THMC constitutive model for membrane geomaterials based on Mixture Coupling Theory
Introduction
Very low permeability geomaterials can act as actual semi-permeable membranes, having good functions for chemical retardation or sorption (Chen & Hicks, 2013). They are widely used in engineering applications such as nuclear waste disposal, carbon capture and storage, landfill etc (Chen et al., 2016). Due to the low permeability, hydraulic flow is not the dominant form of fluid movement (Ghassemi & Diek, 2003). Thermal and chemical gradients will induce fluid flux into or out of the formation, leading to thermal and chemical osmosis (Schlemmer et al., 2003). The chemical osmosis flow direction is from lower chemical concentration to higher chemical concentration, and maybe opposite to the pressure gradient-induced flow direction, thereby reducing the flow velocity (Ghaffour et al., 2013). Similarly, temperature gradient would also cause a thermal osmosis flow, which has been observed in different experiments (Dirksen, 1969, Srivastava & Avasthi, 1975). This kind of flow may occur from high temperature to low temperature or in the opposite direction (Goncalves et al., 2012), depending on the entropy difference between water in the membrane and external to the membrane(Kim and Mench, 2009).
Coupled thermal (T), hydraulic (H), mechanical (M) and chemical processes (C) have been studied mainly by three theoretical approaches, namely: the mechanics approach, the mixture theory approach, and Mixture Coupling Theory (Chen & Hicks, 2011, Chen et al., 2013, Chen et al., 2018). The mechanics approach is based on the classic consolidation theory of Terzaghi (Terzaghi, 1943) and Biot (Biot, 1962, Biot & Temple, 1972). This approach focuses on the macroscopic process of THMC (e.g. pressure/displacement/concentration/temperature). This makes it very practical since the equations may be specially developed for the intended specific application without deep understanding of the microscopic mechanisms. A lot of research has been done using this approach (Seetharam et al., 2007, Huyghe & Janssen, 1999, Graziani & Boldini, 2011, Lewis & Schrefler, 1987). However, the theoretical foundation of the mechanics approach has led to the difficulties of coupling of chemical processes (micro-process dominated), due to the gap between geophysics and geochemistry. The mechanics approach has tried to borrow uncoupled equations from other disciplines to form new governing equations to overcome the challenge. However, such governing equations are highly semi-empirical and rely heavily on experiments, hence they are not rigorously mathematically derived. Mixture theory was firstly developed by Truesdell (Truesdell, 1957) and further extended by Bowen (Bowen, 1984, Bowen, 1980) and Rajagopal & Tao (Rajagopal & Tao, 1995, Rajagopal & Tao, 2005, Rajagopal, 2007). This approach gives detailed couplings between solids and fluids. Mixture theory maintains the individuality of the constituents, which has led to the difficulties of obtaining detailed interaction information between constituents and therefore restricted its application.
Mixture coupling theory originates from mixture theory, but adopts Biot's poroelasticity viewing a fluid-infiltrated rock/soil as a single continuum and employs thermodynamic force-flux couplings, rather than introducing body forces between the constituents in the constituent equilibrium equations (or constituent equations of motion in the general case) as in classic mixture theory (Heidug & Wong, 1996). This approach combines Biot's theory and non-equilibrium thermodynamics. It simplifies the variables of interactions between solids particles which are normally difficult to obtain in geomaterials, and enables incorporating the well-developed continuum mechanics for solids deformation. By using fundamental principles of non-equilibrium thermodynamics (e.g. entropy), mixture coupling theory is capable of mathematically building the coupling between energy and dynamics in the mixture system, and has the potential to smoothly bridge geomechanics and geochemistry (Chen & Hicks, 2013, Chen et al., 2016, Chen, 2010, Chen, 2013, Chen et al., 2009).
In this paper, a new coupled THMC formulation has been developed by extending mixture coupling theory. Classic Darcy's law has been extended to include coupled chemical osmosis and thermal osmosis through using standard arguments of non-equilibrium thermodynamics. Helmholtz free energy has been used to derive the relationship between solid and fluid phase and thermal behaviour. A simple numerical model has been given to illustrate the influence of chemical osmosis and thermal osmosis.
Section snippets
Balance and conservation equations
The mixture within a porous medium contains states of matter which may include solids (denoted as subscript s), liquids (l), and gases (g); constituents () which may include examples as water (denoted as w) or chemicals (as c in general). One state of matter may consist of multiple constituents, if there is only one constituent in a matter state, it leads to a simplified . V is a selected microscopic volume of an arbitrary domain within the porous medium and S is its boundary that
Constitutive relations
Following the discussion of the balance equations in section 2, this section will establish the coupled relationship between the solid/liquid and the stress, strain and temperature response, using the dissipation function.
