A unified temporal-spatial scaling law for hydraulic fracturing in layered rock formations

This work presents a unified scaling law for a plane strain hydraulic fracture propagation in layered rocks, where fracturing behavior is influenced by both layer thickness and property contrasts across interfaces, leading to non-self-similar propagation over time. The singular integral balance equation is derived by introducing a kernel function that accounts for the varying elastic modulus and a modified load term that incorporates in-situ stress. The layered distribution of elastic modulus and fracture energy is considered using dynamic tip asymptotics. The governing equations and boundary conditions are then made dimensionless by new characteristic scales of fracture opening, fluid pressure, and fracture length, which are proposed to depict the spatial relationship between the interface and fracture. Solutions are obtained through a decoupled approach to match the fracture morphology and tip boundary conditions. This model captures a novel time-sensitive propagation mode of a hydraulic fracture, that can be dominated by toughness in the tip region and viscosity dissipation at the interface region, especially in multilayer rocks. Consequently, the temporal scale of injection time and the spatial scale of layer thickness are integrated into the characteristic scales. A scaling law is thus proposed to fully consider the variations of elastic modulus, fracture energy, fluid injection rate, injection time, and layer thickness on propagation behavior. The law spans the temporal-spatial parameter space within a general framework that quantifies the evolution of viscosity dissipation induced by interfaces, summarizing four specific cases: the homogeneous model, single-interface model, multi-layer model, and homogenized model. The proposed law provides a universal measure for modeling hydraulic fracturing of layered heterogeneous rocks with arbitrary thickness and propagation stage.