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Computer Methods in Applied Mechanics and Engineering manuscript No.

(will be inserted by the editor)

Asynchronous phase ﬁeld fracture model for porous media with thermally1

non-equilibrated constituents2

Hyoung Suk Suh ·WaiChing Sun3

4

Received: September 2, 2021/ Accepted: date5

Abstract This paper presents the mathematical framework and the asynchronous ﬁnite element solver6

that captures the brittle fractures in multi-phase ﬂuid-inﬁltrating porous media at the mesoscale where the7

constituents are not necessarily in a thermal equilibrium state. To achieve this goal, we introduce a dual-8

temperature effective medium theory in which the distinct constituent temperatures are homogenized in-9

dependently whereas the heat exchange among the constituents is captured via phenomenological heat10

exchange laws in analog to the dual-permeability theory. To handle the different growth rates of the bound-11

ary layers in a stable and computationally efﬁcient manner, an asynchronous time integrator is proposed12

and implemented in an operator-split algorithm that updates the displacement, pore pressure, phase ﬁeld,13

and temperature of each constituent in an asynchronous manner. Numerical examples are introduced to14

verify the implementation and compare the path-dependent behaviors predicted by the two-temperature15

and one-temperature models.16

Keywords local thermal non-equilibrium, brittle fracture, porous media, hydraulic fracture17

1 Introduction18

The thermo-hydro-mechanical responses of porous media are critical for many geothermal and geome-19

chanics applications such as underground radioactive waste disposal, geothermal energy recovery, oil pro-20

duction, and CO2geological storage [Al-Hadhrami and Blunt,2001,Rybach,2003,Pusch,2009,Shukla21

et al.,2010,Shaik et al.,2011,Dai et al.,2016,Sun et al.,2017,Salimzadeh et al.,2018,Bahmani and Sun,22

2021]. For instance, frictional heating may lead to the temperature increase of both the solid skeleton and23

the pore ﬂuid through heat exchanges [Bryant and Sun,2021]. Geological storage of CO2and oil recovery24

often require the injection of the pore ﬂuid in a supercritical state such that the thermal convection may25

play an important role both for the ﬂuid transport and the ﬂuid-driven fracture. The combination of tem-26

perature, pressure, and loading rate are also critical for the brittle-ductile transition of geological materials27

[Byerlee,1968,Paterson and Wong,2005,Choo and Sun,2018a]. Heat exchange is an important mechanism28

for selecting the candidate materials for the nuclear waste geological disposal such as clay and salt. Long-29

term disposal such as the Yucca Mountain Project in New Mexico, for instance, relies on the combination30

of low permeability, high thermal conductivity, and self-healing mechanisms to ensure the isolation of the31

radioactive wastes [Hansen and Leigh,2011,Na and Sun,2018,Ma and Sun,2020].32

Traditionally, large-scale reservoir simulators that simulate thermo-hydro-mechanical responses of porous33

media, such as TOUGH-FLAC [Rutqvist,2011] and OpenGeoSys [Kolditz et al.,2012], often assume that34

different constituents of the porous media share the same temperature. This assumption could be valid35

when (1) the representative elementary volume is sufﬁciently large for the macroscopic porous continua36

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

to function as an effective medium for the multiphase materials, and (2) when the time scale considered in37

the simulations is much larger than the time it takes for the constituents to reach equilibrium locally. This38

assumption may lead to discrepancy to experimental observation when the constituents have signiﬁcant39

difference in speciﬁc heat capacity and thermal conductivity, and when the temporal and spatial scales of40

interest are sufﬁciently small such that the ”homogenized” temperature may yield erroneous results that41

violates the thermodynamic principle [Nozad et al.,1985a,b,McTigue,1986,Kurashige,1989,Zimmerman,42

2000,Belotserkovets and Prevost,2011,Sun,2015,Na and Sun,2017,Kim,2018,Noii and Wick,2019].43

Exemplary unit cell

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Low conductivity

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High conductivity

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Low

High

Thermal

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equilibrium

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Thermal non-equilibrium

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t= 50.0

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t= 100.0

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t= 2500.0

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t= 1000.0

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t=1

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Fig. 1: Longitudinal heat transfer on an exemplary unit cell that consists of two different materials with

different thermal conductivities.

As a thought experiment, we construct a heat conduction problem of which the domain is occupied by a44

two-phase material where the host matrix exhibits four orders higher thermal conductivity than that of the45

inclusion. As demonstrated in Fig. 1, this leads to a period of transition in which the heat transfer among the46

constituents dominates the overall thermal responses. While it is possible to obtain effective thermal con-47

ductivity through homogenization, doing so may not be suitable to capture the thermo-hydro-mechanical48

responses where the difference in the thermal expansion among the constituents and the thermal-softening49

of the solid skeleton may both affect the residual stress of the constituents. This issue has also been captured50

in experiments. For example, Truong and Zinsmeister [1978] showed that the one-temperature approach51

may not yield physically consistent results if thermal conductivities of the constituent differ signiﬁcantly,52

which corroborates our simulation in Fig. 1, while He et al. [2012] pointed out that the local thermal equi-53

librium assumption is only valid when the interstitial heat transfer coefﬁcient between the constituents is54

sufﬁciently large or the speciﬁc surface area of the porous medium is high enough. Furthermore, Jiang et al.55

