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International Journal for Numerical and Analytical Methods in Geomechanics manuscript No.
(will be inserted by the editor)
Multi-phase-field microporomechanics model for simulating ice lens1
growth in frozen soil2
Hyoung Suk Suh ·WaiChing Sun3
4
Received: April 22, 2022/ Accepted: date5
Abstract This article presents a multi-phase-field poromechanics model that simulates the growth and6
thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In7
this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by8
the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respec-9
tively. The evolution of two phase fields is induced by their own driving forces that capture the physical10
mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled11
with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack12
growth, and phase transition can be observed numerically. Unlike phenomenological approaches that in-13
directly captures the freezing influence on the shear strength, the multi-phase-field model introduces an14
immersed approach where both the homogeneous freezing and the ice lens growth are distinctively cap-15
tured by the freezing characteristic function and the driving force accordingly. Verification and validation16
examples are provided to demonstrate the capacities of the proposed models.17
Keywords ice lens, phase field, frozen soil, thermo-hydro-mechanics, phase transition18
1 Introduction19
Ice lens formation at the microscopic scale is a physical phenomenon critical for understanding the physics20
of frost heave and thawing settlement occurred at the field scale under the thermal cycles. Since ice lens21
may affect the freeze-thaw action and cause frost heave and thawing settlement sensitive to the changing22
climate and environment conditions, knowledge on the mechanism for the ice lens growth is of practical23
value for many civil engineering applications in cold regions [1,2,3,4,5]. For example, substantial heaving24
and settlement caused by the sequential formations and thawing of ice lenses lead to uneven deformation25
of the road which also damages the tires, suspension, and ball joints of vehicles. In the United States alone,26
it was estimated that two billion dollars had been spent annually to repair frost damage of roads [6].27
Moreover, extreme climate change over the last few decades has brought increasing attention to permafrost28
degradation, since unusual heat waves may cause weakening of foundations and increase the likelihood29
of landslides triggered by the abrupt melting of the ice lens [7,8,9,10,11]. Under these circumstances,30
both the fundamental understanding of the ice lens growth mechanisms and the capacity to predict and31
simulate the effect beyond the one-dimensional models becomes increasingly important.32
Since the pioneering work on the ice lens by Stephan Taber in the early 20th century [12,13], there has33
been a considerable amount of progress in the geophysics and fluid mechanics community to elucidate34
the mechanisms in the ice segregation process (e.g., [14] and references cited therein). During the freezing35
phase, it is now known that the crystallized pore ice surrounded by a thin pre-melted water film develops a36
Corresponding author: WaiChing Sun
Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, 614 SW Mudd, Mail
Code: 4709, New York, NY 10027 Tel.: 212-854-3143, Fax: 212-854-6267, E-mail: wsun@columbia.edu
2 Hyoung Suk Suh, WaiChing Sun
suction pressure (i.e., cryo-suction) that attracts the unfrozen water towards the freezing front [15,16,17].37
These films remain unfrozen below the freezing temperature and form an interconnected flow network38
that supplies water to promote ice crystal growth. Accumulation of pore ice crystals accompanies the void39
expansion and micro-cracking of the host matrix, which may result in the formation of a horizontal lens40
of segregated ice. However, despite these substantial amounts of work, the criterion for the ice lens ini-41
tiation and its detailed mechanism still remains unclear. Based on the thermo-hydraulic model proposed42
by Harlan [18], Miller [19,20,21] introduces a concept of stress partitioning and assumed that an ice lens43
starts to form if the solid skeleton experiences tensile stress. This idea has been further adopted and fur-44
ther generalized in [22,23] via an asymptotic method. Gilpin [24] suggests that the ice lens formation takes45
place when the ice pressure reaches the particle separation pressure depending on the particle size and46
the interfacial tension between the water and ice, whereas Zhou and Li [25] propose the idea of separation47
void ratio as a criterion for the ice lensing. Konrad and Morgenstern [26] present an alternative approach48
that can describe the formation and growth of a single ice lens based on segregation potential, of which the49
applicability has been demonstrated in [27,28,29]. On the other hand, Rempel [30,31] develops regime50
diagrams that delineate the growth of a single lens, multiple lenses, and homogeneous freezing. In this51
line of work, the one-dimensional momentum and mass equilibrium equations are coupled with the heat52
flow in a step-freezing Stefan configuration to calculate the intermolecular force that drives the premelted53
fluid to the growing ice lenses. While the proposed method is helpful for estimating the lens thickness and54
spacing, the one-dimensional setting is understandably insufficient for the geo-engineering applications55
that require understanding of the implication of ice lenses on the shear strength. More recently, Style et al.56
[32] propose a new theory on the ice lens nucleation by considering the cohesion of soil and the geometric57
supercooling of the unfrozen water in the pore space. Although the aforementioned studies formed the58
basis to shed light on explaining the ice lens formation, they are limited to the idealized one-dimensional59
problems and often idealized soil as a linear elastic material and hence not sufficient for applications that60
require a more precise understanding of the constitutive responses of the ice-rich soil.61
Meanwhile, within the geomechanics and geotechnical engineering community, a number of theories62
and numerical modeling frameworks have been proposed based on the mixture theory and thermodynam-63
ics principles [33,34,35,36] with a variety of complexities and details. By adopting the premelting theory64
and considering the frozen soil as a continuum mixture of the solid, unfrozen water, and ice constituents,65
the freezing retention behavior of frozen soil can be modeled in a manner similar to those for the unsatu-66
rated soil. The resultant finite element implementation of these models enables us to simulate freeze-thaw67
effects in two- or three-dimensional spaces often with more realistic predictions on the solid constitutive68
responses. Nevertheless, the presence of crystal ices in the pores and that inside the expanded ice lens are69
often represented via phenomenological laws [36,37]. Since the morphology, physics, and the mechanisms70
as well as the resultant mechanical characteristics of the ice lens and ice crystals in pores are profoundly71
different, it remains difficult to develop a predictive phenomenological constitutive law for an effective72
medium that represents the multi-constituent frozen soil with ice lenses [38].73
This study is an attempt to reconcile the fluid mechanics and geotechnical engineering modeling ef-74
forts on modeling the frozen soil under changing climates. Our goal is to (1) extend the field theory for75
ice lens such that it is not restricted to one-dimensional problems and (2) introduce a framework that may76
incorporate more realistic path-dependent constitutive laws. As such, the coupling mechanism among77
phase transition, fluid diffusion, heat transfer, and solid mechanics can be captured without solely relying78
on phenomenological material laws. In particular, we introduce a mathematical framework and a corre-79
sponding finite element solver that may distinctively capture the physics of ice lens and freezing/thawing.80
We leverage the implicit representation of complex geometry afforded by a multi-phase-field framework81
to first overcome the difficulty of capturing the evolving geometry of the ice lens. By considering the ice82
lens as segregated bulk ice inside the freezing-induced fracture, we adopt two phase field variables that83
represent the state of the fluid phase constituent and the regularized crack topology, respectively. This84
treatment enables us to take account of the brittle fracture that may occur during ice lens growth and85
explicitly incorporate the addition and vanishing shear strength and bearing capacity of the ice lens un-86
der different environmental conditions. The phase transition of the fluid is modeled via the Allen-Cahn87
equation [39,40], while we adopt the phase field fracture framework to model brittle cracking in a solid88
matrix [41,42,43]. The resultant framework may provide a fuller picture to analyzing the growth of the89
Multi-phase-field model for ice lens growth 3
ice lens in the frozen soil, while verification exercises also confirm that the model may reduce to a classical90
thermo-hydro-mechanical model and isothermal poromechanics model under limited conditions.91
The rest of the paper is organized as follows. Section 2summarizes the necessary ingredients for the92
mathematical framework, while we present the multi-phase-field microporomechanics model that de-93
scribes the coupled behavior of a fluid-saturated phase-changing porous media in Section 3. For complete-94
ness, the details of the finite element formulation and the operator splitting solution strategy are discussed95
in Section 4. Finally, numerical examples are given in Section 5to verify, validate, and showcase the model96
capacity, which highlights its potential by simulating the growth and melting of multiple ice lenses.97
As for notations and symbols, bold-faced and blackboard bold-faced letters denote tensors (including98
vectors which are rank-one tensors); the symbol ’·’ denotes a single contraction of adjacent indices of two99
tensors (e.g., a·b=aibior c·d=cij djk); the symbol ‘:’ denotes a double contraction of adjacent indices of100
tensor of rank two or higher (e.g., C:ε=Cijk l εkl ); the symbol ‘⊗’ denotes a juxtaposition of two vectors101
(e.g., a⊗b=aibj) or two symmetric second-order tensors [e.g., (α⊗β)ijkl =αij βkl ]. We also define identity102
tensors: I=δij,I=δikδjl , and ¯
I=δil δjk, where δij is the Kronecker delta. As for sign conventions, unless103
specified, tensile stress and dilative pressure are considered positive.104
2 Kinematics and effective stress principle for frozen soil with ice lens105
In this section, we introduce the ingredients necessary to derive the field theory for the phase field mod-106
eling of frozen soil presented later in Section 3. Similar to the treatments in [33], [34], and [35], we first107
assume that the frozen soil is fully saturated with either water or ice and therefore idealize the frozen soil108
as a three-phase continuum mixture that consists of solid, water, and ice phase constituents whereas the ice109
lens is a special case in which the solid skeleton no longer holds bearing capacity. This treatment enables110
us to formulate a multi-phase-field approach to employ two phase field variables as indicator functions111
