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A more generalized model for relative permeability prediction in unsaturated fractal porous media
Journal of Natural Gas Science and Engineering 67 (2019) 82–92
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A more generalized model for relative permeability prediction in unsaturated fractal porous media
T
Fuyong Wanga,∗, Liang Jiaoa, Jiuyu Zhaoa, Jianchao Caib,∗∗ a b
Research Institute of Enhanced Oil Recovery, China University of Petroleum, Beijing, 102249, PR China Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan, 430074, PR China
ARTICLE INFO
ABSTRACT
Keywords: Relative permeability Fractal porous media Tortuosity Fractal dimension Capillary pressure
Although lots of models for relative permeability prediction have been developed in literature, the effect of pore geometric tortuosity on relative permeability has not been well discussed, as usually the tortuosity of pores with various sizes is assumed the same. However, there is a high possibility that tortuosity is unequal and increases with the decrease in pore size. A new and more generalized analytical model for relative permeability prediction has been developed based on the fractal characteristics of pore size distribution (PSD) and tortuosity in porous media. The analytical expressions of relative permeability for wetting phase and nonwetting phase have been derived, which are the function of normalized wetting phase saturation, PSD fractal dimension, tortuosity fractal dimension and the ratio of the entry capillary pressure to the maximum capillary pressure. The studies show that the increasing PSD fractal dimension and tortuosity fractal dimension can significantly reduce wetting phase relative permeability, but the increase in nonwetting phase relative permeability is insignificant. The developed model has been validated with relative permeability experimental data and compared with other classic relative permeability models. The comparison results demonstrate the reliability of the relative permeability model developed in this paper.
1. Introduction
of relative permeability of wetting phase and nonwetting phase equals one, which is invalid in most porous media. Li and Horne (2006) found that the relative permeability for wetting phase predicted by the Purcell model well fits the experimental data, but the predicted nonwetting phase relative permeability using the Purcell model is far from experimental results. Based on the Purcell model, Burdine (1953) introduced tortuosity factors to account for the effect of nonuniform fluid distribution on relative permeability. However, Li and Horne (2006), Li (2010a) believed that the tortuosity factor of the wetting phase is no necessary for predicting wetting phase relative permeability. Ever since, lots of modifications have been made on the Purcell model and Burdine model to characterize the multiphase flow behaviour through porous media (Corey, 1954; Brooks and Corey, 1966; He and Hua, 1998; Li, 2008, 2010a; Zhang et al., 2017). With a linear relation between 1/ Pc2 and normalized wetting phase saturation SwD , Corey (1954) developed the Corey relative permeability model based the Burdine approach, and the relative permeability of wetting phase and nonwetting phase is only the function of SwD . Brooks and Corey (1966) proposed a more generalized capillary pressure function where a PSD index was introduced, and then a more generalized relative permeability model was
Relative permeability plays a crucial role in petroleum engineering, geothermal exploration, textile engineering, chemical engineering, and many other areas, as it provides a means to fully characterize the nature of two-phase fluid flow in porous media (Li, 2010a). Relative permeability can be estimated by laboratory experiments (Egermann and Lenormand, 2005; Reynolds and Krevor, 2015; Jackson et al., 2018) and model prediction methods (Tao and Watson, 1984; Li, 2010a; Sun et al., 2018). However, the relative permeability of some unconventional reservoir rock samples cannot be directly obtained by experiments due to ultra-low permeability or complex pore structures (Zhang et al., 2017). Therefore, the methods of predicting relative permeability from mathematical models are widely used. Many models have been proposed in literature to predict relative permeability in porous media. Some classic models have been summarized and shown in Table 1. Purcell (1949) proposed a method for calculating permeability from capillary pressure data, and this method was extended to estimate relative permeability (Gates and Lietz, 1950). However, the Purcell model is too ideal for field application, as the sum
∗
Corresponding author. Corresponding author. E-mail addresses: [email protected] (F. Wang), [email protected] (J. Cai).
∗∗
https://doi.org/10.1016/j.jngse.2019.04.019 Received 29 November 2018; Received in revised form 18 April 2019; Accepted 19 April 2019 Available online 25 April 2019 1875-5100/ © 2019 Elsevier B.V. All rights reserved.
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Table 1 Several classic relative permeability models in literature. Model
Formula
Purcell model (1949) Burdine model (1953)
Krw =
Krw = (
Corey model (1954) Brooks-Corey model (1966)
rw )
Sw 2 0 dSw / pc Krnw 1 2 0 dSw / pc
Sw dS / p 2 w c 2 0 Krnw 1 2 0 dSw / pc
4 Krw = SwD Krnw = (1
(2 + 3 Krw = SwD
He-Hua model (1998)
)/
11 3Df 3 Df
Krw = SwD Li model (2010a)
Krw =
1 1
Comments
2+ (Swe) 2+
=
1 2 Sw dSw / pc 1 2 0 dSw / pc
=(
rnw )
1 dSw / pc2 2 Sw 1 dS / p2 w c 0
SwD ) 2 (1
2 SwD )
Krnw = (1
SwD ) 2 (1
(2 + SwD
Krnw = (1
SwD ) 2(1
SwD
Krnw =
(Swe )
2+
1
● the sum of Krw and Krnw at a specific saturation equals 1 ● the calculated Krnw is far from the experimental results
)/
5 Df 3 Df
(1
● ● ●
)
)
2+ 2+
●
SwD ) 2
● ● ● ● ● ●
rw
is tortuosity ratio of wetting phase, and
rw
=
Sw Sm ; Sm 1 Sm
is the minimum wetting phase
saturation 1 Sw Sm ; Se is the equilibrium rnw is tortuosity ratio of nonwetting phase, and rnw = 1 Sm Se saturation of the nonwetting phase based on the Burdine model with the assumption of 1/Pc2 = CSwD S S SwD is normalized wetting phase saturation, and SwD = w w c where Swc is the minimum 1 Swc wetting phase saturation. is the PSD index the capillary pressure function can be expressed asPc = pe (SwD ) 1/ where pe is the entry capillary pressure fractal distribution of pore size Df in three-dimensional space is used fractal distribution of tortuosity is not considered p p Swe = 1 bS wD ; b = 1 ( e ) ; = ( e ) pmax
pmax
● In fractal porous media = 3 Df and Df in tree-dimensional space is used ● Swe = 1 bS wD prediction is based on the Purcell approach and Krnw prediction is based on the Burdine approach
derived. The Corey model is only a special case of the Brooks-Corey model when is equal to 2. Fractal theory was firstly proposed by Mandelbrot (1982) and has been widely used to describe complex geometry with self-similarity. Natural porous rocks have been demonstrated to have fractal characteristics (Katz and Thompson, 1985; Krohn, 1988). Numerous scholars have applied fractal theory to predict (relative) permeability of porous media with mathematical modelling and numerical simulation (Yu and Cheng, 2002; Li, 2010b; Cai et al., 2012, 2015; Xiao et al., 2012, 2017; Xu et al., 2013; Tan et al., 2015; Wang et al., 2017). He and Hua (1998) found that the PSD index can be expressed with fractal dimension Df as = 3 Df , and they developed a relative permeability model where the relative permeability of wetting phase and nonwetting phase are only the function of normalized wetting phase saturation and fractal dimension. Li (2010a) developed a more generalized relative permeability model based on fractal assumption of porous media, and he found that the shape of relative permeability is also affected by the ratio of the entry capillary pressure to the maximum capillary pressure besides fractal dimension. Xiao et al. (2012) and Xu et al. (2013) used fractal theory and Monte Carlo simulation to predict relative permeability in unsaturated porous media. Effective prediction of relative permeability of porous media is still challenging due to complex pore structures. One of the characteristics is the path along fluid flowing is not straight, but rather tortuous or meandering. Carman (1937) introduced the concept of tortuosity, and it has been demonstrated that tortuosity has great impact on electrical, hydraulic and diffusive properties of porous media (Scheidegger, 1958; Sahimi, 1993; Clennell, 1997; Matyka et al., 2008). Yu and Cheng (2002) considered fractal distribution of tortuosity and developed a permeability model for fractal porous media. Wang et al. (2019) derived a correlation between flow rate and pressure drop in a single capillary tube with fractal distribution of tortuosity, and found that the increasing tortuosity fractal dimension can significantly reduce permeability of porous media. Although Burdine (1953) introduced the tortuosity factors to modify the Purcell model, the introduced tortuosity factors are used to characterize the flow resistance caused by the nonuniform distribution of fluids rather than the variable geometry tortuosity. The effects of complex distribution of pore geometry tortuosity on relative permeability of unsaturated porous media are unclear. Based on fractal characteristics of PSD and tortuosity in porous media, this paper presents a more generalized analytical model to predict relative permeability in unsaturated porous media. To validate the newly developed
model, experiments of relative permeability measurements with four carbonate core plugs were conducted, and the predicted relative permeability results were compared with the experimental results. Pore structure parameters used for model predicting were derived from highpressure mercury intrusion (HPMI) test. Finally, the factors influencing relative permeability were quantitatively analysed. 2. Model description 2.1. Fractal theory This paper assumes that the porous media is made up of a bundle of tortuous capillary tubes with variable pore sizes and tortuous lengths; the nonwetting phase fluid occupies large pores while the wetting phase fluid flows in small pores; the size and tortuous length of capillary tubes have self-similarity characteristics, and can be described with fractal theory. The correlation between the number of capillary tubes N (a r ) and the size of capillary tubes r can be described as follows (Mandelbrot, 1982):
N (a
r) =
rmax r
Df
(1)
where rmax is the maximum pore diameter; Df is the PSD fractal dimension and its range is 0 < Df < 2 and 0 < Df < 3 in two-dimensional space and three-dimensional space, respectively. The Df in two-dimensional space is used here, as it reflects PSD in the cross-section of a tortuous capillary tube model. Assuming that the straight length of a selected representative unit of porous media is L0 , and the actual length of tortuous capillary tube is LT (r ) , the relationship between LT (r ) and L0 can be described as (Yu and Cheng, 2002):
LT (r ) = (2r )1
DT L DT 0
(2)
where DT is tortuosity fractal dimension, and its range is 1 DT < 2 and 1 DT < 3 in two-dimensional space and three-dimensional space, respectively. Tortuosity fractal dimension DT indicates the heterogeneity of tortuosity distribution. The case of DT = 1 means capillary tubes are straight, and a larger value of DT means the longer capillary tubes (Yu and Cheng, 2002). The flow rate q in a tortuous capillary tube with fractal distribution of tortuosity can be derived from Newton's second law as (Wang et al., 2019): 83
Journal of Natural Gas Science and Engineering 67 (2019) 82–92
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q=
22
P r DT + 3 + 3) µL0DT
DT (D T
Krnw =
(3)
In the porous media with fractal distributions of pore size and tortuosity, a new correlation of wetting phase saturation Sw and capillary pressure pc can be expressed as: 1
Sw 1
Swc Swc
(12)
SwD
In this case, the relative permeability of each phase is the linear function of its own saturation. When DT + Df is less than 3 and pe / pmax 0 , Eqs. (9) and (10) can be simplified as: 3 + DT Df 3 DT Df
(13)
Krw = SwD
3 + DT Df 3 DT Df
Krnw = 1
2.4. Relative permeability model based on the Burdine approach Similar to the Burdine model (Burdine, 1953), tortuosity factors of wetting phase and nonwetting phase are introduced to Eqs. (7) and (8), and the equations for predicting relative permeability of wetting phase and nonwetting phase can be derived as follows: SwD 0
2 Krw = SwD
(6)
dS pc2DT
1 dS 0 p2DT c
where Snwr is the residual nonwetting phase saturation.
