Analytical solution to one-dimensional consolidation in unsaturated soils

Abstract

This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund's one-dimensional consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy's law and Fick's law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soil from analytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct.

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

“…X(z, s) at any depth By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified Fredlund's governing equations of water and air, Darcy's law, and Fick's law, the transfer function between the state vectors X(0, s) at the top and X(z, s) at any depth is constructed as follows [11] :…”

“…[11] for Case 1. The semi-analytical solutions are compared with the results of the finite difference method proposed by Fredlund and Hasan [4] for Case 2.…”

“…Fredlund's consolidation theory [5] is one of the most popular theories in the geotechnical engineering community. Qin et al [11] have obtained the analytical solutions to the one-dimensional consolidation theory for unsaturated soils under the large-area uniform instantaneous loading, while the soil top surface is permeable and the bottom surface is impermeable to water and air (one surface which is permeable to air and water) based on Fredlund's one-dimensional consolidation theory in unsaturated soils. For simple transform problems, we can obtain the analytical expressions of inverse transform by using the table of the Laplace transform.…”

Comparison of Overconsolidated Clay Settlements Calculated by Analytical 1d Terzaghi Consolidation and Biot Numerical Analysis

“…Besides that, there are several other assumptions: (1) soil is homogenous, (2) soil in saturated condition, (3) solid and water particle are not easy to be compressed, (4) air obeys Boyle's law, (5) Darcy's Law is applied along hydraulic gradient, (6) permeability coefficient and volumetric coefficient are constant during consolidation process, (7) the effects of temperature change, air dissolved in water, air diffusion, and generation and diffusion of vapor are disregarded (Qin et al, 2008). In order to calculate the settlement, thickness (H), coefficient of compression (C c ) and coefficient of recompression (C r ) are needed.…”

Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition

“…Qin et al (2008) adopted the Laplace transform method and gave an analytical solution for an unsaturated single-layer soil subjected to a large-area uniform load, with the top surface being permeable and the bottom surface being impermeable to air and water. Qin et al (2010) obtained an analytical solution for unsaturated single-layer soil subjected to a gradually-increasing load using the same method and employing the same boundary condition as Qin et al (2008). Shan et al (2012) used the separation of variables method to give exact solutions for unsaturated single-layer soil subjected to an arbitrary load and with three types of boundary conditions.…”