Accepted Manuscript
Numerical simulation of displacement characteristics of CO2 injected in pore-scale
porous media
Qianlin Zhu, Qianlong Zhou, Xiaochun Li
PII:
S1674-7755(15)00107-9
DOI:
10.1016/j.jrmge.2015.08.004
Reference:
JRMGE 194
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 18 May 2015
Revised Date:
18 August 2015
Accepted Date: 24 August 2015
Please cite this article as: Zhu Q, Zhou Q, Li X, Numerical simulation of displacement characteristics of
CO2 injected in pore-scale porous media, Journal of Rock Mechanics and Geotechnical Engineering
(2015), doi: 10.1016/j.jrmge.2015.08.004.
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Numerical simulation of displacement characteristics of CO2 injected in pore-scale porous
media
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Qianlin Zhua, b, *, Qianlong Zhoua, Xiaochun Lib
a. Key Laboratory of Coal-based CO2 Capture and Geological Storage, China University of Mining and Technology, Xuzhou, 221008, China
b. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071,
China
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Received 18 May 2015; received in revised form 18 August 2014; accepted 24 August 2015
Abstract: Pore structure of porous media, including pore size and topology, is rather complex. In immiscible two-phase displacement process, the capillary force
affected by pore size dominates the two-phase flow in the porous media, affecting displacement results. Direct observation of the flow patterns in the porous
media is difficult, and therefore knowledge about the two-phase displacement flow is insufficient. In this paper, a two-dimensional (2D) pore structure was
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extracted from a sandstone sample, and the flow process that CO2 displaces resident brine in the extracted pore structure was simulated using the Navier-Stokes
equation combined with the conservative level set method. The simulation results reveal that the pore throat is a crucial factor for determining CO2 displacement
process in the porous media. The two-phase meniscuses in each pore throat were in a self-adjusting process. In the displacement process, CO2 preferentially broke
through the maximum pore throat. Before breaking through the maximum pore throat, the pressure of CO2 continually increased, and the curvature and position
of two-phase interfaces in the other pore throats adjusted accordingly. Once the maximum pore throat was broken through by the CO2, the capillary force in the
other pore throats released accordingly; subsequently, the interfaces withdrew under the effect of capillary fore, preparing for breaking through the next pore
throat. Therefore, the two-phase displacement in CO2 injection is accompanied by the breaking through and adjusting of the two-phase interfaces.
Key words: level set method; displacement; pore-scale porous media; numerical simulation
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1. Introduction
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The increase of greenhouse gas emissions to the atmosphere during
the 20th century is considered to have contributed to global warming. A
reduction in the released rate of CO2 to the atmosphere is essential for
mitigating greenhouse effects. One way of achieving this is to inject
CO2 into deep formations (Yang et al., 2015). Such formations include
oil and gas reservoirs, saline aquifers and unminable coal seams, of
which the deep saline aquifers are widely distributed with the largest
CO2 storage capacity (IPCC, 2005). In China, the storage capacity of
deep saline aquifers accounts for 88% of the total geological storage
capacity (Li et al., 2009).
CO2 that is injected into a saline aquifer displaces the resident brine.
Different to single-phase flow, the flow process that CO2 displaces
brine is an immiscible two-phase flow, in which the capillary force
between the CO2 and brine plays an important role. The capillary force
is inversely proportional to interface curvature and directly proportional
to interface tension. Pore diameter of rock is commonly smaller than
millimeter-scale, and the rock is commonly hydrophilic. Therefore, the
capillary force is an important factor that may control the displacement
of the immiscible two-phase flow. However, tempo-spatial variation of
two-phase interfaces between the immiscible two-phase fluids is
difficult to be observed in porous media (Liu et al., 2012). In addition,
methods based on Darcy’s law are effective in investigating macroscale
two-phase flow but cannot reveal the flow dynamics in the immiscible
displacement (Christensen, 2006). Therefore, the dynamics of the
immiscible two-phase flow process needs to be understood from a pore*Corresponding author. Tel: +86 56 83883501; E-mail: zhuql@cumt.edu.cn
scale viewpoint, which is important but very complicated due to a large
number of factors influencing the flow, such as fluid density, fluid
viscosity, surface tension, flow rates, surface wettability, pore geometry
and medium heterogeneity (Pan, 2003; Liu et al., 2013).
