Journal of Unconventional Oil and Gas Resources 7 (2014) 49–54
Contents lists available at ScienceDirect
Journal of Unconventional Oil and Gas Resources
journal homepage: www.elsevier.com/locate/juogr
Slip in natural gas flow through nanoporous shale reservoirs
Akand Islam ⇑, Tad Patzek
Department of Petroleum & Geosystems Engineering, The University of Texas at Austin, TX 78712, USA
a r t i c l e
i n f o
Article history:
Received 6 December 2013
Revised 30 April 2014
Accepted 11 May 2014
Available online 16 June 2014
Keywords:
Slip flow
Knudsen diffusion
Nanopore
Shale reservoir
Hagen–Poiseuille equation
a b s t r a c t
It is observed that to assess the shale gas flow in nanopores the recent literature relies on the flow
regimes discovered by Tsien (1946). Tsien classified fluid flow systems based on the range of Knudsen
number (Kn), the ratio of the mean free path to average pore diameter. The flow regimes are: continuum
flow for Kn < 0.01, slip flow for 0.001 < Kn < 0.1, transition regime for 0.1 < Kn < 10, and free molecule flow
for Kn > 10. This scale was originally developed from the physics of rarefied gas flow. Is it then appropriate to use the classical Kn scale to develop models of shale gas flow in tight reservoirs where the nanopores are in the range of 1–1000 nm, and pore pressures can be as high as 10,000 psi? The present work
explores answers to this question. We provide an analysis based on classical slip flow model. We validate
the Kn scale incorporating PVT (Pressure–Volume–Temperatures) schemes. Our results show that in very
tight shale (order of 1 nm pore size) there can be substantial slip flow based on the characteristics of pore
walls in the reservoirs of high temperatures and low pressures. In the case of large pore size (1000 nm)
there is zero slip flow irrespective of temperature and pressure. The Kn scale which was designed for rarefied gases cannot be true for the natural gas flow regimes at all temperatures and pressures. Therefore
we must be careful in referring this scale to model the shale gas flows. Results presented here from simple calculations agree with those obtained from expensive molecular dynamics (MD) simulations and
laboratory experiments.
Ó 2014 Elsevier Ltd. All rights reserved.
Introduction
Gas bearing shale strata are important energy resources in
North America and they are becoming increasingly important
across the world. Shale gas resources in US comprise over 20 trillion cubic meters equivalent to one-third of the total natural gas
reserve (US Energy Information Administration, 2011). However,
gas production in these formations is difficult justifying their
classification as ‘‘unconventional’’ (Passey et al., 2010; Shabro
et al., 2011). Shales are fine-grained sedimentary rocks containing
high-volume fractions of silt, clay minerals and some organic matter (Tyson, 1995). Shale gas reservoirs customarily have extremely
low permeability. Pore size is usually in the range of nanometers,
10–300 nm, pores are poorly connected, and therefore the permeability is in the range of nanodarcy. Sondergeld et al. (2010)
analyzed two-3D volumes and found that the pore volume is dominated by the smallest pores with larger pores associated with the
better reservoir rock. The mean pore volumes observed were
2188 nm3 and 8053 nm3, respectively, corresponded to characteristic dimensions of 13 and 20 nm. Wang and Reed (2009) reported
⇑ Corresponding author. Tel.: +1 5124365343.
E-mail address: [email protected] (A. Islam).
http://dx.doi.org/10.1016/j.juogr.2014.05.001
2213-3976/Ó 2014 Elsevier Ltd. All rights reserved.
the least pore size as 5 nm. Pore throat diameter in typical shale
gas formations can be even as low as 0.5 nm (Devegowda et al.,
2012). In conventional systems, e.g., rocks with micron size pores,
Darcy permeability is used to compute gas flow. However, the
Darcy equation cannot be used for fine-grained shale strata
because of nanopores. In the case of micropores (conventional reservoirs) and nanopores (unconventional resources) both noncontinuum effects and dominant surface interactive forces become
important reserving the use of Darcy flow (Cooper et al., 2003;
Roy and Raju, 2003; Karniadakis et al., 2005; Hadjicontantinou,
2006; Javadpour et al., 2007; Hornyak et al., 2008; Javadpour,
2009). It is expected that in nanopores the measured flow can be
several orders-of-magnitude higher than the predictions of continuum hydrodynamics models (Majumder et al., 2005; Holt et al.,
2006; Gault and Scotts, 2007; Freeman et al., 2010; Kale et al.,
2010; Sondergeld et al., 2010). In order to capture pore scale flow
mechanisms in shale reservoirs in almost all studies performed,
Darcy equation is modified by incorporating the physics at molecular level to account for higher equivalent permeability. To list,
Ertekin et al. (1986) showed that Kinkenberg’s slippage effect
(Klinkenberg, 1941) is responsible for the higher than predicted
value of production in tight/shale reservoirs; Javadpour’s (2009)
observation was that both slippage and Knudsen diffusion become
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A. Islam, T. Patzek / Journal of Unconventional Oil and Gas Resources 7 (2014) 49–54
Fig. 1. (a–h) Flow rate at different nanopores. Axis representing ‘f’ is the fraction of gas molecules diffusely reflected in pore walls. More is discussed in ‘Governing equations’
section.
