Upscaling Immiscible Gas Displacements: Quantitative Use of Fine Grid Flow Data in Grid Coarsening Schemes
Authors: Darman N.H. , Durlofsky L.J. , Pickup G.E. 
Published: December 2007

Abstract
Grid coarsening schemes based on the quantitative use of fine scale two-phase flow information are presented and assessed. The basic approach is motivated from a volume average analysis of the fine scale saturation equation including gravitational effects. Extensive results for layered systems are presented. It is shown that coarse grid simulation error correlates closely with specific sub-grid quantities involving higher moments of fine grid variables, which can be computed from the fine scale simulations. By forming a coarse grid that minimises the appropriate sub-grid quantity, optimal coarse scale descriptions can be generated. The overall approach is shown to be applicable to coarse scale descriptions using either rock or pseudo relative permeability curves. The accuracy of the coarse grid calculations is, however, significantly better when pseudo functions are used. The method is applied to determine the optimal number and configuration of coarse grid layers in more general cases and it is shown that coarse grid results do not always improve as the number of coarse layers is increased.

Introduction
In modern reservoir characterisation, the spatial resolution that may be incorporated into geological models often exceeds the computational capabilities of fluid flow simulators by a significant margin. Therefore, some level of upscaling must be applied to the fine scale geological models before they can be used for practical flow calculations. This upscaling may be a simple block averaging of the single-phase permeability or it may involve the application of a complex upscaling procedure. When the degree of upscaling is very large, the use of a dynamic technique, which may involve the generation of upscaled or pseudo relative permeabilities, is generally required.
Several such upscaling methods have been developed and described in the literature; e.g. the Kyte and Berry1, Stone2, Vertical Equilibrium (VE)3 and TW4,a methods. Hewett and coworkers have also suggested approaches based on streamline methods5,6. In general, all of these dynamic methods (except VE) involve some procedure for using the fine grid flows to generate modified pseudo relative permeability and capillary pressure curves at the coarse block scale.
When successfully applied, these pseudo functions will accurately incorporate the interaction between small-scale multi-phase fluid flow and heterogeneity, as well as correcting for the numerical dispersion in the coarse grid models. The principal metric is that the upscaling method provides a coarse-scale flow model that accurately reproduces the results (recovery profiles, breakthrough times, etc.) computed using the fine grid model. Pseudo functions do have costs and limitations associated with them, however, and these must be considered when such an approach is applied. Pseudo relative permeabilities may be subject to so-called process dependence, meaning the coarse scale pseudo functions vary with varying global boundary conditions. This in turn can result in a lack of robustness in the coarse scale model. In addition, when pseudo functions are generated through the simulation of a global fine scale flow problem, the computational requirements can be excessive in some cases.