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This paper presents an algorithm for definition of fracture sets based on a probabilistic, geological approach to fracturing. The algorithm asserts that fracture sets should be defined as groups of fractures with statistically homogeneous properties. There properties include orientation, but also emphasize geological information such as termination modes, displacements, striation, and mineralization. The sets are defined by assigning fractures to sets based on how similar all the characteristics of the fractures are to those of the other fractures in the set. The probability that a fracture is a member of a set is used to assign the fractures to sets. The properties of the sets are then calculated based on the statistics of the fractures assigned to the sets. Verification of the algorithm, and an example application are provided.
Rock mechanics analyses require definition of fracture sets based on geometric, geological, mechanics, and hydrological fracture properties. However, the most common technique for fracture set definition, stereonet contouring or cluster analysis (e.g., Shanley and Mahtab, 1974), relies exclusively on fracture orientations (see e.g., Fisher et al, 1987). In addition, contouring based algorithms assume that fractures are always assigned to the closest set pole, effectively truncating orientation distributions in areas of overlap. This can lead to biased estimates of variance. This paper presents a stochastic algorithm for fracture set definition, which has been successfully verified even for fractures sets with significant overlaps in orientation. This approach recognizes that every fracture identified in the field has a finite probability of membership in any given fracture set. Thus, the boundaries between fracture sets are probabilistic rather than rigid.
The algorithm for fracture set definition is based on iterative, stochastic reassignment of fractures to sets, using updated estimates of a priori distributions for the properties of interest. At the outset, the a priori distributions of the properties of interest for each fracture set must be defined. For example, set one might be defined by a near horizontal orientation, with 0% termination at intersections, while set two might be defined by fractures with 100% termination at intersections. The algorithm must then evaluate every fracture and determine the probability that it is a member of either of the two sets. Fractures are assigned to sets based on their similarity to fractures already assigned to that set. This is achieved by assigning fractures based on their probability of being members of that set. This is referred to below as the "component membership probability, PJ k." The probability pj k denotes the component membership probability for fracture property j with respect to the a priori k-the fracture set. The algorithm begins by requiring weighting factors, Wj, for each of the HJ fracture characteristics of concern, and initial guesses for the properties of the each set k. For each fracture i, the probability that the fracture is a member of set k is calculated using the value of the probability density function for that property and set, taken at the value of that fracture.
