\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 14, 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 (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/14\hfil Short-time filtration] {Homogenized models for a short-time filtration in elastic porous media} \author[A. M. Meirmanov\hfil EJDE-2007/14\hfilneg] {Anvarbek M. Meirmanov} \address{Anvarbek M. Meirmanov \newline Department of mahtematics\\ Belgorod State University\\ ul. Pobedi 85, 308015 Belgorod, Russia} \email{meirmanov@bsu.edu.ru} \thanks{Submitted August 27, 2007. Published January 31, 2008.} \subjclass[2000]{35M20, 74F10, 76S05} \keywords{Stokes equations; Lam\'{e}'s equations; hydraulic fracturing; \hfill\break\indent two-scale convergence; homogenization of periodic structures} \begin{abstract} We consider a linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic and a characteristic time of processes is small enough. Such kind of models may describe, for example, hydraulic fracturing or acoustic or seismic waves propagation. As the results, we derive homogenized equations involving non-isotropic Stokes system for fluid velocity coupled with two different types of acoustic equations for the solid component, depending on ratios between physical parameters, or non-isotropic Stokes system for one-velocity continuum. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} \label{Introduction} In the present paper we consider a problem of joint motion of a deformable solid (elastic skeleton), perforated by system of pores (pore space) and a fluid, occupying pore space. In dimensionless variables (without primes) $$\mathbf x'=L \mathbf x, \quad t'=\tau t,\quad \mathbf w'=\frac{L^{2}}{g\tau^{2}} \mathbf w, \quad \rho'_s= \rho_0 \rho_s,\quad \rho'_f =\rho_0 \rho_f,\quad \mathbf F'=g\mathbf F,$$ differential equations of the problem in a domain $\Omega \in \mathbb{R}^3$ for the dimensionless displacement vector $\mathbf{w}$ of the continuum medium have the form \begin{gather}\label{0.1} \bar{\rho} \frac{\partial^2\mathbf w}{\partial t^2}= \mathop{\rm div} \mathbb{P} +\bar{\rho}\mathbf F, \\ \label{0.2} \mathbb {P} =\bar{\chi}\mathbb {P}^{f}+(1-\bar{\chi})\mathbb {P}^{s} ,\\ \label{0.3} \mathbb {P}^{f}=\alpha_\mu D\bigl(x,\frac{\partial\mathbf w}{\partial t}\bigr)-p_{f}\mathbb I, \\ \label{0.4} \mathbb {P}^{s}=\alpha_\lambda \mathbb {D}(x,\mathbf{w})+\alpha_\eta (\mathop{\rm div}\mathbf{w})\mathbb I, \\ \label{0.5} p_{f}+\bar{\chi} \alpha_p \mathop{\rm div}\mathbf{w}=0. \end{gather} Hereafter we use notation $$\mathbb {D}(x,\mathbf u)=(1/2)\left(\nabla\mathbf u + (\nabla\mathbf u)^T\right),\quad \bar{\rho}=\bar{\chi}\rho_f +(1-\bar{\chi})\rho_s,$$ The vector $\mathbb I$ is a unit tensor, the given function $\bar{\chi}(\mathbf x)$ is a characteristic function of the pore space, the given function $\mathbf F(\mathbf x,t)$ is a dimensionless vector of distributed mass forces, $\mathbb {P}^{f}$ is a liquid stress tensor, $\mathbb {P}^{s}$ is a stress tensor in a solid skeleton and $p_{f}$ is a liquid pressure. These differential equations (\ref{0.1})--(\ref{0.5}) mean that the the displacement vector $\mathbf{w}$ satisfies the Stokes equations in the pore space $\Omega_{f}$ and the Lame's equations in the solid skeleton $\Omega_{s}$. On the common boundary $\Gamma$ "solid skeleton-pore space" the displacement vector $\mathbf{w}$ and the liquid pressure $p_{f}$ satisfy the usual continuity condition $$\label{0.6} [\mathbf w](\mathbf x_0,t)=0,\quad \mathbf x_0\in \Gamma,\; t\geq 0$$ and the momentum conservation law in the form $$\label{0.7} [\mathbb {P}\cdot \mathbf{n}](\mathbf x_0,t)=0, \quad \mathbf x_0\in \Gamma ,\; t\geq 0,$$ where $\mathbf{n}(\mathbf x_0)$ is a unit normal to the boundary at the point $\mathbf x_0\in \Gamma$ and \begin{gather*} [\varphi](\mathbf x_0,t)=\varphi_{(s)}(\mathbf x_0,t) -\varphi_{(f)}(\mathbf x_0,t),\\ \varphi_{(s)}(\mathbf x_0,t) =\lim_{\mathbf x\to \mathbf x_0,\; \mathbf x\in \Omega_s} \varphi(\mathbf x,t),\\ \varphi_{(f)}(\mathbf x_0,t) =\lim_{\mathbf x\to \mathbf x_0,\; \mathbf x\in \Omega_f} \varphi(\mathbf x,t). \end{gather*} The problem is endowed with homogeneous initial and boundary conditions \begin{gather} \label{0.8} \mathbf w(\mathbf{x},0)=0,\quad \frac{\partial \mathbf w}{\partial t}(\mathbf{x},0)=0,\quad \mathbf x\in \Omega , \\ \label{0.9} \mathbf w(\mathbf{x},t)=0,\quad \mathbf x \in S=\partial \Omega, \quad t\geq 0. \end{gather} The dimensionless constants $\alpha_i$ $(i=\tau,\nu,\ldots)$ are defined by the formulas $$\alpha_\mu =\frac{2\mu \tau}{L^{2}\rho_0},\quad \alpha_\lambda =\frac{2\lambda \tau^{2}}{L^{2}\rho_0},\quad \alpha_p =\rho _{f}c^{2}\frac{\tau^{2}}{L^{2}}, \quad \alpha_\eta =\frac{\eta\tau^{2}}{L^{2}\rho_0},$$ where $\mu$ is the viscosity of fluid, $\lambda$ and $\eta$ are elastic Lam\'{e}'s constants, $c$ is a speed of sound in fluid, $L$ is a characteristic size of the domain in consideration, $\tau$ is a characteristic time of the process, $\rho_f$ and $\rho_s$ are respectively mean dimensionless densities of liquid and rigid phases, correlated with mean density of water and $g$ is the value of acceleration of gravity. The corresponding mathematical model, describing by the system \eqref{0.1}--\eqref{0.9} is commonly accepted (see \cite{B-K,S-P}) and contains a natural small parameter $\varepsilon$, which is a characteristic size of pores $l$ divided by the characteristic size $L$ of the entire porous body: $$\varepsilon =\frac{l}{L}.$$ Our aim is to derive all possible limiting regimes (homogenized equations) as $\varepsilon\searrow 0$. Such an approximation significantly simplifies the original problem and at the same time preserves all of its main features. But even this approach is too hard to work out, and some additional simplifying assumptions are necessary. In terms of geometrical properties of the medium, the most appropriate is to simplify the problem postulating that the porous structure is periodic. We accept the following constraints \begin{assumption} \label{assumption1} \rm The domain $\Omega =(0,1)^3$ is a periodic repetition of an elementary cell $Y^\varepsilon =\varepsilon Y$, where $Y=(0,1)^3$ and quantity $1/\varepsilon$ is integer, so that $\Omega$ always contains an integer number of elementary cells $Y^\varepsilon$. Let $Y_s$ be a "solid part" of $Y$, and the "liquid part" $Y_f$ -- is its open complement. We denote as $\gamma =\partial Y_f \cap \partial Y_s$ and $\gamma$ is a Lipschitz continuous surface. A pore space $\Omega ^{\varepsilon}_{f}$ is the periodic