A Depth-Dependent Framework for Grouting Parameter Selection in Anisotropic Strata: From Diffusion Mechanism to Pareto-Optimal Decision-Making

Grouting is a critical technique for ground reinforcement and seepage control in underground engineering. However, traditional empirical designs often neglect the anisotropic diffusion mechanism in layered fractured rock and the increasing failure risk in deep, high-pressure environments. This study elucidated the "horizontal-then-vertical" diffusion pathway of the Bingham fluid in layered jointed rock masses. A three-dimensional (3D) spatial diffusion model was established to derive the attenuation law of upward grout climb height. A reliability-based framework was developed using Monte Carlo Simulation (MCS) to account for the spatial variability of fracture apertures. The framework defines grouting failure is quantitatively using a filling rate threshold of η ≥ 80%). Furthermore, a multi-objective optimization model was developed to address the conflict between sealing reliability and comprehensive cost, including drilling footage and operational time. The results demonstrated that hydrostatic pressure in deep zones significantly restricted the effective injection pressure, which caused non-linear increase in failure probability ( P f ). Pareto Front analysis revealed a distinct "Pareto shift" with increasing depth. This shift indicates a necessary transition from a "long-stage, loose-spacing" strategy to a "short-stage, dense-spacing" strategy. A region-wide optimal borehole spacing of R d = 6.0 m was identified as the most robust baseline. Accordingly, the recommended stage length ( L ) decreased from 15.1 m in shallow zones (0–50 m) to 5.1 m in deep zones (150–200 m). This study provides a quantitative decision-making tool for dynamic grouting design and supports high construction efficiency without compromising sealing integrity.