Computational Geosciences

, Volume 17, Issue 6, pp 1055–1078 | Cite as

A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocityand continuous capillary pressure

  • Todd Arbogast
  • Mika Juntunen
  • Jamie Pool
  • Mary F. Wheeler


We consider the slightly compressible two-phase flow problem in a porous medium with capillary pressure. The problem is solved using the implicit pressure, explicit saturation (IMPES) method, and the convergence is accelerated with iterative coupling of the equations. We use discontinuous Galerkin to discretize both the pressure and saturation equations. We apply two improvements, which are projecting the flux to the mass conservative H(div)-space and penalizing the jump in capillary pressure in the saturation equation. We also discuss the need and use of slope limiters and the choice of primary variables in discretization. The methods are verified with two- and three-dimensional numerical examples. The results show that the modifications stabilize the method and improve the solution.


Finite element method FEM Discontinuous Galerkin DG Flow in porous media Capillary pressure IMPES 

Mathematics Subject Classification (2010)

65N30 76S05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arbogast, T., Pencheva, G., Wheeler, F.M., Yotov, I.: A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6(1), 319–346 (2007). (electronic)CrossRefGoogle Scholar
  2. 2.
    Bastian, P., Rivière, B.: Superconvergence and H(div) projection for discontinuous Galerkin methods. Int. J. Numer. Meth. Fl. 42, 1043–1057 (2003)CrossRefGoogle Scholar
  3. 3.
    Bear, J., Cheng, A.H.-D.: Modeling Groundwater Flow and Contaminant Transport. Springer, New York (2010)CrossRefGoogle Scholar
  4. 4.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, vol. 15, Springer Series in Computational Mathematics. Springer, New York (1991)CrossRefGoogle Scholar
  5. 5.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media, Computational Science and Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)CrossRefGoogle Scholar
  6. 6.
    Class, H., Ebigbo, A., Helmig, R., Dahle, H., Nordbotten, J., Celia, M., Audigane, P., Darcis, M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S., Jin, M., Krug, S., Labregere, D., Naderi Beni, A., Pawar, R., Sbai, A., Thomas, S., Trenty, L., Wei, L.: A benchmark study on problems related to co2 storage in geologic formations. Comput. Geosci. 13(4), 409–434 (2009)CrossRefGoogle Scholar
  7. 7.
    Dawson, C.N., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193(23–26), 2565–2580 (2004)CrossRefGoogle Scholar
  8. 8.
    Epshteyn, Y., Rivière, B.: Fully implicit discontinuous finite element methods for two-phase flow. Appl. Numer. Math. 57(4), 383–401 (2007)CrossRefGoogle Scholar
  9. 9.
    Ern, A., Mozolevski, I., Schuh, L.: Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures. Comput. Methods Appl. Mech. Eng. 199(23-24), 1491–1501 (2010)CrossRefGoogle Scholar
  10. 10.
    Ern, A., Mozolevski, I., Schuh, L.: Accurate velocity reconstruction for Discontinuous Galerkin approximations of two-phase porous media flows. C. R. Acad. Sci. Paris 347(9–10), 551–554 (2009)CrossRefGoogle Scholar
  11. 11.
    Ern, A., Nicaise, S., Vohralík, M.: An accurate flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Acad. Sci. Paris 345(12), 709–712 (2007)CrossRefGoogle Scholar
  12. 12.
    Hoteit, H., Firoozabadi, A.: Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Adv. Water Resour. 31(1), 56–73 (2008)CrossRefGoogle Scholar
  13. 13.
    Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Pub. Co., Amsterdam (1977)Google Scholar
  14. 14.
    Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43(1), 195–219 (2005)CrossRefGoogle Scholar
  15. 15.
    Sun, S., Wheeler, M.F.: Projections of velocity data for the compatibility with transport. Comput. Methods Appl. Mech. Eng. 195(7–8), 653–673 (2006)CrossRefGoogle Scholar
  16. 16.
    Wheeler, M., Xue, G., Yotov, I.: A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numerische Mathematik 121(1), 165–204 (2012)CrossRefGoogle Scholar
  17. 17.
    Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091–3120 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Todd Arbogast
    • 1
  • Mika Juntunen
    • 2
  • Jamie Pool
    • 1
  • Mary F. Wheeler
    • 1
  1. 1.Institute for Computational Engineering and SciencesThe University of TexasAustinUSA
  2. 2.Department of Mathematics and Systems AnalysisAalto University School of ScienceAaltoFinland

Personalised recommendations