Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
NEGF2015/5
INVESTIGATION OF VARIOUS PERMEABILITY LAWS IN TIGHT
POROUS MEDIA
S. Kokou Dadzie, William G. Tubby, Chariton Christou
School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS,
Scotland, UK
k.dadzie@hw.ac.uk
KEY WORDS
Micro/nano fluid dynamics - Mean free path - Shale gas
ABSTRACT
Permeability in shale strata is poorly understood and cannot be predicted using conventional continuum-flow
equations such as Darcy’s law. Literature focuses on gas flow regimes based on the range of Knudsen number
(the ratio of mean free path to pore throat diameter) to assess gas flows in shale nanopores. This paper presents
an analytical approach towards investigating the matrix permeability of low pressure shale gas reservoirs in the
transition flow regime. It is reported that the valuation of the mean free path of a gas, and thus the Knudsen
number, affects the interpretation of experimental data for permeability laws. An analysis of the Knudsen
number based upon the experimental data by (Cooper et al., 2004) in characterizing gas flows in
nanotubes/shale-strata is conducted and the implications are explored regarding different existing Knudsen
number dependent permeability correlations. Results show a considerable increase in permeability values which
corroborate the enhanced permeability phenomenon reported in real field data.
1.
INTRODUCTION
Shale (natural) gas emits less environmentally harmful pollutants per unit of energy produced relative
to other fossil fuels and it is in abundant supply [1]. Despite predictions that shale gas is set to be the
energy solution of the future, full understanding of gas transport mechanisms in shale formations has
not yet been achieved causing difficulties in long term reservoir production and economic forecasting.
Gas permeability in shale using conventional reservoir models tends to be several orders of magnitude
lower than permeability observed from real field data [2]. Understanding the geological makeup and
pore structure characteristics of shale formations which comprise of complex pore networks is critical
for the true measurement of permeability. Shales are sedimentary rocks, composed of fine-grained
material such as clay minerals, quartz, silt, and other organic matter which typically leads to pore sizes
in the nanometer range [3]. This along with the tortuous nature of shale pores lead to uncertainties in
permeability estimated based upon Navier-Stokes approaches [4].
Basic continuum theory such as Darcy’s law is widely used to predict permeability and fluid flows in
conventional reservoir rocks. The empirical Klinkenberg model is often implemented to account for
gas slippage – a process which occurs under low pressure conditions in tight porous media. The ultratight nanopores in shale strata, however, cause additional phenomena such as Knudsen diffusion to
become significant [5]. In order to simulate fluid flows in shale formations more accurately,
implementation of new flow equations are required.
1
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
Javadpour, Beskok and Karniadakis proposed models for fluid flow in nanochannels [6, 7]. Beskok
and Karniadakis corrected the Navier-Stokes solution by modifying the second-order slip approach.
This allowed Civan and Florence to model gas flows in shale strata [8, 9]. Durabi proposed a
permeability model for gas flows in ultra-tight natural porous media which accounts for Knudsen
diffusion, slip, and also surface roughness [5]. Note that as pore throat sizes increase (into the
micrometre range) all these models tend toward the basic continuum theory. Dadzie and Brenner
provided a model for gas flows in micro/nanochannels based upon their volume diffusion theory that
also accounts for slip and Knudsen diffusion [10].
2.
KNUDSEN NUMBER AND FLUID FLOW REGIMES
Knudsen was the first to report data suggesting anomalous behaviour of gas flowing in nanochannels
[10]. The Knudsen number is a way to characterise the gas transport regime in a shale [4]. The
Knudsen number is a dimensionless parameter and is the ratio of the mean free path of the gas 𝜆 to
average pore throat diameter 𝑑𝑝 [2].
𝜆
𝐾𝑛 =
(1)
𝑑𝑝
The mean free path of a gas is defined as the average distance a molecule travels before it collides with
another molecule [11]. Expressions for the mean free path of a gas differ between authors causing
variation in the value of the Knudsen number which ultimately impacts flow rate and permeability
calculations. The relationship between Knudsen number, pore throat diameter and pressure is
displayed in Fig. 1 highlighting pressures and pore throat sizes typical of shale gas formations.
