Abstract
This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain, taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast dependence in the convergence. The method’s convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.
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09 March 2019
This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast.
09 March 2019
This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast.
References
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. SIAM J. Multiscale Model. Simul. 2, 421–439 (2004)
Aarnes, J.E., Efendiev, Y., Jiang, L.: Analysis of multiscale finite element methods using global information for two-phase flow simulations. SIAM J. Multiscale Model. Simul. 7, 2177–2193 (2008)
Aarnes, J.E., Krogstad, S., Lie, K.-A.: A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform grids. SIAM J. Multiscale Model. Simul. 5(2), 337–363 (2006)
Arbogast, T., Pencheva, G., Wheeler, M.F., Yotov, I.: A multiscale mortar mixed finite element method. SIAM J. Multiscale Model. Simul. 6(1), 319–346 (2007)
Barker, J.W., Thibeau, S.: A critical review of the use of pseudorelative permeabilities for upscaling. SPE Reservoir Eng. 12, 138–143 (1997)
Bush, L., Ginting, V.: On the application of the continuous Galerkin finite element method for conservation problems. SIAM J. Sci. Comput. 35(6), A2953—A2975 (2013)
Chan, H. Y., Chung, E. T., Efendiev, Y.: Adaptive mixed GMsFEM for flows in heterogeneous media. Numer. Math. Theory, Methods and Appl. 9(4), 497–527 (2016)
Chen, F., Chung, E., Jiang, L.: Least-squares mixed generalized multiscale finite element method. Comput. Methods Appl. Mech. Eng. 311, 764–787 (2016)
Chen, Y., Durlofsky, L., Gerritsen, M., Wen, X.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26, 1041–1060 (2003)
Chen, Z., Hou, T. Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72(242), 541–576 (2003)
Christie, M., Blunt, M.: Tenth SPE comparative solution project: A comparison of upscaling techniques. SPE Reser. Eval. Eng. 4, 308–317 (2001)
Chung, E., Efendiev, Y., Hou, T. Y.: Adaptive multiscale model reduction with generalized multiscale finite element methods. J. Comput. Phys. 320, 69–95 (2016)
Chung, E. T., Efendiev, Y., Lee, C. S.: Mixed generalized multiscale finite element methods and applications. Multiscale Model. Simul. 13(1), 338–366 (2015)
Chung, E. T., Efendiev, Y., Leung, W. T.: Constraint energy minimizing generalized multiscale finite element method. arXiv preprint. arXiv:1704.03193 (2017)
Chung, E. T., Fu, S., Yang, Y.: An enriched multiscale mortar space for high contrast flow problems. arXiv preprint. arXiv:1609.02610 (2016)
Chung, E. T., Leung, W. T., Vasilyeva, M.: Mixed GMsFEM for second order elliptic problem in perforated domains. J. Comput. Appl. Math. 304, 84–99 (2016)
Cortinovis, D., Jenny, P.: Iterative Galerkin-enriched multiscale finite-volume method. J. Comput. Phys. 277, 248–267 (2014)
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.: Coarse scale models of two-phase flow in heterogeneous reservoirs: Volume averaged equations and their relation to existing upscaling techniques. Comput. Geosci. 2, 73–92 (1998)
Efendiev, Y., Galvis, J.: A domain decomposition preconditioner for multiscale high-contrast problems. In: Huang, Y., Kornhuber, R., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XIX, volume 78 of Lect. Notes in Comput. Science and Eng., pages 189–196. Springer-Verlag (2011)
Efendiev, Y., Galvis, J., Hou, T.: Generalized multiscale finite element methods. J. Comput. Phys. 251, 116–135 (2013)
Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESIAM : M2AN 46, 1175–1199 (2012)
Efendiev, Y., Hou, T.: Multiscale finite element methods: Theory and applications. Springer, Berlin (2009)
Efendiev, Y., Hou, T., Wu, X.H.: Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37, 888–910 (2000)
Efendiev, Y., Iliev, O., Vassilevski, P.S.: Mini-workshop: Numerical upscaling for media with deterministic and stochastic heterogeneity. Oberwolfach Reports 10(1), 393–431 (2013)
Hajibeygi, H., Kavounis, D., Jenny, P.: A hierarchical fracture model for the iterative multiscale finite volume method. J. Comput. Phys. 230(4), 8729–8743 (2011)
Hou, T., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)
Hou, T., Zhang, P.: private communications
Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127, 387–401 (1995)
Jenny, P., Lee, S.H., Tchelepi, H.: Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47–67 (2003)
Lie, K.-A., Møyner, O., Natvig, J.R., et al.: A feature-enriched multiscale method for simulating complex geomodels. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (2017)
Lunati, I., Jenny, P.: Multi-scale finite-volume method for highly heterogeneous porous media with shale layers. In: Proceedings of the 9th European Conference on the Mathematics of Oil Recovery (ECMOR), Cannes, France (2004)
Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comput. 83(290), 2583–2603 (2014)
Odsæter, L.H., Wheeler, M.F., Kvamsdal, T., Larson, M.G.: Postprocessing of non-conservative flux for compatibility with transport in heterogeneous media. Comput. Methods Appl. Mech. Eng. 315, 799–830 (2017)
Owhadi, H.: Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games. SIAM Rev. 59(1), 99–149 (2017)
Owhadi, H., Zhang, L., Berlyand, L.: Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM: Math. Model. Numer. Anal. 48(2), 517–552 (2014)
Peszyńska, M.: Mortar adaptivity in mixed methods for flow in porous media. Int. J. Numer. Anal. Model. 2(3), 241–282 (2005)
Peszyńska, M., Wheeler, M., Yotov, I.: Mortar upscaling for multiphase flow in porous media. Comput. Geosci. 6(1), 73–100 (2002)
Wu, X.H., Efendiev, Y., Hou, T.Y.: Analysis of upscaling absolute permeability. Discrete and Continuous Dynamical Systems, Series B. 2, 158–204 (2002)
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The research of Eric Chung is supported by Hong Kong RGC General Research Fund (Project 14317516). YE would like to thank the partial support from NSF 1620318, the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DEFG02-13ER26165 and National Priorities Research Program grant NPRP grant 7-1482-1278 from the Qatar National Research Fund. YE would also like to acknowledge the support of Mega-grant of the Russian Federation Government (N 14.Y26.31.0013).
Appendices
Appendix
An existence of global and multiscale basis functions
In this appendix, we prove the existence of the problem (??)–(??). It suffices to prove the following. There exists a constant \(\widetilde {C}\) such that for all \((u,p)\in V_{0}(K^{+}_{i})\times Q(K^{+}_{i})\), there exist a pair \((v,q)\in V_{0}(K^{+}_{i})\times Q(K^{+}_{i})\) such that
where
and
First, it is clear that
By the inf-sup condition (??), there exist a w such that ∥w∥V = ∥(I − π)p∥s and
Therefore,
and
Next, we take \((v,q)=(u+\beta w,p+\tilde {\kappa }^{-1}\nabla \cdot u)\), so
Finally, we choose β = C, we have
and
This completes the proof.
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Chung, E., Efendiev, Y. & Leung, W.T. Constraint energy minimizing generalized multiscale finite element method in the mixed formulation. Comput Geosci 22, 677–693 (2018). https://doi.org/10.1007/s10596-018-9719-7
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DOI: https://doi.org/10.1007/s10596-018-9719-7