Stochastic three dimensional joint geometry: Modeling and verification.

Abstract

Eight 3D (three dimensional) rock joint geometry modeling schemes which investigate statistical homogeneity, and incorporate corrections for sampling biases and applications of stereology are presented. A procedure for verification of the developed models also is presented. In this study, shape of the joints was assumed as circular. The models provide the number of joint sets, and for each joint set, the intensity, orientation, spacing, location and diameter distributions. Miller's method (1983) with new interpretations (Kulatilake et al., 1990b) and equal area polar plots were used together to identify the largest statistically homogenous region around the ventilation drift, Stripa mine, data of which were used for both modeling and verification. Four joint sets were found in this region. A general vector approach to correct sampling bias on joint orientation is presented. Corrected data as well as raw data were subjected to chi-square goodness-of-fit tests to check the suitability of hemispherical normal and Bingham distributions in representing orientation of joint sets. Only raw data of joint set 4 followed Bingham distribution. Therefore, joint set orientations were best represented as empirical distributions. Two methods are presented for the modeling of joint spacing, linear intensity and location. In each method, spacing distributions of joint sets were best represented by exponential distributions. Then, joint intensity and location distributions are represented by Poisson and uniform distributions respectively. Correction of sampling bias on joint spacing also is presented. Joint size modeling was carried out using two methods: area sampling survey method and scanline sampling survey method. In these two methods, corrections of sampling biases associated with joint size modeling are presented. 3D joint sizes were inferred from 2D trace length measurements using geometrical probability and conditional probability concepts. In both methods diameter distributions are represented by gamma distributions. For verification, joints were generated in a volume according to the statistical models, using Monte-Carlo simulation. This volume was intersected by planes to obtain joint traces on exposures of size and shape similar to the ones used to obtain field data. Characteristics of these predicted joint traces were compared with the field data in a statistical sense. For the rock mass under this study, the modeling scheme 3 was found to be the most suitable scheme.