Phase-field hydraulic fracturing operator network based on En-DeepONet with integrated physics-informed mechanisms

Hydraulic fracturing in porous media, driven by fluid injection, presents a formidable computational challenge due to the intricate interplay of fluid flow and fracture mechanics. The phase-field method offers a powerful approach for modeling such complex phenomena, but its high computational demands limit its practical application in large-scale scenarios. This work introduces a phase-field hydraulic fracturing operator network aimed at efficiently predicting fracture propagation and facilitating in the design of fracturing strategies. We develop a multi-input, multi-physics operator network based on the Enriched-DeepONet framework, incorporating multiple root networks to simultaneously handle diverse physics fields while integrating physical laws into the training process. The governing physical equations are formulated using the widely recognized phase-field hydraulic fracturing model, with Darcy's law describing fluid flow in both fractures and the surrounding porous media. The hydraulic response across different computational domains is captured through interpolation of Darcy's parameters using an indicator function derived from the phase-field variable. This methodology allows for the comprehensive representation of hydraulic fracturing processes through coupled partial differential equations, enabling the solution within the operator network framework. By embedding physical constraints into the loss function, the proposed model achieves enhanced convergence and accuracy during training. The effectiveness of the proposed approach is demonstrated through three numerical experiments varying in permeability, in-situ stress, critical energy release rate, and Young's modulus. The results underscore the critical importance of integrating physical constraints to improve the accuracy of the training process. Our findings indicate that the developed phase-field hydraulic fracturing operator network is a promising advancement for enhancing the simulation capabilities of hydraulic fracturing processes.