Mixture distribution model for three-dimensional geometric attributes of multiple discontinuity sets based on trace data of rock mass

Abstract

The geometric attributes of discontinuities are critical to an accurate discrete fracture network (DFN) model of rock mass. It is difficult to estimate three-dimensional (3D) geometric parameters of multiple groups of discontinuities by observation data. The study proposes a mixture distribution model for the 3D geometric attributes of multiple discontinuity sets. The method of estimating the mixture distributions of 3D geometric attributes of all discontinuity sets from the two-dimensional (2D) trace data in a sampling window is established and modified, including discontinuity diameter distribution estimation, volume density estimation, orientation distribution estimation, and distribution mixing. Discontinuities are assumed to be thin disks uniformly distributed in space, and their roughness and aperture are ignored. Discontinuity diameter and orientation follow lognormal distribution and bivariate normal distribution, respectively. Orientation distributions defined on the definitional domain of the geological coordinate system are derived to represent the real frequency distributions of orientation measurements in 3D rock mass space. The mixture distribution model is verified by using Monte Carlo method to generate DFNs with different volume densities, and distribution parameters of orientation and diameter. The maximum error of volume density is <0.5 m−3. The estimated distributions of diameter and orientation are in good agreement with the true distributions, while there are limited symmetry estimation errors of the discontinuity sets with orientation samples covering both sides of the 0° or 180° polar axis. The model is applied to Sejilashan tunnel, and the result indicates that the surrounding rock of two excavation faces have a high probability to be rated as the V classification according to the engineering classification of rock mass in China, which is in good agreement with the reality according to the geological survey and field observation.