# Determination of Geometrical REVs Based on Volumetric Fracture Intensity and Statistical Tests
## 1 Introduction
[PERSON] [25] suggested that REV size should be three times greater than the typical joint trace length and proposed to determine REV size by using a crack tensor. According to [PERSON] and [PERSON] [26], an REV size should be approximately five times the average crack length. Another size was proposed by [PERSON] and [PERSON] [27]: ten times the mean joint length. On the other hand, the relationship between REV size and joint spacing was established by other researchers [13; 24; 28]. The complexity of rock mass structure characteristics (such as anisotropy, inhomogeneity, and discontinuity) and the limitation of acquiring several parameters (such as the number of fractures, fracture orientation, fracture trace length, fracture width, fracture fill, and ground water flow) have restricted the development of rock mechanics. However, with the development of the computer and mathematical theory, the discontinuities of rock masses have been described in detail with 3D fracture network modeling technology [29; 30; 31]. Subsequently, some scholars have conducted research on geometrical REVs using 3D fracture network models. [PERSON] et al. [32; 33; 34] introduced some methods to determine the geometrical REV size using a 3D fracture network model and statistical test methods: A 3D fracture network model was established, and the simulated rock masses were divided into cubes of the same size, which varied from 1 to 20 m. Then, the parameters of the rock mass were statistically calculated. Finally, the REV size was determined by the Kolmogorov-Smirnov (KS) and the Wilcoxon rank-sum tests. [PERSON] et al. [35] utilized a contingency table method to analyze the geometrical parameters of fractures (such as fracture density, length, dip direction, and dip angle) and to ascertain the geometrical REV size based on a 3 Dfracture numerical network modeling. [PERSON] et al. [1] and [PERSON] et al. [36] considered the discontinuity connectivity to analyze the geometrical REV size at the Songta dam site in China. In this study, the joint data were collected from the abutment rock mass of the Maji hydropower station. The field survey shows that the abutment rock masses are relatively intact. The fractures with one end censored and both ends observable are the most common and the fracture diameters are relatively small. The rock masses were not strongly affected by a certain tectonic activity. According to the abovementioned fact-finding, the regional fractures were clearly divided into four sets. The K-S goodness-of-fit tests indicate that the fracture spacing of each fracture set obeys a negative exponential distribution. Therefore, a homogeneous Poisson progress could simulate the locations of fracture centroids. To comprehensively consider the anisotropy of rock masses, simplicity of calculation and comprehensive differences between different methods, the volumetric fracture intensity P\({}_{32}\) and two statistical test methods were employed to estimate the REV sizes. The purpose of this paper is to introduce a new method for determining the geometrical REV size of the entire rock mass based on the volumetric fracture intensity P\({}_{32}\) and statistical tests. The volumetric fracture intensity P\({}_{32}\) was calculated based on the randomness of the distribution of the cubes in a 3D fracture network model. The probability distribution, size effect, and spatial effect of the volumetric fracture intensity P\({}_{32}\) at the Maji dam site in China were studied. Finally, the size of the geometrical REV was assessed based on the P\({}_{32}\) values, likelihood ratio test and Wald-Wolfowitz runs test in the study area.