Krnw = (1
2.3. Relative permeability model based on the Purcell approach
Sw 0
dS pc2DT
1 dS 0 p 2DT c
Krw =
(7)
1 dS Sw p2DT c
Krnw =
1
(Swe ) 1
1 dS 0 p2DT c
(16)
1
(Swe ) 1 SwD )2
2DT +
2DT +
(17)
(Swe )
2DT +
2DT + 2DT +
(18)
1. For the Eqs. (17) and (18) include an implicit condition that case = 1 which means capillary pressure pc is independent of wetting phase saturation Sw , Eqs. (15) and (16) can be reduced to: (19)
3 Krw = SwD
Krnw = (1
SwD )3
For the case DT + Df < 3 and pe / pmax be simplified as:
2DT +
2DT +
1 dS SwD p 2DT c
1
(8)
The detailed derivation process can be found in Appendix B. Different from the Purcell model presented in Table 1, the tortuosity fractal dimension DT is included. For the case where DT is equivalent to 1, Eqs. (7) and (8) can be simplified as the Purcell model, which means the Purcell model is a special case of the newly developed model. Submitting Eq. (4) into Eqs. (7) and (8), the relative permeability of wetting phase and nonwetting phase can be predicted with:
Krw =
2 SwD
Krnw = (1
1 dS 0 p 2DT c
SwD
)2
(15)
When DT is equivalent to 1, Eqs. (15) and (16) can be simplified to the Burdine model (1953). Submitting Eq. (4) into Eqs. (15) and (16), a generalized relative permeability model with fractal distribution of tortuosity based on the Burdine approach has been derived as follows:
Based on the Purcell approach (Purcell, 1949), a model for predicting relative permeability of wetting phase and nonwetting phase in porous media with fractal distribution of tortuosity has been derived as follows:
Krw =
(14)
SwD
Different from Li model (Li, 2010a), in addition to normalized wetting phase saturation SwD and PSD fractal dimension Df , relative permeability is also affected by tortuosity fractal dimension DT .
(5)
Sw Swc Swc Snwr
1
(11)
Krnw = 1
(4)
where Swc is the minimum wetting phase saturation. For the imbibition case, the normalized wetting phase saturation can be calculated from (Li and Horne, 2006; Li, 2010a):
SwD =
(10)
Krw = SwD
where = 3 DT Df ; b = 1 (pe / pmax ) ; pe is the entry capillary pressure; pmax is the maximum capillary pressure. The detailed derivation process of Eq. (4) can be found in Appendix A. Eq. (4) has the same form with the capillary pressure function proposed by Li (2010a), but in the Li model = 3 Df and pore tortuosity is assumed as constant. In Eq. (4), as fractal distribution of tortuosity is considered and the tortuosity fractal dimension DT is included. It must be noted that the two-dimensional Df should be used in Eq. (4), rather than threedimensional Df used in the Li model (Li, 2010a). When there is residual wetting phase saturation in porous media, the wetting phase saturation Sw can be replaced with the normalized wetting phase saturation SwD . For the displacement process, it can be defined as (Burdine, 1953):
SwD =
2DT +
where Swe = 1 bSwD and = (pe / pmax ) . Eqs. (9) and (10) are a generalized relative permeability model based on the Purcell approach, and it is similar to the model presented by Li (2010a) but the tortuosity fractal dimension DT is here included. When DT is equivalent to 1, Eqs. (9) and (10) can be simplified to the Li model (Li, 2010a). 1. For the There is an implicit condition in Eqs. (9) and (10), i.e. case = 1 , which means pe = pmax and the pore radius is the same, the capillary pressure pc is independent of wetting phase saturation Sw , and then Eqs. (7) and (8) can be reduced to:
2.2. A new capillary pressure model with tortuosity fractal dimension
bSw )
2DT +
1
where P is the pressure difference between two ends of the capillary tube; µ is the fluid viscosity. The detail derivation of Eq. (3) can be found in Appendix B of the paper presented by Wang et al. (2019). Eq. (3) is a modified Hagen-Poiseuille equation for the fluid flow in a capillary tube with fractal distribution of tortuosity.
pc = pmax (1
2DT +
(Swe )
9 DT 3Df 3 DT Df
(9)
Krw = SwD 84
(20)
0 , Eqs. (17) and (18) can
(21)
Journal of Natural Gas Science and Engineering 67 (2019) 82–92
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Table 2 Porosity and permeability of four carbonate core plugs used in experiments. Core No.
Porosity (%)
Air permeability K air (mD)
Klinkenberg permeability KL (mD)
Water permeability Kw (mD)
Relative permeability experiment
48H 280H 145H 192H
14.564 7.813 18.278 12.325
7.578 1.924 33.899 3.454
4.81 1.177 18.491 2.815
1.11 0.038 24.44 0.60
Gas-water Gas-water Gas-oil Gas-oil
Krnw = (1
SwD )2 1
of these four core plugs is 1.5 inch with about 2 inch long. The porosity of core plugs varies from 7.813% (280H) to 18.278% (145H), with air permeability changes from 1.924 mD to 33.899 mD. Klinkenberg permeability was also measured to correct gas slippage effect. The water permeability is less than the Klingenberg permeability might be caused by the adoption of water (Wang et al., 2019). Unsteady-state method based on Buckley-Leverett displacement theory was used for measuring gas-water/oil relative permeability. Fig. 1 shows the schematic diagram of unsteady-state displacement measurement, and air was used as the gas source. The procedures of gas-water relative permeability measurement are introduced as follows. Firstly, the clean core plugs were fully saturated with formation water, and water permeability was measured. Then the air was injected into the confined core plugs with constant pressure, and meanwhile the gas and water production were monitored and recorded with time. For the gas-oil relative permeability measurement, oil was injected into the water-saturated core plugs until no water was produced. After that air was injected into core plugs, gas and oil production were recorded. Finally, the recorded gas and water/oil production data from displacement experiments were used to calculate the gas-water/oil relative permeability. Johnson, Bossler, and Naumann (JBN) method is an explicit method and has been widely used to calculate relative permeability from displacement experiment based on Buckley-Leverett displacement theory (Tao and Watson, 1984; Sun et al., 2018). However, JBN method needs tedious calculations and errors may be introduced during the process of estimating the derivatives of measured data. Jones and Roszelle (1978) proposed a modified JBN method based on graphical techniques, and Jones-Roszelle method was used to derive relative permeability from displacement experiments in this paper. The derived gas-water/oil relative permeability results are shown in Fig. 2.
3 + DT Df 3 DT Df
SwD
(22)
According to Li and Horne (2006) and Li (2010a), the Purcell relative permeability is the best fit to the wetting phase relative permeability measured from experiments, and the Brooks–Corey model well fits the experimental data of nonwetting phase relative permeability. Therefore, in this paper the wetting phase relative permeability is predicted with Eq. (9), and the nonwetting phase relative permeability is calculated with Eq. (18). The pore structure parameters, including PSD fractal dimension Df , tortuosity fractal dimension DT , and the ratio of the entry capillary pressure to the maximum capillary pressure pe / pmax can be calculated from HPMI test. The detailed calculation processes were demonstrated in the model validation section. 3. Relative permeability experiment and model validation The newly developed relative permeability model has been validated with gas-water/oil relative permeability experiment data measured from four carbonate core plugs (Wang et al., 2016). Then the HPMI tests were conducted to measure pore structure parameters of each core plugs, and the values of Df , DT , and pe / pmax were calculated. Finally, the calculated relative permeability using the newly developed model was verified with the measured relative permeability data from experiments. The model prediction results were also compared with the results calculated from other classic relative permeability models, including the Purcell model, the Burdine model, the Brooks-Corey model and the Corey model. 3.1. The relative permeability experiments
3.2. HPMI tests
The permeability and porosity of these four carbonate core plugs are shown in Table 2. 48H and 280H core plugs were used for gas-water relative permeability measurement, and 145H and 192H core plugs were used for gas-oil relative permeability measurement. The diameter
After displacement experiment, pore structure parameters of the four carbonate core plugs were measured from HPMI tests. The obtained mercury intrusion capillary pressure (MICP) curves are shown in
Fig. 1. Schematic diagram of gas-water/oil relative permeability experiment (1: air supply; 2: pressure gauge; 3: balance chamber; 4: core Holder; 5: quizix pump; 6: valve 7: pressure pump; 8: water/oil volume gauge; 9: gas flow meter; 10: formation water/oil). 85
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Fig. 2. The measured relative permeability from experiments (a: gas-water relative permeability of 48H; b: gas-water relative permeability of 280H; c: gas-oil relative permeability of 145H; gas-oil relative permeability of 192H).
Fig. 3. The pore size and distribution can be derived from capillary pressure with the Washburn Equation (Washburn, 1921) given as: r = 2 cos / Pc . The mercury surface tension is 0.48 N/m, and the contact angle between mercury and air is 140°. The maximum MICP is about 275 MPa, and the corresponding minimum pore radius which can be measured by HPMI is about 2.7 nm. Some pore structure parameters derived from HPMI are shown in Table 3. According to Eq. (1), PSD fractal dimension Df can be calculated from the correlation between capillary tube number N (r ) and capillary tube sizer . The capillary tube number N (r ) can be calculated with the following equation (Li, 2010b):
N (r ) =
VHg (r ) r 2l
(23)
where VHg is the accumulative injected mercury volume when the measured capillary tube radius is r . Plotting the capillary tube number N (r ) versus capillary tube size r in a log-log figure, the PSD fractal dimension Df can be obtained from the slope of the lgN (r ) lgr curve S by Df = S according to Eq. (1). The processes of Df calculation for each core plugs are shown in Fig. 4. The calculated Df reflects the fractal characteristics of PSD in three-dimensional space. However, the Df in two-dimensional space should be used in the developed model. It is generally accepted that the Df in three-dimensional space and twodimensional space differs by one (Wang et al., 2018). Therefore, the obtained Df in Fig. 4 subtracting one was used for relative permeability prediction. Taken the core plug 48H as an example, the slope of the lgN (r ) lgr S is −2.38 and therefore the PSD fractal dimension in three-dimensional space is Df = S = 2.38. Then the PSD fractal dimension in two-dimensional space can be inferred as Df = 2.38 1 = 1.38.
