scaling invariant effects on the permeability of fractal

8/16/2019 Scaling Invariant Effects on the Permeability of Fractal

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Transp Porous Med (2015) 109:433–453

DOI 10.1007/s11242-015-0527-4

Scaling Invariant Effects on the Permeability of Fractal

Porous Media

Y. Jin1,2 · Y. B. Zhu1,2 · X. Li1,2 · J. L. Zheng1,2 ·

J. B. Dong1,2

Received: 25 November 2014 / Accepted: 3 June 2015 / Published online: 17 June 2015© Springer Science+Business Media Dordrecht 2015

Abstract Porous media are interconnected systems, in which the distribution of pore sizes

might follow scaling invariant property and will affect the fluid flow through it significantly.

Thus, except for a detailed understanding of the fundamental mechanism at pore scale,

hard is it to determine the appropriate relationship between the permeability and the basic

properties. In this study, in terms of the size distribution and spatial arrangement of the

pores, we analytically derived a permeability model using series–parallel flow resistance

mode firstly. And then, together with the scaling invariant characteristics of the porosity,specific area and hydraulic tortuosity, the analytical permeability model is reformulated into

a fractal permeability–pore structure relationship. The results indicate that: (1) the square of

the porosity (ϕ) is proportional to the permeability in a fractal porous media, not the cubic

law described in Kozeny–Carman (KC) equation; (2) the hydraulic tortuosity is a power

law model of the minimum particle size with the exponent ( Df − d ), where Df and d arethe pore size fractal dimension and Euclidean space dimension, respectively, while is a

parameter characterizing the spatial arrangement of pores; (3) the KC numerical prefactor

is not a constant in fractal porous media. Its value, however, increases linearly with the size

ratio of the minimum to the maximum pores but decreases exponentially with Df . More

importantly, it is found to be a parameter characterizing the difference of fluid flow in porous

media from that in a straight tube described by the Poiseuille law. The performance of the

new fractal permeability–pore model is verified by lattice Boltzmann simulations, and the

numerical prefactor universality is examined as well.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 41102093,

41472128), and CBM Union Foundation of Shanxi Province of China (Grant No. 2012012002).

B Y. Jin

[email protected]

1 School of Resource and Environment, Henan Polytechnic University, Jiaozuo 454003, China

2 Collaborative Innovation Center of Coalbed Methane and Shale Gas for Central Plains Economic

Region, Henan Province, Jiaozuo 454003, China

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434 Y. Jin et al.

Keywords Fractal porous media · Tortuosity–porosity model · Lattice Boltzmann method ·Kozeny–Carman constant · Permeability–pore model

1 Introduction

Investigation of fluid flow in porous media is of paramount importance in different areas.

As one of the basic transport properties, permeability describes how easily the flow passes

through the porous media (Jin et al. 2013; Matyka et al. 2008; Costa 2006), and is a dependent

quantity on some fundamental and well-defined parameters determined solely by the geom-

etry of porous media, such as the porosity, the specific surface area, the hydraulic tortuosity

and the shape factor. In the past, a lot of semiempirical relationships between the permeability

and the structure parameters were proposed experimentally or theoretically (Collins 1961;

Duda et al. 2011; Bear 1972; Dullien 1991), among them the modified KC equation (Carman

1937, 1939; Kozeny 1927) should be the most well-known one for solid particles packing

bed.

Natural porous media microstructure might be disordered and complicated, with

pores/particles distributed scaling invariantly (Adler and Thovert 1998; Smidt and Monro

1998; Thovert et al. 1990; Young and Crawford 1991; Krohn and Thompson 1986). Substan-

tial difference has been observed in the measured permeability even when porous media share

the same statistical quantity of fundamental properties (such as porosity), and the so-called

KC constant in the modified KC equation is actually an empirical parameter, which will alter

with pore structures (Costa 2006; Ahmadi et al. 2011; Cai et al. 2010; Panda and Lake 1994;

Rahli et al. 1997; Xu and Yu 2008). Thus, one is hard to determine the appropriate relation-ship between the permeability and the basic properties because of the large number of related

parameters, except for a detailed understanding of size distribution and spatial arrangement

of pores and particles in porous media (Costa 2006).

To improve the estimation accuracy, various approaches were employed to investigate the

permeability–pore structure (short by permeability–pore later) relationship, approximately

catalogued into three types: experiment-based analysis, analytical derivations and numerical

simulations. The experimental results are usually influenced by testing techniques, scales,

experimental environments, etc. Meanwhile, owing to the potential continuous assumptions

underlying in experiments, the microstructural effects on fluid flows may be thus ignored

(Costa 2006; Yu et al. 2003). That will lead to some unexplained items, always called semi-empirical parameters.

Following the analytical approaches, a mathematical framework can be established with

clear physical meanings for a problem at hand by some simplifications, guiding us to adapt the

inherent coefficients to suit experiments (Ahmadi et al. 2011). Pitchumani and Ramakrishnan

(1999) proposed a permeability model for fractal porous media by assuming them as fractal

tube bundles other than the idealized arrangements of tube bundles with same length (Kozeny

1927; Happel 1959; Sangani and Acrivos 1982). However, the model presented in Pitchumani

and Ramakrishnan (1999) presents unreasonable results, interested readers may consult the

comments by Yu (2001) for detail. And then, Yu et al. (2003, 2005) analytically derived apermeability–porosity relationship, assuming that the size distribution of pores follows fractal

statistics as well as that the relationship between the diameter and the length of capillary tubes

admits a fractal scaling law (Wheatcraft and Tyler 1988). Later, these authors developed a

new form of permeability and KC constant, and demonstrated how KC constant alters with

the range of pore size (Xu and Yu 2008). Other than the accumulation method based on the

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Scaling Invariant Effects on the Permeability of Fractal Porous… 435

nonintersect assumption of the capillary tubes, another idea is to reformulate the classical

KC equation directly, by inserting the scaling invariant relationships between fundamental

parameters and the microstructure of porous media with special settings (Sahimi 1993; Rawls

et al. 1993; Guarracino 2007; Nasta et al. 2013). Accordingly, Costa (2006) proposed a two-

parameter permeability–porosity equation, and then Henderson et al. (2010) analyticallyderived a three-parameter permeability model. Jin et al. (2013) found that the permeability

is a function of the size of the largest pore, the porosity, the pore size fractal dimension ( Df )

and a semiempirical parameter. Based on fractal geometry and Poiseuille’s law, Wang et al.

(2014) analytically derived a semiempirical fractal permeability model, indicating that the

porosity to the power of (4 − Df )/(2 − Df ) is proportional to the permeability of a two-dimensional porous medium. Obviously, direct investigations are more reliable than those

obtained on the assumption of the capillary tubes apart from each other (Jin et al. 2013; Costa

2006; Wang et al. 2014).

