N M C T

N M C T
NUMERICAL MODELING OF C OUPLED
THERMO-HYDRO-MECHANICAL PROCESSES
IN GEOLOGICAL P OROUS M EDIA
Fuguo Tong
January 2010
TRITA-LWR PhD Thesis1055
ISSN 1650-8602
ISRN KTH/LWR/PHD 1055-SE
ISBN 978-91-7415-554-9
FuguoTong
TRITA LWR PhD 1055
© Fuguo Tong 2010
Doctoral Thesis
Engineering Geology and Geophysics Group
Department of Land and Water Resources Engineering
Royal Institute of Technology (KTH)
SE-100 44 STOCKHOLM, Sweden
Reference should be written as: Tong, F.G. (2010) Numerical modeling of coupled
thermo-hydro-mechanical processes in geological porous media, KTH. TRITA-LWR
PhD Thesis 1055:84
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
A B STR A C T
Coupled Thermo-Hydro-Mechanical (THM) behavior in geological
porous media has been a subject of great interest in many geoengineering 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, geothermal energy extraction, and safety assessment of repositories for
radioactive waste and spent nuclear fuel. This thesis presents a new
numerical modeling approach and a new computer code for simulating coupled THM behavior in geological porous media in general, and
compacted bentonite clays in particular, as buffer materials in underground radioactive waste repositories.
New governing equations were derived according to the theory of
mixtures, considering interactions among solid-phase deformation,
flows of water and gases, heat transport, and phase change of water.
For three-dimensional problems, eight governing equations were
formulated to describe the coupled THM processes.
A new thermal conductivity model was developed to predict the
thermal conductivity of geological porous media as composite mixtures. The proposed model considers the combined effects of solid
mineral composition, temperature, liquid saturation degree, porosity
and pressure on the effective thermal conductivity of the porous media. The predicted results agree well with the experimental data for
MX80 bentonite.
A new water retention curve model was developed to predict the suction-saturation behavior of the geological porous media, as a function
of suction, effective saturated degree, temperature, porosity, pore-gas
pressure, and the rate of saturation degree change with time. The
model was verified against experimental data of the FEBEX bentonite, with good agreement between measured and calculated results.
A new finite element code (ROLG) was developed for modeling fully
coupled thermo-hydro-mechanical processes in geological porous
media. The new code was validated against several analytical solutions
and experiments, and was applied to simulate the large scale in-situ
Canister Retrieval Test (CRT) at Äspö Hard Rock Laboratory, SKB,
Sweden, with good agreement between measured and predicted results. The results are useful for performance and safety assessments of
radioactive waste repositories.
Key words: Thermo-hydro-mechanical processes; Porous geological
media; Numerical modeling; FEM; Multiphase flow; Effective thermal conductivity; Water retention curve; Radioactive waste repositories; Bentonite.
Title : N umerical modeling of coupl ed th ermo-hydro-mechani cal
processes in geologi cal porous media
Auth or: Fuguo Tong
Department: Division of Engineerin g Ge ology and Geophysi cs,
Department of Land and Water Resources Engineer ing, School of
Archite cture an d th e Built Environm ent, KTH
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FuguoTong
TRITA LWR PhD 1055
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
A B STR A K T
Kopplade termo-hydralisk-mekaniska (THM) egenskaper i
geologiska porösa medier har orsakat stort intresse inom många av
de tekniskt geologiska disciplinerna. Många försök har gjorts för
att utveckla möjliga numeriska förutsägelser gällande olika
områden såsom rörelse av föroreningsplymer, gasinjektioner,
energiförvaring,
geotermisk
energiutvinning,
och
säkerhetsbedömning av kärnbränsleförvaring. Denna avhandling
presenterar en ny numerisk modell med en ny infallsvinkel och
kod för att simulera den kopplade THM uppförande i geologiska
porösa medier, generellt, men även särskilt i kompakt bentonitlera,
som kan användas som buffermaterial omkring radioaktivt avfall
placerade i bergrum.
Nya styrande ekvationer erhölls ifrån teorin om blandning, med
hänseende på interaktionen mellan solid-fas deformation, flöde av
vatten och gaser, värmetransport, och fasändringar av vatten. Åtta
styrande ekvationer blev formulerade för att beskriva den kopplade
THM processen för 3-dimensionella problem.
En ny termisk konduktivitetsmodell utvecklades för att kunna
prognostisera den termiska konduktiviteten av geologiska porösa
medier som sammansatta blandningar. Den föreslagna modellen
tar hänsyn till den kombinerande effekten av solida
mineralsammansättningar,
temperatur,
vätskemättnadsgrad,
porositet och tryck på den effektiva termiska konduktiviteten av
det porösa mediet. Resultaten överenstämmer väl med de
experimentella datan från MX80 bentonit.
En ny vattenkvarhållningskurvmodell utvecklades att för att
prediktera insugnings-mättnadsuppförandet av det geologiska
porösa mediet, som en funktion mellan insugning, effektiv
mättnadsgrad, temperatur, porositet, por-gastryck, och
mättnadsgradens ändring över tid. Modellen verifierades gentemot
experimentella data från FEBEX bentonit, med god
överenstämmelse mellan mätta och beräknade resultat.
En ny finit-element kod (ROLG) utvecklades för att modellera
fullt kopplade THM processer i geologiska porösa medier. Den
nya koden kontrollerades gentemot flera analytiska lösningar och
experiment, och användes även till att simulera storskalig in-situ
Canister Retrieval Test (CRT) vid Äspölaboratoriet, SKB, Sverige,
med god korrelation mellan mätta och prognostiserade resultat.
Resultaten är användbara för utveckling och säkerhetsbedömning
av kärnbränsleförvaring i underjorden.
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TRITA LWR PhD 1055
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
A C K N O WLEDGM E NTS
I wish to express my gratitude to all the people who have helped
me during the three years I have been working on this thesis and
on the THERESA project. This thesis would not be possible
without their support.
First, I would like to thank to my supervisor, Dr. Lanru Jing, for
his guidance, valuable suggestions and comments during my study
at KTH
I would like to express my appreciation to my assistant supervisor,
Prof. Robert Zimmerman, previously at KTH and currently in Imperial College London, for his valuable suggestions and comments
during my study. As my former first supervisor, he not only gave
me an opportunity to study at KTH, but also provided me a
chance to study at Imperial College, London, as a visiting PhD
student.
I wish to thank Prof. Bin Tian in China Three Gorges University.
He led me into the scientific research field, with guidance and help.
I want to express my appreciation to Prof. Defu Liu in China
Three Gorges University, for his encouragement and help. I also
want to thank to Prof. Weiya Xu in Hohai University, China, for
his supporting my study in Hohai University before coming to
KTH.
I would like to thank my colleagues, Alireza Baghbanan, Thushan
Ekneligoda, Tomofumi Koyama, Zhihong Zhao, Mimmi Magnusson, Solomon Tafesse and Zairis Coello for their understanding,
help and fruitful discussions. I want to thank Colin Leung, and
Emmanuel David, for their help during the period of my study in
Imperial College London.
I also wish to express my gratitude to Dr. Joanne Fernlund and Dr.
Katrin Grünfeld at the Group of Engineering Geology, for their
encouragement and help. I also like to thank other members of the
Department of Land and Water Resources Engineering, KTH, especially Britt Chow, Aira Saarelainen and Jerzy Buczak for their effective and kind help with all administrative matters.
Financial support provided by the European Commission through
the THERESA project (contract no. 6FP-036458) is greatly acknowledged.
Last but not least, I thank my wife, Bin Tan for her support and
help. Without her encouragement and understanding it would have
been impossible for me to finish this work.
Fuguo Tong
Stockholm, December 2009
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FuguoTong
TRITA LWR PhD 1055
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
L I ST OF PA PER S
This thesis is based on the following papers, which are appended at the end of the thesis
and referred in the text of the thesis by their Roman numbers (I- IV):
I: Tong FG, Jing L, Zimmerman RW. A fully coupled thermo-hydro-mechanical model for
simulating multiphase flow, deformation and heat transfer in buffer material and rock
masses. Int J Rock Mech Min Sci 2010; 47:205-217.
II: Tong FG, Jing L, Zimmerman RW. An effective thermal conductivity model of geological porous media for coupled thermo-hydro-mechanical systems with multiphase flow.
Int J Rock Mech Min Sci 2009; 46:1358-1369.
III: Tong FG, Jing L. A water retention curve model for coupled thermo-hydromechanical processes of geological porous media. Submitted to Transport in Porous Media,
2009 (under review)
IV: Tong FG, Jing L, Zimmerman RW. A 3D FEM simulation of the buffer and bufferrock interface behaviour of the Canister Retrieval Test(CRT) at Äspö HRL. In: Proceedings Conference of impact of THMC processes on the safety of underground radioactive
waste repositories, 29 Sep-01 Oct 2009, Luxembourg. European Commission, to be
printed in 2010, 5 pp.
The following publications are related to my research and were written during the period of
my PhD study, but are not appended in this thesis:
V: Tong FG, Jing L, Zimmerman RW. Modeling multiphase flow, deformation and heat
transfer in buffer. In: Proceedings SINOROCK 2009, 18-20 May 2009, Hong Kong, China.
Hudson J.A., Tham L.G., Feng X., Kwong A.K.L., eds. Electronically published by CD, 5
pp.
VI: Tong FG, Jing L, Zimmerman RW. A numerical study of viscous fluid flow in a single
fracture. New development in rock mechanics and engineering. In: Proceedings Conference
on New Direction in Rock Mechanics, 2009, Sanya, China. Feng X., Wang S., Lin Y., eds.
Published by Liberty Culture & Technology, pp 165-170.
VII: Tong FG, Jing L, Zimmerman RW. A fully-coupled finite element code for modeling
thermo-hydro-mechanical processes in porous geological media. In: Proceedings 43rd US
Rock Mechanics Symposium, 28-30 June 2009 Asheville, NC, USA,. Smeallie P, ed. Electronically published by CD, 10 pp.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
T A B LE O F C O NTEN TS
Abstract...................................................................................................................... iii
Acknowledgments .....................................................................................................vii
List of papers ..............................................................................................................ix
Table of contents ........................................................................................................xi
Nomenclature .......................................................................................................... xiii
1 Introduction ..............................................................................................................1
1.1 Background and motivation of this study .................................................................... 1
1.2 Objectives of this thesis................................................................................................ 3
1.3 Layout of this thesis...................................................................................................... 4
2 Governing equations for coupled THM processes of geological porous
media ...........................................................................................................................5
2.1 Introduction.................................................................................................................. 5
2.2 General assumptions.................................................................................................... 6
2.3 General conservation equations ................................................................................... 6
2.3.1 Mass conservation equation .................................................................................................. 6
2.3.2 Momentum conservation equation ........................................................................................ 7
2.3.3 Energy conservation equation ............................................................................................... 7
2.4 Governing equations of coupled THM processes ....................................................... 8
2.4.1 Static equilibrium equation.................................................................................................... 8
2.4.2 Liquid flow equation........................................................................................................... 10
2.4.3 Gas flow equation .............................................................................................................. 11
2.4.4 Vapor flow equation........................................................................................................... 12
2.4.5 Solid mass conservation equation ........................................................................................ 13
2.4.6 Heat transport equation ...................................................................................................... 14
2.5 Summary .................................................................................................................... 14
3 Constitutive models of geological porous media for coupled THM processes.....15
3.1 Constitutive models for mechanical process.............................................................. 15
3.1.1 Nonlinear elasticity model................................................................................................... 15
3.1.2 Elastoplastic model ............................................................................................................ 16
3.2 Constitutive models for liquid and gas flow process ................................................. 17
3.2.1 Water retention curve ......................................................................................................... 17
3.2.2 Relative permeability of liquid ............................................................................................. 32
3.2.3 Relative permeability of gas................................................................................................. 33
3.3 Constitutive models of heat transport process........................................................... 33
3.3.1 Thermal conductivity model development............................................................................ 33
3.3.2 Thermal conductivity model validation ................................................................................ 37
3.3.3 Discussion of the effective thermal conductivity model behavior........................................... 40
3.4 Summary .................................................................................................................... 42
4 Numerical solution of governing equations for coupled THM processes ............43
4.1 FEM method for solving the governing equations .................................................... 43
4.1.1 The spatial discretization with FEM..................................................................................... 43
4.1.2 The temporal discretization................................................................................................. 47
4.2 The solution strategies ............................................................................................... 47
4.3 The computer code .................................................................................................... 48
5 Code validation and application for field scale in-situ problems...........................51
5.1 Validations against analytical solutions ..................................................................... 51
5.1.1 A poroelastic problem ........................................................................................................ 51
5.1.2 A thermoelastic problem..................................................................................................... 53
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5.2 Validation against laboratory experiments on bentonite ........................................... 57
5.2.1 Validation with laboratory benchmark test 1......................................................................... 57
5.2.2 Validation with laboratory benchmark test 2......................................................................... 60
5.3 The simulation of the large scale Canister Retrieval Test (CRT).............................. 63
5.3.1 Introduction of the Canister Retrieval Test (CRT) ................................................................ 63
5.3.2 Comparison of measured and modeling results..................................................................... 65
5.3.3 Predicted results at steady state............................................................................................ 67
5.4 Summary .................................................................................................................... 69
6 Discussion and conclusions ...................................................................................71
6.1 Summary conclusions on scientific achievements ..................................................... 71
6.2 Discussion on outstanding issues.............................................................................. 72
7 Recommendation for future studies .......................................................................76
References..................................................................................................................77
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
N O MENC LA TUR E
α
αB
bα
βα
where phase α = s, l, g ,v denote solid, liquid, gas and vapor
the Biot coefficient
the body force per unit mass of phase α
the volumetric thermal expansion coefficient of phase α
the specific heat of phase α
cα
δ1
χ
D
Dve
the shape coefficient of air/gas bubble
Bishop’s effective stress parameter
the elastic modulus tensor of the underlying drained solid
the tensor of the molecular diffusivity of vapor
ε v , ε v , ε v the total, elastic and plastic volumetric strains
e
p
ε vp
p
the plastic volumetric strain associated with LC yield surface
ε sp
p
the plastic volumetric strain associated with SI yield surface
ετ , ετ , ετ
e
p
the total, elastic and plastic deviatoric strains
εv
ε′
φ
the volumetric strain
the swelling volumetric strain of the solid skeleton
the porosity
ϕα
Fb
G
g
H
I
K
Kα
k αT
k
kT
the volume fractions of phase α
the body force vector
the shear modulus
the gravitational acceleration vector
the coefficient of solubility of gas in liquid defined by Henry’s law
the identity tensor
the bulk modulus
the bulk modulus of phase α
the thermal conductivity tensor of phase α
the intrinsic permeability tensor
the thermal conductivity tensor
klr
the liquid relative permeability
k
r
g
the gas relative permeability
l
T
g
T
the coefficient tensor of thermal coupling for liquid flux.
k
k
the coefficient tensor of thermal coupling for gas flux.
κ
the elastic stiffness parameter for changes in net mean stress
λ
the effective thermal conductivity of a three-phase mixture
λT
the surface tension of water
λa , λ g , λ w , λ s the thermal conductivity of air, gas, water and solid
λLs− g , λUs− g the lower and upper bounds of the effective thermal conductivity of
a solid-gas mixture
λL , λU the lower and upper bound of the effective thermal conductivity of a
three-phase mixture
n
the unit outward normal vector to the boundary ΓV
xiii
FuguoTong
pa
pg
pl
p sv
pv
pc
π̂ α
Q′Tα
TRITA LWR PhD 1055
the air pressure
the gas pressure
the liquid pressure
the saturated vapor pressure.
the vapor pressure
the reference stress
the momentum supply to phase α by the rest of the mixture
the heat source of phase α
Qα
the mass source Qα of phase α
QT
the heat source of the multiphase media
q
the heat flux vector of phase α
θ
the rate of effective saturation degree change with time
θ
the dimensionless rate of effective saturation degree change with time
ρ a , ρ w the densities of air and water
α
ρα
ρα
Sr
Se
s
se
σ
σ′
σ
σα
T
T
Tα
ΓV
the apparent mass density of phase α
the intrinsic mass density of phase α
the saturation degree
the effective saturation degree
the suction
the minimal suction (air entry value at which the air enters the pores)
the total stress tensor
the effective stress tensor
the effective mean-stress
the partial Cauchy stress tensor of phase α
temperature
the dimensionless temperature
the temperature of phase α
the boundary of V
vα
u
ui
µ
µl
µg
the velocity field of particles of phase α
the total displacement vector
the displacement along direction i
the Poisson’s ratio
the viscosity of liquid
the viscosity of gas
v si
w
ω
Xα
ς
the velocity of solid phase in the i-th direction.
the water content
the liquid phase transfer coefficient
a material point of the α phase.
the pore density
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
1 I N TR O DUC TIO N
1.1 Background and motivation of this study
The couplings between the processes of heat transfer, fluid flow
and stress/deformation in porous geological media has become an
increasingly important subject in many engineering disciplines. In
the last few decades, many experimental and theoretical studies on
coupled thermo-hydro-mechanical (THM) behavior of geological
porous media have been made to develop prediction capabilities
for safety assessment of the waste repository for radioactive waste
and spent nuclear fuel, geothermal energy extraction, gas/oil recovery, contaminant transport analysis and environment impact
evaluation (Stephansson et al. (2003); Tsang et al. (2004)).
The term ‘coupled thermo-hydro-mechanical processes’ implies
that one process affects the initiation and progress of others.
Therefore, the geological medium’s response to natural or manmade perturbations, such as construction and operation of a nuclear waste repository, cannot be predicted with adequate confidence by considering each process independently (Jing, 2000). The
