# Determination of Geometrical REVs Based on Volumetric Fracture Intensity and Statistical Tests
## 4 Determination of the Geometrical REV
### P\({}_{32}\) Characteristics  
The volumetric fracture intensity, which is the most useful measure of discontinuity intensity, indicated by the quantity P\({}_{32}\), is a fraction expressing the area of fractures per unit volume of rock mass and is used in the 3D fracture network model [54]. In this work, the Delaunay triangulations algorithm was utilized to calculate the P\({}_{32}\) value, and the calculation procedure was described in detail by [PERSON] et al. [61].  
In this paper, considering the influence of the boundary effect, a 140 m \(\times\) 70 m \(\times\) 40 m fracture network model was applied to calculate the volumetric fracture intensity P\({}_{32}\) and to determine the REV size based on the statistical tests. A total of 100 sets of cubes were selected from the generated 3D fracture network model (Figure 6), and the cube size of each set ranged from 1 to 20 m, changing at an increment of 1 m; then, the P\({}_{32}\) values were calculated using the method of [PERSON] et al. [61]. In this study, the P\({}_{32}\) values from cubes of the same side length were selected as a dataset in the 3D fracture network model. As a result, 20 random datasets, which included 100 random data points (P\({}_{32}\)) in each dataset, were established. The dataset with cube size of \(i\) m was defined as sample \(i\). The selected random cube sizes varied from 1 to 20 m. Therefore, the 20 samples, from 1 to 20, were generated based on the aforementioned procedure. The chi-square test was used to evaluate whether the P\({}_{32}\) values for each sample obey a certain distribution. Several probability density distribution forms (such as uniform, normal, negative exponential, lognormal, gamma, beta, binomial, Poisson, and triangle distribution) were checked using the chi-square goodness-of-fit tests to obtain the optimal distribution of the P\({}_{32}\) values. The results of the chi-square goodness-of-fit tests are illustrated in Table 3. Table 3 shows that the distribution forms of the P\({}_{32}\) values for each sample are different at a significance level of 0.05. The frequency histograms of the P\({}_{32}\) values for cube sizes of 3, 10, and 15 m are shown in Figure 7. As shown in Figure 7a, the P\({}_{32}\) values follow a gamma distribution when the cube size is 3 m, and Figure 7b,c shows that a normal distribution can be used for cube sizes of 10 m and 15 m.  
The 100 selected sets of P\({}_{32}\) values for cube sizes ranging from 1 to 20 m are shown in Figure 8. The size effects were clear, and the P\({}_{32}\) values change with the spatial positions of cubes. When the cube size is smaller, the distribution of the volumetric fracture intensity P\({}_{32}\) values is discrete, the difference between the maximum and minimum values is significant, the fluctuation in P\({}_{32}\) values is greater, and the spatial effect is obvious. When the cube size increases, the maximum value of the P\({}_{32}\) decreases and the minimum value of P\({}_{32}\) increases, the difference between the maximum and minimum values gradually decreases, the fluctuation in P\({}_{32}\) values significantly decreases, and the spatial effect considerably weakens. The fluctuation in the P\({}_{32}\) values tends to stabilize when the cube size is greater than or equal to a critical value.  
Figure 6: The randomly distributed cubes in a fractured rock mass.  
The fractures were randomly distributed in the generated 3D fracture network model. When we evaluate the structural properties of the rock masses, the spatial effects should be considered [34]. Figure 8 shows that, in different regions, the P\({}_{32}\) values from the same cube size differ within a certain range, and the spatial effects weaken as the cube size increases. Therefore, considering both the size effect and the spatial effect is necessary to determine the REV size.  
Figure 7: The frequency histograms of the volumetric fracture intensity P\({}_{32}\) for three cube sizes. (**a**) cube size of 3 m. (**b**) cube size of 10 m. (**c**) cube size of 15 m.