1887

Abstract

Summary

A new Hybrid Finite Volume discretization is proposed in this work for two-phase Darcy flow in Discrete Fracture Matrix (DFM) models accounting for nonlinear transmission conditions at matrix fracture (mf) interfaces. This type of model is more accurate than alternative hybrid-dimensional two-phase Darcy flow models based either on continuous phase pressures at the mf interfaces assuming fractures acting as drains, or based on the elimination of the mf interface phase pressures by harmonic transmissibility. On the other hand, keeping the pressure and saturation unknowns and the nonlinear flux continuity equations at the mf interfaces increases the difficulty to solve the nonlinear and linear systems due to the highly contrasted permeabilities, capillary pressures, and scales between the fractures and the matrix. In order to solve efficiently the nonlinear systems arising at each time step from the fully implicit time integration, a Newton solver with linear elimination and nonlinear update of the mf interface unknowns is derived. Numerical experiments show the efficiency of our approach on several 2D test cases including an anisotropic matrix permeability and a large fracture network.

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2018-09-03
2024-03-28
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References

  1. J.Aghili, K.Brenner, J.Hennicker, R.Masson, and L.Trenty
    : Two-phase discrete fracture matrix models with nonlinear transmission conditions, <mo>https://hal.archives-ouvertes.fr/hal-01764432</mo>, 2018.
  2. Alboin, C., Jaffré, J., Roberts, J., Serres, C.
    : Modeling fractures as interfaces for flow and transport in porous media. Fluid flow and transport in porous media295, 13–24, 2002.
    [Google Scholar]
  3. Ahmed, E., M., Jaffré, J., Roberts, J. E.
    , A reduced fracture model for two-phase flow with different rock types, Mathematics and Computers in Simulation, 7, pp. 49–70, 2017.
    [Google Scholar]
  4. Ahmed, R., Edwards, M.G., Lamine, S., Huisman, B.A.H.
    , Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model, J. Comp. Physics, 462–489, Vol. 284, 2015.
    [Google Scholar]
  5. Angot, P., Boyer, F., Hubert, F.
    Asymptotic and numerical modeling of flows in fractured porous media, M2AN, 43,2, pp; 239–275, 2009.
    [Google Scholar]
  6. R.G.Bentsen and J.Anli
    . Using parameter estimation techniques to convert centrifuge data into a capillary-pressure curve, SPE Journal, 17(1):57–64, 1977.
    [Google Scholar]
  7. Bogdanov, I., Mourzenko, V., Thovert, J.-F., Adler, P. M.
    , Two-phase flow through fractured porous media, Physical Review E68, 026703, 2003.
    [Google Scholar]
  8. K.Brenner; J.Hennicker; R.Masson; P.Samier
    . Gradient discretization of hybrid-dimensional Darcy flow in fractured porous media with discontinuous pressures at matrix-fracture interfaces, IMA Journal of Numerical Analysis, 37,3, pp. 1551–1585, 2016.
    [Google Scholar]
  9. K.Brenner, J.Hennicker, R.Masson, P.Samier
    . Hybrid-dimensional modeling of two-phase flow through fractured porous media with enhanced matrix fracture transmission conditions. J. Comp. Physics, 100–124, Vol. 357, 2018.
    [Google Scholar]
  10. Brenner, K., Groza, M., Guichard, C., Lebeau, G. and Masson, R.
    Gradient discretization of Hybrid-Dimensional Darcy Flows in Fractured Porous Media. Numerische Mathematik, 134,3, pp. 569–609, 2016.
    [Google Scholar]
  11. Brenner, K., Groza, M., Jeannin, L., Masson, R., Pellerin, J.
    Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media, Computational Geosciences, online july 2017.
    [Google Scholar]
  12. Brenner, K., Groza, M., Guichard, C., Masson, R.
