Adaptive Discontinuous Galerkin Methods for Flow in Porous Media

  • Birane Kane
  • Robert Klöfkorn
  • Andreas Dedner
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


We present an adaptive Discontinuous Galerkin discretization for the solution of porous media flow problems. The considered flows are immiscible and incompressible. The fully adaptive approach implemented allows for refinement and coarsening in both the element size, the polynomial degree and the time step size.



Birane Kane would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. Robert Klöfkorn acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, Aker BP ASA, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS, DEA Norge AS of The National IOR Centre of Norway for support. The authors would like to thank the reviewers for helpful comments to improve this work.


  1. 1.
    P. Bastian, A fully-coupled discontinuous galerkin method for two-phase flow in porous media with discontinuous capillary pressure. Comput. Geosci. 18(5), 779–796 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.H. Brooks, A.T. Corey, Hydraulic properties of porous media and their relation to drainage design. Trans. ASAE 7(1), 26–0028 (1964)CrossRefGoogle Scholar
  3. 3.
    D.A. Di Pietro, M. Vohralík, A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences. Oil Gas Sci. Technol. - Revue dIFP Energies nouvelles 69(4), 701–729 (2014)CrossRefGoogle Scholar
  4. 4.
    Y. Epshteyn, B. Rivière, Fully implicit discontinuous finite element methods for two-phase flow. Appl. Numer. Math. 57(4), 383–401 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Kane, Using dune-fem for adaptive higher order discontinuous galerkin methods for strongly heterogenous two-phase flow in porous media. Arch. Numer. Softw. 5:1 (2017).Google Scholar
  6. 6.
    B. Kane, R. Klöfkorn, C. Gersbacher, hp–adaptive discontinuous galerkin methods for porous media flow, in International Conference on Finite Volumes for Complex Applications (Springer, Cham, 2017), pp. 447–456.zbMATHGoogle Scholar
  7. 7.
    W. Klieber, B. Rivière, Adaptive simulations of two-phase flow by discontinuous galerkin methods, Comput. Methods Appl. Mech. Eng. 196(1), 404–419 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    S. Sun, M.F. Wheeler, L 2 (H 1) norm a posteriori error estimation for discontinuous galerkin approximations of reactive transport problems. J. Sci. Comput. 22(1), 501–530 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. Van Duijn, J. Molenaar, M. De Neef, The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media. Transp. Porous Media 21(1), 71–93 (1995)CrossRefGoogle Scholar
  10. 10.
    M. Vohralík, M.F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. Comput. Geosci. 17(5), 789–812 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Wolff, Y. Cao, B. Flemisch, R. Helmig, B. Wohlmuth, Multi-point flux approximation l-method in 3d: numerical convergence and application to two-phase flow through porous media, in Simulation of Flow in Porous Media: Applications in Energy and Environment. Radon Series on Computational and Applied Mathematics, vol. 12 (De Gruyter, Berlin, 2013), pp. 39–80Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany
  2. 2.International Research Institute of StavangerStavangerNorway
  3. 3.Mathematics Institute, University of WarwickCoventryUK

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