Numerische Mathematik

, Volume 129, Issue 4, pp 691–722 | Cite as

A combined finite volume–nonconforming finite element scheme for compressible two phase flow in porous media

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Abstract

We propose and analyze a combined finite volume–nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on its own pressure.

Mathematics Subject Classification

35K57 35K55 92D30 
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References

  1. 1.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers LTD, London (1979)Google Scholar
  2. 2.
    Barrett, J.W., Knabner, P.: Finite element approximation of the transport of reactive solutes in porous media. ii. error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34(2), 455–479 (1997)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bendahmane, M., Khalil, Z., Saad, M.: Convergence of a finite volume scheme for gas water flow in a multi-dimensional porous media. Math. Models Methods Appl. Sci. 24(1), 145–185 (2014)Google Scholar
  4. 4.
    Brenier, Y., Jaffré, J.: Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28, 685–696 (1991)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Brezis, H.: Analyse fonctionnelle. Collection of Applied Mathematics for the Master’s Degree, Theory and applications. Masson, Paris (1983)Google Scholar
  6. 6.
    Caro, F., Saad, B., Saad, M.: Two-component two-compressible flow in a porous medium. Acta Appl. Math. 117(1), 15–46 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows Through Porous Media. Elsevier Science Publishers B. V., North Holland (1986)Google Scholar
  8. 8.
    Chen, Z., Ewing, R.E.: Degenerate two-phase incompressible flow. iii. Sharp error estimates. Numer. Math. 90(2), 215–240 (2001)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, North-Holland (1991)Google Scholar
  10. 10.
    Coudiére, Y., Vila, J.P., Villedieu, P.: Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. M2AN. Math. Model. Numer. Anal. 33(3), 493–516 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Dawson, C.: Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35(5), 1709–1724 (1998)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Debiez, C., Dervieux, A., Mer, K., Nkonga, B.: Computation of unsteady flows with mixed finite volume/finite element upwind methods. Internat. J. Numer. Methods Fluids 27(1–4, Special Issue), 193–206 (1998)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Evans, L.: Partial Differential Equations. American Mathematical Society, Washington, D.C (2010)MATHGoogle Scholar
  14. 14.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, North-Holland (2000)Google Scholar
  15. 15.
    Eymard, R., Gallouët, T., Herbin, R.: A finite volume scheme for anisotropic diffusion problems. C. R. Math. Acad. Sci. Paris 339(4), 299–302 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme. Math. Model. Numer. Anal. 37(6), 937–972 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Eymard, R., Hilhorst, D., Vohralik, M.: A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105, 73–131 (2006)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Faille, I.: A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Eng. 100(2), 275–290 (1992)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Feistauer, M., Felcman, J., Lukáčová, M.: On the convergence of a combined finite volume-finite element method for nonlinear convection. Numer. Methods Partial Differ. Equ. 13(2), 163–190 (1997)CrossRefMATHGoogle Scholar
  20. 20.
    Galusinski, C., Saad, M.: Two compressible immiscible fluids in porous media. J. Differ. Equ. 244, 1741–1783 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Khalil, Z., Saad, M.: Solutions to a model for compressible immiscible two phase flow in porous media. Electron. J. Differ. Equ. 2010(122), 1–33 (2010)MathSciNetGoogle Scholar
  22. 22.
    Michel, A.: A finite volume scheme for the simulation of two-phase incompressible flow in porous media. SIAM J. Numer. Anal. 41, 1301–1317 (2003)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Nochetto, R.H., Schmidt, A., Verdi, C.: A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69(229), 1–24 (2000)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing, Amsterdam (1977)Google Scholar
  25. 25.
    Rulla, J., Walkington, N.J.: Optimal rates of convergence for degenerate parabolic problems in two dimensions. SIAM J. Numer. Anal. 33(1), 56–67 (1996)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Saad, B., Saad, M.: Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media. SIAM J. Numer. Anal. 51(1), 716–741 (2013)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Vohralik, M.: On the discrete poincaré-friedrichs inequalities for nonconforming approximations of the sobolev space h1. Numer. Funct. Anal. Optim. 26(78), 925–952 (2005)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Division of Computer, Electrical and Mathematical Sciences and EngineeringKing Abdullah University of Science and TechnologyThuwalKingdom of Saudi Arabia
  2. 2.Ecole Centrale de Nantes, Département d’ Informatique et MathématiquesLaboratoire de Mathématiques Jean Leray (UMR 6629 CNRS)NantesFrance

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