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Numerical modelling of time-dependent thermally induced excess pore
fluid pressures in a saturated soil
Wenjie Cui
Research Associate, Dept. of Civil and Environmental Engineering, Imperial College London,
London SW7 2AZ, U.K. (Corresponding Author: [email protected])
Aikaterini Tsiampousi
Lecturer, Dept. of Civil and Environmental Engineering, Imperial College London, London SW7
2AZ, U.K. E-mail: [email protected]
David M. Potts
GCG Professor of Geotechnical Engineering, Dept. of Civil & Environmental Engineering,
Imperial College London, London SW7 2AZ, U.K. E-mail: [email protected]
Klementyna A. Gawecka
Teaching Fellow, Dept. of Civil & Environmental Engineering, Imperial College London, London
SW7 2AZ, U.K. E-mail: [email protected]
Lidija Zdravković
Professor of Computational Geomechanics, Dept. of Civil & Environmental Engineering, Imperial
College London, London SW7 2AZ, U.K. E-mail: [email protected]
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Abstract
A temperature rise in soils is usually accompanied by an increase in excess pore fluid pressure
due to the differential thermal expansion coefficients of the pore fluid and the soil particles. To
model the transient behaviour of this thermally induced excess pore fluid pressure in
geotechnical problems, a coupled THM formulation was employed in this study, which accounts
for the non-linear temperature-dependent behaviour of both the soil permeability and the
thermal expansion coefficient of the pore fluid. Numerical analyses of validation exercises
(where there is an analytical solution), as well as of existing triaxial and centrifuge heating tests
on Kaolin clay, were carried out in the current paper. The obtained numerical results exhibited
good agreement with the analytical solution and experimental measurements respectively,
demonstrating good capabilities of the applied numerical facilities and providing insight into
the mechanism behind the observed evolution of the thermally induced pore fluid pressure. The
numerical results further highlighted the importance of accounting for the temperature-
dependent nature of the soil permeability and the thermal expansion coefficient of the pore
fluid, commonly ignored in geotechnical numerical analysis.
Keywords: Finite element methods; Consolidation; Thermal effects; Clays
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Introduction
Soils may be exposed to significant temperature variations in many geotechnical engineering
problems, such as in the vicinity of thermo-active structures or in the disposal of radioactive
waste. When a thermal load is applied to the soil surrounding a geothermal structure, a rise in
pore fluid pressure is generally observed due to the fact that the thermal expansion coefficients
of the pore fluid and the soil particles are different, the former being much larger than the latter.
If there is insufficient drainage, this thermally induced excess pore fluid pressure may become
significant, thus reducing the effective stresses in the ground and consequently the stability of
existing neighbouring structures and may even result in thermal failure of the soil (Gens, 2010).
Experimental investigations of thermally induced excess pore fluid pressures have been carried
out extensively over the past decades. To study the thermo-mechanical behaviour of an oil sand,
undrained triaxial heating tests were performed by Agar et al. (1986) and excess pore fluid
pressures were measured at different elevated temperatures. Similar undrained tests on Boom
and soft Bangkok clays were conducted by Hueckel & Pellegrini (1992) and Abuel-Naga et al.
(2007a) respectively, where notable pore fluid pressure changes were observed when the
temperature of the sample was increased. The time-dependent behaviour of thermally induced
pore fluid pressures was firstly reported by Britto et al. (1989) and Savvidou & Britto (1995),
who undertook centrifuge and triaxial heating tests on saturated Kaolin clay respectively.
Subsequently, a number of laboratory (e.g. Lima et al., 2010; Mohajerani et al., 2012) and in situ
(Gens et al., 2007; François et al., 2009) tests was conducted where the evolution with time of
both excess pore fluid pressure and temperature in soils were monitored.
Various mechanical constitutive models (e.g. Vaziri & Byrne, 1990; Laloui & Francois, 2009;
Abuel-Naga et al., 2007b) have been shown to be able to simulate the increase in pore fluid
pressures measured at elevated temperatures in undrained triaxial heating tests, by adopting an
additional equation describing pore pressure variation with temperature change. However, to
model the time-dependent behaviour of thermally induced pore fluid pressures, where both
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transient heat transfer and consolidation are involved, appropriate numerical tools, which are
able to simulate the fully coupled thermo-hydro-mechanical (THM) behaviour of soils, are
required. In recent years, a number of such numerical models have been developed (e.g. Lewis &
Schrefler, 1998; Thomas et al., 2009; Abed & Sołowski, 2017; Cui et al., 2018) and extensive
numerical studies have also been conducted in which the temperature effect on the behaviour of
the pore fluid pressure in geotechnical engineering is considered. Booker & Savvidou (1985)
presented the governing equations for a transient coupled THM problem of soils and
subsequently derived a closed form solution for the thermally induced pore fluid pressure
around a point heat source. An approximate solution was also derived for a cylindrical heat
source by integrating the point source solutions. Alternatively, the Finite Element (FE) method
has been extensively employed to model transient coupled THM phenomena in soils with
varying degrees of success. Britto et al. (1992) presented a coupled FE formulation for soils
which was used by Britto et al. (1989) and Savvidou & Britto (1995) to simulate the transient
heat transfer and consolidation in triaxial and centrifuge heating tests, respectively. A
satisfactory match between numerical and experimental results was obtained. However,
constant values of permeability and thermal expansion coefficients were adopted in the
simulations, although both properties are known to vary significantly with temperature (Delage
et al., 2000; Çengel & Ghajar, 2011). This simplification was compensated for by adopting
different values of the hydraulic permeability for the Kaolin clay in Savvidou & Britto (1995)
when simulating undrained (7.0×10−9m/s) and drained (2.5×10−9 m/s) triaxial heating tests.
The centrifuge test reported by Britto et al. (1989) was also modelled by Vaziri (1996), however
using a thermally induced structural reorientation coefficient to account for rotation of soil
particles and consequently the generation of excess pore pressures, instead of the thermal
expansion coefficients of the pore fluid and the soil particles. This artificial parameter, as
introduced by Vaziri (1996), is non-linear and may start from a negative value in an analysis,
become positive after a certain temperature is reached and finally reduces to zero. To study the
behaviour of a clay around a cylindrical heat source, Seneviratne et al. (1994) presented a
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coupled THM formulation and carried out a series of FE parametric studies with material
properties similar to those listed by Britto et al. (1989). Both the hydraulic permeability and the
thermal expansion coefficient of the pore fluid were considered as temperature-dependent
variables in that study. Temperature-dependent permeability was adopted by Gens et al. (2007)
and François et al. (2009) to simulate in situ heating tests. However, constant values of the
thermal expansion coefficient of the pore fluid were adopted in their analyses.
