepsilon}=\mathbb{E}_{\Omega^{\varepsilon}_{s}}
\big(\mathbf{w}^{\varepsilon}\big)$ 
(see \cite[Appendix  B, Lemma B.4.2]{AM1}) from the domain 
$\Omega^{\varepsilon}_{s}$ onto the domain $Q$ with
Friedrichs-Poincar\'{e}'s inequality:
\[
I_1=\int_{\Omega}(1-\chi^{\varepsilon})|\mathbf{w}^{\varepsilon}|^2dx=
\int_{\Omega}(1-\chi^{\varepsilon})|\mathbf{w}_{s}^{\varepsilon}|^2dx\leqslant
C\,\int_{\Omega}(1-\chi^{\varepsilon})|\nabla \mathbf{w}_{s}^{\varepsilon}|^2dx,
\]
and Korn's  inequality
\begin{align*}
\int_{\Omega}(1-\chi^{\varepsilon})|\nabla \mathbf{w}_{s}^{\varepsilon}|^2dx
&\leqslant C\,\int_{\Omega}(1-\chi^{\varepsilon})
|\mathbb{D}(x,\mathbf{w}_{s}^{ \varepsilon})|^2dx\\
&=C\int_{Q}(1-\chi^{\varepsilon})(1-\zeta)
|\mathbb{D}(x,\mathbf{w}^{\varepsilon})|^2dx.
\end{align*}
Finally one has
\[
I\leqslant\frac{(\varrho_{s}-\varrho_{f})^2}{2\delta}|\Omega|+
C\,\frac{\delta}{2}\int_{Q}(1-\chi^{\varepsilon})(1-\zeta)
|\mathbb{D}(x,\mathbf{w}^{\varepsilon})|^2dx,
\]
which together with \eqref{3.2} prove \eqref{2.6} for terms, containing 
in \eqref{3.1} and \eqref{3.2}.

The pressure $p$ is estimated as a linear functional 
(\cite[Chapter 3, Theorem 3.1]{AM1}).

\section{Proof of Theorem \ref{theorem2}}

Here we use the method of two-scale convergence, suggested by 
 Nguetseng \cite{NGU2}. This method states that any bounded in 
$L_2(Q_T)$ sequence $\{u_{n}\}$ contains two-scale convergent subsequence 
$\{u_{n_{k}}\}$, such that
\[
\int_{Q_T}u_{n_{k}}(\mathbf{x},t)\varphi
(\mathbf{x},t,\frac{\mathbf{x}}{\varepsilon})\,dx\,dt 
\to \int_{Q_T}\Big(\int_{Y}U(\mathbf{x},t,\mathbf{y})
\varphi(\mathbf{x},t,\mathbf{y})dy\Big)\,dx\,dt
\]
as $\varepsilon\to 0$ for any smooth 1--periodic in $\mathbf{y}$ 
function $\varphi(\mathbf{x},t,\mathbf{y})$. The 1-periodic in 
$\mathbf{y}$ function $U(\mathbf{x},t,\mathbf{y})\in L_2(Q_T\times Y)$ 
is called a two-scale limit of the sequence $\{u_{n_{k}}\}$.

