Lattice Boltzmann Simulation of Fluid Flow in
Complex Porous Media Based on CT Image
Zi-Sen Wu China University of Petroleum, Beijing, China
Email: [email protected]
Ping-Chuan Dong, Gang Lei, Shu Yang, and Nai Cao China University of Petroleum,Beijing, China
Email: [email protected], [email protected], [email protected], [email protected]
Abstract—In this paper, we simulate single phase and two-
fluid-phase flow in realistic porous media using a lattice
Boltzmann(LB). We used X-ray tomographic sandstones.
These are binary digital models describing the complex pore
space. First we show lattice boltzmann simulation results of
single phase flow. We calculate the flow field and
permeability of the micromodel and find agreement with
experiment. Second, we simulate two-fluid-phase flow at the
pore scale using the Shan-Chen model lattice boltzmann
method. First, we calculate relative permeability for each
fluid. Then we observe that the relative permeability of non-
wetting phase increases with pressure gradient, while that of
wetting phase stays almost the same.
Index Terms—lattice boltzmann, porous media, simulation,
relative permeability
I. INTRODUCTION
Fluid flow in porous media is of crucial important role
in many technological and environmental processes, such
as the recovery of hydrocarbons from oil reservoirs. To
characterize fluid flow in porous media, the experimental
analysis is often used. However it is difficult and time-
consuming. More recently, to simulate flow at the pore
scale, it is viable to use simplified representations of
porous media, such as physical micromodels, which can
be constructed in the form of pseudo-two-dimensional
capillary networks. Especially, the lattice-Boltzmann
method has shown great potential for solving the flow in
complex geometries and it has been successfully used in
the study of fluid flow in porous media at the pore scale .
Reference [1], [2] simulate single phase by LBM and
calculate the permeability of porous media. N.S. Martys
and H. Chen [3] applied the LBM to simulate
multicomponent flow in complex three dimensional
geometries. Reference [4] introduced the first detailed
survey of LBE theory and its major applications to date.
There have been many studies on single phase and two-
fluid-phase flow in porous media. X. Shan and H. Chen
[5] discussed a LB model which has the capability of
simulating multiphase and multicomponent immiscible
Manuscript received January 10, 2015; revised May 20, 2015.
fluids. Reference [6] quantitatively evaluated the
capability and accuracy of the lattice Boltzmann equation
(LBE) for modeling flow through porous media. M.
Venturoli and E.S. Boek [7] compared two-dimensional
and three dimensional lattice-Boltzmann simulations of
fluid flow in a pseudo-2D micromodel. In two
dimensions, lattice-Boltzmann simulations have been
used to investigate viscous fingering of binary immiscible
fluids. Reference [8] applied a reactive transport lattice
Boltzmann model developed in previous studies to study
thedissolution-induced changes in permeability and
porosity of two porous media at the pore scale. Reference
[9] performed two-phase flow simulation on digital rock
and discussed the effect of different initial saturation
distribution of two fluids and relative permeability nuder
differnet pressure gradient. Jianhui Yang and Edo S.
Boek [10] compared three different multi-component
Lattice Boltzmann models to explore their capability of
describing binary immiscible fluid fluids. The
multicomponent multiphase LBM [11] proposed by Shan
and Chen (SC) is a more popular model to simulate two-
fluid-phase. The LBM method was used to calculate the
relative permeability of sandstone and compared with
experiment [12]. LBM simulations can also calculate
flow in realistic porous media and compute relative
permeability [13], [14]. In addition, Reference [15], [16]
discussed the equations of state in the Lattice Boltzmann
models.
This paper is organized in the following way. Firstly,
the LB method and the basic concept will be introduced.
Secondly, the permeability of a 2D micromodel of
sandstone will be calculated and the calculation will be
compared with the experimental data. Besides, we present
results of simulations of multicomponent fluid in porous
media. During this process, the relative permeability
curve of a sandstone porous medium based on a CT
Image will be calculated. Finally, the effect of ration of
pressure gradient and surface tension on relative
permeability will be analyzed.