Except for the solutions mentioned above, more and more efforts are now devoted to

direct numerical simulations because of their advantages in understanding the basic physics

of a certain problem (Croce et al. 2007). In numerical experiments, one can easily select

or neglect any relevant effects, and reduce the uncertainty from coupling effects as far as

possible. Moreover, the computational fluid dynamics (CFD) models are not subjected to

any experimental techniques, scales or environments. Among these CFD models, the newly

developed lattice Boltzmann method (LBM) has drawn broad and special attention (Chen

and Doolen 1998; Kandhai et al. 1999; Koponen et al. 1997; Ladd 1994; Succi 2001), and

has been proven to be a powerful tool to explore the controlling mechanism of complex flow

problems (Nithiarasu and Ravindran 1998; Degruyter et al. 2010; Vita et al. 2012). However,

different relationships can be derived from the same numerical results if the backgroundguiding models are different. Thus, to understand the complex fluid flow process in porous

media mechanistically, we need to establish a mathematical framework in advance, and then

investigate the basic physics by numerical simulations to reduce the uncertainty in it.

Based on the analytical–numerical coupling solution, here we first review some classi-

cal permeability equations. Under the hypothesis of fractal pore-space geometry and using

series–parallel flow resistance mode, we then analytically derive a permeability estimation

model with clear physical background. With the help of LBM, we investigate the basic

physics of the geometry effects on the fluid flow. Finally, we establish a permeability–pore

relationship for fractal porous media, and demonstrate its validity via comparisons between

the existing correlations and the numerical results.

2 Theories and Methodologies

2.1 The Fluid Flow Resistance Model of Porous Media

Since the pioneer work by Kozeny (1927), great efforts have been devoted to establishing

the estimation models of the permeability K of porous media. Carman (1937, 1939, 1956)

modified the original Kozeny’s equation as follows:

K = ϕ3

k (1 − ϕ)2 S 2 , (1)

where ϕ is the porosity, S is the specific surface and k = C f τ 2 is the KC constant, in which τ is the hydraulic tortuosity, and C f is a constant shape factor dependent on the capillary pore

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436 Y. Jin et al.

shape. Assuming the porous media is deposited by monosized solid particles, substituting

for S in Eq. (1) leads to

K

=

ϕ3

C 0τ 2

(1 − ϕ)2

δ2, (2)

where δ is the particle size, e.g., the side length of a cubic block or the diameter of spherical

particles. C 0 has been considered to be a constant relative to the shape factor of the solid

particles which are monosized filled in the porous media.

As pointed out by Bear and Verruijt (1987), the flow rate of a porous media can be

expressed as a function of its flow resistance:

Q = P R

, (3)

where Q is the total flow rate, R is the flow resistance and P denotes the pressure difference

between the inlet and outlet boundary. Similar to Ohm’s law, R follows the parallel and series

calculation models.

Combining Darcy’s law with the definition of flow resistance [see Eq. (3)], the relationship

R = µ L/( AK ) is obtained, where µ is the fluid dynamic viscosity, A and L is the cross-sectional area and sample length of the porous medium, respectively. By borrowing the

concept of electrical resistivity, the fluid flow resistivity Rf can be defined as the quantity

of flow resistance of certain porous media with unit sample length and cross-sectional area,

which yields Rf = µ/K . Together with Eq. (2), the flow resistivity of porous media filledwith monosized solid particles reads:

Rf =C 0(1 − ϕ)2µ

ϕ3

τ 2

δ2 , (4)

which implies that Rf is the function of τ and δ if porosity ϕ and the shape of solid particles

are same.

2.2 Lattice Boltzmann Methods

LBM is actually a mesoscopic description of microscopic physics, and has been used widely

to study the microstructural effect on the fluid flow at pore scale. For simplicity and conve-

nience, the model description and fluids simulations are presented using the classical lattice

Bhatnagar–Gross–Krook (BGK) model over a D2Q9 lattice structure.

In LBM framework, based on the principle of effective conversion between physical and

lattice systems, the real pore space Dp is discretized in terms of a regular lattice with spacing

δ x , time t in terms of a time step δt and velocity space in terms of a small set of velocities cito ensure that ci δt is a vector connecting two adjacent lattice sites (Succi 2001; Dünweg et al.

2007; Sukop and Thorne 2007). For the employed D2Q9 lattice structure (Qian et al. 1992),

a simple-cubic lattice has a set of probability functions f i , representing the mass density of

fluid particles going through one of the 9 discrete velocities ci . Thus, the local mass density

ρ and velocity u at each lattice position x and time t are given by

ρ(x, t ) =8

i=0 f i (x, t ), u(x, t ) =

8i=0 f i (x, t )

ρ(x, t ). (5)

By Chapman–Enskog expansion of the Boltzmann equation, the time evolution of fluid

flow is described by the lattice Boltzmann equation (LBE) (Sukop and Thorne 2007), as:

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f i (x, t ) + Ω i = f i (x + ci δt , t + δt ), (6)where Ω i is the collision operator modifying the populations at x according to the mass and

momentum conservative requirements of

8i=0 Ω i =

8i=0 Ω i ci .

According to BGK model, the collision operator takes the single-relaxation-time approx-

imation (Chen and Doolen 1998; Succi 2001; Qian et al. 1992),

Ω i =δt

τ lbm

f

eqi (x, t ) − f i (x, t )

, (7)

where τ lbm is a dimensionless relaxation time, and f eqi (x, t ) is a quasi-equilibrium distribu-

tion function. And then, its discrete velocities ci are defined as

ci = c ×

(0, 0), i = 0(cos θ, sin θ ), i = 1 : 4, θ = i−1

2 π

√ 2(cos θ, sin θ ) i = 5 : 8, θ = 2i

−9

4 π

, (8)

to recover the Navier–Stokes (NS) equation for the fluid flow, f eqi is constructed by Eq. (9),

and the kinetic viscosity of the fluid (ν) is given by Eq. (10), respectively.

f eqi (x, t ) = ωi ρ(x, t )

1 + 3 u · ci

c2 + 9(u · ci )

2

2c4 − 3u

2

2c2

, (9)

ν = (τ lbm − 0.5)δ2 x

3δt . (10)

with c = δ x /δt and lattice sound speed c2s = c

2

/3 due to the constrains of conservation andisotropy. Equation (10) imposes a constraint on the choice of τ lbm being greater than 0.5 for

a physically correct viscosity (Sukop and Thorne 2007), here we choose τ lbm = 1. And theweight assignments follow ω0 = 4/9, ω1−4 = 1/9, and ω5−8 = 1/36, to satisfy

ωi = 1

for symmetry reasons.

In the complex fractal porous media, there is almost no net fluid motion that exists (Succi

2001). The boundary condition at solid–fluid interfaces can, therefore, physically approxi-

mate the no-slip boundary condition (Jin et al. 2013; Wang et al. 2014; Chen et al. 2013).

For simplicity and without loss of generality, the complete bounce-back scheme is adopted

in our flow simulations.

