Anisotropic Permeability Estimation by Horizontal Well Tests
Authors: Zhang L. , Dusseault M.B. 
Published: December 2007

Abstract
A horizontal well drilled along the principal permeability kx direction is often assumed in horizontal well test interpretation. However, the orientation of a well affects well test interpretation significantly, particularly in horizontally anisotropic reservoirs. A new form of horizontal well solution which incorporates the well orientation effects is presented in this paper. The permeability anisotropy usually forms an ellipse in the horizontal plane and an ellipsoid in the 3D space, with the principal permeabilities as its three axis. By using the new solution, the true permeability anisotropy can be obtained for an arbitrarily oriented horizontal well. Two methods are demonstrated for determination of the true reservoir anisotropy (permeability ellipse), a graphical method and a numerical method. Two tests from two horizontal wells of different azimuths are needed in determining the reservoir anisotropy by these methods.

Introduction
Interpretation of a horizontal well test is more difficult than for a vertical well test because of the 3D geometrical nature and its evolution into different flow regimes. Horizontal well test results will not only depend on the reservoir characteristics but also the well geometry and orientation. Even though a number of well defined horizontal well pressure analyses have been reported in the literature, some of the issues are still not clearly resolved. The effect of well orientation in the horizontal plane is one such example. Most of the existing horizontal well test solutions assume that the well was drilled along the kx direction, therefore leading to the conclusion that by using the vertical radial flow regime the combined permeability kykz can be determined, and by using the linear flow regime the minimum principal permeability ky can be determined. This conclusion, however, is only reasonable in an horizontally isotropic reservoir or if the well is truly drilled along the kx direction; otherwise, errors are introduced, and the magnitude of these errors increases with increasing anisotropy.
In this paper a new solution form which incorporates the true anisotropy effect and the interpretation methods using the new solution. By using the new methods, accelerate well test simulations can be obtained for all possible horizontal well orientations. Also, if two well tests are carried out in two differently orientated wells from the same reservoir, then the permeability anisotropy can be determined uniquely from these tests.

Mathematical Description
The horizontal well problem discussed here is similar to that discussed by others except that an arbitrary horizontal well is drilled at a degrees from the kx direction. Other assumptions are as follows:the reservoir is an infinite, horizontal reservoir with constant thickness h;the reservoir has no-flow upper and lower boundaries;the reservoir has anisotropic permeability kx, ky and kz which coincide with the three axes of the coordinate system and,a well of length L is located at a distance b from the lower boundary (Fig. 1).
The following Green's equation can be used to model this problem:
(1)
where ki (i=x, y, z) are the primary permeabilities in the three-dimensional space, ri are coordinates and r'i describe the position of the well, is viscosity, is porosity, cm is compressibility, p = pi - pwf is defined as the pressure drop between wellbore and far field, q (t) is flow rate per unit well length and t is time.
The general solution for the problem can be obtained in the form of dimensionless variables Pd and td, which is (equation A13 of Appendix):
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