Abstract
In this article, we discuss the application of multiscale finite element method (MsFEM) to groundwater flow in heterogeneous porous media. We investigate the ability of MsFEM in qualifying the flow uncertainty. Monte Carlo simulation is employed to implement the stochastic analysis, and MsFEM is used to avoid a full resolution to the spatial variable conductivity field. Large-scale flow with high variability is investigated by inspecting the single realization as well as the probability distribution functions of head and velocity. The numerical results show that the performance of MsFEM depends on the ratio between the correlation length and the coarse element size. An accurate prediction to the velocity requires a much lower ratio than the head. The MsFEM has different convergence rates for the head and the velocity, while the convergence rates do not deteriorate as the variance grows.
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References
Aarnes JE (2004) 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
Aarnes JE, Efendiev Y, Jiang L (2008) Mixed multiscale finite element methods using limited global information. Multiscale Model Simul 7(2):655–676
Ababou R, McLaughlin D, Gelhar LW, Tompson AFB (1989) Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media. Transp Porous Media 4:549–565
Arbogast T, Bryant SL (2002) A two-scale numerical subgrid technique for waterflood simulations. SPE J 7:446–457
Arbogast T, Wheeler MF, Yotov I (1997) Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J Numer Anal 34(2):828–852
Bellin A, Salandin P, Rinaldo A (1992) Simulation of dispersion in heterogeneous porous formations: statistics, first-order theories, convergence of computations. J Hydrol 28(9):2211–2227
Chen Z, Hou TY (2002) A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math Comput 72(242):541–576
Chen Y, Durlofsky LJ, Gerritsen M, Wen XH (2003) A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv Water Resour 26:1041–1060
Chen M, Keller AA, Lu Z (2009) Stochastic analysis of transient three-phase flow in heterogeneous porous media. Stoch Environ Res Risk Assess 23(1):93–109
Chu J, Efendiev Y, Ginting V, Hou TY (2008) Flow based oversampling technique for multiscale finite element methods. Adv Water Resour 31:599–608
Chu CC, Graham IG, Hou TY (2010) A new multiscale finite element method for high-contrast elliptic interface problems. Math Comput 79(272):1915–1955
Cordes C, Kinzelbach W (1992) Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements. Water Resour Res 28:2903–2911
Efendiev Y, Hou TY (2007) Multiscale finite element methods for porous media flows and their applications. Appl Numer Math 57:577–596
Efendiev YR, Hou TY, Wu XH (2000) The convergence of nonconforming multiscale finite element methods. SIAM J Numer Anal 37:888–910
Efendiev Y, Ginting V, Hou TY, Ewing R et al (2006a) Accurate multiscale finite element methods for two-phase flow simulations. J Comput Phys 220:155–174
Efendiev Y, Hou TY, Luo W (2006b) Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM J Sci Comput 28(2):776–803
Engquist WEB (2003) The heterogeneous multiscale methods. Comm Math Sci 1(1):87–133
Ganapathysubramanian B, Zabaras N (2007) Modeling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multiscale method. J Comput Phys 226(1):326–353
Ghanem RG (1998) Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour Res 34:21–23
Ginn TR (2004) On the application of stochastic approaches in hydrogeology. Stoch Environ Res Risk Assess 18(4):282–284
He X, Ren L (2006a) A multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media. Water Resour Res 42:W08417. doi:10.1029/2006WR004905
He X, Ren L (2006b) A modified multiscale finite element method for well-driven flow problems in heterogeneous porous media. J Hydrol 329(3–4):674–684
Hou TY, Wu XH (1997) A multiscale finite element method for elliptic problems in composite materials and porous media. J Comp Phys 134:169–189
Hou TY, Wu XH, Cai Z (1999) Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math Comp 68:913–943
Hughes TJR (1995) Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods. Comp Meth Appl Mech Eng 127:387–401
Jenny P, Lee SH, Tchelepi HA (2005) Adaptive multi-scale finite volume method for multi-phase flow and transport in porous media. Multiscale Model Simul 3:30–64
Jikov VV, Kozlov SM, Oleinik OA (1994) Homogenization of differential operators and integral functionals. Springer, Berlin
Lin G, Tartakovsky A (2009) An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Adv Water Resour 32(5):712–722
Mose R, Siegel P, Ackerer P (1994) Application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity. Water Resour Res 30(11):3001–3012
Rubin Y (1995) Flow and transport in bimodal heterogeneous formations. Water Resour Res 31:2461–2468
Salandin P, Fiorotto V (1998) Solute transport in highly heterogeneous aquifers. Water Resour Res 34(5):949–961
Shi L, Yang J, Zhang D (2009) A stochastic approach to nonlinear unconfined flow subject to multiple random fields. Stoch Environ Res Risk Assess 23(6):823–835
Shi L, Zhang D, Lin L, Yang J (2010) A multiscale probabilistic collocation method for subsurface flow in heterogeneous Media. Water Resour Res. doi:10.1029/2010WR009066
Simunek J, Vogel T, van Genuchten M (1992) The SWMS_2D code for simulating water flow and solute transport in two-dimensional variably saturated media. Version 1.1, Research Report No. 126, US Salinity Lab
Tartakovsky DM (2000) Real gas flow through heterogeneous porous media: theoretical aspects of upscaling. Stoch Environ Res Risk Assess 14:109–122
Van Lent T, Kitanidis PK (1996) Effects of first-order approximations on head and specific discharge covariance in high-contrast log conductivity. Water Resour Res 32(5):1197–1207
Wu XH, Efendiev YR, Hou TY (2002) Analysis of upscaling absolute permeability. Discrete Continuous Dyn Syst B 2(2):185–204
Xiu D, Karniadakis GE (2002) The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24:619–644
Zhang D (2002) Stochastic methods for flow in porous media: copying with uncertainties. Academic Press, San Diego
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This work is partially supported by Natural Science Foundation of China through grants 51009110, 41072189 and 51079101, and 973 Program through grant 2010CB42880204.
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Shi, L., Yang, J. & Zeng, L. Application of multiscale finite element method in the uncertainty qualification of large-scale groundwater flow. Stoch Environ Res Risk Assess 26, 393–404 (2012). https://doi.org/10.1007/s00477-011-0507-3
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DOI: https://doi.org/10.1007/s00477-011-0507-3