3 Characteristics of Fractal Porous Media and Their Flow Resistance

Porous media in nature always consist of numerous irregular pores of different sizes spanning

several orders of magnitude in length scales, such as soil, sandstones in oil reservoir, matrix

pores in coal, packed beds in chemical engineering, fabrics used in liquid composite molding

and wicks in heat pipes. These media possess pore microstructure, including both pore sizes

and pore interfaces, and exhibit fractal characteristics, thus called fractal porous media (Adler

and Thovert 1998; Smidt and Monro 1998; Thovert et al. 1990; Krohn and Thompson 1986).

3.1 Fractal Characteristics of Porous Media

In a fractal pore space, the cumulative distribution of pore size λ has been proven to follow

fractal scaling law. Obviously, a natural porous medium is composed of one or more rep-

resentative units of linear size L at which the fractal behavior starts. These representative

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438 Y. Jin et al.

units share the same physical properties, such as porosity, pore structure, pore size range and

transport property as the porous medium in a statistical mean, but contain only one pore/solid

of the largest size. And, the fractal dimension of the pore size distribution, Df , is a function

of the porosity and the ratio of the lower limit to upper limit of self-similar regions ( Yu and

Li 2001):

ϕ(λ ≥ r ) = r

L

d − Df , (11)

where r is the scale, d is the Euclidean space dimension, the fractal dimension Df is in the

range of 0 < Df


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By comparison, Eq. (14) is consistent with what proposed is in Yu et al. (2009) because

these authors assumed that the size of the maximum pore λmax is equal to the linear size of

the representative region/unit of the porous media, and the size of the minimum pores λminis equal to the measuring scale r .

Consequently, the specific area of fractal porous media is expressed by:

S (δ) = As(δ) L d

= ζβs1

L

δmax

L

d −1 δ

δmax

(d −1− Df ), (15)

Equation (15) indicates that the pore–particle surface area admits a fractal scaling law

with δ. Meanwhile, the number-size distributions of the pore–particle surface area and the

solid particles are similar power laws with identical exponent because of the same scaling

behaviors. Instantaneously, the fractal dimensions of pore–particle surface area, the poresand solid particles are the same in a fractal porous medium, being Df .

According to the independent measuring results, some authors considered that the frac-

tal behaviors of the pores, solid particles and pore–particle surface area in a fractal porous

media are different (Perrier and Bird 2002; Perrier et al. 1999; Dathe and Thullner 2005).

It must be noted that the fractal dimension is just a number by which to quantify the scal-

ing invariant degree of a pattern in geometrical or statistical scenes, meaning that just by

the fractal dimension, the geometries of fractal porous media could not be uniquely deter-

mined. Thus, to accurately describe the self-similar property of a porous media, except for

the fractal dimension, the spatial or statistical pattern which scales must be accompanied,

such as the generators of Menger sponge and the PSF model (Perrier et al. 1999). Oth-erwise, by measuring approaches ignoring the spatial arrangement of pores and particles

completely, such as the box-counting method, one will obtain different fractal dimensions

for the physical properties even if they possess the same scaling behaviors actually, as the

results in Perrier and Bird (2002), Perrier et al. (1999), Dathe and Thullner (2005) and Zhou

et al. (2010).

When the porous medium is filled with monosized solid particles, there will be no fractal

behavior of pore/particle sizes. In such a condition, F λ → 0+, thus Df will tend to be −∞theoretically because of Df = log(F λ)/ log(Pλ). Obviously, this is consistent with whatpointed out is in Ghanbarian-Alavijeh and Hunt (2012), in which the authors indicated that

Df = −∞ represents a uniformly grain/pore size distributed porous medium. Denoting byS 0 the specific area of such a medium, from Eq. (15) we get Eq. (16) instantaneously because

of δ = δmax.

S 0 = ζβsδd −1max L d

. (16)

For a two-dimensional nonfractal porous medium, the boundary length of a solid particle

is equal to 4δmax, also βsδd −1max , thus the specific area takes the form of Eq. (16).

As aforementioned, at a measuring scale of pore size r , the minimum measurable size of

solid particles δ takes the value ξ r . Thus, together with Eq. (11) and with respect to the poresize, Eq. (15) is then rearranged into:

S (λ ≥ r ) = S (δ) = ζβsξ Df 1 ξ d −1− Df ϕ

r = C s

ϕ

r . (17)

where ξ 1 = δmax/ L, being a constant in a fractal porous media. For example, ξ 1 = 1/3 instandard Sierpinski carpet.

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440 Y. Jin et al.

Fig. 1 Basic construction of the VmSqLnRl-type fractal porous media. The solid and void phases are denoted

by black and white, respectively. In a space of Euclidean dimension d , the initiator (a) of linear size l defines

the representative region/unit of a porous media, divided into m2

equal parts. At the first iteration step, thegenerator (b) divides the m2 parts into two sets of q2 (solid ) and m2 − q2 (void ) subregions. At the next step,each of the void subregions is replaced by a reduced replicate of the generator

3.2 Construction of Porous Media with Arbitrary Fractal Dimension

For simplicity and without loss of generality, our attention is focused on the permeability–pore

relationship of fractal porous media in two-dimensional context. In our previous studies (Jin

et al. 2013; Wang et al. 2014), an algorithm was proposed to construct a self-similar fractal

object with arbitrary fractal behavior and without blind pores, denominated as VmSqLnRl-type fractals. The modeling algorithm is briefly as following: (1) define the representative

( R) region of linear size l as an initiator in a space of Euclidean dimension d (Fig. 1a). This

region is then divided into m × m small subregions of linear size l / m and set to be void phase(V ); (2) solidify (S ) the very central region with size q × l/m (Fig. 1b); (3) repeat step (2)in the reset void subregions for n times until the predefined porosity is achieved. The basic

construction of a V5S3L2Rl-type porous media is demonstrated in Fig. 1, where Fig. 1a is an

initiator or the representative region of a fractal porous media, Fig. 1b is the generator and

Fig. 1c is a V5S3L2Rl-type fractal porous medium with Df = log 14/ log 5.It is noticeable that, with successive iterations, pores and solid particles keep their sizes

reduced, while their number increased (Adler and Thovert 1993; Tarafdar et al. 2001). In

a representative region of linear size l, λmax = l and λmin = l are satisfied at the verybeginning. After one subdivision, λmin (also r ) turns into l /m, δmin = δmax = lq/m; after ntimes of iterations, λmin = lm−n, δmin = lq/mn . Hence, for a VmSqLnRl-type fractal porousmedium, λmax and λmin are l and l m

−n , respectively, while δmax = lq/m and δmin = qλmin.In these media, ξ and ξ 1 are expressed by

ξ = q, ξ 1 =q

m, (18)

and the Frequency of pore growth (F λ) and the Lacunarity of pore size ( Pλ) are expressedby

Pλ = md , F λ = md − qd . (19)

Thus, the fractal dimension of pore area and porosity are given by Eqs. (20) and (21),

respectively.