A. Islam, T. Patzek / Journal of Unconventional Oil and Gas Resources 7 (2014) 49–54
important at the shale nanopore scale; Ozkan et al. (2010) found
role of concentration driven diffusion in addition to Darcy flow;
Civan (2010) and Civan et al. (2011) applied the Beskok and
Karniadakis (1999) model to simulate gas flow in tight porous
media; Clarkson et al. (2011) modified the Darcy’s permeability
applying Ertekin’s dynamic slippage concept and investigated its
applicability in the rate transient analysis. To match production
data Swami and Settari (2012) manipulated reservoir and stimulated reservoir volume parameters. Darabi et al. (2012) formulated
a pressure-dependent permeability function assuming Knudsen
diffusion and slip flow. Monteiro et al. (2012) hypothesis was that
the permeability is dependent on pressure gradient and so they
used simple power rule.
While modeling shale gas flow in nanopores the aforementioned published papers have incorporated the flow regimes
discovered by Tsien (1946). Tsien and further slightly redefined
by Karniadakis et al. (2005) classified fluid flow systems based
on the range of Knudsen number (Kn). To recap, Kn is the ratio of
the mean free path to average pore diameter. The flow regimes
are: continuum flow for Kn < 0.01, slip flow regime for
0.001 < Kn < 0.1, transition regime for 0.1 < Kn < 10, and free
molecule flow for Kn > 10. This scale was originally developed
implementing only the physics of rarefied gas flows. This has
raised our attention: Is it appropriate to use this scale for developing models of gas flow in shale reservoirs?
This paper attempts to answer this question. Unlike the case of
rarefied gases the shale gas reservoir pressures are between a few
bars to several hundred bars (Ertekin et al., 1986; Javadpour, 2009;
Ozkan et al., 2010; Shabro et al., 2011). To understand gas flow in
nanopores molecular dynamics (MD) is believed to be a useful tool
(Gad-el-Hak, 1999; Roy and Raju, 2003; Darabi et al., 2012). However, the most significant limitations of MD simulations are CPU
time and memory (Koplik and Banavar, 1995; Gad-el-Hak, 1999;
Mao and Sinnot, 2001; Roy and Raju, 2003; Nie et al., 2004). The
MD calculations are restricted to femtosecond time steps restricting the results to picoseconds to nanoseconds. 1 s real time simulation of complex molecular interaction using MD will execute in
thousands of years of CPU time. Therefore it is unfeasible to
simulate reasonable practical problems with MD.
As mentioned, in shale gas reservoirs the flow mechanism of gas
does not follow the conventional Darcy flow equation due to the
presence of nanopores. To our knowledge, no articles have shown
the examination whether or not gas flow in shale strata is slip flow,
or if it is slip flow how much is the effect, other than directing to
Tsien’s (1946) Kn flow regime scale. In this exploratory short
communication we show the analysis conducting simple slip flow
calculations (Brown and Dinardo, 1946). We also validate the Kn
scale for shale gas flows incorporating PVT (Pressure–Volume–
Temperatures) calculations.
51
Fig. 2. (a, b) Density and viscosity change with temperatures and pressures.
Experimental (exp) data are taken from Gonzalez et al. (1970); ‘cal’ means
calculated.
Governing equations
The classical Hagen–Poiseuille equation with slip correction is
given by
Q¼
pR4 2 2 2
km
:
1
P P2 1 þ 4
f
16lL 1
R
ð1Þ
Here Q is the volumetric flow rate; R is the pore radius; P1 and
P2 are the downstream and upstream pressures, respectively; and
l is the dynamic gas viscosity. f is the fraction of gas molecules diffusely reflected in pore walls (Maxwell, 1890). (1 f ) Represents
the fraction which reflects specularly. For instance, f = 0.5 means
the pore surface acts as if it is half perfectly reflecting and half perfectly absorbent. The condition of the molecules stricken is somewhat in between that of evaporated and reflected gas, approaching
Fig. 3. Changes of mean free path with temperature and pressure.
most nearly to evaporated gas at normal incidence and most nearly
to reflected at grazing conditions. f is to be determined from experimental data reduction. Physically f should not vary with flow conditions, however, Brown and Dinardo (1946) reported f = 0.77 on
flow of air, H2, O2, N2, and CO2 at low pressures (rarefied condition)
in round glass tubes and f = 0.84 at higher pressures. In a recent
study by Javadpour (2009) showed 0.80 for Ar to model gas flow
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A. Islam, T. Patzek / Journal of Unconventional Oil and Gas Resources 7 (2014) 49–54
Fig. 4. (a–h) Calculation of F for different pore different pore sizes at different temperatures and pressures. Axis representing ‘f’ is the fraction of gas molecules diffusely
reflected in pore walls. More is discussed in ‘Governing equations’ section.
