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

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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.
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Physics of Fluids manuscript No.
(will be inserted by the editor)
An immersed phase field fracture model for microporomechanics with1
Darcy-Stokes flow2
Hyoung Suk Suh ()·WaiChing Sun ()3
4
Received: December 14, 2020/ Accepted: date5
Abstract This paper presents an immersed phase field model designed to predict the fracture-induced6
flow due to brittle fracture in vuggy porous media. Due to the multiscale nature of pores in vuggy porous7
material, crack growth may connect previously isolated pores which lead to flow conduits. This mecha-8
nism has important implications for many applications such as disposal of carbon dioxide and radioactive9
materials, hydraulic fracture and mining. To understand the detailed microporomechanics that causes the10
fracture-induced flow, we introduce a new phase field fracture framework where the phase field is not only11
used as an indicator function for damage of the solid skeleton, but also as an indicator of the pore space.12
By coupling the Stokes equation that governs the fluid transport in the voids, cavities and cracks, and the13
Darcy’s flow in the deformable porous media, our proposed model enables us to capture the fluid-solid14
interaction of the pore fluid and solid constituents during the crack growth. Numerical experiments are15
conducted to analyze how presence of cavities affects the accuracy of the predictions based on homoge-16
nized effective medium during crack growth.17
Keywords Biot-Stokes model, coupled Stokes-Darcy flow, vuggy porous media, immersed phase field,18
brittle fracture19
1 Introduction20
Geomaterials such as carbonate rocks, sandstone or limestone often contain geometrical features such as21
cracks, joints, vugs or cavities. When the defects are partially or fully saturated with pore fluid, the geome-22
try of the features may affect effective stiffness, permeability, water retention characteristics and drained or23
undrained shear strength of the material [Juanes et al.,2006,Sun et al.,2011b,Kang et al.,2016,Suh et al.,24
2017,Selvadurai et al.,2017,Wang and Sun,2017,Sun and Wong,2018]. Furthermore, brittle fracture in25
materials that possess geometrical features may lead to pore fluid in cavities migrate into the flow channels26
and cause flow conduits that lead to often undesirable outcomes. Modeling geometrical features in porous27
media are thus highly important and at the same time challenging subject for the hydromechanically cou-28
pled analysis in geomechanics problems like hydrocarbon resources recovery or development of enhanced29
geothermal energy reservoirs [Paterson and Fermigier,1997,Class et al.,2002,Rutqvist et al.,2007,Grant,30
2013,Wagner et al.,2015,Heider and Markert,2017,Suh and Sun,2020].31
One possible modeling choice is to consider a fictitious effective medium at a scale where represen-32
tative elementary volume exists. In this case, the geometrical features of the material are not explicitly33
modeled but the influences of the these geometrical features are incorporated in the constitutive relations34
by treating defects as a different pore system that interacts with the matrix pores [Choo and Borja,2015,35
Choo et al.,2016,Liu and Abousleiman,2017,Wang and Sun,2018]. The upshot of the multi-porosity and36
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
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
2
multi-permeability models is mainly the simple numerical treatment since there is no need for complex37
meshing techniques or embedded strong discontinuities, and the computational efficiency compared to38
pore-scale models that require extremely large domain in order to reproduce hydromechanical behavior39
at large scales [Ghaboussi and Barbosa,1990,Spaid and Phelan Jr,1997,Blunt et al.,2002,Tang et al.,40
2005,Arson and Pereira,2013,Pereira and Arson,2013,Suh and Yun,2018]. However, the drawback of41
this approach is that the homogenized effective medium may not sufficiently represent the microstruc-42
tural details. This makes the identification of material parameters more complicated since the effective43
permeability of multiple interacting systems are not isotropic and the constitutive law for the fluid mass44
exchanges inherently depends on the microstructure.45
Another common alternative to model the interaction between the cavities and the crack growth is to46
conduct simulations via a fracture network model [Ozkan et al.,2010,Leung and Zimmerman,2012,Fu47
et al.,2013,Hyman et al.,2015]. However, the obvious drawback is that the fracture in those models must48
either be straight line (in the two-dimensional case) or a plane (in the three-dimensional case) and hence49
the geometrical effect on the porous media can not be captured precisely.50
In this research, we introduce a phase field framework that allows us to enable a unified treatment to51
simulate the evolving geometry of cracks and the cavities. By introducing the phase field as an unified52
representation of the void space that is not suitable to be treated as as an effective medium, we introduce53
a framework that enables us to analyze how crack propagation in vuggy porous media may affect the54
flow mechanism differently than the porous media with pores well distributed in the host matrix. Our55
result indicates that interaction between the propagating cracks of the cavities is important for capturing56
the hydromechanical responses of the porous media and that existing effective medium approach which57
characterizes the pore space with a single hydraulic model such as cubic law and Kozeny-Carmen model58
may not be sufficient to capture the cavity-crack-host-matrix interactions.59
As for notations and symbols, bold-faced and blackboard bold-faced letters denote tensors (including60
vectors which are rank-one tensors); the symbol ’·’ denotes a single contraction of adjacent indices of two61
tensors (e.g., a·b=aibior c·d=cijdjk); the symbol ‘:’ denotes a double contraction of adjacent indices62
of tensor of rank two or higher (e.g., C:ε=Cijk l εkl); the symbol ‘’ denotes a juxtaposition of two vectors63
(e.g., ab=aibj) or two symmetric second order tensors [e.g., (αβ)ijkl =αij βkl]. We also define identity64
tensors: I=δij and I=δik δjl , where δij is the Kronecker delta. As for sign conventions, unless specified,65
the directions of the tensile stress and dilative pressure are considered as positive.66
2 The model problem67
We consider a fully saturated Biot-Stokes system (Fig. 1) that consists of two regions (intact porous matrix68
BD, and cracks or cavities BS) separated by the sharp interface Γ, where we assume that the solid phase69
in BDforms a deformable porous matrix while solid particles in BSare in suspension. In this case, both the70
solid and fluid phases coexist in both regions. By considering our material of interest as a multi-phase con-71
tinuum, we utilize the effective stress principle for the intact porous matrix where the fluid flow is modeled72
with the Darcy’s law, while the motion of solid-fluid mixture is modeled by the Stokes equation [Li et al.,73
2018]. Two distinct regions are then coupled by properly imposing three transmissibility conditions at the74
interface. The model problem with the sharp interface will be later on extended into a diffuse Biot-Stokes75
model by introducing the phase field in Section 3.76
2.1 Continuum representation77
Although Biot-Stokes system only contains two immiscible solid and fluid phases, for mathematical con-78
venience, we idealize the material of interest as a three-phase continuum where each constituent [i.e., solid79
(s), pore fluid (fD), and free fluid ( fS)] occupies a fraction of volume at the same material point. By let-80
ting dV =dVs+dVfdenote the representative elementary volume of the material, we define the volume81
fractions for the constituents as,82
φα:=dVα
dV ;α={s,f}, (1)
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
3
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Solid skeleton
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Pore fluid
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Liquefied solid
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Free fluid
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n
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m
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BD
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@BD
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@BS
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Intact porous matrix
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Solid-fluid mixture
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Fig. 1: Schematic representation of Biot-Stokes system that possesses sharp interface Γ.
