Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure

$: Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure$
SCIENCE CHINA Earth Sciences

© Science China Press and Springer-Verlag Berlin Heidelberg 2013 earth.scichina.com www.springerlink.com

*Corresponding author (email: [email protected])

• RESEARCH PAPER • January 2013 Vol.56 No.1: 1–12

doi: 10.1007/s11430-013-4643-0

Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure

JIN Yi1,2*, SONG HuiBo1,2, HU Bin1,2, ZHU YiBo1 & ZHENG JunLing1

1 School of Resources and Environment, Henan Polytechnic University, Jiaozuo 454000, China; 2 Key Laboratory of Biogenic Traces & Sedimentary Minerals of Henan Province, Jiaozuo 454000, China

Received November 6, 2012; accepted March 28, 2013

The influences of fractal pore structure in coal reservoir on coalbed methane (CBM) migration were analyzed in detail by cou-pling theoretical models and numerical methods. Different types of fractals were generated based on the construction thought of the standard Menger Sponge to model the 3D nonlinear coal pore structures. Then a correlation model between the permea-bility of fractal porous medium and its pore-size-distribution characteristics was derived using the parallel and serial modes and verified by Lattice Boltzmann Method (LBM). Based on the coupled method, porosity (), fractal dimension of pore structure (Db), pore size range (rmin, rmax) and other parameters were systematically analyzed for their influences on the perme-ability () of fractal porous medium. The results indicate that: ① the channels connected by pores with the maximum size (rmax) dominate the permeability , approximating in the quadratic law; ② the greater the ratio of rmax and rmin is, the higher is; ③ the relationship between Db and follows a negative power law model, and breaks into two segments at the position where Db≌2.5. Based on the results above, a predicting model of fractal porous medium permeability was proposed, formu-

lated as maxnCfr , where C and n (approximately equal to 2) are constants and f is an expression only containing parameters

of fractal pore structure. In addition, the equivalence of the new proposed model for porous medium and the Kozeny-Carman model =Crn was verified at Db=2.0.

fractal pore structure, porous media, lattice Boltzmann model, coalbed methane (CBM)

Citation: Jin Y, Song H B, Hu B, et al. Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure. Science China: Earth Sci-ences, 2013, doi: 10.1007/s11430-013-4643-0

With the rapid development of industrial extraction and commercial exploitation of the CBM in China, more and more attention has been paid to the course of its genesis and migration law in coals.

As a dual-porosity medium, coal reservoir’s pore spaces which is a kind of matrix porous medium coupled with fractured network, dominates the storage and recoverability of CBM (Clarkson et al., 1999; Gilman et al., 2000; Kara-can et al., 2001; Yao et al., 2007, 2009). Due to various causes, the microstructures of pores in coal reservoir are

always disordered and extremely complicated. There is now considerable evidence at home and abroad showing that coal reservoir is a fractal porous medium (Fu et al., 2001, 2005; Yao et al., 2008, 2009; Wang et al., 2002; Zhang et al., 2008, 2009).

Since the microstructures of the real porous media in coal are usually disordered and extremely complicated, this makes it very difficult to find the permeability of the media analytically and access the transport property of CBM ac-curately. Over the last several decades, the migration law of CBM in the fractal porous medium in coals has been inves-tigated both experimentally and theoretically by many au-

2 Jin Y, et al. Sci China Earth Sci January (2013) Vol.56 No.1

thors (Cai et al., 2011; Connell et al., 2010; Fu et al., 2009; Jacob, 1972; Kaviany, 1995; Liu et al., 2011; Liu et al., 2012; Mitra et al., 2012; Pan et al., 2012; Xu et al., 2011). But, experimental study is influenced heavily by many fac-tors, such as experiment condition, scale, and testing envi-ronment and so on. At the same time, the underlying con-tinuous media assumption makes the controlling mechanism of fluid flow hard to explore. As to the theoretical analyses based on semi-empirical models, they always fail to predict the macro transport property resulting from the collective chaotic behaviors of fluid particles in such a complicated porous medium in coal (Cai et al., 2010).

Nowadays, much effort has been devoted to numerical simulations. The computational approach can, thus, be use-ful to understand the basic physics of the problem, since one can easily select or neglect any of the relevant effects (such as viscous dissipation or fractal pore structure), and analyze every single fact of the problem (Croce et al., 2007). At the same time, such approaches are unsubjected to the experi-mental technique, level, and environment.