[2014] showed that the effect of local thermal non-equilibrium becomes more signiﬁcant in fractured media56

where the convective heat ﬂow inside the fracture plays a crucial role in its coupled behavior. Accordingly,57

it has been widely recognized that the thermo-hydro-mechanical responses of the porous media can be58

captured more precisely for the problems at the temporal and spatial scales if the heat transfer among the59

constituents can be explicitly captured [Quintard et al.,1997,Hsu,1999,Minkowycz et al.,1999,Nakayama60

et al.,2001,Fourie and Du Plessis,2003a,b,Virto et al.,2009]. This is particularly important for strain lo-61

calization and fracture simulations where small perturbations in temperature may lead to signiﬁcantly62

different path-dependent behaviors.63

Solving the thermo-hydro-mechanical problems with non-equilibrated constituents is nevertheless not64

trivial. Thermal convection, Soret diffusion, and the orders of difference in thermal diffusivities all require65

complicated and sophisticated treatments in designing the algorithm. While stabilization procedures such66

as streamline upwind Petrov-Galerkin (SUPG) scheme may help resolving the numerical issues related67

to the sharp gradient of temperature and/or pore pressure, capturing the boundary layers of multiple68

constituent temperature ﬁelds remain a great numerical challenge [Hughes et al.,1986,1989,Choo and69

Sun,2018b,Na and Sun,2017] and yet may have a profound impact on the brittle or quasi-brittle fracture70

of porous media [Miehe et al.,2010a,Borden et al.,2012,Suh et al.,2020].71

The goal of this study is to ﬁll this knowledge gap by (1) proposing a thermo-hydro-mechanical theory72

for deformable porous media with non-equilibrated constituents and (2) introducing an asynchronous73

operator-split framework that enables us to capture the coupling mechanisms among the constituents74

without spurious numerical oscillations or over-diffusion. To achieve the ﬁrst goal, we hypothesize that75

the existence of an effective medium where constituents may exhibit different temperature while the heat76

Asynchronous phase ﬁeld fracture 3

transfer among different constituents are captured by an interface constitutive law in analog with the dual-77

permeability theory. Meanwhile, we assume that the material may exhibit fracture and this fracture is cap-78

tured by a phase ﬁeld that provides a diffusive representation of the crack location and therefore does79

not require embedded discontinuities. The governing equations derived from the balance principles of the80

solid and ﬂuid constituents are then discretized in both spatial and time domains to establish numerical al-81

gorithm for computer simulations. In particular, we adopt the staggered scheme for the phase ﬁeld fracture82

while enabling an asynchronous dual-temperature isothermal splitting scheme that updates the displace-83

ment, the pore pressure, the constituent temperature, and the phase ﬁeld sequentially and asynchronously.84

The rest of the paper is organized as follows. In Section 2, we introduce the theoretical framework that85

enables us to consider the heat transfer among constituents in a two-phase effective medium or mixture.86

We present the balance principles (Section 3) and constitutive relations (Section 4) that describe the thermo-87

hydro-mechanically coupled behavior of ﬂuid-saturated porous media undergoing brittle fracture. We then88

propose a special time integration scheme that updates the ﬁeld variables in an asynchronous manner in89

Section 5. Finally, numerical examples are given in Section 6to highlight the computational efﬁciency of90

the proposed scheme, and to showcase the model capacity by simulating the mechanically driven and hy-91

draulically induced fracture propagation during the transient period where the solid and ﬂuid constituents92

are thermally non-equilibrated.93

As for notations and symbols, bold-faced and blackboard bold-faced letters denote tensors (including94

vectors which are rank-one tensors); the symbol ’·’ denotes a single contraction of adjacent indices of two95

tensors (e.g., a·b=aibior c·d=cij djk); the symbol ‘:’ denotes a double contraction of adjacent indices of96

tensor of rank two or higher (e.g., C:ε=Cijk l εkl ); the symbol ‘⊗’ denotes a juxtaposition of two vectors97

(e.g., a⊗b=aibj) or two symmetric second-order tensors [e.g., (α⊗β)ijkl =αij βkl ]. We also deﬁne identity98

tensors: I=δij,I=δikδjl , and ¯

I=δil δjk, where δij is the Kronecker delta. As for sign conventions, unless99

speciﬁed, the directions of the tensile stress and dilative pressure are considered as positive.100

2 Modeling approaches101

In this section, we introduce the necessary ingredients for the conservation laws and the constitutive re-102

lations that will be presented later in Sections 3and 4. We ﬁrst present the homogenization strategy for103

the solid and ﬂuid temperatures that allows us to consider non-isothermal effects in a two-phase porous104

medium with thermally non-equilibrated constituents. Kinematic assumptions based on the mixture the-105

ory are also stated, where thermal expansion of the solid skeleton is considered to be isotropic and solely106

depends on the solid temperature. We then summarize the smooth approximation of crack topology that107

adopts an implicit function, in which the phase ﬁeld variable serves as a damage parameter while the reg-108

ularization length scale parameter controls the size of the diffusive crack zone. Based on this setting, we109

deﬁne the effective stress by following the scenario from Miehe and Mauthe [2016], which decomposes the110

free energy functional into multiple parts including the effective strain energy stored in the solid skeleton111

and the energy stored in the pore ﬂuid.112

2.1 Kinematics and homogenization strategy113

Consider a fully saturated porous element Ωcomposed of solid (s) and ﬂuid ( f) constituents, i.e., Ω=114

