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Multiphase-Flow and Reactive-Transport
Validation Studies at the Pore Scale by
Use of Lattice Boltzmann Computer
Simulations
Edo S. Boek, University of Cambridge; Ioannis Zacharoudiou, Farrel Gray, Saurabh M. Shah, and
John P. Crawshaw, Imperial College London; and Jianhui Yang, Total E&P UK
Summary
We describe the recent development of lattice Boltzmann (LB)
and particle-tracing computer simulations to study flow and reac-
tive transport in porous media. First, we measure both flow and
solute transport directly on pore-space images obtained from
micro-computed-tomography (CT) scanning. We consider rocks
with increasing degree of heterogeneity: a bead pack, Bentheimer
sandstone, and Portland carbonate. We predict probability
distributions for molecular displacements and find excellent
agreement with pulsed-field-gradient (PFG) -nuclear-magnetic-
resonance (NMR) experiments. Second, we validate our LB
model for multiphase flow by calculating capillary filling and cap-
illary pressure in model porous media. Then, we extend our mod-
els to realistic 3D pore-space images and observe the calculated
capillary pressure curve in Bentheimer sandstone to be in agree-
ment with the experiment. A process-based algorithm is intro-
duced to determine the distribution of wetting and nonwetting
phases in the pore space, as a starting point for relative permeabil-
ity calculations. The Bentheimer relative permeability curves for
both drainage and imbibition are found to be in good agreement
with experimental data. Third, we show the speedup of a
graphics-processing-unit (GPU) algorithm for large-scale LB cal-
culations, offering greatly enhanced computing performance in
comparison with central-processing-unit (CPU) calculations.
Finally, we propose a hybrid method to calculate reactive trans-
port on pore-space images by use of the GPU code. We calculate
the dissolution of a porous medium and observe agreement with
the experiment. The LB method is a powerful tool for calculating
flow and reactive transport directly on pore-space images of rock.
Introduction
In previous SPE papers (Ramstad et al. 2009; Boek 2010), the
LB method has been introduced as an efficient model to solve the
Navier-Stokes equations directly for flow in complex geometries,
such as pore-scale images of real rock samples obtained from
micro-CT and confocal microscopy imaging (Shah et al. 2013).
Because of its simple algorithms and local operations, the method
is ideally suited for parallel computing. The LB method has been
widely used for flow in porous-media studies, including single-
phase flow (Ferre´ol et al. 1995; Pan et al. 2006). Various models
have been proposed for multiphase flow (Gunstensen et al. 1991;
Shan and Chen 1993; Swift et al. 1995, 1996). These models have
been used in a wide variety of flow in porous-media studies
(Martys and Chen 1996; Pan et al. 2004; Li et al. 2005; Ahrenholz
et al. 2008; Porter et al. 2009; Middleton et al. 2012; Liu et al.
2014). Boek (2010) gave an overview of the different models used
for multicomponent fluids, such as oil/water and carbon dioxide
(CO
2
)/brine mixtures. Recently, we have made a direct compari-
son between these models (Yang and Boek 2013). It was observed
that the color-gradient (CG) model (Gunstensen et al. 1991) is
very efficient for large-scale calculations. The best part of our
results described in this paper will be dependent on the CG
method. However, under certain conditions—such as large viscos-
ity ratios between displacing and displaced fluids—the free-energy
model (Swift et al. 1995; 1996) offers enhanced stability. In addi-
tion, the free-energy model can also describe (partially) miscible
binary systems, such as CO
2
/brine mixtures under particular ther-
modynamic conditions. Such miscible systems are not accessible
by use of the immiscible CG method. For this reason, we will
describe here initial validation results for capillary-filling proc-
esses by use of this model. Calculations on realistic pore-space
images by use of the free-energy method are currently in progress
(Zacharoudiou and Boek 2016). Recently, LB flow in porous-
media studies have been extended to calculation on GPUs (Jiang
et al. 2014; Chen et al. 2016). Here we describe initial results
showing the excellent scaling behavior of LB calculations per-
formed on multiple GPUs. Finally, we use the GPU code to study
reactive-flow problems. Reactive-flow studies have been described
in the recent literature (Verberg and Ladd 2002; Kang et al. 2002,
2006, 2007, 2001, 2014; Verhaeghe et al. 2006; Huber et al. 2014;
Yu and Ladd 2010; Yoon et al. 2012). Here, we describe initial
results by use of a new hybrid flow-reaction model applied to
solid- and porous-media dissolution under flowing conditions.
Multiple-Relaxation-Time (MRT) Method
One problem with the standard single-relaxation-time LB ap-
proach (Boek and Venturoli 2010) is that the permeability meas-
ured depends on the fluid viscosity (Pan et al. 2006; Venturoli and
Boek 2006). In addition, for multiphase flow, unphysical spurious
velocities near boundaries may emerge (Pooley et al. 2009). We
choose here to use an MRT approach (d’Humieres et al. 2002;
Ginzburg 2005; To¨lke et al. 2013) to eliminate these spurious
velocities. The main idea of the MRT method is that different
relaxation parameters are used for different linear combinations
of the distribution functions. In particular, we set the relaxation
parameters responsible for generating the viscous terms in the
Navier-Stokes equation to s
f
,those connected to conserved quan-
tities (density and momentum) to 1, and all others to unity.
