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A simulation method for permeability of porous media based on multiple fractal model
International Journal of Engineering Science 95 (2015) 76–84
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
A simulation method for permeability of porous media based on multiple fractal model Xiao-Hua Tan a,⇑, Jian-Yi Liu a, Xiao-Ping Li a, Lie-Hui Zhang a, Jianchao Cai b,⇑ a b
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Xindu Road 8, Chengdu 610500, China Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 9 December 2014 Received in revised form 20 February 2015 Accepted 29 June 2015
Keywords: Simulation method Multiple fractal model The pore size distribution Permeability Porous media
a b s t r a c t Fluid flow in fractal porous media is a ubiquitous natural phenomenon which has received much attention over three decades in a wide variety of fields. In order to find a relationship between the pore of distribution and the permeability of porous media, a simulation method for the permeability of porous media is proposed based on the multiple fractal model. The pore size distributions and the permeabilities of the porous medium simulated by the presented method are compared with available experimental data. The validity of simulation method is obtained by the good agreement between the simulated results and the experimental data. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since Mandelbrot’s pioneer work (Mandelbrot, 1983), flow phenomena in fractal porous media have been studied extensively for more than three decades. Katz and Thompson (1985) proved the pore spaces are fractal geometries in several sandstones by using scanning electron microscopy and optical data. Jacquin and Adler (1987) measured some geological structures and found them were fractal. The consequences of these measurements can be used on the analysis of transport processes in porous media. Sahimi (1993) considered that fractal and percolation concepts play important roles in the characterization of porous medium, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Li and Logan (2001) demonstrated that cluster-fractal model was more practical to predict the permeability of the porous media than single-particle-fractal model. The cluster-fractal model has been used widely in many fields. Karacan and Halleck (2003) presented a fractal model for predicting permeability around perforation tunnels. The model provided an easier and cheaper way of mapping the permeability distribution around the perforation tunnels compared to the viscous fluid injection and pressure transient measurement techniques. Cai, Yu, Zou and Luo (2010) developed an analytical model for characterizing spontaneous imbibitions of wetting liquid vertically into gas-saturated porous media, based on the fractal character of pores in porous media. Shou, Ye, and Fan (2014) designed a single-layer porous structure to achieve the fastest capillary flow under gravity. The theoretical results obtained can be used for the optimization of porous architectures, achieving excellent liquid management properties. Xiao, Fan, and Ding (2014) obtained an analytical model for effective
⇑ Corresponding authors. Tel.: +86 13438368455 (X.-H. Tan), +86 18971193160 (J. Cai). E-mail addresses: [email protected] (X.-H. Tan), [email protected] (J.-Y. Liu), [email protected] (X.-P. Li), [email protected] (L.-H. Zhang), [email protected] (J. Cai). http://dx.doi.org/10.1016/j.ijengsci.2015.06.007 0020-7225/Ó 2015 Elsevier Ltd. All rights reserved.
X.-H. Tan et al. / International Journal of Engineering Science 95 (2015) 76–84
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thermal conductivity of nanofluids, considering the effect of Brownian motion of nanoparticles. The researches of flow in fractal porous media have never been stopped from Mandelbrot’s work (Perfect, 2005). It is convenient to describe the characters of porous media and explain the relationship among the parameters of porous media using the fractal theory (Jiang, Wang, Hou, & He, 2013). Hansen and Skjeltorp (1988) thought most porous medium were very complex systems consisted of a range of irregular grain sizes, and that made it be hard to find the relationship between permeability and porosity. They considered that the fractal theory was helpful for the research and proposed a relationship equation between permeability and porosity using the fractal theory. Pape, Clauser, and Iffland (1999) presented a relation linking porosity to permeability based on a fractal model for the internal structure of the porous media. Good agreement between the predictions of the relation and the experimental data was obtained. Park, Lee, and Lee (2006) developed a model based on fractal theory and incorporated a cake collapse effect to predict the porosity and permeability of the cake layers. Costa (2006) derived a permeability–porosity equation using the classical Kozeny–Carman approach and a fractal pore-space geometry assumption. The equation was simple and capable to describe permeabilities of different non-granular porous media. Most studies of permeability of fractal porous media expounded the relationship between permeability and porosity (Cai, Perfect, Cheng, & Hu, 2014; Cihan, Perfect, & Tyner, 2007; Tan, Li, Zhang, Liu, & Cai, 2015). Researchers found that the natural porous media, such as rock, soil and biological tissue, is a very complex structure, and the multifractal theory is a practical way to describe the natural porous media (Stanley & Meakin, 1988). Saucier (1992) calculated the scaling exponents of the effective absolute permeability in multifractal porous media and discuss the implications of the results on the understanding of fluid flow in oil reservoirs. Rigby and Gladden (1996) proposed a multifractal description of porous media, and represented the macroscopic heterogeneity associated with the pore-size distribution and the fractal characteristics of the microscopic pore structure. Martı´n and Montero (2002) analyzed the characterization of dry volume-size distributions in soils using the application of laser diffraction and multifractal theory. The result of analysis showed that multifractal is a useful mathematical tool for the characterization of porous media. Hunt and Gee (2002) proposed a ‘‘dual’’ fractal model using two types of fractal porous medium. The pore size distributions of some porous media were described by the ‘‘dual’’ fractal model. When the pore size distribution of porous media is complex, it is tough to be described accurately by two types of fractal porous medium. Perfect, Gentry, Sukop, and Lawson (2006) derived an approximate analytical expression for the effective hydraulic conductivity of multifractal porous media based on the generation of Sierpinski carpet. They found geometrical multifractals were good to simulate distinct facies or transport abilities of porous media. Vázquez et al. (2008) analyzed the pore size distributions of soils affected by rainfall based on multifractal theory. Pore size distributions of aggregates from both the reference soil surface and the soil surface disturbed by rain exhibited multifractal behavior. Paz-Ferreiro and Vázquez (2014) characterized pore size distributions from tropical soils using the multifractal approach and found that multifractal analyses of soil pore size distributions are useful for analyzing the soil porous system. In this paper, we attempt to establish a method to describe the pore size distribution and the permeability of porous media at the same time assuming that the porous media are consisted by some types of fractal porous media. To this end, multiple fractal model will be derived in Section 2, and then a simulation method for the pore size distribution and the permeability of porous media is proposed in Section 3, finally in Section 4 relevant results and discussions are demonstrated and compared. Conclusions are given in Section 5. 2. The multiple fractal model The natural porous media are composed of different types of porosities. For example, the main porosity types of sedimentation rocks are now described and illustrated in Fig. 1. Six main types of pore systems have been show in Fig. 1, including primary intergranular porosity, primary intragranular porosity, secondary intergranular porosity, secondary intragranular porosity, vuggy porosity and fracture porosity. So it is reasonable to assume that the porous media is composed of some types of porous medium who exhibit fractal behavior in the range of the minimum to the maximum pore size (Fig. 2). Each type of the fractal porous medium has independent fractal parameters, such as the maximum pore diameter, kmax i, the minimum pore diameter, kmin i and the pore fractal dimension, Dfi, of the ith porous media. The cumulative pore numbers, Ni, of each porous medium whose diameters are greater than or equal to the pore diameter k follow the fractal scaling law (Mandelbrot, 1983).