Constitutive equations structure
For reasons of convenience, the dual potential (the solid deformation energy) is used as
By substituting eqn. (38) into the time derivative of , it satisfies the relationship which indicates that is a function of , and , and expressions for, and may be obtained.
Since the following equations are obtained:
Solids
The non-linearity of the equations is of a geometrical nature and associated with large deformations. For isotropic materials, the tensors and are diagonal; that is, they can be written in the form of scalars and , as follows: and the elastic stiffness can be formed as a fourth-order isotropic tensorwhere is the rock shear modulus and the bulk modulus.
With the assumption of small strains, the governing stress (eqn. (43))
Numerical results for coupled thermal and chemical osmosis
This section focuses on the influence of coupled chemical osmosis and thermal osmosis induced flow, and their consequent influence on THMC processes. The governing eqns. (54), 62, 67, (70) are solved by using the classic finite element method (Lewis & Schrefler, 1987) for variables of the displacement vector
, liquid pressure , chemical concentration , and temperature .
A simple numerical model has been established to simulate the mechanical behaviour of an unsaturated very-low permeability
Conclusion
In this paper, a new THMC model has been presented incorporating coupled chemical osmosis and thermal osmosis based on mixture coupling theory. Classic Darcy's law has been extended considering the respective osmotic flux components. The numerical model has illustrated the influence of chemical osmosis and thermal osmosis on the mechanical behaviour of unsaturated rock. Chemical osmosis and thermal osmosis have both been found to induce fluid flux movement and alter the pressure distribution in
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The first author acknowledges the CERES studentship from the university of Leeds, the second author acknowledges the financial support by a University of Leeds Research Grant (40711500). The third and fifth authors acknowledge the financial support by the Welsh European Funding Office (WEFO) through the FLEXIS project.
References (51)
- et al.
Unsaturated hydro-mechanical-chemo coupled constitutive model with consideration of osmotic flow
Comput Geotech
(2013) - et al.
Unsaturated hydro-mechanical–chemical constitutive coupled model based on mixture coupling theory: Hydration swelling and chemical osmosis
International Journal of Engineering Science
(2016) - et al.
Linear chemo-poroelasticity for swelling shales: theory and application
Journal of Petroleum Science and Engineering
(2003) - et al.
Technical review and evaluation of the economics of water desalination: Current and future challenges for better water supply sustainability
Desalination
(2013) - et al.
Non-equilibrium thermodynamics of thermo-osmosis of water through kaolinite
J Hydrol
(1975) - et al.
Importance of thermo-osmosis for fluid flow and transport in clay formations hosting a nuclear waste repository
Earth and Planetary Science Letters
(2012) - et al.
Investigation of temperature-driven water transport in polymer electrolyte fuel cell: Thermo-osmosis in membranes
Journal of Membrane Science
(2009) - et al.
A constitutive model based on modified mixture theory for unsaturated rocks
Comput Geotech
(2011) - et al.
Coupled thermo-hydro-mechanical model with consideration of thermal-osmosis based on modified mixture theory
Int J Eng Sci
(2013) Incompressible Porous-Media Models by Use of the Theory of Mixtures
Int J Eng Sci
(1980)
On the propagation of waves through porous solids
International Journal of Non-Linear Mechanics
Constitutive unsaturated hydro-mechanical model based on modified mixture theory with consideration of hydration swelling
International Journal of Solids and Structures
A fully coupled thermo-hydro-mechanical model for unsaturated porous media
J Rock Mech Geotech Eng
Vertical and horizontal land deformation in a desaturating porous medium
Adv. Water Res.
Numerical analysis of dual porosity coupled thermo-hydro-mechanical behaviour during CO2 sequestration in coal
International Journal of Rock Mechanics and Mining Sciences
Analyzing wellbore stability in chemically-active anisotropic formations under thermal, hydraulic, mechanical and chemical loadings
Journal of Natural Gas Science and Engineering
Chemical osmosis in compacted clayey material and the prediction of water transport
Engineering Geology
Chemical osmosis, shale, and drilling fluids
Spe Drilling & Completion
Thermo-Osmosis Through Compacted Saturated Clay Membranes1
Soil Science Society of America Journal
A new matrix for multiphase couplings in a membrane porous medium
Int J Numer Anal Methods Geomech
Theory of consolidation
Theoretical Soil Mechanics
Mechanics of deformation and acoustic propagation in porous media
Journal of applied physics
Theory of finite deformations of porous solids
Indiana University Mathematics Journal
Coupled thermo/hydro/chemical/mechanical model for unsaturated soils—Numerical algorithm
International Journal for Numerical Methods in Engineering
Thermo-chemo-electro-mechanical formulation of saturated charged porous solids
Transport in Porous Media
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