Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

“Hanging” discontinuities in the saturation with the filtration of immiscible liquids in inhomogenous media

The article discusses the flow of two incompressible immiscible liquids in a porous medium. A study is made of “hanging” discontinuities of the saturation arising at the points of the breakaway of the absolute permeability, taking account of capillary forces and of the force of gravity.     

Percolation with a limiting gradient in a medium with random inhomogeneities

The coverage of a medium by percolation and the effective permeability of a medium with stagnant zones are determined. It is shown that effective permeability is a function of external conditions, particularly the average pressure gradient. Three-, two-, and one-dimensional flows are discussed. The theory of overshoots of random functions and fields beyond a prescribed level [1, 2] is used for the investigation. Overshoots of elements of the percolation field in media with random inhomogeneities are studied. Overshoots of energy being dissipated in a volume are discussed in particular; this permits an approximate determination of the coverage of an inhomogeneous porous medium by migration during percolation with a limiting gradient, i.e., in the case of formation of stagnant zones chaotically disseminated in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–165, September–October, 1970.The authors thank V. M. Entov for discussing the article and useful comments.     

One-dimensional filtration of immiscible liquids in a nonuniform porous medium

As is known, the differential equation for two-phase filtration with account for capillarity was obtained in [1], and later integrated numerically for the case of a uniform stratum of finite length in [2]. Other versions of the solution of the Rapoport-Leas equation or the system which is equivalent to it are known [3, 4]. This article presents the results of a numerical solution of an analogous problem with account for nonuniform permeability of the stratum.The authors wish to thank T. V. Startsev and L. Kh. Aminov for assistance in performing the calculations on the Ural-3 computer.     

The source-type solution in the problem of unsteady filtration in a medium with random nonuniformity

The fundamental problems of the theory of filtration in media with random nonuniformities were formulated in [1] and methods of solution were indicated. Primary attention was devoted to the steadystate filtration processes. In the following we solve one of the most important unsteady problems and indicate the connection of the result obtained with the widely used methods of determining strata parameters from the curves of the pressure variation in nonflowing wells. We note that the interpretation of the results of such measurements is usually carried out with the aid of the solution of the corresponding problem for a homogeneous stratum or for a stratum whose nonuniformity has a regular nature (for example, [2]), which definitely limits the possibilities of the method. At the same time it is obvious that the solution of these problems for irregular media and particularly the determination of their effective characteristics requires the use of statistical methods of computation.It is also not difficult to see that the results obtained below may be used for the solution of the corresponding problems of heat conduction, diffusion, etc.