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Multi-phase-field microporomechanics model for simulating ice lens growth and thaw in frozen soil

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Abstract

This article presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields are driven by the driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly captures the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models.
Multi-phase-field microporomechanics model for
simulating ice lens growth and thaw in frozen soil
Hyoung Suk Suh*WaiChing Sun
December 1, 2021
Abstract
This article presents a multi-phase-field poromechanics model that simulates the growth and
thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen
soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implic-
itly represented by the evolution of two phase fields that indicate the locations of segregated ice
and the damaged zone, respectively. The evolution of two phase fields are driven by the driving
forces that capture the physical mechanisms of ice and crack growths respectively, while the phase
field governing equations are coupled with the balance laws such that the coupling among heat
transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed nu-
merically. Unlike phenomenological approaches that indirectly captures the freezing influence on
the shear strength, the multi-phase-field model introduces an immersed approach where both the
homogeneous freezing and the ice lens growth are distinctively captured by the freezing character-
istic function and the driving force accordingly. Verification and validation examples are provided
to demonstrate the capacities of the proposed models.
1 Introduction
Ice lens formation at the microscopic scale is a physical phenomenon critical for understanding the
physics of frost heave and thawing settlement occurred at the field scale under the thermal cycles.
Since ice lens may affect the freeze-thaw action and cause frost heave and thawing settlement sensi-
tive to the changing climate and environment conditions, knowledge on the mechanism for the ice
lens growth is of practical value for many civil engineering applications in cold regions [Palmer and
Williams,2003,Zhang et al.,2016,Li et al.,2017,Lake et al.,2017,Ji et al.,2019]. For example, sub-
stantial heaving and settlement caused by the sequential formations and thawing of ice lenses lead
to uneven deformation of the road which also damages the tires, suspension, and ball joints of ve-
hicles. In the United States alone, it was estimated that two billion dollars had been spent annually
to repair frost damage of roads [DiMillio,1999]. Moreover, extreme climate change over the last few
decades have brought increasing attention to permafrost degradation, since unusual heat waves may
cause weakening of foundations and increase the likelihood of landslides triggered by the abrupt
melting of the ice lens [Nelson et al.,2001,2002,Streletskiy et al.,2012,Leibman et al.,2014,Mithan
et al.,2021]. Under these circumstances, both the fundamental understanding of the ice lens growth
mechanisms and the capacity to predict and simulate the effect beyond the one-dimensional models
becomes increasingly important.
*Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027.
h.suh@columbia.edu
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027.
wsun@columbia.edu (corresponding author)
1
arXiv:2111.14983v1 [physics.geo-ph] 29 Nov 2021
Since the pioneering work on the ice lens by Stephan Taber in the early 20th century [Taber,1929,
1930], there has been a considerable amount of progress in the geophysics and fluid mechanics com-
munity to elucidate the mechanisms in the ice segregation process (e.g., [Peppin and Style,2013] and
references cited therein). During the freezing phase, it is now known that the crystallized pore ice sur-
rounded by a thin pre-melted water film develops a suction pressure (i.e., cryo-suction) that attracts
the unfrozen water towards the freezing front [Wilen and Dash,1995,Dash et al.,1995,2006]. These
films remain unfrozen below the freezing temperature and form an interconnected flow network that
supplies water to promote ice crystal growth. Accumulation of pore ice crystals accompanies the void
expansion and micro-cracking of the host matrix, which may result in the formation of a horizontal
lens of segregated ice. However, despite these substantial amounts of works, the criterion for the ice
lens initiation and its detailed mechanism still remain unclear. Based on the thermo-hydraulic model
proposed by Harlan [Harlan,1973], Miller [Miller,1972,1977,O’Neill and Miller,1985] introduces a
concept of stress partitioning and assumed that an ice lens starts to form if the solid skeleton expe-
riences tensile stress. This idea has been further adopted and further generalized in [Fowler,1989,
Fowler and Krantz,1994] via an asymptotic method. Gilpin [Gilpin,1980] suggests that the ice lens
formation takes place when the ice pressure reaches the particle separation pressure depending on
the particle size and the interfacial tension between the water and ice, whereas Zhou and Li [Zhou
and Li,2012] propose the idea of separation void ratio as a criterion for the ice lensing. Konrad and
Morgenstern [Konrad and Morgenstern,1980] present an alternative approach that can describe the
formation and growth of a single ice lens based on segregation potential, of which the applicability
has been demonstrated in [Nixon,1982,Konrad and Shen,1996,Tiedje and Guo,2012]. On the other
hand, Rempel [Rempel et al.,2004,Rempel,2007] develops regime diagrams that delineate the growth
of a single lens, multiple lenses, and homogeneous freezing. In this line of work, the one-dimensional
momentum and mass equilibrium equations are coupled with the heat flow in a step-freezing Stefan
configuration to calculate the intermolecular force that drives the premelted fluid to the growing ice
lenses. While the proposed method is helpful for estimating the lens thickness and spacing, the one-
dimensional setting is understandably insufficient for the geo-engineering applications that require
understanding on the implication of ice lenses on the shear strength. More recently, Style et al. [Style
et al.,2011] propose a new theory on the ice lens nucleation by considering the cohesion of soil and
the geometric supercooling of the unfrozen water in the pore space. Although the aforementioned
studies formed the basis to shed light on explaining the ice lens formation, they are limited to the
idealized one-dimensional problems and often idealized soil as a linear elastic material and hence
not sufficient for applications that require a more precise understanding of the constitutive responses
of the ice-rich soil.
Meanwhile, within the geomechanics and geotechnical engineering community, a number of the-
ories and numerical modeling frameworks have been proposed based on the mixture theory and
thermodynamics principles [Nishimura et al.,2009,Zhou and Meschke,2013,Na and Sun,2017,
Michalowski and Zhu,2006] with a variety of complexities and details. By adopting the premelt-
ing theory and considering the frozen soil as a continuum mixture of the solid, unfrozen water, and
ice constituents, the freezing retention behavior of frozen soil can be modeled in a manner similar to
those for the unsaturated soil. The resultant finite element implementation of these models enables
us to simulate freeze-thaw effects in two- or three-dimensional spaces often with more realistic pre-
dictions on the solid constitutive responses. Nevertheless, the presence of crystal ices in the pores and
that inside the expanded ice lens are often represented via phenomenological laws [Michalowski and
Zhu,2006,Ghoreishian Amiri et al.,2016]. Since the morphology, physics, and the mechanisms as
well as the resultant mechanical characteristics of the ice lens and ice crystals in pores are profoundly
different, it remains difficult to develop a predictive phenomenological constitutive law for an effec-
tive medium that represents the multi-constituent frozen soil with ice lenses [Wettlaufer and Worster,
2006].
This study is an attempt to reconcile the fluid mechanics and geotechnical engineering modeling
efforts on modeling the frozen soil under changing climates. Our goal is to (1) extend the field theory
for ice lens such that it is not restricted to one-dimensional problems and (2) introduce a framework
2
that may incorporate more realistic path-dependent constitutive laws. As such, the coupling mech-
anism among phase transition, fluid diffusion, heat transfer, and solid mechanics can be captured
without solely replying on phenomenological material laws. In particular, we introduce a mathemat-
ical framework and a corresponding finite element solver that may distinctively capture the physics
of ice lens and freezing/thawing. We leverage the implicit representation of complex geometry af-
forded by a multi-phase-field framework to first overcome the difficulty on capturing the evolving
geometry of the ice lens. By considering the ice lens as segregated bulk ice inside the freezing-induced
fracture, we adopt two phase field variables that represent the state of the fluid phase constituent and
the regularized crack topology, respectively. This treatment enables us to take account of the brittle
fracture that may occur during ice lens growth and explicitly incorporate the addition and vanishing
shear strength and bearing capacity of the ice lens under different environmental conditions. The
phase transition of the fluid is modeled via the Allen-Cahn equation [Allen and Cahn,1979,Boet-
tinger et al.,2002], while we adopt the phase field fracture framework to model brittle cracking in a
solid matrix [Bourdin et al.,2008,Miehe et al.,2010a,Borden et al.,2012]. The resultant framework
may provide a fuller picture to analyzing the growth of the ice lens in the frozen soil, while verifica-
tion exercises also confirm that the model may reduce to a classical thermo-hydro-mechanical model
and isothermal poromechanics model under limited conditions.
The rest of the paper is organized as follows. Section 2summarizes the necessary ingredients for
the mathematical framework, while we present the multi-phase-field microporomechanics model that
describes the coupled behavior of a fluid-saturated phase-changing porous media in Section 3. For
completeness, the details of the finite element formulation and the operator splitting solution strategy
are discussed in Section 4. Finally, numerical examples are given in Section 5to verify, validate, and
showcase the model capacity, which highlights its potential by simulating the growth and melting of
multiple ice lenses.
As for notations and symbols, bold-faced and blackboard bold-faced letters denote tensors (in-
cluding vectors which are rank-one tensors); the symbol ’·’ denotes a single contraction of adjacent
indices of two tensors (e.g., a·b=aibior c·d=cijdjk); the symbol ‘:’ denotes a double contrac-
tion of adjacent indices of tensor of rank two or higher (e.g., C:ε=Cijkl εkl ); the symbol ‘’ de-
notes a juxtaposition of two vectors (e.g., ab=aibj) or two symmetric second-order tensors [e.g.,
(αβ)ijkl =αij βkl]. We also define identity tensors: I=δij ,I=δik δjl , and ¯
I=δil δjk , where δij is
the Kronecker delta. As for sign conventions, unless specified, the directions of the tensile stress and
dilative pressure are considered as positive.
