## 3 Numerical Model for Multi-Cluster Hydraulic Fracturing in Shale Reservoirs with Natural Fractures
### Multi-Cluster Hydraulic Fracturing in Shale Reservoirs with Natural Fractures  
During the process of multi-cluster fracturing, the hydraulic fractures may intersect each other while propagating, which can significantly influence the trends in the fracture propagation and initiation pressure [21]. The objective of this study was to investigate the perforation fracturing process in shale horizontal wells with multiple clusters of natural fractures. A three-dimensional finite element model has been established for a subsurface setting approximately 1305 m deep, incorporating three horizontal wells and one natural fracture. As shown in Figure 9 below, the specific model had dimensions of 20 m \(\times\) 20 m \(\times\) 10 m and exhibited natural fracturing with a horizontal tilt angle of 15\({}^{\circ}\) and a length of 18 m (red area in Figure 3). Injection wells (Wells 1, 2, and 3) and natural fractures were preset. The spacing of two adjacent injection wells was 5 m, the water injection well was located at the central position of the model, and the three wells were injected simultaneously. The model was meshed using 16,710 elements, and COH3D8P elements (the COH3D8P element is a 12-node three-dimensional cohesive element) were set to simulate the 3D nonlinear fracture propagation on the planes vertical to the direction of the lowest horizontal ground stress of the reservoir. Figure 8: Quadratic nominal stress criterion. Figure 9: Multi-cluster hydraulic fracturing model for shale reservoirs with natural fractures. Based on the mechanical parameters of the shale, such as elastic compression modulus, Poisson's ratio, compressive strength, tensile strength, and tensile modulus, obtained in Tables 2-5 and actual engineering test data [24; 30], Table 6 presents the hydraulic fracturing simulation parameters for multi-cluster hydraulic fractures in the shale reservoirs with natural fractures. To investigate the effects of the variables associated with hydraulic fracturing, the case in Table 6 is called Case 1. Here, the vertical ground stress of the reservoir was 32 MPa, the maximum horizontal ground stress of the reservoir was 30 MPa, and the lowest ground stress in the horizontal direction of the reservoir was 28 MPa. In Case1, each well has an injection rate of 60 L/min, with all three wells injecting simultaneously. The other cases are listed in Table 7, which shows only the parameters that are different from Case 1. In the fracture propagation simulation, the hydraulic fractures may interact with the natural fractures. Therefore, it is necessary to apply a special treatment to the CZM elements to ensure the continuity of the fluid flow at the intersections of 3D fractures. Figure 10a shows the grid diagram at the intersection of these fractures; Figure 10b shows the arrangement diagram of the pore pressure nodes at the intersection between the 3D hydraulic fractures and natural fractures. Notably, each fracture element comprises nodes representing the pore pressure, indicated by the blue dots in Figure 10b. To achieve continuity, the pore pressure nodes located at the intersection of the fractures are combined, as depicted in Figure 10c. Specifically, the pore pressure node 1 of the natural fracture element is merged with the pore pressure node 3 of the adjacent natural hydraulic element. Similarly, the pore pressure node 2 of the natural fracture element is combined with the pore pressure node 4 of the adjacent hydraulic fracture element. By combining the pore pressure nodes at the intersection of the fractures, the pressure is transferred within the fluid. Consequently, the flow continuity at the fracture intersection is ensured. This enables the simulation of the propagation of 3D fractures at the intersection points.