Several approaches have been applied to simulate multiphase flow at
pore-scale in porous media, in which the pore-network model is a
conventional method. Based on two-dimensional (2D) model that is
simplified by ball and column connection, Lenormand and Touboul
(1988) studied the immiscible two-phase flow using the pore-network
model, and established a phase-diagram to identify the flow patterns. In
this phase-diagram, the flow patterns were identified by two
dimensionless parameters (i.e. capillary number and viscosity ratio) that
are determined by fluid viscosity, flow velocity, contact angle and
interfacial tension. The capillary number does not refer to the pore
structure. Therefore, the phase-diagram is difficult to be applied but
plays an instructive role in analysis of two-phase flow in many natural
porous media. Blunt et al. (2002) reviewed the pore-network simulation
in pore-scale flow. The pore-network model simplifies porous media by
connection model of pore body and pore throat with idealized
geometries, acquiring certain prediction effects. However, it is difficult
to extract a pore-network that topologically and geometrically
represents the complex pore geometry and physics, and therefore the
pore-network model cannot reveal the fluid dynamics mechanism
(Ramstad et al., 2012). In addition to pore-network model,
computational fluid dynamics methods, including the lattice Boltzmann
method (Liu et al., 2012), level set method (Gunde et al., 2010),
volume-of-fluid method (Raeini et al., 2012) and phase field method
(Anderson et al., 1998), have been developed in pore-scale simulation.
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Simulating pore-scale two-phase flow by computational fluid dynamics
methods can reveal flow characteristics.
In this paper, a computational fluid dynamics method with the level
set method is used to simulate movement characteristics of the twophase interfaces between the CO2 and resident brine in the displacement
process at pore-scale, and the displacement mechanisms are analyzed,
which may improve our knowledge on CO2 displacement flow in
porous media.
simulation that uses fluid velocity u, the level sets close to the zero
level set move with velocities that considerably distort the level set
function. In this case, the level set function will cease to be an exact
signed distance function after a few of time steps, affecting numerical
calculation at the next time step (Sussman and Puckett, 2000).
Therefore, a process called re-initialization is needed, which is used to
stop the level set calculation at some points in time and to rebuild a
level set function corresponding to the signed distance function. There
are several ways to accomplish this re-initialization. The most
2. Methodology
commonly used method among them is the method introduced by
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Sussman et al. (1994). Its virtue is that the level set function is re-
2.1. Geometric model
A sandstone sample is chosen from a shallow saline aquifer at a
depth of about 200 m, which is located about 41 km northeast of
Tongliao, Inner Mongolia, China. In this saline aquifer, a small-scale
CO2 shallow injection experiment is conducted (Zhu et al., 2015). The
pore structure (Fig. 1) is extracted from a microgram of sandstone slice.
initialized without explicitly finding the zero level set. This reinitialization method is used to solve Eq. (3) to steady state ( τ → ∞ ):
φτ =
φ0
φ + h2
2
0
(1 − ∇φ )
(3)
where φ0 is the level set function distribution calculated by Eq. (2)
before the re-initialization; h is a parameter of amplitude that is a
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function of grid size; the subscript τ denotes a partial derivative with
respect to the re-initialization time τ that is different from the time t.
Calculating Eq. (3) to steady state will provide a new value for φ with
a property that ∇φ = 1 . In addition, this re-initialization calculation
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ensures that the zero level set does not move (Sussman et al., 1994).
However, a main defect of this level set method is that the advection of
φ , including the re-initialization steps, is not done in a conservative
way, not even for divergence-free velocity fields. This implies that the
area bounded by the zero level set is not conserved (Sussman and
Puckett, 2000; Olsson and Kreiss, 2005). In order to ensure mass
conserved, Olsson and Kreiss (2005) and Olsson et al. (2007) modified
Fig. 1. Pore structure extracted from a sandstone slice.
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2.2. Numerical method
According to the continuum assumption of fluid, a crucial issue for
numerically simulating immiscible two-phase displacement is to track
the movement of interfaces in the two-phase flow. The level set method,
an interface capture method, was firstly proposed by Osher and Sethian
(1988). In this method, the flow field is covered by a level set function,
and the interfaces are expressed with contour lines or surfaces of the
level set function. The level set function moves with the flow just as the
interfaces do, and the level set function remains a smooth function
when it evolves. The level set function performs naturally for problems
with singularities, large distortion or interface breakup, etc. Unlike the
volume-of-fluid method, the level set function is a continuous function.
Therefore, there are more choices for numerical discretization. In
addition, some geometric parameters such as the interface normal
vector, tangent vector and curvature can be derived from simple
derivation.