where the index srefer to the solid phase and findicates the fluid phase. Since the sharp interface separates83
our system of interest into two regions, the volume fraction of the pore and free fluids can be expressed as:84
φf= (1HΓ)dVfD
dV
| {z }
:=φfD
+HΓ
dVfS
dV
| {z }
:=φfS
, (2)
where HΓis the Heaviside function that satisfies,85
HΓ=(0 in BD,
1 in BS.(3)
In addition, by letting ρsand ρfdenote the intrinsic mass densities of solid and fluid, respectively, the86
partial mass densities for each constituent (ρα, where α=s,fD,fS) are given by,87
ρs:=φsρs;ρfD:=φfDρf;ρfS:=φfSρf;ρ:=ρs+ρfD+ρfS, (4)
where ρis the mass density of the entire system. In this study, we assume that both the solid and fluid88
phases are incompressible, so that intrinsic mass densities ρsand ρfare considered as constants.89
2.2 Governing equations90
This section briefly reviews the balance principles, constitutive laws in the bulk volume of a porous medium91
(Section 2.2.1), the region where solid-fluid mixture flows freely (Section 2.2.2), and the sharp interface be-92
tween two regions (Section 2.2.3).93
2.2.1 Conservation laws for an intact porous matrix94
For the region where the solid forms an intact porous matrix, we adopt the effective stress principle [Lade
and De Boer,1997,Borja,2006] so that the external loading imposed on the matrix is assumed to be carried
by both the solid skeleton and the pore fluid. In this case, the region BDis governed by the following
system of equations [Borja and Alarc´
on,1995,White and Borja,2008,Sun et al.,2013]:
∇·(σ0Bp fDI) + ρg=0in BD, (5)
∇·vs+∇· wfD+1
M˙
p=0 in BD, (6)
where σ0is the effective stress, B=1K/Ksis the Biot’s coefficient, Mis the Biot’s modulus, pfDis the95
pore pressure, gis the gravitational acceleration, vαis the intrinsic velocity of constituent α, and wfD=96
φfD(vfDvs)is the Eulerian relative flow vector of the pore fluid (i.e., Darcy’s velocity). Here, we assume97
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4
that B1 and 1/M0 to simplify the formulation. Note that the Biot’s coefficient of many sandstone98
and shale specimens are often less than one, whereas it is more reasonable to assume Biot’s coefficient99
equal to 1 for granite (e.g. Westerly granite) [Berryman and Wang,1995,Zimmerman,2000,Jaeger et al.,100
2009]. In either cases, the damage of the solid skeleton may reduce the elastic bulk modulus of the solid101
skeleton. Therefore both the Biot’s coefficient and modulus may evolve according to the solid deformation.102
This nonlinear effect is not considered in this study but will be considered in the future. We also assume103
that the behavior of intact matrix in BDis linear and isotropic elastic and hence only two independent104
elastic modulii are needed to capture the elastic response. The constitutive relation for the solid skeleton105
can therefore be written as follows:106
σ0
0=λtr (ε)I+2µεin BD, (7)
where σ0
0indicates the effective stress of the undamaged matrix. The actual and undamaged effective107
stress are related by a degradation function, which will later be discussed in Section 3.2. Furthermore,108
ε= (us+uT
s)/2 is the infinitesimal solid strain tensor that depends on the solid displacement us,109
and parameters λand µare the Lam´
e constants. For the constitutive equation that describes laminar pore110
fluid flow in BD, we use the generalized Darcy’s law that linearly relates the relative velocity wfDand pore111
pressure gradient pfD, i.e.,112
wfD=k
µf
(pfDρfg)in BD, (8)
where µfis the dynamic viscosity of the pure fluid phase, and kis the effective permeability of the porous113
matrix. Additionally, in order to incorporate the effect of deformation of the matrix on the porous medium114
flow [Mauran et al.,2001,Schutjens et al.,2004], this study adopts the Kozeny-Carman equation to empir-115
ically capture the porosity-permeability relation [Chapuis and Aubertin,2003,Costa,2006,Wang and Sun,116
2017]. Note that the Kozeny-Carmen equation is often considered a rough approximation of the porosity-117
permeability relation. A more precise predictions of permeability may requires new geometrical attributes118
such as tortuosity [Sun et al.,2011b,a], formation factor [Worthington,1993,Sun and Wong,2018], and119
percolation threshold [Mavko and Nur,1997]. This extension is out of the scope of this study but will be120
considered in future work.121
Recall Section 2.1 that φs+φfD=1 in BD. Then, by letting φ:=φfDthe porosity of the matrix, the122
Kozeny-Carman equation reads,123
k=k0"(1φ0)2
φ3
0#φ3
(1φ)2in BD, (9)
where k0and φ0denote the reference permeability and porosity, respectively.124
2.2.2 Conservation laws for solid-fluid mixture125
This study attempts to model suspension flow in BS, where mass and linear momentum balances for both
solid and fluid phases should be satisfied. We therefore write the governing balance equations for BSas,
∇·σfS+ρfSg=0in BS, (10)
∇·σs+ρsg=0in BS, (11)
∇·vs+∇· wfS=0 in BS, (12)
where σαis the Cauchy stress tensor of αconstituent, and the relative flow vector of the free fluid can be126
defined as wfS=φfS(vfSvs). By assuming that the free fluid resides in BSwith low Reynolds number127
(i.e., Re 1), we adopt a simplified version of the Navier-Stokes model, i.e., the Stokes equation. The128
Stokes model for the steady-state motion of an incompressible fluid yields the following relationship for129
the free fluid stress tensor σfSas,130
σfS=pfSI+µeff(vfS+vT
fS)in BS, (13)
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5
where pfSis the free fluid pressure and µeff is the effective viscosity of the solid-fluid mixture [Mooney,131
1951,Cheng and Law,2003], i.e.,132
µeff =µfexp 2.5c
1c/cmax in BS, (14)
where c:=1φfSindicates the solid particle concentration, and cmax denotes its upper bound. Again,133
notice that we introduce only one solid constituent for the entire system since it is convenient for us to later134
on impose interface conditions and further adopt the phase field fracture model that simulates evolving135
interface. This approach may not be suitable for modeling complete suspension flow where vs=vfS.136
However, we assume that solid particles in BSfollows the same constitutive relations as the free fluid in137
order to replicate the suspension flow as close as possible, i.e.,138
σs=pfSI+µeff(vs+vT
s)in BS. (15)
2.2.3 Conservation laws for the sharp interface between intact matrix and solid-fluid mixture139
In order to properly model the interaction between the porous matrix (BD) and the vugs or cavities (BS),140
complete mass conservation and force equilibrium for the entire system should be satisfied. Since we have141
two different constituents for the same type of fluid (fDand fS) while considering only one solid con-142
stituent (s), coupling two subsystems thus requires the enforcement of fluid transmissibility conditions at143
the sharp interface Γthat models the coupled Stokes-Darcy flow [Arbogast and Lehr,2006,Arbogast and144
Brunson,2007,Badia et al.,2009,Wu and Mirbod,2018,Bergkamp et al.,2020].145
The first interface condition is the fluid continuity that ensures the mass conservation. Since we assume146
that the fluid phase is incompressible, the interfacial fluid fuxes for each subsystem (M
fDand M
fS) can147
be expressed as follows:148
M
fD=ZΓwfD·n
D
|{z }
:=m
fD
dΓ;M
fS=ZΓwfS·n
S
| {z }
:=m
fS
dΓ, (16)
where n
Dand n
Sdenote the outward-oriented normal vectors from BDand BS, respectively. From Eq. (16),149
mass continuity (M
fD+M
fS=m
fD+m
fS=0) yields the following transmissibility condition:150
wfD·n
D+wfS·n
S= (wfSwfD)·n=0 on Γ, (17)
where we take n=n
S=n
Dfor notational convenience (Fig. 1). Here, Eq. (16) implies that the normal151
component of the fluid velocities (wfSand wfD) should be identical in order to guarantee that the exchange152
of fluid mass between BSand BDis conservative.153
The second condition is the force equilibrium at the interface Γ. From each subsystem, total forces154
acting on the interface (F
fDand FfS) may be written as,155
F
fD=ZΓpfDn
|{z}
:=t
fD
dΓ;F
fS=ZΓσfS·n
|{z }
:=t
fS
dΓ, (18)
where t
fDand t
fSindicate the tractions at the interface. The force equilibrium requires F
fD+F
fS=156
t
fD+t
fS=0, implying that the normal and shear components should be balanced at the same time. By157
decomposing the traction vectors as,158
t
i= (t
i·n)n+
2
j=1
(t
i·m
j)m
j;i={fD,fS}, (19)
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PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
6
where m
1and m
2are the interfacial tangent vectors, we get two more transmissibility conditions that
describe normal and shear force equilibrium, respectively:
t
fS·n+pfD=0 on Γ, (20)
t
fS·m
j+µf
αSD
k(wfSwfD)·m
j=0 on Γ. (21)
Eq. (21) is the Beavers-Joseph-Saffman condition [Beavers and Joseph,1967,Saffman,1971,Layton et al.,159
2002,Arbogast and Brunson,2007]. This idealized condition relates the slip velocity and the shear stress160
through the dimensionless slippage coefficient αSD, which depends on the microstructural attributes of161
the interfaces, such as surface roughness, irregular patterns, as well as the flow velocity [Beavers and162
Joseph,1967,Terzis et al.,2019,Guo et al.,2020]. The validity and limitations of the Beavers-Joseph-Saffman163
condition are documented in a number of literature such as Auriault [2010], Mikelic and J¨
ager [2000],164
Monchiet et al. [2019] and will not repeated here. Possible extensions of the interface conditions to turbulent165
and multiphase flows are an active research area that is clearly out of the scope of this study but will be166
considered in the future.167
3 The phase field Biot-Stokes model with evolving fractures168
This section introduces the mathematical model that uses smooth implicit function, i.e., the phase field, to169
approximate evolving sharp interfaces due to damage. We first review the general procedure that employs170
an implicit function to approximate sharp interfaces (Section 3.1) shown in Fig. 2. Since the phase field171
is a smooth representation of the Heaviside function, we derive the corresponding mathematical model172
that approximates interfacial transmissibility conditions suitable for the diffuse representation of the inter-173
face. To capture crack growth according to the Griffith’s theory, we adopt the classical variational fracture174
model to allow crack growth represented by the evolution of the phase field defined over the spatial do-175
main (Section 3.2). These techniques are then applied into the derivation shown in Section 3.3 in which176
a mathematical model to capture the hydromechanical coupling of pore fluid flows in both the host ma-177
trix and evolving interfaces in brittle porous media. The resultant model does not require locally defined178
enrichment function or remeshing and can be implemented in a standard finite element or finite elemen-179
t/volume solver.180
0 0.2 0.4 0.6 0.8 1
0
0.5
1
0
0.5
1
BD
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BS
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BD
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undamaged
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damaged
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undamaged
<latexit sha1_base64="63c5nUOtBc/j946gm+AgK1ZYFM8=">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</latexit>
Fig. 2: Diffuse representation of the interface where exemplary 1D domain consists of BSin x/L[0.4, 0.6]
sandwiched between undamaged porous matrix BD.
