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Simplified 3D Numerical Method to Simulate Thawing Permafrost:
Validation and Case Study
Marianna Loli, PhD, Grid Engineers, Pampouki 3, N. Psychiko, Greece, mariannaloli@yahoo.gr
Angelos Tsatsis, PhD, Grid Engineers, Pampouki 3, N. Psychiko, Greece, ag_tsa@yahoo.gr
Rallis Kourkoulis, PhD, Grid Engineers, Pampouki 3, N. Psychiko, Greece, rallisko@gridengineers.com
Ioannis Anastasopoulos, professor, ETH Zürich, Switzerland, ioannis.anastasopoulos@igt.baug.ethz.ch
Abstract
Thawing weakens the soil supporting buildings and lifelines from Siberia to Alaska, while the
consequent consolidation settlements may reach up to hundreds of millimetres during a single
season. A simplified sequentially–coupled numerical approach is presented and implemented
in a general purpose commercially available FE code. It is intended to provide a reasonably
accurate computational tool for the analysis of structures on degrading permafrost, allowing
modelling of complex 3D geometries and boundary/loading conditions. The method permits
simulation of the temperature dependent thermal and mechanical properties of soils in an
approximate manner. It accommodates large strain consolidation theory, allowing also use of
plasticity constitutive relationships. A thorough validation study is presented involving
comparison with: (a) monotonic and cyclic thawconsolidation element tests; (b) analytical
solutions; and (c) a welldocumented case study of an unstable roadway embankment. The
latter is complemented with an investigation of thawsettlement remediation solutions using
thermosyphons with emphasis on 3D response and their transverse spacing.
Keywords: consolidation; settlement; numerical modelling; snow, ice & frost
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List of Symbols
A
temperature amplitude
Sm
specific heat of the mixture ( soilair
water/ice)
Ci
conductivity of ice
Smax
ultimate settlement (after consolidation)
ck
hydraulic conductivity change index
Sr
degree of saturation
Cm
conductivity of the mixture ( soilair
water/ice)
Ss
specific heat of solids
Cmdry
conductivity of dry soil
St
settlement at the end of thawing
Cmsat
conductivity of fully saturated soil
Sw
specific heat of water
Cs
conductivity of solids
T
temperature
cv
coefficient of consolidation
T0
average temperature
Cw
conductivity of water
V
total volume
e
void ratio
Va
volume of air
e0
initial (t = 0) void ratio
Vs
volume of solids
Gs
specific gravity
Vw
volume of water
h
soil height
w
water content in thawed soil
k
permeability
X
depth of thaw plane
k0
initial thawed permeability
ys
thermosyphon spacing
Lm
latent heat of the mixture ( soilair
water/ice)
α
thermal constant
Lw
latent heat of water
γ
unit weight
m
total mass
Δu
excess pore water pressure
M
slope of the critical state line in the qp
space
Δu0
excess pore water pressure at the thaw
front
ms
mass of solids
θi
volumetric fraction of ice
mw
mass of water
θs
volumetric fraction of solids
mα
mass of air
θu
volumetric fraction of unfrozen water
p
mean stress
θw
volumetric fraction of water
P0
overburden stress
κ
compression index
p0’
initial (t = 0) effective stress
ν
Poisson’s ratio
q
deviatoric stress
ρm
density of the (thermally equivalent)
mixture
R
thawconsolidation ratio
ρs
bulk dry density of soil
s
settlement
ρw
density of water
τ
freezethaw period
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1. Introduction
Resource and transport development around the Arctic have been thriving since the second
half of the 20th century despite constantly facing the variety of challenges associated with
permafrost engineering. Construction on frozen soil generally suffers from differential ground
movements. Being the natural result of the periodic freeze–thaw response of the active layer,
they are often exaggerated by the locally increased average ground temperatures due to
human (construction/service) activity. In fact, even in the free field (in the absence of
structures), active layer movements can reach up to 200 mm during a single season solely due
to the concurrent actions of water phase change, water migration, and the soil weakening
associated with freezing and thawing [Andersland & Ladanyi, 1994]. The effect can be
cumulative through the years and is frequently evidenced by massive slope failures, such as
the one shown in Fig. 1a, in cases where the phenomenon is aggravated by even moderate
ground inclinations.
Soils may have dramatically different mechanical properties depending on their thermal
state. Continually frozen ground can provide excellent bearing capacity for the support of
structural loads. Yet, seasonal cycles of freezing–thawing and the consequent redistribution
of pore water may impose differential foundation movements leading to permanent
distortion and severe damage of structures (Fig. 1b). Numerous examples of structural
failures, ranging from fracturing to partial collapse, have been reported in cities above the
Arctic [Goldman, 2002].
When frozen, soils tend to be stronger thanks to the pore ice serving as a bonding agent
for the particles. Upon thawing, water is liberated, generating pore pressures, and the soil
skeleton weakens adapting to a new equilibrium void ratio. Settlement comes as a result of
consolidation (drainage of excess pore water) and icetowater phase change (ΔV ≈ 9%).
Thawing of icerich soils at a fast enough rate (in comparison to draining time) can cause pore
pressure buildup and weaken the soil to such an extent, that even relatively lightweight
structures (e.g., pavements or pipelines) may be subjected to bearing capacity failure. Owing
to their continuity and significant length, lifelines are susceptible to differential displacements;
thus, the associated failure patterns, cracking and/or buckling, have been commonplace all
over the Arctic (Figs. 1c and 1d).
The importance of having models that delineate soil response to freezing and thawing
is reflected on the tremendous amount of relevant research studies. Advanced constitutive
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relationships have been developed to describe, with differing degrees of sophistication, the
interrelated thermal–hydraulic–mechanical (THM) processes that govern the response of
porous materials under frost actions. They can be categorized as: rigidice models [O’Neill &
Miller, 1985; Nixon, 1991], hydrodynamic models [Harlan, 1973; Guymon & Luthin, 1974;
Jame & Norum, 1980]; semiempirical models [Han & Goodings, 2006]; fullycoupled THM
formulations [Li et al. 2000; 2002; Nishimura et al., 2009; Thomas et al., 2009; Gens, 2010; Liu
& Yu, 2011; 2014; Zhang & Michalowski, 2015]; and semicoupled THM formulations
[Selvadurai et al., 1999].
THM models have been primarily implemented in studies of frost heaving effects, while
comparatively few studies have addressed thawing of frozen soils and the consequent thaw
consolidation settlements. Valuable insight as to the physics of the latter is gained by
considering the onedimensional (1D) theoretical formulation of Morgenstern & Nixon [1971].
Merging heat conduction theory with Terzaghi’s linear consolidation theory, they produced a
closed form solution to the idealized moving boundary consolidation problem where a
thawing front propagates within a semiinfinite homogeneous mass of frozen soil due to a
step increase in the surface temperature. Recently, Dumais & Konrad [2018] reported a
theoretical model for coupled 1D thaw consolidation in view of large strain soil response.
Additionally, numerical methods using finite element (FE) or finite difference (FD) schemes
have been utilized to account for more complex geometrical boundaries, temperature
dependent material properties and soil plasticity [Foriero & Ladanyi, 1995; Xu et al. 2009; Qi
et al., 2012; Yao et al., 2012; Zhang & Michalowski, 2015].
