Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium

Abstract

The paper considers a model for filtering a viscous incompressible fluid in a deformable porous medium. The filtration process can be described by a system consisting of mass conservation equations for liquid and solid phases, Darcy's law, rheological relation for a porous medium, and the law of conservation of balance of forces. This paper assumes that the poroelastic medium has both viscous and elastic properties. In the one-dimensional case, the transition to Lagrange variables allows us to reduce the initial system of governing equations to a system of two equations for effective pressure and porosity, respectively. The aim of the work is a numerical study of the emerging initial-boundary value problem. Paragraph 1 gives the statement of the problem and a brief review of the literature on works close to this topic. In paragraph 2, the initial system of equations is transformed, as a result of which a second-order equation for effective pressure and the first-order equation for porosity arise. Paragraph 3 proposes an algorithm to solve the initial-boundary value problem numerically. A difference scheme for the heat equation with the righthand side and a Runge–Kutta second-order approximation scheme are used for numerical implementation.

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

“…Близкие по структуре системы уравнений рассматривались в работах [6][7][8][9][10][11][12][13][14]. В работе [6] выполнено численное решение неизотермической фильтрации вязкой несжимаемой жидкости.…”

“…В [7] для системы уравнений одномерного нестационарного движения жидкости в теплопроводной вязкой пористой среде доказана разрешимость начально-краевой задачи. В работе [8] проведено численное исследование одномерной изотермической задачи фильтрации жидкости. В [9] предложен алгоритм численного решения начальнокраевой задачи в неизотермическом случае.…”