Ni ðl P kÞ ¼
D kmax i fi k
ð1Þ
where 0 < Dfi < 2 in the two-dimensional space and 0 < Dfi < 3 in the three-dimensional Euclidian space. Differentiating Eq. (1) with respect to k the pore number of the ith porous media lying between k to k + dk on cross-sectional area can be obtained. D
ðDfi þ1Þ fi dNi ¼ Dfi kmax dk ik
ð2Þ
It should be noted that the negative sign implies that pore number of the ith porous media decreases with the increase in pore diameter.
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Fig. 1. Main categorization of pores in sedimentary rocks.
Fig. 2. The porous media composed of some types of fractal porous media.
Fig. 3. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 1).
X.-H. Tan et al. / International Journal of Engineering Science 95 (2015) 76–84
79
Since the capillaries in porous media are tortuous, the tortuosity length, Lt, of the capillaries is often defined by Bear (1972)
Lt ¼ sL0
ð3Þ
where s is the tortuosity and L0 is the straight length of capillaries in porous media. If the flow path is straight, s = 1. The flow rate through a single tortuous capillary with hydraulic diameter k is governed by the Hagen–Poiseulle equation.
pk4 Dp 128lLt
qðkÞ ¼
ð4Þ
where l is the fluid viscosity and Dp is pressure gradient along a tortuous capillary. Inserting Eq. (3) into Eq. (4) yields:
pk4 Dp 128lsL0
qðkÞ ¼
ð5Þ
The flow rate, Qi, of the ith porous media can be given by the integral of all flow rates from the minimum to maximum capillaries in the ith porous media.
Z
Qi ¼
kmax i
kmin i
"
pDfi k4max i Dp kmin i qðkÞ dNi ¼ 1 128lsL0 ð4 Dfi Þ kmax i
4Dfi #
ð6Þ
Since 0 < Dfi < 2 in two dimensions, the exponent 4 – Dfi > 2, in general kmin i =kmax i < 102 , thus ðkmin i =kmax i Þ4Dfi 1. Then, Eq. (7) can be reduced to
pDfi k4max i Dp 128ð4 Dfi ÞlsL0
Qi ¼
ð7Þ
Based on the microstructural images of natural porous media, we find that the numbers of pores with maximum diameter in each porous medium are not equal. So we introduce a parameter, Mi, the number of pores with maximum diameter in the ith porous media, to obtain the total flow rate, Q, of the whole porous medium.
Q¼
n n X X Mi Q i ¼ i¼1
i¼1
pMi Dfi k4max i Dp 128lsL0 ð4 Dfi Þ
ð8Þ
The permeability of porous media can be expressed by Darcy.
K¼
Q lL0 A Dp
ð9Þ
where A is the cross-sectional area of the whole porous medium. Inserting Eq. (8) into Eq. (9), the permeability can be expressed as:
K¼
n X
pMi Dfi k4max i 128sAð4 Dfi Þ i¼1
ð10Þ
When n = 1 and Mi = 1 in Eq. (10), the expression of permeability predicted by the multiple fractal model reduces to that predicted by the monofractal model (Yu & Cheng, 2002). In Eq. (10), when all the pore fractal dimensions, Dfi, for each type porous medium tend to 0, the permeability, K, of the whole porous medium approaches to 0. In this case, there is no pore in the whole porous medium. When all the Dfi tends to 2, the whole porous medium is completely occupied by pores and the K approaches its maximum value. This is consistent with the physical situation. The pore cross-sectional area, Api, of the ith porous media can be calculated by
Api ¼
Z
kmax i
kmin i
pk2
pDfi k2max i
"
kmin i dNi ¼ 1 4 4ð2 Dfi Þ kmax i
2Dfi #
ð11Þ
The total pore cross-sectional area, Ap, of the whole porous medium can be obtained as the sum of the pore cross-sectional area of all porous medium.
" 2Dfi # n n X X pMi Dfi k2max i kmin i Ap ¼ M i Api ¼ 1 4ð2 Dfi Þ kmax i i¼1 i¼1
ð12Þ
The total cross-sectional area, A, of the whole porous medium can be given by
A¼
" 2Dfi # n Ap X pMi Dfi k2max i kmin i ¼ 1 / 4/ð2 Dfi Þ kmax i i¼1
where / is the area porosity of the cross-sectional area in the whole porous media.
ð13Þ
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h i 2Dfi min i in Eq. (13) can be expressed as ð1 /Þ (Yu & Li, 2001), and the expression of When n = 1 and Mi = 1, the 1 ðkkmax Þ i total cross-sectional area predicted by the multiple fractal model reduces to that predicted by the monofractal model.
