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A residual-based a posteriori error estimator for single-phase Darcy flow in fractured porous media

Abstract

In this paper we develop an a posteriori error estimator for a mixed finite element method for single-phase Darcy flow in a two-dimensional fractured porous media. The discrete fracture model is applied to model the fractures by one-dimensional fractures in a two-dimensional domain. We consider Raviart–Thomas mixed finite element method for the approximation of the coupled Darcy flows in the fractures and the surrounding porous media. We derive a robust residual-based a posteriori error estimator for the problem with non-intersecting fractures. The reliability and efficiency of the a posteriori error estimator are established for the error measured in an energy norm. Numerical results verifying the robustness of the proposed a posteriori error estimator are given. Moreover, our numerical results indicate that the a posteriori error estimator also works well for the problem with intersecting fractures.

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References

  1. 1.

    Achdou, Y., Bernardi, C., Coquel, F.: A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96, 17–42 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Adams, R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  3. 3.

    Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  4. 4.

    Ainsworth, M.: A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements. SIAM J. Numer. Anal. 30, 189–204 (2007)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    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 (2002)

  6. 6.

    Babuška, I., Gatica, G.N.: A residual-based a posteriori error estimator for the Stokes–Darcy coupled problem. SIAM J. Numer. Anal. 48, 498–523 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Barenblatt, G., Zheltov, Y., Kochina, I.: Basic concepts in the theory of seepage of homogeneous fluids in fissured rocks. J. Appl. Math. Mech. 24, 1286–1303 (1960)

    Article  MATH  Google Scholar 

  8. 8.

    Baca, R., Arnett, R., Langford, D.: Modeling fluid flow in fractured porous rock masses by finite element techniques. Int. J. Num. Methods Fluids 4, 337–348 (1984)

    Article  MATH  Google Scholar 

  9. 9.

    Barrios, T.P., Bustinza, R.: An a posteriori error analysis of an augmented discontinuous Galerkin formulation for Darcy flow. Numer. Math. 120, 231–269 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  12. 12.

    Byfut, A., Gedicke, J., Günther, D., Reininghaus, J., Wiedemann, S., et al.: FFW Documentation. Humboldt University of Berlin, Germany (2007)

    Google Scholar 

  13. 13.

    Cai, Z., Zhang, S.: Recovery-based error estimators for interface problems: mixed and nonconforming finite elements. SIAM J. Numer. Anal. 48, 30–52 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Cai, J., Sun, S.: Fractal analysis of fracture increasing spontaneous imbibition in porous media saturated with gas. Int. J. Mod. Phys. C 24, 1350056 (2013)

    Article  Google Scholar 

  15. 15.

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

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Chen, H., Salama, A., Sun, S.: Adaptive mixed finite element methods for Darcy flow in fractured porous media. Water Resour. Res. (2016). doi:10.1002/2015WR018450

    Google Scholar 

  17. 17.

    Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193, 2565–2580 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Demkowicz, L.: Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations. Lecture Notes in Mathematics. Springer, Berlin (2008)

  19. 19.

    Dong, C., Sun, S., Taylor, G.A.: Numerical modeling of contaminant transport in fractured porous media using mixed finite element and finite volume methods. J. Porous Media. 14, 219–242 (2011)

    Article  Google Scholar 

  20. 20.

    Ervin, V.J., Jenkins, E.W., Sun, S.: Coupled generalized non-linear stokes flow with flow through a porous medium. SIAM J. Numer. Anal. 47, 929–952 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

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

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

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

  24. 24.

    Gebauer, S., Neunhäuserer, L., Kornhuber, R., Ochs, S., Hinkelmann, R., Helmig, R.: Equidimensional modelling of flow and transport processes in fractured porous systems I. Dev. Water Sci. 47, 335–342 (2002)

    Article  Google Scholar 

  25. 25.

    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  26. 26.

    Gopalakrishnan, J., Qiu, W.: Partial expansion of a Lipschitz domain and some applications. Front. Math. China 7, 249–272 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Hoteit, H., Firoozabadi, A.: Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media. Water Resour. Res. 41, W11412 (2005)

    Article  Google Scholar 

  28. 28.

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

    Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75, 1659–1674 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26, 1667–1691 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Moortgat, J., Sun, S., Firoozabadi, A.: Compositional modeling of three-phase flow with gravity using higher-order finite element methods. Water Resour. Res. 47, W05511 (2011)

    Article  Google Scholar 

  33. 33.

    Noorishad, J., Mehran, M.: An upstream finite element method for solution of transient transport equation in fractured porous media. Water Resour. Res. 18, 588–596 (1982)

    Article  Google Scholar 

  34. 34.

    Pruess, K., Narasimhan, T.: A practical method for modeling fluid and heat flow in fractured porous media. SPE J. 25, 14–26 (1985)

    Article  Google Scholar 

  35. 35.

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

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Song, P., Sun, S.: Contaminant flow and transport simulation in cracked porous media using locally conservative schemes. Adv. Appl. Math. Mech. 4, 389–421 (2012)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 2, 245–269 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43, 195–219 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Sun, S., Wheeler, M.F.: L2(H1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems. J. Sci. Comput. 22, 501–530 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Tang, T., Xu, J.: Adaptive Computations: Theory and Algorithms. Science Press, Beijing (2007)

    Google Scholar 

  41. 41.

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

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

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

    MATH  Google Scholar 

  43. 43.

    Warren, J., Root, P.: The behavior of naturally fractured reservoirs. SPE J. 3, 245–255 (1963)

  44. 44.

    Wu, Y.S., Pruess, K.: A multiple-porosity method for simulation of naturally fractured petroleum reservoirs. SPE Reserv. Eng. 3, 327–336 (1988)

    Article  Google Scholar 

  45. 45.

    Zidane, A., Firoozabadi, A.: An efficient numerical model for multicomponent compressible flow in fractured porous media. Adv. Water Resour. 74, 127–147 (2014)

    Article  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referees for their valuable comments and suggestions that led to an improved presentation of this paper.

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Correspondence to Shuyu Sun.

Additional information

Huangxin Chen would like to thank the support from the King Abdullah University of Science and Technology where this work was carried out during his visit, and he also thanks the supports from the NSF of China (Grant No. 11201394), the Fundamental Research Funds for the Central Universities (Grant No. 20720150005) and Program for Prominent Young Talents in Fujian Province University. The work of Shuyu Sun was supported by King Abdullah University of Science and Technology (KAUST) through the Grant BAS/1/1351-01-01.

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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). https://doi.org/10.1007/s00211-016-0851-9

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Mathematics Subject Classification

  • 65N12
  • 65N15
  • 65N30