A coupled THM formulation is introduced in the current paper, which is capable of recovering
the pore fluid pressures induced by the difference in thermal expansion coefficients of the pore
fluid and the soil particle, has been developed and implemented into the bespoke FE software
ICFEP (Potts & Zdravković, 1999) employed in this study. The non-linear temperature-
dependent behaviour of both the soil permeability, k f , and the thermal expansion coefficient of
the pore fluid, αT , f , are taken into account, with their values updated using the value of the
current temperature, T , during the iteration process of each increment in the analysis. The
paper starts with a brief presentation of the new formulation, which was firstly applied to
simulate a simple problem of consolidation around a cylindrical heat source, adopting constant
values of the soil permeability and thermal expansion coefficient of the pore fluid, to which
approximate analytical solutions exist. An excellent match was obtained between numerical
predictions and the existing approximate analytical solutions. Subsequently, existing triaxial
heating tests as well as a centrifuge heating test were modelled, employing non-linearly varying
k f and α f ,T . The comparison between numerical predictions and experimental results
demonstrates the importance of considering the non-linear temperature-dependent behaviour
of both the soil permeability and the thermal expansion coefficient of the pore fluid, when the
time-dependent thermally induced behaviour of soils is modelled. Moreover, the mechanism
behind the generation and the dissipation of the excess pore fluid pressure in both triaxial and
centrifuge heating tests is demonstrated, and recommendations on the essential aspects of
numerical modelling of thermally induced pore pressures are provided. A tension-positive sign
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convention is adopted to derive the presented formulation, while the numerical predictions are
converted into a compression-positive sign convention, as applicable in soil mechanics.
Numerical formulation for a coupled THM problem
Hydraulic governing formulation
For a fully saturated soil, applying the principle of mass conservation for the fluid phase leads to
the following expression:
∂ (n ρf dV )∂ t
+[∇ ∙ ( ρf v f )−ρf Qf ]dV=0 ( 1 )
where ρ f is the density of the pore fluid, v f represents the vector of the seepage velocity, ∇ ∙ is
the symbol of divergence defined as ∇ ∙Θ=∂Θ∂ x
+ ∂Θ∂ y
+ ∂Θ∂ z , dV is an infinitesimal volume of the
soil, n is porosity, Qf represents any pore fluid sources and/or sinks, and t is time. Following the
procedure detailed in the Appendix, Eq. ( 1 ) can be rewritten as:
∇ ∙ v f−nK f
∂ p f∂ t
+3n (α T−αT ,f ) ∂T∂t −Q f=−∂ (ε v−ε vT)
∂ t( 2 )
where K f is the bulk modulus of the pore fluid, α T and α T , f are the linear thermal expansion
coefficients of the soil skeleton and the pore fluid respectively, T is temperature, ε v is the total
volumetric strain and ε vT is the thermal volumetric strain. In Eq. ( 2 ), the first two terms on the
left-hand side represent the flow of pore fluid into and out of the soil element and the changes in
the volume of the pore fluid due to its compressibility, respectively, while the third term
denotes the change in volume of the pore fluid generated by the difference in thermal expansion
coefficients between the soil particles and the pore fluid. It should be noted that Eq. ( 2 ) is the
same as that obtained by Lewis & Schrefler (1998), who followed a different approach which
combines the mass balance equation for the solid phase with the mass balance equation for the
fluid phase. Also, it is shown by Cui et al. (2018) that adopting the principle of volume
conservation of the pore fluid can lead to the same hydraulic governing equation as Eq. ( 2 ). Eq.
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( 1 ) was adopted by Thomas & He (1997) as their starting point, however, the third term on the
left-hand side of Eq. ( 2 ) (i.e. 3n (α T−αT ,f ) ∂T∂ t ) is missing in their final form of the hydraulic
governing equation due to the fact that d V s and ρ f are assumed to be constant in their
derivation.
Adopting the generalised Darcy’s law leads to the expression of the seepage velocity v f in Eq. ( 2
) as:
v f=k f (∇ pfγf +iG) ( 3 )
where k f is the permeability matrix of the soil, ∇ p f represents the gradient of pore fluid
pressure, the vector iGT = {iGx iGy iGz} is the unit vector parallel, but in the opposite direction, to
gravity, and γf is the specific weight of the pore fluid. k fcan be further expressed as:
k f=ρf gμ f (T )
k ( 4 )
where g represents gravity, k is the intrinsic permeability and μf (T ) is the pore fluid viscosity,
which varies with temperature change under non-isothermal conditions. For a soil saturated
with water, the expression of μf (T ) can be approximated by (Al-Shemmeri, 2012):
μf (T )=2.414×10−5×10A ( 5 )
where A=247.8/(T +133.15) if T is defined in degrees Celsius. Since changes in fluid viscosity
dominate the observed changes in permeability with temperature, Eq. ( 4 ) can be further
written as:
k f=μ f (T k f)μf (T )
k f , r=10Bk f ,r ( 6 )
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where k f , r is a reference permeability matrix at the temperature of T k f , and
B=247.8×( 1Tk f+133.15
−1
T+133.15 ) if T is defined in degrees Celsius.
In a heating test on soil, both αT and α T , f in Eq. ( 2 ) are observed to vary with temperature
change, which, as noted above, should be taken into account in order to accurately model the
thermally induced pore fluid pressure. It is noted that a substantial variation (i.e. from
2.93×10−5 m/(m K) to 2.51×10−4 m/(m K) in the temperature range of 10 - 100°C) in the
linear thermal expansion coefficient of pore water, α T , f , with temperature is documented in the
literature (Çengel & Ghajar, 2011). However, a substantially smaller variation in the linear
thermal expansion coefficient of the soil skeleton, α T , has been suggested by Campanella and
Mitchell (1968) and has been observed in some drained heating/cooling tests of
overconsolidated clay samples (Baldi et al., 1991; Abuel-Naga et al., 2007b), and hence is
neglected here. To simulate the variation of α T , f with temperature, a third-order polynomial
function has been established which fits the existing experimental data for temperatures in the
interval between 0 and 100 ˚C provided by Cengel & Ghajar (2011), as shown in Fig. 1. If T is
defined in degrees Celsius, this function can be expressed as:
αT , f (T )=1.48×10−10T3−3.64×10−8T 2+4.88×10−6T−2.02×10−5 ( 7 )
Thermal governing formulation
Adopting the law of energy conservation gives the governing equation of heat transfer in a fully
saturated soil as:
∂ {[n ρf C pf+(1−n )ρ sC ps ] (T−T r )dV }∂ t
+{∇ ∙ [ ρf C pf v f (T−T r ) ]−∇ ∙ (kT ∇T )−QT }dV=0( 8 )
where Cpf and Cps are the specific heat capacities of the pore fluid and soil particles respectively,
ρs is the density of the soil particles, Tr is a reference temperature, kT is the thermal conductivity
matrix and QT represents any heat source and/or sink. The first term in Eq. ( 8 ) denotes the
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heat content of the soil per unit volume, while the second term expresses the heat flux per unit
volume including both heat diffusion and heat advection. Applying the principle of mass
conservation for each phase and following a similar procedure to that for the hydraulic
equation, yields:
[n ρ fC pf+(1−n ) ρsC ps ] ∂T∂t +ρ f Cpf (T−T r )∂e∂t
+∇ ∙ [ρf C pf v f (T−T r ) ]−k T∇ ∙ (∇T )=QT( 9 )
Mechanical governing formulation
Under non-isothermal conditions, the incremental total strain Δε can be expressed as the sum
of the incremental strain due to stress change (mechanical strain), Δεσ, and the incremental
strain due to temperature change (thermal strain), ΔεT:
∆ ε=∆εσ+∆ε T ( 10 )
where ∆ εTT={αT∆T αT ∆T αT ∆T 0 0 0 }. Applying the principle of effective stress, the
total stress can therefore be given as:
∆ σ=D' (∆ε−∆ εT )+∆σ f ( 11 )
where∆ σ fT= {∆ p f ∆ pf ∆ p f 0 0 0 } and D' is the effective constitutive matrix which
depends on the adopted constitutive relations (e.g. linear elastic, non-linear, elasto-plastic).
