A hybrid-dimensional discrete fracture model for non-isothermal two-phase flow in fractured porous media

  • Dennis GläserEmail author
  • Bernd Flemisch
  • Rainer Helmig
  • Holger Class
Original Paper
Part of the following topical collections:
  1. Numerical methods for processes in fractured porous media


We present a hybrid-dimensional numerical model for non-isothermal two-phase flow in fractured porous media, in which the fractures are modeled as entities of codimension one embedded in a bulk domain. Potential fields of applications of the model could be radioactive waste disposal or geothermal energy production scenarios in which a two-phase flow regime develops or where \(\hbox {CO}_2\) is used as working fluid. We test the method on synthetic test cases involving compressible fluids and strongly heterogeneous, full tensor permeability fields by comparison with a reference solution obtained from an equi-dimensional discretization of the domain. The results reveal that especially for the case of a highly conductive fracture, the results are in good agreement with the reference. While the model qualitatively captures the involved phenomena also for the case of a fracture acting as both hydraulic and capillary barrier, it introduces larger errors than in the highly-conductive fracture case, which can be attributed to the lower-dimensional treatment of the fracture. Finally, we apply the method to a three-dimensional showcase that resembles setups for the determination of upscaled parameters of fractured blocks.


Porous media Fractures Finite volumes Multi-point flux approximation 

Mathematics Subject Classification




The research has received funding from the European Community Horizon 2020 Research and Innovation Programme under Grant Agreement No. 636811 in the scope of the FracRisk Project. Furthermore, the authors would like to thank the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart for the support.


