• Developed a nonlinear hydraulic fracture propagation criterion that incorporates the fracture process zone (FPZ) using the cohesive zone model (CZM).
• Derived an analytical model for FPZ length and established a relationship between FPZ length and breakdown pressure.
• The criterion accurately predicts breakdown pressure in deep reservoirs, overcoming limitations of linear elastic fracture mechanics (LEFM).
• Revealed that fracture propagation initiates when apparent stress intensity reaches apparent fracture toughness or in-situ stress intensity reaches in-situ fracture toughness.