Analytical model for uncertainty characterization of fracture intensity measurement in rock masses

This paper presents an analytical model for quantifying the uncertainty of three-dimensional fracture intensity measurements (P₃₂), which measure fracture area per sampling volume, and for determining the geometrical representative elementary volume (REV) of fractured rock. The analytical model for estimating the variance of P₃₂ is derived from the probability of a given number of fractures in a sample of rock. This model is validated through numerical simulations of fractured rock masses based on the discrete fracture network method. A series of parametric studies are conducted considering the dip angle, dip direction, Fisher constant, size and shape of the rock mass space, specimen volume, fracture diameter, and P₃₂. Based on the results of the analytical model and the parametric studies, a simple model to quantify the coefficient of variation (COV) of P₃₂ is established, which requires only the data of specimen volume, fracture diameter, and P₃₂. Four engineering applications of the proposed analytical model are presented, including two case studies of estimating the geometrical REV at a given COV, one of assessing the variation of mechanical properties, and another of determining the variation of permeability.