Finite element formulation and solution scheme
Applying the standard finite element discretisation to Eq. ( 2 ), Eq. ( 9 ) and Eq. ( 11 )
(Zienkiewicz et al., 2005) and the time marching method (Potts & Zdravković, 1999), the
coupled THM finite element formulation for fully saturated soils can be derived as:
[ KG LG −MG
LGT −β1 ΔtΦG−SG −ZG
Y G −β2∆ t ΩG XG+α 1∆t ΓG]{ΔdnG∆ pf nGΔT nG
}={ΔRG
ΔFG
ΔHG} ( 12 )
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where α 1, β1 and β2 are time integration parameters, the values of which should be between 0.5
to 1.0 to ensure the stability of the marching process. The matrices in Eq. (12) are the same as
those detailed in Cui et al. (2017), where the hydraulic formulation was derived using the law of
volume conservation.
It should be noted that both the hydraulic permeability, k f , and the linear thermal expansion
coefficient of the pore fluid, α T , f , may be set to vary with temperature in a coupled THM
analysis. If so, the values of these temperature-dependent variables are updated accordingly (i.e.
Eq. ( 6 ) and Eq. ( 7 )) in the analysis even during the iteration process of each increment.
Compared to the approach of using the initial value at the beginning of an increment throughout
the incremental iterations, the adopted numerical scheme can potentially produce more
accurate solutions especially when the variation of these non-linear properties is significant
over an increment.
The fully coupled THM equations described above have been implemented into the bespoke FE
software ICFEP (Potts & Zdravković, 1999, 2001), which is employed to carry out all of the FE
analyses presented subsequently in this paper.
Verification exercise
Numerical modelling of consolidation around a cylindrical heat source
To demonstrate the capability of the proposed THM formulation in simulating thermally-
induced pore fluid pressures, a series of axisymmetric benchmark analyses, representing the
example of elastic consolidation around a cylindrical heat source proposed by Booker &
Savvidou (1985), has been performed with constant values of the soil permeability and thermal
expansion coefficient of pore fluid. It should be noted that this validation exercise was also used
as a benchmark by Lewis et al. (1986), Britto et al. (1992), and Vaziri (1996) for checking their
FE formulations.
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The same material properties as those from Lewis et al. (1986) were employed (see Table 1),
ensuring the same conditions adopted in the numerical example illustrated by Booker &
Savvidou (1985). The adopted FE mesh is shown in Fig. 2, employing 8-noded quadrilateral
elements, with displacement, pore fluid pressure and temperature degrees of freedom at all
element nodes, leading to the same displacement, pore fluid pressure and temperature shape
functions as those employed by Lewis et al. (1986). A domain of 8 m × 16 m was used and was
shown to be sufficiently large such that the heat front did not reach the boundary during the
analysis. The cylindrical heat source has a length of 2l0 and a diameter of 2r0. As part of the
study, the value of l0 was varied. For convenience, a value of r0=0.16 m is adopted to ensure that
the simulation time in the analysis, t, is the same as the time term
~t (~t=[n ρ fC pf+(1−n) ρsC ps]ro2 t) used in the solutions by Booker & Savvidou (1985). All the
boundaries were assumed to be impermeable and insulated and a constant heat input of 1000
W was prescribed over the elements representing the cylindrical heat source. The pore fluid
pressure was assumed to be initially hydrostatic with a zero pore fluid pressure specified over
the top boundary of the mesh, and a time-step of 0.1 s was used in the analysis.
Numerical results
The changes in temperature and pore fluid pressures at three different points on the plane z=0,
i.e. A (r0, 0), B (2r0, 0) and C (5r0, 0), were monitored throughout the analysis. To compare the
numerical results to the existing solutions approximated by integrating the closed form
solutions of a point heat source (Booker & Savvidou, 1985), the predicted temperature change,
∆T, was normalised with respect to the final temperature change obtained at point A (i.e. the
maximum value in the mesh), ∆TA. The predicted pore fluid pressure change was normalised by
the change of pore fluid pressure, ∆pf,N, at point A assuming that the soil was impermeable. The
expression of ∆pf,N was given by Booker & Savvidou (1985) as:
∆ p f , N=E
1−2v {1−v1+v [3nαT , f+3(1−n)αT ]−α T , f } ( 13 )
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As shown in Fig. 3 and Fig. 4, very good agreement was found in both temperature and pore
fluid pressure changes between the approximate analytical solutions and numerical predictions
with a ratio of l0/r0=10.0 in this study. Conversely, the numerical results obtained by Britto et al.
(1992) with their FE program HOT CRISP (value of l0/r0 was not given) showed larger
differences compared to the approximate analytical solutions. However, it should be noted that
the size of the heat source, i.e. the ratio of l0/r0, was found to significantly affect the numerical
results. Nonetheless, this term was not considered when the solutions were approximated by
Booker & Savvidou (1985). As shown in Fig. 5 and Fig. 6, good agreement between numerical
and approximate analytical results was found only for values l0/r0 between 7.5 and 10.0 at point
C (5r0, 0). The same conclusion also applies to the variations of temperature and pore fluid
pressure changes with time at points A (r0, 0) and B (2r0, 0), which are not shown here for
brevity.
Numerical modelling of thermally induced pore pressures in
triaxial tests
Experimental procedure
A series of triaxial tests were performed by Savvidou & Britto (1995) to investigate the
generation of excess pore water pressures due to a temperature increase under both undrained
and drained conditions. Fully saturated Speswhite Kaolin clay samples, with a diameter of 102
mm and a height of 200 mm, were used in the tests. The samples were firstly one-dimensionally
consolidated at ambient room temperature to a vertical stress σ vmax of 300 kPa for the
undrained test and 400kPa for the drained test. Subsequently, the samples were transferred to a
triaxial cell and isotropically consolidated to p '0 of 100 kPa for the undrained test and 317 kPa
for the drained test at a temperature of approximately 20°C, resulting in overconsoilidation
ratios (OCR) of 3 and 1.26 for the undrained and drained cases, respectively.
A water circulation system was used to heat the confining fluid (water) in the cell and thus the
sample. The temperature of the circulated water in the cell was controlled and monitored
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during the test. Two polycarbonate plates were placed at the top and bottom of the sample to
keep the sample thermally insulated and drainage was allowed through these plates during the
drained test. Temperature and pore fluid pressure were measured by transducers at two
positions: A, located at mid height and 35 mm away from the central line of the sample, and B,
positioned also at mid height but 5 mm away from the central line of the sample, as shown in
Fig. 7.