We implement this by replacing the term 1
sf½f0
iðr;tÞfeq
iðr;tÞ
of the LB equation with the term M1SMðffeqÞ, where the dis-
tribution functions fand f
eq
are now written as column vectors
and Mis a matrix that performs a change of basis. The new basis
is designed in such a way to contain more physically relevant var-
iables. Sis a diagonal matrix that defines different relaxation
times for different modes. More details on the MRT approach are
provided by d’Humieres et al. (2002), Premnath and Abraham
(2007), Yang and Boek (2013), and Yang et al. (2013).
Single-Phase Flow and Transport in Pore-Space
Images: Propagator Distributions
We have calculated flow and solute transport directly on pore-
space images of different rock cores with increasing degree of
Copyright V
C2017 Society of Petroleum Engineers
This paper (SPE 170941) was accepted for presentation at the SPE Annual Technical
Conference and Exhibition, Amsterdam, 27–29 October 2014, and revised for publication.
Original manuscript received for review 4 January 2015. Revised manuscript received for
review 15 July 2016. Paperpeer approved 30 August 2016.
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2017 SPE Journal 1
heterogeneity: a bead pack, Bentheimer sandstone, and Portland
carbonate. First, the flow field is calculated by use of the MRT LB
method (Yang et al. 2013). The MRT method has significantly
improved numerical stability compared with the single-relaxa-
tion-time LB model (Lallemand and Luo 2000) and eliminates the
unfavorable viscosity dependency of computed velocity on relax-
ation time (Pan et al. 2006). We show the geometry of the Port-
land carbonate, obtained from micro-CT scanning, in Fig. 1. The
corresponding LB velocity distribution is also shown in Fig. 1.
To solve the solute dispersion, we treat the solute as a passive
scalar for which the advection diffusion is simulated by a second
distribution function, corresponding closely to the normal fluid-
distribution function, except with a simpler D3Q7 equilibrium dis-
tribution (Sukop and Thorne 2005). The D3Q7 equilibrium distri-
bution is sufficient to accurately describe the solute dispersion,
although it is computationally less demanding than the D3Q19 dis-
tribution used for the fluid. We have used this method, coupled to
LB flow simulations, for the first time to calculate probability dis-
tributions for tracer displacements (or propagator distributions) in
realistic porous media of increasing heterogeneity. The propagator
distributions for the three different porous media are shown in
Fig. 2 in direct comparison with the experimental PFG/NMR
results. Good agreement with the NMR experiments (Scheven
et al. 2005) and previous simulations (Bijeljic et al. 2013) is
observed. The magnitude of the flowing distribution reduces and
the stagnant peak increases as the degree of heterogeneity
increases. We quantitatively measure the trapping capability of
three rock samples of increasing heterogeneity: the homogeneous
bead pack does not trap any solute, whereas the Bentheimer sand-
stone and Portland carbonate retain 1.49 and 8.13% of solute,
respectively, in stagnant pores. The experimental surface-relaxa-
tion problem, leading to an undercount of stagnant spins—as
found in the NMR experiments—is removed by use of the LB sim-
ulations. Compared with existing transport-simulation tools such
as streamline-based algorithms, the LB method is easy to imple-
ment and handles the problem in an efficient way with very-good
accuracy. This method can be widely used to study transport prob-
lems in porous media with applications in hydrology, chemical en-
gineering, carbon storage, and enhanced oil recovery (EOR).
Capillary Filling by Use of a Free-Energy
MRT LB Algorithm
To validate multiphase flow in porous media, we first investigate
the dynamics of capillary filling as a validation of the free-energy
LB algorithm (Swift et al. 1995, 1996; Boek, 2010). The free
energy that describes the binary fluid is the Landau free-energy
functional (Kendon et al. 2001):
W¼ð
V
wbþj
2ð@a/Þ2
hi
dVþð
S
wsdS;ð1Þ
where w
b
is the bulk free-energy density,
wb¼A
2/2þB
2/4þc2
3qlnq;ð2Þ
where /is the concentration or order parameter, qis the fluid-
mass density, and cis a lattice velocity parameter. The constants
Aand Bin Eq. 2 allow binary-phase separation into two phases if
A<0 and B>0. This functional gives phase separation in two
phases, according to
/eq ¼6ffiffiffiffiffiffiffiffiffiffiffiffi
A=B
q;ð3Þ
and the surface tension can be expressed as (Kendon et al. 2001)
r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8kA3=9B2
p:ð4Þ
.............
..................
..........................
.........................