A¼
pDf k2max ð1 /Þ
ð14Þ
4/ð2 Df Þ
Eq. (14) is the same as the expression proposed by Yun, Yu, and Cai (2008). The pore volume, Vpi, of the ith porous media can be calculated by
V pi ¼ Lt Api
ð15Þ
Inserting Eqs. (3) and (11) into Eq. (15), the pore volume of the ith porous media can be expressed as:
V pi ¼
psL0 Dfi k2max i
" 1
4ð2 Dfi Þ
kmin i kmax i
2Dfi # ð16Þ
The total pore volume, Vp, of the whole porous medium can be given by
" 2Dfi # n n X X psL0 Mi Dfi k2max kmin i Vp ¼ Mi V pi ¼ 1 4ð2 Dfi Þ kmax i i¼1 i¼1
ð17Þ
When n = 1 and Mi = 1 in Eq. (17), the expression of pore volume predicted by the multiple fractal model reduces to that predicted by the monofractal model (Tan, Li, Liu, Zhang, & Zhang, 2014). When all the pore fractal dimensions, Dfi, for each type porous medium tend to 0, the whole porous media do not contain any pore and the total pore volume Vp approaches to 0. The Vp approaches its maximum value as all the Dfi tends to 2, which means the whole porous medium is completely occupied by pores. This is consistent with the physical situation. The pore volume, Vpi(k), of capillaries whose diameters are k in the ith porous media can be obtained using the integral of pore volume from the k to k þ Dk capillaries as
V pi ðkÞ ¼
Z
kþDk
pk2 4
k
Lt dN i ¼
h fi psL0 Dfi kDmax i 4ð2 Dfi Þ
ðk þ DkÞ2Dfi k2Dfi
i
ð18Þ
The pore volume, Vp(k), of capillaries whose diameters are k in the whole porous medium can be obtained as
V p ðkÞ ¼
n X
Mi V pi ðkÞ ¼
i¼1
n X psL0 Mi Dfi kDfi
max i
i¼1
h
4ð2 Dfi Þ
ðk þ DkÞ2Dfi k2Dfi
i
ð19Þ
The pore size distribution is defined as:
f ðkÞ ¼
V p ðkÞ Vp
ð20Þ
Inserting Eqs. (18) and (19) into Eq. (20), the pore size distribution of the whole porous medium can be expressed as:
Pn f ðkÞ ¼
D
fi M i Dfi kmax i ð2Dfi Þ
h
ðk þ DkÞ2Dfi k2Dfi 2Dfi Pn Mi Dfi k2max i min i 1 kkmax i¼1 ð2D Þ i
i¼1
i ð21Þ
fi
where the value range of the pore diameter is
kmin i 6 k 6 kmax i
16i6n
ð22Þ
In Eq. (21), the expression of pore size distribution predicted by the multiple fractal model reduces to that predicted by the monofractal model, as n = 1 and Mi = 1.
f ðkÞ ¼
1 1
k kþDk
kmin kmax
2Df
2Df
ð23Þ
The expression of pore size distribution predicted by the monofractal model (Eq. (23)) is a monotonically decreasing function. The pore size distribution decreases as an increase of pore diameter k. When multiple fractal model is considered, the expression of pore size distribution become complex with undulations.
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X.-H. Tan et al. / International Journal of Engineering Science 95 (2015) 76–84 Table 1 The parameters for the simulation of sample 1. i
kmin i (107 m)
kmax i (107 m)
Dfi
Mi
1 2 3 4 5 6
2.522 6.486 117.560 182.690 262.408 342.940
6.486 342.940 262.408 262.408 342.940 789.972
1.1 1.2 1.2 0.5 1.8 0.1
2 3 1 2 1 6
Table 2 Comparison of the permeability of the porous media between the simulated results and experimental data. Sample
1 2 3 4 5 6
K (1013 m2) Simulated results
Experimental data
1.78 1.39 1.51 3.58 1.07 2.13
1.52 1.38 1.52 3.73 1.22 2.71
Fig. 4. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 2).
Fig. 5. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 3).
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Fig. 6. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 4).
3. Simulation method Theoretically, we can obtain all the fractal parameters of the proposed model based on sets of microstructural images with different scales. Because the experimental methods for obtaining fractal parameters have not been matured, so optimizing fractal parameters to match the experimental curves is needed. The target of the simulated method is to establish a porous medium by the presented multiple fractal model. The pore size distribution and the permeability of the established porous media should be consistent with those of the real porous media, respectively. It is noted that multiple solutions exist and any solution meet the requirements can be used for the simulation method. The main steps of the simulation method are as follows: 1. Analyze the pore size distribution of experimental data, determine the number, n, of the types of the porous medium needed and confirm the pore maximum diameter, dmax i and the pore minimum diameter, dmin i for each type of porous medium. 2. Adjust the pore fractal dimension, Dfi and the number, Mi for each type porous medium and calculate the pore size distribution by the presented model (Eq. (21)). 3. Compare the pore size distribution calculated by the presented model and that of experimental data. If the difference between those is not tolerated, repeat steps 1 and 2 until the difference is small enough. 4. Calculate the permeability of the whole porous medium by the presented model (Eq. (10)). 5. Compare the permeability calculated by the presented model and that of experimental data. If the difference between those is not acceptable, repeat steps 1, 2, 3 and 4 until the good agreement between the permeability of the presented model and that of experimental data is obtained.
Fig. 7. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 5).
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83
Fig. 8. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 6).
4. Results and discussion The validity of the simulation method for the permeability of porous media is discussed. The pore size distributions and permeability of porous media simulated by the presented method are compared to those from the available experimental data, (Pia & Sanna, 2014) respectively. The pore size distribution of the sample 1 simulated by the presented method and that from the experiment are shown in Fig. 3. The square points present the pore size distribution of the experimental data and show irregularity with the pore diameter. To begin with, the proportion of the pore increases rapidly with the increase of the pore diameter, when the pore diameter is small than 6.5 106 m. After that, the proportion of the pore rises slowly with undulations until the pore diameter reaches to 2.4 105 m. Finally, the proportion of the pore drops plummets from the highest point 5.5% to the lowest point 0.9% and then grows slowly. In contrast, the curve presents the pore size distribution simulated by the presented method. Six types of porous medium are used for the simulation. The parameters of the porous medium are shown in Table 1. Like the pore size distribution of the experimental data, the pore size distribution simulated by the presented method is irregular with the pore diameter either. Good agreement between the pore size distribution simulated by the presented method and that of experimental data is obtained. The permeability from the experimental data and that simulated by the presented method are shown in Table 2. The difference between those is acceptable. The comparisons of the pore size distribution and the permeability of the samples 2–6 between simulated results and experimental data are revealed in Figs. 4–8 and Table 2. The simulated results present good agreement with available experimental data and these verify the presented simulation method.
