Analysis of permeabilities for slug flow in fractal porous media

International Communications in Heat and Mass Transfer 88 (2017) 194–202

Contents lists available at ScienceDirect

International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Analysis of permeabilities for slug flow in fractal porous media a,b

Tongjun Miao a b

a

b

MARK

a,⁎

, Zhangcai Long , Aimin Chen , Boming Yu

School of Physics, Huazhong University of Science and Technology,1037 Luoyu Road, Wuhan, PR China College of Physics and Electronic Engineering, Xinxiang University, Xinxiang 453003, Henan, PR China

A R T I C L E I N F O

A B S T R A C T

Keywords: Slug flow Permeability Porous media Fractal

Slug flow is one of types of flow in two-phase flow in porous media, and this type of flow widely exists in oil and gas pipelines, underground water reservoirs, and nuclear reactor cooling systems, etc. Study of the mechanisms and characteristics of slug flow in porous media has the great significance in the reservoir engineering, power engineering, aerospace engineering, and chemical engineering etc. In this paper, we propose analytical models for seepage characteristics, both permeabilities and relative permeabilities, for slug flow in a capillary by unit cell approach. Then, we extend the methodology to analyze the seepage characteristics of slug flow in fractal porous media. The proposed relative permeabilities for slug flow in porous media are expressed as a function of micro-structural parameters of porous media and fluid properties, such as maximum and minimum capillary sizes, fractal dimensions, the surface tension, as well as capillary numbers. The parametrical effects on the relative permeabilities are also investigated. The validity of the proposed model for slug flow is verified by comparing the model predictions with the available experimental data.

1. Introduction Slug flow is one of the most likely existing forms of two-phase flows, and slug flow widely exists in oil and gas pipelines, water/oil reservoirs, nuclear reactor cooling systems, heat and mass transfer in fuel cells, etc. [1–4]. Study of the mechanisms and characteristics of slug flow in porous media will help us to understand the flow behaviors in power engineering, aerospace engineering, chemical engineering and petroleum engineering etc. Many researchers carried out a lot of theoretical and experimental investigations on flow characteristics of slug flow in single pipe [5–10]. Bretherton [11] studied the motion of infinitely long bubbles in a circular horizontal capillary with a matched-asymptotic method. Nickin et al. [12] proposed a drift flux model for slug flow in a vertical capillary. Later, Bendisken [13] investigated the motion of long bubbles in inclined tubes by experiments. He found that the Nickin's model agreed well with the experimental data in all inclination angles. Ratulowski and Chang [14] studied the flow characteristics of slug flow in ordered porous media with simple and regular microstructures by using analytical and numerical methods. Stark and Manga [15] investigated the motion of discrete bubbles in ‘realistic’ porous media, which were modeled as a planar network comprised of straight capillaries with different radii, by numerical method based on the extension of slug flow in single straight tube. Chen [16] extended the Bretherton's model to long and homogeneous bubbles and obtained the expressions for the

⁎

Corresponding author. E-mail address: [email protected] (B. Yu).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.09.002

0735-1933/ © 2017 Elsevier Ltd. All rights reserved.

relative permeabilities for steady-state slug flow of liquid and gas phases in a straight capillary tube. His model is related to the structural parameters of bubbles and viscosities of liquid and gas phases. However, the flow characters of slug flow in irregular/disordered porous media were rarely studied. The reason may be that the microstructure of porous media is extremely complicated, and the shape/size of pores and the flow pathways are random. This brings great difficulty in investigating the slug flow in porous media. With the development of computational methods and experimental technologies, a large number of numerical methods were applied such as lattice gas automata, lattice Boltzmann method, and Monte Carlo method etc. [17–19], and new experimental technologies such as laser velocimetry, wavelet analysis, chaotic analysis, as well as nuclear magnetic resonance (NMR) [20–22] were used to study the single-phase and multiphase flows in porous media. However, the results of numerical simulations and experiments were usually reported in forms of graphs or correlations with one or more empirical constants, behind which flow mechanisms in such correlations were often ignored. In 1982, Mandelbrot [23] founded the well-known fractal geometry, which is one of nonlinear sciences and is different from the Euclidean geometry, for describing the characteristics of irregular and disordered objects such as islands on earth and pores in porous media. In the past decades, fractal geometry has been widely applied in many areas in science and engineering since objects and processes in nature as well as in many engineering fields are found to have self-similar or statistically

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

Table 1 A summary of fractal models for permeabilities of two-phase flows. References

Year

Objectives of the model

Results of the study

Yu et al. [36]

2003

Establishment of annular flow model to obtain the permeability of fractal porous media.

Liu et al. [37]

2007

Wang and Yu [38]

2011

Xu et al. [39]

2013

Guarracino et al. [40]

2014

Development of the permeability of annular flow in porous media with the capillary pressure included. Study of the fluid flow characteristics in fractal-like tree networks for two-phase flows. Investigation of the permeability in unsaturated porous media by assuming that all capillaries with the radii less than a critical radius are saturated, and the others with the radii larger than the critical radius are unsaturated. Development of the relative permeabilities with the temporal evolution for porosity of fractal porous media.

The obtained fractal model has no empirical constant. The relative permeabilities of water-gas phases are functions of fractal dimensions of pore area, tortuosity, saturation as well as micro-structural parameters of porous media. They found the capillary pressure has a significant effect on the seepage in unsaturated porous media at low saturation. With capillary pressure, the analytical model for relative permeabilities of fractal-like tree branching network were obtained by fractal method. The results indicate that the fractal permeability model of multiphase fluid related with geometrical parameters and saturation which have role in the multiphase flow in fractal porous media.

Miao et al. [41]

2014

Study of the spherical seepage in unsaturated porous media with the capillary pressure included.

Xiao et al. [42]

2014

Development of the permeability of water and gas in proton exchange membrane fuel cells with fractal geometry methods and Monte Carlo method.

and differentiable function. Therefore, differentiating Eq. (1) with respect to λ results in the number of pores/capillaries in the pore size interval of λ to λ + dλ,

self-similar fractal characteristics at different scales. For example, the fractal geometry theory has been shown to be powerful in characterization of porous media [24–25], and some investigators successfully applied the fractal geometry theory for permeabilities [26–29], spontaneous imbibitions [30,31] and thermal conductivities [32–35]. In addition, many investigators proposed some fractal models for twophase porous media to predict the seepage characteristics in unsaturated porous media. Table 1 shows a summary of some investigations reported in literature. These studies were carried out based on annular flow or division phase flow. However, analysis of slug flow characteristics in porous media by applying the fractal geometry theory was not reported in literature to the best our knowledge. Considering the effect of gas-liquid coupling, the purpose of the present work is to establish the theoretical models for seepage characteristics, permeability and relative permeability of slug flow in single capillary and in porous media based on the fractal geometry theory and technique. In the next section, the fractal geometry theory and technique for fractal porous media is briefly introduced.

