×

A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media. (English) Zbl 1453.76069

Summary: In this paper, we propose a local-global multiscale mortar mixed finite element method (MMMFEM) for multiphase transport in heterogeneous media. It is known that, in the efficient numerical simulations of this problem, one important step is a fast solution of the pressure equation, which is required to be solved in each time step. Thus, some types of efficient numerical methods, such as multiscale methods, are crucial for this problem. To present our main concepts, we take the two-phase flow system as an example. In our proposed method, the pressure equation is solved via the multiscale mortar mixed finite element method. Using this approach, a mass conservative velocity field can be obtained. Next, we use an explicit finite volume method to solve the saturation equation. The key ingredient of our proposed method is the choice of mortar space for the MMMFEM. We will use both polynomials and multiscale basis functions to form the coarse mortar space. The multiscale basis functions used are the restriction of the global pressure field obtained at the previous time step on the coarse interface. To initialize the simulations, we solve the pressure equation on the fine grid. We will present several numerical experiments on some benchmark 2D and 3D heterogeneous models to show the excellent performance of our method.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76T06 Liquid-liquid two component flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs

Software:

MMMFEM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[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, 421-439 (2004) · Zbl 1181.76125
[2] Aarnes, J. E.; Hou, T., Multiscale domain decomposition methods for elliptic problems with high aspect ratios, Acta Math. Appl. Sin. Engl. Ser., 18, 63-76 (2002) · Zbl 1003.65142
[3] Alpak, Faruk O.; Pal, Mayur; Lie, Knut-Andreas, A multiscale adaptive local-global method for modeling flow in stratigraphically complex reservoirs, SPE J., 17, 04, 1-056 (2012)
[4] Araya, R.; Harder, C.; Paredes, D.; Valentin, F., Multiscale hybrid-mixed method, SIAM J. Numer. Anal., 51, 6, 3505-3531 (2013) · Zbl 1296.65152
[5] Arbogast, T., Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42, 2, 576-598 (2004), (electronic) · Zbl 1078.65092
[6] Arbogast, T.; Cowsar, L. C.; Wheeler, M. F.; Yotov, I., Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37, 4, 1295-1315 (2000) · Zbl 1001.65126
[7] Arbogast, T.; Pencheva, G.; Wheeler, M. F.; Yotov, I., A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6, 1, 319-346 (2007), (electronic) · Zbl 1322.76039
[8] Arbogast, T.; Xiao, H., A multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems, SIAM J. Numer. Anal., 51, 1, 377-399 (2013) · Zbl 1267.65192
[9] Arbogast, T.; Xiao, H., Two-level mortar domain decomposition preconditioners for heterogeneous elliptic problems, Comput. Methods Appl. Mech. Eng., 292, 221-242 (2015) · Zbl 1425.65042
[10] Chan, Ho Yuen; Chung, Eric; Efendiev, Yalchin, Adaptive mixed gmsfem for flows in heterogeneous media, Numer. Math., Theory Methods Appl., 9, 4, 497-527 (2016) · Zbl 1399.65322
[11] Chen, Fuchen; Chung, Eric; Jiang, Lijian, Least-squares mixed generalized multiscale finite element method, Comput. Methods Appl. Mech. Eng., 311, 764-787 (2016) · Zbl 1433.76073
[12] Chen, Y.; Durlofsky, Louis J.; Gerritsen, M.; Wen, Xian-Huan, A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations, Adv. Water Resour., 26, 10, 1041-1060 (2003)
[13] Chen, Z.; Hou, T. Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comput., 72, 541-576 (2002) · Zbl 1017.65088
[14] Chung, E.; Efendiev, Y.; Lee, C., Mixed generalized multiscale finite element methods and applications, Multiscale Model. Simul., 13, 1, 338-366 (2015) · Zbl 1317.65204
