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 lower-dimensional 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 three-point-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.

层状页岩的裂缝特征决定了水力压裂开采页岩气的效率。一种有效且稳健的数值方法可能在预测层状页岩裂纹扩展路径方面发挥重要作用。在这项研究中，开发了低维界面元素和切向导数变量来描述相场近似框架内界面处相场的演变。变分相场模型是基于新的能量框架推导出来的。低维界面与岩石矩阵一起离散化。采用分离耦合策略求解耦合系统，在每个时间步内依次求解方程。该模型通过进行两个 2D 经典基准测试和一个 3D 三点弯曲实验进行验证。此外，所提出的方法适用于对受拉力作用的二维带缺口方形页岩样本中的层状页岩破坏进行建模。硬到硬、软到硬和硬到软配置旨在研究具有完美结合和弱结合界面的层状岩石的断裂行为。结果表明，所提出的方法具有极高的鲁棒性，可用于评估界面断裂和裂纹-界面相互作用。硬到软配置旨在研究具有完美结合和弱结合界面的层状岩石的断裂行为。结果表明，所提出的方法具有极高的鲁棒性，可用于评估界面断裂和裂纹-界面相互作用。硬到软配置旨在研究具有完美结合和弱结合界面的层状岩石的断裂行为。结果表明，所提出的方法具有极高的鲁棒性，可用于评估界面断裂和裂纹-界面相互作用。