Based on microscopic models or macroscopic kinetic equations for fluids, the LBM simulates fluid flows by fol-lowing the evolution of a prescribed Boltzmann equation instead of solving the Navier–Stokes equation (Nithiarasu et al., 1997, 1998), and has received more and more attention compared to some conventional CFD techniques, such as the finite-difference, finite-volume, and finite-element methods (Chen et al., 1998; Kandhai et al., 1998; 1999; Keehm, 2003; Koponen et al., 1997; Ladd, 1994a, 1994b; Nourgaliev et al., 2003; Sangani et al., 1982; Succi, 2001). An important advantage of the LBM is that microscopic physical interaction among fluid particles, such as mass transport, chemical reaction and diagenesis, etc., can be conveniently incorporated into the model (Succi, 2001). The LBM has been applied successfully to a lot of fluid dynam-ics, including fluid flows in porous medium, thermal two-phase flow, diffusion in the multi-component fluids, heat conduction problem and multi-scale flow (Cai et al., 2010; Guo et al., 2005; Malaspinas et al., 2010; Qian et al., 2004; Vita et al., 2012). And now, the LBM has proven to be a powerful tool to investigate the controlling mechanism behind the complex flow problems.

In recent years, some Chinese scholars have analyzed the migration law of the CBM focused on different aspects (Jin, 2011; Lu, 2010; Teng et al., 2007, 2008; Xing, 2009; Zhu et al., 2007), but all these investigations are carried out on mesoscopic scale under the continuous media assumption, few on pore scale. Recently, Jin (Jin et al., 2013) modified the QSGS algorithm to generate fractal porous medium to mimic coal’s microstructures statistically, and analyzed the fluid flow’s spatial-temporal evolution pattern in such me-dia based on Lattice Boltzmann simulations. But, the report is few on the systematic analysis of the influences of fractal pore structure parameters on its transport property.

So, to fully understand the response mechanism of fluid

flow in fractal porous medium, different types of fractals were generated to represent coal media and the controlling influences on CBM’s migration were investigated in detail by coupling theoretical models and Lattice Boltzmann sim-ulations. Based on the results above, the permeability model of fractal porous medium was proposed, and the equivalence was verified between new model and Kozeny-Carman’s.

1 Materials and methods

1.1 Characteristics of coal’s microscopic pore-structure and its 3D representation

If the pore size distribution ranged from rmin to rmax in po-rous medium exhibits fractal characteristics, such medium is called fractal porous medium. For such medium, the cumu-lative size-distribution of pores (N(r)), whose sizes are equal to or greater than the size r , should follow the fractal scaling law as N(r)rD, where represents proportionality and D is the pore volume fractal dimension. Taking a cube with side length rl as measurement unit, the media’s pore volume (Vm) will result in

3( ) .m l lV N r r (1)

Incorporating the fractal scaling law “N(r)rD” of pore size distribution into eq. (1), the pore volume can be expressed as

3 .Dm lV r (2)

Differentiating eq. (2) with respect to rl results in the volume (dVm) of pores whose sizes are within the infinites-imal range rl to rl+drl,

2d.

dDm

ll

Vr

r (3)

Eq. (3) indicates that the relationship between volume in-crement (Vm or dVm/dr) and pore size also follows the fractal scaling law (Qin et al., 1995).

In practice, Menger sponge fractals are always taken to mimic the microstructures of coal medium (Fu et al., 2001). The construction process of Menger Sponge fractal is as follows: ① Divide equally the initial cube with side length R into m3 smaller cubes with side length R/m, and remove a part of such smaller cubes according to a certain rule, leav-ing Nbl smaller cubes; ② repeat step ① for each of the remaining small cubes, and continue to iterate ad infinitum. With the continuous iteration, the size of remaining cubes reduces continuously and their number increases on and on (Adler et al., 1993; Tarafdar et al., 2001; Yuan et al., 1986). After the kth iteration, the side length (rk) of the remaining cubes will be rk=R/mk, but the total number 1.k

bk bN N

Based on the construction process of Menger Sponge frac-

Jin Y, et al. Sci China Earth Sci January (2013) Vol.56 No.1 3

tals, Nbk can be rewritten as

,b

b

b

b b

D DD

bk kD Dk k k

R R CN Cr

r r r (4)

where 1lg / lg ,b bD N m the pore volume fractal dimen-

sion. Based on eq. (4), the correlation between pore volume (Vk) of coal and the measurement pore size (rk) follows the

relationship 3 .bDk kV r Differentiating the relationship

with respect to rk results in the volume (dVk) of pores whose sizes are within the infinitesimal range rk to rk+drk

2d.

dbDk

kk

Vr

r (5)

Comparing eq. (3) with eq. (5), D=Db is concluded. Based on the conclusion above, Menger sponge fractal can be employed to model the homogeneous coal media fully (Friesen et al., 1987).