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3.1 Diffuse interface approximation181
This study employs a diffuse approximation for the sharp interface Γby introducing a phase field variable182
d[0, 1]which varies smoothly from 0 in BDto 1 in BS. Specifically, we approximate the interfacial area183
AΓas AΓ
d, which can be expressed in terms of volume integration of surface density functional Γ
d(d,d)184
over B=BDBS[Miehe et al.,2010,Borden et al.,2012,Suh and Sun,2019,Suh et al.,2020]:185
AΓAΓ
d=ZB
Γ
d(d,d)dV. (22)
Here, the size of diffusive zone [i.e., transition zone where d(0, 1)] is controlled by the regularization186
length scale parameter lsuch that AΓ
dΓ-converge to AΓ[Mumford and Shah,1989], i.e.,187
AΓ=lim
l0AΓ
d. (23)
Based on this approach, phase field dand its gradient dcan be regarded as smooth approximations
of the Heaviside function HΓand the Dirac delta function δΓ, respectively [Stoter et al.,2017,Suh and
Sun,2020]. Therefore, the volume integrals of an arbitrary function ˜
Gover BDand BScan respectively be
approximated as,
ZBD
˜
G dV =ZB
˜
G(1HΓ)dV =lim
l0ZB
˜
G(1d)dV ZB
˜
G(1d)dV, (24)
ZBS
˜
G dV =ZB
˜
GHΓdV =lim
l0ZB
˜
Gd dV ZB
˜
Gd dV. (25)
Similarly, the surface integral of the function ˜
Galong the sharp interface Γcan be approximated as,188
ZΓ
˜
G dΓ=ZΓ
˜
GδΓdV =lim
l0ZB
˜
GkdkdV ZB
˜
GkdkdV, (26)
and we also approximate the normal vector nas,189
n≈ − d
kdk. (27)
3.2 Crack growth approximated by evolving phase field190
For completeness, this section reviews the phase field model for brittle fracture. We consider the following191
surface density functional, which is widely used in modeling brittle or quasi-brittle fracture [Bourdin et al.,192
2008,Miehe et al.,2010,Borden et al.,2012,Bryant and Sun,2018,Suh et al.,2020] that possesses quadratic193
local dissipation function:194
Γ
d(d,d) = d2
2l+l
2(d· d). (28)
At this point, we highlight that the evolution of the phase field (i.e., propagation of cavities or cracks) is195
a mechanical process driven by the effective stress σ0. In other words, we assume that the solid skeleton196
is completely damaged in the liquefied zone BS, whereas in BD, the solid skeleton remains undamaged.197
We thus omit the terms that are unrelated to the deformation and fracture in this section. Having critical198
energy Gcthat is required to create new free surfaces, potential energy density ψreads,199
ψ=g(d)ψ+
e(ε) + ψ
e(ε)
| {z }
ψbulk(ε,d)
+GcΓ
d(d,d), (29)
where ψbulk(ε,d)is the degrading elastic bulk energy and g(d)=(1d)2is the degradation function
that induces energy dissipation. Following Amor et al. [2009], we adopt additive decomposition scheme
that splits the elastic energy ψeinto compressive (ψ
e), and tensile and deviatoric (ψ+
e) modes, where we
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only degrade ψ+
ein order to avoid crack propagation under compression [Na and Sun,2018,Bilgen and
Weinberg,2019,Heider and Sun,2020], i.e.,
ψ+
e=1
2KDεvolE2
++µ(εdev :εdev), (30)
ψ
e=1
2KDεvolE2
, (31)
where K=λ+2µ/3 is the bulk modulus of the porous matrix, and hi±= (•±|•|)/2 is the Macaulay200
bracket operator. In this case, the effective stress tensor σ0can also be decomposed as follows:201
σ0=g(d)σ0+
0+σ0−
0, (32)
where σ
0=∂ψ±
e/εis the fictitious undamaged effective stress, in which we previously assumed σ0
0to202
be linear elastic [Eq. (7)].203
Based on the fundamental lemma of calculus of variations, the damage evolution equation can be ob-204
tained by seeking the stationary point where the functional derivative of Eq. (29) with respect to dvanishes,205
i.e.,206
∂ψ
d∇· ψ
d=0, (33)
where:207
∂ψ
d=g0(d)ψ+
e+Gc
ld;∇· ∂ψ
d=Gcl2d. (34)
Here, the superposed prime denotes derivative with respect to dand 2() = ∇·()is the Laplacian208
operator. Furthermore, by following the treatment used in Miehe et al. [2010], we introduce a history func-209
tion Hwhich is the pseudo-temporal maximum of the positive energy density (ψ+
e) in order to ensure210
crack irreversibility constraint:211
H=max
τ[0,t]ψ+
e. (35)
By replacing ψ+
ein Eq. (34) with H, Eq. (33) finally yields the following phase field equation that governs212
the evolution of the interface:213
g0(d)H+Gc
l(dl22d) = 0 in B. (36)
Note that we can obtain the diffuse representation of the interface by solving Eq. (36), as shown in Fig. 2.214
In this study, we leverage the phase field not only as an indicator function for the location of cracks but215
also for other defects such as cavities or geometrically complicated voids that does not fit for computational216
homogenization. This approach may efficiently couple the Stokes flow inside the vugs (BS) that interact217
with pore fluid in the intact porous matrix (BD) while both regions are evolving due to the crack growth. A218
major advantage of this work is that free flow inside the fracture is explicitly replicated and hence there is219
no need to introduce permeability enhancement models (e.g., cubic law) [Witherspoon et al.,1980,Konzuk220
and Kueper,2004,Jin and Arson,2020]. This explicit treatment enables the simulations to remain physical221
even in the situations (e.g., high Reynolds number, rough surface, aperture variation) where the validity of222
the cubic law is questioned [Witherspoon et al.,1980,Miehe and Mauthe,2016,Heider and Markert,2017,223
Wang and Sun,2017,Sun et al.,2017,Choo and Sun,2018,Chukwudozie et al.,2019,Wang and Sun,2019].224
Verification and experimental validation of the phase field fracture model for brittle solid has been well225
documented in the literature. For brevity, similar studies are not provided in this paper. Interested readers226
may refer to, for instance, such as Nguyen et al. [2016] and Pham et al. [2017].227
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3.3 Variational formulation of the phase field Biot-Stokes model228
We present a immersed phase field Biot-Stokes model designed to simulate the coupled hydro-mechanical229
behaviors of flow of vuggy porous media with evolving fractures in the brittle regime. This section omits230
the gravitational effects for brevity (i.e., g=0).231
The model problem with the sharp interface (Section 2) in which the system possesses two distinct