II. THE LATTICE BOLTZMANN METHOD
The lattice boltzmann method consists of two
processes. The first process describes particle advection
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Journal of Industrial and Intelligent Information Vol. 4, No. 1, January 2016
© 2016 Journal of Industrial and Intelligent Informationdoi: 10.12720/jiii.4.1.65-68
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Journal of Industrial and Intelligent Information Vol. 4, No. 1, January 2016
and the second describes particle collision on a regular
lattice. The lattice Boltzmann advection and collision can
be written in the following form [15]:
1
, , , ,eq
i i i i if x c t t t f x t f x t f x t
(1)
where ( , )if x t is the number density distribution in the
i th velocity direction for the fluid at position x and
time t , and t is the time increment, and is the
relaxation time in the lattice unit, ( , )eq
if x t is the
corresponding equilibrium distribution function. For the
LBM model simulated in this work, we used the D2Q9
lattice structure, where D is the dimension and Q is the
number of velocity direction. Then ( , )eq
if x t has the
following form [15]
2 2
2 4 2
9 21 3
2 2
ieq ii i
e u u ue uf
c c c
(2)
In the above equation, ie is the discrete velocities,
which can be written as
0,0 0
1 1cos ,sin 1,2,3,4
2 2
1 12 cos ,sin 5,6,7,8
2 4 2 4
i
i
i ie c i
i ic i
(3)
is the molecular mass, which is defined through 8
1
i
i
f
, and the fluid velocity was calculated after
every time step at each in the fluid volume [16]
8 8
1 1
i i i
i i
uu e f f
(4)
To simulate two-phase flow in porous media through
SC LB model, we need to consider the fluid-fluid
interaction force and fluid-solid interaction force. In the
S-C model [16], the interparticle forces ( )k
f fF x, the
fluid-fluid interaction force on k th fluid at site x is the
sum of the force between the k th fluid particle at site x
and the k th fulid particles at the sites x
( ) ( ) ( , ) ( )( )k k k
f f kk
x
F x x G x x x x x
(5)
where k is a function of local density and kkG
represents the strength of the interparticle force. In this
paper, ( )k kx is used for simplicity.
The interaction force between the k th fluid at site x
and the solid wall at site x is defined as [16]:
( ) ( ) ( , ) ( )k k
f s ks
x
F x x G x x s x x
(6)
where ksG represent the interactive strength between the
k th fluid and the solid phase. The parameter s is 1 for a
solid and 0 for a pore.
Figure 1. The segment recognation image of a sandstone from Berea.
In image, black zones are grains, and white zones are pores and throat. The size is 1418 μm ×1744 μm
Figure 2. The water’ s velocity field under steady state, the color
coding runs from blue for lower to red for higher flow rates.
III. SINGLE PHASE FLOW SIMULATION
We simulate the flow in a 2D micromodel of Berea
sandstone, which is presented in Fig. 1. We discretize on
a lattice and bit-map to create the matrix for the LB
simulation. The permeability of a porous medium can be
calculated by the empirical Darcy’s law.
It is well known that the flow rate is proportional to the
driving force, the permeability of the medium and the
dynamic viscosity of the fluid. Darcy’s law can be
defined as the follows
0ik P k
JL L
(7)
where J is the rate per unit area of cross section, k is the
permeability, P is the pressure gradient. is the
dynamic viscosity of the fluid. iP and
0P are the pressures
at the inlet and outlet, respectively. L is the distance
between the inlet and outlet.
© 2016 Journal of Industrial and Intelligent Information
To simulate the flow, we used periodic boundary
conditions at the inlet and outlet, and no-slip boundary
condition at all other boundaries. We performed
simulations by 5 different values of the body force.
Finally, we got the permeability k =417 mD, agreement
with the experimental value ( k =445±35mD) [13].
Fig. 2 shows the details of the flow in the Berea
sandstone. We can see that the velocity under steady state
is different at different field. The higher flow rates almost
concentrate in the core throat. The lower flow rates
almost concentrate in the pore. This phenomenon is
consistent with the test [13].
