A unified Bingham–Forchheimer theory for grout transport in saturated porous media and its implications for representative strata

Grouting is widely used for ground reinforcement and seepage control, yet the transition of grout transport in saturated porous media from permeation to fracture-driven diffusion remains insufficiently quantified. This study develops a unified Bingham–Forchheimer formulation that simultaneously accounts for yield stress, inertial loss, and a threshold pressure gradient, and embeds it into a three-stage coupled framework spanning unsteady seepage, post-stagnation pressure redistribution, and subsequent fracture propagation. The governing equations are solved using an in-house numerical implementation, and the solution procedure is assessed through stability analysis. The framework is applied to two representative saturated strata at a depth of 50 m, corresponding to a high-permeability sandy formation and a low-permeability clayey aquiclude, to reveal contrasting transport regimes and controllability. Results indicate that, in sandy formations, grout migration is permeation-dominated up to front stagnation: increasing injection pressure enlarges the stagnation radius but with diminishing returns, higher yield stress elevates the effective threshold and promotes earlier stagnation, while viscosity primarily stretches the time scale of pressure redistribution rather than the ultimate reach. In clayey aquicludes, permeation is strongly threshold-controlled and tends to remain confined; once stagnation occurs, fractures initiate rapidly near the borehole, followed by channelization and outward growth that redistribute the near-well pressure peak and partially reactivate the previously stagnated zone. These findings provide design implications for tailoring injection pressure and grout rheology, by which formation-specific, precision-controlled grouting and desired reinforcement outcomes can be achieved.