D

f − dN = Df λ max λ−(Df + 1) dλ

D

f − dN Nt = Df λ min λ−(Df + 1) dλ = f (λ ) dλ

(4a)

where f(λ) is the probability density function for size distribution, i.e. D

f f (λ ) = Df λ min λ−(Df + 1)

(4b)

and it should satisfy the following normalization conditions:

∫λ

λmax

min

f (λ ) dλ = 1 − (λ min λ max ) Df = 1

(5)

It is clear that Eq. (5) holds if and only if the following equation is satisfied [44]

(λ min λ max ) Df ≅ 0

It has been shown [24–26] that pore size distribution in natural porous media follows the fractal scaling law, and these media are often called fractal porous media. According to the fractal geometry theory for porous media, in a representative unit cell in a porous medium the cumulative number of pores/capillaries, whose sizes are greater than or equal to λ, can be described by the following fractal scaling law [26,43,44]

(6) −2

−2

or < 10 , and Eq. In general, in porous media λmin/λmax ~ 10 (6) can be regarded as a criterion whether the fractal geometry theory and technique can be used in porous media. The relationship among the porosity ϕ, the fractal dimension Df and the ratio λmin/λmax in porous media is given by [26,44]

ln ϕ ln(λ min λ max )

Df = dE −

(1)

(7)

where dE is the Euclid dimension, and dE = 2 in two dimensions and dE = 3 in three dimensions. According to Eq. (4b), the average pore size can be expressed as [44]

where N is the number of capillaries and λmaxis the maximum capillary diameter, respectively. Df is the fractal dimension for pore space, generally, 0 < Df < 2 in two dimensions and 0 < Df < 3 in three dimensions. If λ = λmin in Eq. (1), Eq. (1) yields the total number of pores in a fractal set of pores/capillaries in a porous medium, i.e.

Nt (L ≥ λ min ) = (λ max λ min ) Df

(3)

where − dN > 0, implying that the number of pores/capillaries decreases with the increase of sizes. Dividing Eq. (3) by Eq. (2) gives:

2. Fractal geometry theory for porous media

N (L ≥ λ ) = (λ max λ ) Df

The analytical expressions of multiphase flow models were obtained and the validly was verified by comparing well with available experimental data. The results shows that the contributions for rative permeabilities of the spherical seepage from capillary pressure can be ignored when pcav/ pm < 0.01. With the effect of capillary pressure and tortuosity of capillaries, the rative permeabilities were studied. They found that the phase fractal dimensions strongly depend on porosity.

λav =

(2)

∫λ

λmax

λf (λ ) dλ =

min

1 − Df Df λ min ⎡ ⎤ ⎛ λ min ⎞ 1 − ⎥ 2(Df − 1) ⎢ ⎝ λ max ⎠ ⎦ ⎣ ⎜

⎟

(8)

Based on Eq. (3), the total pore area for the representative unit cell is given by

where Nt is the total number of pores/capillaries, and λmin is the minimum diameter. Since, in general, there are numerous pores in a set of fractal pores/ capillaries in a porous medium, Eq. (1) can be considered as continuous

Ap = −

195

∫λ

λmax

min

2 πDf λ max (1 − ϕ) πλ2 dN = 4 4(2 − Df )

(9)

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

The total cross-sectional area of the representative unit cell for the capillaries is obtained by

A=

Ap

=

ϕ

2 πDf λ max

4(2 − Df )

1−ϕ ϕ

q1 =

A

1 4μ w 1 4μ w

r2 Δpi li Δp

< r < r1. (r12 − r12) = 0 ,

δ 2 = 1.34Ca3 r1

Δpπr 2 = −μeff 2πrli

velocity velocity

dv = −

(18)

Δp rdr 2μeff li

(19)

Integrating Eq. (19) yields

P2

Δp 2 r +C 4μeff li

(20)

where the integrating constant C can be easily determined by setting r = r2 in Eq. (20), then compared to Eq. (13), i.e. Δp 1 Δp v (r2) = − 4μ l r22 + C = 4μ l i (r12 − r22) , from which the integrating eff i

2r2

w

i

constant C can be found to be

C= Non-wetting phase

(17)

where μw , μnw are wetting and non-wetting fluid viscosity coefficients, and ϕw , ϕnw are volume fractions for the wetting and non-wetting phase fluids, respectively. Eq. (17) can be rewritten as

v (r ) = −

2r1

dv dr

μeff = μnw ϕnw + μ w ϕw

i v (r2) = (r12 − r22) , which is the velocity of the interface between li wetting phase and mixed phase (bubble and the fluid between adjacent bubbles). Then, based on Eq. (13), the flow rate for wetting phase between bubbles and the wall is obtained as:

Wetting phase

(16)

where Δp is the pressure difference across the slug unit of length (li) for slug flow (mixed phase), and μeff is the fluid viscosity coefficient for mixed phase, which can be obtained by [51]

(13)

P1

(15)

where δ = r1 − r2 is liquid film thickness. Although Eqs. (15) and (16) were used to study the infinitely long bubbles, Olbricht [47] and Chen [15] successfully applied this expression for the bubbles with finite lengths and with the volume as small as 0.95(4π/3)r3, where r is the radius of capillary. In this study, Eq. (16) is also used to analyze the flow characteristic of slug flow with long bubbles by assuming that the length of a bubble is far greater than the diameter of a capillary [48,49]. In general, the mixed phase in slug flow includes the bubbles, and slug flow is discontinuous at instantaneous scale. However, the mixed phase may be considered as continuous on average at the macroscopic scale when the gas phase moves as a homogeneous bubble train [16]. Moreover, the mixed phase composed of bubbles and fluid can be considered as homogeneous model according to the experiment results [50]. As a result, approximately laminar flow is assumed, i.e. quasisteady flow, then we have

(12)

When r = r1, the and when r = r2, the

σ (3Ca)2 3, as Ca → 0 r2

where σ is interface tension and Ca is capillary number. In addition, the relationship between liquid film thickness and the radius of a capillary in slug flow for infinitely long bubble can be obtained by [11,46]

where Δpi = Pi1 − Pi2 is the pressure drop, li is the length across which the pressure difference Δpi is applied, and μw is fluid viscosity for wetting phase. Based on the non-slip boundary condition, the velocity of fluid on wall is zero, then the velocity can be obtained from Eq. (12), i.e.

where v (r1) =

(14)

Δpb ≅ 3.58

Ratulowski and Chang [14] found that each bubble behaves independently for a chain of identical bubbles at low capillary number. According to this observation, in this paper, we suppose that there is a chain consisting of many identical bubbles in a capillary, and then we analyze the dynamic behavior of the slug unit. Suppose that r1 is the radius of capillary, a long bubble or oil droplet (for simplicity, it is called bubble) is assumed to be a cylinder of radius r2, and the two ends of a bubble exhibit the hemispherical, and its length is lig. The length of liquid between adjacent bubbles is lif, and the length of the slug unit is li. See Fig. 1 for detail. For the slug unit in single straight capillary, fluid in the capillary is divided into two parts: one is wetting phase fluid (such as water) between bubbles and the wall (i.e. the wetting phase is called as liquid film), the other is the slug flow fluid between grey bubbles (see Fig. 1). The fluid between the bubbles and wall is continuous, and it satisfies the quasi-steady flow conditions and the force balance equation is given by [45]

1 Δpi 2 v (r ) = (r1 − r 2) 4μ w li

2

r4 π Δpi ⎛ r14 − r22 r12 + 2 ⎞⎟ ⎜ 4μ w li ⎝ 2 2⎠

(11)

3. Model for steady state slug flow in single straight capillary

dv dr

v (r )⋅2πrdr =

Pressure drop across a bubble is an important physical parameter in slug flow. Bretherton [11] found that the pressure difference across an infinitely long bubble is in the following form.

