# Determination of Geometrical REVs Based on Volumetric Fracture Intensity and Statistical Tests
## 4 Determination of the Geometrical REV
### REV Size Based on Statistical Tests  
In a fractured rock mass, the number of fractures increases with the sample size [23]. The relationship between the area of fractures and the volume of the rock mass was quantified by the volumetric fracture intensity \(\mathrm{P_{32}}\). Therefore, it is necessary to determine the geometrical REV size by considering the fracture intensity characteristics in the fractured rock mass. The REVs were obtained from 100 random regions by using the method of [PERSON] et al. [62]. As shown in Figure 9, the REV values fluctuate with the study regions. Generally, the REV value in a certain region cannot exactly represent the REV of the entire rock mass. To obtain a more reasonable REV size, the anisotropy of rock masses and comprehensive differences between different methods should be seriously considered. Therefore, using several statistical methods to comprehensively determine the REV size is necessary [34]. In this paper, a method was presented to estimate the geometrical REV size based on the volumetric fracture intensity \(\mathrm{P_{32}}\) and statistical tests. The properties of the size effect and spatial effect for fractured rock masses were studied using the calculated \(\mathrm{P_{32}}\) values, which depended on the fracture disc characteristics in the 3D fracture network model. To describe the uncertainty in our reported \(\mathrm{P_{32}}\) values, the confidence interval of each dataset is calculated. As shown in Figure 10, the error bars represent a particular confidence interval with a 95% confidence level. The fluctuation of \(\mathrm{P_{32}}\) values significantly decreases with cube sizes varying from 1 to 20 m. Subsequently, the REVs were estimated by calculating \(\mathrm{P_{32}}\) values and the two statistical test methods reported in Section 3. The \(\mathrm{P_{32}}\) values for sample 20 compared with those of the sample sizes varying from 1 to 19, and a 5% significance level was used to determine whether two samples are statistically similar. Figure 8: The volumetric fracture intensity for different cube sizes. \begin{table}
\begin{tabular}{c} **Distribution** \\ **Form** \\ \end{tabular} \\ \hline
1 & 21.0 & 19.1 & Lognormal & 11 & 21.0 & 19.1 & Lognormal \\
2 & 21.0 & 15.0 & Gamma & 12 & 21.0 & 17.3 & Lognormal \\
3 & 21.0 & 13.1 & Gamma & 13 & 21.0 & 18.7 & Normal \\
4 & 21.0 & 16.9 & Lognormal & 14 & 21.0 & 13.1 & Normal \\
5 & 21.0 & 10.9 & Lognormal & 15 & 21.0 & 14.9 & Normal \\
6 & 21.0 & 11.3 & Lognormal & 16 & 21.0 & 12.3 & Lognormal \\
7 & 21.0 & 11.2 & Lognormal & 17 & 21.0 & 10.5 & Lognormal \\
8 & 21.0 & 16.1 & Normal & 18 & 19.7 & 6.3 & Triangle \\
9 & 21.0 & 8.6 & Lognormal & 19 & 19.7 & 14.4 & Triangle \\
10 & 21.0 & 9.5 & Normal & 20 & 19.7 & 11.2 & Triangle \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results of the chi-square goodness-of-fit tests for each sample. The likelihood ratio test and Wald-Wolfowitz runs test were used to determine the geometrical REV size based on the P\({}_{32}\) values for different sample sizes. The results of the statistical tests for the volumetric fracture intensity P\({}_{32}\) values are summarized in Table 4. The null hypothesis does not lead to a rejection when the \(p\) value is greater than or equal to the significance level of 0.05. The larger the \(p\) value is, the greater the likelihood that the two compared samples are statistically similar. Table 4 shows that the \(p\) value increased with the cube size of the sample, the similarity degree between two samples increased with the cube size of the sample, and the fluctuation in the volumetric fracture intensity P\({}_{32}\) values decreased significantly.