Multi-scale finite-volume method for elliptic problems in subsurface flow simulation.

*(English)*Zbl 1047.76538Summary: 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.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76S05 | Flows in porous media; filtration; seepage |

86A05 | Hydrology, hydrography, oceanography |

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\textit{P. Jenny} et al., J. Comput. Phys. 187, No. 1, 47--67 (2003; Zbl 1047.76538)

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##### References:

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