## 4 Numerical Results and Discussion
### Flow Behaviours in DFNs
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Elastic** & **Normal** & **Cohesion** & **Friction** & **Poisson’s** & **Density** \\  & **Modulus** & **Stiffness** & **Stiffness** & **Angle** & **Ratio** & **Density** \\ \hline Matrix & 10 GPa & / & 1.6 MPa & 45\({}^{\circ}\) & 0.3 & 2000 kg/m\({}^{3}\) \\ Fractures & / & 40 MPa/mm & / & / & 0.25 & 1500 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material parameters of the matrix and fractures. Figure 14: Model of a DFN. which cannot be ignored during this stage. The transmissivity decreases significantly with the hydraulic gradient, which indicates that there is a strong inertial region in which the cubic law is inapplicable. Figure 16 also shows that with an increase in in-situ stress, the permeability of the fracture network decreases because the larger normal stress will lead to a smaller hydraulic aperture. However, due to only a small change in aperture under the normal stress, there is no change in the magnitude of the critical hydraulic gradient (it remains between approximately \(10^{-3}\) and \(10^{-2}\)). It was also proven that three regions from linear to nonlinear exist with an increase in the hydraulic gradient in the numerical study carried out by [PERSON] et al. [50] and [PERSON] et al. [57], which is similar to the results of this paper. Figure 17 shows that with increasing confining pressure, both the linear coefficient A and nonlinear coefficient B increase. However, with an increasing fracture aperture, both the linear coefficient and nonlinear coefficient decrease. The fracture fits more closely due to the increase in confining pressure; larger confining stress leads to greater closure of a fracture, and the smaller the aperture is, the smaller the inherent permeability will be, which leads to an increase in Coefficient A. The close fitting of the fracture surface increases the difficulty of water flow, and the increase in contact area leads to the increase in water flow curvature, which increases Coefficient B. Similarly, an increase in the initial fracture aperture weakens this effect. The nonlinear coefficient increases more than the linear coefficient when the confining pressure increases or the fracture aperture decreases, showing that the nonlinear term is more sensitive to changes in the confining pressure and aperture than the linear term. Through the nonlinear flow experiment in a single fracture, carried out by [PERSON] et al. [53], the influence of confining pressure on nonlinear flow characteristics was studied. The results show that Co  
Figure 16: Relationship between Q/J and J.  
Figure 15: Relationship between J and Q.  
increase with an increase in confining pressure, which is consistent with the study results obtained by the numerical simulation of a fracture network in this paper. Figure 18 shows the local velocity contour and vector diagram of the fracture intersection area. Figure 18a indicates that the existence of the fracture intersection affects the continuous distribution of velocity in the fracture. As shown in Figure 18b, the direction and velocity of the flow at the fracture intersection will be determined according to the length and hydraulic aperture of fractures and the pressure gradient, and its direction is no longer consistent with the direction of the fracture. A deviation from the direction will increase the energy loss caused by water flow and enhance its inertia, which will accelerate the occurrence of nonlinear flow. Figure 17: (**a**) Relationship between the linear coefficient A and confining pressure; (**b**) relationship between the nonlinear coefficient B and confining pressure. Figure 18: _Cont._