Analytical solution for one-dimensional consolidation of unsaturated soil deposit subjected to step loading

Abstract

This paper discusses a simple yet precise analytical solution for one-dimensional (1-D) consolidation of an unsaturated soil deposit subjected to a step loading. This solution is derived from nonlinear governing equations of flow using eigenfunction expansions and Laplace transform techniques. In addition, the mathematical development adopts one-way drainage condition for the unsaturated soil, in which the top boundary is permeable to the air and water phases whereas the base is impervious to these phases. Eigenfunctions and eigenvalues are parts of the general solution and can be obtained based on the proposed drainage boundary condition. Furthermore, uniformly distributed initial pore pressures can be used to determine the initial generalised Fourier coefficients. The Laplace transform method is adopted to solve the first-order differential equations. Once the equations with transformed domain are obtained, the final solutions, which are proposed to be functions of time (t) and depth (z), can be achieved by taking an inverse Laplace transform. A worked example is provided to present the consolidation characteristics of unsaturated soils based on the proposed solution. Significance of air to water permeability ratio on the excess pore-water and pore-air pressure dissipation and compression is investigated and discussed.