Unsaturated Soils: Numerical and Theoretical Approaches

We review the governing conservation laws for three-phase partially saturated media using mixture theory including finite deformation effects and kinetic energy production. Under the assumption of barotropic flows we derive the mass balance equations in their most general form, including the compressibilities of the constituent phases. We then derive the momentum balance equations including the rates of change of linear momentum in Eulerian and Lagrangian descriptions. Next we write the balance of energy equation and illustrate the conjugate relationship of the partial stress tensor with the rate of deformation of the corresponding constituent phase. Using balance of mass and balance of momentum, we rewrite the balance of energy equation once again in an equivalent form showing the conjugate relationship of an effective constitutive stress with the rate of deformation of the solid matrix. This effective constitutive stress is analogous to Bishop’s effective stress for partially saturated soils, and to Terzaghi’s effective stress for fully saturated soils.

This paper reviews the thermo-hydro-mechanical behaviour of a cementitious porous material containing a pore space that is incompletely saturated. The incomplete saturation is interpreted in terms of the alteration of the compressibility of the fluid in the porous space rather than the presence of distinct regions of a fluid phase and a gas phase. The paper examines both the experimental and computational modelling of the heating of the plane boundary of a cylinder made of a cementitious material. The parametric evaluations of the computational results point to the appreciable influences of the near saturation compressibility effects on the thermally-induced pore pressure response within the cementitious medium.

A coupled finite element formulation for the hydro-thermo-mechanical behaviour of a water saturated and partially saturated porous material has been presented. This model is obtained as a result of a research in progress on the thermo-hydro-mechanical modelling for multiphase geomaterials undergoing inelastic strains. Numerical results of strain localisation in globally undrained samples of dense and medium dense sands have been presented. Vapour pressure below the saturation water pressure (i.e. water cavitation) develops at localisation in case of dense sands, as experimentally observed. A case of strain localisation induced by a thermal load in a sample where the displacements are constrained and evaporation takes place is also analysed.

Object-oriented (OO) methods become more and more important in order to meet scientific computing challenges, such as the treatment of coupled non-linear multi-field problems with extremely high resolutions. This two-part paper introduces an object-oriented concept for numerical modelling multi-process systems in porous media (Part 1). The C++ implementation of the OO design for process objects (PCS) as a class is described and illustrated with several applications. Due to the importance of the encapsulation of processes as individual PCS objects we denote our concept as an processoriented approach. The presented examples (Part 2) are dealing with thermal (T), hydraulic (H), mechanical (M) and componental processes (C) in bentonite materials, which are used as buDer material for the isolation of hazardous waste in geologic barriers. In particular, we are interested in coupling phenomena such as thermally induced desaturation, non-isothermal consolidation, swelling/shrinking phenomena as well as in a better understanding of the coupled, non-linear THM system.

Unsaturated soils are characterized by hydraulic hysteresis. Hysteresis can be recorded during cycles of wetting and drying. This feature isn’t accounted for in most of the classical constitutive models. In this paper, an extended Cam-clay model for unsaturated soils is applied in the Finite Element (FE) analysis of an ideal geotechnical problem. The extended Cam-clay can be coupled both with an hysteretic equation of the water retention curve (WRC) and with an “elastic” equation (i.e. the main drying curve the main wetting curve coincide). These two constitutive frameworks are adopted to model the construction stages and the impounding of an ideal dike. The differences produced by hydraulic hysteresis in the distribution of pore water pressure, displacement and stress fields are discussed.

In this study, with the distinct element method analysis in threedimensional state, simulations of the true triaxial test for spherical granular material is carried out. In this analysis, the influence of meniscus water, which is mainly cause of the complicated behavior of unsaturated soil, is expressed by introducing a constant intergranular adhesive force that acts perpendicular to the tangential plane at contact point. The influence of the intergranular adhesive force on the failure criterion and deformation are examined. From the analytical results, it is found out that not only the failure criterion but also deformation characteristic in shear process are obey to the Lade failure criterion.

