Global Scale-up on Reservoir Models with Piecewise Constant Permeability Field

Abstract

Global scale-up was first proposed in the 1980s, and its benefits are well-described in the literature. However, global scale-up has not been widely applied in practice due to significant technical challenges. In this paper, we analyze some of the theoretical and numerical difficulties and present practical resolutions. First, we derive the flux and energy formulations for scale-up by using a Multiscale Finite Element framework. These formulations can be applied to local, extended local and global scale-up. Then we present a new method to improve scale-up accuracy for geo-cellular models with piecewise constant permeability. On these models, large jumps in the permeability field lead to well-known singularities in the flow solutions, making them difficult to be calculated accurately. We show that inaccurate flow solutions can lead to large scale-up errors. To mitigate the effect of the singularities and improve the scale-up accuracy, we have developed a hybrid method for solving flows which utilizes the fact that the Continuous Galerkin (CG) finite elment method and the Mixed finite element method provide upper and lower bounds, respectively, for the effective permeability. The new method uses CG to compute the pressure solution followed by a weighted L² projection of the numerical fluxes to enforce local mass conservation. Numerical examples are presented to demonstrate the effectiveness of the hybrid method.

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