A three‐phase model for unsaturated soils

A comprehensive framework to define the constitutive behaviour of unsaturated soils is developed within the theory of mixtures applied to three‐phase porous media. Each of the three phases is endowed with its own strain and stress. Elastic and elastic–plastic constitutive equations are developed. Particular emphasis is laid on the interactions between the phases both in the elastic and plastic regimes. Nevertheless, the clear structure of the constitutive equations requires a minimal number of material parameters. Their identification is provided: in particular, it incorporates directly the soil–water characteristic curve. Crucial to the formulation is an appropriate definition of the effective stress. The coupled influence of this effective stress and of suction makes it possible to describe qualitatively many of the characteristic features observed in experiments, e.g. for normally consolidated soils, a plastic behaviour up to air entry followed by an elastic behaviour at increasing suctions, and, on the way back, an elastic behaviour, unless compression is applied in which case plastic collapse occurs. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical analysis of a one‐dimensional infiltration problem in unsaturated soil by a seepage–deformation coupled method

A multiphase coupled elasto‐viscoplastic finite element analysis formulation, based on the theory of porous media, is used to describe the rainfall infiltration process into a one‐dimensional soil column. Using this framework, we have numerically analyzed the generation of pore water pressure and deformations when rainfall is applied to the soil. A parametric study, including rainfall intensity, soil–water characteristic curves, and permeability, is carried out to observe their influence on the changes in pore water pressure and volumetric strain. From the numerical results, it is shown that the generation of pore water pressure and volumetric strain is mainly controlled by material parameters α and n′ that describe the soil–water characteristic curve. A comparison with the laboratory results shows that the proposed method can describe very well the characteristics observed during the experiments of one‐dimensional water infiltration into a layered unsaturated soil column. Copyright © 2010 John Wiley & Sons, Ltd.     

A density‐dependent elastoplastic hydro‐mechanical model for unsaturated compacted soils

This paper presents a three‐dimensional elastoplastic constitutive model for predicting the hydraulic and mechanical behaviour of unsaturated soils. It is based on experimental results obtained from a series of controlled‐suction triaxial tests on unsaturated compacted clay with different initial densities. Hydraulic hysteresis in the water‐retention behaviour is modelled as an elastoplastic process, with the elastic part modelled by a series of scanning curves and the elastoplastic part modelled by the main drying and wetting curves. The effect of void ratio on the water‐retention behaviour is studied using data obtained from controlled‐suction wetting–drying cyclic tests on unsaturated compacted clay with different initial densities. The effect of the degree of saturation on the stress–strain‐strength behaviour and the effect of void ratio on the water‐retention behaviour are considered in the model, as is the effect of suction on the hydraulic and mechanical behaviour. The initial density dependency of the compacted soil behaviour is modelled by experimental relationships between the initial density and the corresponding yield stress and, thereby, between the initial density and the normal compression line. The model is generalized to three‐dimensional stress states by assuming that the shapes of the failure and yield surfaces in the deviatoric stress plane are given by the Matsuoka–Nakai criterion. Model predictions of the stress–strain and water‐retention behaviour are compared with those obtained from triaxial tests with different initial densities under isotropic compression, triaxial compression and triaxial extension, with or without variation in suction. The comparisons indicate that the model accurately predicts the hydraulic and mechanical behaviour of unsaturated compacted soils with different initial densities using the same material constant. Copyright © 2006 John Wiley & Sons, Ltd.     

A hierarchical thermo‐hydro‐plastic constitutive model for unsaturated soils and its numerical implementation

Unsaturated soils are highly heterogeneous 3‐phase porous media. Variations of temperature, the degree of saturation, and density have dramatic impacts on the hydro‐mechanical behavior of unsaturated soils. To model all these features, we present a thermo‐hydro‐plastic model in which the hydro‐mechanical hardening and thermal softening are incorporated in a hierarchical fashion for unsaturated soils. This novel constitutive model can capture heterogeneities in density, suction, the degree of saturation, and temperature. Specifically, this constitutive model has 2 ingredients: (1) it has a “mesoscale” mechanical state variable—porosity and 3 environmental state variables—suction, the degree of saturation, and temperature; (2) both temperature and mechanical effects on water retention properties are taken into account. The return mapping algorithm is applied to implement this model at Gauss point assuming an infinitesimal strain. At each time step, the return mapping is conducted only in principal elastic strain space, assuming no return mapping in suction and temperature. The numerical results obtained by this constitutive model are compared with the experimental results. It shows that the proposed model can simulate the thermo‐hydro‐mechanical behavior of unsaturated soils with satisfaction. We also conduct shear band analysis of an unsaturated soil specimen under plane strain condition to demonstrate the impact of temperature variation on shear banding triggered by initial material heterogeneities.     

