Skip to main content
SpringerLink
Log in

An adaptive discontinuous Galerkin method for the Darcy system in fractured porous media

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Modeling flows in fractured porous media is important in applications. One main challenge in numerical simulation is that the flow is strongly influenced by the fractures, so that the solutions typically contain complex features, which require high computational grid resolutions. Instead of using uniformly fine mesh, a more computationally efficient adaptively refined mesh is desirable. In this paper we design and analyze a novel residual-type a posteriori error estimator for staggered DG methods on general polygonal meshes for Darcy flows in fractured porous media. The method can handle fairly general meshes and hanging nodes can be simply incorporated into the construction of the method, which is highly appreciated for adaptive mesh refinement. The reliability and efficiency of the error estmator are proved. The derivation of the reliability hinges on the stability of the continuous setting in the primal formulation. A conforming counterpart that is continuous within each bulk domain for the discrete bulk pressure is defined to facilitate the derivation of the reliability. Finally, several numerical experiments including multiple non-intersecting fractures are carried out to confirm the proposed theories.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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), volume 295 of Contemp. Math., pp 13–24. Amer. Math. Soc., Providence (2002)

  2. Alonso, A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)

    Article  Google Scholar 

  3. Antonietti, P.F., Facciolà, C., Russo, A., Verani, M.: Discontinuous Galerkin approximation of flows in fractured porous media on polytopic grids. SIAM J. Sci. Comput. 41, A109–A138 (2019)

    Article  Google Scholar 

  4. Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzotti, N.: Mimetic finite difference approximation of flows in fractured porous media. ESAIM Math. Model. Numer. Anal. 50, 809–832 (2016)

    Article  Google Scholar 

  5. Arbogast, T., Correa, M.R.: Two Families of H(div) mixed finite elements on quadrilaterals of minimal dimension. SIAM J. Numer. Anal. 54, 3332–3356 (2016)

    Article  Google Scholar 

  6. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Meth. Appl. Sci. 23, 199–214 (2013)

    Article  Google Scholar 

  7. Beirão da Veiga, L., Manzini, G.: Residual a posteriori error estimation for the virtual rlement method for elliptic problems. ESAIM Math. Model. Numer. Anal. 49, 577–599 (2015)

    Article  Google Scholar 

  8. Berrone, S., Borio, A.: A residual a posteriori error estimate for the virtual rlement method. Math. Models Meth. Appl. Sci. 27, 1423–1458 (2017)

    Article  Google Scholar 

  9. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  Google Scholar 

  10. Beirão da Veiga, L., Manzini, G.: An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems. Int. J. Numer. Meth. Engng. 76, 1696–1723 (2008)

    Article  Google Scholar 

  11. Babuška, I., Rheinboldt, W.C.: A posteriori error estimates for the fintie element method. Int. J. Numer. Methods Engrg. 12, 1597–1615 (1978)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)

    Article  Google Scholar 

  14. Berre, I., Boon, W.M., Flemisch, B., Fumagalli, A., Gläser, D, Keilegavlen, E., Scotti, A., Stefansson, I., Tatomir, A., Brenner, K., Burbulla, S., Devloo, P., Duran, O., Favino, M., Hennicker, J., Lee, I-H., Lipnikov, K., Masson, R., Mosthaf, K., Nestola, M.G.C., Ni, C.-F., Nikitin, K., Schädle, P., Svyatskiy, D., Yanbarisov, R., Zulian, P.: Verification benchmarks for single-phase flow in three-dimensional fractured porous media. Adv. Water Resour. 147 (2021)

  15. Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comp. 77, 651–672 (2008)

    Article  Google Scholar 

  16. Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)

    Article  Google Scholar 

  17. Carstensen, C.: A posteriori error estimates for the mixed finite element method. Math. Comp. 66, 465–476 (1997)

    Article  Google Scholar 

  18. Carstensen, C., Kim, D., Park, E.-J.: A priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer. Anal. 49, 2501–2523 (2011)

    Article  Google Scholar 

  19. Cangiani, A., Georgoulis, E.H., Pryer, T., Sutton, O.J.: A posteriori error estimates for the virtual element method. Numer. Math. 137, 857–893 (2017)

    Article  Google Scholar 

  20. Chave, F., Di Pietro, D.A., Formaggia, L.: A hybrid high-order method for Darcy flows in fractured porous media. SIAM J. Sci. Comput. 40, A1063–A1094 (2018)

    Article  Google Scholar 

  21. Chen, H., Salama, A., Sun, S.: Adaptive mixed finite element methods for Darcy flow in fractured porous media. Water Resour. Res. 52, 7851–7868 (2016)

    Article  Google Scholar 

  22. Chen, H., Sun, S.: A residual-based a posteriori error estimator for single-phase Darcy flow in fractured porous media. Numer. Math. 136, 805–839 (2017)

    Article  Google Scholar 

  23. Cheung, S.W., Chung, E., Kim, H.H., Qian, Y.: Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 302, 251–266 (2015)

    Article  Google Scholar 

  24. Chung, E.T., Ciarlet, P. Jr., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phy. 235, 14–31 (2013)

    Article  Google Scholar 

  25. Chung, E.T., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)

    Article  Google Scholar 

  26. Chung, E.T., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. Part II: The Stokes flow. J. Sci. Comput. 66, 870–887 (2016)

    Article  Google Scholar 

  27. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for wave propagation. SIAM J. Numer. Anal. 44, 2131–2158 (2006)

    Article  Google Scholar 

  28. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)

    Article  Google Scholar 

  29. Chung, E.T., Kim, H.H., Widlund, O.B.: Two-level overlapping schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51, 47–67 (2013)

