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Real gas transport in shale matrix with fractal structures
Fuel 219 (2018) 353–363
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Full Length Article
Real gas transport in shale matrix with fractal structures a
a,⁎
a
b
a
c
a,⁎
T
Jinze Xu , Keliu Wu , Ran Li , Zhandong Li , Jing Li , Qilu Xu , Zhangxin Chen a b c
Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada College of Petroleum Engineering Institute, Northeast Petroleum University, Daqing, Heilongjiang 163318, China China University of Geosciences-Beijing, School of Earth Sciences and, Beijing 100083, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Shale gas Pore size distribution Fractal structure Shale matrix
A real gas transport model in shale matrix with fractal structures is established to bridge a pore size distribution and multiple transport mechanisms. This model is well validated with experiments. Results indicate that different pore size distributions lead to various transport efficiencies of shale matrix. A larger fractal dimension of the pore size and a smaller minimum pore size yield higher frequency of occurrence of small pores and a lower free gas transport ratio, which further results in lower transport efficiency. Gas transport efficiency due to pore size distribution parameters (a fractal dimension and a minimum pore size) varies with different porosities and pressures. Increasing fractal dimension and decreasing minimum pore size result in a higher contribution of Knudsen diffusion to the total gas transport. Decreased pressure and increased porosity enhance the sensitivity of gas transport efficiency to a pore size distribution. The relationship between apparent permeability and porosity based on different pore size distributions is also established for industrial application.
1. Introduction
Fractal model is an effective method to describe the pore size distribution, and advantages are concluded as follows: (1) the fractal theory has been an effective method to study a wide range of pore size distributions in experimental studies. Clarkson et al. (2013) [13] studied the pore size data from currently active shale gas plays including the Barnett, Marcellus, Haynesville, Eagle Ford, Woodford, Muskwa, and Duvernay shales. Their study indicated a fractal geometry (power law scattering) for a wide range of pore sizes. Further studies by Lee et al. (2014) [14], Bahadur et al. (2015) [15] and Wang et al. (2015) [16] also showed that the fractal geometry is suitable to describe a pore size distribution for Baltic Basin in United Kingdom, Second White Specks and Belle Fourche formations in Canada, and Songliao Basin in China; (2) fractal theory is able to bridge the pore size distribution and gas transport mechanisms based on link of porosity. Effects of pore size distribution on gas transport behavior and efficiency can thus be studied. We thus select fractal model as part of our models. In industry, different kinds of experimental methods may be performed to test the porosity of shale samples[17]. Mercury intrusion (MICP), low-pressure adsorption, and small-angle and ultra-small-angle neutron scattering (SANS/USANS) techniques have been wide used to measure porosity in experiments. MICP cannot access the porosity with pore diameter less than 3 nm in shale gas reservoir, and the risk to damage the pore structure exists. Low-pressure adsorption method must be combined with MICP to investigate the pore size spectrum due to the variable access of the probe molecules. SANS/USANS is able to detect a
As one of the clean energy resources, shale gas significantly reduces greenhouse gas emissions [1]. Owning to technological advancements in horizontal drilling and hydraulic fracturing, the shale gas is developing very fast in North America [2], which has received much attention from all over the world. However, there are still many challenges regarding the development of shale gas sources [3]. For instance, the gas transport behavior varies considerably in pores of shale matrix [4]. Studies on gas transport in shale matrix indicate the mechanisms of the transport behavior and provide a basis for evaluating well productivity, which contributes to the research on the development of shale gas resources. Shale rocks are formed by compaction and solidification under high pressure in deep underground over a long time [5]. The shale matrix consisting of pores exhibits low porosity (1%–15%) and low permeability (nanodarcy to microdarcy) [6]. Chalmers et al. (2012) [7] investigated a range of pore sizes based on samples from the Barnett, Marcellus, Woodford, and Haynesville gas shales in the United States and the Doig Formation of northeastern British Columbia, Canada. They claimed that the pore size in shale matrix ranges from 1 nm to 100 µm. Experimental observations reported by Utpalendu and Prasad (2013) [8], Ye et al. (2015) [9], Zhang et al. (2015) [10], Chalmers and Bustin (2017) [11] and Wu et al. (2017) [12] also indicated a wide range of pore size distributions in shale matrix.
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Corresponding authors. E-mail addresses: [email protected] (K. Wu), [email protected] (Z. Chen).
https://doi.org/10.1016/j.fuel.2018.01.114 Received 18 October 2017; Received in revised form 2 January 2018; Accepted 26 January 2018 0016-2361/ © 2018 Elsevier Ltd. All rights reserved.
Fuel 219 (2018) 353–363
J. Xu et al.
Nomenclature
Vm Z
molar volume, m3/mol ; gas compressibility factor, dimensionless.
Roman Symbols Greek Symbols
At Ap C Dp Dτ dp dm Fp fp ΔH H () KA k L Lt M Np NA n o1 o2 o3 P Pc PL Pr Q q T Tc Tr
cross-sectional area of the sample, m2 ; total cross-sectional area of pores, m2 ; conductance, (mol·m)/(Pa·s) ; fractal dimension of the pore size, dimensionless; fractal dimension of the tortuosity, dimensionless; equivalent pore diameter, m ; gas molecular diameter, m ; cumulative probability of pore size distribution, dimensionless; probability density function of pore size; isosteric adsorption heat at θ = 0 , J/mol ; Heaviside step function; apparent permeability, m2 ; Boltzmann constant, = 1.38 × 10−23J/K ; straight length of the pore, m ; tortuous length of the pore, m ; gas molecular weight, kg/mol ; total number of pores with the equivalent pore diameter larger than dp in shale matrix, dimensionless; Avogadro constant, = 6.02 × 1023mol−1; molecular number density, m−3 ; fitting coefficient, = 7.9; fitting coefficient, = 9 × 10−6 ; fitting coefficient, = 0.28; pressure, Pa ; critical pressure, Pa ; Langmuir pressure, Pa ; P reduced pressure, dimensionless, = P ; c transport molar rate, mol/s ; transport molar rate in a single pore, mol/s ; temperature, K ; critical temperature, K ; T reduced temperature, dimensionless, = T ;
α β γ χ Ψ Γ κ ω ξ ϕ Ds θ ρ μ μ0 η
function of gas properties and temperature in the SRK EOS, dimensionless; fracture dip, dimensionless; aspect ratio, dimensionless; repulsion parameter in the SRK EOS, (Pa·m3)/mol ; ratio between the blockage rate constant and the forward migration rate constant, dimensionless; gas transport ratio, dimensionless; attraction parameter in the SRK EOS, m3/mol ; acentric factor, dimensionless; taper ratio, dimensionless; porosity, dimensionless; surface diffusion coefficient, m2/s ; gas coverage, dimensionless; gas density, kg/m3; gas viscosity, Pa·s ; gas viscosity at P = 1.01325 × 105 Pa and T = 423 K, = 2.31 × 10−5Pa·s ; ratio between dpmin and dpmax in shale matrix, dimensionless, = dpmin/ dpmax .
Subscripts
a b p min max k s t v
adsorbed gas; free gas; pore; minimum; maximum; Knudsen diffusion; surface diffusion; total; viscous flow.
cz
production optimization. Many models have been proposed in the past decade to study the gas transport behavior in shale matrix. Javadpour (2009) [18] and Rahmanian et al. (2013) [19] proposed an ideal free gas transport model based on the sum of viscous flow and Knudsen diffusion in single pores. Sheng et al. (2015) [20] further took the surface diffusion into
wide range of pore sizes and good to be used at reservoir pressure and temperature conditions [17]. The theoretical relationship between porosity and permeability based on pore size distribution benefits in improving efficiency of estimating porosity or permeability. The mechanisms of change of transport efficiency with pore size distribution and porosity are also need to be indicated to better understand
Table 1 Review of existing gas transport models in shale rocks [18–25]. Model
Description
Comments
Javadpour, 2009 [18]
Ideal gas EOS; Linear superposition of viscous flow and Knudsen diffusion based on the slip boundary condition Ideal gas EOS; Weighted superposition of viscous flow and Knudsen diffusion based on slip boundary condition Ideal gas EOS; Weighted superposition of viscous flow, Knudsen diffusion and surface diffusion based on slip boundary condition Ideal gas transport model for a bundle of pores in fractal rocks
Only for ideal gas in single pores; Neither adsorption nor surface diffusion Only for ideal gas in single pores; Neither adsorption nor surface diffusion Only for ideal gas in single pores
Rahmanian et al., 2013 [19] Sheng et al., 2015 [20] Miao et al., 2015 [22] Li et al., 2016 [23] Wu et al., 2016 [21]
Ren et al., 2016 [24]
Xu et al., 2017 [25]
Consideration of real gas effect based on empirical equation of methane; Weighted superposition of viscous flow, Knudsen diffusion and surface diffusion based on the slip boundary condition Consideration of real gas effect based on empirical equation of methane; Linear supervision of viscous flow and Knudsen diffusion based on non-slip boundary Real gas transport for tapered non-circular pores in shale rocks
354
Only for ideal gas; Knudsen diffusion and surface diffusion are not considered Empirical equation to describe real gas effect; single pores (elliptical cross section is omitted); Compressibility factor is always larger than 1 which is not real under low pressure Empirical equation to describe real gas effect; Single pores; Neither adsorption nor surface diffusion Only apply for single pores
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where Npt is the total number of pores, dimensionless; dpmin is the minimum equivalent diameter of pores, m . If we differentiate Eq. (1) on both sides and couple it with Eq. (2), the following equation can be obtained:
consideration and the adsorbed gas was considered in their models. Wu et al. (2016) [21] further considered the real gas effect and the surface diffusion in the gas transport behavior in single pores. Miao et al. (2015) [22] applied the fractal geometry theory to the expression of viscous flow in shale rocks, which can reveal more mechanisms of seepage characteristics than traditional models. Li et al. (2016) [23] showed that the results obtained from a fractal model with viscous flow in shale rocks can well match those from numerical simulation. Ren et al. (2016) [24] included the slippage into the Knudsen diffusion for a model of free gas transport. Xu et al. (2017) [25] proposed a real gas transport model in tapered noncircular nanopores in shale matrix. However, there are still many limitations for existing models as summarizing as Table 1 [18–25]. The previous fractal models in shale matrix are limited to ideal gas transport under a low Knudsen number (viscous flow), and the impacts of a pore size distribution on gas transport behavior are not studied. A relationship between permeability and porosity based on different pore size distributions in shale matrix also needs to be established for industrial applications [26–28]. In order to better understand the gas transport behavior in shale matrix, an analytical model based on the fractal theory is proposed to bridge the pore size distribution and multiple transport mechanisms (viscous flow, Knudsen diffusion and surface diffusion). The proposed model is well validated with the available experimental data, and effects of pore size distribution on gas transport efficiency are further revealed.
