# Topological Representative Element Volume of Fractured Rock Mass
## 5 Comparison between TREV and PREV  
The calculation methods for TREV and PREV of the fractured rock mass are introduced here. They are defined in order to homogenize and simplify the fractured rock mass, and reflect the scale effect of rock mass properties. The difference between them lies in their physical meaning. The TREV reflects the topological properties of the fracture network, which focuses on the topological structure and is affected by the geometric distribution and connection characteristics of fractures. The PREV focuses on the permeability of the fracture network, which is affected by the distribution and connection characteristics of the fractures. Both TREV and PREV are affected by the distribution and connection characteristics of the fractures, which provides the possibility to use TREV to estimate PREV. The PREV of a given rock mass is a relatively fixed value. Many scholars have studied PREV based on the geometric characteristics of fractures [17, 20]. However, little research has paid attention to the topological structure of fracture networks. After comparing TREV and PREV, we will discuss the feasibility of using TREV to estimate PREV. Because TREV does not consider the connectivity of the fracture network, it must be modified when estimating PREV. The critical connectivity, \(N_{c}\), can be used to judge the connectivity of the fracture network, which is the average number of intersections per line [34]. Unlike the fracture connection rate, \(T_{\text{B}}\), \(N_{\text{c}}\) does not consider the existence of V-nodes in the definition, and V-nodes are counted with Y-nodes. For a fracture network with fixed fracture length and equal probability distribution in two directions, the critical connectivity, \(N_{c}\), is 3.11 [35]. The result of the connectivity judgment shows that a TREV for rocks number 3, 7, 11, and 12 does not exist. The TREV and PREV curves have similar characteristics, and their size expands with the increase of fracture spacing for fracture networks with the same fracture length. The REV does not exist when the fracture spacing is too large. For five types of fractured rock mass with different fracture discontinuity, when the fracture spacing is very close the size of TREV and PREV is in very close proximity. When the fractures are long and the spacing is wide (rock mass numbers 22, 23, and 24), TREV size is greatly affected by accidental factors. The fracture length has little effect for fracture networks with very close  
Figure 9: The REV of the fractured rock mass. The shape of the point indicates the fracture spacing, and the vertical bar area shows the fracture length. \({}^{\text{a}}\) After Zhang [33]. -fracture spacing; however, it dramatically influences fracture networks with extremely wide spacing. Comparison of the size of TREV and PREV found that they are essentially the same for the same rock. We further believe TREV can estimate PREV. Although we have studied only 23 types of rock mass, it is reasonable to believe that this method applies to all rock mass types in Table 1. Because TREV and PREV are closer when the fracture spacing is smaller, it is reasonable to expect that this method is suitable for the unstudied rock masses in Table 1 with extremely close fracture spacing. For the unstudied rock masses in the table with wide fracture spacing, it can be directly judged that they cannot form a connected fracture network and there is no PREV. As Table 1 covers all kinds of fracture lengths and spacing arrangements, this method is suitable for all rock masses with an orthogonal fracture network. It should be noted that this method is not suitable for fracture permeability varying with time, because the permeability of all fractures is assumed to be the same and does not change. Through TREV, we can obtain PREV without complex seepage calculation. TREV is only related to the topological structure of the fracture network. The estimation results obtained with this method have high credibility for many kinds of rocks discussed in this paper.