for the state of the pore fluid (in ice or water form) [40,44,45] and that of the solid skeleton (in damage112
or intact form) [41,42,43]. We then extend the effective stress theory originated from damage mechanics113
[46] to incorporate the internal stress of ice lenses caused by the deformation of the effective medium into114
the Bishop’s effective stress principle for frozen soil where the introduction of phase field provide smooth115
transition of the material states for both the pore fluid and the solid skeleton. This procedure allows us to116
incorporate both the capillary pressure of the ice crystal surrounded by the water thin film as well as the117
volumetric and deviatoric stresses triggered by the deformation of the ice lens.118
2.1 Continuum representation and kinematics119
Based on the mixture theory, we idealize our target material as a multiphase continuum where the solid,120
water, and ice phase constituents are overlapped. For simplicity, this study assumes that there is no gas121
phase inside the pore such that the pore space is either occupied by water or ice. The volume fractions of122
each phase constituent are defined as,123
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 =124
dVs+dVw+dVidenote the total elementary volume of the mixture. Note that an index used as a subscript125
indicates the intrinsic property of a phase constituent, while it is used as a superscript when referring to126
a partial property of the entire mixture. By letting ρs,ρw, and ρidenote the intrinsic mass densities of the127
solid, water, and ice, respectively, the partial mass densities of each phase constituent are given by,128
ρs=φsρs;ρw=φwρw;ρi=φiρi;ρs+∑
α={w,i}
ρα=ρ, (2)
4 Hyoung Suk Suh, WaiChing Sun
where ρis the total mass density of the entire mixture. We also define the saturation ratios for the fluid129
phase constituents α={w,i}as:130
Sw=φw
φ;Si=φi
φ;∑
α={w,i}
Sα=1, (3)
where φ=1−φsis the porosity.131
Since the solid (s), water (w), and ice (i) phases do not necessarily follow the same trajectory, each132
constituent possesses its own Lagrangian motion function that maps the position vector of the current133
configuration xat time tto their reference configurations. In this study, we adopt a kinematic description134
that traces the motion of the solid matrix by following the classical theory of porous media [47,48,49,50].135
Hence, the motion of the solid phase is described by using the Lagrangian approach via its displacement136
vector u(x,t), whereas the fluid phase (α={w,i}) motions are described by the modified Eulerian ap-137
proach via relative velocities ˜vwand ˜vi, instead of their own velocity fields vwand vi, i.e.,138
˜vα=vα−v, (4)
where v=˙uis the solid velocity, while ˙
(•) = d(•)/dtis the total time derivative following the solid139
matrix.140
2.2 Multi-phase-field approximation of freezing-induced crack141
In this current study, we assume that the path-dependent constitutive responses of the frozen soil is due142
to the fracture in the brittle regime and the growth/thaw of the ice lens in the void space that could be143
opened by the expanded ice. While plasticity of the solid skeleton as well as the damage and creeping of144
the segregated ice may also play important roles on the mechanisms of the frost heave and thaw settlement,145
they are out of the scope of this study. As such, this study follows Miller’s theory which assumes that146
a new ice lens may only form if and only if the compressive effective stress becomes zero or negative147
[19,20,21,51]. Since opening up the void space is a necessary condition for the ice lens to grow inside, we148
introduce a phase field model that captures the crack growth potentially caused by the ice lenses growth. In149
this work, our strategy is to adopt diffuse approximations for both the phase transition of the pore fluid and150
the crack topology, where each requires a distinct phase field variable. As illustrated in Fig. 1, introducing151
two phase fields not only enables us to distinguish the homogeneous freezing from the ice lens growth but152
also leads to a framework that can be considered as a generalization of a thermo-hydro-mechanical model.153
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
Fig. 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 freezing154
of water (melting of ice) in a regularized manner [44,45]. In other words, we employ a diffuse representa-155
tion of the ice-water interface using variable cthat is a function of xand t:156
c=c(x,t)with
c=0 : completely frozen,
c=1 : completely unfrozen,
c∈(0, 1): diffuse ice-water interface,
(5)
Multi-phase-field model for ice lens growth 5
which is the solution of the Allen-Cahn phase field equation [39,40] that will be presented later in Section157
3.1. Based on this setting, we consider the degree of saturation of water as an interpolation function of the158
phase field c, i.e., Sw=Sw(c), that monotonically increases from 0 to 1 as,159
Sw(c) = c3(10 −15c+6c2), (6)
which guarantees smooth variation of different material properties between ice and water and at the same160
time enables us to properly include the latent heat effect in the energy balance equation in Section 3.1.1.161
Note that the evolution of the phase field variable citself does not necessarily imply the ice lens growth162
since both the homogeneously frozen region and segregated ice can reach c=0, regardless of the level of163
the effective stress or stored energy that drives the crack growth (Fig. 1).164
The second phase field variable d∈[0, 1]adopted in this study is a damage parameter that treats the165
sharp discontinuity as a diffusive crack via implicit function [41,42,43,52]. In particular, we have:166
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 density167
Γd(d,∇d)over a body B, i.e.,168
AΓ≈AΓd=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 crack169
resistance force Rdcan be expressed as,170
Rd=∂Wd
∂d−∇·∂Wd
∂∇d;Wd=GdΓd(d,∇d), (9)
where Gdis the critical energy release rate that quantifies the resistance to cracking. As hinted in Fig. 1, in or-171
der to guarantee crack irreversibility, the thermodynamic restriction ˙
Γd≥0 must be satisfied [42,53,54,55]172
unlike the reversible freezing and thawing process. In other words, we require non-negative crack driving173
force Fdbased on the microforce balance. Among multiple options, this study adopts the most widely174
used quadratic degradation function gd(d) = (1−d)2following [53], that reduces the thermodynamic175
restriction into ˙
d≥0 [56,57] and satisfies the following conditions:176
gd(0) = 1 ; gd(1) = 0 ; ∂gd(1)
∂d=0 ; ∂gd(d)
∂d≤0 for d∈[0, 1]. (10)
Based on this setting, we define an indicator function χi∈[0, 1]for the segregated ice inside the freezing-177
induced fracture as follows:178
χi(c,d) = [1−Sw(c)][1−gd(d)], (11)
such that χi=1 implies the formation of the ice lens, which is different from the in-pore crystallization of179
the ice phase constituent.180
2.3 Effective stress principle181
Leveraging the similarities between freezing/thawing and drying/wetting processes, Miller and co-workers182
[19,20,21,51] proposed the concept of neutral stress that partitions the net pore pressure ¯
pinto the pore183
water and pore ice pressures (pwand pi), respectively:184
¯
p=Sw(c)pw+ [1−Sw(c)]pi. (12)
Clearly, Eq. (12) alone cannot capture the deviatoric stress induced by the deformation of the ice lens. Pre-185
vious efforts on modeling frozen soil often relies on an extension of critical state theory that evolves the186
6 Hyoung Suk Suh, WaiChing Sun
yield function according to the degree of saturation of ice (and therefore introduces the dependence of the187
tensile and shear strength on the presence of ice) [33,35]. However, this treatment is not sufficient to con-188
sider the soil that may become brittle at low temperature due to the low moisture content and the influence189
of ice lens on the elasticity. Hence, this study extends Miller ’s approach into a phase field framework by190
decomposing the effective stress tensor ¯σ0into two partial stresses for the solid and ice lens via the damage191
phase field doubled as a weighting function, i.e.,192
¯σ0=gd(d)σ0
int + [1−gd(d)]σ0
dam. (13)
where the second term on the right hand side of Eq. (13) depends on the saturation Sw(c). Specifically,193
the effective stress contribution from the solid skeleton σ0
int degrades due to the damage when the ice194
lens grows, but may also evolve by the change of σ0
dam in the presence of ice lens [for instance, see Eq.195
(29) in Section 3.2]. From a physical point of view, we propose Eq. (13) based on the assumption that196
there is no relative motion between the solid skeleton and the ice lens in the sense that the ice lenses197
cannot be squeezed out from the host matrix, which also has a benefit of ensuring continuous displacement198
field. Similar models that capture the constituent responses of porous media consisting of multiple solid199
constituents can also be found in [58]. In addition, this study also considers the volumetric expansion due200
to the phase transition from water to ice while neglecting the thermal expansion or contraction of each201
phase constituent. Specifically, we incorporate an additional term for the total Cauchy stress tensor σthat202
describes phase-transition-induced volumetric expansion, which stems from the Helmholtz free energy203
functions of the solid and ice phase constituents postulated in [59,60]. Hence, similar to [33,35], as a204
modification of the Bishop’s equation, the total Cauchy stress tensor can be expressed as follows:205
σ=¯σ0−¯
pI−φ[1−Sw(c)]¯
αvKiI, (14)
where ¯
αv=gd(d)αv,int + [1−gd(d)]αv,dam is the net volumetric expansion coefficient which is influenced206
by the evolution of the fracture. In particular, we assume that the volumetric expansion coefficient of the207
ice lens αv,dam is greater than that of the pore ice crystal αv,int due to the degradation of the solid skeleton.208
3 Multi-phase-field microporomechanics model for phase-changing porous media209
This section presents the balance principles and constitutive laws that capture the thermo-hydro-mechanical210
behavior of the phase-changing porous media. We first introduce the coupled field equations that govern211
the heat transfer and the ice-water phase transition processes which involve the latent heat effect. Unlike212
previous studies that model the phase transition of the pore fluid by using the semi-empirical approach213
which links either the Gibbs-Thomson equation [34] or the Clausius-Clapeyron equation [33,35] with the214
van Genuchten curve [61], we adopt the Allen-Cahn type phase field model [39,40] with a driving force215
that depends both on the temperature and the damage. We then present microporomechanics and phase216
field fracture models that complete the set of governing equations, which is not only capable of simulating217
freeze-thaw action but also the freezing-induced or hydraulically-driven fractures. The implications of our218
model will be examined via numerical examples in Section 5.219
3.1 Thermally induced phase transition220
3.1.1 Heat transfer221
Since underground freezing and thawing processes may span over long temporal scales, this study em-222
ploys a single temperature field θby assuming that all the phase constituents reach a local thermal equilib-223
rium instantly [57]. We also neglect thermal convection by considering the case where the target material224
possesses low permeability. Let edenote the internal energy per unit volume and qthe heat flux. Then, the225
