## 4 Numerical Results and Discussion
### Case Study for Underground Engineering  
Considering the influence of nonlinear flow on the underground excavation process, to accurately predict the flow discharge in the excavation process, a 2D network model of fractured rock masses with tunnel excavation is shown in Figure 19. The statistical distribution parameters of the two groups of fractures are shown in Table 7. The burial depth of the tunnel is 500 m, the boundary length of the model is 200 m, and the diameter of the tunnel is 16 m. The in-situ stress of the calculation model is determined according to the rock density and buried depth. The horizontal in-situ stress \(\sigma_{\rm x}\) and vertical in-situ stress \(\sigma_{\rm y}\) are 10 MPa, and the lateral pressure coefficient is 1. Table 8 lists the material parameter values of the surrounding rock matrix and fracture. The material parameters are assigned to CPE6 element for the rock matrix and CPE6 MP element for a fracture, respectively, before solving by ABAQUS. As shown in Table 9, the fracture apertures are set to 0.1 mm, 0.5 mm, and 1 mm. The upper and lower boundaries of the model are set to have a water pressure of 0.1-1 MPa. The water pressure of the excavation face is set to zero, and the left and right boundaries are impermeable. When the water pressure increases from 0.1 MPa to 1 MPa, the flow rates, calculated based on the developed model for fracture apertures of 0.1 mm, 0.5 mm, and 1 mm, are in the ranges of 9.77 \(\times\) 10\({}^{-5}\)-8.07 \(\times\) 10\({}^{-4}\) m\({}^{3}\)/s, 6.26 \(\times\) 10\({}^{-4}\)-5.19 \(\times\) 10\({}^{-3}\) m\({}^{3}\)/s, and 2.39 \(\times\) 10\({}^{-3}\)-1.47 \(\times\) 10\({}^{-2}\) m\({}^{3}\)/s, respectively. According to Figure 20, higher water pressure and a larger aperture lead to a greater flow rate. When the fracture aperture is 0.1 mm and 0.5 mm, the flow increases almost linearly with an increase in water pressure. However, the flow no longer increases linearly with an increase in water pressure when the aperture is 1 mm, and the increase rate decreases gradually. Figure 19: Two-dimensional model of fractured rock mass. Figure 18: **(a)** The contour of the velocity; **(b)** the vector diagram of the velocity. As Figure 21 shows, when the water pressure increases from 0.1 MPa to 1 MPa, the flow rates, calculated based on the developed model for fracture apertures of 0.1 mm, 0.5 mm, and 1 mm, are in the ranges of 98-807, 626-5191, and 2386-14,740, respectively. The Reynolds number increases gradually with an increase in water pressure. When the hydraulic aperture are 0.1 mm, 0.5 mm, and 1 mm, the water pressure corresponding to the critical Reynolds number is about 0.38 MPa, 0.2 MPa, and 0.09 MPa, respectively, which \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Mean Value (Expectation)** & **Standard Variance** \\ \hline Trace length (m) & 20 & 4 \\
18 & 2 \\ \hline Dip (\({}^{\circ}\)) & 30 & 5 \\
120 & 7 \\ \hline Spacing & \multicolumn{2}{c}{Random distribution} \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Distribution parameters of the two groups of fractures. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Elastic** & **Normal** & **Cohesion** & **Friction** & **Poisson’s** & **Density** \\  & **Modulus** & **Stiffness** & **Angle** & **Ratio** & **Density** \\ \hline Matrix & 10 GPa & / & 1.5 MPa & 42\({}^{\circ}\) & 0.27 & 2150 kg/m\({}^{3}\) \\ Fractures & / & 40 MPa/mm & / & / & 0.25 & 1850 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Material parameters of the rock matrix and fractures. Figure 20: Relationship between P and Q.