one-way coupling mechanism is a simple case that reflects the continuing one-directional effect of one process on the others,
whereas the two-way coupling process reveals continuing reciprocal (feed back) interactions among different processes in a complex way (Fig. 1.1, Hart et al., 1986).
THERMAL
gy
vec
tio
n
Ph
ase
osm cha
osi nge,
s, v the
isc rm
osi oty
n
sio
pan
x
s/e
on
res
l st
ner
al e
nic sion
cha ver
Me con
He
at c
a
erm
Th
Heat conduction and
convection
Change of porosity/permeability
HYDRAULIC
MECHANICAL
Water flow
Water pressure
Stress-starin
Figure 1.1. Coupled THM processes in porous media
The numerical simulations of the coupled THM processes of geological materials are based on the theories of porous media. The
first such theory was Terzaghi’s 1-D consolidation theory of soils
(von Terzaghi, 1923), followed later by Biot’s theory of isothermal
consolidation of elastic porous media, a phenomenological approach of poroelasticity (Biot, 1941, 1956), and mixture theory as
described in Morland (1972), Goodman and Cowin (1972), Sampaio and Williams (1979), Bowen (1982) and others. Nonisothermal consolidation of deformable porous media is the basis
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TRITA LWR PhD 1055
of more recent coupled THM models, using either the averaging
approach as proposed first by Hassanizadeh and Gray (1979, 1980,
1990) and Achanta et al. (1994), or an extension of Biot’s phenomenological approach with a thermal component (de Boer,
1998). Both approaches are applicable for modeling coupled THM
processes in geological materials. In theory, the theory of mixtures
is more suitable for understanding the microscopic behavior of the
porous media, whereas poroelasticity theory is better suited for
macroscopic description and computer modeling (Jing, 2000). In
practice, however, the two conceptual approaches can be developed to have the same usefulness for both micro-and macroscopic
descriptions and modeling.
For safe disposal of radioactive wastes and spent nuclear fuel in
Sweden, Finland and some other countries, a repository in crystalline rocks with multiple engineered and natural barriers to protect
the environment against the radioactive nuclides transport process
is required (Fig. 1.2). The spent fuel assemblies will be encapsulated in metal canisters, then placed in deposition holes several
hundred meters deep in the bedrocks, and the space between rock
and canisters will be filled with compacted expansive bentonite
clay (buffer) for canister protection. The long-term safety of geological nuclear waste repositories is therefore based on the functions of the superposed natural (rock) and engineered (canister and
bentonite) barriers (Horseman and McEwen, 1996). Each successive barrier represents an additional restraint to the movement of
radionuclides from the waste towards the biosphere. The interactions between barriers need to be fully understood in order to predict the long term behavior of the geological nuclear waste repository (Muñoz, 2007).
Figure 1.2. The conceptual scheme of nuclear waste repository in
crystalline rocks
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Bentonites are composed of clay minerals of the smectite group,
whose physical and chemical properties, such as low permeability,
high swelling potential, high capacity of water retention, plasticity
and mechanical strength, make them proper buffer materials (Astudillo-Pastor, 2001). Bentonite is a porous medium planned to be
used as the engineered buffer material between the host rock and
waste canisters in repositories in Sweden, Finland and other countries (Pusch and Young, 2003; Tang et al., 2008).
The work of this thesis was a part of the THERESA project for
numerical modelling of coupled THM processes for safety assessment of radioactive waste repositories, which is an international
co-operative project funded by European Commission’s 6th
Framework Euratom Program (2007-2009). The THERESA project aims to develop a scientific methodology for evaluating the
capabilities of mathematical models and computer codes for performance and safety assessments of geological nuclear waste repositories, based on the scientific principles governing coupled
thermo-hydro-mechanical and chemical processes in geological
systems and geo-materials. The project was made of five work
packages (WP1-WP5), and part of the research presented in this
thesis belongs to WP4 that concerns with development and validation of constitutive models of buffer material and rock-buffer or
canister-buffer interfaces considering coupled THM processes and
computer codes, and applications of the developed models and
codes for simulation of an existing large scale test case. For WP4,
this test case is the full scale in-situ Canister Retrieval Test (CRT)
at the Äspö Hard Rock Laboratory, run by the Swedish Nuclear
Fuel and Waste Management Company (SKB), in southern Sweden.
1.2 Objectives of this thesis
The general goal of this thesis is to develop a new numerical
method of modeling coupled THM processes for porous geological media, with particular applications to compacted bentonite
clays used as buffer materials in underground radioactive waste repositories. To meet also the requirement of THERESA project, it
includes the following specific objectives:
1) At the level of governing equations, to develop a more comprehensive mathematical model representing the coupled THM processes of geological porous media in general, and of the compacted
bentonite materials in particular. These mathematical models
should explicitly describe the gas flow and vapor flow processes,
and phase change phenomena of water should also be treated efficiently, besides commonly considered equations on stress (deformation), liquid flow and heat transfer. Porosity changes that occur
with deformation should be explicitly considered according to
mass conservation equations of the solid phase, instead of only
depending on bulk deformation of solid skeletons as often used in
other THM models of geological meterials.
2) To develop a Finite Element Method (FEM) code based on the
newly developed governing equations for practical applications.
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Special solution strategies and technologies should be developed to
obtain stable and efficient numerical simulations, which are often
the most challenging issue for modeling coupled THM processes
with FEM, due to high nonlinearity of the coupled equations and
intrinsic numerical ill-conditioning caused by the coupling terms
between the phases (solid, liquid, vapor and gas).
3) To develop advanced constitutive models for porous geological
media, especially for simulating coupled THM behavior of bentonite. In view that there are many aspects that need to be addressed by constitutive models and it is impossible to address all of
them in one study, in this thesis, only a thermal conductivity model
and a water retention curve model were developed, validated and
applied for the CRT test case simulations because these two aspects are two of the most important coupling aspects of constitutive behaviors of porous geological media.
4) To validate the new computer code against several analytical solutions and several sets of experiment data, and then to apply this
code to simulate the full scale in-situ CRT experiment for evaluating the predicting capability of the code for modeling long-term
coupled THM behaviors in engineered buffer.
1.3 Layout of this thesis
The thesis is organized in seven chapters, and the contents of each
Chapter are summarized as follows:
• Chapter 1 introduces the general background of research first,
followed by the motivations and objectives of this thesis.
• Chapter 2 describes the development of the equations governing the coupled THM process in porous geological media.
• Chapter 3 presents the development of new constitutive models for heat transfer and unsaturated flow in porous geological
media in general, and compacted bentonite in particular.
• Chapter 4 describes the numerical solution of the governing
equations for the coupled THM processes and a new computer
code using the Finite Element Method. Special solution strategies and technologies are described for handing the issues of
convergence and numerical stability of the solution process.
• Chapter 5 presents the validation of the developed mathematical
models and computer code, against analytical solutions for
problems of poroelasticity and thermoelasticity, and two laboratory benchmark experiments on bentonite, followed by an application of the code for simulation of the SKB’s large scale insitu CRT experiment and predictions for its long-term steadystate behavior.
• Chapter 6 presents summary conclusions on scientific achievements and discussions on outstanding issues for modeling coupled THM processes in porous geological media, based on experiences obtained from the research.
• Chapter 7 simply suggests some topics for further research.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
2 G O V ER NIN G EQUA TION S F OR C O UPLED
G EOLO G ICA L POR O US M EDIA
THM PR O C ESSES OF
The derivation of the governing equations for coupled THM processes was presented in Paper I in detail, and only a summary description is given in this Chapter. Physical meanings of most of the
major variables and/or parameters are explained in the nomenclature list. The rest are given in the text following their appearance in
associated equations.
2.1 Introduction
Different sets of governing equations have been developed for
modeling coupled THM process of geological porous media, such
as for research computer codes ROCMAS (Noorishad & Tsang,
1996), THAMES (Ohnishi & Kobayashi, 1996), FRACON
(Nguyen, 1996), COMPASS (Thomas at al., 1996) and
CODE_BRIGHT (Olivella, 1996). Commercial codes have also
been used, such as ABAQUS (Börgesson et al., 1996). Some of
these modeled fluid flow in unsaturated media by using Richard’s
equation (Richards, 1931), but gas movement was often not rigorously considered, since the gas pressure was often assumed to be
small and constant in many formulations. Some of these codes,
such as COMPASS and CODE_BRIGHT, can simulate twophase (gas and liquid) fluid flow in unsaturated soils, coupled with
heat transport and mechanical response (Rutqvist, 2001). However,
the advective flow of vapor due to phase change of water was often not well described, including heat transfer between the liquid
and gas phases. Khalili and Loret (2001) stated that, only if one can
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. (Schrefler & Pesavento,
2004). This approach, however, 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. This simplification can reduce the total number of equations, but may break the consistency condition among the three
phases (solid, liquid and gas) and reduce the stability of numerical
simulation process.
Fluid transport in and the porosity (density) change of bentonite
usually play important roles in performance and safety assessments
of geological nuclear waste repositories; thus it is necessary to develop a more comprehensive set of governing equations and a
more robust and efficient computer code that can more accurately
simulate coupled THM processes in natural and engineered barriers with porosity changes included.
The theory of mixture is suitable for developing coupled THM
models in deforming porous materials with explicit representations
of liquid, gas, vapor flows; thus it was adopted to derive new gov5
FuguoTong
TRITA LWR PhD 1055
erning equations of the coupled THM processes for geological porous media as presented in this Chapter.
2.2 General assumptions
The basic material assumption adopted in this research is that the
geological materials (rocks, buffer materials) were assumed to be
continuous, homogeneous and isotropic porous media. The problems were defined in three dimensions with small strains.
The adoption of the theory of mixtures indicated other assumptions for developing the coupled thermo-hydro-mechanical equations for deformable porous geological media. Four of them are of
particular interest: 1) the partially saturated medium was treated as
a multiphase system of solid, liquid and gas. The voids of the solid
skeleton were filled partially with liquid water, and partially with
gas. The gas phase was modeled as an ideal gas mixture composed
of dry air and water vapor. The liquid phase consists of water and
dissolved air. 2) The multiphase medium was considered as a homogeneous mixture. Every phase is continuous and each spatial
point of the mixture was occupied simultaneously by a material
point of every phase. The balance laws for the mixture as a whole
have the same form as the equations for the balance laws of a single phase material (Truesdell & Toupin, 1960). 3) The mechanical
behavior of the multiphase medium considers the gas, liquid and
solid responses to local pressure and the overall responses to the
effective stresses. 4) The fluid flow is a multiphase flow in a deformable porous medium under nonisothermal conditions.
2.3 General conservation equations
An unsaturated porous medium, in the context of theory of mixtures, is viewed as a mixed continuum of three independent overlapping phases of solid, liquid and gas (Bear & Bachmat, 1991).
For every phase, its conservation equations can be obtained according to the principles of continuum mechanics and mixture
theory. Relevant literatures can be found in Bowen (1976), Bear &
Bachmat (1991), Li et al. (1993) and Lewis (1998).
2.3.1 Mass conservation equation
Given an arbitrary spatial volume V, the time rate of change of the
total mass of the α phase in V must be equal to the flux of that
phase across the boundary ΓV of volume V, plus the source Qα.
The conservation law of mass for the α phase is written as:
d
ρ α dV = − ∫ ρ α v α ⋅ ndS + ∫ Qα dV
(2.1)
∫
dt V
ΓV
V
Using the divergence theorem in the usual manner, one obtains the
following local form of the mass conservation law:
Dtα ρα + ρα ∇ ⋅ vα − Qα = 0
(2.2)
The material derivative relative to the α phase is given by
Dtα (⋅) = ∂t (⋅) + ∇(⋅) ⋅ vα
(2.3a)
α
The velocity vα is related to the position vector x and is written as
6
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
vα = ∂t xα (Xα ,t ) Xα
(2.3b)
The apparent density of α phase is expressed as
ρ α = ρα ϕ α
(2.3c)
Inserting eqs.(2.3a,b,c) into eq.(2.2), the mass balance equation for
the α phase takes the form
∂ t ( ρ α ϕ α ) + ∇ ⋅ ( v α ρα ϕ α ) − Q α = 0
(2.4)
2.3.2 Momentum conservation equation
The conservation of linear momentum for the α phase of the
mixture is the same as that for a single phase material. In any given
spatial volume V of boundary Γv, the conservation of linear momentum for the α phase (assuming no chemical reactions) (Bowen,
1976) is written as
d
ρ α v α dV = − ∫ ρ α v α ( v α ⋅ n)dS + ∫ σ α ⋅ ndS + ∫ ρ α b α dV
dt V∫
ΓV
ΓV
V
(2.5)
+ ∫ πˆ α dV
V
Using the divergence theorem again in the usual manner, and recalling eq.(2.2), one obtains the following local form of the momentum conservation equation:
ρ α ∂ t v α + ρ α (∇v α ) ⋅ v α = ∇ ⋅ σ α + ρ α b α + πˆ α
(2.6)
Summing up the momentum equations of the three phases, and
accounting for the constraint condition of momentum supplies,
that is ∑ πˆ α = 0 (Eringen & Ingram, 1965), the momentum
α = s ,l , g
conservation equation of mixture is written as:
∑ρ
α = s ,l , g
α
∂ t vα +
∑ ρ (∇v ) ⋅ v
α
α
α = s ,l , g
α
=
∑ (∇ ⋅ σ
α = s ,l , g
α
)+
∑ρ
α = s ,l , g
α
b α (2.7)
2.3.3 Energy conservation equation
Given any spatial volume V, the time rate of change of the total
internal energy of phase α in volume V must be equal to the heat
flux of that phase across the boundary Γv of volume V, plus the
heat source Q′Tα and the energy supply εˆα to the α phase due to
the interactions with the other phases (Bowen, 1976; Bear &
Bachmat, 1991). The energy conservation equation can then be
written as
d
(
ρ α cα Tα )dV = − ∫ q α ⋅ ndS + ∫ QT′α dV + ∫ εˆ α dV
(2.8)
∫
dt V
ΓV
V
V
In a differential form, this relation may be expressed by
Dt ρ α cα Tα = −∇ ⋅ q α + QT′α + εˆα
(
)
(2.9)
Generally, the heat flux vector qα can include heat conduction,
convection and heat radiation terms. Here we only consider heat
7
FuguoTong
TRITA LWR PhD 1055
conduction and convection. The heat flux vector qα can be expressed as
qα = −k αT (∇T ) + ρ α cα Tα v α
(2.10a)
The heat source Q′ in eqs. (2.9) is expressed as
α
T
QT′α = QTα + Q1αT − Q 2αT
(2.10b)
α
T
where Q is the other potential heat sources that may occur from
other physical or chemical processes, such as electric or magnetic
processes and chemical reactions. Symbol Q1αT represents the heat
energy that is produced by the work by external forces. For elastic
behavior, it is always equal to zero. Symbol Q2αT is the heat energy
reduction due to thermal power during the course of the bulk deformation (Boley & Weiner, 1960; Nowacki, 1986; Zimmerman,
2000), and is expressed as:
Q 2Tα = K α β α Tα (∇ ⋅ v α )
(2.10c)
Substituting eqs. (2.10a,b,c) into eq. (2.9), the energy conservation
equation for the α phase is obtained as
Dt ( ρ α cα Tα ) = ∇ ⋅ [k αT (∇Tα )] − ∇ ⋅ (v α ρ α cα Tα ) + QTα
+ Q1αT − K α βα Tα ∇ ⋅ v α + εˆα
(2.11)
Summing up energy equations of the three phases, and accounting
for the constraint condition: εˆ g + εˆ l + εˆ s = 0 , the energy conservation equation of the mixture is given by
∑ D ( ρ c T ) − ∑ ∇ ⋅ [k (∇T )] + ∑ ∇ ⋅ (v
+ ∑ K β T (∇ ⋅ v ) − ∑ Q − ∑ Q1 = 0
α
α = s ,l , g
α α
t
α = s,l , g
α
α
αT
α = s ,l , g
α
α
α = s ,l , g
α
α
T
α = s ,l , g
α = s ,l , g
α
T
α
ρ α cα Tα )
(2.12)
Equations (2.4), (2.7) and (2.12) are the general conservation equations of porous media as mixtures. The specific governing equations for specific porous geological materials must be developed by
introducing specific constitutive laws into these general conservation equations.
2.4 Governing equations of coupled THM processes
2.4.1 Static equilibrium equation
Adopting the assumptions of small deformation and small velocity,
then v α ≈ 0 , eq. (2.7) can then be simplified into
∇ ⋅ (σ s + σ l + σ g ) +
∑ρ
α = s ,l , g
α
bα = 0
(2.13)
According to the mixture theory and the concept of effective
stress (Bishop, 1959; Carslaw & Jaeger, 1959; Jennings & Burland,
1962; Bishop & Blight, 1963; Burland, 1965; Nur & Byerlee, 1971;
Bowen, 1976; Loret & Khalili, 2000), the partial Cauchy stress tensors of solid, liquid and gas have the following expressions:
σ s = σ − ϕ s α B [S r pl + (1 − S r ) p g ]I − KβTI − Kε ′ I
8
(2.14a)
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
σ l = −ϕ l α B p l I
(2.14b)
σ g = −ϕ g α B p g I
(2.14c)
where α B is the Biot coefficient, expressed as α B = 1 − K K S .
Here, traditional effective stress tensor is defined
as σ s + σ l + σ g = σ ′ . Therefore, eqs. (2.14a,b,&c) lead to the effective stress tensor written as
(2.14d)
σ′ = σ − α B pg I + S r α B ( pg − pl )I − KβTI − Kε ′ I
Under fully saturated condition (ignoring the temperature and
swelling strain terms), it can be simplified as σ − α B p l I , which is
Terzaghi’s effective stress.
Substituting eqs. (2.14a,b,c) into eq. (2.13), and considering
σ = D : ∇u , ϕ l = (1 − ϕ s ) S r and ϕ g = (1 − ϕ s )(1 − S r ) , the static
equilibrium equation can be obtained as
[
] ∑ρ
∇ ⋅ D: ∇u −α B pg + S r α B ( pg − pl ) − KβT − Kε ′ +
α =s , l , g
α
bα = 0 (2.15)
Defining χ = S rα B , and considering α B p g ≈ p g , eq. (2.15) can be
rewritten in a time derivative form as:
∂p g
∂p
∂u
∇ ⋅ (D : ∇ ) − ∇ ⋅ [(1 − χ )
] − ∇ ⋅ [χ l ]
∂t
∂t
∂t
(2.16)
∂T
∂ε ′ ∂Fb
− ∇ ⋅ [ Kβ
] − ∇ ⋅ [K
]+
=0
∂t
∂t
∂t
where Fb =
∑ρ
α = s ,l , g
α
b α is body force vector.
The definition of the Bishop’s parameter χ is motivated by experimental and theoretical arguments. Bishop’s parameter describes the contribution of matric suction to the effective stress; it
may be regarded as a scaling factor averaging matric suction from a
pore scale level to a macroscopic level over the representative elementary volume (Loret & Khalili, 2000). For the purposes of numerical simulation, Bishop’s parameter is usually considered to be
an experimental parameter, instead of χ = S rα B defined above.
The reason for such approximation is that the real porous media
may not strictly obey the assumptions of mixture theory. Particularly in the unsaturated state, significant discrepancies may occur if
Bishop’s parameter is simply considered as a linear function of the
saturation degree.
In this research, Bishop’s parameter is taken to be a function of
suction (Khalili & Khabbaz, 1998), and written as
χ = 1 for s ≤ s e ; χ = ( s e s) 0.55 for s ≥ s e
(2.17)
Considering the compressibility of the solid constituents, Bishop’s
parameter is given in the present work by
χ = (1 − K K s ) for s ≤ se ; χ = (1 − K K s )(se s) 0.55 for s ≥ se
where se is the minimum suction.
9
(2.18)
FuguoTong
TRITA LWR PhD 1055
The swelling is involved in eqs. (2.14a) and (2.16) refers to swelling
strain. Swelling is a special phenomenon of bentonite or other
buffer materials, which represents a change of volume (expansion)
in expansive clays or bentonite when water is present in them, and
a pressure generated when the expansion (swelling) is constrained.
The swelling strain depends on the type of soils (clays), as different
soils have completely different swelling behaviors. For clay, some
swelling models were developed by, e.g. Barbour & Fredlund
(1989), Peters & Smith (2004), Rao & Thyagaraj (2007) among
others. For bentonite, some research was done by, e.g. Lajudie et al.
(1994), Börgesson et al. (1995), Villar (2002, 2003), Musso & Romero (2003) and Imbert (2004), among others.
Villar (2002, 2003) performed four swelling tests (EDN-4-9,
EDN-4-10, EDN-2-13 and EDN-2-14) under constant load (0.1
MPa) in an oedometer cell. According to the results of transient
evolution of vertical strain during equilibration phases of test
EDN_4_10, an empirical formula for the swelling strain of bentonite was established and is written in terms of the suction, as
(2.19)
ε ′ = −0.00015ln(s) 
2.4.2 Liquid flow equation
Neglecting inertial and viscous effects (Khalili & Loret, 2001), the
advective flux of liquid phase includes a pressure driving component and a thermo-osmosis component, expressed as
v lr = −
k lr k
(∇pl + ρ l g ) − k Tl ∇T
µl
(2.20a)
The relative velocity of the liquid can also be written as
v lr = ϕ l ( v l − v s )
(2.20b)
Substituting eq. (2.20a) into eq. (2.20b) yields
 k lr k

