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Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Heterogeneous Medium

  • D. SpiridonovEmail author
  • M. Vasilyeva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

We consider a mathematical model for simulation of the unsaturated flow problems in heterogeneous porous medium that describes by the Richards equation. To resolve all heterogeneity, we construct fine grid and construct finite element approximation. For dimension reduction of the discrete system, we construct multiscale solver for coarse grid solution using Generalized Multiscale Finite Element Method (GMsFEM). We generate multiscale basis functions by solution of the local spectral problems. We present numerical result and compare relative error for different number of the multiscale basis functions for 2D and 3D model problems.

Keywords

Heterogeneous media Unsaturated filtration Richards equation Multiscale method GMsFEM 

Notes

Acknowledgments

Work is supported by the mega-grant of the Russian Federation Government (N 14.Y26.31.0013 and RFBR N 17-01-00732).

References

  1. 1.
    Celia, M.A., Bouloutas, E.T., Zarba, R.L.: A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26(7), 1483–1496 (1990)CrossRefGoogle Scholar
  2. 2.
    Celia, M.A., Binning, P.: A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow. Water Resour. Res. 28(10), 2819–2828 (1992)CrossRefGoogle Scholar
  3. 3.
    Ginting, V.E.: Computational upscaled modeling of heterogeneous porous media flow utilizing finite volume method. Ph.D. thesis, Texas A and M University (2004)Google Scholar
  4. 4.
    Haverkamp, R., Vauclin, M., Touma, J., Wierenga, P.J., Vachaud, G.: A comparison of numerical simulation models for one-dimensional infiltration. Soil Sci. Soc. Am. J. 41(2), 285–294 (1977)CrossRefGoogle Scholar
  5. 5.
    Hou, T., Efendiev, Y.: Multiscale Finite Element Methods: Theory and Applications. STAMS, vol. 4, 2nd edn. Springer, New York (2009).  https://doi.org/10.1007/978-0-387-09496-0CrossRefzbMATHGoogle Scholar
  6. 6.
    Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M., Wang, Y.: Multiscale model reduction for shale gas transport in poroelastic fractured media. J. Comput. Phys. 353, 356–376 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chung, E.T., Leung, W.T., Vasilyeva, M., Wang, Y.: Multiscale model reduction for transport and flow problems in perforated domains. J. Comput. Appl. Math. 330, 519–535 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chung, E.T., Efendiev, Y., Lee, C.S.: Mixed generalized multiscale finite element methods and applications. Multiscale Model. Simul. 13(1), 338–366 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chung, E.T., Efendiev, Y., Li, G., Vasilyeva, M.: Generalized multiscale finite element methods for problems in perforated heterogeneous domains. Appl. Anal. 95(10), 2254–2279 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Efendiev, Y., Hou, T.Y., Ginting, V.: Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2(4), 553–589 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Software GMSH. http://geuz.org/gmsh/
  13. 13.
    Logg, A., Mardal, K.-A., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. LNCSE, vol. 84. Springer, Berlin (2011).  https://doi.org/10.1007/978-3-642-23099-8CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Multiscale Model Reduction Laboratory, North-Eastern Federal UniversityYakutskRussia
  2. 2.Institute for Scientific ComputationTexas A&M UniversityCollege StationUSA

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