Analytical solution for one-dimensional consolidation of soft soils under a partially permeable boundary condition and a time-dependent loading considering the threshold hydraulic gradient

The placement of an incompressible cushion on the top of soft soil deposits affects the boundary drainage conditions. It has been well documented in the literature that the seepage through soft clays deviates from Darcy's law, which affects the dissipation of excess pore water pressure. However, the studies on the soil consolidation behavior under the partially permeable boundary condition induced by the incompressible cushion underlain by soil deposits considering the non-Darcian flow with the threshold hydraulic gradient have rarely been reported. In this study, a one-dimensional consolidation model under the partially permeable boundary condition and the time-dependent loading is developed. The effect of the threshold hydraulic gradient on the water flow through the soft soil is taken into account. The analytical solution is derived for the proposed consolidation model based on the eigen value method and Laplace transform. Based on the derived analytical solution, the sensitivity analysis has been conducted for the consolidation behavior under various boundary conditions, threshold hydraulic gradients and loading rates. It indicates that the smaller the partially permeable boundary parameter phi (i.e., the ratio of the product of the coefficient of permeability of the incompressible cushion and the thickness of the clay deposit to the product of the coefficient of permeability of the clay deposit and the thickness of the incompressible cushion, which indicates the permeability of the partially permeable boundary)is, the slower the dissipation of excess pore water pressure and the longer the duration of consolidation process are; the larger phi is, the faster the dissipation of excess pore pressure and the shorter the duration of consolidation process are. The larger the dimensionless variable R (the ratio of the product of the threshold hydraulic gradient, the unit weight of water and the thickness of soil layer to the surcharge) is, the longer the time when the seepage front (i.e., moving boundary) reaches the bottom of soil layer is, and the greater the residual excess pore water pressure at the end of consolidation is. At a given time, the settlement-based average degree of consolidation increases with an increase in the dimensionless variable R. On the contrary, the excess pore water pressure-based average consolidation degree decreases as the dimensionless variable R increases. The loading rate significantly influences the consolidation behavior of soils with threshold hydraulic gradient under the partially permeable boundary condition. It is found that the slower the loading is, the lower the peak value of excess pore water pressure is, and the longer the time it takes to reach its peak is. However, the loading rate has negligible influence on the residual excess pore water pressure.