A unified multi-phase-field model for Rayleigh-Damköhler fluid-driven fracturing

In geological systems where fractures are driven by low-viscosity reactive fluids (e.g., CO₂ fracturing), the leak-off of the reactive fluid from fractures into the rock matrix induces Rayleigh-Taylor instability, leading to the formation of fingering invasion regions that undergo chemical damage, thereby destabilizing fracture propagation. The fracture propagation is strongly coupled with the heterogeneous chemical damage. The significant variability of Rayleigh number (buoyancy-driven convection / diffusion) and Damköhler number (chemical reaction / advection) within a wide range causes various flow and fracture patterns. Based on the principle of virtual work, a unified multi-phase-field model is proposed to model the mechanics enhanced chemical damage and dissolution-assisted fracturing process. The distinct fracture (${\varnothing }_f$) and chemical damage (${\varnothing }_d$) phase field order parameters are introduced to characterize fracture energy, chemical free energy and dissolution interfacial energy. The two phase fields are tightly linked through a synergistic degradation of mechanical energy. The governing equations for the Rayleigh-Damköhler fluid-driven fracturing are derived from the variational formulation of the free energy and micro-force balance. Based on the model, dimensional analysis is employed to establish the scaling laws for rock failure modes. When leak-off fluid flow aligns with fracture propagation, critical curves distinguishing different damage morphology are identified in the phase diagram using penetration lengths. In scenarios where gravity induces a misalignment between leak-off fluid flow and fracture direction, the normalized fracture number (${\Pi }_f$) and chemical damage number (${\Pi }_d$) are summarized to construct a comprehensive phase diagram encompassing various unstable fluid leak-off structures and rock failure modes.