Exact solution to predict excess pore pressures and settlement of unsaturated soil deposit due to uniform loading

Liem Ho, Behzad Fatahi, Hadi Khabbaz

Dans les comptes rendus d’articles de la conférence: GeoMontréal 2013: 66th Canadian Geotechnical Conference; 11th joint with IAH-CNC

Session: Soil and Rock Mechanics III

ABSTRACT: This paper explains a simple yet precise analytical solution for the nonlinear governing equations for one-dimensional consolidation of an unsaturated soil deposit using eigenfunction expansions and Laplace transform techniques. The mathematical development adopts two-way drainage condition for the unsaturated soil, in which the permeable top and base boundaries allow free dissipation of pore-air and pore-water pressures under uniform loading. 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 () and depth (), 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 permeability to water permeability ratio on the excess pore water and air pressure dissipation rates is investigated and discussed.

RÉSUMÉ: Ce papier présente une solution analytique simple et précise des équations non linéaires pour étudier la consolidation unidimensionnelle d'un dépôt de sol non saturé en utilisant des extension de fonction propre et des techniques de transformation de Laplace. Le développement mathématique adopte le drainage dans les deux sens pour les sols non saturés; les surfaces supérieure et inférieure sont perméables et permettent d™annuler les pressions d™air et d'eau interstitielles sous chargement uniforme. Les fonctions propres et les valeurs propres sont des parties de la solution générale et peuvent être obtenues sur la base des conditions aux limites. En outre, la pression interstitielle initiale uniforme peut être utilisée pour déterminer les coefficients initiaux généralisés de Fourier. Par ailleurs, la méthode de transformation de Laplace est adoptée pour résoudre les équations différentielles du premier ordre. Une fois les équations avec domaine transformé sont obtenues, les solutions finales, qui sont proposées pour être fonction du temps (t) et de la profondeur (z), peuvent être obtenues en adoptant une transformation de Laplace inversée. Un exemple pratique est fourni pour présenter les caractéristiques de consolidation des sols non saturés basé sur la solution proposée. L™importance de la perméabilité à l'air par rapport à celle à l'eau sur la surpression d™eau et sur les taux de dissipation de la pression d'air est étudiée et discutée. 1. INTRODUCTION The application of loads to unsaturated soil strata results in the emergence of excess pore-air () and pore-water () pressures. During the loading process, these excess pore pressures dissipate towards permeable surfaces and such activity eventually results in a gradual settlement of the soil. This phenomenon is called consolidation. Over the past five decades, the field of unsaturated soil mechanics, particularly consolidation theory, has gradually formed a solid background through numerous studies and experimental investigations. Some original research in this field was conducted by Scott (1963) and Barden (1965). In late 1970s, Fredlund and Hassan (1979) successfully proposed coupled governing equations describing the flow of air and water phases within a soil element. This pioneered work has inspired numerous investments on the consolidation settlement prediction of unsaturated soils. The last two decades have seen a robust growth in analytical and numerical methods to predict the dissipation rate of excess pore pressures and vertical settlement, which were developed by Chen (1993a & b), Qin et al. (2008, 2009 & 2010), Shan et al. (2011 & 2013), Zhou and Tu (2012), Zhou and Zhao (2013), and numerous others. Apart from noticeable numerical developments proposed by Wong et al. (1998), Conte (2004), and Zhou and Zhao (2013), the analytical methods have also been recently updated. Chen (1993a & b) has established the basic constitutive model for unsaturated soils referring to the effective stress concept introduced by Skempton (1960) and the Curie symmetry principle by Degroot and Mazur (1962). These principles were used to derive a model predicting consolidation behaviour of unsaturated soils. At the end of 2000s, Qin et al. (2008, 2010) included the concepts of water velocity () and mass rate of air () in the mathematical procedure to obtain a one-dimensional consolidation model based on the nonlinear governing equations proposed by Fredlund and Hassan (1979). The authors adopted Laplace transform and Cayley-Hamilton techniques to solve the nonlinear inhomogeneous partial differential equations (PDEs). Although the results gave a good agreement with the numerical prediction, the analytical method is relatively inconvenient for use due to its complex solutions. Shan et al. (2011), on the other hand,

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