Correction of the probabilistic density function of discontinuities spacing considering the statistical error based on negative exponential distribution

The mechanical and hydraulic properties of fractured rock masses are generally controlled by the distribution characteristics of discontinuities developed in the rock masses. In practical measurement on exposures, the spacing data collected frequently contains some statistical errors due to the spacing of small discontinuities, and micro-cracks being ignored. In this study, a correction model aiming to eliminate the statistical error is proposed based on the negative exponential distribution of trace length and spacing, to describe the distribution regularity of the spacing data obtained from outcrops or exposures. Based on the model, a corrected probabilistic density function that can describe the distribution regularity of the spacing data containing the statistical error is developed; and a new method is further presented to determine the true distribution parameter of spacing of all discontinuities in rock masses. The sensitivity analysis indicates that the true distribution parameter λ of all spacing is moderately sensitive to the μ (reciprocal of the mean trace length) and the critical trace length l0; and completely insensitive to the maximum spacing of small discontinuities x0. Finally, the correction theory is verified by a simple 2D model with one set of discontinuities and a complex 2D model with four sets of discontinuities, generated using Monte Carlo method.