A Finite Volume Scheme for Darcy-Brinkman’s Model of Two-Phase Flows in Porous Media

  • Houssein Nasser El Dine
  • Mazen Saad
  • Raafat Talhouk
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

In this paper, we are interested in the displacement of two incompressible phases in a Darcy-Brinkman flow in a porous media. The equations are obtained by the conservation of the mass and by considering the Brinkman regularization velocity of each phase. This model is treated in its general form with the whole nonlinear terms. This paper deals with construction and convergence analysis of a finite volume scheme together with a phase-by-phase upstream according to the total velocity. Finally, numerical tests illustrate the behavior of the solutions of this proposed scheme.

References

  1. 1.
    Auriault, J.L., Geindreau, C., Boutin, C.: Filtration law in porous media with poor separation of scales. Transp. Porous. Med. 60, 89–108 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aziz, A., Settari, K.: Treatment of nonlinear terms in the numerical solution of partial differential equations for multiphase flow in porous media. Int. J. Multiphase Flow 1, 817–844 (1975)CrossRefMATHGoogle Scholar
  3. 3.
    Brenner, K., Cancès, C., Hilhorst, D.: Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure. Comput. Geosci. 17, 573–597 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows Through Porous Media. Elsevier, Amsterdam (1986)MATHGoogle Scholar
  5. 5.
    Chen, Z., Ewing, R.E., Espedal, M.: Multiphase flow simulation with various boundary conditions. Comput. Methods Water Resour. 925–932 (1994)Google Scholar
  6. 6.
    Coclite, G.M., Mishra, S., Risebro, N.H.: Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media. Comput. Geosci. 18, 637–659 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme. ESAIM-Math. Model. Numer. Anal. 37, 937–972 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hannukainen, A., Juntunen, M., Stenberg, R.: Computations with finite element methods for the Brinkman problem. Comput. Geosci. 15, 155–166 (2011)CrossRefMATHGoogle Scholar
  9. 9.
    Misiats, O., Lipnikov, K.: Second-order accurate monotone finite volume scheme for Richards equation. J. Comput. Phys. 239, 123–137 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Paloka, E., Pazanin, I., Marusic, S.: Comparison between Darcy and Brinkman laws in a fracture. Appl. Math. Comput. 14, 7538–7545 (2012)MathSciNetMATHGoogle Scholar
  11. 11.
    Salinger, A.G., Aris, R., Derby, J.: Finite element formulations for large scale, coupled flows in adjacent porous and open fluid domains. Int. J. Numer. Methods Fluids 18, 1185–1209 (1994)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Houssein Nasser El Dine
    • 1
    • 2
  • Mazen Saad
    • 1
  • Raafat Talhouk
    • 2
  1. 1.École Centrale de Nantes. UMR 6629 CNRS, Laboratoire de Mathématiques Jean LerayNantesFrance
  2. 2.Doctoral School of Sciences and Technology (EDST), Laboratory of MathematicsLebanese UniversityBeirut, HadathLebanon

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