The above Eqs. (1)–(11) form the theoretical base of the present work.

Δpi πr 2 = −μ w 2πrli

r1

(10)

And the length of the representative unit cell is

L=

∫r

lig

lif

Δp 2 1 Δpi 2 (r1 − r22) + r2 4μ w li 4μeff li

(21)

Inserting Eq. (21) into Eq. (20), the velocity of mixed phase is obtained as

li Flow direction

v (r ) =

Fig. 1. Description of long and homogeneous slug flow in single straight capillary, where P1 and P2 are respectively the mechanical pressures of two ends of a capillary.

1 Δp 2 1 Δpi 2 (r2 − r 2) + (r1 − r22) 4μeff li 4μ w li

(22)

Based on Eq. (21), the flow rate for mixed phase is obtained as 196

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

q2 =

∫0

r2

π Δp 4 π Δpi 2 r2 + (r1 − r22 ) r22 8μeff li 4μ w li

v (r )⋅2πrdr =

cylindrical bubbles. The volume fractions (ϕw and ϕnw) of wetting and non-wetting phases for mixed phase in the slug unit are defined as

(23)

For the length (li) of the slug unit for mixed phase, the total pressure difference across the two ends of the length (li) should include the dynamic pressure difference (Δpi = pi1 − pi2) and pressure difference (Δpb) across a bubble, i.e. the total pressure difference is Δp = Δpi + Δpb [15,16]. For the length (li) of the slug unit of mixed phase, Eqs. (14) and (23) can be respectively rewritten as

ϕw = 1 − ϕnw = 1 −

r4 π Δpi ⎛ r14 − r12 r22 + 2 ⎞⎟ ⎜ 4μ w li ⎝ 2 2⎠

q2 =

π Δpi + Δpb 4 π Δpi 2 r2 + (r1 − r22 ) r22 8μeff li 4μ w li

π Δpi 4 r 8μeff li 1

(24b)

0 ϕnw = Snw

(

ϕnw

+ μw

ϕw

π

+

(

8 μnw

ϕnw

+ μw

ϕw

)

)

(26)

Then, according to Eqs. (23) and (26), the flow rate of wetting phase in the slug unit can be written as

qw = q1 + q2w

+

(

π

(

8 μw +

ϕnw ϕw

μnw

)

w

)

ϕnw =

Δpi 4 r2 li

π 8

(

ϕnw

μ w + μnw

π

+ 8

(

ϕw ϕnw

μ w + μnw

π Δpi 2 (r1 − r22 ) r22 ϕw + 4μ w li

)

)

(31)

(32)

r2 = κr1 0 1 − Sw ϕnw

where, κ =

and the flow rate of non-wetting phase is ϕw

1 − Sw0 (1 − 1.34Ca2 3)2

For simplicity, Eq. (30b) can be written as

3.58σ (3Ca)2 3 3 3 π Δpi 2 r2 r2 + (r1 − r22 ) r22 ϕw li 4μ w li (27a)

qnw = q2⋅ϕnw =

(30b)

Eqs. (30a) and (30b) indicate that if Sw0 = 1, ϕnw = 0 and r2 = 0. This corresponds to the case of completely wetting phase in the capillary; if Sw0 = 0, ϕnw = 1 and r2 = r1, this means that there is no wetting phase in the capillary. Eq. (30b) also indicates that r2 decreases with increase the saturation Sw0 of wetting phase. This is consistent with the physical condition. Based on Eqs. (16) and (30b), the relationship between the saturations of wetting phase and the volume fraction of non-wetting phases can be obtained as

3.58σ (3Ca)2 3 3 r2 li

r4 π Δpi ⎛ r14 π = − r22 r12 + 2 ⎞⎟ + ⎜ ϕ 4μ w li ⎝ 2 2⎠ 8 μ w + ϕnw μnw

(30a)

1 − Sw0 r1 ϕnw

r2 =

Δpi 4 r2 li

π Δpi 2 (r1 − r22 ) r22 ϕw + 4μ w li

r12 r2 = (1 − Sw0 ) 12 r22 r2

which can be rearranged as

π 3.58σ (3Ca)2 3 3 r2 . 8μeff li

8 μnw

(29b)

where Sw0, Snw0 and Vw, Vnw are, respectively, the saturations and volumes of wetting phase and non-wetting phase. Clearly, Sw0 + Snw0 = 1, where the superscript 0 represents the parameter in a single capillary. Due to Eqs. (28b), (29a) and (29b), the relation between saturations and fractions are obtained as

Based on Eqs. (15), (18) and (24b), the flow rate of wetting phase in mixed phase can be expressed as

q2w = q2⋅ϕw =

r22 lig Vw V − Vnw = i =1− 2 Vi Vi r1 li

Sw0 =

r24 ⎞

π

(29a)

and

π π Δpi 4 π 3.58σ (3Ca)2 3 3 − ⎟+ r2 + r2 ⎜ 4μ w li ⎝ 2 2⎠ 8μeff li 8μeff li (25)

+

(28b)

πr22 lig r22 lig Vnw = = 2 2 Vi πr1 li r1 li

0 Snw =

Eq. (25) is the expression for flow rate of quasi-steady slug flow in a capillary. If r2 = 0, this indicates that there is no non-wetting phase, and π Δp then Eq. (25) is reduced to: q = 8μ l r14 , and this is the well-known w i Hagen-Poiseuille equation. If r2 = r1, this case corresponds to the flow pattern: wetting phase and non-wetting phase are separated in a capillary, then Eq. (25) can be reduced to:

q=

li

(24a)

where Δpi, Δpb and li are the dynamic pressure difference, capillary pressure difference and the length of slug unit, respectively. Due to Eqs. (15), (24a) and (24b), the total flow rate in single straight capillary can be obtained as

q = q1 + q2 =

lig

=

πr22 li

(28a)

li

In the slug unit, the saturations of non-wetting phase and wetting phase are respectively defined by

q1 =

Δpi ⎛ r14

πr22 lig

ϕnw =

lig

, 0 ≤ κ ≤ 1, and the parameter κ is a function of sa-

turation Sw0 of wetting phase and volume fraction ϕnw of non-wetting phase. Due to Eqs. (17) and (32), the total flow rate in slug flow for bubbles in single straight capillary is obtained as

