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

May 2022
International Journal for Numerical and Analytical Methods in Geomechanics 46(100)

DOI:10.1002/nag.3408

Project: Multiscale Computational Poromechanics

Authors:

Hyoung Suk Suh

Columbia University

Waiching Sun

Columbia University

This article presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields is induced by their own driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly capture the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models.

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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

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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c=0

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d=1

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d=0

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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−ciθm(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

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

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−1pw−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) = ρiLθ1−θ

θm. (25)

0 0.2 0.4 0.6 0.8 1

0.02

0.04

0.06

0.08

(a)

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

(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) = ρiLθ1−θ

θm+ [1−gd(d)]K∗

c1−θ

θmg∗

, (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)

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Damaged (d= 1)

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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

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t= 50 s

t= 100 s

t= 200 s

t= 400 s

t= 800 s

0.001 m

<latexit sha1_base64="jL3c0/T/famkD/KIg5t3S4d7znM=">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</latexit>

0.001 m

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 −11

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 −11

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

10 106

mvG =0.8,n

vG =2.0

<latexit sha1_base64="j+Shg9YGl9Y987V81wHSGdHB+Bc=">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</latexit>

bB=0.55 K1,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

(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=ZBξφ˙

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 +ZBζφ[(ρwcw−ρici)(θ−θm) + ρiLθ]˙

Sw(c)dV

−ZB∇ζ·qdV −ZBζˆ

r dV −Z∂Bq

ζˆ

q dΓ=0,

Gc=ZBγ1

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







pw,n

θn







Gd=0

−−−−−−−−−−−−−−→

δu=0,δpw=0, δθ=0, δc=0







pw,n

θn

dn+1







| {z }

Linear solver

Iterative solver

z }| {

Gθ=Gc=0

−−−−−−−−−−−→

δu=0,δpw=0, δd=0







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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<latexit sha1_base64="oN3aao1D0tQe4BckF/G1gAyxnbc=">AAAEo3iclVNNbxMxEHXbBUr4auHIJRAhcYqyKBKcUKVWAgkhStW0kepQeb2zqYk/VrY3pbJ85A9whT/Gv8HOplK6YZGwtLOjN/Oe7ZlxVnJm7GDwe2NzK7l1+8723c69+w8ePtrZfXxiVKUpjKjiSo8zYoAzCSPLLIdxqYGIjMNpNtuP8dM5aMOUPLZXJUwEmUpWMEpsgMbYwjfr9v35Tm/QHyxWd91Jl04PLdfh+e7Wd5wrWgmQlnJijCPaMsrBdzu4MlASOiNTOKts8WbimCwrC5L61ZgjwghiL9bAQklr1lBzJbKbYCwPk9NGapS0SvG/wNoUDdQyAQ2o4IrYAEm4pEoIInMXqiSN0v4snTgc3EpDlHM4UzyP51Lc9VLvG6yvlShrDs90UMfPlk63l3axvsZqp8ElHLStybFFi0Y7DblfbNTBBxCqruFjOManEjSxIYynmoQELEnGSVtOOO5q2pda3xnfqpqzMEDX+ZjmyrZlvmsKx6p1x63KB6vKq4R/7hJu4F00bfHZZYgH0xK3Ycvwtd0W5t5F06Y+lUE9mEW/7AWoMAxOi5l3R2EqdGykhtVWzoH+7+SEnHXKYoJF/c+KmhYfbtp8puvOyat+OuwPPw97e2+XT3gbPUXP0UuUotdoD71Hh2iEKOLoB/qJfiUvkg/JUXJcp25uLDlP0I2VTP4A/nSlEw==</latexit>

0.03 m

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0.03 m

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

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This study

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0 0.5 1 1.5 2

105

235

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285

0 0.5 1 1.5 2

105

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275

285

0 0.5 1 1.5 2

105

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285

0 0.5 1 1.5 2

105

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0 0.5 1 1.5 2

105

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0 0.5 1 1.5 2

105

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Lackner et al. [2005]

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Sweidan et al. [2020]

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(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

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<latexit sha1_base64="t1lo9wUujzFBVB8BceLG3pLBo/o=">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</latexit>

10 m

<latexit sha1_base64="wr1/FGfkbNfh2uTsHyidjXukkqU=">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</latexit>

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

t= 50 s

<latexit sha1_base64="x2PdnzJWtPVbxm31F2ysH6JPias=">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</latexit>

t= 150 s

<latexit sha1_base64="L0YJvBs2pe2xP4ftbI7u9K3URoQ=">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</latexit>

t= 350 s

<latexit sha1_base64="VGV186D2Fq+HqQPoHH0maNiwY+s=">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</latexit>

t= 750 s

<latexit sha1_base64="j3T0cLXw4NB/qhbp7OtDmZ+f1cQ=">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</latexit>

