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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

Authors:

Hyoung Suk Suh

Case Western Reserve 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=">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</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=">AAAEnHiclVNNbxMxEHXbBUr4auGIhAK5cIqyKBIcqzYSSKiiRSSNVIfK651N3fhjZXtTIstHTlzhx/Fv8GZTKd2wSFja2dGbec9jzzjJOTO21/u9tb0T3bl7b/d+68HDR4+f7O0/HRlVaApDqrjS44QY4EzC0DLLYZxrICLhcJbMjsr42Ry0YUp+sYscJoJMJcsYJTZAp4uLvU6v21uu9qYTr5wOWq2Ti/2d7zhVtBAgLeXEGEe0ZZSDb7dwYSAndEamcF7Y7N3EMZkXFiT16zFHhBHEXm6AmZLWbKBmIZLbYHkvTE5rqaWkVYr/BdYmq6GWCahBGVfEBkjCNVVCEJk6HGo3SvvzeOJwcAsNpZzDieJpWZfirhN7X2NdFSKvODzRQR2/XDntTtzG+garnBqXcNC2Ilv4ZpcddhpSv9yohQcQbl3DcSjjUw6a2BDGU01CApYk4aQpJ5S7nva10nfGN6qmLEzOTT6mqbJNme/rwuWttceNyoN15XXCP3cJJ/CuNE3x2XWIB9MQt2HL8DWdFubelaZJfSqDejDLftlLUGEYnBYz7z6HqdBlIzWst3IO9H8nJ+RsUpYTLKp/klU0Hx5uXH+mm87oTTfud/un/c7B4eoJ76Ln6BV6jWL0Fh2gD+gEDRFFgH6gn+hX9CIaRB+j4yp1e2vFeYZurWj0B7BBoe4=</latexit>

10 m

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

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=">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</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=">AAAEpXiclVNNbxMxEHXbhZbw1cKRSyAHOEVZFAlOVaVWgktFQUkaUYfI651NTfyxsr0pleUj/4Ar/C/+Dd5sKqUbFglLOzt6M+/ZnhknOWfG9nq/t7Z3ojt3d/fute4/ePjo8f7Bk5FRhaYwpIorPU6IAc4kDC2zHMa5BiISDufJ/LiMny9AG6bkwF7nMBFkJlnGKLEB+owtfLNucNz30/1Or9tbrvamE6+cDlqts+nBznecKloIkJZyYowj2jLKwbdbuDCQEzonM7gobPZ24pjMCwuS+vWYI8IIYi83wExJazZQcy2S22BZICZntdRS0irF/wJrk9VQywTUoIwrYgMk4YoqIYhMXaiTNEr7i3jicHALDaWcw4niaXkuxV0n9r7G+lqIvOLwRAd1/HzltDtxG+sbrHJqXMJB24pcNmnZaqch9cuNWvgEQtU1nIZjfMhBExvCeKZJSMCSJJw05YTjrqd9qfSd8Y2qKQsjdJOPaapsU+a7unBZtfa4UflkXXmd8M9dwg28K01TfH4V4sE0xG3YMnxNt4WFd6VpUp/JoB7Msl/2ElQYBqfF3LtPYSp02UgN661cAP3fyQk5m5TlBIvqn2QVrXy4cf2Zbjqj19243+1/7HeODldPeA89Qy/QKxSjN+gIvUdnaIgokugH+ol+RS+j02gQjarU7a0V5ym6taLpH5Japa8=</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=">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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

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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

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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 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

0.01

0.02

0.03

0.04

0.05

Foam 2

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Foam 1

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(a)

0 60 120 180

260

265

270

275

280

285

290

TC2

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TC4

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TC3

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(b)

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

4106

Freezing direction

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(a)

0 0.01 0.02 0.03 0.04 0.05

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

21−2ν

1+νexp [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

ˆpw=0

0.1m

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0.04 m

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✓=ˆ

✓(t); ˆpw=0

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0.3 0.4 0.5

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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✓(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

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6hr

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8hr

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10 hr

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12 hr

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14 hr

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Conﬁguration

Final

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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

0.02

0.04

0.06

0.08

0.1

(a)

0 200 400 600

0.02

0.04

0.06

0.08

0.1

(b)

270 272 274 276

0.02

0.04

0.06

0.08

0.1

(c)

0 0.5 1

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

0.02

0.04

0.06

0.08

0.1

(a)

0 20 40 60 80

0.02

0.04

0.06

0.08

0.1

(b)

270 272 274 276

0.02

0.04

0.06

0.08

0.1

(c)

0 0.5 1

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

0.4

0.8

1.2

1.6

Frost heave

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Thawing settlement

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Ice lens formation

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Ice lens formation

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=">AAAEqXiclVNNb9QwEHXbAGX5auHIJbAXJKTVploJjpWoBBdEi9h2RbOtHGeyddcfke1sqSwf+Q9c4V/xb7A3qbTNEiQsZTJ6M+/ZnhlnJaPaDIe/Nza3ojt3723f7z14+Ojxk53dp8daVorAmEgm1STDGhgVMDbUMJiUCjDPGJxk83chfrIApakUX8x1CVOOZ4IWlGDjobPhYLgXpwa+GRtzd77T98ByxetO0jh91KzD892t72kuScVBGMKw1hYrQwkDF/fSSkOJyRzP4LQyxduppaKsDAjiVmMWc82xuVgDCymMXkP1Nc9ug6FIVMxaqUHSSMn+AitdtFBDObSggklsPCTgikjOscitL5PQUrnTZGpT71YKgpxNM8nycC7JbD9xrsW6rHhZc1imvHr6onHifhKn6garnRYXM1CmJoceLdttFeRuuVEvPQBfdQUf/TE+laCw8eF0prBPSAXOGO7K8cddTTur9a12nao59WN0k5+SXJquzPdt4VC1eNKpfLCqvEr45y7+Bs4G0xWfX/m4Nx1x47f0X9dtYeFsMF3qM+HVvVn2y1yA9MNgFZ87+9lPhQqNVLDaygWQ/50cn7NOWU4wr/9ZUdPCw03az3TdOd4bJKPB6GjU3x81T3gbPUcv0SuUoDdoH31Ah2iMCFLoB/qJfkWvo6NoEn2tUzc3Gs4zdGtF5A/fwKZv</latexit>

<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

<latexit sha1_base64="zTGb5/6r9AiiBk0FKRO6ie4S28U=">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</latexit>

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)

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

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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

21+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

0.2

0.4

0.6

0.8

(a)