v
In standard level set method, the level set function φ ( x , t ) is defined
to be a signed distance function:
v
v
v v
φ ( x , t ) = d ( x ) = min( x − xi )
(1)
xi∈i
v
where x is the point coordinates, t is the time, i is the interface domain,
v
v
φ ( x , t ) >0 on one side of the interface, φ ( x , t ) <0 on the other side,
v
v
and φ ( x , t ) =0 on the interface. The level set function φ ( x , t ) evolves
under advection velocity field as
φt + u ⋅ ∇φ = 0
(2)
where u is the velocity field, and therefore is the movement velocity on
the interface; the subscript t, which denotes the time, represents a
partial derivative with respect to time; ∇ is the Laplace operator. In a
this level set function by replacing the signed distance function with a
v
smeared-out Heaviside function φ% ( x , t ) :
v
[φ ( x , t ) < −ε ]
0

v
v
φ

πφ
φ% ( x , t ) =  1 +
+ 1 sin
[ − ε < φ ( x, t ) < ε ]
(4)
2
2
ε
2π
ε

v
[φ ( x , t ) > ε ]
1
where ε corresponds to half of the interface thickness, which can be
taken from half of the characteristic dimension of the smallest cell in
the mesh of computational geometry (Gunde et al., 2010). After the
modification, the interface location is represented by the contour
v
φ% ( x , t ) = 0.5 . In this paper, this modified level set method is chosen to
be the interface tracking method in the CO2 displacement simulation.
The normal vector n and curvature k of the interface can easily be
obtained from the level set function field as (the subscript of modified
function φ% will be omitted in the following part):
∇φ
∇φ



k = −∇ ⋅ n φ = 0.5 
n=
φ = 0.5
(5)
The momentum equation of fluid is described by the incompressible
Navier-Stokes equation as
ρ ut + ρ (u ⋅ ∇)u = ∇ ⋅ {− pI +η[∇u+(∇u)T ]} + Fsv
(6)
and the continuity equation as
∇⋅u = 0
(7)
where ρ is the fluid density, η is the dynamic viscosity, I is the identity
tensor, p is the pressure field, and Fsv is the term representing
interfacial tension. The level set function evolving under the effect of
fluid velocity can be described as
1
φt + u ⋅ ∇φ = ∇ ⋅ [ε∇φ − φ (1 − φ )n]
(8)
µ
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where µ denotes the amount of re-initialization or stabilization of the
level set function φ , and ε determines the interface thickness between
the two fluids. The density ρ and viscosity η are related to the level set
φ in the region of the two-phase flow:
ρ = ρl + ( ρg − ρl )φ 
(9)

η = ηl + (ηg − ηl )φ 
where the subscripts l and g refer to the liquid and gas phases,
respectively. The interfacial tension Fsv is calculated according to the
continuum surface tension model proposed by Brackbill et al. (1992) as
where σ is the surface tension coefficient; and δ is the Dirac delta
function, which can be written as
δ = 6 ∇φ φ (1 − φ )
(11)
This function ensures the following equation:
1
∫0 Fsvdφ = σn
(12)
In addition, the continuum surface tension model is independent of
grid size.
The inlet velocity is set to 0.5 cm/s. According to conditions of the
shallow saline aquifer, the density and viscosity of the water are set to
1×103 kg/m3 and 1.34×10−3 Pa s, respectively. The density and viscosity
of the CO2 are set to 38.5 kg/m3 and 1.47×10−5 Pa s, respectively. The
interfacial tension between water and CO2 is set to 44.57×10−3 N/m
(Chalbaud et al., 2010).
The finite element method is used to solve the coupled equation
system. The geometric model is discretized with 2D unstructured
triangle meshes, and boundaries with large curvature are discretized
with finer meshes as shown in Fig. 3. The number of the mesh is
251,327. A linear system solver PARDISO (parallel direct solver) is
used. A relative tolerance of 0.01 and an absolute tolerance of 0.1 are
assumed for the time discretization scheme. The time-stepping method
uses the generalized alpha method to determine each time step.
An initial condition given by an assumed phase distribution may not
be reasonable. Therefore, before the displacement simulation, the flow
field is calculated with zero inlet velocity within a short time to make
the assumed phase distribution to reach a reasonable pattern. The
calculated initial phase distribution is shown in Fig. 4.
At the beginning of the injection, the injected CO2 displaces the
resident water in the porous media through a relatively large channel, as
shown by the arrow in Fig. 5. At this time, the CO2 in relatively narrow
pores does not show obvious flow. When the CO2 in the relatively large
channel reaches a narrow pore throat as shown by the arrow in Fig. 6a,
the CO2 will stop displacing through this channel, and the CO2 in
relatively narrow pores begins to displace the water as shown by the
arrows in Fig. 6b.
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3. Numerical solution and discussion
Fig. 2. Porous media with a buffer region.
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follows:
v
Fsv ( x ) = σδ (−∇ ⋅ n)n
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The left and right sides of the porous media shown in Fig. 1 are taken
as inlet and outlet boundary conditions, respectively; the lower and
upper boundaries are taken as a symmetric boundary condition. Pores
are considered as water wettable with a contact angle of 180°. The CO2
injected into porous media will automatically choose flow channels.
Therefore, a buffer region, where the CO2 can automatically choose
flow channels, is connected to the inlet boundary of the geometric
model. The modified geometric model is shown in Fig. 2.
