On poroelastic strain energy degradation in the variational phase--field models for hydraulic fracture

Though a number of formulations have been proposed for phase--field models for hydraulic fracture, the definition of the degraded poroelastic strain energy varies from one model to another. This study explores previously proposed forms of the poroelastic strain energy with diffused fracture and assesses their ability to recover the explicit fracture opening aperture. We then propose a new form of degraded poroelastic strain energy derived from micromechanical analyses. Unlike the previously proposed models, our poroelastic strain energy degradation depends not only on the phase--field variable (damage) but also on the type of strain energy decomposition. Comparisons against closed form solutions suggest that our proposed model can recover crack opening displacement more accurately irrespective of Biot's coefficient or the pore--pressure distribution. We then verify our model against the plane strain hydraulic fracture propagation, known as the KGD fracture, in the toughness dominated regime. Finally, we demonstrate the model's ability to handle complex hydraulic fracture interactions with a pre--existing natural fracture.