−
Npt
−(Dp + 1)
D
p dp = Dp dpmin
d(dp) = fp d(dp)
dp
⎧ ∫d
pmin
Fp =
fp d (dp),dp ⩾ dpmin
⎨ 0,dp < dpmin ⎩
(4)
where Fp is the cumulative probability of pores with their equivalent pore diameter less than dp. By coupling Eqs. (3) and (4), we can have the expression of Fp as follows:
( )
D
p dpmin ⎧ ,dp ⩾ dpmin ⎪1− dp Fp = ⎨ 0,d < d pmin ⎪ p ⎩
(5)
The total cross-sectional area of pores (Ap ) is calculated as follows:
Ap = −
∫d
dpmax
pmin
πdp2 4
dNp
(6)
m2 .
where Ap is the total cross-sectional area of pores, By coupling Eqs. (1) and (6), Ap is expressed as follows:
The proposed model for gas transport in shale matrix is developed based on the fractal theory, which accounts for a pore size distribution and different transport mechanisms as shown in Fig. 1. Generally, the classical Knudsen number is employed to divide the flow regime as continuum flow (Kn < 0.001), laminar slip flow (0.001 < Kn < 0.1), transition flow (0.1 < Kn < 10), and free-molecular flow (Kn > 10). However, considering the existence of adsorbed gas transport in shale matrix, classical flow regime classification cannot be well applied in the real gal transport in shale matrix [3]. Both transport mechanisms of the free gas transport (viscous flow and Knudsen diffusion) and the adsorbed gas transport (surface diffusion) are taken into consideration in a bundle of pores in this study. Surface diffusion exists in the pore, which happens in the adsorbed layer formed on walls of pores. For the single adsorbed layer, we apply Langmuir’s law with considering real gas effect to calculate the gas coverage (θ) for the adsorbed gas. (see Appendix B). In this study, we assume instant equilibrium of the gas exchange between adsorption and desorption in pores [25]. This model upscale the gas transport model from pore scale to reservoir scale based on pore size distribution in fractal shale matrix. The boundary condition of the proposed analytical model refers to constant pressure. Physical properties of different gases are calculated based on Soave-Redlich-Kwong (SRK) equation of state (EOS). Detailed equations and procedures are shown in Appendix B. For a fractal shale matrix, the total number of pores (Np) with an equivalent pore diameter larger than dp is as follows (see Appendix A):
Ap =
2 Dp (1−η2 − Dp) πdpmax
2−Dp
4
(7)
where η is the ratio between dpmin and dpmax in shale matrix, dimensionless, = dpmin/ dpmax . Previous experimental studies by Clarkson et al. (2011) [17], Yan et al. (2017) [29] and Pang et al. (2017) [30] indicate the effective porosity dominates the gas transport efficiency in shale matrix and the sample with highest ratio of effective porosity owns the highest permeability. Existing models [18–25] with focus on effective porosity have also shown a convincing ability to deal with gas transport behavior and efficiency in shale rocks. This study is thus focused on gas transport in connected pores, and we only consider the effective porosity in the model. The porosity of shale rock (ϕ) is thus calculated as follows:
ϕ=
Ap (8)
At
where At is the cross-sectional area of the core, m2 .
Dp
(1)
where Np is the total number of pores with their equivalent pore diameter larger than dp , dimensionless; dp is the equivalent diameter of pores in shale matrix, m , (see Appendix B); dpmax is the maximum equivalent diameter of pores, m ; Dp is the fractal dimension of pore sizes, dimensionless. Let dp = dpmin ; we can have the expression of the total number of pores (Npt ) in shale matrix as follows:
dpmax ⎞ Npt = ⎜⎛ ⎟ ⎝ dpmin ⎠
(3)
where fp is the probability density function of pore sizes. The cumulative probability Fp is as follows:
2. Model establishment
dpmax ⎞ Np = ⎜⎛ ⎟ ⎝ dp ⎠
dNp
Dp
Fig. 1. Schematic view of the model.
(2) 355
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By coupling Eqs. (10) and (11), we can have the expressions for the total transport rates as follows:
The tortuous length of pores (Lt ) is calculated as follows [22,23]:
Lt =
LDτ dp1 − Dτ
(9)
where Lt is the tortuous length of pores, L is the straight length of pores, Dτ is the fractal dimension of tortuosity. Based on previous studies, three well-known types of diffusion (Knudsen diffusion, molecular diffusion and surface diffusion) exist in shale matrix. When the mean free path of gas molecules is comparable to the pore dimension in shale matrix, Knudsen diffusion, which is a typical kind of free-molecule diffusion becomes important. The Knudsen diffusion is dominated by molecule-wall collisions. Generally, the Knudsen diffusion is significant only at low pressures and small pore diameters. Molecular diffusion is dominated by molecule-molecule collisions. Concentration gradient is the driving force, and the Fick’s first law of diffusion is often applied to model this process. According to the experimental investigation on the characteristics of gas diffusion by Kim et al. in 2015 [31], the molecular diffusion coefficient remains 5.068 × 10−4 cm2/s while the Knudsen diffusion coefficient ranges from 0.02 to 1.252 cm2/s for nanopores with pore radius of 5–300 nm in shale rocks. Experiment performed by Javadpour et al. (2007) [32] shows that the Knudsen diffusion coefficient is 0.04 cm2/s and the molecular diffusion coefficient in kerogen bulk is 2 × 10−6 cm2/s for the nanopore with the radius of 35 nm. Thus we select the Knudsen diffusion as the main free gas diffusion mechanism in the present model. The total flow rates of viscous flow, Knudsen diffusion and surface diffusion can be obtained as follows [22,23]:
Qv = −
∫d
dpmax
Qk = −
∫d
dpmax
Qs = −
∫d
pmin
qv dNp
(10-a)
qk dNp
(10-b)
qs dNp
(10-c)
pmin
dpmax
pmin
Qv =
qv =
qk =
− 43
( ) 2ξ 2 ε+1
( )
3 + Dτ πϕb ρdpmax ΔP + ξ + 1 128μMLDτ − 1 L
Qk =
2 + Dτ Dp (1−η2 − Dp + Dτ ) πϕb vdpmax ΔP 2−Dp + Dτ 12ZRTLDτ − 1 L
Qs =
1 + Dτ Dp (1−η1 − Dp + Dτ ) ⎛ 2ξ 2 ⎞− 3 ϕa θDs dpmax ΔP ξ⎜ ⎟ 3 Dτ − 1 L 1−Dp + Dτ ⎝ ξ + 1 ⎠ NA dm PL
(12-a)
(12-b)
(12-c)
(13)
Qb = Q v + Qk
where Qb is the total free gas transport molar rate, mol/ s . Surface diffusion only occurs in the adsorbed gas, and the total adsorbed gas transport molar rate (Qa ) is thus calculated as follows: (14)
Qa = Qs
where Qa is the total adsorbed gas transport molar rate, mol/s . The total gas transport molar rate (Qt ) is calculated as follows: (15)
Qt = Qa + Qb
where Qt is the total gas transport molar rate, mol/ s .If the molar rate is divided by the pressure gradient, the viscous flow conductance (Cv ) , Knudsen diffusion conductance (Ck ) , surface diffusion conductance (Cs ) , adsorbed gas transport conductance (Ca) , free gas transport conductance (Cb) , and total gas transport conductance (Ct ) in the fractal shale matrix can be obtained as follows:
Cv =
Dp
(1−η3 − Dp + Dτ )
3−Dp + Dτ
3ξ 3 ξ2
− 43
( ) 2ξ 2 ε+1
3 + Dτ πϕb ρdpmax
+ ξ + 1 128μMLDτ − 1
Ck =
2 + Dτ Dp (1−η2 − Dp + Dτ ) πϕb vdpmax ΔP 2−Dp + Dτ 12ZRTLDτ − 1 L
Cs =
1 + Dτ Dp (1−η1 − Dp + Dτ ) ⎛ 2ξ 2 ⎞− 3 ϕa θDs dpmax ΔP ξ⎜ ⎟ 3 Dτ − 1 L 1−Dp + Dτ 1 ξ + ⎝ ⎠ NA dm PL
(16-a)
(16-b)
2
(11-a)
πϕb vdp3 ΔP (11-b)
12ZRT Lt −2 3
ξ2
− 43
2ξ 2 ε+1
We apply the approach of linear superposition of viscous flow, Knudsen diffusion and surface diffusion for gas transport behavior, which has been well validated with simulation and experimental data in previous studies [24,25]. As the Knudsen diffusion and viscous flow only exist in the free gas, the total free gas transport molar rate (Qb ) is calculated as follows:
πρϕb dp4
ΔP ξ 2 + ξ + 1 128μM Lt
2ξ 2 ⎞ qs = ξ ⎜⎛ ⎟ ξ ⎝ + 1⎠
3−Dp + Dτ
3ξ 3
2
where qv is the viscous flow molar rate in a single pore, mol/s ; qk is the Knudsen diffusion molar rate in a single pore, mol/s ; qs is the surface diffusion molar rate in a single pore, mol/s ; Q v is the total viscous flow molar rate, mol/s ; Qk is the total Knudsen diffusion molar rate, mol/s ; Qs is the total surface diffusion molar rate, mol/s . Considering the tapering effect, the molar rates for the viscous flow flux, Knudsen diffusion flux and surface diffusion flux in a single nanopore are as follows [25]:
3ξ 3
Dp
(1−η3 − Dp + Dτ )
θϕa Ds dp2 ΔP NA dm3 P Lt
(16-c)
Ca = Cs
(16-d)
Cb = Cv + Ck
(16-e)
Ct = Ca + Cb
(16-f)
where Cv is the viscous flow conductance, (mol·m)/(Pa·s ) ; Ck is the Knudsen diffusion conductance, (mol·m)/(Pa·s ) ; cs is the surface diffusion conductance, (mol·m)/(Pa·s ) ; Ca is the adsorbed gas transport conductance, (mol·m)/(Pa·s ) ; Cb is the free gas transport conductance, (mol·m)/(Pa·s ) ; Ct is the total gas transport conductance, (mol·m)/(Pa·s ) .The free gas transport ratio (Γb) and adsorbed gas transport ratio (Γa) are calculated as follows:
(11-c)
where ϕb is the effective porosity for free gas transport, dimensionless (see Appendix B); ρ is the gas density, kg/m3 (see Appendix B); μ is the gas viscosity, Pa·s (see Appendix B); ξ is the taper ratio, dimensionless; M is the molecular weight, kg/mol ; P is pressure, Pa ; Z is the gas compressibility factor, dimensionless (see Appendix B); dm is the molecular diameter, m ; ϕa is the effective porosity for adsorbed gas transport, dimensionless (see Appendix B); Dτ is the fractal dimension of the tortuosity of pores, dimensionless; θ is the gas coverage, dimensionless (see Appendix B); Ds is the surface diffusion coefficient, m2/s (see Appendix B).
