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Multi-scale structures of porous media and the flow prediction
Journal of Natural Gas Science and Engineering 21 (2014) 986e992
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Multi-scale structures of porous media and the flow prediction G. Lei*, P.C. Dong, Z.S. Wu, S.H. Gai, S.Y. Mo, Z. Li Key Laboratory for Petroleum Engineering of the Ministry of Education, China University of Petroleum, Beijing 102249, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 September 2014 Received in revised form 25 October 2014 Accepted 30 October 2014 Available online 7 November 2014
For the same porous media, the pore structures on different scales and the corresponding fluid flow have different features. The pore structure ultimately affects the fluid flow property. So the study of multiscale pore structure of the given sample is important for oil/gas reservoir engineering. In this paper, the lognormal distribution characteristics of the multi-scaled porous media have been described and the analytical expressions for capillary pressure and relative permeability for each scaled porous media are developed based on the lognormal distribution function. It is shown that the pore-throat size distribution density functions of the sample in different analysis scales are lognormal. The predictions of capillary pressure for higher scaled porous media conform to the trend of variation with the available experimental data at lower mercury saturation. However, the variation trend of predictions for lower scaled porous media is in agreement with experimental data only at higher mercury saturation. Compared to the fractal model, our new model for relative permeability has a wider tolerance for changes of pore-throat radius distribution. Our model is also well suitable for those non-fractal porous media, for which the fractal model is not applicable. The predictions of relative permeability by the proposed model are all consistent with experimental data. The effects of irreducible wetting fluid saturation, lognormal distribution parameters a and s on the relative permeability also have been discussed. © 2014 Elsevier B.V. All rights reserved.
Keywords: Multi-scale Microstructure Lognormal distribution Mathematical modeling Relative permeability
1. Introduction For the petroleum industry, it is a constant challenge to understand the interactions between the micro-pore structure of porous media and the fluid flow (Neimark, 1989; Xu et al., 1997; Wu et al., 2007). Many scholars have studied the influence of pore-throat distribution features on the fluid diffusion phenomena. Capillary tube analogs and statistically reconstructed models are the two most important methods to study the relation between pore-throat size distribution characteristics and macroscopic flow characteristics. These two models are useful, but the predictions by these two models are largely affected by the pore-throat size. Besides, the pore structures and the fluid flow of the multi-scaled porous media are different for a given sample. As a result, it is not very meaningful to quantify the grid spacing and pore-throat distribution without considering the multiple-scales pore-throat distribution characteristics of porous media. In 1989, Neimark proposed the multi-scale concept which produced important advance in the techniques to compute the * Corresponding author. E-mail addresses: [email protected], (G. Lei).
[email protected]
http://dx.doi.org/10.1016/j.jngse.2014.10.033 1875-5100/© 2014 Elsevier B.V. All rights reserved.
transport properties. Xu et al. (1997) suggested a given medium can be considered on widely varying scales, and the rules conditioning the interactions between the properties on different scales can be enunciated. Based on bond percolation in a cubic network, Xu et al. (1997) studied the multi-scale structure to facilitate the ensuing computation of the transport coefficients. Radlinski et al. (2002) obtained the entire distribution of pore length scales by combining small-angle scattering and BSEM imaging data. Amirtharaj et al. (2003) employed Mercury intrusion porosimetry (MIP) to access structural information in the submicrometric range and also obtained the entire distribution of pore length scales. They obtained the pore-to-throat aspect ratio and pore accessibility by comparing the complete pore volume distribution to the MIP data. Keehm and Mukerji (2004) suggested the parameters grid spacing and the representative elementary volume with different scales largely influenced the simulated results. They found that permeability is consistently overestimated with the increase of grid spacing. Wu et al. (2007) reconstructed the pore space of multiple-scales of pore systems from appropriately-imaged thin section and made predictions of the mercury injection characteristics of each scaled porous media using the invasion percolation algorithm. They found
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
that the simulated mercury injection curves were quite different for different scaled porous media. Moreover, they had the conclusion that separate mercury injection curves represented distinct pore systems in the samples and the combined mercury injection curve could represent a plausible prediction of the bulk flow characteristics of the sample. Sun et al. (2011) presented the multiple-scales framework to calculate macroscopic effective permeability at a scale relevant to engineering applications. A number of literatures suggest that the both pore radius density distribution function and throat radius density distribution function are lognormal (Han, 1997; Qi and Cui, 2000; Jiao et al., 2010). Han (1997) suggested that the genesis of pore-throat radius density distribution function was lognormal. Qi and Cui (2000) described the distribution characteristics on the frequency histogram of some geotechnical parameters and found some of them were of lognormal distribution. Jiao et al. (2010) stated both the pore size and throat size were lognormal distribution according to central limit theorem stated by Li (2004) and discussed the reason why the pore radius distribution and throat radius distribution were lognormal distribution. In this paper, we show that pore-throat radius distribution density functions of the cores in different analysis scales are lognormal. Then, theoretical models for capillary pressure and relative permeability are attempted to be built based on the lognormal distribution function of multi-scaled porous media. For this purpose, section 2 will introduce the characteristics of lognormal distribution in multi-scaled porous media. In section 3, a novel model for capillary pressure and relative permeability of the scaled porous has been derived. In section 4, the correctness of our model is validated with the available experimental data and the fractal model. Finally, relevant results and discussions are demonstrated in section 5, and conclusions are presented in section 6. 2. The description of multi-scale structures of porous media Porous media consists of numerous irregular pores and throats with different sizes and the pore structure is complex. Jiao et al. (2010) stated the pore-throat size was lognormal distribution according to central limit theorem stated by Li (2004). The expression of lognormal distribution function can be written as
8 > > > <
1 f ðrÞ ¼ pffiffiffiffiffiffiffi 2p sr > > > : 0
ðln r aÞ2 2 2s e
be considered as circles with different radius r. For simplicity, the circular capillaries are assumed. Xu et al. (2013) suggested the nonwetting fluid occupy the pores greater than the critical value rc at a certain capillary pressure pc. The capillary pressure at the capillary radius rc can be expressed as
pc ¼
2ε cos q rc
, Zrmax
Zrmax 2
Snw ¼ rc
The flow paths in porous media can be regarded as a bundle of tortuous capillary channels and the cross sections of capillaries can
re
ðlnraÞ2 2s2
, Zrmax dr
rc
ðlnraÞ2 2s2
re
dr
rmin
(3) In practice, the non-wetting fluid saturation should be lower than 1Sw0, where Sw0 is the irreducible wetting fluid saturation. The relation between Sw0 and irreducible wetting fluid radius rc0 can be written as
Zrc0 Sw0 ¼
re
ðln raÞ2 2s2
, Zrmax dr
rmin
re
ðln raÞ2 2s2
dr
(4)
rmin
Eqs. (2)e(4) reveal that the capillary pressure is the function of pore-throat size and pore-throat distribution function of porous media. Since the pore-throat size and pore-throat distribution functions are different for different resolution research scales, the capillary pressure curves for different scales may be different.