Fig. 3. Plots of MICP curves of four carbonate plugs.
The tortuosity fractal dimension DT can be calculated with the following equation (Yu, 2005):
DT = 1 +
ln ln(L 0 /2r )
(24)
where r is the average pore radius of porous media, and in this paper the media pore radius r50 derived from HPMI is used; ¯ is the average tortuosity, and it can be estimated from HPMI with the following equation (Comiti and Renaud, 1989):
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Table 3 The pore structure parameters calculated from HPMI tests. Core No.
Maximum pore radius rmax (μm)
Median pore radius r50 (μm)
Average tortuosity ¯
PSD fractal dimension (two-dimensional) Df
Tortuosity fractal dimension DT
The value of pe /pmax
48H 280H 145H 192H
0.783 3.676 7.828 14.159
0.326 1.938 1.791 0.558
1.79 2.04 1.70 1.86
1.38 1.14 1.22 1.29
1.22 1.3 1.18 1.13
1/270 1/1200 1/2600 1/4900
Fig. 4. Correlations between lgN (r ) and lgr for four core plugs.
= 1+ 0.41ln(1/ )
(25)
and Corey model were also utilized to predict relative permeability. For the Purcell model, Burdine model, the MICP curves from HPMI test were utilized to predict relative permeability. Compared with other four classic relative permeability models, the predicted relative permeability using our model is much close to the measured relative permeability from the experiments. To quantitatively evaluate the results of relative prediction with different models, the correlation coefficients R2 between predicted values and measured results were calculated, as shown in Table 4. The new model has the best performance in predicting relative permeability of wetting phase and nonwetting phase, and the correlation coefficients R2 are more than 0.98. Brooks-Corey model and Corey model also have satisfying results of relative permeability prediction, and the correlation coefficients R2 are more than 0.93 for both wetting phase and nonwetting phase, and some correlation coefficients close to 1. Purcell model has the poorest performance in predicting nonwetting phase relative permeability, which is far from the experimental results, and the correlation coefficients R2 of the predicted results and experiment results are less than 0.26. Thus, Purcell model is not recommended for predicting the nonwetting phase
The calculated values of Df , DT and pe / pmax of these four core plugs are shown in Table 3. 3.3. Model validation The relative permeability of these four carbonate core plugs were predicted with newly developed model and compared with experimental results. As gas is the nonwetting phase compared with the water and oil, the gas relative permeability was predicted with Eq. (18), and the water/oil relative permeability was predicted with Eq. (9). The relative permeability for wetting phase and nonwetting phase under different normalized wetting phase saturation were calculated and compared with the experiment results. Figs. 5 and 6 present comparisons of predicted gas-water/oil relative permeability using the model developed in this paper and the measured relative permeability. Besides, four classic relative permeability models—Purcell model, Burdine model, Brooks-Corey model
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Fig. 5. The predicted gas-water relative permeability using different models and compared with experimental results for the 48H and 280H core plugs.
Fig. 6. The predicted gas-oil relative permeability using different models and compared with experimental results for the 145H and 192H core plugs.
phase relative permeability prediction by Burdine model are usually worse than Purcell model. It can verify that wetting phase relative permeability prediction based on Purcell approach and nonwetting phase relative permeability prediction based on Burdine model are reasonable (Li and Horne, 2006; Li, 2010a). It must be noted that the limited MICP data can cause the calculation errors of Purcell model and Burdine model. 4. Sensitivity analysis of influencing parameters 4.1. PSD fractal dimension PSD Fractal dimension Df is an important parameter to describe the distribution and heterogeneity of pore structures in porous media. The effects of Df on relative permeability of wetting phase and nonwetting phase were studied. Fig. 7 shows the effect of Df on relative permeability with constant values of DT and pe / pmax . With Df increasing from 1.1 to 1.8, the relative permeability of wetting phase decreases dramatically, but the nonwetting phase relative permeability increases little. In other words, Df has much more effect on relative permeability of wetting phase than nonwetting phase. The reason is that the increasing PSD fractal dimension means the increase proportion of small pores. As wetting phase flows in small pores, the increasing proportion of small pores will decrease the flowability of wetting phase at the same saturation. Besides, according to Eq. (2), the tortuous lengths of capillary tube increase with the decrease in pore size, which also reduces the wetting phase relative permeability.
Fig. 7. The predicted relative permeability with different values of Df for DT = 1.18 and pe / pmax = 1/2600 .
relative permeability. Compared with bad performance of nonwetting phase relative permeability prediction, wetting phase relative permeability prediction by Purcell model is acceptable, and the correlation coefficients are more than 0.59. Compared with predicted oil relative permeability, the performance of water relative permeability prediction is better, and the correlation coefficients are more than 0.81. Burdine model has better performance of nonwetting phase relative permeability prediction than Purcell model, but the performances of wetting
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Table 4 The correlation coefficients of relative permeability predicted from different models. Model
The correlation coefficient R 2 48H
New model Purcell model Burdine model Brooks-Corey model Corey model
280H
145H
192H
Krw
Krnw
Krw
Krnw
Krw
Krnw
Krw
Krnw
0.9942 0.8143 0.7686 0.9655 0.9401
0.9823 0.2248 0.4530 0.9769 0.9746
0.9902 0.9150 0.9232 0.9868 0.9393
0.9959 0.1931 0.2432 0.9929 0.9848
0.9962 0.5936 0.5222 0.9841 0.9994
0.9918 0.1744 0.7522 0.9919 0.9932
0.9977 0.7850 0.7512 0.9880 0.9967
0.9996 0.2545 0.9626 0.9992 0.9935
4.3. The ratio of the entry capillary pressure to the maximum capillary pressure The value of pe / pmax represents the ratio of the smallest pore radius to the maximum pore radius rmin / rmax according to Washburn equation. Fig. 9 shows the effect of pe / pmax on the relative permeability of wetting phase and nonwetting phase with constant values of Df and DT . With pe / pmax decreasing from 0.1 to 0.001, the relative permeability of wetting phase under the same saturation decreases, but the increase in nonwetting phase relative permeability is insignificant. The reason is that the decreasing pe / pmax means the increasing distribution range of the capillary tube size, and wetting phase flowability will be reduced correspondingly. Compared with the work presented by Li (2010a), the novelty of the relative permeability model is the tortuosity fractal dimension DT was introduced in this paper. The assumption that every capillary tube has the same tortuous length in conventional relative permeability models (Corey, 1954; Brooks and Corey, 1966; He and Hua, 1998; Li, 2008, 2010a) might be too ideal for the porous media with complex pore structures, such as tight sandstones/carbonates or shales, and tortuosity fractal dimension DT can provide another parameter to characterize complex pore structures besides PSD fractal dimension Df . When DT is equivalent to 1, our model can be simplified to the Li model (Li, 2010a). Therefore, this newly developed model is a more generalized model for predicting relative permeability in unsaturated fractal porous media.
Fig. 8. The predicted relative permeability with different values of DT for Df = 1.22 and pe / pmax = 1/2600 .
5. Conclusions With the assumption of fractal distributions of pore size and tortuosity, this paper presents a new generalized relative permeability model for unsaturated fractal porous media. The new relative permeability model is the function of the normalized wetting phase saturation SwD , PSD fractal dimension Df , tortuosity fractal dimension DT and the ratio of entry capillary pressure to maximum capillary pressure pe / pmax . With the fractal parameters derived from HPMI tests, the model prediction results were verified with gas-water/oil relative permeability experiments. PSD fractal dimension Df and tortuosity fractal dimension DT have significant influence on relative permeability of wetting phase. The increasing values of Df and DT can significantly reduce relative permeability of wetting phase, but the effect on nonwetting phase relative permeability is insignificant. In addition to Df and DT , the shapes of relative permeability curves are also affected by pe / pmax . The decreasing values of pe / pmax can reduce the relative permeability of wetting phase, but the increase in nonwetting phase relative permeability is insignificant.
Fig. 9. The predicted relative permeability with different values of pe / pmax for Df = 1.22 and DT = 1.18.
4.2. Tortuosity fractal dimension Tortuosity fractal dimension DT reflects the distribution of tortuosity with different pore size. When DT equals 1, the tortuosity of each capillary tube is the same. Increasing DT means the tortuosity of the capillary tube with small radius increases dramatically according to Eq. (2). Fig. 8 shows the effect of DT on relative permeability with constant values of Df and pe / pmax . The increasing DT can significantly reduce relative permeability of wetting phase, but the increase in relative permeability of the nonwetting phase is insignificant. The reason is that the increasing DT means the increase in tortuous length of small pores according to Eq. (2), which will increase resistance of wetting phase flow. As nonwetting phase flows in large pores, whose tortuous length changes little, the increase in nonwetting phase relative permeability is insignificant.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 51604285, 51874320, 41722403), and Scientific Research Foundation of China University of Petroleum, Beijing (No. 2462017BJB11). 89
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F. Wang, et al.
Nomenclature cross-sectional area constant PSD fractal dimension tortuosity fractal dimension pressure difference capillary pressure entry capillary pressure maximum capillary pressure flow rate absolute permeability nonwetting phase relative permeability wetting phase relative permeability gas phase relative permeability water phase relative permeability representative length tortuous length of capillary tubes pore radius average pore radius the median pore radius the largest pore radius the minimum pore radius capillary tube number mercury saturation irreducible water saturation residual nonwetting phase saturation wetting phase saturation normalized wetting phase saturation capillary volume all the volumes of capillary tubes constant constant porosity mercury interfacial tension, and = 0.48N/m contact angle between rock and mercury, and = 140° average tortuosity
A b Df DT P Pc Pe Pmax q K Krnw Krw Krg Krw L0 LT r r r50 rmax rmin N (r ) SHg Scw Snwr Sw SwD Vi Vp
¯
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jngse.2019.04.019. Appendix A. Derivation of a new capillary pressure model with tortuosity fractal dimension In a capillary tube model with fractal distributions of PSD and tortuosity, the total pore volume can be expressed as: rmax
Vp =
r 2LT (r ) dN
rmin
(A-1)
Differentiate Eq. (1), and the number of pores with size varying from r and r+ dr can be expressed as (Yu and Cheng, 2002): D
f dN = Df rmax r
(A-2)
(Df + 1) dr
Submitting Eq. (A-2) and Eq. (2) into Eq. (A-1), the total pore volume Vp can be expressed as:
Vp =
21
DT L DT 0
3
D
f Df rmax
DT
Df
3 D
(rmax T
Df
3 DT Df
rmin
)
(A-3)
The pore volume with pore radius less than r can be expressed as:
V ( < r) =
21
DT L DT 0
3
DT
D
f Df rmax
Df
(r 3
DT Df
3 DT Df
rmin
)
(A-4)
The wetting phase saturation can be expressed as:
Sw =
r 3 DT V = 3 DT Vp rmax
3 DT Df
Df
rmin
Df
rmin
3 DT Df
(A-5)
Replace pore radius r with capillary pressure Pc using Eq. (B-2), and Eq. (A-5) becomes: 90
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F. Wang, et al.
