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Computational Geosciences

, Volume 12, Issue 3, pp 377–398 | Cite as

A comparison of multiscale methods for elliptic problems in porous media flow

  • Vegard Kippe
  • Jørg E. Aarnes
  • Knut-Andreas Lie
Original paper

Abstract

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient, approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition, the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh. We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method – the adaptive local–global upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure. We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three multiscale methods to a dynamic two-phase flow case and demonstrate that high efficiency and accurate results can be obtained when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the simulation.

Keywords

Porous media flow Multiscale methods Upscaling Numerical comparisons 

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References

  1. 1.
    Aarnes, J.E.: On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul. 2(3), 421–439 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aarnes, J.E., Krogstad, S., Lie, K.A.: A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul. 5(2), 337–363 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aarnes, J.E., Krogstad, S., Lie, K.A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. doi:10.1007/s10596-007-9072-8 (2008)
  4. 4.
    Aarnes, J.E., Lie, K.A.: Toward reservoir simulation on geological grid models. In: Proceedings of the 9th European Conference on the Mathematics of Oil Recovery. EAGE, Cannes, France (2004)Google Scholar
  5. 5.
    Arbogast, T.: Numerical subgrid upscaling of two-phase flow in porous media. In: Numerical treatment of multiphase flows in porous media (Beijing, 1999). Lecture Notes in Phys., vol. 552, pp. 35–49. Springer, Berlin (2000)CrossRefGoogle Scholar
  6. 6.
    Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6(3-4), 453–481 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Arbogast, T.: Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42(2), 576–598 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Arbogast, T., Minkoff, S.E., Keenan, P.T.: An operator-based approach to upscaling the pressure equation. In: V.N.B. et al. (ed.) Computational Methods in Water Resources XII, vol. 1, pp. 405–412. Southampton, U.K. (1998)Google Scholar
  9. 9.
    Brezzi, F.: Interacting with the subgrid world. In: Numerical analysis 1999 (Dundee), pp. 69–82. Chapman & Hall/CRC, Boca Raton, FL (2000)Google Scholar
  10. 10.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15. Springer-Verlag, New York (1991)Google Scholar
  11. 11.
    Brezzi, F., Jr., J.D., Marini, L.D.: Two families of mixed elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen, A., Hou, T.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72(242), 541–576 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen, Y., Durlofsky, L.J.: Adaptive local-global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media 62(2), 157–182 (2006)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26(10), 1041–1060 (2003)CrossRefGoogle Scholar
  15. 15.
    Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: A comparison of upscaling techniques. SPE Reservoir Eval. Eng. 4(4), 308–317 (2001). http://www.spe.org/csp/ Google Scholar
  16. 16.
    Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical software library and user’s guide, 2nd edn. Oxford University Press, New York (1998)Google Scholar
  17. 17.
    Durlofsky, L.J.: Numerical calculations of equivalent gridblock permeability tensors for heterogeneous porous media. Water Resour. Res. 27(5), 699–708 (1991)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Efendiev, Y., Ginting, V., Hou, T., Ewing, R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220(1), 155–174 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Gautier, Y., Blunt, M.J., Christie, M.A.: Nested gridding and streamline-based simulation for fast reservoir performance prediction. Comput. Geosci. 3(3–4), 295–320 (1999)zbMATHCrossRefGoogle Scholar
  20. 20.
    Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hughes, T., Feijoo, G., Mazzei, L., Quincy, J.: The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166, 3–24 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003)zbMATHCrossRefGoogle Scholar
  23. 23.
    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 (2004/05)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive fully implicit multi-scale finite-volume methods for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys. 217(2), 627–641 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kippe, V., Aarnes, J., Lie, K.A.: Multiscale finite-element methods for elliptic problems in porous media flow. In: CMWR XVI – Computational Methods in Water Resources. Copenhagen, Denmark, June, 2006 (2006). http://proceedings.cmwr-xiv.org/
  26. 26.
    Lunati, I., Jenny, P.: Multiscale finite-volume method for compressible multiphase flow in porous media. J. Comput. Phys. 216(2), 616–636 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Matache, A.M., Schwab, C.: Homogenization via p-FEM for problems with microstructure. In: Proceedings of the Fourth International Conference on Spectral and High Order Methods (ICOSAHOM 1998) (Herzliya), vol. 33, pp. 43–59 (2000)Google Scholar
  28. 28.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)Google Scholar
  29. 29.
    Sangalli, G.: Capturing small scales in elliptic problems using a residual-free bubbles finite element method. Multiscale Model. Simul. 1(3), 485–503 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Stüben, K.: Multigrid, chap. Algebraic Multigrid (AMG): An Introduction with Applications. Academic (2000)Google Scholar
  31. 31.
    Wen, X.H., Durlofsky, L.J., Chen, Y.: Efficient 3D imple mentation of local-global upscaling for reservoir simulation. SPE J. 11(4), 443–453 (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Vegard Kippe
    • 1
    • 2
  • Jørg E. Aarnes
    • 1
  • Knut-Andreas Lie
    • 1
  1. 1.Department of Applied MathematicsSINTEF ICTOsloNorway
  2. 2.StatoilHydro Research CentreTrondheimNorway

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