-10 -5 0 5 10

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

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

References678

1. Palmer Andrew C, Williams Peter J. Frost heave and pipeline upheaval buckling Canadian Geotechnical679

Journal. 2003;40:1033–1038.680

2. Zhang Sheng, Sheng Daichao, Zhao Guotang, Niu Fujun, He Zuoyue. Analysis of frost heave mecha-681

nisms in a high-speed railway embankment Canadian Geotechnical Journal. 2016;53:520–529.682

28 Hyoung Suk Suh, WaiChing Sun

3. Li Anyuan, Niu Fujun, Zheng Hao, Akagawa Satoshi, Lin Zhanju, Luo Jing. Experimental measure-683

ment and numerical simulation of frost heave in saturated coarse-grained soil Cold Regions Science and684

Technology. 2017;137:68–74.685

4. Lake Craig B, Yousif Mohammed Al-Mala, Jamshidi Reza J. Examining freeze/thaw effects on perfor-686

mance and morphology of a lightly cemented soil Cold Regions Science and Technology. 2017;134:33–44.687

5. Ji Yukun, Zhou Guoqing, Zhou Yang, Vandeginste Veerle. Frost heave in freezing soils: A quasi-static688

model for ice lens growth Cold Regions Science and Technology. 2019;158:10–17.689

6. DiMillio Al F. A quarter century of geotechnical research tech. rep.Turner-Fairbank Highway Research690

Center 1999.691

7. Nelson Frederick E, Anisimov Oleg A, Shiklomanov Nikolay I. Subsidence risk from thawing per-692

mafrost Nature. 2001;410:889–890.693

8. Nelson Frederick E, Anisimov Oleg A, Shiklomanov Nikolay I. Climate change and hazard zonation694

in the circum-Arctic permafrost regions Natural Hazards. 2002;26:203–225.695

9. Streletskiy Dmitry A, Shiklomanov Nikolay I, Nelson Frederick E. Permafrost, infrastructure, and696

climate change: a GIS-based landscape approach to geotechnical modeling Arctic, Antarctic, and Alpine697

Research. 2012;44:368–380.698

10. Leibman Marina, Khomutov Artem, Kizyakov Alexandr. Cryogenic landslides in the West-Siberian699

plain of Russia: classiﬁcation, mechanisms, and landforms in Landslides in cold regions in the context of700

climate change:143–162Springer 2014.701

11. Mithan HT, Hales TC, Cleall PJ. Topographic and Ground-Ice Controls on Shallow Landsliding in702

Thawing Arctic Permafrost Geophysical Research Letters. 2021;48:e2020GL092264.703

12. Taber Stephen. Frost heaving The Journal of Geology. 1929;37:428–461.704

13. Taber Stephen. The mechanics of frost heaving The Journal of Geology. 1930;38:303–317.705

14. Peppin Stephen SL, Style Robert W. The physics of frost heave and ice-lens growth Vadose Zone Journal.706

2013;12.707

15. Wilen LA, Dash JG. Frost heave dynamics at a single crystal interface Physical Review Letters.708

1995;74:5076.709

16. Dash JG, Fu Haiying, Wettlaufer JS. The premelting of ice and its environmental consequences Reports710

on Progress in Physics. 1995;58:115.711

17. Dash JG, Rempel AW, Wettlaufer JS. The physics of premelted ice and its geophysical consequences712

Reviews of modern physics. 2006;78:695.713

18. Harlan RL. Analysis of coupled heat-ﬂuid transport in partially frozen soil Water Resources Research.714

1973;9:1314–1323.715

19. Miller RD. Freezing and heaving of saturated and unsaturated soils Highway Research Record.716

1972;393:1–11.717

20. Miller RD. Lens initiation in secondary heaving in Proceedings of the International Symposium on Frost718

Action in Soils;2:68–74Lule˚

a Alltryck AB Lule˚

a, Sweden 1977.719

21. O’Neill Kevin, Miller Robert D. Exploration of a rigid ice model of frost heave Water Resources Research.720

1985;21:281–296.721

22. Fowler AC. Secondary frost heave in freezing soils SIAM Journal on Applied Mathematics. 1989;49:991–722

1008.723

23. Fowler Andrew C, Krantz William B. A generalized secondary frost heave model Siam Journal on724

Applied Mathematics. 1994;54:1650–1675.725

24. Gilpin R.R. A model for the prediction of ice lensing and frost heave in soils Water Resources Research.726

1980;16:918–930.727

25. Zhou Jiazuo, Li Dongqing. Numerical analysis of coupled water, heat and stress in saturated freezing728

soil Cold Regions Science and Technology. 2012;72:43–49.729

26. Konrad Jean-Marie, Morgenstern Norbert R. A mechanistic theory of ice lens formation in ﬁne-730

grained soils Canadian Geotechnical Journal. 1980;17:473–486.731

27. Nixon John F. Field frost heave predictions using the segregation potential concept Canadian Geotech-732

nical Journal. 1982;19:526–529.733

28. Konrad J-M, Shen M. 2-D frost action modeling using the segregation potential of soils Cold regions734

science and technology. 1996;24:263–278.735

Multi-phase-ﬁeld model for ice lens growth 29

29. Tiedje E, Guo P. Frost heave modeling using a modiﬁed segregation potential approach in Cold Regions736

Engineering 2012: Sustainable Infrastructure Development in a Changing Cold Environment:686–696 2012.737

30. Rempel Alan W, Wettlaufer JS, Worster M Grae. Premelting dynamics in a continuum model of frost738

heave Journal of ﬂuid mechanics. 2004;498:227–244.739

31. Rempel AW. Formation of ice lenses and frost heave Journal of Geophysical Research: Earth Surface.740

2007;112.741

32. Style Robert W, Peppin Stephen SL, Cocks Alan CF, Wettlaufer John S. Ice-lens formation and geo-742

metrical supercooling in soils and other colloidal materials Physical Review E. 2011;84:041402.743

33. Nishimura S, Gens Antonio, Olivella Sebasti`

a, Jardine RJ. THM-coupled ﬁnite element analysis of744

frozen soil: formulation and application G´eotechnique. 2009;59:159–171.745

34. Zhou MM, Meschke G. A three-phase thermo-hydro-mechanical ﬁnite element model for freezing746

soils International journal for numerical and analytical methods in geomechanics. 2013;37:3173–3193.747

35. Na SeonHong, Sun WaiChing. Computational thermo-hydro-mechanics for multiphase freezing and748

thawing porous media in the ﬁnite deformation range Computer Methods in Applied Mechanics and749