On the basis of estimates \eqref{2.6} and Nguetseng's theorem we 
conclude that as $\varepsilon\to 0$,
\begin{gather*}
p^{\varepsilon}\to p(\mathbf{x},t) 
\quad\text{weakly and two-scale in } L_2(\Omega^0_T), 
\\
p^{\varepsilon}\chi^{\varepsilon}\to p^{(f)}(\mathbf{x},t)\chi(\mathbf{y})
 \quad\text{two-scale in }L_2(\Omega_T), 
\\
p^{\varepsilon}\chi^{\varepsilon}\rightharpoonup mp^{(f)}(\mathbf{x},t)
\quad\text{weakly in }L_2(\Omega_T),\; m=\int_{Y}\chi(\mathbf{y})dy,
\\
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}
\rightharpoonup\mathbf{v}\quad\text{weakly in }\mathbf{L}_2(Q_T),
\\
\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}
\rightharpoonup \frac{\partial\mathbf{v}}{\partial t}(\mathbf{x},t) 
\quad\text{weakly in }\mathbf{L}_2(Q_T),
\\
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}
\to \mathbf{V}(\mathbf{x},t,\mathbf{y})
\quad\text{two-scale in }\mathbf{L}_2(Q_T),
\\
\frac{\partial^2\mathbf{w}^{\varepsilon}}{\partial t^2}
\to \frac{\partial\mathbf{V}}{\partial t}(\mathbf{x},t,\mathbf{y}) 
\quad\text{two-scale in }\mathbf{L}_2(Q_T),
\\
\varepsilon\mathbb{D}\big(x,\frac{\partial\mathbf{w}^{\varepsilon}}
{\partial t}\big)\to\mathbb{D}\big(y,\mathbf{V}
(\mathbf{x},t,\mathbf{y})\big) \quad\text{two-scale in }\mathbf{L}_2(Q_T),
\\
\mathbf{w}_{s}^{\varepsilon}\to 0 \quad\text{strongly in }
\mathbf{W}^{1,0}_2(\Omega_T).
\end{gather*}
Moreover,
\begin{equation}\label{4.1}
\int_{Q_T}\int_{Y}\big(\tau_0|\mathbf{V}|^2+
\tau_0^2|\frac{\partial\mathbf{V}}{\partial t}|^2+
|\mathbb{D}\big(y,\mathbf{V})|^2\big)dy \,dx\,dt\leqslant C_0,
\end{equation}
where $C_0$ is  independent of $\tau_0$.

A general theory says that the two-scale limit usually depends on all 
variables $\mathbf{x},t,$ and $\mathbf{y}$. A detailed analysis for the 
case $\mu_0=0$ (\cite[Chapter 1]{AM1}, \cite{SP}) shows that the two-scale 
limit of pressures does not depend on the variable $\mathbf{y}$ in 
$\Omega^0_T$ and has a special form in $\Omega_T$.

The similar analysis shows that
\begin{equation}\label{4.2}
\zeta\,\mathbf{V}=\zeta\,\mathbf{v}(\mathbf{x},t).
\end{equation}
In fact, the two-scale limit in \eqref{2.1} as $\varepsilon\to 0$ with 
test functions $\mathbf{\varphi}=\mathbf{\varphi}_0
(\frac{\mathbf{x}}{\varepsilon})\psi(\mathbf{x},t)$,
where $\mathbf{\varphi}_0(\mathbf{y})$ is 1-periodic in $\mathbf{y}$ function
with $\operatorname{supp}\mathbf{\varphi}_0\subset Y_{f}$, such that
$\nabla_{y}\,\cdot\mathbf{\varphi}_0=0$ for $\mathbf{y}\in Y$  and
$\psi$ is a smooth function with $\operatorname{supp}\psi\subset\Omega^0_T$ results in
\[
\int_{\Omega^0_T}\big((A-\varrho_{f}\,\mathbf{e}\cdot\mathbf{a})\,\psi-
p\,\mathbf{a}\cdot\nabla\psi\big)\,dx\,dt=0,
\]
where
\[
A=\int_{Y}\tau_0\varrho_{f}\frac{\partial\mathbf{V}}{\partial t}\cdot
\mathbf{\varphi}_0dy
= \langle\frac{\partial\mathbf{V}}{\partial t}\cdot\mathbf{\varphi}_0\rangle.
\]
Following \cite[Appendix B, Lemma B.5.3]{AM1} for any unit vector 
$\mathbf{a}$ there exists a smooth function $\mathbf{\varphi}_0(\mathbf{y})$ 
with $\operatorname{supp}\mathbf{\varphi}_0\subset Y_{f}$, such that
$\nabla_{y}\,\cdot\mathbf{\varphi}_0=0$ for $\mathbf{y}\in Y$ and 
$\langle\mathbf{\varphi}_0\rangle=\mathbf{a}$.

Due to arbitrary choice of the test function $\psi$ and vector $\mathbf{a}$ 
the last identity means the existence of $\nabla p\in L_2(\Omega^0_T)$ and 
it may be rewritten as
\[
\int_{\Omega^0_T}\big(A-(\varrho_{f}\,\mathbf{e}-\nabla p)
\cdot\mathbf{a}\big)\,\psi \,dx\,dt=0.
\]
After reintegration with respect to variables $(\mathbf{x},t)$ 
we arrive to the integral identity
\[
\int_{Y}(\tau_0\varrho_{f}\frac{\partial\mathbf{V}}{\partial t}-
\varrho_{f}\,\mathbf{e}+\nabla p)\cdot\mathbf{\varphi}_0dy=0,
\]
which is equivalent to the differential equation
\[
\tau_0\varrho_{f}\frac{\partial\mathbf{V}}{\partial t}(\mathbf{x},t,\mathbf{y})=
\varrho_{f}\,\mathbf{e}-\nabla p(\mathbf{x},t).
\]
It proves our statement and, at the same time, the first equation in \eqref{2.7}.