0.008
0.004
0
Y
X
Z
Fig. 1—(left) Pore-space image (pore is green, solid is blue) and (right) velocity distribution of Portland carbonate. Green and blue
indicate high and low velocities, respectively. Units of velocity are lattice units. The sample size in pixels is 640 33203320 and the
voxel size is 9 lm. This figure is reprinted from Yang et al. (2013) with permission.
–1
00 0
0.5
1
1.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0.4
0.6
0.8
1
1.2
–0.5 0 0.5 1 1.5
ζ/(ζ)0
P(ζ)×(ζ)0
ζ/(ζ)0ζ/(ζ)0
2 2.5 3 3.5 4 –1012345–2–10123456
Fig. 2—Probability of molecular displacement as a function of rescaled mean Darcy displacement in different porous media: (left)
bead pack, (middle) Bentheimer sandstone, and (right) Portland carbonate for time t51.0 seconds. Simulation results (red dotted
solid line) are compared with NMR experimental data (blue dashed line) from Scheven et al. (2005). The Pe´clet number is 18 in the
LB simulations. This figure is reprinted from Yang et al. (2013) with permission.
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2 2017 SPE Journal
The corresponding contact angle is then derived from the surface
tension by use of Young’s law.
Capillary-filling simulations have been reported in the litera-
ture by use of the recoloring CG algorithm (Liu et al. 2014) and
the free-energy model (Liu et al. 2013). Both the Liu et al. (2013)
paper and our current capillary-filling simulations are dependent
on the free-energy LB model of Lee and Liu (2010). We use a
density ratio of unity, whereas Liu et al. (2013) report higher den-
sity ratios by a modification of the particle-distribution functions.
Liu et al. (2013) consider 2D systems with a single-relaxation-
time approximation, whereas we consider 3D systems by use of
the MRT approach, offering enhanced stability. In addition, for
the Liu et al. (2013) approach to work with real rock systems, it
must be demonstrated that the permeability does not become de-
pendent on viscosity, which is a known problem regarding the sin-
gle-relaxation-time approximation.
Our simulation results are compared with the well-known
Washburn (1921) law, which predicts that the filled length of the
capillary scales with the square root of time. We find that the nu-
merical method used can capture the correct scaling behavior for
sufficiently high viscosity ratio [the Washburn (1921) law], but
also for the case when the viscosity of the resident fluid in the
capillary is not negligible. In this paper, we will verify that the
filled length of the capillary, in the case of viscosity ratio unity,
scales linearly with time t, as expected. Moreover, we will investi-
gate the dependence of the advancing contact angle on the inter-
face velocity.
First, we investigate the dynamics of a fluid penetrating a
channel under the action of capillary filling. The wetting fluid
penetrates the hydrophilic channel because of the Laplace pres-
sure across the interface or, equivalently, because of the decrease
in the free energy as the wetting fluid wets the walls of the chan-
nel. Balancing the driving force (2rcosha) and the viscous drag of
the fluid column [12glðdl=dlÞ=h2] gives the well-known Wash-
burn (1921) law:
l2¼rhcosha
3gðtþt0Þ;ð5Þ
where hais the advancing dynamic contact angle, gis the dynamic
viscosity of the fluid, and his the width of the channel. Here, we
use the dynamic contact angle ha, because this is different from the
equilibrium contact angle heq and controls the Laplace pressure
across the interface. Moreover, it is expected to be higher than the
equilibrium contact angle, reflecting the fact that less energy is
available to do useful work than the energy liberated from wetting
the channel’s surface because of viscous dissipation.
The geometry examined consists of a capillary with length
L¼640 and height h¼50. Reservoirs (460 200) of the two
components are attached at the inlet and outlet of the channel.
Periodic boundary conditions are imposed in the x-direction to
ensure that the reservoirs are connected and have the same pres-
sure. The density ratio between the two fluids is unity. The relaxa-
tion times used for the two fluids are sf;w¼2:5 and sf;nw ¼0:502,
leading to a viscosity ratio rn¼gw=gnw ¼1;000, whereas the
equilibrium contact angle is set to heq ¼60For this case of high
viscosity ratio, for which we can neglect the resistance of the resi-
dent nonwetting fluid to the penetration of the wetting fluid,
we verify the Washburn (1921) law, as can be seen from Figs. 3
and 4. Please note that the slight divergence from the analytical
result in Fig. 4 for early times is because of inertial effects.
We proceed to examine the scaling of this dynamic process for
varying ratio of viscosities of the two fluids rn¼gw=gnw.We
decrease the viscosity ratio by increasing the viscosity of the non-
wetting fluid. This increases the viscous dissipation in the fluid
column ahead of the fluid/fluid interface and results in a decrease
in the advancing interface’s velocity, because less energy is avail-
able for filling the channel. Fig. 3 shows the results from the LB
simulations for the length of the wetting-fluid column as a func-
tion of time, verifying the previous argument.
Moreover, for the case of rn¼1, the scaling of the length lof
the column of the penetrating wetting fluid in the channel as a
function of time is now linear with time (length is approximately
equal to time), as can be seen from Fig. 3. This is expected
because viscous dissipation occurs at approximately the same rate
at any given point down the channel, and therefore, the interface
position as it advances in the channel does not affect the results.