Ωs∪Ωf. In small scale, the spatial distribution of each constituent in Ωcan be represented by indicator115

functions rs(y)and rf(y):116

rs(y) = (1 if y∈Ωs,

0 otherwise, ;rf(y) = (1 if y∈Ωf,

0 otherwise, (1)

where ydenotes the position vector associated with small-scale conﬁguration. By assuming that Ωcan be117

regarded as a representative volume element (RVE), the mixture theory states that the material of interest118

can be idealized as a homogenized continuum mixture Bin which the solid and ﬂuid constituents occupies119

4 Hyoung Suk Suh, WaiChing Sun

a fraction of volume at the same material point P[Bachmat and Bear,1986,Coussy et al.,1998,Dormieux120

et al.,2006]. In this case, the volume fractions of each phase constituent are deﬁned as,121

φs=dVs

dV =1

dV ZΩrs(y)dΩ;φf=dVf

dV =1

dV ZΩrf(y)dΩ, (2)

where dV =RΩ[rs(y) + rf(y)] dΩindicates the total elementary volume of the mixture, such that φs+122

φf=1. Similarly, the total mass of the mixture at Pis deﬁned by the mass from each constituent, i.e.,123

dM =d Ms+dMf, and the intrinsic mass densities for the i-phase is given by ρi=dMi/dVi. Hence, the124

total mass density of the mixture reads,125

ρ=ρs+ρf=φsρs+φfρf, (3)

where ρi=dMi/dV is the partial mass density for the i-phase constituent.126

While previous studies on thermo-hydro-mechanics often employ a single temperature ﬁeld at meso- or127

macro-scales (θm) [Zimmerman,2000,Belotserkovets and Prevost,2011,Sun,2015,Na and Sun,2017,Kim,128

2018,Noii and Wick,2019], this study adopts a different homogenization strategy for each constituent. This129

approach not only allows us to model detailed non-isothermal processes in porous media but also to avoid130

the need to construct the mapping between small- and large-scale temperature ﬁelds. Having deﬁned the131

indicator functions in Eq. (1), we deﬁne the intrinsic large-scale solid and ﬂuid temperatures (θsand θf,132

respectively) as follows:133

θs=1

dVsZΩrs(y)θ(y)dΩ;θf=1

dVfZΩrf(y)θ(y)dΩ, (4)

where θ(y)is the small-scale temperature ﬁeld. Here, if the solid and ﬂuid temperatures at the same ma-134

terial point are different from each other, the constituents are said to be in local thermal non-equilibrium135

(LTNE), where the heat exchange between two phases should be taken into account [Gray,1975,Amiri136

et al.,1995,Alazmi and Vafai,2001,Fourie and Du Plessis,2003a,b]. On the other hand, for the case where137

two temperatures are identical to each other at the same material point, two constituents are said to be in138

local thermal equilibrium (LTE), implying a zero heat exchange between the phases. Note that the previous139

works that adopt a single temperature ﬁeld (i.e., one-temperature model) often rely on the assumption that140

the solid and ﬂuid temperatures reach a local equilibrium instantly (i.e., θs=θf=θm). In this case, the141

homogenized large-scale temperature θmmay no longer depend on the volume fraction of the constituents142

nor their microstructural attributes, i.e.,143

θm=1

dV ZΩθ(y)dΩ. (5)

For the kinematic assumptions, we follow the classical theory of porous media [Bowen,1980,Zienkiewicz144

et al.,1999,Ehlers,2002,Coussy,2004,De Boer,2012] and directly adopt the macroscopic descriptions.145

Speciﬁcally, we assume that the solid constituent forms a deformable skeletal structure (i.e., solid skeleton146

or solid matrix) at the RVE scale so that the evolution of our target porous material can be described in147

terms of the deformation of its solid skeleton. Since this study considers distinctive temperature ﬁelds for148

each phase constituents, the volume-averaged thermal expansion of the constituents is not used to com-149

pute thermal expansion of the porous medium Preisig and Pr´

evost [2011], Rutqvist et al. [2001], Na and150

Sun [2016]. Instead, we assume that the solid skeleton is linear thermoelastic, while the thermal expansion151

of solid skeleton solely depends on the solid temperature θs. Considering a body of two-phase continuum152

mixture Bwith material points identiﬁed by the large-scale position vectors x∈ B, we denote the dis-153

placement of the solid skeleton by u(x,t)at time t, so that the strain measure εcan be deﬁned as follows:154

ε=εe+εθs=1

2∇u+∇uT, (6)

where εeis the elastic component of the strain tensor and εθs=αs(θs−θs,ref)Iis its thermal component,155

where θs,ref is the reference temperature and αsis the linear thermal expansion coefﬁcient of the solid156

constituent. Notice that, as pointed out in Khalili et al. [2010], the linear thermal expansion coefﬁcient157

of the solid skeleton is solely controlled by and is equivalent to that of solid phase constituent. In other158

words, by revisiting the homogenization strategy in Eq. (4), it implies that the macroscopic temperature of159

the solid phase θscan be considered to be equivalent to that of solid skeleton.160