(2.21)
(∇pl + ρ l g ) − k Tl ∇T  + v s
−
 µl

Considering the solubility of gas into liquid, the volumetric liquid
content can be expressed by
ρ (1 + H )S r − ρ v
ϕl = φ l
≡ φ S re
(2.22)
ρl − ρ v
vl =
1
ϕl
where the degree of saturation S r is defined as the volume of water
divided by the volume of void space, and can be expressed as:
ϕ l ρl + ϕ g ρv
Sr =
ρ l (1 + H )φ
(2.23)
Considering only the rate of moisture transfer between the liquid
and the gas phases as the source term, according to Dalton’s equation the phase change flux can be described as (Marshall &
Holmes, 1988)
q p = ρ l ω ( p sv − p v )
(2.24)
10
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
where ω >0 is the liquid phase transfer coefficient. The phase
change flux q p is negative for condensation and positive for
evaporation.
Substituting eqs. (2.21), (2.22), and (2.23) into eq. (2.4), and neglecting the smaller terms, the differential equation governing the
non-isothermal liquid flow through an unsaturated porous medium
is obtained finally as:
C Tl
∂p g
∂p
∂T
∂u
∂φ
+ C gl
+ C ll l + C ml ∇ ⋅
+ S re
∂t
∂t
∂t
∂t
∂t
 k lr k

l
−∇⋅
(∇p l + ρ l g)  − ∇ ⋅ (k T ∇T ) + ω ( p sv − p v ) = 0
 µl

(2.25)
where the coefficients in the above equation are defined by :
CTl = −φS re β l
(2.26a)
φ∂S re
∂s
φ
S
φ∂S re
C ll = re −
Kl
∂s
C gl =
(2.26b)
(2.26c)
C ml = φS re
(2.26d)
2.4.3 Gas flow equation
The gas phase is a mixture of dry air and water vapor. Its averaged
advective velocity with respect to the solid phase is due to the gas
pressure and temperature gradient, and is written as
vrg = −
k gr k
(∇p g + ρ g g) − kTg ∇T
µg
(2.27a)
The relative velocity of gas can also be expressed as
v rg = ϕ g ( v g − v s )
(2.27b)
Substituting eq. (2.27a) into eq. (2.27b) yields
 k gr k

(∇p g + ρ g g ) − k Tg ∇T  + v s
(2.28)
−
 µ g

Substituting eq. (2.28) into eq. (2.4), and neglecting the smaller
terms, the equation of gas flow in porous media is obtained as:
1
vg = g
ϕ
CTg
∂p g
∂p
∂T
∂u
∂φ
+ C gg
+ Clg l + Cmg ∇ ⋅
+ Cng
∂t
∂tr
∂t
∂t
∂t
kgk
g
− ∇ ⋅[
(∇p g + ρ g g)] − ∇ ⋅ [k T ∇T] − ω( psv − pv ) = 0
µg
(2.29)
where the coefficients in the above equation are defined by
φ (1 − S re )
T
φ
(
1
−
S
φ∂S re
re )
C gg =
−
pg
∂s
CTg = −
11
(2.30a)
(2.30b)
FuguoTong
TRITA LWR PhD 1055
φ∂S re
∂s
g
C m = φ (1 − S re )
(2.30d)
C = 1 − S re
(2.30e)
C lg =
(2.30c)
g
n
2.4.4 Vapor flow equation
The vapor is mixed with other gases, so the volumetric content of
vapor should be equal to that of gases, i.e. ϕ v = ϕ g . The relative
velocity of vapor vrv includes both the relative velocity of the
whole gas vrg and the relative velocity with respect to its moving
components v dv because of molecular diffusion, i.e.
vrv = vrg + v dv
(2.31a)
Following the Fickian concept of mass transport by molecular diffusion, the diffusive relative velocity of vapor due to the vapor
pressure gradient is written as
v vd = −ϕ v D ve ∇p v
(2.31b)
where Dve is the tensor of the molecular diffusivity of vapor, with
a unit of m2/s Pa , and can be expressed as
1
(2.31c)
D ve = IDv
ρw RT
where Dv is the molecular diffusivity of vapor in pore gas, with
units of m2/s, and ρ w is the water density.
The relative velocity of vapor is written as
v rv = ϕ v ( v v − v s )
(2.31d)
Similar to the derivation of the gas velocity, the velocity of vapor is
finally obtained as
vv =

1  k rg k
−
(∇p g + ρ g g ) − k Tg ∇T  − D ve ∇p v + v s
g 
ϕ  µ g

(2.32)
Substituting eq. (2.32) into eq. (2.4), and neglecting the smaller
terms, the equation of vapor flow in the porous medium is obtained as
∂pg
∂p
∂p
∂T
∂u
∂φ
CTv
+ Cvv v + Cgv
+ Clv l + Cmv ∇ ⋅ + Cnv
∂t r
∂t
∂t
∂t
∂t
∂t
kg k
g
v
− ∇ ⋅[
(∇p g + ρ g g)] − ∇ ⋅ [k T ∇T ] − ∇ ⋅ (Cd Dve∇pv ) − ω( psv − pv ) = 0
µg
(2.33)
The coefficients are defined by:
φ (1 − S re )
CTv = −
(2.34a)
T
φ (1 − S re )
C vv =
(2.34b)
pv
12
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
φ∂S re
∂s
φ ∂ S re
C lv =
∂s
v
v
C m ≡ C d = φ (1 − S re )
C gv = −
(2.34c)
(2.34d)
(2.34e)
C = (1 − S re )
(2.34f)
v
n
2.4.5 Solid mass conservation equation
The consistency of the three phases (solid, liquid and gas) depends
on the variations of saturation degree and porosity. Thus, a new
governing equation involving variable porosity is needed, which
can be obtained by considering the solid mass conservation under
varying stress and fluid pressure. Here we assume that the density
of the solid phase is a function of temperature, liquid pressure, gas
pressure, and the effective mean-stress. This then leads to
 ∂ρ ∂T ∂ρ s ∂p l ∂ρ s ∂p g ∂ρ s ∂σ 
s
∂ t ( ρ sϕ s ) = ϕ s  s
+
+
+
 + ρs ∂tϕ
∂σ ∂t 
 ∂T ∂t ∂pl ∂t ∂p g ∂t
(2.35a)
and
s
 ∂ρ ∂T ∂ρ s ∂pl ∂ρ s ∂p g ∂ρ s ∂σ 
 + vsi ρ s ∂ϕ
∇ ⋅ (v s ρ sϕ s ) = ρ sϕ s ∂ t ε b + vsiϕ s  s
+
+
+
 ∂T ∂x ∂p ∂x ∂p ∂x
∂σ ∂xi 
∂xi
i
l
i
g
i

(2.35b)
Substituting eqs. (2.35a) and (2.35b) into eq. (2.4), and neglecting
the smaller terms and the source term, yields
 ∂ρ ∂T ∂ρ s ∂pl ∂ρ s ∂p g ∂ρ s ∂σ 
s
ϕs  s
+
+
+
 + ρ s ∂ tϕ
∂σ ∂t 
 ∂T ∂t ∂pl ∂t ∂p g ∂t
(2.36)
∂ϕ s
s
+ ρ sϕ ∂ t ε b + v si ρ s
=0
∂xi
Because ϕ s = 1 − φ , and vsi = ∂ui / ∂t , the above equation can be
written as
∂T 1
βs
+
∂t ρ s
∂ρ s ∂pl ∂ρ s ∂pg ∂ρs 
∂u 
∂T 
+
+
K (∇ ⋅ ) − 3Kβ 


∂t 
∂t 
 ∂pl ∂t ∂pg ∂t ∂σ 
∂pg
 ∂pl

− χ ∂t − (1 − χ ) ∂t )

∂[ln(1 − φ)]
∂u ∂ui ∂[ln(1 − φ)]
+
+∇ +
=0
∂t
∂t ∂t
∂xi
(2.37)
The determination of terms ∂ρs / ∂pl , ∂ρs / ∂p g and ∂ρs / ∂σ in
unsaturated states is a difficult task. In the saturated state,
∂ρs / ∂p g = 0, and the values of ∂ρs / ∂pl and ∂ρs / ∂σ are given
by Zimmerman (2000). When the bulk modulus of the solid constituent is much larger than the bulk modulus of the porous medium (as assumed in this research), i.e. K s >> K , then the values
of ∂ρs / ∂pl , ∂ρs / ∂p g and ∂ρs / ∂σ can be assumed to be zero.
Eq. (2.37) can then be simplified as
∂T ∂ ln(1 − φ )
∂u ∂u ∂[ln(1 − φ )]
(2.38)
βs
+
+∇⋅ + i
=0
∂t
∂t
∂t ∂t
∂xi
13
FuguoTong
TRITA LWR PhD 1055
2.4.6 Heat transport equation
It was assumed that the temperature of the solid, liquid and gas
phases are in a local thermal equilibrium, i.e. Ts ≡ Tl ≡ Tg ≡ T . Defining following relations: kT ≡ ksT +klT +kgT and Q≡QTs +QTl +QTg ,
substituting eqs. (2.21) and (2.28) into eq. (2.12), and neglecting the
smaller terms and Q1αT , which is usually very small with small
strain assumption, the heat transport equation through an unsaturated porous medium is obtained finally as
C cT
∂T
− ∇ ⋅ (CTT ∇T ) − ∇ ⋅ [CTg (∇p g + ρ g gC)]
∂t
∂u
− ∇ ⋅ [CTl (∇pl + ρ l g)] + C sT ∇ ⋅
−Q = 0
∂t
(2.39)
where the coefficients are defined by:
C cT = (1 − φ ) ρ s c s + φS re ρ l cl + φ (1 − S re ) ρ g c g
CTT = k T + ρ l cl Tk Tl + ρ g c g Tk Tg + K l β l T
+ K g β gT
1
k Tg
φ (1 − S re )
1 l
kT
φS re
k rg k
k rg k
1
C = ρ g cgT
+ K g β gT
µg
φ (1 − S re ) µ g
T
g
CTl = ρ l cl T
k rl k
1 k rl k
+ Kl βlT
µl
φS re µ l
CsT = [(1−φ)ρs cs + ρl cl + ρ g cg ]T + 3KβT + Kl βl T + K g β g T
(2.40a)
(2.40b)
(2.40c)
(2.40d)
(2.40e)
2.5 Summary
The governing equations are based on the theory of mixtures applied to the multiphysics of geological porous media, considering
solid-phase deformation, liquid-phase flow, gas flow, vapor flow,
heat transport and phase change of water. The vapor transport
process includes molecular diffusion due to the vapor density gradient, and advection due to the bulk flow of the gas. Based on the
conservation of solid mass, this model also accounts for changes in
porosity by stress and fluid pressure.
For three-dimensional problems, the eight equations as derived
above are required to describe the coupled THM processes for
eight primary variables: the three displacement components, the
temperature, the water pressure, the gas pressure, the vapor pressure and the porosity. The completeness of governing equations is
not only necessary to maintain consistency of three phases (solid,
liquid and gas), but it is also needed to increase the stability and efficiency of the numerical simulations. In addition, the density, viscosity, thermal expansion coefficient and bulk modulus of water
and gas can be expressed in terms of these basic variables, so that a
reduced number of material parameters is possible for practical
applications.
14
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
3 C ON ST ITUT IV E M ODEL S O F G EO LO GI C A L PO R O US MEDIA FOR
C OUPLED THM P RO C ESS ES
Besides the governing equations, the problem of constitutive models is the key component of the mathematical model system, for
specific geomaterials. In practice, however, development of the
constitutive models is a problem that can never be completely
solved, and has to be improved step by step by approximations
and validations. Their improvements are always related to the quality of experiment designs and measurement technologies.
There are many constitutive relations developed for modeling
coupled THM processes of geological porous media, and it is impossible to cover all of them in a thesis. In this thesis, besides the
description of two adopted existing mechanical constitutive models of nonlinear elasticity and elastoplasticity, the research was focused on developing a new effective thermal conductivity model
and a new water retention curve model, to improve our modeling
capacity for complex THM behavior of geological porous media
with coupled water, vapor and gas flows. These constitutive models are described below in necessarily different degrees of details.
3.1 Constitutive models for mechanical process
The mechanical constitutive model, i.e. the stress-strain relation, of
partially saturated porous media is a complicated issue, and numerous models have been developed in the past. The elastic-plastic
constitutive relationships are widely used to describe the stressstrain behavior of partially saturated soils, which are derived using
the critical state concept (Schofield & Wroth, 1968), and a generalized modified Cam-Clay model is often used as the plastic driver.
The generalization of the Cam-Clay model was required to fit
quantitatively the experimental data in the saturated range. This
modification is also adopted in the unsaturated range (Loret &
Khalili, 2000). The yield function is defined in the effective stressspace, but is affected by both the plastic volumetric change and
suction, as observed in experiments by, e.g., Alonso et al.(1990),
Maatouk et al. (1995), Wheeler and Sivakumar (1995) and Cui and
Delage (1996).
In this thesis, we adopted two mechanical constitutive models for
representing stress-strain behaviors of compacted bentonite. One
is a simple nonlinear elasticity model and another is an elastoplastic
model developed by Alonso et al. (1990), applied for different
problems.
3.1.1 Nonlinear elasticity model
In laboratory tests, such as the benchmark test problems considered in WP4 of THERESA project, the compacted bentonite
samples were confined in quite limited space, and the deformation
was small. Thus a nonlinear elastic stress-strain model can be
equally applicable as other more complex models.
Therefore, in this thesis, a simple nonlinear elastic model was introduced for simulation of a series of laboratory benchmark tests
15
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TRITA LWR PhD 1055
as validation examples. The elastic model represents isotropic and
homogeneous elastic behavior characterized by two elastic constants, the bulk modulus and shear modulus. A nonlinear stressstrain relation is adopted (Loret & Khalili, 2000):
ε 0 = κ ln(σ / p c )
(3.1)
The bulk modulus can be obtained as
dσ σ
K=
=
dε v κ
(3.2)
The shear modulus G is related to the bulk modulus K and the
Poisson’s ratio µ through
G=K
3(1 − 2 µ )
2(1 + µ )
(3.3)
3.1.2 Elastoplastic model
When the bentonite is placed into the deposition holes, the space
containing the bentonite material may not as confined as in laboratory tests, and irreversible deformation may occur. Therefore we
assumed an elastoplastic behavior for the bentonite. Among all
the elastoplastic relationships for unsaturated porous media, the
pioneering Barcelona Basic Model (BBM) developed by Alonso et
al. (1990) remains to date an important work. Based on an extension of the modified Cam-Clay model (Roscoe and Burland, 1968),
it gives a simple representation of the behavior of unsaturated porous media, by use of relatively fewer parameters, and most of
them are directly deducible from laboratory tests (Vaunat et al.,
2000).
In this thesis, Barcelona Basic Model was adopted to simulate the
large scale in-situ CRT experiment. There are two yield surfaces
which have been respectively defined as:
f1 ( p′, q, s, p0 ) ≡ q 2 − M 2 ( p′ + k s s)( p0 − p′) = 0
(3.4)
f 2 ( s , s0 ) ≡ s − s0 = 0
(3.5)
*
The flow rules include two parts. The plastic strain increment associated with yield surface f1 is given by
dε vp = µ1 and dε s p = µ1 [2qα / M 2 ( 2 p ′ + k s s − p 0 )]
p
(3.6a)
The plastic strain increment associated with yield surface f 2 is
written as:
dε vs = µ 2
(3.6b)
where µ1 and µ 2 can be obtained through plastic consistency
condition.
p
The total volumetric strain increment can be expressed as:
16
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
dεv = dεv + dεv
e
p
p
= dεv + dεvp + dεvs
(3.7)
κ dp′ κs ds  λ(s) − ks dp0  λ(s) − κs ds0 
=
+
+
+

p0   v
s0 + pat 
 v p′ v s + pat   v
e
p
The total shear strain increment can be written as:
dε τ = dε τe + dε τp
1
= Gdq + dετp
3
 λ ( s) − k s dp0 
1
2qα
= Gdq + 