IV. TWO PHASE FLOW SIMULATION
The relative permeability of a phase is given as a ratio
of single-phase permeability at a given saturation to the
absolute permeability of the phase for a fully saturated
porous media. Reference [17], [18] gived some methods
to compute relative permeability. In order to calculate the
relative permeability of a phase using the LBM method,
we adopt a method [17] of normalizing the total
momentum of a phase at a given saturation to the single-
phase momentum, the formulation can be given as
(8)
In (8), π denotes the total momentum of each phase at
a given wetting-phase saturation Sw, and it can be
calculated according to (4).
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Wetting Phase Saturation
Relative permeability
Wetting phaseNonwetting phase
Figure 3. Relative permeability of the Berea sandstone
Values of relative permeability for different saturation
of wetting and non-wetting fluids are shown in Fig 3,
from which we can see that the relative of non-wetting
fluid is very high and almost no permeability for wetting
fluid at low wetting fluid saturation, as we wanted. And
at high wetting fluid saturation, there are still
considerable amount of relative permeability of non-
wetting phase.
Fig. 4 shows the relative permeability curves obtained
for three different ratios. It can be observed that the
change in the ratio of the surface tension with the
pressure gradient does not produce much change in the
relative permeability of wetting fluid. But, the relative
permeability of non-wetting fluid is affected significantly
by the increase of pressure gradient.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Wetting phase saturation
Rela
tive
per
meab
ilit
y
C=1C=1C=5C=5C=10C=10
Figure 4. Relative permeability of the Berea sandstone at different
ratios of pressure gradient to surface tension.
We study the effect of pressure gradient (ΔP) and
surface tension (σ) on relative permeability. The ratio of
the pressure gradient and surface tension was defined as
follows
PC
(9)
V. CONCLUSIONS
In this study, we performed single and two-phase flow
simulation on Berea sandstone using LBM method. First,
we calculate the single phase permeability, which is good
agreement with experiment. And we show the details of
the flow in the Berea sandstone. In addition, we simulate
two-phase flow and get the relative permeability curves,
we found that the increase of pressure gradient cause the
increase of the relative permeability of non-wetting fluid,
But it is no effect on wetting phase. Finally, the LBM is a
viable numerical tool to simulate single and two-phase
flow in complex porous media at the pore scale and
investigate fluid properties .
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Journal of Industrial and Intelligent Information Vol. 4, No. 1, January 2016
( ) ( 1)
( 1) ( )
w
r
w
S P Sk
S P S
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Zi-Sen. Wu and was born in Shandongprovince in 1984. He is a PhD degree
candidate at China University of Petroleum-
Beijing. E-mail:[email protected]. He
holds a MS degree from China University of Petroleum-Huadong (2011). His research
interests include theory and laboratory studies
of the fundamental properties and Numerical Rocks, theory of the lattice Boltzmann and
modeling of multiphase flow in porous media and enhanced oil recovery of low permeability reservoirs.
Ping-Chuan Dong is a professor and director of the department of petroleum engineering at China University of Petroleum-Beijing. He
holds a PhD degree from Northeastern University (2000). Dong’s research interests include unconventional reservoirs description and
development, and enhanced oil recovery of low permeability reservoirs.
Gang Lei was born in Hubei province in 1987. He is a PhD degree
candidate at China University of Petroleum-Beijing. E-mail:[email protected]. He holds a MS degree from
China University of Petroleum-Beijing (2012). His research interests
include theory and laboratory studies of the fundamental properties and behavior of tight sandstone reservoirs, numerical modeling of fractured
horizontal wells and enhanced oil recovery of low permeability reservoirs.
Shu Yang was born in Jilin province in 1987. She is a PhD degree candidate at China University of Petroleum-Beijing. E-mail:
[email protected]. She holds a MS degree from China University of Petroleum-Beijing (2013). Her research interests include theory and
laboratory studies of the fundamental properties and behavior of tight
sandstone reservoirs, Numerical Rocks, and enhanced oil recovery of low permeability reservoirs.
Nai Cao was born in Shandong province in 1988. She is a PhD degree
candidate at China University of Petroleum-Beijing. E-mail:
[email protected]. She holds a MS degree from China University of
Petroleum-Beijing (2014). Her research interests include theory and laboratory studies of the fundamental properties and behavior of Shale
gas, and numerical modeling of fractured horizontal wells.
© 2016 Journal of Industrial and Intelligent Information