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Abstract

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.
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= 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., ab=aibj) or two symmetric second-order tensors [e.g., (αβ)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 composed of solid (s) and fluid ( f) constituents, i.e., =114
sf. In small scale, the spatial distribution of each constituent in can be represented by indicator115
functions rs(y)and rf(y):116
rs(y) = (1 if ys,
0 otherwise, ;rf(y) = (1 if yf,
0 otherwise, (1)
where ydenotes the position vector associated with small-scale configuration. By assuming that can 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 Zrs(y)d;φf=dVf
dV =1
dV Zrf(y)d, (2)
where dV =R[rs(y) + rf(y)] dindicates 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
dVsZrs(y)θ(y)d;θf=1
dVfZrf(y)θ(y)d, (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 Zθ(y)d. (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
2u+uT, (6)
where εeis the elastic component of the strain tensor and εθs=αs(θsθs,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
AΓAΓd=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:ε3αsK(θsθs,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 ϑis 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) = 1K(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=1K/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=1g(d)(1φf
ref)(1∇· u), (14)
where φf
ref is the reference porosity. We also define ϕ=ϑB(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(θiθi,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=eiθiηi, (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}
(eiθiηi) = eθmη. (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: ˙εηs+∂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˙
ηsDs
int +∇·qsχsˆ
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˙
ϕAf·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=Mϕand ηf=∂ψ f/θf. Thus,290
from Eqs. (23) and (31), the fluid energy balance equation reads,291
˙
ψf˙
ef+˙
θfηf=θf˙
ηf+Af+∇·qfχfˆ
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˙
d0, (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)=(1d)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
00. In this case, the thermo-313
dynamic restriction in Eq. (35) becomes:314
˙
d0. (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
RcFc=g0(d)ψ0
0+Gc
lc
(dl2
c2d) = 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(θsθs,ref )hεvoli+, (40)
ψ0
0=1
2hεvoli2
3αsK(θsθs,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+ (dl2
c2d) = 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(θsθs,ref ) + Bpf, (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(vfv)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
˙
ps3αs˙
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
˙
ps3αsφs˙
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+ (Bφf)˙
pf. (52)
By substituting Eq. (52) into Eq. (50), the solid phase mass balance equation now reads,349
˙
φs=Bφf
Ks
˙
pf3αsφsK
Ks˙
θs+φsK
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
˙
pf3αfφf˙
θf+φf∇·v+·w=ˆ
s, (54)
where Kfis the bulk modulus of the fluid, w=φf(vfv)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˙
pf3αs(Bφf)˙
θs3αfφf˙
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˙
pf3αm˙
θm+B∇·v+·w=0, (56)
where αm= (Bφf)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