A Practical Approach to Scaling-Up Permeability and Relative Permeabilities in Heterogeneous Permeable Media
Authors: Malik M.A. , Lake L.W. 
Published: December 2007

Abstract
With modern reservoir characterization techniques, fine-scale permeability and porosity distributions can run into millions of gridblocks in a typical reservoir simulation. However, simulations on so many blocks remain prohibitively expensive, and scale-up of fine-scale permeability and porosity distributions is required to bring the number of blocks to the order of a few thousand.
In this work, we have applied four methods for independent scale-up of permeability from fine to a practical field simulation scale. These methods include independent fine-scale simulation (IFS), the electrical network method (ENM), geometric averaging (GA) and the Cardwell-Parsons (CP) method. Permeability is also scaled-up by a dependent fine-scale simulation (DFS) scheme that acknowledges flow through neighboring coarse blocks.
Results from independent scale-up of conditionally simulated, realistic permeability cross sections show that IFS is an overall preferable scale-up method because of good accuracy and flexibility in application. Investigation of cross flow effects in DFS shows that, especially in low-permeability blocks, IFS and DFS estimates of scaled-up permeability can be significantly different.
In a scaled-up field, treatment of relative permeabilities is an equally important issue. Laboratory measured relative permeability curves, commonly referred to as rock curves, are assumed to apply to the fine-scale simulations. The rock curves may or may not be applicable in coarse-scale simulations.
In this research, we have studied the scaled-up of relative permeabilities by steady-state as well as dynamic fluid-flow simulations. Results show that steady-state scaled-up curves are almost the same as the rock curves, while dynamic scaled-up curves are different. Comparison of results from fluid-flow simulations from fine and corresponding coarse cross sections shows that rock curves perform better than dynamic scaled-up curves in coarse simulations, provided the dimensionality of the flow field is preserved and adequate heterogeneity is retained in the field. We have also investigated the effects of numerical dispersion in the scale-up process.

Introduction
Reservoirs and aquifers are inherently heterogeneous with variation of properties, like permeability and porosity, from the pore to the field scale. Permeability characterization in subsurface reservoirs is specially challenging because it cannot be mapped with any of the existing logging or geophysical surveying techniques. In a typical field, reliable permeability data, measured from cores in the laboratory, are generally limited to a few wells. Geostatistical techniques, developed mainly in the past decade, are now commonly used to generate fine-scale, interwell permeability distributions. For dependable reservoir performance prediction, fluid-flow simulations on a fine-scale grid are the best choice, but typical reservoirs can run into millions of gridblocks on this scale. Despite advances in computation, reservoir simulations on such a large number of blocks remain prohibitively expensive and scale-up of fine-scale permeability is required to bring the number of blocks to the order of a few thousand to economically perform fluid-flow simulations.
A closely related issue in scale-up process is the treatment of relative permeabilities. Usually, only a handful of relative, permeability curves, commonly referred to as rock curves, are determined for a field by laboratory measurement, and these are assumed to apply to the fine-scale reservoir simulation. In a scaled-up field, the rock curves may or may not be applicable.
The first half of this paper is devoted to scale-up of permeability. Therefore, further discussion on relative permeabilities is postponed till the later half of this article.

Permeability Scale-Up
Scale-up of permeability has been addressed in several studies. For layered arrangements, the effective permeability is the arithmetic average for parallel layers and the harmonic average for series layers. P. 485^