- Email: [email protected]
A permeability model for power-law fluids in fractal porous media composed of arbitrary cross-section capillaries
Physica A xx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
A permeability model for power-law fluids in fractal porous media composed of arbitrary cross-section capillaries Q1
Shifang Wang a,b,∗ , Tao Wu c , Hongyan Qi b , Qiusha Zheng b , Qian Zheng d a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, PR China
b
Department of Physics and mechanical & electrical engineering, Hubei University of Education, Wuhan 430205, PR China
c
Laboratory of Optical Information and Technology, School of Science, Wuhan Institute of Technology, Wuhan 430074, PR China
d
School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, PR China
highlights • • • • •
A permeability model for power-law fluids in porous media is developed. The permeability is a function of fluid property and structure parameters of media. The effects of structure parameters on permeability and velocity are discussed. The average flow velocity is compared with existing macroscopic model. Our permeability model can be considered as an extension of Yu and Cheng’s model.
article
info
Article history: Received 10 March 2015 Received in revised form 23 April 2015 Available online xxxx Keywords: Fractal theory Porous media Permeability Power-law fluids
abstract The fractal theory and technology has been applied to determine the flow rate, the average flow velocity, and the effective permeability for the power-law fluid in porous media composed of a number of tortuous capillaries/pores with arbitrary shapes, incorporating the tortuosity characteristic of flow paths. The fractal permeability and average flow velocity expressions are found to be a function of geometrical shape factors of capillaries, material constants, the fractal dimensions, microstructural parameters. The effect of the porosity, the tortuosity fractal dimension, material constants, and geometrical shape factors on the effective permeability is also analyzed in detail. To verify the validity of the present model, our proposed model is compared with the available macroscopic model and experimental data and there is good agreement between them. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Modeling of flow and transport properties in porous media is an active field of interest for a variety of industrial applications, such as oil reservoirs [1–7], textile industry [8–10], and biological tissues and organs [11,12]. The macroscopic transport properties of porous media such as the flow rate, the average velocity, and the permeability mainly depend on the microstructure of porous media. Since the 1880s, large numbers of literatures have begun to investigate flow properties of porous media. For instance, early as in 1856, the preliminary experimental measurement for Newtonian fluids flow through porous media was undertaken by Darcy [13]. To better understand the mechanisms of hydrodynamics of liquid
∗
Corresponding author at: Department of Physics and mechanical & electrical engineering, Hubei University of Education, Wuhan 430205, PR China. E-mail address: [email protected] (S. Wang).
http://dx.doi.org/10.1016/j.physa.2015.05.089 0378-4371/© 2015 Elsevier B.V. All rights reserved.
1
2 3 4 5 6 7
2
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
Fig. 1. A sketch of porous media composed of a bundle of tortuous capillaries with various shapes.
36
flows in smooth wall microchannels (relative roughness <0.5%), Baviere R. et al. [14] performed the experiment on water flow through smooth rectangular microchannels and proved that a friction factor was correctly predicted by Navier–Stokes equation at the small scales. Alexandre lavro [15] studied the power-law fluid flow in a rough fracture of regular or irregular topography by using numerical simulations. Yazdchi K. [16] investigated the macroscopic permeability of fibrous porous media taking its microstructure into account by the finite element method. However, the results obtained from these experimental measurements and numerical simulations are usually correlated as curves or empirical correlations which contained one or several empirical constants, and could not reveal some basic physical mechanisms for flow in porous media. The capillaries or pores in real porous media are usually non-uniform in size, irregular, and randomly distributed. Fortunately, some published literatures have shown that the fractal geometry may have the potential in analysis of flow and transport properties in porous media [8,17–21]. Yu and Cheng [22] developed the fractal permeability model for bidispersed porous media based on the fractal characteristic of porous media. In order to investigate the starting pressure gradient in dual-porosity medium, Wang et al. [23] also applied the fractal theory and technique to obtain an analytical expression of the starting pressure gradient for Bingham fluids in porous media embedded with randomly distributed fractal-like tree networks. Xu et al. [24] developed a probability model for Newtonian fluids radial flow in fractured porous media based on fractal theory and Monte Carlo simulation. Recently, Shou et al. [9] obtained the fractal model for gas diffusion across nanoscale and microscale fibrous media based on fractal theory and capillary model. Luis Guarracino [25] presented a physically-based theoretical model which described the temporal evolution of porosity, saturated and relative permeabilities, retention curve and diffusion coefficient during rock dissolution based on the assumption that rocks consisted of cylinder capillaries with fractal tortuosity and cumulative pore size distribution. However, all the abovementioned models were developed based on porous media composed of a bunch of parallel cylindrical capillaries. But lately, on the basis of the parallel plate model and the fractal scaling law of length distribution of fractures, a fractal model for permeability for fractured rocks was proposed by Miao et al. [26]. Although Miao’s model was in good agreement with numerical simulations, it is not applicable to Non-Newtonian fluids flow through porous media consisting of arbitrary cross-sectional capillaries. In reality, however, most real porous media in the nature is tremendous complex and consists of numerous tortuous capillaries/pores with various shapes. Therefore, it is very meaningful to develop a more general permeability model by considering pores of various geometrical shapes, which can reveal the flow problems of NonNewtonian fluids in real porous media. In this article, we aim to develop the fractal permeability model for porous media composed of arbitrary cross sectional capillaries/pores, incorporating tortuous effects for fracture flow of Non-Newtonian power-law fluids. First, the flow rate for Non-Newtonian power-law fluids flow through a single duct of arbitrary cross section is introduced. Then based on the fractal theory, the flow rate, the average flow velocity as well as the fractal permeability model for Non-Newtonian powerlaw fluids in fractal porous media composed of arbitrary cross-sectional capillaries are derived. Meanwhile, the proposed model is verified by comparing its results with the available macroscopic models. Finally, we also rigorously examine our fractal permeability model, and come to a conclusion that the permeability model of Yu and Cheng [22] can be considered as a special case of our present permeability model.
37
2. The flow rate for Non-Newtonian power-law fluids in porous media composed of arbitrary shape capillaries
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
38 39
40 41 42 43 44 45
In the following section, some assumptions of the present model for Non-Newtonian power-law fluids flow in porous media are summarized as: (i) (ii) (iii) (iv) (v) (vi)
incompressible, full developed, steady state, continuum, and laminar flow; negligible surface effect and body forces such as gravity, Coriolis, no slip boundary condition; a single duct of constant cross-sectional area A and constant perimeter C ; all ducts have no branch and no intersection; power-laws fluids flow through ducts in one direction; porous media composed of a bundle of tortuous capillaries with various shapes as illustrated in Fig. 1.
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
3
Table 1 Geometric parameters a, b, c for various cross-sectional ducts [27]. Geometric parameters
a
Circular Square Equilateral triangular Semicircular
2 π 3.7471 4.3821 4.1798
b
√
√
c
√
π
3 π 5.8606 6.8785 6.6023
1.7330 1.0389 1.3344
Based on the above assumptions, the flow for Non-Newtonian power-law fluids in a straight duct of arbitrary cross section A can be expressed as [27] 3
q(A) =
1
F (n)A 2 + 2n
dp 1n − dL
(1)
µ1/n
where dL , A, and F (n) are the pressure gradient along the axial length of the duct, the cross-sectional area of the duct, and the shape factor function, respectively. The parameters µ and n are called material constants of power-law fluids, and represent the consistency index and the power index, respectively. Eq. (1) denotes the flow rate for the power-law fluid through a duct of the arbitrary cross section. Note that Eq. (1) is valid for various duct geometries [27]. The shape factor function F (n) depends on the duct geometry shape and is defined as dp
1
F (n) = a n
1 2
3
4 5 6 7 8
−1
b
+c
(2)
9
where a, b, and c are three constants, which depend on the shape of ducts. The three geometric parameters a, b, c are listed
10
n
√
√
√
in Table 1 [27]. Inserting a = 2 π, b = π, c = 3 π , and A = π4λ into Eq. (1), where λ is the diameter of a circular duct, the model for power-law fluids flow in a straight duct of arbitrary cross section reduces to the model for power-law fluids flow in a circular straight duct q(A) =
1
πλ3+ n
n
−dp
3n + 1 2 2n +3 µ1/n
1/n
dL
2
.
