The Competing Effects of Discretization and Upscaling – A Study Using the q-Family of CVD-MPFA

Families of flux-continuous, locally conservative, finite-volume schemes have been developed for solving the general tensor pressure equation on structured and unstructured grids [1,2]. A family of flux-continuous schemes is quantified by a quadrature parameterization [3]. Improved convergence has been observed for certain quadrature points [4]. In this paper a q-family of flux-continuous (CVD-MPFA) schemes are used as a part of numerical upscaling procedure for upscaling fine scale permeability fields on to coarse grid scales. A series of data sets [5] are tested where the upscaled permeability tensor is computed on a sequence of grid levels. The upscaling sequence is repeated for three distinct quadrature points (q=0.1, q=0.5, q=1) belonging to the family of schemes. Three types of refinement study are presented: 1. Refinement study with invariant permeability distribution with respect to grid level, involving a classical mathematical convergence study. The same coarse scale underlying permeability map (and therefore the problem) is preserved on all grid levels including the fine scale reference solution. 2. Refinement study with renormalized permeability. In this study, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing the effect of locally upscaled permeability, compared to that obtained directly from the fine scale solution. 3. Reservoir field refinement study, in this study the permeability distribution for each grid level is obtained by upscaling directly from the fine scale permeability field as in standard simulation practice. 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 the study and superconvergence is observed. REFERENCES 1. M.G. Edwards, C.F.Rogers. Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geo. (1998). 2. M.G.Edwards, Unstructured, control-volume distributed, full-tensor finite volume schemes with flow based grids. Comput.Geo. (2002). 3. M. Pal, M.G. Edwards and A.R. Lamb., Convergence study of a family of flux-continuous, finite-volume schemes for the general tensor pressure equation. IJNME, 2005. 4. M. Pal, Families of Control-Volume Distributed CVD (MPFA) Finite Volume Schemes for the Porous Medium Pressure Equation on Structured and Unstructured Grids. PhD Thesis, Swansea University, 2007. 5. M.A.Christie, SPE, Herriot-Watt University, and M.J Blunt, Imperial College, Tenth SPE comparative Solution Project: A Comparison of Upscaling Techniques, 2001.