## 2 Research Methodology
### Model Configuration  
The target formation is located in the western region of China. The lithology of this reservoir is highly variable, composed of various mineral components. The upper part of the formation consists mainly of litharenite, lithic feldspar fine sandstone, litharenite sandstone, and argillaceous fine sandstone. The target interval is distributed between 3100 m and 3350 m. Based on well logging data and rock mechanics experiments, the factors influencing fracture propagation during the multi-cluster fracturing process were studied, and a model was established as shown in Figure 2.  
During the actual hydraulic fracturing process, rocks are subjected to highly complex geological conditions. In order to facilitate calculation and analysis in establishing the hydraulic fracturing model, the following assumptions need to be made:  
1. The formation rock is assumed to be a homogeneous and isotropic porous medium, and it is in a linear elastic state. 2. The influence of temperature field on fracture propagation is neglected. 3. The inertial effects of the fluid are disregarded. 4. The fracturing fluid is fully saturated and incompressible. During the fracturing process, the physical and chemical interactions between the fracturing fluid and the surrounding rock formation are not considered. A three-dimensional model was created using the finite element method (FEM). The initial dimensions of the model were set to 120 m 60 m 120 m. Three injection points, or perforations, were initially set up with a spacing of 20 m between each point. The formation consists of three layers, with the mid layer being the reservoir with a thickness of 10 m, and the upper and lower layers acting as interlayers with a thickness of 25 m each. The x direction represents the direction of maximum principal stress, the y direction represents the direction of minimum principal stress, and the z direction represents the direction of vertical stress. The fractures propagate in the direction of maximum principal stress. Therefore, cohesive elements were embedded in the xoz plane to simulate the initiation and propagation of fractures in the reservoir. The model was refined with a finer mesh in the mid reservoir layer, and to account for the influence of the interlayer, a 5 m region near the reservoir on both sides of the interlayer was also refined, while the rest of the mesh was relatively sparse, totaling 93,000 grid cells. The rock elements of the formation were set as C3D8P, and the cohesive elements representing the pre-existing fractures were set as COH3D8P. Displacement constraints of 0 were applied in all directions. The total duration of the hydraulic fracturing process was set to 400 s. The simulation of the hydraulic fracturing process mainly consists of two steps: (1) the equilibrium stage of the in situ stresses, and (2) the hydraulic fracturing stage. The basic calculation parameters are shown in Table 1.  
The PKN (Perkins-Kern-Nordgren) model [14; 15; 34] is one of the classic models used to describe crack propagation during hydraulic fracturing. It assumes that the expansion of cracks can be approximated by an ellipsoid or ellipsoidal volume. The core idea behind this model is the pattern of crack expansion under the influence of hydraulic fracturing, particularly applicable when the cracks have a relatively large length-to-width ratio. Before initiating multi-fracture propagation simulations, we conducted simulations of a single crack and compared the results with those calculated using the PKN model, as shown in Figure 3.  
To ensure the convergence of the model, a ramp-up injection amplitude was applied to the volume before the start of fracturing, gradually increasing the fracturing volume from 0 to the set value within the first 20 s. Therefore, the results obtained from the simulation before 20 s show relatively small differences but are generally  
Figure 2: Three-dimensional model establishment. (**a**) A multi-cluster three-dimensional hydraulic fracturing model.