3
v
p0  (2 p′ + ps − p0 )M 2

(3.8)
The parameters used in eqs. (3.4), (3.5), (3.6a), (3.6b), (3.7) and (3.8)
are listed below:
λ (s) = λ (0)[(1 − rM ) exp(−α s s) + rM ]
(3.9a)
p ′ = (σ 1 + 2σ 2 ) / 3 − p g
(3.9b)
q = (σ 1 − σ 3 )
(3.9c)
p0 = p c ( p0* p
α=
)
λ ( 0 )−κ
c λ ( s )−κ
M (M − 9)( M − 3)
1
9 (6 − M )
1 − κ λ (0)
v = 1+
φ
1−φ
(3.9d)
(3.9e)
(3.9f)
where α s is the parameter controlling the rate of increase of a
medium’s stiffness with suction, k s is the parameter describing the
increase in cohesion with suction, λ (s ) is the stiffness parameter
for changes in net mean stress for virgin states of a medium, M is
the slope of the critical state lines, p′ is the excess of mean stress
over air pressure (net mean stress), q is the deviatoric stress, pat
is the atmospheric pressure, p0 is the preconsolidation stress, s0 is
the hardening parameter of the suction increase yield curve, and
rM is the parameter defining the maximum soil stiffness.
3.2 Constitutive models for liquid and gas flow process
3.2.1 Water retention curve
• Introduction and objective
The water retention curve (WRC) is one of the key hydraulic constitutive relations for representing the liquid flow process at variable saturated states. It is usually expressed as a mathematical relation between water content (or saturation degree) and suction (also
known as capillary pressure). This curve is expressed differently for
different types of geological porous media (such as soils, clays,
compacted bentonite as buffer materials for radioactive waste re17
FuguoTong
TRITA LWR PhD 1055
positories in crystalline rocks or sealing liners of landfills), and is
also called the soil-water characteristic curve (SWCC) in some literatures. Since the first water retention curve created by Buckingham (1907), different WRC models have been developed, improved and applied in geotechnical engineering practice. Many of
these models are empirical models obtained through curve-fittings
against experimental data, using some fitting parameters whose
physical meanings may or may not be clearly defined, such as
models by Brooks & Corey (1964), van Genuchten (1980), and
Russo (1988). These models have played important roles for modeling unsaturated groundwater flows and are continuously applied
in research and engineering practices, and contributed greatly to
our current understanding of water physics in geological porous
media.
Suction and saturation degree in geological porous media are usually related through different variables and parameters. Besides the
types of media (soil, clay or bentonite, for example), temperature
and porosity (related to deformation) play important roles, and
drying and wetting processes exhibit different behaviors. Hopmans
and Dane (1986), She and Sleep (1998), Bachmann et al. (2002),
among others, observed that WRC changes with temperature
variations, and predicted the temperature dependence on the suction. It is also well known that water flow and solute transport in
structured soils depend not only on porosity and soil texture, but
also on the geometrical shapes and the connectivity of the pores
(Horton et al., 1994). The pores in geological porous media are
generally not rigid, but change with processes of deformation, heat
transfer, geochemical or biochemical reactions and time, with differences between wetting and drying process (Horn, 2004; Stange
& Horn, 2005).
Generally, for small temperature variations and small deformations,
traditional WRC models can be used to simulate single fluid flow
processes with acceptable reliability. However, their validity may
be questionable when the changes of temperature and/or deformation are sufficiently large to cause fluid phase change or significant porosity variations. Such issues are especially important when
fully coupled THM processes of bentonite (as buffer materials)
need to be considered for design, construction, operation and performance/safety assessments of nuclear waste repositories, where
the processes of mechanical deformation, heat transfer and fluid
transport must be considered in coupling terms (Stephansson et al.,
1996; Yu & Neretnieks, 1997; Villar, 2002; Villar et al., 2005; Hudson et al., 2005). Therefore, there is a need to include the coupling
mechanisms between hydraulic, thermal and mechanical processes
in the WRC models. The difference between drying and wetting
processes must also be considered, since such behaviors affect important measures needed for performance and safety assessments
of radioactive waste repositories, such as the saturation time of the
near-field. Based on the above motivation, a water retention curve
model was developed in this thesis for simulating coupled THM
18
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
processes of geological porous media in general, but compacted
bentonite in particular.
• Theoretical model
Geological porous media contain mineral grains and pores of different sizes and geometrical shapes, with complex patterns of connectivity and distribution. Fluids (groundwater, air and gases)
move through connected pores and/or interfaces between grain
boundaries (Fig. 3.1a). For the derivation of new model, it was assumed that the medium is a continuous, isotropic and homogeneous geological porous medium, and the distribution of the pores
filled with water and gas is also isotropic and homogeneous at the
macroscopic scale. From this assumption, on a unit area of 2D
cross-sections produced by any arbitrarily oriented planes cutting
through the medium, the total area of pores ( Ap ), the total area of
water in the pores ( Aw ), and the total area occupied by air/gas in
the pores ( Ag ) should be constant. Thus the macroscopic porosity
(φ ) and saturation degree ( S r ) of the medium can be expressed as:
φ=
Vp
=
Ap
(3.10)
V
A
Vw Aw
Sr =
=
V p Ap
(3.11)
where A is the total area of the cross-section, V p is the total volume of pores, Vw is the total volume of water, and V is the total
volume of the medium.
The real distribution of the pores in a cross-section (Fig. 3.1a) is
more complicated, and it is impossible to construct theoretical
models considering shapes and sizes of every pore in details. The
theoretical idealization, which is based on the equivalence of macroscopic physical effects, is necessary for the theoretical derivation.
In this research, the distribution of pores on the section is idealized as n identical elliptic pores randomly distributed on the crosssection, containing n identical elliptic air/gas-bubbles in them (Fig.
3.1b). The idealized pore structure must meet the following macroscopic equivalency requirements: 1) the total area of pores minus
the total area of air/gas-bubbles is equal to the actual measured total area covered by water on the section; 2) the total length of perimeters of gas-bubbles is equal to the measured total length of
water-air interfaces on the section.
For the idealized elliptic pores air/gas-bubbles on the cross-section,
their shapes can be represented by the major and minor semiaxis a0
and b0 for the pores, and a1 and b1 for the air/gas bubbles, respectively (Fig.3.2).
19
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TRITA LWR PhD 1055
Figure 3.1. The pore structure geometry and distribution of solid,
water and gas phases on an arbitrary 2D cross-section of a macroscopically isotropic and homogeneous geological porous medium.
Figure 3.2. The size and shape of an equivalent pore and the
equivalent gas-bubble.
The detailed derivation of new water retention curve is presented
in paper III. Final theoretical model includes the effects of deformation, liquid saturation degree, gas pressure, and temperature.
The WRC behavior differs between wetting and drying processes.
It is expressed by a single equation consisting of several simple
functions of saturation degree, temperature, porosity, gas pressure,
and the rate of saturation degree change with time, and is written
as:
0.5
[
]
0.5
ς 
0.5
−0.5
−1.5
−1.5 −0.5 (1− Se )
s = pg +   (4π −8π )δ1 + 8π δ1
λT (3.12)
Se
φ 
Symbol ς is the pore density defined by ς = n / A, which represents the number of equivalent pores on a unit area of the crosssection. Shape coefficient δ1 is defined by δ 1 = b1 a1 , which is a
parameter describing the shape of the air/gas bubble in a pore.
Symbol λT is the water surface tension. Symbol Se is the effective
saturation degree, defined as:
Se =
S r − S r min
S r max − S r min
(3.13)
where S r max is the maximum saturation degree when suction is
non zero and positive, and S r min is the minimum residual saturation degree when the suction is at its maximum. Note that S r max
20
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
and S r min are not constant, but vary with mechanical and thermal
processes.
• Analysis of the theoretical WRC model
For convenience, eq. (3.12) can be rewritten simply as
s = p g + f1 (ς , ϕ ) f 2 (δ 1 ) f 3 ( S e )λT
(3.14)
with
0.5
ς 
f 1 (ς , φ ) =  
φ 
0.5
−0.5
f 2 (δ1 ) = ( 4π −0.5 − 8π −1.5 )δ1 + 8π −1.5δ1
(3.15)
(3.16)
(1 − S e )
(3.17)
Se
The functions f1 (ς , φ ) , f 2 (δ1 ) and f3 (S e ) are called the characteristic functions of pore density and porosity, shape coefficient of
air/gas bubble, and saturation, respectively.
f 3 (S e ) =
0.5
1) Pore density and porosity characteristic function: f1 (ς , φ )
The function f1 (ς , φ ) is only related to the equivalent size of pores,
not to the saturation degree. It is not dimensionless, but has units
of 1/m. It shows that suction decreases with the increase of porosity (φ ) for a given pore density (ς ), or increases with the increase
of pore density (ς ) for a given porosity (φ ). In other words, if
temperature, the effective saturation degree and porosity are kept
constant, the value of suction will become larger when the size of
pore decreases.
For a given medium, the change of pore structure is related to the
change of porosity, expressed as ς = ς (φ ) . For situations with a
small strain assumption as considered in this thesis, the changes of
porosity and pore structure are also small. Therefore, the pore
density can be approximated by a simple empirical function of porosity ς (φ ) = d1φ d 2 . Then, f1 (ς , ϕ ) can be expressed as
f1(ς,ϕ) = (d1φ d2 φ) 0.5 = ξ1φξ 2
(3.18)
where ξ1 and ξ 2 are experimental constants for a given geological
porous medium, and the units of ξ1 should be 1/m, since porosity
φ and the constant ξ 2 are dimensionless.
2) Air/gas bubble characteristic function: f 2 (δ1 )
Function f 2 (δ1 ) represents the effect of air/gas bubble shapes. In
order to use the least squares fitting method for parameter identification, it can be approximately expressed, through a good curvefitting (Fig. 3.3), by a simple power law function:
f 2 (δ1 ) = 2.1δ1
−0.364
(3.19)
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TRITA LWR PhD 1055
12
0.5
-0.5
f2(δ1) =(4π-0.5 8π -1.5) δ1 +8π-1.5δ1
10
-0.364
f2(δ1) =2.1δ1
8
f2(δ1)
6
4
2
0
0
0.2
0.4
0.6
0.8
1
δ1
Figure 3.3. The theoretical and approximate functions of f 2 (δ 1 )
with respect to the shape coefficient of air/gas-bubble ( δ 1 ).
Figure 3.3 shows that value of f 2 (δ1 ) decreases with an increase
of δ1 . In other words, the closer the shape of the air/gas-bubble is
to a circle, the smaller the suction becomes. Figure 3.3 also shows
that the value of f 2 (δ 1 ) varies in a small range of [2.26, 3.58] if the
shape coefficient of pores ( δ1 ) is larger than 0.2. This indicates
that although we may not know the exact form of function f 2 (δ 1 )
in reality, it varies in a small range in many cases in practice with
shape coefficient of pores ranges within [0.2, 1.0].
In theory, δ1 should also change with deformation, temperature,
saturation degree, the rate of saturation degree change with time,
and is the main THM coupling term in unsaturated situations. In
practice, it is difficult to measure δ1 directly. Therefore, more simplification, modification and extension of this term are needed. As
derived in details in paper III, we assume that δ1 is related to the
porosity, temperature and effective saturation degree by a power
ξ +ξ φ +ξ T +ξ θ
law function S e 3 4 5 6 , then f 2 (δ 1 ) can be expressed as:
f 2 (δ 1 ) = 2 .1( S e
ξ 3 + ξ 4φ + ξ 5T + ξ 6θ
) −0.364
(3.20)
In eq. (3.20), coefficients ξ 3 , ξ 4 , ξ 5 and ξ 6 are experimental constants of the geological porous media concerned. The symbol T is
a dimensionless temperature defined by T Tr , where Tr is a reference temperature, and chosen as Tr = 647.096 oK in this thesis;
θ is defined by θ θr , where θ is the rate of effective saturation
degree change with time, θ = dS e / dt ; and θ r is a reference values
of θ . If θ r >0, then θ >0 when the saturation degree increases
(wetting), or θ <0 when the saturation degree decreases (drying).
The wetting and drying processes is then represented by the rate of
change of the effective saturation degree with time. According to
eq. (3.20), the value of f 2 (δ1 ) will be different between the wetting and the drying processes, even when the values of porosity,
22
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
saturation degree and temperature are unchanged. If ξ 6 <0, the
suction of drying process is always larger than that of wetting
process in the case of the same saturation, porosity and temperature.
3) The saturation function: f3 (S e )
Function f3 (S e ) is a purely theoretical derivation without empirical simplifications or modifications, and it is a kind of ’shape function’ of the WRC since it decides the general trend of change of
suction curve with respect to saturation degree. The value
of f 3 ( S e ) , which is dimensionless, generally decreases with the increase of the effective saturation degree (Fig. 3.4). Besides f 2 (δ1 ) ,
the shape of the WRC mainly depends on f 3 ( S e ) because it is
also a function of effective saturation degree, and this is why water
retention curve models have similarities in shapes.
100000
1000
f3(Se)
10
0.1
0.001
0
0.2
0.4
0.6
0.8
1
Se
Figure 3.4. The relation between f 3 ( S e ) and effective saturation
degree.
4) The water surface tension: λT
According to the International Association for the Properties of
Water and Steam (IAPWS, 1994), the surface tension of ordinary
water substance can be expressed as a function of temperature,
and a recommended interpolating equation was given as follows:
λT = 0.2358(1 − T )1.256 [1 − 0.625(1 − T )]
(3.21)
The unit of λT is N/m. Equation (3.21) shows that water surface
tension decreases with increase in temperature, indicating that suction decreases with the increase of temperature.
• The final form of the WRC model
Substituting eqs. (3.18) and (3.20) into eq.(3.14) yields the final
mathematical form of the new WRC model:
[
s = p g + ξ1φ ξ2 2.1( S e
ξ 3 +ξ 4φ +ξ 5T +ξ 6θ
]
 (1 − S e ) 0.5 
) −0.364 
 λT
Se


(3.22)
where the set { ξ1 , ξ 2 ξ 3 , ξ 4 , ξ 5 , ξ 6 } are experimental coefficients
representing coupling effects of deformation, temperature, poros23
FuguoTong
TRITA LWR PhD 1055
ity, and degree of saturation, on suction and have to be obtained
by calibrations or curve-fittings based on experiments. The units
of symbols s , p g , ξ1 , and λT are N/m2, N/m2, 1/m, and N/m,
respectively, and other coefficients and variables are dimensionless.
Coefficients { ξ1 , ξ 2 ξ 3 , ξ 4 , ξ 5 , ξ 6 } also make the model complete
and flexible for their experimental determinations. For example,
coefficient ξ 6 may be set as zero if the difference between wetting
and dry process is negligible; coefficients ξ 2 and ξ 4 can be assumed to be zero if the influence of deformation is ignored (rigid
media or very stiff solid phase); and coefficient ξ 5 can be set to
zero if the temperature effect can be ignored. Such simplifications
can be used to identify laboratory experiments to determination of
these coefficients separately with simpler initial and boundary conditions.
• The verification of the new WRC model
The FEBEX bentonite is a commonly studied buffer material for
nuclear waste repositories in Spain and other European countries.
It was chosen to verify the new WRC model due to its good data
base from experiments. The main experimental data were extracted from Figure A1.1 of Gens (2007) and Figure. 5 of Villar &
Lloret (2004), and they were used to define the values of coefficients in eq. (3.22) as
ξ1 =4.0997×109, ξ2 =2.4077498,
ξ3 =89.3353518, 
ξ4 =-196.181368, ξ5 =-19.2133356, ξ6 =-0.2642839.
The effective saturation degree ( S e ) was expressed using water
content terms, as
Se =
w − wmin
w
≈
wmax − wmin 0.1318 + 0.90808φ − 0.03073T
(3.23)
1) The comparison between calculated and measured results
The comparison between measured and calculated water contents
is shown in Figure 3.5. The predicted water contents are generally
close to the experimental data, and the discrepancy with most of
the experimental data is less than 3%. The comparisons between
measured and calculated suctions by using eq. (3.22) are shown in
Figure 3.6 and Figure 3.7, respectively, with good agreements,
again through using parameter calibrations. Results with differences between wetting and drying processes as tested in Villar &
Lloret (2004) at constant temperatures were calculated using the
new WRC model, and are shown in Figure 3.7. The agreement between calculated and experimental data is reasonable for the wetting and drying processes. In general, both figures show that the
new WRC model (3.22) has adequate capability and flexibility for
parameter calibrations through fitting with the experimental data
with acceptable accuracy.
24
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Predicted water content(%)
30
3%
25
-3%
20
15
10
10
15
20
25
30
Measured water content (%)
Figure 3.5. Calculated versus measured water contents (Gens, 2007).
Suction, s(Mpa)
1000
Measured
Predicted
Measured
Predicted
100
10
a) T=22oC, ρd=1.60g/cm3
b) T=22oC, ρd=1.65g/cm3
1
Suction, s(Mpa)
1000
Measured
Predicted
Measured
Predicted
100
10
c) T=22oC, ρd=1.70g/cm3
d) T=40oC, ρd=1.65g/cm3
1
Suction, s(Mpa)
1000
Measured
Predicted
Measured
Predicted
100
10
e) T=40oC, ρd=1.70g/cm3
f) T=60 oC, ρd=1.65g/cm3
1
10
15
20
Water content, w(%)
25
15
20
25
Water content, w(%)
Figure 3.6. Comparison between calculated results by using Eq.
(3.22) and measured data from (Gens, 2007).
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FuguoTong
TRITA LWR PhD 1055
Suction, s(Mpa)
140
Measured (Wetting)
120
Measured (drying)
Predicted (Wetting)
100
Predicted (drying)
80
60
40
20
15
16
17
18
19
20
21
22
Water content, w(%)
a) T=20o C, ρd=1.65g/cm3
Suction, s(Mpa)
160
Measured (Wetting)
140
120
Measured (drying)
Predicted (Wetting)
100
Predicted (drying)
80
60
40
20
0
12
14
16
18
20
22
24
Water content, w(%)
b) T=40o C, ρd=1.65g/cm3
Figure 3.7. Comparison between calculated results by using Eq.
(3.22) and measured data from (Villar & Lloret, 2004) for drying
and wetting processes at a constant temperature.
2) The dependence on deformation (through porosity)
Figure 3.8 presents the sensitivity of suction, on effective saturation (Fig. 3.8a and 3.8c) or water content (Fig. 3.8b and 3.8d) with
different porosities, when temperature was kept constant at 20oC
and 60oC, respectively. Significant variations of suction with
changes in porosity are highlighted, but with a general trend of decrease in suction with increase of porosity. The results also show
that the smaller the effective degree of saturation (or water content)
is, the stronger effect is caused by the change of porosity.
Figure 3.9 presents the sensitivity of the derivative of the effective
saturation degree (or water content) with respect to suction, as a
function of effective saturation (Fig. 3.9a and 3.9c) or water content (Fig. 3.9b and 3.9d) with different porosities, when temperature was kept at 20oC and 60oC, respectively. The effective saturation degree was assumed not to change with time, for clarity. The
figure shows that the derivative decreases with increasing porosity
generally, but increases steeply when approaching full saturation
(Fig. 3.9a and 3.9c), or the maximum water content that varies
with change of porosity (Fig. 3.9b and 3.9d). Such dependence becomes more obvious with increasing temperature. Results also
show that the dependence on porosity (or indirectly deformation)
is small when effective saturation degree (or water content) is small.
26
Suction, s(MPa)
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Suction, s(MPa)
Effective degree of saturation, Se (%)
a) T=20oC
Suction, s(MPa)
Water content, w (%)
b) T=20oC
Suction, s(MPa)
Effective degree of saturation, Se (%)
c) T=60 oC
Water content, w (%)
d) T=60oC
Figure 3.8. The suction variations as functions of effective saturation degree or water constant, with different porosity values (with
θ =0).
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TRITA LWR PhD 1055
∂Se/∂s (1/MPa)
FuguoTong
∂w/∂s (1/MPa)
Effective degree of saturation, Se (%)
a) T=20oC
∂Se/∂s (1/MPa)
Water content, w (%)
b) T=20oC
∂w/∂s (1/MPa)
Effective degree of saturation, Se (%)
c) T=60oC
Water content, w (%)
d) T=60oC
Figure 3.9. The variations of ∂S e ∂s (or ∂w ∂s ) as functions of
effective saturation degree or water content, with different porosity values(with θ =0).
28
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
3) The dependence on temperature
Figure 3.10 shows the sensitivity of suction (as a function of effective saturation (Figs. 3.10a and 3.10c)) or water content (Figs.
3.10b and 3.10d) with different temperature and porosity values
for the FEBEX bentonite. The effective saturation degree was also
assumed not to vary with time, for simplicity. It shows that the
suction, or the maximum water content, decreases with an increase
of temperature, especially when the porosity increases. The dependence of suction on temperature is more obvious at smaller
ranges of effective saturation or water content.
Figure 3.11 shows that the value of ∂S e ∂s (or ∂w ∂s ) generally
decreases with increase of temperature, and there is also a similar
steep increase of its value close to the full saturation (Fig.3.11a,
and 3.11c) or maximum water content (Fig.3.11b and 3.11d). The
effect of temperature is small when the effective saturation degree
or water content is very small.
4) The difference between wetting and drying processes
Both wetting and drying processes can be handled by the same
new WRC equation (3.22) and described by the variable θ , the
rate of effective saturation degree change with time, related to a
change of the shape coefficient of the air/gas bubble ( δ1 ). According to the definition of θ , the value of θ is non-zero and positive
in a wetting process, and is non-zero and negative in a drying
process. Such a mathematical representation is continuous, compact and without need for changing equations when changing wetting (drying) to drying (wetting) processes. The maximum value of
the shape coefficient ( δ 1 ) is 1.0; consequently, there is a lower
bound of suction according to eq. (3.22), during a wetting process.
A numerical sensitivity analysis was performed using the WRC
model to test the different effects of θ on suction during wetting
and drying processes. The temperature and porosity were set to be
constant at 40oC and 0.375, respectively. The results, shown in
Figure 3.12, indicate that the suction increases with a decrease of
θ , especially when Se is small. The curve with θ =0 indicates a
state of no water movement, and is a boundary between wetting
and drying processes. Due to the existence of a lower bound of
suction (during wetting with δ1=1.0), the suction curves of the
wetting process are restricted to the area between the lower bound
curve and the no water movement state curve, with θ =0.
Similarly, Figure 3.13 shows that ∂S e ∂s generally increases with
the decrease of θ . For real geological porous media, the suction
decreases with increase of saturation degree. Hence, the value of
∂S e ∂s is negative. A state of ∂S e ∂s =0 represents the upper
bound of ∂S e ∂s . On the other hand, there is a lower bound
when the shape coefficient ( δ 1 ) is 1.0; thus the curves of ∂S e ∂s
are restricted to the area between the lower and upper bounds, in
any case.
29
TRITA LWR PhD 1055
Suction, s(MPa)
FuguoTong
Suction, s(MPa)
Effective degree of saturation, Se (%)
a) φ =0.25
Suction, s(MPa)
Water content, w (%)
b) φ =0.25
Suction, s(MPa)
Effective degree of saturation, Se (%)
c) φ =0.35
Water content, w (%)
d) φ =0.35
Figure 3.10. The variations of suction with different temperature
(with θ =0) as functions of effective saturation or water constant
with different porosity values.
30
∂Se/∂s (1/MPa)
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
∂w/∂s (1/MPa)
Effective degree of saturation, Se (%)
a) φ =0.25
∂Se/∂s (1/MPa)
Water content, w (%)
b) φ =0.25
∂w/∂s (1/MPa)
Effective degree of saturation, Se (%)
c) φ =0.35
Water content, w (%)
d) φ =0.35
Figure 3.11. The variation of ∂S e ∂s (or ∂w ∂s ), as functions of
effective saturation or water content, with different temperature
and porosity (for θ =0).
31
TRITA LWR PhD 1055
Suction, s (MPa)
FuguoTong
Effective degree of saturation, Se (%)
∂Se/∂s (1/MPa)
Figure 3.12. The comparison of suction with different θ , and T =
40oC , φ = 0.375.
Effective degree of saturation, Se (%)
Figure 3.13. The comparison of ∂S e ∂s with different θ , and T
= 40oC , φ = 0.375.
3.2.2 Relative permeability of liquid
Relative permeability of liquid klr is involved in liquid flow and
heat transport processes, see eqs. (2.25) and (2.40d). Representative researches on the liquid relative permeability are reported in
Burdine (1953), Brooks & Corey (1964), Baker (1988), Delshah &
Pope(1989), Honarpoor & Mahmood (1988), Demond et al.(1993),
Avraam & Payatakes (1995), Couture et al. (1996), Oostrom &
Lenhard (1998), among others. In this research, the relative permeability k rl of water was assumed to be a function of the degree
of saturation and the porosity, and is expressed by (Hadas, 1964)
S −Sr
k = k k =  r r r
 1 − Sr
l
r
l
rS
l
rn



3
φ

 φ0



7
(3.24)
where S rr is a residual degree of water saturation, and φ0 the initial
porosity.
32
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
3.2.3 Relative permeability of gas
The relative permeability of gas k gr is involved in gas flow, vapor
flow and heat transport processes, see eqs. (2.29), (2.33) and
(2.40c). Relevant contributions to its definition and applications
can be found in Oak et al. (1990), Stylianou & DeVantier (1995),
Avraam & Payatakes (1995), Dana & Skoczylas (1999), Laloui et al.
(2003), Feng et al.(2004), among others. In this study, the relative
permeability of gas, k gr , was assumed to be a function of the degree of saturation only, expressed by Laloui et al. (2003)
k gr
  S − S r 3
r
r 
= 1− 

  1− Srr 
(3.25)
3.3 Constitutive models of heat transport process
This section presents the development of a new thermal conductivity model for bentonite. Besides the thermal conductivity, there
are other commonly used constitutive properties important for
heat transport equation, such as specific heat capacities, thermal
expansion coefficients and bulk modulus of solids, water and gas.
They are less complex to some extent, and were given in paper I
without further development.
3.3.1 Thermal conductivity model development
• Introduction and objective
Thermal conductivity is often considered as a constant for many
dry engineering materials, but in many subsurface engineering projects, due to the coupling effect between the fluid flow and heat
transfer processes, the heat conductivity of the porous geological
media is, in reality, not a constant, but a function of many properties such as porosity, water content or saturation degree, phase
change of water, and above all, the pore structure of the media and
temperature.
Many thermal conductivity models have been proposed for common rocks, soils, clays and engineered geological buffer materials
(Johansen, 1975; Sepaskhan & Boersma, 1979; Ayers & Perumpral,
1982; Farouki, 1986; Zimmerman, 1989; Sorour et al., 1990; Becker, 1992; Sakashita & Kumada, 1998; Abu-Hamdeh, 2000; Hiraiwa
& Kasubuchi, 2000; Abu-Hamdeh et al., 2001; Pusch & Young,
2003; Tang et al., 2008). Most of them were characterized by one
single value of the thermal conductivity, under conditions of being
fully saturated or dry. The dependence of the thermal conductivity
on temperature, moisture and porosity was not simultaneously
contained in the most of the thermal conductivity models.
In this thesis, a new effective thermal conductivity model for
three-phase mixtures was developed to predict the thermal conductivity of geological porous media as mixtures in general, and
buffer materials for radioactive waste repositories in particular.
The proposed model considers the effect of saturation, porosity,
33
FuguoTong
TRITA LWR PhD 1055
temperature and pressure. The detailed derivation is presented in
paper II.
• Derivation of the new effective thermal conductivity model
The derivation is divided into three steps. First, the individual
thermal conductivities of gas, liquid (water) and solid are defined,
respectively, followed by the derivation of the effective thermal
conductivity of a two-phase solid-gas mixture, and ending with the
effective thermal conductivity of three-phase (solid-liquid(water)gas) mixture.
1) Thermal conductivity of gas, λ g
In geological porous media, gas is commonly considered as a mixture of dry air and water vapor. A comprehensive thermal conductivity equation of dry air was given by Lemmon & Jacobsen (2004),
which consists of several functions with many coefficients. Considering smaller roles played by gas for the overall thermal conductivity of porous geological media, simplifications were made to the
original equations given by Lemmon & Jacobsen (2004) so that the
thermal conductivity of the dry air, λa , is a function of density and
temperature of the air, approximately expressed as
[
]
λa = 4.55+ 0.072T + (36.17− 0.016T )ρa + (47.4+ 0.121T )ρ a2 ×10−3
(3.26)
where λa is defined in [W/moK], ρ a is in [kg/m3], and T is in
[oK].
The density of air can be expressed as an implicit function of temperature and pressure according to the equation of state of air.
Lemmon et al. (2000) presented a comprehensive equation of state
of air, which is still valid in case of high pressures, but it causes
significant difficulty for the numerical solution. In this research, it
was assumed that the equation of state of ideal gas could be used
to approximately represent that of dry air to avoid such numerical
difficulties. Based on this assumption, the thermal conductivity of
the gas is also a function of air pressure and temperature, expressed as
pa 

4.55 + 0.072T + (36.17− 0.016T ) 286.9T 
2
λa = 
 ×10−3
p


a

+ (47.4 + 0.121T ) 



 286.9T 
(3.27)
where pa is defined in unit of [kPa]. The relation in eq. (3.27) can
provide accurate estimations of thermal conductivity of dry air in
the range 200oK < T < 500oK for air temperature and 0 < ρ a <
0.2 kg/m3 for air density.
In order to consider the effects of high vapor concentration that
may occur in natural or engineering environments with phase
changes, according to Lemmon & Jacobsen (2004), the model of
the effective thermal conductivity of gas, λg, is expressed as a func-
34
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
tion of temperature (T), gas pressure ( p g ) and vapor pressure
( pv ), as
λg =
(1 − C)λa
Cλv
+
(1 − C) + CΦav C + (1 − C)Φva
(3.28)
where symbols C , Φ av , Φ va and λv are coefficients, and are defined further in paper II.
2) Thermal conductivity of water, λw
The thermal conductivity models of water have been reported by,
Kestin et al. (1984), Abdulagatov et al. (2005), Valyashko (2009),
and others. In this thesis, a more recent empirical interpolating
equation was adopted to calculate the thermal conductivity of water as recommended by the International Association for the
Properties of Water and Steam (1998), defined by
(3.29)
λ w = λ∗ [ λ 0 (T ) + λ1 ( ρ ) + λ 2 (T , ρ )]
where functions λ0 (T ) , λ1 ( ρ ) and λ2 (T , ρ ) are defined in paper
II. λ* is the reference thermal conductivity, and given as λ∗ = 1.0
W/moK. Symbols ρ and T are dimensionless variables, and were
defined as ρ = ρ ρ∗ = 0.0031476ρw and T = T T ∗ = 0 .001545T .
The units of water thermal conductivity, temperature and density
are W/moK, Kelvin and kg/m3, respectively.
3) Thermal conductivity of solid phase, λ s
The thermal conductivity of a solid phase depends on its mineral
composition. There is not a universal relation valid for all kinds of
solids. In this research, solid phase are the soils or buffer materials.
According to Johansen (1975), Farouki (1986) and Tang (2008),
the thermal conductivity of soils or buffer materials can be determined by
ς
1−ς
λs = λq λo
(3.30)
where λ q and λ o are the thermal conductivities of quartz and
other minerals, respectively, and ς is the quartz volume fraction.
λ q and λ o can be considered as functions of temperature, but
not stress because the compressibility of the solid constituents is
usually small. The dependence of thermal conductivity of the solid
phase on temperature of quartz can be seen in Yoon et al. (2004).
4) The effective thermal conductivity of the mixture of solid and gas, λ s− g
The mixture of solid and gas is usually called a dry soil. According
to the Wiener bounds (Wiener, 1912) for the series and parallel
models of dry soils (Fig. 3.14), the lower and upper limits of the
effective thermal conductivity of a solid-gas mixture are given by:
35
FuguoTong
TRITA LWR PhD 1055
λ
L
s−g
1 − φ φ 
=
+ 
λ g 
 λ s
−1
(3.31a)
λUs− g = φλg + (1 − φ )λs
(3.31b)
The composite solid-gas mixture was considered as an effective
mixture with mixed serial and parallel connections of solid and gas
phases, and its effective thermal conductivity can be expressed as
λs−g = (1 −η1 )λLs−g +η1λUs−g
(3.32)
where the coefficient η1 depends only on the pore structure of the
solid-gas mixture, and should be a function of porosity. Its value
should be in the range of 0 < η 1 (φ ) < 1 , according to the Wiener
bounds. Substituting eqs. (3.31a, b) into eq. (3.32), the effective
thermal conductivity of a solid-gas mixture is expressed as
λs λg
(3.33)
λ s − g = η1 (1 − φ )λ s + η1φλ g + (1 − η1 )
φλs + (1 − φ )λ g
Figure 3.14. Wiener bounds of thermal conductivity of porous
media under solid and gas conditions. Series and parallel connection models correspond to the lower and upper Wiener bounds.
5). The effective thermal conductivity of the three-phase mixture, λ
From eq. (3.33), one can see that the thermal conductivity of the
solid phase with parallel connection (the first term at the right
hand side (RHS) of eq. (3.33)) does not change with air saturation,
but the thermal conductivity of the gas phase in parallel connection (the 2nd term at the RHS of eq. (3.33)) and the solid-gas mixture in serial connection (the 3rd term at the RHS of eq.(3.33)) are
affected by gas saturation. Because the volume fraction of the first
term at the RHS of eq. (3.33) is φ1 = η1 (1 − φ ) , the volume fraction
of the remainder is φ 2 = 1 − φ1 = 1 − η1 (1 − φ ) . Thus, according to
the Wiener bounds for combined serial and parallel connection
models, the upper and lower limits, λL and λU , of a three-phase
mixture are obtained similarly as
36
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
−1
(1 − φ)(1−η1 ) 1 φSr 1 φ(1− Sr ) 1 
+
+
λ = φ1λs + φ2 

φ2
λs φ2 λw
φ2 λg 

−1

φSr φ(1 − Sr ) 
2 (1 − φ )(1 −η1 )
= η1 (1− φ)λs + [1 −η1 (1 − φ)] 
+
+

λs
λw
λg 

L
(3.34a)
For serial connections among solid, liquid and gas phases, and
 (1 − φ )(1 − η1 )
φS
φ (1 − S r ) 
λU = φ1λs + φ2 
λs + r λw +
λg 
φ2
φ2
φ2


[
= η1 (1 − φ )λs + (1 − φ )(1 −η1 )λs + φS r λw + φ (1 − S r )λg
(3.34b)
]
For parallel connections among solid, liquid and gas phases, respectively. The symbol S r is the degree of saturation.
Note that in eqs. (3.34a, b), the solid phase is split into two portions. One portion, represented by the term η1 (1 − φ ) of the first
term at the RHS in eqs.(3.34a,b), is constant due to its parallel
connection, and the other portion, represented by the term
(1 − φ )(1 − η1 ) is in parallel/serial connections with water and gas
phases.
In general, the effective thermal conductivity of the whole mixture
can be expressed as
λ = (1 − η 2 )λL + η 2λU
(3.35)
where the coefficient η 2 should be a function of pore structure,
saturation degree and temperature. Its magnitude should be in the
range of 0 < η 2 (φ , S r , T ) < 1 , as constrained by Wiener’s bounds
for anisotropic mixtures. The effective thermal conductivity of the
isotropic mixtures should be just a special case of this general anisotropic model and falls within the Hashin-Shtrikman bounds.
Substituting eqs. (3.34a, b) into eq. (3.35), the effective thermal
conductivity of the three-phase mixture is finally expressed as
[
]
λ = λ φ, Sr , T, λg (T, pg ),λw (T, pw ),λs (λq (T ),λo (T ),ς ),η1 (φ),η2 (φ, Sr ,T )
φS φ(1 − Sr ) 
2  (1 − φ)(1 −η1 )
= η1 (1 − φ)λs + (1 −η2 )[1 −η1 (1 − φ)] 
+ r+

λs
λw
λg 

[
+ η2 (1 − φ)(1 −η1 )λs + φSr λw + φ(1 − Sr )λg
−1
(3.36)
]
3.3.2 Thermal conductivity model validation
The effective thermal conductivity model expressed by eq. (3.36) is
a general model developed for porous geological media. To verify
this model, the experimental data measured from a buffer material
MX80 was chosen, as reported in Pusch & Young (2003) and Tang
et al. (2008), and was used in WP4 of the THERESA project.
1) The selection of parameters η1 and η 2
Parameter η1 usually can be obtained by a series of experiments
that require the samples of different porosities but with zero saturation (dry samples). However, there is no such experimental data
37
FuguoTong
TRITA LWR PhD 1055
available for MX80. It was indirectly obtained according to Tang et
al. (2008) and Ould-lahoucine et al. (2002), and given as:
η1 = 0.0692φ −0.7831
(3.37)
In theory, η 2 should be a function of porosity, saturation degree
and temperature. In this research, the effect of temperature was
neglected because of a lack of experimental data, and an empirical
relation between η 2 and saturation degree S r was proposed as
bφ + c
. Using the experimental data in Tang et al. (2008),
η 2 = aS r
coefficients a, b and c were obtained by a least square fitting, and
the parameter η 2 can be approximated as
η 2 = 0.59S r
1.487φ − 0.0404
(3.38)
2) Validation against Hashin-Shtrikman bounds
For isotropic porous media, the effective thermal conductivity
should also fall within the Hashin-Shtrikman bounds. In this thesis,
the predicted values by the new model are compared with the
Hashin-Shtrikman bounds for MX80 bentonite, using the same
parameters as before, based on data from Tang et al. (2008). Figure
3.15 compares the predicted values of the effective thermal conductivity of the MX80 and the Hashin-Shtrikman bounds and
Wiener bounds when the saturation is equal to 0.1, 0.5 and 0.9. It
shows that the predicted effective thermal conductivity falls within
both the Hashin-Shtrikman bounds and Wiener bounds, throughout a wide range of values φ ∈ [0.04, 0.97] and S r ∈ [0,1].
3) Validation against a laboratory benchmark test of bentonite
The developed effective thermal conductivity model was validated
against a laboratory benchmark test of FEBEX bentonite. The test
was named benchmark test 1 in this thesis, and will be used for
more validations in Chapter 5.
The measured and simulated temperature evolutions match almost
perfectly with negligible discrepancies (Fig. 3.16). For comparison,
the results of prediction using the original effective thermal conductivity model expressed by eq. (3.39), as specified in Gens (2007)
and Villar (2002) are shown in Fig.3.17, with larger discrepancies
when the temperature is higher than 40oC. The main reason is that
the effective thermal conductivity model given by eq. (3.39) does
not consider effects of temperature.
λ = 1.28 −
0.71
1 + exp(10 S r − 6.5)
38
(3.39)
Thermal conductivity
(W/m.oK)
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Thermal conductivity
(W/m.oK)
Porosity
a) Sr =10%
Thermal conductivity
(W/m.oK)
Porosity
b) Sr =50%
Porosity
c) Sr =90%
Figure 3.15. Comparison with Hashin-Shtrikman bounds and
Wiener bounds for MX80 bentonite tested in (Tang et al. (2008)).
39
TRITA LWR PhD 1055
Temperature (oC)
FuguoTong
Elapsed time (hours)
Temperature (oC)
Figure 3.16. The comparison between simulated temperatures
(solid curves) by Eq. (3.36) versus measured values (symbols).
(Note: coordinate x in legend is the distance of the sensor from
the heater).
Elapsed time (hours)
Figure 3.17. The comparison between simulated temperatures
(solid curves) by Eq. (3.39) versus measured values (symbols).
3.3.3 Discussion of the effective thermal conductivity model behavior
According to eq. (3.36), the effective thermal conductivity of a
buffer material is a function of saturation degree, porosity, temperature, water pressure, vapor pressure, thermal conductivities of
quartz and other minerals, and the volume fraction of quartz. For
the convenience of analysis, here we chose p w = 0.0, p g =100 kPa,
pv ≈ 0, T = 300oK, λs = 2.08 W/moK. The general behavior of
the effective thermal conductivity and saturation with different po-
40
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Thermal conductivity
(W/m.oK)
rosity values is shown in Figure 3.18, and thermal conductivity
changes with porosity and saturation is shown in Figure 3.19.
Figure 3.18 shows that the smaller the porosity is, the smaller the
influence of saturation on thermal conductivity is, as expected
from the basic principles of effective thermal conductivity theories.
When the porosity is close to 1.0, for example, φ = 0.9, the effective thermal conductivity of buffer material mainly depends on the
thermal conductivity of gas and water, but it is always larger than
the thermal conductivity of gas and less than that of water. Figure
3.19 indicates that when the porosity is close to zero, for instance
φ = 0.1, the thermal conductivity of buffer mainly depends on the
thermal conductivity of solid material, and the influence of saturation is very weak.
Saturated degree
Thermal conductivity
(W/m.oK)
Figure 3.18 Thermal conductivity versus saturation, according to
Eq. (3.36).
Porosity
Figure 3.19. Thermal conductivity versus porosity, according to
Eq.(3.36).
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FuguoTong
TRITA LWR PhD 1055
3.4 Summary
This chapter mainly presents two new constitutive models, a new
water retention curve model and a new thermal conductivity
model, based on earlier developments reported in literature. There
still are necessary researches to be conducted in order to further
improve their performance for more reliable modeling of coupled
THM processes, as briefly summarized in Chapter 7.
42
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
4 N UMER IC A L SO LUT IO N OF G OV ER NIN G EQUA TION S F OR
C OUPLED THM P RO C ESS ES
The governing equations for coupled THM processes of porous
geological media are partial differential equations that have to be
solved by using numerical methods via spatial and temporal discretizations. Analytical solutions in closed-forms do not exist in general for most problems. This chapter presents the basic steps taken
to solve the governing equations developed in Chapter 2 and
Chapter 3. It was assumed that the problems are transient (timedependent) problems in three dimensions.
4.1 FEM method for solving the governing equations
The numerical methods adopted for solving the governing equations developed in Chapters 2 and 3 are the Finite Element
Method (FEM) for spatial discretization and Finite Difference
Method (FDM) for temporal discretization. Both are well established, and are the most commonly applied numerical methods in
engineering sciences. The popularity of FEM is largely due to its
flexibility in handling material inhomogeneity, anisotropy and
complex boundary conditions, for both static and dynamic problems. The FDM is more convenient for simulating transient problems by a time-marching process using time steps.
4.1.1 The spatial discretization with FEM
The spatial discretization is based on a Galerkin formulation of
FEM (Zienkiewicz & Taylor, 2000), a weighted residual method in
which trial functions themselves serve as weighting functions. The
primary variables are the vector of three displacement components
( u ), liquid pressure ( pl ), gas pressure ( p g ), vapor pressure ( p v ),
porosity (φ ) and temperature (T ).
Assuming that the whole space is divided by elements with m
nodes for each element, and the whole FEM mesh has ne elements
and n nodes. The following interpolation function matrix is then
defined:
N2
...Ni ...
Nn
N1


[N 1] = 
N1
N2
...Ni ...
Nn

N1
N2
...Ni ...
Nn 

(4.1a)
N1

N2
...Nj ...
Nm


[N ] =
N1
N2
...Nj ...
Nm


N1
N2
...Nj ...
Nm

(4.1b)
1
e
[ N 2 ] = [N1
N2
N 3 ...N i ... N n ]
(4.1c)
[ N e2 ] = N1
N2
N 3 ...N j ... N m
(4.1d)
[
]
where N i and N j are respectively the interpolation function of ith node and the interpolation function of j-th node of an element.
They are both functions of spatial coordinate. [ N 1 ] and [ N e1 ] are
respectively the global interpolation function matrix and elemental
43
FuguoTong
TRITA LWR PhD 1055
interpolation function matrix for displacement. [ N 2 ] is the global
interpolation function matrix for pressure, temperature and porosity, and [ N e2 ] is element interpolation function matrix for pressure,
temperature and porosity.
Defining elemental vectors of displacement, water pressure, gas
pressure, vapor pressure, temperature and porosity as
 pl1 
 pg1 
 pv1 
 u1 
T1 
φ1 
l
g




 pv2 
u2 
T2 
φ2 
p2
p2
 ...  l  ...  g  ...  v  ...  T  ...
 
u
φ
Xe =   , Xe =  l  , Xe =  g , Xe =  v  , Xe =   , Xe =  ... 
u
T
φ
p j 
p j 
p j 
 j
 j
 j
 ... 
 ... 
 ... 
 ... 
 ...
 ... 
pl m
pgm
pvm
um 
Tm 
φm 
And defining global vectors of displacement, water pressure, gas
pressure, vapor pressure, temperature and porosity as
 pl1 
 pg1
 pv1 
u1 
φ1 
T1 
 pl 2 
pg2
pv2 
u2 
φ2 
T2 







 



Xu =  ... , Xl =  ...l  , Xg =  ...g , Xv =  ...v  , XT =  ... , Xφ = ...
u
φ
T
p i 
p i 
p i 
 i
 i
 i
...
...
...
...
...
 
 
 
 
...
 
l
g
v
u
T

p


p


p

n
n
n
 n
φn 
 n
 
 
 
The primary variables are respectively expressed by
u x (t ) 
u(t ) = u y (t ) = [ N 1 ] ⋅ X u = [ N e1 ] ⋅ X ue


 u z (t ) 
(4.2a)
pl (t ) = [ N 2 ] ⋅ X l = [ N e2 ] ⋅ X le
(4.2b)
p g (t ) = [ N ] ⋅ X = [ N ] ⋅ X
2
g
2
e
pv (t ) = [ N ] ⋅ X = [ N ] ⋅ X
2
v
2
e
T (t ) = [ N ] ⋅ X = [ N ] ⋅ X
2
T
2
e
φ
2
e
φ (t ) = [ N ] ⋅ X = [ N ] ⋅ X
2
v
e
g
e
(4.2c)
(4.2d)
T
e
(4.2e)
φ
e
(4.2f)
A volume integration of all governing equations then leads to a
weighted residual approximation to the governing equations, based
on the Galerkin method. For example, the equilibrium equation
can be discretized as follows:

∂u
∂pl
∂p g 
∇ ⋅ (D∇ ) − ∇ ⋅[χ ] − ∇ ⋅ [(1− χ)
]

1 T
∂t
∂t
∂t dV = 0
∫∫∫Ω[N ] 
∂T
∂ε ′ ∂F
− ∇⋅ [Kβ ] − ∇ ⋅ [K ] + b

∂t
∂t
∂t


(4.3)

∂u
∂pl
∂p g 
ne
∇⋅ (D∇ ∂t ) − ∇ ⋅ [χ ∂t ] − ∇⋅ [(1− χ) ∂t ]
⇒ ∑∫∫∫ [N1e ]T 
dV = 0
Ωe
∂T
∂ε ′ ∂F
e=1
− ∇ ⋅ [Kβ ] − ∇⋅ [K ] + b

∂t
∂t
∂t


44
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
∂ ne
∑ [ N 1e ]T {∇ ⋅ [D∇([ N e1 ] ⋅ X ue )]}dV
∂t e =1 ∫∫∫Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T {∇ ⋅ ( χ [ N e2 ] ⋅ X le )}dV
∂t e =1 Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T {∇ ⋅ [(1 − χ )[ N e2 ] ⋅ X eg ]}dV
∂t e =1 Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T [∇ ⋅ ( Kβ [ N e2 ] ⋅ X Te )]dV
∂t e =1 Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T [∇ ⋅ ( Kε ′)]dV
∂t e =1 Ω e
∂ ne
+ ∑ ∫∫∫ [ N 1e ]T Fb dV = 0
∂t e =1 Ωe
⇒
(4.4)
where [ N 1 ]T and [ N 1e ]T are respectively the transpose of [ N 1 ] and
[ N 1e ] .If surface force at an arbitrary point on boundary Γe is p , and
n is the normal vector of boundary Γe at that point, then the first
term of eq. (4.4) can be expressed as:
∂ ne
∑ [N1e ]T {∇ ⋅[D∇([Ne1]⋅ Xue )]}dV
∂t e=1 ∫∫∫Ωe
∂ ne
= ∑ ∫∫∫ ∇ ⋅ [ N1e ]T D∇([Ne1] ⋅ Xue ) dV − ∫∫∫ (∇ ⋅[ N1e ]T )[D∇([Ne1] ⋅ Xue )]dV
Ωe
∂t e=1 Ωe
∂ ne
∂ ne
u
1T
1
= ∑ ∫∫ Ne D∇([Ne ] ⋅ Xe ) ⋅ ndS − ∑ ∫∫∫ (∇ ⋅[ N1e ]T )[D∇([Ne1] ⋅ Xue )]dV
∂t e=1 Ωe
∂t e=1 Γe
ne
T
∂
∂ ne
= ∑ ∫∫ Ne1 pdS − ∑ ∫∫∫ (∇ ⋅ [ N1e ]T )[D∇([Ne1] ⋅ Xue )]dV
∂t e=1 Ωe
∂t e=1 Γe
{
[
}
]
[ ]
[ ]
(4.5)
Substituting eq. (4.5) in eq. (4.4) leads to
∂ ne
∑ (∇ ⋅ [ N 1e ]T )[D∇([ N e1 ] ⋅ X ue )]dV
∂t e=1 ∫∫∫Ωe
∂ ne
− ∑ ∫∫∫ [ N 1e ]T {∇ ⋅ ( χ [ N e2 ] ⋅ X le )}dV
∂t e=1 Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T {∇ ⋅ [(1 − χ )[ N e2 ] ⋅ X eg ]}dV
∂t e=1 Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T [∇ ⋅ ( Kβ [ N e2 ] ⋅ X Te )]dV
∂t e=1 Ω e
∂ ne
− ∑ ∫∫∫ [ N 1e ]T [∇ ⋅ ( Kε ′)]dV
∂t e=1 Ω e
∂ ne
+ ∑ ∫∫∫ [ N 1e ]T Fb dV
∂t e =1 Ω e
T
∂ ne
+ ∑ ∫∫ N e1 pdS = 0
∂t e =1 Γe
Above equation can be rewritten in matrix form as:
⇒
(4.6)
[ ]
A uu
∂X u
∂t
+ A ul
∂X l
∂t
+ A ug
∂X g
∂t
With
45
+ A uT
∂X T
∂t
= Fu
(4.7)
FuguoTong
TRITA LWR PhD 1055
ne
A uu = ∑ ∫∫∫ (∇ ⋅ [ N 1e ]T ){D∇ ([ N e1 ]}dV
e =1
Ωe
(4.8a)
ne
A ul = ∑ ∫∫∫ [ N 1e ]T {∇ ⋅ ( χ [ N e2 ])}dV
e =1
(4.8b)
Ωe
[
]
(4.8c)
A uT = ∑ ∫∫∫ [ N 1e ]T {∇ ⋅ ( Kβ [ N e2 ])}dV
(4.8d)
ne
A ug = ∑ ∫∫∫ [ N 1e ]T ∇ ⋅ {(1 − χ )[ N e2 ]} dV
e =1
Ωe
ne
e =1
Fu = −
Ωe
ne
∂ ne
1 ]T [∇ ⋅ ( Kε′)]dV + ∂
[N
∑
∑ ∫∫∫ [N 1]T FbdV
∫∫∫
∂t e=1 Ω e e
∂t e=1 Ω e e
+
(4.8e)
∂
∑ ∫∫ N e1 pdS
∂t e=1 Γe
ne
[ ]
T
Similarly, the other governing equations can be discretized, and the
final system of algebraic equations can be expressed in matrix form
as
A
∂X
+ BX = F
∂t
(4.9)
where A and B are coefficient matrices, X is the vector of nodal
primary variables, and F is the vector of nodal loads, expressed by
 A uu
A
 vu
A
A =  gu
 Alu
A
 Tu
 Aφu
0 A ug Aul AuT
A vv A vg Avl A vT
0 Agg Agl AgT
0 Alg All AlT
0
0
0 ATT
0 Aφg Aφl AφT
0
0
0
0 0
0 Bvv Bvg 0 BvT
0 0 B
0 BgT
gg
B=
0 Bll BlT
0 0
0 0 BTg BTl BTT
0 0
0
0
0
Xu 
 Xl 
 g
X = X v 
X
XT 
 φ
X 
0 
A vφ 

Agφ 
Alφ 
0 

Aφφ 
0
0
0

0
0
0
F u 
Fl 
 g
F = F v  .
F
F T 
 φ
F 
(4.10c) ;
46
(4.10a)
(4.10b)
(4.10d)
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
4.1.2 The temporal discretization
The method used for the time discretization is a one-dimensional
finite difference scheme. Within each time step, the variation of
vectors X and F is assumed to follow linear variations, i.e.
X = N1X tk + N 2 Xtk +∆t
(4.11a)
F = N1F tk + N 2 F t k + ∆t
(4.11b)
where N1 = 1 − η , N 2 = η , and η = (t − t k ) / ∆t . Parameter η may
take any value from 0 to 1, to generate different FDM schemes.
The values of η = 0 , η = 0.5 and η = 1 correspond to the three
standard FDM schemes, i.e., forward difference (Euler), centraldifference (Crank-Nicholson) and backward difference.
The time derivatives of vectors X and F are then written, in difference terms, as
∂X ∂N1 tk ∂N 2 tk +∆t 1
(4.11c)
=
X +
X
= [ X t k + ∆t − X tk ]
∂t
∂t
∂t
∆t
∂F ∂N1 t k ∂N 2 tk + ∆t 1 tk +∆t
=
F +
F
= [F
− F tk ]
(4.11d)
∂t
∂t
∂t
∆t
Substituting eqs. (4.11a,b,c,d) into eq. (4.9) yields the final matrix
equation system to be solved:
 1
 tk +∆t  1

= A − (1 −η)BXtk + [(1 −η)Ftk +ηFtk +∆t ] (4.12)
A ∆t +ηBX
 ∆t

The value of η = 0.667 corresponds to the Galerkin difference
scheme that was adopted in this thesis. In order to start the calculation of eq. (4.12), the initial condition is specified as
 X0 u 
 l
 X 0g 
X 
(4.13)
X t0 = X 0 =  0 v 
X
 0T 
X 0 
 X 0 φ 
where X0 represents the initial values of the primary variables.
4.2 The solution strategies
In general, eq. (4.12) is a large matrix equation system, and has
three main characteristics. First of all, it is a highly nonlinear set of
equations, with all coefficient matrices as functions of the primary
variables, and can not be solved in closed-form. Secondly, it is an
asymmetric set of matrix equations, a challenge for data storage
capacities during the process of solution steps. Traditional solution
methods of FEM may not be adequate for solving such equations.
Thirdly, many of the main matrix [A ∆t + ηB ] ’s diagonal terms are
very small in magnitude, which tend to make the matrix
[A ∆t + ηB] ill-conditioned and may reduce the convergence rate
of solution during time-marching. It may also cause numerical in-
47
FuguoTong
TRITA LWR PhD 1055
stability of the solution process, and leads to no solution or solutions with wrong results.
In order to solve the matrix equation (4.12) successfully, some
strategies need to be developed and tested. Firstly, equations need
to be appropriately rearranged in order to keep the desired predominance of diagonal coefficients in matrix [A ∆t + ηB ] before
FEM discretization. This is necessary to maintain numerical stability and to increase the rate of convergence. For example, if vapor
pressure is located at the diagonal position, the vapor flow equation shown in eq. (4.14) needs to be rearranged as eq. (4.15)
φ(1− Sre ) ∂pv
k gk
∂φ
1
+ (1− Sre ) − ∇ ⋅[ r (∇pg + ρg g)]− qp = 0
pv
µg
ρv
∂t
∂t
φ(1 − S re )
(4.14)

∂pv 
kgk
∂φ
1
+ (1− S re ) − ∇ ⋅[ r (∇p g + ρ g g)] − q p  pv = 0 (4.15)
∂t 
∂t
µg
ρ v 
in order to reach a diagonal dominance of vapor pressure in matrix
[A ∆t + ηB] .
The coefficients of high nonlinearity in matrix [A ∆t + ηB ] are
always functions of multiple primary variables, representing the
coupling terms in the governing equations. To calculate such
highly non-linear coefficients, the initial values of the variables
should be properly estimated before each time step.
The coefficients of the equations change with changing values of
primary variables during the solution process. This means that a
fixed single solution method may be too rigid to be suitable for the
whole solution process. In order to increase the flexibility of solution method, the solution process is divided into several steps using suitable but different techniques at different time-marching
stages. The optimal choice of solution method needs to be decided
at each time step, according to the character of the equations at
that state.
Some other important strategies include: 1) special techniques to
avoid unnecessary iterations during the iterative solution process; 2)
appropriate methods to optimize the storage of the non-zero matrix components. These strategies have been helpful for increasing
the computational efficiency.
The above solution strategies were tested for solving the validation
and application problems as reported in Chapter 5, but are still being further tested for more matrix equation complexities.
4.3 The computer code
A computer code ROLG of three-dimensional FEM method was
developed to solve the governing equations for modeling fully
coupled thermo-hydro-mechanical processes associated with underground nuclear waste repositories. The code was intended initially for modeling the behavior of buffer materials, but can also be
applied for simulating THM behaviors of rocks. The code ROLG
was written in FORTRAN language, and has other functions for
simulating excavation sequences and material heterogeneity, han48
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
dling complex initial and boundary conditions, and built-in graphical presentation functions of results.
ROLG is a new code and needs systematic validation tests before
being able to solve practical engineering problems. Besides standard trend tests and symmetry tests used for testing basic characteristics of FEM codes, and comparison tests against analytical
solutions of problems, a series of Bench Mark Test (BMT)
problems on coupled THM processes of bentonite were
performed, and a large scale in-situ THM experiment was
simulated, as part of the WP4 of the THERESA project, with
results presented in Chapter 5.
49
FuguoTong
TRITA LWR PhD 1055
50
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
5 C ODE V A LIDA TIO N A N D A PPLIC A T ION FO R FIELD SC A LE IN SIT U PR OB LEM S
A new numerical method for modeling coupled THM processes
needs to be validated against analytical solutions and well controlled laboratory experiments, in order to establish its validity and
reliability for solving practical engineering problems. Such validation exercises against two analytical solutions of problems of
poroelasticity and thermoelasticity are presented first, followed by
validation modelling of two laboratory experiments on bentonite
involving coupled THM processes. After the validations, the code
ROLG was then applied to simulate a full scale in-situ Canister Retrieval Test (CRT), a coupled THM experiment of canister-bufferrock system in realistic scale and settings, at Äspö Hard Rock
Laboratory, southern Sweden, run by Swedish Nuclear Fuel and
Waste Company (SKB). These validation exercises and the final
application on CRT modelling are presented below. More details
are presented in Gens (2008) and Tong (2009).
5.1 Validations against analytical solutions
5.1.1 A poroelastic problem
• Problem description
A closed-form solution for a problem of compaction of a saturated poroelastic cylinder sandwiched by two impermeable, rigid,
and frictionless plates (Fig.5.1) was obtained by Detournay &
Cheng (1993). The cross section of the cylinder was assumed to be
circular. The pure liquid diffusion solution (without effect of elasticity) is characterized by a monotonic decline of the pore pressure,
whereas the poroelastic solution predicts a rise of pressure above
its initial value before its dissipation. At the start, the water pressure was set to zero, afterwards a constant confining pressure P*
was suddenly applied on the boundary. This phenomenon is
known as the Mandel-Cryer effect in the soil mechanics literature
(Detournay & Cheng, 1993; Cui & Abousleiman, 2001).This problem was simulated as a validation exercise for code ROLG, focusing mainly on the pressure history at the center of the cylinder.
The main parameters used are: height of cylinder = 0.9 m, radius =
0.3 m, permeability = 5×10-12 m2, Young’s modulus = 1875 MPa
and P* = 1.0MPa. In total 2880 hexahedron elements were used to
build the 3D FEM mesh (Fig. 5.1c), with 3249 nodes.
• Comparison of simulated and analytical results
Figure 5.2 presents the simulated result of the pressure at the center of the cylinder varying with time. Figure 5.3 shows the pressure
history at the center of cylinder, from the closed-form solution. In
general, the simulated results agreed well with the analytical ones,
with the slight difference because no parameter values were given
in Detournay & Cheng (1993) and they had to be estimated with
possible small differences.
51
FuguoTong
TRITA LWR PhD 1055
a)
b)
c)
Figure. 5.1. Schematic view of a compaction test of a sandwiched
cylinder Cui & Abousleiman (2001). a) Problem geometry; b)
Loading conditions, and c) FEM mesh
Figure. 5.2. Simulated pressure at the cylindrical center, for the
problem shown in Fig. 5.1.
Figure. 5.3. Same as Fig. 5.2, showing results from Detournay &
Cheng (1993).
52
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
5.1.2 A thermoelastic problem
• Problem description
This problem description and analytical solutions were taken from
Lutz & Zimmerman (1996). The research object is an inhomogeneous elastic solid sphere whose radius is a , and whose elastic
moduli and thermal expansion coefficient vary linearly with the radius. The temperature increases from T0 to T , where, T0 is a reference temperature at which the stresses are zero if the sphere is
undeformed. The thermal expansion coefficient and elastic parameters are expressed as:
λ (r ) = λ0 [1 + (γεr / a )]
µ (r ) = µ 0 [1 + (εr / a )]
(5.1)
(5.2)
α (r ) = α 0 [1 + (δr / a )]
(5.3)
where λ and µ are the Lame’s elasticity parameters, and α 0 is the
thermal expansion coefficient. The subscript 0 refers to the center
of the sphere. Symbols ε and δ are parameters that measure the
degree of inhomogeneity in the elastic moduli and thermal expansion coefficient, respectively. In the ROLG simulation, the values
of parameters in eqs. (5.1, 5.2 and 5.3) were set as: γ = 1,
λ 0 = 2298 ΜPa , µ 0 = 0.25, α 0 = 2.81×10−6, T0 = 0oC , T = 30 oC.
The radius of the sphere is 1.0 m. In total, 4580 hexahedron elements were used to build the 3D FEM mesh (Fig.5.4), with 5050
nodes.
Figure 5.4. The 3-D FEM mesh
• Comparison of simulated and analytic results
1) Displacements
The radial displacement as a function of the radius is shown in
Figure 5.5. Figure 5.5a shows the simulated results, and Figure
5.5b shows the analytic results taken from Lutz & Zimmerman
(1996). The agreement is close.
53
TRITA LWR PhD 1055
Normalized displacement, u/α0(T-T0)a
FuguoTong
Normalized displacement, u/α0(T-T0)a
Normalized radius, r/a
a) Simulated results
Normalized radius, r/a
b) Analytic results
Figure.5.5. The comparison between simulated and analytic radial displacement as a function of radius.
2) Stress
As functions of radius, the radial and tangential normal stresses are
shown in Figure 5.6 and Figure 5.7, respectively. Figure 5.6a
andFigure 5.7a are the simulated results, and Figure 5.6b and Fig54
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Radial stress, Trr /3 (T-T0) α0Κ0
ure5.7b are the analytical results from Lutz & Zimmerman (1996).
The simulated stresses are almost identical to the analytical stresses.
A slight difference occurs in the region near the centre of the
sphere because of the effect of the finite sizes of elements of the
FEM model mesh.
Radial stress, Trr /3 (T-T0) α0Κ0
Normalized radius, r/a
a) Simulated results
Normalized radius, r/a
b) Analytic results
Figure.5.6. The comparison between simulated and analytical radial normal stresses as a function of radius.
55
TRITA LWR PhD 1055
Hoop stress, Tφφ /3 (T-T0) α0Κ0
FuguoTong
Hoop stress, Tφφ /3 (T-T0) α0Κ0
Normalized radius, r/a
a) Simulated results
Normalized radius, r/a
b) Analytic results
Figure. 5.7. The comparison between simulated and analytical
tangential normal stresses as a function of radius.
3) Effective thermal expansion coefficient
The results of effective thermal expansion coefficient are shown in
Figure 5.8. Figure 5.8a shows the simulated results, which agree
well with analytic results shown in Figure 5.8b by Lutz & Zimmerman (1996). The two sets of results agree well.
56
Thermal exp.coff.αeff/α0
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Thermal exp.coff.αeff/α0
Inhomogeneity parameter, δ
a) Simulated results
Inhomogeneity parameter, δ
b) Analytic results
Figure. 5.8. The comparison between simulated and analytical effective thermal expansion coefficients of the inhomogeneous sphere.
5.2 Validation against laboratory experiments on bentonite
5.2.1 Validation with laboratory benchmark test 1
• Description of the laboratory benchmark test 1
A laboratory Benchmark test (Fig.5.9) was performed at the Polytechnic University of Catalonia (UPC), Spain, to investigate coupled THM behavior of bentonite (Gens, 2007;Villar,2002). A
constant power of 2.17 W was supplied for 7 days by a heater at
the central interface of two cylindrical samples of compacted bentonite which were symmetrically placed with respect to the heater.
For the two samples, their inner end surfaces close to the heater
were subjected to the prescribed heat flow from the heater, and
their outer end surfaces were kept to a constant temperature of
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FuguoTong
TRITA LWR PhD 1055
30oC. The temperatures at various points throughout the test, the
water content at the end of the test, and the change of the diameters of the samples at the end of the test, were measured.
Figure. 5.9. Conceptual scheme of the UPC heating test (Gens,
2007).
The FEM model geometry is the same as that of the samples, with
a diameter of 38 mm and a height of 76 mm, respectively. 900
hexahedron elements were used to build the 3D FEM mesh, with
1000 nodes(Fig.5.10). The temperature on the two outer end surfaces of the model was kept constant at 30oC. All surfaces are impermeable. The two outer end surfaces of the model were fixed in
their normal directions as roller boundaries, and other surfaces
were free. The initial conditions were specified by the following
parameter values: initial porosity = 0.4, initial saturation degree =
0.63, initial temperature = 22oC, initial relative humidity = 0.42,
initial stresses = 0.0 MPa. The main mechanical parameters were
given as: the accommodation coefficient = 0.00002, the dry density = 1.791 (kg/m3), the elastic stiffness parameter for changes in
net mean stress = 0.0024, the Poisson’s ratio = 0.3, the thermal
expansion coefficient = 1.0×10-5 (1/K), the specific heat capacity
of solid = 920 (J/kg), the permeability = 1.0 × 10-21(m2), the
thermo-osmosis coefficient of liquid= 5.0×10-12 (m2/s.K), the residual degree of water saturation = 0.001, the minimal suction (air
entry value) = 1.0 (KPa), and the bulk modulus of the solid constituent= 20.9 (GPa), respectively. More details of the simulation
are available in Tong & Jing (2008).
Figure 5.10. 3D FEM Mesh, only a half of the complete model is
displayed.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
• Comparison of measured and simulated results
Temperature (oC)
1) Temperature
Figure 5.11 shows that the measured and simulated results match
almost perfectly, with slight discrepancies at sensors located at x =
38 mm and x = 60 mm, at the initial stage. The reason is that the
initial temperature of the samples was given as 22oC, and the temperature at boundary was specified as 30oC. Therefore, it presents
an incompatibility that makes it difficult to give a proper initial
temperature near the boundaries, where the simulated temperatures are slightly higher than the measured ones.
Elapsed time (hours)
Figure. 5.11. The comparison between simulated temperatures
(solid curves) versus measured values (symbols).
Water content (%)
2) Water content
Figure 5.12 shows the comparison between the simulated and
measured values of the water content at the end of the test. The
water content increases with the increase of the distance, as a result of thermal-osmosis. The simulated results versus distance are
roughly linear in trend, and agree well with the measured results.
Distance to heater, x (mm)
Figure.5.12. The comparison between simulated and measured
water content at the end of the test.
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TRITA LWR PhD 1055
Diameter increment (mm)
3) Deformation
Figure 5.13 shows the comparison between the simulated and
measured diameter increase at the end of the test. The relation between simulated results and distance is roughly linear, but the relation between the measured results and distance is nonlinear to
some extent. The reason is that the stress-strain relation used in
the simulations is nonlinear elasticity that maybe is a slight oversimplification compared with the real bentonite behavior. However, the magnitudes of the measured and simulated results are
very close.
Distance to heater, x (mm)
Figure. 5.13. Comparison between simulated and measured diameter increment at the end of the test.
5.2.2 Validation with laboratory benchmark test 2
• Description of the laboratory benchmark test 2
A benchmark laboratory test (Fig. 5.14) on the MX-80 bentonite
was performed by French Commission of Atomic Energy (CEA)
from 2003 to 2004, for validating computer models for bentonite
behavior, the so-called bentonite MOCK-UP test (Gatabin et al.,
2005). Samples of bentonite had a diameter and a height of 203
mm, and were tightly enclosed in a PTFE sleeve insulated with a
heatproof envelope to minimize heat losses, but not gas tight. Heat
was applied at the bottom plate where hydration proceeded from
the top of the samples. The test was composed of two phases. In
Phase 1, heat was applied to one end of the sample while the temperature at the other end was kept at 20ºC. A maximum temperature of 150ºC was applied. Phase 2 started after a thermal equilibrium was achieved with gradual hydration of the sample. A
constant water pressure was applied to the end opposite to the one
where the temperature variation was prescribed. Constant volume
conditions were ensured in the two phases of the test. The details
of the simulation, such as FEM mesh, boundary conditions, initial
conditions and parameters, are available in Paper I.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Figure. 5.14 The CEA bentonite mock-up experiment (Gatabin et
al., 2005).
• Comparison of measured and simulated results
Temperature (oC)
1) Temperature
Figure 5.15 shows the comparison between measured and simulated temperature as functions of time. At the bottom end of the
sample (from the bottom of the sample to the top, the sensors are
T0 ,T1 ,T2 ,T3 …T13, T14, respectively), the measured and simulated temperatures agree well. At the top end of the sample, the
simulation underestimates the temperature slightly at the initial
heating stage, but the trends are generally the same.
Elapsed time (days)
Figure 5.15. Comparison of results between the simulated and
measured temperature at different locations (heights) of the
specimen as functions of time.
2) Relative humidity and vapor pressure
In some THM models, the vapor flow equation is neglected. In
such cases, the simulated results exclude the effect of vapor pressure; thus, the calculation of relative humidity RH must depend on
empirical models (Schrefler & Pesavento, 2004; Villar et al.,
2005;Nguyen et al.,2005)), for example as given by
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RH = exp(αs / ρ l RvT )
(5.4)
where s is the suction, ρl is the density of water, Rv is the specific gas constant for water vapor, T is the temperature, and α is
an empirical coefficient.
In this thesis, the THM model includes the vapor flow equation
explicitly, and vapor pressure is adopted as a primary variable that
can be obtained directly. Thus, the relativity humidity can be calculated simply and directly by adopting its original physical definition,
without relying on empirical models, as given by
p
RH = v × 100%
(5.5)
p sv
Relative humidity
Figure 5.16 shows a comparison between measured and simulated
relative humidity evolution during the test. The general trends between measured and simulated results agree well, with marginal
differences in magnitude. The reason for the small discrepancies
may be that the adopted thermo-osmosis coefficient and accommodation coefficient, both of which are taken to be constant, may
not be flexible enough to account for the influences of saturation
degree, temperature and pressure during the test.
Elapsed time (days)
Figure 5.16. Comparison between simulated and measured relative humidity.
2) Axial stress
The measured results of radial stress have certain uncertainties
during the experiment and cannot be used for code verification.
Therefore, only the measured and simulated axial stresses are
compared in Fig. 5.17. The general trends of the measured and
simulated results are almost the same, and the discrepancies in
magnitude are small or moderate. Reasons for the slight differences may be that the assumed mechanical parameters may not be
accurate enough to represent the actual behavior of the bentonite
tested. The spike in the measured values is very likely an error of
some sort during the test.
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Axial stress (MPa)
Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Elapsed time (days)
Figure 5.17. Comparison of results between simulated and measured axial stresses vs. time.
5.3 The simulation of the large scale Canister Retrieval Test (CRT)
5.3.1 Introduction of the Canister Retrieval Test (CRT)
The Canister Retrieval Test (CRT) was conducted at Äspö Hard
Rock Laboratory of SKB, Sweden, in order to demonstrate the capability to retrieve deposited nuclear waste if a better disposal solution is found. It started on October 26, 2000 when the gap between buffer and rock was filled with pellets of bentonite, and it
ended on October 11, 2005 when the electric power to the canister
was switched off. The CRT experiment basically aimed at demonstrating the readiness for recovering emplaced canisters, but also at
examining the time when the bentonite reached full saturation with
maximum swelling pressure. The test was a full scale in-situ coupled THM experiment for studying the coupled thermal, hydraulic
and mechanical evolution in the buffer (bentonite) from start until
full water saturation (Gens, 2008).
The CRT was located in the main test area of the Äspö Hard Rock
Laboratory at the -420 m level. The test period was separated into
three stages. Stage I: Boring of deposition hole and installation of
instrumented bentonite blocks and canister with electric heaters.
Stage II: Saturation of the bentonite and starting the heating, with
measurement of thermal, hydraulic and mechanical variables. Stage
III: a) Excavation of the upper half of the buffer in the deposition
hole and extensive sampling and laboratory testing of density, water ratio and other properties of the buffer. b) Test of freeing the
canister from the lower half of the buffer with salt water dissolution.
The buffer was installed in the form of cylinder-shaped and ringshaped blocks of highly compacted bentonite, with a full diameter
of 1.65 m and a nominal height of 0.5 m. Instruments for measuring temperature, relative humidity, total pressure and pore pressure
were installed in the bentonite in many of the bentonite blocks.
When the stack of bentonite blocks was 6 m in height the canister
equipped with the electrical heaters was lowered down in the cen63
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TRITA LWR PhD 1055
tre. Cables to heaters, thermocouples in the rock and strain gauges
in the rock were connected, and additional bentonite blocks were
emplaced until the hole was filled up to 1 m from the tunnel floor.
At the top the hole was sealed with a plug made of concrete and a
steel plate. The plug was secured against heave caused by the swelling bentonite, with 9 cables anchored to the rock. The tunnel was
left open for access and inspections of the plug support. Figure
5.18 shows a sketch of the CRT test, for the dimensions and relative locations of canisters, deposition hole, interfaces and other
components of the test. Besides the cylinder and ring-shaped
blocks, the bentonite components included also bentonite bricks
and pellets filling the gap between bentonite buffer and rock (Börgesson, 2007), which is an important issue to be addressed in the
modeling.
The FEM model geometry is shown in Figure 5.19 (three-fourth
of the complete model is displayed), with 11130 8-noded elements
and 10620 nodes, respectively. The elastoplastic Barcelona Basic
Model (Alonso et al., 1990) was adopted as the mechanical constitutive model. The parameters used and the initial and boundary
conditions were presented in the Report on Simulation of CRT
(Tong, 2009) and Paper IV
Figure 5.18. Design of the Canister Retrieval Test, Äsp ö HRL of
SKB (Gens, 2008)
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Figure 5.19 The 3D FEM model of the CRT experiment.
5.3.2 Comparison of measured and modeling results
Temperature (oC)
1) Temperature
Figure 5.20 shows temperature evolution measured by sensor
T127 located at bentonite ring-shaped block 10 (R10 in Fig.5.18 b),
which was 685 mm away from the centre of the canister. The
simulated and measured temperature evolution agrees well during
the whole process of the CRT test. Analysis of the results shows
that the saturation degree and its evolution play an important role
in accurate simulations of temperature evolution.
Time (days)
Figure 5.20. Comparison of the simulated and measured temperature at the location of sensor T127 as functions of time.
2) Relative humidity
Figure 5.21 shows the simulated and measured relative humidity at
locations of sensors W151 and W153, which were located in bentonite block cylinder 3 (block C3 in Fig. 5.18b), and were respectively 50 mm and 685 mm away from the centre of the canister.
The trends between measured and simulated results are very similar, but more different in magnitude at the initial stage. Besides the
possible effects of measuring techniques, the main reason may be
the uncertainty of the initial conditions of gas and vapor flows,
which was not measured in the test but included in the simulation.
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Relative humidity
FuguoTong
Time (days)
Figure 5.21. Comparison of the simulated and measured relative
humidity at the locations of sensor W151 and W153 as functions of
time.
Dry density (g/cm3)
3) Dry density of bentonite
Figure 5.22 shows the simulated and measured density of the bentonite cylinder block 3 (block C3 in Fig.5.18b). The density of
buffer decreases with increase of time in a general trend. The
simulated density agrees well with measured density in magnitude
at the end of the test. Figure 5.22 also shows simulated and measured density of the bentonite pellets in the gap (the interface between bentonite buffer and rock was filled with bentonite pellets),
where the overall average density of the bentonite in the gap increases with the increase of elapsed time. However, at the end of
test, the simulated bentonite density in the gap is smaller than the
measured density of bentonite. The reason may be that the
adopted mechanical parameters related to swelling deformation of
the bentonite can not accurately describe the actual swelling deformation of bentonite as a result of the variation of water content,
and the small strain assumption may not be suitable for pellets
compaction process.
Radius (mm)
Figure 5.22. Comparison of results between the simulated and
measured dry density of block 3 at the end of the test.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Vertical stress (MPa)
4) Stress
Figure 5.23 presents the simulated and measured vertical stress at
the location of sensor P119 which was in the ring block R10 and
685 mm away from the center of canister. There is a general
agreement in trend and magnitude between calculated and measured vertical stress evolution. However, the simulated vertical
stress is smaller than the measured ones in magnitudes in the latter
half of the test. The reason may be that the associated parameters
of the adopted BBM model might not be accurately calibrated for
describing the real mechanical behavior of the bentonite with
change of density with time.
Time (days)
Figure 5.23 Comparison of the simulated and measured vertical
stresses at the locations of sensor P119 as functions of time.
In general, the heat and fluid transport processes were well predicted with higher degree of agreement between measured and simulated results. The evolution of stress (deformation) could be
better simulated with more improvements to parameterization of
the mechanical constitutive model of the bentonite, especially the
pellets filled gap between buffer and rock.
5.3.3 Predicted results at steady state
In order to connect the modeling of the CRT test more closely to
Performance Assessment (PA), the modeling was continued for
predicting the full CRT until steady state conditions of the system
were reached. The following boundary conditions were used: a)
the heater power was held constant to 1150 W after day 1596
(March 10, 2005). b) The water pressure was held constant and
equal to 0 after day 1598 (March 12, 2005). The results required
for PA are the saturation time of bentonite, radial and axial stresses
in the bentonite, and average dry density (porosity), permeability
(or hydraulic conductivity) and temperature of the bentonite. The
results are given below.
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1) Saturation time of the bentonite
The saturation time is related to the definition of full saturation.
However, it should be pointed out that it is impossible to measure
full saturation when gas and vapor flows are included. Therefore,
in this research, the full saturation was defined as the maximum
saturation degree when the liquid flow reaches a steady state. The
simulated result shows the degree of saturation no longer changes
after 13000 days (about 35 years). The average degrees of saturation of bentonite of different types are listed in Table 5.1.
Table 5.1 The average degree of saturation at steady state
Bentonite Ring shape Cylinder shape Bricks Pellets Whole buffer
Average
saturation
0.99881
0.99937
0.99951 0.99908
0.99912
2) Radial and axial stresses in the bentonite
Generally, the stresses, which are related to swelling pressure of
bentonite, no longer change after 13000 days (about 35 years), and
are different in different regions of buffer system. In the bentonite
components, the maximum radial stress is about 5.4 MPa, and the
maximum axial stress is about 5.2 MPa. The average stresses of
bentonite of different types are listed in Table 5.2
Table 5.2 The average stress at steady state
Bentonite Ring shape Cylinder shape Bricks Pellets Whole buffer
Average radial
stress(MPa)
Average axial
stress(MPa)
0.831
2.869
3.360
0.945
1.854
1.334
3.857
4.962
1.864
2.661
3) Average dry density (porosity) of the bentonite
The dry density no longer changes after 9000 days (about 25 years),
with the maximum dry density of 1.735 g/cm3, and the minimum
dry density of 1.4 g/cm3, respectively. The average dry densities of
bentonite of different types are listed in Table 5.3.
Table 5.3 The average dry density at steady state
Bentonite
Ring shape Cylinder shape Bricks Pellets Whole buffer
Average dry
density (g/cm3)
1.675
1.567
1.479 1.526
1.602
4) Average liquid permeability of the bentonite
The liquid permeability of bentonite no longer changes after 9000
days (about 25 year). The maximum liquid permeability is about
9.6×10-21 m2, and the minimum liquid permeability is about
6.0×10-22 m2. The average liquid permeability values of bentonite
of different types are listed in Table 5.4.
Table 5.4 The average permeability at steady state
Bentonite Ring shape Cylinder shape Bricks Pellets Whole buffer
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
Average permeability (m2×10-21)
1.015
2.74
5.097 3.954
2.28
5) Average temperature of the bentonite
The temperature in bentonite no longer changes after 3000 days
(about 8 years) and is different in different regions. The average
temperatures of bentonite of different types are listed in Table 5.5.
Table 5.5 The average temperature at steady state
Bentonite Ring shape Cylinder shape Bricks Pellets Whole buffer
Average temperature (oC)
46.55
34.22
47.63 38.85
40.02
5.4 Summary
This chapter presents firstly the validation of new numerical model
and computer code ROLG for modeling coupled THM processes
of geological porous media. Besides two analytical solutions of
poroelasticity and thermoelasticity, two laboratory benchmark tests
of bentonite materials were used as validation problems. The
mathematical model and FEM code ROLG performed well for all
these validation exercises and established its applicability for real
scale engineering problems.
The ROLG code was then applied to simulate the coupled THM
processes of a full scale in-situ Canister Retrieval Test (CRT) experiment at the Äspö Hard Rock Laboratory, with acceptable degrees of agreements between the measured and predicted evolutions of temperature, stresses, relative humidity and dry density of
bentonite. A potential steady state of the CRT system was then
predicted to show representative values of some key variables of
importance for safety assessment of repositories: saturation time,
stresses, average dry density (porosity), permeability (or hydraulic
conductivity) and temperature of the bentonite. Although these
values cannot be verified in practice, they may provide supports
for design, operation and safety assessment of potential nuclear
waste repository in Sweden.
A number of outstanding issues were also identified during above
validation and application modeling, as discussed in Chapter 6,
which also serve as the basis for proposed future research topics as
described in Chapter 7.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
6 D ISC USSION A N D C O N CL US IO N S
6.1 Summary conclusions on scientific achievements
FEM modeling of coupled THM processes of geological porous
media is not a new research subject, with many schemes of governing equation derivation (under guidance of conservation laws of
mass, momentum and energy of continuum mechanics), FEM
formulation approaches and solution techniques of equation systems. However, non-linearity and ill-conditioning of the resultant
governing equations are the unavoidable aspects caused by THM
coupling mechanisms. Such complexity increases with increasing
scope and depth of coupling terms, especially when gas and vapor
flow processes are included and when partial saturation and nonisothermal conditions are present. Therefore, special attention
needs to be paid to the choice of primary variables and arrangement of equations in the final equation systems in order to reduce
such complexity, and use proper constitutive models to represent
the coupled physical processes more realistically and with better
possibilities for laboratory parameter identification. In these regards, the research presented in this thesis has a few special
achievements, with scientific originality for some of them:
1) The new THM model includes the vapor flow equation explicitly, and vapor pressure is adopted as an independent primary variable that can be evaluated directly; thus the relative humidity can
be calculated simply and directly using its original physical definition, without relying on empirical models that may contain certain
degree of uncertainty.
2) Porosity is no longer considered as a constant, or a simple function of bulk strain of the porous media concerned. It was taken as
an independent primary variable, and a conservation equation of
solid mass was used to describe the change of porosity. This
choice improved completeness and compactness of the system of
the governing equations, and helped in reducing numerical divergence and instability during solution process.
3) From above choices, the density, viscosity, thermal expansion
coefficient and bulk modulus of water and gas can be expressed in
terms of the basic variables, with reduced number of material parameters that needs to be identified for practical applications.
4) A new water retention curve model was developed and represented as a single equation, with considerations for dependences of
suction on temperature and deformation (via porosity), for both
wetting and drying processes. The equation was made of four simple continuous functions of temperature, porosity, water surface
tension and saturation degree. The derivative of the effective saturation degree (or water content) with respect to the suction can be
directly obtained, and be expressed as a continuous function of
temperature, porosity and saturation degree. This is very helpful to
keep the stability of the solution of fluid flow equations and cannot be obtained when pure empirical models of water retention
curves were used. The model has adequate capability and flexibility
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for parameter calibrations through fitting with the experimental
data with acceptable accuracy.
5) A new effective thermal conductivity model considering the effect of water content, porosity, temperature, pressure and mineral
composition of the porous geological media was developed, and
was expressed as a function of temperature, saturation degree,
pressure of gas and vapor. The model meets the limits defined by
the Wiener bounds and Hashin-Shtrikman bounds in wide parameter ranges of porosity and saturation degree, and was defined
by only two experimental parameters, η1 and η2, which reflect the
different weighting for the contributions of different phases (solid,
liquid and gas) to the heat transport process.
6) Besides above achievements of fundamental nature, an efficient
FEM solution scheme was developed and implemented in the
computer code ROLG for modeling coupled THM processes of
porous geological media in three-dimensional spaces, and under
partially saturated and non-isothermal conditions, while keeping
numerical stability and convergence of results for such equation
systems of high nonlinearity and ill-conditionedness.
7) The newly developed constitutive models of water retention
curve and effective thermal conductivity, and the computer code
ROLG, were systematically and successfully validated against analytical solutions of poroelasticity and thermoelasticity problems
and laboratory benchmark test problems of bentonite, therefore
establishing the scientific and computational reliability and flexibility of models and the code.
8) The code ROLG was applied to simulate the full scale in-situ
Canister Retrieval Test (CRT) experiments at Äspö Hard Rock
Laboratory, southern Sweden, run by Swedish Nuclear Fuel and
Waste Company (SKB), with good agreements in water and heat
transport processes, and reasonable results of mechanical process.
The objective of the thesis has for the most part been achieved.
On the other hand, a number of outstanding issues still exist, as
exhibited by challenges met during development and applications
of the models and the code. These issues are discussed below.
6.2 Discussion on outstanding issues
Numerical modeling is an essential requirement for understanding
and predicting coupled THM behaviors of geological porous media, especially for long-term problems of nuclear waste repositories.
However, the numerical solutions of coupled THM governing equation contain always some uncertainties at both fundamental and
application levels, reflected as outstanding issues. The key points in
such issues are often, if not always, associated with system characterization, basic material assumptions, constitutive models,
parameter identification, and, after all, stability and convergence of
the solutions. The discussion below describes outstanding issues
associated with each of them.
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
1) System characterization:
For porous geological media, the most important issue of system
characterization is pore structure representation. In this thesis, an
idealized elliptical shape of pore/gas bubbles of uniform sizes on
cross sections was assumed for deriving the largely analytical water
retention curve model. The model is a good approximation for
homogeneous and isotropic porous media (such as the engineered
bentonite as studied in this thesis), but may not be so suitable for
other geological materials of heterogeneity and anisotropy, and
with distributional pore sizes. The linear combination of the influences on saturation from porosity, temperature and wetting/drying
processes (cf. Eq. (3.20)) is a crude assumption with equal weight
for all affecting parameters, which may or may not be valid for
other geological porous media, such as loose sands. Different parameter combinations and powers may be used for different media,
according to the needs.
Another issue of system characterization is the choices of processes and primary variables. Three phase flow (liquid (water), gas
and vapor) was adopted to improve modeling unsaturated flow
problem under non-isothermal conditions, which caused increased
ill-conditionedness and non-linearity of the final governing equations, an additional challenge for solution convergence and stability.
It was found that choosing porosity as an independent primary
variable has an advantage of better maintaining mass conservation
and improving the numerical stability of the solutions. However, it
added additional calculation efforts, and without constraints from
boundary conditions.
2) Basic material assumptions
Besides the basic assumptions of homogeneity and isotropy, a
small deformation assumption was adopted as universally valid for
deformation analysis in this thesis. This assumption, however,
proves flawed for modeling the pellets filled gap (interface) between the buffer and rock of the CRT problem described in Section 5.3. As shown in Fig. 5.22, a more than 50% increase of dry
density of the pellet material (assumed as an equivalent continuum)
occurred and was measured in the CRT test, which cannot be approximated reasonably by the FEM model and code ROLG. The
reason is that small deformation assumption may not be valid and
a large (or finite) deformation assumption is needed. This, however, will cause a different kind of difficulties in deriving the governing equations and further increasing their non-linearity and illconditionedness. How to use a large deformation scheme without
increasing the solution complexity is a challenging problem.
3) Constitutive models
Due to the limitations of time and resources, research efforts on
this issue was focused on the effective thermal conductivity and
water retention curve. Both efforts reached their expected degrees
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of success, even though they still need continuous improvements.
The challenge came from the mechanical constitutive model of
bentonite. Both existing nonlinear elasticity and elastoplastic models were adopted for modeling stress-strain behaviors of bentonite,
but with less successful results in validation exercises and application for modeling the CRT test. Besides the limitation of small deformation assumption as mentioned above, the limitation of
nonlinear elasticity model (which cannot evaluate irreversible displacements and tends to overestimate stresses) and the challenges
for identifying the parameters in the elastoplastic Barcelona Basic
Model are the main reasons for less desirable agreements between
the measured and predicted results of stress, displacement and
density changes. This limitation is the most important outstanding
issue of the whole research presented in this thesis.
4) Parameter identification
Parameter identification is a challenging issue in numerical modeling, since it is related to the basic physical processes and their governing equations, basic material assumptions and constitutive
models, besides experiments required. In one word, parameter
identification is the most important task of modeling after the establishments of governing equations, constitutive models and
computer codes. However, for modeling coupled THM processes
of porous media, two of the most important uncertainties are the
uniqueness of the set of material parameters and their values, either directly measured or indirectly calibrated during experiments.
For example, it is difficult to find proper experiments to directly
measure vapor and gas flow parameters in unsaturated and nonisothermal conditions, since direct measurement of relative humidity in porous media is a difficult laboratory task. Indirect calibration then has to be applied but the results thus obtained may or
may not have required uniqueness. Similar difficulties exist also for
mechanical constitutive models when large deformation is required,
since ordinary strain measurement sensors may not be able to
measure the large strain gradients thus induced. The key issue is to
use integrated numerical modeling and specially designed and controlled laboratory experiments for model validation and parameter
calibration.
5) Stability and convergence of the solutions.
The coupled THM governing equations are partial differential
equations, and are converted into a large matrix equation after
discretization and integration. It is a highly nonlinear and
asymmetric matrix equation, and many of its diagonal elements are
very small in magnitude, which tend to make the main coefficient
matrix ill-conditioned, especially when vapor and gas flow are
included. Convergence, numerical stability and computational
efficiency are then three key issues for obtaining reliable numerical
solutions of coupled THM equations. In this thesis, alternative
rearrangement of flow equations and adaptive evaluation of key
primary variables were adopted to improve the behavior of the
equation solver, but these measures are based more on experiences
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Numerical modeling of coupled thermo-hydro-mechanical processes in geological porous media
these measures are based more on experiences through trial-anderror processes accumulated over the years, not from a fundamental theory of mathematical approximation or linear algebra. Such
challenges will remain in foreseeable future, even with rapid progress in computing technology. Better techniques should be investigated to improve our numerical solution capabilities since this issue is currently the main bottleneck affecting successful modeling
of coupled THM processes of porous media or fractured rocks.
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7 R EC O MM ENDA TION FO R FUT UR E ST UDIE S
In order to improve the numerical modeling of the coupled THM
processes in porous geological media, some topics for future studies are suggested, partially based on the outstanding issues identified in Section 6.2 and partially based on the estimated future
trends of relevant research subjects:
1) To develop a more robust mechanical constitutive model of
bentonite in unsaturated state, which considers the effects of fluid
flow, suction, temperature, and for both small and large deformation assumptions. The effective stress model based on Bishop’s
parameter should be included for further research as well.
2) To improve the current water retention curve model , with focuses on more appropriate function forms for the definition of the
parameters ς and δ1 , based on well-controlled experiments of the
drying and/or wetting processes of carefully chosen geological materials.
3) In order to avoid the contradiction of different parameters, for a
given material, it is necessary to research on possible relation(s)
among relative perm abilities of liquid, vapor and gas, and water
retention curve through an integrated modeling and experiment
platform.
4) Comprehensive model development and validation on thermofluid coupling, which considers the effect of temperature, saturation and other factors on coupled heat and flow (liquid, vapor and
gas) processes, through integrated numerical modeling and experimental studies.
5) Continued improvements on solution techniques for maintaining convergence, numerical stability and computational efficiency
of the numerical solution of the coupled THM processes of geological materials.
6) Development of FEM models for representing different discontinuities for coupled THM processes, such as faults, joints, and different types of structural interfaces of underground engineering
structures. Such functions are lacking in the current code ROLG,
but are important for future research and applications.
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Achanta S., Cushman J.H., Okos M.R. On multicomponent, multiphase thermomechanics with interfaces. Int J Eng Sci 1994; 32:1717-1738.
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