5. Conclusions Firstly, the previous researches for permeability and porosity of fractal porous media are reviewed. And then, the multiple fractal model is derived assuming that the porous media are consisted by some types of fractal porous media. Next, a simulation method for the pore size distribution and the permeability of porous media is proposed based on the multiple fractal model. After that, the pore size distributions and the permeability of the porous media simulated by the presented method are compared with available experimental data. Good agreement between the simulated results and the experimental data is obtained. This verifies the validity of simulation method for the permeability of porous media. In the end, taking into account the fractal characters of microstructure of the porous media, a widely used simulation method for permeability of porous media may be developed using numerical simulation technology. Acknowledgements This work was jointly supported by the National Science Fund for Distinguished Young Scholars of China (51125019), the National Natural Science Foundation of China (51474181, 41102080), the 2014 Australia China Natural Gas Technology Partnership Fund Top Up Scholarships, and Open Fund (PLN1406) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).
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References Bear, J. (1972). Dynamics of fluids in porous media. New York: Elsevier. Cai, J., Yu, B., Zou, M., & Luo, L. (2010). Fractal characterization of spontaneous co-current imbibition in porous media. Energy Fuels, 24(3), 1860–1867. Cai, J., Perfect, E., Cheng, C.-L., & Hu, X. (2014). Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir, 30(18), 5142–5151. Cihan, A., Perfect, E., & Tyner, J. S. (2007). Water retention models for scale-variant and scale-invariant drainage of mass prefractal porous media. Vadose Zone Journal, 6(4), 786–792. Costa, A. (2006). Permeability–porosity relationship: A reexamination of the Kozeny–Carman equation based on a fractal pore-space geometry assumption. Geophysical Research Letters, 33, 2. Hansen, J., & Skjeltorp, A. (1988). Fractal pore space and rock permeability implications. Physical Review B, 38(4), 2635. Hunt, A. G., & Gee, G. W. (2002). Application of critical path analysis to fractal porous media: Comparison with examples from the Hanford site. Advances in Water Resources, 25(2), 129–146. Jacquin, C. G., & Adler, P. (1987). Fractal porous media II: Geometry of porous geological structures. Transport in Porous Media, 2(6), 571–596. Jiang, X., Wang, J., Hou, B., & He, G. (2013). Progress in the application of fractal porous media theory to property analysis and process simulation in melt crystallization. Industrial & Engineering Chemistry Research, 52(45), 15685–15701. Karacan, C. Ö., & Halleck, P. M. (2003). A fractal model for predicting permeability around perforation tunnels using size distribution of fragmented grains. Journal of Petroleum Science and Engineering, 40(3), 159–176. Katz, A. J., & Thompson, A. (1985). Fractal sandstone pores: Implications for conductivity and pore formation. Physical Review Letters, 54(12), 1325. Li, X.-Y., & Logan, B. E. (2001). Permeability of fractal aggregates. Water Research, 35(14), 3373–3380. Mandelbrot, B. B. (1983). The fractal geometry of nature. Macmillan. Martı´n, M. Á, & Montero, E. S. (2002). Laser diffraction and multifractal analysis for the characterization of dry soil volume-size distributions. Soil and Tillage Research, 64(1), 113–123. Pape, H., Clauser, C., & Iffland, J. (1999). Permeability prediction based on fractal pore-space geometry. Geophysics, 64(5), 1447–1460. Park, P.-K., Lee, C.-H., & Lee, S. (2006). Permeability of collapsed cakes formed by deposition of fractal aggregates upon membrane filtration. Environmental Science & Technology, 40(8), 2699–2705. Paz-Ferreiro, J., & Vázquez, E. V. (2014). Pore size distribution patterns in tropical soils obtained by mercury intrusion porosimetry: The multifractal approach. Vadose Zone Journal, 13, 6. Perfect, E. (2005). Modeling the primary drainage curve of prefractal porous media. Vadose Zone Journal, 4(4), 959–966. Perfect, E., Gentry, R., Sukop, M., & Lawson, J. (2006). Multifractal Sierpinski carpets: Theory and application to upscaling effective saturated hydraulic conductivity. Geoderma, 134(3), 240–252. Pia, G., & Sanna, U. (2014). An intermingled fractal units model and method to predict permeability in porous rock. International Journal of Engineering Science, 75, 31–39. Rigby, S., & Gladden, L. (1996). NMR and fractal modelling studies of transport in porous media. Chemical Engineering Science, 51(10), 2263–2272. Sahimi, M. (1993). Flow phenomena in rocks: From continuum models to fractals, percolation, cellular automata, and simulated annealing. Reviews of Modern Physics, 65(4), 1393. Saucier, A. (1992). Effective permeability of multifractal porous media. Physica A: Statistical Mechanics and its Applications, 183(4), 381–397. Shou, D., Ye, L., & Fan, J. (2014). The fastest capillary flow under gravity. Applied Physics Letters, 104(23), 231602. Stanley, H. E., & Meakin, P. (1988). Multifractal phenomena in physics and chemistry. Nature, 335(6189), 405–409. Tan, X.-H., Li, X.-P., Liu, J.-Y., Zhang, G.-D., & Zhang, L.-H. (2014). Analysis of permeability for transient two-phase flow in fractal porous media. Journal of Applied Physics, 115(11), 113502. Tan, X.-H., Li, X.-P., Zhang, L.-H., Liu, J.-Y., & Cai, J. (2015). Analysis of transient flow and starting pressure gradient of power-law fluid in fractal porous media. International Journal of Modern Physics C, 26(4), 1550045. Vázquez, E. V., Ferreiro, J. P., Miranda, J., & González, A. P. (2008). Multifractal analysis of pore size distributions as affected by simulated rainfall. Vadose Zone Journal, 7(2), 500–511. Xiao, B., Fan, J., & Ding, F. (2014). A fractal analytical model for the permeabilities of fibrous gas diffusion layer in proton exchange membrane fuel cells. Electrochimica Acta, 134, 222–231. Yu, B., & Cheng, P. (2002). A fractal permeability model for bi-dispersed porous media. International Journal of Heat and Mass Transfer, 45(14), 2983–2993. Yu, B., & Li, J. (2001). Some fractal characters of porous media. Fractals, 9(3), 365–372. Yun, M., Yu, B., & Cai, J. (2008). A fractal model for the starting pressure gradient for Bingham fluids in porous media. International Journal of Heat and Mass Transfer, 51(5–6), 1402–1408.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
A simulation method for permeability of porous media based on multiple fractal model Xiao-Hua Tan a,⇑, Jian-Yi Liu a, Xiao-Ping Li a, Lie-Hui Zhang a, Jianchao Cai b,⇑ a b
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Xindu Road 8, Chengdu 610500, China Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 9 December 2014 Received in revised form 20 February 2015 Accepted 29 June 2015
Keywords: Simulation method Multiple fractal model The pore size distribution Permeability Porous media