3.2. The relative permeability Based on HagenePoiseulle equation, the total volumetric flow rates for wetting fluid and non-wetting fluid through the porous media can be written respectively (Xu et al., 2013).
Zrc rc0
pDpw r 4 NpDpw pffiffiffiffiffiffiffi f dr ¼ 8mw L 8mw L 2ps
Zrmax Qnw ¼ N
3.1. The capillary pressure
pr f dr¼ rmin
others
3. The flow prediction
Zrmax 2
pr f dr
(1)
where f is the lognormal distribution function, which is a function of pore-throat radius r. a and s are the relevant parameters. rmin and rmax are the maximum and the minimum pore-throat radius, respectively. When the parameters a or s is different, lognormal distribution density function f(r) is different. In this study, the porethroat density distribution functions of different scaled cores (sandstone) in X area (a block in Changqing oilfield) were fitted by lognormal distribution function (see Fig. 1). It was shown that the pore-throat size distributions in different analysis scales of the cores in X area are lognormal. The data of the parameters were shown in Table 1.
(2)
where ε is the surface tension and q is the contact angle. Under this capillary pressure, the non-wetting fluid saturation Snw can be calculated by Xu et al. (2013)
Qw ¼ N rmin r rmax
987
rc
Zrc
ðln raÞ2 2s2
r 3 e
dr
(5a)
rmin
pDpnw 4 NpDpnw pffiffiffiffiffiffiffi r f dr ¼ 8mnw L 8mnw L 2ps
Zrmax
r 3 e
ðln raÞ2 2s2
dr
(5b)
rc
where Q (cm3/s) is flow rates, m (mPa$s) is the viscosity of the fluid, Dp (MPa) is the pressure drop and L (cm) is the length of the capillary, the subscripts w and nw denote the wetting and nonwetting phases, respectively. The flow rates through the twophase system can be also given by Darcy's extended law (Yu and Liu, 2004; Xu et al., 2013)
Qw ¼
Kw A Dpw mw 1 Snw L0
(6a)
Knw A Dpnw mnw Snw L0
(6b)
Qnw ¼
where K is the effective permeability of the medium, L0 is the core length and A is the total cross area. The effective permeability for each phase can be obtained from Eqs. (5) and (6) as:
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G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
Fig. 1. The comparison of the calculated pore-throat size density function curve from Eq. (1) and the experimental data, (a) core 1, (b) core 2, (c) core 3.
Kw ¼
Knw
Zrc
pNð1 Snw Þ pffiffiffiffiffiffiffi 8A 2pst
pNSnw pffiffiffiffiffiffiffi ¼ 8A 2pst
r 3 e
ðln raÞ2 2s2
dr
(7a)
t ¼ L=L0
rc0
Zrmax
3
r e
ðln raÞ2 2s2
dr
(7b)
rc
and the non-wetting fluid saturation can be determined by Eq. (3). Based on Eq. (7) the absolute permeability for saturated porous media (all the capillaries are filled with a single-phase fluid) can be expressed as:
where t is the tortuosity (Bear, 1972) is defined as
K¼
Table 1 Parameters of the scaled cores (sandstone). Core number
Scale
1
Scale Scale Scale Scale Scale Scale Scale Scale Scale
2 3
(8)
1 2 1 2 1 2 3 4 5
rmax
rmin
rmin/rmax
a
s
12.5 mm 21.385 mm 0.845 mm 1.86 mm 1.14 mm 1.4 mm 1.415 mm 2.17 mm 3.15 mm
0.25 mm 1.965 mm 0.02 mm 0.385 mm 0.255 mm 0.325 mm 0.475 mm 0.585 mm 0.97 mm
0.02 0.09 0.024 0.207 0.22 0.23 0.34 0.27 0.31
0.7 1.5 3 0.46 0.7 0.42 0.08 0.03 0.65
0.27 0.27 0.21 0.2 0.17 0.22 0.16 0.19 0.19
pN pffiffiffiffiffiffiffi 8A 2pst
Zrmax
r 3 e
ðln raÞ2 2s2
dr
(9)
rc0
Muskat and Meres (2004) recommended that each phase permeability be given by
Krw ¼ Kw =K
(10a)
Krnw ¼ Knw =K
(10b)
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
140
Fractal model The proposed model
100
f(r)
80 60 40 20 0 -5
-4
-3
-2
-1
0
ln(r) Fig. 2. The comparison of the calculated pore-throat size density function curve from Eq. (1) and the fractal model.
