Finite element simulations of chemo-mechanical coupling in elastic–plastic homoionic expansive clays

Abstract

Chemically active saturated clays are considered in a two-phase framework. The solid phase contains clay particles, absorbed water and a single salt. The fluid phase, or pore water, contains free water and salt. Water, and possibly salt, can transfer between the two phases. In addition, part of both species diffuse through the porous medium. A global understanding of all phenomena, mass transfer, diffusion/advection and deformation is provided. The coupled constitutive equations associated to these three phenomena are developed. Emphasis is laid on the chemo-mechanical constitutive equations in an elastic–plastic setting. A finite element formulation embodying all the above phenomena is proposed and simulations of oedometer tests are presented and commented.

Introduction

Swelling is an important property of natural or engineered soils used in foundations, contamination barriers, and petroleum engineering. Swelling may cause a non-uniform soil heave, resulting in damage of over-structures, or weakening of bore walls in petroleum drillings. On the other hand, swelling of bentonite is taken advantage of to build barriers to water flow, or contaminant transport in hazardous and nuclear waste disposal technologies.

Swelling of soils is generally understood to be driven by capillary forces and/or by chemical gradients. The former one, termed matric swelling, occurs when soil is unsaturated. The latter one, termed chemical swelling, is characteristic of so-called expansive clays, rich in mineral smectite. Saturation of pore space with water and salt is assumed here, and predominantly Na-Montmorillonite clays are considered. The presence of several cations in the clay clusters requires modifications with respect to this framework: then the electrolyte nature of pore water has to be accounted for (see [12]).

Saturated clay is viewed as a porous deformable continuum consisting of two overlapping phases, each phase containing several species. Phase identification follows a kinematic criterion and attributes the absorbed water to the solid phase based on the affinity of their velocities [15].

Modeling deformable porous media is mainly concerned with the coupling of the deformation of pore space in soil and the concomitant in- or out-flow of pore liquid. In chemically sensitive soils, this coupling is additionally affected by chemical potentials of the species of the solid and fluid phases. Thus, mechanics of the medium, e.g. balance of momentum, is considered at the phase level, whereas chemical processes, i.e. balances of masses, concern the species. The link between the two levels is obtained through energetic considerations. Whether or not the chemical potentials are in equilibrium, water absorbed in the clay platelets can transfer into free pore water, or conversely, depending on the chemical composition of the clay and pore water phase, and on the mechanical conditions in terms of volume and pressure. The chemo-mechanical elastic–plastic constitutive equations involve the species present in the solid phase but treat the fluid phase as a whole. The species of the latter only diffuse through the porous medium, obeying generalized diffusion equations. The transfer of absorbed water and species between the solid and fluid phases involves a fictitious membrane surrounding the clay platelets, which may be permeable to the chemical species at various degrees.

In LHG,¹ this theoretical framework was illustrated by simulations of typical phenomena observed during laboratory experiments. Parameters involved in the model were calibrated. Although many phenomena are involved, the number of parameters is quite limited, and their range for different clays can be evaluated. This is an advantage, which is to be traced to the strong structuring of the model, which derives from a thermodynamical approach. Mechanical, chemical and chemo-mechanical loading and unloading paths were considered. Increase of the salinity of pore water at constant confinement was shown to lead to a volume decrease, so-called chemical consolidation. Subsequent exposure to a distilled water solution displayed swelling: however, the latter was smaller than the chemical consolidation so that the chemical loading cycle results in a net contractancy whose amount increases with the confinement. With respect to previous publications, the key feature of the present framework is that it is comprehensive and not only qualitative but able to simulate a rather extensive number of experimental data which have been recently published.

Although attention in LHG was paid mainly to the elastic–plastic constitutive equations, the latter are shown now to be embedded in a general formulation where both mass transfer and diffusion processes can be considered in initial and boundary value problems that can be formulated within a mixed finite element method.

The issues addressed here involve three aspects, none of them being susceptible to be neglected, namely

• the seepage of water and the water-advected diffusion of salt through the porous medium. Coupling between these two flows is introduced by osmotic efficiency whose intensity strongly decreases with salt concentration;

• the physico-chemical reaction picturing absorption of water from the fluid phase to clay clusters, and the converse desorption;

• the strong influence of the above mass of absorbed water on the elastic and elastic–plastic properties of solid phase.

Therefore, the analysis encompasses both the hydrogeologic point of view which addresses the two first aspects only and the conventional geomechanical perspective where only hydro-mechanical coupling is accounted for.

As for the geomechanical models of the issues addressed here, the ones we are aware of consider deformable materials which are linear elastic. While valuable, as it allows for analytical developments, such an approach is really limited in the interpretation of laboratory tests or field events. To our knowledge no one of the works mentioned below has provided a calibration of their parameters using either laboratory or field instrumented data, nor have been simulations of controlled situations presented. For analytical results to be obtained, simplifications were adopted a priori, especially by neglecting some couplings. The only case where the elastic assumption can be used with confidence concerns strongly overconsolidated materials. In fact, it is argumented here that couplings between elastic–plastic properties and chemistry are strong, and most experimental paths that we have simulated contain at least a portion of plastic loading.