To explore the control influences on the porous medi-um’s transport property from fractal’s parameters fully and avoid blink pores, this paper proposes a new Menger Sponge generator to construct fractal porous medium, named “SmVq” Menger Sponge model (as shown in Figure 1). The construction process of “SmVq” Menger Sponges is as follows: ① Divide the initial cube equally with side length R into m3 smaller cubes with side length R/m, and remove q×q smaller cubes which are along with the three main axes in the very center of the larger cube; ② Repeat step ① for each of the remaining small cubes, and contin-ue to iterate ad infinitum. Figure 1 demonstrates the two-dimensional pore structure of SmVq Menger Sponge model, where the white part represents pores and the black denotes solid matrix.

Figure 1 Pore structure diagram of SmVq model.

In the SmVq Menger Sponge, Nb1 and Db can be written as eqs. (6) and (7), respectively.

3 2 31 (3 2 ),bN m mq q (6)

3 2 3

1lg lg( 3 2 ).

lg lgb

b

N m mq qD

m m

(7)

However, the pore size distribution can only be within a certain range for real coal media in nature. Based on the fractal character of porous medium (Yu et al., 2001) and eq. (7), the porosity () of a fractal porous medium with pore size within the range from rmin to rmax can be obtained by eq. (8).

max minlg lg1

lg1 min

max

1 1 ,b

r r d Dm

bd

N r

mrm

(8)

where d is the Euclidean dimension, and d=2 and 3 in the two- and three-dimensional spaces, respectively.

Eq. (8) implies that if the pore size distribution of a po-rous medium follows the fractal scaling law, its porosity

is determined by and only by Db and max minlog .r rm On one

hand, as the ratio of rmax to rmin decreases, its porosity de-creases for porous medium with the same Db. On the other hand, the porosity increases as Db decreases for those with the same pore size range and m (Huai et al., 2007; Yu et al., 2001).

1.2 Lattice Boltzmann model

1.2.1 Lattice Boltzmann equation

The LB algorithm is a full discretization version of the Boltzmann equation. Because the algorithm describes mac-roscopic fluid flow by the collective behavior of many ficti-tious molecules but not by full molecular dynamics, it is referred to as a mesoscopic description of microscopic physics. Space x is discretized in terms of a regular (usually simple-cubic) lattice with spacing b, time t in terms of a time step h, and velocity space in terms of a small set of velocities ic

that are chosen such that ic h

is a vector

which connects two nearby lattice sites (D U Nweg et al., 2007, 2009a, 2009b). For example, the popular D3Q19 model (Qian et al., 1992) employs nineteen velocities, cor-responding to zero and the six nearest and twelve next-nearest neighbors on a simple-cubic lattice. The central quantities on which the algorithm operates are the popula-tions fi(x, t), representing the mass density corresponding to velocity ic

, such that the local mass density ( , )x t at the

site x at time t is given by

( , ) ( , ).ii

x t f x t (9)

Similarly, the momentum density ( , )M x t

is obtained as


the first velocity moment,

( , ) ( , ) ,i ii

M x t f x t c

(10)

and the average flow velocity of lattice is given by

( , )( , ) .

( , )

M x tu x t

x t (11)

The algorithm is then described by the lattice Boltzmann equation

*( , ) ( , ) ( , ) ({ ( , )})i i i i if x c h t h f x t f x t f x t . (12)

The collision operator i modifies the populations on the site ({fi} denotes the set of all populations on the site), such that mass and momentum are conserved. Energy conserva-tion is not taken into account, since we are here interested in an isothermal version, where the temperature instead of the energy is fixed (formally, this corresponds to a system with infinite heat conductivity). The conservation equations therefore read

0i i ii i

c . (13)