energy balance of the entire mixture can be expressed as [57,62],226
˙
e=−∇·q+ˆ
r, ; e=es+∑
α={w,i}
eα, (15)
Multi-phase-field model for ice lens growth 7
where ˆ
rindicates the heat source/sink, es=ρscsθand eα=ραcαθare the partial energies for the solid227
and fluid phase constituents, respectively, while csand cαare their heat capacities. Although the freezing228
temperature of water (melting temperature of ice) depends on the curved phase boundaries due to the229
intermolecular forces, i.e., freezing point depression [63], for simplicity, we assume that the freezing tem-230
perature of water remains constant θm=273.15 K, so that the internal energy of the entire mixture ein231
Eq. (15) can be rewritten as,232
e=ρscsθ+ (ρwcw+ρici)(θ−θm) + (ρwcw+ρici)θm. (16)
From the relations shown in Eqs. (1)-(3), substituting Eq. (16) into Eq. (15) yields the following:233
(ρscs+ρwcw+ρici)˙
θ+φ[(ρwcw−ρici)(θ−θm) + ρiLθ]˙
Sw(c) + ∇·q=ˆ
r, (17)
where:234
Lθ=ρw
ρi
cw−ciθm(18)
is the latent heat of fusion which is set to be Lθ=334 kJ/kg for pure water [33,44,64,65]. Notice that235
the second term on the left-hand side of Eq. (17) describes the energy associated with the phase change236
of the fluid phase constituent α={w,i}, which is responsible for the constant temperature during the237
transformation processes, i.e., where cis changing with time since ˙
Sw(c) = {∂Sw(c)/∂c}˙
c. For the consti-238
tutive model that describes the heat conduction, this study adopts Fourier’s law where the heat flux can be239
written as the dot product between the effective thermal conductivity and the temperature gradient, i.e.,240
q=−
φsκs+∑
α={w,i}
φακα
·∇θ, (19)
where κsand καdenote the intrinsic thermal conductivities of the solid and fluid phase constituents, re-241
spectively. This volume-averaged approach, however, is only valid for the case where all the phase con-242
stituents are connected in parallel. Although there exists alternative homogenization approaches such as243
Eshelby’s equivalent inclusion method [66,67,68], determination of correct effective thermal conductiv-244
ity often requires knowledge of the pore geometry and topology [67,69,70]. Since the information is not245
always readily approachable, this extension will be considered in the future.246
3.1.2 Phase transition247
By using the phase field variable cdefined in Eq. (5), we adopt the Allen-Cahn model that is often used to248
simulate dendrite growth or multi-phase flow [39,71,72]. Following [40], we consider one of the simplest249
forms of the Gibbs free energy functional Ψc:250
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 ec
251
is the gradient energy coefficient. From Eq. (20), we consider the evolution of the phase field cover time,252
which yields the well-known Allen-Cahn equation or time-dependent Ginzburg-Landau equation, i.e.,253
−1
Mc
˙
c=∂ψc
∂c−∇·∂ψc
∂∇c=∂fc
∂c−e2
c∇2c, (21)
where ∇2(•) = ∇·∇(•)is the Laplacian operator and Mcis the mobility parameter. Since this study does254
not consider solute transport or any other chemical effects, we focus on the pure water-ice phase transition255
such that the free energy density fc(θ,c)can be written as,256
fc=Wcgc(c) + Fc(θ)pc(c), (22)
8 Hyoung Suk Suh, WaiChing Sun
where gc(c) = c2(1−c)2is the double well potential [Fig. 2(a)] that can be regarded as an energy barrier at257
the ice-water interface with the height of Wc, and pc(c) = Sw(c) = c3(6c2−15c+10)is the interpolation258
function [Fig. 2(b)] that ensures minima of the free energy density fcat c=0 and c=1, respectively.259
The driving force Fc(θ)that induces ice-water phase transition should describe the thermodynamically260
equilibrated state of water and ice phase constituents, which can be derived from the following relation261
[73]:262
dpi=ρi
ρw
dpw−ρiLθ
dθ
θ. (23)
Then, integrating Eq. (23) yields the Clausius-Clapeyron equation:263
pi−pw=ρi
ρw−1pw−ρiLθln θ
θm. (24)
Eq. (24) suggests that the surface tension develops along the ice-water interface, establishing the relation264
among water pressure (pw), ice pressure (pi), and temperature (θ). However, as pointed out in [33], the265
ice-water phase transition is mainly governed by the temperature while the influence of pressure on the266
ice saturation Siis relatively minor. Hence, for simplicity, we define the driving force Fc(θ)as an approx-267
imation of the pressure difference, by neglecting the effect of pore water pressure and adopt its first-order268
Taylor approximation following [40] as follows:269
pi−pw≈ Fc(θ) = ρiLθ1−θ
θm. (25)
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)
Fig. 2: (a) The double well potential gc(c), and (b) the interpolation pc(c)functions. Thin colored curves
correspond to the values outside the range of the phase field c.
As pointed out in [40,44], since Eq. (21) captures the evolution of the regularized ice-water interface,270
numerical parameters ec,Wc, and Mccan be related to the ice-water surface tension γiw , the interface271
thickness δc, and the kinetic coefficient νcas,272
ec=p6γiw δc;Wc=3γiw
δc;Mc=νcθm
6ρiLθδc, (26)
where the procedure that yields the relationships among the parameters are summarized in Appendix273
A. However, physical range of the width of the ice-water interface is at the atomic scale, i.e., 10−10 m,274
which makes macro-scale simulations unfeasible [45,74]. In addition to the interfacial tension γiw, this275
Multi-phase-field model for ice lens growth 9
study therefore treats the interface thickness δcand the gradient energy coefficient ecas input material276
parameters, since they could be increased according to the mesh size without significantly influencing the277
interface evolution [44,75,76].278
Furthermore, since the existence of segregated ice governs the heave rate of frozen soil [36,77], this279
study considers different rates between homogeneous freezing and ice lens growth. Specifically, while280
employing different volumetric expansion coefficients for the in-pore crystallization and the formation of281
ice lens [Eq. (14)], we replace the driving force Fc(θ)with F∗
c(θ,d)that contains an additional term that282
describes the intense growth of ice lenses similar to the kinetic equation proposed by Espinosa et al. [78],283
which is often used to model salt crystallization in porous media [54,79,80]:284
F∗
c(θ,d) = ρiLθ1−θ
θm+ [1−gd(d)]K∗
c1−θ
θmg∗
c
, (27)
where K∗
c>0 and g∗
c>0 are the kinetic parameters. The effect of the additional term in Eq. (27) is285
illustrated in Fig. 3, where we simulate the water-ice phase transition by placing a heat sink at the center286
while the kinetic parameters are set to be K∗
c=5.0 GPa and g∗
c=1.2. By considering two different cases287
where the entire 1 mm2large water-saturated square domain remains intact and is completely damaged,288
Fig. 3shows that the modified driving force F∗
cis capable of capturing different growth rates depending289
on the damage parameter d.290
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=">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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=">AAAErniclVNNj9MwEPXuBljKVwtHLoFeOFUJqgQX0EqsBBfEgmi30qZUjjNpTf0R2U7LyvKRn8EVfhP/BqfpSt2UIGEpk9Gbec/2zDgtGNUmin4fHB4FN27eOr7duXP33v0H3d7DsZalIjAikkk1SbEGRgWMDDUMJoUCzFMG5+nyTRU/X4HSVIrP5rKAKcdzQXNKsPHQrNsz4atwGEVhYuCbsaF2s24/GkSbFe478dbpo+06m/WOvieZJCUHYQjDWlusDCUMXNhJSg0FJks8h4vS5C+nloqiNCCI241ZzDXHZrEH5lIYvYfqS55eB6tKUTFvpFaSRkr2F1jpvIEayqEB5Uxi4yEBayI5xyKzvkxCS+Uu4qlNvFsqqORskkqWVeeSzPZj5xqsryUvag5LlVdPnmydsB+HibrCaqfBxQyUqclVjzY9twoyt9mok5yCr7qC9/4YHwpQ2PhwMlfYJyQCpwy35fjj7qZ9qfWtdq2qGfWzdJWfkEyatsy3TeGqauGkVfl0V3mX8M9d/A2crUxbfLn2cW9a4sZv6b+228LK2cq0qc+FV/dm0y+zAOmHwSq+dPaTnwpVNVLBbitXQP53cnzOPmUzwbz+p3lNqx5u3Hym+874+SAeDoYfh/2T19snfIweo6foGYrRC3SC3qEzNEIErdEP9BP9CqJgHEyDWZ16eLDlPELXVrD4A4owp5M=</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>
Fig. 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=10−10 m2.
3.2 Freezing-induced fracture in microporoelastic medium291
3.2.1 Microporomechanics of the phase-changing porous medium292
Focusing on the ice lens formation that involves a long period of time up to annual scales [81,82], this293
study neglects the inertial effects such that the balance of linear momentum for the three-phase mixture294
can be written as,295
∇·σ+ρg=0. (28)
Based on the observation that geological materials remain brittle at a low temperature [83,84], we assume296
that the evolution of the damage parameter dreplicates the mechanism of brittle fracture. In this case,297
undamaged effective stress σ0
int can be considered linear elastic, while the stress tensor inside the damaged298
zone should remain σ0
dam =0unless the temperature is below θmto form bulk ice. Moreover, since the ice299
flow with respect to the solid phase is negligible compared to that of water [34,35], both σ0
int and σ0
dam can300
10 Hyoung Suk Suh, WaiChing Sun
be related to the strain measure ε= (∇u+∇uT)/2 by approximating ˜vi≈0. Given these considerations,301
we define the constitutive relations for σ0
int and σ0
dam as,302
σ0
int =KεvolI+2Gεdev ;σ0
dam = [1−Sw(c)](KiεvolI+2Giεdev ), (29)
where εvol =tr (ε)and εdev =ε−(εvol/3)I, while Kand Kiare the bulk moduli; and Gand Giare303
the shear moduli for the solid skeleton and the ice, respectively. Based on this approach, σ0
dam can be304
interpreted as a developed stress due to the ice lens growth, since it not only depends on the fracturing305
process but also on the state of the fluid phase. The net pore pressure ¯
p, on the other hand, is a driver of306
deformation and fracture due to the formation of ice crystal that exerts significant excess pressure on the307
premelted water film. This pressure is referred to as cryo-suction scryo that induces the ice pressure pito308
be far greater than the water pressure pw. As shown in Eqs. (12) and (24), the net pore pressure can be309
rewritten as ¯
p= [1−Sw(c)]scryo −pw, while scryo =pi−pwcan be determined based upon the Clausius-310
Clapeyron equation. In practice, however, the Clausius-Clapeyron equation is typically replaced by an311
empirical model, such as the exponential [85] or the van Genuchten [61] curves, which is considered to be312
more accurate, since freezing retention characteristics are affected by both the pore size distribution and313
the ice-water interfacial tension [86,87,88,89]:314
Sw∗=exp (bBhθ−θmi−);s∗
cryo =pref {Sw(c)}−1
mvG −11
nvG , (30)
where bB,pref,mvG , and nvG are empirical parameters while h•i±= (•±|•|)/2 is the Macaulay bracket.315
Note that we use a superscripted symbol ∗to indicate that the corresponding variables are empirically de-316
termined. Yet, these empirical models still yield unrealistic results in some cases. For example, the deriva-317
tive of the exponential model possesses a discontinuity at the freezing temperature θm, while s∗
cryo ap-318
proaches infinity if Sw(c)→0 if adopting the van Genuchten model. Hence, in this study, we combine the319
two models to obtain the freezing retention curve that bypasses such issues (Fig. 4):320
s∗
cryo =pref [{exp (bBhθ−θmi−)}]−1
mvG −11
nvG , (31)
and we replace scryo with s∗
cryo for the net pore pressure such that: ¯
p= [1−Sw(c)]s∗
cryo −pw. For all the321
numerical examples presented in Section 5, we adopt the same values used in [35,89]: bB=0.55 K−1,322
pref =200 kPa, mvG =0.8, and nvG =2.0.323
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>
Fig. 4: Freezing characteristic function [Eq. (31)] used in this study.