356
Γb =
Cb × 100% Ct
(17-a)
Γa =
Ca × 100% Ct
(17-b)
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4. Results and discussions
where Γb is the free gas transport ratio, dimensionless; Γa is the adsorbed gas transport ratio. The Darcy molar rate (Qd ) is obtained as follows:
Qd = −
KA ρAt ∇P μM
4.1. Impacts of pore size distributions on adsorbed and free gas transport Impacts of a pore size distribution on gas transport behavior are first explored. In this study, the pore size distribution in fractal shale matrix is determined by fractal dimension and minimum pore size. The connection between pore size distribution and flow feature is the porosity as shown in Eqs. (7) and (8). The connection between the pore size distribution fractal feature and flow features is thus presented based on effects of fractal dimension and minimum pore size on transport behavior and efficiency. The cumulative probability curves of pore diameters with different Dp under the same dpmin are plotted in Fig. 3(a). Fig. 3(a) indicates that with an increase in Dp , the small pores occur more frequently. Specially, the cumulative probability of pores with a pore diameter less than 5 nm (Fp (dp ⩽ 5nm)) is 98.2% for the Dp = 2.5 case, 98.7% for the Dp = 2.7 case and 99.1% for the Dp = 2.9 case. Fig. 3(b) indicates that with a decrease in Dp , the adsorbed gas transport ratio (Γa) increases and the free gas transport ratio (Γb) decreases; this is because a smaller Dp leads to more frequent occurrence of small pores, which shrinks the space of free gas transport. Free gas transport better benefits in the transport efficiency of nanopores [25], and thus the apparent permeability (KA ) increases with an increase in Dp as shown in Fig. 3(c). These show that a larger Dp yields to a larger free gas transport ratio and better transport efficiency. The cumulative probability curves with different dpmin under the same Dp are shown in Fig. 3(d). With a smaller dpmin , the frequency of occurrence of small pore increases. For instance, the cumulative probability of pores with the pore diameter less than 5 nm (Fp (dp ⩽ 5nm)) is 99.1% for the dpmin = 1nm case, 93.0% for the dpmin = 2nm case and 77.3% for the dpmin = 3nm case. More frequent occurrence of large pores yields a higher free gas transport ratio as shown in Fig. 3(e). When dpmin increases from 1 nm to 3 nm, KA increases with one order of magnitude, indicating that a larger minimum pore size represented by dpmin under the same Dp leads to better transport efficiency of shale matrix. Fig. 3(a)–(f) also indicate that even if the porosity is the same, the gas transport efficiency is significantly affected by the pore size distribution.
(18)
where Qd is the Darcy molar flow rate, mol/ s ; KA is the total apparent permeability, m2 . Considering Qd = Qt , the apparent permeability KA is calculated as follows [24,25]:
KA =
μMCt ρAt
(19)
Higher apparent permeability indicates better transport efficiency [24,25].
3. Model Validation In this section, we apply the model to calculate the pore size distribution and the apparent permeability, and their correctness is validated against the data from shale sample experiments. The distribution of pore sizes and apparent permeability are examined based on the experimental data from an Eagle Ford shale sample reported by Alnoaimi et al. (2013) [33]. The sample of diameter 2.5 cm by 4.6 cm in length is selected for the experiment. The pore size distribution of the shale sample is first measured by Alnoaimi et al. with non-localized density functional theory (NLDFT) method. After the determination of the pore size distribution, the apparent permeability is then tested. The apparent permeability is tested with helium as the probing gas. The permeability from the cited experiments refers to the apparent permeability instead of intrinsic permeability [33]. The intrinsic permeability is the property of rock itself, and does not change with the pressure and type of gas flowing within it. Apparent permeability is determined based on the Darcy’s law and equivalent to Darcy permeability [33]. Owing to the non-zero velocity of gas molecules on surfaces of pore walls, the apparent permeability will change with the pressure during the measurement [33]. Apparent permeability in shale gas systems is thus always larger than intrinsic permeability. In this study, we adjust the fractal dimensions, which is one of the most uncertain parameters, to achieve the match between model results and experimental data points. The pore size distribution and apparent permeability are also calculated based on our proposed model. From Fig. 2, we can see that the match with the experiment is excellent, which shows the industrial practicability of the proposed model.
4.2. Effects of pore size distribution on Knudsen diffusion Compared with conventional gas reservoir, Knudsen diffusion is an important mechanism for gas transport in shale matrix. Shale rocks are formed in deep underground by going through compaction and solidification with increasing pressure and temperature over a long period of time, which process provides a high-pressure and high-temperature condition for the gas transport in shale matrix. Effects of pore size Fig. 2. Validation with experimental data from Eagle Ford shale sample reported by Alnoaimi et al. (2013) [33]: (a) relationship between pore size and cumulative probability; (b) relationship between pressure and apparent permeability. T = 300K , L = 4.6cm , ϕ = 0.08 , Dp = 2.98 , dpmin = 4.9nm , Dτ = 1.1, ξ = 1.2 .
357
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Fig. 3. Impacts of pore size distribution on gas transport behavior: (a) cumulative probability curves of pore diameter under different fractal dimensions, dpmin = 1nm ; (b) relationship between fractal dimension and transport ratio for free gas and adsorbed gas, dpmin = 1nm ; (c) relationship between fractal dimension and apparent permeability, dpmin = 1nm ; (d) cumulative probability curves of pore diameter under different minimum pore diameters, Dp = 2.9 ; (e) relationship between minimum pore diameter and transport ratio for free gas and adsorbed gas, Dp = 2.9 ; (f) relationship between minimum pore diameter and apparent permeability, Dp = 2.9 . For (a)–(f), ϕ = 0.07 ,
L = 3cm , P = 25MPa , T = 350K , Dτ = 1.3 , ξ = 1.2 , At = 5cm2 .
4.3. Sensitivities of transport efficiencies to pore size distribution parameters
distribution on Knudsen diffusion are thus studied under a wide pressure and temperature range as shown in Fig. 4. From Fig. 4, we can find that under lower pressure and temperature, the contribution of Knudsen diffusion to total gas transport is higher. This is because under a lower pressure and temperature, the ratio of molecule-wall collision is higher, which leads to a higher contribution of Knudsen diffusion [34]. Fig. 4(a) and (b) indicate a higher Dp leads to a higher Knudsen diffusion ratio; this is because the occurrence frequency of small pores is lower with a higher Dp , which shrinks the distance of molecule-wall collision and increases the molecule-wall collision ratio. Fig. 4(c) and (d) indicate a higher minimum pore size leads to a lower Knudsen diffusion ratio; this is because a higher minimum pores results in a more frequent occurrence of large pores, which increases the distance of molecule-wall collision and decreases the molecule-wall collision ratio.
As the transport efficiency is affected by a pore size distribution, sensitivities of transport efficiency to the pore size distribution parameters are further studied. The slope (dKA/ dDp) of apparent permeability versus Dp is applied to qualify the sensitivity of transport efficiency to the fractal dimension; a larger absolute value of dKA/ dDp indicates higher sensitivity of apparent permeability to Dp . As shown in Fig. 5(a)–(c), the sensitivity of apparent permeability to Dp decreases with an increase in Dp ; this is because the free gas mainly contributes to the transport efficiency [25], and the ratio of free gas decreases with the increasing Dp (Fig. 3(b)). Fig. 5(a) indicates that the sensitivity of apparent permeability to Dp tends to be higher with an increase in porosity, which is due to the increase of the free gas transport ratio with an increase in porosity; that is, the apparent efficiency of more porous
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Fig. 4. Effects of pore size distribution on ratio of Knudsen diffusion: (a) relationship between pressure and ratio of Knudsen diffusion under different fractal dimensions, dpmin = 1nm ; (b) relationship between temperature and ratio of Knudsen diffusion under different fractal dimensions, dpmin = 1nm ; (c) relationship between pressure and ratio of Knudsen diffusion under different minimum pore sizes, Dp = 2.9 ; (d) relationship between temperature and ratio of Knudsen diffusion under different minimum pore sizes, Dp = 2.9 . For (a)–(d), ϕ = 0.07 , L = 3cm , P = 25MPa , T = 350K , Dτ = 1.3 , ξ = 1.2 , At = 5cm2 .
Fig. 5. (a) relationship between d (KA)/ d (Dp) and fractal dimension under different porosities, dpmin = 1nm , and P = 25MPa ; (b) relationship between
d (KA)/ d (Dp) and fractal dimension under different minimum pore diameters, ϕ = 0.07 , andP = 25MPa ; (c) relationship between d (KA)/ d (Dp) and fractal dimension under different pressures, ϕ = 0.07 , dpmin = 1nm .
L = 3cm , T = 350K , Dτ = 1.3 , ξ = 1.2 , and At = 5cm2 .