While authors generally agree that the Knudsen number limit for Darcy flow is 0.01, it should be noted
that some researchers limit it to 0.001 [12]. Each flow regime is described as follows:
Darcy Flow
In this region, the Knudsen number is so small that the mean free path of gas molecules is negligible
relative to the pore throat diameter and thus basic continuum theory applies. Intermolecular collisions
are important, whereas molecular collisions with pore walls are negligible and gas flow velocity
directly adjacent to the pore wall is zero [12]. Conventional reservoirs typically fall into this category
since they contain pore diameters in the micrometre range resulting in very low Knudsen numbers.
Slip Flow
The gas flow velocity directly adjacent to the pore wall is not zero in this flow regime and therefore
modification of basic continuum theory is required to accommodate the slip condition. The mean free
path of the gas is significant relative to the pore throat diameter and thus molecular collisions with the
pore walls become important in comparison to intermolecular collisions [12].
Transition Flow
Transition regime flows are of particular interest to researchers within the petroleum industry as most
gas shale formations fall into this category. They are the most challenging to model/simulate [12]. In
addition to slip phenomena, other physical phenomenon such as Knudsen diffusion occurs. The
present study takes an analytical approach towards investigating flow in the transition regime.
Knudsen Flow
2
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
In this region, Knudsen numbers are large and gas composition has no significance [13]. As
demonstrated in Fig. 1, shale formations do not typically fall into this category and thus researchers
within the petroleum industry have tended to pay less attention to this flow regime relative to the other
three.
As pressure (gas density) decreases the mean free path of the gas increases and therefore low pressure
gas shales result in higher Knudsen numbers. Shale gas formations tend to lay within slip, transition
and Darcy flow regimes.
Figure 1: Knudsen number versus pore diameter (nm) at different pressures for methane at 100˚C [13]
3.
PERMEABILITY
In 1856, French engineer Henry Darcy first provided a relation which permits the measurement of
fluid flow through porous media [14]. Darcy acknowledged that the rate of fluid flow through rock
formations varies directly with a numerical quantity, now commonly known as permeability. The
ability of a fluid to pass through the interconnected pore spaces of a rock denotes the degree of
permeability possessed by the rock formation and it should be noted that Darcy permeability is purely
a function of pore geometry [15]. Darcy also recognized that for the measurement of fluid flow
through porous media, the force which causes the fluid to flow (pressure) and the viscosity 𝜇 of the
flowing fluid must also be taken into consideration. Darcy’s law is given as:
𝑄=
𝑘𝐴𝑥𝑠𝑎 𝛥𝑃
𝜇𝐿
(
(2)
Klinkenberg was the first person in the oil industry to identify the phenomena of slip [16]. He
discovered that the rate of fluid flow through porous media at very low pressures (such as in low
pressure gas shales) was actually more than predicted by Darcy’s law. He hypothesized that this was
because under these conditions the velocity of molecules along the pore walls is not zero and this
occurs when the diameter of the pore approaches the mean free path of the gas. Klinkenberg proposed
the following equation for apparent (effective) gas permeability 𝑘𝑎 which takes into account the
additional consideration of gas slippage:
𝑏𝑘
𝑘𝑎 = 𝑘∞ (1 + )
(3)
𝑃
where 𝑘∞ is the reference/intrinsic permeability, which is purely a function of pore geometry, and is
given by:
𝜙𝑟𝑝 2
𝑘∞ =
(4)
8𝜏
3
(
(
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
with 𝜙 the porosity and 𝜏 the tortuosity factors [8]. The slippage factor 𝑏𝑘 relates to the mean free path
of the gas and is given by:
4𝑐𝜆𝑃
𝑏𝑘 =
(5)
𝑟𝑝
where 𝑐 ≈ 1.
(
Klinkenberg approach generally works well in slip flow regimes. Additional phenomena such as
Knudsen diffusion are, however, often taken into account for transition flow regimes. Few studies
have been undertaken investigating fluid flows in the transition region. In 1999, Beskok and
Karniadakis published their work which looked at rarefied gas flows in a wide range of Knudsen
number (0 ≤ 𝐾𝑛 < ∞) [7]. This enabled Civan and Florence to approximate apparent permeability as
a function of Knudsen number. Civan’s equation is as follows [8]:
𝑘𝑎 = 𝑘∞ (1 + 𝛼𝑜 (
𝐾𝑛𝐵
4𝐾𝑛
) 𝐾𝑛) (1 +
)
𝐵
𝐴 + 𝐾𝑛
1 + 𝐾𝑛
(
(6)
Extracting the data provided by Civan for flows in the transition region: A = 0.178, B = 0.4348 and 𝛼0
= 1.358.
The model derived by Florence for transition flow can be expressed as [9]:
(
𝑘𝑎 = 𝑘∞ (1 + 4𝐾𝑛)
(7)
It is clear that, from Eq. (6) and Eq. (7), apparent gas permeability in porous media depends on
Knudsen number.
The present study takes an analytical approach towards generating a consistent Knudsen number
which complies with Cruden’s experimental data and thus providing enhanced solutions to Knudsen
number dependent apparent gas permeability correlations.
4.
THE MEAN FREE PATH ANALYSIS
Expressions for the mean free path of a gas differ between researchers, although similarities do exist.
The mean free path expressions investigated in this paper all contain the following common factor:
𝜇√2𝑅𝑇
𝜆=
(8)
𝑃
Maxwell provides two correlations for the mean free path of a gas. The first is [17]:
(
1
2𝑃 2 𝜇
𝜆 = 2( ) .
𝜋𝜌
𝑃
The density in Eq. (9) can be eliminated in favour of the pressure via use of the ideal gas law:
𝑃
𝑃 = 𝜌𝑅𝑇 → 𝜌 =
𝑅𝑇
Therefore, Maxwell’s first expression becomes:
(
(9)
(
(10)
1
2𝑃𝑅𝑇 2 𝜇
2
𝜇√2𝑅𝑇
𝜆 = 2(
) . =
×
𝜋𝑃
𝑃 √𝜋
𝑃
The second correlation Maxwell provides is as follows:
1
(
(11)
3
𝜋 2
𝜆 = 𝜇(
)
(12)
2 2𝑃𝜌
Again, eliminating the density in favour of the pressure via the ideal gas law furnishes the expression:
4
(
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
1
3 𝜋𝑅𝑇 2
𝜆 = 𝜇 ( 2)
2 2𝑃
Multiplying Eq. (13) by √2⁄√2 to obtain 𝜆 yields:
(
(13)
3√𝜋 𝜇√2𝑅𝑇
×
(14)
4
𝑃
Roy and Bird both provide a correlation for the mean free path of a gas using the Chapman-Enskog
expansion for the coefficient of viscosity in a hard sphere intermolecular collision model gas [18].
Roy’s expression is as follows [19]:
𝜇
𝜋
𝜆= √
(15)
𝜌 2𝑅𝑇
𝜆=
(
(
Application of the ideal gas law to eliminate the density yields:
𝜆=
𝜇𝑅𝑇
𝜋
𝜋 √𝑅𝑇
√
=√ 𝜇
𝑃
2𝑅𝑇
2
𝑃
(
(16)
Multiplying Eq. (16) by √2⁄√2 to obtain 𝜆 gives:
√𝜋 𝜇√2𝑅𝑇
×
2
𝑃
(17)
1
16𝜇
(2𝜋𝑅𝑇)−2 /𝜌
5
(18)
𝜆=
(
Bird’s expression is as follows [20]:
𝜆=
(
Applying ideal gas law yields:
𝜆=
16𝜇
𝑅𝑇
16𝜇 √𝑅𝑇
=
5√2𝜋𝑅𝑇 𝑃
5√2𝜋 𝑃
(
(19)
Multiplying Eq. (19) by √2⁄√2 to obtain 𝜆 gives:
𝜆=
16
𝜇√2𝑅𝑇
8
𝜇√2𝑅𝑇
=
×
𝑃
5(2√𝜋) 𝑃
5√𝜋
Ewart also uses a mean free path expression similar to that of the hard sphere model [21]:
𝜇
𝜆 = 𝑘𝜆 √2𝑅𝑇
𝑃
where
√𝜋
𝑘𝜆 =
2
leading to:
(
(20)
(
(21)
(
(22)
(
√𝜋 𝜇√2𝑅𝑇
×
(23)
2
𝑃
The various mean free path expressions are summarised in Table. 2 and presented schematically in
Fig. 2 using Cruden’s experimental data set which is described in the following section.
𝜆=
As demonstrated in Fig. 2, the Knudsen number follows a similar trend for each author – the Knudsen
number steadily decreases as pressure increases. However, the Knudsen number range varies
significantly for Cruden’s experimental data set. The ratio of highest (Maxwell (2)) to lowest (Roy) is
3:2. The same ratio can be obtained using the values for the highest and lowest mean free path
coefficient 𝑘𝜆 presented in Table 2.
