Scale-Averaged Effective Flow Properties for Coarse-Grid Reservoir Simulation
Authors: Hewett Thomas A. , Archer Rosalind A. 
Published: December 2007

Abstract
Unambiguous rules for computing effective flow properties for coarse-grid simulations based on the results of fine-grid simulations are presented. These rules are based on the concept of computing integrated phase mobilities in streamtube segments whose geometry is defined by the fine-grid cells at the coarse-grid block outlet face. The individual phase mobilities are calculated in the segments of each streamtube which run between the iso-potential lines through the centers of adjacent coarse-grid blocks. Gravity and capillary pressure are accounted for by the introduction of an effective capillary pressure defined by associating the capillary pressures at the coarse-grid block centers with the average saturations of the coarse-grid blocks and the addition of the conventional gravity term based on the actual phase density differences and the difference in elevation of the coarse-grid block centers. Validation runs for waterfloods in homogeneous and heterogeneous reservoir models with varying degrees of dip show a good match between the fine-grid and coarse-grid results.

Introduction
Recently, Hewett and Yamada introduced a semi-analytical method for calculating effective relative permeabilities to represent the effects of sub-grid scale permeability heterogeneity in coarse-grid reservoir simulations. This method was based on the use of stream tubes derived from a numerical solution of the steady-state, single-phase flow equations. In their approach, the total mobility of each streamtube carrying flow between coarse-grid blocks was calculated between the isobars through the coarse-grid block centers. They also showed that, by adopting this conceptual approach for calculating effective relative permeabilities, the definition of the correct averaging rules for deriving them from fine-grid simulation results could be obtained unambiguously. Their results, however, did not include provision for the effects of gravity or capillary pressure. This paper addresses the modifications required to include the effects of gravity and capillary pressure in the definition of effective flow properties for coarse grids.
Calculating Scale-Averaged Effective Relative Permeabilities We wish to define scale-averaged properties for use in coarse-grid reservoir simulations which account for the effects of fine-scale permeability variations included in finely gridded reservoir models. Typically, this is done to reduce the dimensionality of the problem from three to two, or from two to one. At the same time, the grid is frequently coarsened in the remaining dimensions. Figure 1 shows several coarse-grid blocks superimposed on a finer grid. The aim is to define effective flow properties for the coarse grid which reproduce the interblock flows on the coarse grid which were observed in the fine-grid simulation. The equations which are solved on the fine grid are
(1)
(2)
(3)
where q1 = qo + qw is the total volumetric flowrate, Tn is the single-phase transmissibility between grid blocks n and n+1, kro, n and krw, n are the oil and water relative permeabilities of grid block n, o and w are oil and water phase viscosities, ro, n, and rw, n, are the oil and water relative mobilities for grid block n, and o, n and w, n, are the oil and water phase potential differences between grid blocks n and n+ 1.
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