## 4 Discussion
### Effect of Cluster Quantity  
Keeping other parameters constant, when the inter-cluster spacing is 20 m, the number of perforations is set to 1, 2, 3, 4, and 5, respectively. Different numbers of pre-existing fractures are created, with the fractures propagating in the same direction as the maximum principal stress. The morphology and extension of the fractures under different cluster quantities are analyzed and studied. The simulation results are shown in Figures 8 and 9. Figure 8 reflects the relationship between the width and height of edge/mid fractures. Since there is no clear division between edge and mid fractures when there is one cluster or two clusters, a comparison is made between the fracture morphology in these two cases and the other cases. It can be observed that for edge fractures, the height of the fracture reaches the maximum in all five cases. The width is maximum when there is  
Figure 8: Illustration of the relationship between fracture width and height. (**a**) Relationship between width and height of the edge fracture. (**b**) Relationship between width and height of the mid fracture. Figure 9: Illustration of the relationship between fracture width and half-length. (**a**) Relationship between width and half-length of the edge fracture. (**b**) Relationship between width and half-length of the mid fracture. a normal extension behavior of the fracture without the influence of other fractures. When there are two clusters, the width is minimum, as the two fractures expand simultaneously and interact with each other. The remaining three cases have curves located between the curves of one cluster and two clusters. The expansion of the front end of the mid fracture hinders the extension of the front end of the edge fracture due to the loss of fracturing fluid into the formation. The fracturing fluid in the edge fracture mainly affects the width expansion, resulting in a larger maximum width compared to one cluster and two clusters. The other three curves overlap, indicating consistent influence on the edge fracture, with a maximum difference of 13% between the curves. For mid fracture, the maximum width decreases in the order of one cluster, two clusters, four clusters, three clusters, and five clusters. The width of three clusters is smaller than four clusters because the mid fracture in three clusters is influenced by the two adjacent edge fractures, while the mid fracture in four clusters is influenced by one edge fracture on one side and another mid fracture on the other side, with the influence of the other mid fracture being smaller than that of the edge fracture. Therefore, the mid fracture in three clusters is more affected by the two edge fractures and has a smaller width compared to four clusters. The height of the fractures in four clusters and five clusters does not reach the maximum, with a maximum difference of 56% between the curves. Figure 9 reflects the relationship between the width and half-length of edge/mid fractures. The expansion pattern of width-half-length for edge fractures is consistent with the expansion pattern of width-height, and the width of the fracture increases while the half-length decreases. When there is one cluster or two clusters, the width of the mid fracture decreases with the increase in the half-length. When there are three clusters, four clusters, and five clusters, the width of the fracture initially increases and then decreases with the increase in the half-length. This indicates that the mid fracture in three clusters, four clusters, and five clusters is initially more influenced by the edge fracture, resulting in a smaller width. The fracturing fluid mainly promotes the forward expansion of the fracture, and as the half-length of the fracture increases, the width increases. When the mid fracture is no longer influenced by the edge fracture, the width of the fracture increases. When the cluster spacing is 20 m with three clusters, interactions between the edge and central clusters begin to affect each other.