Multi-phase-field model for ice lens growth 11
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, (32)
˙
ρw+ρw∇·v+∇·ρw˜vw=˙
mw, (33)
˙
ρi+ρi∇·v+∇·ρi˜vi=˙
mi, (34)
where ˙
ms,˙
mw, and ˙
miindicate the mass production rate for each phase constituent [34,35,54]. Here, we324
assume that only the water and ice phase constituents exchange mass among constituents (i.e., ˙
ms=0 and325
˙
mw=−˙
mi). Hence, summation of Eqs. (33) and (34) yields:326
˙
φ{Sw(c)ρw+ [1−Sw(c)]ρi}+φ˙
Sw(c)(ρw−ρi) + φ{Sw(c)ρw+ [1−Sw(c)]ρi}∇·v+∇·ρw˜vw=0, (35)
since ˜vi≈0, while Eq. (32) can be rewritten as,327
˙
φ= (1−φ)∇·v. (36)
Substituting Eq. (36) into (35) yields the mass balance equation for the three-phase mixture:328
φ˙
Sw(c)(ρw−ρi) + {Sw(c)ρw+ [1−Sw(c)]ρi}∇·v+∇·ρw˜vw=0. (37)
In this study, we focus on the case where the water flow inside both the porous matrix and the fracture329
obeys the generalized Darcy’s law while considering the pore blockage due to the water-ice phase transi-330
tion [90,91,92]. In other words, we adopt the following constitutive relation between ˜vwand pw:331
ww=−krk
µw
(∇pw−ρwg), (38)
where ww=φ˜vwis Darcy’s velocity, kis the permeability tensor, µwis the water viscosity, and kris the332
saturation dependent relative permeability:333
kr=Sw(c)1/2 n1−h1−Sw(c)1/mvG imv G o2. (39)
Remark 1.Note that the linear elasticity model in Eq. (29) is insufficient to accurately predict the elasto-334
plastic behaviors during the thawing. A more comprehensive approach to capture the thawing process335
must take into account of the healing of the soil (e.g. [93]), the evolution of the hydraulic conductivity, the336
changes of the compressibility due to the reduction of over-consolidation ratio due to the effective stress337
built up during the thawing, as well as the geometric nonlinear due to the substantial settlement of the soil338
[33,35,94,95]. Incorporating these extensions with the phase field ice lens model will be considered in the339
future but is out of the scope of this study.340
3.2.2 Damage evolution341
Following [57], this study interprets cracking as the fracture of the solid skeleton. In other words, we define342
the crack driving force Fd≥0 as,343
Fd=−∂gd(d)
∂dψ0
int ;ψ0
int =1
2K(εvol)2+G(εdev :εdev ), (40)
such that the damage evolution equation can be obtained from the balance between the crack driving force344
Fdand the crack resistance Rd[57,96,97]:345
Rd−Fd=∂gd(d)
∂dψ0
int +Gd
ld
(d−l2
d∇2d) = 0. (41)
Recall Section 2.2 that our choice of degradation function gd(d)reduces the thermodynamic restriction into346
˙
d≥0, which requires additional treatment to ensure monotonic crack growth. In this study, we adopt the347
12 Hyoung Suk Suh, WaiChing Sun
same treatment used in [56,98]. By considering the homogeneity ∇d=0, Eq. (41) yields the following348
expression:349
˙
d=2
(1+2H)2˙
H ≥ 0 ; H=ψ0+
int
Gd/ld
, (42)
implying that non-negative ˙
dis guaranteed if ˙
H ≥ 0. Here, notice that we adopt the volumetric-deviatoric350
splitting scheme proposed by Amor et al. [99] to avoid crack growth under compression. Specifically, we351
decompose the elastic strain energy into two parts, i.e., ψ0
int =ψ0+
int +ψ0−
int,352
ψ0
int =1
2Khεvoli2
++G(εdev :εdev);ψ0
int =1
2Khεvoli2
−, (43)
and only degrade the expansive volumetric and deviatoric parts, while h•i±= (•+|•|)/2. To ensure353
˙
H ≥ 0, as a simple remedy, we replace Hwith H∗which is defined as the pseudo-temporal maximum of354
normalized strain energy, while considering a critical value Hcrit that restricts the crack to initiate above a355
threshold strain energy [56,100,101,102]:356
H∗=max
τ∈[0,t]hH − Hcriti+, (44)
such that Eq. (41) accordingly becomes:357
∂gd(d)
∂dH∗+ (d−l2
d∇2d) = 0. (45)
For either partially or fully saturated soils, crack healing may occur during the thawing process. In spe-358
cific, when ice lenses melt in a highly plastic clayey soil, cracks may heal due to the interactions between359
water molecules, whereas in a less cohesive soil, relocation of eroded particles result in the clogging of360
cracks or cavities [93,103,104]. One possible approach to model the crack healing process is to allow crack361
driving force to decrease and incorporate constitutive model that can capture the thaw-weakening process362
properly. For example, Ma and Sun [105] assumed that the healing process is activated when the material363
experiences volumetric compression, while the stiffness recovery rate becomes slower along the healing364
process. This extension is out of scope of this study, and hence, we assume that cracking is irreversible.365
In order to model the fracture flow in a fluid-infiltrating porous media, we adopt the permeability en-366
hancement approach that approximates the water flow inside the fracture as the flow between two parallel367
plates [106,107,108,109]:368
k=kmat +kd=kmatI+d2kd(I−nd⊗nd), (46)
where kmat is the effective permeability of the undamaged matrix, nd=∇d/k∇ dkis the unit normal of369
crack surface, and kd=w2
d/12 describes the permeability enhancement due to the crack opening which370
depends on the hydraulic aperture wdbased on the cubic law. However, freezing-induced fracture involves371
different situations where the pore ice crystal growth drives fracture but at the same time blocks the pore372
that may hinder the water flow therein. Hence, we adopt the approach used in [54] which assumes a linear373
relationship between the hydraulic aperture wdand the water saturation Sw(c):374
wd=Sw(c)l⊥(nd·ε·nd), (47)
where l⊥is the characteristic length of a line element perpendicular to the fracture which is often assumed375
to be equivalent to the mesh size [106,110]. Furthermore, by assuming that the crack opening leads to376
complete fragmentation of the solid matrix, we adopt the following relation for the porosity [57,111]:377
φ=1−gd(d)(1−φ0)(1− ∇·u), (48)
such that the porosity approaches 1 if the solid skeleton is completely damaged.378
Multi-phase-field model for ice lens growth 13
Remark 2.Fragmentation and damage of the solid constituent may alter the microstructure of the solid379
skeleton. Nevertheless, if the constituent remains incompressible, then damage (e.g., split of incompress-380
ible particles) should not change the volume of the solid constituent constituted by a controlled mass and381
hence should not change the porosity. The only exception is when the fragmented particles eroded and flow382
inside the void space in which case a portion of solid mass is lost due to the damage (e.g., [112,113,114]).383
In our case, we are using a regularized phase field to implicitly represent the crack surfaces and hence the384
dependence of damage in Eq. (48) is used to capture the erosion. Note that a more precise predictions may385
require a function difference from gd(d)to establish the relation between erosion and damage as well as386
the calculation of effective viscosity due to the erosion (see [115]), which are out of the scope of this study387
but will be considered in the future.388
4 Finite element implementation389
This section presents a finite element discretization of the set of governing equations described in Section390
3, and the solution strategy for the resulting discrete system. We first formulate the weak form of the391
field equations by following the standard weighted residual procedure. In specific, we adopt the Taylor-392
Hood element for the displacement and pore water pressure fields, while employing linear interpolation393
functions for all other variables in order to remove spurious oscillations. We then describe the operator394
split solution scheme that separately updates {θ,c}and {u,pw}, while the damage parameter dis updated395
in a staggered manner for numerical robustness.396
4.1 Galerkin form397
Let domain Bpossesses boundary surface ∂Bcomposed of Dirichlet boundaries (displacement ∂Bu, pore398
water pressure ∂Bp, and temperature ∂Bθ) and Neumann boundaries (traction ∂Bt, water mass flux ∂Bw,399
and heat flux ∂Bq) that satisfies:400
∂B=∂Bu∪∂Bt=∂Bp∪∂Bw=∂Bθ∪∂Bq;∅=∂Bu∩∂Bt=∂Bp∩∂Bw=∂Bθ∩∂Bq. (49)
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,
(50)
where nis the outward-oriented unit normal on the boundary surface ∂B. Meanwhile, the following401
boundary conditions on ∂Bare prescribed for the phase fields cand d:402
∇c·n=0 ; ∇d·n=0. (51)
For model closure, the initial conditions for the primary unknowns {u,pw,θ,c,d}are imposed as:403
u=u0;pw=pw0;θ=θ0;c=c0;d=d0, (52)
at time t=0. We also define the trial spaces Vu,Vp,Vθ,Vc, and Vdfor the solution variables as,404
Vu=nu:B → R3|u∈[H1(B)]3,u|∂Bu=ˆuo,
Vp=npw:B → R|pw∈H1(B),pw|∂Bp=ˆ
pwo,
Vθ=nθ:B → R|θ∈H1(B),θ|∂Bθ=ˆ
θo,
Vc=nc:B → R|c∈H1(B)o,
Vd=nd:B → R|d∈H1(B)o,
(53)
14 Hyoung Suk Suh, WaiChing Sun
which is complimented by the admissible spaces:405
Vη=nη:B → R3|η∈[H1(B)]3,η|∂Bu=0o,
Vξ=nξ:B → R|ξ∈H1(B),ξ|∂Bp=0o,
Vζ=nζ:B → R|ζ∈H1(B),ζ|∂Bθ=0o,
Vγ=nγ:B → R|γ∈H1(B)o,
Vω=nω:B → R|ω∈H1(B)o,
(54)
where H1indicates the Sobolev space of order 1. By applying the standard weighted residual procedure,406
the weak statements for Eqs. (17), (21), (28), (37), and (45) are to: find {u,pw,θ,c,d} ∈ Vu×Vp×Vθ×Vc×407
Vdsuch that for all {η,ξ,ζ,γ,ω} ∈ Vη×Vξ×Vζ×Vγ×Vω,408
Gu=Gp=Gθ=Gc=Gd=0, (55)
where:
Gu=ZB∇η:σdV −ZBη·ρgdV −Z∂Bt
η·ˆ
tdΓ=0,
Gp=ZBξφ˙
Sw(c)(ρw−ρi)dV +ZBξ{Sw(c)ρw+ [1−Sw(c)]ρi}∇·vdV
−ZB∇ξ·(ρwww)dV −Z∂Bw
ξ(ρwˆww)dΓ=0,
Gθ=ZBζ(ρscs+ρwcw+ρici)˙
θdV +ZBζφ[(ρwcw−ρici)(θ−θm) + ρiLθ]˙
Sw(c)dV
−ZB∇ζ·qdV −ZBζˆ
r dV −Z∂Bq
ζˆ
q dΓ=0,
Gc=ZBγ1
Mc
˙
c dV +ZBγ∂fc
∂cdV +ZB∇γ·(e2
c∇c)dV =0,
Gd=ZBω∂gd(d)
∂dH∗dV +ZBωd dV +ZB∇ω·(l2
d∇d)dV =0.