While cognizant of the reciprocal coupling between the thermal–hydraulic–mechanical
processes involved in the freezing and thawing of soil, this paper proposes a simplified
sequentiallycoupled numerical method. This method is suitable for simulation of problems
which are dominated by thaw consolidation (degrading permafrost) and it is bound by
considerable limitations (discussed in the respective section of this paper). Nevertheless,
having been implemented in the general purpose Abaqus FE package, where 3D geometries
and complex loading/boundary conditions can be easily accommodated, it may be particularly
useful in cases where consideration of such attributes is important. After presentation of an
extensive validation study involving (i) laboratory experiments; (ii) theoretical predictions; and
(iii) a welldocumented case study of an excessively settling roadway embankment in Canada,
the paper presents an example where appropriate consideration of outofplane response is
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critical. Studying the potentials of a remediation solution using thermosyphons against thaw
degradation, it highlights the critical effect of their transverse (outofplane) spacing on the
rate of permafrost degradation and, most importantly, on the deformation performance of
the soil surface.
2. Formulation of Numerical Method
Computationally, the proposed approach is based on the sequentially coupled algorithm
described by the flow chart of Fig. 2. Thermal (step 1) and fluid–mechanical (step 2) equations
are solved independently, through two different analytical steps, with temperature field
output incrementally passed from the first step to the second. This is a simplification of actual
soil behaviour in thermal consolidation problems, which could be better approximated
through full coupling between temperature, hydraulic pressure and mechanical deformation.
Yet, such sequentially coupled models have significant advantages in terms of computational
efficiency, while providing reasonably accurate results for engineering purposes as indicated
by the following validation study.
All of the calculations were carried out using the Abaqus (v. 6.16) program (Dassault Systèmes,
2016).
Step 1: Heat Transfer Analysis
Neglecting coupling between mechanical and thermal processes, the first step of the analysis
considers heat transfer exclusively due to conduction with temperature (state) dependent
conductivity and phase change (latent heat effect). The complex, multiphase solidliquidgas
material is here reduced to an idealised solid continuum (mixture) of virtually equivalent
thermal properties. Consideration of energy balance in 3 dimensions yields the governing
equation:
(1)
where, T is temperature; Q represents the applied boundary heat flux; ρm, Sm, and Cm stand
for the mass density, the specific heat and the conductivity of the idealised soil mixture,
respectively; and Lm is the total internal energy released or absorbed during phase change of
a unit volume.
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Assuming a partially saturated mass of soil with known void ratio (e) and water content
(w), the total volume V is the sum of the volume of solids (Vs), the volume of water (Vw), and
the volume of air (Va). Likewise, the total mass m is the sum of ms, mw, and mα. The thermal
properties of the idealised equivalent continuum are calculated with respect to the properties
of its constituents on the basis of the assumptions detailed below.
Bulk density
Neglecting the very small contribution of air in the heat transfer process, the bulk density ρm
of a thermally equivalent continuum can be calculated as:
(2)
where ρs is the bulk dry density of soil.
Conductivity
Denotes the amount of heat transferred through a unit area of substance in unit time under a
unit temperature gradient. As such, it increases with increasing soil density, the water content
or the degree of saturation. Furthermore, it is undeniably influenced by temperature,
particularly with respect to the amount, phase and condition of pore water.
The herein presented simulations have conveniently made use of available
measurements of thermal conductivities from laboratory tests conducted on frozen and
unfrozen samples of the involved soil materials (referenced in the respective sections). Yet, in
absence of such data, it is possible to estimate this parameter with respect to empirical
relationships available in the literature [Johansen, 1975; Farouki, 1981; 1982; Côté and
Konrad, 2005]. The normalized thermal conductivity concept proposed by Johansen [1975]
correlates the thermal conductivity of a partially saturated mixture (Cm) with the limit
conductivity values corresponding to dry and fullysaturated conditions (Cmdry and Cmsat,
respectively)
(3)
where kr, Cmsat, and Cmdry can be estimated, depending on porosity, the degree of saturation,
and soil particle characteristics thanks to the generalized framework proposed by Côté and
Konrad [2005].
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The effect of the water state (and hence temperature) arises owing to Cmsat being
dependent on the volumetric fractions of water and ice. Specifically it is defined as:
(4)
where Cs is the conductivity of the solids; Ci is the conductivity of ice (equal to 2.24 W/m°C);
and Cw is the conductivity of water (0.6 W/m°C); while the exponents θ represent the
volumetric fractions of the three soil constituents.
The volumetric fraction of the solid particles is only dependent on the void ratio
(5)
and the volumetric fraction of ice is given as
(6)
where, importantly, θw depends on temperature:
(7)
θu refers to the portion of water within the soil that remains unfrozen even at temperatures
below the water freezing point. This portion may be considerable, especially in frost
susceptible materials such as clays and silts. Based on experimental evidence, past studies
have proposed nonlinear relationships correlating the unfrozen water content with
temperature [Nixon, 1991; Michalowski, 1993]. Yet, to avoid excessive computational cost,
the herein presented calculations have adopted a simplified approach assuming a linear phase
composition law:
(8)
where Tf is a temperature level below which the entire quantity of water is assumed frozen
(i.e. for T ≤ Tf, θu = 0).
The following simulations have postulated a constant value for Tf, namely 0.5 °C. Hence,
conductivity measurements from laboratory tests on frozen samples are assumed descriptive
of the soil behaviour for T ≤ 0.5 °C. Measurements from thawed samples are used for T > 0
°C and linear interpolation is implemented in the intermediate temperature regime.
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Mass specific heat
Specific heat is defined as the ratio of the heat capacity of soil (i.e., the heat required to raise
the temperature of a unit mass by 1oC) to the heat capacity of water. It may be computed in
view of the contributions of the different constituents as:
(9)
where Ss, Si and Sw are the specific heats of solids, ice and water, respectively. Eq. (9) can be
rewritten as:
(10)
As previously, linear interpolation is assumed for the temperature range: 0≤ T < Tf , where Tf
has been assumed equal to 0.5 °C.
Latent heat
Latent heat determines the amount of energy absorbed or released during the change of
phase. 333.7 kJ is the amount of heat released when 1 kg of water freezes into ice and the
same amount is absorbed when it melts (Lw = 333.7 kJ /kg). For soil, the total energy involved
in the phasechange process depends on the water content, and hence the equivalent latent
heat of the mixture (Lm) can be calculated as follows:
(11)
Finite Element (FE) modelling
Figure 3a displays the details of the FE model used for the heat transfer analyses involved in
the reproduction of laboratory element tests. Eightnoded linear brick diffusive elements
(DC3D8), which are readily available in the Abaqus library, were used and assigned the
previously discussed thermal properties. In this example, thawing occurs as a result of
applying appropriate conduction boundary conditions and the computed temperature field
(Fig. 3b) is stored to serve as input for the second step.
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Step 2: Consolidation Analysis
Effective stress analysis is carried out to simulate the hydromechanical behaviour of the
thawed soil with respect to Gibson’s [1981] consolidation theory. This step of the analysis
considers nonlinear geometry effects (largedisplacements) and hence accounts for the
variation of permeability and compressibility during consolidation. Figure 3c illustrates select
attributes of the FE mesh used in the consolidation step of the following element test
simulations.