Engineering. 2017;318:667–700.750

36. Michalowski Radoslaw L, Zhu Ming. Frost heave modelling using porosity rate function International751

journal for numerical and analytical methods in geomechanics. 2006;30:703–722.752

37. Ghoreishian Amiri SA, Grimstad G, Kadivar M, Nordal S. Constitutive model for rate-independent753

behavior of saturated frozen soils Canadian Geotechnical Journal. 2016;53:1646–1657.754

38. Wettlaufer JS, Worster M Grae. Premelting dynamics Annu. Rev. Fluid Mech.. 2006;38:427–452.755

39. Allen Samuel M, Cahn John W. A microscopic theory for antiphase boundary motion and its applica-756

tion to antiphase domain coarsening Acta metallurgica. 1979;27:1085–1095.757

40. Boettinger William J, Warren James A, Beckermann Christoph, Karma Alain. Phase-ﬁeld simulation758

of solidiﬁcation Annual review of materials research. 2002;32:163–194.759

41. Bourdin Blaise, Francfort Gilles A, Marigo Jean-Jacques. The variational approach to fracture Journal760

of elasticity. 2008;91:5–148.761

42. Miehe Christian, Hofacker Martina, Welschinger Fabian. A phase ﬁeld model for rate-independent762

crack propagation: Robust algorithmic implementation based on operator splits Computer Methods in763

Applied Mechanics and Engineering. 2010;199:2765–2778.764

43. Borden Michael J, Verhoosel Clemens V, Scott Michael A, Hughes Thomas JR, Landis Chad M. A765

phase-ﬁeld description of dynamic brittle fracture Computer Methods in Applied Mechanics and Engi-766

neering. 2012;217:77–95.767

44. Warren James A, Boettinger William J. Prediction of dendritic growth and microsegregation patterns768

in a binary alloy using the phase-ﬁeld method Acta Metallurgica et Materialia. 1995;43:689–703.769

45. Sweidan Abdel Hassan, Heider Yousef, Markert Bernd. A uniﬁed water/ice kinematics approach for770

phase-ﬁeld thermo-hydro-mechanical modeling of frost action in porous media Computer Methods in771

Applied Mechanics and Engineering. 2020;372:113358.772

46. Chaboche Jean-Louis. Continuum damage mechanics: Part I—General concepts 1988.773

47. Bowen Ray M. Incompressible porous media models by use of the theory of mixtures International774

Journal of Engineering Science. 1980;18:1129–1148.775

48. Zienkiewicz Olgierd C, Chan AHC, Pastor M, Schreﬂer BA, Shiomi T. Computational geomechanics;613.776

Citeseer 1999.777

49. Ehlers Wolfgang. Foundations of multiphasic and porous materials in Porous media:3–86Springer 2002.778

50. Coussy Olivier. Poromechanics. John Wiley & Sons 2004.779

51. O’Neill Kevin. Numerical solutions for a rigid-ice model of secondary frost heave;82. US Army Corps of780

Engineers, Cold Regions Research & Engineering Laboratory 1982.781

52. Suh Hyoung Suk, Sun WaiChing, O’Connor Devin T. A phase ﬁeld model for cohesive fracture in782

micropolar continua Computer Methods in Applied Mechanics and Engineering. 2020;369:113181.783

53. Miehe Christian, Welschinger Fabian, Hofacker Martina. Thermodynamically consistent phase-ﬁeld784

models of fracture: Variational principles and multi-ﬁeld FE implementations International journal for785

numerical methods in engineering. 2010;83:1273–1311.786

54. Choo Jinhyun, Sun WaiChing. Cracking and damage from crystallization in pores: Coupled chemo-787

hydro-mechanics and phase-ﬁeld modeling Computer Methods in Applied Mechanics and Engineering.788

2018;335:347–379.789

30 Hyoung Suk Suh, WaiChing Sun

55. Heider Yousef. A review on phase-ﬁeld modeling of hydraulic fracturing Engineering Fracture Mechan-790

ics. 2021;253:107881.791

56. Bryant Eric C, Sun WaiChing. A mixed-mode phase ﬁeld fracture model in anisotropic rocks with792

consistent kinematics Computer Methods in Applied Mechanics and Engineering. 2018;342:561–584.793

57. Suh Hyoung Suk, Sun WaiChing. Asynchronous phase ﬁeld fracture model for porous media with794

thermally non-equilibrated constituents Computer Methods in Applied Mechanics and Engineering.795

2021;387:114182.796

58. Borja Ronaldo I, Yin Qing, Zhao Yang. Cam-Clay plasticity. Part IX: On the anisotropy, heterogeneity,797

and viscoplasticity of shale Computer Methods in Applied Mechanics and Engineering. 2020;360:112695.798

59. Bluhm Joachim, Ricken Tim. Modeling of freezing and thawing processes in liquid ﬁlled thermo-799

elastic porous solids Transport in Concrete: Nano-to Macrostructure, edited by M. Setzer (Aediﬁcatio Pub-800

lishers, Freiburg, 2007). 2007:41–57.801

60. Ricken Tim, Bluhm Joachim. Modeling ﬂuid saturated porous media under frost attack GAMM-802

Mitteilungen. 2010;33:40–56.803

61. Van Genuchten M Th. A closed-form equation for predicting the hydraulic conductivity of unsatu-804

rated soils Soil science society of America journal. 1980;44:892–898.805

62. Gelet Rachel, Loret Benjamin, Khalili Nasser. A thermo-hydro-mechanical coupled model in local806

thermal non-equilibrium for fractured HDR reservoir with double porosity Journal of Geophysical Re-807

search: Solid Earth. 2012;117.808

63. Liu Zhihong, Muldrew Ken, Wan Richard G, Elliott Janet AW. Measurement of freezing point depres-809

sion of water in glass capillaries and the associated ice front shape Physical Review E. 2003;67:061602.810

64. Loginova Irina, Amberg Gustav, ˚

Agren John. Phase-ﬁeld simulations of non-isothermal binary alloy811

solidiﬁcation Acta materialia. 2001;49:573–581.812

65. Alexiades Vasilios, Solomon Alan D. Mathematical modeling of melting and freezing processes. Routledge813

2018.814

66. Eshelby John Douglas. The determination of the elastic ﬁeld of an ellipsoidal inclusion, and re-815

lated problems Proceedings of the royal society of London. Series A. Mathematical and physical sciences.816