Finally, we measure the dynamic contact angle and examine
its dependence on the velocity of the advancing fluid/fluid inter-
face. We verify the theoretical prediction for the dynamic advanc-
ing contact angle as a function of the capillary number Ca (Latva-
Kokko and Rothman 2007):
cosðhaÞ¼cosðheqÞCalnðKL=lsÞ;ð6Þ
where Kis a constant, Lis the length scale of the system, and l
s
is
the effective slip length at the contact line. The advancing contact
angle decreases with decreasing capillary number Ca ¼gwU
r,as
can be seen from Fig. 5. By extrapolating to Ca ¼0, we find
ha
Ca¼0¼60:3in good agreement with heq ¼60.
Multiphase Flow in Porous Media:
Capillary Pressure
Ramstad et al. (2009) computed capillary pressure in a square
tube at strongly water-wet conditions in comparison with the ana-
lytical solution. In this paper, we extend this to an idealized
porous medium consisting of eight spheres with equal radius
.......................
...............
0
0
100
200
300
400
2 4 6 8 10 12 14
Time (×105)
Distance Down Capillary
rη = 1,000
rη = 100
rη = 10
rη = 4
rη = 1
Fig. 3—The variation of the length lof the column of the pene-
trating wetting fluid in the channel as a function of time for
varying viscosity ratio. Time is given in dimensionless lattice
units.
rη = 1,000
Linear fit
00 200
Time0.5
Distance Down Capillary
400 600 800 1,000 1,200
100
200
300
400
Fig. 4—Variation of column length lof the penetrating wetting
fluid in the channel as a function of (time)
0.5
for viscosity ratio
r
g
51,000.
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2017 SPE Journal 3
(Fig. 6). The analytical solution of the equilibrium capillary pres-
sure relation is given by Mason and Morrow (1991) as
Pc¼rð1þ2ffiffiffiffiffiffiffi
pG
pÞcosh
RFðh;GÞ;ð7Þ
where Ris the inscribed radius given by R¼ð ffiffiffiffiffiffiffiffiffiffiffi
21
pÞRgrain, and
G¼A/O
2
, where Ais the cross-sectional area and Ois the perime-
ter length. F(h,G) is close to unity for small contact angles. The
size of the sphere pack is 50 2020 with a sphere radius of
nine. The entry pressure is computed from the Young-Laplace
equation. As an alternative, morphological methods could be used
to achieve this because this not only tests the entry pressure but
the curve itself (Hilpert and Miller 2001). The nondimensional
pressure P
c
R/cas a function of wetting-phase saturation for pri-
mary drainage is shown in Fig. 7. We started from zero capillary
pressure, which was achieved by setting equal inlet and outlet
pressure. Next, the density/pressure of the outlet was incremen-
tally decreased to achieve the desired capillary pressure. When
the saturation reached the steady state, the wetting saturation was
recorded. The surface tension was set as 0.01 lattice units, and an
equal viscosity for wetting and nonwetting phase of 0.02 lattice
units was used in the calculation. As can be seen, the LB simula-
tion result for the entry pressure is in good agreement with the an-
alytical solution. The simulated saturation increases slowly with
the increase of pressure. When the pressure is increased beyond
the entry pressure, the saturation increases dramatically to a high
value and the nonwetting phase was found to enter the pore/throat.
This entry value was found to match the analytical prediction by
the Young-Laplace law (blue line in Fig. 7). According to the
results, the discretization does not affect the simulation results
significantly because all the cases predict the entry pressure accu-
rately. Snapshots of the primary drainage in the idealized porous
medium are shown in Fig. 8.
After validating the capillary pressure in an idealized porous
medium, we now consider the capillary pressure for a realistic po-
rous medium. Porter et al. (2009) reported capillary pressure cal-
culations for an idealized glass-bead porous medium by use of the
Shan-Chen multicomponent model (Shan and Chen 1993). It
should be noted that the Shan-Chen model suffers from large
fluid/fluid interfacial widths (10 lattice units), which makes it dif-
ficult to study realistic porous media respecting the pore-size dis-
tribution (Yang and Boek 2013). Here, we study a realistic porous
medium, in this case a Bentheimer sandstone. We use the CG
multiphase model, which has a smaller interfacial width of 2 to 3
lattice units, suitable for realistic porous-media studies. Here we
calculate the capillary pressure for a Bentheimer sandstone (Fig.
1) numerically by use of LB calculations on a micro-CT image.
For this purpose, we have used our in-house laboratory micro-CT
scanner, the Xradia Versa XRM-500. The system size is
512 256256 voxels, resulting from mirroring in the x-direction
and leading to periodic boundary conditions. The voxel size is
4.9 mm and porosity is 0.234. The single-phase permeability is
4,755 md, determined from LB calculations. A drainage process
is simulated to mimic the experimental procedure. A buffer layer
with a width of 10 lattice sites is added at the inlet of the sample
for injecting nonwetting phase. A porous plate is added at the out-
let of the sample to prevent the nonwetting phase from flowing
out. This configuration is consistent with the experimental setup.