    Article  Google Scholar 

  30. Chung, E.T., Lam, C.Y., Qian, J.: A staggered discontinuous Galerkin method for the simulation of seismic waves with surface topography. Geophysics 80, T119–T135 (2015)

    Article  Google Scholar 

  31. Chung, E.T., Park, E.-J., Zhao, L.: Guaranteed a posteriori error estimates for a staggered discontinuous Galerkin method. J. Sci. Comput. 75, 1079–1101 (2018)

    Article  Google Scholar 

  32. Chung, E.T., Qiu, W.: Analysis of an SDG method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 55, 543–569 (2017)

    Article  Google Scholar 

  33. 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, 465–489 (2012)

    Article  Google Scholar 

  34. Ern, A., Vohralík, M.: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53, 1058–1081 (2015)

    Article  Google Scholar 

  35. 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  Google Scholar 

  36. Formaggia, L., Scotti, A., Sottocasa, F.: Analysis of a mimetic finite difference approximation of flows in fractured porous media. ESAIM Math. Model. Numer. Anal. 52, 595–630 (2018)

    Article  Google Scholar 

  37. Frih, N., Roberts, J.E., Saada, A.: Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12, 91–104 (2008)

    Article  Google Scholar 

  38. Fumagalli, A., Scotti, A.: A mathematical model for thermal single-phase flow and reactive transport in fractured porous media. J. Comput. Phys. 434 (2021)

  39. Hoteit, J., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31, 891–905 (2008)

    Article  Google Scholar 

  40. Kadeethum, T., Nick, H.M., Lee, S., Ballarin, F.: Flow in porous media with low dimensional fractures by employing enriched Galerkin method. Adv. Water Resour. 142 (2020)

  41. Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)

    Article  Google Scholar 

  42. Keilegavlen, E., Berge, R., Fumagalli, A., Starnoni, M., Stefansson, I., Varela, J., Berre, I.: PorePy: An open-source software for simulation of multiphysics processes in fractured porous media. Comput. Geosci. 25, 243–265 (2021)

    Article  Google Scholar 

  43. Kim, K.Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76, 43–66 (2007)

    Article  Google Scholar 

  44. Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327–3350 (2013)

    Article  Google Scholar 

  45. Kim, D., Park, E.-J.: A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems. Comput. Methods Appl. Mech. Engrg. 197, 806–820 (2008)

    Article  Google Scholar 

  46. Kim, D., Park, E.-J.: A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations. SIAM J. Numer. Anal. 48, 1186–1207 (2010)

    Article  Google Scholar 

  47. Larson, M.G., Målqvist, A.: A posteriori error estimates for mixed finite element approximations of elliptic problems. Numer. Math. 108, 487–500 (2008)

    Article  Google Scholar 

  48. Lee, J.J., Kim, H.H.: Analysis of a staggered discontinuous Galerkin method for linear elasticity. J. Sci. Comput. 66, 625–649 (2016)

    Article  Google Scholar 

  49. Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flows in porous media. SIAM J. Sci. Comput. 26, 1667–1691 (2006)

    Article  Google Scholar 

  50. Monteagudo, J., Firoozabadi, A.: Control-volume method for numerical simulation of two-phase immiscible flow in two- and three-dimensional discrete-fractured media. Water Resour. Res. 40, W07405 (2004)

    Article  Google Scholar 

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

    Google Scholar 

  52. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)

    Article  Google Scholar 

  53. Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)

    Article  Google Scholar 

  54. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, Stuttgart (1996)

    Google Scholar 

  55. Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)

    Article  Google Scholar 

  56. Vohralík, M.: Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comp. 79, 2001–2032 (2010)

    Article  Google Scholar 

  57. Zhao, L., Park, E.-J.: A staggered discontinuous Galerkin method of minimal dimension on quadrilateral and polygonal meshes. SIAM J. Sci. Comput. 40, A2543–A2567 (2018)

    Article  Google Scholar 

  58. Zhao, L., Park, E.-J., Shin, D.-w.: A staggered DG method of minimal dimension for the Stokes equations on general meshes. Comput. Meth. Appl. Mech. Eng. 345, 854–875 (2019)

    Article  Google Scholar 

  59. Zhao, L., Park, E.-J.: A lowest-order staggered DG method for the coupled Stokes-Darcy problem. IMA J. Numer. Anal 40, 2871–2897 (2020)

    Article  Google Scholar 

  60. Zhao, L., Chung, E.T., Lam, M.F.: A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits. Comput. Meth. Appl. Mech. Eng. 364 (2020)

  61. Zhao, L., Chung, E.T., Park, E.-J., Zhou, G.: Staggered DG method for coupling of the stokes and Darcy–Forchheimer problems. SIAM J. Numer. Anal. 59, 1–31 (2021)

    Article  Google Scholar 

  62. Zhao, L., Kim, D., Park, E.-J., Chung, E.: Staggered DG method with small edges for Darcy flows in fractured porous media. J. Sci. Comput. 90 (2022)

Download references

Acknowledgments

The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project numbers 14304719 and 14302018), CUHK Faculty of Science Direct Grant 2019-20 and NSFC/RGC Joint Research Scheme (Project number HKUST620/15).

Author information

Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, China

    Lina Zhao

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China

    Eric Chung

Authors
  1. Lina Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric Chung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Chung.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, L., Chung, E. An adaptive discontinuous Galerkin method for the Darcy system in fractured porous media. Comput Geosci 26, 1581–1596 (2022). https://doi.org/10.1007/s10596-022-10171-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-022-10171-5

Keywords

Search

Navigation