repetition of the elementary cell $\varepsilon Y_f$, and a solid skeleton $\Omega ^{\varepsilon}_{s}$ is the periodic repetition of the elementary cell $\varepsilon Y_s$. A Lipschitz continuous boundary $\Gamma^\varepsilon =\partial \Omega_s^\varepsilon \cap \partial \Omega_f^\varepsilon$ is the periodic repetition in $\Omega$ of the boundary $\varepsilon \gamma$. The solid skeleton" $\Omega _{s}^\varepsilon$ and the pore space" $\Omega ^{\varepsilon}_{f}$ are connected domains and an intersection $\Omega ^{\varepsilon}_{f}$ with any plane \{x_{i}=\mbox{constant}, \;00;\quad {\mathbf\varphi}(\mathbf{x},T)=\frac{\partial {\mathbf \varphi}}{ \partial t}(\mathbf{x},T)=0,\;\mathbf{x}\in \Omega . $$In this definition we changed the form of representation of the stress tensor \mathbb {P} in the integral identity (\ref{1.4}) by introducing new unknown function p_{s}^{\varepsilon}, which in a certain way has a sense of pressure. In what follows we call this function p_{s}^{\varepsilon} as a solid pressure and equations (\ref{1.2}) and (\ref{1.3})-- as continuity equations. We also introduced functionals$$ \beta^{\varepsilon}= \int_{\Omega}\chi^{\varepsilon}\mathop{\rm div}\mathbf w^{\varepsilon}dx \mbox{ if } p_*+\eta_{0}=\infty \quad\mbox{and}\quad \beta^{\varepsilon}=0 \mbox{ if } p_*+\eta_{0}<\infty, which have been chosen from the conditions $$\label{1.5} \int_{\Omega}p_{f}^{\varepsilon}dx=\int_{\Omega}p_{s}^{\varepsilon}dx=0,$$ if p_*+\eta_{0}=\infty. This special choice of continuity equations permits to estimate pressures, even if p_*=\infty (incompressible liquid) or \eta_{0}=\infty (incompressible solid) and simplifies the use of homogenization procedure. In (\ref{1.3}) by A:B we denote the convolution (or, equivalently, the inner tensor product) of two second-rank tensors along the both indexes, i.e., A:B=\mbox{tr\,} (B^*\circ A)=\sum_{i,j=1}^3 A_{ij} B_{ji}. The following two theorems are the main results of the paper. \begin{theorem} \label{theorem1} Let \mathbf F and \partial \mathbf F / \partial t are bounded in L^2(\Omega_{T}). Then for all \varepsilon >0 on the arbitrary time interval [0,T] there exists a unique generalized solution of the problem \eqref{0.1}--\eqref{0.9} and \begin{gather} \label{1.6} \max_{0\leq t\leq T}\| \frac{\partial ^{2}\mathbf w^{\varepsilon}}{\partial t^{2}}(t) \|_{2,\Omega}+\|\chi^{\varepsilon}\sqrt{\alpha_\mu} \nabla \frac{\partial ^{2}\mathbf w^\varepsilon}{\partial t^{2}}|\|_{2,\Omega _{T}} \leq C_{0} , \\ \label{1.7} \max_{0\leq t\leq T}\|\chi^\varepsilon \sqrt{\alpha_\mu}|\nabla_x \frac{\partial \mathbf w^{\varepsilon}}{\partial t}(t)|+(1-\chi^\varepsilon) \sqrt{\alpha_\lambda}|\nabla_x\frac{\partial \mathbf w^{\varepsilon}}{\partial t}(t)| \|_{2,\Omega} \leq C_{0} , \\ \label{1.8} \max_{0\leq t\leq T}\||p_{f}^{\varepsilon}(t)|+ |p_{s}^{\varepsilon}(t)|\|_{2,\Omega}\leq C_{0}, \end{gather} where C_{0} does not depend on the small parameter \varepsilon . \end{theorem} \begin{theorem} \label{theorem2} Assume that the hypotheses in theorem \ref{theorem1} and restrictions \eqref{0.10} hold. Then functions \partial \mathbf w^{\varepsilon} /\partial t admit an extension \mathbf v^{\varepsilon} from \Omega_f^\varepsilon \times (0,T) into \Omega_{T} such that sequence \{\mathbf v^{\varepsilon}\} converges strongly in L^{2}(\Omega_{T}) and weakly in L^{2}((0,T);W^1_2(\Omega)) to the function \mathbf v. At the same time, sequences \{\mathbf w^\varepsilon\}, \{(1-\chi^\varepsilon)\mathbf w^\varepsilon\}, \{p_{f}^{\varepsilon}\} and \{p_{s}^{\varepsilon}\} converge weakly in L^{2}(\Omega_{T}) to \mathbf w, \mathbf u_{s}, p_{f} and p_{s}, respectively. (I) If \lambda _{1}=\infty , then \partial\mathbf u_{s}/\partial t=(1-m)\mathbf v=(1-m)\partial\mathbf w/\partial t and weak and strong limits p_{f}, p_{s} and \mathbf v satisfy in \Omega_{T} the initial-boundary value problem \begin{gather} \label{1.9} \begin{aligned} &\hat{\rho}\frac{\partial \mathbf v}{\partial t}+\nabla(p_{f}+p_{s})-\hat{\rho}\mathbf F\\ &= \mathop{\rm div}\{\mu_{0}\mathbb A^{f}_{0} :\mathbb {D}(x,\mathbf v)+\mathbb B^{f}_{0}p_{s} +\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+ \int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau\}, \end{aligned} \\ \label{1.10} \begin{aligned} &p_{*}^{-1}\partial p_{f} /\partial t+\mathbb C^{f}_{0}:\mathbb{D}(x,\mathbf v) + a^{f}_{0}p_{s} \\ &+ (a^{f}_{1}+m)\mathop{\rm div}\mathbf v +\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau=0, \end{aligned}\\ \label{1.11} \frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t} +\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial t}+\mathop{\rm div}\mathbf v=0, \end{gather} where \hat{\rho}=m \rho_{f} + (1-m)\rho_{s} is the average density of the mixture, m=\int_{Y}\chi dy is a porosity and the symmetric strictly positively defined constant fourth-rank tensor \mathbb A^{f}_{0}, matrices \mathbb C^{f}_{0}, \mathbb B^{f}_{0}, \mathbb B^{f}_{1} and \mathbb B^{f}_{2}(t) and scalars a^{f}_{0}, a^{f}_{1} and a^{f}_{2}(t) are defined below by formulas \eqref{0.10}, \eqref{4.32} and \eqref{4.34}, where \mathbb B^{f}_{1}=0, a^{f}_{1}=0 if p_{*}<\infty, and \mathbb B^{f}_{2}=0, a^{f}_{2}=0 if p_{*}=\infty. Differential equations \eqref{1.9} are endowed with homogeneous initial and boundary conditions $$\label{1.12} \mathbf v(\mathbf x,0)=0,\quad \mathbf x\in \Omega; \quad \mathbf v(\mathbf x,t)=0, \quad \mathbf x\in S, \quad t>0.$$ (II) If \lambda _{1}<\infty , then weak and strong limits \mathbf u_{s}, p_{f}, p_{s} and \mathbf v satisfy in \Omega_{T} the initial-boundary value problem, which consists of Stokes like system \begin{gather}\label{1.13} \begin{aligned} & \rho_{f}m\frac{\partial \mathbf v}{\partial t}+\rho_{s}\frac{\partial ^2\mathbf u_{s}}{ \partial t^2} + \nabla (p_{f}+p_{s})-\hat{\rho}\mathbf F\\ &=\mathop{\rm div}\{\mathbb B^{f}_{0}p_{s}+\mu_{0}\mathbb A^{f}_{0}:\mathbb {D}(x,\mathbf v)+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v +\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau\}, \end{aligned} \\ \label{1.14} \begin{aligned} &p_{*}^{-1}\partial p_{f} /\partial t+\mathbb C^{f}_{0}:\mathbb {D}(x,\mathbf v)+ a^{f}_{0}p_{s}\\ &+ (a^{f}_{1}+m)\mathop{\rm div}\mathbf v +\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau=0, \end{aligned} \end{gather} for the liquid component, coupled with the continuity equation $$\label{1.15} \frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t} +\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial t}+\mathop{\rm div} \frac{\partial\mathbf u_{s}}{\partial