Frequency Distribution Of Permeability In Nonhomogeneous Reservoirs
Authors: Bennion D. W. 
Published: March 2010

Abstract
The object of this study was to investigate the frequency distribution of permeability for various types of reservoirs. Permeability was considered both as a scalar and tensor quantity which varied in the lateral and vertical directions. Monte Carlo methods were used to generate the samples. The following cases were studied:A circular reservoir with permeability decreasing toward the edge in a linear, quadratic, cubic or exponential manner,A circular reservoir with the permeability decreasing in the lateral and vertical directions in a linear, quadratic, cubic or exponential relationship,A circular reservoir with permeability decreasing in the lateral direction but composed of several layers eachwith a different mean permeability,Reservoirs with constant directional permeability,Reservoirs with variable directional permeability.
The shape of these distributions was compared to those from actual reservoirs and suggestions for sampling are discussed in this paper.

Introduction
Generally several wells in most reservoirs are cored and measurements on the cores are made. From these measurements inferences concerning the properties of the reservoir are then made. The object of this study was to investigate the statistical distribution of one of these parameters (permeability).
Since only a very small portion of the reservoir is sampled, statistical methods should be employed to make decisions concerning the properties of the reservoir. Several investigatorsl – 3 have suggested that permeability has a log normal distribution, others have found from using large numbers of samples that this may not be so.4,5 For this study samples were generated for hypothetical reservoirs using Monte Carlo techniques.
In these models permeability was treated both as a tensor and a scalar quantity. The tensor theory of permeability has been discussed by several investigators.6–8 The investigation employed a circular reservoir and various functional assumptions concerning the reservoir's permeability were made.

RESERVOIR MODELS
A circular reservoir was selected for this study and the following five cases studied. In the first case the reservoir was homogeneous in the vertical direction with the highest permeability at the center of the reservoir and decreasing towards the edge in the following four functional relationships:
Equation (1) (Available in full paper)
Equation (2) (Available in full paper)
Equation (3) (Available in full paper)
Equation (4) (Available in full paper)
In the second case the highest permeability was at the center and decreasing in both the lateral and vertical direction in the following manner:
Equation (5) (Available in full paper)
Equation (6) (Available in full paper)
Equation (7) (Available in full paper)
Equation (8) (Available in full paper)
Equation (9) (Available in full paper)
Equation (10) (Available in full paper)
Equation (11) (Available in full paper)
Equation (12) (Available in full paper)
Model number four was a reservoir with directional permeability which did not decrease laterally or vertically; two and three-dimensional models were used. The permeability in anisotropic porous media has the form of a symmetric tensor. Directional variations of permeability have been studied by several investigators;6–9 Scheidegger6 and Greenkorn8 have discussed two types of directional permeability measurements.