Computational Geosciences

, Volume 19, Issue 6, pp 1219–1230 | Cite as

Dimensionally reduced flow models in fractured porous media: crossings and boundaries

  • Nicolas Schwenck
  • Bernd Flemisch
  • Rainer Helmig
  • Barbara I. Wohlmuth


For the simulation of fractured porous media, a common approach is the use of co-dimension one models for the fracture description. In order to simulate correctly the behavior at fracture crossings, standard models are not sufficient because they either cannot capture all important flow processes or are computationally inefficient. We propose a new concept to simulate co-dimension one fracture crossings and show its necessity and accuracy by means of an example and a comparison to a literature benchmark. From the application point of view, often the pressure is known only at a limited number of discrete points and an interpolation is used to define the boundary condition at the remaining parts of the boundary. The quality of the interpolation, especially in fracture models, influences the global solution significantly. We propose a new method to interpolate boundary conditions on boundaries that are intersected by fractures and show the advantages over standard interpolation methods.


Reduced flow models Fractured porous media Boundary conditions Fracture crossings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdelaziz, Y., Hamouine, A.: A survey of the extended finite element. Comput. Struct. 86(11–12), 1141–1151 (2008)CrossRefGoogle Scholar
  2. 2.
    Acosta, M., Merten, C., Eigenberger, G., Class, H., Helmig, R., Thoben, B., Müller-Steinhagen, H.: Modeling non-isothermal two-phase multicomponent flow in the cathode of PEM fuel cells. J. Power Sources 159 (2), 1123–1141 (2006)CrossRefGoogle Scholar
  3. 3.
    Alboin, C, Jaffré, J, Roberts, J E, Serres, C: Modeling fractures as interfaces for flow and transport in porous media. In: Fluid Flow and Transport in Porous Media, Mathematical and Numerical Treatment: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Fluid Flow and Transport in Porous Media, Mathematical and Numerical Treatment, June 17–21, 2001, Mount Holyoke College, South Hadley, Massachusetts, American Mathematical Soc., vol. 295, pp. 13–25 (2002)Google Scholar
  4. 4.
    Angot, P., Boyer, F., Hubert, F., et al: Asymptotic and numerical modelling of flows in fractured porous media. Model. Math. Anal. Numér. 23(2), 239–275 (2009)CrossRefGoogle Scholar
  5. 5.
    Assteerawatt, A.: Flow and transport modelling of fractured aquifers based on a geostatistical approach. PhD thesis, Universität Stuttgart (2008)Google Scholar
  6. 6.
    Bear, J., Tsang, C.F., Marsily, G.: Flow and contaminant transport in fractured rocks. Academic, San Diego (1993)CrossRefGoogle Scholar
  7. 7.
    Berkowitz, B.: Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour. 25(8–12), 861–884 (2002)CrossRefGoogle Scholar
  8. 8.
    Berrone, S., Pieraccini, S., Scialò, S: On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35(2), A908–A935 (2013)CrossRefGoogle Scholar
  9. 9.
    D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: Math. Model. Numer. Anal. 46(02), 465–489 (2012)CrossRefGoogle Scholar
  10. 10.
    Dietrich, P., Helmig, R., Sauter, M., Hötzl, H, Köngeter, J, Teutsch, G: Flow and transport in fractured porous media. Springer, Berlin (2005)CrossRefGoogle Scholar
  11. 11.
    Dogan, M.O., Class, H., Helmig, R.: Different concepts for the coupling of porous-media flow with lower-dimensional pipe flow. CMES: Comput. Model. Eng. Sci. 53(3), 207–234 (2009)Google Scholar
  12. 12.
    Dolbow, J.: An extended finite element method with discontinuous enrichment for applied mechanics. PhD thesis, Northwestern University (1999)Google Scholar
  13. 13.
    Dolbow, J, Moës, N, Belytschko, T: Discontinuous enrichment in finite elements with a partition of unity method. Finite Elem. Anal. Des. 36(3–4), 235–260 (2000)CrossRefGoogle Scholar
  14. 14.
    Erbertseder, K., Reichold, J., Helmig, R., Jenny, P., Flemisch, B.: A coupled discrete/continuum model for describing cancer therapeutic transport in the lung. PLoS One 7(3), e31,966 (2012)CrossRefGoogle Scholar
  15. 15.
    Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. MOX report 32 (2012)Google Scholar
  16. 16.
    Fumagalli, A., Scotti, A.: An efficient XFEM approximation of Darcy flows in fractured porous media. MOX report 53 (2012)Google Scholar
  17. 17.
    Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193(33), 3523–3540 (2004)CrossRefGoogle Scholar
  18. 18.
    Hansbo, P.: Nitsche’s method for interface problems in computational mechanics. GAMM-Mitteilungen 28(2), 183–206 (2005)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    Matthäi, S K, Belayneh, M: Fluid flow partitioning between fractures and a permeable rock matrix. Geophys. Res. Lett. 31(7), 7602–6 (2004)CrossRefGoogle Scholar
  22. 22.
    Mohammadi, S.: Extended Finite Element Method. Wiley, New York (2008)CrossRefGoogle Scholar
  23. 23.
    Neumann, S.P.: Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol. J. 13, 124–147 (2005)CrossRefGoogle Scholar
  24. 24.
    Neunhäuserer, L: Diskretisierungsansätze zur Modellierung von Strömungs- und Transportprozessen in geklüftet-porösen Medien. PhD thesis, Universität Stuttgart. (2003)
  25. 25.
    Nordbotten, J., Celia, M., Dahle, H., Hassanizadeh, S.: Interpretation of macroscale variables in Darcy’s law. Water Resour. Res. 43(8) (2007)Google Scholar
  26. 26.
    Swedish Nuclear Power Inspectorate (SKI): The International Hydrocoin Project–Background and Results. Organization for Economic Co-operation and Development, Paris (1987)Google Scholar
  27. 27.
    Tsang, Y.W., Tsang, C.: Channel model of flow through fractured media. Water Resour. Res. 23(3), 467–479 (1987)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Nicolas Schwenck
    • 1
  • Bernd Flemisch
    • 1
  • Rainer Helmig
    • 1
  • Barbara I. Wohlmuth
    • 2
  1. 1.IWS, Department of Hydromechanics and Modelling of HydrosystemsUniversity of StuttgartStuttgartGermany
  2. 2.Institute for Numerical MathematicsTechnische Universität MünchenGarchingGermany

Personalised recommendations