A. Islam, T. Patzek / Journal of Unconventional Oil and Gas Resources 7 (2014) 49–54
in mudrocks (shales and siltstones). km is the mean free path of the
gas molecules at the mean pressure, Pm ¼ 12 ðP1 þ P2 Þ. From the
kinetic theory and Maxwell’s distribution of molecular velocities
this reads,
km ¼
1
p 2l
:
2q P m
ð2Þ
Here q is the gas density at Pm. The slip correction in Eq. (1) is
termed as F expressed by
2
km
F ¼1þ4
1
:
f
R
ð3Þ
53
previously, for 1000 nm F is 1 indicating no slip flow. Limitation
of the calculations shown is availability of no clue of the value of f
of any particular wall material. This caveats further extensive
research. The calculations presented are also checked for natural
gas mixtures (Gonzalez et al., 1970). We have observed results
similar to CH4, and therefore are not discussed repeatedly.
Now we validate the Kn scale. From Fig. 3 we see that when
R = 1 nm, Kn and km are within 1.5 and 1.5 nm, respectively. The
diameter of CH4 molecule is 0.38 nm. Therefore 1 nm pore is too
small to pass by two molecules freely at temperatures 100–250 F
and pressures 1000–5000 psi. This, at best, can be the single free
molecule flow. However, in contrast, the Kn scale shows free
molecular flow only when 10 6 Kn 6 1.
3. Results and discussions
4. Conclusions
In order to investigate the slippage effect first flow rates of CH4
are generated from Eq. (1). Fig. 1 shows the results for different
pore sizes (R), temperatures (T), and pressures (Pm). q is calculated
by Peng–Robinson equation of state (Peng and Robinson, 1976). For
critical properties Sutton’s (1985) correlations are used. l is
obtained from Lee et al. (1966) formula. The detailed calculations
are shown in Frank et al. (2014). We performed calculations temperatures of 100 and 250 F and pressures of 1000, 3000, and
5000 psi. It is observed that when the shale is extreme tight
(R = 1 nm) the tendency of the gas to be reflected randomly in
the pore walls is higher. In another words, interactions of gas molecules with pore wall increase as the temperature is increased and
pressure is decreased: More slip flow is expected in the reservoirs
at high temperatures and low pressures. When the pore radius is
1000 nm almost there is no variation of flow rate at any temperature and pressure. This means there is no slip (F ffi 1). Actually
when the pore size is large interparticle interactions increase dramatically, suggesting rapid equilibrium of reflected particles with
the pore phase. As a consequence, the slip velocity is reduced substantially (Bhatia and Nicolson, 2003).
According to the PVT computations shown in Fig. 2, the relative
change of density with temperature and pressure is 4 times larger
than that of the viscosity. As temperature increases and pressure
decreases mean free path of gas increases (see Fig. 3). However,
the exceedingly tiny pore restricts the free movement of gas
molecules resulting in more slip flow. On the other hand, as the
pore size increases the slippage effect tends to vanish irrespective
of temperature and pressure. At low temperatures the variation of
mean free path is small, and slip flow is almost independent of
temperature.
Fig. 4 shows the calculations of F for different pore sizes. From
the results it is observed that the total flow at low temperatures
and high pressures can be severely underestimated by disregarding
slip flow, provided that f is below 0.2. The correction F in Eq. (1) can
be 4–100 for 1 nm pore size indicating orders-of-magnitude flow
differences due to slip. Considering pore walls are smooth
(f < 0.2) the slip flow through 1 nm pore can be around 100 times
higher than that in 100 nm pore with no slip (F = 1) at 250 F. Even
in uneven or rough pores (f 0.8) the difference is around 10
times. The results are consistent not only qualitatively but also
quantitatively with those obtained from MD simulations and
experimental findings (Holt et al., 2006; Didar and Akkutlu,
2013; Roy and Raju, 2003; Sokhan et al., 2002; QiXin et al.,
2012). For instance, Holt et al. (2006) reported gas flow through
2 nm carbon nanotube exceeded by more than one order-of-magnitude. QiXin et al. (2012) found 9 times excess flow near the wall
of 21.3 nm pore. The measured flow rate through 10 nm pore
found by Roy and Raju (2003) was more than a magnitude larger
because of slip flow. When the temperature is low (100 F)
slippage nature remains invariant at high pressures. As observed
In conclusion, the classical Hagen–Poiseuille flow equation with
slip correction, and general PVT behavior have been used to investigate slip in natural gas flow through nanopores. This study provides us with key answers without performing expensive and
time consuming MD and lab experiments:
For a very tight shale (<100 nm in pore diameter) slip flows are
experienced in the reservoirs of high temperature and low pressure provided the diffusive deflection fraction, f, is below 0.5.
The slip factor F, can be 4–100. In rough pore walls (f = 0.8) slip
flows can still vary by couple of times (F < 4).
For large pore size (1000 nm) there is no slip flow irrespective of
temperature and pressure.
The slip effect diminishes as the pore size increases irrespective
of temperature and pressure.
Mean free path changes little at low temperatures and the slip
flow variation is minimal.
The classical Kn scale which was designed for rarefied gases
cannot be true representative for natural gas flow regimes at
all temperatures and pressures.
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