Numerical modelling
A number of axi-symmetric fully coupled THM analyses were carried out with the THM
formulation described above to model the triaxial heating tests. The adopted mesh consisted of
20×80 8-noded elements (i.e. width = 2.55 mm and height = 2.5 mm), with displacement
degrees of freedom (DOF) at all nodes and both pore fluid pressure and temperature DOFs only
at the corner nodes (see Fig. 7). The displacement in the z direction on the top surface of the
mesh was tied to simulate the top cap placed on the sample. All boundaries of the mesh were
modelled as impermeable in the undrained test, while a zero change of pore fluid pressure was
applied at the top and bottom boundaries (Lines 3-4 and 1-2 in Fig. 7) in the drained case. The
monitored temperature variation of the circulated water in the triaxial cell (see Fig. 8) was
prescribed at the surface of the soil sample (Line 2-3) as a temperature boundary condition,
while the top and bottom boundaries were modelled as thermally insulated. A value of 0.8 was
used for all time marching parameters and the time-step size was chosen arbitrarily as 10
seconds. A hydrostatic initial pore fluid pressure condition with zero pore fluid pressure
specified at the top surface, as well as an initial temperature of 20°C, was adopted in the
modelling.
The material properties adopted in the analysis are listed in Table 2. Savvidou & Britto (1995)
reported the thermal properties determining the heat capacity of Kaolin clay (i,e. ρ f , ρ s, C pf , and
C ps), as well as the thermal conductivity. The same values were used in this work. There is a
lack of experimental data in the literature regarding the thermo-mechanical behaviour of the
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type of soil used in the tests (i.e. Speswhite Kaolin), and, therefore, the value of the linear
thermal expansion coefficient of the soil skeleton, αT , is not known. Triaxial cooling tests (where
a linear elastic volumetric behaviour is observed) on Soft Bangkok clay have yielded a value of
α T=1.43×10−5 m/(m K) (Abuel-Naga et al., 2007b) and on Boom clay a value of
α T=3.30×10−5 m/(m K) (Baldi et al., 1991). A value of α T=2.0×10
−5 m/(m K) is considered
to be appropriate for Kaolin clay here. It should be highlighted that the value of the thermal
expansion coefficient of water, α T , f , can be chosen to vary with temperature in the analysis, as
shown in Fig. 1.
Although soil permeability has a negligible effect on the generation of excess pore fluid
pressures under undrained conditions, it affects significantly the numerical simulation of
drained tests (Seneviratne et al., 1994). Surprisingly, different values of the hydraulic
permeability, k f , were reported by Savvidou & Britto (1995) for the undrained (k f=7.0×10−9
m/s) and drained (k f=2.5×10−9 m/s) cases. As the difference between the two values
reported is not negligible, both values were disregarded and the value adopted in this study was
instead obtained from the experimental data on Kaolin clay reported by Al-Tabbaa & Wood
(1987). For a void ratio range between 1.0 and 1.4, which is the case in this study, the measured
permeability of Kaolin clay at room temperature was observed to vary from 1.0×10−9 to
2.0×10−9 m/s. For simplicity, an average value of 1.5×10−9 m/s at room temperature was
employed in the analysis. Consequently, a varying permeability with temperature, as
determined by Eq. ( 6 ) and shown in Fig. 9, was used in the analysis for both the undrained and
drained cases.
The modified Cam-clay (MCC) model, with all the parameters reported by Savvidou & Britto
(1995), was adopted in the analysis. The reduction in mean effective stress, due to the
significant increase in the pore fluid pressures, ensures that the effective stress paths observed
both in the undrained and drained tests lie inside the yield locus throughout the analysis,
indicating that elastic soil behaviour is dominant in this study. The same observation was made
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by Seneviratne et al. (1994) who conducted parametric numerical analyses to investigate the
thermally induced pore pressures around a buried canister of hot radioactive waste. Therefore,
it is thought that the conventional MCC model is adequate in the present study, although a more
sophisticated thermo-plastic constitutive model can be used in the future to verify the above
observation.
Numerical results
Undrained triaxial test
Fig. 10(a) and Fig. 10(b) compare the measured and predicted temperature changes and excess
pore fluid pressures at positions A and B (see Fig. 7) in the undrained triaxial test, respectively.
A good match is found in the temperature evolutions at both transient and steady state stages at
both positions. The slight difference may be due to the error related to the exact positions of the
transducers in the sample, as noted by Savvidou & Britto (1995). However, the predicted excess
pore fluid pressures increased at a slower rate at both positions A and B compared to those
observed in the test, although similar steady state values were achieved.
It should be noted that the evolution of the measured excess pore fluid pressure with time, as
illustrated in Fig. 10(a) and Fig. 10(b), does not follow the corresponding temperature variation
measured in the test. The excess pore fluid pressures measured at both positions A and B
reached their peak values long before the maximum temperature at the corresponding position
was reached. Interestingly, the measured excess pore fluid pressures reached a plateau as soon
as the temperature of the confining fluid in the cell reached an almost steady value. However, no
further explanation was given in the literature regarding this observation, hence the mechanism
behind it remains unclear and further experimental investigation with modern techniques is
required.
The excess pore fluid pressures predicted using the presented THM formulation increased
proportionally to the temperature rise at the corresponding position, with peak values being
achieved simultaneously (see also Fig. 10). It should be noted that a decrease in excess pore
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fluid pressures was predicted at position B at the beginning of the analysis, which is thought to
be caused by the coupled thermal and mechanical effect on the hydraulic behaviour. To
investigate this further, additional analyses were carried out. In one analysis, an extremely low
initial permeability of k f 0=1.0×10−20 m/s was adopted so that no re-distribution of pore fluid
within the soil was allowed and the excess pore fluid pressure near the axis of symmetry (e.g.
position B) is solely induced by the mechanical volumetric change (i.e. thermal expansion). In
the other simulation, a higher initial permeability of k f 0=1.5×10−8 m/s was employed
ensuring a much quicker pore fluid pressure re-distribution. Fig. 11 compares the excess pore
fluid pressure distributions at the beginning of the analysis (t=3.5min which refers to the
maximum predicted tensile excess pore fluid pressure in Fig. 10(b)) along the radial direction of
the sample with different initial values of permeability. A tensile excess pore fluid pressure of
around 1.7 kPa was predicted around the axis of symmetry in the case of extremely low
permeability, while an almost uniform compressive pore fluid pressure distribution in the radial
direction could be observed when a higher initial permeability of k f 0=1.5×10−8 m/s was
applied. Therefore, it is evident that the expansive thermal volumetric change at the outer
boundary of the sample leads to the generation of tensile excess pore fluid pressure next to the
axis of symmetry. Simultaneously, due to the difference between the thermal expansion
coefficients of the pore fluid and the soil particles, a compressive excess pore fluid pressure is
generated in the outer part of the sample which tends to drive the pore fluid flow towards the
axis of symmetry. Fig. 12 demonstrates the influence of the adopted values of permeability on
the predicted excess pore fluid pressure at position B. Although increasing the initial
permeability (k f 0=1.5×10−8 m/s) leads to a quicker pore fluid pressure re-distribution, the
predicted excess pore fluid pressure still reached its peak much later compared to the measured
one. It should be noted that further increasing the permeability has negligible influence on the
predicted evolution of excess pore fluid pressure.
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In the modelling of the undrained triaxial heating test, a significant difference was also observed
when different thermal expansion coefficients of the pore fluid were adopted. When a constant
value of αT , f=1.0×10−4 m/(m K) (corresponding to the value of α T , f at approximately 30°C)
was employed, which is the same as that adopted in the numerical analysis performed by
Savvidou & Britto (1995), the peak pore fluid pressure was underestimated compared to that
using a temperature dependent α T , f (see Fig. 13). The numerical results highlight the
importance of adopting a variable thermal expansion coefficient for the pore fluid.
Smooth ends, i.e. no radial restrains at the top and bottom surfaces of the triaxial sample, as
suggested by Savvidou & Britto (1995), were assumed in the above analyses. To further
investigate the end effects in a triaxial heating test, an additional analysis was carried out where
rough end restraints were applied (i.e. by restricting radial movements along the boundaries 1-
2 and 3-4 in Fig. 7). As shown in Fig. 14 (a) and Fig. 14 (b), a peak excess pore fluid pressure,
which is approximately 6% higher, was observed when rough ends were adopted, highlighting
the influence of end effects in a triaxial heating test.
Drained triaxial test
As shown in Fig. 15, a good match was found between numerical predictions and experimental
data in both temperature and excess pore fluid pressure evolutions with time for the drained
triaxial test. Under drained conditions, the excess pore fluid pressures initially increased rapidly
with increasing temperature, but started dissipating before the temperature reached its peak
value. The maximum excess pore fluid pressure was achieved immediately after the prescribed
temperature at the boundary reached its plateau, while the associated temperature at that
position was still rising.
In the modelling of the triaxial drained heating test, the adopted values of both the permeability
and thermal expansion coefficient of the pore fluid were found to significantly influence the
predicted variation of excess pore fluid pressures, as illustrated in Fig. 16. When a constant
permeability at room temperature, i.e. k f=1.5×10−9 m/s, was employed throughout the
17
analysis, the peak pore fluid pressure was overestimated and a much slower dissipation rate
was observed compared to the measured results. In contrast, employing a constant value of
α T , f=1.0×10−4 m/(m K) (corresponding to the value at around 30°C), which is the same as
that adopted by Savvidou & Britto (1995), leads to a slight underestimation of the peak pore
fluid pressure. The numerical results highlight the significance of adopting both variable
hydraulic permeability for the soil and variable thermal expansion coefficient for the pore fluid.
Modelling of thermally induced pore pressures in centrifuge
tests
Experimental procedure
To investigate the behaviour of coupled heat flow and consolidation around a nuclear waste
canister, a series of 100g centrifuge tests were carried out by Maddocks & Savvidou (1984),
where a 6 mm in diameter and 60 mm long model cylinder was buried in a steel tub containing
fully saturated Kaolin clay. Two 5 mm thick sand layers were placed in the clay at some distance
below and above the canister to help accelerate the consolidation process. A constant power
supply was applied to the model canister after its installation to heat it up. The temperature and
pore fluid pressure changes in the clay surrounding the canister were monitored by transducers
(i.e. thermocouple on the surface of the canister and pore pressure transducer adapted to also
measure temperature in the clay).
A representative centrifuge test (CS5), with all the experimental results detailed in Britto et al.
(1989), was chosen for the numerical case study using the presented coupled THM formulation.
In this test the Kaolin clay was normally consolidated and a constant power of 13.9 W was
supplied to heat the canister.
Numerical modelling
A prototype axi-symmetric finite element analysis was performed to model the centrifuge test
CS5. It should be noted the scaling law from Kutter (1992), as listed in Table 3, was adopted for
18
the numerical modelling, where N denotes the scaling factor from model to prototype analysis
(N=100 in this study). Therefore, a finite element mesh, which is 52.5m×35.0m in dimension
as shown in Fig. 17, was used in the analysis.
Except for a slightly higher thermal conductivity which is the same as that reported by Britto et
al. (1989), all the hydraulic and thermal material properties of the Kaolin clay employed in the
numerical analysis of the centrifuge test were the same as those adopted above in the modelling
of the triaxial heating tests (see Table 4). It should be noted that as the diffusion time in the
numerical simulation was scaled by N2, the diffusion coefficients, i.e. the hydraulic permeability
and the thermal conductivity, were not scaled in the analysis. The modified cam-clay model,
with all the parameters measured and presented by Maddocks & Savvidou (1984), was adopted
in the analysis of the centrifuge test to model the mechanical behaviour of the normally
consolidated Kaolin clay. In the numerical analysis, plastic volumetric strains were only
observed in very small zones of the clay below and above the canister due to the thermal
expansion of the canister, while stress paths in other parts of the clay were found to lie within
the yield locus due to the significant rise in the excess pore fluid pressure. As the sand layers
were extremely thin and had negligible mechanical effect in the analysis, a simple linear elastic
model with a Young’s module E=3.0×105 kPa and a Poisson’s ratio υ=0.25 was employed for
modelling the mechanical behaviour of the sand, while typical values of thermal and hydraulic
properties of a sand, listed in Table 4, were used. All boundaries of the mesh were assumed to
be smooth. As shown in Fig. 17, no lateral displacement is allowed at both the axis of symmetry
and the right vertical boundary, while a no vertical movement boundary condition is prescribed
at the bottom boundary. The two vertical and top horizontal boundaries were impermeable and
water can only leave the mesh from the bottom boundary where a zero change in the pore fluid
pressure was prescribed. A constant scaled heat flux of 0.819 kW/m3, equivalent to the power
supply of 13.9 W in the centrifuge test, was applied to the canister throughout the analysis and
all boundaries of the mesh were assumed to be thermally insulated.
19
The same initial stress profile as that presented in Britto et al. (1989) was used for this study,
comprising a hydrostatic initial pore fluid pressure profile over the depth of the mesh with, a
zero value at the top boundary, as well as an initial temperature of 20°C. The initial vertical
effective stresses can be determined by σ v'=γsat h−γwh, where h represents the depth from the
top boundary of the mesh, and γw and γsat are the specific unit weight of water and saturated
soil respectively. A value of γsat=16.7 kN/m3 (no scaling is needed) for the Kaolin clay was
deduced from the in situ initial stress profile provided by Britto et al. (1989). The initial
horizontal effective stress profile was obtained from σ h'=K0σ v
' adopting a value of K 0=0.69, as
suggested by Britto et al. (1989).
8-noded quadrilateral elements were employed in the numerical analysis, where each node has
displacement DOFs and the corner nodes also have temperature and pore fluid pressure DOFs.
The adopted time marching scheme for the prototype modelling is listed in Table 5, with a value
of 0.8 applied to all time marching parameters. The variation in temperature was monitored at
the surface of the canister by a thermocouple, while variations in both temperature and excess
pore fluid pressure were monitored at the position where the pore pressure and temperature
transducer was placed (see Fig. 17).
