Implementation of Mixed-Dimensional Models for Flow in Fractured Porous Media

  • Eirik KeilegavlenEmail author
  • Alessio Fumagalli
  • Runar Berge
  • Ivar Stefansson
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


Models that involve coupled dynamics in a mixed-dimensional geometry are of increasing interest in several applications. Here, we describe the development of a simulation model for flow in fractured porous media, where the fractures and their intersections form a hierarchy of interacting subdomains. We discuss the implementation of a simulation framework, with an emphasis on reuse of existing discretization tools for mono-dimensional problems. The key ingredients are the representation of the mixed-dimensional geometry as a graph, which allows for convenient discretization and data storage, and a non-intrusive coupling of dimensions via boundary conditions and source terms. This approach is applicable for a wide class of mixed-dimensional problems. We show simulation results for a flow problem in a three-dimensional fracture geometry, applying both finite volume and virtual finite element discretizations.



We acknowledge financial support from the Research Council of Norway, project no. 244129/E20 and 250223.


  1. 1.
    L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM: Math. Model. Numer. Anal. 50(3), 727–747 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    W.M. Boon, J.M. Nordbotten, J.E. Vatne, Mixed-dimensional elliptic partial differential equations. arXiv:1710.00556 (2017)Google Scholar
  3. 3.
    W.M. Boon, J.M. Nordbotten, I. Yotov, Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56(4), 2203–2233 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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 J. Numer. Anal. 37(3), 1551–1585 (2017).MathSciNetzbMATHGoogle Scholar
  5. 5.
    F.A. Chave, D. Di Pietro, L. Formaggia, A hybrid high-order method for Darcy flows in fractured porous media. Tech. report, HAL archives:hal-01482925, 2017Google Scholar
  6. 6.
    C. D’Angelo, A. Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: Math. Model. Numer. Anal. 46(2), 465–489 (2012)CrossRefGoogle Scholar
  7. 7.
    A. Fumagalli, E. Keilegavlen, Dual virtual element methods for discrete fracture matrix models. Tech. report, arXiv:1711.01818, 2017Google Scholar
  8. 8.
    A. Fumagalli, A. Scotti, An efficient XFEM approximation of Darcy flows in arbitrarily fractured porous media. Oil Gas Sci. Technol. Revue d’IFP Energies Nouvelles 69(4), 555–564 (2014)CrossRefGoogle Scholar
  9. 9.
    E. Keilegavlen, A. Fumagalli, R. Berge, I. Stefansson, I. Berre, Porepy: an open source simulation tool for flow and transport in deformable fractured rocks. Tech. report, arXiv:1712.00460, 2017Google Scholar
  10. 10.
    V. Martin, J. Jaffré, J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    N. Schwenck, B. Flemisch, R. Helmig, B. Wohlmuth, Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19(6), 1219–1230 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L. Formaggia, A. Scotti, F. Sottocasa, Analysis of a mimetic finite difference approximation of flows in fractured porous media. ESAIM: Math. Model. Numer. Anal. 52, 595–630 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Eirik Keilegavlen
    • 1
    Email author
  • Alessio Fumagalli
    • 1
  • Runar Berge
    • 1
  • Ivar Stefansson
    • 1
  1. 1.University of BergenDepartment of MathematicsBergenNorway

Personalised recommendations