Statistical features of rock fracture such as tortuously of crack path, scale effect and scatter of critical fracture parameters are well documented. The apparent randomness of the process is closely associated with the distribution of inhomogeneities on various scales within a solid. The phenomenological description of inhomogeneities is presented by a random field of specific fracture energy following the framework of Statistical Fracture Mechanics (SFM). A brief review of SFM is presented and application to modeling of hydraulic fracture is discussed.
The analysis of hydraulic fracture is plagued by the question of the validity of conventional fracture parameters such as critical stress intensity factor K1c and energy release rate G1c of linear fracture mechanics or J1c and R-curve of non-linear analysis (Shlyapobersky 1985; Shlyapobersky and Chudnovsky 1992). These parameters can be directly linked to the micromechanisms of hydraulic fracture. Fracture surface is one of the sources of information regarding these micromechanisms. Fig. 1 displays a top view of the random (fractal) pattern of a crack formed in hydraulic fracture test in polyaxial cell. The pattern closely resembles ones generated in some of diffusion limited aggregation models (Bunde and Havlin 1991). The random nature of rock fracture and its dependence on inhomogeneities on various scales is illustrated in Fig. 2. It is reported that the fracture toughness parameters (K1c, G1c, etc.) are related to the roughness of fracture surface (Mandelbrot, Passoja and Paullay 1984; Saouma, Barton and Gamaleldin 1990, Issa, Hammad and Chudnovsky 1993). The first theory that proposed a way to relate the fracture toughness and roughness of crack path within the continuum mechanics framework was formulated in (Chudnovsky 1973) and later evolved into Statistical Fracture Mechanics (SFM) (Chudnovsky and Kunin 1987; Chudnovsky and Kunin 1992). SFM bridges conventional Fracture Mechanics with the Weakest Link theory. It addresses the problem of brittle fracture, when a crack propagation is controlled by a pre-existing field of defects and does not cause noticeable changes to this field. In this case the random location and orientation of the individual inhomogeneities result in an irregular, stochastic crack trajectory, scatter of the main fracture parameters and scale effect. SFM explicitly incorporates the fractographic information, e.g. fractal characterization of fracture surfaces, in the probabilistic description of brittle fracture. SFM is based on a few natural assumptions concerning brittle crack growth: 1. The crack path is random, i.e., the crack randomly selects a path of least resistance through the material. Since the material is considered to be a continuum, an innumerable set of all the paths that possess certain statistical features constitute a set Ù of virtual fracture trajectories.
For the purpose of these studies, approximate solution for the ERR along the irregular crack trajectory is utilized (see (Gorelik 1993) for details). Fig. 5 shows the results of study of the evolution of the statistical distribution of elastic ERR as a function of crack penetration depth (horizontal projection of crack path).