boundaries BDand BSthat can both be decomposed into Dirichlet (Bu
D,Bp
D,Bw
Sand Bp
S) and Neu-
mann (Bt
D,Bq
D,Bt
Sand Bq
S) boundaries satisfying,
BD=Bu
DBt
D=Bp
DBq
D;=Bu
DBt
D=Bp
DBq
D, (37)
BS=Bw
SBt
S=Bp
SBq
S;=Bw
SBt
S=Bp
SBq
S, (38)
where the union of BSand BDis Band the boundary domain follows the same treatment. Here we cap-
ture the transition of the constitutive responses of the solid constituent in the intact and liquefied states
through a partition of unity argument in the local constitutive responses. As such, we adopt only one solid
constituent and the balance of linear momentum equations in the sub-domains BDand BS[Eqs (5) and
(11)] are combined into one set of equations over the domains B. The governing equations for the model
problem are summarized as follows:
(1d)h∇·(σ0pfDI)i+d(∇· σs)=0in B,
∇·vs+∇· wfD=0 in BD,
∇·σfS+ρfSg=0in BS,
∇·vs+∇· wfS=0 in BS,
g0(d)H+Gc
l(dl22d) = 0 in B,
(wfSwfD)·n=0 on Γ,
t
fS·n+pfD=0 on Γ,
t
fS·m
j+µf
αSD
k(wfSwfD)·m
j=0 on Γ,
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
where the natural and essential boundary conditions are not included for brevity. Following the standard
weighted residual procedure, we multiply Eqs. (39)-(43) with proper weight functions (ηs,ξfD,ηfS,ξfS
and ζ), and integrating over their corresponding domain. The resultant weighted-residual statement reads
[Badia et al.,2009,Stoter et al.,2017],
ZBηs:(σ0pfDI)(1d)dV +ZBηs:σsd dV ZBt
D
ηs·ˆ
tDdΓ=0, (47)
ZBD
ξfD(∇· ˙us)dV ZBDξfD·wfDdV ZBq
D
ξfDˆ
qDdΓ+ZΓξfDwfD·(n)
| {z }
=m
fD
dΓ=0, (48)
ZBSηfS:σfSdV ZBt
S
ηfS·ˆ
tSdΓZΓηfS·σfS·n
|{z }
=t
fS
dΓ=0, (49)
ZBS
ξfS(∇· ˙us)dV +ZBS
ξfS(∇·wfS)dV =0, (50)
ZBζg0(d)H+Gc
lddV +ZBζ· Gcld dV =0, (51)
where ˆ
tDand ˆ
qDis the prescribed traction and flux at the porous matrix, respectively; and ˆ
tSis the fluid232
traction. Then, we directly impose the interfacial transmissibility conditions [Eqs. (44)-(46)] into the field233
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equations Eqs. (48) and (50). Due to the fluid mass continuity (i.e., m
fD+m
fS=0), the fourth term on the234
left hand side of Eq. (48) becomes:235
ZΓξfDwfD·(n)dΓ=ZΓξfDwfS·ndΓ, (52)
while normal and shear force equilibrium (i.e., t
fD+t
fS=0) can be imposed at the third term on the left236
hand side of Eq. (49), i.e.,237
ZΓηfS·σfS·ndΓ=ZΓηfS·(pfDn)dΓ+
2
j=1ZΓηfS·µf
αSD
k(wfSwfD)·m
jm
jdΓ. (53)
Finally, we apply Eqs. (24)-(27) in order to convert subdomain integrals (BDand BS) into integral over238
the entire domain (B), and to also transform the interface equations [Eqs. (52)-(53)] into a set of immersed239
boundary conditions. As a result, we get the weak statements for a phase field Biot-Stokes model, which is240
to: find {us,pfD,wfS,pfS,d}such that for all {ηs,ξfD,ηfS,ξfS,ζ},241
Gu=Gp
D=Gw
S=Gp
S=Gd=0, (54)
where:
Gu=ZBηs:(σ0pfDI)(1d)dV +ZBηs:σsd dV ZBt
D
ηs·ˆ
tDdΓ, (55)
Gp
D=ZBξfD(∇· ˙us)(1d)dV ZBξfD·wfD(1d)dV +ZBξfD(wfS· d)dV ZBq
D
ξfDˆ
qDdΓ, (56)
Gw
S=ZBηfS:σfSd dV ZBηfS·(pfDd)dV
+
2
j=1ZBηfS·µf
αSD
k(wfSwfD)·m
jm
jkdkdV ZBt
S
ηfS·ˆ
tSdΓ, (57)
Gp
S=ZBξfS(∇· ˙us)d dV +ZBξfS(∇· wfS)d dV, (58)
Gd=ZBζg0(d)H+Gc
lddV +ZBζ· Gcld dV. (59)
Here, as pointed out in Stoter et al. [2017], the Γ-convergence ensures that the immersed boundary condi-242
tions imposed in Eqs. (56)-(57) are consistent with the interface conditions [Eqs. (44)-(46)] if l0, which243
in turn confirms the mass conservation and force equilibrium for the entire system B.244
4 Numerical examples245
This section highlights the capacities of the immersed phase field model to capture the hydromechanical246
interactions among the pore fluid in the cavities, cracks and the homogenized pore space and the host ma-247
trix in two numerical experiments. Our focus is on modeling the problems that involve the mechanically-248
driven pore fluid migration due to deformation and crack growth inside the solid skeleton. The first ex-249
ample simulates the consolidation process of the porous material that contains a semi-circular cavity at250
the bottom that serves as a pore fluid outlet, while the second problem showcases the fracture-induced251
Stokes-Darcy flow in vuggy porous medium.252
In order to solve Eqs. (55)-(59) numerically, we adopt standard finite element method where the so-253
lution procedure is based on the operator-split [Miehe et al.,2010,Heister et al.,2015,Suh et al.,2020]254
that successively updates the field variables. In other words, the phase field dis updated first by solving255
Gd=0, while all other field variables are held fixed, and the solver then advances the remaining variables256
by solving {Gu,Gp
D,Gw
S,Gp
S}T=0. The implementation of our proposed model including finite element257
discretization and the solution scheme relies on the finite element package FEniCS [Logg et al.,2012a,b,258
Alnæs et al.,2015]. It is noted that there exists multiple different strategies to solve the same system of259
equations. Since the exploration of different solution schemes are out of the scope of this study, we omit260
the details for the implementation for brevity.261
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4.1 Consolidation of porous matrix with a semi-circular cavity262
We first simulate a consolidation problem, which has always been one of the key problems in geotechnical263
engineering. While classical consolidation problem considers time-dependent water expulsion from the264
homogeneous porous material, as illustrated in Fig. 3, this numerical example explores the case where the265
system includes a cavity at the bottom that serves as a pore fluid outlet. This specific setting is designed to266
simulate mechanically driven Stokes-Darcy flow without significant changes in microstructural attributes.267
1m
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2m
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0.2m
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ˆ
tD
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ˆpfS=0
<latexit sha1_base64="7PextX8SXjo9pAcQCXXe/o9gosY=">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</latexit>
Fig. 3: Schematic of geometry and boundary conditions for the consolidation problem.