Using this fact and the strong convergence of the sequence 
$\{\frac{\partial\mathbf{w}_{s}^{\varepsilon}}{\partial t}\}$ 
to zero we obtain that
\begin{align*}
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}
&= \zeta\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}+
(1-\zeta)\chi^{\varepsilon}\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}+
(1-\zeta)(1-\chi^{\varepsilon})\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}\\
&\to \zeta\mathbf{v}(\mathbf{x},t)+
(1-\zeta)\chi(\mathbf{y})\mathbf{V}(\mathbf{x},t,\mathbf{y})
\end{align*}
two-scale in  $L_2(Q_T)$ as $\varepsilon \to 0$, or
\begin{equation}\label{4.3}
\mathbf{V}(\mathbf{x},t,\mathbf{y})=\zeta\mathbf{v}(\mathbf{x},t)+
(1-\zeta)\chi(\mathbf{y})\mathbf{V}(\mathbf{x},t,\mathbf{y}).
\end{equation}
Thus, $\mathbf{V}=0$ in $Y_{s}$ and we may apply Friedrichs-Poincar\'{e}'s 
inequality, which together with \eqref{4.1} imply
\begin{equation}\label{4.4}
\int_{Q_T}\int_{Y}|\mathbf{V}|^2dy\,dx\,dt\leqslant C_0
\int_{Q_T}\int_{Y}|\mathbb{D}\big(y,\mathbf{V})|^2\big)\,dx\,dt\leqslant C_0.
\end{equation}
The two-scale limit in \eqref{2.1} as $\varepsilon\to 0$ with test functions 
$\mathbf{\varphi}=\mathbf{\varphi}_0
(\frac{\mathbf{x}}{\varepsilon})\psi(\mathbf{x},t)$,
where $\mathbf{\varphi}_0(\mathbf{y})$ is 1-periodic in $\mathbf{y}$ function, 
such that
$\nabla_{y}\cdot\mathbf{\varphi}_0=0$ for
$\mathbf{y}\in Y$, and $\operatorname{supp}\mathbf{\varphi}_0\subset
Y_{f}$ and $\psi$ is a smooth function, vanishing  at $t=T$ and at $S^2$, 
results in
\begin{equation}\label{4.5}
\begin{aligned}
&\int_{Q_T}\mathbf{a}\cdot\nabla(\,p^0\,\psi)\,dx\,dt+
\int_{\Omega^0_T}\big((\tau_0\varrho_{f}\frac{\partial\mathbf{v}}{\partial t}-
\varrho_{f}\,\mathbf{e})\cdot\mathbf{a}\,\psi-
p\,\mathbf{a}\cdot\nabla\psi)\big)\,dx\,dt\\
&+\int_{\Omega_T}\Big(\tau_0\varrho_{f}
\frac{\partial\mathbf{v}^{(f)}}{\partial t}-
\hat{\varrho}\,\mathbf{e})\cdot\mathbf{a}\,\psi+B\,\psi-
p^{(f)}\,\mathbf{a}\cdot\nabla\psi\Big)\,dx\,dt=0.
\end{aligned}
\end{equation}
Here
\[
\mathbf{a}=\langle\mathbf{\varphi}_0\rangle, \quad
\mathbf{v}^{(f)}=\langle\mathbf{V}\chi\rangle,\quad
B=\mu_1\langle\mathbb{D}\big(y,\mathbf{V}):
\mathbb{D}\big(y,\mathbf{\varphi}_0)\rangle.
\]
As before, we conclude that $\nabla p\in L_2(\Omega^0_T)$, 
$\nabla p^{(f)}\in L_2(\Omega_T)$:
\begin{equation}\label{4.6}
\int_{\Omega^0_T}|\nabla p(\mathbf{x},t)|^2\,dx\,dt+
\int_{\Omega_T}|\nabla p^{(f)}(\mathbf{x},t)|^2\,dx\,dt\leqslant C_0,
\end{equation}
hold true the continuity condition \eqref{2.10} on the common boundary $S^0$, 
boundary conditions \eqref{2.12} and \eqref{2.14} on the outer boundary 
$S$, and initial condition \eqref{2.15}.

