Coupled chemo-mechanics of intergranular contact: Toward a three-scale model
Introduction
Most of natural and engineering processes in mechanics of geomaterials involve intergranular forces and displacements, which at the continuum level are represented by stress and strain fields. A well developed continuum mechanics theory of granular medium would usually allow one to competently deal with most soil mechanics problems. However, as grain contact phenomena, especially for fluid saturated soils, are often substantially affected by mineral alterations at the micro-scale, purely mechanical considerations may not be sufficient over larger time scales. Such alterations are mainly due to chemical phenomena between solid and fluid phases and driven by chemical variables, which from the mechanics point of view, are internal variables and hence, uncontrollable. A better understanding of what mechanisms control the contact process at the micro-scale will allow us to more completely represent this process at the macroscopic scale.
Diverse processes in geomechanics are known to directly depend on the microscopic level dissolution of minerals at the stressed intergranular contact. They include: structuration or post-sedimentary development of a secondary micro-structure in natural soils, aging of materials in laboratory tests, intergranular cementation, and compaction of oil bearing sediments via induced pressure solution, as well as the degradation processes of weathering, micro-erosion, desiccation cracking, etc. While the time scale and contribution of different mechanisms constituting these processes may be different in each of the phenomena mentioned, one feature that these processes appear to have in common, is the redistribution of mass of solids within the soil. The specific mechanisms of this redistribution are a subject of a more or less intense debate in the respective communities and a subject of both theoretical and experimental research. The redistribution is often linked to a series of processes such as dissolution of compounds and minerals, diffusion, formation of gel in pores, precipitation of mass on exposed (free) surfaces. It often produces changes in mechanical properties of material at a macro-scale. While some geochemical mechanisms of the above lists have been analyzed using micro-scale models, their outcome, such as alteration of the mechanical properties of material at a macro-scale is often unnoticed, with the exception of cementation effect.
In this paper, we further develop an earlier concept of coupled chemo-plasticity [1], [2] by linking irreversible dilatancy and the increase of the free surface area to the increased dissolution of minerals per unit volume, enhancing in turn the chemical softening of the material [3].
It seems clear from the above considerations that the chemo-mechanical behavior of the grain assembly cannot be confidently dealt with at a single length scale only. While the ultimate scale and the corresponding variables are macroscopic and continual, phenomena that cause the involved processes and the corresponding variables are quantified at the (chemistry) molecular scale. To characterize the chemically induced changes in mechanical properties it may appear necessary to identify idealized meso-structures that undergo such an evolution and mechanisms determining the end result: variation in the continual material properties.
Therefore, in this paper for the particular scales we choose representative elementary volumes, which contain: for the micro-scale deformation and intra-granular transport: a microscopic continuum of the material of an individual grain encompassing a representative volume of micro-cracks of few tens of a micron; for the meso-scale inter-granular transport and deformability evolution of a grain system: a characteristic unit of four solid grains endowed with initially free surface of grains, on which precipitation occurs forming a thin coating, and the same unit surrounding an inter-grain pore and deforming around it as a hollow cylinder; for the macro-scale: a continuum of assemblies of grains characterized by a Cam-clay type chemo-plasticity.
At each scale the boundary value problems formulated is obviously different as they describe different mechanisms. An important part of such a multiple scale modeling is the cross-scale transfer functions. In this paper they are formulated for the macro-scale constitutive functions based on averaging of meso-scale and micro-scale variables of mass change of the same mineral but engaged in distinctly different lower-scale phenomena.
Section snippets
Macroscopic observations on soils exposed to aging
One of the areas, where a proper understanding of intergranular contact phenomena is crucial for the industrial decision-making is water, petroleum or methane gas extraction from underground reservoirs. A key element of this decision-making is the assessment of reservoir compaction and a potential for land subsidence. Especially in coastal areas, even a small amount of subsidence may cause flooding and severe damage to the urban infrastructures.
One of the principal factors of a fundamental
Geochemical evidence of intergranular aging
The results obtained also indicate that the increase of the material stiffness is clearly a time dependent phenomenon [4]. However, they also suggest that the time scale of the underlying physico-chemical processes is that of week or months, rather than millennia or more. This latter observation differentiates them from the processes understood as pressure solution occurring over the geological ages. Both soil aging and pressure solution seem to encompass the evolution of physical and
Microscopic scenarios of a submerged intergranular contact process
A series of qualitative experiments using chalk bars immersed in a light salt solution were performed [44], visualizing the chemically enhanced damage. Identical chalk bars were used with fluid of different salinities or without fluid. Indentation was conducted using metal indenter, wooden indenter and chalk-to-chalk indentation. Chalk was selected as a modeling material for two reasons: a high rate of reaction; and a good representation of the brittle behavior, typical of silica or calcite.
Micro-scale model for the coupled damage – dissolution–diffusion
The question at hand is: what is the effect that the irreversible damage developed at an intergranular contact may have on dissolution of minerals from the damage area, and vice versa, what is the effect of mineral dissolution on the strength reduction and hence further mechanical damage, but also on a macro-void filling via long range precipitation and the resulting strength increase at the macro-scale? We consider indentation of a harder (undeformable) grain into a (softer) grain of quartz
Inter-grain pore transport
We keep in mind one possible engineering application of the presented study, which is the prediction of compaction of soils and sediments at a constant in situ stress, prior, during and after the pore fluid (water, oil, natural gas) extraction. The above-mentioned aspects of the solute fate play a cardinal role in the extraction technologies and the consequent subsidence of soil/sediment masses.
As mentioned earlier the fate of the dissolved mineral remains largely a mystery. Apparently
A macro-scale constitutive law: effects of the micro-scale mass removal and precipitation. Cross-scale transfer functions
We will employ previously proposed macroscopic chemo-plasticity theory [1], [2], which is an extension of Cam-clay theory of hardening plasticity [60]. Its main tool is the yield limit usually expressed in terms of mean normal stress and deviatoric stress invariant expressed as . Variables marked with a top bar refer to macro-scale, if they were previously used at micro- or meso-scaleThe hardening variable pc, describing the size of the yield locus and
Conclusions
In this paper, dissolution of minerals at a stressed intergranular contact and its consequence are investigated via scenarios of chemo-mechanical processes hypothesized to develop during soil or sediment compaction. Some qualitative experiments on submerged solid grains of chalk in contact illustrate a damage-enhanced dissolution. These experiments together with geological evidence in the literature provide a motivation for the main hypothesis that the chemo-mechanical damage in the contact
Acknowledgement
This work was partially supported by US National Science Foundation grant #0324543, Geomechanics and Geotechnical Systems Program.
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