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A Unified Model for Gas Transfer in
Nanopores of Shale-Gas Reservoirs:
Coupling Pore Diffusion and
Surface Diffusion
Keliu Wu, University of Calgary and China University of Petroleum; Xiangfang Li, China University of Petroleum;
Chaohua Guo, Missouri University of Science and Technology; and
Chenchen Wang and Zhangxin Chen, University of Calgary
Summary
A model for gas transfer in nanopores is the basis for accurate nu-
merical simulation, which has important implications for eco-
nomic development of shale-gas reservoirs (SGRs). The gas-
transfer mechanism in SGRs is significantly different from that of
conventional gas reservoirs, which is mainly caused by the nano-
scale phenomena and organic matter as a medium of gas sourcing
and storage. The gas-transfer mechanism includes bulk-gas trans-
fer and adsorption-gas surface diffusion in nanopores of SGRs,
where the bulk-gas-transfer mechanism includes continuous flow,
slip flow, and Knudsen diffusion. First, a model for bulk-gas
transfer in nanopores was established, which was dependent on
slip flow and Knudsen diffusion. The total gas flux in the bulk
phase is not a simple sum of slip-flow flux and Knudsen-diffusion
flux but a weighted sum on the basis of corresponding contribu-
tions. The weighted factors are primarily controlled by the mutual
interaction between slip flow and Knudsen diffusion, which is
determined by probabilities between gas molecules colliding with
each other and colliding with nanopore surface in this newly pro-
posed model. Second, a model for adsorbed-gas surface diffusion
in nanopores was established, which was modeled after the
Hwang and Kammermeyer (1966) model and considered the
effect of gas coverage under a high-pressure condition. Finally,
with the combination of these two models, a unified model for gas
transport in nanopores of SGRs was established, and this model
was validated through molecular simulation and experimental
data. Results show that:
•Slip flow makes a great contribution to gas transfer under
the condition of meso/macropores (pore radius greater than
2 nm) and high pressure.
•Knudsen diffusion makes an important contribution to gas
transfer under the condition of macropores (pore radius
greater than 50 nm) and less than 1 MPa in pressure,
whereas it can be ignored in other cases.
•A surface-diffusion coefficient is comparable with a pore-
diffusion coefficient, and gas transfer is always dominated
by surface diffusion over all the ranges of pressure in micro-
pores (pore radius 2 nm).
•Surface-diffusion contribution decreases with an increase in
pore size, isosteric sorption heat, pressure, and temperature
in SGRs.
Introduction
A model for gas transfer in nanopores is the basis for numerical
simulation (Xiong et al. 2012) and is important for gas-production
forecasting, well placement, and configuration optimization in
shale-gas reservoirs (SGRs) (Tinni et al. 2012; Civan et al. 2013).
Gas-Transfer Mechanism in Shale Nanopores. Scanning-elec-
tron-microscope images with nano-resolution show that organic
matter is abundant in the matrix of shale. Moreover, a major por-
tion of total porosity is contained in pores of the organic matter
(Ambrose et al. 2010; Passey et al. 2010).
Organic pores are generally smaller than 10 nm in diameter
(Xiong et al. 2012; Firouzi et al. 2014b), which is similar to the
order of gas-molecule free path. Compared with the collision
between gas molecules, the collision between gas molecules and
nanopore walls is also important and must be considered. There-
fore, Knudsen diffusion plays an important role in gas transfer in
nanopores (Xiao and Wei 1992; Wang and Li 2003). Roy et al.
(2003) and Holt et al. (2006) reported that Knudsen diffusion is
the dominant gas-transport mechanism under a large Knudsen
number through laboratory experiments. Darabi et al. (2012) also
indicated that the contribution of Knudsen diffusion to cumulative
production at the given conditions (typical SGR conditions) is up
to 20%. Therefore, in addition to continuum flow and slip flow,
the Knudsen diffusion is also very prominent in SGRs (Firouzi
et al. 2014a).
Organic matter and clay minerals both adsorb gas (Chalmers
and Bustin 2007; Yan et al. 2013; Zhang et al. 2013); especially
nanopores of organic matter with a large surface area (Shabro
et al. 2011, 2012; Mosher et al. 2013) and mostly oil-wet (Passey
et al. 2010; Sondergeld et al. 2010) contain a significant amount
of adsorbed gas in SGRs. Lu et al. (1995) studied 24 samples of
Devonian shale and showed that the adsorbed gas can be an aver-
age of 61% of total gas volume. Desorption of the adsorbed gas
plays an important part in gas transfer (Javadpour et al. 2007; Cui
et al. 2009). As adsorbed gas desorbs, it increases pores’ hydraulic
diameter, reduces tortuosity, and causes extra gas slippage at the
boundary, thereby leading to multiple-fold increase of the matrix
permeability (Swami et al. 2013).
In addition to desorption, the surface diffusion of adsorbed gas
also occurs under a concentration gradient, which plays an impor-
tant role in gas transfer in nanopores (Darabi et al. 2012). Mobil-
ity of physically adsorbed molecules was first demonstrated by
Volmer and Adhikari (1925) for molecules of benzophenone.
They also demonstrated that the migration of benzophenone was a
result of a surface-concentration gradient. Later, Wentworth
(1944) observed that the measured fluxes were higher than those
predicted for the continuum-, slip-, or diffusion-flow regimes, and
explained that the surface diffusion is a supplementary mecha-
nism of gas transfer. Kilislioglu and Bilgin (2003) also showed
that the gas-effective-diffusion coefficient should be a sum of a
bulk-gas-diffusion coefficient and an adsorbed-gas-diffusion coef-
ficient. The main explanation of surface diffusion is the hopping
mechanism. A gas molecule is adsorbed on a low-energy site and
vibrates. The activated molecules can then hop between fixed sites
with a specific velocity, thus contributing to the surface migration
in the adsorbed phase. A molecule returns to the gaseous phase
when it acquires a sufficient amount of energy to escape from the
surface. The existence of the net flux on the surface is a probabil-
istic phenomenon that depends on a local concentration gradient
Copyright V
C2016 Society of Petroleum Engineers
This paper (SPE 1921039) was accepted for presentation at the SPE/AAPG/SEG
Unconventional Resources Technology Conference, Denver, 25–27 August 2014, and
revised for publication. Original manuscript received for review 7 September 2014. Revised
manuscript received for review 31December 2014. Paper peer approved 9 April 2015.
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2016 SPE Journal 1
and temperature, as well as the molecule type. Thus, the surface
diffusion makes an additional contribution to the gas-phase trans-
fer (Roaue-Malherbe 2007). Surface diffusion is a very-important
transfer mechanism in nanopores of organic matter with a large
surface area (Kang et al. 2011; Xiong et al. 2012; Etminan et al.
2014; Fathi and Akkutlu 2014). The Knudsen diffusion of the
bulk gas and surface diffusion of adsorbed gas can increase the
apparent permeability in nanopores of SGRs. Some authors also
showed that the predicted value of apparent permeability was 10
times that of conventional hydrodynamic methods (Darabi et al.
2012), or even higher, to several orders of magnitude (Holt et al.
2006; Rahmanian et al. 2010).
Review of Gas-Transfer Models in Nanopores of SGRs. Bulk-
Gas-Transfer Models. Transfer mechanisms of the bulk gas in
shale nanopores include continuum flow, slip flow, and transition
flow (Aguilera 2010; Civan 2010, Civan et al. 2011, 2012, 2013;
Rahmanian et al. 2010; Yuan et al. 2014). According to Gad-el-
Hak (1999), gas-transfer mechanisms can be divided into two pos-
sible kinds of model concepts: a molecule model considering the
gas-molecule nature (Gad-el-Hak 1999; Malek and Coppens
2002) or a macro model ignoring the gas-molecule nature (Klin-
kenberg 1941; Ertekin et al. 1986; Arkilic 1994; Beskok and Kar-
niadakis 1999; Jang et al. 2002; Liu et al. 2002; Maurer et al.
2003; Colin et al. 2004; Hsieh et al. 2004; Colin 2005; Ewart
et al. 2006; Stevanovic 2007; Graur et al. 2009; Javadpour 2009;
Pitakarnnop et al. 2010; Yamaguchi et al. 2011; Darabi et al.