Sw =
pc
(3 DT Df )
(3 pmax
pe
(3 DT Df )
(3 DT Df ) pmax
DT Df )
(A-6)
where pe is the entry pressure, corresponding to the maxim pore radiusrmax . Assuming
Sw =
pc
pmax
pe
pmax
=3
DT
Df , Eq. (A-6) can be expressed as: (A-7)
Rearrange Eq. (A-6), and expression of pc becomes:
pc = [pmax
(pmax
( ) pe
Assume b = 1
pc = pmax (1
1
pe ) Sw]
(A-8)
, Eq. (A-8) can be rearranged into the similar form with the Li model (2010a):
pmax
1
bSw )
(A-9)
Appendix B. Derivation of a modified Purcell model with tortuosity fractal dimension The volume of a capillary tube with radius r is given as:
V = r 2LT =21
DT
r3
(B-1)
DT L DT 0
The radius r can be obtained from mercury intrusion test with Washburn Equation as (Washburn, 1921):
2 cos pc
r=
(B-2)
where pc is the mercury capillary pressure; is the mercury interfacial tension; is the contact angle between the rock and the mercury. Substituting Eqs. (B-1) and (B-2) into Eq. (3), the flow rate q in a capillary tube is expressed as:
P ( cos )2DT V + 3) µL02DT pc2DT
q=
23 4DT (DT
(B-3)
The total flow rate in the bundle of capillary tubes is the sum of flow rate in each capillary tube and can be expressed as: n
Q=
qi = i=1
P ( cos )2DT + 3) µL02DT
23 4DT (DT
n i=1
Vi pci2DT
(B-4)
According to Darcy's law, the absolute permeability K is expressed as:
K=
( cos )2DT 23 4DT (DT + 3) AL 02DT
n 1
i=1
Vi pci2DT
(B-5)
When the porous media is filled with a single-phase fluid, the capillary tube volume Vi with the pore radius ri can be replaced with the fluid saturation Si by: (B-6)
Vi = Vp Si where Vp is the total pore volume and is a constant. Submit Eq. (B-6) into Eq. (B-5):
K=
n
( cos )2DT Vp 23 4DT (DT
+
3) AL 02DT 1 i = 1
Si pci2DT
(B-7)
When n approaches infinity, the summation in Eq. (B-7) can be replaced with the integral:
K=
( cos )2DT Vp + 3) AL 02DT
23 4DT (DT
1 1
0
dS pc2DT
(B-8)
Eq. (B-8) is the expression of absolute permeability of porous media. When there is two-phase flowing in porous media, the wetting phase flows in small pores and its permeability Kw is given by:
Kw =
( cos ) 2DT Vp 23 4DT (DT
+
3) AL02DT 1
Sw 0
dS pc2DT
(B-9)
The nonwetting phase flows in large pores and its permeability Knw is given by:
Knw =
( cos )2DT Vp 23 4DT (DT
+
3) AL02DT 1
1 Sw
dS pc2DT
(B-10)
The relative permeability of wetting phase and nonwetting phase Krw , Krnw can be obtained with the following equations respectively:
Krw
K = w = K
Sw 0
dS pc2DT
1 dS 0 p2DT c
(B-11)
91
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F. Wang, et al.
Krnw
K = nw = K
1 dS Sw p 2DT c 1 dS 0 p 2DT c
(B-12)
Mandelbrot, B.B., 1982. The Fractal Geometry of Nature, vol. 2 WH freeman, New York. Matyka, M., Khalili, A., Koza, Z., 2008. Tortuosity-porosity relation in porous media flow. Phys. Rev. 78 (2) 026306. Purcell, W.R., 1949. Capillary pressures-their measurement using mercury and the calculation of permeability therefrom. J. Pet. Technol. 1 (02), 39–48. Reynolds, C.A., Krevor, S., 2015. Characterizing flow behavior for gas injection: relative permeability of CO2‐brine and N2‐water in heterogeneous rocks. Water Resour. Res. 51 (12), 9464–9489. Sahimi, M., 1993. Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65 (4), 1393. Scheidegger, A., 1958. The Physics of Flow through Porous Media. University Of Toronto Press, London. Sun, X., Yao, Y., Ripepi, N., Liu, D., 2018. A novel method for gas–water relative permeability measurement of coal using NMR relaxation. Transp. Porous Media (1241), 73–90. Tan, X.H., Liu, J.Y., Li, X.P., Zhang, L.H., Cai, J., 2015. A simulation method for permeability of porous media based on multiple fractal model. Int. J. Eng. Sci. 95, 76–84. Tao, T.M., Watson, A.T., 1984. Accuracy of JBN estimates of relative permeability: part 1error analysis. Soc. Petrol. Eng. J. 24 (02), 209–214. Wang, F., Li, Y., Tang, X., Chen, J., Gao, W., 2016. Petrophysical properties analysis of a carbonate reservoir with natural fractures and vugs using X-ray computed tomography. J. Nat. Gas Sci. Eng. 28, 215–225. Wang, F., Liu, Z., Jiao, L., Wang, C., Guo, H.U., 2017. A fractal permeability model coupling boundary-layer effect for tight oil reservoirs. Fractals 25 (05), 1750042. Wang, F., Jiao, L., Liu, Z., Tan, X., Wang, C., Gao, J., 2018. Fractal analysis of pore structures in low permeability sandstones using mercury intrusion porosimetry. J. Porous Media 21 (11), 1097–1119. Wang, F., Jiao, L., Lian, P., Zeng, J., 2019. Apparent gas permeability, intrinsic permeability and liquid permeability of fractal porous media: carbonate rock study with experiments and mathematical modelling. J. Pet. Sci. Eng. (173), 1304–1315. Washburn, E.W., 1921. The dynamics of capillary flow. Phys. Rev. (173), 273–283. Xiao, B., Fan, J., Ding, F., 2012. Prediction of relative permeability of unsaturated porous media based on fractal theory and Monte Carlo simulation. Energy Fuels 26 (11), 6971–6978. Xiao, B., Chen, H., Xiao, S., Cai, J., 2017. Research on relative permeability of nanofibers with capillary pressure effect by means of Fractal-Monte Carlo technique. J. Nanosci. Nanotechnol. 17 (9), 6811–6817. Xu, P., Qiu, S., Yu, B., Jiang, Z., 2013. Prediction of relative permeability in unsaturated porous media with a fractal approach. Int. J. Heat Mass Transf. 64, 829–837. Yu, B., Cheng, P., 2002. A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf. 45 (14), 2983–2993. Yu, B., 2005. Fractal character for tortuous streamtubes in porous media. Chin. Phys. Lett. 22 (1), 158–160. Zhang, T., Li, X., Sun, Z., Feng, D., Miao, Y., Li, P., Zhang, Z., 2017. An analytical model for relative permeability in water-wet nanoporous media. Chem. Eng. Sci. 174, 1–12.
References Brooks, R.H., Corey, A.T., 1966. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. 92 (2), 61–90. Burdine, N., 1953. Relative permeability calculations from pore size distribution data. J. Pet. Technol. 5 (3), 71–78. Cai, J., You, L., Hu, X., Wang, J., Peng, R., 2012. Prediction of effective permeability in porous media based on spontaneous imbibition effect. Int. J. Mod. Phys. C 23 (07), 1250054. Cai, J., Luo, L., Ye, R., Zeng, X., Hu, X., 2015. Recent advances on fractal modeling of permeability for fibrous porous media. Fractals 23 (01), 1540006. Carman, P., 1937. Fluid flow through granular beds. Chem. Eng. Res. Des. 75 (1), S32–S48. Clennell, M.B., 1997. Tortuosity: a guide through the maze. Geol. Soc. Lond. Special Publications 122 (1), 299–344. Comiti, J., Renaud, M., 1989. A new model for determining mean structure parameters of fixed beds from pressure drop measurements: application to beds packed with parallelepipedal particles. Chem. Eng. Sci. 44 (7), 1539–1545. Corey, A.T., 1954. The interrelation between gas and oil relative permeabilities. Prod. Mon. 19 (1), 38–41. Egermann, P., Lenormand, R., 2005. A new methodology to evaluate the impact of localized heterogeneity on petrophysical parameters (kr, Pc applied to carbonate rocks. Petrophysics 46 (05). Gates, J.I., Lietz, W.T., 1950. Relative permeabilities of California cores by the capillarypressure method. In: Drilling and Production Practice. American Petroleum Institute. He, C.Z., Hua, M.Q., 1998. Fractal geometry description of reservoir pore structure. Oil Gas Geol. 19 (1), 15–23. Jackson, S.J., Agada, S., Reynolds, C.A., Krevor, S., 2018. Characterizing drainage multiphase flow in heterogeneous sandstones. Water Resour. Res. 54 (4), 3139–3161. Jones, S.C., Roszelle, W.O., 1978. Graphical techniques for determining relative permeability from displacement experiments. J. Pet. Technol. 30 (05), 807–817. Katz, A.J., Thompson, A.H., 1985. Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54 (12), 1325. Krohn, C.E., 1988. Fractal measurements of sandstones, shales, and carbonates. J. Geophys. Res.: Solid Earth 93 (B4), 3297–3305. Li, K., Horne, R.N., 2006. Comparison of methods to calculate relative permeability from capillary pressure in consolidated water‐wet porous media. Water Resour. Res. 42 (6), W06405. Li, K., 2008. A new method for calculating two-phase relative permeability from resistivity data in porous media. Transp. Porous Media 74 (1), 21–33. Li, K., 2010a. More general capillary pressure and relative permeability models from fractal geometry. J. Contam. Hydrol. 111 (1–4), 13–24. Li, K., 2010b. Analytical derivation of Brooks–Corey type capillary pressure models using fractal geometry and evaluation of rock heterogeneity. J. Pet. Sci. Eng. 73 (1–2), 20–26.
