Estimation of Reservoir Anisotropy From Production Data
Authors: Arnold Marion D. , Gonzalez H.J. , Crawford Paul B. 
Published: December 2007

Abstract
A method is presented for estimating the effective directional permeability ratio and the direction of maximum and minimum permeabilities in anisotropic oil reservoirs. The method is based on the principle that production from a well in an anisotropic reservoir results in elliptical isopotentials about the well, rather than circular. Bottom-hole pressure data from three observation wells surrounding a producing well are required to apply the method. The method involves fitting field pressure data to a set of general charts of isopotentials and making a few simple calculations until a solution is found. The method is based on a steady-state equation for homogeneous fluid flow. In addition to the method, a brief discussion of the theory underlying it is presented.

Introduction
The existence of a different permeability in one direction than another in oil reservoirs has been mentioned in several papers. Hutchinson reported laboratory tests on 10 limestone cores and pointed out that one-half of them showed significant, preferential, directional permeability ratios, the average being about 16:1. Johnson and Hughes-reported a permeability trend in the Bradford field in the northeast-southwest direction with flow being 25 to 30 per cent greater in that direction. Barfield, Jordan and Moore reported an effective permeability ratio of 144:1 in the Spraberry. Crawford and Landrum showed that sweep efficiencies could often vary by a factor of two to four, and sometimes considerably more, due to variations in flooding direction and patterns in anisotropic media. These findings indicate that the possibility of anisotropy may be worthy of consideration in the development of an oil field. In considering this, it should first be determined if anisotropy exists. If it does, the direction of the maximum and minimum permeabilities and the ratio of their magnitudes are quantities which can be of value in planning the most efficient well-spacing patterns. Past methods of determining these quantities have included analysis of oriented cores and analysis of flooding performance of pilot injection patterns. In recent work, Elkins and Skov presented an analysis of the pressure behavior in the Spraberry which accounted for anisotropic permeability. This work was based on the transient pressure distribution in a porous and permeable medium, with the solution expressed as an exponential integral function involving rock and fluid properties. The purpose of this study is to provide a method, based on steady-state equations, of estimating the direction and relative magnitude of permeabilities in an oil reservoir from field pressure data and well locations only. The method presented is based on work by Muskat which shows that Laplace's equation represents the steady-state pressure distribution for homogeneous fluid flow in homogeneous, anisotropic media if the coordinates of the system are shrunk or expanded by replacing x with x/sqaure root of k(x), y with y/sqaure root of k(y) and z with a/sqaure root of k(z). In many cases it is desirable that data be obtained early in the history of a field because knowledge of an anisotropic condition would allow new wells to be spaced in such a manner that reservoir development and subsequent secondary recovery programs could be planned more efficiently.

THEORETICAL CONSIDERATIONS
A brief discussion of the theoretical basis on which the graphical solution was developed is presented in this section. Muskat's two-dimensional solution for the pressure distribution in an homogeneous, anisotropic medium with an homogeneous fluid flowing can be algebraically manipulated to show that the isobaric lines are perfect ellipses. The ratio of the major axis to the minor axis, a/b, is related to the permeability ratio, k(y)/k(x), as follows.
a/b = .........................(1)
It can also be shown that the pressure varies linearly with logarithm of the radial distance from the producing well. However, the gradient along any ray is a function of the orientation of that ray, and a variable is present when anisotropy exists which cancels out for a radial (isotropic) system. For a system such as that described, a dimensionless pressure-drop ratio was developed which is completely independent of the actual magnitude of the pressures. This was done by arranging Muskat's solution in such a way that all variables cancelled out except k(y)/k(x), and well positions. However, this solution depends on having a co-ordinate system with axes coinciding with the major and minor axes of the elliptical isobars. Thus, it was necessary to introduce a co-ordinate system rotation factor. The two unknown variables are then k(y)/k(x) and, and the two measured dimensionless pressure-drop ratios are related to the unknown variables as follows.
JPT
P. 909^