The Influence of Fracture Toughness on the Geometry of Hydraulic Fractures

Abstract This paper contains an assessment of the importance of fracture toughness in hydraulic fracturing. The results given in this paper show that fracture toughness can significantly affect both the height growth and the propagation of hydraulic fractures. Most researchers have considered the effect of fracture toughness on the geometry of the fracture to be negligible except in the cases of small fractures, during injection of very low viscosity fluids, or during fracture closure1. In addition, energy considerations for twodimensional fractures indicated that the energy losses associated with fluid flow were one to two orders of magnitude higher than the energy used in creating new fracture surfaces,γF.2,3 Because γF is directly related to fracture toughness, the conclusion that fracture toughness has a negligible effect on fracture propagation was reaffirmed. However, neglecting the effects of fracture toughness in hydraulic fracturing needs to be reconsidered, because the magnitudes used in earlier calculations may not be representative of in-situ values, and because the development of fully threedimensional models makes possible a more complete study of the problem. The definition of fracture toughness is obtained from the concept of the stress intensity factor, developed in linear elastic fracture mechanics. Preexisting defect are assumed to exist and to induce high stress concentrations in their vicinity, becoming sites for crack initiation and propagation. A single such defect may be represented by a sharp line crack (known as a Griffith crack) in a linear elastic medium. Irwin4 demonstrated that, for a linear elastic homogeneous material, the stresses in the vicinity of a Griffith crack tip vary as 1/r, where r is measured from the crack tip. For a fracture in opening mode (or mode I, see Figure 1), the stresses are given by: