A fully coupled thermo-hydro-mechanical model for simulating multiphase flow, deformation and heat transfer in buffer material and rock masses

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Abstract

This paper presents a numerical method for modeling coupled thermo-hydro-mechanical processes of geomaterials with multiphase fluid flow. A FEM code has been developed and validated for modeling the behavior of porous geological media, and is equally applicable for modeling coupled THM processes in rocks. The governing equations are based on the theory of mixtures applied to the multiphysics of porous media, considering solid phase deformation, multiphase fluid flow, and heat transport. New numerical techniques have been developed for more efficient FEM formulation and equation solution for modeling saturated or partially saturated water flow, gas flow and heat transfer in deformable porous media, as are commonly encountered in performance and safety assessment of underground radioactive repositories. The code has been validated against an experimental benchmark test, which involves bentonite under laboratory conditions, with good results. Several critical outstanding issues for modeling coupled processes of geomaterials are discussed in depth.

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 vα0, Eq. (8) can be simplified to·(σs+σl+σg)+α=s,l,gραbα=0

According to the concept of effective stress and mixture theory [25], [29], [30], [31], [32], [33], [34], [35], one has the following expressions:σs=σφsαB[Srpl+(1Sr)pg]IKβTIKεIσl=φlαBplIσg=φgαBpgIwhere 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.

References (72)

  • N. Khalili et al.

    An elasto-plastic model for non-isothermal analysis of flow and deformation in unsaturated porous media: formulation

    Int J Solids Struct

    (2001)
  • B.A. Schrefler et al.

    Multiphase flow in deforming porous material

    Comput Geotech

    (2004)
  • R.M. Bowen

    Theory of mixtures

  • A.C. Eringen et al.

    A continuum theory for chemically reacting media–I

    Int J Eng Sci

    (1965)
  • S.M. Rao et al.

    Role of direction of sal migration on the swelling behaviour of compacted clays

    Appl Clay Sci

    (2007)
  • G. Musso et al.

    The role of structure in the chemically induced deformations of FEBEX bentonite

    Appl Clay Sci

    (2003)
  • G.P. Peters et al.

    The influence of advective transport on coupled chemical and mechanical consolidation of clays

    Mech Maters

    (2004)
  • R.T. Walczak et al.

    Modeling of soil water retention curve using soil solid phase parameters

    J Hydrol

    (2006)
  • M. Nuth et al.

    Advances in modeling hysteretic water retention curve in deformable soils

    Comp Geotech

    (2008)
  • L.F. Pires et al.

    Soil porous system changes quantified by analyzing soil water retention curve modifications

    Soil Tillage Res

    (2008)
  • S. Hwang et al.

    Use of a lognormal distribution model for estimating soil water retention curve from particle-size distribution data

    J Hydrol

    (2006)
  • E. Romero et al.

    Water permeability, water retention and microstructure of unsaturated compacted Boom clay

    Eng Geol

    (1999)
  • V. Guvanasen et al.

    A three-dimensional numerical model for thermohydromechanical deformation with hysteresis in a fractured rock mass

    Int J Rock Mech Min Sci

    (2000)
  • E. Dana et al.

    Gas relative permeability and pore structure of sandstones

    Int J Rock Mech Min Sci

    (1999)
  • H. Feng et al.

    Intrinsic and relative permeability for flow of humid air in unsaturated apple tissues

    J Food Eng

    (2004)
  • A.M. Tang et al.

    A study on the thermal conductivity of compacted bentonites

    Appl Clay Sci

    (2008)
  • B.R. Becker et al.

    Development of correlations for soil thermal conductivity

    Int Commun Heat Mass Transfer

    (1992)
  • N.H. Abu-Hamdeh et al.

    A comparison of two methods used to evaluate thermal conductivity for some soils

    Int J Heat Mass Transfer

    (2001)
  • R.W. Zimmerman

    Coupling in poroelasticity and thermoelasticity

    Int J Rock Mech Min Sci

    (2000)
  • M.V. Villar et al.

    State of the bentonite barrier after five years operation of an in situ test simulating a high level radioactive waste repository

    Eng Geol

    (2005)
  • T.S. Nguyen et al.

    Modelling the FEBEX THM experiment using a state surface approach

    Int J Rock Mech Min Sci

    (2005)
  • K. von Terzaghi

    Die berechnug der Durchlässigkeitsziffer desTones aus dem Verlauf der hydrodynamischen Spannungserscheinungen

    Sitzungsber Akad Wiss Math-Naturwiss Section IIa

    (1923)
  • M.A. Biot

    General theory of three-dimensional consolidation

    J Appl Phys

    (1941)
  • M.A. Biot

    General solution of the equation of elasticity and consolidation for a porous material

    J Appl Mech

    (1956)
  • L.W. Morland

    A simple constitutive theory for fluid saturated porous solids

    J Geophys Res

    (1972)
  • M. Hassanizadeh et al.

    Mechanics and theormodynamics of multiphase flow in porous media including interphase transport

    Adv Water Resourc

    (1990)
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