Mathematical models of joint filtration of liquids are the main part of mathematical models of oil displacement by suspension. Since mining is a very important and urgent economic task, exact modeling of joint filtration of two different fluids is also an urgent economic task. For example, mathematical models of oil displacement by suspension are needed to create a hydrodynamic simulator of oil by suspension. All the existing simulators are based on the macroscopic Buckley-Leverett model, which does not distinguish between the free boundary separating liquids and the details of liquid interaction. All these fundamental processes occur at a microscopic level corresponding to the average size of pores, while all proposed macroscopic models operate on completely different orders of magnitude and do not distinguish between free boundaries or the characteristics of fluid interactions and are simply a set of axioms. Exact modeling involves describing the process using the equations of classical Newtonian continuum mechanics at the microscopic level (average size of tens of micrometers). The only obstacle to using such models is that any numerical implementation for domains hundreds of meters in size would take years. A solution to this problem (the homogenization method) was proposed in the works of J. Keller and E. Sánchez-Palencia.

We consider a free boundary problem for a one-dimensional system of Buckley-Leverett equations, describing the displacement of oil by a suspension. For this problem we formulated conditions for the strong decay of the discontinuity of the initial oil concentration. We will prove that the phenomenological Buckley-Leverett model does not adequately describe the physical process under consideration. To do this, we will study the problem of the decay of a discontinuity in the initial concentration of oil, when at rest in one half of the domain there is oil, and in the other half of the domain there is a suspension, and these domains are separated by an impenetrable partition. At the initial moment of time, the partition is removed and a non-negative suspension velocity is maintained at the injection wells. An accurate analysis of the unique solution to the Buckley-Leverett model shows that at the initial moment of time, oil begins to displace the suspension, resulting in the formation of a zone of mixing of oil and suspension. If the velocity of the suspension at the injection wells is high enough, then at some point in time the natural option of displacing oil by the suspension begins to be realized.}\keywords{Free boundary problems, transport equations, displacement of oil by suspension, strong discontinuity conditions.

An initial boundary value problem for the in situ leaching is considered. We describe physical processes at the microscopic level with a pore size epsilon << 1 by model A epsilon , where dynamics of the incompressible solid skeleton is described by the Lam & eacute; equations and the physical process in the pore space by the Stokes equations for the incompressible fluid with diffusion equations for the concentration of acid and product of chemical reactions. Since the solid skeleton changes its geometry upon dissolution, the "pore space - solid skeleton" boundary is a free boundary. The goal of the present manuscript is a model IHI, which is the homogenization of the model A epsilon . That is, the limit as epsilon tend to zero, of the model A epsilon . As usual, free boundary problems are only solvable locally in time. On the other hand, in situ leaching has a very long process duration and there is still no correct microscopic model that describes this process for an arbitrary time interval. To avoid this contradiction, we propose correct approximate microscopic models B epsilon (r) for this process with a given solid skeleton structure depending on some function r from the set M (0 ,T ) . Problem B epsilon(r) is the model A epsilon without an additional boundary condition at the free boundary that defines this boundary, but with some additional terms in the Stokes and Lame equations that depend linearly on the velocities and disappear upon homogenization. To derive a macroscopic mathematical model IHI(r) and separately the additional boundary condition at free boundary we use Nguenseng's two-scale convergence method as epsilon tends to zero. As a result, we obtain a homogenized model IHI(r) and an additional equation, possesses construct an operator, which fixed point uniquely defines function r & lowast; from the set M (0 ,T ) and prove the existence and uniqueness theorem for the macroscopic mathematical model IHI.

МЕТОД ДВУХМАСШТАБНОГО РАЗЛОЖЕНИЯ В ЗАДАЧЕ

In the article we deal with some physical processes in rock mechanics, which are described by free-boundary problems. Some of them are well known (Muskat problems), some of them are completely new (in-situ leaching and dynamics of cracks in underground rocks).

We consider the initial-boundary value problem for in-situ leaching process of rare metals at the microscopic level. This physical process is described by the Stokes equations for the liquid component coupled with the Lamé equations for the solid skeleton and the diffusion–convection equations for acid concentration. Due to the dissolution of the solid skeleton, the pore space has an unknown (free) boundary. We establish the existence and uniqueness of the classical solution to the initial-boundary value problem.

We consider the homogenization of diffusion-convective problems with given divergence-free velocities in nonperiodic structures defined by sequences of characteristic functions (the first sequence). The sequence of concentration (the second sequence) is uniformly bounded in the space of square-summable functions with square-summable derivatives with respect to spatial variables. At the same time, the sequence of time-derivative of product of these concentrations on the characteristic functions, that define a nonperiodic structure, is bounded in the space of square-summable functions from time interval into the conjugated space of functions depending on spatial variables, with square-summable derivatives. We prove the strong compactness of the second sequences in the space of quadratically summable functions and use this result to homogenize the corresponding boundary value problems that depend on a small parameter.

