Analytical Solutions for the Characteristic Size Distribution of the Elliptical Model in Fractured Rock Mass

Fracture characteristics have a significant effect on the mechanical and hydraulic properties of rock mass. In this paper, before modeling a discrete fracture network, the fracture is simplified as a non-similar ellipse. First, the multifactor coupling stereological relationship of the sampling trace length distribution is established, which proves that the trace length distribution is independent of the fracture occurrence distribution. Using the Volterra integral equation method, the stereological formula of the trace length is inversely solved to obtain the distribution expressions of the major axis and axial ratio of the elliptical fractures. The analytical solutions of the probability density function (PDF) of the characteristic size of the elliptical fractures are derived for cases in which the trace length follows a uniform distribution, fractal distribution, and polynomial distribution. Second, for cases in which the trace length conforms to a negative exponential distribution, gamma distribution, chi-square distribution, and lognormal distribution, the statistical eigenvalues of the major axis and axial ratio of the elliptical fractures are deduced. Finally, a Monte Carlo statistical simulation is performed using the Rock Mass Joint Network Simulation (RJNS3D) toolkit to verify the applicability of the derivation process and the correctness of fitting a polynomial function to the trace length distribution to solve the PDF of the characteristic size of the elliptical fractures. The proposed method can better predict the distribution of the required parameters, such as the major axis and axial ratio of the elliptical fractures, according to the typical two-dimensional data in the sampling windows. This method can be further applied to reconstruct fracture networks in practical engineering tasks and lay a foundation for the analysis of rock mass strength and deformation, representative elementary volume (REV), seepage, and surrounding rock stability.