On The Computation Of The Three-Dimensional Geometry Of Hydraulic Fractures
Authors: Clifton R.J. , Abou-Sayed A.S. 
Published: December 2007

Abstract
A computational method is outlined for modelling the three-dimensional development of hydraulic fractures due to the injection of a non-Newtonian fluid at the well bore. The rock formation is modelled as an infinite, homogeneous, isotropic, elastic solid with in situ stresses that vary with depth. The three dimensional problem is made two-dimensional by assuming that the velocity profile through the thickness of the crack opening is the same as for flow between parallel plates and by reducing the elasticity problem to an integral equation that relates pressure on the crack faces to crack openings. Crack openings for a given crack geometry and pressure distribution are obtained by using properties of two-dimensional Chebyshev polynomials to properties of two-dimensional Chebyshev polynomials to facilitate inversion of the integral equation. Two-dimensional fluid flow between the crack faces is analyzed using a finite element method.

Introduction
Current computational procedures for predicting vertical hydraulic fractures are based on an assumed height of the fracture and on fluid flow in the horizontal direction only. These assumptions, while necessary and useful in many cases, are clearly not fully satisfactory since the height of the fracture is an important quantity that one would like to predict from the computations. Furthermore, knowledge of the two-dimensional flow should be helpful in predicting proppant transport. In addition, knowledge of the proppant transport. In addition, knowledge of the pressure-time history predicted at the well bore in a well pressure-time history predicted at the well bore in a well formulated simulation of hydraulic fracturing should allow field pressure-time records to be interpreted with greater insight and confidence.
Experience with computations of one-dimensional hydraulic fractures suggests that fluid flow and elastic stiffness characteristics are of primary importance whereas fracture mechanics considerations affect only the length of a small cracked region between the fluid front and the crack tip. This situation results because the weight function that relates pressure on the crack face to the stress intensity factor at the crack tip has a square root singularity at the crack tip. Consequently, the in situ compressive stress tending to close the crack can, when not nullified by fluid pressure on the crack faces near the crack tip, offset major changes in loading at positions distant from the crack front. These observations suggest that in two-dimensional hydraulic fracturing the height-length ratio of the crack may be determined primarily by considerations of fluid flow and elastic stiffness. Thus, the analysis outlined herein gives primary attention to these aspects of hydraulic fracturing.

ELASTICITY OF THE FORMATION
Consider the rock formation to be an infinite, isotropic elastic body under an initial stress fieldo ij. Let this body be subjected to an additional stress field 1 ij corresponding to reducing the initial stress o zz (x,y,O) on a region B bounded by(x,y) = 0 to a pressure p(x,y). (See Figure 1.) Then, the stress field 1 ij is obtained as the solution to the problem of an infinite medium with pressurep {p(x,y) - ozz (x,y,0)} acting on the planar region B. The other boundary conditions on B are assumed to be 1 xz = 1 yz = 0, corresponding to being a surface on which o xz = o yz = 0.The pressure p(x,y) can be related to the crack opening w(x,y) by making use of the fundamental solution for the stress field due to an infinitesimal segment of a dislocation line. For this, consider a dislocation segment with Burger's vector b = bh k and~ z ~ length dx' at position (x',y') as shown in-Figure 2. The normal stress on the plane z = 0 due to the dislocation segment is (Hirth and Lothe, p. 125)
(1)
where
(2)
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