THM coupling
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]
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
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 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 (m/s) and
drained ( 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
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, , and the thermal expansion coefficient
of the pore fluid, are taken into account, with their values
updated using the value of the current temperature, , 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 and . 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 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 problemHydraulic
governing formulation
For a fully saturated soil, applying the principle of mass
conservation for the fluid phase leads to the following
expression:
( 1 )
where is the density of the pore fluid, represents the vector of
the seepage velocity, is the symbol of divergence defined as , is
an infinitesimal volume of the soil, n is porosity, 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:
( 2 )
where is the bulk modulus of the pore fluid, and are the linear
thermal expansion coefficients of the soil skeleton and the pore
fluid respectively, T is temperature, is the total volumetric
strain and 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. ( 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. ) is missing in their final form of the hydraulic
governing equation due to the fact that and are assumed to be
constant in their derivation.
Adopting the generalised Darcy’s law leads to the expression of
the seepage velocity in Eq. ( 2 ) as:
( 3 )
where is the permeability matrix of the soil, represents the
gradient of pore fluid pressure, the vector T = {iGx iGy iGz} is
the unit vector parallel, but in the opposite direction, to
gravity, and is the specific weight of the pore fluid. can be
further expressed as:
( 4 )
where represents gravity, is the intrinsic permeability and is
the pore fluid viscosity, which varies with temperature change
under non-isothermal conditions. For a soil saturated with water,
the expression of can be approximated by (Al-Shemmeri, 2012):
( 5 )
where 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:
( 6 )
where is a reference permeability matrix at the temperature of ,
and if T is defined in degrees Celsius.
In a heating test on soil, both and 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 m/(m K) to m/(m K) in the temperature range of
10 - 100°C) in the linear thermal expansion coefficient of pore
water, , 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, ,
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 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:
( 7 )
Thermal governing formulation
Adopting the law of energy conservation gives the governing
equation of heat transfer in a fully saturated soil as:
( 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, is the thermal
conductivity matrix and QT represents any heat source and/or sink.
The first term in Eq. ( 8 ) denotes the 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:
( 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:
( 10 )
where . Applying the principle of effective stress, the total
stress can therefore be given as:
( 11 )
where and 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:
( 12 )
where , and 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, , and
the linear thermal expansion coefficient of the pore fluid, , 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 exerciseNumerical 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.
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 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:
( 13 )
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 testsExperimental 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 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 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 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 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 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. , , ,
and ), 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 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, , is not known. Triaxial cooling tests (where a linear
elastic volumetric behaviour is observed) on Soft Bangkok clay have
yielded a value of m/(m K) (Abuel-Naga et al., 2007b) and on Boom
clay a value of m/(m K) (Baldi et al., 1991). A value of 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, , 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, , were reported by Savvidou
& Britto (1995) for the undrained ( m/s) and drained ( 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 to m/s. For simplicity, an average value of
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 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 resultsUndrained 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 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 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 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 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 ( 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.
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 m/(m K) (corresponding to the value of 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 (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. m/s, was employed throughout the 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 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
testsExperimental 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 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 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 and a Poisson’s ratio 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.
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 , where represents the depth from the top boundary of
the mesh, and and are the specific unit weight of water and
saturated soil respectively. A value of 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 adopting a
value of , 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 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. 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 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 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. m/(m K) and 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 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 , 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 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 , it is not always possible to obtain a good estimate of excess
pore pressures overall in the FE mesh.
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.
(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, , in the single element modelling of a heating test, or at a
single point in boundary value problems, adopting a constant value
of 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:
( 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:
( A-2 )
where dVs0 is the initial infinitesimal volume of the soil
particles and 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, 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:
(A-3)
The pore fluid density can be expressed by a function of
temperature, T, and pore pressure, pf, as (Fernandez, 1972):
(A-4)
where is the reference pore fluid density under the
corresponding reference pore pressure and reference temperature ,
is the bulk modulus of the pore fluid and is the linear thermal
expansion coefficient of pore fluid. Differentiating Eq. (A-4) with
respect to time yields:
(A-5)
Noting that where is the linear thermal expansion coefficient of
the soil skeleton, substituting Eq. (A-5) into Eq. (A-3)
yields:
(A-6)
Assuming that the changes in void ratio are only a result of the
mechanical volumetric strain (i.e. ) and ignoring the effect of
pore fluid buoyancy (, Eq. (A-6) can be further derived as:
(A-7)
Assuming that and eliminating lead to:
(A-8)
References
Abed, 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.
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.
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.
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.
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
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
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 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
for modelling centrifuge test
Fig. 21. Comparison between numerical predictions with constant
and variable for modelling centrifuge test at: (a) canister
surface; (b) transducer
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10(a)
Figure 10(b)
Figure 11
Figure 12
Figure 13
(a) (b)
Figure 14
Figure 15 (a)
Figure 15(b)
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
(a) (b)
Figure 21
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
Table 2. Material properties for modelling triaxial heating
tests
Thermal and thermo-mechanical properties
Linear thermal expansion coefficient of soil skeleton, (m/(m
K))
Linear thermal expansion coefficient of water, (m/(m K))
From Eq. Error! Reference source not found.
Density of water, (kg/m3)
1000
Density of soil particles, (kg/m3)
2610
Specific heat capacity of water, (kJ/(kg K))
4.2
Specific heat capacity of soil particles, (kJ/(kg K))
0.94
Thermal conductivity, (kJ/(s m K))
Hydraulic properties
Permeability, at room temperature (m/s) (from Al-Tabbaa &
Wood (1987))
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
Table 3 Scaling law for modelling centrifuge test
Quantity
Scaling law
Length
N
Volume
N3
Stress
1
Time (Dynamic)
N
Time (Diffusion)
N2
Table 4 Material properties for modelling the centrifuge
test
Material properties
Kaolin clay
Sand
Thermal properties
Linear thermal expansion coefficient of soil skeleton, (m/(m
K))
Linear thermal expansion coefficient of water, (m/(m K))
From Eq. (7)
From Eq. (7)
Density of water, (kg/m3)
1000
1000
Density of soil particles, (kg/m3)
2610
2650
Specific heat capacity of water, (kJ/(kg K))
4.2
4.2
Specific heat capacity of soil particles, (kJ/(kg K))
0.94
0.83
Thermal conductivity, (kJ/(s m K))
Hydraulic properties
Permeability, at room temperature (m/s) (from Al-tabbaa &
Wood (1987))
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
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
101-200
10.0
201-1000
100.0
0
21