Finite element formulation and algorithms for unsaturated soils. Part II: Verification and application

The finite‐element formulation and integration algorithms developed in Part I are used to analyse a number of practical problems involving unsaturated and saturated soils. The formulation and algorithms perform well for all the cases analysed, with the robustness of the latter being largely insensitive to user‐defined parameters such as the number of coarse time steps and error control tolerances. The efficiency of the algorithms, as measured by the CPU time consumed, does not depend on the number of coarse time steps, but may be influenced by the error control tolerances. Based on the analyses presented here, typical values for the error control tolerances are suggested. It is also shown that the constitutive modelling framework presented in Part I can, by adjusting one constitutive equation and one or two material parameters, be used to simulate soils that expand or collapse upon wetting. Treating the suction as a strain variable instead of a stress variable proves to be an efficient and robust way of solving suction‐dependent plastic yielding. Moreover, the concept of the constitutive stress is a particularly convenient way of handling the transition between saturation and unsaturation. Copyright © 2003 John Wiley & Sons, Ltd.     

Consolidation in spatially random unsaturated soils based on coupled flow‐deformation simulation

This paper integrates random field simulation of soil spatial variability with numerical modeling of coupled flow and deformation to investigate consolidation in spatially random unsaturated soil. The spatial variability of soil properties is simulated using the covariance matrix decomposition method. The random soil properties are imported into an interactive multiphysics software COMSOL to solve the governing partial differential equations. The effects of the spatial variability of Young's modulus and saturated permeability together with unsaturated hydraulic parameters on the dissipation of excess pore water pressure and settlement are investigated using an example of consolidation in a saturated‐unsaturated soil column because of loading. It is found that the surface settlement and the pore water pressure profile during the process of consolidation are significantly affected by the spatially varying Young's modulus. The mean value of the settlement of the spatially random soil is more than 100% greater than that of the deterministic case, and the surface settlement is subject to large uncertainty, which implies that consolidation settlement is difficult to predict accurately based on the conventional deterministic approach. The uncertainty of the settlement increases with the scale of fluctuation because of the averaging effect of spatial variability. The effects of spatial variability of saturated permeability k_sat and air entry parameters are much less significant than that of elastic modulus. The spatial variability of air entry value parameters affects the uncertainties of settlement and excess pore pressure mostly in the unsaturated zone. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical solution to 1D coupled water infiltration and deformation in two‐layer unsaturated soils

An analytical solution to 1D coupled water infiltration and deformation in layered soils is derived using a Laplace transformation. Coupling between seepage and deformation, and initial conditions defined by arbitrary continuous pore‐water pressure distributions are considered. The analytical solutions describe the transient pore‐water pressure distributions during 1D, vertical infiltration toward the water table through two‐layer unsaturated soils. The nonlinear coupled formulations are first linearized and transformed into a form that is solvable using a Laplace transformation. The solutions provide a reliable means of comparing the accuracy of various numerical methods. Parameters considered in the coupled analysis include the saturated permeability (k_s), desaturation coefficient (α), and saturated volumetric water content (θ_s) of each soil layer, and antecedent and subsequent rainfall infiltration rates. The analytical solution demonstrates that the coupling of seepage and deformation plays an important role in water infiltration in layered unsaturated soils. A smaller value of α or a smaller absolute value of the elastic modulus of the soil with respect to a change in soil suction (H) for layered unsaturated soils means more marked coupling effect. A smaller absolute value of H of the upper layer soil also tends to cause more marked coupling effect. A large difference between the saturated coefficients of permeability for the top and bottom soil layers leads to reduced rainfall infiltration into the deep soil layer. The initial conditions also play a significant role in the pore‐water pressure redistribution and coupling effect. Copyright © 2011 John Wiley & Sons, Ltd.     

ACMEG‐TS: A constitutive model for unsaturated soils under non‐isothermal conditions

This paper introduces an unconventional constitutive model for soils, which deals with a unified thermo‐mechanical modelling for unsaturated soils. The relevant temperature and suction effects are studied in light of elasto‐plasticity. A generalized effective stress framework is adopted, which includes a number of intrinsic thermo‐hydro‐mechanical connections, to represent the stress state in the soil. Two coupled constitutive aspects are used to fully describe the non‐isothermal behaviour. The mechanical constitutive part is built on the concepts of bounding surface theory and multi‐mechanism plasticity, whereas water retention characteristics are described using elasto‐plasticity to reproduce the hysteretic response and the effect of temperature and dry density on retention properties. The theoretical formulation is supported by comparisons with experimental results on two compacted clays. Copyright © 2008 John Wiley & Sons, Ltd.     

