## 3 Results and Analysis  
### Effect of Fracturing Cluster Number  
To investigate the behavior of multi-cluster fracturing under reservoir conditions in the target area, the fracture propagation results under different cluster numbers were first compared, and the results are shown in Figure 3. With inter-cluster distance increases, the fracturing simulation results of different numbers of clusters show different fracture propagation characteristics. As the number of clusters increases, the number of fracture elements and total area show an increasing trend. However, from single-cluster fracturing to triple-cluster fracturing, the increase in the number and area of fractures is relatively small. This indicates that as the number of clusters increases, the propagation of fractures is influenced by inter-cluster interactions. Concurrently, as the inter-cluster distance increases and the number of clusters is augmented, a reduction in fracture volume is observed. Intriguingly, the proportion of tensile failure within the fracture network shows a significant increase. This phenomenon suggests that single-cluster fracturing faces challenges in creating large-scale fractures. The injected fluid is primarily used to create and lengthen fractures. In contrast, multi-cluster fracturing facilitates the development of a more intricate \begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Value** \\ \hline Young’s modulus (GPa) & 40 \\ Poisson’s ratio & 0.22 \\ Permeability coefficient (m/s) & \(1\times 10^{-7}\) \\ Porosity & 0.04 \\ Tensile strength of natural fractures (MPa) & 2 \\ Critical damage displacement of natural fractures (m) & 0.0001 \\ Tensile strength of matrix interfaces (MPa) & 6 \\ Critical damage displacement of matrix interfaces (m) & 0.001 \\ Injection rate (m\({}^{3}\)/min) & 19–20 \\ Fracturing fluid viscosity (mPa-s) & 1 \\ Pipe roughness (mm) & \(0.015\times 10^{-3}\) \\ Perforation diameter (m) & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters used in simulation models. Figure 2: Schematic diagram of FDEM models with natural fractures: (**a**) 20 natural fractures; (**b**) 30 natural fractures; (**c**) 40 natural fractures; (**d**) 50 natural fractures. fracture network and enhances the overall fracture volume. These observations underscore the influence of cluster configuration on fracture morphology and highlight the potential benefits of using multi-cluster fracturing strategies to optimize reservoir stimulation and enhance hydrocarbon recovery. At a simulation time of 1000 s, the ultimate distribution of fractures is illustrated in Figure 4. It is noteworthy that as the number of clusters rises, the spacing between clusters diminishes within the model. Consequently, the total surface area covered by the induced fractures amplifies, whereas the aperture of fractures emanating from the injection point decreases. This finding indicates that with an increasing number of fracturing clusters, the area of medium-pressure fractures in the reservoir expands. However, as the total injection volume remains constant, there is a slight decreasing trend observed in the fracture aperture. Furthermore, the total volume of visible fractures decreases. These observations suggest that variations in the number of clusters and cluster spacing influence fracture morphology. The model tends to produce a higher quantity of smaller fractures influenced by a combination of factors including the model mesh, the presence of natural fractures, the number of fracturing clusters, and geo-stress conditions. This leads to an increase in the total fracture surface area and a reduction in the visible fracture volume. This behavior highlights the complex interplay between fracturing parameters and the resulting fracture network geometry.