in which Krw (or Krnw) is wetting phase (or non-wetting phase) relative permeability. Based on Eqs. 7e10, the relative permeability for each phase can be respectively determined
Zrc
2 ðln raÞ 2s2
r3 e
Krw ¼ ð1 Snw Þ rc0
Zrmax Krnw ¼ Snw
r e
2 ðln raÞ 2s2
r3 e
dr
(11)
rc0
ðln raÞ 3 2s2
, Zrmax
2
dr
rc
r 3 e
ðln raÞ2 2s2
dr
(12)
rc0
Eqs. (11) and (12) show that the relative permeability Krw (or Krnw) is a function of non-wetting fluid saturation Snw and irreducible wetting fluid saturation as well as multi-scale pore-throat size and multi-scale pore-throat distribution function. 4. Model validation The predictions by the proposed model, namely Eq. (1) and the fractal model (Yu and Cheng, 2002; Xu et al., 2013) have been compared in Fig. 2. It shows clearly in Fig. 2 that our model predictions are all consistent with the fractal model. The relevant parameters applied are listed in Table 2. As shown in Fig. 3, we study the relative permeability by using the parameters in Table 2. And we compare predictions by our current model, namely Eqs. (11) and (12) with the available experimental data and those from the fractal model (see the following Eq. (13)) from Xu et al. (2013). These experimental data are all cited from reference (Xu et al., 2013). The fractal model given by Xu et al. (2013) can be written as
Table 2 Description of the relevant parameters. Parameters
Values
Description
rmax rmin/rmax
0.9 mm 0.01
DT 4
1.10 20% 4.3 0.24
The maximum pore-throat radius The ratio of minimum pore diameter to maximum pore diameter The tortuosity fractal dimension The porosity Lognormal distribution parameters
a s
(13a)
n o Krnw ¼ Snw 1 ½ð1 4Þð1 Snw Þ þ 4Cf
(13b)
In Eq. (13), the fractal factor Cf ¼ (3-Df þ DT)/(2-Df). Df is fractal dimension for pore size and the it is in the range of 0 < Df < 2 in two dimension. From Fig. 3 it is seen that the relative permeability predicted by the proposed model is compared with experimental measurements and the fractal model Eq. (13). It can be clearly seen that our model predictions and that of the fractal model (Xu et al., 2013) have the similar variation trend besides our predictions by the proposed model Eqs. (11) and (12) are closer to the experimental data (Abaci et al., 1992; Piquemal, 1994; Sandberg et al., 1958; Jerauld and Salter, 1990; Satik, 1998; Mahiya, 1999). Fig. 3 also validates the correctness of our model. Compared to the fractal model (Xu et al., 2013), our model has a wider tolerance for changes of pore-throat radius distribution. In our model, there is no limitation with the value of the ratio rmin/rmax. However, the ratio rmin/rmax < 0.01 must be satisfied for fractal analysis of porous media, otherwise the porous medium is a non-fractal or may cause great errors (Xu et al., 2013). For many scaled porous media the ratio rmin/rmax is greater than 0.01 (see Table 1), which implies that the porous medium is a non-fractal and the fractal model is not applicable. 5. Multi-scale analysis results and discussions
, Zrmax dr
Krw ¼ ð1 Snw Þ½ð1 4Þð1 Snw Þ þ 4Cf
In this section, we will study multi-scale capillary pressure and multi-scale relative permeability in detail. Fig. 4 demonstrates a comparison of the multi-scale pore-throat size density function curves between the lognormal model predictions by Eq. (1) and the experimental data (Wu et al., 2007). In the experiment, the core sample is siltstone which contains two scales (micron scale and submicron scale) of pores. The lower scale indicates an average pore-size radius of approximately 0.3 mm. However, the higher scale shows a much larger mode for the pore sizes, in which the siltstone has an average pore-size radius of about 1.25 mm. It can be further seen from Fig. 4 that the distribution density functions predicted by Eq. (1) in which a ¼ 0.75; s ¼ 0.36 for the higher scale (micron) and a ¼ 0.5; s ¼ 0.29 for the lower scale (submicron) are all consistent with the experimental data (Wu et al., 2007). It also
1.6 The proposed model Piquemal 1994 Abaci et al. 1992 Sandberg et al. 1958 Jerauld and Salter 1990 Satik 1998 Mahiya 1999 Fractal model
1.4 1.2 Relative permeability
120
989
1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Wetting fluid saturation Fig. 3. A comparison of the calculated relative permeability curve and available experimental data as well as the fractal model.
990
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
Fig. 4. The comparison of the calculated pore-throat size density function curve from Eq. (1) and the experimental data from Wu et al. (2007), the density function curves are from two scaled siltstone, (a) the higher scale (micron), (b) the lower scale (submicron).
2000 1800 1600 Capillary pressure (psi)
reveals that the distribution density functions of the two scaled siltstone (Wu et al., 2007) are lognormal. Based on the current lognormal distribution parameters a and s, the multi-scale capillary pressure curves are studied in Fig. 5. Fig. 5 demonstrates that the micron scale siltstone has very low capillary pressure, which indicates larger pore sizes. However, the lower scale (higher resolution) siltstone has higher capillary pressure because most small pores can be reproduced at this scale. It is also seen that our model predictions by Eq. (2) are in good agreement with those by the experiment (Wu et al., 2007). Notably, the micron scale capillary pressure curve coincides with the beginning capillary pressure at lower mercury saturation and the submicron scale capillary pressure curve coincides with the capillary pressure at higher mercury saturation, which suggests that the complete capillary pressure curve contain the pore-throat size information for both scaled porous media. Fig. 6 is plotted based on the capillary pressure of five-scaled porous media of core 3 by Eq. (2). The relevant parameters applied are listed in Table 1. From Fig. 6 it is seen that the capillary pressure is related to the pore-throat distribution density function,
1400
Scale 1 Scale 2 Scale 3 Scale 4 Scale 5
1200 1000 800 600 400 200 0 0.0
0.2
0.4
0.6
0.8
1.0
Wetting fluid saturation Fig. 6. The capillary curve for five scaled porous media.
1.0 Scale 1(Krw Krnw) Scale 2(Krw Krnw) Scale 3(Krw Krnw) Scale 4(Krw Krnw) Scale 5(Krw Krnw)
Relative permeability
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
Wetting fluid saturation Fig. 5. The comparison of the calculated capillary curve from Eq. (2) and the experimental data from Wu et al. (2007) for two scaled siltstone porous media.
Fig. 7. The relative permeability for the five-scaled porous media.
1.0
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
1.0 Swc0=0 (Krw Krnw) Swc0=15% (Krw Krnw) Swc0=25% (Krw Krnw) Swc0=35% (Krw Krnw)
Relative permeability
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Wetting fluid saturation Fig. 8. The relative permeability versus wetting fluid saturation at different initial irreducible wetting phase saturation.
and this is expected. Fig. 6 also demonstrates that the lower scale porous media has larger capillary pressure and the higher scale porous media has lower capillary pressure. Eqs. (11) and (12) show that the relative permeability is related to the irreducible wetting fluid saturation, non-wetting fluid saturation as well as pore-throat distribution. So, we analyze the effects of multi-scale pore-throat distribution of porous media and irreducible wetting fluid saturation on the relative permeability curves, as shown in Figs. 7e9. Fig. 7 demonstrates relative permeability versus wetting phase fluid saturation for the five-scaled porous media. It shows that the relative permeability curves vary with the pore-throat distribution density functions of different scaled porous media. The main reason for this is the pore-throat distribution density function reflects the pore structure of porous media. Since relative permeability is related to the pore structure of porous media (Yu and Liu, 2004; Xu et al., 2013), the pore-throat distribution density functions for different scaled porous media have a great significance on relative permeability.