1957;241:376–396.817

67. Hiroshi Hatta, Minoru Taya. Equivalent inclusion method for steady state heat conduction in com-818

posites International Journal of Engineering Science. 1986;24:1159–1172.819

68. Sun WaiChing. A stabilized ﬁnite element formulation for monolithic thermo-hydro-mechanical sim-820

ulations at ﬁnite strain International Journal for Numerical Methods in Engineering. 2015;103:798–839.821

69. Lee Changho, Suh Hyoung Suk, Yoon Boyeong, Yun Tae Sup. Particle shape effect on thermal con-822

ductivity and shear wave velocity in sands Acta Geotechnica. 2017;12:615–625.823

70. Suh Hyoung Suk, Yun Tae Sup. Modiﬁcation of capillary pressure by considering pore throat geom-824

etry with the effects of particle shape and packing features on water retention curves for uniformly825

graded sands Computers and Geotechnics. 2018;95:129–136.826

71. Takaki Tomohiro. Phase-ﬁeld modeling and simulations of dendrite growth ISIJ international.827

2014;54:437–444.828

72. Aihara Shintaro, Takaki Tomohiro, Takada Naoki. Multi-phase-ﬁeld modeling using a conservative829

Allen–Cahn equation for multiphase ﬂow Computers & Fluids. 2019;178:141–151.830

73. Henry Karen S. A review of the thermodynamics of frost heave 2000.831

74. Wheeler Adam A, Boettinger William J, McFadden Geoffrey B. Phase-ﬁeld model for isothermal phase832

transitions in binary alloys Physical Review A. 1992;45:7424.833

75. Caginalp G, Socolovsky EA. Efﬁcient computation of a sharp interface by spreading via phase ﬁeld834

methods Applied Mathematics Letters. 1989;2:117–120.835

76. Caginalp G, Socolovsky EA. Computation of sharp phase boundaries by spreading: the planar and836

spherically symmetric cases Journal of Computational Physics. 1991;95:85–100.837

77. Penner E. Aspects of ice lens growth in soils Cold regions science and technology. 1986;13:91–100.838

78. Espinosa Rosa Maria, Franke Lutz, Deckelmann Gernod. Phase changes of salts in porous materials:839

Crystallization, hydration and deliquescence Construction and Building Materials. 2008;22:1758–1773.840

79. Koniorczyk Marcin, Gawin Dariusz. Modelling of salt crystallization in building materials with841

microstructure–Poromechanical approach Construction and Building Materials. 2012;36:860–873.842

Multi-phase-ﬁeld model for ice lens growth 31

80. Derluyn Hannelore, Moonen Peter, Carmeliet Jan. Deformation and damage due to drying-induced843

salt crystallization in porous limestone Journal of the Mechanics and Physics of Solids. 2014;63:242–255.844

81. Guodong Cheng. The mechanism of repeated-segregation for the formation of thick layered ground845

ice Cold Regions Science and Technology. 1983;8:57–66.846

82. Harris Charles, Arenson Lukas U, Christiansen Hanne H, et al. Permafrost and climate in Europe:847

Monitoring and modelling thermal, geomorphological and geotechnical responses Earth-Science Re-848

views. 2009;92:117–171.849

83. Evans Brian, Fredrich Joanne T, Wong Teng-Fong. The brittle-ductile transition in rocks: Recent exper-850

imental and theoretical progress The Brittle-Ductile Transition in Rocks, Geophys. Monogr. Ser. 1990;56:1–851

20.852

84. Lee Moo Y, Fossum Arlo, Costin Laurence S, Bronowski David. Frozen soil material testing and con-853

stitutive modeling Sandia Report, SAND. 2002;524:8–65.854

85. Anderson Duwayne M, Tice Allen R. Predicting unfrozen water contents in frozen soils from surface855

area measurements Highway research record. 1972;393:12–18.856

86. Koopmans Ruurd Willem Rienk, Miller RD. Soil freezing and soil water characteristic curves Soil857

Science Society of America Journal. 1966;30:680–685.858

87. Black Patrick B, Tice Allen R. Comparison of soil freezing curve and soil water curve data for Windsor859

sandy loam Water Resources Research. 1989;25:2205–2210.860

88. Ma Tiantian, Wei Changfu, Xia Xiaolong, Zhou Jiazuo, Chen Pan. Soil freezing and soil water re-861

tention characteristics: Connection and solute effects Journal of performance of constructed facilities.862

2017;31:D4015001.863

89. Bai Ruiqiang, Lai Yuanming, Zhang Mingyi, Yu Fan. Theory and application of a novel soil freezing864

characteristic curve Applied Thermal Engineering. 2018;129:1106–1114.865

90. Luckner L, Van Genuchten M Th, Nielsen DR. A consistent set of parametric models for the two-phase866

ﬂow of immiscible ﬂuids in the subsurface Water Resources Research. 1989;25:2187–2193.867

91. Seyfried MS, Murdock MD. Use of air permeability to estimate inﬁltrability of frozen soil Journal of868

Hydrology. 1997;202:95–107.869

92. Demand Dominic, Selker John S, Weiler Markus. Inﬂuences of macropores on inﬁltration into season-870

ally frozen soil Vadose Zone Journal. 2019;18:1–14.871

93. Eigenbrod KD. Self-healing in fractured ﬁne-grained soils Canadian Geotechnical Journal. 2003;40:435–872

449.873

94. Foriero A, Ladanyi B. FEM assessment of large-strain thaw consolidation Journal of Geotechnical Engi-874

neering. 1995;121:126–138.875

95. Zhang Yao, Michalowski Radoslaw L. Thermal-hydro-mechanical analysis of frost heave and thaw876

settlement Journal of geotechnical and geoenvironmental engineering. 2015;141:04015027.877

96. Dittmann M, Kr ¨

uger M, Schmidt F, Schuß S, Hesch C. Variational modeling of thermomechanical878

fracture and anisotropic frictional mortar contact problems with adhesion Computational Mechanics.879

2019;63:571–591.880

97. Dittmann M, Aldakheel F, Schulte J, et al. Phase-ﬁeld modeling of porous-ductile fracture in881

non-linear thermo-elasto-plastic solids Computer Methods in Applied Mechanics and Engineering.882

2020;361:112730.883

98. Miehe Christian, Mauthe Steffen, Teichtmeister Stephan. Minimization principles for the coupled884

problem of Darcy–Biot-type ﬂuid transport in porous media linked to phase ﬁeld modeling of fracture885

