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Model reduction and discretization using hybrid finite volumes for flow in porous media containing faults

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Abstract

In this paper, we study two different model reduction strategies for solving problems involving single phase flow in a porous medium containing faults or fractures whose location and properties are known. These faults are represented as interfaces of dimension N − 1 immersed in an N dimensional domain. Both approaches can handle various configurations of position and permeability of the faults, and one can handle different fracture permeabilities on the two inner sides of the fracture. For the numerical discretization, we use the hybrid finite volume scheme as it is known to be well suited to simulating subsurface flow. Some results, which may be of use in the implementation of the proposed methods in industrial codes, are demonstrated.

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References

  1. Aavatsmark, I.: Interpretation of a two-point flux stencil for skew parallelogram grids. Comput. Geosci. 11(3), 199–206 (2007)

    Article  Google Scholar 

  2. Pierre, M.A., Thovert, J.-F.: Fractures and fracture networks. Springer (1999)

  3. Adler, P.M., Thovert, J.-F., Valeri, V.: Mourzenko. Fractured porous media. Oxford University Press (2012)

  4. Alboin, C., Jaffré, J., Roberts, J.E., Wang, X., Serres, C.: Domain decomposition for some transmission problems in flow in porous media, volume 552 of Lecture Notes in Physics, pp. 22–34. Springer, Berlin (2000)

    Google Scholar 

  5. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. M2AN Math. Model. Numer. Anal. 43(2), 239–275 (2009)

    Article  Google Scholar 

  6. Baca, R.G., Arnett, R.C., Langford, D.W.: Modelling fluid flow in fractured-porous rock masses by finite-element techniques. Int. J. Numer. Methods Fluids 4(4), 337–348 (1984)

    Article  Google Scholar 

  7. Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, MG., McInnes, L.C., Smith, BF., Zhang, H.: PETSc users manual. Technical Report ANL-95/11 - Revision 3.4, Argonne National Laboratory (2013)

  8. Bear, J., Tsang, C.-F., de Marsily, G.: Flow and contaminant transport in fractured rock. Academic Press, San Diego (1993)

    Google Scholar 

  9. Berkowitz, B.: Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour. 25(8–12), 861–884 (2002)

    Article  Google Scholar 

  10. Biryukov, D., Kuchuk, F.J.: Transient pressure behavior of reservoirs with discrete conductive faults and fractures. Trans. Porous Med. 95(1), 239–268 (2012)

    Article  Google Scholar 

  11. Brenner, K., Groza, M., Guichard, C., Masson, R.: Vertex approximate gradient scheme for hybrid dimensional two-phase darcy flows in fractured porous media. ESAIM: Math. Modell. Numer. Anal. 49(2), 303–330 (2015)

    Article  Google Scholar 

  12. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, volume 15 of Computational Mathematics. Springer Verlag, Berlin (1991)

    Book  Google Scholar 

  13. D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. Math. Modell. Numer. Anal. 46(02), 465–489 (2012)

    Article  Google Scholar 

  14. Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: A unified approach to mimetic finite difference, hybird finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20(02), 265–295 (2010)

    Article  Google Scholar 

  15. Ern, A., Guermond, J.-L.: Theory and practice of finite elements. Springer (2004)

  16. Eymard, R., Gallouët, T., Herbin, R.: A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. Compt. Rendus Math. 344(6), 403–406 (2007)

    Article  Google Scholar 

  17. Eymard, R., Gallout, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010)

    Article  Google Scholar 

  18. Faille, I., Flauraud, E., Nataf, F., Pégaz-Fiornet, S., Schneider, F., Willien, F.: A new fault model in geological basin modelling, Application of finite volume scheme and domain decomposition methods. In: Finite volumes for complex applications, III (Porquerolles, 2002), pp. 529–536. Hermes Scientific Publications, Paris (2002)

    Google Scholar 

  19. Faille, I., Nataf, F., Saas, L., Willien, F.: Finite Volume Methods on Non-Matching Grids with Arbitrary Interface Conditions and Highly Heterogeneous Media. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T., Kornhuber, R., Hoppe, R., Priaux, J., Pironneau, O., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering, volume 40 of Lecture Notes in Computational Science and Engineering, pp. 243–250. Springer Berlin Heidelberg (2005)

  20. Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. ESAIM: Math. Modell. Numer. Anal. 48, 7 (2014)

    Article  Google Scholar 

  21. Frih, N., Martin, V., Roberts, J.E., Ai, S.: Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16(4), 1043–1060 (2012)