(56)
(57)
(58)
(59)
(60)
4.2 Operator-split solution strategy409
Although one may consider different strategies to solve the coupled system of equations [Eqs. (56)-(60)], the410
solution strategy adopted in this study combines the staggered scheme [42] and the isothermal operator411
splitting scheme [116,117]. Specifically, we first update the damage field dvia linear solver while the412
variables {u,pw,θ,c}are held fixed. We then apply the isothermal splitting solution scheme that iteratively413
solves the thermally-induced phase transition problem to advance {θ,c}, followed by a linear solver that414
updates {u,pw}by solving an isothermal poromechanics problem [57], i.e.,415
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
, (61)
where we adopt an implicit backward Euler time integration scheme. The implementation of the model416
including the finite element discretization and the solution scheme relies on the finite element package417
FEniCS [118,119,120] with PETSc scientific computational toolkit [121].418
Multi-phase-field model for ice lens growth 15
5 Numerical examples419
This section presents three sets of numerical examples to verify (Section 5.1), validate (Section 5.2), and420
showcase (Sections 5.3 and 5.4) the capacity of the proposed model. Since the evolution of two phase421
fields cand drequires a fine mesh to capture their sharp gradients, we limit our attention to one- or two-422
dimensional simulations while considering the diffusion coefficient ecas an individual input parameter423
independent to the interface thickness δcwhich may additionally reduce the computational cost [45,122].424
We first present two examples that simulate the latent heat effect and 1d consolidation to verify the im-425
plementation of our proposed model. As a validation exercise, we perform numerical experiments that426
replicate the physical experiments conducted by Feng et al. [123], which studies the homogeneous freezing427
of a phase change material (PCM) embedded in metal foams. We then showcase the performance of the428
computational model for simulating the ice lens formation and the thermo-hydro-mechanical processes in429
geomaterials undergoing freeze-thaw cycle, and also its capacity to simulate non-planar ice lens growth430
that follows the crack trajectory.431
5.1 Verification exercises: latent heat effect and 1d consolidation432
Our first example simulates one-dimensional freezing of water-saturated porous media to investigate the433
phase transition of the fluid phase α={w,i}and the involved latent heat effect. By comparing the results434
against the models presented by Lackner et al. [124] and Sweidan et al. [45], this example serves as a435
verification exercise that ensures the robust implementation of the heat transfer model involving phase436
transition [i.e., Eqs. (58) and (59)]. Hence, this example considers a rigid solid matrix while neglecting the437
fluid flow, following [124]. As illustrated in Fig. 5(a), the problem domain is a fully saturated rectangular438
specimen with a height of 0.09 m and a width of 0.41 m. While the initial temperature of the entire specimen439
is set to be θ0=283.15 K, the specimen is subjected to freezing with a constant heat flux of ˆ
q=100440
W/m2on the top surface, whereas all other boundaries are thermally insulated. Here, we choose the same441
material properties used in [124] and [45] as follows: φ0=0.42, ρs=2650 kg/m3,ρw=1000 kg/m3,442
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=0.611443
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, and444
ec=1.25 (J/m)1/2 for the Allen-Cahn phase field model, while we use the structured mesh with element445
size of he=0.6 mm and choose the time step size of ∆t=100 sec.446
ˆq= 100W/m2
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A
<latexit sha1_base64="ArbWvUFLhSoey0zXnILkQjT20IQ=">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</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
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0.09 m
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0.41 m
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(a)
0 0.5 1 1.5 2
105
235
245
255
265
275
285
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>
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
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
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)
Fig. 5: (a) Schematic of geometry and boundary conditions for the 1d freezing example; (b) Temperature
evolution at points A, B, and C.
16 Hyoung Suk Suh, WaiChing Sun
As shown in Figure 5(b), measured temperatures at points A, B, and C during the simulation first447
linearly decrease due to the applied heat flux ˆ
quntil they reach the freezing temperature of θm=273.15448
K. As soon as the phase transition starts, the freezing front propagates through the specimen while the449
release of the energy associated with the phase transition prevents the temperature decrease (i.e., latent450
heat effect). Once the phase change is complete, the temperature linearly decreases over time again since451
the heat transfer process is no longer affected by the latent heat. More importantly, a good agreement452
with the results reported in [45,124] verifies that our proposed model is capable of capturing the thermal453
behavior of the phase-changing porous media.454
x
<latexit sha1_base64="+r+6jZP7DgLnuIzSdy3jRGTBv9w=">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</latexit>
y
<latexit sha1_base64="t1lo9wUujzFBVB8BceLG3pLBo/o=">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</latexit>
10 m
<latexit sha1_base64="wr1/FGfkbNfh2uTsHyidjXukkqU=">AAAEp3iclVNNbxMxEHXbBUr4auHIZSEXuERZFAmOlagEF9SCSBupDpXXO5ua+GNle1Mqy0f+Alf4W/wb7GwqpRsWCUs7O3oz79meGecVZ8YOh7+3tneSW7fv7N7t3bv/4OGjvf3HJ0bVmsKYKq70JCcGOJMwtsxymFQaiMg5nObztzF+ugBtmJKf7VUFU0FmkpWMEhsgnA1TbOGbdanw53v94WC4XOmmk62cPlqt4/P9ne+4ULQWIC3lxBhHtGWUg097uDZQETonMzirbflm6pisaguS+vWYI8IIYi82wFJJazZQcyXym2AsEZOzVmqUtErxv8DalC3UMgEtqOSK2ABJuKRKCCILF8okjdL+LJs6HNxaQ5RzOFe8iOdS3PUz71usr7WoGg7PdVDHz1ZO2s9SrK+xxmlxCQdtG3Ls0bLZTkPhlxv18CGEqmv4EI5xVIEmNoTxTJOQgCXJOenKCcddT/vS6DvjO1ULFoboOh/TQtmuzHdt4Vi1dNKpfLiuvE745y7hBt5F0xWfX4Z4MB1xG7YMX9dtYeFdNF3qMxnUg1n2y16ACsPgtJh79ylMhY6N1LDeygXQ/52ckLNJWU6waP552dDiw83az3TTOXk1yEaD0cdR/2C0esK76Cl6jl6gDL1GB+g9OkZjRFGFfqCf6FfyMjlKTpJJk7q9teI8QTdWQv4A+tul/A==</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=">AAAErXiclVNNb9QwEHXbAGX52sKRS2AvnJYNWgQXpEpUgguiILZdqd5WjjNJw/ojsp0tleUj/4Ir/Cf+DfZmK22zBAlLmYzezHu2Z8ZpxUptRqPfW9s70Y2bt3Zv9+7cvXf/QX/v4ZGWtaIwoZJJNU2JBlYKmJjSMJhWCghPGRyn87chfrwApUspvpjLCmacFKLMS0qMh876fRO/iV+OYmzgm7Gxdmf9wWg4Wq5400lWzgCt1uHZ3s53nElacxCGMqK1JcqUlIGLe7jWUBE6JwWc1CZ/PbOlqGoDgrr1mCVcc2LON8BcCqM3UH3J0+tgKFQpilZqkDRSsr/ASuct1JQcWlDOJDEeEnBBJedEZNaXSWip3Ekys9i7tYIgZ3EqWRbOJZkdJM61WF9rXjUcliqvjp+snHiQxFhdYY3T4hIGyjTk0KNly62CzC036uED8FVX8MEf42MFihgfxoUiPgELkjLSleOPu5522uhb7TpVs9KP0lU+ppk0XZnv2sKhavG0U/lgXXmd8M9d/A2cDaYrPr/wcW864sZv6b+u28LC2WC61Avh1b1Z9sucg/TDYBWfO/vZT4UKjVSw3soF0P+dHJ+zSVlOMG/+ad7QwsNN2s900zl6MUzGw/Gn8WB/vHrCu+gxeoqeoQS9QvvoPTpEE0TRAv1AP9Gv6Hk0iXB02qRub604j9C1FRV/AJXYp1A=</latexit>
t= 150 s
<latexit sha1_base64="L0YJvBs2pe2xP4ftbI7u9K3URoQ=">AAAErniclVNNj9MwEPXuBljKVwtHLoFeOFUJKoIL0kqsBBfEgmi30qZUjjNpTf0R2U7LyvKRn8EVfhP/BqfpSt2UIGEpk9Gbec/2zDgtGNUmin4fHB4FN27eOr7duXP33v0H3d7DsZalIjAikkk1SbEGRgWMDDUMJoUCzFMG5+nyTRU/X4HSVIrP5rKAKcdzQXNKsPHQrNsz4eswfhGFiYFvxobazbr9aBBtVrjvxFunj7brbNY7+p5kkpQchCEMa22xMpQwcGEnKTUUmCzxHC5Kk7+aWiqK0oAgbjdmMdccm8UemEth9B6qL3l6HawqRcW8kVpJGinZX2Cl8wZqKIcGlDOJjYcErInkHIvM+jIJLZW7iKc28W6poJKzSSpZVp1LMtuPnWuwvpa8qDksVV49ebJ1wn4cJuoKq50GFzNQpiZXPdr03CrI3GajTnIKvuoK3vtjfChAYePDyVxhn5AInDLcluOPu5v2pda32rWqZtTP0lV+QjJp2jLfNoWrqoWTVuXTXeVdwj938TdwtjJt8eXax71piRu/pf/abgsrZyvTpj4XXt2bTb/MAqQfBqv40tlPfipU1UgFu61cAfnfyfE5+5TNBPP6n+Y1rXq4cfOZ7jvj54N4OBh+HPZPhtsnfIweo6foGYrRS3SC3qEzNEIErdEP9BP9CqJgHEyDWZ16eLDlPELXVrD4A4+xp4s=</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)
Fig. 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 since455
it possesses an analytical solution [125], which can directly be compared with the results obtained via456
poromechanics model [Eqs. (56) and (57)]. Our problem domain shown in Fig. 6(a) consists of a 10 m high457
water-saturated linear elastic soil mass. While a 1 MPa compressive load tyis imposed on the top surface,458
we replicate the single-drained condition by prescribing zero pore water pressure at the top ( ˆ
pw=0) and459
a no-slip condition at the bottom. By assuming that the temperature of the soil column remains constant460
during the simulation (θ=293.15 K), we only focus on its hydro-mechanically coupled response while the461
material parameters are chosen as follows: φ0=0.4, ρs=2650 kg/m3,ρw=1000 kg/m3,K=66.67 MPa,462
G=40 MPa, kmat =10−12 m2, and µw=10−3Pa·s. Here, we choose he=0.1 m and ∆t=20 sec.463
Fig. 6(b) illustrates the pore water pressure profile during the simulation at t=50, 150, 350, and 750 s.464
The results show that the applied mechanical load tybuilds up the pore water pressure, affecting the pore465
water to migrate towards the top surface, which leads to the dissipation of the excess pressure over time466
(i.e., consolidation). By comparing the simulation results (circular symbols) to the analytical solution (solid467
curves), Fig. 6(b) verifies the reliability of our model to capture the hydro-mechanically coupled responses.468
5.2 Validation example: homogeneous freezing469
This section compares the results obtained from the numerical simulation against the physical experiment470
conducted by Feng et al. [123]. This experiment is used as a benchmark since it considers the unidirectional471
freezing of distilled water filled in porous copper foams, which does not involve a fracturing process and472
yields a clear water-ice boundary layer due to the microstructural attributes of the host matrix. As schemat-473
ically shown in Fig. 7(a), a 30 mm wide, 50 mm long water-saturated copper foam is mounted on a 4 mm474
Multi-phase-field model for ice lens growth 17
A0
<latexit sha1_base64="zXFOejrB40RFamCuISH4G7rLpz0=">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</latexit>
A
<latexit sha1_base64="DH2FKiEnN77qmkMMGEBzj5YobRQ=">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</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:
<latexit sha1_base64="r9S9q1YJliSY7piS1xRPe48OttM=">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</latexit>
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=">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</latexit>
ˆ
✓AA’ = 20.8e0.1pt+ 264.75 K
<latexit sha1_base64="ds7AbXgqS/Mvk/wpz9sK4JRdVM8=">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</latexit>
(b)
Fig. 7: (a) Schematic of the experimental setup for the unidirectional freezing test conducted in [123]; (b)
Temperature boundary condition applied at the bottom surface of the copper foam (AA’) for the numerical
simulation.