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pore size distribution on transport efficiency based on the isolines of apparent permeability of 100 nd. Fig. 7(a) indicates that for different shale matrix with the same transport efficiency under the same pressure, the shale matrix with a lower minimum pore size owns a larger porosity. Fig. 7(b) shows that a higher fractal dimension of a pore size pushes the isoline of apparent permeability to the region with higher porosity; this is in agreement with the conclusion in the previous section: a higher fractal dimension of a pore size results in a lower gas transport efficiency in shale matrix. 5. Conclusions An improved gas transport model in fractal shale matrix has been proposed in this study, which bridges a pore size distribution and multiple real gas transport mechanisms. Based on the proposed model, the effects of pore size distribution on gas transport efficiencies are further explored and highlighted. Validations of this model with experimental data have shown the reliability and practicability of the present model. The main conclusions are as follows:
Fig. 6. Relationship between apparent permeability, porosity and pressure. T = 350K , L = 3cm , ϕ = 0.13 , dpmin = 1nm , Dp = 2.9 , Dτ = 1.3, ξ = 1.2 , At = 5cm2 .
(1) A pore size distribution affects the gas transport efficiency by a fractal dimension and a minimum pore size. A larger fractal dimension of a pore size leads to higher frequency of occurrence of small pores and a lower free gas transport ratio, which further results in lower transport efficiency; a larger minimum pore size yields a higher free gas transport ratio and transport efficiency with increased frequency of occurrence of large pores. Contribution of Knudsen diffusion to the total gas transport is higher under lower pressure and temperature. Increasing fractal dimension and decreasing minimum pore size result in a higher Knudsen diffusion contribution. (2) Gas transport efficiency due to pore size distribution parameters (a fractal dimension and a minimum pore size) varies with different porosities and pressures. Decreased pressure and increased porosity enhance the sensitivity of gas transport efficiency to a pore size distribution. (3) A relationship between apparent permeability and porosity based on a pore size distribution is established, from which the apparent permeability can be estimated. For different shale matrix with the same transport efficiency under the same pressure, the shale matrix with a lower minimum pore size owns a larger porosity. A higher fractal dimension of a pore size pushes the isoline of apparent permeability to the region with higher porosity. (4) The developed relationship between permeability and porosity based on different pore size distribution indicates the apparent permeability may be an indicator of the porosity under a given pore size distribution and pressure condition. It helps the industry to
shale matrix is more subject to be affected by Dp . Fig. 5(b) shows that with a larger minimum pore diameter (dpmin ) , the apparent permeability is more sensitive to Dp , which is caused by the increased free gas transport ratio with increasing Dp (Fig. 3(e)). Fig. 5(c) indicates that larger pressure results in lower sensitivity of apparent permeability to Dp ; this is because the gas coverage increases with an increases in pressure, which results in a higher effective porosity for adsorbed gas transport and a lower free gas transport ratio.
4.4. Apparent permeability versus porosity based on pore size distributions The relationship between apparent permeability and porosity based on a pore size distribution is further established as shown in Fig. 6. Fig. 6 indicates that the apparent permeability increases with an increase in porosity and a decrease in pressure; this is because the adsorbed gas transport ratio decreases and the free gas transport ratio increases with increasing porosity and decreasing pressure. Fig. 6 also indicates that the apparent permeability may be an indicator of the porosity under a given pore size distribution and pressure condition. Taking the pressure condition of 1 MPa as an example, we can see: (1) if the apparent permeability is larger than 1000 nd under 1 Mpa, the porosity is higher than 0.13; (2) if the apparent permeability ranges between 100 nd and 1000 nd under 1 MPa, the porosity is between 0.05 and 0.13; (3) for the apparent permeability less than 100 nd under 1 MPa, the porosity is lower than 0.05. Fig. 7 further reveals effects of a
Fig. 7. (a) Isoline of apparent permeability of 100 nd under minimum pore diameters of 1 nm, 2 nm and 3 nm, Dp = 2.9 ; (b) Isoline of apparent permeability of 100 nd under fractal dimension of 2.7, 2.8 and 2.9, dpmin = 1nm . For (a) and (b), T = 350K , L = 3cm ,
ϕ = 0.13 , Dτ = 1.3, ξ = 1.2 , At = 5cm2 .
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Acknowledgments
quick determine the porosity based on apparent permeability based on a given pore size distribution. The apparent permeability can also be estimated with tested porosity under a given pore size distribution.
The authors would like to acknowledge the NSERC/AIEES/ Foundation CMG and Alberta Innovates - Technology Futures Chairs for providing research funding.
Appendix A.A. Fractal Properties of Pores in Shale Matrix For a fractal shale rock with a ratio δ < 1 between adjacent scales, the power-law relationship is as follows [35]:
dp,1 = δdp,0 dp,2 = δdp,1 = δ 2dp,0 dp,3 = δdp,2 = δ 3dp,0
⋮ dp,j = δdp,j − 1 = δ jdp,0 with
ln (dp,j / dp,0)
j=
(A-1)
lnr
where r is a scaling factor. If there are J structures of scale dp,j embedded within each structure of scale dp,j − 1, and J structures of scale dp,j − 1 embedded within each structure of scale dp,j − 2 …, and J structures of scale dp,1 embedded within each structure of scale dp,0 , then the total number of structures (Jr ) with a scale larger than dp,j is as follows:
Jt (dp,j ) = J j = J
dp,j ⎞ ln ⎛⎜ ⎟ / lnr ⎝ dp,0 ⎠
(A-2)
where Jt is the total number of structures with a scale larger than εj , dimensionless. Based on Eq. (A-2), we can have the following expression:
lnJt (εj ) =
ln (dp,j / dp,0) lnr
Let Dp =
lnJ − lnr
dp,0 ⎞ Jt (dp,j ) = ⎜⎛ ⎟ ⎝ dp,j ⎠
lnJ
(A-3)
and the expression of Jt (dpe,j ) is as follows:
Dp
(A-4)
Let dp,0 = dpmax and Jt = Np . We can have the expression of Np as in Eq. (1). Let dp,j = dpmin . We can have the expression of the total number of pores as Eq. (2). Appendix B.B. Real Gas Properties According to the Soave-Redlich-Kwong (SRK) equation of state (EOS), the solution of molar volume (Vm ) considering confinement is as follows [25]:
Vm = −
3P (χ 2 P + χRT −ακ ) + R2T 2 1 ⎛ B+ −RT ⎞ 3P ⎝ B ⎠ ⎜
g+
⎟
(B-1)
g 2−4[3P (χ 2 P + χRT −ακ ) + R2T 2]3
B=
3
g=
−2R3T 3−27ακχP 2−9RPT (χ 2 P
(B-2)
2 + χRT −ακ )
(B-3)
α = [1 + (0.48508 + 1.55171ω−0.15613ω2)(1− Tr )]2
κ = 0.42748
χ = 0.08664
(B-4)
R2Tc2 (B-5)
Pc
RTc Pc
(B-6)
(Pa·m3)/mol ;
m3/mol ;
α is a function of gas κ is an attraction parameter in the SRK EOS, where χ is a repulsion parameter in the SRK EOS, T properties and temperature in the SRK EOS, dimensionless; ω is the acentric factor, dimensionless; Tr is the reduced temperature, dimensionless, = T ; Vm is the molar volume, m3/mol ; Pc is the critical pressure, Pa ; Tc is the critical temperature, K . The molecular number density (n ) under the real gas effect is expressed as follows:
361
c
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n=
NA Vm
(B-7)
m−3 ;
1023mol−1.
NA is the Avogadro constant, = 6.02 × where n is the molecular number density, The mean molecular speed (v ) from the Maxwell-Boltzmann distribution is given below:
2RT πM
v =2
(B-8)
where v is the mean molecular speed from the Maxwell-Boltzmann distribution, m/s . The mean free path (λ ) can be expressed as follows:
λ=
1 πndm2
(B-9)
where λ is the mean free path, m . The density ( ρ ) can be calculated in terms of n as follows:
ρ=
Mn NA
(B-10)
The viscosity ( μ ) is obtained as follows [36]: 2
Pr4 o1 ⎛ ⎞ ⎛ Pr ⎞ ⎛ Pr ⎞ ⎤ μ = μ0 ⎡ ⎢1 + T 5 ⎜ T 20 + P 4 ⎟ + o2 Tr + o3 Tr ⎥ r r r ⎝ ⎠ ⎝ ⎠⎦ ⎝ ⎠ ⎣ ⎜
⎟
⎜
⎟
(B-11)
where μ0 is the gas viscosity at P = 1.01325 × 105 Pa and T = 423 K, = 2.31 × 10−5Pa·s ; Pr is the reduced pressure, dimensionless, =
o3 are the fitting coefficients, o1 = 7.9, o2 = 9 × 10−6 , o3 = 0.28. The compressibility factor (Z ) in terms of n is as follows: Z=
P nkT
P ; Pc
o1, o2 and
(B-12)
10−23J / K .
where k is the Boltzmann constant, = 1.38 × The gas coverage (θ ) is calculated as follows [37]:
P /Z P / Z + PL
θ=
(B-13)
where PL is the Langmuir pressure, Pa . The total cross-sectional area occupied by the free gas ( Apb ) is calculated as follows:
Apb = −
∫d
dpmax
pmin
π (dp−2θdm)2 4
dNp
(B-14)
By coupling Eqs. (1), we can obtain Apb as follows:
Apb =
2 (1−η2 − Dp ) 4θdm dpmax (1−η1 − Dp ) 4θ 2dm2 (1−η−Dp ) ⎤ πDp ⎡ dpmax − − ⎢ ⎥ 4 ⎣ 2−Dp 1−Dp Dp ⎦
(B-15)
The effective porosity for free gas transport (ϕb ) and adsorbed gas transport (ϕa ) are calculated as follows:
ϕb =
Apb Ap
ϕ
(B-16)
ϕa = ϕ−ϕb
(B-17)
The Knudsen number (Kn ) is calculated as follows:
Kn =
λ dpm
(B-18)
where Kn is the Knudsen number, dimensionless; dpm is the mean pore diameter, m, =
Dp
d . Dp − 1 pmin
The surface diffusion coefficient (Ds ) is expressed as follows [38]:
ΔH 0.8 ⎞ 2(1−θ) + Ψθ (2−θ) + [H (1−Ψ)](1−Ψ)Ψθ 2 Ds = 8.29 × 10−7T 0.5exp ⎛− Ψ 2(1−θ + 2 )2 ⎝ RT ⎠ ⎜
⎟
(B-19)
where ΔH is the isosteric adsorption heat at θ = 0 , J/mol ; H () is the Heaviside step function; Ψ is the ratio between the blockage rate constant and the forward migration rate constant, dimensionless.