Periodic-boundary conditions are used for flow calculation, and a
constant-color boundary condition—which converts all entry fluid
into nonwetting fluid—is applied for the inlet. A certain pressure
gradient DPis applied to both phases until the system reaches
pseudosteady state and the saturation becomes constant. Then,
one point on the capillary pressure curve is obtained. More points
on the curve are obtained by applying different pressure gradients
DP. The calculated capillary pressure is converted into the dimen-
sionless Leverett J-function:
JðSÞ¼ Pc
rcoshffiffiffiffi
x
/
r;ð8Þ
where P
c
is capillary pressure, jis absolute permeability, /is the
porosity, ris surface tension, and his the contact angle. The com-
parison of computed and measured capillary pressure curves is
shown in Fig. 9. The agreement between the experimental mea-
surement and simulation is generally good. The predicted irreduc-
ible-wetting-phase saturation (S
wi
¼7%) is similar to the
experimental measurement. However, the calculated capillary
................
.........................
0.00
0.00 0.01
Ca
0.02 0.03
0.10
0.20
0.30
0.40
0.50
rη = 1,000
Linear fit
COS (θa)
Fig. 5—The dynamic contact angle as a function of the capillary
number.
Fig. 6—Idealized porous medium consisting of spheres with
equal radius. A cubic unit cell is shown with spheres centered
on the corners of the cube, and therefore only one-eighth of
each sphere is shown.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.1 0.2 0.3 0.4
Saturation Sw
Nondimensional Pressure (pR/γ)
0.5 0.6 0.7 0.8 0.9 1
Fig. 7—Capillary pressure curve for primary-drainage process
in an idealized porous medium, consisting of eight packed
spheres of equal radius. LBM simulation (red) is compared with
the analytical solution (blue). The blue curve is the analytical
solution for the capillary-entry pressure, rather than the entire
capillary-saturation curve.
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pressure for this saturation is significantly lower than the experi-
mental data. This may be because of resolution effects, because
some very-small pores or throats may not be captured by micro-
CT imaging. Captured small pores or throats in the geometry may
be difficult to handle because these may be smaller than the inter-
facial width in the LB simulation. For this reason, very-high capil-
lary pressures may lead to poor numerical stability. These
difficulties can be overcome by use of high-resolution images,
although this will increase computational cost. In addition, much
of the irreducible wetting saturation observed in experiments is
related to clay-bound fluid. Such pore space is less than the reso-
lution in the digital images presented here and in other studies.
Including clay-bound water in an effective way may contribute to
resolving the discrepancy between calculation and experiment.
Multiphase Flow in Pore-Space Images:
Relative Permeability
For relative permeability calculations of Bentheimer sandstone by
use of the Shan-Chen model (Shan and Chen 1993) for immiscible
fluids, we refer to Boek and Venturoli (2009), Boek (2010), and De
Prisco et al. (2012). In the current study, we use the CG model. We
use the Bentheimer micro-CT image described previously. First, we
show snapshots of the drainage process in Bentheimer sandstones in
Fig. 10. From Fig. 10, we observe that the big pores and channels
are filled with the nonwetting phase first because of low capillary
pressure. Small pores and throats are not filled with the nonwetting
phase because of the high capillary pressure. The fluid in these pores
becomes the residual wetting phase that cannot be drained out
unless a further increase of injection pressure is applied.
Next, we start with the forced-imbibition process, where the
fluid flowing out is recolored to wetting fluid for in-flow. The ter-
minal condition of the simulation is similar to drainage: The simu-
lation stops when the wetting-phase saturation converges to a
constant value. Snapshots of the imbibition process are shown in
Fig. 11.
We observe that the wetting phase contacts the sample surfa-
ces and fills the narrow pores first, whereas the nonwetting phase
preferentially stays in the center of the pores, leading to snapoff
of nonwetting phase. The tendency to snap off depends on the
capillary number and wettability. When the contact angle
increases, a piston-like advance dominates the displacement and
suppresses snapoff, which leads to little or no trapping (Blunt and
Fig. 8—Snapshots of primary drainage in idealized porous medium, from top to bottom and left to right.
0
0
1
2
3
4
5
6
7LB calculations
Experiment data
8
0.1 0.2 0.3 0.4 0.5
Saturation
Leverett J-Function
0.6 0.7 0.8 0.9 1
Fig. 9—Computed capillary pressure (blue) in comparison with
experimental results (red) (Ramstad et al. 2009) for Bentheimer
sandstone.
Fig. 10—Snapshots of the drainage-process simulation in Bentheimer sandstone. The nonwetting phase is shown in red, and the
rock is shown in transparent green. Nonwetting-phase saturation increases from left to right.
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Scher 1995). A quantitative analysis of this process, including the
effects of capillary number and wettability on front propagation,
is currently under way and will be reported in a forthcoming
paper (Boek et al. 2016). Preliminary results of the quantitative
analysis show that the qualitative explanation provided previously
is correct.
Next, we calculate relative permeabilities for our rock sample.
There are not many studies in the open literature on the direct cal-
culation of relative permeability of reservoir rocks (Martys and
Chen 1996; Ramstad et al. 2012; Koroteev et al. 2013). The
effects of capillary number, wettability, and viscosity ratio on the
relative permeability saturation curve was carefully studied (Li
et al. 2005) by use of the Shan-Chen model (Shan and Chen
1993). This type of calculation requires a high level of accuracy,
reliability, and efficiency of the algorithm. In addition, it is diffi-
cult to reproduce the experimental conditions in the simulations.
The discussed publications report calculation of the relative per-
meability on the basis of a random distribution of the nonwetting
phase according to the desired saturation. This initial configura-
tion is easy to set up but has a number of disadvantages.
•It does not recover the experimental measurement
procedures.
•Small or dead pores may be occupied by the nonwetting
phase, which is not the case in the experiments because of
high capillary pressure.
•It is difficult to calculate the imbibition relative permeability
curve because of lack of drainage-imbibition hysteresis.
For this reason, we have developed a process-based approach
for the initial nonwetting-phase distribution, dependent on imbibi-
tion simulations as a starting point for subsequent drainage calcu-
lations. To consider the drainage-imbibition hysteresis, a forced
drainage was performed and followed by a forced imbibition to
generate the distribution of oil and water. This configuration was
then used as the initial distribution for subsequent relative perme-
ability calculations. The rock sample was fully saturated with
water, whereas buffer layers of oil and water were generated near
the inlet and outlet of the sample, respectively. The oil buffer was
injected into the sample until the saturation converges, which
completes the drainage calculation. The distributions of oil and
water at different saturation values were saved for the drainage
relative permeability calculations. To perform the imbibition cal-
culation, the fluid in the buffer layer was then changed into wet-
ting phase, and the forced injection was continued until the
saturation converged again. For further details of this method, we
refer to Yang (2013). In this fashion, we calculate relative perme-
ability curves for both drainage and imbibition. These curves are
shown in Fig. 12.
We observe that the agreement with experimental data is gener-
ally good for both drainage and imbibition. However, the nonwet-
ting-phase relative permeability is slightly overpredicted at high-
wetting-phase saturations for both drainage and imbibition calcu-
lations. This is also true for the drainage case at lower wetting satu-
rations. The relative permeability endpoint for imbibition is also
overpredicted by the LB simulation. These discrepancies may be
because of the limited resolution of the mesh in small pores and
throats, because the thin wetting layers (with a thickness smaller
than the imaging resolution, which in this study is 4.9 mm) near the
rock surfaces and small scale snapped-off nonwetting-phase bub-
bles may not be captured effectively (Yang and Boek 2013). A
potential solution is to use a finer mesh, although this will decrease
the computational efficiency. Second, the capillary number Ca in
our calculations is controlled by the body force and set as 10
–5
.
Lowering the capillary number may further improve the results,
although this will require longer calculations. Finally, the calcula-
tion is probably also influenced by the stability and sharpness of
the fluid/fluid interfaces. We are currently working on an efficient
GPU implementation of the code to solve these problems.
We believe that this new process-based method to predict the
relative permeability for both drainage and imbibition is more
faithful to the experiment than the random initial distribution
12 3 4
Fig. 11—Snapshots of the imbibition-process simulation of Bentheimer sandstone. The nonwetting phase is shown in orange, and
the rock is shown in transparent green. Nonwetting-phase saturation decreases from left to right.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Water Saturation
Relative Permeability
LB water
LB oil
Experiment water
Experiment oil
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Water Saturation
Relative Permeability
LB water
LB oil
Experiment oil
Experiment water
Fig. 12—Steady-state relative permeability simulation for imbibition (left) and drainage (right) in direct comparison with experimen-
tal data (Ramstad et al. 2012) for Bentheimer sandstone.
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6 2017 SPE Journal
approach. The saturation does not need to be renormalized to con-
sider the “irreducible” pores (Ramstad et al. 2009). Although this
new approach comes at a greater computational cost than tradi-
tional methods, we believe it opens the way to more-accurate rep-
resentation of the experimental process. Note that De Prisco et al.
(2012) show how to mimic a steady-state fractional-flow setup to
compute relative permeabilities.
LB Single Phase on the GPU
To address the issue of computational cost, we have developed a
single-phase LB-method code for GPU processing as a basis for a
number of new simulation techniques in fields including represen-
tative-elementary-volume analysis, transport analysis, and
coupled reaction and flow calculations. We use the MRT method
for its enhanced stability when dealing with very-heterogeneous
pore spaces. The calculation itself is performed on the GPU,
which offers greatly enhanced performance over CPU implemen-
tations in terms of computational cost (To¨lke and Krafczyk 2008).