Numerical results
Fig. 18 compares temperature evolutions between numerical and experimental results at both
the canister surface and the transducer (its location is shown in Fig. 17). A good match was
observed at the canister surface which implies that the scaling law applied to the numerical
modelling is appropriate. A slightly higher numerical prediction was found at the transducer,
which may be due to the fact that the boundaries of the centrifuge apparatus were not
completely insulated from the environment and a heat loss existed at those boundaries.
Due to the lack of experimental data at the canister surface in the literature, Fig. 19 only
compares the predicted and measured evolutions of excess pore fluid pressure at the
transducer. Although similar peak values were observed, the measured excess pore fluid
20
pressure reached its maximum value much faster than that predicated by the numerical
analysis. It should be noted that both the measured and the predicted excess pore fluid pressure
started to increase before the heat front arrived at the location of the transducer.
Based on the experimental data on Kaolin clay reported by Al-Tabbaa & Wood (1987), an
average value of initial permeability (i.e. k f 0=1.5×10−9 m/s) at room temperature was
employed in the above analysis for a void ratio range between 1.0 and 1.4. However, when a
value of k f 0=1.0×10−9 m/s at room temperature, which corresponds to the void ratio of 1.0,
was used in the analysis, a higher peak excess pore pressure than the measured one was
obtained compared to that predicted in the previous analysis with k f 0=1.5×10−9 m/s , as
shown in Fig. 20. This suggests that when a temperature dependent permeability is employed in
the numerical modelling of non-isothermal problems, its initial value should be carefully
selected as it may have a significant influence on the predicted thermally induced pore fluid
pressure.
A further investigation was conducted which adopted constant values of the linear thermal
expansion coefficient of the pore fluid in the analysis. Two values, i.e. α T , f=1.0×10−4 m/(m K)
and αT , f=0.81×10−4 m/(m K), were employed, which correspond to the temperatures of 30°C
and 25°C respectively. It can be seen from Fig. 21 that although the analysis with a constant
α T , f=1.0×10−4 m/(m K) predicted similar peak excess pore pressures at the canister surface
compared to the analysis which accounts for the temperature dependent behaviour of αT , f , a
higher peak excess pore pressure was found at the transducer. In contrast, similar maximum
excess pore pressure was observed at the transducer when a constant value of
α T , f=0.81×10−4 m/(m K) was adopted, while it was underestimated at the canister surface.
Therefore, although it may be possible to capture the generation of excess pore pressure at a
single point employing an appropriately determined constant value of α T , f , it is not always
possible to obtain a good estimate of excess pore pressures overall in the FE mesh.
21
Conclusions
This paper briefly presented the numerical facilities necessary for modelling the transient
behaviour of thermally induced pore fluid pressure. The THM formulation takes into account
the non-linear temperature-dependent behaviour of both the soil permeability and the thermal
expansion coefficient of the pore fluid, commonly ignored in geotechnical analysis. To
demonstrate the importance of accounting for the non-linear temperature dependent behaviour
of both the soil permeability and the thermal expansion coefficient of the pore fluid and to help
understand the behaviour of the thermally induced pore fluid pressure, FE analyses of existing
triaxial and centrifuge heating tests on Kaolin clay were carried out. The key conclusions can be
summarised as follows:
(1) An existing analytical solution to the problem of elastic consolidation around the cylindrical
heat source was used to verify the adopted THM formulation. An excellent match was found
between the analytical solution and numerical predictions, demonstrating the capabilities of the
coupled THM finite element formulation in the simulation of thermally induced pore fluid
pressures.
(2) The numerical analyses of drained and undrained triaxial heating tests were repeated
several times with constant, but different, values of the thermal expansion coefficient of the
pore fluid and soil permeability, as well as with different combinations of one of the two
parameters varying with temperature. Consistently good agreement with the experimental data
was obtained only when the variation of both parameters with temperature was accounted for.
(3) Further investigation was carried out showing that the generation of excess pore fluid
pressures in the triaxial heating test is a consequence of both thermo-mechanical volumetric
change of the soil and the fact that the thermal expansion coefficients of the pore fluid and the
soil particle are different. Additional analyses were also performed with different end restraints
demonstrating the significance of end effects in a triaxial heating test.
22
(4) The analyses of the centrifuge test, also repeated several times with different combinations
of one of the two parameters varying with temperature, demonstrated that when the
temperature-dependent behaviour of the soil permeability is taken into account, it is essential
to appropriately determine its initial value at room temperature, as it may have a significant
influence on the predicted thermally induced pore fluid pressures. Furthermore, although it is
possible to recover the peak value of the excess pore fluid pressure with an appropriately
determined constant value of the thermal expansion coefficient of the pore fluid, α T , f , in the
single element modelling of a heating test, or at a single point in boundary value problems,
adopting a constant value of α T , f is not appropriate in the modelling of a boundary value
problem where the excess pore water pressures generated are not expected to be uniform.
Acknowledgements
The research presented in this paper was funded by the post-doctoral Fellowship from the
Geotechnical Consulting Group (GCG) in the UK.
Appendix
Substituting n = e/(1+e), where e is the void ratio, and dV = (1+e)dVs , where dVs is the
infinitesimal volume of the soil particles, into Eq. ( 1 ) yields:
∂ (e ρ f dV s )∂ t
+[∇ ∙ (ρf v f )−ρfQ f ](1+e)dV s=0 ( A-1 )
Under isothermal conditions, dVs is generally assumed to be constant in the analysis, regardless
of the change in effective stresses. Under non-isothermal conditions, however, dVs is
temperature dependent and can be written as:
d V s=(1+ε vT)dV s0 ( A-2 )
where dVs0 is the initial infinitesimal volume of the soil particles and ε vT is the thermal
volumetric strain of the soil particle, which is generally assumed to be equal to that of the soil
skeleton (Campanella & Mitchell, 1968). It is noted that dVs0 is assumed to be constant here,
23
which is different from Thomas et al. (2009) who assume that dVs is constant for a coupled THM
problem. Substituting Eq. (A-2) into Eq. (A-1) and eliminating dVs0 leads to:
∂ [e ρ f (1+ε vT)]∂t
1(1+e )(1+ε vT)
+[∇ ∙ (ρf v f )−ρfQ f ]=0 (A-3)
The pore fluid density can be expressed by a function of temperature, T, and pore pressure, pf, as
(Fernandez, 1972):
ρ f=ρ f 0exp[−1K f( pf−pf 0 )−3α T , f (T−T 0)] (A-4)
where ρ f 0 is the reference pore fluid density under the corresponding reference pore pressure
pf 0 and reference temperature T 0, K f is the bulk modulus of the pore fluid and α T , f is the linear
thermal expansion coefficient of pore fluid. Differentiating Eq. (A-4) with respect to time yields:
∂ ρf∂ t
=ρ f [−1K f
∂ p f∂ t
−3αT , f∂T∂ t ] (A-5)
Noting that ∆ εvT=3α T∆T where αT is the linear thermal expansion coefficient of the soil
skeleton, substituting Eq. (A-5) into Eq. (A-3) yields:
ρ f(1+e )
∂e∂t
+ ρf [−nK f
∂ pf∂t
−3nαT ,f∂T∂ t ]+ρ f 3nαT(1+εvT )
∂T∂ t
+[∇ ∙ (ρf v f )−ρfQ f ]=0(A-6)
Assuming that the changes in void ratio are only a result of the mechanical volumetric strain (i.e.
