Solution of Equations of a One-Dimensional Problem on Two-Phase Filtration in a Porous Medium with Account of Thermodynamical Effects by Using Geometric Methods

One-dimensional problems of two-phase filtration of liquids (water and oil) in porous media are described by the Buckley–Leverett equations, the Darcy law, and the law of conservation of energy under certain initial and boundary conditions. In this paper, we propose an asymptotic method of constructing a solution of the problem and methods for resolution of singularities associated with shock waves that arise in the process. The method proposed is implemented numerically by using the Maple software.