Δpi 4 r2 li

3.58σ (3Ca)2 3 3 3 r2 r2 li

qw (λ ) =

+

(27b)

πλ4 Δpi πλ4 (1 − 2κ 2 + κ 4 ) + ϕ 128μ w li 128 μ w + ϕnw μnw

(

πλ3

(

64 μ w +

For long bubbles, the hemispherical volume at two ends is smaller than that of cylindrical bubble. By using same assumption with Chen's model [16], the volume of the slug liquid can be calculated by

ϕnw ϕw

3.58σ (3Ca)2 3 μnw

)

li

r23 κ 3 +

w

)

Δpi 4 κ li

πλ4

Δpi (1 − κ 2) κ 2ϕw 64μ w li (33a)

197

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

πλ4

qnw (λ ) = 128

(

ϕnw

μ w + μnw

Δpi 4 κ li

)

πλ3

+

(

64 +

ϕw

ϕw ϕnw

μ w + μnw

)

capillary tube and then gave the averaged capillary number in the form of

3.58σ (3Ca)2 3 3 κ li

πλ4 Δpi (1 − κ 2) κ 2ϕnw 64μ w li

(33b)

The total flow rates of wetting phase and non-wetting phase through the cross-sectional area of porous medium can be found respectively by integrating Eqs. (33a) and (33b) from the minimum capillary to the maximum capillary to be

4. Analysis of slug flow in fractal porous media

Qw = −

In this section, we apply the fractal capillary model, i.e. porous medium consists of a bundle of straight capillaries, whose diameter distribution follows the fractal scaling law [19,26,30,31], see also Section 2.

⎜

⎟

⎤ ⎥ ⎦

=

πκ 2l

∫λ

λmax

min

πr22 lig dN = −

4

∫λ

λmax

min

⎡1 − ⎛ λ min ⎞ 2 − Df ⎢ ⎝ λ max ⎠ ⎣ ⎜

⎟

πκ 2λ2 4

lig dN

⎤ ⎥ ⎦

(35a)

+

li − κ 2lig Vw = = 1 − κ 2ϕnw V li

64

⎜

μnw

)

⎤ ⎥ ⎦

4 − Df

⎟

⎤ ⎥ ⎦ 3 − Df

⎡1 − ⎛ λ min ⎞ ⎢ ⎝ λ max ⎠ ⎣ ⎜

⎟

⎤ ⎥ ⎦

λmax

min

qnw (λ ) dN =

(

ϕw ϕnw

μ w + μnw

)

(

ϕw ϕnw

Df Δp π κ 2 (1 − κ 2) i 64μ w 4 − Df li 4 − Df

⎤ ⎥ ⎦ Δp 4 κ 4 i λ max 4 − Df li ⎟

Df

)

μ w + μnw 3 − Df

κ3

⎡1 − ⎛ λ min ⎞ ⎢ ⎝ λ max ⎠ ⎣ ⎜

⎟

3.58σ (3Caav )2 3 3 3 r2 λ max li

4 − Df

⎤ ⎥ ⎦ 3 − Df

⎡1 − ⎛ λ min ⎞ ⎢ ⎝ λ max ⎠ ⎣ ⎜

⎟

⎤ ⎥ ⎦

⎟

(39b)

(35b) −2

in porous where –dN is given by Eq. (3), In general, λmin/λmax < 10 media, and 0 < Df < 2 in the two dimensional cross-section of the set of capillaries, hence, (λmin/λmax)4 − Df < < 1, (λmin/ 3 − Df λmax) < < 1, and Eqs. (39a) and (39b) can be simplified as:

(36a)

Df Δp 4 π (1 − 2κ 2 + κ 4 ) i λ max 128μ w 4 − Df li Df Δpi π 4 2 2 κ (1 − κ ) + ⋅ϕ λ max 64μ w 4 − Df li w Df Δp 4 π κ 4 i λ max + ϕnw 4 li D − f 128 μ w + ϕ μnw

Qw = − (36b)

From Eqs. (36a) and (36b), it is seen that the relationship of Sw + Snw = 1 holds. Inserting Eqs. (36a) and (36b) into the expression κ = (1‐Sw0 ) ϕnw , we have Sw = Sw0. This case implies that the saturations of wetting phase in a capillary and in the porous medium are equal. Based on Eq. (31) and Sw = Sw0, the expression for the equivalent single capillary can be given by

1 − Sw = 2 3 2 (1 − 1.34Caav )

ϕw

π

+

Base on Eqs. (34), (35a) and (35b), the saturations of wetting phase and non-wetting phase are respectively given by

κ 2lig Vnw = = κ 2ϕnw V li

⎡1 − ⎛ λ min ⎞ ⎢ ⎝ λ max ⎠ ⎣

3.58σ (3Caav )2 3 3 3 3 r2 κ λ max 3 − Df li

⎡1 − ⎛ λ min ⎞ 4 ⋅ϕnw λ max ⎢ ⎝ λ max ⎠ ⎣ Df π 128

Snw =

ϕnw

∫λ

Qnw = −

λmax

⎜

ϕnw

(

4 − Df

⎟

Df

π

⎜

πλ2 Vw = − li dN − Vnw λmin 4 2 − Df 2 2 π (li − κ lig ) Df λ max ⎤ ⎡1 − ⎛ λ min ⎞ = ⎥ 4 2 − Df ⎢ ⎝ λ max ⎠ ⎦ ⎣

Sw =

)

4 − Df

Δpi 4 λ max li

⎜

and

and

∫

ϕw

μnw

κ4

⎡1 − ⎛ λ min ⎞ ⎢ ⎝ λ max ⎠ ⎣

(39a)

2 − Df

2 ig Df λ max

ϕnw

⎡1 ⎢ ⎣

(34)

Then the volumes of non-wetting and wetting phases are respectively obtained by

Vnw = −

Df

(

64 μ w + 2 − Df

Df Δp 4 π (1 − 2κ 2 + κ 4 ) i λ max 128μ w 4 − Df li

⎤ ⎥ ⎦

π

+

⎡1 − ⎛ λ min ⎞ ⎢ ⎝ λ max ⎠ ⎣

4 − Df

⎟

128 μ w +

πλ2 ⋅li dN 4

min

qw (λ ) dN =

Df Δp π 4 κ 2 (1 − κ 2) i ⋅ϕw λ max 64μ w 4 − Df li

+

Based on Eq. (3), the total volumes of the slug unit in a porous medium is calculated by

2 πl Df λ max = i 4 2 − Df

λmax

min

⎜

4.1. The saturations of wetting phase and non-wetting phase

λmax

∫λ

λ − ⎛ min ⎞ ⎝ λ max ⎠ +

∫λ

(38)

4.2. The total flow rates of wetting phase and non-wetting phase

where λ is diameter of a capillary and λ = 2r1. From Eqs. (33a) and (33b), it can be seen that the flow rates of wetting and non-wetting phases are functions of the structural parameters (li) of slug unit, volume fractions (ϕw and ϕnw), fluid viscosities (μw, μnw), the interfacial tension (σ) and the capillary number (Ca).