An elastoplastic strainhardening model for soil allowing for hydraulic bonding–debonding effects

The paper presents a strainhardening constitutive model for unsaturated soil behaviour based on energy conjugated stress variables in the framework of superposed continua. The proposed constitutive law deals with hydro‐mechanical coupling phenomena. The main purpose is to develop within a consistent framework a model that can deal with possible mechanical instabilities occurring in partially saturated materials. The loss of capillary effects during wetting processes can, in fact, play a central role in unstable processes. Therefore, it will be shown that the bonding effects due to surface tensions can be described in a mathematical framework similar to that employed for bonded geomaterials to model weathering or diagenesis effects, either mechanically or chemically induced. The results of several simulations of common laboratory tests on partially saturated soil specimens are shown. The calculated behaviour appears to be in good qualitative agreement with that observed in the laboratory. In particular it is shown that volumetric collapse phenomena due to hydraulic debonding effects can be successfully described by the model. Finally, it will be highlighted the ability of the model to naturally capture the transition to a fully saturated condition and to deal with possible mechanical instabilities in the unsaturated regime. Copyright © 2008 John Wiley & Sons, Ltd.     

A thermo‐hydro‐mechanical model for unsaturated soils based on the effective stress concept

A simple thermo‐hydro‐mechanical (THM) constitutive model for unsaturated soils is described. The effective stress concept is extended to unsaturated soils with the introduction of a capillary stress. This capillary stress is based on a microstructural model and calculated from attraction forces due to water menisci. The effect of desaturation and the thermal softening phenomenon are modelled with a minimal number of material parameters and based on existing models. THM process is qualitatively and quantitatively modelled by using experimental data and previous work to show the application of the model, including a drying path under mechanical stress with transition between saturated and unsaturated states, a heating path under constant suction and a deviatoric path with imposed suction and temperature. The results show that the present model can simulate the THM behaviour in unsaturated soils in a satisfactory way. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite element formulation and algorithms for unsaturated soils. Part I: Theory

This paper presents a complete finite‐element treatment for unsaturated soil problems. A new formulation of general constitutive equations for unsaturated soils is first presented. In the incremental stress–strain equations, the suction or the pore water pressure is treated as a strain variable instead of a stress variable. The global governing equations are derived in terms of displacement and pore water pressure. The discretized governing equations are then solved using an adaptive time‐stepping scheme which automatically adjusts the time‐step size so that the integration error in the displacements and pore pressures lies close to a specified tolerance. The non‐linearity caused by suction‐dependent plastic yielding, suction‐dependent degree of saturation, and saturation‐dependent permeability is treated in a similar way to the elastoplasticity. An explicit stress integration scheme is used to solve the constitutive stress–strain equations at the Gauss point level. The elastoplastic stiffness matrix in the Euler solution is evaluated using the suction as well as the stresses and hardening parameters at the start of the subincrement, while the elastoplastic matrix in the modified Euler solution is evaluated using the suction at the end of the subincrement. In addition, when applying subincrementation, the same rate is applied to all strain components including the suction. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical stability for modelling of dynamic two‐phase interaction

Dynamic two‐phase interaction of soil can be modelled by a displacement‐based, two‐phase formulation. The finite element method together with a semi‐implicit Euler–Cromer time‐stepping scheme renders a discrete equation that can be solved by recursion. By experience, it is found that the CFL stability condition for undrained wave propagation is not sufficient for the considered two‐phase formulation to be numerically stable at low values of permeability. Because the stability analysis of the two‐phase formulation is onerous, an analysis is performed on a simplified two‐phase formulation that is derived by assuming an incompressible pore fluid. The deformation of saturated porous media is now captured in a single, second‐order partial differential equation, where the energy dissipation associated with the flow of the fluid relative to the soil skeleton is represented by a damping term. The paper focuses on the different options to discretize the damping term and its effect on the stability criterion. Based on the eigenvalue analyses of a single element, it is observed that in addition to the CFL stability condition, the influence of the permeability must be included. This paper introduces a permeability‐dependent stability criterion. The findings are illustrated and validated with an example for the dynamic response of a sand deposit. Copyright © 2015 John Wiley & Sons, Ltd.     

