Abstract
The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information. The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the fine-scale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308–317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods.
Similar content being viewed by others
References
Aarnes, J.: On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. SIAM MMS 2, 421–439 (2004)
Aarnes, J., Efendiev, Y., Jiang, L.: Mixed multiscale finite element methods using limited global information. SIAM Multiscale Model. Simul. 7(2), 655–676 (2008)
Aarnes, J., Krogstad, S., Lie, K.A.: A hierarchical multiscale method for two-phase flow based on upon mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul. 5(2), 337–363 (2006)
Aarnes, J.E., Hauge, V., Efendiev, Y.: Coarsening of three-dimensional structured and unstructured grids for subsurface flow. Adv. Water Resour. 30(11), 2177–2193 (2007)
Allaire, G., Brizzi, R.: A multiscale finite element method for numerical homogenization. SIAM MMS 4(3), 790–812 (2005)
Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6, 453–481 (2002)
Avellaneda, M., Lin, F.-H.: Compactness method in the theory of homogenization. Commun. Pure Appl. Math. 40(6), 803–847 (1987)
Babus̆ka, I., Osborn, E.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20, 510–536 (1983)
Babus̆ka, I., Caloz, G., Osborn, E.: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31, 945–981 (1994)
Brezzi, F.: Interacting with the subgrid world. In: Numerical Analysis 1999 (Dundee), pp. 69–82. Chapman & Hall/CRC, Boca Raton, FL (2000)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72, 541–576 (2002)
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, 1041–1060 (2003)
Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Evalu. Eng. 4, 308–317 (2001)
Deutsch, C., Journel, A.: GSLIB: Geostatistical Software Library and User’S Guide, 2nd edn. Oxford University Press, New York (1998)
Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27, 699–708 (1991)
Durlofsky, L.J., Efendiev, Y., Ginting, V.: An adaptive local-global multiscale finite volume element method for two-phase flow simulations. Adv. Water Resour. 30, 576–588 (2007)
Efendiev, Y., Hou, T., Ginting, V.: Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2, 553–589 (2004)
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)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Franca, L., Madureira, A., Valentin, F.: Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions. Comput. Methods Appl. Mech. Eng. 194(27–29), 3006–3021 (2005)
Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)
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)
Jikov, V., Kozlov, S., Oleinik, O.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994) (Translated from Russian)
Moskow, S., Vogelius, M.: First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc. R. Soc. Edinb. Sect. A 127(6), 1263–1299 (1997)
Murat, F., Tartar, L.: H-convergence, in topics in the mathematical modeling of composite materials. In: Cherkaev, A., Kohn, R.V. (eds.) Progress in Nonlinear Differential Equations and their Applications. Birkhauser, Boston (1997)
Owhadi, H., Zhang, L.: Metric based up-scaling. Commun. Pure Appl. Math. 60, 675–723 (2007)
Sangalli, G.: Capturing small scales in elliptic problems using a residual-free bubbles finite element method. Multiscale Model. Simul. 1, 485–503 (2003) (electronic)
Wu, X.H., Efendiev, Y., Hou, T.Y.: Analysis of upscaling absolute permeability. Discrete Continuous Dyn. Syst. Ser. B 2(2), 185–204 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, L., Efendiev, Y. & Mishev, I. Mixed multiscale finite element methods using approximate global information based on partial upscaling. Comput Geosci 14, 319–341 (2010). https://doi.org/10.1007/s10596-009-9165-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-009-9165-7