## 4 Analysis of Factors Influencing Hydraulic Fracture Propagation Behavior  
### Influence of Pumping Rate on Multi-Fracture Propagation  
Understanding the influence of fluid flow on fracture propagation is essential for optimizing hydraulic fracturing parameters. The pumping rate is a critical factor affecting the fracturing outcome. In this study, injection rates of 5, 10, 40, 60, and 80 m\({}^{3}\)/h are tested. The corresponding simulation times are set to 480, 240, 60, 40, and 30 min, respectively, to ensure that the total injected fluid volume remains constant across all cases. As shown in Figure 4, with an increase in injection rate, the damage zone around the fractures gradually expands, and the extent of damage increases. This indicates that higher injection rates induce greater stress disturbance in the rock, making microcracks more likely to initiate, propagate, and coalesce, thereby enlarging the damage area around the fractures. Figure 4a shows that at an injection rate of 5 m\({}^{3}\)/h, the fracture development is not clearly visible. This may be due to the low injection rate causing significant fluid loss into the formation. Therefore, the available fluid pressure is insufficient to overcome the in situ stress and rock tensile strength, preventing effective fracture initiation. The distribution of Misses stress under different pumping rates is shown in Figure 5. The results demonstrate that increasing the injection rate enhances stress interactions between fracture clusters, leading to a greater degree of deflection in the central fracture. Under high injection rates, the heterogeneity of fracture damage becomes more pronounced, with localized damage concentration observed. This may be attributed to the uneven distribution of fluid pressure, which increases the stress variation across different locations and lead to a more complex damage distribution. Due to the effects of heterogeneity, the  
Figure 3: Comparison between the results of the numerical model and the KGD model. The distribution of Misses stress under different pumping rate is shown in Figure 5. The results demonstrate that increasing the injection rate enhances stress interactions between fracture clusters, leading to a greater degree of deflection in the central fracture. Under high injection rates, the heterogeneity of fracture damage becomes more pronounced, with localized damage concentration observed. This may be attributed to the uneven distribution of fluid pressure, which increases the stress variation across different locations and lead to a more complex damage distribution. Due to the effects of heterogeneity, the rock's resistance to deformation varies along different hydraulic fracture paths, causing the fracture to expand asymmetrically or even exhibit multiple-wing propagation. Additionally, the stress induced by pre-existing fractures can lead to deflection in the growth direction of subsequent fractures. Figure 6 shows that, as the pumping rate increases, the lengths of the three fractures first increase and then decrease. Among these, the fracture length increased by 98% as the injection rate was raised from 5 to 40 m\({}^{3}\)/h, peaking at the maximum tested rate of 40 m\({}^{3}\)/h. Subsequently, as the rate increased to 80 m\({}^{3}\)/h, the fracture length decreased by 24%. This indicates that a higher injection rate can initially enhance fracture propagation length. However, when the injection rate reaches a certain level, the injection time decreases due to the fixed fluid volume. The reduced duration of fluid application to the fracture results in a shorter fracture length. Figure 4: Fracture propagation characteristics under different pumping rate conditions. The colours indicate the degree of damage. The higher the value, the higher the degree of damage. (**a**) 5 m\({}^{3}\)/h; (**b**) 10 m\({}^{3}\)/h; (**c**) 40 m\({}^{3}\)/h; (**d**) 60 m\({}^{3}\)/h; (**e**) 80 m\({}^{3}\)/h. Figure 7 shows that, as compressive strength increases, the damage zone around fractures shrinks significantly.