Non-parametric estimation of probability density distribution for disc-shaped fracture diameter in rock masses

Fracture diameter and its distribution are key parameters for constructing three-dimensional fracture network models, with significant implications for evaluating the quality and stability of engineering rock masses. To solve the issue that the parametric method relies on the assumption of the diameter distribution, this study proposed a non-parametric method based on the trace length data to estimate the diameter probability density distribution according to the proportion of the chord length intercepted by the fracture disc. Specifically, fractures were assumed to follow the Baecher disc model. Using the measured trace length distribution histogram, the probability of each interval producing a corresponding interval trace length was calculated. The probability density of each interval diameter was derived from the relationship between areal density and volumetric density, and then probability density distribution histogram of fracture diameters was drawn. To validate the method, simulation experiments were conducted using lognormal and negative exponential diameter distributions. The estimated mean value and standard deviation of the diameters were compared with the initial value, and the relative error is within 10%, confirming the accuracy of the approach. Finally, the method was applied to an open-pit mine slope in Manzhouli City, Inner Mongolia. The diameter distribution histogram derived from measured trace lengths yielded simulated trace length distributions consistent with field measurements, Based on the solved fracture parameters, the slope fracture network model is established and its block stability is analyzed. The results show that the single slip surface block accounts for a large proportion in the unstable block of the slope, and the block volume is generally small. This non-parametric method provides an accurate and effective tool for estimating the probability density distribution of fracture diameters, which is of great significance for the fine characterization of three-dimensional fracture networks and the evaluation of engineering rock mass.