Statistical assignment of upscaled flow functions for an ensemble of geological models

Upscaled flow functions are often needed to account for the effects of fine-scale permeability heterogeneity in coarse-scale simulation models. We present procedures in which the required coarse-scale flow functions are statistically assigned to an ensemble of upscaled geological models. This can be viewed as an extension and further development of a recently developed ensemble level upscaling (EnLU) approach. The method aims to efficiently generate coarse-scale flow models capable of reproducing the ensemble statistics (e. g., cumulative distribution function) of fine-scale flow predictions for multiple reservoir models. The most expensive part of standard coarsening procedures is typically the generation of upscaled two-phase flow functions (e. g., relative permeabilities). EnLU provides a means for efficiently generating these upscaled functions using stochastic simulation. This involves the use of coarse-block attributes that are both fast to compute and correlate closely with the upscaled two-phase functions. In this paper, improved attributes for use in EnLU, namely the coefficient of variation of the fine-scale single-phase velocity field (computed during computation of upscaled absolute permeability) and the integral range of the fine-scale permeability variogram, are identified. Geostatistical simulation methods, which account for spatial correlations of the statistically generated upscaled functions, are also applied. The overall methodology thus enables the efficient generation of coarse-scale flow models. The procedure is tested on 3D well-driven flow problems with different permeability distributions and variable fluid mobility ratios. EnLU is shown to capture the ensemble statistics of fine-scale flow results (water and oil flow rates as a function of time) with similar accuracy to full flow-based upscaling methods but with computational speedups of more than an order of magnitude.