Asynchronous phase ﬁeld fracture 5

2.2 Phase ﬁeld approximation of crack topology161

This study adopts the diffuse representation of fracture by using the phase ﬁeld approach [Bourdin et al.,162

2008,Miehe et al.,2010a,Borden et al.,2012]. By letting Γbe the sharp crack surface within a body of163

mixture B, the total fracture surface area AΓcan be approximated as AΓd, which is the volume integral164

over body Bof the surface density Γd(d,∇d). In other words,165

AΓ≈AΓd=ZB

Γd(d,∇d)dV, (7)

where d∈[0, 1]is the phase ﬁeld that serves as a damage parameter in which d=0 indicates the intact166

region while d=1 denotes the completely damaged region. Here, the approximation AΓdmust be able to167

recover AΓby reducing the regularization length scale parameter lcto zero (i.e., Γ-convergence), while the168

generalized form of the corresponding crack density functional [Mumford and Shah,1989] reads,169

Γd(d,∇d)=1

c01

lc

w(d) + lc(∇d·∇d);c0=4Z1

0qw(s)ds, (8)

where c0is the normalization constant, and w(d)is the monotonically increasing local dissipation function170

that controls the shape of the regularized proﬁle of the phase ﬁeld [Clayton and Knap,2011,Mesgarnejad171

et al.,2015,Bleyer and Alessi,2018]. Note that a linear local dissipation along with a quadratic stiffness172

degradation yields a threshold energy model (existence of a linear elastic phase before the onset of dam-173

age), which is contrary to the quadratic model for which damage starts at zero loading. However, the174

threshold energy model can be converted to a critical stress which is dependent of the length scale param-175

eter lc. Both approaches have been used to model brittle fracture as two alternative regularizations of the176

variational theory of brittle fracture of Francfort and Marigo [1998]. Meanwhile, previous work, such as177

Lorentz [2017], Geelen et al. [2019], Suh and Sun [2019], Suh et al. [2020], have used non-quadratic degra-178

dation function which may yield a critical stress independent of lc.179

In this study, we adopt the quadratic local dissipation model, so that the crack resistance force Rccan180

be expressed as [Dittmann et al.,2020]:181

Rc=∂Wc

∂d−∇·∂Wc

∂∇d;Wc=GcΓd(d,∇d)=Gc

lcd2

2+l2

c

2(∇d·∇d), (9)

where Gc=Gc(θs)is the critical energy release rate that quantiﬁes the resistance to cracking, which will be182

explicitly deﬁned in Section 4.2.183

2.3 Free energy and effective stress principle184

We adopt the effective stress principle that decomposes the total macroscopic stress σinto the effective185

stress σ0and the contribution due to the pore ﬂuid pressure pf. As the effective stress is solely caused by186

the macroscopic deformation of the solid skeleton, it constitute a energy-conjugate relationship with the187

strain measure [Borja,2006,Borja and Koliji,2009]. As such, the free energy (ψ) of the porous media may188

take the following form (cf. Miehe and Mauthe [2016]):189

ψ=ψ0(ε,θs,d) + ψ∗(ε,ϑ∗,d) + ψθs(θs) + ψθf(θf). (10)

Note that the energy required for crack growth [i.e., Wcin Eq. (9)] is dissipatve by nature and hence not190

included in this stored energy function ψ[Choo and Sun,2018a,b,Dittmann et al.,2020]. Our deﬁnition of191

free energy will be used for constructing the energy balance equations based on the ﬁrst law of thermody-192

namics in Section 3.1, while this section deﬁnes all the terms in detail ﬁrst, and then presents the effective193

stress principle.194

The effective part of the strain energy density ψ0(ε,θs,d)can be viewed as a stored energy density due to195

the intergranular stress acting on the solid skeleton that leads to its deformation. In particular, we assume196

that the effective part of the strain energy density ψ0(ε,θs,d)is composed of the ﬁctitious undamaged197

6 Hyoung Suk Suh, WaiChing Sun

thermoelastic strain energy ψ0

0(ε,θs)and the degradation function g(d)∈[0, 1][Yang et al.,2006,Miehe198

et al.,2015,Na and Sun,2018], i.e.,199

ψ0(ε,θs,d) = g(d)ψ0

0(ε,θs);ψ0

0(ε,θs) = 1

2ε:Ce:ε−3αsK(θs−θs,ref)tr (ε), (11)

where Ceis the elastic moduli and Kis the bulk modulus of the solid skeleton. This approach allows us to200

interpret the cracking in a saturated porous material as the fracture of the solid matrix.201

Following Miehe and Mauthe [2016], and by assuming that the effect of thermal expansion of the pore202

ﬂuid is negligible (i.e., its thermal expansion coefﬁcient αf=0), the contribution of pore ﬂuid to the free203

energy ψ∗(ε,ϑ∗,d)can be deﬁned as follows:204

ψ∗(ε,ϑ∗,d) = 1

2M∗(d)[B∗(d)tr (ε)−ϑ∗]2;ϑ∗=B∗(d)tr (ε) + pf

M∗(d), (12)

where the expression for ϑ∗is similar to Eq. (2.12) in [Biot,1941], while B∗(d)and M∗(d)are the modiﬁed205