Journal of the Mechanics and Physics of Solids. 2015;82:186–217.886

99. Amor Hanen, Marigo Jean-Jacques, Maurini Corrado. Regularized formulation of the variational brit-887

tle fracture with unilateral contact: Numerical experiments Journal of the Mechanics and Physics of Solids.888

2009;57:1209–1229.889

100. Miehe Christian, Schaenzel Lisa-Marie, Ulmer Heike. Phase ﬁeld modeling of fracture in multi-890

physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation891

in thermo-elastic solids Computer Methods in Applied Mechanics and Engineering. 2015;294:449–485.892

101. Suh Hyoung Suk, Sun WaiChing. An open-source FEniCS implementation of a phase ﬁeld fracture893

model for micropolar continua International Journal for Multiscale Computational Engineering. 2019;17.894

102. Bryant Eric C, Sun WaiChing. Phase ﬁeld modeling of frictional slip with slip weakening/strength-895

ening under non-isothermal conditions Computer Methods in Applied Mechanics and Engineering.896

32 Hyoung Suk Suh, WaiChing Sun

2021;375:113557.897

103. Viklander Peter. Permeability and volume changes in till due to cyclic freeze/thaw Canadian Geotech-898

nical Journal. 1998;35:471–477.899

104. Rayhani MHT, Yanful EK, Fakher A. Physical modeling of desiccation cracking in plastic soils Engi-900

neering Geology. 2008;97:25–31.901

105. Ma Ran, Sun WaiChing. Computational thermomechanics for crystalline rock. Part II: Chemo-902

damage-plasticity and healing in strongly anisotropic polycrystals Computer Methods in Applied Me-903

chanics and Engineering. 2020;369:113184.904

106. Miehe Christian, Mauthe Steffen. Phase ﬁeld modeling of fracture in multi-physics problems. Part905

III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of ﬂuid-saturated porous906

media Computer Methods in Applied Mechanics and Engineering. 2016;304:619–655.907

107. Mauthe Steffen, Miehe Christian. Hydraulic fracture in poro-hydro-elastic media Mechanics Research908

Communications. 2017;80:69–83.909

108. Wang Kun, Sun WaiChing. A uniﬁed variational eigen-erosion framework for interacting brittle frac-910

tures and compaction bands in ﬂuid-inﬁltrating porous media Computer Methods in Applied Mechanics911

and Engineering. 2017;318:1–32.912

109. Suh Hyoung Suk, Sun WaiChing. An immersed phase ﬁeld fracture model for microporomechanics913

with Darcy–Stokes ﬂow Physics of Fluids. 2021;33:016603.914

110. Wilson Zachary A, Landis Chad M. Phase-ﬁeld modeling of hydraulic fracture Journal of the Mechanics915

and Physics of Solids. 2016;96:264–290.916

111. Heider Yousef, Sun WaiChing. A phase ﬁeld framework for capillary-induced fracture in unsaturated917

porous media: Drying-induced vs. hydraulic cracking Computer Methods in Applied Mechanics and En-918

gineering. 2020;359:112647.919

112. Bouddour A, Auriault JL, Mhamdi-Alaoui M. Erosion and deposition of solid particles in porous920

media: Homogenization analysis of a formation damage Transport in porous media. 1996;25:121–146.921

113. Bonelli St´

ephane, Brivois Olivier. The scaling law in the hole erosion test with a constant pressure922

drop International Journal for numerical and analytical methods in geomechanics. 2008;32:1573–1595.923

114. Sterpi Donatella. Effects of the erosion and transport of ﬁne particles due to seepage ﬂow International924

journal of Geomechanics. 2003;3:111–122.925

115. Pope SBj. A more general effective-viscosity hypothesis Journal of Fluid Mechanics. 1975;72:331–340.926

116. Simo JC, Miehe Ch. Associative coupled thermoplasticity at ﬁnite strains: Formulation, numerical927

analysis and implementation Computer Methods in Applied Mechanics and Engineering. 1992;98:41–104.928

117. Nguyen TS, Selvadurai APS. Coupled thermal-mechanical-hydrological behaviour of sparsely frac-929

tured rock: implications for nuclear fuel waste disposal in International journal of rock mechanics and930

mining sciences & geomechanics abstracts;32:465–479Elsevier 1995.931

118. Logg Anders, Wells Garth N. DOLFIN: Automated ﬁnite element computing ACM Transactions on932

Mathematical Software (TOMS). 2010;37:1–28.933

119. Logg Anders, Mardal Kent-Andre, Wells Garth. Automated solution of differential equations by the ﬁnite934

element method: The FEniCS book;84. Springer Science & Business Media 2012.935

120. Alnæs Martin, Blechta Jan, Hake Johan, et al. The FEniCS project version 1.5 Archive of Numerical936

Software. 2015;3.937

121. Abhyankar Shrirang, Brown Jed, Constantinescu Emil M, Ghosh Debojyoti, Smith Barry F, Zhang938

Hong. PETSc/TS: A modern scalable ODE/DAE solver library arXiv preprint arXiv:1806.01437. 2018.939

122. Sweidan AH, Niggemann K, Heider Y, Ziegler M, Markert B. Experimental study and numerical940

modeling of the thermo-hydro-mechanical processes in soil freezing with different frost penetration941

directions Acta Geotechnica. 2021:1–25.942

123. Feng Shangsheng, Zhang Ye, Shi Meng, Wen Ting, Lu Tian Jian. Unidirectional freezing of phase943

change materials saturated in open-cell metal foams Applied Thermal Engineering. 2015;88:315–321.944

124. Lackner Roman, Amon Andreas, Lagger Hannes. Artiﬁcial ground freezing of fully saturated soil:945

thermal problem Journal of Engineering Mechanics. 2005;131:211–220.946

125. Terzaghi Karl, Peck RALPH B, Mesri G. Soil mechanics New York: John Wiley & Sons. 1996.947

126. Amato Giorgia, And`

o Edward, Lyu Chuangxin, Viggiani Gioacchino, Eiksund Gudmund Reinar. A948

glimpse into rapid freezing processes in clay with x-ray tomography Acta Geotechnica. 2021:1–12.949

Multi-phase-ﬁeld model for ice lens growth 33

127. Uyanık Osman. Estimation of the porosity of clay soils using seismic P-and S-wave velocities Journal950

of Applied Geophysics. 2019;170:103832.951

128. Dunn David E, LaFountain Lester J, Jackson Robert E. Porosity dependence and mechanism of brittle952

fracture in sandstones Journal of Geophysical Research. 1973;78:2403–2417.953

129. Karma Alain, Rappel Wouter-Jan. Phase-ﬁeld method for computationally efﬁcient modeling of so-954

lidiﬁcation with arbitrary interface kinetics Physical review E. 1996;53:R3017.955