A dynamic two‐phase flow model for air sparging

Air sparging (AS) is an in situ soil/groundwater remediation technology, which involves the injection of pressurized air/oxygen through an air sparging well below the zone of contamination. Characterizing the mechanisms governing movement of air through saturated porous media is critical for the design of an effective cleanup treatment system. In this research, micromechanical investigation was performed to understand the physics of air migration and subsequent spatial distribution of air at pore scale during air sparging. The void space in the porous medium was first characterized by pore network consisting of connected pore bodies and bonds. The biconical abscissa asymmetric concentric bond was used to describe the connection between two adjacent pore bodies. Then a rule‐based dynamic two‐phase flow model was developed and applied to the pore network model. A forward integration of time was performed using the Euler scheme. For each time step, the effective viscosity of the fluid was calculated based on fractions of two phases in each bond, and capillary pressures across the menisci was considered to compute the pressure field. The developed dynamic model was used to study the rate‐dependent drainage during air sparging. The effect of the capillary number and geometrical properties of the network on the dynamic flow properties of two‐phase flow including residual saturation, spatial distribution of air and water, dynamic phase transitions, and relative permeability‐capillary pressure curves were systematically investigated. Results showed that all the above information for describing the air water two‐phase flow are not intrinsic properties of the porous medium but are affected by the two‐phase flow dynamics and spatial distribution of each phase, providing new insight to air sparging. Copyright © 2012 John Wiley & Sons, Ltd.     

Discussing the definition of the second‐order work for unsaturated soils

The volumetric compaction due to wetting processes is a phenomenon observed quite often in unsaturated soils. Under certain circumstances, saturation events can result into a sudden and unexpected collapse of the system. These phenomena are usually referred to as wetting‐induced collapses, without providing any detailed theoretical justification for this terminology. In order to predict in a general fashion the occurrence of coupled instabilities induced by saturation processes, a generalization of the theoretical approaches usually employed for saturated geomaterials is here provided. More specifically, this paper addresses the problem of hydro‐mechanical instability in unsaturated soils from an energy standpoint. For this purpose, an extension of the definition of the second‐order work is here suggested for the case of unsaturated porous media. On the basis of some examples of numerical simulations of laboratory tests, coupled hydro‐mechanical instabilities are then interpreted in the light of this second‐order energy measure. Finally, the implications of the theoretical results here presented are commented from a constitutive modelling perspective. Two possible alternative approaches to formulate incremental coupled constitutive relations are indeed discussed, showing how the onset of hydro‐mechanical instabilities can be predicted using an extended form of Hill's stability criterion. Copyright © 2010 John Wiley & Sons, Ltd.     

A model for moisture and heat flow in fractured unsaturated porous media

The present study investigates propagation of a cohesive crack in non‐isothermal unsaturated porous medium under mode I conditions. Basic points of skeleton deformation, moisture, and heat transfer for unsaturated porous medium are presented. Boundary conditions on the crack surface that consist of mechanical interaction of the crack and the porous medium, water, and heat flows through the crack are taken into consideration. For spatial discretization, the extended finite element method is used. This method uses enriched shape functions in addition to ordinary shape functions for approximation of displacement, pressure, and temperature fields. The Heaviside step function and the distance function are exploited as enrichment functions for representing the crack surfaces displacement and the discontinuous vertical gradients of the pressure and temperature fields along the crack, respectively. For temporal discretization, backward finite difference scheme is applied. Problems solved from the literature show the validity of the model as well as the dependency of structural response on the material properties and loading. Copyright © 2016 John Wiley & Sons, Ltd.     

A three‐phase thermo‐hydro‐mechanical finite element model for freezing soils

Artificial ground freezing (AGF) is a commonly used technique in geotechnical engineering for ground improvement such as ground water control and temporary excavation support during tunnel construction in soft soils. The main potential problem connected with this technique is that it may produce heave and settlement at the ground surface, which may cause damage to the surface infrastructure. Additionally, the freezing process and the energy needed to obtain a stable frozen ground may be significantly influenced by seepage flow. Evidently, safe design and execution of AGF require a reliable prediction of the coupled thermo‐hydro‐mechanical behavior of freezing soils. With the theory of poromechanics, a three‐phase finite element soil model is proposed, considering solid particles, liquid water, and crystal ice as separate phases and mixture temperature, liquid pressure, and solid displacement as the primary field variables. In addition to the volume expansion of water transforming into ice, the contribution of the micro‐cryo‐suction mechanism to the frost heave phenomenon is described in the model using the theory of premelting dynamics. Through fundamental physical laws and corresponding state relations, the model captures various couplings among the phase transition, the liquid transport within the pore space, and the accompanying mechanical deformation. The verification and validation of the model are accomplished by means of selected analyses. An application example is related to AGF during tunnel excavation, investigating the influence of seepage flow on the freezing process and the time required to establish a closed supporting frozen arch. Copyright © 2013 John Wiley & Sons, Ltd.     

