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

УДК 519.6+536

  • R.A. Virts Altai State University (Barnaul, Russia) Email: virtsrudolf@gmail.com
  • A.A. Papin Altai State University (Barnaul, Russia) Email: papin@math.asu.ru
  • W.A. Weigant Bonn University (Bonn, Germany) Email: w-weigant@t-online.de
DOI_disc_logo https://doi.org/10.14258/izvasu(2020)1-11
Keywords: porosity, filtration, poroelasticity, Darcy law

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.

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Author Biographies

R.A. Virts, Altai State University (Barnaul, Russia)

аспирант факультета математики и информационных технологий

A.A. Papin, Altai State University (Barnaul, Russia)

доктор физико-математических наук, заведующий кафедрой дифференциальных уравнений факультета математики и информационных технологий

W.A. Weigant, Bonn University (Bonn, Germany)

профессор

References

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). DOI: 10.1016/S0985-3111(98)80006-5.

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. DOI: 10.1029/2006JB004701.

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). DOI: 10.1088/0951-7715/20/1/003.

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). DOI: 10.1016/j.chaos.2009.03.078.

Geng Y., Zhang L. Bifurcations of traveling wave solutions for the magma equations // Applied Mathematics and computation, 217(2010). DOI: 10.1016/j.amc.2009.11.035.

Вирц Р.А., Папин А.А., Вайгант В.А. Численное решение одномерной задачи фильтрации несжимаемой жидкости в вязкой пористой среде // Известия Алт. гос. ун-та. 2018. № 4 (102). DOI: 10.14258/izvasu(2018)4-11.

Koleva M.N., Vulkov L.G. Numerical analysis of one dimensional motion of magma without mass forces // Journal of Computational and Applied Mathematics. 2020. Т. 366. DOI: 10.1016/j.cam.2019.07.003.

Токарева М.А., Вирц Р.А. Аналитическое и численное исследование задачи фильтрации в пороупругой среде : c6. трудов Всероссийской конференции по математике "МАК-2016". 2016.

Байкин А.Н. Динамика трещины гидроразрыва пласта в неоднородной пороупругой среде : дисс. ... канд. физико-математических 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. DOI: 10.1088/17426596/722/1/012037.

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. DOI: 10.1088/1742-6596/894/1/012070.

Papin A.A., Tokareva M.A. On Local solvability of the system of the equation of one dimensional motion of magma // Журн. Сиб. федерального ун-та. Серия: Математика и физика. 2017. T. 10. №3. DOI: 10.17516/1997-1397-2017-103-385-395.

Токарева М.А. Конечное время стабилизации уравнений фильтрации жидкости в пороупругой среде // Известия Алт. гос. ун-та. 2015. Т. 2. № 1. DOI: 10.14258/izvasu(2015)1.2-28.

Tokareva M., Papin A. Solvability of the system of equations of one-dimensional movement of a viscous liquid in a deformable viscous porous medium // Journal of Physics: Conference Series. IOP Publishing, 2019. Т. 1268. № 1. DOI: 10.1088/1742-6596/1268/1/012053.

Tokareva M.A., Papin A.A. Global solvability of a system of equations of onedimensional motion of a viscous fluid in a deformable viscous porous medium // Journal of Applied and Industrial Mathematics. 2019. Т. 13. № 2. DOI: 10.1134/S1990478919020169.

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

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

Fowler A. Mathematical Geoscience. Springer-Verlag London Limited, 2011. DOI: 10.1007/s11004-012-9399-0.

Калиткин Н.Н. Численные методы. M., 1978.

Published
2020-03-06
How to Cite
Virts R., Papin A., Weigant W. Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium // Izvestiya of Altai State University, 2020, № 1(111). P. 72-76 DOI: 10.14258/izvasu(2020)1-11. URL: https://izvestiya.asu.ru/article/view/%282020%291-11.
Issue
No 1(111) (2020): Izvestiya of Altai State University
Section
Математика и механика

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