Kaczmarek and Hueckel [18] and Smith [34] have presented one-dimensional linear elastic models intended to analyze the effect of contaminant diffusion on a deformable medium, and the converse effect of material deformation on contaminant migration. Kaczmarek and Hueckel [18] introduce coupling between chemistry and the deformation model by assuming that the change in contaminant concentration affects linearly the porosity. This assumption is quite close, at least at small concentrations, to that of Barbour and Fredlund [1] where the osmotic pressure is involved in place of the concentration. For both Kaczmarek and Hueckel [18] and Smith [34], the volume changes of the solid phase is not affected by chemistry, and therefore the chemo-mechanical coupling is weak. Two cross-couplings are introduced in the diffusion law, which relates fluxes of water and contaminant to gradients of pore pressure and concentration. These couplings are neglected by Smith [34]: in the present thermodynamic analysis, this assumption amounts to a zero osmotic efficiency (Section 3.2). The materials addressed by Kaczmarek and Hueckel [18] are not swelling and there is no mass transfer involved. On the other hand, Smith [34] introduces the possibility of contaminant transfer through an absorption/desorption isotherm, following the hydrogeological approach (see Section 3.3.2). Advection of contaminant by water is neglected by Kaczmarek and Hueckel [18], accounting for the low velocity of water.

Sherwood [29] has formulated a poroelastic constitutive behaviour referred to as inert, where the solid particles constitute the solid phase but the fluid phase contains several species, water and salts: at a fixed chemical potential, the mass of a species squeezed out is proportional to its molar fraction. The model has been used in a petroleum engineering context where, for a sufficient difference in salinity between mud in borehole and rock, the pore pressure in the rock may at equilibrium be forced to be lower than its initial value.

Bennethum and Cushman [3] and Murad [27] have addressed the transfer of water into interlamellar space in clays through a two- or three-spatial scale modeling using homogenization schemes. This type of approach entails a substantial number of constitutive assumptions, requiring sophisticated identification procedures. Absorbed water has been considered as a third phase by Murad [27] and endowed with a partial stress, related thermodynamically to the total fluid volume fraction.

Other models where, on top of the above couplings, the electrolytic nature of the water phase has to be accounted for, have been developed mainly for biological tissues undergoing large elastic strains (e.g. [16], [20], [33]). The model presented in [12] is an extension of the present elastic–plastic framework to heteroionic clays. The related finite element formulation and simulations are in progress.

The paper is comprehensive in order the reader not familiar with physico-chemical mechanisms in deformable materials can find the necessary definitions at one place.

Basic entities pertaining to the two-phase/multi species framework developed here are defined in Section 2, and the local balance equations are stated in view of future use in the finite element simulations. The mass transfer equations, diffusion equations and the elastic–plastic constitutive equations are motivated by satisfaction of a general dissipation inequality that naturally highlights chemo-mechanical couplings (Section 3). Mass transfer is viewed as a physico-chemical reaction. The generalized diffusion equations couple hydraulic and diffusion effects. The influences of these couplings in the balance equations are highlighted and the approach is compared with hydrogeological analyses where mass transfer is introduced through absorption isotherms and the deformability of the medium is neglected. For comprehensiveness, the elastic and elastic–plastic constitutive equations developed in LHG are summarized in 4 Elastic constitutive equations for incompressible species, 5 Elastic–plastic constitutive equations.

The weak form of the field equations and the time-integration procedure to solve the highly non-linear matrix equations through a finite element procedure are provided in Section 6. Non-linearity is contributed by all aspects of the model, inter alia chemical potentials and osmotic efficiency are strongly non-linear functions of salt concentration, and elasto-plasticity introduces another type of strong non-linearity. Non-symmetry of the global diffusion and stiffness arises due to the involvement of the couplings of different natures, even if each of them, taken individually, would lead to symmetry. Indeed, the generalized diffusion law obeys Onsager reciprocity principle, and plasticity can be associative. The cure of spurious oscillations due to advection–diffusion by an implicit SUPG method introduces another source of non-symmetry in the effective diffusion matrix. The choice of the four primary variables, and associated field equations, is motivated by the general perspective of this chemo-mechanical analysis in fluid-saturated porous media. In particular, they have to provide sufficient information to define at each time step the elastic–plastic state of the two-phase medium. These primary variables are the solid displacement vector, the fluid pressure, salt concentration in water phase and content of absorbed water in solid phase.

Simulations of oedometer tests, for which quite an extensive number of tests are available, evidence the influences of the various couplings in transient initial and boundary value problems (Section 7). Notice however that neither the formulation nor the parameter identification are restricted to this one-dimensional setting. Beyond hydro-mechanical coupling, comments focus on various aspects of transport of salt, transfer of water and chemo-mechanical coupling on the spatial and time profiles of strains, pore pressure and concentration during transient processes and at steady states. In fact, transient irreversible inhomogeneities might question the interpretation of laboratory tests.

Notation: Compact or index tensorial notation will be used throughout this note. Vector, matrix and tensor quantities are identified by boldface letters. $\text{I}\text{=(I}{}_{\mathrm{ij}}\text{)}$ is the identity tensor/matrix of appropriate order. Symbols ‘·’ and ‘:’ between tensors of various orders denote their inner product with single and double contraction respectively. tr denotes the trace of a second-order tensor, dev its deviatoric part and div is the divergence operator. Unless stated otherwise, the convention of summation over repeated indices does not apply.