(3)
Eq. (3) coincides with the result of Govier and Aziz [28]. According to the tortuous characteristic of flow path in porous media, Eq. (1) can be modified replacing dL by dLt as 1
3
q(A) =
F (n)A 2 + 2n
1n dp − dL t
µ1/n
.
(5)
where λ denotes the capillary equivalent diameter, and DT is the tortuosity fractal dimension, with 1 < DT < 3 in three dimensions. The larger the fractal dimension DT , the more tortuous capillary in fractal porous media. Since the crosssectional area A of capillaries is proportional to the square of the equivalent diameter, A can be defined as A = (αλ)2 , where α is a geometry correction factor dependent on the shape of the duct. The geometry correction factor α is equal to √ π
for a capillary with a circular cross section, equal to 1 for a square, equal to 2 A = (αλ)2 into Eq. (5), Eq. (5) can be rearranged as Lt (A) =
1
α
1−DT
A
1−DT 2
31/4 2
for an equilateral triangle. Substituting
D
L0 T .
F (n)A q(A) =
α 1−DT 1p A
1−DT D 2 L0 T
µ1/n
14
17
18 19 20
21 22 23 24 25 26 27
(6)
Considering the fractal characteristics of tortuous capillaries, insert Eq. (6) into Eq. (4) and yield 3 1 + 2n 2
13
16
Eq. (4) represents the flow rate for the power-law fluid flow in a tortuous capillary with variable shapes. Assuming that porous media can be considered as a bundle of tortuous capillaries with various shapes, the actual length Lt of a tortuous capillary and the straight distance L0 can be scaled as [22] D
12
15
(4)
Lt (λ) = λ1−DT L0 T
11
28
29
1n .
(7)
Eq. (7) represents the flow rate for power-law fluids through a tortuous and arbitrary cross-section duct. Majumdar and Bhushan [29] described the contact spots on engineering surfaces, which satisfies the fractal scaling law. Since the pores
30
31 32
4
1 2
3
4
5
6 7 8
9
10 11
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
in fractal porous media are analogous to the islands/lakes on earth or spots on engineering surfaces, the cumulative crosssectional area distribution of pores is assumed to obey the following fractal scaling law [17,22], N (A > A) =
Amax
D2f (8)
A
where Amax is the maximum capillary/pore cross-sectional area and the pore area fractal dimension Df is defined as [22] Df = 2 −
ln
√
ln φ
(9)
Amin /Amax
where φ is the porosity of porous media, and Df has a value between 1 and 3 in three dimensions. Since there are numerous pores in fractal porous media, Eq. (8) can be considered as a continuous and differential equation. Differentiating Eq. (8) with respect to A results in the number of pores whose area sizes are within the infinitesimal range A to A + dA, dN = −
Df
Df
−
2
Amax A
2
Df 2
+1
dA.
(10)
The total flow rate Q through all the fractures in fractal porous media can be calculated by integrating Eq. (7) from the minimum pore area Amin to the maximum pore area Amax in a unit cell, i.e., Amax
12
′
Q = −
q(A)dN
Amin
13
14 15 16 17 18 19
20
21
22
23
24
=
F (n) 1
µn
DT 2n
Amax
Df
+ 32
DT n
3 − Df +
α 1−DT 1p D
1n DT /n+23−Df Amin . 1 −
(11)
Amax
L0 T
Eq. (11) indicates that the flow rate in fractal porous media is a function of the geometrical shape factors of capillaries/ pores, the maximum and minimum pore cross-sectional area A, the pore area fractal dimension Df , the tortuosity fractal dimension DT , the pressure gradient and material constants (µ, n). Eq. (11) works for the power-law fluid flow in fractal porous media with variable shape capillaries. The average flow velocity V for Non-Newtonian power-law fluids in fractal porous media with a cluster of fractal arbitrary cross-section capillaries/pores can be obtained by V =
Q
(12)
A0
where A0 is the total cross section area of the unit cell, which can be calculated as follows:
A0 =
Ap
φ
− =
Amax Amin
φ
AdN
=
Df Amax
(2 − Df )φ
1 −
Amin
1− D2f
Amax
.
(13)
Substituting Eqs. (11) and (13) into Eq. (12), we can obtain
V =
F ( n) 1
µn
DT 2n
+ 21
Amax
(2 − Df )φ 3 − Df +
DT n
α 1−DT 1p D
L0 T
1n DT /n+23−Df 1− D2f Amin Amin 1 − 1 − . Amax
Amax
(14)
28
Eq. (14) demonstrates that the average flow velocity is related not only to microstructural parameters, the geometry correction factor, and the shape factor function, but also to material constants of power-law fluids and the pressure gradient. Eq. (14) is applicable for fractal porous media consisting of capillaries with various shapes. There is no empirical constant, and every parameter has clear physical meaning.
29
3. Effective permeability for Non-Newtonian power-law fluids in fractal porous media
25 26 27
30 31 32 33 34
In the following section, a fractal permeability model of Non-Newtonian power law fluids in porous media consisting of a bundle of tortuous capillaries/pores with various shapes will be derived based on the generalized Darcy law and the fractal theory. Consider a unit cell composed of a bundle of parallel and tortuous capillaries with arbitrary cross-section areas. The relationship between the shear stress τ on the wall of a duct and the pressure drop can be given by
τ CLt = 1p · A
(15)
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
5
where C is the perimeter. Assume that the relationship between the perimeter and the cross-sectional area is A = β C 2 , where β is another geometry factor dependent on the shape of the capillary. Inserting Eq. (6) into Eq. (15), the shear stress in a duct of an arbitrary cross section is obtained as 1
τ = α 1−DT β 2 A
DT 2
1p D
L0 T
.
(16)
The total shear stress can be calculated by integral
τall =
DT
Amax
τ dN = α
1−DT
β
1 2
Amin
2 Df Amax 1p
DT − Df LDT 0
1 −
Amin
DT −2 Df
Amax
.
(17)
(18)
Substituting Eq. (18) into the generalized Darcy law: (19)
The effective permeability of porous media for power-law fluids can be expressed as 1 DT −2 Df 1− n Amin DT /n+23−Df 1 − Amax 1 − Amin . DT Amax −1
1− 1n
3
+
DT
2 2 F (n)Amax
1−DT
Df 3 − Df +
L0 DT n
A0
2π
Amax = (αλmax ) = 2
(20)
Amax
√ =
2(1 − φ)
αR
24
2
11
13
14 15 16 17 18 19
2φ 1−φ
.
