Triple-phase-field modeling and simulation for mixed-mode fracture of bedded shale

Abstract

Comprehending the intricate mechanics and fracture characteristics of shale holds significant importance in the realm of controlling the reconstruction of shale gas reservoirs. It is noteworthy that shale typically exhibits a distinctive bedding structure which sets it apart from other geological formations. Previously, single-mode cracks in shale have been widely studied by former researchers. However, its matrix and bedding may undergo various fracture modes under loading due to the bedded structure. To effectively capture the mixed-mode fracture behavior observed, a triple-phase-field model is proposed in this paper. Three phase-field variables are introduced herein to track pure-tension, tensile-shear, and compressive-shear cracks in the matrix and bedding, respectively. The energetic driving forces governing the propagation of pure-tension and tensile-shear cracks are formulated through the spectrum decomposition of positive strain tensor with its volumetric and deviatoric components; while the energetic driving force that propels the progression of compressive-shear cracks is derived by employing the Mohr-Coulomb criterion within the context of the negative strain tensor. For the bedding plane, lower-dimensional interfacial elements are adopted to capture the mixed-mode fracture. Based on the above triple-phase-field model, both 2D and 3D problems are modeled with good results. The heterogeneity of material is also considered. The simulations reveal that the compression strength and fracture mode of bedded shale are highly related to the orientations of the bedding plane and pre-existing fissure.