A constitutive model is presented to describe the stress-strain behaviour of sand while unsaturated. The model is formulated using bounding surface theory and is presented in a critical state framework using the concept of effective stress. The model is validated against triaxial shear and oedometric compression tests. A single set of material parameters are required for the complete constitutive modelling.

A theoretical framework for modeling the elastoplastic constitutive behavior of unsaturated soils is presented. By combining the theory of mixtures with interfaces (TMI) and continuum theory of plasticity, a dissipation mechanism associated with capillary hysteresis is identified. We show that the plastic deformation of the soil matrix can be described by using a pseudo effective stress tensor. In this context the plastic deformation and capillary hysteresis are consistently simulated in a hierarchical and coupled manner. The proposed framework preserves all the advantages of those models based on the effective stress concept, two stress state variables, and mixture theory, while excluding their drawbacks.

Experimental investigations are the basis for the understanding of the soil mechanical and hydraulic properties of unsaturated soils. The requirements for the testing and measurement equipment are relatively high. Important for planning the test procedures and the development of new testing devices is having experience with saturated soils and having carried out tests under simplified conditions. In this paper, basic knowledge of soil mechanical properties of saturated fine grained and non cohesive soils are reconsidered to include some basic considerations into unsaturated soil models. The aim is to incorporate the state parameters of sand and the equivalent stress of fine grained soils. For this reason some modified approaches are presented and requirements for further experimental investigations are specified.

Submerged soils may be considered as an unsaturated porous medium containing gas, water and solids. By using a three-phase model, the mechanical behaviour of such soils can be described by an extended consolidation equation in order to calculate transient pore water pressures, induced by external pressure changes. Microscopic gas bubbles embedded in the pore fluid of the soil skeleton may play a key role in soil behaviour due to the increased gas-water mixture compressibility. Pressure changes applied on such unsaturated submerged soils may cause soil structure deformations such as heaving, settling or even fluidisation. Rapid external pressure reductions such as excavations, draw down loading or ground water level lowering are followed by a delayed pore water pressure propagation. Transient pore water pressures may therefore cause embankment sliding, hydraulic failure and unacceptable deformation. Case studies concerning geotechnical applications are presented. Results from numerical simulations based on Biot’s consolidation equation are compared with computations based on Richards equation. The results are discussed towards formulation of protection measures in order to improve stability requirements.

Field and theoretical efforts indicate that the presence of entrapped gas bubbles below the water table may influence the propagation dynamics of excess pore pressure caused by load changes. The process of pore pressure propagation depends on the volume of the occluded gas phase and on the relation between hydraulic conductivity of the dynamics of (hydraulic or mechanical) loading changes at the boundary. In the context of safety considerations the estimation of excess pressure dissipation dynamics may be essential. Based on Boyle-Mariotte and Henry principles predictions on water saturation versus pore pressure were established and the soil-water characteristic curve was extended from the unsaturated zone throughout the zone of air occlusion. Analogous to unsaturated zone concepts a pore pressure dependent water-capacity relationship was derived for the zone of the residual gas phase and set in 2D FE-model based on Richards equation. The extended model accounting for gas entrapment was evaluated against an explicit 2-phase model, laboratory experiments and field observations. The approach presented enables the inspection of the effect of gas entrapment on pressure propagation considering the transition from saturated to unsaturated state.

In this paper we evaluate a number of model equations for the soil-water characteristic curve (SWCC), provided a relationship exists for each relevant soil. The models are with two parameters and relate suction to volumetric water content for values of suction higher then air entry value. Logarithmic tranformations are applied to the variables in order to find linear patern. Most of generally used models are also discussed and compared to the proposed. An example is given how to assess the experimental data to define the variable transformation and how to choose the most proper model. The procedure for estimating the air entry value is also presented.

An extension of the Distinct Element Method (DEM) in corporated with a capillary water (CW) contact model proposed is used to obtain an insight into the effective stress in the unsaturated granulate from the viewpoint of strength. The disappearances of menisci and air bubbles were numerically simulated in the wetting process. DEM biaxial compression and hydric tests showed that: with the description of net stresses, peak strength envelopes move parallel to each other in the direction of increases of peak shear strength nonlinearly with the increase of suction. There appears a unique peak/residual strength line if it is described with generalized effective stress (called GES here). The GES due solely to CW (called GESS here), is a nonlinear function of suction on a wide range of values suction. GESS vs. suction curve appears to be analogous to the water-retention curve, and to be correlated with the particle gradation.

