Numerical Solution of the One-Dimensional Problem of Filtration of an Incompressible Fluid in a Viscous Porous Medium

Abstract

The process of fluid filtration in a deformable porous medium is described by a system of equations consisting of the equations of mass conservation for the liquid and solid phases, the Darcy law, the Maxwell-type rheological relation, and the law of conservation of the balance of forces. It is assumed that the poroelastic medium has predominantly viscous properties and the phase densities are constant. In the case of one spatial variable, the transition to Lagrange variables makes it possible to reduce the initial system of determining equations to one equation for the required porosity. The paper is aimed to study the initial — boundary value problem numerically. In clause 1, the task is set and a brief review of the literature on related work. In paragraph 2, the system of equations is transformed, resulting in a nonlinear third-order equation for porosity. In paragraph 3 we propose an algorithm for the numerical solution of a one-dimensional initial-boundary problem. For numerical realization, we use a homogeneous difference scheme for a second-order equation with variable coefficients and a Runge — Kutta scheme of second-order approximation. The obtained solution satisfies the physical principle of maximum. In paragraph 4, we consider a more general case of reducing the original system to a single equation.

DOI 10.14258/izvasu(2018)4-11