21 22 23
(21)
24
(22)
25
2
where R is the average radius of particles in porous media, which can be measured by experimentation [22]. The ratio of the minimum cross-section area of the capillary to the maximum cross-section area of the capillary is given by [31] Amin
9
20
3(1 − φ)
8
Df
In this section, the proposed flow velocity model for the power law fluid in fractal porous media is compared with the available macroscopic model [30]. The effective permeability of porous media in Eq. (20) and the average flow velocity in Eq. (14) are calculated by the following structural parameters of porous media, which can be expressed as [22]
√
7
4. The result and discussion
L0 = R
6
12
Eq. (20) is the effective permeability for power-law fluids in fractal porous media with an assembly of fractal arbitrary shape capillaries/pores. The fractal permeability model can be expressed as a function of geometry shape factors of capillaries, material constants of fluids, the minimum and the maximum capillary cross-section area and the fractal dimensions, and has no empirical constant. In addition, the physical mechanism of affecting the permeability of powerlaw fluids in porous media can be clearly revealed. It can be seen from Eqs. (20) and (11), both the permeability and the total flow rate are very sensitive to the maximum cross-section area of the capillary.
4
10
µa L 0 Q . A0 1p
1
3
5
1 DT DT −2 Df 1− n 2 τall A 1 Df Amax 1p 1 1− 1 1 min 1 − = µ n τall n = µ n α 1−DT β 2 . µa = γ˙ DT − Df LDT Amax 0
Ke = α 1−DT β 2
2
According to the constitutive equation τ = µγ˙ n of the power law fluids and the definition of the apparent viscosity µa , one can find
Ke =
1
(23)
Once the average radius of solid particles (R), the porosity (φ ), and geometry factors (α, β, a, b, c) are found, the other parameters such as Df , L0 , Amax can be determined from Eqs. (9), (21) and (22). The rheological properties of power-law
26 27
28
29 30
6
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
Fig. 2. A comparison of the present predicted average flow velocity model equation (14) with the macroscopic model equation (24). The parameters for power-law fluids are: n = 0.3, µ0 = 0.2 Pa·s and the parameters in packed beds of spherical particles are: dp = 6.9×10−4 m, c0 = 25/12, β ′ = 1.49 [28]. 1 2
fluids in packed beds were investigated by Balhoff and Thompson [30]. The relationship between the average flow velocity and the pressure gradient can be expressed as [30]
3
V =
4
µeff
5
6
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33
k 1p
1n (24)
µeff L0 n 1−2 n 3n + 1 kφ =µ . 4n β′
(25)
The permeability of porous media is given by the well known Kozeny–Carman equation k=
d2p φ 3
(26)
72c0 (1 − φ)2
where µeff is the effective viscosity of the power-law fluid, β ′ and c0 are empirical constants, c0 is often reported as 25/12 in the literature, and dp is the diameter of the spherical solid particle. However, two empirical constants (β ′ and c0 ) are introduced in their macroscopic model. The influence of flow tortuosity on the average flow velocity is not taken into account in Eq. (24), so the physical mechanisms which affect the average flow velocity for the power-law fluid flow through porous media could not be sufficiently revealed. Therefore, our fractal average flow velocity model of Eq. (14) is obviously more advantageous than that of macroscopic model. We compare our fractal average flow velocity prediction equation (14) with the average flow velocity equation (24) at given parameters R = 0.3 mm, DT = 1.10, n = 0.3 and µ = 0.2 Pa · s. As we can be seen from Fig. 2, the average flow velocities of both Eqs. (14) and (24) increase nonlinearly with the increase of the pressure gradient. This is in consistence with the practical physical situation. A good agreement can be shown between the present fractal average flow velocity prediction by Eq. (14), whose porous media is assumed to be composed of circular capillaries, and the macroscopic model by Eq. (24). The fractal average flow velocity prediction from Eq. (14) for power-law fluids in porous media consisting of equilateral triangular or semicircular capillaries is slightly lower than that in porous media composed of circular/square capillaries at the large pressure gradient. A possible explanation for this might be that the flow resistance for power-law fluids flow through porous media consisting of circular/square capillaries is lower than that through porous media consisting of equilateral triangular/semicircular capillaries. Fig. 3 illustrates the relationship between the effective permeability and the porosity at different power indexes, for porous media composed of circular capillaries at DT = 1.10. It is apparent from this figure that the permeability increases with the increase of the power index n and the porosity φ . This is also qualitatively in consistence with the practical physical situation. The effect of the tortuosity fractal dimension DT on the effective permeability in Eq. (20) for porous media with circular capillaries is shown in Fig. 4. From Fig. 4, we can see that the effective permeability increases with the increase of porosity and the decrease of the tortuosity fractal dimension at the given parameters Df = 1.4 and n = 1.2. This is because of the larger the tortuosity fractal dimension, the larger flow resistance, causing the lower the permeability. Finally, we compare our present model equation (20) with the available fractal permeability model for fractal porous media with circular cross-section capillary/pores [22]. For n = 1, Eq. (11) reduces to
Q =
1 a(b + c )µ
DT 2
+ 23
Amax
Df 3 − Df + DT
α 1−DT 1p D
L0 T
1−
Amin Amax
DT +3−Df .
(27)
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
7
Fig. 3. The effective permeability in Eq. (20) for porous media composed of circular capillaries versus the porosity at the different power indexes and DT = 1.10.
Fig. 4. Effect of DT on the permeability at Df = 1.4 and n = 1.2.
Eq. (27) indicates that the total flow rate for Newtonian fluids flow through fractal porous media with arbitrary shape
capillaries/pores. Since 1 < DT < 3 and 1 < Df < 3, the exponent 3 + DT − Df > 1 holds. Because of
porous media, ( Q =
)
Amin DT +3−Df Amax DT
1
Df
α 1−DT 1p
3 − Df + DT
L0 T
.
1
∼ 10−2 in
approaches 0. Then, Eq. (27) can be simplified into
+ 32
2 Amax
Amin Amax
2
Q2
3
(28)
4
The average velocity V for Newtonian fluids in fractal porous media with arbitrary shape capillaries/pores can be obtained
5
a(b + c )µ
D
by
6
V =
Q A0
=
1 a(b + c )µ
DT
+ 23
2 Amax
Df
α 1−DT 1p
3 − Df + DT
A0 L0 T
D
.
(29)
Using Darcy’s law, we can obtain the permeability expression for Newtonian fluids flow through fractal porous media with arbitrary shape capillary/pores as follows: K =
µL 0 Q A0 1p
7
8 9
10
1−D
=
α 1−DT D2T + 32 Df L0 T Amax . a(b + c ) 3 − Df + DT A0
(30)
11
It should be noted that Eq. (30) is applicable for Newtonian fluids flow through porous media composed √ of√various shape √ capillaries. For circular cross-section capillaries/pores, the geometric parameters a, b, c are taken as 2 π , π and 3 π ,
13
12
8
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
Fig. 5. A comparison of permeability between the present model by Eq. (30) and experimental data [30].
1 2
3
respectively (see in Table 1). Inserting Amax = π4 λ2max into Eq. (30), the permeability expression for Newtonian fluids in fractal porous media with circular cross-section capillaries/pores can be expressed as K =
π 128
T +3 λDmax
Df 3 − Df + DT
1−DT
L0
A0
.