2 Kinematics and effective stress principle for frozen soil with ice
lens
In this section, we introduce the ingredients necessary to derive the field theory for the phase field
modeling of frozen soil presented later in Section 3. Similar to the treatments in [Nishimura et al.,
2009], [Zhou and Meschke,2013], and [Na and Sun,2017], we first assume that the frozen soil is fully
saturated with either water or ice and therefore idealize the frozen soil as a three-phase continuum
mixture that consists of solid, water, and ice phase constituents whereas the ice lens is a special case
in which the solid skeleton no longer holds bearing capacity. This treatment enables us to formulate
a multi-phase-field approach to employ two phase field variables as indicator functions for the state
of the pore fluid (in ice or water form) [Warren and Boettinger,1995,Boettinger et al.,2002,Sweidan
et al.,2020] and that of the solid skeleton (in damage or intact form) [Bourdin et al.,2008,Miehe et al.,
2010a,Borden et al.,2012]. We then extend the effective stress theory originated from damage me-
chanics [Chaboche,1988] to incorporate the internal stress of ice lenses caused by the deformation of
the effective medium into the Bishop’s effective stress principle for frozen soil where the introduc-
tion of phase field provide smooth transition of the material states for both the pore fluid and the
solid skeleton. This procedure allows us to incorporate both the capillary pressure of the ice crystal
3
surrounded by the water thin film as well as the volumetric and deviatoric stresses triggered by the
deformation of ice lens.
2.1 Continuum representation and kinematics
Based on the mixture theory, we idealize our target material as a multiphase continuum where the
solid, water, and ice phase constituents are overlapped. For simplicity, this study assumes that there
is no gas phase inside the pore such that the pore space is either occupied by water or ice. The volume
fractions of each phase constituent are defined as,
φs=dVs
dV ;φw=dVw
dV ;φi=dVi
dV ;φs+
α={w,i}
φα=1, (1)
where the indices s,w, and irefer to the solid, water, and ice phase constituents, respectively, while
dV =dVs+dVw+dVidenote the total elementary volume of the mixture. Note that an index used as a
subscript indicates the intrinsic property of a phase constituent, while it is used as a superscript when
referring to a partial property of the entire mixture. By letting ρs,ρw, and ρidenote the intrinsic mass
densities of the solid, water, and ice, respectively, the partial mass densities of each phase constituent
are given by,
ρs=φsρs;ρw=φwρw;ρi=φiρi;ρs+
α={w,i}
ρα=ρ, (2)
where ρis the total mass density of the entire mixture. We also define the saturation ratios for the
fluid phase constituents α={w,i}as:
Sw=φw
φ;Si=φi
φ;
α={w,i}
Sα=1, (3)
where φ=1φsis the porosity.
Since the solid (s), water (w), and ice (i) phases do not necessarily follow the same trajectory,
each constituent possesses its own Lagrangian motion function that maps the position vector of the
current configuration xat time tto their reference configurations. In this study, we adopt a kinematic
description that traces the motion of the solid matrix by following the classical theory of porous media
[Bowen,1980,Zienkiewicz et al.,1999,Ehlers,2002,Coussy,2004]. Hence, the motion of the solid
phase is described by using the Lagrangian approach via its displacement vector u(x,t), whereas
the fluid phase (α={w,i}) motions are described by the modified Eulerian approach via relative
velocities ˜vwand ˜vi, instead of their own velocity fields vwand vi, i.e.,
˜vα=vαv, (4)
where v=˙uis the solid velocity, while ˙
() = d()/dtis the total time derivative following the solid
matrix.
2.2 Multi-phase-field approximation of freezing-induced crack
In this current study, we assume that the path-dependent constitutive responses of the frozen soil is
due to the fracture in the brittle regime and the growth/thaw of the ice lens in the void space that
could be opened by the expanded ice. While plasticity of the solid skeleton as well as the damage and
creeping of the segregated ice may also play important roles on the mechanisms of the frost heave
and thaw settlement, they are out of the scope of this study. As such, this study follows Miller’s
theory which assumes that a new ice lens may only form if and only if the compressive effective
stress becomes zero or negative [Miller,1972,1977,O’Neill,1982,O’Neill and Miller,1985]. Since
opening up the void space is a necessary condition for the ice lens to grow inside, we introduce a
4
phase field model that captures the crack growth potentially caused by the ice lenses growth. In this
work, our strategy is to adopt diffuse approximations for both the phase transition of the pore fluid
and the crack topology, where each requires a distinct phase field variable. As illustrated in Fig. 1,
introducing two phase fields not only enables us to distinguish the homogeneous freezing from the
ice lens growth but also leads to a framework that can be considered as a generalization of a thermo-
hydro-mechanical model.
Homogeneously
frozen
Intact and
water-saturated
Ice lens
formation
(Hydraulically)
fractured
c=1
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: Damage evolution (A)
: Phase transition (B)
: A + B
Figure 1: Schematic of multi-phase-field approach coupled with a thermo-hydro-mechanical model.
The first phase field variable c[0, 1]used in this study is an order parameter that models the
freezing of water (melting of ice) in a regularized manner [Warren and Boettinger,1995,Sweidan
et al.,2020]. In other words, we employ a diffuse representation of the ice-water interface using
variable cthat is a function of xand t:
c=c(x,t)with
c=0 : completely frozen,
c=1 : completely unfrozen,
c(0, 1): diffuse ice-water interface,
(5)
which is the solution of the Allen-Cahn phase field equation [Allen and Cahn,1979,Boettinger et al.,
2002] that will be presented later in Section 3.1. Based on this setting, we consider the degree of satu-
ration of water as an interpolation function of the phase field c, i.e., Sw=Sw(c), which monotonically
increases from 0 to 1 as,
Sw(c) = c3(10 15c+6c2). (6)
Note that the evolution of the phase field variable citself does not necessarily imply the ice lens
growth since both the homogeneously frozen region and segregated ice can reach c=0, regardless of
the level of the effective stress or stored energy that drives the crack growth (Fig. 1).
The second phase field variable d[0, 1]adopted in this study is a damage parameter that treats
the sharp discontinuity as a diffusive crack via implicit function [Bourdin et al.,2008,Miehe et al.,
2010a,Borden et al.,2012,Suh et al.,2020]. In particular, we have:
d=d(x,t)with
d=0 : intact,
d=1 : damaged,
d(0, 1): transition zone,
(7)
to approximate the fracture surface area AΓas AΓd, which is the volume integration of crack surface
density Γd(d,d)over a body B, i.e.,
AΓAΓd=ZB
Γd(d,d)dV ;Γd(d,d) = d2
2ld
+ld
2(d· ∇ d), (8)
where ldis the length scale parameter that controls the size of the transition zone. In this case, the
crack resistance force Rdcan be expressed as,
Rd=Wd
d− ∇· Wd
d;Wd=GdΓd(d,d), (9)
5
where Gdis the critical energy release rate that quantifies the resistance to cracking. As hinted in Fig.
1, in order to guarantee crack irreversibility, the thermodynamic restriction ˙
Γd0 must be satisfied
[Miehe et al.,2010a,b,Choo and Sun,2018,Heider,2021] unlike the reversible freezing and thaw-
ing process. A necessary condition for thermodynamic consistency is to adopt a quadratic stiffness
degradation function gd(d) = (1d)2which satisfies the following conditions:
gd(0) = 1 ; gd(1) = 0 ; gd(d)
d0 for d[0, 1]. (10)
Based on this setting, we define an indicator function χi[0, 1]for the segregated ice inside the
freezing-induced fracture as follows:
χi(c,d) = [1Sw(c)][1gd(d)], (11)
such that χi=1 implies the formation of the ice lens, which is different from the in-pore crystalliza-
tion of the ice phase constituent.
2.3 Effective stress principle
Leveraging the similarities between freezing/thawing and drying/wetting processes, Miller and co-
workers [Miller,1972,1977,O’Neill,1982,O’Neill and Miller,1985] propose the concept of neutral
stress that partitions the net pore pressure ¯
pinto the pore water and pore ice pressures (pwand pi),
respectively:
¯
p=Sw(c)pw+ [1Sw(c)]pi. (12)
Clearly, Eq. (12) alone cannot capture the deviatoric stress induced by the deformation of the ice
lens. Previous efforts on modeling frozen soil often relies on a extension of critical state theory that
evolves the yield function according to the degree of saturation of ice (and therefore introduces the
dependence of the tensile and shear strength on the presence of ice) [Nishimura et al.,2009,Na and
Sun,2017]. However, this treatment is not sufficient to consider the soil that may become brittle at
low temperature due to the low moisture content and the influence of ice lens on the elasticity. Hence,
this study extends Miller’s approach into a phase field framework by decomposing the effective stress
tensor ¯
σ0into two partial stresses for the solid and ice lens via the damage phase field doubled as a
weighting function, i.e.,
¯
σ0=gd(d)σ0
int + [1gd(d)]σ0
dam. (13)
where the second term on the right hand side of Eq. (13) depends on the saturation Sw(c). Specifically,
the effective stress contribution from the solid skeleton σ0
int degrades due to the damage when ice lens
grows, but may also evolve by the change of σ0
dam in the presence of ice lens [for instance, see Eq.