The statistical grading entropy of soils S (Lörincz, 1986) consists of two terms: the base entropy S ₀ arising from the difference in the width of the statistical cells in the conventional grading curve and, the entropy increment ΔS due to the mixing of the fractions. The aim of this ongoing research is to examine which part of the entropy plays the role of the “true” entropy in thermodynamic sense (i.e. undergoes an increase during irreversible thermodynamic processes).

Object-oriented (OO) methods become more and more important in order to meet scientific computing challenges, such as the treatment of coupled non-linear multi-field problems with extremely high resolutions. This two-part paper introduces an object-oriented concept for numerical modelling multi-process systems in porous media (Part 1). The C++ implementation of the OO design for process objects (PCS) as a class is described and illustrated with several applications. Due to the importance of the encapsulation of processes as individual PCS objects we denote our concept as an process-oriented approach.

The presented examples (Part 2) are dealing with thermal (T), hydraulic (H), mechanical (M) and componental processes (C) in bentonite materials, which are used as buffer material for the isolation of hazardous waste in geologic barriers. In particular, we are interested in coupling phenomena such as thermally induced desaturation, non-isothermal consolidation, swelling/shrinking phenomena as well as in a better understanding of the coupled, non-linear THM system.

Part 1 of this paper is about design and implementation of processes in an object-oriented way. Here we give numerical examples to show the variety of problems which can be treated based on the process-oriented approach.

The prior objective of the present paper is to predict the formation of desiccation cracks in a mineral liner for landfills. Therefore, a coupled pore water diffusion and stress analysis is made using the finite element method and the numerical program ABAQUS. For this analysis, test results of hydraulic soil relations and tensile behaviour of the investigated cohesive soil are needed. Also the inelastic mechanical properties must be given and implemented in the used numerical model. The present paper shows these test results, the used plasticity model and the geometry, loading and results of the numerical model.

This paper presents a modelling work of the “El Infiernillo” dam which was built in Mexico in the 60’s. A recently developed constitutive model for rockfill has been implemented in a coupled finite element program, which solves the hydromechanical problem for general unsaturated conditions. The performance of the dam during construction and impoundment has been analysed. The agreement between calculated and measured displacements indicates that the capabilities of the constitutive models and computational tools used.

In shallow tunnelling below the groundwater table compressed air can be used for preventing water inflow into the tunnel. Using this method air loss takes place through both the unsupported tunnel face and shrinkage cracks of the shotcrete lining. Until today it is difficult to correctly estimate the amount of air loss during the design phase of a project, although this is a significant factor concerning the total costs of a tunnel. For solving the problem the multi-phase flow in the soil above the tunnel has to be considered. The aim of the conducted research project was to develop a numerical simulation of the air flow in the soil, based on existing unsaturated soil constitutive models. In the first stage large scale laboratory tests were conducted at the Institute for Soil Mechanics and Foundation Engineering in Graz to simulate the air-permeability of the shotcrete lining and the soil. Additionally, the experimental results were simulated numerically. In a second stage the numerical model was extended to a three dimensional simulation of tunnel advance under compressed air. In this contribution the results of the tunnel advance model with respect to the air flow into the soil through the cracked shotcrete lining and the tunnel face are presented and discussed.

For the past few decades, there has been an intense public and scientific concern over the disposal of solid and hazardous waste, which hampers the harmony of the ecosystem. The disposed off waste in the due course of time interacts with rainwater and contaminates the ground water resource. Such a situation calls for proper understanding and modeling of the long-term performance of these wastes and their transport through the porous media. With this in view, an attempt was made to model advective-diffusive transport of a nonreactive contaminant (chloride ion) through the unsaturated silty soil, in a geotechnical centrifuge. The results were compared with those obtained from the finite element analysis using SEEP/W and CTRAN/W, and an excellent matching between the results has been noted.