(31)
17
The above permeability expression Eq. (31) is exactly the same as Eq. (15) of Yu and Cheng [22]. However, the permeability model of Yu and Cheng is valid only for Newtonian fluids in the fractal porous media composed of a bundle of tortuous cylindrical capillaries. Therefore, our present permeability model equation (20) can be regarded as the generalization of Yu and Cheng’s model, which is suitable for both Newtonian and Non-Newtonian fluids flow in fractal porous media composed of arbitrary cross sectional capillaries. Fig. 5 shows a comparison of the permeability for Newtonian fluids between the present model predictions by Eq. (30) and experimental data measured by Chen et al. [32], at the given tortuosity fractal dimension DT = 1.1 and R = 0.3 mm. It is shown that the present fractal permeability model for Newtonian fluids in fractal porous media is in good agreement with experimental results. In the same figure, the Kozeny–Carman Equation given by Eq. (25) is also plotted, which describes the relationship between the porosity and the permeability based on porous bed packed consisting of uniform spherical solid particles. It is evident that the Kozeny–Carman Equation overestimates the permeability. Furthermore, Kozeny–Carman Equation is a semi-empirical relation, which could not reveal the underlying mechanisms for flow in porous media, while our fractal permeability model equation (30) has no empirical constant and reveals the effect of microstructural parameters of porous media on the permeability.
18
5. Concluding remarks
4 5 6 7 8 9 10 11 12 13 14 15 16
28
The fractal average flow velocity and effective permeability models for power-law fluids in porous media, which can be considered as numerous arbitrary cross-section capillaries/pores, have been developed based on the fractal geometry. Considering the effects of tortuous capillaries with arbitrary cross sections, both the fractal flow velocity and effective permeability can be expressed in terms of geometrical shape factors of capillaries, material constants, the fractal dimensions, microstructural parameters of porous media. There are no empirical constants in our model and every parameter has definite physical meaning. In addition, the present model is compared with the available macroscopic model and a good agreement is obtained between them. Meanwhile, our permeability model can be considered as an extension of Yu and Cheng’s model. Nevertheless, it should also be pointed out that our models are presented based on the assumptions that the capillaries are smooth and have no branches. However, the flow and transport property for the power-law fluid in real porous media are very complicated, we should develop a more accurate model. This will be our next work.
29
Acknowledgments
19 20 21 22 23 24 25 26 27
30 31 32 33
This work was supported by Open Fund (PLN1205) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), the National Natural Science Foundation of China under Grant Nos. Q3 11402081 and 11304235 as well as the Scientific Research Foundation of the Education Department of Hubei Province under Grant No. Q20131512.
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
9
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
J. Bear, Dynamics of Fluid in Porous Media, Université of Toronto Press, New York, 1972. M. Tek, K. Coats, D. Katz, The effect of turbulence on flow of natural gas through porous reservoirs, J. Pet. Technol. 14 (1962) 799–806. Y.S. Wu, An approximate analytical solution for non-Darcy flow toward a well in fractured media, Water Resour. Res. 38 (2002) 5-1–5-7. M.I.J. van Dijke, K.S. Sorbie, M. Sohrabi, A. Danesh, Simulation of WAG floods in an oil-wet micromodel using a 2-D pore-scale network model, J. Pet. Sci. Eng. 52 (2006) 71–86. S. Wang, Y. Huang, F. Civan, Experimental and theoretical investigation of the Zaoyuan field heavy oil flow through porous media, J. Pet. Sci. Eng. 50 (2006) 83–101. L.B. Monachesi, L. Guarracino, A fractal model for predicting water and air permeabilities of unsaturated fractured rocks, Transp. Porous Media 90 (2011) 779–789. Y.-S. Wu, B. Lai, J. Miskimins, P. Fakcharoenphol, Y. Di, Analysis of multiphase non-Darcy flow in porous media, Transp. Porous Media 88 (2011) 205–223. B. Yu, L.J. Lee, H. Cao, A fractal in-plane permeability model for fabrics, Polym. Compos. 23 (2002) 201–221. D. Shou, J. Fan, M. Mei, F. Ding, An analytical model for gas diffusion though nanoscale and microscale fibrous media, Microfluid. Nanofluid. 16 (2014) 381–389. E.B. Belov, S.V. Lomov, I. Verpoest, T. Peters, D. Roose, R. Parnas, K. Hoes, H. Sol, Modelling of permeability of textile reinforcements: lattice Boltzmann method, Compos. Sci. Technol. 64 (2004) 1069–1080. B.R. Masters, Fractal analysis of the vascular tree in the human retina, Annu. Rev. Biomed. Eng. 6 (2004) 427–452. L. Li, B. Yu, M. Liang, S. Yang, M. Zou, A comprehensive study of the effective thermal conductivity of living biological tissue with randomly distributed vascular trees, Int. J. Heat Mass Transfer 72 (2014) 616–621. H. Darcy, Les Fontaines Publiques de la Ville de Dijon: Exposition et Application, Victor Dalmont, 1856. R. Baviere, F. Ayela, S. Le Person, M. Favre-Marinet, Experimental characterization of water flow through smooth rectangular microchannels, Phys. Fluids (1994-present) 17 (2005) 098105. A. Lavrov, Numerical modeling of steady-state flow of a non-Newtonian power-law fluid in a rough-walled fracture, Comput. Geotech. 50 (2013) 101–109. K. Yazdchi, S. Srivastava, S. Luding, Microstructural effects on the permeability of periodic fibrous porous media, Int. J. Multiph. Flow 37 (2011) 956–966. B. Yu, Analysis of flow in fractal porous media, Appl. Mech. Rev. 61 (2008) 050801. B.B. Mandelbrot, The Fractal Geometry of Nature, Macmillan, 1983. J. Kou, F. Wu, H. Lu, Y. Xu, F. Song, The effective thermal conductivity of porous media based on statistical self-similarity, Phys. Lett. A 374 (2009) 62–65. 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 (2014) 5142–5151. Q. Zheng, J. Xu, B. Yang, B. Yu, Research on the effective gas diffusion coefficient in dry porous media embedded with a fractal-like tree network, Physica A 392 (2013) 1557–1566. B. Yu, P. Cheng, A fractal permeability model for bi-dispersed porous media, Int. J. Heat Mass Transfer 45 (2002) 2983–2993. S. Wang, B. Yu, Q. Zheng, Y. Duan, Q. Fang, A fractal model for the starting pressure gradient for Bingham fluids in porous media embedded with randomly distributed fractal-like tree networks, Adv. Water Resour. 34 (2011) 1574–1580. P. Xu, B. Yu, X. Qiao, S. Qiu, Z. Jiang, Radial permeability of fractured porous media by Monte Carlo simulations, Int. J. Heat Mass Transfer 57 (2013) 369–374. 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 analysis of permeability for fractured rocks, Int. J. Heat Mass Transfer 81 (2015) 75–80. T.J. Liu, Fully developed flow of power-law fluids in ducts, Ind. Eng. Chem. Fundam. 22 (1983) 183–186. G.W. Govier, K. Aziz, The Flow of Complex Mixtures in Pipes, second ed., Society of Petroleum Engineers, Richardson, TX, 2008. A. Majumdar, B. Bhushan, Role of fractal geometry in roughness characterization and contact mechanics of surfaces, J. Tribol. 112 (1990) 205–216. M.T. Balhoff, K.E. Thompson, A macroscopic model for shear-thinning flow in packed beds based on network modeling, Chem. Eng. Sci. 61 (2006) 698–719. Y. Li, B. Yu, J. Chen, C. Wang, Analysis of permeability for Ellis fluid flow in fractal porous media, Chem. Eng. Commun. 195 (2008) 1240–1256. Z. Chen, P. Cheng, T. Zhao, An experimental study of two phase flow and boiling heat transfer in bi-dispersed porous channels, Int. Commun. Heat Mass Transfer 27 (2000) 293–302.
