Quantitative estimation of three-dimensional fracture density: Insights from the stereological relationship between borehole and universal elliptical DFN

Complex fracture networks significantly affect the deformation, damage, and strength of rock masses. Three-dimensional (3D) fracture density is an important parameter for estimating the mechanical properties and seepage characteristics of rock masses. This study, using a universal elliptical fracture model with all variable parameters that approximates the true fracture shape more closely than a circular disk model, strictly derived the mathematical intersection relationship between 3D elliptical fracture network and the borehole in space from the perspective of stereology. By correcting the influence of borehole size and fully utilizing the fracture trace information on the inner surface of the borehole, an accurate calculation method for rock mass 3D fracture density (P30 and P32) was proposed. A series of 3D universal elliptical discrete fracture networks (DFNs) and boreholes were generated through Monte Carlo simulations to verify the correctness and applicability of the estimation theory. A sensitivity analysis of the impact of different fracture parameters on the estimation of 3D fracture density was conducted, revealing that the influences of the borehole radius, major axis distribution of the elliptical fracture, and 3D fracture density were more significant. Compared with conventional methods for estimating 3D fracture density through borehole data based on the circular disk model and elliptical model, the results showed that the method proposed in this study had greater characterization accuracy. For conventional methods that do not consider borehole size correction, the error increases exponentially with increasing borehole radius. The error of the disk model is always larger than that of the universal elliptical model. Finally, the relationships among P32, P30 and P10 of the universal elliptical fracture model was assessed. This research provides a more comprehensive evaluation of rock mass quality and surrounding rock stability.