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Mixed Multiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

Published online by Cambridge University Press:  20 August 2015

Lijian Jiang  and
Ilya D. Mishev
Lijian Jiang*
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA
Ilya D. Mishev*
Affiliation:
ExxonMobil Upstream Research Company, Houston, TX 77252, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media. Some of the methods developed using the framework are already known [20]; others are new. New insight is gained for the known methods and extra flexibility is provided by the new methods. We give as an example a mixed MsFV on uniform mesh in 2-D. This method uses novel multiscale velocity basis functions that are suited for using global information, which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features. The method efficiently captures the small effects on a coarse grid. We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media. Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.

Keywords

65N3065N2034E13Mixed multiscale finite volume methodselliptic equationstwo-phase flowsheterogeneous porous media
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 1 , January 2012 , pp. 19 - 47
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Aarnes, J. E., Efendiev, Y. and Jiang, L., Mixed multiscale finite element methods using limited global information, Multiscale Model Sim., 7 (2008), 655–676.Google Scholar
[2]Aavatsmark, I., Barkve, T. and Bøe, O., Discretization on unstructured grids for inhomogeneous, anisotropic media, part I: derivation of the methods, SIAM J. Sci. Comput., 19 (1998), 1700–1716.Google Scholar
[3]Arbogast, T., Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow, Comput. Geosci., 6 (2002), 453–481.Google Scholar
[4]Arbogast, T., Pencheva, G., Wheeler, M. F. and Yotov, I., A multiscale mortar mixed finite element method, Multiscale Model Sim., 6 (2007), 319–346.Google Scholar
[5]Babuška, I. and Osborn, E., Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal., 20 (1983), 510–536.CrossRefGoogle Scholar
[6]Brezzi, F., Interacting with the subgrid world, in Numerical Analysis 1999 (Dundee), Chapman & Hall/CRC, Boca Raton, FL, 2000, 69–82.Google Scholar
[7]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.Google Scholar
[8]Chen, Z. and Hou, T. Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comput., 72 (2003), 541–576.Google Scholar
[9]Chen, Q.-Y., Wan, J., Yang, Y. and Mifflin, R. T., Enriched multi-point flux approximation for general grids, J. Comput. Phys., 227 (2008), 1701–1721.CrossRefGoogle Scholar
[10]Christie, M. and Blunt, M., Tenth SPE comparative solution project: a comparison of upscal-ing techniques, SPE Reser. Eval. Eng., 4 (2001), 308–317.Google Scholar
[11]Dawson, C., Sun, S. and Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 193 (2004), 2565–2580.Google Scholar
[12]Deutsch, C. and Journel, A., GSLIB: Geostatistical Software Library and User’s Guide, 2nd edition, Oxford University Press, New York, 1998.Google Scholar
[13]Droniou, J. and Eymard, R., A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math., 105 (2006), 35–71.CrossRefGoogle Scholar
[14]Dubois, O., Mishev, I. D. and Zikatanov, L., Energy minimizing bases for efficient multiscale modelling and linear solvers in reservoir simulation, Paper SPE-119137 presented at 20th SPE Reservoir Simulation Symposium, The Woodlands, Texas, 12-15 February 2009.Google Scholar
[15]Engquist, W.E. and Engquist, B., The heterogeneous multi-scale methods, Commun. Math. Sci., 1 (2003), 87–133.Google Scholar
[16]Edwards, M. G. and Roger, C. F., Finite volume discretization with imposed flux continuity for the general pressure equation, Comput. Geosci., 2 (1998), 259–290.Google Scholar
[17]Edwards, M. G. and Zheng, H., A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support, J. Comput. Phys., 227 (2008), 9333–9364.CrossRefGoogle Scholar
[18]Hou, T. Y. and Wu, X. H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), 169–189.CrossRefGoogle Scholar
[19]Hughes, T., Feijoo, G., Mazzei, L. and Quincy, J., The variational multiscale method–a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., 166 (1998), 3–24.CrossRefGoogle Scholar
[20]Jenny, P., Lee, S. H. and Tchelepi, H., Multi-scale finite volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187 (2003), 47–67.CrossRefGoogle Scholar
[21]Jiang, L., Efendiev, Y. and Mishev, I., Mixed multiscale finite element methods using approximate global information based on partial upscaling, Comput. Geosci., 14 (2010), 319–341.Google Scholar
[22]Jiang, L., Mishev, I. and Li, Y., Stochastic mixed multiscale finite element methods and their applications in random porous media, Comput. Methods Appl. Mech. Eng., 199 (2010), 2721–2740.Google Scholar
[23]Kou, J. and Sun, S., A new treatment of capillarity to improve the stability of IMPES two-phase flow formulation, Comput. Fluids, 39 (2010), 1923–1931.Google Scholar
[24]Mishev, I., Nonconforming finite volume method, Comput. Geosci., 6 (2002), 253–268.CrossRefGoogle Scholar
[25]Mishev, I., Analysis of a new mixed finite volume method, Comput. Methods Appl. Math., 3 (2003), 189–201.Google Scholar
[26]Mishev, I. and Chen, Q. Y., A mixed finite volume method for elliptic problems, Numer. Methods PDE., 23 (2007), 1122–1138.Google Scholar
[27]Mishev, I., Jiang, L. and Dubois, O., Method and system for finite volume simulation of flow, U.S. Patent Application No. 61/330, 012, filed April 30, 2010.Google Scholar
[28]Owhadi, H. and Zhang, L., Metric based up-scaling, Commun. Pure Appl. Math., 60 (2007), 675–723.Google Scholar
[29]Roberts, J. E. and Thomas, J. M., Mixed and Hybrid Methods, Handbook of Numerical Analysis 2, Elsevier, 1991.Google Scholar
[30]Thomas, J.-M. and Trujillo, D., Mixed finite volume methods, Int. J. Numer. Methods Eng., 46 (1999), 1351–1366.Google Scholar
[31]Sun, S. and Geiser, J., Multiscale discontinuous Galerkin and operator-splitting methods for modeling subsurface flow and transport, Int. J. Multiscale Comput. Eng., 6 (2008), 87–101.Google Scholar
[32]Verma, S. K. and Aziz, K., Two- and three-dimensional flexible grids for reservoir simulation, Proceedings of 5th European Conference on the Mathematics of Oil Recovery, Leoben, Austria, 1996.Google Scholar
[33]Xu, J. and Zikatanov, L., On an energy minimizing basis for algebraic multigrid methods, Comput. Visual. Sci., 7 (2004), 21–127.Google Scholar