zbMATH — the first resource for mathematics

Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. (English) Zbl 1047.76538
Summary: In this paper we present a multi-scale finite-volume (MSFV) method to solve elliptic problems with many spatial scales arising from flow in porous media. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of the differential operator. This leads to a multi-point discretization scheme for the finite-volume solution algorithm. Transmissibilities for the MSFV have to be constructed only once as a preprocessing step and can be computed locally. Therefore this step is perfectly suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from the coarse-scale pressure solution. Two sets of locally computed basis functions are employed. The first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed in order to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field. The accuracy and efficiency of our method is demonstrated by various numerical experiments.

76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] T. Arbogast, Numerical subgrid upscaling of two phase flow in porous media, Technical Report, Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, 1999 · Zbl 1072.76560
[2] T. Arbogast, S.L. Bryant, Numerical subgrid upscaling for waterflood simulations, SPE 66375, presented at the SPE Symp. on Reservoir Simulation, Houston, 2001
[3] Beckie, R.; Aldama, A.A.; Wood, E.F., Modeling the large-scale dynamics of saturated groundwater flow using spatial filtering, Water resour. res., 32, 1269-1280, (1996)
[4] Chen, Z.; Hou, T.Y., A mixed finite element method for elliptic problems with rapidly oscillating coefficients, Math. comput., 72, 541-576, (2003) · Zbl 1017.65088
[5] Durlofsky, L.J., Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water resour. res., 27, 699-708, (1991)
[6] Efendiev, Y.; Hou, T.; Wu, X., Convergence of a nonconformal multiscale finite element method, SIAM J. numer. anal., 37, 888-910, (2000) · Zbl 0951.65105
[7] Efendiev, Y.; Wu, X., Multiscale finite element for problems with highly oscillatory coefficients, Numer. math., 90, 459-486, (2002) · Zbl 0997.65129
[8] Hou, T.; Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comp. phys., 134, 169-189, (1997) · Zbl 0880.73065
[9] Jenny, P.; Wolfsteiner, C.; Lee, S.H.; Durlofsky, L.J., Modeling flow in geometrically complex reservoirs using hexahedral multi-block grids, Spe j., 149-157, (2002)
[10] Lee, S.H.; Durlofsky, L.J.; Lough, M.F.; Chen, W.H., Finite difference simulation of geologically complex reservoirs with tensor permeabilities, Spere&e, 567-574, (1998)
[11] Lee, S.H.; Tchelepi, H.; Jenny, P.; Dechant, L., Implementation of a flux continuous finite-difference method for stratigraphic, hexahedron grids, Spe j., 269-277, (2002)
[12] T.C. Wallstrom, T.Y. Hou, M.A. Christie, L.J. Durlofsky, D.H. Sharp, in: Application of a New-phase Upscaling Technique to Realistic Reservoir Crosssections, SPE51939, Presented at the SPE Symp. on Reservoir Simulation, Houston, 1999
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.