Numerical Study on Hydraulic Fracture Propagation in Sand–Coal Interbed Formations

To investigate hydraulic fracture propagation in multi-layered porous media such as sand–coal interbedded formations, we present a new phase-field-based model. In this formulation, a diffuse fracture is activated only when the local element strain exceeds the rock’s critical strain, and the fracture width is represented by orthogonal components in the x and y directions. Unlike common PFM approaches that map the permeability directly from the damage field, our scheme triggers fractures only beyond a critical strain. It then builds anisotropy via a width-to-element-size weighting with parallel mixing along and series mixing across the fracture. At the element scale, the permeability is constructed as a weighted sum of the initial rock permeability and the fracture permeability, with the weighting coefficients defined as functions of the local width and the element size. Using this model, we examined how the in situ stress contrast, interface strength, Young’s modulus, Poisson’s ratio, and injection rate influence the hydraulic fracture growth in sand–coal interbedded formations. The results indicate that a larger stress contrast, stronger interfaces, a greater stiffness, and higher injection rates increase the likelihood that a hydraulic fracture will cross the interface and penetrate the barrier layer. When propagation is constrained to the interface, the width within the interface segment is markedly smaller than that within the coal-seam segment, and interface-guided growth elevates the fluid pressure inside the fracture.