The problem domain is a water-saturated 1 m ×2 m sized rectangular porous matrix (BD) that contains268
a semi-circular cavity (BS) whose diameter is 0.2 m. We prescribe a 1 kPa compressive mechanical traction269
at the top, while zero pressure boundary is imposed at the bottom of the cavity so that the time-dependent270
dissipation of pore pressure can be observed. The material parameters for this example are chosen as271
follows. Intrinsic mass densities for the solid and fluid: ρs=2700 kg/m3and ρf=1000 kg/m3; Young’s272
modulus and Poisson’s ratio of the solid skeleton: E=100 MPa and ν=0.25; initial permeability and273
initial porosity of the matrix: k0=1.0 ×108m2and φ0=0.4; dynamic viscosity of the fluid phase:274
µf=1.0 ×103Pa·sec; slippage coefficient αSD =0; and regularization length for the interface l=0.002275
m. Furthermore, we assume that solid constituent remains intact in BDthroughout the simulation while276
free fluid inside the cavity has zero particle concentration (i.e., c=0).277
Fig. 4shows the spatial distributions for the prime variables at t=1.0 ×103sec. Here, we compute278
fluid pressure and relative fluid velocity for the entire system as: pf= (1d)pfD+dpfSand wf= (1279
d)wfD+dwfS, respectively, since we have separate degrees of freedoms for pore and free fluids residing280
in each regions BDand BS. The results imply that applied mechanical load ˆ
tDat t=0 builds up the pore281
pressure which in turn affects the pore fluid to migrate towards the cavity. Furthermore, free fluid inside282
the cavity tends to exhibit higher velocity and lower pressure compared to those of pore fluid, because of283
different constitutive relations (i.e., Stokes equation and Darcy’s law) in each region. As illustrated in Fig.284
5, we also investigate the time-dependent response of the system that clearly describes the consolidation285
process and at the same time highlights the continuous pressure and velocity fields along y-axis (i.e., from286
the center point of the cavity to the top-central point of the external boundary). At t=0, the entire load287
is taken by the incompressible pore water which triggers the fluid flow inside the medium. This fluid288
flow is accompanied by a dissipation of pore pressure over time and an increase in the compression of the289
entire system, which is consistent with previous studies on homogeneous materials [Li et al.,2004,White290
and Borja,2008,Kim et al.,2009,Wang and Sun,2016]. In addition, the continuous pressure and velocity291
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
12
kusk
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kwfk
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pf
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d
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(a)
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(b)
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(c)
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(d)
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Fig. 4: Spatial distributions of the (a) phase field d; (b) solid displacement kusk[m]; (c) fluid pressure pf=
(1d)pfD+dpfS[Pa]; and (d) relative fluid velocity kwfk=k(1d)wfD+dwfSk[m/s], at t=1.0 ×103
sec.
profiles imply that our model is capable of imposing mass continuity and force equilibrium at the interface292
as a set of immersed boundary conditions, which confirms the validity of the model.293
4.2 Comparison studies on fracture-induced flow in vuggy porous media294
In the second set of experiment, we conduct numerical simulations within two different types of domains295
that possess horizontal edge crack (Fig. 6): one explicitly captures the geometry of the large cavities in296
the porous media; another one captures the influence of the cavities by increasing the porosity of the297
homogenized effective medium. While the former approach adopt a more explicit representation of the298
pore geometry and hence may provide more detailed information on the interactions between the vugs299
and the propagating cracks, the latter approach could be numerically more efficient. Our objective is to300
demonstrate, quantitatively, the difference of the two approaches such that a fuller picture on the trade-off301
between computational efficiency, accuracy and precision of the predictions ca be established.302
4.2.1 Modeling vuggy porous media303
As illustrated in Fig. 6(a), we first consider a domain that consists of porous matrix with explicitly modeled304
cavities. Our first representation consists of total nine cavities with different major and minor radii (Table 1)305
that share the same aspect ratio of 2:1 and are tilted by 45, such that the volume fraction of the cavities θcav
306
is 0.056. Here, we assume that the solid skeleton inside the cavities are completely damaged (i.e., d=1),307
while the porous matrix initially remains completely undamaged (i.e., d=0). The material properties for308
this case is chosen as follows: ρs=2700 kg/m3,ρf=1000 kg/m3,E=20 GPa, ν=0.2, k0=1.0 ×1012
309
m2,µf=1.0 ×103Pa·sec, αSD =0.1, Gc=20 J/m2, and l=0.125 ×103m. In addition, the initial310
particle concentration is chosen as c0=0.6 and its upper bound as cmax =0.7, in order to mimic the311
mudflow inside the cracks or cavities [O’Brien and Julien,1988,Iverson,1997].312
In contrast, our second domain in Fig. 6(b) is a homogenized representation of Fig. 6(a), where all the313
cavities are considered as a part of matrix pores. In this case, the porosity of the homogenized medium314
is determined as: φhom = (1θcav)φ0+θcav =0.433. It is noted the correct homogenized effective prop-315
erties often depend on the geometry of the vugs or inclusions, which can be determined from computed316
tomographic images or directly obtained from the experiment [Sun et al.,2011a,b,Kim et al.,2016,Lee317
et al.,2017]. Since the micro-structural attributes are not always available, this study adopts an alternative318
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
13
0 0.5 1 1.5 2
-1.5
-1
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Consolidation
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(a)
0 0.5 1 1.5 2
0
200
400
600
800
1000
Consolidation
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(b)
0 0.5 1 1.5 2
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
(c)
Fig. 5: Response of the saturated Biot-Stokes system under 1 kPa consolidation pressure. (a) Solid displace-
ment; (b) Fluid pressure; and (c) Fluid velocity.
Index 1 2 3 4 5 6 7 8 9
Major radius ra[mm] 0.400 0.600 0.500 0.500 0.820 0.800 0.580 0.700 0.650
Minor radius rb[mm] 0.200 0.300 0.250 0.250 0.410 0.400 0.290 0.350 0.325
Table 1: The major and minor radii of the explicitly modeled elliptical cavities in Fig. 6(a).
approach where the effective material properties are determined by using the equivalent inclusion method319
[Hashin,1960,Zimmerman,1991,Ramakrishnan and Arunachalam,1993]. Following Ramakrishnan and320
Arunachalam [1993] and by assuming that the matrix shares the same material properties of those chosen321
for Fig. 6(a), the effective bulk modulus (Khom) and shear modulus (µhom ) for the homogenized represen-322
tation [Fig. 6(b)] are determined as follows:323
Khom =K(1θcav)2
1+1+ν
2(12ν)θcav
;µhom =µ(1θcav)2
1+1119ν
4(1+ν)θcav
, (60)
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
14
0.01 m
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0.002 m
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Tension
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Shear
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45
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0.01 m
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ˆpfD=0
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ˆpfD=0
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0=0.4
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6
5
4
7
2
9
3
8
1
(a)
0.01 m
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0.002 m
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Tension
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Shear
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0.01 m
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ˆpfD=0
<latexit sha1_base64="HJjwenlgjVOqNQnc34GxaemsXc4=">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</latexit>
ˆpfD=0
<latexit sha1_base64="HJjwenlgjVOqNQnc34GxaemsXc4=">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</latexit>
hom =0.433
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(b)
Fig. 6: Schematic of geometry and boundary conditions for the fracture problem. (a) The domain with
explicitly modeled cavities; and (b) its homogenized counterpart.