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Full Length Article
Real gas transport in shale matrix with fractal structures a
a,⁎
a
b
a
c
a,⁎
T
Jinze Xu , Keliu Wu , Ran Li , Zhandong Li , Jing Li , Qilu Xu , Zhangxin Chen a b c
Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada College of Petroleum Engineering Institute, Northeast Petroleum University, Daqing, Heilongjiang 163318, China China University of Geosciences-Beijing, School of Earth Sciences and, Beijing 100083, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Shale gas Pore size distribution Fractal structure Shale matrix
A real gas transport model in shale matrix with fractal structures is established to bridge a pore size distribution and multiple transport mechanisms. This model is well validated with experiments. Results indicate that different pore size distributions lead to various transport efficiencies of shale matrix. A larger fractal dimension of the pore size and a smaller minimum pore size yield higher frequency of occurrence of small pores and a lower free gas transport ratio, which further results in lower transport efficiency. Gas transport efficiency due to pore size distribution parameters (a fractal dimension and a minimum pore size) varies with different porosities and pressures. Increasing fractal dimension and decreasing minimum pore size result in a higher contribution of Knudsen diffusion to the total gas transport. Decreased pressure and increased porosity enhance the sensitivity of gas transport efficiency to a pore size distribution. The relationship between apparent permeability and porosity based on different pore size distributions is also established for industrial application.
1. Introduction
Fractal model is an effective method to describe the pore size distribution, and advantages are concluded as follows: (1) the fractal theory has been an effective method to study a wide range of pore size distributions in experimental studies. Clarkson et al. (2013) [13] studied the pore size data from currently active shale gas plays including the Barnett, Marcellus, Haynesville, Eagle Ford, Woodford, Muskwa, and Duvernay shales. Their study indicated a fractal geometry (power law scattering) for a wide range of pore sizes. Further studies by Lee et al. (2014) [14], Bahadur et al. (2015) [15] and Wang et al. (2015) [16] also showed that the fractal geometry is suitable to describe a pore size distribution for Baltic Basin in United Kingdom, Second White Specks and Belle Fourche formations in Canada, and Songliao Basin in China; (2) fractal theory is able to bridge the pore size distribution and gas transport mechanisms based on link of porosity. Effects of pore size distribution on gas transport behavior and efficiency can thus be studied. We thus select fractal model as part of our models. In industry, different kinds of experimental methods may be performed to test the porosity of shale samples[17]. Mercury intrusion (MICP), low-pressure adsorption, and small-angle and ultra-small-angle neutron scattering (SANS/USANS) techniques have been wide used to measure porosity in experiments. MICP cannot access the porosity with pore diameter less than 3 nm in shale gas reservoir, and the risk to damage the pore structure exists. Low-pressure adsorption method must be combined with MICP to investigate the pore size spectrum due to the variable access of the probe molecules. SANS/USANS is able to detect a
As one of the clean energy resources, shale gas significantly reduces greenhouse gas emissions [1]. Owning to technological advancements in horizontal drilling and hydraulic fracturing, the shale gas is developing very fast in North America [2], which has received much attention from all over the world. However, there are still many challenges regarding the development of shale gas sources [3]. For instance, the gas transport behavior varies considerably in pores of shale matrix [4]. Studies on gas transport in shale matrix indicate the mechanisms of the transport behavior and provide a basis for evaluating well productivity, which contributes to the research on the development of shale gas resources. Shale rocks are formed by compaction and solidification under high pressure in deep underground over a long time [5]. The shale matrix consisting of pores exhibits low porosity (1%–15%) and low permeability (nanodarcy to microdarcy) [6]. Chalmers et al. (2012) [7] investigated a range of pore sizes based on samples from the Barnett, Marcellus, Woodford, and Haynesville gas shales in the United States and the Doig Formation of northeastern British Columbia, Canada. They claimed that the pore size in shale matrix ranges from 1 nm to 100 µm. Experimental observations reported by Utpalendu and Prasad (2013) [8], Ye et al. (2015) [9], Zhang et al. (2015) [10], Chalmers and Bustin (2017) [11] and Wu et al. (2017) [12] also indicated a wide range of pore size distributions in shale matrix.
⁎
Corresponding authors. E-mail addresses: [email protected] (K. Wu), [email protected] (Z. Chen).
https://doi.org/10.1016/j.fuel.2018.01.114 Received 18 October 2017; Received in revised form 2 January 2018; Accepted 26 January 2018 0016-2361/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature
Vm Z
molar volume, m3/mol ; gas compressibility factor, dimensionless.
Roman Symbols Greek Symbols
At Ap C Dp Dτ dp dm Fp fp ΔH H () KA k L Lt M Np NA n o1 o2 o3 P Pc PL Pr Q q T Tc Tr
cross-sectional area of the sample, m2 ; total cross-sectional area of pores, m2 ; conductance, (mol·m)/(Pa·s) ; fractal dimension of the pore size, dimensionless; fractal dimension of the tortuosity, dimensionless; equivalent pore diameter, m ; gas molecular diameter, m ; cumulative probability of pore size distribution, dimensionless; probability density function of pore size; isosteric adsorption heat at θ = 0 , J/mol ; Heaviside step function; apparent permeability, m2 ; Boltzmann constant, = 1.38 × 10−23J/K ; straight length of the pore, m ; tortuous length of the pore, m ; gas molecular weight, kg/mol ; total number of pores with the equivalent pore diameter larger than dp in shale matrix, dimensionless; Avogadro constant, = 6.02 × 1023mol−1; molecular number density, m−3 ; fitting coefficient, = 7.9; fitting coefficient, = 9 × 10−6 ; fitting coefficient, = 0.28; pressure, Pa ; critical pressure, Pa ; Langmuir pressure, Pa ; P reduced pressure, dimensionless, = P ; c transport molar rate, mol/s ; transport molar rate in a single pore, mol/s ; temperature, K ; critical temperature, K ; T reduced temperature, dimensionless, = T ;
α β γ χ Ψ Γ κ ω ξ ϕ Ds θ ρ μ μ0 η
function of gas properties and temperature in the SRK EOS, dimensionless; fracture dip, dimensionless; aspect ratio, dimensionless; repulsion parameter in the SRK EOS, (Pa·m3)/mol ; ratio between the blockage rate constant and the forward migration rate constant, dimensionless; gas transport ratio, dimensionless; attraction parameter in the SRK EOS, m3/mol ; acentric factor, dimensionless; taper ratio, dimensionless; porosity, dimensionless; surface diffusion coefficient, m2/s ; gas coverage, dimensionless; gas density, kg/m3; gas viscosity, Pa·s ; gas viscosity at P = 1.01325 × 105 Pa and T = 423 K, = 2.31 × 10−5Pa·s ; ratio between dpmin and dpmax in shale matrix, dimensionless, = dpmin/ dpmax .
Subscripts
a b p min max k s t v
adsorbed gas; free gas; pore; minimum; maximum; Knudsen diffusion; surface diffusion; total; viscous flow.
cz
production optimization. Many models have been proposed in the past decade to study the gas transport behavior in shale matrix. Javadpour (2009) [18] and Rahmanian et al. (2013) [19] proposed an ideal free gas transport model based on the sum of viscous flow and Knudsen diffusion in single pores. Sheng et al. (2015) [20] further took the surface diffusion into
wide range of pore sizes and good to be used at reservoir pressure and temperature conditions [17]. The theoretical relationship between porosity and permeability based on pore size distribution benefits in improving efficiency of estimating porosity or permeability. The mechanisms of change of transport efficiency with pore size distribution and porosity are also need to be indicated to better understand
Table 1 Review of existing gas transport models in shale rocks [18–25]. Model
Description
Comments
Javadpour, 2009 [18]
Ideal gas EOS; Linear superposition of viscous flow and Knudsen diffusion based on the slip boundary condition Ideal gas EOS; Weighted superposition of viscous flow and Knudsen diffusion based on slip boundary condition Ideal gas EOS; Weighted superposition of viscous flow, Knudsen diffusion and surface diffusion based on slip boundary condition Ideal gas transport model for a bundle of pores in fractal rocks
Only for ideal gas in single pores; Neither adsorption nor surface diffusion Only for ideal gas in single pores; Neither adsorption nor surface diffusion Only for ideal gas in single pores
Rahmanian et al., 2013 [19] Sheng et al., 2015 [20] Miao et al., 2015 [22] Li et al., 2016 [23] Wu et al., 2016 [21]
Ren et al., 2016 [24]
Xu et al., 2017 [25]
Consideration of real gas effect based on empirical equation of methane; Weighted superposition of viscous flow, Knudsen diffusion and surface diffusion based on the slip boundary condition Consideration of real gas effect based on empirical equation of methane; Linear supervision of viscous flow and Knudsen diffusion based on non-slip boundary Real gas transport for tapered non-circular pores in shale rocks
354
Only for ideal gas; Knudsen diffusion and surface diffusion are not considered Empirical equation to describe real gas effect; single pores (elliptical cross section is omitted); Compressibility factor is always larger than 1 which is not real under low pressure Empirical equation to describe real gas effect; Single pores; Neither adsorption nor surface diffusion Only apply for single pores
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where Npt is the total number of pores, dimensionless; dpmin is the minimum equivalent diameter of pores, m . If we differentiate Eq. (1) on both sides and couple it with Eq. (2), the following equation can be obtained:
consideration and the adsorbed gas was considered in their models. Wu et al. (2016) [21] further considered the real gas effect and the surface diffusion in the gas transport behavior in single pores. Miao et al. (2015) [22] applied the fractal geometry theory to the expression of viscous flow in shale rocks, which can reveal more mechanisms of seepage characteristics than traditional models. Li et al. (2016) [23] showed that the results obtained from a fractal model with viscous flow in shale rocks can well match those from numerical simulation. Ren et al. (2016) [24] included the slippage into the Knudsen diffusion for a model of free gas transport. Xu et al. (2017) [25] proposed a real gas transport model in tapered noncircular nanopores in shale matrix. However, there are still many limitations for existing models as summarizing as Table 1 [18–25]. The previous fractal models in shale matrix are limited to ideal gas transport under a low Knudsen number (viscous flow), and the impacts of a pore size distribution on gas transport behavior are not studied. A relationship between permeability and porosity based on different pore size distributions in shale matrix also needs to be established for industrial applications [26–28]. In order to better understand the gas transport behavior in shale matrix, an analytical model based on the fractal theory is proposed to bridge the pore size distribution and multiple transport mechanisms (viscous flow, Knudsen diffusion and surface diffusion). The proposed model is well validated with the available experimental data, and effects of pore size distribution on gas transport efficiency are further revealed.
