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

  • Р.А. Вирц Алтайский государственный университет (Барнаул, Россия)
  • А.А. Папин Алтайский государственный университет (Барнаул, Россия)
  • В.А. Вайгант Боннский университет (Бонн, Германия)
Keywords: porosity, filtration, poroelasticity, Darcy law


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


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Bear J. Dynamics of Fluids in Porous Media // Elseiver. — New York, 1972.

Connoly J.A.D., Podladchikov Y.Y. Compaction-driven fluid flow in viscoelastic rock // Geodin. — Acta. — 11 (1998).

Morency S., Huismans R.S., Beaumont C, Fullsack P. A numerical model for coupled fluid flow and matrix deformation with applications to disequilibrium compaction and delta stability // Journal of Geophysical Redearch. — 112 (2007). — B10407.

Нигматулин Р.И. Динамика многофазных сред. — М., 1987. — Ч. 1.

Simpson M., Spiegelman M., Weinstein C.I. Degenerate dispersive equations arising in the stady of magma dynamics // Nonlinearty. — 20 (2007).

Abourabia A.M., Hassan K.M., Morad A.M. Analytical solutions of the magma equations for molten rocks in a granular matrix // Chaos Solutions Fract. — 42 (2009).

Geng Y., Zhang L. Bifurcations of traveling wave solutions for the magma equations // Applied Mathematics and computation. — 217 (2010).

Гоман В.А., Папин А.А., Шишмарев К.А. Численное решение двумерной задачи движения воды и воздуха в тающем снеге // Известия Алтайского государственного университета. — 2014. — № 1-2.

Шишмарев К.А. Тепломассоперенос в тающем снеге // Труды молодых ученых Алтайского государственного университета. — 2011. — № 8.

Токарева М.А., Вирц Р.А. Аналитическое и численное исследование задачи фильтрации в пороупругой среде : cборник трудов Всероссийской конференции по математике. — 2016.

Байкин А. Н. Динамика трещины гидроразрыва пласта в неоднородной пороупругой среде: дис. ... физ.-мат. наук. — Новосибирск, 2016.

Dushin V.R., Nikitin V.F., Legros J.C., Silnikov M.V. Mathematical modeling of flows in porous media // WSEAS Transactions on Fluid Mechanics. — 2014. — T. 9.

Tokareva M.A. Solvability of initial boundary value problen for the equations of filtration poroelastic media // Journal of Physics: Conference Series. — 2016. — T. 722. — № 1.

Papin A.A., Tokareva M.A. Correctness of the initial-boundary problem of the compressible fluid filtration in a viscous porous medium // Journal of Physics: Conference Series. — 2017. — T. 894. — № 1.

Papin A.A., Tokareva M.A. On Local solvability of the system of the equation of one dimensional motion of magma // Журнал Сибирского федерального университета. Серия: Математика и физика. — 2017. — T. 10. — № 3.

Токарева М.А. Конечное время стабилизации уравнений фильтрации жидкости в пороупругой среде // Известия Алтайского государственного университета. — 2015. — Т. 2. — № 1.

Самарский А.А., Гулин А.А. Численные методы. — М., 1989.

Самарский А.А. Теория разностных схем. — М., 1977.

Fowler A. Mathematical Geoscience. — Springer-Verlag London Limited, 2011.

Ларькин Н.А., Новиков В.А., Яненко Н.Н. Нелинейные уравнения переменного типа. — Новосибирск, 1983.
How to Cite
Вирц, Р., Папин, А., & Вайгант, В. (2018). Numerical Solution of the One-Dimensional Problem of Filtration of an Incompressible Fluid in a Viscous Porous Medium. Izvestiya of Altai State University, (4(102), 62-67.
Математика и механика

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