A Three‐Dimensional Spatiotemporal Evolution Model for Two‐Phase Grouting Flow in Water‐Bearing Porous Media Based on the Forchheimer‐Modified Brinkman Equation

ABSTRACT Water‐bearing gravelly sand layers pose significant risks to underground construction due to their loose structure, high permeability, and susceptibility to seepage‐induced failure. To better understand and control grout migration in such formations, a three‐dimensional spatiotemporal model for grouting diffusion in water‐bearing porous media was developed using the Forchheimer‐modified Brinkman equation, with the level‐set method employed to capture the dynamic slurry‐water interface. The model incorporates viscous diffusion (Brinkman term) and inertial resistance (Forchheimer term), allowing characterization of both low‐Reynolds‐number shear‐dominated flow and nonlinear inertial effects at higher Reynolds numbers. The coupled framework simultaneously resolves the flow field, pressure field, and interface migration. Numerical simulations were compared with transparent‐chamber dynamic water grouting experiments and showed reasonable agreement in slurry diffusion morphology, pressure evolution, and asymmetric propagation in the with‐flow and against‐flow directions. Results show that the hydraulic gradient strongly controls slurry diffusion under flowing‐water conditions. The diffusion front advances farther downstream, while upstream propagation is restricted by hydraulic back pressure. The pressure field evolves from an initial transient state to a quasi‐steady regime, with pressure accumulation near the injection point and asymmetric attenuation along the flow direction. Slurry velocity decreases progressively with time, and high‐viscosity zones expand and shift downstream. The proposed model provides a useful numerical tool and theoretical basis for optimizing grouting design and predicting slurry diffusion in complex hydrogeological environments.