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Fig. 3. Pore geometry discretized with unstructured 2D meshes.
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Fig. 5. Injected CO2 flows through the relatively large pore.
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Fig. 4. Initial CO2-water distribution.
Comparing CO2 distribution in Fig. 6b and c, we may see that
although CO2 firstly flows through a relatively large pore, a relatively
narrow throat restricts the CO2 to flow through this channel. Therefore,
pore throat, rather than pore body, is determinant that determines
channel of the CO2 flow. This result indicates that capillary force is
determinant in CO2 flow in porous media.
Distribution of the two-phase interface in Fig. 6c shows that, when
reaching a pore throat, CO2 will displace through another channel.
Consequently, there will be a moment when all two-phase interfaces
reach respective pore throats of potential flow channels. At this time,
once the maximum pore throat is broken through by the CO2,
meniscuses at the other pore throats will retreat quickly; this process
can be shown by comparing Fig. 6c and d. Therefore, the two-phase
interfaces are in a self-adjusting process in the displacement process.
This result also indicates that CO2 may have ever reached a position in
porous media where although the CO2 is not seen at a certain time, such
as the positions shown with pink arrows in Fig. 6d. This push-pull
movement of interfaces may improve the dissolution and reaction of
CO2 in porous media (Li, 2011).
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Fig. 6. Variations of two-phase interface during displacement.
Fig. 8 shows the finally simulated results of two-phase distribution.
This result shows that 47.71% of the water is displaced. Therefore,
more than half of the water remains in the porous media.
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Fig. 7 shows the variation of injection pressure during the
displacement process. The pressure variation indicates that injection
pressure fluctuates throughout the displacement process. The variations
of two-phase distribution and injection pressure reveal that, when CO2
reaches a pore throat and needs to break through it, the injection
pressure needs to increase high enough. Once the CO2 breaks through
the pore throat under effect of the increased injection pressure, the
injection pressure will release quickly, and the stored pressure by the
capillary pressure of meniscus will release with the meniscuses
retreating. Subsequently, the injection pressure gradually increases to
prepare for breaking through the next pore throat.
Fig. 7. Variation of injection pressure in the injection process.
Fig. 8. Simulated results of CO2 distribution.
4. Conclusions
A 2D pore structure is extracted from a sandstone sample, and CO2
displacement in the sandstone is simulated using the Navier-Stokes
equation combined with the conservative level set method. The
simulation results show that pore throat, rather than pore body, is a
crucial factor that determines flow channel of the CO2 displacement in
porous media. The injected CO2 firstly flows through the largest pore.
When reaching a pore throat, the pore throat will restrict the CO2 from
flowing in that channel, and the CO2 will choose to flow through the
second-largest pore, and so on. In this way, CO2 displacement needs to
break through one or more pore throats. At this time, the CO2 will
choose the relatively large pore throat to break through, and therefore
this pore throat determines flow channel of the CO2.
The two-phase meniscuses in each pore throat are in a self-adjusting
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process during the displacement. Before breaking through the relatively
large pore throat, the injection pressure of the CO2 increases gradually,
and the curvature and the position of interfaces in other pore throats
adjust accordingly. Once the maximum pore throat is broken through by
the CO2, the capillary force in other pore throats releases subsequently,
and the interfaces retreat accordingly, preparing for breaking through
the next pore throat. The two-phase displacement in CO2 injection is
accompanied by the breaking through and adjusting of the two-phase
interfaces.
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Conflict of interest
International Conference on Computational Methods in Water Resources. University
The authors wish to confirm that there are no known conflicts of
interest associated with this publication and there has been no
significant financial support for this work that could have influenced its
outcome.
of Illinois at Urbana-Champaign, 2012.
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Acknowledgements
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This work is funded by Key Laboratory of Coal-based CO2 Capture
and Geological Storage, Jiangsu Province, China and US Advanced
Coal Technology Consortium (No. 2013 DFB60140-08). We also thank
Prof. Qi Li for useful discussions.
Pan CX. Use of pore-scale modeling to understand flow and transport in porous media.
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Qianlin Zhu obtained his M.Sc. and Ph.D. degrees in geotechnical engineering from Hohai University in 2008 and Institute of Rock and
Soil Mechanics, Chinese Academy of Sciences in 2011, respectively. He is a research assistant at Institute of Low Carbon Energy, China
University of Mining and Technology (Xuzhou). His research interests include CO2 geological storage, permeability theory in
geotechnical engineering and isotopic hydrology. In recent years, he is striving to study two-phase displacement flow in porous media
and coupled fluid flow-geomechanics characteristics in CO2 saline aquifer storage.
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The authors wish to confirm that there are no known conflicts of interest associated with this publication and there
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has been no significant financial support for this work that could have influenced its outcome.