# Topological Representative Element Volume of Fractured Rock Mass
## 4 Discussion  
### The Topological Representative Element Volume (TREV)  
The equivalent parameters can be considered separately or comprehensively when calculating the REV for a fractured rock mass. Our hope was that TREV would reflect more topology characteristics. Therefore, all topological parameters were considered comprehensively when calculating TREV. The effect of each topological parameter on TREV is discussed below.  
Figure 5: The fracture strength (\(P_{\rm 2X}\)) and the coefficient of variation (\(C_{\rm v}\)) versus sample size.  
Figure 6: The branch strength (\(B_{\rm 2X}\)) and the coefficient of variation (\(C_{\rm v}\)) versus sample size.  
(1) When taking the connection rate, \(T_{\rm x}\), as the equivalent parameter to calculate TREV, \(T_{\rm B}\) has the greatest influence compared with the connection rate series parameters (\(T_{\rm B}\), \(T_{\rm D}\), \(T_{\rm E}\), \(T_{\rm F}\)), which determines the TREV of 22 kinds of fracture network. The parameter \(T_{\rm F}\) only increases the size of TREV from 7 m \(\times\) 7 m to 8 m \(\times\) 8 m in one kind of fracture network (L \(>\) 20, 0.06 \(<\) C \(<\) 0.2). The parameter \(T_{\rm C}\) is not suitable as an equivalent parameter because it has an irregular variation for fracture networks with long fracture length or large fracture density.  
(2) For TREV calculation with the fracture strength \(P_{\rm 2x}\), when taking \(P_{\rm 20}\) and \(P_{\rm 21}\) as equivalent parameters, it is found that the effect of these two parameters varies in different cases; thus, these two parameters should be comprehensively considered in the process of calculating TREV. For certain types of rock mass with long fracture extension and dense cracks, \(P_{\rm 22}\) fluctuates in a disorderly fashion, and it is thus not suitable to be used as the equivalent parameter.  
(3) For TREV calculation using the branch strength, \(B_{\rm 2x}\), considering \(B_{\rm 20}\) and \(B_{\rm 21}\) as the equivalent parameter, it is found that the TREV of all 23 kinds of fracture network depends entirely on \(B_{\rm 20}\), and \(B_{\rm 21}\) does not affect TREV. The parameter \(B_{\rm 22}\) is unsuitable as an equivalent parameter because it increases continuously for those types of rock mass with long fracture length or large fracture density.  
(4) Comparing these three kinds of topological parameters (\(T_{\rm x}\), \(P_{\rm 2x}\), and \(B_{\rm 2x}\)), it is found that the TREV is only controlled by \(T_{\rm x}\) and \(B_{\rm 2x}\) series parameters, and \(P_{\rm 2x}\) series parameters have no effect at all. According to the previous discussion, TREV based on \(T_{\rm x}\) series parameters is controlled by \(T_{\rm B}\), and TREV based on \(B_{\rm 2x}\) series parameters is controlled by \(B_{\rm 20}\). It is concluded that when considering three kinds of topological parameters, TREV only depends on the fracture connection rate, \(T_{\rm B}\), and the aerial branch frequency, \(B_{\rm 20}\). As these two parameters (\(T_{\rm B}\) and \(B_{\rm 20}\)) are wholly determined by the number of intersections, the relationship between TREV and intersections is closer than that between TREV and fracture length. Table 3 shows the TREV of 23 kinds of fracture network calculated by comprehensively considering three kinds of topological structure parameters.