This results in a set of post-collisional populations *if ,

which are then propagated to the neighboring sites. The algorithm thus satisfies important requirements for

simulating fluid flows—mass and momentum conservation, and locality—but lacks Galilean invariance due to the finite number of velocities. Full rotational symmetry is also lost, which makes isotropic momentum transport unable to be recovered. To overcome the shortcomings, Chen and Qian (Chen et al., 1998; Qian et al., 1992) proposed a simplified collision function based on single relaxation time model, and named as Lattice BG Bhatnagar-Gross-Krook Boltz-mann model (LBGK). The BGK collision operator reads as

eq({ ( , )}) [ ( , ) ( , )]i i i

hf x t f x t f x t

, (14)

where is a dimensionless relaxation time, and eq ( , )if x t

is quasi-equilibrium distribution function. In this work three-dimensional LBGK model is used.

This model uses a D3Q19 lattice model with nineteen dis-crete velocities. The velocity vectors are illustrated in Fig-ure 2.

The velocity directions of the D3Q19 Lattice Boltzmann BGK model are defined as

0,0,0 0,

( 1,0,0),(0, 1,0), (0,0, 1) 1, ,6,

( 1, 1,0),( 1,0 1),(0, 1, 1) 7, ,18,

i

i

i

ie (15)

and the equilibrium distribution function eqif is taken as

Figure 2 Velocity directions of D3Q19 lattice model.

2 2eq

2 4 2

( )( , ) ( , ) 1 ,

2 2i i

i i

s s s

u c u c uf x t x t

c c c (16)

where cs is the speed of sound, and the weights i>0 are normalized such that 1.ii

This notation has been

chosen in order to emphasize that, for symmetry reasons, the weights depend only on the absolute values of the speeds ic

, but not on their direction. Furthermore, the

weights are adjusted in such a way that eqif satisfies the

properties

eq ( , ) ( , ),ii

f x t x t (17)

eq ( , ) .i ii

f x t c M

(18)

For D3Q19, this implies 1 3i for the rest popula-

tion, 1 /18i for the nearest neighbors, and 1 / 36i

for the next-nearest neighbors.

1.2.2 Boundary conditions of LBM

Boundary conditions have significant influences on the sta-bility, efficiency, and accuracy of numerical computation. In coals, the physical solid-fluid interface is extremely complex, which is resulted from the disordered and ex-tremely complicated microstructures. Thus, for such sol-id-fluid interfaces, there is almost no net fluid motion at the interfaces almost. So, the physical boundary condition at solid-fluid interfaces in coal medium can physically ap-proximate the no-slip boundary condition.

In LBM, the physical no-slip boundary condition is usu-ally realized as the bounce-back rule (Guo et al., 2009). For simplicity and without loss of generality, the complete bounce-back scheme is used to simulate the no-slip bound-ary condition at solid-fluid interfaces, which requires that when a particle distribution streams to a solid boundary


node, it scatters completely back to the node where it came from with opposite velocity vector, as eq. (19)

0( , ) ( , )i i if x t f x t , (19)

where x0 is the node position on the solid wall and xi is the position of the fluid node which is the closest one to x0 in the direction i.

1.3 Analytical estimation and numerical validation of the permeability of fractal porous medium

1.3.1 Analytical estimation of the permeability

For fractal analysis, that min maxr r must be satisfied in a

fractal porous medium. But in real nature porous medium, such as coal and carbonatite, pore size can distribute only in some certain range. Fortunately, if 2

min max 10r r is satis-

fied, the fractal theory and technique can be used to analyze properties of porous medium (Yu et al., 2001). For coals, there are several orders of magnitude difference between rmin and rmax, so the requirement above can obviously be satisfied.

Before simulation, the porous medium (physical system: PS) should be discretized into lattice system (LBS). Alt-hough the discretization schemes of PS to LBS will change with different simulation objectives, the conversion process should satisfy the conditions below: (1) the LBS must rep-resent the PS equivalently; (2) simulation parameters must reach the requirements of precision, such as the fine lattice and small time step (He et al., 2009).