(27) in Section 3.2]. Similar models that capture the constituent responses of porous media consisting
of multiple solid constituents can also be found in [Borja et al.,2020]. In addition, this study also
considers the volumetric expansion due to the phase transition from water to ice while neglecting the
thermal expansion or contraction of each phase constituent. Hence, the total stress of the mixture σ
can be expressed as,
σ=¯
σ0¯
pIφ[1Sw(c)]¯
αvKiI, (14)
where ¯
αv=gd(d)αv,int + [1gd(d)]αv,dam is the net volumetric expansion coefficient which is in-
fluenced by the evolution of the fracture. In particular, we assume that the volumetric expansion
coefficient of the ice lens αv,dam is greater than that of the pore ice crystal αv,int due to the degradation
of the solid skeleton.
6
3 Multi-phase-field microporomechanics model for phase-changing
porous media
This section presents the balance principles and constitutive laws that capture the thermo-hydro-
mechanical behavior of the phase-changing porous media. We first introduce the coupled field equa-
tions that govern the heat transfer and the ice-water phase transition processes which involve the
latent heat effect. Unlike previous studies that model the phase transition of the pore fluid by using
the semi-empirical approach which links either the Gibbs-Thomson equation [Zhou and Meschke,
2013] or the Clausius-Clapeyron equation [Nishimura et al.,2009,Na and Sun,2017] with the van
Genuchten curve [Van Genuchten,1980], we adopt the Allen-Cahn type phase field model [Allen and
Cahn,1979,Boettinger et al.,2002] with a driving force that depends both on the temperature and
the damage. We then present microporomechanics and phase field fracture models that complete the
set of governing equations, which is not only capable of simulating freeze-thaw action but also the
freezing-induced or hydraulically-driven fractures. The implications of our model will be examined
via numerical examples in Section 5.
3.1 Thermally induced phase transition
3.1.1 Heat transfer
Since underground freezing and thawing processes may span over long temporal scales, this study
employs a single temperature field θby assuming that all the phase constituents reach a local thermal
equilibrium instantly [Suh and Sun,2021a]. We also neglect thermal convection by considering the
case where the target material possesses low permeability. Let edenote the internal energy per unit
volume and qthe heat flux. Then, the energy balance of the entire mixture can be expressed as [Gelet
et al.,2012,Suh and Sun,2021a],
˙
e=− ∇· q+ˆ
r, ; e=es+
α={w,i}
eα, (15)
where ˆ
rindicates the heat source/sink, es=ρscsθand eα=ραcαθare the partial energies for the solid
and fluid phase constituents, respectively, while csand cαare their heat capacities. Assuming that the
freezing temperature of water (i.e., melting temperature of ice) remains constant: θm=273.15 K, the
internal energy of the entire mixture ein Eq. (15) can be rewritten as,
e=ρscsθ+ (ρwcw+ρici)(θθm) + (ρwcw+ρici)θm. (16)
From the relations shown in Eqs. (1)-(3), substituting Eq. (16) into Eq. (15) yields the following:
(ρscs+ρwcw+ρici)˙
θ+φ[(ρwcwρici)(θθm) + ρiLθ]˙
Sw(c) + ∇· q=ˆ
r, (17)
where:
Lθ=ρw
ρi
cwciθm(18)
is the latent heat of fusion which is set to be Lθ=334 kJ/kg for pure water [Warren and Boettinger,
1995,Loginova et al.,2001,Nishimura et al.,2009,Alexiades and Solomon,2018]. Notice that the
second term on the left-hand side of Eq. (17) describes the energy associated with the phase change
of the fluid phase constituent α={w,i}, which is responsible for the constant temperature during
the transformation processes, i.e., where cis changing with time since ˙
Sw(c) = {Sw(c)/c}˙
c. For the
constitutive model that describes the heat conduction, this study adopts Fourier’s law where the heat
flux can be written as the dot product between the effective thermal conductivity and the temperature
gradient, i.e.,
q=
φsκs+
α={w,i}
φακα
· ∇ θ, (19)
7
where κsand καdenote the intrinsic thermal conductivities of the solid and fluid phase constituents,
respectively. This volume-averaged approach, however, is only valid for the case where all the phase
constituents are connected in parallel. Although there exists alternative homogenization approaches
such as Eshelby’s equivalent inclusion method [Eshelby,1957,Hiroshi and Minoru,1986,Sun,2015],
determination of correct effective thermal conductivity often requires knowledge of the pore geome-
try and topology [Hiroshi and Minoru,1986,Lee et al.,2017,Suh and Yun,2018]. Since the informa-
tion is not always readily approachable, this extension will be considered in the future.
3.1.2 Phase transition
By using the phase field variable cdefined in Eq. (5), we adopt the Allen-Cahn model that is often
used to simulate dendrite growth or multi-phase flow [Allen and Cahn,1979,Takaki,2014,Aihara
et al.,2019]. Following [Boettinger et al.,2002], we consider one of the simplest forms of the Gibbs
free energy functional Ψc:
Ψc=ZBψcdV =ZBfc(θ,c) + e2
c
2| ∇ c|2dV, (20)
where fc(θ,c)is the free energy density that couples the heat transport with the phase transition,
while ecis the gradient energy coefficient. From Eq. (20), we consider the evolution of the phase field
cover time, which yields the well-known Allen-Cahn equation or time-dependent Ginzburg-Landau
equation, i.e.,
1
Mc
˙
c=∂ψc
c− ∇· ψc
c=fc
ce2
c2c, (21)
where 2() = ∇·∇()is the Laplacian operator and Mcis the mobility parameter. Since this study
does not consider solute transport or any other chemical effects, we focus on the pure water-ice phase
transition such that the free energy density fc(θ,c)can be written as,
fc=Wcgc(c) + Fc(θ)pc(c), (22)
where gc(c) = c2(1c)2is the double well potential [Fig. 2(a)] that can be regarded as an energy
barrier at the ice-water interface with the height of Wc, and pc(c) = Sw(c) = c3(6c215c+10)
is the interpolation function [Fig. 2(b)], while Fc(θ)is the driving force that is a first-order Taylor
approximation of the cryo-suction (scryo) based on the Clausius-Clapeyron relation [Boettinger et al.,
2002]:
scryo =pipw=ρiLθln θ
θm
≈ Fc(θ) = ρiLθ1θ
θm. (23)
As pointed out in [Warren and Boettinger,1995,Boettinger et al.,2002], since Eq. (21) captures the
evolution of the regularized ice-water interface, numerical parameters ec,Wc, and Mccan be related
to the ice-water surface tension γiw , the interface thickness δc, and the kinetic coefficient νcas,
ec=p6γiw δc;Wc=3γiw
δc;Mc=νcθm
6ρiLθδc. (24)
t Furthermore, since the existence of segregated ice governs the heave rate of frozen soil [Penner,
1986,Michalowski and Zhu,2006], this study considers different rates between homogeneous freez-
ing and ice lens growth. Specifically, while employing different volumetric expansion coefficients for
the in-pore crystallization and the formation of ice lens [Eq. (14)], we replace the driving force Fc(θ)
with F
c(θ,d)that contains additional term that describes the intense growth of ice lenses similar to
the kinetic equation proposed by Espinosa et al. [Espinosa et al.,2008], which is often used to model
salt crystallization in porous media [Koniorczyk and Gawin,2012,Derluyn et al.,2014,Choo and Sun,
2018]:
F
c(θ,d) = ρiLθ1θ
θm+ [1gd(d)]K
c1θ
θmg
c
, (25)
8
0 0.2 0.4 0.6 0.8 1
0
0.02
0.04
0.06
0.08
(a)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
(b)
Figure 2: (a) The double well potential gc(c), and (b) the interpolation pc(c)functions.
where K
c>0 and g
c>0 are the kinetic parameters. The effect of the additional term in Eq. (25)
is illustrated in Fig. 3, where we simulate the water-ice phase transition by placing a heat sink at
the center while the kinetic parameters are set to be K
c=5.0 GPa and g
c=1.2. By considering
two different cases where the entire 1 mm2large water-saturated square domain remains intact and
is completely damaged, Fig. 3shows that the modified driving force F
cis capable of capturing
different growth rates depending on the damage parameter d.
