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SUMMARYTo be able to use a limited number of relatively large grid cells in numerical oil reservoir simulators and groundwater models, upscaling of the absolute permeability is frequently applied. The spatial ÿne-scale permeability distribution, which is generally obtained from geological and geostatistical models, is incorporated in the relatively large grid cells of the numerical model. If the porous medium may be approximated as a periodic medium, upscaling can be performed by the homogenization method. Numerical homogenization gives rise to an approximation error. The complementarity between the conformal-nodal ÿnite element method and the mixed-hybrid ÿnite element method has been used to quantify this error. The two methods yield, respectively, upper and lower bounds for the eigenvalues of the coarse-scale permeability tensor. Results of numerical experiments obtained using tetrahedral meshes are shown both in the far ÿeld and in the near well region.

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This paper presents a 'physics-oriented' approach to approximate the continuum equations governing porous media flow by discrete analogs. To that end, the continuity equation and Darcy's law are reformulated using exterior differential forms. This way the derivation of a system of algebraic equations (the discrete analog) on a finite-volume mesh can be accomplished by simple and elegant 'translation rules.' In the discrete analog the information about the conductivities of the porous medium and the metric of the mesh are represented in one matrix: the discrete dual. The discrete dual of the blockcentered finite difference method is presented first. Since this method has limited applicability with respect to anisotropy and non-rectangular grid blocks, the finite element dual is introduced as an alternative. Application of a domain decomposition technique yields the face-centered finite element method. Since calculations based on pressures in volume centers are sometimes preferable, a volume-centered approximation of the face-centered approximation is presented too.

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