5
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
Figure 2: Variation of Knudsen number across Cruden’s experimental range using the different mean free path
expressions.
Author
kλ
Ewart
√
𝜋
2
8
1.25
Bird
5√ 𝜋
0.90
Roy
√𝜋
2
2
0.89
Maxwell (1)
Maxwell (2)
kλ
[value]
√𝜋
3√ 𝜋
4
1.13
1.33
Table 2: Summary of the different coefficients kλ to the common factor, 𝜆.
5.
A HOMOGENEOUS KNUDSEN NUMBER APPROACH
An expression for the mass flow rate through a rectangular nanochannel based upon volume diffusion
theory accounting for slip and Knudsen diffusion is given in [10]:
𝑀̇ =
𝑤ℎ3 𝑃𝑜2
1
𝑙𝑛𝑃
(𝑃 2 − 1) [1 + 𝐴𝐾𝑛𝑜
+ 𝐵𝐾𝑛𝑜 2
]
24𝐿𝜇𝑔 𝑅𝑇
𝑃+1
𝑃 −1
(24)
where the various coefficients appearing therein are described in Table 3. The mass flux expression
from Eq. (24) is obtained by dividing through by area (ℎ × 𝑤):
𝐽=
ℎ2 𝑃𝑜2
1
𝑙𝑛𝑃
(𝑃 2 − 1) [1 + 𝐴𝐾𝑛𝑜
+ 𝐵𝐾𝑛𝑜 2
]
24𝜇𝐿𝑅𝑇
𝑃+1
𝑃 −1
6
(25)
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
A
B
12𝑘𝑠𝑙𝑖𝑝
P
24
𝑃𝑟𝑘𝜆2
Kno
𝑝𝑖 ⁄𝑝𝑜
Kslip
√𝜋 √2
×
2
𝑘𝜆
𝑘𝜆 𝜆𝑜 ⁄𝑑𝑝
Table 3: Summary of the different coefficients (kλ) to the common factor, 𝜆.
In Eq. (25), the hydraulic radius rh may be used. The hydraulic radius is defined as the flow area
divided by the wetted perimeter of the conduit [22]. If the cross-sectional area of the tube is assumed
to be a square, rh is obtained by the expression:
𝑟ℎ =
ℎ2 ℎ
=
4ℎ 4
(26)
Rearranging equation (26) yields:
(27)
ℎ2 = 16𝑟ℎ2
Therefore, the mass flux of a fluid through a cylindrical nanotube can be calculated using the Dadzie
and Brenner model in the following form:
𝐽=
2𝑟ℎ 2 𝑃𝑜2 2
1
𝑙𝑛𝑃
(𝑃 − 1) [1 + 𝐴𝐾𝑛𝑜
+ 𝐵𝐾𝑛𝑜 2
]
3𝐿𝜇𝑅𝑇
𝑃+1
𝑃 −1
(28)
Cruden provided an experimental data set studying the relationship between mass flux and pressure
drop for argon gas in a straight cylindrical nanotube [23]. The geometrical characteristics of Cruden’s
experimental nanotube are disclosed in table 4:
T
300 [K]
µ
2.22 x 10-5 [Pa s]
rh
117.5 x 10-9 [m]
Po
4800 [Pa]
L
60 x 10-6 [m]
Pr
0.68
Table 4: Summary of fluid (argon) properties and physical coefficients.
Application of the mass flow rate Eq. (28) relative to Cruden’s experimental data allows a consistent
Knudsen number to be established. The solution of Eq. (28) is calculated using the value of kλ for each
author previously obtained. So that the theoretical values for each author lie in agreement with
Cruden’s experimental data, kλ is multiplied by an alteration factor α. Table 5 summarises this
analysis. We obtain as a result 𝛼 × 𝑘𝜆 ≈ 6.0 the new coefficient to 𝜆 for determining the mean free
path that results in the same Knudsen number range among all authors. In other words, the actual
Knudsen number range corresponding to same values among all authors and in agreement with
Cruden’s experiments and Eq. (28) is now 6.69-8.36.
