Revisiting each fracture size and spatial pattern: Inference from rock mass surface to interior

Estimating fracture size is a fundamental aspect of rock engineering. However, determining the most probable diameter (MPD) from a fracture's surface trace remains challenging in the scientific community. The prevailing methodologies typically infer statistical distributions of fracture sizes rather than specific values. This research presents a novel approach to inferring the MPD and the true spatial distribution pattern of each fracture. The challenge lies in linking the inference process with the trace length of each fracture and the statistical characteristics of the entire outcrop. Additionally, it is necessary to address the non-unique inverse problem. The methodology comprises several key steps. Firstly, the issue of censoring bias is addressed by considering the lengths of the traces contained. Secondly, the orientation bias is corrected using the vector method, and the true mean trace length and standard deviation are estimated and derived. Thirdly, assuming a lognormal distribution for fracture sizes, the mean and standard deviation of diameters are derived through a high-order moment relationship between trace lengths and diameters, validated by Crofton's theorem. Finally, the MPDs of all trace samples are determined by relating MPDs to trace lengths and the standard deviation of diameters using stereology techniques. Furthermore, the true fracture spatial patterns are inverted based on spatial geometric relationships. The proposed methodology is validated through rigorous Monte Carlo simulation and applied in a practical engineering case study, demonstrating its potential for use in rock engineering applications.