A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures

  • Markus KöppelEmail author
  • Vincent Martin
  • Jean E. Roberts
Original Paper
Part of the following topical collections:
  1. Numerical methods for processes in fractured porous media


In this work we introduce a stabilized, numerical method for a multidimensional, discrete-fracture model (DFM) for single-phase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the \((n-1)\)-dimensional fracture domain is coupled with that in the n-dimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddle-point problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a weakly consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM’s.


Discrete fracture model Finite element method Stabilized Lagrange multiplier method Penalization Nonconforming grids 

Mathematics Subject Classification

35J50 35J57 65N12 65N85 76M10 76S05 



The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart, and the two referees whose comments helped improving this paper.


  1. Ahmed, E., Jaffré, J., Roberts, J.E.: A reduced fracture model for two-phase flow with different rock types. Math. Comput. Simul. 137, 49–70 (2017)MathSciNetCrossRefGoogle Scholar
  2. Ainsworth, M.: A posteriori error estimation for discontinuous Galerkin finite element approximation. SIAM J. Numer. Anal. 45(4), 1777–1798 (2007)MathSciNetCrossRefGoogle Scholar
  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, Contemporary Mathematics, vol. 295, American Mathematical Society, Providence, RI, pp. 13–24 (2002)Google Scholar
  4. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: M2AN 43(2), 239–275 (2009)MathSciNetCrossRefGoogle Scholar
  5. Antonietti, P.F., Facciolà, C., Russo, A., Verani, M.: Discontinuous Galerkin approximation of flows in fractured porous media on polytopic grids. Technical Report 22/2016, Politecnico di Milano (2016a)Google Scholar
  6. Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzott, N.: Mimetic finite difference approximation of flows in fractured porous media. ESAIM: M2AN 50(3), 809–832 (2016b)MathSciNetCrossRefGoogle Scholar
  7. 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, 337–348 (1984)CrossRefGoogle Scholar
  8. Berrone, S., Pieraccini, S., Scialò, S.: An optimization approach for large scale simulations of discrete fracture network flows. J. Comput. Phys. 256, 838–853 (2014)MathSciNetCrossRefGoogle Scholar
  9. Boon, W., Nordbotten, J., Yotov, I.: Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56(4), 2203–2233 (2018)MathSciNetCrossRefGoogle Scholar
  10. 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. Model. Numer. Anal. 49(2), 303–330 (2015)MathSciNetCrossRefGoogle Scholar
  11. Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005)MathSciNetCrossRefGoogle Scholar
  12. Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Eng. 199(41–44), 2680–2686 (2010a)MathSciNetCrossRefGoogle Scholar
  13. Burman, E., Hansbo, P.: Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems. IMA J. Numer. Anal. 30, 870–885 (2010b)MathSciNetCrossRefGoogle Scholar
  14. Chave, F.A., Di Pietro, D.A., Formaggia, L.: A hybrid high-order method for passive transport in fractured porous media (2018).
  15. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)MathSciNetCrossRefGoogle Scholar
  16. Ding, Z.: A proof of the trace theorem of sobolev spaces on Lipschitz domains. Proc. Am. Math. Soc. 124(2), 591–600 (1996)MathSciNetCrossRefGoogle Scholar
  17. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefGoogle Scholar
  18. Faille, I., Fumagalli, A., Jaffré, J., Roberts, J.E.: Model reduction and discretization using hybrid finite volumes for flow in porous media containing faults. Comput. Geosci. 20(2), 317–339 (2016)MathSciNetCrossRefGoogle Scholar
  19. Flemisch, B., Berre, I., Boon, W., Fumagalli, A., Schwenck, N., Scotti, A., Stefansson, I., Tatomir, A.: Benchmarks for single-phase flow in fractured porous media. Adv. Water Resour. 111, 239–258 (2018)CrossRefGoogle Scholar
  20. Formaggia, L., Scotti, A., Sottocasa, F.: Analysis of a mimetic finite difference approximation of flows in fractured porous media. ESAIM: M2AN 52(2), 595–630 (2018)MathSciNetCrossRefGoogle Scholar
  21. Frih, N., Martin, V., Roberts, J.E., Saâda, A.: Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16(4), 1043–1060 (2012)CrossRefGoogle Scholar
  22. 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), 454–464. (Computational Methods in Geologic CO\(_2\) Sequestration (2013))Google Scholar
  23. Fumagalli, A., Pasquale, L., Zonca, S., Micheletti, S.: An upscaling procedure for fractured reservoirs with embedded grids. Water Resour. Res. 52(8), 6506–6525 (2016)CrossRefGoogle Scholar
  24. Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes–Darcy equations. Electron. Trans. Numer. Anal. 26, 350–384 (2007)MathSciNetzbMATHGoogle Scholar
  25. Geiger, S., Dentz, M., Neuweiler, I.: A novel multi-rate dual-porosity model for improved simulation of fractured and multi-porosity reservoirs. Soc. Petrol. Eng. J. 18(4), 670–684 (2013)Google Scholar
  26. Girault, V., Glowinski, R.: Error analysis of a fictitious domain method applied to a Dirichlet problem. Jpn. J. Ind. Appl. Math. 12(3), 487 (1995)MathSciNetCrossRefGoogle Scholar
  27. Hoteit, H., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31, 891–905 (2008)CrossRefGoogle Scholar
  28. Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. Soc. Petrol. Eng. J. 9(2), 227–236 (2004)Google Scholar
  29. Knabner, P., Roberts, J.E.: Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture. ESAIM: Math. Model. Numer. Anal. 48(5), 1451–1472 (2014)MathSciNetCrossRefGoogle Scholar
  30. Köppel, M.: Flow in Heterogeneous Porous Media: Fractures and Uncertainty Quantification. PhD thesis, University of Stuttgart, Germany (2018)Google Scholar
  31. Köppel, M., Martin, V., Jaffré, J., Roberts, J.E.: A Lagrange multiplier method for a discrete fracture model for flow in porous media (2018 accepted).
  32. Lesinigo, M., D’Angelo, C., Quarteroni, A.: A multiscale Darcy–Brinkman model for fluid flow in fractured porous media. Numer. Math. 117(4), 717–752 (2011)MathSciNetCrossRefGoogle Scholar
  33. 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)MathSciNetCrossRefGoogle Scholar
  34. Massing, A.: A Cut Discontinuous Galerkin method for coupled bulk-surface problems. (2017) ArXiv e-print. arXiv:1707.02153v1 [math.NA]
  35. Pichot, G., Erhel, J., de Dreuzy, J.R.: A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks. SIAM J. Sci. Comput. 34(1), B86–B105 (2012)MathSciNetCrossRefGoogle Scholar
  36. 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
  37. Sandve, T.H., Berre, I., Nordbotten, J.M.: An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations. J. Comput. Phys. 231(9), 3784–3800 (2012)MathSciNetCrossRefGoogle Scholar
  38. Schwenck, N., Flemisch, B., Helmig, R., Wohlmuth, B.I.: Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19(6), 1219–1230 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Markus Köppel
    • 1
    Email author
  • Vincent Martin
    • 2
  • Jean E. Roberts
    • 3
  1. 1.Universtität Stuttgart, IANSStuttgartGermany
  2. 2.Université de Technologie de Compiègne (UTC), LMACCompiègne CedexFrance
  3. 3.INRIA ParisParisFrance

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