Soil compression
Thaw consolidation test results [Morgenstern & Smith, 1973] highlight that assuming linear
elastic soil behaviour essentially underpredicts pore pressures yielding unconservative
settlement predictions. On the other hand, adoption of a nonlinear relation between void
ratio and effective stress is viewed as a better approximation of the actual physical behaviour,
especially when high water contents are considered. Volume changes under isotropic
compression in both thawed and frozen soil obey the effective stress – void ratio relationship:
(12)
where p’ is the mean effective stress, p0’ is the initial effective isotropic stress (further
discussed in the following), e0 is the initial void ratio, and κ is the compression index. The latter
varies significantly with respect to the soil state in such way that soil stiffness increases
substantially with freezing.
Researchers have proposed relationships that can be used to calculate κf with respect
to the pore ice ratio [see Zhang and Michalowski, 2013]. Due to unavailability of the soil data
needed for such modelling, this study has simply stipulated that κf = κth /10. This assumption
was based on the results of a preliminary sensitivity analysis focused on achieving general
agreement with the theoretical DoC–R relationships [Yao et al., 2012], where DoC and R stand
for the degree of consolidation and the thaw consolidation ratio, respectively, as discussed in
the following. Similarly to thermal modelling, the adopted postulation here is: κ = κth for T >
Tf, κ = κf for T ≤ Tf. Therefore, at least for the loading cases of interest, settlement is essentially
dominated by the compression of the thawed domain (see example of deformed model in Fig.
3d).
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Permeability (k)
The evolution of settlements depends on the draining rate and is therefore sensitive to
appropriately modelling permeability, especially its variation with: (i) the soil void ratio and
(ii) temperature. Regarding the former, in thawed soil permeability is defined according to
Konrad [2010]:
(13)
Where k0 is the initial permeability of the thawed soil and ck is the index of hydraulic
conductivity change. To account for temperature dependency, the permeability of the
(practically impermeable) frozen soil is assumed to be 3 orders of magnitude lower than that
of the thawed soil. Again, this assumption was based on a sensitivity analysis of the benchmark
1D thaw consolidation problem which showed that further reduction of the frozen state
permeability causes negligible variation in the results.
Initial (“0”) condition
Upon thawing, water is released and pore pressures are generated. Then, the state of the just
thawed soil is determined by three parameters, namely the initial void ratio e0, the initial
permeability k0, and the initial effective isotropic stress p0’. Also known as residual stress, the
latter is defined as the effective stress in a soil thawed under undrained conditions. Realistic
estimation of its value is critical because thaw consolidation settlement is presumably
(according to Eq. [12]) dependent on the p’– p0’ stress increment. Morgenstern and Nixon
[1971] indicated that in ice rich soils, or soils with high initial void ratio, p0’ could be as low as
zero. However, two years later the same researchers [Nixon and Morgenstern, 1973] showed
experimentally how the particular stress and thermal histories associated with the formation
of a frozen soil specimen can lead to significantly increased values of p0’. Moreover, they
developed an experimental method that allows its measurement in the laboratory.
In the absence of relevant experimental evidence, the following simulations of
monotonic and cyclic laboratory element tests (§3), where soil material behaviour is governed
by porous elasticity, have conservatively assumed the minimum value for p0’ (0.1 Pa instead
of absolute zero to avoid numerical stability issues). The implied premise is that the initial
excess pore water pressure at the thaw front, Δu0, is essentially equal to the overburden stress
P0. Likewise, the same assumption has been made for the simulation of the soil column
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response (Fig. 9) leading to Δu0 being equal to the overburden stress at the respective thaw
depth. On the other hand, the need to involve soil plasticity (according to the modified Cam
Clay model) in the case study of the embankment (§5, 7) has led to assuming a somewhat
larger value for p0’. Specifically, for any element of the frozen soil layer under the embankment
the initial stress field is p0’ = 0.1σV, where σV is the total vertical stress at any point under
geostatic conditions (prior to construction of the embankment). This assumption is in
agreement with evidence indicating that in the field residual stresses increase linearly with
depth [Nixon and Morgenstern, 1973].
It is important to note that the e0 – p0’ condition is herein, in fact, prescribed as the initial
condition for the analysis (t = 0) prior to application of loads and temperature variations.
Nevertheless, owing to the relatively low permeability and compressibility of the frozen soil
there is practically minimal variation e0 in as long as the soil remains frozen. Hence, e0 – p0’
practically constitutes the initial condition for the first thawing cycle while, naturally, quite
different may be the e – p’ condition at the onset of any subsequent thawing period depending
on the exact loading/consolidation history of every soil element.
In all of the simulations (for the 1D as well as the 3D problems considered) the initial
p0’ condition is implemented through a geostatic stress field. The unavoidable associated
assumption is that of a uniform, horizontal layer of frozen soil at t = 0. This constitutes a
limitation of the numerical methodology in its present form (see following discussion on
model limitations).
FE implementation of transition from frozen to thawed compressive behaviour
The abrupt change in the compressibility of a soil element from κf to κth at T = Tf is likely to
cause severe numerical instabilities that in most cases terminate the analysis. To avoid this
problem, the present study introduced a computational strategy that involves the concurrent
simulation of two models: (i) the model of interest and (ii) a “dummy” model of identical
geometry, which is subjected to exactly the same loads and temperature variations. In model
(ii) the permafrost material has a constant compressibility index κ = κf irrespective of T.
Likewise, in model (i) κ = κth. However, every node in the model of interest (i) is engaged to its
homologous node in model (ii) through a set of User Defined multipoint Kinematic
Constraints (UDKCs). UDKCs demand that the displacements of the former (node in model (i))
be equal to the displacements of the latter (node in model (ii)) when T ≤ Tf . As such, the model
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of interest virtually displays the desired compressibility transition: κ = κf for T ≤ Tf and κ = κth
for T > Tf.
However, it is important to note that for every node of the model, the aforementioned
UDKC relationship is cancelled the first time that temperature exceeds the Tf limit (at first
thaw). Thereafter, the specific point of the soil responds with compressibility κ = κth regardless
of the subsequent temperature history (i.e. even if it freezes again). This limitation renders
the methodology suitable only for problems where deformations are dominated by the
thawing process and freezeback effects on soil stiffness can be neglected.
By contrast, the increase in permeability from frozen to thawed state is simulated
uninterruptedly, in every freezethaw cycle. This can be crucial for the realistic estimation of
thawconsolidation settlements under periodic thermal boundaries, especially when the latter
interfere with the draining path, as discussed in the following.
Soil plasticity
When judged necessary, consolidation analysis has accounted for inelastic soil material
behaviour. In fact, such was the case in the analysis of the roadway embankment, because the
monitored settlements were high enough to betray significant plastic deformation.
The Abaqus clay plasticity model implements the modified Cam Clay [Roscoe & Burland,
1968]. The adopted elliptical yield surface in the p’–q space is:
(14)
where q is a deviatoric stress measure, and M is the slope of the critical state line. Plastic
deformation follows an associative flow rule.
3. Validation against Laboratory Element Tests
The proposed simplified sequentially–coupled (2step) approach is validated against published
laboratory element tests [Yao et al., 2016] on frozen soil specimens subjected to monotonic
and periodic (cyclic) thawing. Yao et al. [2016] used the apparatus sketched in Fig. 4 to conduct
monotonic thaw consolidation tests on samples of finegrained soil obtained along the
Qinghai–Tibet highway. From the two different groups of samples used in their study, only the
first soil type (dry unit weight γ = 16.1 kN/m3) is used for the analyses. The measured
permeability and compression properties are summarized in Fig. 5a.
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Finegrained soil samples, reconstituted into cylinders of 100 mm height and diameter,
were installed into Plexiglas tubes. They were frozen quickly at –30⁰C, keeping the tube ends
fixed to ensure no change of volume and hence no change of unit weight due to freezing.
