Upscaling Using a Non-Uniform Coarsened Grid With Optimum Power Average
Authors: Merchan S. , Srinivasan S. 
Published: March 2010

Abstract
Geostatistical modeling techniques are capable of generating high-resolution reservoir models. Since a limited amount of information is available to model the reservoir, geological uncertainty is represented through a suite of equally probable models. Unfortunately, these high-resolution models are often too large to process through numerical flow simulators. Upscaling methods are required that reduce the size of detailed models while preserving the important geological characteristics of the reservoir. Most upscaling methods currently in vogue work with uniform grids. A new upscaling approach based on non-uniform coarsening with optimum power average is presented. The proposed algorithm identifies likely high connectivity regions using streamline simulations, then constructs a non-uniform coarse-scale grid preserving the areas with probable high connectivity and assign equivalent permeability to the coarse grid blocks using an optimum power average technique. The power average exponent is calibrated using the data from a series of single-phase flow simulations.

Introduction
Upscaling is a procedure that transforms a detailed geological model to a coarse grid simulation model such that the flow behaviour in the two systems is similar. Upscaling is required because fine-scale flow simulation of multiple geostatistical realizations can be CPU (computer processing unit) expensive. Any upscaling procedure involves basically two steps:gridding, to define the new coarse blocks, andaveraging or estimation of properties, to preserve the local geologic details. Numerous upscaling methods have been reported in the literature(1). Nevertheless, efficient and accurate estimation of equivalent rock properties of coarse-scale from geological data at fine-scale remains an active area of research.
The simplest numerical procedure for the determination of equivalent permeability involves the solution of the Laplace equation for pressure within the reservoir domain, subject to constant pressure gradient in the direction of flow and no flux perpendicular to it. The limitation of these conditions is that the cross terms of the K tensor (Kxy and Kyx in 2D systems) cannot be determined. Despite this limitation, the approach continues to be used assuming that the diagonal terms of K tensor that are computed are correct and the cross-terms are not important. This is true if the coordinate's direction (i.e. x and y) coincides with the principal directions of the effective permeability tensor. Unfortunately this is not usually the case or known a priori and it can change from one location to another.
To overcome this limitation, Durlofsky(2) presented a numerical procedure for the determination of equivalent grid block permeability tensors. The method entails solution of the fine-scale pressure equation, subject to periodic boundary conditions. Symmetric, positive definite equivalent permeability tensors are obtained. A numerical approach to obtain a full tensor consists of using linear boundary conditions(3,4). A pressure gradient is imposed in the flow direction and a linear pressure profile is enforced on the two other opposite faces. This variation results in a non-symmetrical permeability tensor taking into account the cross-flow term.
Despite improved representation of flow in heterogeneous media using a permeability tensor, the pressure solver techniques employ approximations such as single-phase flow and simplified boundary conditions.