a b s t r a c t Fluid flow in fractal porous media is a ubiquitous natural phenomenon which has received much attention over three decades in a wide variety of fields. In order to find a relationship between the pore of distribution and the permeability of porous media, a simulation method for the permeability of porous media is proposed based on the multiple fractal model. The pore size distributions and the permeabilities of the porous medium simulated by the presented method are compared with available experimental data. The validity of simulation method is obtained by the good agreement between the simulated results and the experimental data. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since Mandelbrot’s pioneer work (Mandelbrot, 1983), flow phenomena in fractal porous media have been studied extensively for more than three decades. Katz and Thompson (1985) proved the pore spaces are fractal geometries in several sandstones by using scanning electron microscopy and optical data. Jacquin and Adler (1987) measured some geological structures and found them were fractal. The consequences of these measurements can be used on the analysis of transport processes in porous media. Sahimi (1993) considered that fractal and percolation concepts play important roles in the characterization of porous medium, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Li and Logan (2001) demonstrated that cluster-fractal model was more practical to predict the permeability of the porous media than single-particle-fractal model. The cluster-fractal model has been used widely in many fields. Karacan and Halleck (2003) presented a fractal model for predicting permeability around perforation tunnels. The model provided an easier and cheaper way of mapping the permeability distribution around the perforation tunnels compared to the viscous fluid injection and pressure transient measurement techniques. Cai, Yu, Zou and Luo (2010) developed an analytical model for characterizing spontaneous imbibitions of wetting liquid vertically into gas-saturated porous media, based on the fractal character of pores in porous media. Shou, Ye, and Fan (2014) designed a single-layer porous structure to achieve the fastest capillary flow under gravity. The theoretical results obtained can be used for the optimization of porous architectures, achieving excellent liquid management properties. Xiao, Fan, and Ding (2014) obtained an analytical model for effective
⇑ Corresponding authors. Tel.: +86 13438368455 (X.-H. Tan), +86 18971193160 (J. Cai). E-mail addresses: [email protected] (X.-H. Tan), [email protected] (J.-Y. Liu), [email protected] (X.-P. Li), [email protected] (L.-H. Zhang), [email protected] (J. Cai). http://dx.doi.org/10.1016/j.ijengsci.2015.06.007 0020-7225/Ó 2015 Elsevier Ltd. All rights reserved.
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thermal conductivity of nanofluids, considering the effect of Brownian motion of nanoparticles. The researches of flow in fractal porous media have never been stopped from Mandelbrot’s work (Perfect, 2005). It is convenient to describe the characters of porous media and explain the relationship among the parameters of porous media using the fractal theory (Jiang, Wang, Hou, & He, 2013). Hansen and Skjeltorp (1988) thought most porous medium were very complex systems consisted of a range of irregular grain sizes, and that made it be hard to find the relationship between permeability and porosity. They considered that the fractal theory was helpful for the research and proposed a relationship equation between permeability and porosity using the fractal theory. Pape, Clauser, and Iffland (1999) presented a relation linking porosity to permeability based on a fractal model for the internal structure of the porous media. Good agreement between the predictions of the relation and the experimental data was obtained. Park, Lee, and Lee (2006) developed a model based on fractal theory and incorporated a cake collapse effect to predict the porosity and permeability of the cake layers. Costa (2006) derived a permeability–porosity equation using the classical Kozeny–Carman approach and a fractal pore-space geometry assumption. The equation was simple and capable to describe permeabilities of different non-granular porous media. Most studies of permeability of fractal porous media expounded the relationship between permeability and porosity (Cai, Perfect, Cheng, & Hu, 2014; Cihan, Perfect, & Tyner, 2007; Tan, Li, Zhang, Liu, & Cai, 2015). Researchers found that the natural porous media, such as rock, soil and biological tissue, is a very complex structure, and the multifractal theory is a practical way to describe the natural porous media (Stanley & Meakin, 1988). Saucier (1992) calculated the scaling exponents of the effective absolute permeability in multifractal porous media and discuss the implications of the results on the understanding of fluid flow in oil reservoirs. Rigby and Gladden (1996) proposed a multifractal description of porous media, and represented the macroscopic heterogeneity associated with the pore-size distribution and the fractal characteristics of the microscopic pore structure. Martı´n and Montero (2002) analyzed the characterization of dry volume-size distributions in soils using the application of laser diffraction and multifractal theory. The result of analysis showed that multifractal is a useful mathematical tool for the characterization of porous media. Hunt and Gee (2002) proposed a ‘‘dual’’ fractal model using two types of fractal porous medium. The pore size distributions of some porous media were described by the ‘‘dual’’ fractal model. When the pore size distribution of porous media is complex, it is tough to be described accurately by two types of fractal porous medium. Perfect, Gentry, Sukop, and Lawson (2006) derived an approximate analytical expression for the effective hydraulic conductivity of multifractal porous media based on the generation of Sierpinski carpet. They found geometrical multifractals were good to simulate distinct facies or transport abilities of porous media. Vázquez et al. (2008) analyzed the pore size distributions of soils affected by rainfall based on multifractal theory. Pore size distributions of aggregates from both the reference soil surface and the soil surface disturbed by rain exhibited multifractal behavior. Paz-Ferreiro and Vázquez (2014) characterized pore size distributions from tropical soils using the multifractal approach and found that multifractal analyses of soil pore size distributions are useful for analyzing the soil porous system. In this paper, we attempt to establish a method to describe the pore size distribution and the permeability of porous media at the same time assuming that the porous media are consisted by some types of fractal porous media. To this end, multiple fractal model will be derived in Section 2, and then a simulation method for the pore size distribution and the permeability of porous media is proposed in Section 3, finally in Section 4 relevant results and discussions are demonstrated and compared. Conclusions are given in Section 5. 2. The multiple fractal model The natural porous media are composed of different types of porosities. For example, the main porosity types of sedimentation rocks are now described and illustrated in Fig. 1. Six main types of pore systems have been show in Fig. 1, including primary intergranular porosity, primary intragranular porosity, secondary intergranular porosity, secondary intragranular porosity, vuggy porosity and fracture porosity. So it is reasonable to assume that the porous media is composed of some types of porous medium who exhibit fractal behavior in the range of the minimum to the maximum pore size (Fig. 2). Each type of the fractal porous medium has independent fractal parameters, such as the maximum pore diameter, kmax i, the minimum pore diameter, kmin i and the pore fractal dimension, Dfi, of the ith porous media. The cumulative pore numbers, Ni, of each porous medium whose diameters are greater than or equal to the pore diameter k follow the fractal scaling law (Mandelbrot, 1983).
Ni ðl P kÞ ¼
D kmax i fi k
ð1Þ
where 0 < Dfi < 2 in the two-dimensional space and 0 < Dfi < 3 in the three-dimensional Euclidian space. Differentiating Eq. (1) with respect to k the pore number of the ith porous media lying between k to k + dk on cross-sectional area can be obtained. D
ðDfi þ1Þ fi dNi ¼ Dfi kmax dk ik
ð2Þ
It should be noted that the negative sign implies that pore number of the ith porous media decreases with the increase in pore diameter.
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Fig. 1. Main categorization of pores in sedimentary rocks.
Fig. 2. The porous media composed of some types of fractal porous media.
Fig. 3. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 1).
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Since the capillaries in porous media are tortuous, the tortuosity length, Lt, of the capillaries is often defined by Bear (1972)
Lt ¼ sL0
ð3Þ
where s is the tortuosity and L0 is the straight length of capillaries in porous media. If the flow path is straight, s = 1. The flow rate through a single tortuous capillary with hydraulic diameter k is governed by the Hagen–Poiseulle equation.
pk4 Dp 128lLt
qðkÞ ¼
ð4Þ
where l is the fluid viscosity and Dp is pressure gradient along a tortuous capillary. Inserting Eq. (3) into Eq. (4) yields:
pk4 Dp 128lsL0
qðkÞ ¼
ð5Þ
The flow rate, Qi, of the ith porous media can be given by the integral of all flow rates from the minimum to maximum capillaries in the ith porous media.
Z
Qi ¼
kmax i
kmin i
"
pDfi k4max i Dp kmin i qðkÞ dNi ¼ 1 128lsL0 ð4 Dfi Þ kmax i
4Dfi #
ð6Þ
Since 0 < Dfi < 2 in two dimensions, the exponent 4 – Dfi > 2, in general kmin i =kmax i < 102 , thus ðkmin i =kmax i Þ4Dfi 1. Then, Eq. (7) can be reduced to
pDfi k4max i Dp 128ð4 Dfi ÞlsL0
Qi ¼
ð7Þ
Based on the microstructural images of natural porous media, we find that the numbers of pores with maximum diameter in each porous medium are not equal. So we introduce a parameter, Mi, the number of pores with maximum diameter in the ith porous media, to obtain the total flow rate, Q, of the whole porous medium.
Q¼
n n X X Mi Q i ¼ i¼1
i¼1
pMi Dfi k4max i Dp 128lsL0 ð4 Dfi Þ
ð8Þ
The permeability of porous media can be expressed by Darcy.
K¼
Q lL0 A Dp
ð9Þ
where A is the cross-sectional area of the whole porous medium. Inserting Eq. (8) into Eq. (9), the permeability can be expressed as:
K¼
n X
pMi Dfi k4max i 128sAð4 Dfi Þ i¼1
ð10Þ
When n = 1 and Mi = 1 in Eq. (10), the expression of permeability predicted by the multiple fractal model reduces to that predicted by the monofractal model (Yu & Cheng, 2002). In Eq. (10), when all the pore fractal dimensions, Dfi, for each type porous medium tend to 0, the permeability, K, of the whole porous medium approaches to 0. In this case, there is no pore in the whole porous medium. When all the Dfi tends to 2, the whole porous medium is completely occupied by pores and the K approaches its maximum value. This is consistent with the physical situation. The pore cross-sectional area, Api, of the ith porous media can be calculated by
Api ¼
Z
kmax i
kmin i
pk2
pDfi k2max i
"
kmin i dNi ¼ 1 4 4ð2 Dfi Þ kmax i
2Dfi #
ð11Þ
The total pore cross-sectional area, Ap, of the whole porous medium can be obtained as the sum of the pore cross-sectional area of all porous medium.
" 2Dfi # n n X X pMi Dfi k2max i kmin i Ap ¼ M i Api ¼ 1 4ð2 Dfi Þ kmax i i¼1 i¼1
ð12Þ
The total cross-sectional area, A, of the whole porous medium can be given by
A¼
" 2Dfi # n Ap X pMi Dfi k2max i kmin i ¼ 1 / 4/ð2 Dfi Þ kmax i i¼1
where / is the area porosity of the cross-sectional area in the whole porous media.
ð13Þ
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h i 2Dfi min i in Eq. (13) can be expressed as ð1 /Þ (Yu & Li, 2001), and the expression of When n = 1 and Mi = 1, the 1 ðkkmax Þ i total cross-sectional area predicted by the multiple fractal model reduces to that predicted by the monofractal model.
A¼
pDf k2max ð1 /Þ
ð14Þ
4/ð2 Df Þ
Eq. (14) is the same as the expression proposed by Yun, Yu, and Cai (2008). The pore volume, Vpi, of the ith porous media can be calculated by
V pi ¼ Lt Api
ð15Þ
Inserting Eqs. (3) and (11) into Eq. (15), the pore volume of the ith porous media can be expressed as:
V pi ¼
psL0 Dfi k2max i
" 1
4ð2 Dfi Þ
kmin i kmax i
2Dfi # ð16Þ
The total pore volume, Vp, of the whole porous medium can be given by
" 2Dfi # n n X X psL0 Mi Dfi k2max kmin i Vp ¼ Mi V pi ¼ 1 4ð2 Dfi Þ kmax i i¼1 i¼1
ð17Þ
When n = 1 and Mi = 1 in Eq. (17), the expression of pore volume predicted by the multiple fractal model reduces to that predicted by the monofractal model (Tan, Li, Liu, Zhang, & Zhang, 2014). When all the pore fractal dimensions, Dfi, for each type porous medium tend to 0, the whole porous media do not contain any pore and the total pore volume Vp approaches to 0. The Vp approaches its maximum value as all the Dfi tends to 2, which means the whole porous medium is completely occupied by pores. This is consistent with the physical situation. The pore volume, Vpi(k), of capillaries whose diameters are k in the ith porous media can be obtained using the integral of pore volume from the k to k þ Dk capillaries as
V pi ðkÞ ¼
Z
kþDk
pk2 4
k
Lt dN i ¼
h fi psL0 Dfi kDmax i 4ð2 Dfi Þ
ðk þ DkÞ2Dfi k2Dfi
i
ð18Þ
The pore volume, Vp(k), of capillaries whose diameters are k in the whole porous medium can be obtained as
V p ðkÞ ¼
n X
Mi V pi ðkÞ ¼
i¼1
n X psL0 Mi Dfi kDfi
max i
i¼1
h
4ð2 Dfi Þ
ðk þ DkÞ2Dfi k2Dfi
i
ð19Þ
The pore size distribution is defined as:
f ðkÞ ¼
V p ðkÞ Vp
ð20Þ
Inserting Eqs. (18) and (19) into Eq. (20), the pore size distribution of the whole porous medium can be expressed as:
Pn f ðkÞ ¼
D
fi M i Dfi kmax i ð2Dfi Þ
h
ðk þ DkÞ2Dfi k2Dfi 2Dfi Pn Mi Dfi k2max i min i 1 kkmax i¼1 ð2D Þ i
i¼1
i ð21Þ
fi
where the value range of the pore diameter is
kmin i 6 k 6 kmax i
16i6n
ð22Þ
In Eq. (21), the expression of pore size distribution predicted by the multiple fractal model reduces to that predicted by the monofractal model, as n = 1 and Mi = 1.
f ðkÞ ¼
1 1
k kþDk
kmin kmax
2Df
2Df
ð23Þ
The expression of pore size distribution predicted by the monofractal model (Eq. (23)) is a monotonically decreasing function. The pore size distribution decreases as an increase of pore diameter k. When multiple fractal model is considered, the expression of pore size distribution become complex with undulations.
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X.-H. Tan et al. / International Journal of Engineering Science 95 (2015) 76–84 Table 1 The parameters for the simulation of sample 1. i
kmin i (107 m)
kmax i (107 m)
Dfi
Mi
1 2 3 4 5 6
2.522 6.486 117.560 182.690 262.408 342.940
6.486 342.940 262.408 262.408 342.940 789.972
1.1 1.2 1.2 0.5 1.8 0.1
2 3 1 2 1 6
Table 2 Comparison of the permeability of the porous media between the simulated results and experimental data. Sample
1 2 3 4 5 6
K (1013 m2) Simulated results
Experimental data
1.78 1.39 1.51 3.58 1.07 2.13
1.52 1.38 1.52 3.73 1.22 2.71
Fig. 4. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 2).
Fig. 5. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 3).
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Fig. 6. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 4).
3. Simulation method Theoretically, we can obtain all the fractal parameters of the proposed model based on sets of microstructural images with different scales. Because the experimental methods for obtaining fractal parameters have not been matured, so optimizing fractal parameters to match the experimental curves is needed. The target of the simulated method is to establish a porous medium by the presented multiple fractal model. The pore size distribution and the permeability of the established porous media should be consistent with those of the real porous media, respectively. It is noted that multiple solutions exist and any solution meet the requirements can be used for the simulation method. The main steps of the simulation method are as follows: 1. Analyze the pore size distribution of experimental data, determine the number, n, of the types of the porous medium needed and confirm the pore maximum diameter, dmax i and the pore minimum diameter, dmin i for each type of porous medium. 2. Adjust the pore fractal dimension, Dfi and the number, Mi for each type porous medium and calculate the pore size distribution by the presented model (Eq. (21)). 3. Compare the pore size distribution calculated by the presented model and that of experimental data. If the difference between those is not tolerated, repeat steps 1 and 2 until the difference is small enough. 4. Calculate the permeability of the whole porous medium by the presented model (Eq. (10)). 5. Compare the permeability calculated by the presented model and that of experimental data. If the difference between those is not acceptable, repeat steps 1, 2, 3 and 4 until the good agreement between the permeability of the presented model and that of experimental data is obtained.
Fig. 7. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 5).
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Fig. 8. Comparison of the pore size distribution of the porous media between simulated result and experimental data (Sample 6).
4. Results and discussion The validity of the simulation method for the permeability of porous media is discussed. The pore size distributions and permeability of porous media simulated by the presented method are compared to those from the available experimental data, (Pia & Sanna, 2014) respectively. The pore size distribution of the sample 1 simulated by the presented method and that from the experiment are shown in Fig. 3. The square points present the pore size distribution of the experimental data and show irregularity with the pore diameter. To begin with, the proportion of the pore increases rapidly with the increase of the pore diameter, when the pore diameter is small than 6.5 106 m. After that, the proportion of the pore rises slowly with undulations until the pore diameter reaches to 2.4 105 m. Finally, the proportion of the pore drops plummets from the highest point 5.5% to the lowest point 0.9% and then grows slowly. In contrast, the curve presents the pore size distribution simulated by the presented method. Six types of porous medium are used for the simulation. The parameters of the porous medium are shown in Table 1. Like the pore size distribution of the experimental data, the pore size distribution simulated by the presented method is irregular with the pore diameter either. Good agreement between the pore size distribution simulated by the presented method and that of experimental data is obtained. The permeability from the experimental data and that simulated by the presented method are shown in Table 2. The difference between those is acceptable. The comparisons of the pore size distribution and the permeability of the samples 2–6 between simulated results and experimental data are revealed in Figs. 4–8 and Table 2. The simulated results present good agreement with available experimental data and these verify the presented simulation method.