−
Npt
−(Dp + 1)
D
p dp = Dp dpmin
d(dp) = fp d(dp)
dp
⎧ ∫d
pmin
Fp =
fp d (dp),dp ⩾ dpmin
⎨ 0,dp < dpmin ⎩
(4)
where Fp is the cumulative probability of pores with their equivalent pore diameter less than dp. By coupling Eqs. (3) and (4), we can have the expression of Fp as follows:
( )
D
p dpmin ⎧ ,dp ⩾ dpmin ⎪1− dp Fp = ⎨ 0,d < d pmin ⎪ p ⎩
(5)
The total cross-sectional area of pores (Ap ) is calculated as follows:
Ap = −
∫d
dpmax
pmin
πdp2 4
dNp
(6)
m2 .
where Ap is the total cross-sectional area of pores, By coupling Eqs. (1) and (6), Ap is expressed as follows:
The proposed model for gas transport in shale matrix is developed based on the fractal theory, which accounts for a pore size distribution and different transport mechanisms as shown in Fig. 1. Generally, the classical Knudsen number is employed to divide the flow regime as continuum flow (Kn < 0.001), laminar slip flow (0.001 < Kn < 0.1), transition flow (0.1 < Kn < 10), and free-molecular flow (Kn > 10). However, considering the existence of adsorbed gas transport in shale matrix, classical flow regime classification cannot be well applied in the real gal transport in shale matrix [3]. Both transport mechanisms of the free gas transport (viscous flow and Knudsen diffusion) and the adsorbed gas transport (surface diffusion) are taken into consideration in a bundle of pores in this study. Surface diffusion exists in the pore, which happens in the adsorbed layer formed on walls of pores. For the single adsorbed layer, we apply Langmuir’s law with considering real gas effect to calculate the gas coverage (θ) for the adsorbed gas. (see Appendix B). In this study, we assume instant equilibrium of the gas exchange between adsorption and desorption in pores [25]. This model upscale the gas transport model from pore scale to reservoir scale based on pore size distribution in fractal shale matrix. The boundary condition of the proposed analytical model refers to constant pressure. Physical properties of different gases are calculated based on Soave-Redlich-Kwong (SRK) equation of state (EOS). Detailed equations and procedures are shown in Appendix B. For a fractal shale matrix, the total number of pores (Np) with an equivalent pore diameter larger than dp is as follows (see Appendix A):
Ap =
2 Dp (1−η2 − Dp) πdpmax
2−Dp
4
(7)
where η is the ratio between dpmin and dpmax in shale matrix, dimensionless, = dpmin/ dpmax . Previous experimental studies by Clarkson et al. (2011) [17], Yan et al. (2017) [29] and Pang et al. (2017) [30] indicate the effective porosity dominates the gas transport efficiency in shale matrix and the sample with highest ratio of effective porosity owns the highest permeability. Existing models [18–25] with focus on effective porosity have also shown a convincing ability to deal with gas transport behavior and efficiency in shale rocks. This study is thus focused on gas transport in connected pores, and we only consider the effective porosity in the model. The porosity of shale rock (ϕ) is thus calculated as follows:
ϕ=
Ap (8)
At
where At is the cross-sectional area of the core, m2 .
Dp
(1)
where Np is the total number of pores with their equivalent pore diameter larger than dp , dimensionless; dp is the equivalent diameter of pores in shale matrix, m , (see Appendix B); dpmax is the maximum equivalent diameter of pores, m ; Dp is the fractal dimension of pore sizes, dimensionless. Let dp = dpmin ; we can have the expression of the total number of pores (Npt ) in shale matrix as follows:
dpmax ⎞ Npt = ⎜⎛ ⎟ ⎝ dpmin ⎠
(3)
where fp is the probability density function of pore sizes. The cumulative probability Fp is as follows:
2. Model establishment
dpmax ⎞ Np = ⎜⎛ ⎟ ⎝ dp ⎠
dNp
Dp
Fig. 1. Schematic view of the model.
(2) 355
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By coupling Eqs. (10) and (11), we can have the expressions for the total transport rates as follows:
The tortuous length of pores (Lt ) is calculated as follows [22,23]:
Lt =
LDτ dp1 − Dτ
(9)
where Lt is the tortuous length of pores, L is the straight length of pores, Dτ is the fractal dimension of tortuosity. Based on previous studies, three well-known types of diffusion (Knudsen diffusion, molecular diffusion and surface diffusion) exist in shale matrix. When the mean free path of gas molecules is comparable to the pore dimension in shale matrix, Knudsen diffusion, which is a typical kind of free-molecule diffusion becomes important. The Knudsen diffusion is dominated by molecule-wall collisions. Generally, the Knudsen diffusion is significant only at low pressures and small pore diameters. Molecular diffusion is dominated by molecule-molecule collisions. Concentration gradient is the driving force, and the Fick’s first law of diffusion is often applied to model this process. According to the experimental investigation on the characteristics of gas diffusion by Kim et al. in 2015 [31], the molecular diffusion coefficient remains 5.068 × 10−4 cm2/s while the Knudsen diffusion coefficient ranges from 0.02 to 1.252 cm2/s for nanopores with pore radius of 5–300 nm in shale rocks. Experiment performed by Javadpour et al. (2007) [32] shows that the Knudsen diffusion coefficient is 0.04 cm2/s and the molecular diffusion coefficient in kerogen bulk is 2 × 10−6 cm2/s for the nanopore with the radius of 35 nm. Thus we select the Knudsen diffusion as the main free gas diffusion mechanism in the present model. The total flow rates of viscous flow, Knudsen diffusion and surface diffusion can be obtained as follows [22,23]:
Qv = −
∫d
dpmax
Qk = −
∫d
dpmax
Qs = −
∫d
pmin
qv dNp
(10-a)
qk dNp
(10-b)
qs dNp
(10-c)
pmin
dpmax
pmin
Qv =
qv =
qk =
− 43
( ) 2ξ 2 ε+1
( )
3 + Dτ πϕb ρdpmax ΔP + ξ + 1 128μMLDτ − 1 L
Qk =
2 + Dτ Dp (1−η2 − Dp + Dτ ) πϕb vdpmax ΔP 2−Dp + Dτ 12ZRTLDτ − 1 L
Qs =
1 + Dτ Dp (1−η1 − Dp + Dτ ) ⎛ 2ξ 2 ⎞− 3 ϕa θDs dpmax ΔP ξ⎜ ⎟ 3 Dτ − 1 L 1−Dp + Dτ ⎝ ξ + 1 ⎠ NA dm PL
(12-a)
(12-b)
(12-c)
(13)
Qb = Q v + Qk
where Qb is the total free gas transport molar rate, mol/ s . Surface diffusion only occurs in the adsorbed gas, and the total adsorbed gas transport molar rate (Qa ) is thus calculated as follows: (14)
Qa = Qs
where Qa is the total adsorbed gas transport molar rate, mol/s . The total gas transport molar rate (Qt ) is calculated as follows: (15)
Qt = Qa + Qb
where Qt is the total gas transport molar rate, mol/ s .If the molar rate is divided by the pressure gradient, the viscous flow conductance (Cv ) , Knudsen diffusion conductance (Ck ) , surface diffusion conductance (Cs ) , adsorbed gas transport conductance (Ca) , free gas transport conductance (Cb) , and total gas transport conductance (Ct ) in the fractal shale matrix can be obtained as follows:
Cv =
Dp
(1−η3 − Dp + Dτ )
3−Dp + Dτ
3ξ 3 ξ2
− 43
( ) 2ξ 2 ε+1
3 + Dτ πϕb ρdpmax
+ ξ + 1 128μMLDτ − 1
Ck =
2 + Dτ Dp (1−η2 − Dp + Dτ ) πϕb vdpmax ΔP 2−Dp + Dτ 12ZRTLDτ − 1 L
Cs =
1 + Dτ Dp (1−η1 − Dp + Dτ ) ⎛ 2ξ 2 ⎞− 3 ϕa θDs dpmax ΔP ξ⎜ ⎟ 3 Dτ − 1 L 1−Dp + Dτ 1 ξ + ⎝ ⎠ NA dm PL
(16-a)
(16-b)
2
(11-a)
πϕb vdp3 ΔP (11-b)
12ZRT Lt −2 3
ξ2
− 43
2ξ 2 ε+1
We apply the approach of linear superposition of viscous flow, Knudsen diffusion and surface diffusion for gas transport behavior, which has been well validated with simulation and experimental data in previous studies [24,25]. As the Knudsen diffusion and viscous flow only exist in the free gas, the total free gas transport molar rate (Qb ) is calculated as follows:
πρϕb dp4
ΔP ξ 2 + ξ + 1 128μM Lt
2ξ 2 ⎞ qs = ξ ⎜⎛ ⎟ ξ ⎝ + 1⎠
3−Dp + Dτ
3ξ 3
2
where qv is the viscous flow molar rate in a single pore, mol/s ; qk is the Knudsen diffusion molar rate in a single pore, mol/s ; qs is the surface diffusion molar rate in a single pore, mol/s ; Q v is the total viscous flow molar rate, mol/s ; Qk is the total Knudsen diffusion molar rate, mol/s ; Qs is the total surface diffusion molar rate, mol/s . Considering the tapering effect, the molar rates for the viscous flow flux, Knudsen diffusion flux and surface diffusion flux in a single nanopore are as follows [25]:
3ξ 3
Dp
(1−η3 − Dp + Dτ )
θϕa Ds dp2 ΔP NA dm3 P Lt
(16-c)
Ca = Cs
(16-d)
Cb = Cv + Ck
(16-e)
Ct = Ca + Cb
(16-f)
where Cv is the viscous flow conductance, (mol·m)/(Pa·s ) ; Ck is the Knudsen diffusion conductance, (mol·m)/(Pa·s ) ; cs is the surface diffusion conductance, (mol·m)/(Pa·s ) ; Ca is the adsorbed gas transport conductance, (mol·m)/(Pa·s ) ; Cb is the free gas transport conductance, (mol·m)/(Pa·s ) ; Ct is the total gas transport conductance, (mol·m)/(Pa·s ) .The free gas transport ratio (Γb) and adsorbed gas transport ratio (Γa) are calculated as follows:
(11-c)
where ϕb is the effective porosity for free gas transport, dimensionless (see Appendix B); ρ is the gas density, kg/m3 (see Appendix B); μ is the gas viscosity, Pa·s (see Appendix B); ξ is the taper ratio, dimensionless; M is the molecular weight, kg/mol ; P is pressure, Pa ; Z is the gas compressibility factor, dimensionless (see Appendix B); dm is the molecular diameter, m ; ϕa is the effective porosity for adsorbed gas transport, dimensionless (see Appendix B); Dτ is the fractal dimension of the tortuosity of pores, dimensionless; θ is the gas coverage, dimensionless (see Appendix B); Ds is the surface diffusion coefficient, m2/s (see Appendix B).
