Skip to main content
SpringerLink
Log in

A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media

  • Chapter
  • First Online:
Book cover Advances in Discretization Methods

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 12))

Abstract

This paper presents a review of the available mathematical models and corresponding non-conforming numerical approximations which describe single-phase fluid flow in a fractured porous medium. One focus is on the geometrical difficulties that may arise in realistic simulations such as intersecting and immersed fractures. Another important aspect is the choice of the approximation spaces for the discrete problem: in mixed formulations, both the Darcy velocity and the pressure are considered as unknowns, while in classical primal formulations, a richer space for the pressure is considered and the Darcy velocity is computed a posteriori. In both cases, the extended finite element method is used, which allows for a complete geometrical decoupling among the fractures and rock matrix grids. The fracture geometries can thus be independent of the underlying grid thanks to suitable enrichments of the spaces that are able to represent possible jumps of the solution across the fractures. Finally, due to the dimensional reduction, a better approximation of the resulting boundary conditions for the fractures is addressed.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abdelaziz, Y., Hamouine, A.: A survey of the extended finite element. Comput. Struct. 86 (11–12), 1141–1151 (2008)

    Article  Google Scholar 

  2. Alboin, C., Jaffré, J., Roberts, J.E., Wang, X., Serres, C.: Domain decomposition for some transmission problems in flow in porous media. In: Numerical Treatment of Multiphase Flows in Porous Media (Beijing, 1999). Lecture Notes in Physics, vol. 552, pp. 22–34. Springer, Berlin (2000)

    Google 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: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Fluid Flow and Transport in Porous Media, Mathematical and Numerical Treatment, Mount Holyoke College, South Hadley, Massachusetts, 17–21 June 2001, vol. 295, pp. 13–25. American Mathematical Society, Providence (2002)

    Google Scholar 

  4. 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 (South Hadley, MA, 2001). Contemporary Mathematics, vol. 295, pp. 13–24. American Mathematical Society, Providence (2002)

    Google Scholar 

  5. Amir, L., Kern, M., Martin, V., Roberts, J.E.: Décomposition de domaine et préconditionnement pour un modèle 3D en milieu poreux fracturé. In: Proceeding of JANO 8, 8th Conference on Numerical Analysis and Optimization (2005)

    Google Scholar 

  6. Angot, P.: A model of fracture for elliptic problems with flux and solution jumps. C. R. Math. 337 (6), 425–430 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. M2AN Math. Model. Numer. Anal. 43 (2), 239–275 (2009)

    Google Scholar 

  8. Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzotti, N.: Mimetic finite difference approximation of flows in fractured porous media. Technical Report, Politecnico di Milano (2015)

    MATH  Google Scholar 

  9. Bear, J.: Dynamics of Fluids in Porous Media. American Elsevier, New York (1972)

    MATH  Google Scholar 

  10. Benedetto, M.F., Berrone, S., Pieraccini, S., Scialò, S.: The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Eng. 280 (0), 135–156 (2014)

    Article  MathSciNet  Google Scholar 

  11. 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), 908–935 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Berrone, S., Pieraccini, S., Scialò, S.: An optimization approach for large scale simulations of discrete fracture network flows. J. Comput. Phys. 256 (0), 838–853 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bonn, W.M., Nordbotten, J.M.: Robust discretization of flow in fractured porous media. arXiv:1601.06977 [math.NA] (2016)

    Google Scholar 

  14. Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  15. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Computational Mathematics, vol. 15. Springer, Berlin (1991)

    Google Scholar 

  16. Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (4), 328–341 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104 (7), 472–501 (2015)

    Article  MathSciNet  Google Scholar 

  18. D’Angelo, C., Scotti, A.: A Mixed Finite Element Method for Darcy Flow in Fractured Porous Media with Non-Matching Grids. Technical Report, MOX, Mathematical Department, Politecnico di Milano, 2010

    MATH  Google Scholar 

  19. D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. Math. Model. Numer. Anal. 46 (02), 465–489 (2012)

    Article  MATH  Google Scholar 

  20. Dassi, F., Perotto, S., Formaggia, L., Ruffo, P.: Efficient geometric reconstruction of complex geological structures. Math. Comput. Simul. 46 (02), 465–489 (2014)

    MathSciNet  Google Scholar 

  21. Del Pra, M., Fumagalli, A., Scotti, A.: Well posedness of fully coupled fracture/bulk Darcy flow with XFEM. Technical Report 25/2015, Politecnico di Milano (2015). Submitted to: SIAM Journal on Numerical Analysis

    Google Scholar 

  22. Dolbow, J.: An extended finite element method with discontinuous enrichment for applied mechanics. Ph.D. thesis, Northwestern University (1999)

    Google Scholar 

  23. 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)

    Article  MATH  Google Scholar 

  24. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer, New York (2004)