991
Fig. 8 demonstrates relative permeability versus wetting phase fluid saturation for the first scaled porous media at different irreducible wetting fluid saturation. It shows that the non-wetting fluid relative permeability increases with irreducible wetting fluid saturation, while the wetting fluid relative permeability decreases with irreducible wetting fluid saturation. The main reason for this is the seepage channels for non-wetting fluid get larger and wetting phase flow channels get lower at the given wetting phase saturation with the increase of irreducible wetting fluid saturation, which leads to the increase of non-wetting fluid relative permeability and the reduce of wetting phase relative permeability. The relative permeabilities for wetting and non-wetting phase at different parameters a and s are plotted in Fig. 9. Fig. 9(a) shows that the wetting phase relative permeability increases with increase of the parameter a at the same saturation, while increased parameter a can reduce the non-wetting fluid relative permeability at lower wetting fluid saturation and enhance the non-wetting fluid relative permeability at higher wetting fluid saturation. This is because increased parameter a implies increases of the amount of larger pores, and the critical porous-throat radius will be increased for a given saturation. As the result, wetting phase relative permeability will be enhanced. Fig. 9(b) illustrates the effect of parameter s on the relative permeability. Although increased parameter s can highly reduce the wetting phase relative permeability, the effect of parameter s on the non-wetting phase relative permeability is not significant. 6. Conclusions In this paper, we have studied the lognormal distribution characteristics of the multi-scaled porous media. The pore-throat radius distributions of the sample in different analysis scales are lognormal. Then an analytical model for capillary pressure and relative permeability for each scaled porous media is developed. Our model predictions of the relative permeability and that of the fractal model have the similar variation trend with available experimental data. However, compared with the fractal model researches, our model has a wider tolerance for changes of porethroat radius distribution. Our model is also well suited for those non-fractal porous media. The predictions of capillary pressure for higher scaled porous media show the same variation trend with the available experimental data at lower mercury saturation. However, the predictions of capillary pressure for lower scaled porous media
Fig. 9. Effect of distribution parameters on relative permeability at: (a) different parameters a and s ¼ 0.22, (b) different parameters s and a ¼ 0.41.
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G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
conform to the available experimental data only at higher mercury saturation. The complete capillary pressure curve contains the entire pore-throat size information. The proposed relative permeability is expressed as a function of the pore-throat distribution parameters a and s, irreducible wetting phase saturation and wetting fluid saturation. The pore-throat distribution functions of the porous media in different analysis scales are lognormal. The parametric study shows that the non-wetting fluid relative permeability increases with irreducible wetting fluid saturation, while the wetting fluid relative permeability decreases with irreducible wetting fluid saturation. The wetting phase relative permeability increases with the parameter a and decreases with the parameter s at the same saturation. However, the effect of parameter a on the non-wetting phase relative permeability is complex and the parameter s on the non-wetting phase relative permeability is not significant. Nomenclature
Latin symbols A The cross sectional area The pore area fractal dimension Df DT The tortuosity fractal dimension f(r) The density distribution function of the radius r K The absolute permeability Kw The wetting phase effect permeability Knw The non-wetting phase effect permeability Krw The wetting phase relative permeability Krnw The non-wetting phase relative permeability L The actual length of a tortuous capillary L0 The representative length of a capillary N The number of pores of a unit cell p The pressure pc The capillary pressure Q The total fluid volumetric flow rate q(r) The volumetric flow rate through a single tortuous pore r The radius of pore/capillary rc The critical radius rc0 The irreducible wetting fluid radius Sw0 The irreducible wetting fluid saturation Sw Wetting fluid saturation Snw Non-wetting fluid saturation Greek symbols a,s The lognormal parameter q The contact angle ε The surface tension t The tortuosity m Fluid viscosity 4 Porosity
Subscripts max Maximum values min Minimum values w Wetting phase nw Non-wetting phase
References Abaci, S., Edwards, J.S., Whittaker, B.N., 1992. Relative permeability measurements for two phase flow in unconsolidated sands [J]. Mine Water Environ. 11 (2), 11e26. Amirtharaj, S.E., Ioannidis, M.A., Macdonald, I.F., 2003. Statistical Synthesis of Image Analysis and Mercury Porosimetry for Multiscale Pore Structure Characterization [D]. University of Waterloo [Department of Chemical Engineering]. Bear, J., 1972. Dynamics of Fluids in Porous Media. Elsevier, New York. Han, Chunming, 1997. The computation of some parameters of logarithmic normal distribution density function and the discussion its genesis [J]. J. Xinjiang Inst. Technol. 18 (3), 222e226. Jerauld, G.R., Salter, S.J., 1990. The effect of pore-structure on hysteresis in relative permeability and capillary pressure: pore-level modeling [J]. Transp. Porous Media 5 (2), 103e151. Jiao, Chunyan, He, Shunli, Gao, Shusheng, et al., 2010. The characteristics of lognormal distribution of pore and throat structure of low permeability core [J]. Pet. Sci. Technol. 31 (8), 856e865. Keehm, Y., Mukerji, T., 2004. Permeability and relative permeability from digital rocks: Issues on grid resolution and representative elementary volume[C]. In: SEG Int'l Exposition and 74th Annual Meeting, Denver, Colorado. Li, Xianping, 2004. Primary of Probability [M]. Higher Education Press, Beijing. Mahiya, G.F., 1999. Experimental Measurement of Steam-water Relative Permeability [D]. Stanford University. Muskat, M., Meres, M.W., 2004. The flow of heterogeneous fluids through porous media [J]. J. Appl. Phys. 7 (9), 346e363. Neimark, A.V., 1989. Multiscale percolation systems [J]. Sov. Phys. JETP 69 (4), 786e791. Piquemal, J., 1994. Saturated steam relative permeabilities of unconsolidated porous media [J]. Transp. Porous Media 17 (2), 105e120. Qi, Shuangyue, Cui, Guimei, 2000. Lognormal distribution characteristics of some geotechnical parameters [J]. Hydrogeol. Eng. Geol. (1), 42e44. Radlinski, A.P., Ioannidis, M.A., Hinde, A.L., et al., Sept. 23-27, 2002. Multiscale characterization of reservoir rock microstructure: combining small angle neutron scattering and image analysis [C]//SCA2002-35. In: Proceedings of 2002 International Symposium of the Society of Core Analysts, Monterey, California, pp. 22e25. Sandberg, C.R., Gournay, L.S., Sippel, R.F., 1958. The effect of fluid-flow rate and viscosity on laboratory determinations of oil-water relative permeabilities [J]. Pet. Trans. AIME 213, 36e43. Satik, C., Horne, R.N., 1998. A measurement of steam-water relative permeability [C]. In: Proceedings of 23rd Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California. Sun, W.C., Andrade, J.E., Rudnicki, J.W., 2011. Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability [J]. Int. J. Numer. Methods Eng. 88 (12), 1260e1279. Wu, K., Jiang, Z., Couples, G.D., et al., 2007. Reconstruction of multi-scale heterogeneous porous media and their flow prediction [C]. In: International Symposium of the Society of Core Analysts Held in Calgary, Canada. Xu, K., Daian, J., Quenard, D., 1997. Multiscale structures to describe porous media part I: theoretical background and invasion by fluids [J]. Transp. Porous Media 26 (1), 51e73. Xu, P., Qiu, S.X., Yu, B.M., et al., 2013. Prediction of relative permeability in unsaturated porous media with a fractal approach [J]. Int. J. Heat Mass Transf. 64, 829e837. Yu, B.M., Cheng, P., 2002. A fractal permeability model for bi-dispersed porous media [J]. Int. J. Heat Mass Transf. 45 (14), 2983e2993. Yu, B.M., Liu, W., 2004. Fractal analysis of permeabilities for porous media [J]. AIChE J. 50 (1), 46e57.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Multi-scale structures of porous media and the flow prediction G. Lei*, P.C. Dong, Z.S. Wu, S.H. Gai, S.Y. Mo, Z. Li Key Laboratory for Petroleum Engineering of the Ministry of Education, China University of Petroleum, Beijing 102249, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 September 2014 Received in revised form 25 October 2014 Accepted 30 October 2014 Available online 7 November 2014
For the same porous media, the pore structures on different scales and the corresponding fluid flow have different features. The pore structure ultimately affects the fluid flow property. So the study of multiscale pore structure of the given sample is important for oil/gas reservoir engineering. In this paper, the lognormal distribution characteristics of the multi-scaled porous media have been described and the analytical expressions for capillary pressure and relative permeability for each scaled porous media are developed based on the lognormal distribution function. It is shown that the pore-throat size distribution density functions of the sample in different analysis scales are lognormal. The predictions of capillary pressure for higher scaled porous media conform to the trend of variation with the available experimental data at lower mercury saturation. However, the variation trend of predictions for lower scaled porous media is in agreement with experimental data only at higher mercury saturation. Compared to the fractal model, our new model for relative permeability has a wider tolerance for changes of pore-throat radius distribution. Our model is also well suitable for those non-fractal porous media, for which the fractal model is not applicable. The predictions of relative permeability by the proposed model are all consistent with experimental data. The effects of irreducible wetting fluid saturation, lognormal distribution parameters a and s on the relative permeability also have been discussed. © 2014 Elsevier B.V. All rights reserved.
Keywords: Multi-scale Microstructure Lognormal distribution Mathematical modeling Relative permeability
1. Introduction For the petroleum industry, it is a constant challenge to understand the interactions between the micro-pore structure of porous media and the fluid flow (Neimark, 1989; Xu et al., 1997; Wu et al., 2007). Many scholars have studied the influence of pore-throat distribution features on the fluid diffusion phenomena. Capillary tube analogs and statistically reconstructed models are the two most important methods to study the relation between pore-throat size distribution characteristics and macroscopic flow characteristics. These two models are useful, but the predictions by these two models are largely affected by the pore-throat size. Besides, the pore structures and the fluid flow of the multi-scaled porous media are different for a given sample. As a result, it is not very meaningful to quantify the grid spacing and pore-throat distribution without considering the multiple-scales pore-throat distribution characteristics of porous media. In 1989, Neimark proposed the multi-scale concept which produced important advance in the techniques to compute the * Corresponding author. E-mail addresses: [email protected], (G. Lei).
[email protected]
http://dx.doi.org/10.1016/j.jngse.2014.10.033 1875-5100/© 2014 Elsevier B.V. All rights reserved.
transport properties. Xu et al. (1997) suggested a given medium can be considered on widely varying scales, and the rules conditioning the interactions between the properties on different scales can be enunciated. Based on bond percolation in a cubic network, Xu et al. (1997) studied the multi-scale structure to facilitate the ensuing computation of the transport coefficients. Radlinski et al. (2002) obtained the entire distribution of pore length scales by combining small-angle scattering and BSEM imaging data. Amirtharaj et al. (2003) employed Mercury intrusion porosimetry (MIP) to access structural information in the submicrometric range and also obtained the entire distribution of pore length scales. They obtained the pore-to-throat aspect ratio and pore accessibility by comparing the complete pore volume distribution to the MIP data. Keehm and Mukerji (2004) suggested the parameters grid spacing and the representative elementary volume with different scales largely influenced the simulated results. They found that permeability is consistently overestimated with the increase of grid spacing. Wu et al. (2007) reconstructed the pore space of multiple-scales of pore systems from appropriately-imaged thin section and made predictions of the mercury injection characteristics of each scaled porous media using the invasion percolation algorithm. They found
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
that the simulated mercury injection curves were quite different for different scaled porous media. Moreover, they had the conclusion that separate mercury injection curves represented distinct pore systems in the samples and the combined mercury injection curve could represent a plausible prediction of the bulk flow characteristics of the sample. Sun et al. (2011) presented the multiple-scales framework to calculate macroscopic effective permeability at a scale relevant to engineering applications. A number of literatures suggest that the both pore radius density distribution function and throat radius density distribution function are lognormal (Han, 1997; Qi and Cui, 2000; Jiao et al., 2010). Han (1997) suggested that the genesis of pore-throat radius density distribution function was lognormal. Qi and Cui (2000) described the distribution characteristics on the frequency histogram of some geotechnical parameters and found some of them were of lognormal distribution. Jiao et al. (2010) stated both the pore size and throat size were lognormal distribution according to central limit theorem stated by Li (2004) and discussed the reason why the pore radius distribution and throat radius distribution were lognormal distribution. In this paper, we show that pore-throat radius distribution density functions of the cores in different analysis scales are lognormal. Then, theoretical models for capillary pressure and relative permeability are attempted to be built based on the lognormal distribution function of multi-scaled porous media. For this purpose, section 2 will introduce the characteristics of lognormal distribution in multi-scaled porous media. In section 3, a novel model for capillary pressure and relative permeability of the scaled porous has been derived. In section 4, the correctness of our model is validated with the available experimental data and the fractal model. Finally, relevant results and discussions are demonstrated in section 5, and conclusions are presented in section 6. 2. The description of multi-scale structures of porous media Porous media consists of numerous irregular pores and throats with different sizes and the pore structure is complex. Jiao et al. (2010) stated the pore-throat size was lognormal distribution according to central limit theorem stated by Li (2004). The expression of lognormal distribution function can be written as
8 > > > <
1 f ðrÞ ¼ pffiffiffiffiffiffiffi 2p sr > > > : 0
ðln r aÞ2 2 2s e
be considered as circles with different radius r. For simplicity, the circular capillaries are assumed. Xu et al. (2013) suggested the nonwetting fluid occupy the pores greater than the critical value rc at a certain capillary pressure pc. The capillary pressure at the capillary radius rc can be expressed as
pc ¼
2ε cos q rc
, Zrmax
Zrmax 2
Snw ¼ rc
The flow paths in porous media can be regarded as a bundle of tortuous capillary channels and the cross sections of capillaries can
re
ðlnraÞ2 2s2
, Zrmax dr
rc
ðlnraÞ2 2s2
re
dr
rmin
(3) In practice, the non-wetting fluid saturation should be lower than 1Sw0, where Sw0 is the irreducible wetting fluid saturation. The relation between Sw0 and irreducible wetting fluid radius rc0 can be written as
Zrc0 Sw0 ¼
re
ðln raÞ2 2s2
, Zrmax dr
rmin
re
ðln raÞ2 2s2
dr
(4)
rmin
Eqs. (2)e(4) reveal that the capillary pressure is the function of pore-throat size and pore-throat distribution function of porous media. Since the pore-throat size and pore-throat distribution functions are different for different resolution research scales, the capillary pressure curves for different scales may be different.
3.2. The relative permeability Based on HagenePoiseulle equation, the total volumetric flow rates for wetting fluid and non-wetting fluid through the porous media can be written respectively (Xu et al., 2013).
Zrc rc0
pDpw r 4 NpDpw pffiffiffiffiffiffiffi f dr ¼ 8mw L 8mw L 2ps
Zrmax Qnw ¼ N
3.1. The capillary pressure
pr f dr¼ rmin
others
3. The flow prediction
Zrmax 2
pr f dr
(1)
where f is the lognormal distribution function, which is a function of pore-throat radius r. a and s are the relevant parameters. rmin and rmax are the maximum and the minimum pore-throat radius, respectively. When the parameters a or s is different, lognormal distribution density function f(r) is different. In this study, the porethroat density distribution functions of different scaled cores (sandstone) in X area (a block in Changqing oilfield) were fitted by lognormal distribution function (see Fig. 1). It was shown that the pore-throat size distributions in different analysis scales of the cores in X area are lognormal. The data of the parameters were shown in Table 1.
(2)
where ε is the surface tension and q is the contact angle. Under this capillary pressure, the non-wetting fluid saturation Snw can be calculated by Xu et al. (2013)
Qw ¼ N rmin r rmax
987
rc
Zrc
ðln raÞ2 2s2
r 3 e
dr
(5a)
rmin
pDpnw 4 NpDpnw pffiffiffiffiffiffiffi r f dr ¼ 8mnw L 8mnw L 2ps
Zrmax
r 3 e
ðln raÞ2 2s2
dr
(5b)
rc
where Q (cm3/s) is flow rates, m (mPa$s) is the viscosity of the fluid, Dp (MPa) is the pressure drop and L (cm) is the length of the capillary, the subscripts w and nw denote the wetting and nonwetting phases, respectively. The flow rates through the twophase system can be also given by Darcy's extended law (Yu and Liu, 2004; Xu et al., 2013)
Qw ¼
Kw A Dpw mw 1 Snw L0
(6a)
Knw A Dpnw mnw Snw L0
(6b)
Qnw ¼
where K is the effective permeability of the medium, L0 is the core length and A is the total cross area. The effective permeability for each phase can be obtained from Eqs. (5) and (6) as:
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Fig. 1. The comparison of the calculated pore-throat size density function curve from Eq. (1) and the experimental data, (a) core 1, (b) core 2, (c) core 3.
Kw ¼
Knw
Zrc
pNð1 Snw Þ pffiffiffiffiffiffiffi 8A 2pst
pNSnw pffiffiffiffiffiffiffi ¼ 8A 2pst
r 3 e
ðln raÞ2 2s2
dr
(7a)
t ¼ L=L0
rc0
Zrmax
3
r e
ðln raÞ2 2s2
dr
(7b)
rc
and the non-wetting fluid saturation can be determined by Eq. (3). Based on Eq. (7) the absolute permeability for saturated porous media (all the capillaries are filled with a single-phase fluid) can be expressed as:
where t is the tortuosity (Bear, 1972) is defined as
K¼
Table 1 Parameters of the scaled cores (sandstone). Core number
Scale
1
Scale Scale Scale Scale Scale Scale Scale Scale Scale
2 3
(8)
1 2 1 2 1 2 3 4 5
rmax
rmin
rmin/rmax
a
s
12.5 mm 21.385 mm 0.845 mm 1.86 mm 1.14 mm 1.4 mm 1.415 mm 2.17 mm 3.15 mm
0.25 mm 1.965 mm 0.02 mm 0.385 mm 0.255 mm 0.325 mm 0.475 mm 0.585 mm 0.97 mm
0.02 0.09 0.024 0.207 0.22 0.23 0.34 0.27 0.31
0.7 1.5 3 0.46 0.7 0.42 0.08 0.03 0.65
0.27 0.27 0.21 0.2 0.17 0.22 0.16 0.19 0.19
pN pffiffiffiffiffiffiffi 8A 2pst
Zrmax
r 3 e
ðln raÞ2 2s2
dr
(9)
rc0
Muskat and Meres (2004) recommended that each phase permeability be given by
Krw ¼ Kw =K
(10a)
Krnw ¼ Knw =K
(10b)
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
140
Fractal model The proposed model
100
f(r)
80 60 40 20 0 -5
-4
-3
-2
-1
0
ln(r) Fig. 2. The comparison of the calculated pore-throat size density function curve from Eq. (1) and the fractal model.