2012; Azom and Javadpour 2012; Xiong et al. 2012; Civan 2010,
Civan et al. 2011, 2012, 2013; Rahmanian et al. 2013; Singh and
Javadpour 2013).
Molecule Models. Description of gas-transfer mechanism
through a molecule model, known as molecule modeling, takes
into account the gas-molecule nature and can accurately simulate
various physical mechanisms in nanopores (Gad-el-Hak 1999;
Malek and Coppens 2002). However, molecule-modeling techni-
ques used in shale-gas simulations require enormous computing
resources and computing time, so their application is limited
(Koplik and Banavar 1995; Gad-el-Hak 1999; Mao and Sinnott
2001; Nie et al. 2004; Coppens and Dammers 2006).
Macro Models. A conventional hydrodynamic-continuity
model cannot properly describe gas-transfer mechanism in or-
ganic nanopores (Darabi et al. 2012). Currently, there are two pos-
sible types of macro models describing gas-transport mechanism
in shale nanopores. One is a hydrodynamic model modifying a
no-slip boundary condition in continuum models by accounting
for a slip boundary condition (Klinkenberg 1941; Beskok and
Karniadakis 1999; Civan 2010, Civan et al. 2011, 2012, 2013;
Xiong et al. 2012). The other model is the combination of differ-
ent transfer mechanisms by use of a certain weight coefficient
(Ertekin et al. 1986; Liu et al. 2002; Javadpour 2009; Azom and
Javadpour 2012; Darabi et al. 2012; Rahmanian et al. 2013; Singh
and Javadpour 2013), as shown in Table 1.
A hydrodynamic model is summarized as follows: Klinken-
berg (1941) proposed an empirical model to consider the Klin-
kenberg effect; on the basis of the empirical Klinkenberg
model, Luffel et al. (1993) and Wu et al. (1998) described gas
transport with a slippage effect under low pressure; Beskok and
Karniadakis (1999) proposed a model to describe all known
gas-transport mechanisms in nanopores, including continuum
flow, slip flow, transition flow, and free molecule flow; Civan
(2010) and Civan et al. (2011, 2013) modeled gas transport in
shale nanopores considering rarefaction and slippage effects on
the basis of the Beskok and Karniadakis (1999) model; and
Xiong et al. (2012) presented a gas-permeability model consid-
ering surface diffusion of adsorbed gas modeled after the
Beskok and Karniadakis (1999) model.
The models of the second category are summarized as follows:
Ertekin et al. (1986) proposed an analytical model with a con-
stant-contribution coefficient for continuum flow and Fick’s diffu-
sion, which did not cover the whole flow-regime spectrum,
especially for transition-flow regimes. Liu et al. (2002) applied
Adzumi’s contribution coefficient (Adzumi 1937a, 1937b, 1937c)
as a weight to balance continuum flow and Knudsen diffusion and
modeled gas slippage in nanopores, where the weight coefficient
of Knudsen diffusion is the ratio of the slip-layer area to a nano-
pore cross-sectional area. However, when the Knudsen number
(Kn, the ratio of the gas-molecule free path to the characteristic
length in a porous medium) is larger than or equal to unity, the
Knudsen-diffusion-weight coefficient becomes smaller than zero,
meaning that there is no Knudsen diffusion, and it does not match
the actual physical process. Javadpour (2009) made a linear super-
position of the Knudsen diffusion and slip flow modeled after the
Maxwell theory. Azom and Javadpour (2012) proposed a gas-
Table 1—Comparison and evaluation of different permeability models.
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transfer model for real gas in nanopores modeled after the Javad-
pour model (2009). Darabi et al. (2012) considered the effect of
wall roughness on the Knudsen diffusion in nanopores on the ba-
sis of the Javadpour model (2009). Singh and Javadpour (2013)
proposed a nonempirical apparent-permeability model modeled
after the analytical model of Veltzke and Tho¨ming (2012), which
is a linear superposition of the convective flow and Fick’s diffu-
sion. Rahmanian et al. (2013) proposed an empirical formula
describing the contributions of continuum flow and Knudsen dif-
fusion by use of the Aguilera (2006) formula. The empirical for-
mula contains unknown weighting coefficients which are obtained
through experiments, resulting in limited applications.