    Book  MATH  Google Scholar 

  25. Faille, I., Flauraud, E., Nataf, F., Pégaz-Fiornet, S., Schneider, F., Willien, F.: A new fault model in geological basin modelling. Application of finite volume scheme and domain decomposition methods. In: Finite Volumes for Complex Applications, III (Porquerolles, 2002), pp. 529–536. Hermes Science Publishing, Paris (2002)

    Google Scholar 

  26. Faille, I., Fumagalli, A., Jaffré, J., Roberts, J.E.: A double-layer reduced model for fault flow on slipping domains with hybrid finite volume scheme. SIAM: J. Sci. Comput. (2015, in preparation). hal-01162048

    Google Scholar 

  27. 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, 1089–1116 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fries, T.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75 (5), 503–532 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Frih, N., Martin, V., Roberts, J.E., Saâda, A.: Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16 (4), 1043–1060 (2012)

    Article  Google Scholar 

  30. Fumagalli, A.: Numerical Modelling of Flows in Fractured Porous Media by the XFEM Method. Ph.D. thesis, Politecnico di Milano (2012)

    Google Scholar 

  31. Fumagalli, A., Scotti, A.: Numerical modelling of multiphase subsurface flow in the presence of fractures. Commun. Appl. Ind. Math. 3 (1) (2011)

    Google Scholar 

  32. Fumagalli, A., Scotti, A.: A reduced model for flow and transport in fractured porous media with non-matching grids. In: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications. Springer, Berlin (2012)

    Google Scholar 

  33. Fumagalli, A., Scotti, A.: An efficient XFEM approximation of Darcy flows in fractured porous media. MOX Report 53 (2012)

    Google Scholar 

  34. 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(0)), 454–464 (2013). Computational Methods in Geologic CO2 Sequestration

    Google Scholar 

  35. Fumagalli, A., Scotti, A.: An efficient XFEM approximation of Darcy flow in arbitrarly fractured porous media. Oil Gas Sci. Technol. - Rev. d’IFP Energies Nouv. 69 (4), 555–564 (2014)

    Article  MathSciNet  Google Scholar 

  36. Hackbusch, W., Sauter, S.A.: Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75 (4), 447–472 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hanowski, K., Sander, O.: Simulation of deformation and flow in fractured, poroelastic materials (2016). Preprint, arXiv:1606.05765

    Google Scholar 

  38. Hansbo, P.: Nitsche’s method for interface problems in computational mechanics. GAMM-Mitteilungen 28 (2), 183–206 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191 (47–48), 5537–5552 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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)

    Article  MathSciNet  MATH  Google Scholar 

  42. 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 

  43. Massing, A., Larson, M.G., Logg, A.: Efficient implementation of finite element methods on non-matching and overlapping meshes in 3d. arXiv preprint arXiv:1210.7076 (2012)

    Google Scholar 

  44. Mohammadi, S.: Extended Finite Element Method. Wiley, New York (2008)

    Book  MATH  Google Scholar 

  45. Olshanskii M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47 (5), 3339–3358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)

    Google Scholar 

  47. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. Lect. Notes Math. 606, 292–315 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  48. Remij, E., Remmers, J., Huyghe, J., Smeulders, D.: The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials. Comput. Methods Appl. Mech. Eng. 286, 296–312 (2015)

    Article  MathSciNet  Google Scholar 

  49. Roberts, J.E., Thomas, J.-M.: Mixed and Hybrid Methods. Handbook of Numerical Analysis, vol. II, pp. 523–639. North-Holland, Amsterdam (1991)

    Google Scholar 

  50. Schwenck, N.: An XFEM-based model for fluid flow in fractured porous media. Ph.D. thesis, Department of Hydromechanics and Modelling of Hydrosystems, University of Stuttgart (2015)

    Google Scholar 

  51. Schwenck, N., Flemisch, B., Helmig, R., Wohlmuth, B.: Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19 (6), 1219–1230 (2015)

    Article  MathSciNet  Google Scholar 

  52. 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 

  53. Zimmerman, R.W., Kumar, S., Bodvarsson, G.S.: Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 28 (4), 325–331 (1991)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Hydromechanics and Modelling of Hydrosystems, University of Stuttgart, Pfaffenwaldring 61, 70569, Stuttgart, Germany

    Bernd Flemisch

  2. MOX, Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133, Milan, Italy

    Alessio Fumagalli & Anna Scotti

Authors
  1. Bernd Flemisch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessio Fumagalli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna Scotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Scotti .

Editor information

Editors and Affiliations

  1. Structural, Building, Geotechnical Eng., Politecnico di Torino, Torino, Italy

    Giulio Ventura

  2. Department of Engineering, University of Ferrara, Ferrara, Italy

    Elena Benvenuti

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Flemisch, B., Fumagalli, A., Scotti, A. (2016). A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media. In: Ventura, G., Benvenuti, E. (eds) Advances in Discretization Methods. SEMA SIMAI Springer Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-41246-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation