# Topological Representative Element Volume of Fractured Rock Mass
## 3 Numerical Experiment  
The feasibility of using topological parameters as the equivalent parameters for TREV was examined based on the generated fracture networks. The fracture network was generated by computer programming using a modification of [PERSON] open-source algorithm [21]. As input, the program requires the statistical parameters of fractures, including the average fracture length, average fracture direction, and average fracture spacing. According to these statistical parameters, the occurrence of a fracture and its location, direction, and trace length can be determined by a random number. The edge length of the generation area is not fixed and is generally set to ten times the average fracture length. For extremely dense or sparse fracture networks, the edge length of the generation area may be reduced or increased to simplify the calculation and ensure representativeness. The fracture is divided into horizontal and vertical groups in a generation area. The number of fractures in a group equals the generation area divided by the average fracture spacing, then by the average fracture length. The fracture location is determined by the coordinates of the center of the fracture. The uniformly distributed random numbers are used as the coordinates of the center of a fracture. Setting the dip angle of a fracture to a random function can generate fractures in any direction. Because the orthogonal fracture network is studied, there is no need to take a random number for the fracture direction; a normal distribution function can determine the fracture length. It is easy to obtain the location of a fracture based on the coordinates of the center, direction, and length. The fracture network can then be produced according to the coordinates of all fractures. Table 1 shows the classification of a fractured rock mass [22; 23]. Twenty-three kinds of rock mass were selected for calculation (the shaded part in Table 1), considered representative fracture networks. The simulated cases are orthogonal fracture networks generated according to vertical and horizontal fractures. The fracture lengths of the two groups follow a normal distribution. The estimation process of TREV can then be introduced, taking the types of fractured rock mass with low fracture length and medium spacing (1 \(<\) L \(<\) 3, 0.2 \(<\) C \(<\) 0.6) as an example. Figure 2 shows the workflow to calculate TREV. First, the parameters of a fracture network, including the generation area, average fracture length, standard deviation of the fracture length, and average fracture spacing were entered into the program to generate a fracture network. Second, the number of fractures and nodes in the selected research area were counted. Then, the topological parameters were calculated according to the fracture length, branch length, and the number of nodes (\(N_{\rm L}\), \(N_{\rm V}\), \(N_{\rm Y}\), \(N_{\rm X}\)). The study area was expanded and the above steps repeated until the study area was large enough. Every type of fracture network was generated five times to ensure its randomness and representativeness. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Average Fracture Spacing (m)** & \multicolumn{5}{c}{**Fracture Length/Persistence (m) (L-Fracture Length)**} \\ \cline{2-6}
**(C-Spacing)** & **Very Low** & **Low** & **Moderate** & **High** & **Very High** \\ \hline \multirow{2}{*}{Extremely closely} & L \(<\) 1 & 1 \(<\) L \(<\) 3 & 3 \(<\) L \(<\) 10 & 10 \(<\) L \(<\) 20 & L \(>\) 20 \\  & C \(<\) 0.02 & C \(<\) 0.02 & C \(<\) 0.02 & C \(<\) 0.02 & C \(<\) 0.02 \\ \hline \multirow{2}{*}{Very closely} & L \(<\) 1 & 1 \(<\) L \(<\) 3 & 3 \(<\) L \(<\) 10 & 10 \(<\) L \(<\) 20 & L \(>\) 20 \\  & 0.02 \(<\) C \(<\) 0.06 & 0.02 \(<\) C \(<\) 0.06 & 0.02 \(<\) C \(<\) 0.06 & 0.02 \(<\) C \(<\) 0.06 & 0.02 \(<\) C \(<\) 0.06 \\Figure 3 shows an example of a fracture network.