For the function $\psi$ with a compact support in $\Omega_T$ \eqref{4.5} 
implies the integral identity
\begin{equation}\label{4.7}
\int_{Y_{f}}\big((\tau_0\varrho_{f}\frac{\partial\mathbf{V}}{\partial t}+
\nabla p^{(f)}-\hat{\varrho}\,\mathbf{e})\cdot\mathbf{\varphi}_0+
\mu_1\mathbb{D}\big(y,\mathbf{V}):
\mathbb{D}\big(y,\mathbf{\varphi}_0)\big)dy=0.
\end{equation}
The reintegration of this identity results in the differential equation
\begin{equation}\label{4.8}
\tau_0\varrho_{f}\frac{\partial\mathbf{V}}{\partial t}-
\mu_1\nabla_{y}\cdot\mathbb{D}\big(y,\mathbf{V})=
-\nabla_{y}\,\Pi-\nabla p^{(f)}+\hat{\varrho}\,\mathbf{e}
\end{equation}
in the domain $Y_{f}$ for $t>0$ and for almost all $\mathbf{x} \in \Omega$.

The term $\nabla_{y}\,\Pi$ has appeared virtue of the condition 
$\nabla_{y}\cdot\mathbf{\varphi}_0=0$ on an arbitrary function $\mathbf{\varphi}_0$.

To derive the limiting continuity equations we rewrite \eqref{1.1} 
in its equivalent form as an integral identity
\begin{equation}\label{4.9}
\int_{Q_T}\nabla\xi\,\cdot\,\Big(\zeta\,
\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}+(1-\zeta)
\big(\chi^{\varepsilon}\frac{\partial\mathbf{w}^{\varepsilon}}{\partial t}+
(1-\chi^{\varepsilon})\, \frac{\partial\mathbf{w}^{\varepsilon}_{s}}
{\partial t}\big)\Big)\,dx\,dt=0,
\end{equation}
which holds true for any smooth function $\xi$ vanishing at the part 
$S^1$ of the boundary $S$.

The limit in \eqref{4.9}  as $\varepsilon \to 0$ results in the integral identity
\begin{equation}\label{4.10}
\int_{Q_T}\nabla\xi\,\cdot\,\big(\zeta\,
\mathbf{v}+(1-\zeta)\mathbf{v}^{(f)}\big)\,dx\,dt=0,
\end{equation}
which is equivalent to the continuity equation in \eqref{2.7}, 
continuity equation \eqref{2.8}, continuity condition \eqref{2.11} 
on the common boundary $S^0$, and boundary condition \eqref{2.13}.

Finally, the limit in \eqref{4.9}  as $\varepsilon \to 0$ with test 
function $\xi$ in the form $\xi=\varepsilon\xi_0(\mathbf{x},t)
\xi_1(\frac{\mathbf{x}}{\varepsilon})$, where $\xi_1(\mathbf{y})$ is 
1-periodic smooth function and $\operatorname{supp}\xi_0\subset \Omega_T$, 
leads to the integral identity
\[
\int_{Q_T}\xi_0(\mathbf{x},t)\,\big(\int_{Y_{f}}
\nabla_{y}\xi_1(\mathbf{y})\cdot\mathbf{V}(\mathbf{x},t,\mathbf{y})dy\big)\,dx\,dt=0,
\]
and, consequently, to the differential equation
\begin{equation}\label{4.11}
\nabla_{y}\cdot\mathbf{V}=0
\end{equation}
in the domain $Y_{f}$.

