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Upscaled Model for Mixed Dimensional Coupled Flow Problem in Fractured Porous Media Using Non-local Multicontinuum (NLMC) Method

  • Maria VasilyevaEmail author
  • Eric T. Chung
  • Yalchin Efendiev
  • Wing Tat Leung
  • Yating Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

In this paper, we consider a mixed dimensional discrete fracture model with highly conductive fractures. Mathematically the problem is described by a coupled system of equations consisting a d - dimensional equation for flow in porous matrix and a \((d-1)\) - dimensional equation for fracture networks with a specific exchange term for coupling them. For the numerical solution on the fine grid, we construct unstructured mesh that is conforming with fracture surface and use the finite element approximation. Fine grid approximation typically leads to very large systems of equations since it resolves the fracture networks, and therefore some multiscale methods or upscaling methods should be applied. The main contribution of this paper is that we propose a new upscaled model using Non-local multi-continuum (NLMC) method and construct an effective coarse grid approximation. The upscaled model has only one additional coarse degree of freedom (DOF) for each fracture network. We will present results of the numerical simulations using our proposed upscaling method to illustrate its performance.

Keywords

Fractured porous media Fluid flow Coupled system Upscaling Multiscale method Non-local multi-continuum method 

Notes

Acknowledgements

MV’s work is supported by the grant of the Russian Scientific Found N17-71-20055. YE’s is supported by the mega-grant of the Russian Federation Government (N 14.Y26.31.0013). EC’s work is partially supported by Hong Kong RGC General Research Fund (Project 14317516) and CUHK Direct Grant for Research 2016-17

References

  1. 1.
    Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M., Wang, Y.: Multiscale model reduction for shale gas transport in a coupled discrete fracture and dual-continuum porous media. J. Nat. Gas Sci. Engin. 48, 65–76 (2017)CrossRefGoogle Scholar
  2. 2.
    Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M.: Multiscale model reduction for shale gas transport in fractured media. Computat. Geosci. 20(5), 1–21 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)CrossRefGoogle Scholar
  4. 4.
    Chung, E.T., Efendiev, Y., Leung, W., Vasilyeva, M.: Coupling of multiscale and multi-continuum approaches. GEM-Int. J. Geomath. 8(1), 9–41 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chung, E.T., Efendiev, Y., Leung, W.: Constraint energy minimizing generalized multiscale finite element method, arXiv preprint arXiv:1704.03193 (2017)
  6. 6.
    Chung, E.T., Efendiev, Y., Leung, W., Vasilyeva, M., Wang, Y.: Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains. Appl. Anal. 96(12), 2002–2031 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chung, E.T., Efendiev, Y., Leung, W., Wang, Y., Vasilyeva, M.: Non-local multi-continua upscaling for flows in heterogeneous fractured media, arXiv preprint arXiv:1708.08379 (2017)
  8. 8.
    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
  9. 9.
    Efendiev, Y., Galvis, J., Hou, T.: Generalized multiscale finite element methods. J. Comput. Phys. 251, 116–135 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Efendiev, Y., Hou, T.: Multiscale Finite Element Methods: Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4. Springer, New York (2009).  https://doi.org/10.1007/978-0-387-09496-0CrossRefzbMATHGoogle Scholar
  11. 11.
    Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Simul. 3(1), 50–64 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tene, M., Al Kobaisi, M.S., Hajibeygi, H.: Algebraic multiscale solver for flow in heterogeneous fractured porous media. In: SPE Reservoir Simulation Symposium, Society of Petroleum Engineers (2015)Google Scholar
  13. 13.
    Vasilyeva, M., Stalnov, D.: A generalized multiscale finite element method for thermoelasticity problems. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2016. LNCS, pp. 713–720. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-319-57099-0_82CrossRefGoogle Scholar
  14. 14.
    Warren, J.E., Root, P.J., et al.: The behavior of naturally fractured reservoirs. Soc. Pet. Eng. J. 3(03), 245–255 (1963)CrossRefGoogle Scholar
  15. 15.
    D’Angelo, C., Quarteroni, A.: On the coupling of 1D and 3D diffusion-reaction equations: application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18(08), 1481–1504 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Maria Vasilyeva
    • 1
    • 2
    Email author
  • Eric T. Chung
    • 3
  • Yalchin Efendiev
    • 1
    • 4
  • Wing Tat Leung
    • 5
  • Yating Wang
    • 4
  1. 1.Institute for Scientific ComputationTexas A&M UniversityCollege StationUSA
  2. 2.North-Eastern Federal UniversityYakutskRussia
  3. 3.Department of MathematicsThe Chinese University of Hong Kong (CUHK)ShatinHong Kong
  4. 4.Department of MathematicsTexas A&M UniversityCollege StationUSA
  5. 5.Institute of Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA

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