A computational method is outlined for modeling the three-dimensional development of hydraulic fractures due to the injection of a non-Newtonian fluid at the well bore. The rock formation is modeled as an infinite, homogeneous, isotropic, elastic solid with in situ stresses that vary with depth. This three- dimensional problem is made two-dimensional by reducing the elasticity problem to an integral equation that relates pressure on the crack faces to crack openings and by pressure on the crack faces to crack openings and by neglecting the component of the fluid velocity in the direction perpendicular to the fracture plane. Variational principles are used to derive the discrete form of the governing equations that are used for the computations. Preliminary computations indicate that the essential features of the approach are suitable for use in a numerical method for predicting the three-dimensional geometry of an advancing crack.
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 propant transport. In addition, knowledge of the propant 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 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, elastic stiffness and in situ stresses. Thus, the analysis outlined herein gives primary attention to these aspects of hydraulic fracturing.
In order to obtain a numerical method that has the potential for obtaining sufficient accuracy with the potential for obtaining sufficient accuracy with the least computations the problem is formulated in a way that keeps the number of unknowns at each cycle of the computations relatively low while maintaining the possibility of accurate modeling of the region near the possibility of accurate modeling of the region near the crack tip where steep gradients of fluid pressure and crack opening occur. To this end the equations of elasticity governing the opening of the crack due to the fluid pressure acting on its faces are formulated as an integral equation over the surface of the crack The crack opening near the crack-tip is modeled accurately by introducing a 'near crack-tip zone' in which the crack opening w(x, y) increases with the square root of distance from the crack-tip as predicted by linear elastic fracture mechanics. An intermediate zone is introduced between the near crack-tip zone an the fracturing fluid front since the crack opening is not expected to be proportional to the square root of distance from the crack tip over the entire distance from the crack front to the front of the fracturing fluid. Steep gradients in pressure within the fracturing fluid and near the front are accommodated by using a mesh with elements that are narrow in the direction perpendicular to the front.
The equations governing the opening of the crack due to the fluid pressure and those governing the flow of the fluid in the fracture are expressed as variational principles. In the modeling of the fluid flow the system of equations to be solved is obtained by a relatively straightforward application of the method of finite elements. The equations used for modeling the elasticity of the formation are obtained by a new approach that is similar to finite elements, but is applicable for cases in which the physical problem is formulated in terms of integral equations instead of differential equations.