A comparison of multiscale methods for elliptic problems in porous media flow
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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.
KeywordsPorous media flow Multiscale methods Upscaling Numerical comparisons
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- 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.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
- 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.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.Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15. Springer-Verlag, New York (1991)Google Scholar
- 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
- 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/
- 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.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
- 30.Stüben, K.: Multigrid, chap. Algebraic Multigrid (AMG): An Introduction with Applications. Academic (2000)Google Scholar
- 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