Influence of Fracture Toughness on the Geometry of Hydraulic Fractures

Abstract

Summary This paper presents an assessment of the importance of fracture toughness in hydraulic fracturing. The results show that fracture toughness can significantly affect the geometry of hydraulic fractures. Introduction Most researchers considered fracture toughness to have a negligible effect in hydraulic fracturing, except for small fractures, during injection of very-low-viscosity fluids or during fracture closure. Calculations of the energy losses associated with fluid flow in two-dimensional (2D) fractures indicated that the energy used per unit area in creating new fracture surface energy, was one to two orders of magnitude smaller than the energy used in overcoming viscous forces. Because, is directly related to fracture toughness, this calculation supports the conclusion that fracture toughness has a negligible effect on fracture propagation. But because the values for fracture toughness used in earlier calculations may not be representative of in-situ values and because the development of fully three-dimensional (3D) models makes a more complete study of the problem possible. The importance of fracture toughness in hydraulic fracturing needs to be reconsidered. In addition, Abe et al. and more recently Shoji et al. published results that indicate that fracture size can be strongly influenced by fracture toughness. Definition of Fracture Toughness Preexisting defects assumed to exist in rock are sites where high stress concentrations occur, leading to 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. Stress-Intensity Factor. The definition of fracture toughness follows from the concept of the stress-intensity factor developed in linear elastic fracture mechanics. Irwins demonstrated that, for a linear elastic homogeneous material, the stresses in the vicinity of a Griffith crack tip vary as, where r is a distance measured from the crack tip. For a fracture in opening mode (or Mode 1, see Fig. 1), the stresses are given by (1) where bounded function of 0 only: r, O=polar coordinates with respect to the crack tip (Fig. 2); and KI = stress-intensity factor in Mode 1. Similar equations exist to define KII and KIII, the stress-intensity factors in Modes 11 (in-plane shear) and III (antiplan shear), respectively. In hydraulic fracturing, a pure Mode I behavior is usually assumed. If the fracture turns away from a principal stress direction because of disturbances in the rock fabric or in the stress field, the fracture propagates in a mixed mode. The stress intensity factors, K, are a measure of the magnitude of the stress singularity at the crack tip and are a function of the loading parameters (such as the pressure in the fracture and the insitu stresses) and the body geometry (including the crack). Fracture Criterion. A well-established fracture criterion exists when the fracture is in Mode I: growth occurs when KI reaches KIc, the critical stress-intensity factor, or fracture toughness of the material. Fracture toughness is a material property that must be evaluated experimentally. For sedimentary rocks, KIc values at ambient conditions range from very low (less than 90 psi [0.1 MPa m] to moderately high (2,700 psi [3 MPa]), with no clear dependence on rock type. A sandstone may have either a higher or lower fracture toughness than an adjacent shale. When the stress-intensity factor exceeds KIc, the fracture becomes unstable and propagates with a high velocity. Dynamic fracture mechanics then applies. In classic hydraulic fracturing, the fracture is assumed to be in a quasistatic situation owing to both the nature of the loading (pressurized fracture with a nearly incompressible fluid) and the coupling of the pressure to the fracture growth rate. Quasistatic propagation occurs when the stress-intensity factor is equal at all times to the fracture toughness. Griffith Criterion. The physical justification for using the stress-intensity factor as a failure criterion has its origin in Griffits work. Historically. Griffith established the foundation of fracture mechanics; he studied propagation by considering the energy used in various parts of the fracturing process. He assumed that the energy, required to create new elementary fracture surfaces., is proportional to the area created: (2) where is the fracture surface energy of the solid, which is the energy per unit area required to create fresh fracture surface (similar to the surface tension of a fluid). Griffith proposed that, for a crack to propagate, the energy consumed by the creation of new surface should be balanced by the change in the potential energy of deformation : (3) During hydraulic fracturing (which is assumed to be a quasistatic process). the kinetic energy and changes in it are negligibly small. If the Strain energy release rate, G, is equal to (4) then, according to the Griffith criterion, propagation will occur if (5) In a linear elastic material (if the fracture propagates in its plane), the strain energy release rate can be related to the stress-intensity factors: e.g., for Mode I deformation in plane strain (6) Eq. 6 shows that the two criteria described above are identical in linear elasticity. The advantage of Irwin's approach, relating Griffith's global energy criterion to a crack-tip parameter. comes from the greater ease with which the stress-intensity factor can be calculated, which is of crucial importance in designing computationally efficient 3D numerical hydraulic fracturing models. Parameters Affecting Fracture Toughness In the definition of KIc it is assumed that the rock is homogeneous and linearly elastic. These conditions are unlikely to hold true in practice; rocks are not perfectly linearly elitstic materials. Because of the nonlinearity associated with fracture growth in rock, an apparent KIc is introduced, the value of which depends on the test geometry and boundary, conditions. SPEPE P. 435^