Optimal Gridding of Stochastic Models for Scale-up

Stochastic models of reservoir properties are routinely scaled-up to reduce the number of grid blocks in the input of the flow simulation model. In this process, the heterogeneities in the stochastic model, represented by small-scale grid blocks, are homogenized into much larger grid blocks for flow simulation. As a result, the volume in the larger simulation grids are represented by a single value, and the correlation structure and variance of the small-scale grids are lost. We have conducted an extensive study on the effects of all the parameters determining effective permeabilities through detailed higher-order finite-difference solution of the flow equations. Our results demonstrate the dependence of effective permeability on variance, permeability in principal flow directions, permeability anisotropy, spatial correlation length in the principal flow directions, and spatial correlation anisotropy. We show that the simulation technique employed here may be required to incorporate all factors that affect the value of effective permeability into account. After the scaleup process for each large-scale grid block, the variance of the small-scale permeabilities is removed and the correlation length is maximized. On the other hand, the variance of the large-scale permeabilities is less than the original smallscale permeability distribution, and the spatial correlation is also changed. For these reasons, the design of the large-scale grids should be such that it minimizes the variance and maximizes correlation length of the small-scale grids in each large-scale grid block, and best preserves the spatial distribution and variance of the small-scale grids. In this paper, an efficient simulated annealing algorithm is described that generates near-optimal large-scale grids for scale-up and best preserves the heterogeneities for flow simulation.