(ε¿¿v−εvT )¿) and ignoring the effect of pore fluid buoyancy (v fT∇ ∙(ρ¿¿ f )¿, Eq. (A-6) can be
further derived as:
ρ f∂(εv−εvT )
∂ t−ρf
nK f
∂ pf∂ t
+ρ f [ 3nαT(1+εvT )−3nαT , f ] ∂T∂ t +ρf ∇ ∙ vf−ρfQ
f=0 (A-7)
Assuming that 1+ε vT≈1 and eliminating ρ f lead to:
24
∇ ∙ v f−nK f
∂ p f∂ t
+3n (α T−αT ,f ) ∂T∂t −Q f=−∂ (ε v−ε vT)
∂ t(A-8)
ReferencesAbed, A. A. & Sołowski, W. T. (2017) A study on how to couple thermo-hydro-mechanical
behaviour of unsaturated soils: Physical equations, numerical implementation and
examples. Computers and Geotechnics, 92 132-155.
Abuel-Naga, H. M., Bergado, D. T. & Bouazza, A. (2007a) Thermally induced volume change and
excess pore water pressure of soft Bangkok clay. Engineering Geology, 89 (1-2), 144-154.
Abuel-Naga, H. M., Bergado, D. T., Bouazza, A. & Ramana, G. V. (2007b) Volume change behaviour
of saturated clays under drained heating conditions: Experimental results and
constitutive modeling. Canadian Geotechnical Journal, 44 (8), 942-956.
Agar, J. G., Morgenstern, N. R. & Scott, J. D. (1986) Thermal expansion and pore pressure
generation in oil sands. Canadian Geotechnical Journal, 23 (3), 327-333.
Al-Shemmeri, T. (2012) Engineering Fluid Mechanics. Ventus Publishing ApS.
Al-Tabbaa, A. & Wood, D. M. (1987) Some measurements of the permeability of kaolin.
Géotechnique, 37 (4), 499-514.
Baldi, G., Hueckel, T., Peano, A. & Pellegrini, R. (1991) Developments in modelling of thermo-
hydro-geomechanical behaviour of Boom clay and clay-based buffer materials.
Commission of the European Communities. Report number: EUR 13365/2.
Booker, J. R. & Savvidou, C. (1985) Consolidation around a point heat source. International
Journal for Numerical and Analytical Methods in Geomechanics, 9 (2), 173-184.
Britto, A. M., Savvidou, C., Gunn, M. J. & Booker, J. R. (1992) Finite element analysis of the
coupled heat flow and consolidation around hot buried objects. Soils and Foundations,
32 (1), 13-25.
Britto, A. M., Savvidou, C., Maddocks, D. V., Gunn, M. J. & Booker, J. R. (1989) Numerical and
centrifuge modelling of coupled heat flow and consolidation around hot cylinders buried
in clay. Géotechnique, 39 (1), 13-25.
25
Campanella, R. G. & Mitchell, J. K. (1968) Influence of temperature variations on soil behaviour.
ASCE Journal Soil Mechanics and Foundation Engineering Division, 4 (3), 709-734.
Çengel, Y. A. & Ghajar, A. J. (2011) Heat and Mass Transfer: Fundamentals and Applications. 4th
Edition. New York, McGraw-Hill.
Cui, W., Potts, D. M., Zdravković, L., Gawecka, K. A. & Taborda, D. M. G. (2018) An alternative
coupled thermo-hydro-mechanical finite element formulation for fully saturated soils.
Computers and Geotechnics, 94 22-30.
Delage, P., Sultan, N. & Cui, Y. J. (2000) On the thermal consolidation of Boom clay. Canadian
Geotechnical Journal, 37 (2), 343-354.
Fernandez, R. T. (1972) Natural convection from cylinders buried in porous media. PhD thesis.
University of California.
François, B., Laloui, L. & Laurent, C. (2009) Thermo-hydro-mechanical simulation of ATLAS in
situ large scale test in Boom Clay. Computers and Geotechnics, 36 (4), 626-640.
Gens, A. (2010) Soil–environment interactions in geotechnical engineering. Géotechnique, 60 (1),
3-74.
Gens, A., Vaunat, J., Garitte, B. & Wileveau, Y. (2007) In situ behaviour of a stiff layered clay
subject to thermal loading: observations and interpretation. Geotechnique, 57 (2), 207-
28.
Hueckel, T. & Pellegrini, R. (1992) Effective stress and water pressure in saturated clays during
heating–cooling cycles. Canadian Geotechnical Journal, 29 (6), 1095-1102.
Kutter, B. L. (1992) Dynamic centrifuge modeling of geotechnical structures. Transportation
Research Record 1336, 89 24-30.
Laloui, L. & Francois, B. (2009) ACMEG-T: Soil thermoplasticity model. Journal of Engineering
Mechanics, 135 (9), 932-944.
Lewis, R. W., Majorana, C. E. & Schrefler, B. A. (1986) A coupled finite element model for the
consolidation of nonisothermal elastoplastic porous media. Transport in Porous Media, 1
(2), 155-178.
26
Lewis, R. W. & Schrefler, B. A. (1998) The finite element method in the static and dynamic
deformation and consolidation of porous media. John Wiley.
Lima, A., Romero, E., Gens, A., Muñoz, J. & Li, X. L. (2010) Heating pulse tests under constant
volume on Boom clay. Journal of Rock Mechanics and Geotechnical Engineering, 2 (2),
124-128.
Maddocks, D. V. & Savvidou, C. (1984) The effects of heat transfer from a hot penetrator
installed in the ocean bed. In: Craig, W. H. (ed.) Proceedings of the Symposium on
Application of Centrifuge Modelling to Geotechnical Design, Manchester, Balkema. pp.
336-355.
Mohajerani, M., Delage, P., Sulem, J., Monfared, M., Tang, A. M. & Gatmiri, B. (2012) A laboratory
investigation of thermally induced pore pressures in the Callovo-Oxfordian claystone.
International Journal of Rock Mechanics and Mining Sciences, 52 112-121.
Potts, D. M. & Zdravković, L. (1999) Finite Element Analysis in Geotechnical Engineering: Theory.
London, Thomas Telford.
Potts, D. M. & Zdravković, L. (2001) Finite Element Analysis in Geotechnical Engineering:
Application. London, Thomas Telford.
Savvidou, C. & Britto, A. M. (1995) Numerical and experimental investigation of thermally
induced effects in saturated clay. Soils and Foundations, 35 (1), 37-44.
Seneviratne, H. N., Carter, J. P. & Booker, J. R. (1994) Analysis of fully coupled thermomechanical
behaviour around a rigid cylindrical heat source buried in clay. International Journal for
Numerical and Analytical Methods in Geomechanics, 18 (3), 177-203.
