Estimating the Pressure-Transient Response for a Horizontal or a Hydraulically Fractured Well at an Arbitrary Orientation in an Anisotropic Reservoir

Summary

This article presents a method for finding the equivalent isotropic system for a horizontal well or a hydraulically fractured well at an arbitrary azimuth in an anisotropic reservoir. The equivalent isotropic system can then be used to calculate the pressure response of the original system.

The horizontal well solution is compared with previous solutions reported in the literature. The general hydraulically fractured well case has not, to our knowledge, been presented before.

This study confirms the observation by Yildiz and Ozkan¹ that small deviations (up to 20°) from the optimal azimuth for a horizontal well do not have a major effect on the pressure response. Because this solution is obtained by finding an equivalent isotropic system, this study also confirms the observation by Earlougher² that an anisotropic system cannot be identified from a single-well test.

Introduction

Permeability anisotropy may occur for a variety of reasons. One of the most important causes of permeability anisotropy is the presence of natural fractures. These reservoirs are often candidates for horizontal drilling. The optimal wellbore orientation is perpendicular to the direction of maximum permeability. Often, the principal axes of permeability are not well characterized during the initial stages of field development. Thus, the wellbore orientation may lie at any angle to the principal axes of permeability.

A similar situation is that of a well with a hydraulic fracture in a naturally fractured reservoir. Although the fracture created will often be parallel to the primary set of natural fractures, this is not necessarily the case. Tectonic stresses control the direction of growth of both natural fracture systems and hydraulic fractures. If the tectonic stresses have changed since the formation of the natural fracture system, the hydraulic fracture may have a different orientation from the natural fractures. This may even occur because of production. Cases have been documented where repeat fracture treatments had a different orientation from the original fractures because of the change in the tectonic stresses caused by reservoir depletion.³

The case of a horizontal well in an anisotropic reservoir with the wellbore parallel to one of the principal axes of permeability has been studied by a number of authors. The case of a horizontal well having the wellbore at an arbitrary orientation with respect to the principal axes of permeability has previously been studied by Besson,⁴ by Zhang and Dusseault,⁵ and by Yildiz and Ozkan.¹ Besson⁴ studied the case of a horizontal or slanted wellbore in a reservoir with horizontal to vertical anisotropy. Although he presented the transformations for the general case of k_x?k_y?k_z, he did not consider the case of areal anisotropy further. Zhang and Dusseault⁵ presented a solution based on transforming the anisotropic system to an equivalent isotropic system. They also proposed graphical and numerical methods for determining the permeability anisotropy from analysis of tests in two horizontal wells having different azimuths. Yildiz and Ozkan¹ likewise presented a solution by defining dimensionless variables that perform an implicit coordinate transformation.

To the best of our knowledge, the fractured well case has been studied only for the situation where the hydraulic fracture is parallel to one of the principal axes of permeability.⁶

We present new analytical solutions for these two situations. The new solutions are obtained by transforming the original problem with anisotropic permeability into an equivalent problem with isotropic permeability. Thus, the pressure transient response in an anisotropic system cannot be distinguished from that of an isotropic system from the shape of the pressure response alone. The major contribution of this article is to define the properties of the equivalent isotropic system in terms of the properties of the original anisotropic system. Any existing solution for an isotropic system can then be used to evaluate the pressure response.

For the hydraulically fractured well case, we assume a finite-conductivity fracture. The infinite-conductivity fracture case may be obtained in the limit as the fracture conductivity increases.

In the next section we discuss a solution for the horizontal well case, while in the following section we discuss the hydraulically fractured well case. The derivations of the equivalent systems for these two cases are given in Appendices B and C, respectively.