However, in practical simulation, if the size of smallest pore of the porous medium is chosen as the lattice unit in LBS, we will fail to convert the PS to its dimensionless one, LBS, which is ascribed to the constraints in computing and storage resources. For that, this paper takes the following two steps to construct SmVq Menger sponge fractals:

① Decompose the SmVq Menger Sponge fractals, the pore sizes range in [rmin, rmax], into uniform-pore-size po-rous medium with pore size 1

min min min, , ,Nr mr m r

maxr respectively and same pore structure S Vm q , as Fig-

ure 3(a)—(c), and name them rPM (where r is the uni-

form pore size in rPM ). Denote the lattice position as P

and its phase as FP, if P is solid, FP=1, and pore FP=0. ② Superpose these rPM s, which have the same poros-

ity and pore structure but different pore sizes, into one po-

rous medium, named max

minr

rPM .

③ Determine the phase value by Boolean AND opera-tion among all PF from different rPM and such opera-

tion goes through the whole space, as is shown in Figure 3 (for simplicity and without loss of generality, two dimen-sional construction process were demonstrated). The super-posed result is SmVq Menger sponge with pore size ranging from rmin to rmax.

The process above separated the complex fractal pore structure into a series of inhomogeneous ones with single pore size and same pore structure effectively. For the next, the key issues are to find out the relationship between the

permeability of superposed model, max

minr

rPM , and that of its

components rPM .

The existing theoretical models show that permeability of composited material can be estimated by its components’ permeability based on two basic modes, named as parallel mode and series mode (Tuncer et al., 2001) (see Figure 4). Figure 4(a) and (b) are composite materials from two kinds of materials with permeability 1 and 2 respectively,

and the compound modes are parallel mode for (a) and se-ries mode for (b). According to the theoretical models, per-meability of material (a) is 1 2( ) / 2 and that of mate-

rial (b) is 1 21 / (1 2 1 2 ) .

Based on the theoretical modes, it is easy to infer the an-alytical permeability of porous media composited from the following spatial combination modes:

① Spatial superposition mode (Pa1): in this combination mode (see Figure 3), assume that the permeability of each superposed porous medium (Figure 3(a)–(c)), is a , b

and c , respectively, then, the permeability d of the

superposed result porous media (Figure 3(d)) should be

a b c .

Figure 3 Schematic diagram of the construction process of Menger Sponge by overlapping successive SmVq models with different pore sizes.


Figure 4 Two basic structures for validation. (a) Parallel mode; (b) series mode.

② Spatial arrangement of the same porous medium (Pa2): in this pattern, taking Figure 4 as example, if

1 2 and the permeability of composite result media

is denoted as ab , then the permeability of the composited

material should be . Based on the results above, the permeability of fractal

porous media (denoted as total) should be the sum of its superposed component’s permeability. Denote the permea-bility as 0 1 2 1, , , ,N N for each superposed porous

medium which contains pores with the same characteristic size 1

min min min max, , ,Nr mr m r r q , and rmax respectively, and

the analytical permeability of fractal porous media is ob-tained.

Total0

.N

ii

(20)

1.3.2 Numerical validation

To validate analytical permeability model of fractal porous medium (eq. (20)), Lattice Boltzmann simulations were carried out to verify the inferences for the permeability of porous medium constructed from mode Pa1 and Pa2.

For Pa1, different XiYjZk spatial arrange modes based on S3V1 Menger Sponge fractal were constructed, and they are X1Y1Z1, X2Y1Z1, X3Y1Z1, X1Y2Z1, X1Y3Z1, X1Y1Z2, X1Y1Z3 and X3Y3Z3 (in this pattern, X, Y and Z represent the three directions of coordinate axis, and i, j and k represent the number of tiled models in different direc-tions). The Lattice Boltzmann simulation results are listed in Table 1.

According to the simulated result, the permeability’s co-efficient of variation (denoted as CVP) of fractal porous me-dia from different spatial arrangement modes is 0.002173. Numerical experiments were also carried out for different SmVq Menger Sponge fractals, and all results show that CVP of the permeability of composite material and that of its component is less than 0.0025. Thus, the inference of mode Pa1 is effective and at the same time, the validity of LBM is demonstrated.

To verify the inference of Pa2, the relationship was ana-

lyzed comparatively between the permeability of different fractal porous medium SmVqLn (denoted as S V L ,m q n

where n is the iteration level) and the accumulative per-meability of their superposed porous medium SmVqL1 (denoted as r ) with different pore sizes (see Figure 5).

In Figure 5, the circle marks represent the relationship

between S V Lm q n and the accumulative permeability max

min

r

ir

from Lattice Boltzmann simulations and the solid curve is its fitted power law model. In the fitted model, the power law coefficient and the power exponent are slightly greater than 1.0, which implies the predicted value from equation (20) is slightly lower than the real fractal media, but the errors are all less than 3%. At the same time, the correlation coefficient is almost equal to 1, which indicates the validity of the analytical permeability model (eq. (20)).

