Variational phase-field model based on lower-dimensional interfacial element in FEM framework for investigating fracture behavior in layered rocks

The fracture characteristics of layered shale rock determine the efficiency of shale gas production through hydraulic fracturing. An effective and robust numerical method may play a significant role in predicting the crack propagation path in layered shale. In this study, a lower-dimensional interfacial element and tangential derivative variable are developed to describe the evolution of the phase-field at the interface within the framework of the phase-field approximation. The variational phase-field model is derived based on a new energetic framework. The lowerdimensional interface is discretized together with the rock matrix. A separated coupling strategy is adopted to solve the coupled system, and the equations are solved sequentially during each time step. The model is validated by conducting two 2D classical benchmark tests and a 3D threepoint-bending experiment. Further, the presented approach is applied to modeling layered shale failure in 2D notched square shale specimens subjected to tension. The stiff-to-stiff, soft-to-stiff, and stiff-to-soft configurations are designed to investigate the fracture behavior in layered rocks with perfectly-bonded and weakly-bonded interfaces. The results indicate that the proposed method exhibits extreme robustness and can be applied to evaluate the interfacial fracture and crack-interface interaction.