The representation \eqref{4.3} and the smoothness of $\mathbf{V}$ evidently 
imply the boundary condition
\begin{equation}\label{4.12}
\mathbf{V}(\mathbf{x},t,\mathbf{y})=0,\quad
\mathbf{y}\in \gamma=\partial Y_{f}\cap Y_{s}.
\end{equation}
To find correctly $\mathbf{V}$ we complete differential equations \eqref{4.8}
 and \eqref{4.11} and boundary condition \eqref{4.12} with initial condition
\begin{equation}\label{4.13}
\mathbf{V}(\mathbf{x},0,\mathbf{y})=0,\,\,\,\mathbf{y}\in Y_{f}.
\end{equation}

Problem \eqref{4.8}, \eqref{4.11}--\eqref{4.13} for
$\tau_0=1$ has been solved in \cite[Chapter 3, Theorem 3.5]{AM1}.
 Therefore, we simply formulate the result.

\begin{lemma}\label{lemma1}
For almost all $\mathbf{x}\in\Omega$ the function
$ \mathbf{v}^{(f)}=\langle\chi\,\mathbf{V}\rangle$
satisfies equation \eqref{2.9}, where
\begin{equation}\label{4.14}
\mathbb {B}^{(f)}(\tau_0;t)=\sum_{i=1}^{3}\big(\int_{Y_{f}}
\mathbf{V}^{(f)}_{i}(\mathbf{y},t)dy\big)
\otimes\mathbf{e}_{i},
\end{equation}
and $\mathbf{V}^{(f)}_{i}$ , $i=1,2,3$, are solutions to the
periodic initial boundary value problem
\begin{gather}\label{4.15}
\tau_0\varrho_{f}\frac{\partial\mathbf{V}^{(f)}_{i}}{\partial t}=
\mu_1\nabla_{y}\cdot\mathbb{D}\big(y,\mathbf{V}^{(f)}_{i})-
\nabla_{y}\Pi^{(f)}_{i},\quad (\mathbf{y},t)\in Y_{f}\times(0,T),\\
\label{4.16}
\nabla_{y}\cdot\mathbf{V}^{(f)}_{i}=0,
\,\,(\mathbf{y},t)\in Y_{f},\quad t>0, \\
\label{4.17}
\tau_0\varrho_{f}\mathbf{V}^{(f)}_{i}(\mathbf{y},0)=
\mathbf{e}_{i},\quad \mathbf{y}\in Y_{f}, \\
\label{4.18}
\mathbf{V}^{(f)}_{i}(\mathbf{y},t)=0,\,\,\mathbf{y}\in \gamma,\,\,t>0.
\end{gather}
\end{lemma}

In \eqref{4.14} $\mathbf{a}\otimes\mathbf{b}$ is a second order tensor, 
such that
\[
(\mathbf{a}\otimes\mathbf{b})\cdot\mathbf{c}=\mathbf{a}(\mathbf{b}\cdot\mathbf{c})
\]
for any vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.
The estimate \eqref{2.16} follows from \eqref{4.1}, \eqref{4.4}, and \eqref{4.6}.

\section{Proof of Theorem \ref{theorem3}}

 Estimates \eqref{2.16}, \eqref{4.1}, \eqref{4.4}, and \eqref{4.6} imply 
the strong convergence of the sequence $\{p^{k}\}$ in  $W^{1,0}_2(\Omega^0_T)$, 
the weak compactness of the sequence $\{p^{(f,k)}\}$ in $W^{1,0}_2(\Omega_T)$, 
the weak compactness of the sequence $\{\mathbf{v}^{(f,k)}\}$ in $L_2(\Omega_T)$,
and the weak compactness of the sequences $\{\mathbf{V}^{k}\}$ and 
$\{\nabla_{y}\mathbf{V}^{k}\}$ in $L_2(\Omega_T\times Y_{f})$.

Let $p$, $p^{(f)}$, and $\mathbf{v}^{(f)}$ be the limits of above mentioned 
sequences. First of all note that in virtue of the continuity condition \eqref{2.10}
 functions $\tilde{p}^{k}$, where $\tilde{p}^{k}=p^{k}$ in $\Omega^0_T$ and 
$\tilde{p}^{k}=p^{f,k}$ in $\Omega_T$, belongs to $W^{1,0}_2(Q_T)$ and uniformly 
bounded there with respect to $k$. Therefore, the weak limit $\tilde{p}$ 
of the sequence $\{\tilde{p}^{k}\}$ in  $W^{1,0}_2(Q_T)$ coincides with $p$ 
in $\Omega^0_T$ and with $p^{(f)}$ in $\Omega_T$. 
It means that these functions $p$ and $p^{(f)}$ still satisfy the continuity 
condition \eqref{2.10} on the common boundary $S^0$ and boundary conditions
 \eqref{2.12} and \eqref{2.14} on the outer boundary $S$.