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
A permeability model for power-law fluids in fractal porous media composed of arbitrary cross-section capillaries Q1
Shifang Wang a,b,∗ , Tao Wu c , Hongyan Qi b , Qiusha Zheng b , Qian Zheng d a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, PR China
b
Department of Physics and mechanical & electrical engineering, Hubei University of Education, Wuhan 430205, PR China
c
Laboratory of Optical Information and Technology, School of Science, Wuhan Institute of Technology, Wuhan 430074, PR China
d
School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, PR China
highlights • • • • •
A permeability model for power-law fluids in porous media is developed. The permeability is a function of fluid property and structure parameters of media. The effects of structure parameters on permeability and velocity are discussed. The average flow velocity is compared with existing macroscopic model. Our permeability model can be considered as an extension of Yu and Cheng’s model.
article
info
Article history: Received 10 March 2015 Received in revised form 23 April 2015 Available online xxxx Keywords: Fractal theory Porous media Permeability Power-law fluids
abstract The fractal theory and technology has been applied to determine the flow rate, the average flow velocity, and the effective permeability for the power-law fluid in porous media composed of a number of tortuous capillaries/pores with arbitrary shapes, incorporating the tortuosity characteristic of flow paths. The fractal permeability and average flow velocity expressions are found to be a function of geometrical shape factors of capillaries, material constants, the fractal dimensions, microstructural parameters. The effect of the porosity, the tortuosity fractal dimension, material constants, and geometrical shape factors on the effective permeability is also analyzed in detail. To verify the validity of the present model, our proposed model is compared with the available macroscopic model and experimental data and there is good agreement between them. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Modeling of flow and transport properties in porous media is an active field of interest for a variety of industrial applications, such as oil reservoirs [1–7], textile industry [8–10], and biological tissues and organs [11,12]. The macroscopic transport properties of porous media such as the flow rate, the average velocity, and the permeability mainly depend on the microstructure of porous media. Since the 1880s, large numbers of literatures have begun to investigate flow properties of porous media. For instance, early as in 1856, the preliminary experimental measurement for Newtonian fluids flow through porous media was undertaken by Darcy [13]. To better understand the mechanisms of hydrodynamics of liquid
∗
Corresponding author at: Department of Physics and mechanical & electrical engineering, Hubei University of Education, Wuhan 430205, PR China. E-mail address: [email protected] (S. Wang).
http://dx.doi.org/10.1016/j.physa.2015.05.089 0378-4371/© 2015 Elsevier B.V. All rights reserved.
1
2 3 4 5 6 7
2
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
Fig. 1. A sketch of porous media composed of a bundle of tortuous capillaries with various shapes.
36
flows in smooth wall microchannels (relative roughness <0.5%), Baviere R. et al. [14] performed the experiment on water flow through smooth rectangular microchannels and proved that a friction factor was correctly predicted by Navier–Stokes equation at the small scales. Alexandre lavro [15] studied the power-law fluid flow in a rough fracture of regular or irregular topography by using numerical simulations. Yazdchi K. [16] investigated the macroscopic permeability of fibrous porous media taking its microstructure into account by the finite element method. However, the results obtained from these experimental measurements and numerical simulations are usually correlated as curves or empirical correlations which contained one or several empirical constants, and could not reveal some basic physical mechanisms for flow in porous media. The capillaries or pores in real porous media are usually non-uniform in size, irregular, and randomly distributed. Fortunately, some published literatures have shown that the fractal geometry may have the potential in analysis of flow and transport properties in porous media [8,17–21]. Yu and Cheng [22] developed the fractal permeability model for bidispersed porous media based on the fractal characteristic of porous media. In order to investigate the starting pressure gradient in dual-porosity medium, Wang et al. [23] also applied the fractal theory and technique to obtain an analytical expression of the starting pressure gradient for Bingham fluids in porous media embedded with randomly distributed fractal-like tree networks. Xu et al. [24] developed a probability model for Newtonian fluids radial flow in fractured porous media based on fractal theory and Monte Carlo simulation. Recently, Shou et al. [9] obtained the fractal model for gas diffusion across nanoscale and microscale fibrous media based on fractal theory and capillary model. Luis Guarracino [25] presented a physically-based theoretical model which described the temporal evolution of porosity, saturated and relative permeabilities, retention curve and diffusion coefficient during rock dissolution based on the assumption that rocks consisted of cylinder capillaries with fractal tortuosity and cumulative pore size distribution. However, all the abovementioned models were developed based on porous media composed of a bunch of parallel cylindrical capillaries. But lately, on the basis of the parallel plate model and the fractal scaling law of length distribution of fractures, a fractal model for permeability for fractured rocks was proposed by Miao et al. [26]. Although Miao’s model was in good agreement with numerical simulations, it is not applicable to Non-Newtonian fluids flow through porous media consisting of arbitrary cross-sectional capillaries. In reality, however, most real porous media in the nature is tremendous complex and consists of numerous tortuous capillaries/pores with various shapes. Therefore, it is very meaningful to develop a more general permeability model by considering pores of various geometrical shapes, which can reveal the flow problems of NonNewtonian fluids in real porous media. In this article, we aim to develop the fractal permeability model for porous media composed of arbitrary cross sectional capillaries/pores, incorporating tortuous effects for fracture flow of Non-Newtonian power-law fluids. First, the flow rate for Non-Newtonian power-law fluids flow through a single duct of arbitrary cross section is introduced. Then based on the fractal theory, the flow rate, the average flow velocity as well as the fractal permeability model for Non-Newtonian powerlaw fluids in fractal porous media composed of arbitrary cross-sectional capillaries are derived. Meanwhile, the proposed model is verified by comparing its results with the available macroscopic models. Finally, we also rigorously examine our fractal permeability model, and come to a conclusion that the permeability model of Yu and Cheng [22] can be considered as a special case of our present permeability model.
37
2. The flow rate for Non-Newtonian power-law fluids in porous media composed of arbitrary shape capillaries
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
38 39
40 41 42 43 44 45
In the following section, some assumptions of the present model for Non-Newtonian power-law fluids flow in porous media are summarized as: (i) (ii) (iii) (iv) (v) (vi)
incompressible, full developed, steady state, continuum, and laminar flow; negligible surface effect and body forces such as gravity, Coriolis, no slip boundary condition; a single duct of constant cross-sectional area A and constant perimeter C ; all ducts have no branch and no intersection; power-laws fluids flow through ducts in one direction; porous media composed of a bundle of tortuous capillaries with various shapes as illustrated in Fig. 1.
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
3
Table 1 Geometric parameters a, b, c for various cross-sectional ducts [27]. Geometric parameters
a
Circular Square Equilateral triangular Semicircular
2 π 3.7471 4.3821 4.1798
b
√
√
c
√
π
3 π 5.8606 6.8785 6.6023
1.7330 1.0389 1.3344
Based on the above assumptions, the flow for Non-Newtonian power-law fluids in a straight duct of arbitrary cross section A can be expressed as [27] 3
q(A) =
1
F (n)A 2 + 2n
dp 1n − dL
(1)
µ1/n
where dL , A, and F (n) are the pressure gradient along the axial length of the duct, the cross-sectional area of the duct, and the shape factor function, respectively. The parameters µ and n are called material constants of power-law fluids, and represent the consistency index and the power index, respectively. Eq. (1) denotes the flow rate for the power-law fluid through a duct of the arbitrary cross section. Note that Eq. (1) is valid for various duct geometries [27]. The shape factor function F (n) depends on the duct geometry shape and is defined as dp
1
F (n) = a n
1 2
3
4 5 6 7 8
−1
b
+c
(2)
9
where a, b, and c are three constants, which depend on the shape of ducts. The three geometric parameters a, b, c are listed
10
n
√
√
√
in Table 1 [27]. Inserting a = 2 π, b = π, c = 3 π , and A = π4λ into Eq. (1), where λ is the diameter of a circular duct, the model for power-law fluids flow in a straight duct of arbitrary cross section reduces to the model for power-law fluids flow in a circular straight duct q(A) =
1
πλ3+ n
n
−dp
3n + 1 2 2n +3 µ1/n
1/n
dL
2
.
