In-Place Determination of Permeability Distribution for Heterogeneous Porous Media Through Analysis of Energy Dissipation
Authors: Nelson R. William 
Published: December 2007

Nelson, R. William,* Member AIME, Battelle Memorial Institute, Richland, Wash.

Abstract
The theoretical basis is presented for energy dissipation methods of measuring permeability in saturated heterogeneous media. Analysis starts with the equation for single-phase flow in heterogeneous porous media which is a first-order partial differential equation in the unknown permeability. This equation is reduced to a system of characteristic equations useful in deriving a differential expression for the permeability as a function of the known potential distribution. For steady flow systems the expression is integrable, giving the permeability integral for direct calculation of the distribution along successive streamlines. The boundary condition in permeability is presented and includes a discussion of the special requirement to assure uniqueness. Theoretical results and necessary computations are incorporated into two general computer programs capable of determining the permeability distribution in two- and three-dimensional steady flow systems. Results are then presented for an energy dissipation analysis using the methods and software to determine a field permeability thickness distribution.

Introduction
Theoretical work leading to the description of fluid flow in heterogeneous porous media has developed slowly and somewhat sporadically, although the areas of engineering applications predominantly involve flow in nonhomogeneous media. The lack of accurate and economical methods for measuring field permeability distributions has been a major limitation to realistic description of natural flow systems. The economical measurement of permeability using energy dissipation methods may overcome this limitation. Among other things, any permeability measurement method involves consideration of energy dissipation; as used here, "energy dissipation method" denotes the following general approach. The areal energy distribution is measured on an existing flow system in porous media. Then through appropriate analysis of the dissipation of energy in the flow system, the permeability distribution is determined throughout the region using the minimum boundary conditions required to assure a mathematically unique result. This paper presents the theoretical basis for such a method, as well as the techniques and computer software needed to carry out the analysis. The method and techniques are then used in a preliminary field application.

EQUATIONS DESCRIBING FLOW IN HETEROGENEOUS MEDIA
Heterogeneous as used here implies differing in kind, having unlike quantities or possessing different characteristics. Accordingly, a heterogeneous porous medium has unlike quantities or differing characteristics at different locations. More precisely, if a porous medium is heterogeneous with respect to some property, then that property is functionally dependent on the spatial location. (Media properties are considered here as macroscopic.) Specifically considering permeability k,** then (1)
or the permeability is a scalar function G of the location (x, y, z). Spatial variation of several other properties, of which porosity, dispersivity and medium compressibility are but a few, is implied by the broad title of heterogeneous porous media. The term will be used hereafter to imply only heterogeneity with respect to permeability. The porous medium is considered isotropic as implied by G being only a scalar function.
SPEJ
P. 33ˆ