Biot’s coefﬁcient and the modiﬁed Biot’s modulus, respectively:206

B∗(d) = 1−K∗(d)

Ks;1

M∗(d)=B∗(d)−φf

Ks

+φf

Kf

. (13)

Here, K∗(d) = g(d)K, while Ksand Kfdenote the bulk moduli of the solid and ﬂuid phases, respectively.207

As shown in Eq. (13), this study assumes that the damage of the solid skeleton degrades the elastic bulk208

modulus K∗(d), so that B∗(d)and M∗(d)may evolve according to the deformation. In other words, if209

the solid skeleton remains undamaged, the modiﬁed coefﬁcient recovers the classical deﬁnition of Biot’s210

coefﬁcient (i.e., B∗=1−K/Ks) that is often less than 1 for rock [Vinck´

e et al.,1998,Zimmerman,2000,211

Jaeger et al.,2009], while we have B∗(1) = 1 for the case where the solid skeleton is completely damaged,212

which has been accepted in previous studies on hydraulic fracture [Miehe and Mauthe,2016,Mauthe and213

Miehe,2017,Ha et al.,2018]. Following Heider and Sun [2020], we assume that crack opening leads to a214

complete fragmentation of solid skeleton, such that we adopt the following relation for the porosity (i.e.,215

the volume fraction of ﬂuid phase constituent φf):216

φf=1−g(d)(1−φf

ref)(1−∇· u), (14)

where φf

ref is the reference porosity. We also deﬁne ϕ=ϑ∗−B∗(d)tr (ε)for convenience, which is related217

to the variation of the ﬂuid content and is the energy conjugate to the pore ﬂuid pressure pf. In this case,218

Eq. (12) can be re-written in a simple quadratic form:219

ψ∗(ε,ϑ∗,d) = ψ∗(ϕ,d) = 1

2M∗(d)ϕ2. (15)

The pure thermal contribution on the stored energy density ψθi(θi)may have the simple form as220

[Lubarda,2004,Yang et al.,2006,Miehe et al.,2015,Na and Sun,2018],221

ψθi(θi) = ρici(θi−θi,ref)−θiln θi

θi,ref , (16)

where i={s,f}, while ciindicates the speciﬁc heat capacity and θi,ref is the reference temperature for the222

i-phase constituent. Note that, as shown in Eq. (16), we simplify the coupled thermo-mechanical-fracture223

problem by assuming that the thermal part of the stored energy densities ψθs(θs)and ψθf(θf)are not224

affected by the fracture (cf. [Miehe and Mauthe,2016,Na and Sun,2018,Dittmann et al.,2020]).225

Having deﬁned all the terms for the free energy, we now present the effective stress principle based on226

the hyperelastic relations. From Eqs. (10), (11), and (12), the total stress σcan be found by taking the partial227

derivative of the total energy density ψwith respect to the strain ε:228

σ=∂ψ

∂ε=∂

∂εψ0(ε,θs,d)

| {z }

=σ0

+∂

∂εψ∗(ε,ϑ∗,d)

| {z }

=−B∗(d)pfI

. (17)

Asynchronous phase ﬁeld fracture 7

A similar decomposition can be found in a number of studies on theories of porous media [Bowen,1980,229

Zienkiewicz et al.,1999,Ehlers,2002,Coussy,2004,De Boer,2012], where the ﬁrst term of the right hand230

side in Eq. (17) becomes the effective stress σ0, while the second term indicates the contribution of the pore231

pressure which is assumed to produce a hydrostatic stress state [Miehe and Mauthe,2016]. From Eqs. (11)232

and (17), the effective stress tensor can also be expressed as,233

σ0=σ+B∗(d)pfI=g(d)σ0

0, (18)

where σ0

0=∂ψ0

0/∂εis the ﬁctitious undamaged effective stress.234

3 Conservation laws for thermally non-equilibrated porous media235

In this section, we derive the balance principles that govern the brittle fracture in saturated porous media236

with constituents of different temperatures. While previous work such as [Miehe et al.,2015,Na and Sun,237

2018,Noii and Wick,2019,Dittmann et al.,2019] has introduced a framework to address the thermal ef-238

fect of brittle or quasi-brittle fracture in porous media, our new contribution here is to introduce the heat239

exchange between the two thermally connected constituents, such that the multi-scale nature of the heat240

transfer can be considered. Since our homogenization strategy enables us to consider two macroscopic241

temperatures for each constituent, we derive two distinct energy balance equations by assuming that the242

thermodynamic state of each phase is measured by their own temperature, internal energy, and entropy.243

Our derivation in Section 3.1 shows that the two-temperature approach can be reduced into a classical heat244

equation with a single temperature ﬁeld if we consider the special case where two constituents are ther-245

mally equilibrated. Then, in addition to two energy equations, we present a thermodynamically consistent246

phase ﬁeld model and the balances of linear momentum and mass, that complete the set of governing247

equations which not only describes the thermo-hydro-mechanical behavior of porous media in local ther-248

mal non-equilibrium, but also the evolution of the fracture.249

3.1 Balance of energy250

In contrast to the models that employ a single temperature ﬁeld [McTigue,1986,Belotserkovets and Pre-251

vost,2011,Kim,2018], our approach requires two energy balance equations for each phase in order to252

account for the transient period, i.e., local thermal non-equilibrium [Fourie and Du Plessis,2003a,b,Gelet253

et al.,2012]. Hence, following Gelet et al. [2012], we assume that thermodynamic states of the solid skeleton254

and pore ﬂuid can respectively be measured by their own temperature θi, internal energy Eiand entropy255

