θ-STOCK, a powerful tool of thermohydromechanical behaviour and damage modelling of unsaturated porous media

Abstract

A brief review of the basic points of a suction-based heat, moisture transfer and skeleton deformation equations for an unsaturated medium is presented. The main issues such as: two temperature-dependent state surfaces of void ratio and degree of saturation which are used to present the coupling effects of temperature, moisture content and deformation of skeleton; the new thermoelastoplastic constitutive law, etc. are briefly mentioned. The Bubnov–Galerkin integral form of field equations has been developed as the basis of spatial and temporal discretized matrix form. The single-step integration in time is described. The numerical solution algorithm of the finite element package, θ-STOCK, is presented. Some application cases are presented and discussed to show the strong ability of presented model and the prepared numerical package.

Introduction

Since the soil is continuously under the effect of temperature changes in its natural environment, a great deal of attention has been paid to the phenomenon of moisture transport due to thermal gradient from at least the beginning decades of the past century. In the earlier investigations the critical role of unsaturated zone near the soil surface in the groundwater recharge, surface runoff and evapo-transpiration of the precipitation was the center of attention, but in the latter studies major attentions have been focused on geothermal energy extraction, contaminant transport and specially the safe disposal of high-level radioactive waste. The engineered clay barriers are currently used for the filling and sealing of the underground nuclear waste repositories. Considering the unsaturated state of such deformable materials, a deep understanding of coupling effects of moisture, heat, air and soil deformation seems to be an absolute necessity.

The phase changes between liquid and gas, evaporation, condensation, induced moisture transfer under thermal and pore pressure gradients and the effects of moisture distribution on heat flow are important aspects in non-deformable unsaturated porous media. If the deformation of porous media, which is significant in engineered clay barriers, is considered, the coupling effects among deformation, moisture, and heat should also be addressed in addition of all above aspects.

Philip and de Vries [1] and de Vries [2] theories assume that the moisture transfer in unsaturated soil occurs in both vapour and liquid phases, under the combined influence of gravity, the gradient of temperature and the gradient of moisture content. This theory makes a consistent distinction between liquid and vapour phases concerning the changes of moisture content.

Another theory for the analysis of coupled mass and heat transfer in porous media was developed by Taylor and Cary [3], who used the general theory of irreversible thermodynamics processes (TIP). Laboratory experiments performed by Cassel et al. [4] showed that this approach underestimated the flow rate by a factor of 10–40. It seems this approach did not consider the problem of integration from the microscopic scale to the macroscopic scale rigorously enough. This is a real lack, since the mechanical equations are currently written by means of this integration.

Dirksen [5] studied the soil moisture movement in a freezing column of soil, in absence of water table. He observed a good agreement with Philip and de Vries’ theory. The laboratory experiments of Cassel et al. [4] proved to be in a close correlation with the Philip and de Vries’ theory.

The theory of Philip and de Vries is now generally accepted in soil sciences and geotechnical engineering studies. However, this theory encompasses some restrictions in geotechnical engineering practice, which should be overcome. One of these limitations lies in the assumption of incompressibility of the soil skeleton. It is not realistic, especially in the case of engineered clay barriers, which are soft and significantly deformable. The θ-based formulation, which was initially chosen by Philip and de Vries for the presentation of their theory, is valid for homogeneous soils and cannot take hysteretic effects into account. In order to overcome these latter restrictions, two attempts have been reported by Sophocleous [6] and Milly [7]. Both of these works have been undertaken with the aim of converting the θ-based formulation of Philip and de Vries to a matric head-based formulation in order to consider soil inhomogeneity and hysteretic effects in desiccation and resaturation conditions.

Sophocleous [6] has begun with Philip and de Vries [1] model and does not consider hysteresis, while Milly [7] considers de Vries [2] model in a more general frame. Milly [7] has criticized Sophocleous’ work [6] and has found apparently two major errors. These are in the equation for the liquid flux and in the expression for the temperature dependence of matric potential.

Thomas [8] has presented a simple numerical solution of the θ-based formulation of Philip and de Vries, ignoring the convection effects in an incompressible soil. Thomas and He [9] have presented the moisture and heat transfer analysis in a deformable unsaturated soil.

