Diffusion Model of Rapid‐Setting Slurry in Micro‐Fractures Considering Spatiotemporal Viscosity Variation Characteristics: Constant Flow and Constant Pressure Grouting Models

In grouting engineering, constant flow and constant pressure modes are widely used, yet existing models struggle to accurately describe grout diffusion, especially with spatiotemporal viscosity variations. To address this, diffusion models for grout under both modes were derived using Newton's fluid mechanics principles and the rheological properties of rapid‐setting slurry. These models were developed through force balance analysis of micro‐elements, mathematical integration, and numerical solutions. The study examines diffusion radius, velocity, and pressure distribution in micro‐fractures, analyzing scenarios with and without viscosity variations. It evaluates the impacts of flow rate, pressure, viscosity, and fracture geometry on diffusion behavior. A three‐dimensional numerical grouting model, employing the two‐phase flow level set method, was developed to simulate the diffusion process, validating the derived models’ reliability. The effects of grouting time, fracture size, flow rate, and pressure on diffusion characteristics were systematically analyzed. A quantitative comparison of constant pressure and constant flow modes was conducted to guide mode selection in grouting engineering. Key findings include: (1) In constant‐flow mode, viscosity changes cause non‐linear pressure decreases as the diffusion radius grows, requiring higher pressures for sustained flow, impacting equipment costs during extended grouting. (2) In constant‐pressure mode, viscosity increases flow resistance, slowing expansion, with radius and velocity stabilizing over time. (3) Constant flow ensures stable diffusion for precise control, while constant pressure enables faster diffusion with significant pressure fluctuations, ideal for rapid filling. This study provides critical insights for optimizing grouting operations and enhancing efficiency in complex geological conditions.