(b)

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

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<latexit sha1_base64="DH2FKiEnN77qmkMMGEBzj5YobRQ=">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</latexit>

✓= 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>

ˆpw=0

<latexit sha1_base64="FjkplLuwcCGAXJK2OwIatMmBMfg=">AAAEqHiclVNNbxMxEHXbhZbw1cKRy0IuiEOURZHgQlWJSnBBBESaoDpEXu9sso0/FtubUFk+8hu4ws/i3+DNplK6YZGwtLOjN/Oex55xnLNMm273987uXnDj5v7BrdbtO3fv3T88enCmZaEoDKhkUo1iooFlAgYmMwxGuQLCYwbDeP66jA8XoHQmxSdzmcOYk6nI0owS46ExnhFjczdZhq/C7uSw3e10VyvcdqK100br1Z8c7X3HiaQFB2EoI1pbokxGGbiwhQsNOaFzMoXzwqQvxzYTeWFAULcZs4RrTsxsC0ylMHoL1Zc8vg6Wd5SJaS21lDRSsr/ASqc11GQcalDKJDEeErCkknMiEot97Voqdx6NLfZuoaCUsziWLCnrksy2I+dqrIuC5xWHxcqr48drJ2xHIVZXWOXUuISBMhXZwDez6rZVkLjVRi18Cv7WFbzzZbzPQRHjw3iqiE/AgsSMNOX4cjfTvlT6VrtG1STzU3SVj2kiTVPmm7pweWvhqFH5dFN5k/DPXfwJnC1NU3y+9HFvGuLGb+m/ptPCwtnSNKlPhVf3ZtUvMwPph8EqPnf2o58KVTZSwWYrF0D/d3J8zjZlNcG8+sdpRXP+4Ub1Z7rtnD3vRL1O70OvfXK8fsIH6BF6gp6iCL1AJ+gt6qMBougr+oF+ol/Bs6AfDIPPVeruzprzEF1bQfwHuWOmbQ==</latexit>

TC1

<latexit sha1_base64="oVQkJqI6Gd3xAm7IXo09OsG2G6U=">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</latexit>

TC2

<latexit sha1_base64="nczcZSVR+p8CsvlbOvKyxbTq+0c=">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</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>

✓AA’ = 20.8e0.1pt+ 264.75 K

<latexit sha1_base64="ds7AbXgqS/Mvk/wpz9sK4JRdVM8=">AAAE5niclVPPb9MwFPa2AKP86uDIJdADSIioqQrbBWkTkwAhxEB0qzRvlZO8tKGOHWynY7J85MoNceU/4K/hwAX+FZwmk7qUIGEpL0/fe99n+/m9IKOJVN3uz5XVNefCxUvrl1tXrl67fqO9cXNf8lyEMAg55WIYEAk0YTBQiaIwzASQNKBwEEyfFvGDGQiZcPZOnWZwlJIxS+IkJMpCo/YLPCFKYzUBRczIOvBR6Z2de8a4T9xe19tySwjMsX7Y9XwsPwillTEPeo/73uajKuy+NKN2p+t158tddvzK6aBq7Y021j7hiId5CkyFlEipiVBJSMG4LZxLyEg4JWM4zFW8daQTluUKWGgWY5qkMiVqsgTGnCm5hMrTNDgPFhVO2LiWWkgqzulfYCHjGqqSFGpQTDlRFmJwEvI0JSwqCsskF+bQP9LYurmAQk7jgNOoOBenuuMbU2O9z9Os5NBAWHV8p3Lcju9icYaVTo1LKAhVkos3mveKFhCZ+UYtvAu26gJe2WO8zkAQZcN4LIhNwIwElDTl2OMuph1XXSNNo2qU2B48y8dhxFVT5rO6cFE1d9iovLuovEj45y72BkYXpik+PbFxaxriym5pv6bbwszowjSpj5lVt2b+Xnb0uG0GLdKp0W9tV4jiIQUsPuUMwv/tHJuzTJl3cFr+g7ikFYPr18d02dnveX7f67/pd7b71Qivo9voLrqPfLSJttFztIcGKETf0Q/0C/12Js5n54vztUxdXak4t9C55Xz7A7fDvhA=</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

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This study

(Feng et al. [2015])

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t= 10 min

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t= 60 min

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t= 120 min

<latexit sha1_base64="F7W95ZfZaMzB+j/pvAKQW9nVovs=">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</latexit>

t= 180 min

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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 th