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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

mailto:[email protected]

66


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 .

REFERENCES

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( ) ( 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.


lattice boltzmann simulation of fluid flow in … · lattice boltzmann simulation of fluid flow in...

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