356
Γb =
Cb × 100% Ct
(17-a)
Γa =
Ca × 100% Ct
(17-b)
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4. Results and discussions
where Γb is the free gas transport ratio, dimensionless; Γa is the adsorbed gas transport ratio. The Darcy molar rate (Qd ) is obtained as follows:
Qd = −
KA ρAt ∇P μM
4.1. Impacts of pore size distributions on adsorbed and free gas transport Impacts of a pore size distribution on gas transport behavior are first explored. In this study, the pore size distribution in fractal shale matrix is determined by fractal dimension and minimum pore size. The connection between pore size distribution and flow feature is the porosity as shown in Eqs. (7) and (8). The connection between the pore size distribution fractal feature and flow features is thus presented based on effects of fractal dimension and minimum pore size on transport behavior and efficiency. The cumulative probability curves of pore diameters with different Dp under the same dpmin are plotted in Fig. 3(a). Fig. 3(a) indicates that with an increase in Dp , the small pores occur more frequently. Specially, the cumulative probability of pores with a pore diameter less than 5 nm (Fp (dp ⩽ 5nm)) is 98.2% for the Dp = 2.5 case, 98.7% for the Dp = 2.7 case and 99.1% for the Dp = 2.9 case. Fig. 3(b) indicates that with a decrease in Dp , the adsorbed gas transport ratio (Γa) increases and the free gas transport ratio (Γb) decreases; this is because a smaller Dp leads to more frequent occurrence of small pores, which shrinks the space of free gas transport. Free gas transport better benefits in the transport efficiency of nanopores [25], and thus the apparent permeability (KA ) increases with an increase in Dp as shown in Fig. 3(c). These show that a larger Dp yields to a larger free gas transport ratio and better transport efficiency. The cumulative probability curves with different dpmin under the same Dp are shown in Fig. 3(d). With a smaller dpmin , the frequency of occurrence of small pore increases. For instance, the cumulative probability of pores with the pore diameter less than 5 nm (Fp (dp ⩽ 5nm)) is 99.1% for the dpmin = 1nm case, 93.0% for the dpmin = 2nm case and 77.3% for the dpmin = 3nm case. More frequent occurrence of large pores yields a higher free gas transport ratio as shown in Fig. 3(e). When dpmin increases from 1 nm to 3 nm, KA increases with one order of magnitude, indicating that a larger minimum pore size represented by dpmin under the same Dp leads to better transport efficiency of shale matrix. Fig. 3(a)–(f) also indicate that even if the porosity is the same, the gas transport efficiency is significantly affected by the pore size distribution.
(18)
where Qd is the Darcy molar flow rate, mol/ s ; KA is the total apparent permeability, m2 . Considering Qd = Qt , the apparent permeability KA is calculated as follows [24,25]:
KA =
μMCt ρAt
(19)
Higher apparent permeability indicates better transport efficiency [24,25].
3. Model Validation In this section, we apply the model to calculate the pore size distribution and the apparent permeability, and their correctness is validated against the data from shale sample experiments. The distribution of pore sizes and apparent permeability are examined based on the experimental data from an Eagle Ford shale sample reported by Alnoaimi et al. (2013) [33]. The sample of diameter 2.5 cm by 4.6 cm in length is selected for the experiment. The pore size distribution of the shale sample is first measured by Alnoaimi et al. with non-localized density functional theory (NLDFT) method. After the determination of the pore size distribution, the apparent permeability is then tested. The apparent permeability is tested with helium as the probing gas. The permeability from the cited experiments refers to the apparent permeability instead of intrinsic permeability [33]. The intrinsic permeability is the property of rock itself, and does not change with the pressure and type of gas flowing within it. Apparent permeability is determined based on the Darcy’s law and equivalent to Darcy permeability [33]. Owing to the non-zero velocity of gas molecules on surfaces of pore walls, the apparent permeability will change with the pressure during the measurement [33]. Apparent permeability in shale gas systems is thus always larger than intrinsic permeability. In this study, we adjust the fractal dimensions, which is one of the most uncertain parameters, to achieve the match between model results and experimental data points. The pore size distribution and apparent permeability are also calculated based on our proposed model. From Fig. 2, we can see that the match with the experiment is excellent, which shows the industrial practicability of the proposed model.
4.2. Effects of pore size distribution on Knudsen diffusion Compared with conventional gas reservoir, Knudsen diffusion is an important mechanism for gas transport in shale matrix. Shale rocks are formed in deep underground by going through compaction and solidification with increasing pressure and temperature over a long period of time, which process provides a high-pressure and high-temperature condition for the gas transport in shale matrix. Effects of pore size Fig. 2. Validation with experimental data from Eagle Ford shale sample reported by Alnoaimi et al. (2013) [33]: (a) relationship between pore size and cumulative probability; (b) relationship between pressure and apparent permeability. T = 300K , L = 4.6cm , ϕ = 0.08 , Dp = 2.98 , dpmin = 4.9nm , Dτ = 1.1, ξ = 1.2 .
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Fig. 3. Impacts of pore size distribution on gas transport behavior: (a) cumulative probability curves of pore diameter under different fractal dimensions, dpmin = 1nm ; (b) relationship between fractal dimension and transport ratio for free gas and adsorbed gas, dpmin = 1nm ; (c) relationship between fractal dimension and apparent permeability, dpmin = 1nm ; (d) cumulative probability curves of pore diameter under different minimum pore diameters, Dp = 2.9 ; (e) relationship between minimum pore diameter and transport ratio for free gas and adsorbed gas, Dp = 2.9 ; (f) relationship between minimum pore diameter and apparent permeability, Dp = 2.9 . For (a)–(f), ϕ = 0.07 ,
L = 3cm , P = 25MPa , T = 350K , Dτ = 1.3 , ξ = 1.2 , At = 5cm2 .
4.3. Sensitivities of transport efficiencies to pore size distribution parameters
distribution on Knudsen diffusion are thus studied under a wide pressure and temperature range as shown in Fig. 4. From Fig. 4, we can find that under lower pressure and temperature, the contribution of Knudsen diffusion to total gas transport is higher. This is because under a lower pressure and temperature, the ratio of molecule-wall collision is higher, which leads to a higher contribution of Knudsen diffusion [34]. Fig. 4(a) and (b) indicate a higher Dp leads to a higher Knudsen diffusion ratio; this is because the occurrence frequency of small pores is lower with a higher Dp , which shrinks the distance of molecule-wall collision and increases the molecule-wall collision ratio. Fig. 4(c) and (d) indicate a higher minimum pore size leads to a lower Knudsen diffusion ratio; this is because a higher minimum pores results in a more frequent occurrence of large pores, which increases the distance of molecule-wall collision and decreases the molecule-wall collision ratio.
As the transport efficiency is affected by a pore size distribution, sensitivities of transport efficiency to the pore size distribution parameters are further studied. The slope (dKA/ dDp) of apparent permeability versus Dp is applied to qualify the sensitivity of transport efficiency to the fractal dimension; a larger absolute value of dKA/ dDp indicates higher sensitivity of apparent permeability to Dp . As shown in Fig. 5(a)–(c), the sensitivity of apparent permeability to Dp decreases with an increase in Dp ; this is because the free gas mainly contributes to the transport efficiency [25], and the ratio of free gas decreases with the increasing Dp (Fig. 3(b)). Fig. 5(a) indicates that the sensitivity of apparent permeability to Dp tends to be higher with an increase in porosity, which is due to the increase of the free gas transport ratio with an increase in porosity; that is, the apparent efficiency of more porous
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Fig. 4. Effects of pore size distribution on ratio of Knudsen diffusion: (a) relationship between pressure and ratio of Knudsen diffusion under different fractal dimensions, dpmin = 1nm ; (b) relationship between temperature and ratio of Knudsen diffusion under different fractal dimensions, dpmin = 1nm ; (c) relationship between pressure and ratio of Knudsen diffusion under different minimum pore sizes, Dp = 2.9 ; (d) relationship between temperature and ratio of Knudsen diffusion under different minimum pore sizes, Dp = 2.9 . For (a)–(d), ϕ = 0.07 , L = 3cm , P = 25MPa , T = 350K , Dτ = 1.3 , ξ = 1.2 , At = 5cm2 .
Fig. 5. (a) relationship between d (KA)/ d (Dp) and fractal dimension under different porosities, dpmin = 1nm , and P = 25MPa ; (b) relationship between
d (KA)/ d (Dp) and fractal dimension under different minimum pore diameters, ϕ = 0.07 , andP = 25MPa ; (c) relationship between d (KA)/ d (Dp) and fractal dimension under different pressures, ϕ = 0.07 , dpmin = 1nm .
L = 3cm , T = 350K , Dτ = 1.3 , ξ = 1.2 , and At = 5cm2 .
