Study of a boundary value transmission problem for two-dimensional flows in a piecewise anisotropic inhomogeneous porous layer

Abstract

We consider a boundary value transmission problem for two-dimensional filtration flows in an anisotropic porous layer consisting of adjacent domains in which the media have essentially different conductivities (permeability and thickness). In general, the layer conductivity is specified by a nonsymmetric second rank tensor whose components are modeled by continuously differentiable functions of coordinates. To study the problem, we use two complex planes, the physical plane and an auxiliary plane, which are related by a homeomorphic (one-to-one and continuous) transformation satisfying an equation of the Beltrami type. On the physical plane, we pose a transmission problem for a rather complicated elliptic system of equations. This problem is reduced on the auxiliary plane to canonical form, which dramatically simplifies the analysis of the problem. Then the problem is reduced to a system of boundary singular integral equations with generalized kernels of the Cauchy type, which are expressed via the fundamental solutions of the main equations. The boundary value transmission problem studied here can be used as a mathematical model of processes arising in the recovery of fluids (water and oil) from natural soil formations of complicated geological structure.