V=−

λav 2L0

Caav ≅

∫λ

λmax

min

(

qw (λ ) dN =

)

w

Df

π

+

(

64 μ w +

ϕnw ϕw

μnw

)

3 − Df

3.58σ (3Caav )2 3 3 3 κ λ max li (40a)

(37) and

where Caav is the average capillary number in porous media. Guillen et al. [52] assumed that porous media consist of a bundle of 198

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

Qnw = −

∫λ

λmax

min

π

+ 128

(

ϕw ϕnw

μ w + μnw

)

Df

π

+ 64

(

ϕw ϕnw

Df Δp π 4 κ 2 (1 − κ 2) i ⋅ϕw λ max 64μ w 4 − Df li Df Δp 4 κ 4 i λ max 4 − Df li

D

4 ⎡ 128π A 4 − fD (1 − 2κ 2 + κ 4 ) λ max Sw ⎤ f ⎥ ⎢ Df π ⎢ 2 (1 − κ 2 ) λ 4 ϕ S ⎥ + κ max w w ⎥ ⎢ 64A 4 − Df ⎥ ⎢ Df πμ w 4 4 ⎢+ κ λ max Sw ⎥ − D 4 ϕ f nw ⎥ ⎢ 128A ⎛μ w + ϕ μnw ⎞ w ⎝ ⎠ ⎦ Kw = ⎣ 1 + Δpbav Δpi

qnw (λ ) dN =

)

μ w + μnw 3 − Df

κ3

⎜

3.58σ (3Caav )2 3 3 λ max li

⎟

(44a)

(40b)

⎡ πμ Df 4 ⎢ nw κ 2 (1 − κ 2) ϕnw λ max (1 − Sw ) ⎢ 64Aμ w 4 − Df ⎣

Eqs. (40a) and (40b) shows that the total flow rates of wetting phase and non-wetting phase are related to the structural parameters (li) of slug unit, volume fractions (ϕw and ϕnw), maximum capillary diameter (λmax), fluid viscosities (μw, μnw), the interfacial tension (σ), the average capillary number (Caav), and the fractal dimension Df. According to Eqs. (4b), (15) and (32), the average pressure difference across bubbles in porous media can be calculate by

Δpbav =

∫λ

λmax

Δpb f (λ ) dλ =

min

+

⎜

Knw =

3.58σ (3Caav )2 3Df k (1 + Df ) λ min

Df

πμnw

4 − Df ϕ 128A ⎛ w μ w + μnw ⎞ ⎝ ϕnw ⎠ ⎟

1 + Δpbav

⎤ 4 κ 4λ max (1 − Sw ) ⎥ ⎥ ⎦ Δpi

(44b)

Inserting Eqs. (36a) and (36b) into Eqs. (44a) and (44b) yields (41)

Kw Eq. (41) implies that the average pressure difference is sensitive to the minimum capillary diameter.

(

This sub-section is devoted to analyzing the relative permeabilities of slug flow in fractal porous media. Chen [16] extended the Darcy's law for slug flow to investigate the relative permeabilities of steam and water. The expression can be written as

(42a)

Δpi + Δpbav Knw A = μnw (1 − Sw ) li

(

Knw (42b)

where Kw and Krw are the permeabilities for wetting and non-wetting phases for slug flow in porous media, respectively. Based on Eqs. (40a), (40b), (42a) and (42b), the permeabilities for wetting and non-wetting phases in slug flow in porous media can be obtained as

K=

(43a)

D

f 4 nw ⎤ ⎡ 64Aμ κ 2 (1 − κ 2) ϕnw λ max (1 − Sw ) w 4 − Df ⎥ ⎢ D πμ f ⎥ ⎢ nw 4λ 4 + − κ (1 S ) w max ⎥ ⎢ 4 − Df ϕ 128A ⎛ w μ w + μnw ⎞ ⎥ ⎢ ϕ ⎝ nw ⎠ ⎥ ⎢ 2(Df + 1) 4 4 πμnw λmin Δpbav ⎥ ⎢+ κ λ max (1 − Sw ) λ − D Δp 3 ϕ max f i ⎥ ⎢ 128A ⎛ w μ w + μnw ⎞ ϕ nw ⎝ ⎠ ⎦ = ⎣ 1 + Δpbav Δpi ⎜

(1 − Sw ) ϕ w ϕnw

Sw ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(45a)

)

Sw + ϕnw − 1 4 ⎤ ⎡ πμnw Df (1 − Sw ) λ max (1 − Sw ) ϕnw ⎥ ⎢ 64Aμ w 4 − Df 2 ⎥ ⎢ Df πμnw 1 − Sw 4 = ⎢+ − λ (1 S ) w ⎥ max 1 − Sw ϕnw ⎛ + Sw − 1 ⎞ 4 − Df ⎥ ⎢ ϕ ⎥ ⎢ 128A ⎜⎜ nw1 − Sw μ w + μnw ⎟⎟ ⎥ ⎢ ⎝ ⎠ ⎦ ⎣ Δp bav ⎞ / ⎛⎜1 + ⎟ Δpi ⎠ ⎝

( )

(45b)

Df π 4 λ max 128A 4 − Df

(46)

Eq. (46) is exactly the absolute permeability for fractal porous media [26,44]. According to the definition of relative permeability and Eqs. (45a), (45b) and (46), the relative permeabilities of slug flow for wetting and non-wetting phase can be respectively obtained as

⎟

⎜

Knw

)

⎟

⎜

πμ

(

If Sw = 1, Krnw = 0, this indicates that there is only wetting phase in the porous medium. For Sw = 0, ϕnw = 1 then Krw = 0. This means that there is only non-wetting phase in the porous medium. In the above two cases, Eqs. (45a) and (45b) can be reduced to

D

4 ⎤ ⎡ 128π A 4 − fD (1 − 2κ 2 + κ 4 ) λ max Sw f ⎥ ⎢ D f π ⎥ ⎢ 2 (1 − κ 2 ) λ 4 ϕ S + κ w max w ⎥ ⎢ 64A 4 − Df ⎥ ⎢ Df πμ w 4λ 4 S ⎥ ⎢+ κ max w − 4 D ϕ f ⎥ ⎢ 128A ⎛μ w + ϕnw μnw ⎞ w ⎝ ⎠ ⎥ ⎢ 2(Df + 1) 4 4 πμ w ⎢ λmin Δpbav ⎥ + κ λ S w max ⎢ 3 − Df λmax Δpi ⎥ ϕ 128A ⎛μ w + nw μnw ⎞ ⎥ ⎢ ϕw ⎝ ⎠ ⎦ ⎣ Kw = 1 + Δpbav Δpi ⎜

4

and

Kw A Δpi + Δpbav μ w Sw li

and

Qnw

2

( )

4.3. Relative permeabilities

Qw =

)