Homogenously frozen samples were produced, without observable ice lenses formation, and
loaded axially using a servo electronic system. Temperatures of the top and bottom caps were
controlled by two cooling baths while the environmental temperature was adjusted by air
conditioning. Thawconsolidation tests were instrumented with thermistors and pore
pressure transducers measuring temperature and excess pore pressure at different depths.
Drainage was permitted only through the top cap, the settlement of which (s) was recorded
by a displacement transducer.
Monotonic Thaw Propagation
After keeping top, bottom and environmental temperatures constant at –1⁰C for at least 4
hours, a surcharge load of 100 kPa was applied and the top cap temperature was reset to
+10⁰C to impose downward thaw propagation in the specimen.
Table 1 lists the soil parameters that are useful for the heat transfer simulation with
reference to Eq. [1] – Eq. [10] as they were measured and reported by Yao et al. [2016]. The
indicated Poisson’s ratio values (νth) have been measured in thawed samples. Lacking the
relevant data for the frozen domain, the present study has assumed νf = 0.15 which is a typical
value for frozen cohesive soils under small strains [Chen et al., 2018]. The same assumption
has been made for all of the following simulations.
Analysis results are shown in Fig. 6a and compared to the temperatures measured at
two different depths in the specimen with respect to time. The agreement between analysis
and experiment is quite satisfactory. The consecutive consolidation step assumes porous
elastic soil material behaviour incorporating the, experimentally measured, k–e and p–e
relationships for thawed soil (Fig. 5a). The constitutive model parameters listed in Table 1
were estimated through appropriate (semilogarithmic) curve fitting to these data, with
reference to Eq. [12] – Eq. [13]. Figure 6b compares the analytical results to the experimental
measurements of Yao et al. [2016]. The sequentially–coupled numerical approach accurately
captures the settlement response both in terms of its residual value and accumulation rate.
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Periodic Thaw–Freeze Action
In order to have practical merit, the proposed method should be able to adequately capture
the thawconsolidation response under periodic (cyclic) sine–type thermal boundaries, which
resemble seasonal climatic changes and are therefore frequently used in the analysis of field
scale problems.
Wang et al. [2015] used the same setup (Fig. 4) and sample preparation method as Yao
et al. [2016] to investigate the response of a slightly stiffer and less permeable soil (Fig. 5b).
Downward thawconsolidation of uniformly frozen (at –1⁰C) samples was induced by
periodically varying the top plate temperature according to a sinusoidal relation, which is
typically used for climate variations:
(15)
where: T0 is the average temperature; A is temperature amplitude; and τ is the freezethaw
period. Different tests were conducted, varying parameters T0, A, and τ. Figure 7 shows
temperature time histories measured at four different depths within the sample during one
of the tests (T0 = 2.5⁰C, A = 4⁰C, and τ = 6 hours). Also plotted in the figure are the results of
the heat transfer analysis (Step 1), which is performed using the measured soil properties
summarized in Table 1. The comparison between analytical and experimental results is quite
satisfactory.
Notably, the temperature measurements indicate a thawing trend: temperatures within
the sample are progressively rising with time (Figs. 7b–7d), although the top is subjected to
periodic freezethaw cycles (Fig. 7a). However, regardless of the state and the corresponding
excess pore water pressures within the thawed part of the specimen, the consolidation rate
(i.e., drainage) is exclusively dependent upon the state, and specifically the permeability of
the topmost layer. When the top layer freezes, it creates an impermeable surface which
impedes drainage (and hence consolidation settlement) for as long as this surface layer
remains frozen. This mechanism is evidenced by the plateaus in the s–t consolidation response
(Step 2) shown in Fig. 8 for two different tests (note that τTest2 = 2τTest 1). The analysis predicts
the measured response with adequate accuracy.
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4. Theoretical verification
As previously mentioned, the 1D thaw consolidation theory formulated by Morgenstern &
Nixon [1971] combines Terzaghi's consolidation theory with a moving boundary, under the
assumption that the rate of thaw propagation is proportional to the square root of time
according to the relationship:
(16)
where: X denotes the depth to the thaw plane; and α is a thermal constant which depends on
the initial temperature of the soil, the magnitude of the applied step temperature, and the
thermal properties of both the thawed and frozen regions. Their analytical solutions, which
were supported by thaw consolidation oedometer test results [Morgenstern & Smith, 1973;
Nixon & Morgenstern, 1974], highlight the dominant role of the thawconsolidation ratio (R)
in the development of excess pore pressures and the degree of consolidation. R signifies how
quickly the thaw front moves in comparison to the time required for consolidation and can be
calculated as:
(17)
where: cv is the coefficient of consolidation.
A numerical parametric study is carried out to theoretically verify the validity of the
proposed numerical method for the entire range of possible R values. Figure 9a shows the FE
model used to study 1D thawconsolidation of a 1 m thick soil column subjected to overburden
stress P0 (100 kPa). Two different approaches were considered, reproducing the assumptions
made in two different theoretical studies:
(a) Nonlinear, porous elasticity [with reference to the theoretical solution for large strain
thaw consolidation by Yao et al., 2012]: the soil material behaviour follows the p’– e
relationship defined by Eq. (12). The model parameters have the same properties as in the
previously described monotonic element tests (Table 1) with the exception of permeability.
The latter varies from 1.42E13 m/s to 3.93E09 m/s, to achieve a range of thaw consolidation
ratios (from R = 10 for the least permeable to R = 0.06 in the opposite case) while the thermal
constant α remains constant and equal to 0.002 m/sec0.5.
(b) Linear elastic behaviour [in accordance with the approach of Morgenstern & Nixon 1971]:
the same soil as in (a) but with a constant Young’s modulus:
16
(18)
Numerical results are summarized in Fig. 9b and compared to the respective theoretical
solutions in view of the degree of consolidation (DoC). The latter is quantified as the ratio of
surface settlement at the end of thawing (St) to the ultimate settlement when all pore
pressures have dissipated (Smax). In a few words, R increases with reducing permeability and
the analytical solutions imply that for relatively impermeable soils the surface settlement at
the end of a thawing period (St) is only a small portion of the ultimate consolidation settlement
(Smax). The opposite is the case for relatively permeable soils.
It is important to note that the assumption of small strains and linear elastic behaviour
[Morgenstern & Nixon 1971] may lead to significant underestimation of excess pore water
pressures and underpredict the rate of settlement. In fact, having accounted for the nonlinear
p–e behaviour, Yao et al. [2012] produced a markedly different DoC–R relationship (Fig. 9b).
Nevertheless, the numerical analysis results conform closely to the theoretical solutions, as
long as the respective soil constitutive modelling assumptions (i.e. linear vs. nonlinear
response) are reproduced.
5. Case Study: a roadway embankment on degrading permafrost
Simulation of a case involving a roadway embankment has been carried out to show that the
proposed numerical method can be useful in the study of actual engineering problems. For
this purpose, Section 391 of a provincial road (PR391) embankment in Manitoba (Canada) is
selected, owing to the systematic and thoroughly documented monitoring programs that
were carried out on the site from 2008 onwards [Batenipour et al., 2014;
Gholamzadehabolfazl, 2015; Flynn et al., 2016]. The site is located about 18 km northwest of
Thompson (Fig. 10a) within a zone of sporadic discontinuous permafrost, where thaw
associated ground displacements constitute a common hazard for the serviceability of
transportation networks [Alfaro et al., 2009].
