Initiation and Extension of Hydraulic Fractures in Rocks
Authors: Haimson Bezalel , Fairhurst Charles 
Published: December 2007

Abstract
A criterion is proposed for the initiation of vertical hydraulic fracturing taking into consideration the three stress fields around the wellbore. These fields arise from
- nonhydrostatic regional stresses in earth
- the difference between the fluid pressure in the wellbore and the formation fluid pressure and
- the radial fluid flow through porous rock from the wellbore into the formation due to this pressure difference.
The wellbore fluid pressure required to initiate a fracture (assuming elastic rock and a smooth wellbore wall) is a function o/ the porous elastic constants of the rock, the two unequal horizontal principal regional caresses, the tensile strength of the rock and the formation fluid pressure. A constant injection rate will extend the fracture to a point where equilibrium is reached and then, to keep the fracture open, the pressure required is a function of the porous elastic constants of the rock, the component of the regional stress normal to the plane of the fracture, the formation fluid pressure and the dimensions of the crack. The same expression may also be used to estimate the vertical fracture width, provided all other variables are known. The derived equations for the initiation and extension pressures in vertical fracturing may be employed to solve for the two horizontal, regional, principal stresses in the rock.

Introduction
Well stimulation by hydraulic fracturing is a common practice today in the petroleum industry. However, this stimulation process is not a guaranteed success; hence, the deep interest shown by the petroleum companies in better 'understanding the mechanism that brings about rock fracturing, fracture extension and productivity increase. Geologists and mining people became interested in hydraulic fracturing from a different point of view: the method may possibly be employed to determine the magnitude and direction of the principal stresses of great depth. Numerous articles in past years have dealt with the theory of hydraulic fracturing, but they all seem to underestimate the effect of stresses around the wellbore due to penetration of some of the injected fluid into the porous formation. Excellent papers on stresses in porous materials due to fluid flow have been published but no real attempt has been made to show the effect of these stresses in the form of a more complete criterion for vertical hydraulic fracturing initiation and extension. This paper is such an attempt.

ASSUMPTIONS
It is assumed that rock in the oil-bearing formation is elastic, porous, isotropic and homogeneous. The formation is under a nonhydrostatic state of regional stress with one of the principal regional stresses acting parallel to the vertical axis of the wellbore. This assumption is justified in areas where rock formations do not dip at steep angles and where the surface of the earth is relatively flat. This vertical principal regional stress equals the pressure of the overlying rock, i.e. S33= -pD where S33 is the total vertical principal stress (positive for tension), p is average density of the overlying material and D is the depth of the point where S 33 is calculated. The wellbore wall in the formation is considered to be smooth and circular in cross-section. The fluid flow through the porous elastic rock obeys Darcy's law. The whole medium is looked upon as an infinitely long cylinder with its axis along the axis of the wellbore. The radius of the cylinder is also very large. Over the range of depth at which the oil-bearing formation occurs, it will be assumed that any horizontal cross-section of the cylinder is subjected to the same stress distribution, and likewise that it will deform in the same manner.