Estimation of Fracture Orientation Distributions from a Sampling Window Based on Geometric Probabilistic Method

Accurate orientation distributions are crucial to generating a reliable discrete fracture network (DFN) model for rock mass, while conventional one-dimensional (1D) and two-dimensional (2D) observation data have significant sampling bias. The study proposes a complete analytical method for estimating the orientation distributions of three-dimensional (3D) fractures in rock mass in conjunction with trace statistics in a sampling window, which is suitable for most continuous distributions by reducing the sampling bias. Traces are divided into three categories to derive the geometric probabilistic relationships between 2D trace statistics and 3D fracture orientation distributions. The moment estimation, number estimation, and normalization error functions are derived, and the distribution parameters are determined by minimizing the total error function. The proposed method is compared with the Terzaghi family methods and validated by multiple sets of stochastic fracture networks with different orientation distributions and sampling windows generated by the Monte Carlo method. The results indicate that the estimated continuous orientation distributions subjected to the error functions from a large single sampling window are well matched with the true distributions after removing the number estimation error functions of trace samples fewer than 20. Daxiagu tunnel is selected as a case study and the distributions estimated by the proposed method are more coincident with the field observation than those fitted by the orientations of the rock outcrops on the excavation face.