## 4 Results and Analysis
### Influence of Ground Stress on Fracture Propagation
#### 4.2.1 Effect of Vertical Ground Stress  
To study the effect of vertical ground stress on fracture propagation, the vertical ground stress was varied from 32 MPa in Case 1 to 34 MPa for Case 2, while the other parameters were kept unchanged. Figure 13 shows the bottom-hole pressure and hydraulic fracture area at a corresponding time during injection for Case 2. In terms of the bottom-hole pressure, the curve of Well 2 initially maintains a higher level than those of Wells 1 and 3. However, in the later stages, the bottom-hole pressure curve of Well 2 becomes lower compared with those of Wells 1 and 3. Moreover, the well-bottom pressure curves of Wells 1 and 3 have similarities. There exists an evident difference in the area of the hydraulic fractures compared with that in Case 1. From the figure, it can be observed that the increase in hydraulic fracture area might influence the pressure distribution among the wells. A larger fracture area could result in more fluids entering the fracture system, thus reducing the bottomhole pressure. The growth of the hydraulic fracture area is not linear but rather exhibits nonlinear behavior, reflecting the complexity of the fracture extension process, such as closure, reopening, or other intricate geological effects. By analyzing these data, injection strategies can be optimized to maximize hydraulic fracture extension and enhance production efficiency. For instance, adjusting the injection rate or pressure can control the fracture's expansion to achieve the most favorable fracture geometry and maximum fracture area. Figure 14 shows the hydraulic fracture shape in Case 2 when the injection times are \(t=109\) s and \(t=140\) s. In Figure 14a, the hydraulic fracture in Well 2 interacts with the natural fracture at \(t=109\) s and expands upward, while the hydraulic fractures in Wells 1 and 3 expand in a downward direction due to the interference from hydraulic fractures. Figure 13: Bottom hole pressure and hydraulic fracture area with time in Case 2. The hydraulic fracture in Well 2, shown in Figure 14b, continues to expand within the existing natural fracture, and the natural fracture remains unopened. Figure 14 shows that the fracture shape in Case 2 is significantly different from that in Case 1, in which the hydraulic fracture induced by Well 3 interacts with the natural fracture, and the natural fracture is opened. Figure 15 shows the volume of the hydraulic fracture during injection under different vertical ground stress conditions in Cases 1 and Case 2; Figure 16 shows the filtration loss over time in the two cases. During the initial injection stage, the varying vertical ground stress does not significantly affect the volume of the hydraulic fracture and the extent of filtration loss. However, during the later stages of injection, the volume growth of the hydraulic fractures decelerates after the opening of the natural fracture in Case 1, whereas in Case 2, a strong linear correlation can be seen with the injection time. This phenomenon can be attributed to the fact that the filtration coefficient of the natural fracture exceeds that of the shale reservoir matrix. After the opening of the natural fracture in Case 1, as time progresses, the filtration loss in Case 1 becomes increasingly greater than that in Case 2, as shown in Figure 16.  
Figure 14: Shape of the hydraulic fractures in Case 2. (**a**) \(t=109\) s (**b**) \(t=140\) s.  
Figure 15: Volume of hydraulic fractures with time in Cases 1 and 2. Notably, in Figure 15, the volume of hydraulic fractures is measured based on the cumulative volume from three wells. Assuming each well has an injection rate of 60 L/min (equivalent to 0.06 m\({}^{3}\)), the combined injection rate for all three wells would be 0.18 m\({}^{3}\). According to this rate, the theoretical volume of fluid injected after 60 s would be 0.18 m\({}^{3}\). However, Figure 11 illustrates that at the 60 s mark, the observed fracture volume is approximately 0.12 m\({}^{3}\).