The stress interference between fracturing clusters in horizontal wells is key in affecting the initiation and propagation of multi-cluster fractures. However, in reservoirs such as shale, there are often changes in the mechanical properties of the rock along the direction of the wellbore. Such changes are often not considered in models for multi-fracture propagation. As a result, the impact of rock mechanical heterogeneity on induced stress is not fully understood. Therefore, in this paper, a prediction model of the three-dimensional induced stress field of hydraulic fractures is established, taking into account mechanical heterogeneity. Combined with the prediction model of fracturing pressure, the changes in induced stress and initiation pressure under different conditions are analyzed. The results indicate a significant difference between the induced stress results considering mechanical heterogeneity and those considering only a single homogeneous region. The change in elastic modulus has a profound influence on induced stress and initiation pressure. Perforating at locations with low elastic modulus in situ can help reduce initiation pressure. Additionally, temporarily blocking the wellbore and reducing the flow rate to decrease the net pressure in the fractures can facilitate more fractures to initiate.