t} +m\mathop{\rm div}\mathbf v=0,$$ the relation $$\label{1.16} \begin{gathered} \frac{\partial \mathbf u_{s}}{\partial t}=(1-m)\mathbf v(\mathbf x,t)+\int_{0}^{t}\mathbb B^{s}_{1}(t-\tau)\cdot \mathbf z(\mathbf x,\tau )d\tau, \\ \mathbf z(\mathbf x,t)=-\frac{1}{1-m}\nabla p_{s}(\mathbf x,t) +\rho_{s}\mathbf F(\mathbf x,t)-\rho_{s}\frac{\partial \mathbf v}{\partial t}(\mathbf x,t) \end{gathered}$$ in the case of \lambda_{1}>0, or the balance of momentum equation in the form $$\label{1.17} \rho_{s}\frac{\partial^{2}\mathbf u_{s}}{\partial t^{2}}=\rho_{s}\mathbb B^{s}_{2}\cdot \frac{\partial \mathbf v}{\partial t}+((1-m)I-\mathbb B^{s}_{2})\cdot(-\frac{1}{1-m}\nabla p_{s}+\rho_{s}\mathbf F)$$ in the case of \lambda_{1}=0 for the solid component. The problem is supplemented by boundary and initial conditions (\ref{1.12}) for the velocity \mathbf v of the liquid component and by the homogeneous initial conditions $$\label{1.18} \mathbf u_{s}(\mathbf x,0)=\frac{\partial\mathbf u_{s}}{\partial t}(\mathbf x,0)=0, \quad (\mathbf x,t) \in \Omega$$ and homogeneous boundary condition $$\label{1.19} \mathbf u_{s}(\mathbf x,t)\cdot \mathbf n(\mathbf x)=0, \quad (\mathbf x,t) \in S, \quad t>0,$$ for the displacements \mathbf u_{s} of the solid component. In \eqref{1.16}--\eqref{1.19} \mathbf n(\mathbf x) is the unit normal vector to S at a point \mathbf x \in S, and matrices \mathbb B^{s}_{1}(t) and \mathbb B^{s}_{2} are given below by \eqref{4.38} and \eqref{4.40}, where the matrix ((1-m)\mathbb I - \mathbb B^{s}_{2}) is symmetric and strictly positively definite. \end{theorem} \section{Preliminaries} \label{Preliminaries} \subsection{Nguetseng's theorem} Justification of theorem \ref{theorem2} relies on systematic use of the method of two-scale convergence, which had been proposed by G. Nguetseng \cite{NGU} and has been applied recently to a wide range of homogenization problems (see, for example, the survey \cite{LNW}). \begin{definition} \label{TS} \rm A sequence \{w^\varepsilon\}\subset L^2(\Omega_{T}) is said to be \textit{two-scale convergent} to a 1- periodic in \mathbf{y} function W (\mathbf{x},\mathbf{y},t)\in L^2(\Omega_{T}\times Y), if and only if for any 1-periodic in \mathbf y function \sigma=\sigma(\mathbf x,t,\mathbf y) $$\label{2.1} \int_{\Omega_{T}} w^\varepsilon(\mathbf x,t) \sigma\big(\mathbf x,t,\frac{\mathbf x}{\varepsilon}\big)\,dx\,dt\to\int _{\Omega_{T}}\int_Y W(\mathbf x,t,\mathbf y)\sigma(\mathbf x,t,\mathbf y)dy \,dx\,dt$$ as \varepsilon\to 0. \end{definition} Existence and main properties of weakly convergent sequences are established by the following fundamental theorem \cite{NGU,LNW}. \begin{theorem}[Nguetseng's theorem] \label{theorem3} \begin{enumerate} \item Any bounded in L^2(\Omega_{T}) sequence contains a subsequence, two-scale convergent to some limit W\in L^2(\Omega_{T}\times Y). \item Let sequences \{w^\varepsilon\} and \{\varepsilon \nabla_x w^\varepsilon\} be uniformly bounded in L^2(\Omega_{T}). Then there exist a 1-periodic in \mathbf y function W=W(\mathbf x,t,\mathbf y) and a subsequence \{w^\varepsilon\} such that W,\, \nabla_y W\in L^2(\Omega_{T}\times Y), and the subsequences \{w^\varepsilon\} and \{\varepsilon \nabla w^\varepsilon\} two-scale converge to W and \nabla_y W, respectively. \item Let sequences \{w^\varepsilon\} and \{\nabla w^\varepsilon\} be bounded in L^2(Q). Then there exist functions w\in L^2(\Omega_{T}) and W \in L^2(\Omega_{T}\times Y) and a subsequence from \{\nabla w^\varepsilon\} such that the function W is 1-periodic in \mathbf y, \nabla w \in L^2(\Omega_{T}), \nabla_y W \in L^2(\Omega_{T}\times Y), and the subsequence \{\nabla w^\varepsilon\} two-scale converge to the function (\nabla w(\mathbf x,t)+\nabla_y W(\mathbf x,t,\mathbf y)). \end{enumerate} \end{theorem} \begin{corollary} \label{corollary2.1} Let \sigma\in L^2(Y) and \sigma^\varepsilon(\mathbf x)=\sigma(\mathbf x/\varepsilon). Assume that a sequence \{w^\varepsilon\}\subset L^2(\Omega_{T}) two-scale converges to W \in L^2(\Omega_{T}\times Y). Then the sequence \{\sigma^\varepsilon w^\varepsilon\} two-scale converges to the function \sigma W. \end{corollary} \subsection{An extension lemma} The typical difficulty in homogenization problems, like problem \eqref{0.1}--\eqref{0.9}, while passing to a limit as \varepsilon \searrow 0 arises because of the fact that the bounds on the gradient of displacement \nabla_x \mathbf w^\varepsilon may be distinct in liquid and rigid components. The classical approach in overcoming this difficulty consists of constructing of extension to the whole \Omega of the displacement field defined merely on \Omega_s or \Omega_f. The following lemma is valid due to the well-known results from \cite{ACE,JKO}. We formulate it in appropriate for us form: \begin{lemma} \label{lemma2.1} Suppose that assumption \ref{assumption1} on geometry of periodic structure holds, w^\varepsilon\in W^1_2(\Omega^\varepsilon_f) and w^\varepsilon =0 on S_{f}^{\varepsilon}=\partial\Omega ^\varepsilon_f \cap \partial \Omega in the trace sense. Then there exists a function w_{f}^\varepsilon \in W^1_2(\Omega) such that its restriction on the sub-domain \Omega^\varepsilon_f coincide with w^\varepsilon, i.e., $$\label{2.2} \chi^\varepsilon(\mathbf x)(w_{f}^\varepsilon(\mathbf x)-w^\varepsilon (\mathbf x))=0,\quad \mathbf x\in\Omega,$$ and, moreover, the estimate $$\label{2.3} \|w_{f}^\varepsilon\|_{2,\Omega}\leq C\|w^\varepsilon\|_{2,\Omega ^{\varepsilon}_{f}} , \quad \|\nabla w_{f}^\varepsilon\|_{2,\Omega} \leq C \|\nabla w^\varepsilon\|_{2,\Omega ^{\varepsilon}_{f}}$$ hold true, where the constant C depends only on geometry Y and does not depend on \varepsilon. \end{lemma} \subsection{Friedrichs--Poincar\'{e}'s inequality in periodic structure} The following lemma was proved by Tartar in \cite[Appendix]{S-P}. It specifies Friedrichs--Poincar\'{e}'s inequality for \varepsilon-periodic structure. We formulate this lemma for our particular case just to estimate functions in the \varepsilon--layer Q^{\varepsilon} of the boundary S. This domain Q^{\varepsilon} consists of all elementary cells \varepsilon Y touching the boundary \partial\Omega. We consider special class of functions w_{f}^\varepsilon, which are extensions of functions w^{\varepsilon}\in W^1_2(\Omega^\varepsilon_f), vanishing on the part S_{f}^{\varepsilon} =\partial\Omega^\varepsilon_f \cap\partial \Omega of the boundary S=\partial\Omega, from subdomain \Omega^\varepsilon_f onto whole domain \Omega (see lemma \ref{lemma2.1}). Due to supposition