International Journal of Rock Mechanics and Mining Sciences
A fully coupled thermo-hydro-mechanical model for simulating multiphase flow, deformation and heat transfer in buffer material and rock masses
Introduction
Coupled thermo-hydro-mechanical (THM) processes are commonly simulated using the theories of porous media. The first such theory was Terzaghi's 1-D consolidation theory of soils [1], followed later by Biot's theory of isothermal consolidation of elastic porous media, a phenomenological approach of poroelasticity [2], [3], and mixture theory as described by Morland [4], Bowen [5] and others. Non-isothermal consolidation of deformable porous media is the basis of modern coupled THM models, using either the averaging approach as proposed first by Hassanizadeh and Gray [6], [7], [8], [9] and Achanta et al. [10], or an extension of Biot's phenomenological approach with a thermal component [11]. Both approaches are suitable for modeling coupled THM processes in geomaterials. In the early literature of coupled THM research, it was often stated that the theory of mixtures is more suitable for understanding the microscopic behavior of porous media, whereas poroelasticity theory is better suited for macroscopic description and computer modeling [12]. In this paper, we show that models based on the mixture theory can also be equally suitable for computer modeling when appropriate numerical techniques are used, and that these models show more flexibility for modeling unsaturated multiphase fluid flow in porous media.
In recent years, coupled thermo-hydro-mechanical behavior in porous media has been a subject of great interest in many engineering disciplines. Many attempts have been made to develop numerical prediction capabilities associated with topics such as the movement of pollutant plumes, gas injection, energy storage, permafrost and frozen soil engineering, safety assessment of waste repositories for radioactive waste and spent nuclear fuel, and geothermal energy extraction. Many computer codes have been applied in modeling coupled THM process, such as ROCMAS [13], THAMES [14], FRACON [15], ABAQUS [16], COMPASS [17] and CODE_BRIGHT [18]. Some of these codes can model fluid flow in unsaturated media by using Richard's equation [19], but gas movement is often not rigorously considered, since the gas pressure is assumed to be small and constant. The codes COMPASS and CODE_BRIGHT can simulate two-phase (gas and liquid) fluid flow with two components (air and water) in partially saturated soils, coupled with heat transport and mechanical response [20]. However, the advective flow of vapor was not well described, including heat transfer between the liquid and gas phases. As pointed out in [21], only if we consider the spontaneous thermo-dynamic equilibrium between the liquid water and the water vapor in the porous media, and treat the vapor pressure as a variable dependent on both suction and temperature, can the flux of vapor and liquid water be properly modeled using a single empirical equation [21], [22]. Unfortunately, this approach makes it impossible to predict vapor pressure evolution accurately, and the relative humidity cannot be calculated according to its physical definition due to the limited validity of its empirical models. Another problem is that the porosity is often considered as a constant, or a simple function of bulk strain, in order to reduce the number of primary variables. This simplification can reduce the total number of equations, but may break the consistency among the three phases (solid, liquid and gas) and reduce the stability and efficiency of the numerical simulation.
The major objective of this paper is to present a more comprehensive modeling strategy for coupled THM processes and the numerical solution method for porous geomaterials. The work is part of work package 4 (WP4) of the THERESA project, which is supported by the European Commission. The objective of the project is to develop, verify and improve the modeling capabilities of coupled THM models and computer codes for performance and safety assessments of the safety of radioactive waste repositories. The main difference between the model proposed in this paper and other coupled THM numerical models mentioned above is that the gas flow and vapor flow processes are described explicitly by their mass conservation equations, so that phase change phenomena can be treated more objectively and efficiently. Another difference is that the mass conservation equation of the solid phase considers porosity changes that occur with deformation. Most importantly, an efficient equation solution strategy is developed to obtain stable and efficient numerical simulations, which is often the most challenging issue for modeling coupled THM processes with FEM, due to intrinsic numerical ill-conditioning caused by the coupling terms between the phases.
In the following sections, the general conservation equations and assumptions are first introduced, followed by development of the numerical model. The FEM formulation and solution strategy is introduced in the third section. The numerical results for the code (named ROLG) verification against a benchmark test problem as defined in WP4 of the THERESA project are presented afterwards, to verify the presented model and method. The paper is completed with a discussion section about outstanding issues and scientific conclusions.
Section snippets
General assumptions
Based on the theory of mixtures, a number of assumptions have been adopted to develop the coupled thermo-hydro-mechanical model for deformable porous geological media. Four of them of particular interest are:
- (1)
The partially saturated medium is treated as a multiphase system (solid, liquid, and gas). The voids of the solid skeleton are filled partially with liquid water, and partially with gas. The gas phase is modeled as an ideal gas mixture composed of dry air and water vapor. The liquid phase
Static equilibrium equation
Adopting the assumption of small the deformation and the assumption of small velocity, then , Eq. (8) can be simplified to
According to the concept of effective stress and mixture theory [25], [29], [30], [31], [32], [33], [34], [35], one has the following expressions:where σ is the total stress tensor, K is the bulk modulus of the skeleton, β is the volumetric thermal expansion coefficient, and Sr is the
FEM formulation
The governing equations (18), (27), (40), (48), (51), (62) are nonlinear differential equations with the following primary variables: displacement (three components), vapor pressure, gas pressure, liquid pressure, porosity and temperature. To solve them, they must be appropriately discretised in space and time. A Galerkin finite element solution approach [68] is used for the spatial discretization. The boundary conditions are introduced into the governing equations naturally. Then the
Concluding remarks
The presented mathematical theory of mixtures, FEM formulation and solution technique are demonstrated to be comprehensive and reliable tools for simulating fully coupled thermo-hydro-mechanical processes of multiphase geological media, which is essential for performance and safety assessments of geological waste repositories. The use of the theory of mixture makes the simulation of the coupled THM processes more process-oriented, without the need for using empirical models for unsaturated
Acknowledgments
The work performed in this paper is sponsored by the European Commission through the THERESA project. The authors are grateful for fruitful discussions with our THERESA project partners and Dr. Yifeng Chen from Wuhan University, China. The authors also appreciate very much the support to this research of professors Tian Bin and Defu Liu of China Three Gorges University.
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