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Fig. 2 Relationship between real specific area and that by Eq. (22) of different VmSqLnRl-type porous media,

where the solid line is y = x for reference

Table 1 Pore structure

parameter ζ of some

VmSqLn-type porous media by

the best-fitted linear model

between βs(q/m)d −

1(ϕ/λmin)

and the real specific area

Porous media type d / Df ζ

V3S1Ln 1.0566 1.6000

V4S2Ln 1.1158 1.5000V5S3Ln 1.1611 1.4545

V7S5Ln 1.2246 1.4118

V9S3Ln 1.0275 1.1429

V11S3Ln 1.0164 1.1089

V11S5Ln 1.0507 1.1294

V11S7Ln 1.1214 1.1803

V25S3Ln 1.0023 1.0423

V45S7Ln 1.0032 1.0233

Df =ln F λ

ln Pλ= ln (m

d − qd )ln m

, (20)

ϕ =

m Df −d n

=

λd max − δd maxλd max

n. (21)

Actually, following the PSF model (Perrier et al. 1999), F λ is ranged in (0, md −qd ]. When

F λ → 0+, Df → −∞ theoretically (Ghanbarian-Alavijeh and Hunt 2012). According toEq. (17), the specific area of VmSqLnRl-type porous media is then expressed by Eq. (22),

which is validated from different VmSqLnRl-type porous media as shown in Fig. 2.

S (λ ≥ λmin) = ζβsqd −1

m Df

ϕ

λmin. (22)

Some pore structure parameter ζ are listed in Table 1. It is noted that ζ will approach to

d / Df as Df increases.

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442 Y. Jin et al.

3.3 Flow Resistance of Fractal Porous Media

According to Eq. (21), the porosity of a VmSqL1Rl-type porous medium is determined only

by the parameters m and q , so is the hydraulic tortuosity due to the spatial arrangement of

void and solid phase determined by the construction process in such type porous medium.And in a VmSqLnRl-type fractal porous medium, the space is filled with VmSqL1R(lm−n )units and solid phase, as shown in Fig. 1. For clarity, the porosity and hydraulic tortuosity of

VmSqL1Rl-type porous media are denoted by ϕ1 and τ 1, respectively. Taking them together,

we can rewrite Eq. (4) into the form of Eq. (23) for certain fluid flow through VmSqL1Rl-type

porous media with determined m and q .

Rf = C 11

δ2 ∝ 1

δ2, (23)

where

∝ stands for “proportional to,” δ

= ξ 1l for porous media of VmSqL1Rl-type, and

constant C 1 reads

C 1 =C 0(1 − ϕ1)2µτ 21

ϕ31. (24)

Equation (23) implies that the spatial distribution of δ−2 satisfies the parallel and seriesmode.

For convenience, denote by R(d /i ) f the flow resistivity of the VmSqLiRl-type porous

medium. Thus, when a VmSqL1Rl-type porous medium turns into the VmSqL2Rl-type after

one iteration, its flow resistivity will change from R(d /1) f to R

(d /2) f . According to Eq. (23),

R(d /1) f takes the value of C 1(qd /m)

−2, while R(d /2) f can be calculated from Eq. (25) by theparallel–series spatial integration in a two-dimensional space.

R(d /2) f = R

(dm −1)/1

f

m − q

m+ q

m − q

. (25)

Replacing the expression (m − q)/m + q/(m − q) by f (m, q) for short, we can expressthe average flow resistivity of VmSqLnRl-type porous media by

R(d /n) f = m

2 f (m, q)n−1

R(d /1) f

= C 1

m2 f (m, q)n−1 1

δ2max. (26)

Substituting δmax = q L/m into Eq. (26) yields

R(d /n) f = C 1

m2 f (m, q)

nq2 f (m, q)

1

L 2. (27)

where L is the linear size of the representative unit of a fractal porous medium, also the size

of the largest pore.

Although Eq. (27) is derived from the regular model, but it is applied to porous media withsolid particles distributed randomly according to the parallel–series model. To explain intu-

itively, we demonstrate a VmSqL2Rl-type model with the first-level solid particle distributed

randomly on porous media composed of VmSqL1Rlm−1-type models, see Fig. 3.Denoting by R1, R2 and R3 the flow resistivity of VmSqL1Rl-, VmSqL1Rl/m- and

VmSqL2Rl-type model, respectively, by parallel–series mode, we can get the relationship

R3 = R2 ((m − q)/m + q/(m − q)) for arbitrary a and b, same as Eq. (25) derived from

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Fig. 3 Demonstration of a

VmSqL2Rl-type model with solid

particle distributed randomly

regular model. So, the flow resistivity model of Eq. (27) is not only applied to regular fractal

porous media, but also to the random ones on average. However, Ghanbarian-Alavijeh and

Hunt (2012) have pointed out that the parallel–series approach does not model connectivity

among pores, which are more descriptive rather than predictive. But we think that the random-

ness effect on the transport property is in the charge of the semiempirical parameter of C 0.

Thus, according to the relationship between the permeability (K ) and flow resistivity

R(d /n) f and taking Eqs. (17), (21) and (27) all together, we obtain the permeability estimation

model of VmSqLnRl-type porous media, reads

K = ζ 2β2s

C 0

ϕ1ϕ2

S (λ)2T 2 (28)

where T 2 =

τ 21 f (m, q)n−1 is introduced as a geometrical parameter accounting for the

hydraulic tortuosity of a fractal porous medium.

4 Results and Discussions

As a general function, Eq. (28) should be able to describe the permeability–pore relationshipfor porous media without fractal behavior. In such a condition, n = 1, ϕ1 = ϕ, and S takesthe value of S 0. Then, Eq. (28) is reduced to

K = ϕ31

C 0τ 21

δ2maxδmax

L

d 2

= ϕ31

C 0τ 21

δ2max

(1

−ϕ1)

2 (29)

which is equal to the modified KC function defined in Eq. (2) for porous media deposited by

monosized particles.

To make Eq. (28) practical, the parameter T must be described by the fundamental prop-

erties contained in Eq. (2). As aforementioned, the hydraulic tortuosity depends only on

the spatial arrangement of solid particles, not on their sizes. That is, there is no effect on

the hydraulic tortuosity for a determined spatial pattern when it is zoomed in or out. For

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444 Y. Jin et al.

Fig. 4 Relationships between ϕ1 and τ 1, ϕ1 and τ 2. The circles represent the experimental data, and the solid

line its best power law-fitted result

example, in a VmSqLnRl-type porous medium, τ will alter with parameters m , q and n , but

not with the linear size of the representative unit L. Meanwhile, a certain spatial pattern

leads to a determined porosity; however, for a porous medium with a determined porosity,

its hydraulic tortuosity will alter with the spatial arrangement of solid phase. As pointed out

in Valdés-Parada et al. (2011), the hydraulic tortuosity cannot be a function of porosity only.