A two-dimensional finite difference model, which solves mixed type of Richards’ equation, whose non-linearity is dealt with modified Picard’s iteration and strongly implicit procedure to solve the resulting equations, is presented. Modeling of seepage flow through heterogeneous soils, which is common in the field is addressed in the present study. The present model can be applied to both unsaturated and saturated soils and can handle very dry initial condition and steep wetting fronts. The model is validated by comparing experimental results reported in the literature. Newness of this two dimensional model is its application on layered soils with transient seepage face development, which has not been reported in the literature. Application of the two dimensional model for studying unconfined drainage due to sudden drop of water table at seepage face in layered soils is demonstrated. In the present work different sizes of rectangular flow domain with different types of layering are chosen. Sensitivity of seepage height due to problem dimension of layered system is studied. The effect of aspect ratio on seepage face development in case of the flow through layered soil media is demonstrated. The model is also applied to random heterogeneous soils in which the randomness of the model parameters is generated using the turning band technique. The results are discussed in terms of phreatic surface and seepage height development and also flux across the seepage face. Such accurate modeling of seepage face development and quantification of flux moving across the seepage face becomes important while modeling transport problems in variably saturated media.

This paper explores the hydro/mechanical behaviour of unsaturated engineered buffer. This is achieved via the assessment of a conceptual model for micromacro moisture flow. In particular the re-saturation behaviour of a large scale insitu experiment performed by Atomic Energy of Canada Limited, (AECL), is examined. Numerical simulations of the test are presented and the conceptual model investigated.

The experiment, known as the isothermal test, examines water inflow, from the surrounding rock, into highly compacted, unsaturated buffer material. Numerical simulations of the experiment are performed using an approach developed to model the thermo/hydraulic/mechanical behaviour of the system. The simulation of the experiment, using a “conventional” approach to describe the hydraulic conductivity of the buffer material, revealed that neither the duration nor the pattern of moisture influx could be modelled accurately. Further study suggested incorporation of a conceptual model for micro/macro moisture flow via the conductivity relationship may explain the experimentally observed behaviour. This approach yielded significant results with both the pattern and the rate of water uptake being simulated with a much greater accuracy.

The computer code CODE-BRIGHT has been applied to simulate thermal-hydro-mechanical (THM) experiments on clays being considered as host rock and as buffer/backfill for the disposal of radioactive wastes. In this paper, two modelling exercises will be presented: (1) prediction of the hydration of bentonite, and (2) scoping calculation of large-scale heating tests to be performed with a bentonite-filled clay cylinder in order to study coupled THM processes taking place in the near-field of heat-generating wastes in drifts and boreholes. The results suggest that the coupled THM phenomena observed in the laboratory can be well represented and interpreted by numerical simulation with the code.

This paper presents a case history of an unsaturated soil seepage analysis for design of a soil cover system to promote saturation within a mine tailings deposit. A gold mine tailings facility in Canada is being closed. The tailings facility consists of perimeter dykes containing tailings with a pond in the central area and a tailings beach along one perimeter dyke. Water quality will be enhanced by reducing the volume of material exposed to oxidation along the tailings beach. An engineered soil cover was considered as an option to maintain tailings saturation and therefore reduce oxidation along the beach. Numerical analyses were conducted to estimate surface water infiltration rates and corresponding saturation and ground water elevations in the tailings. A one-dimensional finite element software, SoilCover, was used to simulate the beach surface infiltration with various cover options. The net infiltration rates obtained from the SoilCover model were then used as input parameters for a subsequent two-dimensional finite element seepage analysis. The 2-D seepage analysis was conducted using a computer software, Seep/W, to determine ground water level and seepage quantities. Transient state analyses were conducted to evaluate ground water levels in response to construction activities and weather condition changes. A practical cover option was selected based on the analysis results. The groundwater levels predicted in the beach and perimeter dyke agreed with the piezometer data, which confirmed the infiltration rate obtained from the SoilCover model and the results from the Seep/W model.