Hiper unit mass. Based on the assumption, the internal energy per unit volume ecan additively be decom-256

posed as follows,257

e=es+ef;ei=ρiEi, (19)

where i={s,f}so that eiis the partial quantity. Similarly, entropy per unit volume of the mixture ηcan258

also be decomposed into,259

η=ηs+ηf;ηi=ρiHi, (20)

where we assume that each partitioned entropies satisfy:260

ηi=−∂ψi

∂θi

. (21)

Here, by revisiting Section 2.3, we deﬁne ψias,261

ψ=ψs+ψf;(ψs=ψ0(ε,θs,d) + ψθs(θs),

ψf=ψ∗(ϕ,d) + ψθf(θf),(22)

such that ψsand ψfare the partial free energy of the solid and ﬂuid phase constituents, respectively. As262

shown in Eq. (22), this study assumes that the effects of the skeletal structure of the solid phase (e.g., effec-263

tive stress and degradation) on the free energy is solely stored in ψs, while ψfonly includes the contribution264

8 Hyoung Suk Suh, WaiChing Sun

of its intrinsic pressure and temperature. Furthermore, we postulate that the partial quantities of internal265

energy eiand entropy ηican be subjected to a Legendre transformation, i.e.,266

ψi=ei−θiηi, (23)

so that the following classical relation [Truesdell and Toupin,1960,Abraham et al.,1978,Holzapfel,2002]267

can be recovered if two constituents are in thermal equilibrium (i.e., θs=θf=θm):268

ψ=∑

i={s,f}

ψi=∑

i={s,f}

(ei−θiηi) = e−θmη. (24)

On the other hand, the energy exchange between the constituents can be described by introducing the rates269

of energy transfer χi, in which energy conservation requires the following constraint to be satisﬁed:270

χs+χf=0. (25)

Based on the ﬁrst law of thermodynamics, the balance of energy for the solid constituent that accounts271

for the ﬂux of thermal energy due to heat conduction (qs), the rate of energy exchange (χs), and the heat272

source (ˆ

rs) can be written as,273

˙

es=σ0: ˙ε−∇·qs+χs+ˆ

rs, (26)

where ˙

(•) = d(•)/dt is the total material time derivative following the solid phase. Although will be dis-274

cussed later in Section 3.2, we brieﬂy show that the second law of thermodynamics (i.e., Clausius–Duhem275

inequality) yields the following expression for the dissipation functional Ds:276

Ds=σ0−∂ψs

∂ε: ˙ε−ηs+∂ψs

∂θs˙

θs−∂ψs

∂d˙

d

| {z }

=Ds

int

−1

θs

qs·∇θs

| {z }

=Ds

con

≥0, (27)

where the entropy input is assumed to be related to the heat ﬂux across the boundary and the heat source277

[Na and Sun,2018,Dittmann et al.,2019]. From the relations deﬁned previously [Eqs. (17) and (21)], dissi-278

pation functional in Eq. (27) can be reduced into,279

Ds=Ds

int +Ds

con ≥0. (28)

Finally, from Eqs. (23) and (26), the solid phase energy balance equation in Eq. (26) becomes:280

˙

ψs−˙

es+˙

θsηs=θs˙

ηs−Ds

int +∇·qs−χs−ˆ

rs=0. (29)

By substituting the explicit expression for ηs[i.e., from Eqs. (20) and (22)], Eq. (29) can be re-written as281

follows, where similar form can be found in [Simo and Miehe,1992,Na and Sun,2017,2018].282

ρscs˙

θs= [Ds

int −Hθs]−∇·qs+χs+ˆ

rs. (30)

In this study, to simplify the equation, we assume that structural heating/cooling is negligible (i.e., Hθs=0)283

compared to the internal dissipation Dint.284

We now repeat the same procedure for the ﬂuid phase. Again, from the ﬁrst law, the internal energy for285

the pore ﬂuid that accounts for the heat ﬂux due to the conduction (qf), the rate of energy exchange (χf),286

the heat convection (Af), and the heat source (ˆ

rf) can be written as,287

˙

ef=−pf˙

ϕ−Af−∇·qf+χf+ˆ

rf, (31)

where we take Af=ρfcf(w· ∇ θf)with wdenoting Darcy’s velocity, by assuming that the advection288

process is governed by the movement of the pore ﬂuid relative to that of the solid skeleton [Gelet et al.,289

2012,Sun,2015]. Recall that from Eqs. (12), (15) and (21) we have: pf=−M∗ϕand ηf=−∂ψ f/∂θf. Thus,290

from Eqs. (23) and (31), the ﬂuid energy balance equation reads,291

˙

ψf−˙

ef+˙

θfηf=θf˙

ηf+Af+∇·qf−χf−ˆ

rf, (32)

