THMC constitutive model for membrane geomaterials based on Mixture Coupling Theory

Abstract

Modelling of coupled thermal (T), hydro (H), mechanical (M) and chemical (C) processes in geomaterials has attracted attention in the past decades due to many significant contemporary engineering applications such as nuclear waste disposal, carbon capture and storage etc. However, in very-low permeability membrane geomaterials, the couplings between chemical osmosis and thermal osmosis and their consequent influence on temperature, water transport and mechanical deformation remain as a long-lasting challenge due to the gap between geomechanics and geochemistry. This paper extends Mixture Coupling Theory by bridging the chemical-thermal field based on non-equilibrium thermodynamics, and develops a new constitutive THMC fully-coupled model incorporating the interactions between chemical and thermal osmosis. Classic Darcy's law has been fundamentally extended with osmosis as the major driving force of the diffusion process. A simple numerical simulation used for the demonstration purpose has illustrated that the couplings between chemical and thermal osmosis will significantly change the water flow directions, consequently influencing the saturation variation and mechanical deformation.

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.