PR391 was originally constructed in the 1960’s as a compacted earthfill road. It was
upgraded to a gravelsurface road in the early 1970’s, and asphalt pavement was added about
a decade later. Thereafter, parts of the road have suffered considerable differential
settlements (especially those where ground ice had been reported during construction),
posing a threat to the safety of drivers and necessitating frequent maintenance. In an attempt
17
to relieve permafrost degradation rates, the site of interest (Fig. 10b) was reverted to gravel
surface in the 1990’s. A number of site investigation campaigns have been launched
thereafter, revealing the existence of a frost bulb beneath the centerline of the embankment.
Ongoing thawconsolidation settlements are the result of the frost bulb constantly shrinking.
More specifically, consecutive drilling below the toe of the embankment encountered frozen
ground between depths of 1.9 m to 10.5 m in 1991, while the frozen region had receded to
a depth of 4.6 m in 2005 and no frozen soil was found there in 2008. Yet, frozen ground was
still present, however limited between depths of 4.5 m and 11 m (along the centerline) in
2012.
Figure 10c depicts a cross section of the site. The 2 m tall embankment consists of a
gravel fill core, which in the centerline reaches up to 6 m depth from the road surface. The
slope and the natural ground surface consist of a shallow (1 m deep) clayey peatsilt layer,
underlain by a layer of silty clay. The surrounding area is poorly drained and the ground water
table remains practically at the natural ground surface all year long.
Modelling and Assumptions
The FE discretization of the cross section of interest is shown in Fig. 11a, highlighting the
different materials and mechanical boundary conditions. Taking advantage of symmetry along
the centreline, only half of the section is modelled. The bottom of the model (z = 18 m) is
assumed impermeable while the highlighted water table (z = 0) is modelled as a free drainage
boundary.
Numerical simulations have considered the entire time period of 25 years, from 1991,
when the gravel core was constructed, until 2016, when the monitoring period ended.
Although very little information was available regarding initial conditions in 1991, a crude
assumption was deemed necessary. In accordance with the previously mentioned
documented drilling data, the sequential (heat transfer, consolidation) analyses take as
starting point a condition where a uniform, horizontal layer of frozen soil exists between
depths of 1.9 m to 10.5 m. Both the underlying soil (from 10.5 m to 18 m) and the 1.9 m
deep surface layer were assumed to have constant temperatures with T ≥ 0⁰C. A preliminary
sensitivity analysis was carried out to determine the temperature values that produce heat
transfer results consistent with the field observations for the time period 2008–2012 (see
18
following thermal analysis results). Figure 11b displays the temperature profile designated as
initial condition (in 1991) and used as input in the analyses.
The simulated thermal history involves temperature variations due to seasonal changes
exclusively and has ignored any additional parameters (such as construction and maintenance
activity, or traffic) that may have also contributed to the thawing process. Specifically, Fig. 12
shows the assumed sine function used to describe the air temperature at the ground surface
for the time period of interest (1991–2014). The function is derived through fitting to daily air
temperature records from the nearby Thompson Airport station, also accounting for an
estimated temperature increase rate of 0.00015 °C/day. Typical values of nfactor indices
[Andersland & Ladanyi, 1994] are used to convert air temperatures to surface ground
temperatures as summarized in Table 2.
Soil model parameters are based on laboratory test results [Kurz et al., 2012; Batenipour
et al., 2014; and Gholamzadehabolfaz, 2015]. Table 3 summarizes the thermal properties used
for the heat transfer analysis. The magnitude of lateral displacements and settlements
recorded in the embankment on a yearly basis imply considerably nonlinear material
response. Soil plasticity was considered through the modified Cam Clay constitutive model
with its parameters (listed in Table 4) being calibrated with respect to documented triaxial
and oedometer test results [Batenipour et al., 2014]. It should be noted that, due to the lack
of specific k – e data, the analysis here has assumed constant permeability values.
Numerical Results vs. Field Data
A twophase monitoring campaign was carried out at the site. The first instruments, including
strings of thermistors, wire piezometers and settlement plates, were installed beneath the toe
and the midslope of the embankment in 2008. Additional instruments were placed under the
shoulder and the centreline in 2012.
Temperature variations
Recorded temperature data for the time periods 2008–2011 and 2012–2014 are summarized
and compared to the analytical predictions in Figs. 13 and 14, respectively. The temperature
time histories indicate a strong degradation of surface periodic temperature variations with
depth. Surface soil layers constitute a means of thermal insulation and practically constant
temperatures have been recorded at a depth of 8 m under the toe and the midslope of the
19
embankment (Fig. 13a,b), and similarly at a depth of 14 m under the centreline and the
shoulder (Fig. 14a,b). This phenomenon, as well as the magnitude and frequency of
temperature fluctuations at shallower depths, is numerically simulated with reasonable
accuracy. Analytical isotherm contours are compared to readings of thermistor arrays on four
indicative dates from January 2009 to October 2014 (Fig. 13c,d; Fig. 14c,d). The computed
temperature contours confirm the presence of the previously discussed frost bulb under the
embankment. Moreover, they indicate its significant size variation due to seasonal changes.
An evident thawing trend (shrinking of the frost bulb) should be noted. Numerical predictions
are consistent with observations from drilling samples and in reasonably good agreement with
recorded temperature data throughout the entire period of monitoring.
Settlement evolution
The second (consolidation) step of the sequential analysis predicts appreciably high
deformations suffered by the embankment throughout the considered lifetime. Figure 15a
plots the numerically computed evolution of settlements for the two monitored points (A at
the centreline; B at the shoulder) indicating cumulative displacements of the order of 1 m. A
zoomed view of the last two years, during which monitoring data is available, shown in Figs.
15b,c reveals satisfactory agreement between analysis and field response. The observed
transient deviations from the recorded data are largely due to the fact that the model fails to
reproduce the heaving effects associated with soil refreezing. However, notwithstanding this
undeniable flaw, the proposed simplified sequentially–coupled approach can be seen to
provide a powerful tool for the simulation of the long term response of the embankment
subjected to a significant number of seasonal permafrost–thawing cycles.
6. Advantages vs. Limitations of the Proposed Methodology
The presented sequentially coupled numerical methodology is bound by the following
limitations:
o In its present form, the method is not suitable for studying frost action, or freezeback
phenomena, and cannot be used to simulate frost heaving effects that are often
encountered in frost susceptible soils. It may be used for simulation of problems were
deformations are dominated by the thawing process (degrading permafrost).
20
o The frozen soil is treated as uniform, isotropic, and homogenously frozen. The method
ignores the effect of ice lenses and shrinkage cracks which can, according to past studies
[e.g. Chamberlain & Gow, 1979; Konrad & Morgenstern, 1980; Eigenbrod, 1996], develop
in fine grained frost susceptible soils due to cyclic freezing–thawing. Moreover, the
method is suitable for simulation of problems where the frozen ground in the beginning
of the analysis (t = 0) may be approximated as a uniform horizontal layer.
o The presence of unfrozen water when temperature drops below the freezing point is an
important parameter that has been here accounted for in a crude manner [Eq. 8].