on the structure of the pore space, the intersection of the boundary of the "liquid part" Y_f with each sides of the boundary \partial Y is a set with nonempty interior and strictly positive measure. Therefore on the each side of the boundary S the function w_{f}^\varepsilon is equal to zero on some set with nonempty interior, periodic structure and strictly positive measure, independent of \varepsilon. \begin{lemma} \label{lemma2.2} Suppose that assumptions on the geometry of \Omega^\varepsilon_f hold true. Then for any function w_{f}^\varepsilon\in W^1_2(\Omega) such that w_{f}^\varepsilon=0 on the part S_{f}^{\varepsilon} =\partial\Omega^\varepsilon_f \cap\partial \Omega of the boundary S, the inequality $$\label{2.4} \int_{Q^{\varepsilon}} |w_{f}^\varepsilon|^2 dx \leq C \varepsilon^2 \int_{Q^{\varepsilon}} |\nabla w_{f}^\varepsilon|^2 dx$$ holds true with some constant C independent of the small parameter \varepsilon. \end{lemma} \subsection{Some notation} Further we denote (1) \begin{gather*} \langle\Phi \rangle_{Y} =\int_Y \Phi dy, \quad \langle\Phi \rangle_{Y_{f}} =\int_Y \chi \Phi dy, \quad \langle\Phi \rangle_{Y_{s}} =\int_Y (1-\chi )\Phi dy, \\ \langle\varphi \rangle_{\Omega } =\int_{\Omega } \varphi dx, \quad \langle\varphi \rangle_{\Omega_{T}} =\int_{\Omega_{T}} \varphi \,dx\,dt. \end{gather*} (2) If \mathbf{a} and \mathbf{b} are two vectors then the matrix \mathbf{a}\otimes \mathbf{b} is defined by the formula(\mathbf{a}\otimes \mathbf{b})\cdot \mathbf{c}=\mathbf{a}(\mathbf{b}\cdot \mathbf{c})$$for any vector \mathbf{c}. (3) If \mathbb B and \mathbb C are two matrices, then \mathbb B\otimes \mathbb C is a forth-rank tensor such that its convolution with any matrix \mathbb A is defined by the formula$$ (\mathbb B\otimes \mathbb C):\mathbb A=\mathbb B (\mathbb C:\mathbb A). $$(4) By \mathbb I^{ij}={\mathbf e}_i \otimes {\mathbf e}_j we denote the 3\times 3-matrix with just one non-vanishing entry, which is equal to one and stands in the i-th row and the j-th column. (5) We also introduce$$ \mathbb J^{ij}=\frac{1}{2}(\mathbb I^{ij}+\mathbb I^{ji}) =\frac{1}{2} ({\mathbf e}_i \otimes {\mathbf e}_j + {\mathbf e}_j \otimes {\mathbf e}_i),\quad \mathbb J=\sum_{i,j=1}^{3}\mathbb J^{ij}\otimes \mathbb J^{ij}, where ({\mathbf e}_1, {\mathbf e}_2, {\mathbf e}_3) are the standard Cartesian basis vectors. \section{Proof of theorem \ref{theorem2}} Estimates (\ref{1.6})-(\ref{1.7}) follow from the energy equality in the form \begin{aligned} &\frac{d}{dt}\{\int_{\Omega}\rho^{\varepsilon}(\frac{\partial ^{2}\mathbf w^\varepsilon}{\partial t^{2}})^{2}dx+ \alpha_\lambda\int_{\Omega}(1-\chi^{\varepsilon})\mathbb D(x,\frac{\partial\mathbf w^{\varepsilon}}{\partial t}):\mathbb D(x,\frac{\partial\mathbf w^{\varepsilon}}{\partial t})dx \\ &+\alpha_p\int_{\Omega}\chi^{\varepsilon}(\mathop{\rm div}\frac{\partial\mathbf w^{\varepsilon}}{\partial t})^{2}dx+\alpha_\eta\int_{\Omega} (1-\chi^{\varepsilon})(\mathop{\rm div}\frac{\partial\mathbf w^{\varepsilon}}{\partial t})^{2}dx\} \\ &+ \alpha_\mu\int_{\Omega}\chi^{\varepsilon}\mathbb D(x,\frac{\partial^{2}\mathbf w^{\varepsilon}}{\partial t^{2}}):\mathbb D(x,\frac{\partial^{2}\mathbf w^{\varepsilon}}{\partial t^{2}})dx\\ &=\int_{\Omega} \frac{\partial \mathbf{F}}{\partial t}\cdot\frac{\partial ^{2}\mathbf w^\varepsilon}{\partial t^{2}}dx\\ &\quad + \frac{\partial \beta^{\varepsilon}}{\partial t}\big(\frac{\alpha_p}{m}\int_{\Omega}\chi^{\varepsilon} \mathop{\rm div}\frac{\partial^{2}\mathbf w^{\varepsilon}}{\partial t^{2}}dx+\frac{\alpha_\eta}{(1-m)}\int_{\Omega} (1-\chi^{\varepsilon})\mathop{\rm div}\frac{\partial^{2}\mathbf w^{\varepsilon}}{\partial t^{2}}dx\big). \end{aligned} \label{3.1} We obtain this equality if we differentiate equation for \mathbf w^{\varepsilon} with respect to time, multiply the result by \partial ^{2} \mathbf w^{\varepsilon}/\partial t^{2} and integrate the product by parts using continuity equations (\ref{1.2}) and (\ref{1.3}). Note, that all terms on the common interface \Gamma^{\varepsilon} "solid skeleton--pore space" disappear due to boundary conditions \eqref{0.6}--\eqref{0.7}. In fact, if p_{*}+\eta_0<\infty (\beta^{\varepsilon}=0), then we just use H\"{o}lder and Gronwall inequalities in (\ref{3.1}) and get \label{3.2} \begin{aligned} &\max_{00\} and functions p_{f}, p_{s} and \mathbf w such that p_{f}^\varepsilon \to p_{f}, \quad p_{s}^\varepsilon \to p_{s},\quad \mathbf w^\varepsilon \to \mathbf w $$weakly in L^2(\Omega_T) as \varepsilon\searrow 0. Relabeling if necessary, we assume that the sequences converge themselves. At the same time $$\label{4.1} (1-\chi^\varepsilon )\alpha_\lambda \mathbb {D}(x,\mathbf w^\varepsilon) \to 0.$$ strongly in L^2(\Omega_T) and the sequence \{\mathop{\rm div}\mathbf w^\varepsilon \} converges weakly in L^2(\Omega_T) to \mathop{\rm div}\mathbf w as \varepsilon\searrow 0. Moreover, due to extension lemma \ref{lemma2.1} there are functions$$ \mathbf v^\varepsilon \in L^\infty (0,T;W^1_2(\Omega)) $$such that \mathbf v^\varepsilon =\partial \mathbf w^\varepsilon / \partial t in \Omega_{f}\times (0,T), v^{\varepsilon}=0 on the part S^{\varepsilon}_{f} of the boundary S and \begin{gather} \label{4.2} \|\frac{\partial\mathbf v^{\varepsilon}} {\partial t}\|_{2,\Omega_{T}}+\|\nabla \frac{\partial\mathbf v^{\varepsilon}}{\partial t}\|_{2,\Omega_{T}}\leq C_{0}, \\ \label{4.3} \max_{0\leq t\leq T}\big(\|\mathbf v^{\varepsilon}(t)\|_{2,\Omega}+\|\nabla \mathbf v^{\varepsilon}(t)\|_{2,\Omega}\big)\leq C_{0}, \end{gather} where C_{0} does not depend on the small parameter \varepsilon . \begin{lemma} \label{lemma4.1} There exist a subsequence of \{\varepsilon>0\} and function$$ \mathbf v\in L^{\infty}\big(0,T;W^1_2(\Omega)\big), $$such that \begin{enumerate} \item \mathbf v^\varepsilon (\,,t)\to \mathbf v(\,,t) weakly in W^1_2(\Omega) as \varepsilon \searrow 0 for all t\in [0,T], and \item \mathbf v(\,,t)\in {\mathaccent"7017 W}_2^1 (\Omega) for all t\in [0,T]. \end{enumerate} \end{lemma} \begin{proof} First of all note, that there are a subsequence of small parameters \{\varepsilon>0\} and function \mathbf v, such that$$ \mathbf v,\frac{\partial\mathbf v}{\partial t}\in L^{2}\big(0,T;W^1_2(\Omega)\big), $$and \mathbf v^\varepsilon (\,,t)\to \mathbf v(\,,t) weakly in L^{2}\big(0,T;W^1_2(\Omega)\big) as \varepsilon \searrow 0. Now, let \mathbf{\varphi}(\mathbf{x}) be an arbitrary smooth function, and$$ J^{\varepsilon}_{\varphi}(t)= \int_{\Omega}\Big(\big(\mathbf v^\varepsilon(\mathbf{x},t) -\mathbf