One-Dimensional Consolidation Analysis of Unsaturated Soils Subjected to Time-Dependent Loading

During consolidation, excess pore-air and pore-water pressures are forced to dissipate through permeable boundaries. This dissipation process inevitably results in the reduction of the soil volume, and thus settlement. Such a phenomenon can be mathematically described by inhomogeneous governing equations of flow according to Fick’s (with respect to air phase) and Darcy’s (with respect to water phase) laws. This paper discusses the dissipation of excess pore-air and pore-water pressures and settlement of an unsaturated soil layer subjected to various time-dependent external loadings. An analytical solution is derived from the governing flow equations with respect to air and water using eigenfunction expansion and Laplace-transform techniques. Eigenfunctions and eigenvalues are parts of the general solution and can be obtained using one-way or two-way drainage boundary conditions. On the other hand, four types of external loadings, namely ramping, asymptotic, sinusoid, and damped sine wave, are mathematically simulated and incorporated in the mathematical procedure. Once the time variable ($t$) in partial differential equations is transformed into the Laplace complex argument ($s$), generalized Fourier coefficients can be obtained by taking a Laplace inverse, and eventually the final solution can be obtained. The consolidation behavior of the unsaturated soil subjected to the previously mentioned types of time-dependent loads will be presented in separate examples. Additionally, the permeability ratio ($k_a/k_w$) and the loading function parameters influencing changes in dissipation rates of excess pore pressures and settlement are investigated and discussed in this paper. It is observed that the increasing permeability ratio and loading parameters have significant effects on the change in the pore pressures.