Table 1 Simulated permeability of models with different arrangement modesa)

Composition pattern Permeability estimated by LBM (dimen-

sionless result) X1Y1Z1 0.705612

X1Y1Z2 0.705615

X1Y1Z3 0.705616

X1Y2Z1 0.705615

X1Y3Z1 0.705616

X2Y1Z1 0.703212

X3Y1Z1 0.70243

X3Y3Z3 0.702439

a) During the LBM simulations, to ensure the numerical precision, the lattice discretization resolution is set to 12×12×12. b) Simulated fluid is single component gas, the dimensionless relaxation time =1.0 and the pressure difference of inlet and outlet takes P=105 Pa. c) Velocity vec-tors on each lattice point were calculated by the lattice Boltzmann method. Absolute permeability of the porous medium could then be obtained by calculating the mean flux from the velocity vectors and with the Darcy’s law.

Figure 5 Relationship between the permeability of SmVqLn porous medium and the accumulative permeability of their superposed porous medium SmVqL1.


2 The results and discussion

2.1 Models of porous medium and their structure pa-rameters

In this paper, we focus our attention on the derivation of a predicted model for the permeability of fractal porous me-dium. So, SmVq Menger sponges with different fractal characteristics, such as porosity, pore structures, fractal di-mension, and pore size ranges etc., were generated to mimic different types of coal rock porous medium. The pore sizes of different SmVq structure Menger sponge fractals were ranged in 4–80 dimensionless lattice units. The pore volume fractal dimension numbers Db of different structures and other relevant parameters of the pore structures, m and q are listed in Table 2.

2.2 Simulation of fluid flow

During the Lattice Boltzmann simulations the mixed gas, CBM, is simplified as single component idea gas, and the LBGK model is employed to simulate the CBM’s migration process in different SmVqL1 models with characteristic pore sizes varying from 4 to 80 lattice dimensionless units, and the dimensionless relaxation time =1.0.

Some results are shown in Figure 6, where (a) and (b)

Table 2 Parameters and fractal dimensions of different SmVq models

SmVq Monger Sponge m q Db

S3V1 3 1 2.726833

S4V2 4 2 2.5

S5V1 5 1 2.931768

S5V3 5 3 2.351249

S6V2 6 2 2.832508

S6V4 6 4 2.246592

S7V1 7 1 2.970715

S7V3 7 3 2.742952

S7V5 7 5 2.168398

S8V2 8 2 2.918296

S8V4 8 4 2.666667

S8V6 8 6 2.107309

S9V1 9 1 2.984118

S9V3 9 3 2.863417

S9V5 9 5 2.601931

S9V7 9 7 2.057955

S10V2 10 2 2.952308

S10V4 10 4 2.811575

S10V6 10 6 2.546543

S10V8 10 8 2.017033

Figure 6 3D flow velocity vector fields and 3D velocity distribution fields in different SmVqL2 models when LB simulations converged.


represent velocity distribution field and velocity trace lines respectively when flow reached the stable state in S3V1L2 structure (in Figure 6(b), the gray part denotes as solid phase); (c) and (d) in S4V2L2 structure; (e) and (f) in S5V3L2 structure. And for comparison, the minimum pore size of different SmVqLn structures was set to 4 lattice di-mensionless units.

The velocity distribution field shows that the transport property of porous medium is dominated by the channels connected by the pores with large size. In narrow and tor-tuous paths formed by small pores, the fluid particles be-have violently and disorderly, which will resist the fluid flow through.

2.3 Influences of pore structures on the permeability

Based on the results of Lattice Boltzmann simulations, the permeability of different SmVqL1 porous medium, whose

characteristic pore sizes vary from 4 to 80 lattice dimen-sionless units, was calculated using Darcy’s law. By com-parison, we found that for the same pore structure SmVqL1 medium, the permeability and the characteristic pore size follow power law relation, n= ar ( 2 0.99R , where represents permeability, a is coefficient of power correlation, r is the characteristic pore size, and n rep-resents power exponent). As for different pore structure SmVq medium, there are large differences among the coef-ficients “ a ” of the fitted power law relations, but the varia-tion of exponents “ n ” is little and approximately equal to 2. The phenomenon above is generally consistent with Kozeny-Carman permeability model (Mavko et al., 1998), and the subtle differences are ascribed to insufficiency of LBS’ discretization precision. Some results are shown in Figure 7.