1 − Sw − ϕnw π Df λ max Sw + ϕnw − 1 4 ⎡ π Df λ max Sw + 64A 4 − D ϕnw ϕnw f ⎢ 128A 4 − Df ⎢ 2 Df πμ w 1 − Sw 4 = ⎢+ λ max Sw ϕnw ⎛ ⎞ 4 − Df ⎢ 1 − Sw ⎜ ⎟ 128 + A μ μ ⎢ w nw 1 − Sw ⎜ ⎟ + Sw − 1 ⎢ ϕnw ⎝ ⎠ ⎣ Δpbav ⎞ / ⎛⎜1 + ⎟ Δpi ⎠ ⎝

(

Krw

⎟

⎟

(43b) and

In this work, since λmin/λmax ≅ 0.001, Eqs. (43a) and (43b) can be reduced as 199

)

2

(

⎡ 1 − Sw − ϕnw Sw + 2 Sw + ϕnw − 1 ϕnw ϕnw ⎢ ⎢ 2 Kw μ w Sw 1 − Sw = = ⎢+ ϕ K nw ⎞ ⎢ ⎛ 1−S ⎢ ⎜μ w + 1 − Sw w μnw ⎟ ⎜ ⎟ S 1 + − w ⎢ ⎝ ϕnw ⎠ ⎣

( )

)

(1 − Sw ) Sw ϕ w ⎤ ϕnw

⎥ ⎥ ⎛ Δpbav ⎞ ⎥/ ⎜1 + ⎟ Δpi ⎠ ⎥ ⎝ ⎥ ⎥ ⎦ (47a)

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

(

Krnw

)

⎡ 2 μnw Sw + ϕnw − 1 (1 − Sw )2 ϕnw ⎢ μw ⎢ Knw μnw 1 − Sw = = ⎢+ 1 − S ϕnw w K ⎞ ⎢ ⎛ ϕnw + Sw − 1 ⎢ ⎜⎜ 1 − Sw μ w + μnw ⎟⎟ ⎢ ⎝ ⎠ ⎣

( )

2

⎤ ⎥ ⎥ Δpbav ⎞ ⎟ (1 − Sw ) ⎥/ ⎛⎜1 + Δpi ⎠ ⎥ ⎝ ⎥ ⎥ ⎦ (47b)

It can be seen from Eqs. (47a) and (47b) that the relative permeabilities of slug flow for wetting and non-wetting phases are functions of fractal dimension Df for pores space, the maximum capillary diameter λmax, the interfacial tension σ, and the average capillary number Caav. If Df = 2, Eqs. (47a) and (47b) are reduced to the relative permeabilities of slug flow in a straight capillary tube and they can be rewritten as

(

Krw

)

2

(

⎡ 1 − Sw − ϕnw Sw + 2 Sw + ϕnw − 1 ϕnw ϕnw ⎢ ⎢ 2 μ w Sw 1 − Sw = ⎢+ ϕ nw ⎞ ⎢ ⎛ 1−S ⎢ ⎜μ w + 1 − Sw w μnw ⎟ ⎜ ⎟ S 1 + − w ⎢ ⎝ ϕnw ⎠ ⎣

( )

)

Fig. 2. Comparison of the predictions by using the present model in a capillary with Chen's model at μw = 0.2703 × 10‐3Pa ⋅ s, μnw = 0.124 × 10− 4Pa ⋅ s, Ca = 10− 4, Δpbav/ Δp = 0.8 [54].

(1 − Sw ) Sw ϕ w ⎤ ϕnw

⎥ ⎥ ⎛ Δpb ⎞ ⎥/ ⎜1 + ⎟ Δpi ⎠ ⎥ ⎝ ⎥ ⎥ ⎦

The figure indicates that our model predictions consist with those from Chen's model, which is a widely used model in literature. However, the relative permeability of Chen's model, which is related to the radius of capillary tube r, is unreasonable, see Eq. (49). Another weakness of Eq. (49) is that it has a fitting constant (0.931), while in our model Eq. (48b) there is no fitting constant and every parameter has clear physical meaning. Fig. 3 shows the averaged capillary number versus porosity of porous media. It is clear that the averaged capillary number increases with increase of porosity. This is because that the higher porosity tends to form higher velocity of wetting phase, leading to the higher capillary number. This is consistent with the physical phenomena. Fig. 4 compares our model predictions with the available experimental data. It is seen that good agreement between model predictions and the available experimental data. In the present comparisons, the experimental data were measured for oil-water [54]. The viscosities of the wetting phase and non-wetting phase are μw = 0.88 × 10‐3Pa ⋅ s , μnw = 1.14 × 10‐3Pa ⋅ s, respectively, and the porosity of porous media is 0.25 [54]. It is seen from Fig. 4 that the relative permeabilities of wetting phase increase with saturation. Conversely, the relative permeabilities of the non-wetting phase decrease with the increase of the wetting phase saturation. Fig. 5 plots the general relationship between the relative permeabilities and the saturation of wetting phase at different porosities and fractal dimensions. Fig. 6 shows that although the fractal dimension Df for pore space depends on porosity (see Eq. (7)), the relative permeabilities are independent of porosity (see Eqs. (47a) and (47b)). This is consistent with the experimental observations [55]. In addition, we can see from Fig. 5 that the relation of Krw + Krnw < 1 holds. The similar results were also observed in several other experiments [55,56]. Fig. 6 shows the relative permeability of wetting phase versus the

(48a) and

(

Krnw

)

⎡ 2 μnw Sw + ϕnw − 1 (1 − Sw )2 ϕnw ⎢ μw ⎢ μnw 1 − Sw = ⎢+ 1 − S ϕnw w ⎞ ⎢ ⎛ ϕnw + Sw − 1 ⎢ ⎜⎜ 1 − Sw μ w + μnw ⎟⎟ ⎢ ⎝ ⎠ ⎣

( )

⎤ ⎥ 2 ⎥ Δpb ⎞ ⎟ (1 − Sw ) ⎥/ ⎛⎜1 + Δpi ⎠ ⎥ ⎝ ⎥ ⎥ ⎦

(48b)

It is clear that the physical quantities ϕnw and Sw in Eqs. (47a), (47b), (48a) and (48b) are related to Eq. (37). 5. Results and discussion In calculation, the parameters are determined by the following procedures: 1). Porosity ϕ and total pressure Δp are based on a real sample and experimental conditions, respectively. 2). The minimum diameter 1 μm [53] of pores is chosen and then determines the maximum diameter λmax from λmin/λmax = 10− 3 [44]. 3). Determine the fractal dimension Df for pore space by Eq. (7). 4). The averaged capillary number Caav is determined by Eqs. (9)–(11) and (38). 5). Then, find average capillary difference Δpbav by Eqs. (35a), (37) and (41), and the mechanical pressure Δpi = Δp − Δpbav. Chen [16] got the expression of the relative permeability of gas phase for steady-state slug flow in a straight capillary for long and homogeneous bubbles by the widely used drift model. i.e. (1 − Sw ) μnw

Krnw =

Sw μ w (1 − w )

1 + 0.931

πr 3 (1 − Sw ) Ca−1 3 Vg (1 − w )2 3

(49)

where Vg is volume of bubble and r is radius of capillary, and 0.931 is a fitting constant, and the physical meaning behind this fitting constant is not clear. The parameters μw = 0.2703 × 10‐3Pa ⋅ s (viscosity of water), μnw = 0.124 × 10− 4Pa ⋅ s (viscosity of steam), Ca = 10− 4 and σ = 0.058N/m in his model are used for comparison. Fig. 2 shows a comparison of the present model in a capillary with Chen's model. From the Fig. 2, good agreement is found between the predictions by using the present model in a straight capillary with those from Chen's model.