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pore size distribution on transport efficiency based on the isolines of apparent permeability of 100 nd. Fig. 7(a) indicates that for different shale matrix with the same transport efficiency under the same pressure, the shale matrix with a lower minimum pore size owns a larger porosity. Fig. 7(b) shows that a higher fractal dimension of a pore size pushes the isoline of apparent permeability to the region with higher porosity; this is in agreement with the conclusion in the previous section: a higher fractal dimension of a pore size results in a lower gas transport efficiency in shale matrix. 5. Conclusions An improved gas transport model in fractal shale matrix has been proposed in this study, which bridges a pore size distribution and multiple real gas transport mechanisms. Based on the proposed model, the effects of pore size distribution on gas transport efficiencies are further explored and highlighted. Validations of this model with experimental data have shown the reliability and practicability of the present model. The main conclusions are as follows:
Fig. 6. Relationship between apparent permeability, porosity and pressure. T = 350K , L = 3cm , ϕ = 0.13 , dpmin = 1nm , Dp = 2.9 , Dτ = 1.3, ξ = 1.2 , At = 5cm2 .
(1) A pore size distribution affects the gas transport efficiency by a fractal dimension and a minimum pore size. A larger fractal dimension of a pore size leads to higher frequency of occurrence of small pores and a lower free gas transport ratio, which further results in lower transport efficiency; a larger minimum pore size yields a higher free gas transport ratio and transport efficiency with increased frequency of occurrence of large pores. Contribution of Knudsen diffusion to the total gas transport is higher under lower pressure and temperature. Increasing fractal dimension and decreasing minimum pore size result in a higher Knudsen diffusion contribution. (2) Gas transport efficiency due to pore size distribution parameters (a fractal dimension and a minimum pore size) varies with different porosities and pressures. Decreased pressure and increased porosity enhance the sensitivity of gas transport efficiency to a pore size distribution. (3) A relationship between apparent permeability and porosity based on a pore size distribution is established, from which the apparent permeability can be estimated. For different shale matrix with the same transport efficiency under the same pressure, the shale matrix with a lower minimum pore size owns a larger porosity. A higher fractal dimension of a pore size pushes the isoline of apparent permeability to the region with higher porosity. (4) The developed relationship between permeability and porosity based on different pore size distribution indicates the apparent permeability may be an indicator of the porosity under a given pore size distribution and pressure condition. It helps the industry to
shale matrix is more subject to be affected by Dp . Fig. 5(b) shows that with a larger minimum pore diameter (dpmin ) , the apparent permeability is more sensitive to Dp , which is caused by the increased free gas transport ratio with increasing Dp (Fig. 3(e)). Fig. 5(c) indicates that larger pressure results in lower sensitivity of apparent permeability to Dp ; this is because the gas coverage increases with an increases in pressure, which results in a higher effective porosity for adsorbed gas transport and a lower free gas transport ratio.
4.4. Apparent permeability versus porosity based on pore size distributions The relationship between apparent permeability and porosity based on a pore size distribution is further established as shown in Fig. 6. Fig. 6 indicates that the apparent permeability increases with an increase in porosity and a decrease in pressure; this is because the adsorbed gas transport ratio decreases and the free gas transport ratio increases with increasing porosity and decreasing pressure. Fig. 6 also indicates that the apparent permeability may be an indicator of the porosity under a given pore size distribution and pressure condition. Taking the pressure condition of 1 MPa as an example, we can see: (1) if the apparent permeability is larger than 1000 nd under 1 Mpa, the porosity is higher than 0.13; (2) if the apparent permeability ranges between 100 nd and 1000 nd under 1 MPa, the porosity is between 0.05 and 0.13; (3) for the apparent permeability less than 100 nd under 1 MPa, the porosity is lower than 0.05. Fig. 7 further reveals effects of a
Fig. 7. (a) Isoline of apparent permeability of 100 nd under minimum pore diameters of 1 nm, 2 nm and 3 nm, Dp = 2.9 ; (b) Isoline of apparent permeability of 100 nd under fractal dimension of 2.7, 2.8 and 2.9, dpmin = 1nm . For (a) and (b), T = 350K , L = 3cm ,
ϕ = 0.13 , Dτ = 1.3, ξ = 1.2 , At = 5cm2 .
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Acknowledgments
quick determine the porosity based on apparent permeability based on a given pore size distribution. The apparent permeability can also be estimated with tested porosity under a given pore size distribution.
The authors would like to acknowledge the NSERC/AIEES/ Foundation CMG and Alberta Innovates - Technology Futures Chairs for providing research funding.
Appendix A.A. Fractal Properties of Pores in Shale Matrix For a fractal shale rock with a ratio δ < 1 between adjacent scales, the power-law relationship is as follows [35]:
dp,1 = δdp,0 dp,2 = δdp,1 = δ 2dp,0 dp,3 = δdp,2 = δ 3dp,0
⋮ dp,j = δdp,j − 1 = δ jdp,0 with
ln (dp,j / dp,0)
j=
(A-1)
lnr
where r is a scaling factor. If there are J structures of scale dp,j embedded within each structure of scale dp,j − 1, and J structures of scale dp,j − 1 embedded within each structure of scale dp,j − 2 …, and J structures of scale dp,1 embedded within each structure of scale dp,0 , then the total number of structures (Jr ) with a scale larger than dp,j is as follows:
Jt (dp,j ) = J j = J
dp,j ⎞ ln ⎛⎜ ⎟ / lnr ⎝ dp,0 ⎠
(A-2)
where Jt is the total number of structures with a scale larger than εj , dimensionless. Based on Eq. (A-2), we can have the following expression:
lnJt (εj ) =
ln (dp,j / dp,0) lnr
Let Dp =
lnJ − lnr
dp,0 ⎞ Jt (dp,j ) = ⎜⎛ ⎟ ⎝ dp,j ⎠
lnJ
(A-3)
and the expression of Jt (dpe,j ) is as follows:
Dp
(A-4)
Let dp,0 = dpmax and Jt = Np . We can have the expression of Np as in Eq. (1). Let dp,j = dpmin . We can have the expression of the total number of pores as Eq. (2). Appendix B.B. Real Gas Properties According to the Soave-Redlich-Kwong (SRK) equation of state (EOS), the solution of molar volume (Vm ) considering confinement is as follows [25]:
Vm = −
3P (χ 2 P + χRT −ακ ) + R2T 2 1 ⎛ B+ −RT ⎞ 3P ⎝ B ⎠ ⎜
g+
⎟
(B-1)
g 2−4[3P (χ 2 P + χRT −ακ ) + R2T 2]3
B=
3
g=
−2R3T 3−27ακχP 2−9RPT (χ 2 P
(B-2)
2 + χRT −ακ )
(B-3)
α = [1 + (0.48508 + 1.55171ω−0.15613ω2)(1− Tr )]2
κ = 0.42748
χ = 0.08664
(B-4)
R2Tc2 (B-5)
Pc
RTc Pc
(B-6)
(Pa·m3)/mol ;
m3/mol ;
α is a function of gas κ is an attraction parameter in the SRK EOS, where χ is a repulsion parameter in the SRK EOS, T properties and temperature in the SRK EOS, dimensionless; ω is the acentric factor, dimensionless; Tr is the reduced temperature, dimensionless, = T ; Vm is the molar volume, m3/mol ; Pc is the critical pressure, Pa ; Tc is the critical temperature, K . The molecular number density (n ) under the real gas effect is expressed as follows:
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n=
NA Vm
(B-7)
m−3 ;
1023mol−1.
NA is the Avogadro constant, = 6.02 × where n is the molecular number density, The mean molecular speed (v ) from the Maxwell-Boltzmann distribution is given below:
2RT πM
v =2
(B-8)
where v is the mean molecular speed from the Maxwell-Boltzmann distribution, m/s . The mean free path (λ ) can be expressed as follows:
λ=
1 πndm2
(B-9)
where λ is the mean free path, m . The density ( ρ ) can be calculated in terms of n as follows:
ρ=
Mn NA
(B-10)
The viscosity ( μ ) is obtained as follows [36]: 2
Pr4 o1 ⎛ ⎞ ⎛ Pr ⎞ ⎛ Pr ⎞ ⎤ μ = μ0 ⎡ ⎢1 + T 5 ⎜ T 20 + P 4 ⎟ + o2 Tr + o3 Tr ⎥ r r r ⎝ ⎠ ⎝ ⎠⎦ ⎝ ⎠ ⎣ ⎜
⎟
⎜
⎟
(B-11)
where μ0 is the gas viscosity at P = 1.01325 × 105 Pa and T = 423 K, = 2.31 × 10−5Pa·s ; Pr is the reduced pressure, dimensionless, =
o3 are the fitting coefficients, o1 = 7.9, o2 = 9 × 10−6 , o3 = 0.28. The compressibility factor (Z ) in terms of n is as follows: Z=
P nkT
P ; Pc
o1, o2 and
(B-12)
10−23J / K .
where k is the Boltzmann constant, = 1.38 × The gas coverage (θ ) is calculated as follows [37]:
P /Z P / Z + PL
θ=
(B-13)
where PL is the Langmuir pressure, Pa . The total cross-sectional area occupied by the free gas ( Apb ) is calculated as follows:
Apb = −
∫d
dpmax
pmin
π (dp−2θdm)2 4
dNp
(B-14)
By coupling Eqs. (1), we can obtain Apb as follows:
Apb =
2 (1−η2 − Dp ) 4θdm dpmax (1−η1 − Dp ) 4θ 2dm2 (1−η−Dp ) ⎤ πDp ⎡ dpmax − − ⎢ ⎥ 4 ⎣ 2−Dp 1−Dp Dp ⎦
(B-15)
The effective porosity for free gas transport (ϕb ) and adsorbed gas transport (ϕa ) are calculated as follows:
ϕb =
Apb Ap
ϕ
(B-16)
ϕa = ϕ−ϕb
(B-17)
The Knudsen number (Kn ) is calculated as follows:
Kn =
λ dpm
(B-18)
where Kn is the Knudsen number, dimensionless; dpm is the mean pore diameter, m, =
Dp
d . Dp − 1 pmin
The surface diffusion coefficient (Ds ) is expressed as follows [38]:
ΔH 0.8 ⎞ 2(1−θ) + Ψθ (2−θ) + [H (1−Ψ)](1−Ψ)Ψθ 2 Ds = 8.29 × 10−7T 0.5exp ⎛− Ψ 2(1−θ + 2 )2 ⎝ RT ⎠ ⎜
⎟
(B-19)
where ΔH is the isosteric adsorption heat at θ = 0 , J/mol ; H () is the Heaviside step function; Ψ is the ratio between the blockage rate constant and the forward migration rate constant, dimensionless.
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