Analysis Of Consolidation Characteristics Of Unsaturated Soils Enhanced By Vertical Drains Under Complicated Conditions

In infrastructure construction,foundation treatment is usually carried out by methods such as vertical drain combined with surcharge preloading,and the deformation of the treated foundation is dominated by consolidation settlement.Most of the existing consolidation settlement deformation calculations generally use Terzaghi’s consolidation theory,while the soils located above the infiltration line are often unsaturated.Therefore,it is necessary to conduct corresponding research on the consolidation theory of unsaturated soils.The very few studies on the consolidation of unsaturated foundations with vertical drains have been conducted under simple loadings,conventional boundaries and initial conditions,which are far from satisfying the analytical requirements of complex situations in practical engineering.Based on the three-dimensional(3-D)consolidation theory of unsaturated soil proposed by Dakshanamurthy et al.and the assumption of equal strain,the analytical solution or semi-analytical solution for the axisymmetric consolidation of unsaturated soil with vertical drains considering various complex factors such as time-varying loadings,well resistance,smear effect,and continuous seepage boundary are solved respectively,for the two cases of considering only radial flow and considering both radial and vertical flow,and the consolidation characteristics are analyzed.Firstly,a set of axisymmetric consolidation controlling equations established in the two cases is integrated and substituted into the radial boundary conditions to obtain the coupled non-homogeneous partial differential governing equations of the radial average excess pore pressures.Secondly,the semi-analytical and analytical solutions for the consolidation of unsaturated foundations with vertical drains under various complex conditions are obtained by applying the Laplace transform,the decoupling technique of differential equations,the inverse Laplace transform(semi-analytical solution),as well as the homogenization of boundary conditions and eigenfunction method(analytical solution),respectively.Then the corresponding procedures are prepared and the reliability of the obtained solutions is verified using degenerate and comparative validation methods.Finally,the consolidation characteristics of the vertical drain foundation are analyzed utilizing typical calculations in unsaturated soils.The main contents of the paper are as follows:(1)According to the three-dimensional consolidation theory of unsaturated soil proposed by Dakshanamurthy et al.,a set of governing equations for axisymmetric consolidation of unsaturated vertical drain foundation under time-varying loadings is derived by combining the basic assumptions,Darcy’s law,and Fick’s law,taking into account only radial flow and combined radial and vertical flow.The correctness of the derivation is verified based on the method of coordinate transformation.(2)Based on a set of controlling equations for unsaturated soil improved by vertical drains considering radial flow subjected to time-varying loadings,the system is combined with the assumption of equal strain,and the constraints when considering well resistance and smear effect are introduced in the solution process.Then,the Laplace transform and decoupling techniques are used to obtain axisymmetric consolidation solutions for unsaturated vertical drain foundation in the Laplace transform domain;and a semi-analytic solution in the time domain is obtained after implementing numerical Laplace inversion by using the Crump method.After verification,the effects of well resistance,smear effect,and loading types on the consolidation characteristics are analyzed by applying examples.The effects of air and water phases well resistance factors,air and water phases smear factors,loading types,and loading parameters on the consolidation characteristics of unsaturated soil enhanced by vertical drains are systematically summarized.(3)By introducing continuous permeable boundary conditions,the analytical solution for the consolidation of unsaturated soil enhanced by vertical drains under instantaneous uniform loading,considering the radial and vertical flow,is solved by using the method of the homogenization of boundary conditions,eigenfunction method,and decoupling technique of intermediate variables.The correctness of the solutions is verified by comparing them with existing analytical solutions.An example is used to simulate the arbitrary distribution of permeability at the upper and lower boundaries,which makes up for the current inability to consider the permeability between permeable and impermeable or asymmetric distribution at the top and bottom boundaries.(4)A semi-analytical solution for 3-D consolidation of unsaturated soil with vertical drains considering both radial and vertical flow under continuous permeability boundaries is obtained by further considering time-varying loadings and smear effect.After the correctness of the solution is verified by programming,the influence of the ratio of radial and vertical permeability coefficient and the parameters of the upper and lower boundary on the dissipation of excess pore pressure is analyzed through examples.