in which Krw (or Krnw) is wetting phase (or non-wetting phase) relative permeability. Based on Eqs. 7e10, the relative permeability for each phase can be respectively determined
Zrc
2 ðln raÞ 2s2
r3 e
Krw ¼ ð1 Snw Þ rc0
Zrmax Krnw ¼ Snw
r e
2 ðln raÞ 2s2
r3 e
dr
(11)
rc0
ðln raÞ 3 2s2
, Zrmax
2
dr
rc
r 3 e
ðln raÞ2 2s2
dr
(12)
rc0
Eqs. (11) and (12) show that the relative permeability Krw (or Krnw) is a function of non-wetting fluid saturation Snw and irreducible wetting fluid saturation as well as multi-scale pore-throat size and multi-scale pore-throat distribution function. 4. Model validation The predictions by the proposed model, namely Eq. (1) and the fractal model (Yu and Cheng, 2002; Xu et al., 2013) have been compared in Fig. 2. It shows clearly in Fig. 2 that our model predictions are all consistent with the fractal model. The relevant parameters applied are listed in Table 2. As shown in Fig. 3, we study the relative permeability by using the parameters in Table 2. And we compare predictions by our current model, namely Eqs. (11) and (12) with the available experimental data and those from the fractal model (see the following Eq. (13)) from Xu et al. (2013). These experimental data are all cited from reference (Xu et al., 2013). The fractal model given by Xu et al. (2013) can be written as
Table 2 Description of the relevant parameters. Parameters
Values
Description
rmax rmin/rmax
0.9 mm 0.01
DT 4
1.10 20% 4.3 0.24
The maximum pore-throat radius The ratio of minimum pore diameter to maximum pore diameter The tortuosity fractal dimension The porosity Lognormal distribution parameters
a s
(13a)
n o Krnw ¼ Snw 1 ½ð1 4Þð1 Snw Þ þ 4Cf
(13b)
In Eq. (13), the fractal factor Cf ¼ (3-Df þ DT)/(2-Df). Df is fractal dimension for pore size and the it is in the range of 0 < Df < 2 in two dimension. From Fig. 3 it is seen that the relative permeability predicted by the proposed model is compared with experimental measurements and the fractal model Eq. (13). It can be clearly seen that our model predictions and that of the fractal model (Xu et al., 2013) have the similar variation trend besides our predictions by the proposed model Eqs. (11) and (12) are closer to the experimental data (Abaci et al., 1992; Piquemal, 1994; Sandberg et al., 1958; Jerauld and Salter, 1990; Satik, 1998; Mahiya, 1999). Fig. 3 also validates the correctness of our model. Compared to the fractal model (Xu et al., 2013), our model has a wider tolerance for changes of pore-throat radius distribution. In our model, there is no limitation with the value of the ratio rmin/rmax. However, the ratio rmin/rmax < 0.01 must be satisfied for fractal analysis of porous media, otherwise the porous medium is a non-fractal or may cause great errors (Xu et al., 2013). For many scaled porous media the ratio rmin/rmax is greater than 0.01 (see Table 1), which implies that the porous medium is a non-fractal and the fractal model is not applicable. 5. Multi-scale analysis results and discussions
, Zrmax dr
Krw ¼ ð1 Snw Þ½ð1 4Þð1 Snw Þ þ 4Cf
In this section, we will study multi-scale capillary pressure and multi-scale relative permeability in detail. Fig. 4 demonstrates a comparison of the multi-scale pore-throat size density function curves between the lognormal model predictions by Eq. (1) and the experimental data (Wu et al., 2007). In the experiment, the core sample is siltstone which contains two scales (micron scale and submicron scale) of pores. The lower scale indicates an average pore-size radius of approximately 0.3 mm. However, the higher scale shows a much larger mode for the pore sizes, in which the siltstone has an average pore-size radius of about 1.25 mm. It can be further seen from Fig. 4 that the distribution density functions predicted by Eq. (1) in which a ¼ 0.75; s ¼ 0.36 for the higher scale (micron) and a ¼ 0.5; s ¼ 0.29 for the lower scale (submicron) are all consistent with the experimental data (Wu et al., 2007). It also
1.6 The proposed model Piquemal 1994 Abaci et al. 1992 Sandberg et al. 1958 Jerauld and Salter 1990 Satik 1998 Mahiya 1999 Fractal model
1.4 1.2 Relative permeability
120
989
1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Wetting fluid saturation Fig. 3. A comparison of the calculated relative permeability curve and available experimental data as well as the fractal model.
990
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
Fig. 4. The comparison of the calculated pore-throat size density function curve from Eq. (1) and the experimental data from Wu et al. (2007), the density function curves are from two scaled siltstone, (a) the higher scale (micron), (b) the lower scale (submicron).
2000 1800 1600 Capillary pressure (psi)
reveals that the distribution density functions of the two scaled siltstone (Wu et al., 2007) are lognormal. Based on the current lognormal distribution parameters a and s, the multi-scale capillary pressure curves are studied in Fig. 5. Fig. 5 demonstrates that the micron scale siltstone has very low capillary pressure, which indicates larger pore sizes. However, the lower scale (higher resolution) siltstone has higher capillary pressure because most small pores can be reproduced at this scale. It is also seen that our model predictions by Eq. (2) are in good agreement with those by the experiment (Wu et al., 2007). Notably, the micron scale capillary pressure curve coincides with the beginning capillary pressure at lower mercury saturation and the submicron scale capillary pressure curve coincides with the capillary pressure at higher mercury saturation, which suggests that the complete capillary pressure curve contain the pore-throat size information for both scaled porous media. Fig. 6 is plotted based on the capillary pressure of five-scaled porous media of core 3 by Eq. (2). The relevant parameters applied are listed in Table 1. From Fig. 6 it is seen that the capillary pressure is related to the pore-throat distribution density function,
1400
Scale 1 Scale 2 Scale 3 Scale 4 Scale 5
1200 1000 800 600 400 200 0 0.0
0.2
0.4
0.6
0.8
1.0
Wetting fluid saturation Fig. 6. The capillary curve for five scaled porous media.
1.0 Scale 1(Krw Krnw) Scale 2(Krw Krnw) Scale 3(Krw Krnw) Scale 4(Krw Krnw) Scale 5(Krw Krnw)
Relative permeability
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
Wetting fluid saturation Fig. 5. The comparison of the calculated capillary curve from Eq. (2) and the experimental data from Wu et al. (2007) for two scaled siltstone porous media.
Fig. 7. The relative permeability for the five-scaled porous media.
1.0
G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
1.0 Swc0=0 (Krw Krnw) Swc0=15% (Krw Krnw) Swc0=25% (Krw Krnw) Swc0=35% (Krw Krnw)
Relative permeability
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Wetting fluid saturation Fig. 8. The relative permeability versus wetting fluid saturation at different initial irreducible wetting phase saturation.
and this is expected. Fig. 6 also demonstrates that the lower scale porous media has larger capillary pressure and the higher scale porous media has lower capillary pressure. Eqs. (11) and (12) show that the relative permeability is related to the irreducible wetting fluid saturation, non-wetting fluid saturation as well as pore-throat distribution. So, we analyze the effects of multi-scale pore-throat distribution of porous media and irreducible wetting fluid saturation on the relative permeability curves, as shown in Figs. 7e9. Fig. 7 demonstrates relative permeability versus wetting phase fluid saturation for the five-scaled porous media. It shows that the relative permeability curves vary with the pore-throat distribution density functions of different scaled porous media. The main reason for this is the pore-throat distribution density function reflects the pore structure of porous media. Since relative permeability is related to the pore structure of porous media (Yu and Liu, 2004; Xu et al., 2013), the pore-throat distribution density functions for different scaled porous media have a great significance on relative permeability.
991
Fig. 8 demonstrates relative permeability versus wetting phase fluid saturation for the first scaled porous media at different irreducible wetting fluid saturation. It shows that the non-wetting fluid relative permeability increases with irreducible wetting fluid saturation, while the wetting fluid relative permeability decreases with irreducible wetting fluid saturation. The main reason for this is the seepage channels for non-wetting fluid get larger and wetting phase flow channels get lower at the given wetting phase saturation with the increase of irreducible wetting fluid saturation, which leads to the increase of non-wetting fluid relative permeability and the reduce of wetting phase relative permeability. The relative permeabilities for wetting and non-wetting phase at different parameters a and s are plotted in Fig. 9. Fig. 9(a) shows that the wetting phase relative permeability increases with increase of the parameter a at the same saturation, while increased parameter a can reduce the non-wetting fluid relative permeability at lower wetting fluid saturation and enhance the non-wetting fluid relative permeability at higher wetting fluid saturation. This is because increased parameter a implies increases of the amount of larger pores, and the critical porous-throat radius will be increased for a given saturation. As the result, wetting phase relative permeability will be enhanced. Fig. 9(b) illustrates the effect of parameter s on the relative permeability. Although increased parameter s can highly reduce the wetting phase relative permeability, the effect of parameter s on the non-wetting phase relative permeability is not significant. 6. Conclusions In this paper, we have studied the lognormal distribution characteristics of the multi-scaled porous media. The pore-throat radius distributions of the sample in different analysis scales are lognormal. Then an analytical model for capillary pressure and relative permeability for each scaled porous media is developed. Our model predictions of the relative permeability and that of the fractal model have the similar variation trend with available experimental data. However, compared with the fractal model researches, our model has a wider tolerance for changes of porethroat radius distribution. Our model is also well suited for those non-fractal porous media. The predictions of capillary pressure for higher scaled porous media show the same variation trend with the available experimental data at lower mercury saturation. However, the predictions of capillary pressure for lower scaled porous media
Fig. 9. Effect of distribution parameters on relative permeability at: (a) different parameters a and s ¼ 0.22, (b) different parameters s and a ¼ 0.41.
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G. Lei et al. / Journal of Natural Gas Science and Engineering 21 (2014) 986e992
conform to the available experimental data only at higher mercury saturation. The complete capillary pressure curve contains the entire pore-throat size information. The proposed relative permeability is expressed as a function of the pore-throat distribution parameters a and s, irreducible wetting phase saturation and wetting fluid saturation. The pore-throat distribution functions of the porous media in different analysis scales are lognormal. The parametric study shows that the non-wetting fluid relative permeability increases with irreducible wetting fluid saturation, while the wetting fluid relative permeability decreases with irreducible wetting fluid saturation. The wetting phase relative permeability increases with the parameter a and decreases with the parameter s at the same saturation. However, the effect of parameter a on the non-wetting phase relative permeability is complex and the parameter s on the non-wetting phase relative permeability is not significant. Nomenclature
Latin symbols A The cross sectional area The pore area fractal dimension Df DT The tortuosity fractal dimension f(r) The density distribution function of the radius r K The absolute permeability Kw The wetting phase effect permeability Knw The non-wetting phase effect permeability Krw The wetting phase relative permeability Krnw The non-wetting phase relative permeability L The actual length of a tortuous capillary L0 The representative length of a capillary N The number of pores of a unit cell p The pressure pc The capillary pressure Q The total fluid volumetric flow rate q(r) The volumetric flow rate through a single tortuous pore r The radius of pore/capillary rc The critical radius rc0 The irreducible wetting fluid radius Sw0 The irreducible wetting fluid saturation Sw Wetting fluid saturation Snw Non-wetting fluid saturation Greek symbols a,s The lognormal parameter q The contact angle ε The surface tension t The tortuosity m Fluid viscosity 4 Porosity
Subscripts max Maximum values min Minimum values w Wetting phase nw Non-wetting phase
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