Generalized Transmission Problem for Two-Dimensional Filtration Flows in an Anisotropic Inhomogeneous Layer

Resumo

We state and study a transmission boundary value problem for two-dimensional filtration flows in a piecewise anisotropic inhomogeneous layer of a porous medium. The layer is characterized by a generally nonsymmetric conductivity (permeability) tensor with components that undergo discontinuity on some smooth curve (the transmission line). The tensor components are modeled by a function of the coordinates that undergoes a discontinuity on the transmission line but is continuously differentiable outside of it. We consider a layer with separated anisotropy and inhomogeneity. Using a nonsingular affine transformation of the coordinates, we state the problem for a complex potential in canonical form, which considerably simplifies the analysis of the problem. The sources–sinks of the flow are set arbitrarily; they do not lie on the transmission line and are modeled by the singular points of the complex potential. The problem is reduced to a system of two singular integral equations if the discontinuity in the layer conductivity along the transmission line is variable and to one singular integral equation if the discontinuity is constant. The problem is of practical interest, for example, in extracting water (or oil) from natural piecewise anisotropic inhomogeneous layers (strata) of soil.