  1. Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3), 405–432 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  2. Aavatsmark, I., Eigestad, G., Mallison, B., Nordbotten, J.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24(5), 1329–1360 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  3. Agélas, L., Di Pietro, D.A., Droniou, J.: The g method for heterogeneous anisotropic diffusion on general meshes. ESAIM Math. Model. Numer. Anal. 44(4), 597–625 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  4. Ahmed, R., Edwards, M., Lamine, S., Huisman, B., Pal, M.: Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model. J. Comput. Phys. 284, 462–489 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ahmed, R., Edwards, M.G., Lamine, S., Huisman, B.A., Pal, M.: CVD-MPFA full pressure support, coupled unstructured discrete fracturematrix Darcy-flux approximations. J. Comput. Phys. 349, 265–299 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  6. Assteerawatt, A.: Flow and transport modelling of fractured aquifers based on a geostatistical approach. Ph.D. Thesis, Universitätsbibliothek der Universität Stuttgart, Stuttgart (2008).
  7. Brenner, K., Groza, M., Guichard, C., Masson, R.: Vertex approximate gradient scheme forhybrid dimensional two-phase darcy flowsin fractured porous media. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, pp. 507–515. Springer, Cham (2014)zbMATHGoogle Scholar
  8. Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media and their relation to drainage design. Trans. ASAE 7(1), 26–0028 (1964)CrossRefGoogle Scholar
  9. Davis, T.A.: Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  10. Droniou, J.: Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24(08), 1575–1619 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  11. Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259–290 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  12. Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., Helmig, R.: DuMu\(^{\rm x}\): DUNE for multi-phase, component, scale, physics,.flow and transport in porous media. Adv. Water Resour. 34, 1102–1112 (2011). CrossRefGoogle Scholar
  13. Friis, H.A., Edwards, M.G.: A family of mpfa finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids. J. Comput. Phys. 230(1), 205–231 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fuchs, A.: Almost regular triangulations of trimmend nurbs-solids. Eng. Comput. 17(1), 55–65 (2001)CrossRefGoogle Scholar
  15. Fumagalli, A., Scotti, A.: A numerical method for two-phase flow in fractured porous media with non-matching grids. Adv. Water Resour. 62, 454–464 (2013). CrossRefzbMATHGoogle Scholar
  16. Geuzaine, C., Remacle, J.F.: Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gläser, D., Helmig, R., Flemisch, B., Class, H.: A discrete fracture model for two-phase flow in fractured porous media. Adv. Water Resour. 110, 335–348 (2017). CrossRefGoogle Scholar
  18. Jaeger, J., Cook, N., Zimmerman, R.: Fundamentals of Rock Mechanics. Wiley, Hoboken (2007)Google Scholar
  19. Jaffré, J., Mnejja, M., Roberts, J.: A discrete fracture model for two-phase flow with matrix–fracture interaction. Procedia Comput. Sci. 4, 967–973 (2011). CrossRefGoogle Scholar
  20. Karimi-Fard, M., Durlofsky, L., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9, 227–236 (2004)CrossRefGoogle Scholar
  21. Kauffman, G.W., Jurs, P.C.: Prediction of surface tension, viscosity, and thermal conductivity for common organic solvents using quantitative structure property relationships. J. Chem. Inf. Comput. Sci. 41(2), 408–418 (2001). (pMID: 11277730)CrossRefGoogle Scholar
  22. Kazemi, H., Merrill Jr., L., Porterfield, K., Zeman, P., et al.: Numerical simulation of water-oil flow in naturally fractured reservoirs. Soci. Pet. Eng. J. 16(06), 317–326 (1976). CrossRefGoogle Scholar
  23. 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). MathSciNetCrossRefzbMATHGoogle Scholar
  24. Matthai, S.K., Mezentsev, A., Belayneh, M.: Finite element—node-centered finite-volume two-phase-flow experiments with fractured rock represented by unstructured hybrid-element meshes. Soc. Pet. Eng. (2007).
  25. Pruess, K.: Brief Guide to the MINC-Method for Modeling Flow and Transport in Fractured Media. United States, Department of Energy, Washington, DC (1992)Google Scholar
  26. Pruess, K.: Enhanced geothermal systems (EGS) using CO\(_{2}\) as working fluida novel approach for generating renewable energy with simultaneous sequestration of carbon. Geothermics 35(4), 351–367 (2006). CrossRefGoogle Scholar
  27. 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)CrossRefGoogle Scholar
  28. Reid, R., Prausnitz, J., Poling, B.: The Properties of Gases and Liquids. McGraw-Hill Inc., New York City (1987)Google Scholar
  29. Sandve, T., Berre, I., Nordbotten, J.: An efficient multi-point flux approximation method for discrete fracture–matrix simulations. J. Comput. Phys. 231(9), 3784–3800 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  30. Schneider, M., Agélas, L., Enchry, G., Flemisch, B.: Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. J. Comput. Phys. 351, 80–107 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  31. Schwenck, N.: An XFEM-based model for fluid flow in fractured porous media. Ph.D. Thesis, Universitätsbibliothek der Universitäat Stuttgart, Stuttgart (2015).
  32. Silberhorn-Hemminger, A.: Modellierung von kluftaquifersystemen: Geostatistische analyse und deterministisch-stochastische kluftgenerierung. Ph.D. Thesis, Universitätsbibliothek der Universität Stuttgart, Stuttgart (2003)Google Scholar
  33. Somerton, W., Keese, A., Chu, L.: Thermal behavior of unconsolidated oil sands. Soc. Pet. Eng. J. 14, 513–521 (1974)CrossRefGoogle Scholar
  34. Tatomir, A.B.: From discrete to continuum concepts of flow in fractured porous media. Ph.D. Thesis, Universitätsbibliothek der Universität Stuttgart, Stuttgart (2013).
  35. Tene, M., Bosma, S.B., Kobaisi, M.S.A., Hajibeygi, H.: Projection-based embedded discrete fracture model (pEDFM). Adv. Water Resour. 105, 205–216 (2017). CrossRefGoogle Scholar
  36. Van Genuchten, M.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils1. Soil Sci. Soc. Am. J. 44, 892 (1980)CrossRefGoogle Scholar
  37. Wagner, W., Kretzschmar, H.J.: IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam. In: International Steam Tables, pp. 7–150. Springer, Berlin (2008)Google Scholar
  38. Warren, J., Root, P.: The behavior of naturally fractured reservoirs. Soc. Pet. Eng. J. 3(03), 245–255 (1963)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Dennis Gläser
    • 1
    Email author
  • Bernd Flemisch
    • 1
  • Rainer Helmig
    • 1
  • Holger Class
    • 1
  1. 1.Department of Hydromechanics and Modelling of HydrosystemsUniversity of StuttgartStuttgartGermany

Personalised recommendations