Undamaged (d= 0)
<latexit sha1_base64="jatIEjRHvcL9aQSkQ76EmiSKfvU=">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</latexit>
Damaged (d= 1)
<latexit sha1_base64="Izm9HIHkUx4ULVYVqMxepFHR6Q8=">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</latexit>
t= 50 s
<latexit sha1_base64="WpIyNHa9rUnpnADiNq5l2qBQ7UY=">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</latexit>
t= 100 s
<latexit sha1_base64="PZF6LEdhty14s3Twiz5DiPGMQbY=">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</latexit>
t= 200 s
<latexit sha1_base64="37JxL0KjqUm4p8+ox5AJbEsaj58=">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</latexit>
t= 400 s
<latexit sha1_base64="VVe6XKCH83+bfd4AgTuWs9qLyG4=">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</latexit>
t= 800 s
<latexit sha1_base64="s5gw1nIpuG/aKs0A27+YQ/H3DOE=">AAAErniclVNNj9MwEPXuBljKVwtHLoFeOFUJWom9gFZiJbggFkS7ldalcpxJ660/IttpWVk58jO4wm/i3+A0XambEiQsZTJ6M+/ZnhknOWfGRtHvvf2D4NbtO4d3O/fuP3j4qNt7PDKq0BSGVHGlxwkxwJmEoWWWwzjXQETC4TxZvK3i50vQhin5xV7lMBFkJlnGKLEemnZ7NnwdHkdRiC18sy405bTbjwbReoW7Trxx+mizzqa9g+84VbQQIC3lxBhHtGWUQxl2cGEgJ3RBZnBR2Ox44pjMCwuSltsxR4QRxM53wExJa3ZQcyWSm2BVKSZnjdRK0irF/wJrkzVQywQ0oIwrYj0kYUWVEESmzpdJGqXLi3jisHcLDZWcw4niaXUuxV0/LssG67IQec3hifbq+NnGCftxiPU1VjsNLuGgbU2uerTuudOQluuNOvgUfNU1fPDH+JiDJtaH8UwTn4AlSThpy/HH3U77Wus7U7aqpszP0nU+pqmybZnvmsJV1cJxq/LptvI24Z+7+BuUrjJt8cXKx71piVu/pf/abgvL0lWmTX0mvbo3637ZOSg/DE6LRek++6nQVSM1bLdyCfR/J8fn7FLWEyzqf5LVtOrhxs1nuuuMXg7io8HRp6P+yZvNEz5ET9Fz9ALF6BU6Qe/RGRoiilboB/qJfgVRMAomwbRO3d/bcJ6gGyuY/wGbQKeX</latexit>
t= 50 s
<latexit sha1_base64="WpIyNHa9rUnpnADiNq5l2qBQ7UY=">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</latexit>
t= 100 s
<latexit sha1_base64="PZF6LEdhty14s3Twiz5DiPGMQbY=">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</latexit>
t= 200 s
<latexit sha1_base64="37JxL0KjqUm4p8+ox5AJbEsaj58=">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</latexit>
t= 400 s
<latexit sha1_base64="VVe6XKCH83+bfd4AgTuWs9qLyG4=">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</latexit>
t= 800 s
<latexit sha1_base64="s5gw1nIpuG/aKs0A27+YQ/H3DOE=">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</latexit>
0.001 m
<latexit sha1_base64="jL3c0/T/famkD/KIg5t3S4d7znM=">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</latexit>
0.001 m
<latexit sha1_base64="jL3c0/T/famkD/KIg5t3S4d7znM=">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</latexit>
Figure 3: Different growth rates of the ice phases when a heat sink of ˆ
r=109W/m3is placed at a
small region at the center with the area of Ac=1010 m2.
3.2 Freezing-induced fracture in microporoelastic medium
3.2.1 Microporomechanics of the phase-changing porous medium
Focusing on the ice lens formation that involves a long period of time up to annual scales [Guodong,
1983,Harris et al.,2009], this study neglects the inertial effects such that the balance of linear momen-
tum for the three-phase mixture can be written as,
∇· (¯
σ0¯
pI) + ρg=0. (26)
9
Based on the observation that geological materials remain brittle at a low temperature [Evans et al.,
1990,Lee et al.,2002], we assume that the evolution of the damage parameter dreplicates the mecha-
nism of brittle fracture. In this case, undamaged effective stress σ0
int can be considered linear elastic,
while the stress tensor inside the damaged zone should remain σ0
dam =0unless the temperature is
below θmto form a bulk ice. Moreover, since the ice flow with respect to the solid phase is negligible
compared to that of water [Zhou and Meschke,2013,Na and Sun,2017], both σ0
int and σ0
dam can be re-
lated to the strain measure ε= (u+uT)/2 by approximating ˜vi0. Given these considerations,
we define the constitutive relations for σ0
int and σ0
dam as,
σ0
int =Kεvol I+2Gεdev ;σ0
dam = [1Sw(c)](Kiεvol I+2Giεdev), (27)
where εvol =tr (ε)and εdev =ε(εvol/3)I, while Kand Kiare the bulk moduli; and Gand Gi
are the shear moduli for the solid skeleton and the ice, respectively. Based on this approach, σ0
dam
can be interpreted as a developed stress due to the ice lens growth, since it not only depends on the
fracturing process but also on the state of the fluid phase. The net pore pressure ¯
p, on the other hand,
is a driver of deformation and fracture due to the formation of ice crystal that exerts significant excess
pressure on the premelted water film. This pressure is referred to as cryo-suction scryo that induces
the ice pressure pito be far greater than the water pressure pw. As shown in Eqs. (12) and (23), the
net pore pressure can be rewritten as ¯
p= [1Sw(c)]scryo pw, while scryo can be determined based
upon the Clausius-Clapeyron equation. In practice, however, the Clausius-Clapeyron equation is
typically replaced by an empirical model, such as the exponential [Anderson and Tice,1972] or the
van Genuchten [Van Genuchten,1980] curves, which is considered to be more accurate, evidenced by
the experiments [Koopmans and Miller,1966,Black and Tice,1989,Ma et al.,2017,Bai et al.,2018]:
Sw=exp (bBhθθmi);s
cryo =pref {Sw(c)}1
mvG 11
nvG , (28)
where bB,pref,mvG , and nvG are empirical parameters while h•i±= (•±|•|)/2 is the Macaulay
bracket. Note that we use a superscripted symbol to indicate that the corresponding variables are
empirically determined. Yet, these empirical models still yield unrealistic results in some cases. For
example, the derivative of the exponential model possesses a big jump discontinuity at the freezing
temperature θm, while s
cryo approaches infinity if Sw(c)0 if adopting the van Genuchten model.
Hence, in this study, we combine the two models to obtain the freezing retention curve that bypasses
such issues (Fig. 4):
s
cryo =pref [{exp (bBhθθmi)}]1
mvG 11
nvG , (29)
and we replace scryo with s
cryo for the net pore pressure such that: ¯
p= [1Sw(c)]s
cryo pw. For all
the numerical examples presented in Section 5, we adopt the same values used in [Na and Sun,2017,
Bai et al.,2018]: bB=0.55 K1,pref =200 kPa, mvG =0.8, and nvG =2.0.
Recall Section 2that our material of interest is a fluid-saturated phase-changing porous media.
Thus, this study considers the balance of mass for three phase constituents (i.e., solid, water and ice)
as follows:
˙
ρs+ρs∇· v=˙
ms, (30)
˙
ρw+ρw∇· v+∇· ρw˜vw=˙
mw, (31)
˙
ρi+ρi∇· v+∇· ρi˜vi=˙
mi, (32)
where ˙
ms,˙
mw, and ˙
miindicate the mass production rate for each phase constituent [Zhou and Meschke,
2013,Na and Sun,2017,Choo and Sun,2018]. Here, we assume that only the water and ice phase
constituents exchange mass among constituents (i.e., ˙
ms=0 and ˙
mw=˙
mi). Hence, summation of
10
260 265 270 275
0
2
4
6
8
10 106
mvG =0.8,n
vG =2.0
<latexit sha1_base64="j+Shg9YGl9Y987V81wHSGdHB+Bc=">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</latexit>
bB=0.55 K1,p
ref = 200 kPa,
<latexit sha1_base64="dskg7pqVGRAXkNDAG+oqaS2KrzQ=">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</latexit>
Figure 4: Freezing characteristic function [Eq. (29)] used in this study.