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A more generalized model for relative permeability prediction in unsaturated fractal porous media
T
Fuyong Wanga,∗, Liang Jiaoa, Jiuyu Zhaoa, Jianchao Caib,∗∗ a b
Research Institute of Enhanced Oil Recovery, China University of Petroleum, Beijing, 102249, PR China Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan, 430074, PR China
ARTICLE INFO
ABSTRACT
Keywords: Relative permeability Fractal porous media Tortuosity Fractal dimension Capillary pressure
Although lots of models for relative permeability prediction have been developed in literature, the effect of pore geometric tortuosity on relative permeability has not been well discussed, as usually the tortuosity of pores with various sizes is assumed the same. However, there is a high possibility that tortuosity is unequal and increases with the decrease in pore size. A new and more generalized analytical model for relative permeability prediction has been developed based on the fractal characteristics of pore size distribution (PSD) and tortuosity in porous media. The analytical expressions of relative permeability for wetting phase and nonwetting phase have been derived, which are the function of normalized wetting phase saturation, PSD fractal dimension, tortuosity fractal dimension and the ratio of the entry capillary pressure to the maximum capillary pressure. The studies show that the increasing PSD fractal dimension and tortuosity fractal dimension can significantly reduce wetting phase relative permeability, but the increase in nonwetting phase relative permeability is insignificant. The developed model has been validated with relative permeability experimental data and compared with other classic relative permeability models. The comparison results demonstrate the reliability of the relative permeability model developed in this paper.
1. Introduction
of relative permeability of wetting phase and nonwetting phase equals one, which is invalid in most porous media. Li and Horne (2006) found that the relative permeability for wetting phase predicted by the Purcell model well fits the experimental data, but the predicted nonwetting phase relative permeability using the Purcell model is far from experimental results. Based on the Purcell model, Burdine (1953) introduced tortuosity factors to account for the effect of nonuniform fluid distribution on relative permeability. However, Li and Horne (2006), Li (2010a) believed that the tortuosity factor of the wetting phase is no necessary for predicting wetting phase relative permeability. Ever since, lots of modifications have been made on the Purcell model and Burdine model to characterize the multiphase flow behaviour through porous media (Corey, 1954; Brooks and Corey, 1966; He and Hua, 1998; Li, 2008, 2010a; Zhang et al., 2017). With a linear relation between 1/ Pc2 and normalized wetting phase saturation SwD , Corey (1954) developed the Corey relative permeability model based the Burdine approach, and the relative permeability of wetting phase and nonwetting phase is only the function of SwD . Brooks and Corey (1966) proposed a more generalized capillary pressure function where a PSD index was introduced, and then a more generalized relative permeability model was
Relative permeability plays a crucial role in petroleum engineering, geothermal exploration, textile engineering, chemical engineering, and many other areas, as it provides a means to fully characterize the nature of two-phase fluid flow in porous media (Li, 2010a). Relative permeability can be estimated by laboratory experiments (Egermann and Lenormand, 2005; Reynolds and Krevor, 2015; Jackson et al., 2018) and model prediction methods (Tao and Watson, 1984; Li, 2010a; Sun et al., 2018). However, the relative permeability of some unconventional reservoir rock samples cannot be directly obtained by experiments due to ultra-low permeability or complex pore structures (Zhang et al., 2017). Therefore, the methods of predicting relative permeability from mathematical models are widely used. Many models have been proposed in literature to predict relative permeability in porous media. Some classic models have been summarized and shown in Table 1. Purcell (1949) proposed a method for calculating permeability from capillary pressure data, and this method was extended to estimate relative permeability (Gates and Lietz, 1950). However, the Purcell model is too ideal for field application, as the sum
∗
Corresponding author. Corresponding author. E-mail addresses: [email protected] (F. Wang), [email protected] (J. Cai).
∗∗
https://doi.org/10.1016/j.jngse.2019.04.019 Received 29 November 2018; Received in revised form 18 April 2019; Accepted 19 April 2019 Available online 25 April 2019 1875-5100/ © 2019 Elsevier B.V. All rights reserved.
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F. Wang, et al.
Table 1 Several classic relative permeability models in literature. Model
Formula
Purcell model (1949) Burdine model (1953)
Krw =
Krw = (
Corey model (1954) Brooks-Corey model (1966)
rw )
Sw 2 0 dSw / pc Krnw 1 2 0 dSw / pc
Sw dS / p 2 w c 2 0 Krnw 1 2 0 dSw / pc
4 Krw = SwD Krnw = (1
(2 + 3 Krw = SwD
He-Hua model (1998)
)/
11 3Df 3 Df
Krw = SwD Li model (2010a)
Krw =
1 1
Comments
2+ (Swe) 2+
=
1 2 Sw dSw / pc 1 2 0 dSw / pc
=(
rnw )
1 dSw / pc2 2 Sw 1 dS / p2 w c 0
SwD ) 2 (1
2 SwD )
Krnw = (1
SwD ) 2 (1
(2 + SwD
Krnw = (1
SwD ) 2(1
SwD
Krnw =
(Swe )
2+
1
● the sum of Krw and Krnw at a specific saturation equals 1 ● the calculated Krnw is far from the experimental results
)/
5 Df 3 Df
(1
● ● ●
)
)
2+ 2+
●
SwD ) 2
● ● ● ● ● ●
rw
is tortuosity ratio of wetting phase, and
rw
=
Sw Sm ; Sm 1 Sm
is the minimum wetting phase
saturation 1 Sw Sm ; Se is the equilibrium rnw is tortuosity ratio of nonwetting phase, and rnw = 1 Sm Se saturation of the nonwetting phase based on the Burdine model with the assumption of 1/Pc2 = CSwD S S SwD is normalized wetting phase saturation, and SwD = w w c where Swc is the minimum 1 Swc wetting phase saturation. is the PSD index the capillary pressure function can be expressed asPc = pe (SwD ) 1/ where pe is the entry capillary pressure fractal distribution of pore size Df in three-dimensional space is used fractal distribution of tortuosity is not considered p p Swe = 1 bS wD ; b = 1 ( e ) ; = ( e ) pmax
pmax
● In fractal porous media = 3 Df and Df in tree-dimensional space is used ● Swe = 1 bS wD prediction is based on the Purcell approach and Krnw prediction is based on the Burdine approach
derived. The Corey model is only a special case of the Brooks-Corey model when is equal to 2. Fractal theory was firstly proposed by Mandelbrot (1982) and has been widely used to describe complex geometry with self-similarity. Natural porous rocks have been demonstrated to have fractal characteristics (Katz and Thompson, 1985; Krohn, 1988). Numerous scholars have applied fractal theory to predict (relative) permeability of porous media with mathematical modelling and numerical simulation (Yu and Cheng, 2002; Li, 2010b; Cai et al., 2012, 2015; Xiao et al., 2012, 2017; Xu et al., 2013; Tan et al., 2015; Wang et al., 2017). He and Hua (1998) found that the PSD index can be expressed with fractal dimension Df as = 3 Df , and they developed a relative permeability model where the relative permeability of wetting phase and nonwetting phase are only the function of normalized wetting phase saturation and fractal dimension. Li (2010a) developed a more generalized relative permeability model based on fractal assumption of porous media, and he found that the shape of relative permeability is also affected by the ratio of the entry capillary pressure to the maximum capillary pressure besides fractal dimension. Xiao et al. (2012) and Xu et al. (2013) used fractal theory and Monte Carlo simulation to predict relative permeability in unsaturated porous media. Effective prediction of relative permeability of porous media is still challenging due to complex pore structures. One of the characteristics is the path along fluid flowing is not straight, but rather tortuous or meandering. Carman (1937) introduced the concept of tortuosity, and it has been demonstrated that tortuosity has great impact on electrical, hydraulic and diffusive properties of porous media (Scheidegger, 1958; Sahimi, 1993; Clennell, 1997; Matyka et al., 2008). Yu and Cheng (2002) considered fractal distribution of tortuosity and developed a permeability model for fractal porous media. Wang et al. (2019) derived a correlation between flow rate and pressure drop in a single capillary tube with fractal distribution of tortuosity, and found that the increasing tortuosity fractal dimension can significantly reduce permeability of porous media. Although Burdine (1953) introduced the tortuosity factors to modify the Purcell model, the introduced tortuosity factors are used to characterize the flow resistance caused by the nonuniform distribution of fluids rather than the variable geometry tortuosity. The effects of complex distribution of pore geometry tortuosity on relative permeability of unsaturated porous media are unclear. Based on fractal characteristics of PSD and tortuosity in porous media, this paper presents a more generalized analytical model to predict relative permeability in unsaturated porous media. To validate the newly developed
model, experiments of relative permeability measurements with four carbonate core plugs were conducted, and the predicted relative permeability results were compared with the experimental results. Pore structure parameters used for model predicting were derived from highpressure mercury intrusion (HPMI) test. Finally, the factors influencing relative permeability were quantitatively analysed. 2. Model description 2.1. Fractal theory This paper assumes that the porous media is made up of a bundle of tortuous capillary tubes with variable pore sizes and tortuous lengths; the nonwetting phase fluid occupies large pores while the wetting phase fluid flows in small pores; the size and tortuous length of capillary tubes have self-similarity characteristics, and can be described with fractal theory. The correlation between the number of capillary tubes N (a r ) and the size of capillary tubes r can be described as follows (Mandelbrot, 1982):
N (a
r) =
rmax r
Df
(1)
where rmax is the maximum pore diameter; Df is the PSD fractal dimension and its range is 0 < Df < 2 and 0 < Df < 3 in two-dimensional space and three-dimensional space, respectively. The Df in two-dimensional space is used here, as it reflects PSD in the cross-section of a tortuous capillary tube model. Assuming that the straight length of a selected representative unit of porous media is L0 , and the actual length of tortuous capillary tube is LT (r ) , the relationship between LT (r ) and L0 can be described as (Yu and Cheng, 2002):
LT (r ) = (2r )1
DT L DT 0
(2)
where DT is tortuosity fractal dimension, and its range is 1 DT < 2 and 1 DT < 3 in two-dimensional space and three-dimensional space, respectively. Tortuosity fractal dimension DT indicates the heterogeneity of tortuosity distribution. The case of DT = 1 means capillary tubes are straight, and a larger value of DT means the longer capillary tubes (Yu and Cheng, 2002). The flow rate q in a tortuous capillary tube with fractal distribution of tortuosity can be derived from Newton's second law as (Wang et al., 2019): 83
Journal of Natural Gas Science and Engineering 67 (2019) 82–92
F. Wang, et al.
q=
22
P r DT + 3 + 3) µL0DT
DT (D T
Krnw =
(3)
In the porous media with fractal distributions of pore size and tortuosity, a new correlation of wetting phase saturation Sw and capillary pressure pc can be expressed as: 1
Sw 1
Swc Swc
(12)
SwD
In this case, the relative permeability of each phase is the linear function of its own saturation. When DT + Df is less than 3 and pe / pmax 0 , Eqs. (9) and (10) can be simplified as: 3 + DT Df 3 DT Df
(13)
Krw = SwD
3 + DT Df 3 DT Df
Krnw = 1
2.4. Relative permeability model based on the Burdine approach Similar to the Burdine model (Burdine, 1953), tortuosity factors of wetting phase and nonwetting phase are introduced to Eqs. (7) and (8), and the equations for predicting relative permeability of wetting phase and nonwetting phase can be derived as follows: SwD 0
2 Krw = SwD
(6)
dS pc2DT
1 dS 0 p2DT c
where Snwr is the residual nonwetting phase saturation.