Thomas, H. R., Cleall, P., Li, Y.-C., Harris, C. & Kern-Luetschg, M. (2009) Modelling of cryogenic
processes in permafrost and seasonally frozen soils. Géotechnique, 59 (3), 173-184.
Thomas, H. R. & He, Y. (1997) A coupled heat–moisture transfer theory for deformable
unsaturated soil and its algorithmic implementation. International Journal for Numerical
Methods in Engineering, 40 (18), 3421-3441.
27
Vaziri, H. H. (1996) Theory and application of a fully coupled thermo-hydro-mechanical finite
element model. Computers & Structures, 61 (1), 131-146.
Vaziri, H. H. & Byrne, P. M. (1990) Numerical analysis of oil sand under nonisothermal
conditions. Canadian Geotechnical Journal, 27 (6), 802-812.
28
List of figure captions
Fig. 1. Variation of linear thermal expansion coefficient of pore water with temperature
Fig. 2. Finite element mesh for modelling consolidation around a cylindrical heat source
(l0/r0=10.0)
Fig. 3. Variation of normalised temperature with time (l0/r0=10.0 for numerical results in this
study)
Fig. 4. Variation of normalised pore fluid pressure with time (l0/r0=10.0 for numerical results in
this study)
Fig. 5. Variation of normalised temperature with time for different l0/r0 at point C
Fig. 6. Variation of normalised pore fluid pressure with time for different l0/r0 at point C
Fig. 7. Geometry for modelling triaxial heating tests
Fig. 8. Prescribed temperature boundary conditions at boundary 2-3 for modelling triaxial
heating tests: (a) Undrained test; (b) Drained test
Fig. 9. Variation of permeability with temperature in modelling of triaxial heating tests
Fig. 10. Comparison between numerical predictions and experimental results for the undrained
triaxial heating test: (a) Position A; (b) Position B
Fig. 11. Comparison in excess pore fluid pressure distributions along the radial axis with
different initial values of permeability at t=3.5min
Fig. 12. Comparison between numerical predictions at Position B with different initial values of
permeability for modelling undrained triaxial heating test
Fig. 13. Comparison between numerical predictions with different αT , f for modelling undrained
triaxial heating test: (a) Position A; (b) Position B
Fig. 14. Comparison between numerical predictions with smooth and rough ends for modelling
undrained triaxial heating test: (a) Position A; (b) Position B
29
Fig. 15. Comparison between numerical predictions and experimental results for the drained
triaxial heating test: (a) Position A; (b) Position B
Fig. 16. Comparison between numerical predictions with constant and variable k f for modelling
drained triaxial heating test: (a) Position A; (b) Position B
Fig. 17. Finite element mesh for modelling a centrifuge test
Fig. 18. Comparison in temperature evolution between numerical predictions and experimental
measurements for the centrifuge test
Fig. 19. Comparison in the evolution of excess pore fluid pressure between numerical
predictions and experimental measurements for the centrifuge test
Fig. 20. Comparison between numerical predictions with different k f 0 for modelling centrifuge
test
Fig. 21. Comparison between numerical predictions with constant and variable αT . f for
modelling centrifuge test at: (a) canister surface; (b) transducer
30
Figure 1
31
Figure 2
32
Figure 3
33
Figure 4
34
Figure 5
35
Figure 6
36
Figure 7
37
Figure 8
38
Figure 9
39
Figure 10(a)
Figure 10(b)
40
Figure 11
41
Figure 12
42
Figure 13
43
(a) (b)
Figure 14
44
Figure 15 (a)
Figure 15(b)
45
Figure 16
46
Figure 17
47
Figure 18
48
Figure 19
49
Figure 20
50
(a) (b)
Figure 21
51
Table 1. Material properties for modelling consolidation around a cylindrical heat source
Young’s modulus, E (Pa) 6.0×103
Poisson ratio, ν (-) 0.4
Permeability, kf (m/s) 3.92×10-5
Initial void ratio, e0 (-) 1.0
ρsCps , ρfCpf (kJ/m3 K) 167.2
Thermal conductivity kT (kJ/m s K) 4.3
Thermal expansion coefficient αT (m/m K) 3.0×10-7
Thermal expansion coefficient of pore fluid αT,f (m/m
K)
2.1×10-6
52
Table 2. Material properties for modelling triaxial heating tests
Thermal and thermo-
mechanical properties
Linear thermal expansion coefficient of soil
skeleton, αT(m/(m K)) 2.0×10−5
Linear thermal expansion coefficient of water, α T , f(m/(m K)) From Eq. ( 7 )
Density of water, ρ f(kg/m3) 1000
Density of soil particles, ρ s(kg/m3) 2610
Specific heat capacity of water, C pf (kJ/(kg K)) 4.2
Specific heat capacity of soil particles, C ps(kJ/(kg
K)) 0.94
Thermal conductivity, kT (kJ/(s m K)) 1.26×10−3
Hydraulic properties Permeability, k f at room temperature (m/s) (from
Al-Tabbaa & Wood (1987)) 1.5×10−9
Mechanical properties Slope of the compression line, λ 0.2
(Modified Cam-clay) Slope of the swelling line, κ 0.03
Angle of friction, φ 23°
Poisson’s ratio, υ 0.25
Specific volume at 1 kPa, v 3.272
53
Table 3 Scaling law for modelling centrifuge test
Quantity Scaling law
Length N
Volume N3
Stress 1
Time (Dynamic) N
Time (Diffusion) N2
54
Table 4 Material properties for modelling the centrifuge test
Material properties Kaolin clay Sand
Thermal properties
Linear thermal expansion coefficient
of soil skeleton, α T(m/(m K))2.0×10−5 2.0×10−5
Linear thermal expansion coefficient
of water, αT , f (m/(m K))From Eq. (7) From Eq. (7)
Density of water, ρ f(kg/m3) 1000 1000
Density of soil particles, ρ s(kg/m3) 2610 2650
Specific heat capacity of water, C pf
(kJ/(kg K))4.2 4.2
Specific heat capacity of soil particles,
C ps(kJ/(kg K))0.94 0.83
Thermal conductivity, kT (kJ/(s m K)) 1.5×10−3 2.0×10−3
Hydraulic properties Permeability, k f at room temperature
(m/s) (from Al-tabbaa & Wood
(1987))
1.50×10−9 1.0×10−5
Mechanical properties Slope of the compression line, λ 0.25
(Modified Cam-clay) Slope of the swelling line, κ 0.05
Angle of friction, φ 23°
Poisson’s ratio, υ 0.25
Specific volume at 1 kPa, v 3.58
Initial in situ state Earth pressure coefficient, Ko 0.69
Saturated specific weight of soil, γsat
(kN/m3)16.7
specific weight of water, γw (kN/m3) 9.81
55
Table 5 Time marching scheme
Increments Time-step size in the test (s) Scaled time-step size in the prototype modelling
(s)
1-100 1.0 1.0×104
101-200 10.0 1.0×105
201-1000 100.0 1.0×106
56