Boundary Value Problem of Two-Dimensional Filtrationof a Liquid in an Inhomogeneous Layer with a Boundary Condition Having Discontinuityof the First Kind

The first boundary value problems (internal and external) for a two-dimensional filtration flow of a liquid in a non-uniform porous layer of variable thickness and permeability are studied. The given discrete sources of the flow are located in the flow region and are modeled by singularities (isolated singular points) of the complex potential. The boundary of the flow region is modeled by an arbitrary smooth (piecewise smooth) closed contour. The pressure distribution on the boundary is characterized by a given function that has discontinuities of the first kind. A method for regularizing (smoothing) the boundary condition is proposed, which allows reducing the problems to a boundary singular integral equation with a weak singularity and with smooth right-hand side. This regularization method is applied to the solution of a boundary value problem modeling the operation of a system of wells in a heterogeneous soil layer (formation) on whose supply contour the given pressure distribution (the generalized potential) experiences discontinuities of the first kind.