so that the effective Young’s modulus Ehom =16.50 GPa and Poisson’s ratio νhom =0.206. Assuming324
that the inclusion permeability is much higher than the matrix permeability, we approximate the effective325
permeability (khom) by following Markov et al. [2010] which is obtained based on the Maxwell’s formula,326
i.e.,327
khom =k0(1+2θcav)
1θcav
=1.18 ×1012 [m2]. (61)
In addition, since all the cavities in Fig. 6(a) are completely isolated, we adopt the following effective328
critical energy Gc,hom proposed by Jelitto and Schneider [2018] for the homogenized representation, which329
depends on the volume fraction of the cavities, i.e.,330
Gc,hom =Gc(1θ2/3
cav ) = 17.07 [J/m2]. (62)
4.2.2 Mechanically driven fracture-induced flow331
As illustrated in Fig. 6, we conduct two different types of simulations within each domain: the tension332
tests with prescribed vertical displacement rate of 0.01 ×103m/s, and the shear tests with horizontal333
displacement rate of 0.01 ×103m/s. In both tension and shear tests, the displacements are prescribed at334
the upper boundary, whereas the bottom part of the domain is held fixed. We also impose hydraulically335
insulated boundary conditions for the left and right boundaries while we permit water intake from the336
upper and lower boundaries by imposing ˆ
pfD=0.337
Fig. 7illustrates the evolution of the phase field for both tension and shear tests in a computational338
domain where the cavities are explicitly modeled, compared with the crack trajectories obtained from the339
homogenized domain. The domain without cavities exhibits the crack patterns that are similar to the results340
of previous studies on homogeneous solids [Miehe et al.,2010,Borden et al.,2012,Bryant and Sun,2018,341
Suh et al.,2020], while the domain with explicitly modeled cavities exhibit distinct crack patterns. More342
importantly, Fig. 8and Fig. 9reveals that neglecting the interaction between the cavity and crack in the343
homogenized model may lead to over-simplified global responses that lacks the distinctive characteristics344
of the cavity-crack coalescence.345
During the numerical experiments, the porous matrix initially undergoes linear elastic deformation346
until the crack nucleation takes place. At this point, since tensile loading directly influences the volume347
change of the material, both specimens under tensile load exhibit higher fluid influx at the top, compared348
to those measured from the shear tests. After the first peaks shown in Fig. 8(a) and Fig. 8(b), cracks start to349
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
15
Initial state
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Shear
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Tension
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t=0.346 sec
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t=0.360 sec
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t=0.476 sec
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t=0.498 sec
<latexit sha1_base64="8BumLWS47aWnqKg/iwkhIsUZWiw=">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</latexit>
t=0.782 sec
<latexit sha1_base64="B1m+1Jk2kW9RhL07qX4StjbHFJ0=">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</latexit>
t=0.852 sec
<latexit sha1_base64="SNogm8ruyuAAX3kS4mjTb4RX/NQ=">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</latexit>
t=0.974 sec
<latexit sha1_base64="C4CDc7pM2HN2Th5i55lI6rRuyso=">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</latexit>
t=1.116 sec
<latexit sha1_base64="sLuTzFRGkmyAlKEMr0OOZQeEoT0=">AAAEe3ichVNbb9MwFPbWAqPcOhBPvATyghCqGlQuL0gTTIIXYCC6VZpL5Tgnnakvke10TJZ/DK/wi/gxSDhtJ3UpAUs5OvrOdz4755IWnBnb7//a2m61L12+snO1c+36jZu3uru3D40qNYUhVVzpUUoMcCZhaJnlMCo0EJFyOEpnr6v40Ry0YUp+tmcFjAWZSpYzSmyAJt27NnoZJb0keRZhC9+siwxQP+nG/V5/caJNJ1k5MVqdg8lu6z3OFC0FSEs5McYRbRnl4KMOLg0UhM7IFI5Lm78YOyaL0oKkfj3miDCC2JMNMFfSmg3UnIn0IlhVi8lpjVpJWqX4X2Bt8hpqmYAalHNFbIAknFIlBJGZC4WSRml/nIwdDm6poZJzOFU8q96luIsT72tZX0tRLHN4qoM6vr9yojiJsD7Hlk4tl3DQdplcdWnRd6ch84uLOngfQtU1vAvP+FCAJjaE8VSTQMCSpJw0ccJz12lflvrO+EbVjIV5OudjminbxHxTF66qFo0alffXldcT/nlL+APvKtMUn52GeDANcRuuDF/T38Lcu8o0qU9lUA9m0S97AioMg9Ni5t2nMBW6aqSG9VbOw379f3LCBib1fdt0Dp/0kkFv8HEQ771a7eIOuoceoIcoQc/RHnqLDtAQUeTQd/QD/Wz9bsftR+3HS+r21irnDrpw2k//AOIqkhw=</latexit>
Shear
<latexit sha1_base64="k7RNpDURj2QmlQnP01BS6mnsLz4=">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</latexit>
Tension
<latexit sha1_base64="pA1pUZQ0Hg00nD1uyFgzeoRaOMQ=">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</latexit>
Explicitly modeled cavities
<latexit sha1_base64="5mMxnQkyqtvG6iYKG0hTSEtojP0=">AAAEiHicjVNNb9NAEN22Bkr4SuHIxTQXTlGMIhVuVSmCC1AQaSPVIVqvJ+mS/bB212mjle/8Gq7wV/g3jBNXSh2MWMmj0Zt5b9Yzs0kmuHW93u+t7Z3g1u07u3db9+4/ePiovff41OrcMBgwLbQZJtSC4AoGjjsBw8wAlYmAs2T2uoyfzcFYrtUXt8hgJOlU8Qln1CE0bu/HDq6cf3OFxRjSF6HUKQhIQ0bn3HGwxbjd6XV7yxNuOlHldEh1TsZ7Ox/iVLNcgnJMUGs9NY4zAUXYinMLGWUzOoXz3E1ejjxXWe5AsWI95qm0krqLDXCilbMbqF3I5CZYto6raS21lHRai7/Axk5qqOMSatBEaOoQUnDJtJRUpR77p6w2xXk08jG6uYFSzseJFml5Ly18JyqKGutbLrMVRyQG1eNnlRN2ojA219jKqXGpAONW5HJ4yyXwBtJiWagVHwN23cB7vMbHDAx1GI6nhmJCrGgiaFMOXnc97etK39uiUTXluFzX+TFLtWvKfFsXLrsWDhuVj9eV1wn/rIJ/UPjSNMVnlxhH0xB3WBK/pr+FeeFL06Q+VaiOZjkvdwEal8EbOSv8Z9wKUw7SwPoo58D+Y3PwBUb197bpnL7oRv1u/1O/c3hUvcVd8pTsk+ckIgfkkLwjJ2RAGPlOfpCf5FfQCnrBQfBqlbq9VXGekBsnOPoDTNeY1Q==</latexit>
Homogenized
<latexit sha1_base64="kaSCryF4TaFekpoNwvl5C17ZHFY=">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</latexit>
Final configuration
<latexit sha1_base64="KGB8/INmHhh+neGJX4H2jwUW/Tg=">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</latexit>
Final configuration
<latexit sha1_base64="KGB8/INmHhh+neGJX4H2jwUW/Tg=">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</latexit>
Final configuration
<latexit sha1_base64="KGB8/INmHhh+neGJX4H2jwUW/Tg=">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</latexit>
Final configuration
<latexit sha1_base64="KGB8/INmHhh+neGJX4H2jwUW/Tg=">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</latexit>
Fig. 7: Evolution of the phase field of the specimens subjected to the numerical tension and shear tests.
initiate from the tips of the pre-existing flaw since they experience higher stress concentration compared350
to the matrix-cavity interface.351
In both tensile and shear experiments performed on the vuggy specimen, the crack nucleation increases352
the surface influx rate at the permeable boundaries as pore fluid starts to leak from the intact matrix to the353
damaged regions regardless of the spatial homogenization. The two numerical specimens, nevertheless,354
begin to behave differently when the cracks propagate towards the adjacent vugs and coalesce with each355
other in both tension and shear tests in the vuggy specimen (Fig. 7) [Qinami et al.,2019,Suh and Sun,356
2020]. These changes in surface influx cannot be replicated in the homogenized porous specimen as the357
homogenization takes away the possibility of simulating the coalescence between the cavity and the crack.358
After the coalescence of the cavity and the crack in the vuggy specimen, the reaction force in both359
cases increases again with lower influx rates until it reaches the second peak (i.e., where crack nucleation360
takes place at the matrix-cavity interface), and the crack eventually reaches both end of the specimens. This361
result implies that the existence of vugs or cavities has a profound impact on the material behavior that362
cannot be easily replicated in the homogenized effective medium. Consequently, either a more effective363
macroscopic theory or a suitable multiscale technique is needed to incorporate the cavity-crack interaction364
into the predictions.365
0123456
10-3
0
20
40
60
80
100
(a)
0 0.003 0.006 0.009 0.012 0.015
0
20
40
60
80
(b)
Fig. 8: Force-displacement curves obtained from (a) tension and (b) shear tests.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
16
0 0.1 0.2 0.3 0.4 0.5 0.6
0
1
2
3
410-8
(a)
0 0.3 0.6 0.9 1.2 1.5
0
2
4
6
810-8
(b)
Fig. 9: Fluid influx at the top surface over time measured from (a) tension and (b) shear tests.