v(\mathbf{x},t)\big)\cdot\mathbf{\varphi}(\mathbf{x}) +\nabla\big(\mathbf v^\varepsilon(\mathbf{x},t) -\mathbf v(\mathbf{x},t)\big)\cdot \nabla\mathbf{\varphi}(\mathbf{x})\Big)dx. $$By construction$$ \int_{0}^{T}J^{\varepsilon}_{\varphi}(t)\psi(t)dt\to 0 $$as \varepsilon \searrow 0 for any \psi\in L^{2}(0,T). The first statement of the lemma means that$$ J^{\varepsilon}_{\varphi}(t)\to 0 $$as \varepsilon \searrow 0 for all t\in [0,T]. Estimates (\ref{4.2}) and (\ref{4.3}) imply$$ \int_{0}^{T}|\frac{dJ^{\varepsilon}_{\varphi}}{dt}(t)|^{2}dt \leq C_{0}^{2}. $$Using this estimate, the initial condition J^{\varepsilon}_{\varphi}(0)=0, and the weak convergence in L^{2}(0,T) of the sequence \{J^{\varepsilon}_{\varphi}\} to zero, one may easily prove that$$ J^{\varepsilon}_{\varphi}(t)\to 0 \quad \mbox{in } C[0,T], $$which proves the first part of the lemma. To prove the second part of the lemma note, that$$ \mathbf v^\varepsilon (\,,t)\to \mathbf v(\,,t)\quad \mbox{ strongly in } \quad L^2(S)\quad \mbox{as} \quad \varepsilon \searrow 0 \quad \mbox{for all} \quad t\in [0,T]. This fact follows from the well-known imbedding theorem, which states that any weakly convergent sequence in W^{1}_{2}(\Omega) converges strongly in L^2(S). Now we use lemma \ref{lemma2.2} and estimate (\ref{2.4}) to conclude that $$\label{4.4} \max_{0\leq t\leq T}\|\mathbf v^{\varepsilon}(t)\|^{2}_{2,S}\leq \varepsilon C_{0}.$$ In fact, we may prove it for each facet separately. Considering, for example, the facet S_{3,0}=\{x_{3}=0, x^{'}= (x_{1},x_{2})\in (0,1)\times(0,1)\} one has \begin{align*} &|\mathbf v^{\varepsilon}(x^{'},0,t)|^{2}\\ &= |\mathbf v^{\varepsilon}(x^{'},x_{3},t)|^{2}+2\int_{0}^{x_{3}} \mathbf v^{\varepsilon}(x^{'},y_{3},t)\frac{\partial\mathbf v^{\varepsilon}}{\partial y_{3}}(x^{'},y_{3},t)dy_{3}\\ &\leq |\mathbf v^{\varepsilon}(x^{'},x_{3},t)|^{2}+ 2\Big(\int_{0}^{\varepsilon}|\mathbf v^{\varepsilon}(x^{'},y_{3},t)|^{2}dy_{3}\Big)^{1/2} \Big(\int_{0}^{\varepsilon}|\frac{\partial\mathbf v^{\varepsilon}}{\partial y_{3}}(x^{'},y_{3},t)|^{2}dy_{3}\Big)^{1/2} \end{align*} and consequently, after integration over S_{3,0} and interval x_{3}\in (0,\varepsilon), \varepsilon \int_{S_{3,0}}|\mathbf v^{\varepsilon}|^{2}dx^{'} \leq \int_{Q^{\varepsilon}}|\mathbf v^{\varepsilon}|^{2}dx+2\varepsilon \Big(\int_{Q^{\varepsilon}}|\mathbf v^{\varepsilon}|^{2}dx\Big)^{1/2} \Big(\int_{\Omega}|\nabla\mathbf v^{\varepsilon}|^{2}dx\Big)^{1/2}. $$Using estimates (\ref{2.4}) and (\ref{4.3}) we finally get estimate (\ref{4.4}), which means that$$ \mathbf v^\varepsilon (\,,t)\to 0 \quad \mbox{ strongly in } \ L^2(S) $$as \varepsilon \searrow 0 for all t\in [0,T] and that \mathbf v=0 on the boundary S. \end{proof} On the strength of Nguetseng's theorem, there exist 1-periodic in \mathbf y functions P_{f}(\mathbf x,t,\mathbf y), P_{s}(\mathbf x,t,\mathbf y), \mathbf W(\mathbf x,t,\mathbf y) and \mathbf V(\mathbf x,t,\mathbf y) such that the sequences \{p_{f}^\varepsilon\}, \{p_{s}^\varepsilon\}, \{\mathbf w^\varepsilon \} and \{\nabla\mathbf v^\varepsilon \} two-scale converge to P_{f}(\mathbf x,t,\mathbf y), P_{s}(\mathbf x,t,\mathbf y), \mathbf W(\mathbf x,t,\mathbf y) and \nabla \mathbf v(\mathbf x,t) +\nabla_{y}\mathbf V(\mathbf x,t,\mathbf y), respectively. \subsection{Micro- and macroscopic equations I} \begin{lemma} \label{lemma4.2} For all \mathbf x \in \Omega and \mathbf y\in Y weak and two-scale limits of the sequences \{p_{f}^\varepsilon\}, \{p_{s}^\varepsilon\}, \{\mathbf w^\varepsilon\}, and \{\mathbf v^\varepsilon\} satisfy the relations \begin{gather}\label{4.5} P_{s}=p_{s}\frac{(1-\chi)}{(1-m)}, \quad P_{f}=\chi P_{f}, \\ \label{4.6} \frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}+m\mathop{\rm div}\mathbf v+\langle \mathop{\rm div}{}_y\mathbf V\rangle_{Y_{f}}=\frac{\partial \beta }{\partial t}, \\ \label{4.7} \frac{1}{p_{*}}\frac{\partial P_{f}}{\partial t}+\chi(\mathop{\rm div}\mathbf v+ \mathop{\rm div}{}_y\mathbf V)=\frac{\chi }{m}\frac{\partial \beta}{\partial t}, \\ \label{4.8} \frac{1}{p_{*}}p_{f}+\frac{1}{\eta_{0}}p_{s}+\mathop{\rm div}\mathbf w=0, \\ \label{4.9} \mathbf w(\mathbf x,t)\cdot \mathbf n(\mathbf x)=0, \quad \mathbf x\in S,\\ \label{4.10} \mathop{\rm div}{}_y \mathbf W=0, \\ \label{4.11} \frac{\partial \mathbf W}{\partial t}=\chi \mathbf v+(1-\chi)\frac{\partial \mathbf W}{\partial t}, \end{gather} where \partial \beta /\partial t =\langle\langle \mathop{\rm div}{}_y\mathbf V\rangle_{Y_{f}}\rangle_{\Omega}, if p_{*}+\eta_{0}=\infty and \beta=0, if p_{*}+\eta_{0}<\infty and \mathbf n(\mathbf x) is the unit normal vector to S at a point \mathbf x \in S. \end{lemma} \begin{proof} To prove (\ref{4.5}), into (\ref{1.4}) we insert a test function {\mathbf \psi}^\varepsilon =\varepsilon {\mathbf \psi}\left(\mathbf x,t,\mathbf x / \varepsilon\right), where {\mathbf \psi}(\mathbf x,t,\mathbf y) is an arbitrary 1-periodic and finite on Y_s function in \mathbf y. Passing to the limit as \varepsilon \searrow 0, we get $$\label{4.12} \nabla_y P_{s}(\mathbf x,t,\mathbf y)=0, \quad \mathbf y\in Y_{s}.$$ Next, fulfilling the two-scale limiting passage in equality$$\chi^{\varepsilon}p_{s}^{\varepsilon}=0$$we arrive at \chi P_{s}=0 which along with (\ref{4.12}) justifies (\ref{4.5}). Equations (\ref{4.6})--(\ref{4.9}) appear as the results of two-scale limiting passages in (\ref{1.2})--(\ref{1.3}) with the proper test functions being involved. Thus, for example, (\ref{4.8}) and (\ref{4.9}) arise, if we consider the sum of (\ref{1.2}) and (\ref{1.3}), $$\label{4.13} \frac{1}{\alpha_{p}}p_{f}^\varepsilon +\frac{1}{\alpha_{\eta}}p_{s}^\varepsilon +\mathop{\rm div}\mathbf w^\varepsilon =\frac{1}{m(1-m)}\beta ^{\varepsilon}(\chi^\varepsilon -m);$$ multiply by an arbitrary function, independent of the fast'' variable \mathbf x/\varepsilon, and then pass to the limit as \varepsilon\searrow 0. In order to prove (\ref{4.10}), it is sufficient to consider the two-scale limiting relations in (\ref{4.13}) as \varepsilon \searrow 0 with the test functions \varepsilon \psi \left(\mathbf x / \varepsilon\right) h(\mathbf x,t), where \psi and h are arbitrary smooth functions. In order to prove (\ref{4.11}) it is sufficient to consider the two-scale limiting relations in$$ \chi ^{\varepsilon}(\frac{\partial \mathbf w^{\varepsilon}}{\partial t}-\mathbf v^{\varepsilon})=0. $$\end{proof} \begin{corollary}\label{corollary4.1} If p_{*}+\eta_{0}=\infty, then weak limits p_{f} and p_{s} satisfy relations $$\label{4.14} \langle p_{f} \rangle _{\Omega}=\langle p_{s}\rangle _{\Omega}=0.