Author
Ewart
Bird
Roy
Maxwell (1)
Maxwell (2)
𝜆 = 𝑘𝜆 × 𝜆
[10-7]
3.28 - 4.09
2.36 - 2.95
2.32 - 2.89
2.94 - 3.38
3.47 - 4.34
Kn Range
α
1.39 - 1.74
1.00 - 1.25
0.98 - 1.23
1.25 - 1.57
1.48 - 1.85
4.80
6.68
6.80
5.34
4.53
λxα
[10-6]
1.57 – 1.97
1.57 – 1.97
1.57 – 1.97
1.57 – 1.97
1.57 – 1.97
New Kn
Range
6.69 – 8.36
6.69 – 8.36
6.69 – 8.36
6.69 – 8.36
6.69 – 8.36
Table 5: Summary of values achieved following correction of the Dadzie model for mass flux, Eq. (28)
Figure 3 presents the resulting mass flux from Eq. (28) and the experimental data. The flux model
provided by Javadpour [24] and the Hagen-Poiseuille flow (K/HP) analytical solution by Guo are also
included in this comparison [25].
7
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
Figure 3: Comparison between K/HP analytical solution, Javadpour’s model (with tangential accommodation
coefficient equals to one), Dadzie and Brenner’s corrected model, and experimental data for pressure difference
versus flux [23]
In order to assess the validity of our correction with regards to the Knudsen number, the corrected
apparent permeability models are compared with original models proposed by Civan and Florence.
This is shown in Fig. 4 and Fig. 5.
Figure 4: Comparison of apparent permeability versus Figure 5: Comparison of apparent permeability versus
radius between corrected models and original models radius between corrected models and original models
in the transition flow regime
Note that the corrected correlations show enhanced permeability measurements at smaller pore throat
diameters relative to the original correlations. This corroborates field data observation for low pressure
gas shales [2]. Furthermore, the apparent gas permeability converges towards basic continuum theory
values with increasing pore throat size.
5. CONCLUSION
In this paper it is reported that the valuation of the mean free path of a gas and thus the Knudsen
number affects the interpretation of experimental data for permeability laws. Various correlations for
quantifying the mean free path of a gas were summarized and compared in this work. Using
experimental data provided by Cruden and, Dadzie and Brenner mass flow rate theoretical model, a
new alteration factor has been introduced for the mean free path to characterize gas flow in
nanotubes/shale-strata in the transition flow region. The effects of this modification were explored
using the Knudsen number dependent permeability correlations postulated by previous investigators.
Results demonstrate a considerable increase in the permeability.
8
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
NOMENCLATURE
=
𝐴
=
𝐴𝑥𝑠𝑎
=
𝐵
=
𝑏𝑘
=
𝑐
=
𝑑𝑝
=
ℎ
=
𝐽
=
𝐾𝑛
=
𝐾𝑛𝑜
=
𝑘
=
𝑘𝑎
=
𝑘∞
=
𝐿
=
𝑃
=
𝑃
=
𝑃𝑖
=
𝑃𝑜
=
𝛥𝑃
=
𝑃𝑟
=
𝑄
=
𝑅
=
𝑟ℎ
=
𝑟𝑝
=
𝑇
=
𝑤
GREEK LETTERS
=
𝛼
=
𝛼𝑜
=
𝛫𝜆
=
𝜆
=
𝜆
=
𝜇
=
𝜌
=
𝜏
=
𝜙
Constant used by Civan, 0.178
Cross-sectional area, m2
Constant used by Civan, 0.4348
Slippage factor
Constant used by Klinkenberg, 1
Pore diameter, m
Height, m
Flux, mol/m2s
Knudsen number
Outlet Knudsen number
Permeability, m2
Apparent permeability, m2
Intrinsic permeability, m2
Length, m
Pressure, Pa (or Torr when specified)
Average pressure, Pa
Inlet pressure, Pa
Outlet pressure, Pa
Pressure change, Pa
Prandtl number
Volume flow rate, m3/s
Ideal gas constant, J/mol.K
Hydraulic radius, m
Pore radius, m
Temperature, K
Width, m
Correction coefficient
Constant used by Civan, 1.358
Mean free path coefficient
Mean free path
Mean free path common factor
Gas viscosity, Pa.s
Gas density, kg/m3
Tortuosity factor
Porosity
REFERENCES
[1] EIA, "Annual Energy Outlook," US Energy Information Administration, Washington DC, 1999.
[2] A. Islam and T. Patzek, "Slip in natural gas flow through nanoporous shale reservoirs," Journal of
Unconventional Oil and Gas Resources, vol. 7, pp. 49-54, 2014.
[3] D. J. Soeder, "Petrophysical Characterization of the Marcellus and Other Gas Shales," in U.S.