Asynchronous phase ﬁeld fracture 9

where we assume that the contribution of the phase ﬁeld on ψfis negligible. Then, by substituting the292

explicit expression for ηf, the ﬂuid phase energy balance equation can be re-written as,293

ρfcf˙

θf=−ρfcf(w·∇θf)− ∇·qf+χf+ˆ

rf, (33)

where Eq. (33) is similar to the form that seen in [D´

orea and De Lemos,2010,Gandomkar and Gray,2018,294

Heinze,2020].295

Remark 1 Eqs. (30) and (33) describes the heat transfer process in porous media under LTNE condition,296

however, one may obtain a different form of governing equations if adopting either different form of the297

free energy functional or different decomposition scheme on the internal energy. Based on our approach,298

for the situation where the material is undamaged (d=0) and is under LTE condition (i.e., θs=θf=θm),299

adding Eqs. (30) and (33) yields the classical one-temperature model [McTigue,1986,Zimmerman,2000,300

Coussy,2004,Belotserkovets and Prevost,2011]:301

ρcm˙

θm=−ρfcf(w·∇θm)− ∇·q+ˆ

r, (34)

where ρcm=ρscs+ρfcf,q=qs+qf, and ˆ

r=ˆ

rs+ˆ

rf. Here, Eq. (34) not only demonstrates the connection302

between one- and two-temperature approaches but also implies that the classical model assumes a special303

case where all the phase constituents instantly reach a local thermal equilibrium.304

3.2 Dissipation inequality and crack evolution305

By revisiting the expression for the dissipation functional Dsin Eq. (28), the following thermodynamic306

restriction must be satisﬁed:307

Ds

int =Fc˙

d≥0, (35)

since the dissipation due to heat conduction Ds

con is guaranteed positive by the Fourier’s law, while:308

Fc=−∂ψs

∂d=−g0(d)ψ0

0(36)

indicates the crack driving force [Dittmann et al.,2019,2020]. Notice that a sufﬁcient condition for the309

inequality in Eq. (35) is that all the components Fcand ˙

dare individually non-negative. By adopting the310

quadratic degradation function, i.e., g(d)=(1−d)2, that satisﬁes the following conditions [Pham and311

Marigo,2013,Suh et al.,2020]:312

g(0) = 1 ; g(1) = 0 ; g0(d)≤0 for d∈[0, 1], (37)

the non-negative crack driving force Fcis automatically guaranteed since ψ0

0≥0. In this case, the thermo-313

dynamic restriction in Eq. (35) becomes:314

˙

d≥0. (38)

While the stored energy functional in the microforce approach often contains the fracture energy [Gurtin,315

1996,Wilson et al.,2013,Na and Sun,2018], recall Section 2.3 that our energy functional ψdoes not include316

the energy used to create a fracture. Again, it allows us to consider crack growth as a fully dissipative pro-317

cess, resulting in the solid phase energy balance equation [Eq. (30)] that contains the internal dissipation318

Ds

int. Based on this setting, we adopt a concept similar to the variational framework for fracture that char-319

acterizes the crack propagation process by energy dissipation [Francfort and Marigo,1998,Bourdin et al.,320

2008,Miehe et al.,2010b]. By assuming that the viscous resistance is neglected, thermodynamic consistency321

requires the balance between the crack driving force Fcin Eq. (36) and the crack resistance Rcin Eq. (9),322

i.e.,323

Rc−Fc=g0(d)ψ0

0+Gc

lc

(d−l2

c∇2d) = 0, (39)

10 Hyoung Suk Suh, WaiChing Sun

where ∇2(•) = ∇·∇ (•)indicates the Laplacian operator. Here, we adopt the volumetric-deviatoric split

proposed by Amor et al. [2009], which is the stored energy that may contribute as the driving force for

crack growth, i.e.,

ψ0

0

+=1

2hεvoli2

++µ(εdev :εdev)−3αsK(θs−θs,ref )hεvoli+, (40)

ψ0

0−=1

2hεvoli2

−−3αsK(θs−θs,ref)hεvol i−, (41)

where εvol =tr (ε),εdev =ε−(εvol/3)I, and h•i±=(• ±|•|)/2 indicates the Macaulay bracket operator.324

To prevent healing of the crack, we adopt a normalized local history ﬁeld H ≥ 0 of the maximum positive325

reference energy, i.e.,326

H=max

τ∈[0,t] ψ0

0

+

Gc/lc!, (42)

which satisﬁes the following Karush–Kuhn–Tucker condition [Borden et al.,2012,Choo and Sun,2018a]:327

W+− H ≤ 0 ; ˙

H ≥ 0 ; ˙

H(W+−H) = 0, (43)

where W+=ψ0

0

+/(Gc/lc)denote the portion of nondimensional ψ0

0that contributes to cracking.328

By replacing the stored energy term in Eq.(39) by (Gc/lc)H, the governing equation for the phase ﬁeld329

dcan be re-written as follows:330

g0(d)H+ (d−l2

c∇2d) = 0. (44)

Note that Eq. (44) is based on balance of the material force (cf. Borden et al. [2012]) and is not a Euler-331