Geraminegad and Saxena [10] may have been the first investigators who have developed a model in which the soil deformation is considered. Their model does not include soil deformations due to external loading. In this formulation, soil deformation was limited to volumetric deformation due to pore air pressure and suction change.

Coleman [11] and Bishop and Blight [12] have found out the basic frame of two stress state variables approaches. Fredlund and Morgenstern [13], [14] have shown that any pair of three stress parameters σ–P_a, σ–P_w, P_a–P_w would be sufficient to describe the mechanical behaviour of unsaturated soils. For the last three decades, net stress σ–P_a, and matrix suction P_a–P_w have been the most commonly used variables. Various constitutive laws have been used, such as the incremental elastic formulation suggested by Coleman [11] and Fredlund [15], and the state surface concept developed in order to describe the volumetric behaviour of soil under the coupled effects of net stress and suction changes. Matyas and Radhakrishna [16] have compared experimental data to the predictions given by the state surfaces of the void ratio and the degree of saturation. Fredlund [15] has proposed explicit mathematical expressions for both state surfaces. Lloret and Alonso [17] have given alternative expressions. Gatmiri has developed an explicit expression of the void ratio state surface, which is compatible with nonlinear elastic (hyperbolic) constitutive laws [19].

Starting from the approach proposed by Alonso et al. [19], [20] and Olivella et al. [21], a nonlinear elastic model of unsaturated soil has been developed at the CERMES since 1989. A new state surface formulation has been performed. This model has been incorporated within U-DAM finite element code, developed in CERMES to model the behaviour and transfer laws characterizing an unsaturated soil [22]. The important aspects of earth dam construction are considered in this code.

Another finite element code with elastoplastic constitutive law has been developed by Gatmiri et al. [23]. Nonlinear elastic models have the advantage to be easily numerically implemented. Considering the hysteretic phenomenon due to desiccation and wetting cycles by a unique state surface, is a restriction. Though, this difficulty can be overcome by using two different formulations of volumetric changes, corresponding to two paths of moisture changes in desiccation and wetting. However, the state surface approach which can be considered equivalent to the nonlinear behaviour in stress–strain path is limited to specific loading especially in drying/wetting paths. As it is mentioned, in order to deal with various loading paths, we have to add more state surfaces and, as such, this approach is eventually not simpler than the standard elastoplastic approach.

The theory of Philip and de Vries [1] and de Vries [2] is a comprehensive theory of moisture and heat movement in an incompressible porous medium. The new suction-based mathematical model presented by Gatmiri et al. [24], [25] and Gatmiri [26] is suitable for analysis of the thermo-hydro-mechanical behaviour of deformable unsaturated media. In this approach, heat and moisture transfer equations are given in an alternative form, based on water and air pressures. This model has been formulated with the two most widely used independent state variables: net stress and suction. It describes the water and air pore pressures distribution, and the deformation of the skeleton. The coupling effects of temperature and moisture content on the deformation of the skeleton, and their inverse effects, are included in the model, via a thermal state surface concept. Temperature-dependent state surface formulations are given for the void ratio and degree of saturation variations within the porous media. A nonlinear constitutive strain–stress relationship is considered. In this new type of formulation, the soil non-homogeneity, as well as hysteretic effects, may be included. The phase change between liquid and vapour phases is taken into account.

A mixed damage model, formulated in net stress and suction, has also been developed in order to represent fracturing around excavation damaged zones. Behaviour laws are derived from a thermodynamic potential, which encompasses localization–regularization terms. Damage rigidities associated with the state variables are computed by applying the principle of equivalent elastic energy, which is widely used in micromechanics. Homogenized cracking parameters are also included in the expression of the intrinsic liquid permeability of the medium, in order to represent the influence of damage on fluid transfers.

Because of the complexity of the established governing partial differential equations, a numerical solution scheme has been developed, by means of the finite element method. Bubnov–Galerkin integral forms of field equations are taken into consideration as the basis for the spatial and temporal discretized matrix form of the equilibrium equations. Implicit integration scheme is used in time. (θ-STOCK) code has been validated by several applications. For the sake of brevity of the text the literature review has been omitted. An exhaustive literature review has been given in [26], [27], [28], [29], [30].