Horizontal Stresses in Anisotropic Formation and Prediction of Hydraulic Fracture Direction
Authors: Garg A. , Desai A.M. , Towler B.F. 
Published: December 2007

Abstract
In this paper we present;a methodology for estimation of horizontal stresses in an anisotropic formation by considering values of Poisson's ratio to be different along the three principal axis;a criteria for artificial fractures to grow with respect to natural fractures based on a relationship between stress anisotropy and strength anisotropy of a medium.

Introduction
Proper profiling of horizontal stresses in an anisotropic formation is very important for the design and analysis of a hydraulic fracture treatment. It is a well established fact that in-situ stress contrast between various rock layers have a marked effect on the fracture propagation and, thus, directly affects fracture length and height.
Hydraulic fracturing is the creation and preservation of a fracture in a reservoir rock. A hydraulic fracture is created and extended by pumping a viscous fluid at a high enough rate to overcome the maximum rate of fluid loss into the medium which is to be fractured. The fluid injection results in a buildup of pressure in the well bore which is large enough to overcome the stresses in the surrounding rock mass and initiate a fracture. The in-situ stress is equivalent to the minimum confining stress and the closure pressure is defined as the fluid pressure at which an artificial fracture closes. To initiate the opening of an existing artificial fracture, the pressure must be greater than or equal to sum of the minimum stress and the tensile strength of a rock.
At present, the only reliable methods of measuring the in-situ stress state at depth is the hydraulic fracturing technique. Currently there are also log analysis techniques available which can provide calculated mechanical properties and in-situ stresses. The log analysis techniques are based on the classic theories of elastic stress-strain relationship. The relationship between vertical stress and in-situ stress often assumes that the two principal horizontal stresses in a rock formation have equal magnitude and the same value of Poisson's ratio along the three principal directions. Such relationship is reliable to some extent in homogeneous, isotropic, and tectonically relaxed formations only. However most geological materials exhibit anisotropy. Using such log analysis techniques to determine in-situ stresses from acoustic logs often produces erroneous results. For determining in-situ stresses m an anisotropic formation, the log derived values must either be calibrated with field measured stresses, or the elastic stress-strain relationship be appropriately modified to account for anisotropy in mechanical properties (e.g. Poisson's ratio) of the rock formation being studied. In this study, we have derived a set of equations to calculate in-situ stresses in an anisotropic formation, by taking into consideration the Poisson's ratio anisotropy.
In most hydraulic fracture operations, artificial fracture generally run parallel to the natural fractures and consequently often fail to completely drain the main natural fractures. We introduce a concept, based on the relationship between strength anisotropy and stress anisotropy of a rock formation, which establishes a criterion for artificial fractures to grow with respect to natural fractures. By considering cases of stress anisotropy versus strength anisotropy a guideline can be established which can provide conditions under which an artificial fracture will propagate either parallel to a natural fracture or perpendicular to it.

Calculated Stresses
There have been many attempts to correlate the minimum principal in-situ stress with rock properties, particularly Poisson's ratio. The common practice in log analysis is to determine Poisson's ratio as a function of depth with a sonic log. Equation 1 is more commonly used for a porous solid;
(1)
Equation 1 is applicable only if the following conditions apply;
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