To explore the tortuosity–porosity relationship, we calculate τ 1 of different VmSqL1-type

porous media by the method in Jin et al. (2015) based on the velocity fields simulated by the

LBM. Meanwhile, inspired by the construction process of fractal porous media, we rewrite

T 2 as τ 22 f (m, q)n , with τ 1/ f (m, q)

0.5 denoted by τ 2, and plot the relationship between ϕ 1and τ 1, τ 2, as well as that between ϕ and f (m, q)

n/2 (see Figs. 4, 5).

In Fig. 4, one is hard to find a clear relation between ϕ1 and τ 1. But the best-fitted

result indicates that τ 2 and ϕ1 follows the relation τ 2 = α1ϕ1 after combining the spatialarrangement. Meanwhile, the relationship between f (m, q)n/2 and ϕ follows f (m, q)n/2 =α2ϕ

− , as shown in Fig. 5. α1 and α2 are the fitted coefficients of power law models and bothapproximate to 1.0; and the fitted exponents are almost the same, denoted by and satisfying

= 0.71.As we all know, the tortuosity was first proposed as a fundamental parameter to predict

the permeability (Carman 1937), as shown in Eq. (1). But, due to the ambiguous concept, the

tortuosity has several definitions, some are based on geometrical approaches, while others

prefer to the hydraulic ones, for details one can refer to the literatures published recently

(Ghanbarian et al. 2013a, b).

Vast investigations have shown that the tortuosity τ is related to the porosity, and one of

the most invoked tortuosity–porosity models is given by:

τ = 1 − pln (ϕ) (30)where p is a coefficient and will be semiempirically modified by the microstructure of

porous media, as the estimations have been reported in various numerical or experimental

results (Matyka et al. 2008; Comiti and Renaud 1989; Mauret and Renaud 1997).

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Fig. 5 Relationship between ϕ and f (m, q)n/2. The circles represent the raw data, and the solid line the best

power law-fitted result

Other investigations showed that the tortuosity–porosity relationship admits a power law

model, which reads:

τ = ϕ−β (31)

where β is an empirical exponent. Mota et al. (2001) found β = 0.4 for binary mixturesof spherical particles when measuring the conductivity of porous media; Liu and Masliyah

(1996a) reported that β = 0.5 for random packs of grains with porosity ϕ > 0.2. Recently,Ghanbarian et al. (2013b) found β = 0.378 in their geometrical tortuosity model for porousmedia whose pore size distribution is narrow.

Actually, the hydraulic tortuosity cannot be a function of porosity only (also shown in

Fig. 4 and pointed out in Valdés-Parada et al. 2011), and β cannot be universal (Ghanbarianet al. 2013a). More generally, Henderson et al. (2010) assumed a fractal scale law as follows:

τ = C τ ϕ− Dτ , (32)

where C τ and Dτ are the fractal coefficient and fractal exponent of τ , respectively.

For the consolidated porous media, Liu and Masliyah (1996b) found C τ = 1.61 and Dτ = 1.15. By means of fractal geometry, together with the scale–measurement relationproposed by Wheatcraft and Tyler (1988) for a fractal capillary tube, Feng and Yu (2007)

derived a geometrical tortuosity estimation model, approximately follows:

τ ≈ Df Df + DT − 1

L

λmin

DT−1(33)

where DT was defined as the tortuosity fractal dimension of the curve in Wheatcraft and Tyler

(1988), and λmin is the diameter of the minimum capillary tube. Actually, Eq. (33) admits a

fractal scaling law between tortuosity and porosity in terms of Eq. ( 11).

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446 Y. Jin et al.

Fig. 6 Relationship between ϕ1/ϕ and T , with T calculated by T =

τ 21 f (m, q)n−1

and ϕ1, ϕ by

Eq. (21). The solid line is the best fitted power law model to the relationship represented by circles

Recently, based on the geometry of standard Sierpinski carpet named by V3S1-type

fractal here, Li and Yu (2011) proposed a tortuosity–porosity model which yields τ =(19/18)ln ϕ/ ln (8/9) (Ghanbarian et al. 2013b). For a V 3S 1-type porous medium, ϕ1 = 8/9according to Eq. (21), thus the tortuosity model of Li and Yu (2011) can be rewritten into

τ = (19/18)ln ϕ/ ln ϕ1 , indicating that the tortuosity is a function of ϕ , ϕ1 and the prefractaldistribution of solid phase. Even though these models are either based on the highly ideal-

ized geometry or from special porous media, they do provide a mathematical framework to

establish the tortuosity–porosity relationship. Of course, all these tortuosity–porosity models

are validated in a certain range of porosity due to the percolation threshold, because at very

low porosity, the system cannot even percolate that makes defining a tortuosity meaningless

(Ghanbarian et al. 2013a). Obviously, the validity of porosity range is beyond our objective

of the present work.

Thus, in terms of the relationship between τ 2 and ϕ1, as well as that between f (m, q)n/2

and ϕ, it is reasonable to consider that T is a function of ϕ1/ϕ, following the power law

relationship expressed in Eq. (34), which is then validated by the best power law-fitted result

in Fig. 6.

T =

ϕ1

ϕ

. (34)

The difference between and the exponents proposed before (Henderson et al. 2010; Mota

et al. 2001; Liu and Masliyah 1996a, b) should be ascribed to the fractal behavior and thegeometrical shape of solid particles.

In terms of the characteristics of fractal porous media, the relationship between ϕ1/ϕ and

the size distribution of particles satisfies:

ϕ1

ϕ=

δmax

δmin

d − Df . (35)

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Taking into account Eqs. (34)and (35), the hydraulic tortuosity for a fractal porous medium

is then defined as:

τ

=C τ T

=C τ

δmax

δmin

(d − Df ), (36)

where C τ is introduced for the geometrical considerations and spatial arrangement of solid

phase, with C τ > 1. In porous media filled with monosized solid particles, the relationship

C τ = τ 1 is satisfied.Equation (36) interrelates the tortuosity with determined parameters with clear physical

meanings, such as ϕ1, ϕ and tortuosity of the prefractal solid distribution τ 1 in a power law

model, which can be used directly for porous media with solid particles randomly distributed

because the derivation process using parallel–series flow resistance mode without special

assumption. And Eq. (36) is in accord with the real situation:

1. If a porous medium is filled with monosized solid particles, its hydraulic tortuosity willbe mainly affected by the spatial arrangement of solid phase;

2. For the fractal porous media with the same Df , the larger the ratio between δmax and δmin,

the higher the value of the hydraulic tortuosity;

3. For any porous media with Df → d , the hydraulic tortuosity τ → 1, because Df → d means that the pore space will be ultimately filled with void phase, consequently resulting

in a straight-channel flow;

4. Because δmax/δmin ≥ 1 and (d − Df ) > 0, that hydraulic tortuosity satisfies τ ≥ 1 isalways true, consistent with the tortuous nature of fluid flow through porous media.

