# Deterministic Discrete Fracture Network Model and Its Application in Rock Mass Engineering
## 2 Acquisition of Structural Surface Parameters
### Structural 3D Modeling of Rock Mass
The expected value of the cosine of the acute angle \(\gamma\) between the normal vector of structural planes and the sampling window plane, approximated using directional samples, is given as follows:  
\[E^{(k)}[cos(\gamma)]=\frac{\sum_{i=1}^{N^{(k)}}\sqrt{1-sin^{2}\alpha_{i}^{(k) }sin^{2}\beta_{i}^{(k)}}}{N^{(k)}} \tag{2}\]  
where \(E^{(k)}[cos(\gamma)]\) denotes the expected value of \(cos(\gamma)\) formed by the kth joint set. \(\alpha_{i}^{(k)}\) and \(\beta_{i}^{(k)}\) represent the dip direction and dip angle of the ith sample in the kth structural plane group, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **VCS1** & **VCS2** & **CS1** & **CS2** & **MS1** & **MS2** & **WS1** & **WS2** & **VWS** \\  & **0.04 m** & **0.06 m** & **0.13 m** & **0.2 m** & **0.4 m** & **0.6 m** & **1.3 m** & **2 m** & **4 m** \\ \hline VHP & & & & & & & & \\
20 m & 0.079 & 0.053 & 0.024 & 0.015 & 0.008 & 0.005 & 0.002 & 0.001 & 0.000 \\ HP & & & & & & & & \\
15 m & 0.141 & 0.094 & 0.043 & 0.028 & 0.014 & 0.009 & 0.004 & 0.002 & 0.001 \\ MP1 & & & & & & & & \\
10 m & & & & & & & & \\ MP2 & 0.753 & 0.502 & 0.231 & 0.150 & 0.075 & 0.050 & 0.023 & 0.015 & 0.007 \\ 6.5 m & & & & & & & & \\ LP1 & & & & & & & & \\
3 m & & & & & & & & \\ LP2 & & & & & & & & \\
2 m & & & & & & & & \\ VLP & & & & & & & & \\
1 m & & & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 1: The Relationship Between Spacing, Persistence, and Volumetric Density. By combining the trace line density \(\rho_{m}^{(k)}\) with the expected value of the cosine of the acute angle \(\gamma\), we can derive the formula for the volumetric density \(\rho_{V}^{(k)}\):  
\[\rho_{V}^{(k)}=\frac{\rho_{m}^{(k)}}{E^{(k)}[cos(\gamma)]\cdot[D^{2}/W+D]} \tag{3}\]  
From this, we can obtain the relationship between the fracture volumetric density and the fracture persistence [29; 30]. By referring to Table 1, which relates volumetric density, spacing, and persistence, we can determine the average size of the disk model. 2 Calculation of the disk center coordinates:  
After obtaining the persistence and using the trace length and the midpoint of the trace obtained previously, we can calculate the center coordinates of the disk using the following formula:  
\[\left\{ \begin{aligned} a&=-\sqrt{\left[(1/2)D^{2} \right]-\left[(1/2)l^{2}\right]}cos(\beta_{i}-\beta_{Q})\\ b&=-\sqrt{\left[(1/2)D^{2}\right]-\left[(1/2)l^{2} \right]}sin(\beta_{i}-\beta_{Q})\\ c&=\sqrt{\left[(1/2)D^{2}\right]-\left[(1/2)l^{2} \right]}cos\alpha_{i}\end{aligned} \right. \tag{4}\]  
Among them, \((a,b,c)\) represents the direction from the center of the disk model to the center of the trace, \(\beta_{Q}\) is the smaller quadrant angle when a structural surface is oriented such that it lies within the first quadrant and its \(\beta_{Q}\) is 0 or 360, \(D\) is the model size, and \(l\) is the trace length. \(\alpha_{i},\beta_{i}\) represent the dip direction and dip angle, respectively. Dividing the equation into quadrants:  
For example, when \(\beta_{i}\in(90,180)\):  
\[\left\{ \begin{aligned} a&=-\sqrt{\left[(1/2)D^{2} \right]-\left[(1/2)l^{2}\right]}cos(\beta_{i}-90)\\ b&=-\sqrt{\left[(1/2)D^{2}\right]-\left[(1/2)l^{2} \right]}sin(\beta_{i}-90)\\ c&=\sqrt{\left[(1/2)D^{2}\right]-\left[(1/2)l^{2} \right]}cos\alpha_{i}\end{aligned} \right. \tag{5}\]  
The relationship between the center coordinates of the disk and the center coordinates of the trace is:  
\[\left\{ \begin{aligned} x_{i}&=x_{j}-\frac{a \sqrt{\left[(1/2)D^{2}\right]-\left[(1/2)l^{2}\right]}}{\sqrt{l^{2}+l^{2}+c^{ 2}}}\\ y_{i}&=y_{j}-\frac{b\sqrt{\left[(1/2)D^{2}\right]- \left[(1/2)l^{2}\right]}}{\sqrt{l^{2}+l^{2}+c^{2}}}\\ z_{i}&=z_{j}-\frac{c\sqrt{\left[(1/2)D^{2}\right]- \left[(1/2)l^{2}\right]}}{\sqrt{l^{2}+l^{2}+c^{2}}}\end{aligned} \right. \tag{6}\]  
\(P_{i}=(x_{i},y_{i},z_{i})\) is the center coordinates of the disk, and \(P_{j}=\left(x_{j},y_{j},z_{j}\right)\) represents the center coordinates of the trace.