## 4 Numerical Results and Discussion
### Influence of In situ Stress  
To investigate the influence of in situ stress on hydraulic fracture propagation, we kept the minimum in situ stress at 50 MPa, while the maximum in situ horizontal stress ranged \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Young’s modulus & 30.2 GPa & Initial pore pressure & 36.5 MPa \\ Poisson’s ratio & 0.2 & Porosity & 0.1 \\ Tensile strength of the rock & 3/6/9 MPa & Permeability & \(2.23\times 10^{-7}\) m/s \\ Cementing strength of & & coefficient & \\ the NFs 1 & 0.5/1/3 MPa & Leak-off coefficient & \(1\times 10^{-13}\) m/(Pa-s) \\ Minimum in situ & 50 MPa & Injection rate & 0.01 m\({}^{3}\)/s \\ Maximum in situ & & & \\ horizontal stress & 65 MPa & Fluid viscosity & 0.001 Pa-s \\ \hline \hline \end{tabular} \({}^{\dagger}\) NFs means natural fractures.  
\end{table} 
Table 1: Relevant parameters for the model. Figure 7: Schematic diagram of the natural fractures network model established using global cohesive elements. (**a**) Mesh generation of the natural fractures network model. (**b**) The mesh is locally enlarged to show the location of the natural fracture more clearly. (**c**) Mesh before embedding cohesive element. (**d**) Mesh after global embedding cohesive elements. Figure 11: The in situ stress difference at 10 MPa. (**a**) Schematic diagram of the fracture morphology. (**b**) The propagation process of the hydrofractures in a natural fracture network. Figure 10: The in situ stress difference at 5 MPa. (**a**) Schematic diagram of the fracture morphology. (**b**) The propagation process of the hydrofractures in a natural fracture network. To understand the propagation pattern of the hydraulic fractures, we analyzed the step-by-step fracture development when the in situ stress differences were 0 MPa, 5 MPa, 10 MPa, and 15 MPa, as shown in Figures 9-12. When the horizontal in situ stress difference was 0 MPa, the hydrofracture in the natural fracture network was no longer a symmetrical bi-wing fracture but complicated and tortuous. It initiated at the ends of the perforation, then expanded along the right-hand natural fractures, and finally formed a main fracture at about 60\({}^{\circ}\) to the vertical. After the main fracture above the perforation developed for a short distance, a bifurcation occurred and, then, stopped cracking quickly. The other branch at the bifurcation continued to propagate in the direction of the right-sloping natural fractures and turned to the 45\({}^{\circ}\) slowly. When it came to the fracture below the perforation, a secondary fracture appeared in the direction of the left-sloping natural fractures with several obvious turns during the propagation. In addition, burry-like microfractures were observed on the edges of major fractures. To figure out the mechanism of fracture propagation in a natural fracture network, we extracted the coordinates of natural cracks and hydraulic fractures at different times, as depicted in Figure 9b. The 0 MPa in situ stress difference means that there was no preferential direction. As weak planes in the rock, the location of natural fractures might observably attract the hydrofracture. Thus, the induced fracture propagated along the nearest right-sloping natural fractures after it initiated, since they were closer than the left-sloping ones. Thereafter, the fracture tip continuously captured the nearest weak plane to extend. At the intersection of natural fractures in different directions, the two natural fractures may be reactivated simultaneously, where fracturing fluid distributed and lead to bifurcation and secondary fracture. When the in situ stress difference was 5 MPa, a tortuous hydraulic fracture was formed with no secondary fracture, as illustrated in Figure 9. Influenced by the difference in in situ stress, the main fracture propagated along the right-sloping natural fractures at the outset but later deviated toward the direction of maximum in situ horizontal stress.