Study of Three-Dimensional Boundary Value Problems of Fluid Filtration in an Anisotropic Porous Medium

Three-dimensional boundary value problems (the first and second boundary value problems and the conjugation problem) of stationary filtration of fluids in anisotropic (orthotropic) and inhomogeneous porous media are posed and studied. A medium is characterized by a symmetric permeability tensor whose components generally depend on the coordinates of points of the space. A nonsingular affine transformation of coordinates is used and the problems are stated in canonical form, which dramatically simplifies their study. In the case of orthotropic and piecewise orthotropic homogeneous medium, the solution of the problem with canonical boundaries (plane and ellipsoid surfaces) can be obtained in finite form. In the general case, where the orthotropic medium is inhomogeneous and the boundary surfaces are arbitrary and smooth, the problem can be reduced to singular and hypersingular integral equations. The problems are topical, for example, in the practice of fluid (water, oil) recovery from natural anisotropic and inhomogeneous soil strata.