An exponential porosity distribution model for analyzing grout penetration in deteriorated subgrades

Fine particle migration induced by rainfall infiltration and traffic loading can increase soil porosity and contribute to subgrade deterioration. However, the continuous variation of porosity with depth caused by this process is rarely quantified as a continuous mathematical function in existing models. This study proposes an exponential model to describe the vertical distribution of porosity in affected subgrades. The model is defined by three parameters: surface porosity (n 0 ), deep undisturbed porosity (n c ), and decay coefficient (λ). Model parameters are estimated using limited field monitoring data to reconstruct the vertical porosity profile. Results indicate that n 0 primarily influences porosity in the shallow layer, n c represents the baseline porosity of deep soil, and λ controls the rate of decrease with depth. The resulting porosity distribution affects grout penetration behavior through its influence on soil permeability. A higher n c increases the horizontal penetration radius (R) and vertical penetration distance (D) while reducing the diffusion cone angle (θ). Increasing n 0 enlarges R and θ, whereas a larger λ decreases R and D and leads to a stabilized θ. The proposed model provides a quantitative description of porosity variation with depth and offers a basis for analyzing grout penetration in subgrade remediation.