Two-Dimensional Boundary Value Problems for Filtration Flows with Randomly Located Sources in an Inhomogeneous Porous Layer

The first and second boundary value problems and the transmission problem for the complex potential of a two-dimensional filtration flow in a generally inhomogeneous (variable thickness and permeability) porous thin layer are studied. The flow sources are arbitrary and discrete and can be located both on the boundaries and outside the boundaries of the flow domain. The boundaries are modeled by arbitrary smooth closed curved lines (contours), and the sources are modeled by singularities (isolated singular points of logarithmic type and poles) of the complex potential. The presence of sources at the boundaries leads to a fundamentally new generalization (complication) of the boundary conditions characterized by given singular functions. Solutions of the problems posed are presented in closed form for layers in certain classes of conductivity; this is demonstrated by the example of a layer with conductivity modeled by a power-law function of the coordinates. In the case where the layer conductivity is modeled by an arbitrary smooth function and the boundaries are arbitrary smooth closed curves, a generalized Cauchy type integral for the complex potential is used. This allows the second boundary value problem and the transmission problem (in the presence of a sink singularity of logarithmic type at the boundaries and arbitrary sources outside the boundaries) to be reduced to boundary singular integral equations with a weak singularity. The problems studied are mathematical models of two-dimensional filtration processes in layered porous media and are of interest, for example, for the practice of oil (water) extraction from natural soil layers of complex geological structure.