Computing and Visualization in Science

, Volume 15, Issue 1, pp 3–20 | Cite as

Numerical simulation of gas migration through engineered and geological barriers for a deep repository for radioactive waste

  • Brahim Amaziane
  • Mustapha El Ossmani
  • Mladen Jurak


In this paper a finite volume method approach is used to model the 2D compressible and immiscible two-phase flow of water and gas in heterogeneous porous media. We consider a model describing water-gas flow through engineered and geological barriers for a deep repository of radioactive waste. We consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. This process can be formulated as a coupled system of partial differential equations which includes a nonlinear parabolic gas-pressure equation and a nonlinear degenerated parabolic water-saturation equation. Both equations are of diffusion-convection type. An implicit vertex-centred finite volume method is adopted to discretize the coupled system. A Godunov-type method is used to treat the convection terms and a conforming finite element method with piecewise linear elements is used for the discretization of the diffusion terms. An averaging technique is developed to obtain an effective capillary pressure curve at the interface of two media. Our numerical model is verified with 1D semi-analytical solutions in heterogeneous media. We also present 2D numerical results to demonstrate the significance of capillary heterogeneity in flow and to illustrate the performance of the method for the FORGE cell scale benchmark.


Finite volume Heterogeneous porous media Hydrogen migration Immiscible compressible  Nuclear waste  Two-phase flow Vertex-centred 

Mathematics Subject Classification

74Q99 76E19 76T10 76M12 76S05 



The research leading to these results has received funding from the European Atomic Energy Community’s Seventh Framework Programme (FP7/2009-2013) under Grant Agreement no230357, the FORGE project. This work was partially supported by the GnR MoMaS (PACEN/CNRS, ANDRA, BRGM,CEA, EDF, IRSN) France, their supports are gratefully acknowledged.


  1. 1.
    ANDRA: Dossier 2005 Argile, les recherches de l’Andra sur le stockage géologique des déchets radioactifs à haute activité à vie longue, Collection les Rapports, Andra, Châtenay-Malabry (2005)Google Scholar
  2. 2.
    Angelini, O., Chavant, C., Chénier, E., Eymard, R., Granet, S.: Finite volume approximation of a diffusion-dissolution model and application to nuclear waste storage. Math. Comput. Simul. 81(10), 2001–2017 (2011)MATHCrossRefGoogle Scholar
  3. 3.
    Bastian, P.: Numerical computation of multiphase flows in porous media. Habilitationsschrift (1999)Google Scholar
  4. 4.
    Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwe, London (1991)MATHCrossRefGoogle Scholar
  5. 5.
    Balay, S., Brown, J., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Web page. (2011)
  6. 6.
    Bourgeat, A., Jurak, M.: A two level scaling-up method for multiphase flow in porous media; numerical validation and comparison with other methods. Comput. Geosci. 14(1), 1–14 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bourgeat, A., Jurak, M., Smaï, F.: Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository. Comput. Geosci. 13(1), 29–42 (2009)MATHCrossRefGoogle Scholar
  8. 8.
    Cancès, C.: Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities, M2AN Math. Model. Numer. Anal. 43(5), 973–1001 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation. North-Holland, Amsterdam (1986)MATHGoogle Scholar
  10. 10.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. SIAM, Philadelphia (2006)MATHCrossRefGoogle Scholar
  11. 11.
    Croisé, J., Mayer, G., Talandier, J., Wendling, J.: Impact of water consumption and saturation-dependent corrosion rate on hydrogen generation and migration from an intermediate-level radioactive waste repository. Transp. Porous Media 90, 59–75 (2011)CrossRefGoogle Scholar
  12. 12.
    Enchery, G., Eymard, R., Michel, A.: Numerical approximation of a two-phase flow problem in a porous medium with discontinuous capillary forces. SIAM J. Numer. Anal. 43(6), 2402–2422 (2006)Google Scholar
  13. 13.
    Fučík, R., Mikyška, J., Beneš, M., Illangasekare, T.H.: Semianalytical solution for two-phase flow in porous media with a discontinuity. Vadose Zone J. 7, 1001–1007 (2008)CrossRefGoogle Scholar
  14. 14.
  15. 15.
    Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin (1997)CrossRefGoogle Scholar
  16. 16.
    Hoteit, H., Firoozabadi, A.: Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Adv. Water Resour. 31, 56–73 (2008)CrossRefGoogle Scholar
  17. 17.
    Huber, R., Helmig, R.: Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media. Comput. Geosci. 4(2), 141–164 (2000)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    McWhorter, D.B., Sunada, D.K.: Exact integral solutions for two-phase flow. Water Resour. Res. 26, 339–413 (1990)CrossRefGoogle Scholar
  19. 19.
  20. 20.
    Niessner, J., Helmig, R., Jacobs, H., Roberts, J.E.: Interface condition and linearization schemes in the Newton iterations for two-phase flow in heterogeneous porous media. Adv. Water Resour. 28, 671–678 (2005)Google Scholar
  21. 21.
    Norris, S.: Summary of gas Generation and migration current state-of-the-art, FORGE D1.2-R (2009). Available online at
  22. 22.
    OECD/NEA: Safety of Geological Disposal of High-level and Long-lived Radioactive Waste in France, An International Peer Review of the “Dossier 2005 Argile” Concerning Disposal in the Callovo-Oxfordian Formation, OECD Publishing (2006)., Available online at
  23. 23.
    Papafotiou, A., Sheta, H., Helmig, R.: Numerical modeling of two-phase hysteresis combined with an interface condition for heterogeneous porous media. Comput. Geosci. 14(2), 273–287 (2010)MATHCrossRefGoogle Scholar
  24. 24.
    Senger, R., Ewing, J., Zhang, K., Avis, J., Marschall, P., Gauss, I.: Modeling approaches for investigating gas migration from a deep low/intermediate level waste repository (Switzerland). Transp. Porous Media 90, 113–133 (2011)CrossRefGoogle Scholar
  25. 25.
    van Duijn, C.J., de Neef, M.J.: Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity. Adv. Water Resour. 21, 451–461 (1998)CrossRefGoogle Scholar
  26. 26.
    van Duijn, C.J., Molenaar, J., Neef, M.J.: The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media. Transp. Porous Media 21(1), 71–93 (1995) Google Scholar
  27. 27.
    Zhang, K., Croisé, J., Mayer, G.: Computation of the Couplex-Gaz exercise with TOUGH2-MP: hydrogen flow and transport in the pore water of a low-permeability clay rock hosting a nuclear waste repository. Nucl. Technol. 174, 364–374 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Brahim Amaziane
    • 1
  • Mustapha El Ossmani
    • 2
  • Mladen Jurak
    • 1
  2. 2.Université Moulay Ismaïl, ENSAMMeknesMorocco

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