thick copper block. While the initial temperature is measured to be θ0=285.55 K, the experiment is per-475
formed by applying a constant temperature of ˆ
θ=264.15 K at the bottom part of the copper block at t=0.476
Temperature measurements during the experiment are made by three thermocouples (TC2-TC4) located at477
10 mm, 28 mm, and 46 mm from the bottom of the foam (AA’), whereas TC1 records the temperature of the478
block. For the numerical simulation, instead of considering the problem domain as a layered material, we479
only focus on the water-saturated copper foam and apply time-dependent Dirichlet boundary condition480
on AA’ by using the temperature measured by TC1 [Fig. 7(b)]. We also assume an unlimited water supply481
from the top surface by imposing ˆ
pw=0 and applying a fixed boundary condition at the bottom part of482
the foam. Moreover, we consider two different types of copper foams (Foam 1 and Foam 2) with different483
initial porosity and thermal conductivity [Fig. 7(a)]. As summarized in Table 1, our numerical simulation484
directly adopts the same thermal properties compared to the physical experiment whereas the solid phase485
thermal conductivities of the foams are computed based upon the effective properties reported in [123]. For486
all other material parameters that are not specified in [123], we choose the properties that resemble those487
of the water-saturated copper foam. In this section, the Allen-Cahn parameters are chosen as: νc=0.0001488
m/s, γc=0.065 J/m2,δc=0.0001 m, and ec=0.75 (J/m)1/2, while adopting a structured mesh with489
he=2.5 mm and ∆t=60 sec.490
Fig. 8illustrates the evolution of the freezing front within a water-saturated copper foam (Foam 2). In491
both the physical and numerical experiments, water freezing starts from the bottom (AA’) and migrates492
towards the upper part of the foam over time, depending on the conductive heat transfer process. While it493
shows a qualitative agreement between the two, Fig. 9quantitatively confirms the validity of our model,494
where we use the circular symbols to indicate the experimental measurements whereas the solid curves495
denote the numerical results. As shown in Fig. 9(a), since Foam 1 possesses higher solidity (lower poros-496
ity) compared to Foam 2, the water-ice interface tends to grow relatively faster because it exhibits higher497
effective thermal conductivity. In addition, temperature variations illustrated in Fig. 9(b) clearly show the498
interplay between the thermal boundary layer growth and the latent heat, resulting in a nonlinear evo-499
lution of the freezing front. Although has not been measured experimentally, we further investigate the500
time-dependent hydro-mechanical response of the specimen from the simulation results shown in Fig. 10.501
Based on the freezing retention curve [Eq. (31)] adopted in this study, positive suction starts to develop502
if θ<θmwhile the region where s∗
cryo >0 evolves over time following the same trajectory of that of the503
18 Hyoung Suk Suh, WaiChing Sun
Parameter Description [Unit] Value Reference
ρsIntrinsic solid mass density [kg/m3] 7800.0 -
ρwIntrinsic water mass density [kg/m3] 1000.0 [123]
ρiIntrinsic ice mass density [kg/m3] 920.0 [123]
csSpecific heat of solid [J/kg/K] 0.385 ×103-
cwSpecific heat of water [J/kg/K] 4.216 ×103[123]
ciSpecific heat of ice [J/kg/K] 2.040 ×103[123]
κsThermal conductivity of solid [W/m/K] 62.855, 44.48 [123]
κwThermal conductivity of water [W/m/K] 0.56 [123]
κiThermal conductivity of ice [W/m/K] 1.90 [123]
KBulk modulus of solid skeleton [Pa] 0.555 ×109-
KiBulk modulus of ice [Pa] 5.56 ×109-
GShear modulus of solid skeleton [Pa] 0.185 ×109-
GiShear modulus of ice [Pa] 4.20 ×109-
φ0Initial porosity [-] 0.96, 0.98 [123]
kmat Matrix permeability [m2] 3.25 ×10−7-
µwViscosity of water [Pa·s] 1.0 ×10−3-
αv,int Volumetric expansion coefficient [-] 5.0 ×10−3-
Table 1: Material parameters for the validation exercise.
Experiment
<latexit sha1_base64="LmlihOf9FPI2cc/NgMc43yE9f5U=">AAAErniclVNNj9MwEPXuBljKVwtHLoVeOFUNqgTHlVgEF8SCaLfSplSOM2lD/RHZTrsrK0d+Blf4Tfwbxk1X6qYECUuZjN7Me7ZnxnHOM2MHg98Hh0fBrdt3ju+27t1/8PBRu/N4bFShGYyY4kpPYmqAZxJGNrMcJrkGKmIO5/HyjY+fr0CbTMkv9iqHqaBzmaUZoxahWbsTWbi07u1lDjoTIG05a/cG/cFmdfedcOv0yHadzTpH36NEscKzGafGOKptxjiU3VZUGMgpW9I5XBQ2fT11mcwLC5KVuzFHhRHULvbAVElr9lBzJeKboK9UJue1VC9pleJ/gbVJa6jF69eglCtqEZKwZkoIKhOH5ZJG6fIinLoI3UKDl3NRrHjiz6W464VlWWN9K0RecXisUT16tnW6vbAb6WuscmpcykHbiux7tem505CUm41a0Slg1TV8wGN8xC5Si+ForikmRJLGnDbl4HF3075W+s6UjapJhrN0nR+xRNmmzHd1YV+17qRR+XRXeZfwz13wBqXzpim+XGMcTUPc4pb4Nd0WVqXzpkl9LlEdzaZfdgEKh8FpsSzdZ5wK7RupYbeVK2D/OzmYs0/ZTLCo/nFa0fzDDevPdN8Zv+yHw/7w07B3Mtw+4WPylDwnL0hIXpET8p6ckRFhZE1+kJ/kVzAIxsE0mFWphwdbzhNyYwWLP6tOqXI=</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
<latexit sha1_base64="F7W95ZfZaMzB+j/pvAKQW9nVovs=">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</latexit>
t= 180 min
<latexit sha1_base64="OQyLP5B2s4Ypa9Avwl6PePYN5aQ=">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</latexit>
Fig. 8: Comparison between the physical and numerical experiments on the evolution of the water-ice
interface.
freezing front [Fig. 10(a)]. This process also involves a volumetric expansion of the specimen that leads to504
an increase of the vertical displacement as shown in Fig. 10(b), due to the difference between water (ρw)505
and ice densities (ρi). Since our framework idealizes the material as a multiphase mixture of the solid,506
water, and ice phase constituents, notice that relatively small displacement compared to the volume ex-507
pansion due to the ice-water phase transition is because of the mechanical properties of the host matrix,508
which is less compressible compared to geological materials. It should be also noted that the freezing front509
always exhibits the largest vertical displacement, implying that the water migration towards the freezing510
front induced by the suction triggers the consolidation process above the frozen area, resulting in a small511
Multi-phase-field model for ice lens growth 19
volumetric compression therein. This observation agrees with the explanation in [126] where the consoli-512
dation front of a frozen soil has been observed experimentally, which corroborates the applicability of our513
proposed model.514
0 60 120 180
0
0.01
0.02
0.03
0.04
0.05
Foam 2
<latexit sha1_base64="BhGQuoLJj8cqVl+uAsoecs+sCA0=">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</latexit>
Foam 1
<latexit sha1_base64="K6el+eUdpEWZI4ipoLiu+k0lhwA=">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</latexit>
(a)
0 60 120 180
260
265
270
275
280
285
290
TC2
<latexit sha1_base64="2Y0fqz8QUc1hNdHT0Jf3by5xkaA=">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</latexit>
TC4
<latexit sha1_base64="ZUkUo7YnE7/wEcBkue+rDKjbC8Q=">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</latexit>
TC3
<latexit sha1_base64="Y5/Rdx0AjZLUGxjAloZ3CHjQyX8=">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</latexit>
(b)
Fig. 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
<latexit sha1_base64="xir51CL+HL5mSWVCIFDfrD4/buE=">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</latexit>
(a)
0 0.01 0.02 0.03 0.04 0.05
0
0.2
0.4
0.6
0.8
110-3
(b)
Fig. 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 soil515
In this section, we showcase the capability of our proposed model by simulating the formation and melting516
of multiple ice lenses inside a heterogeneous clayey soil specimen. As illustrated in Fig. 11(a), the problem517
20 Hyoung Suk Suh, WaiChing Sun
domain is 0.04 m wide and 0.1 m long soil column that possesses a random porosity profile along the518
vertical axis with a mean value of φref =0.4 such that the specimen possesses layered microstructure.519
In addition, we introduce a set of heterogeneous material properties that solely depends on the spatial520
distribution of initial porosity φ0. Specifically, we adopt a phenomenological model proposed by [127] for521
the shear modulus G, while we use a power law for the critical energy release rate Gdsimilar to [108,128]:522
G=3
21−2ν
1+νexp [10(1−φ0)][MPa] ; Gd=Gd,ref 1−φ0
1−φref nφ
. (62)
Here, we assume that the Poisson’s ratio remains constant ν=0.25 throughout the entire domain while523
we set Gd,ref =1.5 N/m and nφ=50. Based on this setting, we attempt to incorporate ice lens initiation524
criterion proposed by Zhou and Li [25], where a separation void ratio determines the positions of the525
ice lenses. For all other material properties that are homogeneous, as summarized in Table 2, we choose526
values similar to those of the clayey soil. It should be noted that we adopt αv,dam =0.08 which is identical527
to the theoretical value of 1 −ρi/ρwfor the expansion coefficient, whereas we set αv,int =0.005 due to the528
existence of thin water film between the intact solid and the pore ice. Meanwhile, the parameters for the529
Allen-Cahn phase field equation are chosen as: νc=0.0001 m/s, γc=0.065 J/m2,δc=0.0001 m, and530
ec=1.0 (J/m)1/2, whereas we set he=0.5 mm and ∆t=60 sec.531
x
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y
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ˆpw=0
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0.1m
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0.04 m
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ˆ
✓=ˆ
✓(t); ˆpw=0
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0.3 0.4 0.5
0
0.02
0.04
0.06
0.08
0.1
(a)
0 5 10 15
268
270
272
274
276
278
✓m= 273.15 K
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freezing
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thawing
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ˆ
✓(t) = 273.15 + 3 cos (2⇡t/15) K
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(b)
Fig. 11: (a) Schematic of geometry and boundary conditions for the numerical freeze-thaw test; (b) Tem-
perature boundary condition applied at the top surface.