Figure 7 demonstrated the relationship between charac-

Figure 7 Relationship between permeability numerically simulated by LBM and the pore size of different SmVqL1 models.


teristic pore size and simulated permeability of different SmVq pore structures on log-log scale, where the horizontal axis represents dimensionless characteristic pore size, which is denoted as r , and vertical axis, permeability (because the unit of pore size is dimensionless, the permeability is dimensionless also). Among them, the circle marks repre-sent the relation of numerical results, and the solid lines represent the corresponding fitted power law trend curves.

Based on the numerical results and the pore-permeability relationships (Figure 7), the permeability of fractal porous medium with pore sizes ranging in min max,r r can be ob-

tained from eq. (20) as

max

min

min max

max 1

nnr

n nTotal n

r

m r rar ar

m

. (21)

From eq. (8), eq. (21) can be rewritten as

Total max

1 (1 ),

1 1

bn d Dn

nar

m

(22)

where a, rmin, rmax and Db are the same as aforementioned. Eqs. (21) and (22) imply that: the transport property of

fractal porous medium is influenced mainly by three factors: the channels connected by the maximum pore aforemen-tioned, the distribution of pore size, and the coefficient of power law a . Among these factors, the channels composited by the maximum pores are the dominant factor, which means that, the larger the maximum pore size is, the higher

Total is.

As for the factor of pore size distribution, because the

porosity of fractal porous media is determined by max minlogr rm

and bD (as eq. 8 shows), the larger max minlogr rm is, the higher

is, which results in the higher permeability Total . On

the other hand, if the parameter m and pore size range are the same, as bD increases, the porosity decreases, which

leads to the lower permeability. In Figure 7, the relationships follow max

nar in all

SmVq Menger Sponge fractals, and the power exponents n are constantly approaching 2, but, the coefficients a change with parameters m and q . In order to explore the

origin of such difference, the relationships between a and the parameters such as porosity , the pore volume fractal

dimension bD , the ratio q m and ( / ) bDq m , were plotted

in Figure 8 on log-log scale. The results show that the cor-relations between a and different parameters are all di-vided into two segments approximately at point 2.5bD .

In Figure 8(a), the relationship between q m and a

can be approximated by a power-law function and is broken into two segments as independent variable q m ap-

proaching 0.5 (where bD approaches 2.5). And when pa-

rameter q m exceeds the value 0.5, the influence obvi-

ously becomes heavier. As Figure 8(b) shows, there is a negative power law cor-

relation between bD and a , which implies that as bD

increases, the transport capacity of fractal porous media is reduced. And as bD approaches 3, a tends to 0, which is

consistent with the physical truth: as bD is negative rela-

tive with q m (eq. (7)), which means that as bD in-

creases, the relative space occupied by solid phase increases, and when 3bD , the space is filled with solid phase, thus

the permeability 0 certainly. The broken behavior

Figure 8 Relationships between a and different parameters. (a) q/ma; (b)Dba; (c) a; (d) (q/m)Dba.


occurs again approximated at point 2.5bD as that in

Figure 8(a), and a decreases as bD increases, but as bD

exceeds the value 2.5, the decline rate of a increases ob-viously.

Figure 8(c) shows that a power law model can approxi-mate the relation between and a , and the model is

decomposed by the porosity value 0.5 approximately. Such decomposition behavior is the result of the differences from pores structures and pore size ranges, because is just the

output of those coupled differences. As aforementioned, for porous medium with the same pore structure (same bD ), if

its porosity is determined, the permeability will depend en-tirely on the pore size range (as eq. (22) shows).

Figure 8(d) demonstrates the power law modeling rela-tionship between (q/m)Db and a , the similar decomposition behavior occurs again when bD and q/m approximately be

the value 2.5 and 0.5 respectively. By comparison, the rela-tivity between (q/m)Db and a is stronger than that of q/ma, Dba and a as the correlation coefficients R demon-strate. At the same time, the parameter (q/m)Db integrates more details of fractal pore structures and thus will benefit the fully exploring the underlying laws.