However, in future studies, the method can be, rather straightforwardly, modified to
incorporate more realistic θu–Τ relationships, as proposed by other researchers [Nixon,
1991; Michalowski, 1993; Dumais & Konrad, 2018].
o The adopted loosely coupled algorithm is a simplification of the actually intricate,
interrelated, thermal–hydraulic–mechanical behaviour. As such, it only accounts for heat
transfer due to conduction, neglecting the contribution of convection and advection.
o Finally, although the 2nd step of the analysis (consolidation) accounts for large
deformation effects, which can lead to significant movements of the ground surface (as
in the case study), these are not incorporated in the 1st step (thermal analysis) due to the
lack of fullcoupling between the two computational steps. Hence, the method cannot
simulate the accelerating effect of consolidation displacements on thaw propagation (i.e.
the fact that surface thermal boundary conditions move towards the thaw front as
consolidation settlements take place), neither can it model comprehensively the
influence of void ratio changes on the thermal properties.
Nevertheless, this numerical method is intended to provide a practical, relatively simple
to use, tool for engineers to simulate thaw dominated problems with sufficient accuracy. For
this purpose, it has been developed within a longestablished, general purpose, program
which provides the ability to concurrently account for structural behaviour as well as complex
3D geometries. This may be essential for the study of problems where 3D effects can be
critical. The following section presents one such example, investigating the potentials of a
retrospective remediation solution for the embankment in Manitoba.
21
7. Thaw Settlement Mitigation using Thermosyphons: The Effect of their Spacing
The developed methodology is here employed in the investigation of the potentials of a
possible remediation solution for the previously studied embankment in Manitoba. This
investigation is in fact retrospective because it involves construction of thermosyphons at the
crest of the embankment slope in 1991, the same time that was postulated as t = 0 in the
previously presented analysis. Nevertheless, it is believed to be useful in illustrating the
occasional importance of considering outofplane response, by engaging 3D numerical
modelling.
Thermosyphons are cooling devises that are frequently used to remedy permafrost
degradation and the associated thaw settlements. Past studies [e.g. Xu et al., 2011; Abdalla
et al., 2016] have dealt with the complex issue of numerically simulating their thermal
response in a faithful manner. Judging that such degree of elaboration exceeds the purposes
of this study, thermosyphons are here modelled simply as columns of square crosssection [0.
3 m x 0.3 m] embedded in the ground. A minimum of two elements has been used to compose
this crosssection (Fig. 16a), for a higher degree of refinement would excessively increase
computational time. The soil–thermosyphon interface has been assigned a heat flux condition
which ensures that the temperature over the thermosyphon surface remains constant and
equal to 1 °C. At the same time, appropriate kinematic constraints were imposed to supress
differential movements (sliding) along this interface. A single row of 4 m deep thermosyphons
is assumed to be installed at the crest of the embankment slope (Fig. 16b). The effect of their
transverse spacing ys has been the focus of this set of analyses, where all of the material
properties and boundary/initial conditions are the same as those in the previously described
case study.
Figure 17 shows the computed thermal condition 23 years after the installation of the
thermosyphons (October, 2014) for two cases of transverse spacing, namely ys = 5 m (Fig. 17a)
and ys = 2.5 m (Fig. 17b). Comparison of the characteristic crosssections with the respective
condition of the original embankment (Fig. 14d), indicates that although none of the
remediating solutions is capable of completely negating permafrost degradation, the size of
the frost bulb at this time is in both cases significantly greater than in the original, un
mitigated, embankment. The presence of the thermosyphons appears effective in delaying
permafrost thawing and, as expected, there is some measurable advantage in having shorter
spacing distance.
22
Similarly, Fig. 18 portrays the thermal response of the embankment 30 years after the
application of mitigation measures (i.e. in October, 2021). Now the advantage of having ys =
2.5 m (Fig. 18b) instead of ys = 5 m (Fig. 18a) is more pronounced, critically influencing the
evolution of thawing under the roadway: by this time the soil under the crest (x = 4 m) has
thawed entirely when ys = 5 m while, by contrast, it remains frozen when ys = 2.5 m.
The effect of ys is further elucidated in Fig. 19, where the comparison is made in terms
of settlements. The 30year history of settlements experienced at the crest (Fig. 19a) and at
the toe (Fig. 19b) reveals that both solutions would greatly benefit the response of the
embankment. Naturally, this benefit becomes more significant as the spacing distance gets
shorter. Yet, it is important to highlight a difference of probably greater importance: In the
case where ys = 2.5 m the embankment surface deforms uniformly, as indicated by its surface
displacement profile illustrated in Fig. 19d. On the other hand, Fig. 19c shows that double
spacing distance (ys = 5 m) would result in significant differential displacements of the surface
(reaching 0.3 m). Such surface deformation would probably constitute this solution ineffective
for the serviceability of the roadway.
23
8. Concluding Remarks
A simplified numerical approach has been developed for the analysis of permafrost ground
subjected to thawing. It enables loosely (sequentially) coupled integration of 3D transient heat
transfer and thaw consolidation. The method comes with limitations which are detailed in the
respective section of this paper. Nevertheless, implementation with due consideration of
these limitations can be useful for the study of various thawdeformation problems in arctic
engineering practice.
The paper has presented a detailed validation study where numerical simulations were
compared to experimental results and theoretical solutions. Simulation of a welldocumented
case study, involving a constantly deforming roadway embankment in Canada, is also
presented with the intention to provide additional confidence in the validity of the proposed
methodology. Finally, two possible remediation solutions are presented involving the
installation of thermosyphons at the crest of the embankment slope. Consideration of the
effect of their spacing highlights the importance of considering outofplane response by
engaging 3D numerical modelling. Having been developed within a general purpose code that
enables simulation of complex 3D geometries and structural loads, the presented
methodology can be useful in conducting design optimization studies for such solutions.
24
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1
Table 1. Simulated properties of the soil samples tested under monotonic [Yao et al., 2016]
and periodic [Wang et al., 2015] thaw propagation.
Property
Units
Monotonic
Periodic
Specific weight
Gs

1.6
1.75
Water content
w

0.24
0.19
Degree of saturation
Sr

0.58
0.59
Thermal Modelling (step 1)
Specific heat
Sm(T > 0 ⁰C)
J/kg/⁰C
1221
1149
Sm(T ≤ Tf)
J/kg/⁰C
1611
1397
Conductivity
Cm(T > 0 ⁰C)
W/m/⁰C
1.28
1.38
Cm(T ≤ Tf)
W/m/⁰C
1.8
1.24
Latent heat
Lw
kJ/kg
333.7
333.7
Hydromechanical Modelling (step 2)
Initial effective stress
p0’
Pa
0.1
0.1
Initial void ratio
e0

0.675
0.56
Initial permeability
k0
m/s
1.82E06
7.02E08
Hydr. conductivity index
ck

0.2
0.17
Thawed compression index
κth

0.029
0.015
Poisson ratio of thawed soil
νth

0.3
0.3
2
Table 2. Freezing and thawing nfactors for each surface type considered in the case study.
Surface
Thawing (nth)
Freezing (nf)
Road
1.3
0.9
Slope
1
0.3
Natural Ground
1.2
0.5
3
Table 3. Soil material properties used in the heat transfer analysis of PR391.
Note that dry gravel refers to the gravel material above the water table (z > 0).
