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

The fracture diameter and its distribution form are the key parameters for establishing a three-dimensional fracture network model, which has an important influence on the quality and stability evaluation of engineering rock mass. In order to solve the problem that the parametric method relies on the assumption of the diameter distribution form, this study proposes 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, it is assumed that the fracture is a Baecher disc. According to the real trace length distribution histogram, the probability of each interval diameter generating a certain interval trace length is calculated. The probability density of each interval diameter is derived from the relationship between areal density and volumetric density, and then probability density distribution histogram of fracture diameters is drawn. In order to verify the effectiveness of this method, a simulation example of diameter obeying the lognormal and negative exponential distribution is designed. The mean value and standard deviation of the estimated diameters are compared with the initial value, and the relative error is within 10%, which proves the correctness of this method. Finally, it is applied to an open-pit mine slope in Manzhouli City, Inner Mongolia. According to the measured trace length distribution histogram, the distribution histogram of fracture diameter is obtained. The simulated trace length and the measured trace length obey the same distribution. 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 method can accurately and effectively estimate the probability density distribution of fracture diameters, which is of great significance for the fine characterization of three-dimensional fracture network and the evaluation of engineering rock mass.