While we set the initial temperature as θ0=276.15 K, the numerical freeze-thaw test is performed532
by applying a time-dependent temperature boundary condition at the top, represented by a sinusoidal533
function. As shown in Fig. 11(b), the freezing process starts at t=3.75 hr and continues until the top534
surface temperature reaches the melting temperature of θm=273.15 K at t=11.25 hr, where the frozen535
soil begins to thaw. During the simulation, the bottom part of the specimen is held fixed while we prescribe536
zero pore water pressure boundaries ( ˆ
pw=0) at both the top and the bottom surfaces. The left and right537
boundaries, on the other hand, are subjected to zero water mass flux and heat flux conditions. Based on538
this setting, the water is supplied from the bottom during the freezing phase, while the water expulsion539
towards the top surface during the melting phase leads to a thawing settlement of the specimen.540
Fig. 12 shows the formation and melting of multiple ice lenses and the evolution of the fracture phase541
field during the numerical freeze-thaw test. Here, we use a scaling factor of 5 while the color bar illustrated542
in Fig. 12(a) represents the value of the indicator function χidefined in Eq. (11). As illustrated in Fig. 13, the543
water freezes from the top to the bottom during the freezing phase (3.75 hr ≤t≤11.25 hr), which leads544
to the development of the cryo-suction and a volumetric expansion due to the phase transition. Since the545
Multi-phase-field model for ice lens growth 21
Parameter Description [Unit] Value
ρsIntrinsic solid mass density [kg/m3] 2650.0
ρwIntrinsic water mass density [kg/m3] 1000.0
ρiIntrinsic ice mass density [kg/m3] 920.0
csSpecific heat of solid [J/kg/K] 0.75 ×103
cwSpecific heat of water [J/kg/K] 4.20 ×103
ciSpecific heat of ice [J/kg/K] 1.90 ×103
κsThermal conductivity of solid [W/m/K] 7.69
κwThermal conductivity of water [W/m/K] 0.56
κiThermal conductivity of ice [W/m/K] 2.25
KiBulk modulus of ice [Pa] 5.56 ×109
GiShear modulus of ice [Pa] 4.20 ×109
φref Reference porosity [-] 0.4
kmat Matrix permeability [m2] 1.0 ×10−13
µwViscosity of water [Pa·s] 1.0 ×10−3
Gd,ref Reference critical energy release rate [N/m] 1.5
ldRegularization length scale parameter [m] 1.0 ×10−3
Hcrit Normalized threshold strain energy [-] 0.05
αv,int Volumetric expansion coefficient (intact) [-] 5.0 ×10−3
αv,dam Volumetric expansion coefficient (damaged) [-] 80.0 ×10−3
K∗
cKinetic parameter [Pa] 5.0 ×109
g∗
cKinetic parameter [-] 1.25
Table 2: Material parameters for the numerical freeze-thaw test.
applied temperature at the top starts to increase after reaching its minimum, s∗
cryo tends to decrease after546
t=7.5 hr due to the freezing characteristic function in Eq. (31) although the freezing front still propagates547
towards the bottom. Also, during the freezing phase, soil specimen tends to exhibit a constant temperature548
distribution at the region below the freezing front due to the latent heat effect, similar to our previous549
example shown in Section 5.2. More importantly, we observe a sequential development of the ice lenses at550
y=0.092 m, y=0.066 m, and y=0.042 m, respectively, which implies that separation void ratio (esep) can551
be approximated as ∼0.75 [25]. This result is expected, since those regions possess relatively high initial552
porosity compared to the other regions [Fig. 11(a)]. If the freezing front reaches the porous zone where the553
critical energy release rate is relatively low, both the cryo-suction and the exerted stress due to the phase554
transition initiate the horizontal crack.555
Once the freezing-induced fracture is developed, segregated bulk ice tends to form inside the opened556
crack at higher growth rates that lead to an abrupt volume expansion therein (Fig. 12). As illustrated in557
Fig. 14, we observe the opposite response during the thawing phase (11.25 hr ≤t≤15 hr). At t=11.25558
hr, once the applied temperature at the top again reaches the melting temperature θm=273.15 K, the soil559
specimen stops freezing and begins to thaw from the top to the bottom. During the thawing process, the560
melting front tends to move downwards whereas the freezing front remains unchanged since the bottom561
surface is thermally insulated. As the melted region where θ>θmevolves, the vertical displacement562
tends to decrease over time due to both the volume contraction during the phase transition and the water563
expulsion towards the top surface.564
Fig. 15 shows the evolution of the vertical displacement of the top surface during the freeze-thaw test565
(black curve). For comparison, we introduce a control experiment where the phase field solvers for both ice566
lens and damage are disabled but otherwise the material parameters are identical (blue curve). Hence, the567
numerical specimen in the control experiment may exhibit homogeneous freezing and thawing but not ice568
22 Hyoung Suk Suh, WaiChing Sun
Initial
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Configuration
<latexit sha1_base64="+N6Ym1mKdO64ZVZC2vpU0Fiau94=">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</latexit>
6hr
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8hr
<latexit sha1_base64="aFAQnmtj3GRZ9tMkXtS1jrh8u74=">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</latexit>
10 hr
<latexit sha1_base64="AndMYxvtslFlKxiNnqwMxge++WM=">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</latexit>
12 hr
<latexit sha1_base64="cL1l4Nf1mvB+QfDFucrlMKJxNqU=">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</latexit>
14 hr
<latexit sha1_base64="c2yK8tdtnLc73ZmNGqTx3nTnzuI=">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</latexit>
Configuration
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Final
<latexit sha1_base64="vNuADMmDA3P921u1LNYmS3Rrjj0=">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</latexit>
Freezing
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Thawing
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(a)
(b)
Fig. 12: (a) Formation and melting of multiple ice lenses and (b) evolution of the fracture phase field d
during the numerical freeze-thaw test.
0 0.5 1 1.5
0
0.02
0.04
0.06
0.08
0.1
(a)
0 200 400 600
0
0.02
0.04
0.06
0.08
0.1
(b)
270 272 274 276
0
0.02
0.04
0.06
0.08
0.1
(c)
0 0.5 1
0
0.02
0.04
0.06
0.08
0.1
(d)
Fig. 13: Thermo-hydro-mechanical response of the specimen during the freezing phase: (a) vertical dis-
placement (uy), (b) cryo-suction (s∗
cryo), (c) temperature (θ), and (d) ice saturation (Si) profiles along the
central axis.
lens formation and melting. The frost heave and thawing settlement for both experiments are compared to569
assess the impact of the ice lenses on the material responses.570
In the prime numerical experiment, ice lenses sequentially develop at y=0.092 m, y=0.066 m, and571
y=0.042 m (see Fig. 12), respectively. Each time the ice lens begins to form, the soil expands more rapidly572
and hence the steeper slope of the black curve, which indicates the rapid expansion of the numerical spec-573
imen, at t=4.2 hr, t=6.2 hr, and t=8.8 hr. During the thawing phase, the prescribed temperature574
of the top surface increase. This temperature increase leads to abrupt settlement within the first 2 hours575
of the thawing phase. As the ice lenses melt and subsequently drain out from the domain, the numerical576
Multi-phase-field model for ice lens growth 23
0 0.5 1 1.5
0
0.02
0.04
0.06
0.08
0.1
(a)
0 20 40 60 80
0
0.02
0.04
0.06
0.08
0.1
(b)
270 272 274 276
0
0.02
0.04
0.06
0.08
0.1
(c)
0 0.5 1
0
0.02
0.04
0.06
0.08
0.1
(d)
Fig. 14: Thermo-hydro-mechanical response of the specimen during the thawing phase: (a) vertical dis-
placement (uy), (b) cryo-suction (s∗
cryo), (c) temperature (θ), and (d) ice saturation (Si) profiles along the
central axis.
specimen shrinks (black curve). In contrast, homogeneous freezing and thawing result in considerably less577
amount of frost heaving and thawing settlement, due to the absence of cracks where ice lenses may form.578
The significant difference between the two simulations has important practical implications. It is pre-579
sumably possible to use an optimization algorithm to identify the material parameters such that the control580
experiment may match better with the observed frost heave and thawing settlements of soil vulnerable to581
ice lens formation. However, the apparent match obtained from such an excessive calibration is fruitless582
as it may lead to material parameters that are not physics and therefore lead to a model weak at forward583
predictions. Results of these numerical experiments again suggest that the ice lenses play a key role in frost584
heaving and the subsequent settlement of soils. This example also highlights that our proposed model is585
capable of simulating the ice lens growth and thaw in a fluid-saturated porous media, which may not be586
easily captured via a classical thermo-hydro-mechanical model.587
0 2 4 6 8 10 12 14 16
0
0.4
0.8
1.2
1.6
2
Frost heave
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Thawing settlement
<latexit sha1_base64="3nhqj15ZSvdWcccKk9VNk0S4YEY=">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</latexit>
Ice lens formation
<latexit sha1_base64="afjdOWYIlipFlv9dzL5TxEhbW+w=">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</latexit>
Ice lens formation
<latexit sha1_base64="afjdOWYIlipFlv9dzL5TxEhbW+w=">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</latexit>
Ice lens formation
<latexit sha1_base64="afjdOWYIlipFlv9dzL5TxEhbW+w=">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</latexit>
Fig. 15: Vertical displacement (uy) evolution of the top surface during the numerical freeze-thaw test. The
black curve is obtained from a thermo-hydro-mechanical simulation that enables ice lensing; the blue curve
is obtained from the control experiment that takes out the ice lensing capacity.