Eqs. (31) and (32) yields:
˙
φ{Sw(c)ρw+ [1Sw(c)]ρi}+φ˙
Sw(c)(ρwρi) + φ{Sw(c)ρw+ [1Sw(c)]ρi}∇· v+· ρw˜vw=0,
(33)
since ˜vi0, while Eq. (30) can be rewritten as,
˙
φ= (1φ)∇· v. (34)
Substituting Eq. (34) into (33) yields the mass balance equation for the three-phase mixture:
φ˙
Sw(c)(ρwρi) + {Sw(c)ρw+ [1Sw(c)]ρi}∇· v+· ρw˜vw=0. (35)
In this study, we focus on the case where the water flow inside both the porous matrix and the fracture
obeys the generalized Darcy’s law while considering the pore blockage due to the water-ice phase
transition [Luckner et al.,1989,Seyfried and Murdock,1997,Demand et al.,2019]. In other words, we
adopt the following constitutive relation between ˜vwand pw:
ww=krk
µw
(pwρwg), (36)
where ww=φ˜vwis Darcy’s velocity, kis the permeability tensor, µwis the water viscosity, and kris
the saturation dependent relative permeability:
kr=Sw(c)1/2 n1h1Sw(c)1/mvG imv G o2. (37)
3.2.2 Damage evolution
Following [Suh and Sun,2021a], this study interprets cracking as the fracture of the solid skeleton. In
other words, we define the crack driving force Fd0 as,
Fd=gd(d)
dψ0
int ;ψ0
int =1
2K(εvol)2+G(εdev :εdev ), (38)
such that the damage evolution equation can be obtained from the balance between the crack driving
force Fdand the crack resistance Rd[Dittmann et al.,2019,2020,Suh and Sun,2021a]:
Rd− Fd=gd(d)
dψ0
int +Gd
ld
(dl2
d2d) = 0. (39)
11
Recall Section 2.2 that our choice of degradation function gd(d)reduces the thermodynamic restric-
tion into ˙
d0 [Bryant and Sun,2018,Suh and Sun,2021a], which requires additional treatment to
ensure monotonic crack growth. In this study, we adopt the same treatment used in [Miehe et al.,
2015a,Bryant and Sun,2018]. By considering the homogeneity d=0, Eq. (39) yields the following
expression:
˙
d=2
(1+2H)2˙
H ≥ 0 ; H=ψ0
int
Gd/ld
, (40)
implying that non-negative ˙
dis guaranteed if ˙
H ≥ 0. As a simple remedy, we replace Hwith H
which is defined as the pseudo-temporal maximum of normalized strain energy, while considering
a critical value Hcrit that restricts the crack to initiate above a threshold strain energy [Miehe et al.,
2015b,Bryant and Sun,2018,Suh and Sun,2019,Bryant and Sun,2021]:
H=max
τ[0,t]hH − Hcriti+, (41)
such that Eq. (39) accordingly becomes:
gd(d)
dH+ (dl2
d2d) = 0. (42)
In order to model the fracture flow in a fluid-infiltrating porous media, we adopt the permeability
enhancement approach that approximates the water flow inside the fracture as the flow between two
parallel plates [Miehe and Mauthe,2016,Mauthe and Miehe,2017,Wang and Sun,2017,Suh and Sun,
2021b]:
k=kmat +kd=kmat I+d2kd(Indnd), (43)
where kmat is the effective permeability of the undamaged matrix, nd=d/k ∇ dkis the unit normal
of crack surface, and kd=w2
d/12 describes the permeability enhancement due to the crack opening
which depends on the hydraulic aperture wdbased on the cubic law. However, freezing-induced
fracture involves different situations where the pore ice crystal growth drives fracture but at the same
time blocks the pore that may hinder the water flow therein. Hence, we adopt the approach used in
[Choo and Sun,2018] which assumes a linear relationship between the hydraulic aperture wdand the
water saturation Sw(c):
wd=Sw(c)l(nd·ε·nd), (44)
where lis the characteristic length of a line element perpendicular to the fracture which is often
assumed to be equivalent to the mesh size [Miehe and Mauthe,2016,Wilson and Landis,2016]. Fur-
thermore, by assuming that the crack opening leads to complete fragmentation of the solid matrix,
we adopt the following relation for the porosity [Heider and Sun,2020,Suh and Sun,2021a]:
φ=1gd(d)(1φ0)(1 ∇· u), (45)
such that the porosity approaches 1 if the solid skeleton is completely damaged.
4 Finite element implementation
This section presents a finite element discretization of the set of governing equations described in
Section 3, and the solution strategy for the resulting discrete system. We first formulate the weak
form of the field equations by following the standard weighted residual procedure. In specific, we
adopt the Taylor-Hood element for the displacement and pore water pressure fields, while employing
linear interpolation functions for all other variables in order to remove spurious oscillations. We
then describe the operator split solution scheme that separately updates {θ,c}and {u,pw}, while the
damage parameter dis updated in a staggered manner for numerical robustness.
12
4.1 Galerkin form
Let domain Bpossesses boundary surface Bcomposed of Dirichlet boundaries (displacement Bu,
pore water pressure Bp, and temperature Bθ) and Neumann boundaries (traction Bt, water mass
flux Bw, and heat flux Bq) that satisfies:
B=BuBt=BpBw=BθBq;=BuBt=BpBw=BθBq. (46)
Then, the prescribed boundary conditions can be specified as,
u=ˆuon Bu,
pw=ˆ
pwon Bp,
θ=ˆ
θon Bθ,
;
σ·n=ˆ
ton Bt,
ww·n=ˆ
wwon Bw,
q·n=ˆ
qon Bq,
(47)
where nis the outward-oriented unit normal on the boundary surface B. Meanwhile, the following
boundary conditions on Bare prescribed for the phase fields cand d:
c·n=0 ; d·n=0. (48)
For model closure, the initial conditions for the primary unknowns {u,pw,θ,c,d}are imposed as:
u=u0;pw=pw0;θ=θ0;c=c0;d=d0, (49)
at time t=0. We also define the trial spaces Vu,Vp,Vθ,Vc, and Vdfor the solution variables as,
Vu=nu:B → R3|u[H1(B)]3,u|Bu=ˆuo,
Vp=npw:B → R|pwH1(B),pw|Bp=ˆ
pwo,
Vθ=nθ:B → R|θH1(B),θ|Bθ=ˆ
θo,
Vc=nc:B → R|cH1(B)o,
Vd=nd:B → R|dH1(B)o,
(50)
which is complimented by the admissible spaces:
Vη=nη:B → R3|η[H1(B)]3,η|Bu=0o,
Vξ=nξ:B → R|ξH1(B),ξ|Bp=0o,
Vζ=nζ:B → R|ζH1(B),ζ|Bθ=0o,
Vγ=nγ:B → R|γH1(B)o,
Vω=nω:B → R|ωH1(B)o,
(51)
where H1indicates the Sobolev space of order 1. Specifically, we employ the inf-sup stable Taylor-
Hood finite element for the displacement and pore water pressure fields, while all other field vari-
ables are discretized with linear functions that eliminates spurious oscillations [Borja et al.,1998,
Zienkiewicz et al.,1999]. By applying the standard weighted residual procedure, the weak state-
ments for Eqs. (17), (21), (26), (35), and (42) are to: find {u,pw,θ,c,d} ∈ Vu×Vp×Vθ×Vc×Vdsuch
that for all {η,ξ,ζ,γ,ω} ∈ Vη×Vξ×Vζ×Vγ×Vω,
Gu=Gp=Gθ=Gc=Gd=0, (52)
13
where:
Gu=ZBη:σdV ZBη·ρgdV ZBt
η·ˆ
tdΓ=0,
Gp=ZBξφ˙
Sw(c)(ρwρi)dV +ZBξ{Sw(c)ρw+ [1Sw(c)]ρi}∇· vdV
ZBξ·(ρwww)dV ZBw
ξ(ρwˆww)dΓ=0,
Gθ=ZBζ(ρscs+ρwcw+ρici)˙
θdV +ZBζφ[(ρwcwρici)(θθm) + ρiLθ]˙
Sw(c)dV
ZBζ·qdV ZBζˆ
r dV ZBq
ζˆ
q dΓ=0,
Gc=ZBγ1
Mc
˙
c dV +ZBγfc
cdV +ZBγ·(e2
cc)dV =0,
Gd=ZBωgd(d)
dHdV +ZBωd dV +ZBω·(l2
dd)dV =0.
(53)
(54)
(55)
(56)
(57)
4.2 Operator-split solution strategy
Although one may consider different strategies to solve the coupled system of equations [Eqs. (53)-
(57)], the solution strategy adopted in this study combines the staggered scheme [Miehe et al.,2010a]
and the isothermal operator splitting scheme [Simo and Miehe,1992,Nguyen and Selvadurai,1995].
Specifically, we first update the damage field dvia linear solver while the variables {u,pw,θ,c}
are held fixed. We then apply the isothermal splitting solution scheme that iteratively solves the
thermally-induced phase transition problem to advance {θ,c}, followed by a linear solver that up-
dates {u,pw}by solving an isothermal poromechanics problem [Suh and Sun,2021a], i.e.,
un
pw,n
θn
cn
dn
Gd=0
δu=0,δpw=0, δθ=0, δc=0
un
pw,n
θn
cn
dn+1
|{z }
Linear solver
Iterative solver
z }| {
Gθ=Gc=0
δu=0,δpw=0, δd=0
un
pw,n
θn+1
cn+1
dn+1
Gu=Gp=0
δθ=0, δc=0, δd=0
un+1
pw,n+1
θn+1
cn+1
dn+1
|{z }
Linear solver
, (58)
where we adopt an implicit backward Euler time integration scheme. The implementation of the
model including the finite element discretization and the solution scheme relies on the finite element
package FEniCS [Logg and Wells,2010,Logg et al.,2012,Alnæs et al.,2015] with PETSc scientific
computational toolkit [Abhyankar et al.,2018].
5 Numerical examples
This section presents three sets of numerical examples to verify (Section 5.1), validate (Section 5.2),
and showcase (Section 5.3) the capacity of the proposed model. Since the evolution of two phase
fields cand drequires a fine mesh to capture their sharp gradients, we limit our attention to one- or
two-dimensional simulations while considering the diffusion coefficient ecas an individual input pa-
rameter independent to the interface thickness δcwhich may additionally reduce the computational
cost [Sweidan et al.,2020,2021]. We first present two examples that simulate the latent heat effect
and 1d consolidation to verify the implementation of our proposed model. As a validation exercise,
we perform numerical experiments that replicate the physical experiments conducted by Feng et al.
14
[Feng et al.,2015], which studies the homogeneous freezing of a phase change material (PCM) em-
bedded in metal foams. Then, our final example showcases the performance of the computational
model for simulating the ice lens formation and the thermo-hydro-mechanical processes in geomate-
rials undergoing freeze-thaw cycle.
5.1 Verification exercises: latent heat effect and 1d consolidation
Our first example simulates one-dimensional freezing of water-saturated porous media to investigate
the phase transition of the fluid phase α={w,i}and the involved latent heat effect. By comparing
the results against the models presented by Lackner et al. [Lackner et al.,2005] and Sweidan et al.