(3)
Eq. (3) coincides with the result of Govier and Aziz [28]. According to the tortuous characteristic of flow path in porous media, Eq. (1) can be modified replacing dL by dLt as 1
3
q(A) =
F (n)A 2 + 2n
1n dp − dL t
µ1/n
.
(5)
where λ denotes the capillary equivalent diameter, and DT is the tortuosity fractal dimension, with 1 < DT < 3 in three dimensions. The larger the fractal dimension DT , the more tortuous capillary in fractal porous media. Since the crosssectional area A of capillaries is proportional to the square of the equivalent diameter, A can be defined as A = (αλ)2 , where α is a geometry correction factor dependent on the shape of the duct. The geometry correction factor α is equal to √ π
for a capillary with a circular cross section, equal to 1 for a square, equal to 2 A = (αλ)2 into Eq. (5), Eq. (5) can be rearranged as Lt (A) =
1
α
1−DT
A
1−DT 2
31/4 2
for an equilateral triangle. Substituting
D
L0 T .
F (n)A q(A) =
α 1−DT 1p A
1−DT D 2 L0 T
µ1/n
14
17
18 19 20
21 22 23 24 25 26 27
(6)
Considering the fractal characteristics of tortuous capillaries, insert Eq. (6) into Eq. (4) and yield 3 1 + 2n 2
13
16
Eq. (4) represents the flow rate for the power-law fluid flow in a tortuous capillary with variable shapes. Assuming that porous media can be considered as a bundle of tortuous capillaries with various shapes, the actual length Lt of a tortuous capillary and the straight distance L0 can be scaled as [22] D
12
15
(4)
Lt (λ) = λ1−DT L0 T
11
28
29
1n .
(7)
Eq. (7) represents the flow rate for power-law fluids through a tortuous and arbitrary cross-section duct. Majumdar and Bhushan [29] described the contact spots on engineering surfaces, which satisfies the fractal scaling law. Since the pores
30
31 32
4
1 2
3
4
5
6 7 8
9
10 11
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
in fractal porous media are analogous to the islands/lakes on earth or spots on engineering surfaces, the cumulative crosssectional area distribution of pores is assumed to obey the following fractal scaling law [17,22], N (A > A) =
Amax
D2f (8)
A
where Amax is the maximum capillary/pore cross-sectional area and the pore area fractal dimension Df is defined as [22] Df = 2 −
ln
√
ln φ
(9)
Amin /Amax
where φ is the porosity of porous media, and Df has a value between 1 and 3 in three dimensions. Since there are numerous pores in fractal porous media, Eq. (8) can be considered as a continuous and differential equation. Differentiating Eq. (8) with respect to A results in the number of pores whose area sizes are within the infinitesimal range A to A + dA, dN = −
Df
Df
−
2
Amax A
2
Df 2
+1
dA.
(10)
The total flow rate Q through all the fractures in fractal porous media can be calculated by integrating Eq. (7) from the minimum pore area Amin to the maximum pore area Amax in a unit cell, i.e., Amax
12
′
Q = −
q(A)dN
Amin
13
14 15 16 17 18 19
20
21
22
23
24
=
F (n) 1
µn
DT 2n
Amax
Df
+ 32
DT n
3 − Df +
α 1−DT 1p D
1n DT /n+23−Df Amin . 1 −
(11)
Amax
L0 T
Eq. (11) indicates that the flow rate in fractal porous media is a function of the geometrical shape factors of capillaries/ pores, the maximum and minimum pore cross-sectional area A, the pore area fractal dimension Df , the tortuosity fractal dimension DT , the pressure gradient and material constants (µ, n). Eq. (11) works for the power-law fluid flow in fractal porous media with variable shape capillaries. The average flow velocity V for Non-Newtonian power-law fluids in fractal porous media with a cluster of fractal arbitrary cross-section capillaries/pores can be obtained by V =
Q
(12)
A0
where A0 is the total cross section area of the unit cell, which can be calculated as follows:
A0 =
Ap
φ
− =
Amax Amin
φ
AdN
=
Df Amax
(2 − Df )φ
1 −
Amin
1− D2f
Amax
.
(13)
Substituting Eqs. (11) and (13) into Eq. (12), we can obtain
V =
F ( n) 1
µn
DT 2n
+ 21
Amax
(2 − Df )φ 3 − Df +
DT n
α 1−DT 1p D
L0 T
1n DT /n+23−Df 1− D2f Amin Amin 1 − 1 − . Amax
Amax
(14)
28
Eq. (14) demonstrates that the average flow velocity is related not only to microstructural parameters, the geometry correction factor, and the shape factor function, but also to material constants of power-law fluids and the pressure gradient. Eq. (14) is applicable for fractal porous media consisting of capillaries with various shapes. There is no empirical constant, and every parameter has clear physical meaning.
29
3. Effective permeability for Non-Newtonian power-law fluids in fractal porous media
25 26 27
30 31 32 33 34
In the following section, a fractal permeability model of Non-Newtonian power law fluids in porous media consisting of a bundle of tortuous capillaries/pores with various shapes will be derived based on the generalized Darcy law and the fractal theory. Consider a unit cell composed of a bundle of parallel and tortuous capillaries with arbitrary cross-section areas. The relationship between the shear stress τ on the wall of a duct and the pressure drop can be given by
τ CLt = 1p · A
(15)
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
5
where C is the perimeter. Assume that the relationship between the perimeter and the cross-sectional area is A = β C 2 , where β is another geometry factor dependent on the shape of the capillary. Inserting Eq. (6) into Eq. (15), the shear stress in a duct of an arbitrary cross section is obtained as 1
τ = α 1−DT β 2 A
DT 2
1p D
L0 T
.
(16)
The total shear stress can be calculated by integral
τall =
DT
Amax
τ dN = α
1−DT
β
1 2
Amin
2 Df Amax 1p
DT − Df LDT 0
1 −
Amin
DT −2 Df
Amax
.
(17)
(18)
Substituting Eq. (18) into the generalized Darcy law: (19)
The effective permeability of porous media for power-law fluids can be expressed as 1 DT −2 Df 1− n Amin DT /n+23−Df 1 − Amax 1 − Amin . DT Amax −1
1− 1n
3
+
DT
2 2 F (n)Amax
1−DT
Df 3 − Df +
L0 DT n
A0
2π
Amax = (αλmax ) = 2
(20)
Amax
√ =
2(1 − φ)
αR
24
2
11
13
14 15 16 17 18 19
2φ 1−φ
.