5. Conclusions Firstly, the previous researches for permeability and porosity of fractal porous media are reviewed. And then, the multiple fractal model is derived assuming that the porous media are consisted by some types of fractal porous media. Next, a simulation method for the pore size distribution and the permeability of porous media is proposed based on the multiple fractal model. After that, the pore size distributions and the permeability of the porous media simulated by the presented method are compared with available experimental data. Good agreement between the simulated results and the experimental data is obtained. This verifies the validity of simulation method for the permeability of porous media. In the end, taking into account the fractal characters of microstructure of the porous media, a widely used simulation method for permeability of porous media may be developed using numerical simulation technology. Acknowledgements This work was jointly supported by the National Science Fund for Distinguished Young Scholars of China (51125019), the National Natural Science Foundation of China (51474181, 41102080), the 2014 Australia China Natural Gas Technology Partnership Fund Top Up Scholarships, and Open Fund (PLN1406) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).
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References Bear, J. (1972). Dynamics of fluids in porous media. New York: Elsevier. Cai, J., Yu, B., Zou, M., & Luo, L. (2010). Fractal characterization of spontaneous co-current imbibition in porous media. Energy Fuels, 24(3), 1860–1867. Cai, J., Perfect, E., Cheng, C.-L., & Hu, X. (2014). Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir, 30(18), 5142–5151. Cihan, A., Perfect, E., & Tyner, J. S. (2007). Water retention models for scale-variant and scale-invariant drainage of mass prefractal porous media. Vadose Zone Journal, 6(4), 786–792. Costa, A. (2006). Permeability–porosity relationship: A reexamination of the Kozeny–Carman equation based on a fractal pore-space geometry assumption. Geophysical Research Letters, 33, 2. Hansen, J., & Skjeltorp, A. (1988). Fractal pore space and rock permeability implications. Physical Review B, 38(4), 2635. Hunt, A. G., & Gee, G. W. (2002). Application of critical path analysis to fractal porous media: Comparison with examples from the Hanford site. Advances in Water Resources, 25(2), 129–146. Jacquin, C. G., & Adler, P. (1987). Fractal porous media II: Geometry of porous geological structures. Transport in Porous Media, 2(6), 571–596. Jiang, X., Wang, J., Hou, B., & He, G. (2013). Progress in the application of fractal porous media theory to property analysis and process simulation in melt crystallization. Industrial & Engineering Chemistry Research, 52(45), 15685–15701. Karacan, C. Ö., & Halleck, P. M. (2003). A fractal model for predicting permeability around perforation tunnels using size distribution of fragmented grains. Journal of Petroleum Science and Engineering, 40(3), 159–176. Katz, A. J., & Thompson, A. (1985). Fractal sandstone pores: Implications for conductivity and pore formation. Physical Review Letters, 54(12), 1325. Li, X.-Y., & Logan, B. E. (2001). Permeability of fractal aggregates. Water Research, 35(14), 3373–3380. Mandelbrot, B. B. (1983). The fractal geometry of nature. Macmillan. Martı´n, M. Á, & Montero, E. S. (2002). Laser diffraction and multifractal analysis for the characterization of dry soil volume-size distributions. Soil and Tillage Research, 64(1), 113–123. Pape, H., Clauser, C., & Iffland, J. (1999). Permeability prediction based on fractal pore-space geometry. Geophysics, 64(5), 1447–1460. Park, P.-K., Lee, C.-H., & Lee, S. (2006). Permeability of collapsed cakes formed by deposition of fractal aggregates upon membrane filtration. Environmental Science & Technology, 40(8), 2699–2705. Paz-Ferreiro, J., & Vázquez, E. V. (2014). Pore size distribution patterns in tropical soils obtained by mercury intrusion porosimetry: The multifractal approach. Vadose Zone Journal, 13, 6. Perfect, E. (2005). Modeling the primary drainage curve of prefractal porous media. Vadose Zone Journal, 4(4), 959–966. Perfect, E., Gentry, R., Sukop, M., & Lawson, J. (2006). Multifractal Sierpinski carpets: Theory and application to upscaling effective saturated hydraulic conductivity. Geoderma, 134(3), 240–252. Pia, G., & Sanna, U. (2014). An intermingled fractal units model and method to predict permeability in porous rock. International Journal of Engineering Science, 75, 31–39. Rigby, S., & Gladden, L. (1996). NMR and fractal modelling studies of transport in porous media. Chemical Engineering Science, 51(10), 2263–2272. Sahimi, M. (1993). Flow phenomena in rocks: From continuum models to fractals, percolation, cellular automata, and simulated annealing. Reviews of Modern Physics, 65(4), 1393. Saucier, A. (1992). Effective permeability of multifractal porous media. Physica A: Statistical Mechanics and its Applications, 183(4), 381–397. Shou, D., Ye, L., & Fan, J. (2014). The fastest capillary flow under gravity. Applied Physics Letters, 104(23), 231602. Stanley, H. E., & Meakin, P. (1988). Multifractal phenomena in physics and chemistry. Nature, 335(6189), 405–409. Tan, X.-H., Li, X.-P., Liu, J.-Y., Zhang, G.-D., & Zhang, L.-H. (2014). Analysis of permeability for transient two-phase flow in fractal porous media. Journal of Applied Physics, 115(11), 113502. Tan, X.-H., Li, X.-P., Zhang, L.-H., Liu, J.-Y., & Cai, J. (2015). Analysis of transient flow and starting pressure gradient of power-law fluid in fractal porous media. International Journal of Modern Physics C, 26(4), 1550045. Vázquez, E. V., Ferreiro, J. P., Miranda, J., & González, A. P. (2008). Multifractal analysis of pore size distributions as affected by simulated rainfall. Vadose Zone Journal, 7(2), 500–511. Xiao, B., Fan, J., & Ding, F. (2014). A fractal analytical model for the permeabilities of fibrous gas diffusion layer in proton exchange membrane fuel cells. Electrochimica Acta, 134, 222–231. Yu, B., & Cheng, P. (2002). A fractal permeability model for bi-dispersed porous media. International Journal of Heat and Mass Transfer, 45(14), 2983–2993. Yu, B., & Li, J. (2001). Some fractal characters of porous media. Fractals, 9(3), 365–372. Yun, M., Yu, B., & Cai, J. (2008). A fractal model for the starting pressure gradient for Bingham fluids in porous media. International Journal of Heat and Mass Transfer, 51(5–6), 1402–1408.