# Topological Representative Element Volume of Fractured Rock Mass
## 1 Introduction  
Fluid flow through a fractured rock mass has always been an important issue within hydrogeology and engineering geology. A fractured rock mass is composed of a porous matrix and a fracture network. The permeability of porous media has been widely discussed, and the concept of Representative Element Volume (REV) has been widely used in the study of porous media [1; 2; 3; 4]. Generally, the Hagen-Poiseuille law can be used to calculate the seepage in the pore [5], and the pore-scale or meso-scale seepage in the matrix can be calculated by the pore network method or Lattice Boltzmann method [6]. Compared with the pore structure of the matrix, large-scale fractures are generally larger and have greater permeability. The cubic law is often used to calculate the seepage in fractures [7]. The main numerical models of fluid flow in fractured rock mass are Discrete Fracture Matrix Models, Discrete Fracture Network models, and Discrete Fracture Matrix Models with dual-continuum models [8]. This paper focuses on the permeability of the fracture network, ignoring the permeability of the porous matrix, which is a common assumption of Discrete Fracture Network models [9].  
The representative element volume (REV) of the fractured rock mass is a fundamental concept in rock mass mechanics. It provides great convenience to the study of deformation and groundwater seepage within the rock mass. The REV of fractured rock mass has been widely discussed, including studies based on two-dimensional [10; 11], three-dimensional [12], and radial flow simulations [13]. The REV size is usually regarded as the minimum volume of heterogeneous material that is sufficiently large to be statisticallyrepresentative of the composite. The REV size is not a constant value for different physical properties of the same rock mass. Based on numerical techniques, REV can be obtained using the following steps. First, determine an equivalent parameter for REV; then, calculate the variation characteristics of the equivalent parameter with the increase of the study area; finally, determine the critical size (REV) of the study area according to a given convergence criterion [5].  
The geometric characteristics of a fracture network (strike, dip direction, opening, density, and connectivity) control the physical properties of the rock mass, such as hardness, mechanical strength, porosity, and permeability [14; 15; 16; 17]. The study of fracture geometry properties has enormous guiding significance for examining the characteristics of the rock mass. Researchers have studied the REV of rock mass according to the individual geometric features of fractures. [PERSON], et al. [18] examined the geometric REV of rock mass in the Brunswick mining area based on fracture strength (\(P_{30}\), \(P_{32}\)). Scholars have proposed REV considering various geometric characteristics of fractures [19]. [PERSON] and [PERSON] [14] reported that the typical structural REV has great practical value and comprehensively considers fracture density, size, inclination, and dip angle.  
It is difficult to obtain the spatial distribution of fractures. The characteristics of fractures obtained in the field mostly derive from a plane surface; thus, it is convenient to analyze the properties of fractured networks in the 2D dimension. Scholars have introduced the concept of topological structure into the analysis of fracture networks and used it to describe the relationship between geometric objects as well as to estimate the properties of the rock mass [14; 20], as the topological features do not change with the deformation of the fracture network compared with other geometric features.  
The REV is discussed here based on the topological structure, focusing on the two-dimensional fracture network. We propose the concept of TREV and verify the feasibility of using it to estimate REV. We generated 23 different kinds of rock mass fracture networks and analyzed the fluctuation of topological structure parameters within the research scale.