A dual--continuum phase-field model for hydraulic fracturing: Viscosity-dominated regime and fluid lag

The phase-field model regularizes sharp fractures into a diffuse representation, blurring the boundary between the fracture and the intact material. This blurring makes it difficult to capture distinct domain processes in hydraulic fracturing, where Reynolds flow governs the fracture and Darcy flow describes the surrounding porous matrix. Consequently, the blurred delineation artificially smears the pressure field across the fracture--matrix interface, which is acceptable in toughness-dominated hydraulic fracturing regimes where pressure drops within the fracture are negligible. However, in viscosity-dominated regimes, typically for actual subsurface injections due to high injection rates, the fluid pressure drops more drastically, and the fluid front may even lag behind the propagating fracture tip, a phenomenon that a smeared pressure field cannot capture. Despite its relevance, the viscosity-dominated regime has not been addressed by any existing phase-field models to date, likely due to its numerical instability. In this study, we propose a dual--continuum phase-field model based on double-porosity microporomechanics that explicitly separates mesoscale crack pressure from micropore pressure. The framework provides a variationally consistent formulation alongside phase-field--dependent poroelasticity. To ensure the numerical stability of the hydromechanical coupling, a fixed-stress split scheme is modified for two independent fluid pressures, while a variational inequality constraint is applied to reproduce fluid lag. Verified against the closed-form solutions in toughness-dominated, viscosity-dominated, and early-time transitional regimes, the model accurately captures complex fluid flow behavior and transient fluid lag within the fracture, and opens a new frontier for applying phase-field models to realistic viscosity-dominated hydraulic fracturing.