So far, although there are many bulk-gas-transfer models pro-
posed like those mentioned, these models cannot depict Knudsen
diffusion; the weight coefficients of these models are given unrea-
sonably or contain empirical coefficients; and the empirical coeffi-
cients obtained by means of experimental data play a key role in
the accuracy of the models. Unfortunately, there are limited reli-
able empirical data available because of the complexity of shale
resulting from different organic materials and mineral types, as
well as different gas components in shale (Singh and Azom 2013).
Adsorbed-Gas Surface-Diffusion Models. Currently, many
theories have been proposed to describe the fluid surface diffusion
in porous media, and can be divided into three classes.
Theory Derived From Hopping Models. This theory is derived
from hopping models. The hopping models assume that the
adsorbed-gas molecule hops from one adsorption site to the neigh-
boring adsorption site on the solid surface, which refers to the
activation process of the adsorbed-gas molecule (Higashi et al.
1963; Yang et al. 1973; Okazaki et al. 1981; Kapoor and Yang
1989; Chen and Yang 1991, 1998). If the adsorbed-gas molecule
could obtain enough energy to hop into the neighboring adsorp-
tion site, then the activation process and surface diffusion occur.
A hopping model is only applicable to gas surface diffusion of
single-layer adsorption.
Theory Derived From Hydrodynamic Models. This theory is
derived from hydrodynamic models, assuming that surface diffu-
sion of adsorbed fluid is caused by viscous flow in surface film.
This theory was first proposed by Gilliland et al. (1958). Another
hydrodynamic model was proposed by Petropoulos (1996) for
mesopores (2 nm <pore radius <50 nm), and then the extensions
of homogeneous hydrodynamic models to heterogeneous surfaces
were proposed (Kainourgiakis et al. 1996, 1998; Kikkinides et al.
1997). This theory is only suitable for fluid surface diffusion for
multilayer adsorption.
Theory Derived From Fickian Models. This theory assumes
that the fluid surface diffusion and bulk-phase transfer in porous
media are independent, and the total flux can be expressed as
Jtot ¼JbþJs;ð1Þ
where J
tot
is the mass flux of the total fluid transfer in kg/(m
2
s);
J
b
is the mass flux of the bulk-phase transfer in kg/(m
2
s); and J
s
is the mass flux of the adsorbed-gas surface diffusion in kg/(m
2
s).
As seen in Eq. 1, the mass flux of surface diffusion can be
obtained when the mass flux of the bulk-phase transfer is sub-
tracted from the mass flux of the total fluid transfer. The mass flux
of the total fluid transfer and bulk-phase transfer could be
obtained theoretically or experimentally, and the mass flux of the
bulk phase is usually obtained by nonadsorption-fluid experi-
ments. This theory ignores the physical phenomenon of fluid
“desorption/bulk phase transfer/readsorption,” which underesti-
mates the bulk transfer and overestimates surface diffusion
(Kapoor et al. 1989).
Many authors have shown that it is more reasonable to investi-
gate shale-gas desorption by use of the Langmuir isotherm with
an assumption of single-layer adsorption (Cui et al. 2009;
Ambrose et al. 2010). Hence, the theory derived from hopping
models is more suitable to surface diffusion of adsorbed gas in
shale rock. Several classical hopping models are introduced.
Hwang and Kammermeyer (1966) derived an analytic model of
surface diffusion in a low-pressure condition from a hopping
model, which was verified by experiments. On the basis of experi-
mental data and a previous analytic model, Guo et al. (2008) fitted
an empirical expression of a surface-diffusion coefficient in a
methane/activated-carbon system, but the influence of pressure
was not considered and cannot be applied for surface-diffusion
calculation of adsorbed gas in a high-pressure condition. Chen
and Yang (1991) derived a surface-diffusion model with consider-
ation of adsorbed-gas coverage from a hopping model.