[15] Chung, Eric; Efendiev, Yalchin; Hou, Thomas Y., Adaptive multiscale model reduction with generalized multiscale finite element methods, J. Comput. Phys., 320, 69-95 (2016) · Zbl 1349.76191
[16] Chung, Eric; Pollock, Sara; Pun, Sai-Mang, Goal-oriented adaptivity of mixed gmsfem for flows in heterogeneous media, Comput. Methods Appl. Mech. Eng., 323, 151-173 (2017) · Zbl 1433.76076
[17] Chung, Eric T.; Efendiev, Yalchin; Tat Leung, Wing; Vasilyeva, Maria; Wang, Yating, Non-local multi-continua upscaling for flows in heterogeneous fractured media, J. Comput. Phys., 372, 22-34 (2018) · Zbl 1415.76449
[18] Chung, Eric T.; Lee, Chak Shing, A mixed generalized multiscale finite element method for planar linear elasticity, J. Comput. Appl. Math., 348, 298-313 (2019) · Zbl 1418.74035
[19] Durfolsky, L. J., Numerical calculation of equivalent grid block permeability tensors of heterogeneous porous media. Water resour res v27, n5, may 1991, p299-708, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 28, 5, Article A350 pp. (1991)
[20] 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, 3, 576-588 (2007)
[21] Efendiev, Y.; Galvis, J.; Wu, X. H., Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys., 230, 937-955 (2011) · Zbl 1391.76321
[22] 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) · Zbl 1158.76349
[23] Efendiev, Y.; Hou, T., Multiscale Finite Element Methods: Theory and Applications, vol. 4 (2009), Springer Science & Business Media · Zbl 1163.65080
[24] Ganis, Benjamin; Yotov, Ivan, Implementation of a mortar mixed finite element method using a multiscale flux basis, Comput. Methods Appl. Mech. Eng., 198, 49, 3989-3998 (2009) · Zbl 1231.76145
[25] Glowinski, R.; Wheeler, M. F., Domain decomposition and mixed finite element methods for elliptic problems, (First International Symposium on Domain Decomposition Methods for Partial Differential Equations (1988)), 144-172 · Zbl 0661.65105
[26] Harder, Christopher; Paredes, Diego; Valentin, Frédéric, A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients, J. Comput. Phys., 245, 107-130 (2013) · Zbl 1349.76214
[27] Kippe, V.; Aarnes, J. E.; Lie, K. A., Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels, Adv. Water Resour., 28, 257-271 (2005)
[28] 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) · Zbl 1047.76538
[29] Mansfield, Lois, Damped Jacobi preconditioning and coarse grid deflation for conjugate gradient iteration on parallel computers, SIAM J. Sci. Stat. Comput., 12, 6, 1314-1323 (1991) · Zbl 0738.65026
[30] Raviart, Pierre-Arnaud; Thomas, Jean-Marie, A mixed finite element method for 2-nd order elliptic problems, (Mathematical Aspects of Finite Element Methods (1977), Springer), 292-315 · Zbl 0362.65089
[31] Vasilyeva, Maria; Chung, Eric T.; Wun Cheung, Siu; Wang, Yating; Prokopev, Georgy, Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media, J. Comput. Appl. Math., 355, 258-267 (2019) · Zbl 1432.76181
[32] Wheeler, M. F.; Xue, G.; Yotov, I., A multiscale mortar multipoint flux mixed finite element method, ESAIM: Math. Model. Numer. Anal., 46, 4, 759-796 (2012) · Zbl 1275.65082
[33] Wu, X.; Efendiev, Y.; Hou, T. Y., Analysis of upscaling absolute permeability, Discrete Contin. Dyn. Syst., Ser. B, 2, 2, 185-204 (2002) · Zbl 1162.65327
[34] Xiao, H., Multiscale Mortar Mixed Finite Element Methods for Flow Problems in Highly Heterogeneous Porous Media (2013), PhD thesis
[35] Yang, Yanfang; Chung, Eric T.; Fu, Shubin, An enriched multiscale mortar space for high contrast flow problems, Commun. Comput. Phys., 23, 2, 476-499 (2018) · Zbl 1488.65477
[36] Yang, Yanfang; Chung, Eric T.; Fu, Shubin, Residual driven online mortar mixed finite element methods and applications, J. Comput. Appl. Math., 340, 318-333 (2018) · Zbl 1432.65172
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.