Property
Gravel (z > 0)
Gravel (z ≤ 0)
Clayey Silt
Silty Clay
Equivalent Density (kg/m3)
ρm
1900
1900
1620
1500
Equivalent Specific Heat
(J/kg/⁰C)
Sm(T > 0 ⁰C)
1160
840
940
1450
Sm(T ≤ Tf)
890
760
800
1020
Equivalent Conductivity
(W/m/⁰C)
Cm(T > 0 ⁰C)
2.66
1.78
0.77
1.35
Cm(T ≤ Tf)
4.12
1.42
0.85
2.00
Equivalent Latent heat
(kJ/kg)
Lm
40660
12850
21850
57850
Figure 1. Examples of failures induced by thawing of permafrost: (a) massive landslide in the Noatak
river, Alaska [source: U.S. National Park Service]; (b) structural damage due to differential
settlements in Norilsk, Russia [Luhn, 2016]; (c) wrinkle on pipeline [Oswell, 2011]; and (d) buckling of
railroad in Alaska [sourse: U.S. Geological Survey].
(a)
(d)
(b)
(c)
Figure 2. Outline of the simplified 2–step decoupled approach.
Step 1:
Heat Transfer Analysis
Problem
Geometry
Initial/
Boundary
Temperature
Conditions
Thermal
Properties
Temperature
Field
T(x,y,z,t)
Mechanical
Properties
f(T)
Stresses

Strains
Step 2:
Consolidation Analysis
Loads,
Kinematic
Boundaries,
Drainage
Figure 3. The simplified 2–step decoupled approach with highlighted select attributes and
boundary conditions: (a) Step 1–heat transfer analysis to calculate the temperature of
every element with respect to time T(t); (b) example of produced temperature field
(indicatively for t≈2 hours), used as the incrementally updated input for the next step; (c)
Step 2–consolidation analysis to calculate the developing stresses; and (d) example of the
resultant displacements.
(c)
P0
Element type:
C3D8RP
Porous Material:
κ(T) λ(T), k(T)
f(T,t) Kinematic
Constraints
STEP 1: Heat Transfer
T= 10 °C
T= 1 °C
T0= 1 °C
Element type:
DC3D8
Properties:
ρm, Sm, Cm, Lm
(a)
(d)
Incrementally Updated
Field Input T(t)
T(°C)
10
5
7
3
0
t= 7170 s
(b)
STEP 2: Consolidation
s(mm)
0
3
6
9
t= 7170 s
free drainage
Soil
Sample
1
2
1/2: Top/bottom cap temperature
controlling system
3: Axial Loading System
4: Top cap
5: Bottom cap
6: Drainage pipe
7: Permeable stone
8: Plexiglas cell
9: Thermistors
10: Ceramic Disks
11: Pore Pressure Transducers
3
4
5
6
7
10
9
11
Figure 4. Schematic of the apparatus and experimental setup used by Yao et al.
[2016]. Note that dimensions are shown in mm.
8
34
66
100
100
0.3
0.5
0.7
1.00E08 1.00E07 1.00E06 1.00E05
0.001 0.01 0.1 1 10 100
Thousands
Figure 5. Experimentally measured permeability (e–k) and compression (e–p) properties
for: (a) monotonic steptype thermal increase tests [Yao et al., 2016], and (b) periodic
thermal boundaries [Wang et al., 2015].
0.3
0.5
0.7
1.00E09 1.00E08 1.00E07 1.00E06
110 100
Thousands
k (m/s) p (kPa)
(a) Monotonic tests
e
0.7
0.5
0.3
10 8 10 7 10 6 10 5
p (kPa)k (m/s)
10 9 10 8 10 7 10 6
e
0.7
0.5
0.3
(b)
(b) Periodic tests
frozen
1
1
3
5
7
0 2 4 6
0
5
10
15
0 2 4 6
T (°C)
t (hours)
s (mm)
(a)
(b)
Figure 6. Validation of the simplified decoupled approach
against the monotonic tests of Yao et al. [2016] in terms of: (a)
evolution of temperature with time at two different depths of
the soil specimen; and (b) settlement at the top.
1 °C
10 °C
1 °C
h
s
Analysis Experiment
1
1
3
5
7
0 2 4 6
Analysis
Experiment
1
1
3
5
7
0 2 4 6
Analysis
Experiment
Figure 7. Validation of the simplified decoupled approach against the periodic tests of Wang et al.
[2015]–Step 1: heat transfer analysis. Temperature timehistories at four different specimen
depths: (a) h=90 mm; (b) h=70 mm;(c) h=50 mm; and (d) h=30 mm.
1
1
3
5
0 4 8 12 16
1
1
3
5
04812 16
T (°C)
t (hours)
1
1
3
5
04812 16
1
1
3
5
0 4 8 12 16
T (°C)
(a) (b)
(c) (d)
t (hours)
frozen
1
1
3
5
7
0 2 4 6
Analysis
Experiment
Analysis
Experiment
𝑻°C= 𝟐. 𝟓 + 𝟒𝒔𝒊𝒏 𝟐𝝅
𝟔𝒕
1 °C
1 °C
h
s
T (°C)
t (hours)
Figure 8. Validation of the simplified decoupled approach against two periodic tests of Wang et
al. [2015]–Step 2: consolidation analysis: (a) timehistory of top plate temperature; and (b)
comparison between measured and computed settlements.
s (mm)
(a)
t (hours)
2
2
6
0 5 10 15
2
2
6
0 5 10 15
frozen
0
4
8
12
0 5 10 15
0
4
8
12
0 5 10 15
(b)
Test 1 Test 2
1
1
3
5
7
0 2 4 6
Analysis
Experiment
Analysis
Experiment
s(t)
0
0.2
0.4
0.6
0.8
1
0.01 0.1 1 10
0
0.2
0.4
0.6
0.8
1
0.01 0.1 110
Degree of Consolidation
Thaw Consolidation Ratio : R
Morgi Yao
LinearElastic PorousElastic
0
0.2
0.4
0.6
0.8
1
0.01 0.1 110
Degree of Consolidation
Thaw Consolidation Ratio : R
Morgi Yao
LinearElastic PorousElastic
Morgenstern & Nixon
0
0.2
0.4
0.6
0.8
1
0.01 0.1 110
Degree of Consolidation
Thaw Consolidation Ratio : R
Morgi Yao
LinearElastic PorousElastic
Analysis
0
0.2
0.4
0.6
0.8
1
0.01 0.1 110
Degree of Consolidation
Thaw Consolidation Ratio : R
Morgi Yao
LinearElastic PorousElastic
Yao et al.
Analysis
Linear Elastic Porous Elastic
Consolidation Degree
Figure 9. Verification of the simplified decoupled approach
to theoretical solutions for 1D thaw consolidation in the
small–strain [Morgenstern & Nixon 1971] and the large–
strain domain [Yao et al., 2012]: (a) numerical model of
thaw propagation in a column of soil; (b) degree of
consolidation with respect to the thaw consolidation ratio R.
T (°C)
10
7.5
5
2.5
0
1 m
P0
X(t)
(a)
(b)
Thaw Consolidation Ratio –R
Figure 10.Case study of an embankment supported upon degrading discontinuous permafrost:
(a) site location [adapted from Flynn et al., 2016]; (b) photo of the site; and (c) soil profile of the
unstable section PR 391 [Gholamzadehabolfazl, 2015].
18
14
10
6
2
2
010 20 30 40
gravel
Clayey Silt
Silty Clay
Bedrock
Monitoring
2008–2010
Monitoring
2012–2014
z (m)
x (m)
(b)
(c)
Continuous Permafrost
Extensive Discontinuous Permafrost
Sporadic Discontinuous Permafrost
Isolated Patches of Permafrost
No Permafrost
(a)
A B
RR391 Thompson
Gillam
Churchill
Figure 11.Numerical simulation of the thaw consolidation
response of section PR 391: (a) FE mesh with boundary
conditions; (b) assumed initial temperature conditions
(setting 1991 as the start time).