$$ \end{corollary} \begin{lemma} \label{lemma4.3} For all (\mathbf x,t) \in \Omega_{T} the relations $$\label{4.15} \mathop{\rm div}{}_y \{\mu_0\chi (\mathbb {D}(y,\mathbf V)+\mathbb {D}(x,\mathbf v))- (P_{f} +\frac{(1-\chi)}{(1-m)}p_{s})\cdot\mathbb {I}\}=0,$$ holds true. \end{lemma} \begin{proof} Substituting a test function of the form {\mathbf \psi}^\varepsilon =\varepsilon {\mathbf \psi}\left(\mathbf x,t,\mathbf x / \varepsilon \right), where {\mathbf \psi}(\mathbf x,t,\mathbf y) is an arbitrary 1-periodic in \mathbf y function vanishing on the boundary S, into integral identity (\ref{1.4}), and passing to the limit as \varepsilon \searrow 0, we arrive at (\ref{4.15}). \end{proof} \begin{lemma} \label{lemma4.4} Let \hat{\rho}=m \rho_{f} + (1-m)\rho_{s}. Then functions \mathbf u_{s}=\langle \mathbf W\rangle _{Y_{s}}, \mathbf v, p_{f} and p_{s} satisfy in \Omega_{T} the system of macroscopic equations $$\label{4.16} \rho_{f}m\frac{\partial \mathbf v}{\partial t}+ \rho_{s}\frac{\partial ^2\mathbf u_{s}}{\partial t^2}-\hat{\rho}\mathbf F =\mathop{\rm div}\{\mu _{0}(m\mathbb {D}(x,\mathbf v)+\langle \mathbb {D}(y,\mathbf V)\rangle _{Y_{f}})-(p_{f}+p_{s})\cdot\mathbb {I}\},$$ and the homogeneous initial conditions $$\label{4.17} \mathbf u_{s}(\mathbf{x},0)=\rho_{f}m\mathbf v(\mathbf{x},0)+\rho_{s}\frac{\partial \mathbf u_{s}}{\partial t}(\mathbf{x},0)=0, \quad \mathbf{x}\in \Omega .$$ \end{lemma} \begin{proof} Equations (\ref{4.16}) and initial conditions (\ref{4.17}) arise as the limit of (\ref{1.4}) with test functions being independent of \varepsilon in \Omega_T. \end{proof} \subsection*{ Micro- and macroscopic equations II} \begin{lemma}\label{lemma4.5} If \lambda_{1}=\infty, then the weak limits of \{\mathbf v^\varepsilon\} and \{\partial \mathbf w^\varepsilon /\partial t\} coincide and$$ (1-m)\mathbf v=\frac{\partial\mathbf u_{s}}{\partial t}. $$\end{lemma} \begin{proof} Let \Psi(\mathbf x,t,\mathbf y) be an arbitrary smooth function periodic in \mathbf y. The sequence \{\sigma_{ij}^{\varepsilon}\}, where$$ \sigma_{ij}^{\varepsilon}=\int_{\Omega}\sqrt{\alpha_{\lambda}} \frac{\partial w_{i}^\varepsilon}{\partial x_{j}} (\mathbf x,t)\Psi(\mathbf x,t,\mathbf x /\varepsilon )dx, \quad \mathbf w^\varepsilon=(w_{1}^\varepsilon, w_{2}^\varepsilon, w_{3}^\varepsilon ) $$is uniformly bounded in \varepsilon. Therefore,$$ \int_{\Omega}\varepsilon \frac{\partial w_{i}^\varepsilon} {\partial x_{j}} (\mathbf x,t)\Psi(\mathbf x,t,\mathbf x /\varepsilon )dx=\frac{\varepsilon}{\sqrt{\alpha_{\lambda}}} \sigma_{ij}^{\varepsilon}\to 0 $$as \varepsilon\searrow 0, which is equivalent to$$ \int_{\Omega}\int_{Y} W_{i}(\mathbf x,t,\mathbf y) \frac{\partial\Psi}{\partial y_{j}}(\mathbf x,t,\mathbf y)dxdy=0, \quad \mathbf W=(W_{1}, W_{2}, W_{3}), $$or \mathbf W(\mathbf x,t,\mathbf y)=\mathbf w(\mathbf x,t). Therefore, taking the two-scale limit as \varepsilon\searrow 0 in the equality$$ \chi^{\varepsilon}(\mathbf v^\varepsilon- \frac{\partial\mathbf w^\varepsilon}{\partial t})=0 we arrive at the first statement of the lemma. The last statement follows from the definition of \mathbf u_{s}. \end{proof} \begin{lemma} \label{lemma4.6} Let \lambda_1 <\infty. Then the weak and two-scale limits p_{s} and \mathbf W satisfy the microscopic relations \begin{gather}\label{4.18} \rho_{s}\frac{\partial ^{2}\mathbf W}{\partial t^{2}}= \lambda_{1}\Delta_y \mathbf W -\nabla_y R -\frac{1}{1-m}\nabla p_{s} +\rho_{s}\mathbf F, \quad \mathbf y \in Y_{s}, \\ \label{4.19} \frac{\partial \mathbf W}{\partial t}=\mathbf v, \quad \mathbf y \in \gamma \end{gather} in the case \lambda_{1}>0, and relations \begin{gather}\label{4.20} \rho_{s}\frac{\partial ^{2}\mathbf W}{\partial t^{2}}= -\nabla_y R -\frac{1}{1-m}\nabla p_{s}+\rho_{s}\mathbf F, \quad \mathbf y \in Y_{s}, \\ \label{4.21} (\frac{\partial \mathbf W}{\partial t}-\mathbf v)\cdot{\mathbf n}=0, \quad \mathbf y \in \gamma \end{gather} in the case \lambda_{1}=0. Differential equations (\ref{4.18}) and (\ref{4.20}) are endowed with homogeneous initial conditions $$\label{4.22} \mathbf W(\mathbf y,0)=\frac{\partial \mathbf W}{\partial t}(\mathbf y,0)=0, \quad \mathbf y \in Y_{s}.$$ In (\ref{4.21}), {\mathbf n} is the unit normal to \gamma. \end{lemma} \begin{proof} Differential equations (\ref{4.18}), (\ref{4.20}) and initial conditions (\ref{4.22}) follow as \varepsilon\searrow 0 from integral equality (\ref{1.4}) with the test function {\mathbf \psi}={\mathbf \varphi}(x\varepsilon^{-1})\cdot h({\mathbf x},t), where {\mathbf \varphi} is solenoidal and finite in Y_{s}. Boundary condition (\ref{4.19}) is a consequence of the two-scale convergence of \{\sqrt{\alpha_{\lambda}}\nabla\mathbf w^{\varepsilon}\} to the function \sqrt{\lambda_{1}}\nabla_y\mathbf W(\mathbf x,t,\mathbf y). On the strength of this convergence, the function \nabla_y \mathbf W (\mathbf x,t,\mathbf y) is L^2-integrable in Y. The boundary condition (\ref{4.21}) follows from Eqs. (\ref{4.10})-(\ref{4.11}). \end{proof} \subsection{Homogenized equations I} In this section we derive homogenized equations for the liquid component. \begin{lemma} \label{lemma4.7} If \lambda_1 =\infty then \partial \mathbf w / \partial t=\mathbf v and the weak limits \mathbf v, p_{f} and p_{s} satisfy in \Omega_{T} the initial-boundary value problem \begin{gather}\label{4.23} \begin{aligned} &\hat{\rho}\frac{\partial \mathbf v}{\partial t}+\nabla(p_{f}+p_{s})-\hat{\rho}\mathbf F\\ &= \mathop{\rm div}\{\mu _{0}\mathbb A^{f}_{0}:\mathbb {D}(x,\mathbf v) +\mathbb B^{f}_{0}p_{s}+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+ \int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau\}, \end{aligned}\\ \label{4.24} \begin{aligned} &p_{*}^{-1}\partial p_{f} /\partial t+\mathbb C^{f}_{0}:\mathbb {D}(x,\mathbf v)+ a^{f}_{0}p_{s} +(a^{f}_{1}+m)\mathop{\rm div}\mathbf v\\ &+\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau=0, \end{aligned}\\ \label{4.25} \frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t} +\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial t}+\mathop{\rm div}\mathbf v=0, \end{gather} where the symmetric strictly positively defined constant fourth-rank tensor \mathbb A^{f}_{0}, matrices \mathbb C^{f}_{0}, \mathbb B^{f}_{0}, \mathbb B^{f}_{1} and \mathbb B^{f}_{2}(t) and scalars a^{f}_{0}, a^{f}_{1} and a^{f}_{2}(t) are defined below by formulas \eqref{0.10}, \eqref{4.32} and \eqref{4.34}, where \mathbb B^{f}_{1}=0, a^{f}_{1}=0 if p_{*}<\infty, and \mathbb B^{f}_{2}=0, a^{f}_{2}=0 if p_{*}=\infty. Differential equations (\ref{4.23}) are endowed with homogeneous initial and boundary conditions $$\label{4.26} \mathbf v(\mathbf x,0)=0,\quad \mathbf x\in \Omega, \quad \mathbf v(\mathbf x,t)=0, \quad \mathbf x\in S, \quad t>0.