[Sweidan et al.,2020], this example serves as a verification exercise that ensures the robust imple-
mentation of the heat transfer model involving phase transition [i.e., Eqs. (55) and (56)]. Hence, this
example considers a rigid solid matrix while neglecting the fluid flow, following [Lackner et al.,2005].
As illustrated in Fig. 5(a), the problem domain is a fully saturated rectangular specimen with a height
of 0.09 m and a width of 0.41 m. While the initial temperature of the entire specimen is set to be
θ0=283.15 K, the specimen is subjected to freezing with a constant heat flux of ˆ
q=100 W/m2on the
top surface, whereas all other boundaries are thermally insulated. Here, we choose the same material
properties used in [Lackner et al.,2005] and [Sweidan et al.,2020] as follows: φ0=0.42, ρs=2650
kg/m3,ρw=1000 kg/m3,ρi=913 kg/m3,cs=740 J/kg/K, cw=4200 J/kg/K, ci=1900 J/kg/K,
κs=7.694 W/m/K, κw=0.611 W/m/K, and κi=2.222 W/m/K. In addition, we set νc=0.001 m/s,
γc=0.03 J/m2,δc=0.005 m, and ec=1.25 (J/m)1/2 for the Allen-Cahn phase field model, while we
use the structured mesh with element size of he=0.6 mm and choose the time step size of t=100
sec.
ˆq= 100W/m2
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A
<latexit sha1_base64="ArbWvUFLhSoey0zXnILkQjT20IQ=">AAAEo3iclVNNbxMxEHXbBUr4auHIJRAhcYqyKBKcUBGVQEKIUjVtpDpUXu9sauKPle1NqSwf+QNc4Y/xb7CzqZRuWCQs7ezozbxne2aclZwZOxj83tjcSm7cvLV9u3Pn7r37D3Z2Hx4bVWkKI6q40uOMGOBMwsgyy2FcaiAi43CSzd7G+MkctGFKHtnLEiaCTCUrGCU2QGNs4Zt1b/zZTm/QHyxWd91Jl04PLdfB2e7Wd5wrWgmQlnJijCPaMsrBdzu4MlASOiNTOK1s8WrimCwrC5L61Zgjwghiz9fAQklr1lBzKbLrYCwPk9NGapS0SvG/wNoUDdQyAQ2o4IrYAEm4oEoIInMXqiSN0v40nTgc3EpDlHM4UzyP51Lc9VLvG6yvlShrDs90UMdPlk63l3axvsJqp8ElHLStybFFi0Y7DblfbNTB+xCqruFjOManEjSxIYynmoQELEnGSVtOOO5q2pda3xnfqpqzMEBX+ZjmyrZlvmsKx6p1x63K+6vKq4R/7hJu4F00bfHZRYgH0xK3Ycvwtd0W5t5F06Y+lUE9mEW/7DmoMAxOi5l3h2EqdGykhtVWzoH+7+SEnHXKYoJF/c+KmhYfbtp8puvO8Yt+OuwPPw97e6+XT3gbPUZP0XOUopdoD71HB2iEKOLoB/qJfiXPkg/JYXJUp25uLDmP0LWVTP4A9gKlEQ==</latexit>
B
<latexit sha1_base64="v9WXVlT38yX+ssQZyC/l0WNtutw=">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</latexit>
C
<latexit sha1_base64="oN3aao1D0tQe4BckF/G1gAyxnbc=">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</latexit>
0.03 m
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0.03 m
<latexit sha1_base64="ro84BnM7sIr5OU7FjIhDXFi/hF8=">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</latexit>
0.09 m
<latexit sha1_base64="kyFrWLUQ5kOzvZXgDIvrZVsj+hs=">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</latexit>
0.41 m
<latexit sha1_base64="aGo8LE382rG6zOOrZtwCP39USrk=">AAAEqXiclVNNb9QwEHXbAGX5auHIJbAXJKTVBq0EJ1SJSnBBtIhtVzTbynEmW3f9EdnOlsrykf/AFf4V/wZ7k0rbLEHCUiajN/Oe7ZlxVjKqzXD4e2NzK7p1+8723d69+w8ePtrZfXykZaUIjIlkUk0yrIFRAWNDDYNJqQDzjMFxNn8X4scLUJpK8cVclTDleCZoQQk2HjodDkZJnBr4ZmzM3dlOfzgYLle87iSN00fNOjjb3fqe5pJUHIQhDGttsTKUMHBxL600lJjM8QxOKlO8mVoqysqAIG41ZjHXHJvzNbCQwug1VF/x7CYYikTFrJUaJI2U7C+w0kULNZRDCyqYxMZDAi6J5ByL3PoyCS2VO0mmNvVupSDI2TSTLA/nksz2E+darIuKlzWHZcqrp88aJ+770qtrrHZaXMxAmZocerRst1WQu+VGvXQffNUVfPTH+FSCwsaH05nCPiEVOGO4K8cfdzXttNa32nWq5tSP0XV+SnJpujLft4VD1eJJp/L+qvIq4Z+7+Bs4G0xXfH7p4950xI3f0n9dt4WFs8F0qc+EV/dm2S9zDtIPg1V87uxnPxUqNFLBaisXQP53cnzOOmU5wbz+Z0VNCw83aT/Tdefo1SAZDUaHo/7e2+YJb6On6Dl6gRL0Gu2hD+gAjRFBCv1AP9Gv6GV0GE2ir3Xq5kbDeYJurIj8Ae+Mpnw=</latexit>
(a)
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A
<latexit sha1_base64="9qFjtcfYEKnTLh1vvPQ/vzj/g4w=">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</latexit>
B
<latexit sha1_base64="V0P8Bh1WblEgH7ZvSw4s+V0VQ7U=">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</latexit>
C
<latexit sha1_base64="Cpwpg6WqNsLkMhQgXq3XDjaaS0M=">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</latexit>
This study
<latexit sha1_base64="PEWR9MREEjYDIQMHpHsZixPt4rk=">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</latexit>
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285
0 0.5 1 1.5 2
105
235
245
255
265
275
285
0 0.5 1 1.5 2
105
235
245
255
265
275
285
0 0.5 1 1.5 2
105
235
245
255
265
275
285
0 0.5 1 1.5 2
105
235
245
255
265
275
285
Lackner et al. [2005]
<latexit sha1_base64="ILpy4HXmYAYdInuA4gG0xh6inxc=">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</latexit>
Sweidan et al. [2020]
<latexit sha1_base64="hFP7oyTZI5Nufa751Y1E7hSpb2U=">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</latexit>
(b)
Figure 5: (a) Schematic of geometry and boundary conditions for the 1d freezing example; (b) Tem-
perature evolution at points A, B, and C.
As shown in Figure 5(b), measured temperatures at points A, B, and C during the simulation first
linearly decrease due to the applied heat flux ˆ
quntil they reach the freezing temperature of θm=
273.15 K. As soon as the phase transition starts, the freezing front propagates through the specimen
while the release of the energy associated with the phase transition prevents the temperature decrease
(i.e., latent heat effect). Once the phase change is complete, the temperature linearly decreases over
time again since the heat transfer process is no longer affected by the latent heat. More importantly, a
good agreement with the results reported in [Lackner et al.,2005,Sweidan et al.,2020] verifies that our
proposed model is capable of capturing the thermal behavior of the phase-changing porous media.
15
x
<latexit sha1_base64="+r+6jZP7DgLnuIzSdy3jRGTBv9w=">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</latexit>
y
<latexit sha1_base64="t1lo9wUujzFBVB8BceLG3pLBo/o=">AAAEnHiclVNNbxMxEHXbBUr4auGIhAK5cIqyKBIcqzYSSKiiRSSNVIfK651N3fhjZXtTIstHTlzhx/Fv8GZTKd2wSFja2dGbec9jzzjJOTO21/u9tb0T3bl7b/d+68HDR4+f7O0/HRlVaApDqrjS44QY4EzC0DLLYZxrICLhcJbMjsr42Ry0YUp+sYscJoJMJcsYJTZAp4uLvU6v21uu9qYTr5wOWq2Ti/2d7zhVtBAgLeXEGEe0ZZSDb7dwYSAndEamcF7Y7N3EMZkXFiT16zFHhBHEXm6AmZLWbKBmIZLbYHkvTE5rqaWkVYr/BdYmq6GWCahBGVfEBkjCNVVCEJk6HGo3SvvzeOJwcAsNpZzDieJpWZfirhN7X2NdFSKvODzRQR2/XDntTtzG+garnBqXcNC2Ilv4ZpcddhpSv9yohQcQbl3DcSjjUw6a2BDGU01CApYk4aQpJ5S7nva10nfGN6qmLEzOTT6mqbJNme/rwuWttceNyoN15XXCP3cJJ/CuNE3x2XWIB9MQt2HL8DWdFubelaZJfSqDejDLftlLUGEYnBYz7z6HqdBlIzWst3IO9H8nJ+RsUpYTLKp/klU0Hx5uXH+mm87oTTfud/un/c7B4eoJ76Ln6BV6jWL0Fh2gD+gEDRFFgH6gn+hX9CIaRB+j4yp1e2vFeYZurWj0B7BBoe4=</latexit>
10 m
<latexit sha1_base64="wr1/FGfkbNfh2uTsHyidjXukkqU=">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</latexit>
ˆ
ty= 106Pa
<latexit sha1_base64="eaZJNgVFBa/h4COHHSYl8aVoQPo=">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</latexit>
ˆpw=0
<latexit sha1_base64="FVoCPhLScbutGJhqH31VZOUmSF4=">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</latexit>
(a)
0246810
105
0
2
4
6
8
10
t= 50 s
<latexit sha1_base64="x2PdnzJWtPVbxm31F2ysH6JPias=">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</latexit>
t= 150 s
<latexit sha1_base64="L0YJvBs2pe2xP4ftbI7u9K3URoQ=">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</latexit>
t= 350 s
<latexit sha1_base64="VGV186D2Fq+HqQPoHH0maNiwY+s=">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</latexit>
t= 750 s
<latexit sha1_base64="j3T0cLXw4NB/qhbp7OtDmZ+f1cQ=">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</latexit>
(b)
Figure 6: (a) Schematic of geometry and boundary conditions for Terzaghi’s problem; (b) Time-
dependent pore water pressures along the height of the specimen.