Department of Energy , Arlington, 2011.
[4] M. Firouzi, K. Alnoaimi, A. Kovscek and J. Wilcox, "Klinkenberg effect on predicting and
measuring helium permeability in gas shales," International Journal of Coal Geology, vol. 123,
pp. 62-68, 2014.
9
Proceedings of the 2nd European Conference on Non-equilibrium Gas Flows – NEGF15
December 9-11, 2015 – Eindhoven, the Netherlands
[5] H. Darabi, A. Ettehad, F. Javadpour and K. Sepehrnoori, "Gas flow in ultra-tight shale strata,"
Journal of Fluid Mechanics, vol. 710, pp. 641-658, 2012.
[6] F. Javadpour, "Nanopores and apparent permeability of gas flow in mudrocks (shales and
siltstone)," Journal of Canadian Petroleum Technology, vol. 48, no. 8, pp. 16-21, 2009.
[7] A. Beskok and G. E. Karniadakis, "A Model for Flows in Channels, Pipes, and Ducts at Micro
and Nano Scales," Microscale Thermophysical Engineering, vol. 3, no. 1, pp. 43-78, 1999.
[8] F. Civan, "Effective correlation of apparent gas permeability in tight porous media," Transport in
Porous Media, vol. 82, no. 2, pp. 375-384, 2010.
[9] F. A. Florence, R. J. A., K. E. Newsham and T. A. Blasingame, "Improved Permeability
Prediction Relations for Low Permeability Sands," Colorado, 2007.
[10] S. K. Dadzie and H. Brenner, "Predicting Enhanced Mass Flow rates in Gas Microchannels Using
Nonkinetic Models," Physical Review E, vol. 86, no. 3, 2012.
[11] R. D. Knight, Physics for Scientists and Engineers: A strategic Approach, Addison-Wesley, 2003.
[12] V. Swami, C. Clarkson and S. T. , "Non Darcy Flow in Shale Nanopores: Do We Have a Final
Answer?," Calgary, 2012.
[13] R. Heller, J. Vermylen and M. Zoback, "Experimental investigation of matrix permeability of gas
shales," AAPG Bulletin, vol. 98, no. 5, pp. 975-995, 2014.
[14] H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Paris, 1856, p. 647 p. & atlas.
[15] J. M. Somerville, Introduction to Petroleum Engineering, course notes, p427, Edinburgh: Heriot
Watt, 2001.
[16] L. Klinkengerg, The Permeability of Porous Media to Liquids and Gases, New York: American
Petroleum Institute, 1941, pp. 200-213.
[17] J. C. Maxwell, "On Stresses in Rarified Gases Arising from Inequalities of Temperature,"
Philosphical Transactions of the Royal Society of London, vol. 170, pp. 231-256, 1879.
[18] S. Chapman and T. G. Cowling, The Mathematical Theory Of Non-Uniform Gases, Cambridge:
Cambridge Mathematical Library, 1991.
[19] S. Roy, R. Raju, H. F. Chuang, B. A. Cruden and M. Meyyappan, "Modeling Gas Flow Through
Micro-channels and Nano-pores," Journal of Applied Physics, vol. 93, pp. 4870-4879, 2003.
[20] G. A. Bird, "Definition of Mean Free Path for Real Gases," Phys. Fluids, vol. 26, no. 11, pp.
3222-3223, 1983.
[21] T. Ewart, P. Perrier, I. Graur and J. M. Meolans, "Mass flow rate measurements in a
microchannel, from hydrodynamic to near free molecular regimes," Journal of Fluid Mechanics,
vol. 584, pp. 337-356, 2007.
[22] Neutrium,
"Hydraulic
Diameter
–
Neutrium,"
2012.
[Online].
Available:
https://neutrium.net/fluid_flow/hydraulic-diameter/.
[23] S. M. Cooper, B. A. Cruden, M. Meyyappan, R. Raju and S. Roy, "Gas Transport Characteristics
through a Carbon Nanotubule," Nano Letters, vol. 4, no. 2, pp. 377-381, 2004.
[24] F. Javadpour and D. Fisher, "Nanoscale Gas Flow in Shale Gas Sediments," Journal of Canadian
Petroleum Technology, vol. 46, no. 10, 2007.
[25] C. Guo, B. Bai, M. Wei, X. He and Y. Wu, "Study on Gas Permeability in Nano Pores of Shale
Gas Reservoirs," Calgary, 2013.
10