Fig. 10 illustrates the pressure (pf) and x-directional velocity (wfS) fields from a domain with explicitly366
modeled cavities under tensile and shear loadings, at t=0.476 sec and t=1.012 sec, respectively, where367
cracks start to propagate from the cavities. Here, the superimposed arrows in Fig. 10 indicate the direction368
of the velocity vector wf= (1d)wfD+dwfS. In both cases, the leakage of pore fluid takes place towards369
the interconnected cracks and cavities at the middle, while free fluid in BStends to migrate towards the370
center, the region where large crack opening displacement occurs. However, it is worthy to note that the371
fluid flow occurs from the region that has negative pressure to the damaged zone where pfS0. Unlike372
previous studies that use the cubic law to predict the hydraulic responses of the flow conduit [Mauthe and373
Miehe,2017,Heider and Markert,2017,Wang and Sun,2017,Chukwudozie et al.,2019], the pore pressure374
distribution inside the void space is governed by the Stokes equation directly. This set of numerical exper-375
iments again highlight that our proposed model is capable of simulating fracture-cavity interaction with376
evolving interface, which may not be easily captured either by using hydraulic phase field fracture models377
or by adopting classical Biot-Stokes model with sharp interface.378
To assess the computational efficiency of the proposed model, we record the CPU time for both sim-379
ulations. A laptop with a Intel Core i9-9880H Processor CPU with 16 GB memory at 2667MHz (DDR4) is380
used to run both simulation on a single core. Both simulations are solved by the same Scalable Nonlinear381
Equation Solver (SNES) available in FEniCS. In the case where vuggy pores are explicitly modeled, the382
time taken to assemble the system of equation is 1.13 second and the averaged time taken to advance one383
time step with (on average) 5 Newton-Raphson iteration is 35.69 seconds. Meanwhile, in the homogenized384
case, it takes 1.17 second to assemble the system of equation and 33.34 seconds to advance one time step385
with also (on average) 5 Newton-Raphson iteration. In general, simulations with the explicitly captured386
vuggy pores require about 7% more CPU time to run the same simulation.387
Future work may consider flow with higher Reynold’s number suitable for the Navier-Stokes equation388
in the fluid domain. Such an extension is nevertheless out of the scope of the current study.389
5 Conclusion390
This article presents a new immersed phase field model that captures the hydro-mechanical coupling391
mechanisms in vuggy porous media where brittle cracks filled with water may coalescence with pores392
that trigger both redistribution of flow and macroscopic softening that cannot be captured without the393
Stokes-Darcy flow. By generalizing the phase field as an indicator of defects, we introduce a simple and394
unified treatment to handle the evolving geometries due to crack growths and the resultant changes of395
constitutive responses without the need of re-meshing or introduction of enrichment functions. By directly396
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
17
Shear
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Tension
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wfS|x
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pf
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wfS|x
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pf
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Fig. 10: Snapshots of the pressure pf= (1d)pfD+dp fS[Pa] and velocity wfS[m/s] fields obtained from
the tension (t=0.476 sec) and shear (t=1.012 sec) tests.
simulating the flow inside the cracks, we bypass the need of introducing phenomenological permeabil-397
ity enhancement model to replicate the flow conduit. This explicit approach can be advantageous over398
the embedded discontinuity approach when there is a substantial crack opening and a flow near the lo-399
cations with void-crack interaction where a homogenized pore pressure jump would not be sufficient to400
capture the pattern of the pore pressure field in the defects. Future work may include the extension of401
the proposed model to three-dimensional cases as well as extending the Stokes-Darcy flow model for the402
generalized Navier-Stokes-Darcy flow for injection and other problems with higher Reynolds numbers.403
6 Acknowledgments404
The suggestions of the two anonymous reviewers are gratefully acknowledged. The authors are thank-405
ful for the fruitful discussion with Professor Robert Zimmerman while the second author visited London406
in 2017. The first author is supported by the Earth Materials and Processes program from the US Army407
Research Office under grant contract W911NF-18-2-0306. The second author is supported by by the NSF408
CAREER grant from Mechanics of Materials and Structures program at National Science Foundation under409
grant contract CMMI-1846875, the Dynamic Materials and Interactions Program from the Air Force Office410
of Scientific Research under grant contracts FA9550-17-1-0169. These supports are grate- fully acknowl-411
edged. The views and conclusions contained in this document are those of the authors, and should not be412
interpreted as representing the official policies, either expressed or implied, of the sponsors, including the413
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
18
Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and414
distribute reprints for Government purposes notwithstanding any copyright notation herein.415
7 Data availability416
The data that support the findings of this study are available from the corresponding author upon request.417
References418
Martin Alnæs, Jan Blechta, Johan Hake, August Johansson, Benjamin Kehlet, Anders Logg, Chris Richard-419
son, Johannes Ring, Marie E Rognes, and Garth N Wells. The fenics project version 1.5. Archive of420
Numerical Software, 3(100), 2015.421
Hanen Amor, Jean-Jacques Marigo, and Corrado Maurini. Regularized formulation of the variational brit-422
tle fracture with unilateral contact: Numerical experiments. Journal of the Mechanics and Physics of Solids,423
57(8):1209–1229, 2009.424
Todd Arbogast and Dana S Brunson. A computational method for approximating a darcy–stokes system425
governing a vuggy porous medium. Computational geosciences, 11(3):207–218, 2007.426
Todd Arbogast and Heather L Lehr. Homogenization of a darcy–stokes system modeling vuggy porous427
media. Computational Geosciences, 10(3):291–302, 2006.428
Chlo´
e Arson and Jean-Michel Pereira. Influence of damage on pore size distribution and permeability of429
rocks. International Journal for Numerical and Analytical Methods in Geomechanics, 37(8):810–831, 2013.430
Jean-Louis Auriault. About the beavers and joseph boundary condition. Transport in porous media, 83(2):431
257–266, 2010.432
Santiago Badia, Annalisa Quaini, and Alfio Quarteroni. Coupling biot and navier–stokes equations for433
modelling fluid–poroelastic media interaction. Journal of Computational Physics, 228(21):7986–8014, 2009.434
Gordon S Beavers and Daniel D Joseph. Boundary conditions at a naturally permeable wall. Journal of fluid435
mechanics, 30(1):197–207, 1967.436
EA Bergkamp, CV Verhoosel, JJC Remmers, and DMJ Smeulders. A staggered finite element procedure for437
the coupled stokes-biot system with fluid entry resistance. Computational Geosciences, pages 1–26, 2020.438
James G Berryman and Herbert F Wang. The elastic coefficients of double-porosity models for fluid trans-439
port in jointed rock. Journal of Geophysical Research: Solid Earth, 100(B12):24611–24627, 1995.440
Carola Bilgen and Kerstin Weinberg. On the crack-driving force of phase-field models in linearized and441