Krnw = (1
2.3. Relative permeability model based on the Purcell approach
Sw 0
dS pc2DT
1 dS 0 p 2DT c
Krw =
(7)
1 dS Sw p2DT c
Krnw =
1
(Swe ) 1
1 dS 0 p2DT c
(16)
1
(Swe ) 1 SwD )2
2DT +
2DT +
(17)
(Swe )
2DT +
2DT + 2DT +
(18)
1. For the Eqs. (17) and (18) include an implicit condition that case = 1 which means capillary pressure pc is independent of wetting phase saturation Sw , Eqs. (15) and (16) can be reduced to: (19)
3 Krw = SwD
Krnw = (1
SwD )3
For the case DT + Df < 3 and pe / pmax be simplified as:
2DT +
2DT +
1 dS SwD p 2DT c
1
(8)
The detailed derivation process can be found in Appendix B. Different from the Purcell model presented in Table 1, the tortuosity fractal dimension DT is included. For the case where DT is equivalent to 1, Eqs. (7) and (8) can be simplified as the Purcell model, which means the Purcell model is a special case of the newly developed model. Submitting Eq. (4) into Eqs. (7) and (8), the relative permeability of wetting phase and nonwetting phase can be predicted with:
Krw =
2 SwD
Krnw = (1
1 dS 0 p 2DT c
SwD
)2
(15)
When DT is equivalent to 1, Eqs. (15) and (16) can be simplified to the Burdine model (1953). Submitting Eq. (4) into Eqs. (15) and (16), a generalized relative permeability model with fractal distribution of tortuosity based on the Burdine approach has been derived as follows:
Based on the Purcell approach (Purcell, 1949), a model for predicting relative permeability of wetting phase and nonwetting phase in porous media with fractal distribution of tortuosity has been derived as follows:
Krw =
(14)
SwD
Different from Li model (Li, 2010a), in addition to normalized wetting phase saturation SwD and PSD fractal dimension Df , relative permeability is also affected by tortuosity fractal dimension DT .
(5)
Sw Swc Swc Snwr
1
(11)
Krnw = 1
(4)
where Swc is the minimum wetting phase saturation. For the imbibition case, the normalized wetting phase saturation can be calculated from (Li and Horne, 2006; Li, 2010a):
SwD =
(10)
Krw = SwD
where = 3 DT Df ; b = 1 (pe / pmax ) ; pe is the entry capillary pressure; pmax is the maximum capillary pressure. The detailed derivation process of Eq. (4) can be found in Appendix A. Eq. (4) has the same form with the capillary pressure function proposed by Li (2010a), but in the Li model = 3 Df and pore tortuosity is assumed as constant. In Eq. (4), as fractal distribution of tortuosity is considered and the tortuosity fractal dimension DT is included. It must be noted that the two-dimensional Df should be used in Eq. (4), rather than threedimensional Df used in the Li model (Li, 2010a). When there is residual wetting phase saturation in porous media, the wetting phase saturation Sw can be replaced with the normalized wetting phase saturation SwD . For the displacement process, it can be defined as (Burdine, 1953):
SwD =
2DT +
where Swe = 1 bSwD and = (pe / pmax ) . Eqs. (9) and (10) are a generalized relative permeability model based on the Purcell approach, and it is similar to the model presented by Li (2010a) but the tortuosity fractal dimension DT is here included. When DT is equivalent to 1, Eqs. (9) and (10) can be simplified to the Li model (Li, 2010a). 1. For the There is an implicit condition in Eqs. (9) and (10), i.e. case = 1 , which means pe = pmax and the pore radius is the same, the capillary pressure pc is independent of wetting phase saturation Sw , and then Eqs. (7) and (8) can be reduced to:
2.2. A new capillary pressure model with tortuosity fractal dimension
bSw )
2DT +
1
where P is the pressure difference between two ends of the capillary tube; µ is the fluid viscosity. The detail derivation of Eq. (3) can be found in Appendix B of the paper presented by Wang et al. (2019). Eq. (3) is a modified Hagen-Poiseuille equation for the fluid flow in a capillary tube with fractal distribution of tortuosity.
pc = pmax (1
2DT +
(Swe )
9 DT 3Df 3 DT Df
(9)
Krw = SwD 84
(20)
0 , Eqs. (17) and (18) can
(21)
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Table 2 Porosity and permeability of four carbonate core plugs used in experiments. Core No.
Porosity (%)
Air permeability K air (mD)
Klinkenberg permeability KL (mD)
Water permeability Kw (mD)
Relative permeability experiment
48H 280H 145H 192H
14.564 7.813 18.278 12.325
7.578 1.924 33.899 3.454
4.81 1.177 18.491 2.815
1.11 0.038 24.44 0.60
Gas-water Gas-water Gas-oil Gas-oil
Krnw = (1
SwD )2 1
of these four core plugs is 1.5 inch with about 2 inch long. The porosity of core plugs varies from 7.813% (280H) to 18.278% (145H), with air permeability changes from 1.924 mD to 33.899 mD. Klinkenberg permeability was also measured to correct gas slippage effect. The water permeability is less than the Klingenberg permeability might be caused by the adoption of water (Wang et al., 2019). Unsteady-state method based on Buckley-Leverett displacement theory was used for measuring gas-water/oil relative permeability. Fig. 1 shows the schematic diagram of unsteady-state displacement measurement, and air was used as the gas source. The procedures of gas-water relative permeability measurement are introduced as follows. Firstly, the clean core plugs were fully saturated with formation water, and water permeability was measured. Then the air was injected into the confined core plugs with constant pressure, and meanwhile the gas and water production were monitored and recorded with time. For the gas-oil relative permeability measurement, oil was injected into the water-saturated core plugs until no water was produced. After that air was injected into core plugs, gas and oil production were recorded. Finally, the recorded gas and water/oil production data from displacement experiments were used to calculate the gas-water/oil relative permeability. Johnson, Bossler, and Naumann (JBN) method is an explicit method and has been widely used to calculate relative permeability from displacement experiment based on Buckley-Leverett displacement theory (Tao and Watson, 1984; Sun et al., 2018). However, JBN method needs tedious calculations and errors may be introduced during the process of estimating the derivatives of measured data. Jones and Roszelle (1978) proposed a modified JBN method based on graphical techniques, and Jones-Roszelle method was used to derive relative permeability from displacement experiments in this paper. The derived gas-water/oil relative permeability results are shown in Fig. 2.
3 + DT Df 3 DT Df
SwD
(22)
According to Li and Horne (2006) and Li (2010a), the Purcell relative permeability is the best fit to the wetting phase relative permeability measured from experiments, and the Brooks–Corey model well fits the experimental data of nonwetting phase relative permeability. Therefore, in this paper the wetting phase relative permeability is predicted with Eq. (9), and the nonwetting phase relative permeability is calculated with Eq. (18). The pore structure parameters, including PSD fractal dimension Df , tortuosity fractal dimension DT , and the ratio of the entry capillary pressure to the maximum capillary pressure pe / pmax can be calculated from HPMI test. The detailed calculation processes were demonstrated in the model validation section. 3. Relative permeability experiment and model validation The newly developed relative permeability model has been validated with gas-water/oil relative permeability experiment data measured from four carbonate core plugs (Wang et al., 2016). Then the HPMI tests were conducted to measure pore structure parameters of each core plugs, and the values of Df , DT , and pe / pmax were calculated. Finally, the calculated relative permeability using the newly developed model was verified with the measured relative permeability data from experiments. The model prediction results were also compared with the results calculated from other classic relative permeability models, including the Purcell model, the Burdine model, the Brooks-Corey model and the Corey model. 3.1. The relative permeability experiments
3.2. HPMI tests
The permeability and porosity of these four carbonate core plugs are shown in Table 2. 48H and 280H core plugs were used for gas-water relative permeability measurement, and 145H and 192H core plugs were used for gas-oil relative permeability measurement. The diameter
After displacement experiment, pore structure parameters of the four carbonate core plugs were measured from HPMI tests. The obtained mercury intrusion capillary pressure (MICP) curves are shown in
Fig. 1. Schematic diagram of gas-water/oil relative permeability experiment (1: air supply; 2: pressure gauge; 3: balance chamber; 4: core Holder; 5: quizix pump; 6: valve 7: pressure pump; 8: water/oil volume gauge; 9: gas flow meter; 10: formation water/oil). 85
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Fig. 2. The measured relative permeability from experiments (a: gas-water relative permeability of 48H; b: gas-water relative permeability of 280H; c: gas-oil relative permeability of 145H; gas-oil relative permeability of 192H).
Fig. 3. The pore size and distribution can be derived from capillary pressure with the Washburn Equation (Washburn, 1921) given as: r = 2 cos / Pc . The mercury surface tension is 0.48 N/m, and the contact angle between mercury and air is 140°. The maximum MICP is about 275 MPa, and the corresponding minimum pore radius which can be measured by HPMI is about 2.7 nm. Some pore structure parameters derived from HPMI are shown in Table 3. According to Eq. (1), PSD fractal dimension Df can be calculated from the correlation between capillary tube number N (r ) and capillary tube sizer . The capillary tube number N (r ) can be calculated with the following equation (Li, 2010b):
N (r ) =
VHg (r ) r 2l
(23)
where VHg is the accumulative injected mercury volume when the measured capillary tube radius is r . Plotting the capillary tube number N (r ) versus capillary tube size r in a log-log figure, the PSD fractal dimension Df can be obtained from the slope of the lgN (r ) lgr curve S by Df = S according to Eq. (1). The processes of Df calculation for each core plugs are shown in Fig. 4. The calculated Df reflects the fractal characteristics of PSD in three-dimensional space. However, the Df in two-dimensional space should be used in the developed model. It is generally accepted that the Df in three-dimensional space and twodimensional space differs by one (Wang et al., 2018). Therefore, the obtained Df in Fig. 4 subtracting one was used for relative permeability prediction. Taken the core plug 48H as an example, the slope of the lgN (r ) lgr S is −2.38 and therefore the PSD fractal dimension in three-dimensional space is Df = S = 2.38. Then the PSD fractal dimension in two-dimensional space can be inferred as Df = 2.38 1 = 1.38.
