## 2 Characterization of Random Fracture Networks in Broken Rock Masses
### Mathematical Description of Stochastic Model for Geometric Parameters of Structural Surfaces
#### 2.2.1 Shape and Size of Structural Surfaces  
In the literature, most studies assumed a circular or elliptical shape for the structural surfaces for simplicity. Nonetheless, some research considered a polygonal shape [25]. [PERSON] et al. [26] proposed the use of a circular shape to describe structural surfaces in block rock masses and a rectangular shape for those in layered rock masses. In homogeneous rock masses, it is reasonable to represent a single structural surface by a circle or an ellipse. However, in broken rock masses, due to the criss-crossing and interacting cracks, the shape of the structural surfaces is polygonal. Therefore, this paper uses planar quadrilateral (rectangle) to simulate the shape of the structural surfaces in broken rock masses. The dimensions of structural surfaces are generally described by the fracture trace length, which is the intersection of the fracture surface with the exposed rock surface. As of now, it is not possible to directly measure the size of fracture surface in three-dimensional space under field conditions. The size can only be derived from the fracture trace length obtained from a field survey based on relevant theories. From the perspective of probability statistics, the distribution models of fracture trace length mainly include log-normal distribution [27] and negative exponential distribution [28]. The density functions of these two distributions are shown in Figure 3. It can be seen from Figure 3a that, when adopting log-normal distribution to describe the fracture trace length, i.e., when \(x\) equals the fracture trace length, the probability of occurrence for very short fracture trace lengths approaches zero. This can be attributed to the limitations of existing observation methods in measuring fine cracks, which may not accurately represent real cases, as a large number of laboratory experiments have observed numerous micro-cracks in the rock masses. To better describe the distribution of fracture trace length in broken rock masses, this study adopted the negative exponential distribution, as illustrated in Figure 3b. The figure demonstrates a notable increase in the number of cracks as the fracture trace length decreases, aligning more closely with the experimental results. Figure 2: Schematic diagram of rock mass structural plane collected by statistical window method. To establish the dimension of structural surfaces, a rectangular statistical window \(a\times b\) is first set up on the rock outcrop face (see Figure 2). Assuming that the numbers of structural surfaces whose traces are in cutting, intersecting, and containing relationships with the statistical window are \(N_{0}\), \(N_{1}\) and \(N_{2}\), respectively, and the angle between the traces of this group of structural surfaces and the length direction of the statistical window is \(\theta\), the average trace length of this group of structural surfaces, \(l\), can be expressed as:  
\[l=[ab(1+N_{0}/N)-N_{2}/N]/[(1-N_{0}/N+N_{2}/N)\left(aA+bB\right)] \tag{1}\]  
where \(N=N_{0}+N_{1}+N_{2}\), \(A=\int_{\theta_{1}}^{\theta_{n}}f(\theta)\sin\theta d\theta\), and \(B=\int_{\theta_{n}}^{\theta_{n}}f(\theta)\cos\theta d\theta\). \(f(\theta)\) is the probability density function of \(\theta\), and \(\theta_{l}\) and \(\theta_{u}\) are the lower and upper values of \(\theta\), respectively. [PERSON] and [PERSON] provided a relationship between the structural surface trace length and the diameter of circular structural surface [29]. Based on this relationship, the mean and variance values of the structural surface diameter, denoted as \(D\) and \({\sigma_{D}}^{2}\), can be expressed as the following equation.