$$ \end{lemma} \begin{proof} First note that \mathbf v =\partial \mathbf w / \partial t due to lemma \ref{lemma4.5}. The homogenized equations (\ref{4.23}) follow from the macroscopic equations (\ref{4.16}), after we insert in them the expression\mu_{0}\langle \mathbb {D}(y,\mathbf V)\rangle _{Y_{f}}= \mu_{0}\mathbb A^{f}_{1}:\mathbb {D}(x,\mathbf v)+\mathbb B^{f}_{0}p_{s}+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+ \int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau+\mathbb{A}(t).$$In turn, this expression follows by virtue of solutions of (\ref{4.7}) and (\ref{4.15}) on the pattern cell Y_{f}. In fact, if p_{*}<\infty , then setting \begin{gather*} \mathbf V=\sum_{i,j=1}^{3}\mathbf V^{(ij)}(\mathbf y)D_{ij} +\mathbf V^{(0)}(\mathbf y)p_{s}+\int_{0}^{t}\mathbf V^{(2)}(\mathbf y,t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau, \\ P_{f} =\mu_{0}\sum_{i,j=1}^{3}P^{ij}(\mathbf y)D_{ij} +P^{0}(\mathbf y)p_{s}+\int_{0}^{t}P^{(2)}(\mathbf y,t-\tau) \mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau, \end{gather*} where$$ D_{ij}(\mathbf x,t)=\frac{1}{2}(\frac{\partial v_{i}} {\partial x_{j}}(\mathbf x,t)+ \frac{\partial v_{j}}{\partial x_{i}}(\mathbf x,t)), $$we arrive at the following periodic-boundary value problems in Y: \begin{gather}\label{4.27} \mathop{\rm div}{}_y\{\chi \mathbb D(y,\mathbf V^{(ij)})-\chi P^{(ij)}\mathbb I+\chi J^{ij}\}=0, \quad \chi\mathop{\rm div}{}_y \mathbf V^{(ij)} =0; \\ \label{4.28} \mathop{\rm div}{}_y \{\mu_{0}\chi \mathbb D(y,\mathbf V^{(0)}) -\big(\chi P^{(0)}+\frac{1-\chi}{1-m}\big)\mathbb I \}=0, \quad \chi\mathop{\rm div}{}_y \mathbf V^{(0)} =0; \\ \mathop{\rm div}{}_y \{\mu_{0}\chi \mathbb D(y,\mathbf V^{(2)}) -\chi P^{(2)}\mathbb I \}=0, \\ \label{4.29} \frac{1}{p_{*}}\frac{\partial P^{(2)}}{\partial t} +\chi\mathop{\rm div}{}_y \mathbf V^{(2)} =0, \, \frac{1}{p_{*}}P^{(2)}(\mathbf y,0)=-\chi(\mathbf y). \end{gather} For the case p_{*}=\infty we put \begin{gather*} \mathbf V=\sum_{i,j=1}^{3}\mathbf V^{(ij)}(\mathbf y)D_{ij} +\mathbf V^{(0)}(\mathbf y)p_{s}+\mathbf V^{(1)}(\mathbf y)\mathop{\rm div}\mathbf v,\\ P_{f} =\sum_{i,j=1}^{3}P^{ij}(\mathbf y)D_{ij} +P^{0}(\mathbf y)p_{s}+P^{(1)}(\mathbf y)\mathop{\rm div} \mathbf v, \end{gather*} where functions \mathbf V^{(1)} and P^{(1)} satisfy in Y the following periodic-boundary value problem in Y: $$\label{4.30} \mathop{\rm div}{}_y \{\mu_{0}\chi \mathbb D(y,\mathbf V^{(1)}) -\chi P^{(1)}\mathbb I \}=0,\, \chi(\mathop{\rm div}{}_y \mathbf V^{(1)}+1) =0.$$ Note, that for all cases the functional \beta is equal to zero due to the special choice of the function \mathbf{V}, boundary condition (\ref{4.26}) for the function \mathbf{v} and conditions (\ref{4.14}). On the strength of the assumptions on the geometry of the pattern liquid'' cell Y_{f}, problems (\ref{4.27})--(\ref{4.30}) have unique solution, up to an arbitrary constant vector. In order to discard the arbitrary constant vectors we demand$$ \langle\mathbf V^{(ij)}\rangle _{Y_{f}}=\langle\mathbf V^{(0)}\rangle_{Y_{f}} =\langle\mathbf V^{(1)}\rangle_{Y_{f}}=\langle\mathbf V^{(2)}\rangle_{Y_{f}}=0. $$Thus \begin{gather}\label{4.31} \mathbb A^{f}_{0}=m\mathbb J+\mathbb A^{f}_{1}, \quad \mathbb A^{f}_{1}=\sum_{i,j=1}^{3}\langle \mathbb {D}(y,\mathbf V^{(ij)})\rangle _{Y_{f}}\otimes \mathbb J^{ij}, \\ \label{4.32} \mathbb B^{f}_{i}=\mu_{0}\langle \mathbb {D}(y,\mathbf V^{(i)})\rangle _{Y_{f}}, \quad i=0,1,2. \end{gather} Symmetry of the tensor \mathbb A^{f}_{0} follows from symmetry of the tensor \mathbb A^{f}_{1}. And symmetry of the latter one follows from the equality $$\label{4.33} \langle \mathbb {D}(y,\mathbf V^{(ij)})\rangle _{Y_{f}} : J^{kl} =-\langle \mathbb {D}(y,\mathbf V^{(ij)}) : \mathbb {D}(y,\mathbf V^{(kl)})\rangle_{Y_{f}}$$ which appears by means of multiplication of (\ref{4.27}) for \mathbf V^{(ij)} by \mathbf V^{(kl)} and by integration by parts. This equality also implies positive definiteness of the tensor \mathbb A^{f}_{0}. Indeed, let \mathbb Z=(Z_{ij}) be an arbitrary symmetric matrix. Setting \mathbf{Z}=\sum_{i,j=1}^{3}\mathbf V^{(ij)}Z_{ij} and taking into account (\ref{4.33}) we get$$ \langle \mathbb {D}(y,\mathbf{Z})\rangle _{Y_{f}}:\mathbb Z =-\langle \mathbb {D}(y,\mathbf{Z}): \mathbb {D}(y,\mathbf{Z})\rangle_{Y_{f}}. $$This equality and the definition of the tensor A_{0}^f give$$ (\mathbb A_{0}^f:\mathbb Z):\mathbb Z= \langle(\mathbb {D}(y,\mathbf{Z})+\mathbb Z): (\mathbb {D}(y,\mathbf{Z})+\mathbb Z)\rangle_{Y_{f}}. $$Now the strict positive definiteness of the tensor \mathbb A_{0}^{f} follows immediately from the equality above and the geometry of the elementary cell Y_{f}. Namely, suppose that (\mathbb A_{0}^{s}:\mathbb Z):\mathbb Z=0 for some matrix \mathbb Z, such that \mathbb Z:\mathbb Z=1. Then (\mathbb {D}(y,\mathbf{Z})+\mathbb Z)=0, which is possible if and only if \mathbf{Z} is a linear function in \mathbf y. On the other hand, all linear periodic functions on Y_{f} are constant. Finally, the normalization condition \langle\mathbf V^{(ij)}\rangle_{Y_{f}} =0 yields that \mathbf{Z}=0. However, this is impossible because the functions \mathbf V^{(ij)} are linearly independent. Equations (\ref{4.24}) and (\ref{4.25}) for the pressures follow from (\ref{4.6}), (\ref{4.8}) and equality$$ \langle \mathop{\rm div}{}_y\mathbf V\rangle_{Y_{f}} =\mathbb{C}^{f}_{0}:\mathbb {D}(x,\mathbf v)+ a^{f}_{0}p_{s} +a^{f}_{1}\mathop{\rm div}\mathbf v+ \int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau with $$\label{4.34} \mathbb C^{f}_{0}=\sum_{i,j=1}^{3}\langle \mathop{\rm div}{}_y\mathbf V^{(ij)}\rangle _{Y_{f}}\mathbb{J}^{ij}, \quad a^{f}_{i}=\langle\mathop{\rm div}{}_y\mathbf V^{(i)}\rangle _{Y_{f}},\quad i=0,1,2.