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

The article continues the series of the authors’ papers devoted to averaging of mathematical models describing the isothermal acoustic processes in a heterogeneous medium with two components separated by a common boundary. One of these components is an elastic body, and the other is a poroelastic continuum. Poroelastic medium is understood to mean an elastic body permeated with a system of pores filled with liquid. The exact mathematical model constructed proceeding from the classical laws of continuum mechanics is used as the initial model. The model’s differential equations contain rapidly oscillating coefficients that appear in making transition to dimensionless variables. It is assumed that there are finite or infinite limits of these coefficients as the small parameter “epsilon” tends to zero. The small parameter “epsilon” is taken equal to the ratio of the average pore size to the characteristic size of the region under consideration. It should be noted that the coefficients of differential equations and the geometry of the considered region both depend on these parameters. Various averaged (limit) models that do not contain rapidly oscillating coefficients have been derived. For the possibility of using the averaging theory and the known averaging results, simplifying geometric assumptions about the periodicity and connectivity of the pore space and elastic skeleton are added. Averaged models are understood to mean such boundary value problems for equations or systems with relatively slowly changing characteristics that the solutions of boundary value problems for the initial models converge (in a sense) to the solution of the corresponding equations for the averaged model when the period ε of the considered periodic structure tends to zero. Depending on the characteristics of the continuum (whether the fluid is viscous, low-viscous, compressible, or incompressible; whether the skeleton is highly deformable, elastic, perfectly rigid, etc.), different limit modes are obtained. One of cases involving weakly compressible and low viscous liquid and a weakly deformable elastic skeleton in one region and an elastic body in the other region is investigated. The original mathematical model reflects the real physical process in a fairly accurate manner, but is so complex that the standard averaging scheme does not work for it. Therefore, the two-scale convergence method is used as the main tool. On the one hand, it is often impossible to calculate the model’s limit modes even in terms of weak convergence, but it is possible to do so in terms of two-scale convergence. On the other hand, the sequence of solutions is usually bounded but not compact, and in this case the sequence weak limit is not a satisfactory approximation to the solution of the initial mathematical model, and it is more preferable to use a two-scale limit. The results for an individually taken poroelastic region or for a region occupied by an elastic body were presented in the authors’ previous papers. In the case considered, the joint motion of an elastic body and porous elastic medium is studied, and the main problem lies in deriving the conditions at the common boundary of the elastic and poroelastic regions.

Рассматривается начально-краевая задача, описывающая фильтрацию слабо вязкой жидкости в двух различных пористых средах с общей границей. Доказывается на микроскопическом уровне теорема существования и единственности обобщенного решения задачи о совместном движении двух несжимаемых упругих пористых (пороупругих) тел с различными постоянными Ламе, с различной микроструктурой и вязкой несжимаемой поровой жидкости. При различных предположениях на данные задачи выводятся усредненные модели фильтрации несжимаемой слабовязкой жидкости в двух различных пористых упругих или абсолютно твердых средах, имеющих общую границу. Библиография: 21 наименование.

A homogenized model of filtration of a viscous fluid in two domains with common boundary is deduced on the basis of the method of two-scale convergence. The domains represent an elastic medium with perforated pores. The fluid, filling the pores, is the same in both domains, and the properties of the solid skeleton are distinct.

In the paper, we consider the evolution of the free boundary separating two immiscible viscous fluids with different constant densities in an absolutely rigid solid body and in an elastic skeleton. The motion of the liquids is described by the Stokes equations driven by the input pressure and the force of gravity. For flows in a bounded domain, we prove the existence and uniqueness of classical solutions and emphasize the study of the properties of the moving boundary separating the two fluids.

A mathematical model describing the processes of isothermal acoustics in a heterogeneous medium with two components separated by a common boundary is studied. One of the components is an elastic body, and the other one is a poroelastic medium (for example, it may be a liquid-saturated soil). The poroelastic medium is permeated with a system of pores filled with viscous weakly compressible liquid. The differential equations of the model describing the motion of an elastic body and the joint motion of a solid skeleton and liquid in the pores are based on the classical laws of continuous medium mechanics and adequately reflect the physical processes. However, these equations contain rapidly oscillating coefficients that depend on a small parameter equal to the ratio of the mean pore size to the size of the region under study. The existence of such coefficients prevents the use of the model for carrying out numerical calculations. The generalized solution of the initial boundary-value problem is given; and the theorem for existence and uniqueness of the generalized solution is presented together with its a priori estimates. For performing the homogenization procedure, the standard assumption about the periodicity of the pore space and solid skeleton is adopted. The obtained a priori estimates and the N. Nguetseng's two-scale convergence method were used as a basis for deriving the averaged equations and the initial boundary conditions (that is, the limit equations with the small parameter tending to zero). Different limiting modes depending on the continuous medium parameters are obtained. An averaged model for a special case that does not contain rapidly oscillating coefficients and can be used for numerical calculations is presented.

Two major arrangements describing the oil-by-water displacement process, i.e., Muskat’s model without taking into account the surface tension on the free boundary and the Buckley–Leverett model based on the surface tension, are considered. These arrangements were subject to theoretical and numerical study, which made it possible to uncover their self-contradictoriness. The focus was on construction of a numerical solution at the microlevel using the direct method with pressure correction.

In the present paper we consider elastic and poroelastic media having a common interface. We derive the macroscopic mathematical models for seismic wave propagation through these two different media as a homogenization of the exact mathematical model at the microscopic level. They consist of seismic equations for the each component and boundary conditions at the common interface, which separates different media. To do this we use the two-scale expansion method in the corresponding integral identities, defining the weak solution. Our results we illustrate with the numerical implementations of the inverse problem for the simplest model.

We consider some mathematical model of isothermal acoustics in a composite medium consisting of two different porous soils (poroelastic domains) separated by a common boundary. Each of the domains has its own characteristics of the solid skeleton; the liquid filling the pores is the same for both domains. The differential equations of the exactmodel contain some rapidly oscillating coefficients. The averaged equations (i.e., without rapidly oscillating coefficients) are derived.