The effects of control-volume distributed multi-point flux approximation (CVD-MPFA) on upscalingu-A study using the CVD-MPFA schemes

A family of flux-continuous, locally conservative, finite-volume schemes has been developed for solving the general geometry-permeability tensor (petroleum reservoir-simulation) pressure equation on structured and unstructured grids and are control-volume distributed (textit Comput. Geo. 1998; 2:259290; Comput. Geo. 2002; 6:433452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir-simulation schemes (two-point flux approximation) when applied to full tensor flow approximation. The family of flux-continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51:11771203). Improved convergence (for two- and three-dimensional formulation) using the quadrature parameterization has been observed for the family of flux-continuous control-volume distributed multi-point flux approximation (CVD-MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007). In this paper family of flux-continuous (CVD-MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine-scale grid information (permeability) onto a coarse grid scale. A series of data-sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve: (i) Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine-scale permeability field as in standard simulation practice. (ii) Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine-scale solution. (iii) Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse-scale underlying permeability map is preserved on all grid levels including the fine-scale reference solution. The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright (C) 2010 John Wiley & Sons, Ltd.