×

New scheme of finite difference heterogeneous multiscale method to solve saturated flow in porous media. (English) Zbl 1474.76049

Summary: A new finite difference scheme, the development of the finite difference heterogeneous multiscale method (FDHMM), is constructed for simulating saturated water flow in random porous media. In the discretization framework of FDHMM, we follow some ideas from the multiscale finite element method and construct basic microscopic elliptic models. Tests on a variety of numerical experiments show that, in the case that only about a half of the information of the whole microstructure is used, the constructed scheme gives better accuracy at a much lower computational time than FDHMM for the problem of aquifer response to sudden change in reservoir level and gives comparable accuracy at a much lower computational time than FDHMM for the weak drawdown problem.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dykaar, B. B.; Kitanidis, P. K., Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach 2: results, Water Resources Research, 28, 4, 1167-1178 (1992) · doi:10.1029/91WR03083
[2] Hou, T. Y.; Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134, 1, 169-189 (1997) · Zbl 0880.73065 · doi:10.1006/jcph.1997.5682
[3] Hou, T. Y.; Wu, X.-H.; Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation, 68, 227, 913-943 (1999) · Zbl 0922.65071 · doi:10.1090/S0025-5718-99-01077-7
[4] E, W.; Engquist, B., The heterogeneous multiscale methods, Communications in Mathematical Sciences, 1, 1, 87-132 (2003) · Zbl 1093.35012 · doi:10.4310/CMS.2003.v1.n1.a8
[5] Efendiev, Y.; Pankov, A., Numerical homogenization of nonlinear random parabolic operators, Multiscale Modeling & Simulation, 2, 2, 237-268 (2004) · Zbl 1181.76113 · doi:10.1137/030600266
[6] Babuška, I., Solution of interface problems by homogenization. I, SIAM Journal on Mathematical Analysis, 7, 5, 603-634 (1976) · Zbl 0343.35022 · doi:10.1137/0507048
[7] Babuška, I.; Hubbard, B., Homogenization and its application. Mathematical and computational problems, Numerical Solution of Partial Differential Equations. III, 89-116 (1976), New York, NY, USA: Academic Press, New York, NY, USA
[8] Babuška, I., Solution of interface problems by homogenization. III, SIAM Journal on Mathematical Analysis, 8, 6, 923-937 (1977) · Zbl 0402.35046 · doi:10.1137/0508071
[9] Durlofsky, L. J.; Efendiev, Y.; Ginting, V., An adaptive local-global multiscale finite volume element method for two-phase flow simulations, Advances in Water Resources, 30, 3, 576-588 (2007) · doi:10.1016/j.advwatres.2006.04.002
[10] Ye, S.; Xue, Y.; Xie, C., Application of the multiscale finite element method to flow in heterogeneous porous media, Water Resources Research, 40 (2004) · doi:10.1029/2003WR002914
[11] Chen, Z.; Hou, T. Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Mathematics of Computation, 72, 242, 541-576 (2003) · Zbl 1017.65088 · doi:10.1090/S0025-5718-02-01441-2
[12] He, X.; Ren, L., Finite volume multiscale finite element method for solving the groundwater flow problems in heterogeneous porous media, Water Resources Research, 41, 10 (2005) · doi:10.1029/2004WR003934
[13] E, W.; Ming, P.; Zhang, P., Analysis of the heterogeneous multiscale method for elliptic homogenization problems, Journal of the American Mathematical Society, 18, 1, 121-156 (2005) · Zbl 1060.65118 · doi:10.1090/S0894-0347-04-00469-2
[14] Ming, P.; Zhang, P., Analysis of the heterogeneous multiscale method for parabolic homogenization problems, Mathematics of Computation, 76, 257, 153-177 (2007) · Zbl 1129.65067 · doi:10.1090/S0025-5718-06-01909-0
[15] Ming, P.; Yue, X., Numerical methods for multiscale elliptic problems, Journal of Computational Physics, 214, 1, 421-445 (2006) · Zbl 1092.65102 · doi:10.1016/j.jcp.2005.09.024
[16] Yue, X.; E, W., Numerical methods for multiscale transport equations and application to two-phase porous media flow, Journal of Computational Physics, 210, 2, 656-675 (2005) · Zbl 1089.76049 · doi:10.1016/j.jcp.2005.05.009
[17] Rotersa, F.; Eisenlohra, P.; Hantcherlia, L., Overview of constitutive laws, kinematics, homogenization and multiscalemethods in crystal plasticity finite-element modeling: theory, experiments, applications, Acta Materialia, 58, 1152-1211 (2010)
[18] Principe, J.; Codina, R.; Henke, F., The dissipative structure of variational multiscale methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, 199, 13-16, 791-801 (2010) · Zbl 1406.76034 · doi:10.1016/j.cma.2008.09.007
[19] John, V.; Kindl, A., Numerical studies of finite element variational multiscale methods for turbulent flow simulations, Computer Methods in Applied Mechanics and Engineering, 199, 13-16, 841-852 (2010) · Zbl 1406.76029 · doi:10.1016/j.cma.2009.01.010
[20] Abdulle, A.; E, W., Finite difference heterogeneous multi-scale method for homogenization problems, Journal of Computational Physics, 191, 1, 18-39 (2003) · Zbl 1034.65067 · doi:10.1016/S0021-9991(03)00303-6
[21] E, W.; Engquist, B.; Huang, Z., Heterogeneous multiscale method: a general methodology for multiscale modeling, Physical Review B, 67 (2003) · doi:10.1103/PhysRevB.67.092101
[22] Chen, F.; Ren, L., Application of the finite difference heterogeneous multiscale method to the Richards’ equation, Water Resources Research, 44 (2008) · doi:10.1029/2007WR006275
[23] Wen, X.; Gomez-Hernandez, J. J., Upscaling hydraulic conductivities in heterogeneous media: an overview, Journal of Hydrology, 183, 1-2
[24] Renard, P.; de Marsily, G., Calculating equivalent permeability: a review, Advances in Water Resources, 20, 5-6, 253-278 (1997)
[25] Du, R.; Ming, P., Heterogeneous multiscale finite element method with novel numerical integration schemes, Communications in Mathematical Sciences, 8, 797-1091 (2010)
[26] de Zeeuw, P. M., Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, Journal of Computational and Applied Mathematics, 33, 1, 1-27 (1990) · Zbl 0717.65099 · doi:10.1016/0377-0427(90)90252-U
[27] Gelhar, L. W.; Axness, C. L., Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resources Research, 19, 1, 161-180 (1983)
[28] Cao, H.; Yue, X., The discrete finite volume method on quadrilateral mesh, Journal of Suzhou University, 10, 6-10 (2005)
[29] Mantoglou, A.; Wilson, J. L., The turning bands method for simulation of random fields using line generation by a spectral method, Water Resources Research, 18, 5, 1379-1394 (1982)
[30] Wang, H. F.; Anderson, M. P., Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods (1982), San Francisco, Calif, USA: W. H. Free-Man and Company, San Francisco, Calif, USA
[31] He, X.; Ren, L., A modified multiscale finite element method for well-driven flow problems in heterogeneous porous media, Journal of Hydrology, 329, 3-4, 674-684 (2006) · doi:10.1016/j.jhydrol.2006.03.018
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.