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Preprint

Full-text available

Dec 2024
INT J HEAT MASS TRAN

A coupled one-dimensional (1D) thermo-mechanical model is developed to explore the ice lens formation in porous building materials, specifically sandstone, under changing climatic conditions. Heat conduction is coupled to an ice lens model to predict the formation and growth of ice lenses with the change of the temperature profile. Simultaneously, changes in the effective properties of the system due to evolving ice lenses are considered for calculating the temperature profile. Monotonic cooling and cyclic climatic conditions are applied to a sandstone wall to investigate the effects of different material parameters and boundary conditions on ice lensing. Triggering conditions for periodic ice lensing are identified. Finally, the periodic evolution of ice lenses (i.e., growing, melting, and re-freezing) is shown in the simulations under cyclic climatic conditions.

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Article

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Mar 2025

Frost heave occurs when the ground swells during freezing conditions due to the growth of ice lenses in the subsurface. The mechanics of ice-infiltrated sediment, or frozen fringe, influences the formation and evolution of ice lenses. As the frozen fringe thickens during freezing, progressive unloading can result in dilation of the pore space and the formation of new ice lenses. Compaction can also occur as water is expelled from the pore space and freezes onto the ice lenses. We introduce a mathematical model for compaction within frozen fringe to explore how internal variability influences the fundamental characteristics of frost heave cycles. At faster freezing rates, compaction below ice lenses can generate complex dynamics and chaos when the frozen fringe follows a consolidation law based solely on the sediment yield stress. The complex oscillations arise because a downward water flux below the compacting zone generates a distributed zone of weakening. We introduce viscous effects into the compaction law through a bulk viscosity to determine how the cycles could be influenced by the creep of ice through the pore space. Localized zones of decompaction in the viscoplastic model can prevent the feedback mechanisms that cause complex oscillations in the perfectly plastic model.

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Conference Paper

Full-text available

Jan 2025

The freezing of water in porous media depends on various characteristics such as pore size, distribution of grains or boundary conditions. In this contribution we describe a numerical model of freeze/thaw process of a soil sample on the centimeter scale (laboratory sample of repacked sand with relatively low porosity and water) using the finite element method. The model is based on the Štefan problem with a modified latent heat release depending on the water content and is treated in axial symmetry the sample domain. We investigate the sensitivity of this model on initial conditions, material properties and boundary conditions. This contributes to a more profound understanding of freeze/thaw processes observed under laboratory conditions.

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Article

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Oct 2024
COMPUT MECH

We propose a novel continuum damage model for describing the propagation of compaction bands formed by grain crushing in porous sedimentary rock using the kernel-based approximation and discretization of the meshfree Lagrangian smoothed particle hydrodynamics (SPH) method. In the model, damage is assumed to be caused by grain crushing, and the effects of plasticity are incorporated through a critical state type model which depends on a degradation function designed to capture the abrupt onset of pore collapse following grain crushing. In doing so, we account for the two main forces understood to drive compaction band formation, brittle failure at the grain scale, and plastic dissipation. Through the smoothing length parameter, the SPH method possesses nonlocal properties intrinsic to the method, allowing for the development of a nonlocal integral form of the equivalent strain variable used to compute the damage in addition to a distinct gradient enhanced approach that when discretized using SPH is endowed with two characteristic length scales. We compare and contrast both formulations, as well as the roles of the different length scales. We evaluate our model by performing numerical experiments on Bentheim and Berea sandstone as well as on Tuffeau de Maastricht, in both notched and unnotched specimens, matching experimentally observed results such as a transition from high-angled bands to horizontal bands with increasing confining pressure, and similar compaction band styles to those visible in the field. We lastly consider the effects of material heterogeneity on samples of Tuffeau de Maastricht noting that our model produces precursory low-magnitude localization events which may develop into persistent compaction bands involving localized grain crushing, which is consistent with our understanding of compaction band formation.

Experimental and numerical investigation of water freezing and thawing in fully saturated sand

Article

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Aug 2024

This paper presents an experimental and numerical study of the freezing-thawing behavior of water in fully saturated sand. A relatively inexpensive and easily replicable experimental procedure was developed to simulate the freezing-thawing cycles in a medium-sized sand sample placed in a modified commercial freezer. By insulating the sides and bottom of the sample well, while allowing good thermal conductivity at the top of the sample, a nearly vertical advance of the freezing and thawing front was achieved. A series of freeze-thaw cycles were performed with higher and lower temperature gradients. A numerical multiphysics model, assuming an axially symmetric geometry based on the transient heat transfer during the phase transition, used a parametric approach to estimate the effective thermal properties of the sand-water-ice system. A good agreement between experimental and modelling results was shown, but slightly different parameter sets were obtained for each temperature gradient. The presented method could be a simple way to characterize the freeze-thaw process in natural and artificial porous materials.

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Full-text available

Oct 2024

Permanent deformation and uplift caused by fault rupture is one of the most significant hazards posed by earthquakes on the built environment. In this paper, we use smoothed particle hydrodynamics (SPH) to explore the effects of soil layering or stratification on the trajectories and deformation patterns caused by rupturing reverse faults in bedrock, as well as in the foundations of engineered earth structures. SPH is a continuum meshfree numerical method highly adept at modeling large deformation problems in geotechnics. Through the use of constitutive models involving softening behavior as well as critical state type models, we isolate the effects of rigid body rotation from critical state behavior of soil in helping explain the frequently observed rotation of shear bands emanating from the bedrock fault. This analysis is facilitated by the fact that the SPH method allows us to track the propagation of shear bands over substantial amounts of vertical uplift (more than 50% of the total height of the soil deposit), far beyond many previous computational studies employing the finite element method (FEM). We observe and characterize various emergent features including fault bifurcations, stunted faults, and tension cracking, while providing insights into practical guidelines regarding the potential surface distortion width, and the critical amount of fault displacement required for surface rupture depending on the multilayered constitution of the soil deposit. Finally, we predict the expected amount of surface distortion and internal damage to earthen embankments depending on varying fault location and soil makeup.

Diffuse interface modeling of non-isothermal Stokes-Darcy flow with immersed transmissibility conditions

Article

Full-text available

Aug 2024
INT J NUMER METH ENG

Hyoung Suk Suh

The coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid‐soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non‐isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non‐isothermal Stokes‐Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen‐Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity.