Fig. 3. Plots of MICP curves of four carbonate plugs.
The tortuosity fractal dimension DT can be calculated with the following equation (Yu, 2005):
DT = 1 +
ln ln(L 0 /2r )
(24)
where r is the average pore radius of porous media, and in this paper the media pore radius r50 derived from HPMI is used; ¯ is the average tortuosity, and it can be estimated from HPMI with the following equation (Comiti and Renaud, 1989):
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Table 3 The pore structure parameters calculated from HPMI tests. Core No.
Maximum pore radius rmax (μm)
Median pore radius r50 (μm)
Average tortuosity ¯
PSD fractal dimension (two-dimensional) Df
Tortuosity fractal dimension DT
The value of pe /pmax
48H 280H 145H 192H
0.783 3.676 7.828 14.159
0.326 1.938 1.791 0.558
1.79 2.04 1.70 1.86
1.38 1.14 1.22 1.29
1.22 1.3 1.18 1.13
1/270 1/1200 1/2600 1/4900
Fig. 4. Correlations between lgN (r ) and lgr for four core plugs.
= 1+ 0.41ln(1/ )
(25)
and Corey model were also utilized to predict relative permeability. For the Purcell model, Burdine model, the MICP curves from HPMI test were utilized to predict relative permeability. Compared with other four classic relative permeability models, the predicted relative permeability using our model is much close to the measured relative permeability from the experiments. To quantitatively evaluate the results of relative prediction with different models, the correlation coefficients R2 between predicted values and measured results were calculated, as shown in Table 4. The new model has the best performance in predicting relative permeability of wetting phase and nonwetting phase, and the correlation coefficients R2 are more than 0.98. Brooks-Corey model and Corey model also have satisfying results of relative permeability prediction, and the correlation coefficients R2 are more than 0.93 for both wetting phase and nonwetting phase, and some correlation coefficients close to 1. Purcell model has the poorest performance in predicting nonwetting phase relative permeability, which is far from the experimental results, and the correlation coefficients R2 of the predicted results and experiment results are less than 0.26. Thus, Purcell model is not recommended for predicting the nonwetting phase
The calculated values of Df , DT and pe / pmax of these four core plugs are shown in Table 3. 3.3. Model validation The relative permeability of these four carbonate core plugs were predicted with newly developed model and compared with experimental results. As gas is the nonwetting phase compared with the water and oil, the gas relative permeability was predicted with Eq. (18), and the water/oil relative permeability was predicted with Eq. (9). The relative permeability for wetting phase and nonwetting phase under different normalized wetting phase saturation were calculated and compared with the experiment results. Figs. 5 and 6 present comparisons of predicted gas-water/oil relative permeability using the model developed in this paper and the measured relative permeability. Besides, four classic relative permeability models—Purcell model, Burdine model, Brooks-Corey model
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Fig. 5. The predicted gas-water relative permeability using different models and compared with experimental results for the 48H and 280H core plugs.
Fig. 6. The predicted gas-oil relative permeability using different models and compared with experimental results for the 145H and 192H core plugs.
phase relative permeability prediction by Burdine model are usually worse than Purcell model. It can verify that wetting phase relative permeability prediction based on Purcell approach and nonwetting phase relative permeability prediction based on Burdine model are reasonable (Li and Horne, 2006; Li, 2010a). It must be noted that the limited MICP data can cause the calculation errors of Purcell model and Burdine model. 4. Sensitivity analysis of influencing parameters 4.1. PSD fractal dimension PSD Fractal dimension Df is an important parameter to describe the distribution and heterogeneity of pore structures in porous media. The effects of Df on relative permeability of wetting phase and nonwetting phase were studied. Fig. 7 shows the effect of Df on relative permeability with constant values of DT and pe / pmax . With Df increasing from 1.1 to 1.8, the relative permeability of wetting phase decreases dramatically, but the nonwetting phase relative permeability increases little. In other words, Df has much more effect on relative permeability of wetting phase than nonwetting phase. The reason is that the increasing PSD fractal dimension means the increase proportion of small pores. As wetting phase flows in small pores, the increasing proportion of small pores will decrease the flowability of wetting phase at the same saturation. Besides, according to Eq. (2), the tortuous lengths of capillary tube increase with the decrease in pore size, which also reduces the wetting phase relative permeability.
Fig. 7. The predicted relative permeability with different values of Df for DT = 1.18 and pe / pmax = 1/2600 .
relative permeability. Compared with bad performance of nonwetting phase relative permeability prediction, wetting phase relative permeability prediction by Purcell model is acceptable, and the correlation coefficients are more than 0.59. Compared with predicted oil relative permeability, the performance of water relative permeability prediction is better, and the correlation coefficients are more than 0.81. Burdine model has better performance of nonwetting phase relative permeability prediction than Purcell model, but the performances of wetting
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Table 4 The correlation coefficients of relative permeability predicted from different models. Model
The correlation coefficient R 2 48H
New model Purcell model Burdine model Brooks-Corey model Corey model
280H
145H
192H
Krw
Krnw
Krw
Krnw
Krw
Krnw
Krw
Krnw
0.9942 0.8143 0.7686 0.9655 0.9401
0.9823 0.2248 0.4530 0.9769 0.9746
0.9902 0.9150 0.9232 0.9868 0.9393
0.9959 0.1931 0.2432 0.9929 0.9848
0.9962 0.5936 0.5222 0.9841 0.9994
0.9918 0.1744 0.7522 0.9919 0.9932
0.9977 0.7850 0.7512 0.9880 0.9967
0.9996 0.2545 0.9626 0.9992 0.9935
4.3. The ratio of the entry capillary pressure to the maximum capillary pressure The value of pe / pmax represents the ratio of the smallest pore radius to the maximum pore radius rmin / rmax according to Washburn equation. Fig. 9 shows the effect of pe / pmax on the relative permeability of wetting phase and nonwetting phase with constant values of Df and DT . With pe / pmax decreasing from 0.1 to 0.001, the relative permeability of wetting phase under the same saturation decreases, but the increase in nonwetting phase relative permeability is insignificant. The reason is that the decreasing pe / pmax means the increasing distribution range of the capillary tube size, and wetting phase flowability will be reduced correspondingly. Compared with the work presented by Li (2010a), the novelty of the relative permeability model is the tortuosity fractal dimension DT was introduced in this paper. The assumption that every capillary tube has the same tortuous length in conventional relative permeability models (Corey, 1954; Brooks and Corey, 1966; He and Hua, 1998; Li, 2008, 2010a) might be too ideal for the porous media with complex pore structures, such as tight sandstones/carbonates or shales, and tortuosity fractal dimension DT can provide another parameter to characterize complex pore structures besides PSD fractal dimension Df . When DT is equivalent to 1, our model can be simplified to the Li model (Li, 2010a). Therefore, this newly developed model is a more generalized model for predicting relative permeability in unsaturated fractal porous media.
Fig. 8. The predicted relative permeability with different values of DT for Df = 1.22 and pe / pmax = 1/2600 .
5. Conclusions With the assumption of fractal distributions of pore size and tortuosity, this paper presents a new generalized relative permeability model for unsaturated fractal porous media. The new relative permeability model is the function of the normalized wetting phase saturation SwD , PSD fractal dimension Df , tortuosity fractal dimension DT and the ratio of entry capillary pressure to maximum capillary pressure pe / pmax . With the fractal parameters derived from HPMI tests, the model prediction results were verified with gas-water/oil relative permeability experiments. PSD fractal dimension Df and tortuosity fractal dimension DT have significant influence on relative permeability of wetting phase. The increasing values of Df and DT can significantly reduce relative permeability of wetting phase, but the effect on nonwetting phase relative permeability is insignificant. In addition to Df and DT , the shapes of relative permeability curves are also affected by pe / pmax . The decreasing values of pe / pmax can reduce the relative permeability of wetting phase, but the increase in nonwetting phase relative permeability is insignificant.
Fig. 9. The predicted relative permeability with different values of pe / pmax for Df = 1.22 and DT = 1.18.
