Stress-Intensity Factor and Fracture Cross-Sectional Shape Predictions From a Three-Dimensional Model for Hydraulically Induced Fractures
Authors: Morita Nobuo , Whitfill Donald L. , Wahl Harry A. 
Published: December 2007

Summary
An efficient three-dimensional (3D) fracture model was developed that can be applied to arbitrary planar cracks with Mode 1 stress singularity at the fracture tip. Surface stress, thermally induced stress, and poroelasticity were included. In addition, the elasticity constants can be varied between elements. A parameter study was conducted with the model to evaluate the stress-intensity factor, the evolving shape of the fracture, and fracture width. The study includes the borehole effect, fracture migration, fracture barriers, fracture cross-sectional shape, fracture front shape and its stress-intensity factor, and evaluation of the validity of some simplified models. Further work on the details of the theory, algorithm, accuracy, and efficiency of the 3D fracture model is in progress.

Introduction
To improve the accuracy of the Christianovich-Geertsma-deKlerk1 (CGD) and Perkins-Kern-Nordgren2 (PKN) -type models, the pseudo-3D hydraulic3 (P3DH) concept and the simplified boundary-element technique4,5 (no interface elements between layers) have been introduced. The main problem with the P3DH concept is that the accuracy is significantly degraded for uncontained fractures, although it gives acceptable accuracy for well-contained fractures. The main problem with the simplified-boundary-element technique is that it cannot rigorously handle regular fracturing problems where the formation consists of at least three layers with a different modulus. It can, however, handle one- or two-layer problems without interface elements. Another problem of the boundary-element technique is that it requires high order and fine meshes6–9 to achieve acceptable accuracy, but it is very inefficient if the number of nodes is increased. Because of these problems with the P3DH and boundary-element models, rigorous analyses are still needed to study the effects of mechanical factors on fracture geometry.
Fracturing fluids usually contain materials to control fluid loss. These fine materials form bridges and seal gaps that are several times larger than pore size. This sealing mechanism not only controls fluid loss but also seals the narrow fracture-tip zone to prevent fracture propagation. It creates a large geometrical fracture toughness (or may be called pseudofracture toughness) that is significantly larger than the actual fracture toughness of rock. Such a mechanical factor significantly affects fracture-propagation pressure and width.
Several codes6–11 have been developed to study such effects with the finite- and boundary-element techniques. These codes potentially give a reasonably accurate stress-intensity factor and fracture width with fine meshes if boundary conditions are specified for arbitrarily shaped fractures. The computation time is several hours for each run, however, even with the largest computer. The codes are not suited for a parameter study of mechanical factors affecting the stress-intensity factor and geometry, which requires hundreds of calculations by changing the fracture shape evolving during fracture propagation.
A very efficient 3D finite-element code was developed in this work that increases the computation speed hundreds of times by using the singular-force technique, equivalent-nodal-force technique, and one-plane-block frontal successive overrelaxation (SOR) technique especially designed to model elongated fractures. With the model, stress-intensity factors and fracture cross-sectional shapes were calculated for various 3D fractures observed during hydraulic-fracturing operations. The evolution of fracture geometry and fracture pressure were also calculated by incrementally evaluating the stress-intensity factor and the cross-sectional shape.
To simulate a 3D fracture geometry, both mechanical factors and flow character should be taken into account. The flow-character effect has been sufficiently studied1–3,5,12 because the flow problem can be rigorously handled with reasonable efficiency. The mechanical factors have been oversimplified, however, because of the extended calculations if a strict mathematical treatment is imposed. The present work was intended to update the effect of mechanical factors on the fracture geometry and the stress-intensity factor. Further studies are necessary for those items dominantly affected by fluid flow, however, because the present work neglects the fluid interaction.

Fracture Initiation From a Borehole
Fracture initiation is affected by the wellbore conditions. The factors affecting fracture initiation are the cement bond, the directional in-situ stress, and wellbore diameter and perforation depth.
Three cement-bond conditions can be assumed when a fracture is initiated:borehole with open gaps and channels (borehole pressure acts directly on borehole surface);borehole with sealed gaps filled with solid-like dehydrated mudcake (the hydrostatic mud weight during casing setting acts on borehole surface; hence, the borehole-surface load is close to the in-situ stress at the depth); andborehole perfectly bonded to rigid cement casing.
A fracture initiated at borehole surface or from a perforation hole grows into a semielliptical shape if the borehole surface has a gap without being fixed to a rigid casing. A fracture initiated at the borehole surface or from a perforation, however, cannot propagate along the borehole surface if it is perfectly bonded to a rigid casing and results in a cone-shaped fracture.
Fig. 1 shows the normalized stress-intensity factors for a small fracture initiated at the borehole surface, where ?pf=pw for a borehole with open channels and ?pw=0 for a borehole with sealed channels. Figs. 2 through 4 show the normalized stress-intensity factors for intermediate fractures initiated at a borehole or a perforation with the three types of cement-bond conditions. These three figures show that the cement bond affects the magnitude and distribution of stress-intensity factors; hence, it affects evolving fracture shape and fracture-propagation pressure.