# Topological Representative Element Volume of Fractured Rock Mass
## 6 Conclusions  
The topological representative element volume (TREV) is proposed based on the structural characteristics of the fracture network. Because the topological structure contains many characteristics of the fracture network, TREV can reflect comprehensive properties of the fracture network such as connectivity, continuity, etc. The feasibility of using TREV to estimate the permeability representative element volume (PREV) is discussed, and the following conclusions obtained.  
(1) There is a noticeable scale effect for the topological parameter of a two-dimensional fracture network. When the study area is small, the topological parameters fluctuate violently. The variability of the topological parameters decreases gradually with the increase of the study area. When the research area is large enough, the topological parameters are stable. Therefore, topological parameters can be used as equivalent parameters for TREV.  
(2) TREV considers various topological characteristics of the fracture network. After analyzing eleven topological parameters, two of them, fracture connection rate (\(T_{\text{B}}\)) and aerial branch frequency (\(B_{20}\)) were selected as equivalent parameters to estimate TREV. These two topological parameters are only related to the node system and can be calculated directly from the number of nodes in the fracture network.  
(3) The size of TREV and PREV is essentially the same for the fractured rock discussed here. TREV is only related to the topological structure of the fracture network. Using TREV to estimate PREV can avoid complicated seepage calculation. The estimation results using this method have high credibility for many kinds of rocks. Particular attention should be paid to the connectivity of a fracture network when using TREV to estimate PREV.  
The calculations in this paper focus on rock mass types with orthogonal fracture networks, which are composed of horizontal and vertical fractures. Orthogonal fracture networks are often used in related research [35; 36] and are common in nature, including three types of squares [37] as well as brick [38] and pavement [20; 39]. The Representative Element Volumes of the fractured rock masses provided in this paper (Figure 9) represent a basis for the study of seepage in the corresponding rock masses. Based on the analysis of the topological structure and permeability characteristics of the fracture networks, the feasibility of using topological structure to study permeability is verified by the comparison of TREV and PREV. This new idea of establishes the relationship between the topological structure of fractures and their permeability through the concept of representative element volume. Similarly, future studies might attempt to study the relationship between topological structure and the mechanical properties of different rock types.  
Conceptualization, M.L. and Z.Z.; Data curation, M.L.; Funding acquisition, Z.Z.; Methodology, M.L.; Project administration, Z.Z.; Software, M.L.; Supervision, Z.Z.; Validation, M.L., M.C. and J.W.; Visualization, M.C. and J.W.; Writing--original draft, M.L.; Writing--review & editing, Z.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Major Research Program of the National Natural Science Foundation of China (Grant No. 91747204), the Research Program of China Three Gorges Corporation (Grant No. XLD/2118), and the China Scholarship Council (Grant No. 202006710075).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interests.