Numerical Solution of a Two-Dimensional Problem of Fluid Filtration in a Deformable Porous Medium

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible fluid in a deformable porous medium. The model is based on the equations of conservation of mass for liquid and solid phases, Darcy’s law, the rheological relationship for a porous medium, and the law of conservation of the balance of forces. In this article, the equation of the balance of forces is taken in full form, i.e. the viscous and elastic properties of the medium are taken into account. The aim of the work is a numerical study of a model initial-boundary value problem. Section 1 gives a statement of the problem and a brief review of the literature on works related to this topic. In item 2, the original system of equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and the first-order equation for porosity. Section 3 proposes an algorithm for the numerical solution of the resulting initial-boundary value problem. For the numerical implementation, a variable direction scheme for the heat equation with variable coefficients is used, as well as the Runge — Kutta scheme of the fourth order of approximation.