We are able to compute flow in large domains in multi-GPU set-
tings and have obtained consistently high device-memory band-
width and computational efficiency, as well as excellent scaling
behavior by use of fully asynchronous (overlapped with computa-
tion) memory transfer between devices (Fig. 13).
Obrecht et al. (2013) have investigated the scaling behavior of
their LB code on multiple GPUs. They observed a strong scaling
exponent of –0.8 in comparison with –1 for ideal scaling behavior
for up to 100 GPUs. For our code, we observe strong scaling
behavior close to ideal scaling for up to 4 GPUs. An extension to
larger number of GPUs is currently in progress. The calculation
of single-phase flow field and permeability on a 700 7001,400
pore-scale image takes 6 hours on a current high-end dual-CPU
system, compared with fewer than 10 minutes on a four-GPU
workstation. Our current program of work is toward implementing
multicomponent LB models for the GPU architecture in a similar
way. This will extend the capability of our current CPU imple-
mentations to longer time scales and larger domains. The GPU
code described previously has been used to enable the reactive-
transport simulations in the next section.
Reactive Flow in Porous Media for
CO
2
Storage and EOR
Finally, we describe preliminary results of reactive-flow studies in
porous media by use of the GPU LB code described previously.
These studies were performed to obtain a better understanding of
coupled reaction and flow processes in CO
2
storage and CO
2
EOR
operations. First, we consider injection of CO
2
into a brine aqui-
fer, producing carbonic acid
CO2þH2O!
k1HþþHCO
3:ð9Þ
Here, we consider the flow and diffusion of the H
þ
produced,
which leads to reaction with the solid CaCO
3
(calcite) in the po-
rous medium:
CaCO3þHþ!
k2Ca2þþHCO
3:ð10Þ
This, in turn, leads to dissolution of the rock and alters its pore
structure. Experimental results suggest that at high flow rates,
wormlike-dissolution patterns emerge, whereas at low flow rates,
the dissolution pattern is more homogeneous (Fredd and Fogler
1998). However, how the pore structure is altered as a function of
flow rate and pH is poorly understood. A more-detailed under-
standing of this process is of great importance to the industry.
First, we validate the reaction model without flow by investi-
gating the dissolution of a solid sphere and find good agreement
with the analytical solution (Rice and Do 2006). In Fig. 14, we
show that the volume change as a function of time is in good
agreement with the analytical solution (Rice and Do 2006). Then
we calculate the dissolution of a solid cube (Fig. 15) under quies-
cent conditions. The final state is total dissolution of the cube,
although this is not presented here. After validating the dissolu-
tion model, we couple this to a single-phase-flow LB method by
use of the GPU algorithm described previously, which computes
the flow as it evolves with the dissolving pore space. This method
is very computationally efficient and a moderate-sized domain of
a few hundred lattice units cubed can be simulated in a matter of
hours on a single workstation. As an example, we show the differ-
ent dissolution regimes in a carbonate rock at different flow rates.
At low flow rates, the dissolution pattern is homogeneous and
...................
................
0
1
2
3
4
5
012345
Relative Performance
Number of GPUs
Ideal
Scaling
Fig. 13—Scaling behavior of single-phase MRT LB model run-
ning on a 4 NVIDIA-GTX-Titan-Black workstation. The geometry
used was a bead pack with lattice dimensions of 512 3
2563256.
Analytical solution
0
–0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
–1
–1.5
–2
–2.5
–3
–3.5
–4
–4.5
–5
d(V/V0)dt
d/dτ(V/V0)
τ
Fig. 14—Volume change of a sphere as a function of time in
comparison with the analytical solution.
Fig. 15—Dissolution of solid cube by use of the transport model in combination with chemical reaction.
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2017 SPE Journal 7
shows a flat-face dissolution front (Fig. 16). At high flow rates, on
the other hand, we observe the formation of wormholes, leading
to inhomogeneous dissolution (Fig. 16, bottom). This confirms the
experimental observations reported (Fredd and Fogler 1998). We
are currently quantifying the dissolution process in more detail.
This work will lead to greater understanding of the mechanisms
of carbonate dissolution in carbon-capture and storage (CCS) CO
2
injection and acidizing behavior in EOR applications. More
details on the numerical methods used and results obtained will
be provided in a forthcoming paper (Gray et al. in process).
Conclusions
We present an overview of recent developments of LB codes to
study flow and reactive transport in porous media. First, single-
phase flow and solute transport were calculated directly on pore-
space images obtained from micro-CT scanning. The influence of
increasing degree of heterogeneity on flow properties was investi-
gated for a bead pack, Bentheimer sandstone, and Portland car-
bonate. We predict probability distributions for molecular
displacements and find excellent agreement with PFG/NMR
experiments. Second, we perform multiphase-flow validation cal-
culations by use of the free-energy model. Unlike the CG model,
the free-energy method can tackle both immiscible and partially
miscible fluids, such as CO
2
/brine mixtures. We validate our
model by calculating capillary filling and capillary pressure in
model porous media. Then, we consider the CG model for oil/
water immiscible displacement in realistic 3D pore-space images.