Deformation analysis of tunnel excavation below existing pipelines in multi‐layered soils based on displacement controlled coupling numerical method

The effect of tunneling on surrounding environments, especially on existing buried pipelines is a problem that engineers designing and practicing in urban geotechnical environments encounter more frequently than in the past. However, previous studies are usually based on the assumption that the soil is homogeneous. How to reflect soil stratification is the main focus for the problem of tunneling in multi‐layered soils. A displacement controlled coupling numerical method is presented for the displacement analysis of tunnel excavation below existing pipelines in multi‐layered soils. On the basis of the layered soil model, to consider the soil nonhomogeneous characteristic, the finite element method and boundary element method are coupled to simulate the deformation of existing pipelines induced by tunneling. The solutions indicate that good agreements are obtained between the proposed coupling numerical method and the commercial software. The accuracy of the proposed numerical method is better than the two stages method based on the existing closed‐form solutions. Moreover, the results discussed in this paper show that the error obtained by the previous method of weighted average on the basis of homogeneous half space converted from layered soils is not negligible for the obvious difference of elastic parameters among successive layers. Copyright © 2012 John Wiley & Sons, Ltd.     

A quasistatic analysis of a plate on consolidating layered soils by analytical layer‐element/finite element method coupling

In this paper, a coupling method between finite element and analytical layer‐elements is utilized to analyze the time‐dependent behavior of a plate of any shape and finite rigidity resting on layered saturated soils. Based on the integral transform techniques together with the aid of an order reduction method, an analytical layer‐element solution is derived from the governing equations for three‐dimensional Biot consolidation with respect to a Cartesian coordinate system and then extended to be the fundamental solution for the layered saturated soil under a point load. The Mindlin plate is modeled by eight‐noded isoparametric elements. The governing equations of the interaction between soil and plate in the Laplace‐Fourier transformed domain are deduced by referring to the coupling theory of FEM/BEM, and the final solution is obtained by applying numerical inversion. Numerical examples concerned with the time‐dependent response of a plate are performed to demonstrate the influence of soil and plate properties on the interaction process. Copyright © 2014 John Wiley & Sons, Ltd.     

A non‐linear coupled finite element–boundary element model for the prediction of vibrations due to vibratory and impact pile driving

This paper presents a non‐linear coupled finite element–boundary element approach for the prediction of free field vibrations due to vibratory and impact pile driving. Both the non‐linear constitutive behavior of the soil in the vicinity of the pile and the dynamic interaction between the pile and the soil are accounted for. A subdomain approach is used, defining a generalized structure consisting of the pile and a bounded region of soil around the pile, and an unbounded exterior linear soil domain. The soil around the pile may exhibit non‐linear constitutive behavior and is modelled with a time‐domain finite element method. The dynamic stiffness matrix of the exterior unbounded soil domain is calculated using a boundary element formulation in the frequency domain based on a limited number of modes defined on the interface between the generalized structure and the unbounded soil. The soil–structure interaction forces are evaluated as a convolution of the displacement history and the soil flexibility matrices, which are obtained by an inverse Fourier transformation from the frequency to the time domain. This results in a hybrid frequency–time domain formulation of the non‐linear dynamic soil–structure interaction problem, which is solved in the time domain using Newmark's time integration method; the interaction force time history is evaluated using the θ‐scheme in order to obtain stable solutions. The proposed hybrid formulation is validated for linear problems of vibratory and impact pile driving, showing very good agreement with the results obtained with a frequency‐domain solution. Linear predictions, however, overestimate the free field peak particle velocities as observed in reported field experiments during vibratory and impact pile driving at comparable levels of the transferred energy. This is mainly due to energy dissipation related to plastic deformations in the soil around the pile. Ground vibrations due to vibratory and impact pile driving are, therefore, also computed with a non‐linear model where the soil is modelled as an isotropic elastic, perfectly plastic solid, which yields according to the Drucker–Prager failure criterion. This results in lower predicted free field vibrations with respect to linear predictions, which are also in much better agreement with experimental results recorded during vibratory and impact pile driving. Copyright © 2008 John Wiley & Sons, Ltd.