Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Gradient discretization of hybrid dimensional Darcy flows in fractured porous media

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

This article deals with the discretization of hybrid dimensional Darcy flows in fractured porous media. These models couple the flow in the fractures represented as surfaces of codimension one with the flow in the surrounding matrix. The convergence analysis is carried out in the framework of gradient schemes which accounts for a large family of conforming and nonconforming discretizations. The vertex approximate gradient scheme and the hybrid finite volume scheme are extended to such models and are shown to verify the gradient scheme framework. Our theoretical results are confirmed by numerical experiments performed on tetrahedral, Cartesian and hexahedral meshes in heterogeneous isotropic and anisotropic porous media.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces, Pure and Applied Mathematics, vol. 140. Academic Press, New York (1978)

  2. 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. Comput. Phys. 284, 462–489 (2015)

  3. Alboin, C., Jaffré, J., Roberts, J., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. Fluid Flow Transp. Porous Media 295, 13–24 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. ESAIM Math. Model. Numer. Anal. 23, 239–275 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner, K., Masson, R.: Convergence of a vertex centered discretization of two-phase Darcy flows on general meshes. Int. J. Finite, Vol. Methods 10, 1–37 (2013)

    Google Scholar 

  6. 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, 303–330 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)

  8. 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), 1533–1552 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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(2), 465–489 (2012)

    Article  MATH  Google Scholar 

  11. Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: Gradient schemes for elliptic and parabolic problems (2014). (Personal communication)

  12. 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 Methods Appl. Sci. 20(2), 265–295 (2010)

  13. Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. 23(13), 2395–2432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3D schemes for diffusive flows in porous media. Math. Model. Numer. Anal. 46, 265–290 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eymard, R., Herbin, R., Guichard, C., Masson, R.: Vertex centered discretization of compositional multiphase Darcy flows on general meshes. Comput. Geosci. 16(4), 987–1005 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Flauraud, E., Nataf, F., Faille, I., Masson, R.: Domain decomposition for an asymptotic geological fault modeling. Comptes Rendus à l’académie des Sci. de Mécanique 331, 849–855 (2003)

    Article  MATH  Google Scholar 

  18. Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. ESAIM Math. Model. Numer. Anal. 48(4), 1089–1116 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grisvard, P.: Elliptic Problems on Non Smooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston, MA (1985)

  20. Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. (2004)

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  23. Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378(1), 324–342 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Monteagudu, J., Firoozabadi, A.: Control-volume model for simulation of water injection in fractured media: incorporating matrix heterogeneity and reservoir wettability effects. SPE J. 12, 355–366 (2007)

    Article  Google Scholar 

  25. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for multiphase flow in fractured porous media. Adv. Water Resour. 29(7), 1020–1036 (2006)

    Article  Google Scholar 

  26. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  27. Saad, Y.: http://www-users.cs.umn.edu/~saad/software/SPARSEKIT/index.html. Accessed 10 Dec 2014

  28. Sandve, T.H., Berre, I., Nordbotten, J.M.: An efficient multi-point flux approximation method for discrete fracture-matrix simulations. J. Comput. Phys. 231, 3784–3800 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Si, H.: http://tetgen.org

  30. Tunc, X., Faille, I., Gallouët, T., Cacas, M.C., Havé, P.: A model for conductive faults with non matching grids. Comput. Geosci. 16, 277–296 (2012)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank GDFSuez EP and Storengy for partially supporting this work, and Robert Eymard for fruitful discussions during the elaboration of this work.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Konstantin Brenner, Mayya Groza, Gilles Lebeau & Roland Masson

  2. Team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose, 06108, Nice Cedex 02, France

    Konstantin Brenner, Mayya Groza & Roland Masson

  3. Laboratoire Jacques-Louis Lions, UMR 7598, CNRS, UPMC Univ Paris 06, Sorbonne Universités, 75005, Paris, France

    Cindy Guichard

  4. INRIA, ANGE Project-Team, Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Cindy Guichard

  5. CEREMA, ANGE Project-Team, 134 rue de Beauvais, 60280, Margny-Lès-Compiègne, France

    Cindy Guichard

Authors
  1. Konstantin Brenner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mayya Groza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cindy Guichard
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gilles Lebeau
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Roland Masson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Masson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brenner, K., Groza, M., Guichard, C. et al. Gradient discretization of hybrid dimensional Darcy flows in fractured porous media. Numer. Math. 134, 569–609 (2016). https://doi.org/10.1007/s00211-015-0782-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-015-0782-x

Mathematics Subject Classification

Search

Navigation