    Vertex Approximate Gradient Scheme for Hybrid-Dimensional Two-Phase Darcy Flows in Fractured Porous Media. ESAIM Mathematical Modeling and Numerical Analysis, 49, pp. 303–330, 2015.
    [Google Scholar]
  13. Brezzi, F., LipnikovK., SimonciniV.
    , A family of mimetic finite difference methods on polygonal and polyhedral meshes, Mathematical Models and Methods in Applied Sciences, vol. 15, 10, 2005, 1533–1552.
    [Google Scholar]
  14. R. H.Brooks and A. T.Corey
    . Hydraulic properties of porous media and their relation to drainage design. Transactions of the ASAE, 7(1):0026–0028, 1964.
    [Google Scholar]
  15. Droniou, J., Eymard, R., Gallouët, T., Herbin, R.
    : A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods. Math. Models and Methods in Appl. Sci. 20,2, 265–295 (2010).
    [Google Scholar]
  16. Eymard, R., Gallouët, T., Herbin, R.
    : Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30, 4, 1009–1043 (2010).
    [Google Scholar]
  17. R.Eymard, T.Gallouët, C.Guichard, R.Herbin, and R.Masson
    . TP or not TP, that is the question. Comput. Geosci., 18:285–296, 2014.
    [Google Scholar]
  18. Flauraud, E., Nataf, F., Faille, I., Masson, R.
    Domain Decomposition for an asymptotic geological fault modeling, Comptes Rendus à l’Académie des Sciences, Mécanique, 331, pp 849–855, 2003.
    [Google Scholar]
  19. FlemischB., BerreI., BoonW., FumagalliA., SchwenckN., ScottiA., StefanssonI., TatomirA.
    Benchmarks for single-phase flow in fractured porous media, Advances in Water Resources239–258, Vol. 111, 2018.
    [Google Scholar]
  20. Hoteit, H., Firoozabadi, A.
    Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Advanced Water Resources31, pp. 56–73, 2008.
    [Google Scholar]
  21. Jaffré, J., Martin, V., Roberts, J. E.
    Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput. 26,5, pp. 1667–1691, 2005.
    [Google Scholar]
  22. Jaffré, J., Mnejja, M., Roberts, J. E.
    , A discrete fracture model for two-phase flow with matrix-fracture interaction, Procedia Computer Science4, pp. 967–973 (2011)
    [Google Scholar]
  23. Karimi-Fard, M., Durlovski, L.J., Aziz, K.
    An efficient discrete-fracture model applicable for general-purpose reservoir simulators, SPE journal, June 2004.
    [Google Scholar]
  24. Lacroix, S., Vassilevski, Y. V., Wheeler, M. F.
    : Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS). Numerical Linear Algebra with Applications, 8, 537–549 (2001)
    [Google Scholar]
  25. Monteagudo, J.E.P. and Firoozabadi, A.
    Control-Volume Model for Simulation of Water Injection in Fractured Media: Incorporating Matrix Heterogeneity and Reservoir Wettability Effects, SPE Journal12, 3, 2007.
    [Google Scholar]
  26. Mualem, Y.
    A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12, 513–522, 1976
    [Google Scholar]
  27. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.
    : A mixed-dimensional finite volume method for multiphase flow in fractured porous media. Adv. Water Resources29, 7, pp. 1020–1036, 2006.
    [Google Scholar]
  28. Scheichl, R., Masson, R., Wendebourg, J.
    : Decoupling and block preconditioning for sedimentary basin simulations. Computational Geosciences, 7, 295–318 (2003)
    [Google Scholar]
  29. Sandve, T.H., Berre, I., Nordbotten, J.M.
    An efficient multi-point flux approximation method for Discrete Fracture-Matrix simulations, JCP231 pp. 3784–3800, 2012.
    [Google Scholar]
  30. Tunc, X., Faille, I., Gallouët, T., Cacas, M.C., Havé, P.
    A model for conductive faults with non matching grids, Comp. Geosciences, 16, pp. 277–296, 2012.
    [Google Scholar]
  31. Van Genuchten, M.T.
    A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils, Soil Sci. Soc. Am. J. 1980, 44, 892–898.
    [Google Scholar]
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