Lagrangian equation obtained from the minimization of an energy functional.332

3.3 Balance of linear momentum333

By neglecting the inertial force, the balance of linear momentum for the solid-ﬂuid mixture can be written334

as,335

∇·σ+ρg=0, (45)

where σ=σs+σfis the total Cauchy stress that can be obtained from the sum of partial stresses σifor336

i-phase constituents [Atkin and Craine,1976,Pr´

evost,1980]. Hence, from Eq. (18), the mean pressure pcan337

be expressed as:338

p=−1

3tr (σ) = φsps+φfpf=−K∗∇·u+3αsK∗(θs−θs,ref ) + B∗pf, (46)

where psand pfare the intrinsic pressures deﬁned in dVsand dVf, respectively, while the detailed consti-339

tutive model for the solid skeleton will be presented in Section 4.2.340

3.4 Balance of mass341

Assuming that there is no phase transition between two constituents, the balance of mass for the solid

skeleton and the pore ﬂuid reads,

˙

ρs+ρs∇·v=0, (47)

˙

ρf+ρf∇·v+∇·hρf(vf−v)i=ρfˆ

s, (48)

where ρfˆ

sis the rate of prescribed ﬂuid mass source/sink per unit volume, while vand vfindicate the342

solid and ﬂuid velocities, respectively. Since the change of dVsdepends on both the intrinsic pressure ps

343

and the temperature θs, the total time derivative of partial density ρscan be expanded as,344

˙

ρs=˙

φsρs=˙

φsρs+φsdρs

dps

˙

ps+dρs

dθs

˙

θs=˙

φsρs+φsρs1

Ks

˙

ps−3αs˙

θs, (49)

Asynchronous phase ﬁeld fracture 11

so that the solid phase mass balance equation in Eq. (47) can be re-expressed as,345

−˙

φs=φs

Ks

˙

ps−3αsφs˙

θs+φs∇·v. (50)

Also, from Eq. (46), the total time derivative of mean pressure pyields:346

˙

p=˙

φsps+φfpf=−K∗∇·v+3αsK∗˙

θs+B∗˙

pf. (51)

Following Sun et al. [2013], we assume that the change of porosity at an inﬁnitesimal time is small (i.e.,347

˙

φipiis relatively small compared to φi˙

pi), so that Eq. (51) reduces into,348

φs˙

ps=−K∗∇·v+3αsK∗˙

θs+ (B∗−φf)˙

pf. (52)

By substituting Eq. (52) into Eq. (50), the solid phase mass balance equation now reads,349

−˙

φs=B∗−φf

Ks

˙

pf−3αsφs−K∗

Ks˙

θs+φs−K∗

Ks∇·v. (53)

Similar to Eq. (50), the ﬂuid phase mass balance equation in Eq. (48) can also be expanded as,350

˙

φf+φf

Kf

˙

pf−3αfφf˙

θf+φf∇·v+∇·w=ˆ

s, (54)

where Kfis the bulk modulus of the ﬂuid, w=φf(vf−v)indicates Darcy’s velocity, and αfis the linear351

thermal expansion coefﬁcient of the pore ﬂuid which has been assumed to be zero in Section 2.3. Recall352

that Eq. (2) yields the condition φs+φf=1, which leads to: ˙

φf=−˙

φs. Thus, we substitute Eq. (53) into353

the ﬁrst term in Eq. (54) that gives the following expression for the ﬂuid phase mass balance equation:354

1

M∗˙

pf−3αs(B∗−φf)˙

θs−3αfφf˙

θf+B∗∇·v+∇·w=ˆ

s. (55)

Remark 2 If we assume that the solid and ﬂuid temperatures are locally equilibrated (i.e., θs=θf=θm)355

and ˆ

s=0, Eq. (55) can be reduced into a similar form that is shown in [Coussy,2004,Belotserkovets and356

Prevost,2011,Na and Sun,2016]:357

1

M∗˙

pf−3αm˙

θm+B∗∇·v+∇·w=0, (56)

where αm= (B∗−φf)αs+φfαfis the coefﬁcient of linear thermal expansion for thermally equilibrated358

medium. Furthermore, if we consider a special case where thermal expansion is negligible and each con-359

stituent is incompressible (i.e., Ki→∞), Eq. (56) further reduces to the form identical to that seen in [Borja360

and Alarc´

on,1995,Sun et al.,2013,Sun,2015]:361

∇·v+∇·w=0, (57)

since B∗=1 and 1/M∗=0 in this case, regardless of the damage parameter d.362

4 Constitutive responses363

The goal of this section is to identify constitutive relations that capture thermo-hydro-mechanically cou-364

pled behavior of the material of interest. We begin this section by the constitutive relationships for partial365

heat ﬂuxes for each phase, where we assume both the solid and ﬂuid constituents obey Fourier’s law. We366

also present the explicit expression for the heat exchange χibetween the solid skeleton and pore ﬂuid367

based on Newton’s law of cooling. We then brieﬂy summarize the linear thermoelasticity for the undam-368

aged solid skeleton, while the hydraulic responses in both the bulk and crack regions are modeled by the369

Darcy’s law, where we adopt permeability enhancement approach in order to account for the anisotropy370

due to the crack opening. In addition, this study adopts an empirical two-parameter model for the pore371

ﬂuid viscosity, which is capable of predicting the temperature-dependent viscosity of typical liquids in372

geomaterials.373