In uence of numerical diffusion on the growth rate of viscous ngers in the numerical implementation of the Peaceman model by the finite volume method

Cover Page

Cite item


A numerical model of oil displacement by a mixture of water and polymer based on the Peaceman model is considered. Numerical experiments were carried out using the DuMux package, which is a software library designed for modeling nonstationary hydrodynamic problems in porous media. The software package uses the vertex-centered variant of finite volume method. The effect of diffusion on the growth rate of ''viscous fingers'' has been studied. The dependencies of the leading front velocity on the value of model diffusion are obtained for three viscosity models. It is shown that the effect of numerical diffusion on the growth rate of ''viscous fingers'' imposes limitations on calculations for small values of model diffusion.

About the authors

D. E. Apushkinskaya

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.

G. G. Lazareva

Peoples’ Friendship University of Russia (RUDN University)


V. A. Okishev

Peoples’ Friendship University of Russia (RUDN University)



  1. Азиз Х., Сеттари Э. Математическое моделирование пластовых систем. - М.-Ижевск: Инст. комп. иссл., 2004.
  2. Ламб Г. Гидродинамика. - М.-Л.: Гостехиздат, 1947.
  3. Логвинов О. А. Об устойчивости боковой поверхности вязких пальцев, образующихся при вытеснении жидкости из ячейки Хеле-Шоу// Вестн. Моск. ун-та. Сер. 1. Мат. Мех. - 2011. - № 2. - С. 40-46.
  4. Тятюшкина Е. С., Козелков А. С., Куркин А. А., Курулин В. В., Ефремов В. Р., Уткин Д. А. Оценка численной диффузии метода конечных объемов при моделировании поверхностных волн// Вычисл. техн. - 2019. - 24, № 1. - С. 106-119.
  5. Arun R., Dawson S. T. M., Schmid P. J., Laskari A., McKeon B. J. Control of instability by injection rate oscillations in a radial Hele-Shaw cell// Phys. Rev. Fluids. - 2020. - 5. - 123902.
  6. Bakharev F., Campoli L., Enin A., Matveenko S., Petrova Y., Tikhomirov S., Yakovlev A. Numerical investigation of viscous ngering phenomenon for raw eld data// Transp. Porous Med. - 2020. - 132.- С. 443-464.
  7. Bakharev F., Enin A., Groman A., Kalyuzhnuk A., Matveenko S., Petrova Yu., Starkov I., Tikhomirov S. S. Velocity of viscous ngers in miscible displacement// J. Comput. Appl. Math. - 2022. - 402. - 113808.
  8. Booth R. J. S. Miscible ow through porous media. - Канд. дисс., 2008.
  9. Booth R. J. S. On the growth of the mixing zone in miscible viscous ngering// J. Fluid Mech. - 2010. - 655. - С. 527-539.
  10. Chen Ch., Yang X. A second-order time accurate and fully-decoupled numerical scheme of the Darcy- Newtonian-Nematic model for two-phase complex uids con ned in the Hele-Shaw cell// J. Comput. Phys. - 2022. - 456. - 111026.
  11. De Wit A., Homsy G. M. Viscous ngering in reaction-diusion systems// J. Chem. Phys. - 1999. - 110.- С. 8663-8675.
  12. DuMux Code Documentation (doxygen), Ver. 3.5. -, 2022.
  13. DuMux Handbook, Ver. 3.5. -, 2022.
  14. Fontana J., Juel A., Bergemann N., Heil M., Hazel A. Modelling nger propagation in elasto-rigid channels// J. Fluid Mech. - 2021. - 916. - A27.
  15. Karimi F., Maleki Jirsaraei N., Azizi S. Simulation of viscous ngering due to Sa man-Taylor instability in Hele-Shaw cell// J. Nanoelectron. Materials. - 2019. - 12, № 3. - С. 309-318.
  16. Kupervasser O. Laplacian growth without surface tension in ltration combustion: analytical pole solution// В сб.: «Pole solutions for ame front propagation. Mathematical and analytical techniques with applications to engineering». - Cham: Springer, 2015. - С. 85-107.
  17. Lu D., Municchi F., Christov I. C. Computational analysis of interfacial dynamics in angled Hele-Shaw cells: instability regimes// Transp. Porous Med. - 2020. - 131. - С. 907-934.
  18. Lustri Ch. J., Green Ch. C., McCue S. W. Hele-Shaw bubble via exponential asymptotics// SIAM J. Appl. Math. - 2020. - 80, № 1. - С. 289-311.
  19. Noskov M. D., Istomin A. D., Kesler A. G. Stochastic-deterministic modeling of the development of hydrodynamic instability in ltration of mixing uids// J. Eng. Phys. Thermophys. - 2002. - 75. - С. 352- 358.
  20. Sa man P. G., Taylor G. The penetration of a uid into a porous medium or a Hele-Shaw cell containing a more viscous uid// Proc. Roy. Soc. London. A. - 1958. - 245. - С. 312-329.
  21. Singh P., Lalitha R., Mondal S. Sa man-Taylor instability in a radial Hele-Shaw cell for a shear-dependent rheological uid// J. Non-Newtonian Fluid Mech. - 2021. - 294. - 104579.
  22. Skopintsev A. M., Dontsov E. V., Kovtunenko P. V., Baykin A. N., Golovin S. V. The coupling of an enhanced pseudo-3D model for hydraulic fracturing with a proppant transport model// Eng. Fracture Mech. - 2020. - 236. - 107177.
  23. Smirnov N. N., Kisselev A. B., Nikitin V. F., Zvyaguin A. V., Thiercelin M., Legros J. C. Hydraulic fracturing and ltration in porous medium// SPE Russian Oil and Gas Technical Conference and Exhibition, Moscow, Russia, October 2006.
  24. Smirnov N. N., Nikitin V. F., Maximenko A., Thiercelin M., Legros J. C. Instability and mixing ux in frontal displacement of viscous uids from porous media// Phys. Fluids. - 2005. - 17. - 084102. Contemporary Mathematics. Fundamental Directions, 2022, Vol. 68, No. 4, 553-563 561
  25. Sorbie K. S. Polymer-improved oil recovery. - Dordrecht: Springer, 1991.
  26. Tan C. T., Homsy G. M., Stability of miscible displacements in porous media: rectilinear ow// Phys. Fluids. - 1986. - 29, № 11. - С. 3549-3556.
  27. Yang X. Fully-discrete, decoupled, second-order time-accurate and energy stable nite element numerical scheme of the Cahn-Hilliard binary surfactant model con ned in the Hele-Shaw cell// ESAIM Math. Model. Numer. Anal. - 2022. - 56, № 2. - С. 651-678.

Copyright (c) 2022 Apushkinskaya D.E., Lazareva G.G., Okishev V.A.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies