Estimation of three-dimensional diameter distributions of multiple fracture sets clustered by a multi-level clustering method

Fracture diameter is an important factor to evaluate rock mass quality accurately, while there is a lack of effective methods to estimate three-dimensional diameter of multiple groups of fractures by observation data. The study proposes a density-based multi-level clustering method to cluster the two-dimensional outcrop fracture data according to both fracture orientation and size. The method considers the five types of boundary problems of the orientation domains and is applied to estimate diameter distributions of multiple fracture sets by clustering fracture orientation, merging misclustered fracture clusters, and classifying fractures based on the estimated fracture size. The K -means clustering by fast search and find of density peaks is established to intuitively determine the centers of fracture sets and the independent fractures, and assign the fracture samples to the appropriate clusters. The orientation cluster differentiation coefficient (OCDC) is defined to derive the subcluster merging conditions and center merging function for the fracture sets near the boundaries of the orientation domains, and the empirical formula of OCDC versus the pooled standard deviations of orientation samples is derived. A simplified calculation of the most probable diameter of a trace is derived according to the estimation formula of diameter standard deviation versus trace length. The multi-level clustering method is verified and discussed by Monte Carlo simulation. The proposed method is applied to an excavation face from Sejilashan tunnel. The result indicates that the clustered outcrop fractures are similar to the field observation, and the estimated three-dimensional diameter distributions of multiple fracture sets are in good agreement with the geological survey.