# Topological Representative Element Volume of Fractured Rock Mass
## 2 The Topological Structure of Fracture Networks  
The topological structure of a fracture network consists of branches and nodes. The topological structure can reflect various characteristics, and provides a good tool for studying the fracture network [14]. The nodes in a fracture network contain the ends of fractures and the intersection points between fractures, which can be subdivided into four types: isolated points (I-nodes), lap points (V-nodes), adjacent points (Y-nodes), and crossing points (X-nodes). The fractures between various nodes (including isolated fractures) are called branches. The two ends encounter to form a V-node. The meeting of an \"end\" and a branch forms a Y-node. The intersection of two branches forms an X-node. Figure 1 shows a typical fracture network, in which the rectangular range represents a study area which contains undivided fractures (I\({}_{2}\)-Y\({}_{1}\), I\({}_{3}\)-I\({}_{4}\), I\({}_{7}\)-V\({}_{1}\), I\({}_{9}\)-V\({}_{1}\)) and parts of fractures (I\({}_{1}\)-I\({}_{8}\), I\({}_{5}\)-I\({}_{6}\)). The fracture divided by the boundary of the study area is seen as a complete fracture when counting the number of fractures in the study area. The number of different nodes in the fracture network reflects its basic information. The symbols \(N_{\rm I}\), \(N_{\rm V}\), \(N_{\rm Y}\), \(N_{\rm X}\) represent the number of each of the four kinds of nodes I, V, Y, and X, within a study area, for example, \(N_{\rm I}=9\), \(N_{\rm V}=1\), \(N_{\rm Y}=1\), \(N_{\rm X}=1\) in Figure 1.  
For the number of fractures, \(N_{\rm L}\), the fracture starts and ends in one of the three nodes I, V, and Y; thus, the \(N_{\rm L}\) can be calculated as:  
\[N_{\rm L}=\frac{N_{\rm I}+2N_{\rm V}+N_{\rm Y}}{2}, \tag{1}\]  
There are six fractures in Figure 1, and the same result can be obtained using Formula (1). For the number of branches, \(N_{\rm B}\), the characteristics of the connection between nodes and branches can be described as follows: one branch connects two nodes, one I-node connects one branch, one V-node connects two branches, one Y-node connects tree branches, and one X-node connects four branches. The number of branches is given by  
\[N_{\rm B}=\frac{N_{\rm I}+2N_{\rm Y}+3N_{\rm Y}+4N_{\rm X}}{2}, \tag{2}\]  
The number of fracture branches is \(N_{\rm B}=9\) in Figure 1, according to Equation (2). It should be noted that the points I\({}_{1}\), I\({}_{6}\), and I\({}_{8}\) were generated by the delimitation of the study area in Figure 1. These points must be taken into account when calculating the number of fractures. Considering that this kind of point is most consistent with the definition of I-nodes, and that taking them into I-nodes will not cause a calculation error of the number of cracks and branches, these kinds of points are regarded as Insolated point I-nodes. In addition, the condition that three or more fractures intersect at the same point is not considered in this work. The proportion of the four types of nodes can reflect certain fracture network properties. [PERSON] and [PERSON] [20] proposed several topological parameters. The definition of a total of eleven parameters in three categories is described below. The first category contains five connection rate series parameters that reflect the fracture connection \(T_{\rm B}\), \(T_{\rm C}\), \(T_{\rm D}\), \(T_{\rm E}\), \(T_{\rm E}\). The fracture connection rate, \(T_{\rm B}\), is the ratio of the frequency of connected fractures to the total number of fractures. The connection nodes between fractures belong to three kinds of nodes, V-nodes, Y-nodes, and X-nodes, and each node is connected with two fractures. Taking into consideration the above connection relationship, the fracture connection rate can be written as  
\[T_{\rm B}=4\frac{N_{\rm V}+N_{\rm Y}+N_{\rm X}}{N_{\rm I}+2N_{\rm V}+N_{\rm Y}}, \tag{3}\]  
The equivalent fracture connection rate, \(T_{\rm C}\), is defined similarly to the fracture connection rate, \(T_{\rm B}\); however, the difference lies in dealing with the V-nodes and Y-nodes.