# 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
For the trace length simulation, due to the sampling deviations of the trace lengths in the field, the observed trace lengths should be corrected before generating the disc diameter by using the method introduced by [PERSON] and [PERSON] [37]:  
\[\mu=\frac{wh(1+R_{0}-R_{2})}{(1-R_{0}+R_{2})(w\int\limits_{0}^{\pi/2}g(\theta )\sin\theta d\theta+h\int\limits_{0}^{\pi/2}g(\theta)\cos\theta d\theta)} \tag{2}\]  
where \(w\) and \(h\) represent the width and height of the sampling window, respectively; \(R_{0}\) and \(R_{2}\) are the fractions of joints with both ends unobservable and both ends observable in the sampling window, respectively; \(\theta\) is the dip angle of the joint disc; and \(g(\theta)\) is the probability density function of dip angle \(\theta\). A goodness-of-fit test (Kolmogorov-Smirnov) was utilized to determine the optimal probability distribution of the observed trace length. In this section, we assume that the same distribution form and standard deviation can be used for the observed trace length and the corrected trace length. In this paper, for the simulation of the fracture diameter, the fracture is assumed to be a thin planar disc. Based on the stereological relationship between the distribution of the corrected trace length and disc diameter [48], the fracture diameter of each fracture set was identified using the method proposed by [PERSON] and [PERSON] [49]. The results are shown in Table 1.  
5. Simulation of the discontinuity intensity and location  
The 2D fracture network was generated based on the fracture information (such as joint trace coordinate, dip direction, and dip angle) obtained from the sampling window. The normal fracture frequency of each fracture set could be calculated based on scanlines generated in the 2D fracture network [50]. The volumetric fracture intensity of each fracture set could be determined [49; 51; 52; 53; 54] (Table 1). Based on the KS goodness-of-fit tests, a negative exponential distribution could accurately fit the fracture spacing of each fracture set. As a result, the location of each fracture centroid could be obtained by using a homogeneous Poisson progress [46; 53; 55; 56]. 6. Monte-Carlo simulation and validation  
A 150 m \(\times\) 80 m \(\times\) 50 m 3D model size was designed for engineering purposes. The 3D fracture network model was established based on the volumetric fracture intensity P\({}_{32}\) and the model size. Based on the abovementioned parameters (such as fracture orientation, fracture diameter, and fracture location) and the corresponding statistical distributions, more 3D fracture network models were generated according to the Monte-Carlo simulation method [46]. An important concern is whether the generated fracture networks can represent the actual field conditions. Therefore, a model validation procedure was implemented based on the fracture parameters in the 2D cut plane to determine the optimal model from the many generated network models. Finally, an optimal model was selected from the many accepted models according to the method of [PERSON] et al. [23], as reported in Table 2.