21 22 23
(21)
24
(22)
25
2
where R is the average radius of particles in porous media, which can be measured by experimentation [22]. The ratio of the minimum cross-section area of the capillary to the maximum cross-section area of the capillary is given by [31] Amin
9
20
3(1 − φ)
8
Df
In this section, the proposed flow velocity model for the power law fluid in fractal porous media is compared with the available macroscopic model [30]. The effective permeability of porous media in Eq. (20) and the average flow velocity in Eq. (14) are calculated by the following structural parameters of porous media, which can be expressed as [22]
√
7
4. The result and discussion
L0 = R
6
12
Eq. (20) is the effective permeability for power-law fluids in fractal porous media with an assembly of fractal arbitrary shape capillaries/pores. The fractal permeability model can be expressed as a function of geometry shape factors of capillaries, material constants of fluids, the minimum and the maximum capillary cross-section area and the fractal dimensions, and has no empirical constant. In addition, the physical mechanism of affecting the permeability of powerlaw fluids in porous media can be clearly revealed. It can be seen from Eqs. (20) and (11), both the permeability and the total flow rate are very sensitive to the maximum cross-section area of the capillary.
4
10
µa L 0 Q . A0 1p
1
3
5
1 DT DT −2 Df 1− n 2 τall A 1 Df Amax 1p 1 1− 1 1 min 1 − = µ n τall n = µ n α 1−DT β 2 . µa = γ˙ DT − Df LDT Amax 0
Ke = α 1−DT β 2
2
According to the constitutive equation τ = µγ˙ n of the power law fluids and the definition of the apparent viscosity µa , one can find
Ke =
1
(23)
Once the average radius of solid particles (R), the porosity (φ ), and geometry factors (α, β, a, b, c) are found, the other parameters such as Df , L0 , Amax can be determined from Eqs. (9), (21) and (22). The rheological properties of power-law
26 27
28
29 30
6
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
Fig. 2. A comparison of the present predicted average flow velocity model equation (14) with the macroscopic model equation (24). The parameters for power-law fluids are: n = 0.3, µ0 = 0.2 Pa·s and the parameters in packed beds of spherical particles are: dp = 6.9×10−4 m, c0 = 25/12, β ′ = 1.49 [28]. 1 2
fluids in packed beds were investigated by Balhoff and Thompson [30]. The relationship between the average flow velocity and the pressure gradient can be expressed as [30]
3
V =
4
µeff
5
6
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33
k 1p
1n (24)
µeff L0 n 1−2 n 3n + 1 kφ =µ . 4n β′
(25)
The permeability of porous media is given by the well known Kozeny–Carman equation k=
d2p φ 3
(26)
72c0 (1 − φ)2
where µeff is the effective viscosity of the power-law fluid, β ′ and c0 are empirical constants, c0 is often reported as 25/12 in the literature, and dp is the diameter of the spherical solid particle. However, two empirical constants (β ′ and c0 ) are introduced in their macroscopic model. The influence of flow tortuosity on the average flow velocity is not taken into account in Eq. (24), so the physical mechanisms which affect the average flow velocity for the power-law fluid flow through porous media could not be sufficiently revealed. Therefore, our fractal average flow velocity model of Eq. (14) is obviously more advantageous than that of macroscopic model. We compare our fractal average flow velocity prediction equation (14) with the average flow velocity equation (24) at given parameters R = 0.3 mm, DT = 1.10, n = 0.3 and µ = 0.2 Pa · s. As we can be seen from Fig. 2, the average flow velocities of both Eqs. (14) and (24) increase nonlinearly with the increase of the pressure gradient. This is in consistence with the practical physical situation. A good agreement can be shown between the present fractal average flow velocity prediction by Eq. (14), whose porous media is assumed to be composed of circular capillaries, and the macroscopic model by Eq. (24). The fractal average flow velocity prediction from Eq. (14) for power-law fluids in porous media consisting of equilateral triangular or semicircular capillaries is slightly lower than that in porous media composed of circular/square capillaries at the large pressure gradient. A possible explanation for this might be that the flow resistance for power-law fluids flow through porous media consisting of circular/square capillaries is lower than that through porous media consisting of equilateral triangular/semicircular capillaries. Fig. 3 illustrates the relationship between the effective permeability and the porosity at different power indexes, for porous media composed of circular capillaries at DT = 1.10. It is apparent from this figure that the permeability increases with the increase of the power index n and the porosity φ . This is also qualitatively in consistence with the practical physical situation. The effect of the tortuosity fractal dimension DT on the effective permeability in Eq. (20) for porous media with circular capillaries is shown in Fig. 4. From Fig. 4, we can see that the effective permeability increases with the increase of porosity and the decrease of the tortuosity fractal dimension at the given parameters Df = 1.4 and n = 1.2. This is because of the larger the tortuosity fractal dimension, the larger flow resistance, causing the lower the permeability. Finally, we compare our present model equation (20) with the available fractal permeability model for fractal porous media with circular cross-section capillary/pores [22]. For n = 1, Eq. (11) reduces to
Q =
1 a(b + c )µ
DT 2
+ 23
Amax
Df 3 − Df + DT
α 1−DT 1p D
L0 T
1−
Amin Amax
DT +3−Df .
(27)
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
7
Fig. 3. The effective permeability in Eq. (20) for porous media composed of circular capillaries versus the porosity at the different power indexes and DT = 1.10.
Fig. 4. Effect of DT on the permeability at Df = 1.4 and n = 1.2.
Eq. (27) indicates that the total flow rate for Newtonian fluids flow through fractal porous media with arbitrary shape
capillaries/pores. Since 1 < DT < 3 and 1 < Df < 3, the exponent 3 + DT − Df > 1 holds. Because of
porous media, ( Q =
)
Amin DT +3−Df Amax DT
1
Df
α 1−DT 1p
3 − Df + DT
L0 T
.
1
∼ 10−2 in
approaches 0. Then, Eq. (27) can be simplified into
+ 32
2 Amax
Amin Amax
2
Q2
3
(28)
4
The average velocity V for Newtonian fluids in fractal porous media with arbitrary shape capillaries/pores can be obtained
5
a(b + c )µ
D
by
6
V =
Q A0
=
1 a(b + c )µ
DT
+ 23
2 Amax
Df
α 1−DT 1p
3 − Df + DT
A0 L0 T
D
.
(29)
Using Darcy’s law, we can obtain the permeability expression for Newtonian fluids flow through fractal porous media with arbitrary shape capillary/pores as follows: K =
µL 0 Q A0 1p
7
8 9
10
1−D
=
α 1−DT D2T + 32 Df L0 T Amax . a(b + c ) 3 − Df + DT A0
(30)
11
It should be noted that Eq. (30) is applicable for Newtonian fluids flow through porous media composed √ of√various shape √ capillaries. For circular cross-section capillaries/pores, the geometric parameters a, b, c are taken as 2 π , π and 3 π ,
13
12
8
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
Fig. 5. A comparison of permeability between the present model by Eq. (30) and experimental data [30].
1 2
3
respectively (see in Table 1). Inserting Amax = π4 λ2max into Eq. (30), the permeability expression for Newtonian fluids in fractal porous media with circular cross-section capillaries/pores can be expressed as K =
π 128
T +3 λDmax
Df 3 − Df + DT
1−DT
L0
A0
.
