Production Engineering and Research - Average Permeabilities of Heterogeneous Oil Sands (T. P. 1852, Petr. Tech., March 1945)

This paper discusses the practical problem of estimating a single equivalent permeability for an oil reservoir, or a portion thereof, whose actual permeability varies in an irregular manner. Limiting averages for general types of permeability variation are developed, and illustrated by examples involving important, specific types of variation. Introduction The theory of the flow of fluids through porous media1"4 is becoming increasingly important in predictions of oil-reservoir behavior. In practical applications, however, reservoirs are seldom found to which simple theory strictly applies. Actual reservoirs have complicated shapes and nonuniform permeabilities and porosities. This paper discusses the problem of estimating a single equivalent permeability for an oil reservoir, or a segment of an oil reservoir, whose actual permeability varies in an irregular manner. The equivalent permeability of a reservoir segment is defined as the permeability of a homogeneous segment of the same dimensions that would pass the same flux under the same pressure drop- It is the permeability value that can be used in simple theoretical formulas to calculate the reservoir behavior. To the authors' knowledge, previous theoretical calculationst on systems of nonuniform permeability have been carried out only by Muskat'"; and he has not dealt with irregular variations, except in writing down the general differential equation for the pressure in variably permeable systems.lb In practical work, many calculations necessarily have been made to estimate equivalent permeabilities from the permeabiiity profiles obtained by core analyses. Regarding such calculations, Johnston and Sherborne6 say: Simple arithmetic and weighted averages have been tried on many wells and it has been found that, where frequent sampling has occurred, the arithmetic average is as satis- factory as a weighted average. It is possible that the application of statistical methods to the analysis of permeability data as recently presented by Law8 may prove fruitful. L,, has made a valuable contribution in showing how statistical analyses may aid in picturing the characteristics of a reservoir from those of a necessarily limited number of core samples. It is believed however that the particular problem of the estimation of equivalent permeabilities can be most directly approached from a fluid dynamical viewpoint as given in the present paper. An example is presented of the use of the conclusions herein in conjunction with Law's method. Theory In order to approach the problem of heterogeneity simply, it is interesting to consider a square block composed of four smaller, homogeneous, square blocks of porous medium, two of which have a