finite elasticity. Computer Methods in Applied Mechanics and Engineering, 353:348–372, 2019.442
Martin J Blunt, Matthew D Jackson, Mohammad Piri, and Per H Valvatne. Detailed physics, predictive443
capabilities and macroscopic consequences for pore-network models of multiphase flow. Advances in444
Water Resources, 25(8-12):1069–1089, 2002.445
Michael J Borden, Clemens V Verhoosel, Michael A Scott, Thomas JR Hughes, and Chad M Landis. A446
phase-field description of dynamic brittle fracture. Computer Methods in Applied Mechanics and Engineer-447
ing, 217:77–95, 2012.448
Ronaldo I Borja. On the mechanical energy and effective stress in saturated and unsaturated porous con-449
tinua. International Journal of Solids and Structures, 43(6):1764–1786, 2006.450
Ronaldo I Borja and Enrique Alarc´
on. A mathematical framework for finite strain elastoplastic consol-451
idation part 1: Balance laws, variational formulation, and linearization. Computer Methods in Applied452
Mechanics and Engineering, 122(1-2):145–171, 1995.453
Blaise Bourdin, Gilles A Francfort, and Jean-Jacques Marigo. The variational approach to fracture. Journal454
of elasticity, 91(1-3):5–148, 2008.455
Eric C Bryant and WaiChing Sun. A mixed-mode phase field fracture model in anisotropic rocks with456
consistent kinematics. Computer Methods in Applied Mechanics and Engineering, 342:561–584, 2018.457
Robert P Chapuis and Michel Aubertin. On the use of the kozeny carman equation to predict the hydraulic458
conductivity of soils. Canadian Geotechnical Journal, 40(3):616–628, 2003.459
Nian-Sheng Cheng and Adrian Wing-Keung Law. Exponential formula for computing effective viscosity.460
Powder Technology, 129(1-3):156–160, 2003.461
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
19
Jinhyun Choo and Ronaldo I Borja. Stabilized mixed finite elements for deformable porous media with462
double porosity. Computer Methods in Applied Mechanics and Engineering, 293:131–154, 2015.463
Jinhyun Choo and WaiChing Sun. Cracking and damage from crystallization in pores: Coupled chemo-464
hydro-mechanics and phase-field modeling. Computer Methods in Applied Mechanics and Engineering, 335:465
347–379, 2018.466
Jinhyun Choo, Joshua A White, and Ronaldo I Borja. Hydromechanical modeling of unsaturated flow in467
double porosity media. International Journal of Geomechanics, 16(6):D4016002, 2016.468
Chukwudi Chukwudozie, Blaise Bourdin, and Keita Yoshioka. A variational phase-field model for hy-469
draulic fracturing in porous media. Computer Methods in Applied Mechanics and Engineering, 347:957–982,470
2019.471
Holger Class, Rainer Helmig, and Peter Bastian. Numerical simulation of non-isothermal multiphase mul-472
ticomponent processes in porous media.: 1. an efficient solution technique. Advances in Water Resources,473
25(5):533–550, 2002.474
Antonio Costa. Permeability-porosity relationship: A reexamination of the kozeny-carman equation based475
on a fractal pore-space geometry assumption. Geophysical research letters, 33(2), 2006.476
Pengcheng Fu, Scott M Johnson, and Charles R Carrigan. An explicitly coupled hydro-geomechanical477
model for simulating hydraulic fracturing in arbitrary discrete fracture networks. International Journal478
for Numerical and Analytical Methods in Geomechanics, 37(14):2278–2300, 2013.479
Jamshid Ghaboussi and Ricardo Barbosa. Three-dimensional discrete element method for granular mate-480
rials. International Journal for Numerical and Analytical Methods in Geomechanics, 14(7):451–472, 1990.481
Malcomm Grant. Geothermal reservoir engineering. Elsevier, 2013.482
Chunsheng Guo, Yanwei Li, Xianbo Nian, Minghai Xu, Hai Liu, and Yaan Wang. Experimental study on483
the slip velocity of turbulent flow over and within porous media. Physics of Fluids, 32(1):015111, 2020.484
Zvi Hashin. The elastic moduli of heterogeneous materials. Technical report, HARVARD UNIV CAM-485
BRIDGE MA, 1960.486
Y Heider and B Markert. A phase-field modeling approach of hydraulic fracture in saturated porous media.487
Mechanics Research Communications, 80:38–46, 2017.488
Yousef Heider and WaiChing Sun. A phase field framework for capillary-induced fracture in unsaturated489
porous media: Drying-induced vs. hydraulic cracking. Computer Methods in Applied Mechanics and Engi-490
neering, 359:112647, 2020.491
Timo Heister, Mary F Wheeler, and Thomas Wick. A primal-dual active set method and predictor-corrector492
mesh adaptivity for computing fracture propagation using a phase-field approach. Computer Methods in493
Applied Mechanics and Engineering, 290:466–495, 2015.494
Jeffrey D Hyman, Satish Karra, Nataliia Makedonska, Carl W Gable, Scott L Painter, and Hari S495
Viswanathan. dfnworks: A discrete fracture network framework for modeling subsurface flow and496
transport. Computers & Geosciences, 84:10–19, 2015.497
Richard M Iverson. The physics of debris flows. Reviews of geophysics, 35(3):245–296, 1997.498
John Conrad Jaeger, Neville GW Cook, and Robert Zimmerman. Fundamentals of rock mechanics. John Wiley499
& Sons, 2009.500
Hans Jelitto and Gerold A Schneider. A geometric model for the fracture toughness of porous materials.501
Acta materialia, 151:443–453, 2018.502
Wencheng Jin and Chlo´
e Arson. Fluid-driven transition from damage to fracture in anisotropic porous503
media: a multi-scale xfem approach. Acta Geotechnica, 15(1):113–144, 2020.504
Ruben Juanes, EJ Spiteri, FM Orr Jr, and MJ Blunt. Impact of relative permeability hysteresis on geological505
co2 storage. Water resources research, 42(12), 2006.506
Dong Hun Kang, Tae Sup Yun, Kwang Yeom Kim, and Jaewon Jang. Effect of hydrate nucleation mecha-507
nisms and capillarity on permeability reduction in granular media. Geophysical Research Letters, 43(17):508
9018–9025, 2016.509
Jihoon Kim, Hamdi A Tchelepi, Ruben Juanes, et al. Stability, accuracy and efficiency of sequential meth-510
ods for coupled flow and geomechanics. In SPE reservoir simulation symposium. Society of Petroleum511
Engineers, 2009.512
Kwang Yeom Kim, Hyoung Suk Suh, Tae Sup Yun, Seong-Woo Moon, and Yong-Seok Seo. Effect of particle513
shape on the shear strength of fault gouge. Geosciences Journal, 20(3):351–359, 2016.514
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0035602
20
Julie S Konzuk and Bernard H Kueper. Evaluation of cubic law based models describing single-phase flow515
through a rough-walled fracture. Water Resources Research, 40(2), 2004.516
PV Lade and R De Boer. The concept of effective stress for soil, concrete and rock. Geotechnique, 47(1):61–78,517
1997.518
William J Layton, Friedhelm Schieweck, and Ivan Yotov. Coupling fluid flow with porous media flow.519
SIAM Journal on Numerical Analysis, 40(6):2195–2218, 2002.520
Changho Lee, Hyoung Suk Suh, Boyeong Yoon, and Tae Sup Yun. Particle shape effect on thermal conduc-521
tivity and shear wave velocity in sands. Acta Geotechnica, 12(3):615–625, 2017.522
Colin TO Leung and Robert W Zimmerman. Estimating the hydraulic conductivity of two-dimensional523
fracture networks using network geometric properties. Transport in porous media, 93(3):777–797, 2012.524
Chao Li, Ronaldo I Borja, and Richard A Regueiro. Dynamics of porous media at finite strain. Computer525
methods in applied mechanics and engineering, 193(36-38):3837–3870, 2004.526
Zhilin Li, Ming-Chih Lai, Xiaofei Peng, and Zhiyue Zhang. A least squares augmented immersed interface527
method for solving navier–stokes and darcy coupling equations. Computers & Fluids, 167:384–399, 2018.528
Chao Liu and Younane N Abousleiman. Shale dual-porosity dual-permeability poromechanical and chem-529
ical properties extracted from experimental pressure transmission tests. Journal of Engineering Mechanics,530
143(9):04017107, 2017.531
Anders Logg, Kent-Andre Mardal, and Garth Wells.