For visual comparison, the streamlines of some VmSq-type porous media are demonstratedin Fig. 7. In the LBM simulations, no-slip boundary conditions were imposed on the top and

bottom walls and periodic boundary conditions were assumed at the inlet (left wall) and outlet

(right wall). The flow was driven by an external force field whose magnitude was chosen

so that the Reynolds number Re < 1 to ensure the Darcy’s flow. For short, the hydraulic

tortuosity of fluid flow is denoted by T VmSqLnRl .

In Fig. 7, the streamlines indicate that: (1) T V3S1L3Rl > T V3S1L2Rl > T V3S1L1Rl, which

is due to that the ratio of δmax to δmin in V3S1L3Rl-type porous media is larger than that

in V3S1L2Rl-type porous media, even if they share the same Df ; (2) T V3S1L2Rl > T V9S3L2Rl

although the porosity of V3S1L2Rl-and V9S3L2Rl-type porous media is same. That is becausethe hydraulic tortuosity will alert with Df , and a large fractal dimension will result in a small

hydraulic tortuosity, as aforementioned; (3) For a porous medium filled with monosized

solid particles, as that in Fig. 7a, it could be denominated by any kind of V3xSxL1Rl-type

porous media, such as V3S1L1Rl-, V6S2L1Rl- and V9S3LiRl-type as well. So, the hydraulic

tortuosity of VmSqL1Rl-type porous media is determined by q /m, because q /m determines

the spatial arrangement of solid phase.

Obviously, Eq. (36) is consistent with the numerical result. Substituting Eqs. (17) and (36)

into Eq. (28) gives:

K = 1C 0τ

21

ϕ3

−2

1 ϕ2

(1 − ϕ1)2δ2min (37)

According to Eq. (35), we can then rewrite Eq. (37) as

K = 1C 0τ

21

ϕ3−21 ϕ2

(1 − ϕ1)2

ϕ1

ϕ

2 Df −d

δ2max. (38)

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448 Y. Jin et al.

Fig. 7 (Color) The fluid flow streamlines of some VmSqLn-type porous media when the LBM simulations

reached a stable condition. a–d are the flow streamlines of V3S1L1Rl-, V3S1L2Rl-, V3S1L3Rl- and V9S3L2Rl-

type porous media, respectively. The color represents the relative magnitude of the dimensionless velocity

which was normalized by the max velocity of a flow

The correlation between the analytical permeability ( K as) by Eq. (38) without parameterC 0 and that calculated from the LBM simulations is investigated for different VmSqLn-

type porous media of various linear sizes ( K ns, calculated from the LBM simulations).

The relationships between K ns and C 0 K as all yield highly linear correlation(not shown

here), but the coefficient C 0 will alter with the pore structure characterized by m, q and

n. Substituting the fitted C 0 into Eq. (38), we note that the values of permeability found

in LBM simulations are in excellent agreement with those estimated by Eq. (38), some

small deviation should be ascribed to the numerical precision and calculation errors (Fig. 8).

The corresponding parameters of the porous media for comparison in Fig. 8 are listed in

Table 2.

As aforementioned, parameter C 0 has been considered to be a constant in the porousmedia filled by monosized particles. But in fractal porous media, C 0 is found to alert with the

porosity and pore structure, as shown in Fig. 9. The fitted result shows that C 0 is a function

of the porosity ϕ and Df , approximately follows:

C 0 ≈ 32ϕ1

d − Df (39)

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Fig. 8 (Color) Relationship between K as and K ns. The symbols represent the relationship between K as and

K ns from different VmSqLn-type porous media, and the solid line serves as a reference, where y = x indicatesthat K as has an identical value to K ns. l.u. is the dimensionless lattice unit

Table 2 The corresponding

parameters of the VmSqLn-type

porous media in Fig. 8 for

validation of Eq. (38)

Porous media type m q Df λmax/λmin n

V3S1 3 1 1.893 3–35 1–5

V4S2 4 2 1.792 4–44 1–4

V5S1 5 1 1.975 5–53 1–3

V6S2 6 2 1.934 6–63 1–3

V9S3 9 3 1.946 9–92 1–2

In a fractal porous medium, that Df = d means the Euclidean space is fully occupied byvoid phase. In such a condition, the flow rate Q in a circular capillary is usually represented

by the classical H–P equation, reads:

Q = π λ4P

128µ L, (40)

where λ is the capillary diameter. Meanwhile, according to the Darcy’s law, Q interrelates

the permeability K by:

Q = K APµ L

. (41)

where A the total cross-sectional area. Substituting Eq. (40) into Eq. (41) yields:

K = π λ4

128 A. (42)

Because that the Euclidean space is fully occupied by void space, thus A = π λ2/4 andλ = L are satisfied. Consequently, Eq. (42) is rearranged into K = L 2/32. For a VmSqLnRl-type porous medium, when Df = d , the specific area S = βs L

d −1 Ld

, T = 1 (because τ 1 = 1and

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450 Y. Jin et al.

Fig. 9 Relationship between ϕ1

d − Df and the fitted coefficients C 0. Symbol markers represent the fitted C 0from different porous models, and the solid line is the best-fitted power law model

f (m, q) = 1), ζ 2 = 1 (see Table 1, and because Pλ → 0 when Df → d ), and ϕ1 = ϕ = 1are all satisfied. Together with Eq. (28), the relation C 0

=32 is obtained, which is obviously

consistent with the fitted result that is approximately expressed by Eq. (39), where the smallerrors should be ascribed to the numerical precision.

In the meantime, taking Eqs. (11) and (39) both into account, we can get C 0 ∝ λmin/ L ,which is consistent with that pointed out by Xu and Yu (2008).

Together with Eqs. (28), (34) and (35), the KC constant k approximately yields:

k ≈ 32 λmin L

ϕ1

ϕ

2

= 32 P

2( Df −d )λ

λmin L

1+2( Df −d )

. (43)

Equation (43) indicates that the KC constant k is a function of the fractal dimension, range

and the Lacunarity of pore size.

1. If Df → d , then k → 32, meaning the fluid flow in porous media is approaching aPoiseuille flow;

2. If the scaling characteristics is weak or there is no fractal behavior of the pore size

distribution. Thus, ϕ → ϕ1 and then k → 32 λmin L ;3. If the size range of pores is fixed, because Pλ <

Lλmin

and Df ≤ d are always satisfied,meaning that k will decrease with the fractal dimension Df ; in terms of Eq. (11), theporosity increases with fractal dimension monotonically if the pore size range is fixed

(Jin et al. 2013), thus the smaller the porosity is, the larger is the value of k .

The variation characteristic indicates that the KC constant is not a constant, which charac-

terizes the difference of fluid flow in a porous media between that in a straight tube. A large

deviation of a fluid flow from the Poiseuille’s flow will result in a small k , vice versa.