For the second verification exercise, we choose the classical Terzaghi’s 1d consolidation problem
since it possesses an analytical solution [Terzaghi et al.,1996], which can directly be compared with
the results obtained via poromechanics model [Eqs. (53) and (54)]. Our problem domain shown in
Fig. 6(a) consists of a 10 m high water-saturated linear elastic soil mass. While a 1 MPa compressive
load tyis imposed on the top surface, we replicate the single-drained condition by prescribing zero
pore water pressure at the top ( ˆ
pw=0) and a no-slip condition at the bottom. By assuming that the
temperature of the soil column remains constant during the simulation (θ=293.15 K), we only focus
on its hydro-mechanically coupled response while the material parameters are chosen as follows:
φ0=0.4, ρs=2650 kg/m3,ρw=1000 kg/m3,K=66.67 MPa, G=40 MPa, kmat =1012 m2, and
µw=103Pa·s. Here, we choose he=0.1 m and t=20 sec.
Fig. 6(b) illustrates the pore water pressure profile during the simulation at t=50, 150, 350,
and 750 s. The results show that the applied mechanical load tybuilds up the pore water pressure,
affecting the pore water to migrate towards the top surface, which leads to the dissipation of the
excess pressure over time (i.e., consolidation). By comparing the simulation results (circular symbols)
to the analytical solution (solid curves), Fig. 6(b) verifies the reliability of our model to capture the
hydro-mechanically coupled responses.
5.2 Validation example: homogeneous freezing
This section compares the results obtained from the numerical simulation against the physical exper-
iment conducted by Feng et al. [Feng et al.,2015]. This experiment is used as a benchmark since it
considers the unidirectional freezing of distilled water filled in porous copper foams, which does not
involve a fracturing process and yields a clear water-ice boundary layer due to the microstructural
attributes of the host matrix. As schematically shown in Fig. 7(a), a 30 mm wide, 50 mm long water-
saturated copper foam is mounted on a 4 mm thick copper block. While the initial temperature is
measured to be θ0=285.55 K, the experiment is performed by applying a constant temperature of
ˆ
θ=264.15 K at the bottom part of the copper block at t=0. Temperature measurements during the
experiment are made by three thermocouples (TC2-TC4) located at 10 mm, 28 mm, and 46 mm from
the bottom of the foam (AA’), whereas TC1 records the temperature of the block. For the numerical
simulation, instead of considering the problem domain as a layered material, we only focus on the
16
A0
<latexit sha1_base64="zXFOejrB40RFamCuISH4G7rLpz0=">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</latexit>
A
<latexit sha1_base64="DH2FKiEnN77qmkMMGEBzj5YobRQ=">AAAEo3iclVNNbxMxEHXbBUr4auHIZSFC4hRlUSQ4oSIqgYQQpWraSHWovN7Z1MQfK9ubUlk+8ge4wh/j3+DNJlKyYZGwtLOjN/Oe7ZlxWnBmbL//e2t7J7px89bu7c6du/fuP9jbf3hqVKkpDKniSo9SYoAzCUPLLIdRoYGIlMNZOn1bxc9moA1T8sReFzAWZCJZziixARphC9+se+Mv9rpJrz9fcX/DWYa6aLGOLvZ3vuNM0VKAtJQTYxzRllEOPu7g0kBB6JRM4Ly0+auxY7IoLUjqV2OOCCOIvdwAcyWt2UDNtUjXwao8TE4aqZWkVYr/BdYmb6CWCWhAOVfEBkjCFVVCEJm5UCVplPbnydjh4JYaKjmHU8Wz6lyKu27ifYP1tRRFzeGpDur4ycKJu0mM9RKrnQaXcNC2Jlctmjfaacj8fKMOPoRQdQ0fwzE+FaCJDWE80SQkYElSTtpywnFX077U+s74VtWMhQFa5mOaKduW+a4pXFUtHrUqH64qrxL+uUu4gXeVaYtPr0I8mJa4DVuGr+22MPOuMm3qExnUg5n3y16CCsPgtJh6dxymQleN1LDayhnQ/52ckLNJmU+wqP9pXtPWHm67c/qilwx6g8+D7sHrxRPeRY/RU/QcJeglOkDv0REaIoo4+oF+ol/Rs+hDdByd1KnbWwvOI7S2ovEf+jalEg==</latexit>
ˆ
= 264.15 K
<latexit sha1_base64="1+jTMvWJ8gPQKfBAOnOyfYTnBt8=">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</latexit>
30 mm
<latexit sha1_base64="NG8xmSN/KlU4e40XagTz1NKDc38=">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</latexit>
4mm
<latexit sha1_base64="X+Fcld7YuL40lrmdduvZAfCB260=">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</latexit>
50 mm
<latexit sha1_base64="g1Addfoys1wI9TNo1pmCVGF2lzc=">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</latexit>
x
<latexit sha1_base64="+r+6jZP7DgLnuIzSdy3jRGTBv9w=">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</latexit>
y
<latexit sha1_base64="t1lo9wUujzFBVB8BceLG3pLBo/o=">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</latexit>
ˆpw=0
<latexit sha1_base64="FjkplLuwcCGAXJK2OwIatMmBMfg=">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</latexit>
TC1
<latexit sha1_base64="oVQkJqI6Gd3xAm7IXo09OsG2G6U=">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</latexit>
TC2
<latexit sha1_base64="nczcZSVR+p8CsvlbOvKyxbTq+0c=">AAAEpXiclVNNbxMxEHXbBUr4auHIJZADnKJsFQlOqFIrwaWioCSN6IbI651N3fhjZXtTKstH/gFX+F/8G+xsKqUbFglLOzt6M+/ZnhmnBaPa9Hq/t7Z3ojt37+3ebz14+Ojxk739pyMtS0VgSCSTapxiDYwKGBpqGIwLBZinDM7S+VGIny1AaSrFwFwXMOF4JmhOCTYe+pIY+Gbs4OjATfc6vW5vudqbTrxyOmi1Tqf7O9+TTJKSgzCEYa0tVoYSBq7dSkoNBSZzPIPz0uRvJ5aKojQgiFuPWcw1x+ZiA8ylMHoD1dc8vQ2GAlExq6UGSSMl+wusdF5DDeVQg3ImsfGQgCsiOccis75OQkvlzuOJTbxbKghyNkkly8K5JLOd2Lka67LkRcVhqfLqyYuV0+7E7UTdYJVT42IGylTk0KRlq62CzC03aiXH4Kuu4MQf42MBChsfTmYK+4RE4JThphx/3PW0r5W+1a5RNaN+hG7yE5JJ05T5vi4cqtYeNyofryuvE/65i7+Bs8E0xedXPu5NQ9z4Lf3XdFtYOBtMk/pMeHVvlv0yFyD9MFjF585+9lOhQiMVrLdyAeR/J8fnbFKWE8yrf5pXtPBw4/oz3XRGB9243+1/6ncO362e8C56jl6i1yhGb9Ah+oBO0RARJNAP9BP9il5FJ9EgGlWp21srzjN0a0XTP4nopa0=</latexit>
TC3
<latexit sha1_base64="MaFUDIRksUgQ3/FpRgl7fzZSLPI=">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</latexit>
TC4
<latexit sha1_base64="b2KKISDBU/IAfGcI5REm1n9DFd8=">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</latexit>
Foam 1:
<latexit sha1_base64="Cb1sBudyS+aL399EljpoY2xX6q0=">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</latexit>
Foam 2:
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0=0.96
s= 62.855 W/m/K
<latexit sha1_base64="tpfiNUEGvmtRq9mrOqTky85951w=">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</latexit>
0=0.98
s= 44.48 W/m/K
<latexit sha1_base64="cVGtG2sEJsXMPM2TMsz4pzGZdVA=">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</latexit>
(a)
0 60 120 180
260
265
270
275
280
285
290
TC1
<latexit sha1_base64="5wInNpOZkbCz4z3w1RvcKmKmQsw=">AAAEpXiclVNNbxMxEHXbBUr4auHIJZADnKIsigTHSq1ULhUFJWlEHSKvdzZ144+V7U1bWXvkH3CF/8W/wc6mUrphkbC0s6M3857tmXGSc2Zsr/d7a3snunf/we7D1qPHT54+29t/PjKq0BSGVHGlxwkxwJmEoWWWwzjXQETC4SyZH4b42QK0YUoO7E0OE0FmkmWMEuuhr9jCtXWDw7ic7nV63d5ytTedeOV00GqdTvd3vuNU0UKAtJQTYxzRllEOZbuFCwM5oXMyg/PCZh8mjsm8sCBpuR5zRBhB7MUGmClpzQZqbkRyFwwFYnJWSw2SVin+F1ibrIZaJqAGZVwR6yEJV1QJQWTqfJ2kUbo8jycOe7fQEOQcThRPw7kUd524LGusy0LkFYcn2qvjVyun3YnbWN9ilVPjEg7aVuTQpGWrnYa0XG7Uwkfgq67hxB/jUw6aWB/GM018ApYk4aQpxx93Pe1bpe9M2aiaMj9Ct/mYpso2ZR7XhUPV2uNG5aN15XXCP3fxNyhdME3x+ZWPe9MQt35L/zXdFhalC6ZJfSa9ujfLftkLUH4YnBbz0n3xU6FDIzWst3IB9H8nx+dsUpYTLKp/klW08HDj+jPddEbvunG/2//c7xz0V094F71Er9FbFKP36AB9RKdoiCiS6Af6iX5Fb6KTaBCNqtTtrRXnBbqzoukfgq2log==</latexit>
ˆ
AA’ = 20.8e0.1pt+ 264.75 K
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(b)
Figure 7: (a) Schematic of the experimental setup for the unidirectional freezing test conducted in
[Feng et al.,2015]; (b) Temperature boundary condition applied at the bottom surface of the copper
foam (AA’) for the numerical simulation.