Pore‐Morphology‐Based Estimation of the Freezing Characteristic Curve of Water‐Saturated Porous Media

Article

Full-text available

Aug 2024
WATER RESOUR RES

Assessment of freezing effects on soil requires estimating the soil freezing characteristic curve (SFCC)—the variation of unfrozen water content with temperature. The existing methods for obtaining SFCCs often involve either costly experiments or heuristic inference from water retention data. Here, we propose a pore‐morphology‐based method for simple and efficient estimation of the freezing characteristic curve of water‐saturated porous media, whereby the pore‐scale configurations of water and ice phases are simulated in a digital image of porous microstructure. Idealizing the pore space as a system of overlapping spherical pores, the method simulates the freezing process with the Gibbs‐Thomson equation that can consider freezing‐point depression—a decrease in the freezing point due to spatial confinement—based on thermodynamics. For validation, we apply the proposed method to estimate the SFCC of a field soil for which the experimental freezing characteristics data are available. Results show that even with a digital pore image extracted from a surrogate discrete‐element packing of the soil, the proposed method provides an SFCC very close to the experimental data.

State of the Art of Coupled Thermo–Hydro-Mechanical–Chemical Modelling for Frozen Soils

Article

Full-text available

Jul 2024

Numerous studies have investigated the coupled multi-field processes in frozen soils, focusing on the variation in frozen soils and addressing the influences of climate change, hydrological processes, and ecosystems in cold regions. The investigation of coupled multi-physics field processes in frozen soils has emerged as a prominent research area, leading to significant advancements in coupling models and simulation solvers. However, substantial differences remain among various coupled models due to the insufficient observations and in-depth understanding of multi-field coupling processes. Therefore, this study comprehensively reviews the latest research process on multi-field models and numerical simulation methods, including thermo-hydraulic (TH) coupling, thermo-mechanical (TM) coupling, hydro-mechanical (HM) coupling, thermo–hydro-mechanical (THM) coupling, thermo–hydro-chemical (THC) coupling and thermo–hydro-mechanical–chemical (THMC) coupling. Furthermore, the primary simulation methods are summarised, including the continuum mechanics method, discrete or discontinuous mechanics method, and simulators specifically designed for heat and mass transfer modelling. Finally, this study outlines critical findings and proposes future research directions on multi-physical field modelling of frozen soils. This study provides the theoretical basis for in-depth mechanism analyses and practical engineering applications, contributing to the advancement of understanding and management of frozen soils.

Asynchronous phase field fracture model for porous media with thermally non-equilibrated constituents

Article

Full-text available

Dec 2021
COMPUT METHOD APPL M

This paper presents the mathematical framework and the asynchronous finite element solver that captures the brittle fractures in multi-phase fluid-infiltrating porous media at the mesoscale where the constituents are not necessarily in a thermal equilibrium state. To achieve this goal, we introduce a dual-temperature effective medium theory in which the distinct constituent temperatures are homogenized independently whereas the heat exchange among the constituents is captured via phenomenological heat exchange laws in analog to the dual-permeability theory. To handle the different growth rates of the boundary layers in a stable and computationally efficient manner, an asynchronous time integrator is proposed and implemented in an operator-split algorithm that updates the displacement, pore pressure, phase field, and temperature of each constituent in an asynchronous manner. Numerical examples are introduced to verify the implementation and compare the path-dependent behaviors predicted by the two-temperature and one-temperature models.

A review on phase-field modeling of hydraulic fracturing

Article

Full-text available

Aug 2021
ENG FRACT MECH

Yousef Heider

Motivated by the successful implementation of the phase-field method (PFM) to simulate complicated fracture patterns at moderate computational costs in solid materials, many research groups have started since 2012 applying the PFM to model hydraulic fracturing, especially that occurs in porous geomaterials. These research works have contributed to the development of the PFM from different perspectives, especially in connection with the mathematical formulations of the hydro-mechanical processes and the numerical algorithms to solve the emerging coupled problems. In this regard, the underlying paper aims to review the significant scientific works that utilized the PFM to model fracturing caused mainly by fluid injection in a certain porous domain and, less common, by fluid extraction (e.g., drying) from a certain porous domain. This includes reviewing different approaches for deriving the phase-field evolution formulation (e.g. Ginzburg–Landau approach, thermodynamically consistent approaches, and microforce-based approach) and reviewing several formulations for the stiffness degradation function and that of the crack driving force. Besides, the paper will go through several methods to estimate the crack aperture width, in addition to reviewing different numerical approaches and implementations. The paper will be concluded by presenting a number of open topics and challenges to be addressed in future works.

Topographic and Ground‐Ice Controls on Shallow Landsliding in Thawing Arctic Permafrost

Article

Full-text available

Jul 2021
GEOPHYS RES LETT

Plain Language Summary With a rapidly warming climate, landslide activity is increasing in the Arctic. Warmer Arctic temperatures thaw the permanently frozen soil (permafrost) that underlies hillslopes. Thawing creates meltwater, decreasing the strength of the soil, and in some cases initiating landsliding. We mapped the location of shallow landslides in Alaska. These landslides demonstrate a clear spatial pattern, forming above the channel network in depressions called zero‐order drainage basins. The similarity between the spatial patterns of permafrost landslides and rainfall‐triggered landslides suggests that the topography, hydrology, and spatial distribution of ground ice act to enhance landslide potential in zero order drainage basins. Modeling of hydrology and ground ice effects on slope stability, suggests a weak hydrologic control and a strong ground ice control on landslide triggering. Here we argue that topography affects the distribution of ground ice, in particular its concentration at potential landslide failure planes. We conclude that the strong relationship between topography and landsliding in the Arctic could improve estimates of shallow landslide susceptibility.