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Even though the variation trend of the KC constant proposed here is consistent with that

in Xu and Yu (2008), their estimation expressions are not the same. The main difference is

that we use intersected pore space other than the nonintersecting assumption of fluid paths.

5 Conclusions

In this study, we investigate the scaling invariant characteristics of the fractal porous media.

Following the analytical–numerical coupling solution, the scaling effects of pores on the per-

meability, hydraulic tortuosity and the KC constant are fully analyzed, and some conclusions

are drawn as follows:

(1) In fractal porous media, the permeability–porosity relationship follows power law cor-

relation with exponent being 2, not the cubic law described in KC equation;

(2) The hydraulic tortuosity of the fractal porous medium is a function of the size range of

solid particles, fractal dimension of pore size and the prefractal solid distribution; the

tortuosity–porosity relation admits a fractal scaling law with tortuosity fractal dimension

approximately being 0.71 in two-dimensional context;

(3) The KC constant is not a constant, which characterizes the derivation of the fluid flow

from the Poiseuille flow. If the pore size fractal dimension is deterministic, a large value

of λmin/ L will yield a small KC constant; and KC constant will decrease with the fractal

dimension if the range of pore size is fixed;

(4) When the Euclidean space is occupied fully by the void phase, the fluid flow in porous

media will turn into the Poiseuille flow; while if there is no fractal behavior of poressize, the media’s permeability follows KC equation characterizing the fluid flow in the

pore space filled by monosized particles. The shift of these two kinds of fluid flow is

controlled by the fractal dimension and the range of pore size distribution.

References

Adler, P.M., Thovert, J.F.: Fractal porous media. Transp. Porous Med. 13, 41–78 (1993)

Adler, P.M., Thovert, J.: Real porous media: local geometry and macroscopic properties. Appl. Mech. Rev.

51, 537–585 (1998)Ahmadi, M.M., Mohammadi, S., Hayati, A.N.: Analytical derivation of tortuosity and permeability of mono-

sized spheres: a volume averaging approach. Phys. Rev. E 83, 026312 (2011)

Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)

Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution. Springer, New York (1987)

Cai, J.C., Yu, B.M., Zou, M.Q., Mei, M.F.: Fractal analysis of surface roughness of particles in porous media.

Chin. Phys. Lett. 27, 024705 (2010)

Carman, P.C.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166 (1937)

Carman, P.C.: Permeability of saturated sands, soils and clays. J. Agric. Sci. 29, 262–273 (1939)

Carman, P.C.: Flow of Gases Through Porous Media. Butterworths Scientific Publications, London (1956)

Chen, Q., Zhang, X.B., Zhang, J.F.: Improved treatments for general boundary conditions in the lattice Boltz-

mann method for convection–diffusion and heat transfer processes. Phys. Rev. E 88, 033304 (2013)

Chen, S.Y., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364(1998)

Collins, R.E.: Flow of Fluids Through Porous Materials. Reinhold Pub. Corp, New York (1961)

Comiti, J., Renaud, M.: A new model for determining mean structure parameters of fixed beds from pressure

drop measurements: application to beds packed with parallelepipedal particles. Chem. Eng. Sci. 44,

1539–1545 (1989)

Costa, A.: Permeability–porosity relationship: a reexamination of the Kozeny–Carman equation based on a

fractal pore-space geometry assumption. Geophys. Res. Lett. 33, L2318 (2006)

1 3


20/21

452 Y. Jin et al.

Croce, G., D’Agaro, P., Nonino, C.: Three-dimensional roughness effect on microchannel heat transfer and

pressure drop. Int. J. Heat Mass Transf. 50, 5249–5259 (2007)

Dathe, A., Thullner, M.: The relationship between fractal properties of solid matrix and pore space in porous

media. Geoderma 129, 279–290 (2005)

Degruyter, W., Burgisser, A., Bachmann, O., Malaspinas, O.: Synchrotron X-ray microtomography and lattice

Boltzmann simulations of gas flow through volcanic pumices. Geosphere 6, 470–481 (2010)Duda, A., Koza, Z., Matyka, M.: Hydraulic tortuosity in arbitrary porous media flow. Phys. Rev. E 84, 036319

(2011)

Dullien, F.A.: Porous Media: Fluid Transport and Pore Structure. Academic Press, Salt Lake City (1991)

Dünweg, B., Schiller, U.D., Ladd, A.J.C.: Statistical mechanics of the fluctuating lattice Boltzmann equation.

Phys. Rev. E 76, 036704 (2007)

Feng, Y.F., Yu, B.M.: Fractal dimension for tortuous streamtubes in porous media. Fractals 15, 385–390 (2007)

Ghanbarian, B., Hunt, A.G., Ewing, R.P., Sahimi, M.: Tortuosity in porous media: a critical review. Soil Sci.

Soc. Am. J. 77, 1461–1477 (2013)

Ghanbarian, B., Hunt, A.G., Sahimi, M., Ewing, R.P., Skinner, T.E.: Percolation theory generates a physically

based description of tortuosity in saturated and unsaturated porous media. Soil Sci. Soc. Am. J. 77,

1920–1929 (2013)

Ghanbarian-Alavijeh, B., Hunt, A.G.: Comments on More general capillary pressure and relative permeabilitymodels from fractal geometry by Kewen Li. J. Contam. Hydrol. 140–141, 21–23 (2012)

Ghanbarian-Alavijeh, B., Hunt, A.: Unsaturated hydraulic conductivity in porous media: percolation theory.

Geoderma 187–188, 77 (2012)

Guarracino, L.: Estimation of saturated hydraulic conductivity Ks from the van Genuchten shape parameter.

Water Resour. Res. 43, W11502 (2007)

Happel, J.: Viscous flow relative to arrays of cylinders. AiChE J. 5, 174–177 (1959)

Henderson, N., Brêttas, J.C., Sacco, W.F.: A three-parameter Kozeny–Carman generalized equation for fractal

porous media. Chem. Eng. Sci. 65, 4432–4442 (2010)

Jin, Y., Song, H.B., Hu, B., Zhu, Y.B., Zheng, J.L.: Lattice Boltzmann simulation of fluid flow through coal

reservoir’s fractal pore structure. Sci. China Earth Sci. 56, 1519–1530 (2013)

Jin, Y., Dong, J.B., Li, X., Wu, Y.: Kinematical measurement of hydraulic tortuosity of fluid flow in porous

media. Int. J. Mod. Phys. C 26, 1550017 (2015)Kandhai, D., Vidal, D.J.E., Hoekstra, A.G., Hoefsloot, H., Iedema, P., Sloot, P.M.A.: Lattice-Boltzmann and

finite element simulations of fluid flow in a SMRX Static Mixer Reactor. Int. J. Numer. Methods Fluids

31, 1019–1033 (1999)

Koponen, A., Kataja, M., Timonen, J.: Permeability and effective porosity of porous media. Phys. Rev. E 56,

3319–3325 (1997)

Kozeny, J.: Uber Kapillare Leitung Des Wassers in Boden. Stizungsber. Akad. Wiss. Wien 136, 271–306

(1927)

Krohn, C.E., Thompson, A.H.: Fractal sandstone pores: automated measurements using scanning-electron-

microscope images. Phys. Rev. B. 33, 6366 (1986)

Ladd, A.J.C.: Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2.