$$ Finally note, that initial conditions (\ref{4.26}) follow from initial conditions (\ref{4.17}) and lemma \ref{lemma4.5}. \end{proof} \subsection{Homogenized equations II} We complete the proof of theorem \ref{theorem2} with homogenized equations for the solid component. Let \lambda_{1}<\infty. In the same manner as above, we verify that the limit \mathbf v of the sequence \{\mathbf v^\varepsilon\} satisfies the initial-boundary value problem likes (\ref{4.23})-- (\ref{4.25}). The main difference here that, in general, the weak limit \partial\mathbf w / \partial t of the sequence \{\partial\mathbf w^\varepsilon /\partial t\} differs from \mathbf v. More precisely, the following statement is true. \begin{lemma} \label{lemma4.8} Let \lambda_{1}<\infty. Then the weak limits \mathbf v, \mathbf u_{s}, p_{f}, and p_{s} of the sequences \{\mathbf v^\varepsilon\}, \{(1-\chi^{\varepsilon})\mathbf w^\varepsilon\}, \{p_{f}^\varepsilon\}, and \{p_{s}^\varepsilon\} satisfy the initial-boundary value problem in \Omega_T, consisting of the balance of momentum equation \label{4.35} \begin{aligned} &\rho_{f}m\frac{\partial \mathbf v}{\partial t}+\rho_{s} \frac{\partial ^2\mathbf u_{s}}{\partial t^2} + \nabla(p_{f}+p_{s}) -\hat{\rho}\mathbf F\\ &= \mathop{\rm div}\{\mu_{0}\mathbb A^{f}_{0}:\mathbb {D}(x,\mathbf v) + \mathbb B^{f}_{0}p_{s}+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau\}, \end{aligned} and the continuity equation (\ref{4.24}) for the liquid component, where \mathbb A^{f}_{0}, \mathbb B^{f}_{0}-- \mathbb B^{f}_{2} are the same as in (\ref{4.23}), the continuity equation $$\label{4.36} \frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}+ \frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial t}+\mathop{\rm div} \frac{\partial\mathbf u_{s}}{\partial t} +m \mbox {div}\mathbf v=0,$$ the relation \begin{gather}\label{4.37} \frac{\partial \mathbf u_{s}}{\partial t}=(1-m)\mathbf v(\mathbf x,t)+\int_{0}^{t}\mathbb B^{s}_{1}(t-\tau)\cdot \mathbf z(\mathbf x,\tau )d\tau , \\ \mathbf z(\mathbf x,t)=-\frac{1}{1-m}\nabla_x p_{s} (\mathbf x,t)+\rho_{s}\mathbf F(\mathbf x,t)- \rho_{s}\frac{\partial \mathbf v}{\partial t}(\mathbf x,t) \end{gather} in the case \lambda_{1}>0, or the balance of momentum equation in the form $$\label{4.38} \rho_{s}\frac{\partial^{2}\mathbf u_{s}}{\partial t^{2}}=\rho_{s}\mathbb B^{s}_{2}\cdot \frac{\partial \mathbf v}{\partial t}+((1-m)I-\mathbb B^{s}_{2})\cdot(-\frac{1}{1-m}\nabla p_{s}+\rho_{s}\mathbf F)$$ in the case of \lambda_{1}=0 for the solid component. The problem is supplemented by boundary and initial conditions (\ref{4.26}) for the velocity \mathbf v of the liquid component and by homogeneous initial conditions and the boundary condition $$\label{4.39} \mathbf u_{s}(\mathbf x,t)\cdot \mathbf n(\mathbf x)=0, \quad (\mathbf x,t) \in S, \quad t>0,$$ for the displacement \mathbf u_{s} of the solid component. In Eqs. (\ref{4.37})--(\ref{4.39}) \mathbf n(\mathbf x) is the unit normal vector to S at a point \mathbf x \in S, and matrices \mathbb B^{s}_{1}(t) and \mathbb B^{s}_{2} are given below by Eqs. (\ref{4.41}) and (\ref{4.43}). \end{lemma} \begin{proof} The boundary condition (\ref{4.39}) follows from (\ref{4.9}), the equality \frac{\partial\mathbf w}{\partial t} =\frac{\partial\mathbf u_{s}}{\partial t}+m\mathbf v, $$and the homogeneous boundary condition for \mathbf v. The same equality and (\ref{4.8}) imply (\ref{4.36}). The homogenized equations of balance of momentum (\ref{4.35}) derives exactly as before. Therefore we omit the relevant proofs now and focus ourself only on derivation of homogenized equation of the balance of momentum for the solid displacements \mathbf u_{s}. (a) If \lambda_{1}>0, then the solution of the system of microscopic equations (\ref{4.10}), (\ref{4.18}), and (\ref{4.19}), provided with the homogeneous initial data (\ref{4.22}), is given by formula \begin{gather*} \mathbf W=\int_{0}^{t}( \mathbf v(\mathbf x,\tau )+ \sum_{i=1}^{3}\mathbf W^{i}(\mathbf y,t-\tau)\otimes {\mathbf e}_{i}\cdot\mathbf z(\mathbf x,\tau ))d\tau , \\ R=\int_{0}^{t}\sum_{i=1}^{3}R^{i}(\mathbf y,t-\tau){\mathbf e}_{i}\cdot\mathbf z(\mathbf x,\tau )d\tau , \end{gather*} in which functions \mathbf W^{i}(\mathbf y,t) and R^{i}(\mathbf y,t) are defined by virtue of the periodic initial-boundary value problem $$\label{4.40} \begin{gathered} \rho_{s}\frac{\partial ^{2} \mathbf W^{i}}{\partial t^{2}}-\lambda_{1}\Delta \mathbf W^{i} +\nabla R^{i} =0, \quad \mathop{\rm div}{}_y \mathbf W^{i} =0, \quad \mathbf y \in Y_{s},\;t>0, \\ \mathbf W^{i}=0, \quad \mathbf y \in \gamma ,\; t>0, \\ \mathbf W^{i}(y,0)=0, \quad \rho_{s}\frac{\partial\mathbf W^{i}}{\partial t}(y,0)={\mathbf e}_{i},\quad \mathbf y \in Y_{s}. \end{gathered}$$ In \eqref{4.40}, {\mathbf e}_{i} is the standard Cartesian basis vector. Therefore, $$\label{4.41} B^{s}_{1}(t)= \sum_{i=1}^{3}\langle \frac{\partial \mathbf W^{i}}{\partial t}\rangle _{Y_{s}}\otimes {\mathbf e}_{i}(t).$$ Note, that differential equations in \eqref{4.40} are understood in the sense of distributions (the compatibility conditions on the boundary \gamma at t=0 have no place) and therefore the functions \partial\mathbf W^{i}/\partial t have no time derivative at t=0. (b) If \lambda_{1}=0 then in the process of solving the system (\ref{4.10}), (\ref{4.18}), and (\ref{4.19}) we firstly find the pressure R(\mathbf x,t,\mathbf y) by virtue of solving the Neumann problem for Laplace's equation in Y_{s} in the form$$ R(\mathbf x,t,\mathbf y)=\sum_{i=1}^{3} R_{i}(\mathbf y) {\mathbf e}_{i}\cdot \mathbf z(\mathbf x,t), $$where R^{i}(\mathbf y) is the solution of the problem $$\label{4.42} \Delta_y R_{i}=0,\quad \mathbf y \in Y_{s}; \quad \nabla_y R_{i}\cdot \mathbf n =\mathbf n\cdot{\mathbf e}_{i}, \quad \mathbf y \in \gamma .$$ Formula (\ref{4.35}) appears as the result of homogenization of (\ref{4.18}) and $$\label{4.43} B^{s}_{2}=\sum_{i=1}^{3}\langle \nabla R_{i}(\mathbf y)\rangle _{Y_{s}}\otimes {\mathbf e}_{i},$$ where the matrix ((1-m)I - B^{s}_{2}) is symmetric and positively definite. In fact, let \tilde{R}=\sum_{i=1}^{3}R_{i}\mathbf xi_{i} for any unit vector \xi. Then$$ (B\cdot\xi)\cdot\xi=\langle(\xi-\nabla\tilde{R})^{2}\rangle_{Y_{f}}>0$due to the same reasons as in lemma \ref{lemma4.6}. On the strength of the assumptions on the geometry of the pattern solid'' cell$Y_{s}\$, problem \eqref{4.40} has unique solution and problem (\ref{4.42}) has unique solution up to an arbitrary constant. \end{proof} \begin{thebibliography}{00} \bibitem{ACE} E. Acerbi, V. Chiado Piat, G. Dal Maso and D. Percivale; \emph{An extension theorem from connected sets and homogenization in general periodic domains}, Nonlinear Anal., 18 (1992), pp.~481--496. \bibitem{B-K} R. Burridge R and J.~B. Keller; \emph{Poroelasticity equations derived from microstructure}, J. Acoust. Soc. Am.,70, N4 (1981), pp.~1140--1146. \bibitem{G-M2} Th. Clopeau, J.~L. Ferrin, R.~P. Gilbert and A. Mikeli\'{c}; \emph{Homogenizing the acoustic properties of the seabed: Part II}, Mathematical and Computer Modelling, 33 (2001), pp.~821--841. \bibitem{JKO} V.~V. Jikov, S.~M. Kozlov and O.~A. 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