24 Hyoung Suk Suh, WaiChing Sun
5.4 Vertical ice lens growth in edge notched specimen588
While numerical example presented in Section 5.3 demonstrated horizontal ice lens formation perpendic-589
ular to the freezing direction, in this section, we simulate vertical ice lens growth which is parallel to the590
freezing direction, by leveraging the proposed driving force [Eq. (27)] for the Allen-Cahn equation. Specifi-591
cally, our objective is to demonstrate the formation of an ice lens that follows the crack trajectory that leads592
to a non-planar ice growth. Hence, as shown in Fig. 16, the problem domain is a 0.06 m wide and 0.02593
m long rectangular globally undrained porous specimen that contains a 0.005 m long initial vertical edge594
notch along the central axis, while considering an ideal case where σ0
dam =0and ¯
αv=0 to focus on the ice595
lens growth along the crack by decoupling the interactions between the two. By setting the initial temper-596
ature as θ0=274.15 K, the numerical experiment is performed by applying a constant heat flux of ˆ
q=25597
W/m2that induces conductive vertical cooling from the top surface, with prescribed vertical displacement598
ˆuat a rate of −10−6mm/s to promote crack growth from the notch tip. Here, we assume that the material is599
homogeneous while the material parameters are chosen as follows: φ0=0.2, ρs=2500 kg/m3,ρw=1000600
kg/m3,ρi=920 kg/m3,E=2.5 GPa, ν=0.3, kmat =10−15 m2,µw=10−3Pa·s, cs=0.9 ×103J/kg/K,601
cw=4.2 ×103J/kg/K, ci=1.9 ×103J/kg/K, κs=7.55 W/m/K, κw=0.5 W/m/K, κi=2.25 W/m/K,602
Gd=2.25 N/m, ld=1.0 ×10−3m, K∗
c=5.0 ×109Pa, and g∗
c=1.25. In addition, we set νc=0.0001,603
γc=0.05 J/m2,δc=0.0001 m, and ec=0.5 (J/m)1/2 for the Allen-Cahn phase field model while adopting604
the structured mesh with element size of he=0.25 mm and the time step size of ∆t=1 min.605
0.06 m
<latexit sha1_base64="a+Onjw/FPIh9U6rGTGACT3Iheqs=">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</latexit>
0.02 m
<latexit sha1_base64="m9eX+1N+y62/UC28jhzeUuWJ+Vk=">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</latexit>
ˆ
u
<latexit sha1_base64="QZ3fow5qu4FI5S3X94sk9SWJKWc=">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</latexit>
0.005 m
<latexit sha1_base64="vtu5ndmzcj8Kp8/6+uGifVTUzEY=">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</latexit>
ˆq= 25 W/m2
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Fig. 16: Schematic of geometry and boundary conditions for the single edge notched test.
Based on this setting, as shown in Fig. 17, prescribed compression results in tensile stresses perpen-606
dicular to the loading direction that stimulates crack growth, while permeability enhancement [Eq. (46)]607
and relative permeability [Eq. (39)] yield relatively low pore water pressure inside the notch similar to the608
results shown in [109]. The phase transition process of pore water begins once the temperature at the top609
surface reaches the freezing temperature θmin both the damaged and undamaged regions, however, since610
the proposed driving force for the Allen-Cahn equation in Eq. (27) leads to an intense growth of ice inside611
the fracture (i.e., ice lens) such that the phase field ctends to evolve faster inside the damaged region [Fig.612
17(a) and Fig. 17(d)].613
As evidenced in Fig. 17(d), ice phase tends to continuously grow along the pre-existing notch until crack614
initiates from the tip. Then, as illustrated in Fig. 18, once crack starts to propagate due to the combined615
effect of ice-water phase transition and the applied load, ice lens tends to follow the crack trajectory. We616
can also see from Fig. 18 that crack opening leads to complete fragmentation of the solid matrix due to617
the relation shown in Eq. (48), which results in more realistic ice lens simulations since it possesses zero618
solidity, i.e., 1 −φ. More importantly, the results indicate that our proposed framework is not only restricted619
to simulating planar ice lenses but also capable of modeling non-planar ice lenses that are not necessarily620
perpendicular to the freezing direction, which may have a more profound impact on microporomechanical621
problems that involve water adsorption processes which can affect microscopic fluid motion inside the622
heterogeneous matrix and hence the freezing patterns.623
Multi-phase-field model for ice lens growth 25
(a) (b)
(c) (d)
Fig. 17: Transient response of porous specimen at t=45 min. (a) fracture phase field d; (b) x-displacement
ux; (c) pore water pressure pw; and (d) Allen-Cahn phase field c.
6 Conclusions624
In this work, we introduce a multi-phase-field microporomechanics theory and the corresponding finite625
element solver to capture the freeze-thaw action in a frozen/freezing/thawing porous medium that may626
form ice lenses. By introducing two phase field variables that indicate the phase of the ice/water and dam-627
aged/undamaged material state, the proposed thermo-hydro-mechanical model is capable of simulating628
the freezing-induced fracture caused by the growth of the ice lens as segregated ice. We also extend the629
Bishop’s effective stress principle for frozen soil to incorporate the effects of damage and ice growth and630
distinguish them from those of the freezing retention responses. This treatment enables us to take into631
account the shear strength of the ice lenses and analyzes how the homogeneous freezing process and the632
ice lens growth affect the thermo-hydro-mechanical coupling effects in the transient regime. The model633
is validated against published freezing experiments. To investigate how the formation and thawing of634
ice lens affect the frost heave and thaw settlement, we conduct numerical experiments that simulate the635
climate-induced frozen heave and thaw settlement in one thermal cycle and compare the simulation re-636
sults with those obtained from a thermo-hydro-mechanical model that does not explicitly capture the ice637
lens. The simulation results suggest that explicitly capturing the growth and thaw of ice lens may provide638
more precise predictions and analyses on the multi-physical coupling effects of frozen soil at different time639
scales. Accurate and precise predictions on the frozen heave and thaw settlement are crucial for many640
modern engineering applications, from estimating the durability of pavement systems to the exploration641
of ice-rich portions of Mars. This work provides a foundation for a more precise depiction of frozen soil642
by incorporating freezing retention, heat transfer, fluid diffusion, fracture mechanics, and ice lens growth643
in a single model. More accurate predictions nevertheless may require sufficient data to solve the inverse644
problems and quantify uncertainties as well as optimization techniques to identify material parameters645
from different experiments. Such endeavors are important and will be considered in the future studies.646
26 Hyoung Suk Suh, WaiChing Sun
t= 70 min
<latexit sha1_base64="7LcDZeLqSOalSfrHH04kbrsdca4=">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</latexit>
t= 100 min
<latexit sha1_base64="oZJqcN4yoejopCwXyi4FZWKlMnM=">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</latexit>
t= 130 min
<latexit sha1_base64="+LSalf2nEmYFQdda3U7UJfICQmo=">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</latexit>
Fig. 18: Evolution of ice lens (χi) and porosity (φ) during vertical freezing of edge notched specimen.
7 Acknowledgments647
This work is primarily supported by the Earth Materials and Processes program from the US Army Re-648
search Office under grant contract W911NF-18-2-0306, with additional time of the PI supported by the649
NSF CAREER grant from Mechanics of Materials and Structures program at National Science Foundation650
under grant contract CMMI-1846875. These supports are gratefully acknowledged. The views and conclu-651
sions contained in this document are those of the authors, and should not be interpreted as representing652
the official policies, either expressed or implied, of the sponsors, including the Army Research Labora-653
tory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for654
Government purposes notwithstanding any copyright notation herein.655
Appendix A Relationship among the Allen-Cahn model parameters and physical properties656
In this section, we consider an idealized one-dimensional transition zone between water (c=1) and ice657
(c=0) phase constituents to obtain the relationship among the parameters [Eq. (26)] for the Allen-Cahn658
phase field equation. Here, we assume planar ice-water transition zone, where the phase field cvaries659
along the xdirection. Since ∇2c=∂2c/∂x2in one-dimensional setting, an equilibrium solution ( ˙
c=0) at660
the freezing temperature θ=θmcan be obtained as follows:661
c(x) = 1
21+tanh x
2δc, (A1)
where δcis a measure of interface thickness which can be expressed as [40,44]:662
δc=ec
√2Wc
. (A2)
As pointed out in Boettinger et al. [40], the interface thickness δcbalances two opposing effects. The tran-663
sition zone tends to become narrow depending on the energy hump parameter Wc, and at the same time664
tends to be diffusive in order to reduce the energy associated with ∇c, based on the coefficient ec. From665
Multi-phase-field model for ice lens growth 27
Eqs. (A1) and (A2), Fig. 19 illustrates the variations of the phase field cand the integrand of Eq. (20) nor-666
malized by Wcalong the distance xacross a flat ice-water interface at the freezing temperature. Since the667
area under the curve shown in Fig. 19(b) is the ice-water interface energy (i.e., interfacial tension γiw) [39],668
we can obtain the first two expressions in Eq. (26) by taking the limit δc→0 while keeping γiw fixed.669
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
(a)
-10 -5 0 5 10
0
0.02
0.04
0.06
0.08
0.1
iw
<latexit sha1_base64="LcXu/oE4W0mJZ2OiQ1S4oJL63NM=">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</latexit>
(b)
Fig. 19: Variations of (a) phase field variable cand (b) the normalized energy density with distance x, across
a flat ice-water interface.
On the other hand, to investigate a moving interface in one-dimensional space, we consider the case670
where cmoves at a constant velocity v0while neglecting the diffusion process. In this case, we can assume671
that:672
˙
c=−v0
∂c
∂x, (A3)
and substituting Eq. (A3) into Eq. (21) yields the following expression:673
v0
Mc∂c
∂x=∂fc
∂c. (A4)
Note that Eq. (A4) has no solution if θ=θm, however, a solution does exist for a small δcif the temperature674
is given by [40,129],675
θ=θm−v0
νc. (A5)
In this case, the kinetic coefficient νccan be estimated as [74]:676
νc=6McρiLθec
θm√2Wc
. (A6)
By rearranging Eq. (A6), the expression for the parameter Mcin Eq. (26) can be obtained.677
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