Numerical solution to the problem of two-phase filtration with heterogeneous coefficients by the finite element method

We consider the process of filtration of a two-phase fluid in a porous, heterogeneous medium. This process is described by a coupled system of equationsfor saturation, filtration rate, and pore pressure. We consider mathematical models with and without capillary forces, in the presence of which, for saturation, we have a nonstationary convection-diffusion equation. Since this process is characterized by a significant predominance of the convective term in the equation for saturation, counter current approximations are used by adding non-uniform artificial diffusion. Speed and pressure are approximated using a mixed finite element method. The results of numerical calculations for a two-dimensional case with strongly heterogeneous permeability coefficients of a porous medium are presented. Several cases of relative fluid permeability associated with linear and nonlinear coefficients and the presence of capillary forces are considered.