4.2. Tortuosity fractal dimension Tortuosity fractal dimension DT reflects the distribution of tortuosity with different pore size. When DT equals 1, the tortuosity of each capillary tube is the same. Increasing DT means the tortuosity of the capillary tube with small radius increases dramatically according to Eq. (2). Fig. 8 shows the effect of DT on relative permeability with constant values of Df and pe / pmax . The increasing DT can significantly reduce relative permeability of wetting phase, but the increase in relative permeability of the nonwetting phase is insignificant. The reason is that the increasing DT means the increase in tortuous length of small pores according to Eq. (2), which will increase resistance of wetting phase flow. As nonwetting phase flows in large pores, whose tortuous length changes little, the increase in nonwetting phase relative permeability is insignificant.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 51604285, 51874320, 41722403), and Scientific Research Foundation of China University of Petroleum, Beijing (No. 2462017BJB11). 89
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Nomenclature cross-sectional area constant PSD fractal dimension tortuosity fractal dimension pressure difference capillary pressure entry capillary pressure maximum capillary pressure flow rate absolute permeability nonwetting phase relative permeability wetting phase relative permeability gas phase relative permeability water phase relative permeability representative length tortuous length of capillary tubes pore radius average pore radius the median pore radius the largest pore radius the minimum pore radius capillary tube number mercury saturation irreducible water saturation residual nonwetting phase saturation wetting phase saturation normalized wetting phase saturation capillary volume all the volumes of capillary tubes constant constant porosity mercury interfacial tension, and = 0.48N/m contact angle between rock and mercury, and = 140° average tortuosity
A b Df DT P Pc Pe Pmax q K Krnw Krw Krg Krw L0 LT r r r50 rmax rmin N (r ) SHg Scw Snwr Sw SwD Vi Vp
¯
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jngse.2019.04.019. Appendix A. Derivation of a new capillary pressure model with tortuosity fractal dimension In a capillary tube model with fractal distributions of PSD and tortuosity, the total pore volume can be expressed as: rmax
Vp =
r 2LT (r ) dN
rmin
(A-1)
Differentiate Eq. (1), and the number of pores with size varying from r and r+ dr can be expressed as (Yu and Cheng, 2002): D
f dN = Df rmax r
(A-2)
(Df + 1) dr
Submitting Eq. (A-2) and Eq. (2) into Eq. (A-1), the total pore volume Vp can be expressed as:
Vp =
21
DT L DT 0
3
D
f Df rmax
DT
Df
3 D
(rmax T
Df
3 DT Df
rmin
)
(A-3)
The pore volume with pore radius less than r can be expressed as:
V ( < r) =
21
DT L DT 0
3
DT
D
f Df rmax
Df
(r 3
DT Df
3 DT Df
rmin
)
(A-4)
The wetting phase saturation can be expressed as:
Sw =
r 3 DT V = 3 DT Vp rmax
3 DT Df
Df
rmin
Df
rmin
3 DT Df
(A-5)
Replace pore radius r with capillary pressure Pc using Eq. (B-2), and Eq. (A-5) becomes: 90
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Sw =
pc
(3 DT Df )
(3 pmax
pe
(3 DT Df )
(3 DT Df ) pmax
DT Df )
(A-6)
where pe is the entry pressure, corresponding to the maxim pore radiusrmax . Assuming
Sw =
pc
pmax
pe
pmax
=3
DT
Df , Eq. (A-6) can be expressed as: (A-7)
Rearrange Eq. (A-6), and expression of pc becomes:
pc = [pmax
(pmax
( ) pe
Assume b = 1
pc = pmax (1
1
pe ) Sw]
(A-8)
, Eq. (A-8) can be rearranged into the similar form with the Li model (2010a):
pmax
1
bSw )
(A-9)
Appendix B. Derivation of a modified Purcell model with tortuosity fractal dimension The volume of a capillary tube with radius r is given as:
V = r 2LT =21
DT
r3
(B-1)
DT L DT 0
The radius r can be obtained from mercury intrusion test with Washburn Equation as (Washburn, 1921):
2 cos pc
r=
(B-2)
where pc is the mercury capillary pressure; is the mercury interfacial tension; is the contact angle between the rock and the mercury. Substituting Eqs. (B-1) and (B-2) into Eq. (3), the flow rate q in a capillary tube is expressed as:
P ( cos )2DT V + 3) µL02DT pc2DT
q=
23 4DT (DT
(B-3)
The total flow rate in the bundle of capillary tubes is the sum of flow rate in each capillary tube and can be expressed as: n
Q=
qi = i=1
P ( cos )2DT + 3) µL02DT
23 4DT (DT
n i=1
Vi pci2DT
(B-4)
According to Darcy's law, the absolute permeability K is expressed as:
K=
( cos )2DT 23 4DT (DT + 3) AL 02DT
n 1
i=1
Vi pci2DT
(B-5)
When the porous media is filled with a single-phase fluid, the capillary tube volume Vi with the pore radius ri can be replaced with the fluid saturation Si by: (B-6)
Vi = Vp Si where Vp is the total pore volume and is a constant. Submit Eq. (B-6) into Eq. (B-5):
K=
n
( cos )2DT Vp 23 4DT (DT
+
3) AL 02DT 1 i = 1
Si pci2DT
(B-7)
When n approaches infinity, the summation in Eq. (B-7) can be replaced with the integral:
K=
( cos )2DT Vp + 3) AL 02DT
23 4DT (DT
1 1
0
dS pc2DT
(B-8)
Eq. (B-8) is the expression of absolute permeability of porous media. When there is two-phase flowing in porous media, the wetting phase flows in small pores and its permeability Kw is given by:
Kw =
( cos ) 2DT Vp 23 4DT (DT
+
3) AL02DT 1
Sw 0
dS pc2DT
(B-9)
The nonwetting phase flows in large pores and its permeability Knw is given by:
Knw =
( cos )2DT Vp 23 4DT (DT
+
3) AL02DT 1
1 Sw
dS pc2DT
(B-10)
The relative permeability of wetting phase and nonwetting phase Krw , Krnw can be obtained with the following equations respectively:
Krw
K = w = K
Sw 0
dS pc2DT
1 dS 0 p2DT c
(B-11)
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Krnw
K = nw = K
1 dS Sw p 2DT c 1 dS 0 p 2DT c
(B-12)
Mandelbrot, B.B., 1982. The Fractal Geometry of Nature, vol. 2 WH freeman, New York. Matyka, M., Khalili, A., Koza, Z., 2008. Tortuosity-porosity relation in porous media flow. Phys. Rev. 78 (2) 026306. Purcell, W.R., 1949. Capillary pressures-their measurement using mercury and the calculation of permeability therefrom. J. Pet. Technol. 1 (02), 39–48. Reynolds, C.A., Krevor, S., 2015. Characterizing flow behavior for gas injection: relative permeability of CO2‐brine and N2‐water in heterogeneous rocks. Water Resour. Res. 51 (12), 9464–9489. Sahimi, M., 1993. Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65 (4), 1393. Scheidegger, A., 1958. The Physics of Flow through Porous Media. University Of Toronto Press, London. Sun, X., Yao, Y., Ripepi, N., Liu, D., 2018. A novel method for gas–water relative permeability measurement of coal using NMR relaxation. Transp. Porous Media (1241), 73–90. Tan, X.H., Liu, J.Y., Li, X.P., Zhang, L.H., Cai, J., 2015. A simulation method for permeability of porous media based on multiple fractal model. Int. J. Eng. Sci. 95, 76–84. Tao, T.M., Watson, A.T., 1984. Accuracy of JBN estimates of relative permeability: part 1error analysis. Soc. Petrol. Eng. J. 24 (02), 209–214. Wang, F., Li, Y., Tang, X., Chen, J., Gao, W., 2016. Petrophysical properties analysis of a carbonate reservoir with natural fractures and vugs using X-ray computed tomography. J. Nat. Gas Sci. Eng. 28, 215–225. Wang, F., Liu, Z., Jiao, L., Wang, C., Guo, H.U., 2017. A fractal permeability model coupling boundary-layer effect for tight oil reservoirs. Fractals 25 (05), 1750042. Wang, F., Jiao, L., Liu, Z., Tan, X., Wang, C., Gao, J., 2018. Fractal analysis of pore structures in low permeability sandstones using mercury intrusion porosimetry. J. Porous Media 21 (11), 1097–1119. Wang, F., Jiao, L., Lian, P., Zeng, J., 2019. Apparent gas permeability, intrinsic permeability and liquid permeability of fractal porous media: carbonate rock study with experiments and mathematical modelling. J. Pet. Sci. Eng. (173), 1304–1315. Washburn, E.W., 1921. The dynamics of capillary flow. Phys. Rev. (173), 273–283. Xiao, B., Fan, J., Ding, F., 2012. Prediction of relative permeability of unsaturated porous media based on fractal theory and Monte Carlo simulation. Energy Fuels 26 (11), 6971–6978. Xiao, B., Chen, H., Xiao, S., Cai, J., 2017. Research on relative permeability of nanofibers with capillary pressure effect by means of Fractal-Monte Carlo technique. J. Nanosci. Nanotechnol. 17 (9), 6811–6817. Xu, P., Qiu, S., Yu, B., Jiang, Z., 2013. Prediction of relative permeability in unsaturated porous media with a fractal approach. Int. J. Heat Mass Transf. 64, 829–837. Yu, B., Cheng, P., 2002. A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf. 45 (14), 2983–2993. Yu, B., 2005. Fractal character for tortuous streamtubes in porous media. Chin. Phys. Lett. 22 (1), 158–160. Zhang, T., Li, X., Sun, Z., Feng, D., Miao, Y., Li, P., Zhang, Z., 2017. An analytical model for relative permeability in water-wet nanoporous media. Chem. Eng. Sci. 174, 1–12.
References Brooks, R.H., Corey, A.T., 1966. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. 92 (2), 61–90. Burdine, N., 1953. Relative permeability calculations from pore size distribution data. J. Pet. Technol. 5 (3), 71–78. Cai, J., You, L., Hu, X., Wang, J., Peng, R., 2012. Prediction of effective permeability in porous media based on spontaneous imbibition effect. Int. J. Mod. Phys. C 23 (07), 1250054. Cai, J., Luo, L., Ye, R., Zeng, X., Hu, X., 2015. Recent advances on fractal modeling of permeability for fibrous porous media. Fractals 23 (01), 1540006. Carman, P., 1937. Fluid flow through granular beds. Chem. Eng. Res. Des. 75 (1), S32–S48. Clennell, M.B., 1997. Tortuosity: a guide through the maze. Geol. Soc. Lond. Special Publications 122 (1), 299–344. Comiti, J., Renaud, M., 1989. A new model for determining mean structure parameters of fixed beds from pressure drop measurements: application to beds packed with parallelepipedal particles. Chem. Eng. Sci. 44 (7), 1539–1545. Corey, A.T., 1954. The interrelation between gas and oil relative permeabilities. Prod. Mon. 19 (1), 38–41. Egermann, P., Lenormand, R., 2005. A new methodology to evaluate the impact of localized heterogeneity on petrophysical parameters (kr, Pc applied to carbonate rocks. Petrophysics 46 (05). Gates, J.I., Lietz, W.T., 1950. Relative permeabilities of California cores by the capillarypressure method. In: Drilling and Production Practice. American Petroleum Institute. He, C.Z., Hua, M.Q., 1998. Fractal geometry description of reservoir pore structure. Oil Gas Geol. 19 (1), 15–23. Jackson, S.J., Agada, S., Reynolds, C.A., Krevor, S., 2018. Characterizing drainage multiphase flow in heterogeneous sandstones. Water Resour. Res. 54 (4), 3139–3161. Jones, S.C., Roszelle, W.O., 1978. Graphical techniques for determining relative permeability from displacement experiments. J. Pet. Technol. 30 (05), 807–817. Katz, A.J., Thompson, A.H., 1985. Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54 (12), 1325. Krohn, C.E., 1988. Fractal measurements of sandstones, shales, and carbonates. J. Geophys. Res.: Solid Earth 93 (B4), 3297–3305. Li, K., Horne, R.N., 2006. Comparison of methods to calculate relative permeability from capillary pressure in consolidated water‐wet porous media. Water Resour. Res. 42 (6), W06405. Li, K., 2008. A new method for calculating two-phase relative permeability from resistivity data in porous media. Transp. Porous Media 74 (1), 21–33. Li, K., 2010a. More general capillary pressure and relative permeability models from fractal geometry. J. Contam. Hydrol. 111 (1–4), 13–24. Li, K., 2010b. Analytical derivation of Brooks–Corey type capillary pressure models using fractal geometry and evaluation of rock heterogeneity. J. Pet. Sci. Eng. 73 (1–2), 20–26.
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