Numerical results. J. Fluid Mech. 271, 311–339 (1994)

Liu, S., Masliyah, J.H.: Single fluid flow in porous media. Chem. Eng. Commun. 148, 653–732 (1996a)Liu, S., Masliyah, J.H.: Steady developing laminar flow in helical pipes with finite pitch. Int. J. Comput. Fluid

Dyn. 6, 209–224 (1996b)

Li, J., Yu, B.: Tortuosity of flow paths through a Sierpinski carpet. Chin. Phys. Lett. 28, 34701 (2011)

Mandelbrot, B.B.: The Fractal Geometry of Nature. Macmillan, New York (1983)

Matyka, M., Khalili, A., Koza, Z.: Tortuosity–porosity relation in porous media flow. Phys. Rev. E 78, 026306

(2008)

Mauret, E., Renaud, M.: Transport phenomena in multi-particle systems. Limits of applicability of capillary

model in high voidage beds-application to fixed beds of fibers and fluidized beds of spheres. Chem. Eng.

Sci. 52, 1807–1817 (1997)

Mortensen, N.A., Okkels, F., Bruus, H.: Reexamination of Hagen–Poiseuille flow: shape dependence of the

hydraulic resistance in microchannels. Phys. Rev. E 71, 057301 (2005)

Mota, M., Teixeira, J.A., Bowen, W.R., Yelshin, A.: Binary spherical particle mixed beds porosity and perme-ability relationship measurement. Trans. Fit. Soc. 1, 101–106 (2001)

Nasta, P., Vrugt, J.A., Romano, N.: Prediction of the saturated hydraulic conductivity from Brooks and Corey’s

water retention parameters. Water Resour. Res. 49, 2918–2925 (2013)

Nithiarasu, P., Ravindran, K.: A new semi-implicit time stepping procedure for buoyancy driven flow in a fluid

saturated porous medium. Comput. Methods Appl. Mech. 165, 147–154 (1998)

1 3


21/21


Panda, M.N., Lake, L.W.: Estimationof single-phase permeability from parameters of particle-size distribution.

AAPG Bull. 78, 1028–1039 (1994)

Perrier, E., Bird, N., Rieu, M.: Generalizing the fractal model of soil structure: the pore solid fractal approach.

Geoderma 88, 137–164 (1999)

Perrier, E.M.A., Bird, N.R.A.: Modelling soil fragmentation: the pore solid fractal approach. Soil Tillage Res.

64, 91–99 (2002)Pitchumani, R., Ramakrishnan, B.: A fractal geometry model for evaluating permeabilities of porous preforms

used in liquid composite molding. Int. J. Heat Mass Transf. 42, 2219–2232 (1999)

Qian, Y.H., D’Humières, D., Lallemand, P.: Lattice BGK models for Navier–Stokes equation. Europhys. Lett.

17, 479–488 (1992)

Rahli, O., Tadrist, L., Miscevic, M., Santini, R.: Fluid flow through randomly packed monodisperse fibers: the

Kozeny–Carman parameter analysis. J. Fluids Eng. 119, 188–192 (1997)

Rawls, W.J., Brakensiek, D.L., Logsdon, S.D.: Predicting saturated hydraulic conductivity utilizing fractal

principles. Soil Sci. Soc. Am. J. 57, 1193–1197 (1993)

Sahimi, M.: Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and

simulated annealing. Rev. Mod. Phys. 65, 1393–1534 (1993)

Sangani, A.S., Acrivos, A.: Slow flow past periodic arrays of cylinders with application to heat transfer. Int.

J. Multiph. Flow 8, 193–206 (1982)Smidt, J.M., Monro, D.M.: Fractal modeling applied to reservoir characterization and flow simulation. Fractals

6, 401–408 (1998)

Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford, New York (2001)

Sukop, M.C., Thorne, D.T.: Lattice Boltzmann Modeling : An Introduction for Geoscientisits and Engineers.

Springer, New York (2007)

Tarafdar, S., Franz, A., Schulzky, C., Hoffmann, K.H.: Modelling porous structures by repeated Sierpinski

carpets. Phys. A 292, 1–8 (2001)

Thovert, J.F., Wary, F., Adler, P.M.: Thermal conductivity of random media and regular fractals. J. Appl. Phys.

68, 3872–3883 (1990)

Valdés-Parada, F.J., Porter, M.L., Wood, B.D.: The role of tortuosity in upscaling. Transp. Porous Med. 88,

1–30 (2011)

Vita, M.C., De Bartolo, S., Fallico, C., Veltri, M.: Usage of infinitesimals in the Mengers Sponge model of porosity. Appl. Math. Comput. 218, 8187–8195 (2012)

Wang,B.Y., Jin, Y., Chen, Q., Zheng, J.L.,Zhu, Y.B.,Zhang, X.B.: Derivation of permeability–pore relationship

for fractal porous reservoirs using series-parallel flow resistance model and lattice Boltzmann method.

Fractals 22, 1440005 (2014)

Wheatcraft, S.W., Tyler, S.W.: An explanation of scale-dependent dispersivity in heterogeneous aquifers using

concepts of fractal geometry. Water Resour. Res. 24, 566–578 (1988)

Xu, P., Yu, B.M.: Developing a new form of permeability and Kozeny–Carman constant for homogeneous

porous media by means of fractal geometry. Adv. Water Resour. 31, 74–81 (2008)

Young, I.M., Crawford, J.W.: The fractal structure of soil aggregates: its measurement and interpretation. Eur.

J. Soil. Sci. 42, 187–192 (1991)

Yu, B.M.: Comments on A fractal geometry model for evaluating permeabilities of porous preforms used in

liquid composite molding. Int. J. Heat Mass Transf. 44, 2787–2789 (2001)Yu, B.M., Li, J.H., Li, Z.H., Zou, M.Q.: Permeabilities of unsaturated fractal porous media. Int. J. Multiph.

Flow 29, 1625–1642 (2003)

Yu, B., Zou, M., Feng, Y.: Permeability of fractal porous media by Monte Carlo simulations. Int. J. Heat Mass

Transf. 48, 2787–2794 (2005)

Yu, B., Cai, J., Zou, M.: On the physical properties of apparent two-phase fractal porous media. Vadose Zone

J. 8, 177–186 (2009)

Yu, B.M., Li, J.H.: Some fractal characters of porous media. Fractals 9, 365–372 (2001)

Zhou, H., Perfect, E., Li, B., Lu, Y.: Comments on On the physical properties of apparent two-phase fractal

porous media. Vadose Zone J. 9, 192 (2010)

scaling invariant effects on the permeability of fractal

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