Based on the results above, the relationship between a and the parameters of fractal pore structures can be mathe-matically described as

, 2.5

, 2.5

l b

h b

n D

l b

n D

h b

C q m Da

C q m D (23)

where lC and ln are the coefficient and exponent of

the power law correlation between (q/m)Db and a respec-tively when 2.5bD , and when bD exceeds the value

2.5, lC and ln change into hC and hn .

2.4 Quantitative permeability estimation model for fractal porous medium

For coal porous medium or those really existing in nature, there are several orders of magnitude differences in sizes between the minimum pore and the maximum one. Thus, eq. (21) can be simplified into Total max ( ( 1))n n nar m m , and

replace a by eq. (23) in the simplified form, eq. (21) can be rewritten as

Total max 1

bn D nn

n

q mC r

m m, (24)

where C represents lC or hC , and n , ln or hn .

As n is a constant approximately equal to 2, and the spatial discretization resolution follows 3m , so we get

the close approximation / ( 1) 1n nm m . For that, eq. (24)

turns into

Total max .bn D

n qC r

m (25)

Eq. (25) implies that, the larger bD is, the lower per-

meability of fractal porous medium is, and the permeability will exponentially decline with bD .

However, eq. (25) differs largely from Kozeny-Carman model in form. For Kozeny-Carman predicted model, it was obtained on the assumption that pore sizes are almost the same, like the SmVqL1 model with porosity

2 3(3 2 ) /mq q m . As for eq. (25), if no fractal behavior

exists in pore structures which means bD is equal to 2, the

relationship 2( ) ( )k b kn D nq m q m can be obtained. So, the

difference between equation (25) and Kozeny-Carman model turns into the relationship between and

2( ) nq m , which is plotted in Figure 9.

In Figure 9, the horizontal axis represents porosity of porous medium, calculated by 2 3(3 2 ) /mq q m in

SmVqL1 model, and the vertical axis represents parameter 2( / ) nq m . By linear fitting, a strong linear relativity exists

between and 2(1 / ) nm no matter 2.5bD or

2.5bD . If the experimental and fitting errors are elimi-

nated, we can express the relationship in Figure 9 as the

approximate linear form 2( ) nC q m .

Based on the linear relationship between and

2( / ) nq m , eq. (25) turns into Total max maxn nC r C C r .

As the porosity of non-fractal porous medium is invar-

iant, eq. (25) can be simplified as eq. (26), which is the mathematical expression of Kozeny-Carman model.

Figure 9 Relationship between porosity of porous media represented by SmVqL1 model and (q/m)2n.


TotalnCr . (26)

If there is no fractal behavior in the pore structures, eq. (25) is equivalent to the Kozeny-Carman model as n is a constant approximately equal to 2. But for fractal porous medium, the Kozeny-Carman model should be modified by an operator, named f , as eq. (27) shows.

Total maxnCfr , (27)

where ( ) bn Df q m .

In conclusion, compared to the Kozeny-Carman model, the parameters in the new proposed model have definite physical meanings.

3 Conclusions

① Transport property of coal porous media with fractal pore structures is controlled by the maximum pores, the pore size distribution, and the fractal dimension number of pore structure. The channels connected by the maximum aperture dominate the permeability of fractal porous medi-um. The permeability increases as the size ratio of rmax and rmin increases, and the fractal dimension, Db, influences the permeability negatively following a power law model. Alt-hough a positive correlation exists between porosity and permeability, porosity is the result of pore size range, fractal dimension of pore structure, so we do not take porosity as the basic control factor of porous medium’s transport prop-erty of porous media.

② In fractal porous medium, the relationship between fractal dimension and permeability will be divided into two parts at point near 2.5bD . When 2.5bD , the negative

influence on permeability from bD is increased signifi-

cantly. ③ To estimate the permeability of fractal porous me-

dium, Kozeny-Carman model needs to be modified by a

multiplier operator named as f, where ( / ) k bn Df q m . And

thus, the modified model can unify permeability prediction model for porous medium, which are fractals or not.

We are grateful to the three anonymous reviewers for their constructive comments, and thank Prof. Miao Desui for correcting the syntax errors. This work was supported by the National Natural Science Foundation of China (Grant Nos. 41102093 & 41072153), CBM Union Foundation of Shanxi Province (Grant Nos. 2012012002), and Doctoral Scientific Foun-dation of Henan Polytechnic University (Grant Nos. 648706).

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Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure

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