# Determination of Geometrical REVs Based on Volumetric Fracture Intensity and Statistical Tests
## 2 The Establishment of a 3D Fracture Network Model
### 3D Fracture Network Modeling  
A 3D fracture network model was generated based on the following two assumptions [38; 39; 40]:  
1. The discontinuities are thin planar discs with a random distribution. 2. The fractures are not open; water and fracture fill can be disregarded. The simulation procedure was described as follows:  
1. Determination of the statistical homogeneity  
Because limited joints are located in the tunnel, the division of statistical homogeneity would reduce the number of statistical samples and affect the accuracy of the fracture simulation results. Furthermore, the 3D fracture network model generated by the inhomogeneity of the joint data can meet engineering needs. Therefore, the entire joint dataset is considered to be statistically homogeneous to generate a 3D fracture network. 2. The discontinuity orientation data simulation  
The joint data were divided into four sets based on the fracture orientations in tunnel PD222 using the modified K-means algorithm introduced by [PERSON] et al. [41]. The results are shown in Table 1 and Figure 3. As shown in Table 1, some of the fracture sets were identified based on the relatively few fracture data points available. To ensure the reasonableness and reliability of the identified fracture sets, the regional fracture data (i.e., the fractures collected from five tunnels near tunnel PD222) were divided into four different fracture sets using the method proposed by [PERSON] et al. [41]. As shown in Figure 4, the dataset of the fractures with steep dip angles is larger than that of the fractures with low dip angles. The mean orientation of each fracture set for tunnel PD222 and that of the corresponding regional fracture set are similar. Subsequently, the statistical similarity between the regional joint data of each fracture set and the corresponding joint data obtained from PD222 was determined by using KS tests. As a result, the identified fracture sets based on the data obtained from tunnel PD222 were reasonable, reliable, and statistically significant. Figure 2: The illustration of the 2D joint traces. In theory, the joint orientations of each fracture set can be determined by a Fisher distribution, Bingham distribution, bivariate normal distribution, or bivariate empirical distribution [40; 42; 43; 44; 45]. However, due to the complexity of the geological structure and investigation error, it is very difficult to determine a distribution that precisely fits the fracture set data. Therefore, considering goodness-of-fit and simple calculation, the Fisher distribution and bivariate empirical distribution were generally used to describe the distribution of the fracture orientations. In this work, the bivariate empirical distribution was used to depict the fracture orientation distributions of each fracture set [46]. 3. Bias correction on orientations of the discontinuities  
The weighting function \(W_{i}\) of the discontinuities was calculated to correct the orientation sampling bias of the sampling windows [47]. Figure 4: Equal-area projection diagram for fracture sets located in the dam site area. Figure 3: The poles of the fracture orientations on the upper hemisphere for each fracture set. \[W_{i}=\left[\left(\cos^{2}\theta_{i}\sin^{2}\beta_{i}+\cos^{2}\beta_{i}\right)^{1/ 2}+\frac{\pi d_{i}}{4h}\cos\beta_{i}\sin\theta_{i}+\frac{\pi d_{i}}{4w}\cos \theta_{i}\right]^{-1} \tag{1}\]  
where \(w\) and \(h\) represent the width and height of the sampling window, respectively; \(\theta_{i}\) represents the dip angle of the discontinuity \(i\); \(\beta_{i}\) represents the angle between the strike of the sample plane and the dip direction of the discontinuity \(i\); and \(d_{i}\) represents the diameter of discontinuity \(i\). 4. Discontinuity geometry simulation  
The discontinuity geometry simulation includes a trace length simulation and diameter simulation.