Experimental study and numerical modeling of the thermo-hydro-mechanical processes in soil freezing with different frost penetration directions

Article

Full-text available

Apr 2021

This research work presents an experimental and numerical study of the coupled thermo-hydro-mechanical (THM) processes that occur during soil freezing. With focusing on the artificial ground freezing (AGF) technology, a new testing device is built, which considers a variety of AGF-related boundary conditions and different freezing directions. In the conducted experiments, a distinction is made between two thermal states: (1) The thermal transient state, which is associated with ice penetration, small deformations, and insignificant water suction. (2) The thermal (quasi-) steady state, which has a much longer duration and is associated with significant ice lens formation due to water suction. In the numerical modeling, a special focus is laid on the processes that occur during the thermal transient state. Besides, a demonstration of the micro-cryo-suction mechanism and its realization in the continuum model through a phenomenological retention-curve-like formulation is presented. This allows modeling the ice lens formation and the stiffness degradation observed in the experiments. Assuming a fully saturated soil as a biphasic porous material, a phase-change THM approach is applied in the numerical modeling. The governing equations are based on the continuum mechanical theory of porous media (TPM) extended by the phase-field modeling (PFM) approach. The model proceeds from a small-strain assumption, whereas the pore fluid can be found in liquid water or solid ice state with a unified kinematics treatment of both states. Comparisons with the experimental data demonstrate the ability and usefulness of the considered model in describing the freezing of saturated soils.

An immersed phase field fracture model for microporomechanics with Darcy-Stokes flow

Article

Full-text available

Dec 2020
PHYS FLUIDS

This paper presents an immersed phase field model designed to predict the fracture-induced flow due to brittle fracture in vuggy porous media. Due to the multiscale nature of pores in vuggy porous material, crack growth may connect previously isolated pores which lead to flow conduits. This mechanism has important implications for many applications such as disposal of carbon dioxide and radioactive materials, hydraulic fracture and mining. To understand the detailed microporomechanics that causes the fracture-induced flow, we introduce a new phase field fracture framework where the phase field is not only used as an indicator function for damage of the solid skeleton, but also as an indicator of the pore space. By coupling the Stokes equation that governs the fluid transport in the voids, cavities and cracks, and the Darcy's flow in the deformable porous media, our proposed model enables us to capture the fluid-solid interaction of the pore fluid and solid constituents during the crack growth. Numerical experiments are conducted to analyze how presence of cavities affects the accuracy of the predictions based on homogenized effective medium during crack growth.

Phase field modeling of frictional slip with slip weakening/strengthening under non-isothermal conditions

Article

Full-text available

Dec 2020

Cracks, veins, joints, faults, and ocean crusts are strong discontinuities of different length scales that can be found in many geological formations. While the constitutive laws for the frictional slip of these interfaces have been the focus of decades-long geophysical research, capturing the evolving geometry such as branching, coalescence, and the corresponding interplay between frictional slip and the mode II crack growth in compression remains a challenging task. This work employs a phase field framework for frictional contact originated in Fei and Choo [2020] , s.t. these strong discontinuities are represented by implicit functions and the frictional responses of the transitional damage zone is approximated by a diffusive constitutive law that captures the coupling between the bulk and interfacial plasticity. To replicate the rate dependence and size effects commonly exhibited in frictional interfaces, we propose a regularized constitutive law for the slip weakening/strengthening at different loading rates and temperature regimes. Numerical examples are provided to show that the regularized model may converge into the strong-discontinuity counterpart and capture the frictional response along geometrically complex interfaces.

A unified water/ice kinematics approach for phase-field thermo-hydro-mechanical modeling of frost action in porous media

Article

Full-text available

Dec 2020
COMPUT METHOD APPL M

This research work introduces a novel phase-field thermo-hydro-mechanical (P-THM) modeling approach that allows to deeply understand and model the freezing–thawing cyclic process in a fluid-saturated porous medium. In this, a biphasic macroscopic, non-isothermal porous media model, augmented by the phase-field method (PFM), is applied to account for the temperature development, the interstitial pore-fluid flow, and the volumetric deformations due to ice formation (phase change). Utilizing the theory of porous media (TPM) in the continuum mechanical formulation provides a well-founded basis for the description of deformable, fluid-saturated, non-isothermal porous solid materials. Of particular importance in the underlying work is the unified kinematics treatment of the ice and water constituents as a single pore-fluid, where the PFM is employed for the description of the phase transition between both constituents. The PFM is a diffuse-interface approach that relies on the specification of the free energy density function as the main driving force in the phase transition. It employs a scalar-valued, phase-field variable to indicate the state of the pore-fluid, i.e., a solid (ice) or a liquid (water). A significant virtue of using the PFM approach is its viability in the implementation within standard finite element frameworks, as no need to explicitly track the moving boundaries (interfaces) of the phase-change constituent. The numerical examples and comparisons presented at the end of the manuscript demonstrate the ability, reliability and usefulness of the proposed modeling framework in describing the freezing–thawing process in a saturated porous solid under thermal loading within an elastic deformation limit.

Computational thermomechanics for crystalline rock. Part II: Chemo-damage-plasticity and healing in strongly anisotropic polycrystals

Article

Full-text available

May 2020
COMPUT METHOD APPL M

We present a new thermal-mechanical-chemical-phase field model that captures the multi-physical coupling effects of precipitation creeping, crystal plasticity, anisotropic fracture, and crack healing in polycrystalline rock at various temperature and strain-rate regimes. This model is solved via a fast Fourier transfer solver with an operator-split algorithm to update displacement, temperature and phase field, and chemical concentration incrementally. In nuclear waste disposal in salt formation, brine inside the crystal salt may migrate along the grain boundary and cracks due to the gradient of interfacial energy and pressure. This migration has a significant implication on the permeability evolution, creep deformation, and crack healing within rock salt but is difficult to incorporate implicitly via effective medium theories compared with computational homogenization. As such, we introduce a thermodynamic framework and a corresponding computational implementation that explicitly captures the brine diffusion along the grain boundary and crack at the grain scale. Meanwhile, the anisotropic fracture and healing are captured via a high-order phase field that represents the regularized crack region in which a newly derived non-monotonic driving force is used to capture the fracture and healing due to the solution precipitation. Numerical examples are presented to demonstrate the capacity of the thermodynamic framework to capture the multiphysics material behaviors of rock salt.

Computational Geomechanics: Theory and Applications

Book

Apr 2022

A glimpse into rapid freezing processes in clay with x-ray tomography

Article

May 2021

This short paper proposes time-series x-ray tomography as a valuable tool for quantifying freezing processes in a remoulded clay. As an example, three simple closed system (i.e., no water entry) experiments are performed and analysed, all of which were frozen under relatively high but different thermal gradients. The progressive appearance of a zone of icelenses, and the deformation of the sample due to freezing and cryogenic consolidation is quantified with time—with the hope of providing data to THM models, and to this end the data presented are made available online. The rich patterns in the frozen zone are also illustrated and discussed. The measurement technique used has satisfactory performance and further work on open systems, different materials and different thermal gradients is expected to contribute significantly to understanding of the complex phenomena surrounding freezing in soil.

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

Abstract

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