Now we note that due to strong convergence of $\{\frac{1}{k}\mathbf{v}^{k}\}$ 
to zero in $L_2(\Omega^0_T)$ the function $p$ satisfies in $\Omega^0_T$
 hydraulic's equation
\[
\nabla p=\varrho_{f}\,\mathbf{e},
\]
which together with the boundary condition \eqref{2.12} result in the equality
\[
p=p_0(\mathbf{x},t)=p^0(t)-\varrho_{f}x_{3},
\]
and the boundary condition \eqref{2.19}.

We do not know, how to pass to the limit as $k\to\infty$ in the 
equation \eqref{2.9} to get Darcy's law, but we may do it using the 
integral identity \eqref{4.7}, if we rewrite it as
\begin{equation}\label{5.1}
\int_0^{T}\int_{Y_{f}}\Big(\big(\frac{1}{k}\varrho_{f}\mathbf{V}^{k}
\frac{\partial\psi}{\partial t}
+\psi(\nabla p^{(f,k)}-\hat{\varrho}\,\mathbf{e})\big)\cdot\mathbf{\varphi}_0
+\psi\mu_1\mathbb{D}\big(y,\mathbf{V}^{k}):
\mathbb{D}\big(y,\mathbf{\varphi}_0)\Big)dydt=0
\end{equation}
with test function $\psi(t)$ vanishing at $t=0$ and $T=0$.

The limit as $k\to\infty$ results in integral identities
\begin{gather*}
\int_0^{T}\psi\Big(\int_{Y_{f}}\big((\nabla p^{(f)}-\hat{\varrho}\,\mathbf{e})
\cdot\mathbf{\varphi}_0+\psi\mu_1\mathbb{D}\big(y,\mathbf{V}):
\mathbb{D}\big(y,\mathbf{\varphi}_0)\big)dy\Big)dt=0, \\
\int_{Y_{f}}\big((\nabla p^{(f)}-\hat{\varrho}\,\mathbf{e})
\cdot\mathbf{\varphi}_0+\psi\mu_1\mathbb{D}\big(y,\mathbf{V}):
\mathbb{D}\big(y,\mathbf{\varphi}_0)\big)dy=0,
\end{gather*}
and the differential equation
\begin{equation}\label{5.2}
\mu_1\nabla_{y}\cdot\mathbb{D}\big(y,\mathbf{V})=
-\nabla_{y}\,\Pi-\nabla p^{(f)}+\hat{\varrho}\,\mathbf{e}.
\end{equation}
It is obvious that the continuity equation \eqref{4.11} and boundary 
condition \eqref{4.12} for functions $\mathbf{V}^{k}$ will remain 
valid for the limit function $\mathbf{V}$.

Problem \eqref{5.2}, \eqref{4.11}, and \eqref{4.12} is well-known 
(see \cite{SP}, \cite{AM1}) and its solution
$\mathbf{v}^{(f)}=\langle\chi\,\mathbf{V}\rangle$ is given by \eqref{2.18}, where
\begin{equation} \label{5.3}
\mathbb{B}=\sum_{i=1}^{3}\big(\int_{Y_{f}}\boldsymbol{V}^{(i)}(\mathbf{y})dy\big)
\otimes\boldsymbol{e}_{i}=\sum_{i=1}^{3}\langle\chi\boldsymbol{V}^{(i)}\rangle
\otimes\boldsymbol{e}_{i},
\end{equation}
and 1-periodic functions $\boldsymbol{V}^{(i)}(\mathbf{y})$, $i=1,2,3,$ 
solve the periodic boundary value problem
\begin{gather}\label{5.4}
\mu_1\nabla_{y}\cdot\mathbb{D}\big(y,\boldsymbol{V}^{(i)}\big) -
\nabla \Pi^{(i)} =-\boldsymbol{e}_{i}, \\
\label{5.5}
\nabla\cdot\boldsymbol{V}^{(i)}=0, \quad \mathbf{y} \in Y_{f}, \\
\label{5.6}
\boldsymbol{V}^{(i)}=0, \quad \mathbf{y}\in\gamma.
\end{gather}

\subsection*{Acknowledgments}
This research  is partially supported  by Grants N0 0691/GF and N0 0751/GF 
from the Ministry of Education and Science of Kazakhstan.

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