(31)
17
The above permeability expression Eq. (31) is exactly the same as Eq. (15) of Yu and Cheng [22]. However, the permeability model of Yu and Cheng is valid only for Newtonian fluids in the fractal porous media composed of a bundle of tortuous cylindrical capillaries. Therefore, our present permeability model equation (20) can be regarded as the generalization of Yu and Cheng’s model, which is suitable for both Newtonian and Non-Newtonian fluids flow in fractal porous media composed of arbitrary cross sectional capillaries. Fig. 5 shows a comparison of the permeability for Newtonian fluids between the present model predictions by Eq. (30) and experimental data measured by Chen et al. [32], at the given tortuosity fractal dimension DT = 1.1 and R = 0.3 mm. It is shown that the present fractal permeability model for Newtonian fluids in fractal porous media is in good agreement with experimental results. In the same figure, the Kozeny–Carman Equation given by Eq. (25) is also plotted, which describes the relationship between the porosity and the permeability based on porous bed packed consisting of uniform spherical solid particles. It is evident that the Kozeny–Carman Equation overestimates the permeability. Furthermore, Kozeny–Carman Equation is a semi-empirical relation, which could not reveal the underlying mechanisms for flow in porous media, while our fractal permeability model equation (30) has no empirical constant and reveals the effect of microstructural parameters of porous media on the permeability.
18
5. Concluding remarks
4 5 6 7 8 9 10 11 12 13 14 15 16
28
The fractal average flow velocity and effective permeability models for power-law fluids in porous media, which can be considered as numerous arbitrary cross-section capillaries/pores, have been developed based on the fractal geometry. Considering the effects of tortuous capillaries with arbitrary cross sections, both the fractal flow velocity and effective permeability can be expressed in terms of geometrical shape factors of capillaries, material constants, the fractal dimensions, microstructural parameters of porous media. There are no empirical constants in our model and every parameter has definite physical meaning. In addition, the present model is compared with the available macroscopic model and a good agreement is obtained between them. Meanwhile, our permeability model can be considered as an extension of Yu and Cheng’s model. Nevertheless, it should also be pointed out that our models are presented based on the assumptions that the capillaries are smooth and have no branches. However, the flow and transport property for the power-law fluid in real porous media are very complicated, we should develop a more accurate model. This will be our next work.
29
Acknowledgments
19 20 21 22 23 24 25 26 27
30 31 32 33
This work was supported by Open Fund (PLN1205) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), the National Natural Science Foundation of China under Grant Nos. Q3 11402081 and 11304235 as well as the Scientific Research Foundation of the Education Department of Hubei Province under Grant No. Q20131512.
S. Wang et al. / Physica A xx (xxxx) xxx–xxx
9
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
J. Bear, Dynamics of Fluid in Porous Media, Université of Toronto Press, New York, 1972. M. Tek, K. Coats, D. Katz, The effect of turbulence on flow of natural gas through porous reservoirs, J. Pet. Technol. 14 (1962) 799–806. Y.S. Wu, An approximate analytical solution for non-Darcy flow toward a well in fractured media, Water Resour. Res. 38 (2002) 5-1–5-7. M.I.J. van Dijke, K.S. Sorbie, M. Sohrabi, A. Danesh, Simulation of WAG floods in an oil-wet micromodel using a 2-D pore-scale network model, J. Pet. Sci. Eng. 52 (2006) 71–86. S. Wang, Y. Huang, F. Civan, Experimental and theoretical investigation of the Zaoyuan field heavy oil flow through porous media, J. Pet. Sci. Eng. 50 (2006) 83–101. L.B. Monachesi, L. Guarracino, A fractal model for predicting water and air permeabilities of unsaturated fractured rocks, Transp. Porous Media 90 (2011) 779–789. Y.-S. Wu, B. Lai, J. Miskimins, P. Fakcharoenphol, Y. Di, Analysis of multiphase non-Darcy flow in porous media, Transp. Porous Media 88 (2011) 205–223. B. Yu, L.J. Lee, H. Cao, A fractal in-plane permeability model for fabrics, Polym. Compos. 23 (2002) 201–221. D. Shou, J. Fan, M. Mei, F. Ding, An analytical model for gas diffusion though nanoscale and microscale fibrous media, Microfluid. Nanofluid. 16 (2014) 381–389. E.B. Belov, S.V. Lomov, I. Verpoest, T. Peters, D. Roose, R. Parnas, K. Hoes, H. Sol, Modelling of permeability of textile reinforcements: lattice Boltzmann method, Compos. Sci. Technol. 64 (2004) 1069–1080. B.R. Masters, Fractal analysis of the vascular tree in the human retina, Annu. Rev. Biomed. Eng. 6 (2004) 427–452. L. Li, B. Yu, M. Liang, S. Yang, M. Zou, A comprehensive study of the effective thermal conductivity of living biological tissue with randomly distributed vascular trees, Int. J. Heat Mass Transfer 72 (2014) 616–621. H. Darcy, Les Fontaines Publiques de la Ville de Dijon: Exposition et Application, Victor Dalmont, 1856. R. Baviere, F. Ayela, S. Le Person, M. Favre-Marinet, Experimental characterization of water flow through smooth rectangular microchannels, Phys. Fluids (1994-present) 17 (2005) 098105. A. Lavrov, Numerical modeling of steady-state flow of a non-Newtonian power-law fluid in a rough-walled fracture, Comput. Geotech. 50 (2013) 101–109. K. Yazdchi, S. Srivastava, S. Luding, Microstructural effects on the permeability of periodic fibrous porous media, Int. J. Multiph. Flow 37 (2011) 956–966. B. Yu, Analysis of flow in fractal porous media, Appl. Mech. Rev. 61 (2008) 050801. B.B. Mandelbrot, The Fractal Geometry of Nature, Macmillan, 1983. J. Kou, F. Wu, H. Lu, Y. Xu, F. Song, The effective thermal conductivity of porous media based on statistical self-similarity, Phys. Lett. A 374 (2009) 62–65. 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 (2014) 5142–5151. Q. Zheng, J. Xu, B. Yang, B. Yu, Research on the effective gas diffusion coefficient in dry porous media embedded with a fractal-like tree network, Physica A 392 (2013) 1557–1566. B. Yu, P. Cheng, A fractal permeability model for bi-dispersed porous media, Int. J. Heat Mass Transfer 45 (2002) 2983–2993. S. Wang, B. Yu, Q. Zheng, Y. Duan, Q. Fang, A fractal model for the starting pressure gradient for Bingham fluids in porous media embedded with randomly distributed fractal-like tree networks, Adv. Water Resour. 34 (2011) 1574–1580. P. Xu, B. Yu, X. Qiao, S. Qiu, Z. Jiang, Radial permeability of fractured porous media by Monte Carlo simulations, Int. J. Heat Mass Transfer 57 (2013) 369–374. 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 analysis of permeability for fractured rocks, Int. J. Heat Mass Transfer 81 (2015) 75–80. T.J. Liu, Fully developed flow of power-law fluids in ducts, Ind. Eng. Chem. Fundam. 22 (1983) 183–186. G.W. Govier, K. Aziz, The Flow of Complex Mixtures in Pipes, second ed., Society of Petroleum Engineers, Richardson, TX, 2008. A. Majumdar, B. Bhushan, Role of fractal geometry in roughness characterization and contact mechanics of surfaces, J. Tribol. 112 (1990) 205–216. M.T. Balhoff, K.E. Thompson, A macroscopic model for shear-thinning flow in packed beds based on network modeling, Chem. Eng. Sci. 61 (2006) 698–719. Y. Li, B. Yu, J. Chen, C. Wang, Analysis of permeability for Ellis fluid flow in fractal porous media, Chem. Eng. Commun. 195 (2008) 1240–1256. Z. Chen, P. Cheng, T. Zhao, An experimental study of two phase flow and boiling heat transfer in bi-dispersed porous channels, Int. Commun. Heat Mass Transfer 27 (2000) 293–302.
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33