Fig. 3. The averaged capillary number versus porosity of porous media at λmin = 1 μm.

200

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al.

Fig. 6. The relative permeability of wetting phase versus the saturation of wetting phase at different viscosity ratios for wetting phase at ϕ = 0.25,Df = 1.79,Δpbav/Δp = 0.1.

Fig. 4. Comparison of the present model for the relative permeabilities of wetting phase and non-wetting phase with experimental data at μw = 0.88 × 10− 3Pa ⋅ s , μnw = 1.14 × 10− 3Pa ⋅ s , ϕ = 0.25 , Df = 1.79 , Δpbav/Δp = 0.45 [54].

saturation of wetting phase at different viscosity ratios for wetting phase. We see that the relative permeability of wetting phase increases with the increase of viscosity ratio of wetting phase to non-wetting phase. This can be explained by the higher viscosity of non-wetting phase corresponds to the larger viscous force, leading to the lower flow speed of wetting phase in porous media. The effect of gas-liquid coupling is very important for multiphase flow in porous media. Next, we discuss the effect of the relative permeability of wetting phase at different ratios of Δpbav/Δp. Fig. 7 shows the relative permeability of wetting phase increases with the decrease of the ratios of Δpbav/Δp. This may be explained that since the pressure difference across bubbles blocks the fluid flow in porous media. We can also see from Fig. 7 that the permeability contribution from the pressure difference across bubbles can be negligible when Δpbav/Δp < 0.01.

Fig. 7. The relative permeability versus saturation of wetting phase at different ratios of the average capillary pressure difference to the mechanical pressure difference at μw = 1 × 10− 3Pa ⋅ s , μnw = 8 × 10− 4Pa ⋅ s , ϕ = 0.25 , Df = 1.79.

6. Conclusions

agreement between them is found.

In this paper, the analytical models for slug flow have been proposed in a capillary and in porous media by the fractal geometry theory. The proposed models for the relative permeabilities for slug flow in porous media have been derived and are expressed as a function of micro-structural parameters of porous media, slugs and bubbles as well as fluid properties. The relative permeabilities for the wetting phase and non-wetting phase with coupling between them are investigated. The present results show that the relative permeability for non-wetting phase decreases with the increase of wetting phase saturation, and the relative permeability for wetting phase increases with the increase of wetting phase saturation. The predictions from the proposed slug flow models are compared with the available experimental data, and good

Acknowledgment This work was supported by the National Natural Science Foundation of China (through Grant No. 51576077 and 10932010), the Research Key Project of Science and Technology of Education Bureau of Henan Province, China (Grant No. 17A470013 and 14A140030), the Science and Technology Program of Department of Science and Technology of Henan Province, China (through Grant No. 152102210201), the Innovation Talants Program of Science and Technology of Institution of Higher Education of Henan Province, China (Grant No. 14HASTIT044) and the Key Project for Theoretical Physics of Xinxiang University. References [1] G. Gregory, M. Nicholson, K. Aziz, Correlation of the liquid volume fraction in the slug for horizontal gas-liquid slug flow, Int. J. Multiphase Flow 4 (1) (1978) 33–39. [2] B. Bai, X. Zhang, M. Liu, W. Su, Flow regime classification and transition of flow boiling through porous channel, 17th International Conference on Nuclear Engineering, American Society of Mechanical Engineers, 2009, pp. 487–493. [3] J. Zhang, W. Li, Investigation of hydrodynamic and heat transfer characteristics of gas–liquid Taylor flow in vertical capillaries, Int. Commun. Heat Mass Transfer 74 (2016) 1–10. [4] Y. Chen, P. Cheng, Heat transfer and pressure drop in fractal tree-like microchannel nets, Int. J. Heat Mass Transf. 45 (13) (2002) 2643–2648. [5] E. Kordyban, Horizontal slug flow: a comparison of existing theories, J. Fluids Eng. 112 (1) (1990) 74–83. [6] J. Fabre, A. Liné, Modeling of two-phase slug flow, Annu. Rev. Fluid Mech. 24 (1) (1992) 21–46. [7] K. Triplett, S. Ghiaasiaan, S. Abdel-Khalik, D. Sadowski, Gas–liquid two-phase flow in microchannels part I: two-phase flow patterns, Int. J. Multiphase Flow 25 (3) (1999) 377–394. [8] Y. Han, N. Shikazono, Measurement of the liquid film thickness in micro tube slug

Fig. 5. The relative permeabilities versus the saturation of wetting phase at different porosities and fractal dimensions at μw = 1 × 10− 3Pa ⋅ s, μnw = 8 × 10− 4Pa ⋅ s, Δpbav/ Δp = 0.1.

