Phase-Field Modeling of Fracture Propagation Patterns Under Proppant Support in Sequential Hydraulic Fracturing

Numerical simulation of sequential fracturing in horizontal wells for shale gas and oil extraction requires careful consideration of mechanical interactions between proppant and fracture surfaces—a challenge that remains largely unresolved. This study proposes a novel phase-field model featuring a strain-based formulation and a width-dependent proppant reaction force. Unlike previous studies, we integrate an empirical propped force solution, adapted from established work to account for rock properties and proppant support, to capture nonlinear fracture closure. Results show that reaction stress models significantly dictate propped geometry. The model’s fracture length, width, and closure predictions are validated against theoretical solutions. We conducted a sensitivity analysis to evaluate how fracture deflection angles and widths vary with dimensionless fracture spacing, in situ stress contrast, and proppant strength. Numerical results show that proppants induce pronounced morphological asymmetry and distinct geometric discrepancies. Specifically, the heterogeneous support provided by proppants and the resulting stress redistribution alter fracture propagation paths, leading to an 8% reduction in fracture length and a marked difference in fracture orientation of approximately 80° between supported and unsupported fractures, highlighting the important role of proppants in governing fracture geometry. Both dimensionless fracture spacing and in situ stress contrast strongly influence fracture deflection, with proppant strength also contributing. The propped-force formulation is further extended to nonplanar fractures, enabling application to sequential fracturing with multiple fractures. These results highlight fracture propagation mechanisms and demonstrate the robustness of the proposed phase-field model.