## 4 Results and Analysis
### Hydraulic Fracture Propagation in Case 1  
Figure 11 shows the bottom-hole pressure, hydraulic fracture area, and fracture shape at the corresponding time for Case 1. As shown in Figure 11a, the bottom-hole pressures in Wells 1, 2, and 3 do not differ significantly during the initial injection phase. The bottom-hole pressure increases from 12 MPa to 31 MPa and then drops rapidly to 18 MPa. With the decrease in the bottom-hole pressure, the hydraulic fractures enlarge. During this stage, the hydraulic fractures gradually expand, and the PFOPEN (the opening displacements of a crack) in Figure 11b (\(t\) = 20 s) shows that the fracture shape is semi-circular. In Figure 11c (\(t\) = 75 s), the three hydraulic fractures continue to expand and interfere with each other. The hydraulic fracture in Well 3 that extended further upward near the natural fracture in Well 2 expanded vertically along the \(Z\) axis, which represents the maximum horizontal ground stress, while the fracture in Well 1 expanded horizontally along the Y direction. When the injection time reaches \(t\) = 90 s, the total area of the hydraulic fractures quickly increases, and the bottom-hole pressures of Wells 1 and 2 remain the same, while the bottom-hole pressure of Well 3 drops to 17 MPa, as shown in Figure 11a. From Figure 11d, an interaction between the hydraulic fracture from Well 3 and the pre-existing natural fracture can be observed. As the injection time reaches \(t\) = 140 s, the hydraulic fractures continue to spread along the natural fractures, as depicted in Figure 11e. The hydraulic fractures in Well 2 extended in the direction of the highest horizontal ground stress, and the hydraulic fractures in Well 1 expanded vertically in the negative direction of the \(Y\)-axis. The pressure curve and hydraulic fracture morphology obtained from the numerical calculation were in agreement with the experimental results [17]. During the actual process of hydraulic fracturing, it is crucial to highlight the significance regarding the fact that the bottom-hole pressure could be relatively stable after fracture expansion, rather than constantly fluctuating, as shown in Figure 11a. The hydraulic fracture area should also increase linearly rather than in a stepwise manner. This is  
Figure 10: Diagram of shared pore pressure at fracture intersections (a) Grid diagram at the intersection of fractures, (b) Arrangement diagram of the pore pressure nodes at the intersection, (c) Pore pressure nodes located at the intersection of the fractures. because the expansion of the hydraulic fractures by the CZM depends on whether the next cohesive element is damaged, and the damaged cohesive elements are opened stepwise. Figure 12 shows the curve of the hydraulic fracture volume with time during injection. Evidently, the hydraulic fracture volume increases suddenly after the natural fracture opens at around 9 s. This is because the filtration coefficient of the natural fractures is greater than that of the shale reservoir matrix. While the injection rate remains constant, the filtration hydraulic volume undergoes a more rapid propagation, thereby decreasing the volume expansion of the hydraulic fractures. In summary, the graph furnishes valuable insights into the dynamic nature of hydraulic fracturing processes and underscores the necessity of accurately modeling and understanding both artificial and natural fractures. This is essential for optimizing well performance and maximizing hydrocarbon recovery rates. Figure 11 shows that the hydraulic fracture through Well 3 expands the fastest, gradually approaches the natural fracture, and then expands along the natural fracture. Due to the effect of hydraulic fracture interference, the hydraulic fracture in Well 2 expands toward the horizontal direction of the minimum ground stress, whi  
Figure 11: Bottom-hole pressure, hydraulic fracture area, and fracture shape at the corresponding time in Case 1.