A double-porosity phase-field model with independent crack and matrix pressures, fixed-stress splitting, and a variational inequality for non-negative pressure reproduces K-, M-, and O-vertex KGD hydraulic-fracture solutions including fluid lag.
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A dual--continuum phase-field model for hydraulic fracturing: Viscosity-dominated regime and fluid lag
A double-porosity phase-field model with independent crack and matrix pressures, fixed-stress splitting, and a variational inequality for non-negative pressure reproduces K-, M-, and O-vertex KGD hydraulic-fracture solutions including fluid lag.