The calculated capillary pressure curve in Bentheimer sandstone
is observed to be in agreement with experiment. A process-based
algorithm is then introduced to determine the distribution of wet-
ting and nonwetting phases in the pore space as a starting point
for more-realistic relative permeability calculations, including
drainage-imbibition hysteresis. The Bentheimer relative perme-
ability curves for both drainage and imbibition are found to be in
reasonable agreement with experimental data, and suggestions for
improvement are provided. The LB simulations can be used for
the prediction of multiphase-flow properties in pore-space images,
as potential elements of special core analysis, and for EOR opera-
tions. Third, we introduce a GPU algorithm for large-scale LB
calculations, offering greatly enhanced computing performance in
comparison with CPU calculations. The GPU code is used to cal-
culate reactive transport on pore-space images. We first validate
our model by calculating the dissolution of a solid cube in an
acidic solution under quiescent conditions. Next, we extend this
to flowing conditions and observe good agreement with experi-
mental results, showing homogeneous dissolution for low flow
rates and wormhole formation for high flow rates. This opens the
way to calculating dissolution of the pore space in direct compari-
son with micro-CT-imaging experiments for matrix acidizing and
CCS operations. It is concluded that the LB method is a powerful
tool for calculating flow and reactive transport directly on rock-
pore-space images.
Nomenclature
A¼LB cross-sectional area, lattice units [–]
c¼LB lattice velocity parameter, lattice units [–]
h¼LB width of the channel, lattice units [–]
P
c
¼LB capillary pressure, lattice units [–]
R¼LB inscribed radius, lattice units [–]
g¼LB dynamic viscosity of the fluid, lattice units [–]
h
a
¼LB advancing dynamic contact angle, lattice units [–]
q¼LB fluid-mass density, lattice units [–]
r¼LB surface tension, lattice units [–]
s
f
¼LB relaxation time, lattice units [–]
/¼LB concentration or order parameter, lattice units [–]
w
b
¼LB bulk free-energy density, lattice units [–]
Acknowledgments
This work was carried out as part of the activities of the Qatar
Carbonates and Carbon Storage Research Centre (QCCSRC). We
gratefully acknowledge the funding of QCCSRC provided jointly
by Qatar Petroleum, Shell, and the Qatar Science and Technology
Park, and their permission to publish this research. We are grate-
ful to Nathan Welch, Geoff Maitland, and Martin Blunt for help-
ful discussions.
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03.013.
Edo S. Boek is a high-performance-computing expert at Math-
Works, Cambridge. He holds an honorary research fellowship
at the Department of Applied Mathematics and Theoretical
Physics at the University of Cambridge. Previously, Boek
worked as a senior lecturer at Imperial College London and
spent 15 years with Schlumberger in the Cambridge Research
Centre. His research interests include the flow of complex fluids
in porous media. Boek has authored more than 100 technical
papers and holds three patents. He is a member of SPE. Boek
holds a degree in Earth sciences from the University of Utrecht,
The Netherlands, and a PhD degree in chemical engineering
from the University of Twente, The Netherlands.
Ioannis Zacharoudiou is a research associate at Imperial Col-
lege London. His research interests include computational
fluid dynamics, multiphase flow, porous-media flow, and inter-
facial instabilities with applications to geological CO
2
seques-
tration and oil and gas recovery. Zacharoudiou is a member
of SPE. He holds a PhD degree in theoretical physics from the
University of Oxford.
Farrel Gray is a PhD degree candidate at Imperial College
London. His research interests include high-performance com-
puting, LB methods, and reactive-flow modeling. Gray holds a
bachelor’s degree in physics from Imperial College London.
Saurabh M. Shah is QCCSRC experimental officer at Imperial
College London. His current interests include pore-scale and
core-scale imaging, modeling, multiscale imaging, dissolution,
and in-situ flow experiments. Shah is an expert in operating dif-
ferent imaging techniques, such as medical CT, micro-CT,
confocal laser-scanning microscopy, and focused ion beam.
He holds a PhD degree in chemical engineering from Imperial
College London and a master’s degree in medical imaging
from the University of Surrey, UK.
John P. Crawshaw is a research fellow at Imperial College Lon-
don. Previously, he spent 14 years with Schlumberger in the
Cambridge Research Centre. Crawshaw’s research interest is
in the flow of complex fluids in porous media, currently
focused on CO
2
sequestration. He has authored more than 50
technical papers and holds seven patents. Crawshaw holds a
PhD degree in chemical engineering from Nottingham Univer-
sity, UK.
Jianhui Yang is a research reservoir engineer at the Total Ex-
ploration and Production UK Geoscience Research Centre.
Previously, he worked for 3 years as a PhD degree candidate
at QCCSRC at Imperial College London. Yang’s current inter-
ests include multiphase flow in porous media, recovery mech-
anisms, and digital rock physics. He holds a PhD degree in
chemical engineering from Imperial College London.
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