201

International Communications in Heat and Mass Transfer 88 (2017) 194–202

T. Miao et al. flow, Int. J. Heat Fluid Flow 30 (5) (2009) 842–853. [9] B. Bai, M. Liu, W. Su, X. Zhang, An experimental study on air-water two-phase flow patterns in pebble beds, J. Eng. Gas Turbines Power 132 (4) (2010) 042903. [10] R. Gupta, S.S. Leung, R. Manica, D.F. Fletcher, B.S. Haynes, Hydrodynamics of liquid–liquid Taylor flow in microchannels, Chem. Eng. Sci. 92 (2013) 180–189. [11] F. Bretherton, The motion of long bubbles in tubes, J. Fluid Mech. 10 (02) (1961) 166–188. [12] D. Nicklin, J. Wilkes, J. Davidson, Two-phase flow in vertical tubes, Trans. Inst. Chem. Eng. 40 (1) (1962) 61–68. [13] K.H. Bendiksen, An experimental investigation of the motion of long bubbles in inclined tubes, Int. J. Multiphase Flow 10 (4) (1984) 467–483. [14] J. Ratulowski, H.C. Chang, Transport of gas bubbles in capillaries, Phys. Fluids A (1989–1993) 1 (10) (1989) 1642–1655. [15] J. Stark, M. Manga, The motion of long bubbles in a network of tubes, Transp. Porous Media 40 (2) (2000) 201–218. [16] Z. Chen, Liquid-Gas Relative Permeabilities in Fractures: Effects of Flow Structures, Phase Transformation and Surface Roughness, Stanford University, 2005. [17] A. Koponen, M. Kataja, J. Timonen, Permeability and effective porosity of porous media, Phys. Rev. E 56 (3) (1997) 3319. [18] S. Ansumali, I.V. Karlin, Consistent lattice Boltzmann method, Phys. Rev. Lett. 95 (26) (2005) 260605. [19] B. Xiao, J. Fan, F. Ding, Prediction of relative permeability of unsaturated porous media based on fractal theory and Monte Carlo simulation, Energy Fuel 26 (11) (2012) 6971–6978. [20] M. Gopal, W. Jepson, Development of digital image analysis techniques for the study of velocity and void profiles in slug flow, Int. J. Multiphase Flow 23 (5) (1997) 945–965. [21] Y. Cai, M. Wambsganss, J. Jendrzejczyk, Application of chaos theory in identification of two-phase flow patterns and transitions in a small, horizontal, rectangular channel, J. Fluids Eng. 118 (2) (1996) 383–390. [22] M. Turner, L. Knüfing, C. Arns, A. Sakellariou, T. Senden, A. Sheppard, R. Sok, A. Limaye, W. Pinczewski, M. Knackstedt, Three-dimensional imaging of multiphase flow in porous media, Phys. A 339 (1) (2004) 166–172. [23] B.B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, 1982. [24] A.J. Katz, A. Thompson, Fractal sandstone pores: implications for conductivity and pore formation, Phys. Rev. Lett. 54 (12) (1985) 1325. [25] J.M. Smidt, D.M. Monro, Fractal modeling applied to reservoir characterization and flow simulation, Fractals 6 (04) (1998) 401–408. [26] B. Yu, P. Cheng, A fractal permeability model for bi-dispersed porous media, Int. J. Heat Mass Transf. 45 (14) (2002) 2983–2993. [27] T. Miao, B. Yu, Y. Duan, Q. Fang, A fractal analysis of permeability for fractured rocks, Int. J. Heat Mass Transf. 81 (2015) 75–80. [28] R. Liu, Y. Jiang, B. Li, X. Wang, A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks, Comput. Geotech. 65 (2015) 45–55. [29] T. Miao, S. Yang, Z. Long, B. Yu, Fractal analysis of permeability of dual-porosity media embedded with random fractures, Int. J. Heat Mass Transf. 88 (2015) 814–821. [30] J. Cai, X. Hu, D.C. Standnes, L. You, An analytical model for spontaneous imbibition in fractal porous media including gravity, Colloids Surf. A Physicochem. Eng. Asp. 414 (2012) 228–233. [31] J. Cai, E. Perfect, C.-L. Cheng, X. Hu, Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures, Langmuir 30 (18) (2014) 5142–5151. [32] B. Yu, P. Cheng, Fractal models for the effective thermal conductivity of bidispersed porous media, J. Thermophys. Heat Transf. 16 (1) (2002) 22–29. [33] J. Kou, Y. Liu, F. Wu, J. Fan, H. Lu, Y. Xu, Fractal analysis of effective thermal

[34]

[35]

[36] [37] [38]

[39] [40]

[41] [42]

[43] [44] [45] [46] [47] [48] [49] [50]

[51] [52]

[53] [54]

[55] [56]

202

conductivity for three-phase (unsaturated) porous media, J. Appl. Phys. 106 (5) (2009) (054905-054905-054906). B. Xiao, Y. Yang, L. Chen, Developing a novel form of thermal conductivity of nanofluids with Brownian motion effect by means of fractal geometry, Powder Technol. 239 (2013) 409–414. T. Miao, S. Cheng, A. Chen, B. Yu, Fan, Analysis of axial thermal conductivity of dual-porosity fractal porous media with random fractures, Int. J. Heat Mass Transf. 102 (2016) 884–890. B. Yu, J. Li, Z. Li, M. Zou, Permeabilities of unsaturated fractal porous media, Int. J. Multiphase Flow 29 (10) (2003) 1625–1642. Y. Liu, B. Yu, P. Xu, J. Wu, Study of the effect of capillary pressure on permeability, Fractals 15 (01) (2007) 55–62. S. Wang, B. Yu, Study of the effect of capillary pressure on the permeability of porous media embedded with a fractal-like tree network, Int. J. Multiphase Flow 37 (5) (2011) 507–513. P. Xu, S. Qiu, B. Yu, Z. Jiang, Prediction of relative permeability in unsaturated porous media with a fractal approach, Int. J. Heat Mass Transf. 64 (2013) 829–837. L. Guarracino, T. Rötting, J. Carrera, A fractal model to describe the evolution of multiphase flow properties during mineral dissolution, Adv. Water Resour. 67 (2014) 78–86. T. Miao, B. Yu, Y. Duan, Q. Fang, A fractal model for spherical seepage in porous media, Int. Commun. Heat Mass Transfer 58 (2014) 71–78. B. Xiao, J. Fan, F. Ding, A fractal analytical model for the permeabilities of fibrous gas diffusion layer in proton exchange membrane fuel cells, Electrochim. Acta 134 (2014) 222–231. Q. Zheng, B. Yu, A fractal permeability model for gas flow through dual-porosity media, J. Appl. Phys. 111 (2) (2012) (024316-024316-024317). B. Yu, Analysis of flow in fractal porous media, Appl. Mech. Rev. 61 (5) (2008) 050801. M.M. Denn, Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1980. P. Angeli, A. Gavriilidis, Hydrodynamics of Taylor flow in small channels: a review, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 222 (5) (2008) 737–751. W. Olbricht, Pore-scale prototypes of multiphase flow in porous media, Annu. Rev. Fluid Mech. 28 (1) (1996) 187–213. A.E. Dukler, M.G. Hubbard, A model for gas-liquid slug flow in horizontal and near horizontal tubes, Ind. Eng. Chem. Fundam. 14 (4) (1975) 337–347. E. Wren, G. Baker, B. Azzopardi, R. Jones, Slug flow in small diameter pipes and Tjunctions, Exp. Thermal Fluid Sci. 29 (8) (2005) 893–899. K. Triplett, S. Ghiaasiaan, S. Abdel-Khalik, A. LeMouel, B. McCord, Gas–liquid twophase flow in microchannels: part II: void fraction and pressure drop, Int. J. Multiphase Flow 25 (3) (1999) 395–410. E. Aker, K.J. Måløy, A. Hansen, Simulating temporal evolution of pressure in twophase flow in porous media, Phys. Rev. E 58 (2) (1998) 2217. V.R. Guillen, M.I. Romero, M.d.S. Carvalho, V. Alvarado, Capillary-driven mobility control in macro emulsion flow in porous media, Int. J. Multiphase Flow 43 (2012) 62–65. B. Yu, L.J. Lee, H. Cao, A fractal in-plane permeability model for fabrics, Polym. Compos. 23 (2) (2002) 201–221. C. Sandberg, L. Gournay, R. Sippel, The effect of fluid-flow rate and viscosity on laboratory determinations of oil-water relative permeabilities, Soc. Pet. Eng. 213 (1958) 36–43. M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd edition, SpringerVerlag, New York, 1995. J.C. Charpentier, M. Favier, Some liquid holdup experimental data in trickle-bed reactors for foaming and nonfoaming hydrocarbons, AICHE J. 21 (6) (1975) 1213–1218.