A General Analytical Solution for Axisymmetric Consolidation of Unsaturated Soil with Impeded Drainage Boundaries

The consolidation of soil is one of the most common phenomena in geotechnical engineering. Previous studies for the axisymmetric consolidation of unsaturated soil have usually idealized the boundary conditions as fully drained and absolutely undrained, but the boundaries of unsaturated soil are actually impeded drainage in most practical situations. In this study, we present a general analytical solution for predicting the axisymmetric consolidation behavior of unsaturated soil that incorporates impeded drainage boundary conditions in both the radial and vertical directions simultaneously. The impeded drainage boundary is modeled using the third kind boundary, and it can also realize fully drained and absolutely undrained ones by changing the drainage parameter. A general analytical solution is developed to predict the excess pore-air and pore-water pressures as well as the average degree of consolidation in an unsaturated soil stratum using the common methods of eigenfunction expansion and Laplace transform. The newly developed solution is expressed in the product of the terms of time, depth, and radius, which are derived using Laplace transform, usual Fourier, and Fourier-Bessel series, respectively. The eigenfunctions and eigenvalues are evaluated from the impeded drainage boundaries in both radial and depth dimensions. Then, the correctness of the proposed analytical solution is verified against the existing analytical solution for the case of traditional boundaries and against the finite difference solution for the case of general impeded drainage boundaries, and excellent agreements are obtained. Finally, the axisymmetric consolidation behavior of unsaturated soil involving impeded drainage boundaries is demonstrated and analyzed, and the effects of the drainage parameters are discussed. The results indicate that the larger drainage parameter generally expedites the dissipations of the excess pore pressures and further promotes the soil settling process. As the drainage parameter increases, its influence gradually diminishes and even can be neglected when it is larger than 100. The general analytical solution and findings of this study can help for better understanding the axisymmetric consolidation behavior of the unsaturated soil stratum in the ground improvement project with vertical drains as well as the gas-oil gravity drainage mechanism in the naturally fractured reservoirs.

The Effect of Biot Number on a One-Dimensional General Conduction Solution

Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tablular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.