water-saturated copper foam and apply time-dependent Dirichlet boundary condition on AA’ by us-
ing the temperature measured by TC1 [Fig. 7(b)]. We also assume an unlimited water supply from
the top surface by imposing ˆ
pw=0 and applying a fixed boundary condition at the bottom part of
the foam. Moreover, we consider two different types of copper foams (Foam 1 and Foam 2) with dif-
ferent initial porosity and thermal conductivity [Fig. 7(a)]. As summarized in Table 1, our numerical
simulation directly adopts the same thermal properties compared to the physical experiment whereas
the solid phase thermal conductivities of the foams are computed based upon the effective properties
reported in [Feng et al.,2015]. For all other material parameters that are not specified in [Feng et al.,
2015], we choose the properties that resemble those of the water-saturated copper foam. In this sec-
tion, the Allen-Cahn parameters are chosen as: νc=0.0001 m/s, γc=0.65 J/m2,δc=0.0001 m, and
ec=0.75 (J/m)1/2, while adopting a structured mesh with he=2.5 mm and t=60 sec.
Fig. 8illustrates the evolution of the freezing front within a water-saturated copper foam (Foam
2). In both the physical and numerical experiments, water freezing starts from the bottom (AA’) and
migrates towards the upper part of the foam over time, depending on the conductive heat transfer
process. While it shows a qualitative agreement between the two, Fig. 9quantitatively confirms the
validity of our model, where we use the circular symbols to indicate the experimental measurements
whereas the solid curves denote the numerical results. As shown in Fig. 9(a), since Foam 1 possesses
higher solidity (lower porosity) compared to Foam 2, the water-ice interface tends to grow relatively
faster because it exhibits higher effective thermal conductivity. In addition, temperature variations
illustrated in Fig. 9(b) clearly show the interplay between the thermal boundary layer growth and the
latent heat, resulting in a nonlinear evolution of the freezing front. Although has not been measured
experimentally, we further investigate the time-dependent hydro-mechanical response of the speci-
men from the simulation results shown in Fig. 10. Based on the freezing retention curve [Eq. (29)]
adopted in this study, positive suction starts to develop if θ<θmwhile the region where s
cryo >0
evolves over time following the same trajectory of that of the freezing front [Fig. 10(a)]. This process
also involves a volumetric expansion of the specimen that leads to an increase of the vertical displace-
ment as shown in Fig. 10(b), due to the difference between water (ρw) and ice densities (ρi). It should
be also noted that the freezing front always exhibits the largest vertical displacement, implying that
17
Parameter Description [Unit] Value Reference
ρsIntrinsic solid mass density [kg/m3] 7800.0 -
ρwIntrinsic water mass density [kg/m3] 1000.0 [Feng et al.,2015]
ρiIntrinsic ice mass density [kg/m3] 920.0 [Feng et al.,2015]
csSpecific heat of solid [J/kg/K] 0.385 ×103-
cwSpecific heat of water [J/kg/K] 4.216 ×103[Feng et al.,2015]
ciSpecific heat of ice [J/kg/K] 2.040 ×103[Feng et al.,2015]
κsThermal conductivity of solid [W/m/K] 62.855, 44.48 [Feng et al.,2015]
κwThermal conductivity of water [W/m/K] 0.56 [Feng et al.,2015]
κiThermal conductivity of ice [W/m/K] 1.90 [Feng et al.,2015]
KBulk modulus of solid skeleton [Pa] 93.75 ×109-
KiBulk modulus of ice [Pa] 5.56 ×109-
GShear modulus of solid skeleton [Pa] 33.58 ×109-
GiShear modulus of ice [Pa] 4.20 ×109-
φ0Initial porosity [-] 0.96, 0.98 [Feng et al.,2015]
kmat Matrix permeability [m2] 3.25 ×107-
µwViscosity of water [Pa·s] 1.0 ×103-
αv,int Volumetric expansion coefficient [-] 1.0 ×103-
Table 1: Material parameters for the validation exercise.
the water migration towards the freezing front induced by the suction triggers the consolidation pro-
cess above the frozen area, resulting in a small volumetric compression therein. This observation
agrees with the explanation in [Amato et al.,2021] where the consolidation front of a frozen soil has
been observed experimentally, which corroborates the applicability of our proposed model.
Experiment
<latexit sha1_base64="LmlihOf9FPI2cc/NgMc43yE9f5U=">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</latexit>
This study
<latexit sha1_base64="PEWR9MREEjYDIQMHpHsZixPt4rk=">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</latexit>
(Feng et al. [2015])
<latexit sha1_base64="ZGajt3pem05WrihwG3FFhfH7Ldw=">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</latexit>
t= 10 min
<latexit sha1_base64="nSTO3x1zGmMaW9TziSpADIJPz7s=">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</latexit>
t= 60 min
<latexit sha1_base64="dFbcbTl52tHVOGPP8SyZ9j/Tz6M=">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</latexit>
t= 120 min
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t= 180 min
<latexit sha1_base64="OQyLP5B2s4Ypa9Avwl6PePYN5aQ=">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</latexit>
Figure 8: Comparison between the physical and numerical experiments on the evolution of the water-
ice interface.
18
0 60 120 180
0
0.01
0.02
0.03
0.04
0.05
Foam 2
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Foam 1
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(a)
0 60 120 180
260
265
270
275
280
285
290
TC2
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TC4
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TC3
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(b)
Figure 9: (a) Evolution of the freezing front over time; (b) Temperature variation within Foam 2
measured from TC2, TC3, and TC4.
0 0.01 0.02 0.03 0.04 0.05
0
1
2
3
4106
Freezing direction
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(a)
0 0.01 0.02 0.03 0.04 0.05
0
0.5
1
1.5 10-6
(b)
Figure 10: Hydro-mechanical response of Foam 2 subjected to freezing: (a) cryo-suction (s
cryo) and
(b) vertical displacement (uy) profiles.
5.3 Freeze-thaw action: multiple ice lens growth and thawing in heterogeneous
soil
In this section, we showcase the capability of our proposed model by simulating the formation and
melting of multiple ice lenses inside a heterogeneous clayey soil specimen. As illustrated in Fig. 11(a),
the problem domain is 0.04 m wide and 0.1 m long soil column that possesses a random porosity
profile along the vertical axis with a mean value of φref =0.4. In addition, we introduce a set of
heterogeneous material properties that solely depends on the spatial distribution of initial porosity φ0.
Specifically, we adopt a phenomenological model proposed by [Uyanık,2019] for the shear modulus
G, while we use a power law for the critical energy release rate Gdsimilar to [Dunn et al.,1973,Wang
19
and Sun,2017]:
G=3
212ν
1+νexp [10(1φ0)][MPa] ; Gd=Gd,ref 1φ0
1φref nφ
. (59)
Here, we assume that the Poisson’s ratio remains constant ν=0.25 throughout the entire domain
while we set Gd,ref =4.5 N/m and nφ=10. For all other material properties that are homogeneous,
as summarized in Table 2, we choose values similar to those of the clayey soil. It should be noted
that we adopt αv,dam =0.08 which is identical to the theoretical value of 1 ρi/ρwfor the expansion
coefficient, whereas we set αv,int =0.01αv,dam due to the existence of thin water film between the
intact solid and the pore ice. Meanwhile, the parameters for the Allen-Cahn phase field equation are
chosen as: νc=0.0001 m/s, γc=0.65 J/m2,δc=0.0001 m, and ec=1.0 (J/m)1/2, whereas we set
he=0.5 mm and t=60 sec.
0.2 0.4 0.6
0
0.02
0.04
0.06
0.08
0.1
x
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y
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ˆpw=0
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