Evaluation of Permeability Upsealing Techniques and a New Algorithm for Interblock Transmissibilities
Authors: Abbaszadeh Maghsood , Koide Naoki 
Published: December 2007

Abstract
This paper evaluates various published permeability upscaling techniques and provides conditions for their reuse of applicability. Specifically, the important upscaling techniques of power averaging, Cardwell-Parsons, real-space renormalization and numerical pressure solver are applied to 2-D and 3-D heterogeneous gridblocks with different spatial correlation structure and different levels of heterogeneity. It is shown that the most common approach of pressure solver with the harmonic averaging scheme for interblock transmissibility calculation is proned to numerical inaccuracy, especially for high degrees of heterogeneity.
Analytical expressions are derived which show that the harmonic-average weighting transmissibility algorithm is not applicable to 2-D and 3-D heterogeneous systems. A new interblock transmissibility calculation scheme based on upstream gridblock permeability weighting is developed which has an improved numerical accuracy. The algorithm is fast and produces more accurate results for larger heterogeneous domains, a desired feature in field-scale reservoir simulations.

Introduction
All naturally occurring porous media are heterogeneous at different length scales. Each scale of heterogeneity impacts fluid flow and displacement efficiency in a different manner. The challenge in understanding and predicting reservoir performance is to first describe reservoir geologic heterogeneities realistically and quantitatively, and then model the reservoir flow behavior in the presence of all heterogeneities accurately and efficiently. While large-scale reservoir features can be described by deterministic techniques, less correlated medium-scale and more chaotic small-scale heterogeneities may be characterized by geostatistical techniques. These statistical methods generate multiple equ-probable reservoir descriptions at any resolution.
The fundamental question is how to utilize this characterization of heterogeneous reality in reservoir flow models without artificially diminishing their effect on flow. Conventional three-dimensional reservoir simulation models are often inadequate to represent detailed heterogeneity fields generated from geological information. Heterogeneties simply occur at scales much smaller than the smallest gridblock that can be afforded in practical simulation models. Use of large gridblocks smooths out heterogeneities, and detailed geologic information is lost during the flow modeling phase.
Scaleup processes develop procedures to properly represent detailed reservoir descriptions in reservoir flow models. More specifically, appropriate averaging techniques for large-scale flow models are needed in order to correctly represent the impact of heterogeneity on pressure, saturation distribution, breakthrough, sweep and recovery. Such capability is necessary if a reservoir model is to portray fluid flow behavior realistically and serve as a reliable predictive tool.
The transport parameters that need to be averaged (or scaled up) depend on flow and process mechanism. For single-phase flow, as in gas reservoirs or unit-mobility ratio tracer tests, porosity, permeability and hydrodynamic dispersivity are the major parameters. For two-phase immiscible water injections, relative permeabilities, capillary pressure and saturation need to be scaled up in addition to permeability and porosity. For multiphase flows, three-phase relative permeabilities, phase behavior, and fingering factors are added to the list.
This paper critically evaluates the technical advances made so far in the upscaling of permeability heterogeneity under a variety of heterogeneity conditions. Both analytical and numerical methods are reviewed, and their applicability and limitations are identified. A new algorithm for the computation of interblock transmissibility is presented to improve accuracy of the commonly used pressure solver method for upscaling of permeability.