T (°C)
1 0 1
frozen
(a)
Water table: free drainage
Symmetry plane
(b)
Figure 12.Assumed sine function to describe the fluctuations of
air temperature during the years of interest (1991–2014).
30
15
0
15
30
1991 1995 1999 2003 2007 2011
T (°C)
t (years)
frozen
frozen
18
13
8
3
2
010 20 30
18
13
8
3
2
010 20 30
2
0
2
4
6
8
10
7020 7170 7320 7470 7620 7770 7920
2
0
2
4
6
8
10
7020 7170 7320 7470 7620 7770 7920
T (°C)
T (°C)
1.11.08
Date
z (m)
x (m)
(a)
(c)
(b)
(d)
January 2009
April 2009
+1.8
+1.7
+0.2
+2.6
+2.9
+1.8
+0.8
+1.3
+0.7
+1.6
+2.5
+1.9
Figure 13.Comparison of anaylsis with field measurements [Batenipour et al., 2014]–1st monitoring
period (2008–2010). Temperature time–histories at various depths: (a) toe; and (b) middle of the
embankment slope. Temperature spatial variation at two instances: (c) January 2009; and (d) April,
2009.
Temperature (°C)
0 1 2 3
z (m)
31.3.08 28.8.08 25.1.10 24.6.10 21.11.10
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
Measured: –2 m –5 m –8 m
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
Analysis: –2 m –5 m –8 m
18
13
8
3
2
010 20 30
18
13
8
3
2
010 20 30
4
2
0
2
4
6
8
8094 8244 8394 8544 8694 8844
4
2
0
2
4
6
8
8094 8244 8394 8544 8694 8844
July, 2013
October, 2014
+2.1
+1.2
+1.1
+5.2
+2.4
+2.2
0.2
0.2
+0.3
+0.7
0
0.2
+0.5
+0.8
+4.2
+2
+1.4
+6.7
+3.4
+2.5
+0.2
0
+0.6
+1
+1.5
+0.1
+0.8
+1.1
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
Measured: –2 m –5 m –14 m
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
4
2
0
2
4
6
8
10
7819 7969 8119 8269 8419 8569
2m
5 m
14 m
Series4
Series5
Series6
Analysis: –2 m –5 m –14 m
Temperature (°C)
0 3 6 9
x (m)
18.9.12
Date
16.1.13 16.5.13 13.9.13 11.1.14 11.5.14
T (°C)
T (°C)
z (m)
(a)
(c)
(b)
(d)
z (m)
Figure 14.Comparison of anaylsis with field measurements [Batenipour et al., 2014]–2nd monitoring
period (2012–2014). Temperature time–histories at various depths: (a) toe; and (b) middle of the
embankment slope. Temperature spatial variation at two instances: (c) July 2013; and (d) October,
2014.
Figure 15. Settlement response of PR391: (a) numerically computed evolution of settlements
throughout the entire lifetime (25 years) of the embankment and comparison with the field
measurements taken during the 2year long monitoring period; Closer look at: (b) centerline (point
A); and (c) shoulder (point B).
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
1.2
1
0.8
0.6
0.4
0.2
0
22.5 23 23.5 24 24.5 25
1.2
1
0.8
0.6
0.4
0.2
0
22.5 23 23.5 24 24.5 25
t (years)
(a)
A B
1.2
1
0.8
0.6
0.4
0.2
0
22.5 23 23.5 24 24.5 25
Series2
Series3
Point A Point B
1.2
1
0.8
0.6
0.4
0.2
0
22.5 23 23.5 24 24.5 25
Series3
Series2
Measurement
Analysis
1.2
1
0.8
0.6
0.4
0.2
0
22.5 23 23.5 24 24.5 25
Series2
Series3
Monitoring
Period
(b) (c)
s (m)
s (m)
Symmetry plane
Therm. section: 0.4 x 0.4 m
x
y
ys
4 m
T (°C)
0 11
Figure 16. Numerical model of the PR 391 embankment slope
with thermosyphons installed at the crest of its slope: (a) plan,
and (b) 3D views.
(a)
z(b)
18
13
8
3
2
010 20
18
13
8
3
2
010 20
18
13
8
3
2
0 5 10
18
13
8
3
2
0 5 10
18
13
8
3
2
0 5 10
18
13
8
3
2
0 5 10
9
T(°C)
0
3
6
z (m)
Figure 17. Crosssections of embankment–thermosyphon models with superimposed temperature
contours corresponding to the thermal condition in October, 2014 (23 years after installation) for:
(a) thermosyphon spacing ys= 5 m, and (b) ys= 2.5 m.
x (m) x (m)
y (m) y (m)
z (m)
y= 0 y= 2.5 m
x= 0
(a) (b)
y (m)
x= 4 mx= 0
y (m)
x= 4 m
October, 2014
18
13
8
3
2
010 20
18
13
8
3
2
0 5 10
18
13
8
3
2
0 5 10
18
13
8
3
2
010 20
18
13
8
3
2
0 5 10
18
13
8
3
2
0 5 10
z (m)
x (m) x (m)
October, 2021
y (m) y (m)
z (m)
x= 0
(a) (b)
y (m)
x= 4 mx= 0
y (m)
x= 4 m
Figure 18. Crosssections of embankment–thermosyphon models with superimposed temperature
contours corresponding to the thermal condition in October, 2021 (30 years after installation) for:
(a) thermosyphon spacing ys= 5 m, and (b) ys= 2.5 m.
9
T(°C)
0
3
6
thermosyphon
Point C
Point B
1.5
1
0.5
0
010 20 30
1.5
1
0.5
0
010 20 30
1.5
1
0.5
0
010 20 30
Series3
Series1
Series2
Original Embankment
Thermos. (ys= 5m)
Thermos. (ys = 2.5 m)
t (years)
s (m)
s (m)
Figure 19. Effect of thermosyphon spacing on the settlement response of the embankment:
comparison of s–tresponses calculated at (a) the crest (Point B); and (b) at the toe (Point C) of the
embankment; contours of incremental settlement (from year 25 to year 30)of the model surface
for: (c) ys= 5 m and (d) ys= 2.5 m.
(a)
(b) (d)
Point C
Point B
(c)
s(m)
00.40.2
4
Table 4. Thawed (T > 0 ⁰C) soil modelling parameters used in the consolidation analysis of PR391.
Note that k is permeability; v is the Poisson’s ratio; λ the slope of the normal compression line; κ
the slope of the unload reload line; and M the slope of the critical state line in p’ q space.
Property
Gravel
Clayey Silt
Silty Clay
(1 – 2 m)
(2 – 3 m)
(3 – 4.5 m)
(4.5 – 6 m)
(6 – 6.8 m)
(6.8 – 8.3 m)
(8.3 – 9 m)
(9 – 10.5 m)
k (m/s)
1 x 103
1 x 107
1 x 108
1 x 108
1 x 108
1 x 108
1 x 108
1 x 108
1 x 108
1 x 109
ν
0.15
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
κ
0.001
0.005
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
λ



0.15
0.154
0.154
0.11
0.1
0.1
0.1
M



1
1.4
1.7
1.3
1.1
1.2
1.5