    Article  Google Scholar 

  22. Frih, N., Roberts, J.E., Saada, A.: Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12(1), 91–104 (2008)

    Article  Google Scholar 

  23. Fumagalli, A., Scotti, A.: A numerical method for two-phase flow in fractured porous media with non-matching grids. Adv. Water Resour. 62(Part C(0)), 454–464 (2013). Computational Methods in Geologic CO2 Sequestration

    Article  Google Scholar 

  24. Fumagalli, A., Scotti, A.: An efficient xfem approximation of darcy flow in arbitrarly fractured porous media. Oil Gas Sci. Technol. - Revue d’IFP Energ. Nouvelles 69(4), 555–564 (2014)

    Article  Google Scholar 

  25. Hægland, H., Assteerawatt, A., Dahle, H.K., Eigestad, G.T., Helmig, R.: Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture-matrix system. Adv. Water Resour. 32(12), 1740–1755 (2009)

    Article  Google Scholar 

  26. Herbin, R., Hubert, F., et al.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. Finite volumes for complex applications V, 659–692 (2008)

  27. Hoteit, H., Firoozabadi, A.: Multicomponent fluid flow by discontinuous galerkin and mixed methods in unfractured and fractured media. Water Resour. Res. 41(11) (2005)

  28. Hoteit, H., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31(6), 891–905 (2008)

    Article  Google Scholar 

  29. Huang, H., Long, T.A., Wan, J., Brown, W.P.: On the use of enriched finite element method to model subsurface features in porous media flow problems. Comput. Geosci. 15(4), 721–736 (2011)

    Article  Google Scholar 

  30. Jaffré, J., Mnejja, M., Roberts, J.E.: A discrete fracture model for two-phase flow with matrix-fracture interaction. Proced. Comput. Sci. 4, 967–973 (2011)

    Article  Google Scholar 

  31. Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9(2), 227–236 (2004)

    Article  Google Scholar 

  32. Karimi-Fard, M., Firoozabadi, A.: Numerical simulation of water injection in fractured media using the discrete-fracture model and the galerkin method. SPE Reserv. Eval. Eng. 6(2), 117–126 (2003)

    Article  Google Scholar 

  33. Knabner, P., Jean, E.: Roberts. Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with darcy-forchheimer flow in the fracture. 48 9, 1451– 1472 (2014)

    Google Scholar 

  34. Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005)

    Article  Google Scholar 

  35. Monteguado, J.E.P., Firoozabadi, A.: Control-volume method for numerical simulation of two-phase immiscible flow in two- and three-dimensional discrete-fractured media. Water Resour. Res. 40(7), n/a–n/a (2004)

    Article  Google Scholar 

  36. Morales, F., Showalter, R.E.: The narrow fracture approximation by channeled flow. J. Math. Anal. Appl. 365(1), 320–331 (2010)

    Article  Google Scholar 

  37. Morales, F., Showalter, R.E.: Interface approximation of Darcy flow in a narrow channel. Math. Methods Appl. Sci. 35(2), 182–195 (2012)

    Article  Google Scholar 

  38. Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations, volume 23 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1994)

    Google Scholar 

  39. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29(7), 1020–1036 (2006)

    Article  Google Scholar 

  40. Revil, A., Cathles, L.M., III.: Fluid transport by solitary waves along growing faults a field example from the south eugene island basin, gulf of Mexico. Earth and Planetary Science Letters, 321–335 (2002)

  41. Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Handbook of numerical analysis, vol. II, pp. 523–639. North-Holland, Amsterdam (1991)

    Google Scholar 

  42. Tunc, X., Faille, I., Gallouët, T., Cacas, M.C., Havé, P.: A model for conductive faults with non-matching grids. Comput. Geosci. 16, 277–296 (2012)

    Article  Google Scholar 

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Authors and Affiliations

  1. IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852, Rueil-Malmaison, France

    Isabelle Faille & Alessio Fumagalli

  2. INRIA Roquencourt, BP 105, 78153, Le Chesnay, France

    Jérôme Jaffré & Jean E. Roberts

Authors
  1. Isabelle Faille
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  2. Alessio Fumagalli
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  3. Jérôme Jaffré
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  4. Jean E. Roberts
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Correspondence to Alessio Fumagalli.

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Faille, I., Fumagalli, A., Jaffré, J. et al. Model reduction and discretization using hybrid finite volumes for flow in porous media containing faults. Comput Geosci 20, 317–339 (2016). https://doi.org/10.1007/s10596-016-9558-3

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