Research Content and Procedure. In this paper, a new model for
gas transfer in nanopores of SGRs is established. This new model
considers multiple mechanisms, including slip flow, Knudsen diffu-
sion, and surface diffusion. Five parts are included in this paper.
•Part 1: Establishment of a bulk-gas-transfer model in shale
nanopores is presented. In typical SGRs, the bulk-gas-transfer
mechanisms include continuum flow, slip flow, and transition
flow. Continuum flow and slip flow are caused by intermolec-
ular collision, and Knudsen diffusion is caused by collision
between molecules and nanopore walls. Therefore, the bulk-
gas-transfer model is derived by use of superposition of slip
flow and Knudsen diffusion on the basis of weight coeffi-
cients, which are determined by the probabilities of gas mole-
cules colliding with each other and colliding with nanopores’
walls. In addition, the effects of rarefaction, wall roughness,
poromechancial response, and sorption-induced swelling
response on bulk-gas transfer are investigated.
•Part 2: A surface-diffusion model of adsorbed gas in shale
nanopores is proposed, which is modeled after the Hwang
and Kammermeyer (1966) model in a low-pressure condi-
tion, and the influence of adsorbed-gas coverage in a high-
pressure condition is considered.
•Part 3: The combination of the bulk-gas-transfer model and
the surface-diffusion model, a model for gas transfer in shale
nanopores, is established and validated through the pub-
lished molecular simulation and experimental data.
•Part 4: This model is applied for results analysis and discussion.
•Part 5: A final conclusion is given.
Bulk-Gas Transfer in Shale Nanopores
Continuum Flow. When Kn is less than 10
3
, the intermolecular
collisions dominate and gas flow is continuum flow, which satis-
fies the continuity condition and can be expressed by the Hagen-
Poiseuille equation (Choi et al. 2001):
Jv¼fmb
r2p
8gRT
dp
dl;ð2Þ
where J
v
is the continuum-flow flux in mol/(m
2
s); f
mb
is a dimen-
sionless correction factor of apparent permeability in nanopores of
SGRs, and its detailed expression and derivation process are shown
in Appendix A; ris the nanopore radius in m; gis the gas viscosity
in Pas; Ris the universal gas constant in J/(molk); Tis temperature
in K; pis pressure in Pa; and lis the gas-transport distance in m.
The effect of pore structure on gas transfer is significant for
gas-continuum flow. The gas-transfer capacity is proportional to
the square of the nanopore radius, and this capacity increases with
high porosity and low tortuosity. The operation parameters also
affect the gas-transfer process. For example, under low pressure,
the continuum-flow flux is extremely low because of the gradual
transfer from continuum flow to Knudsen diffusion. The gas trans-
fer is sensitive to temperature and inversely proportional to 1.5
power of temperature because the gas viscosity is a function of
0.5 power of temperature.
Slip Flow. When 10
3
<Kn <10
1
, both the intermolecular and
wall/molecule collisions cannot be ignored, and gas molecules slip
on the wall. With a modification of the slip-boundary condition,
the gas-slip flow can be expressed as (Karniadakis et al. 2005):
.............................
.........................
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Jvs ¼fmb
r2p
8gRT ð1þaKnÞ1þ4Kn
1bKn
dp
dl;ð3Þ
where J
vs
is the slip-flow flux in mol/(m
2
s); and bis a gas-slip
constant and dimensionless. If the boundary is a first-order slip
boundary, b¼0; if the boundary is a second-order slip boundary,
b¼–1; Kn is the Knudsen number, a ratio of the gas-mean free
path to the nanopore diameter, and is dimensionless; and ais a
rarefaction coefficient and is dimensionless.
Molecular-simulation results show that the gas-slip constant
b¼–1 is more reasonable within different Knudsen numbers. The
rarefaction coefficient is defined to describe the physical phenom-
enon of gas viscosity decreasing with an increase in the Knudsen
number. It equals zero during the continuum-flow region and
increases to a constant during molecular flow (this value is a spe-
cific constant for a specific pore/gas system). The rarefaction
effect can be influenced by many factors. For example, the effect
decreases with increasing pressure (Civan 2010; Civan et al.
2011; Tinni et al. 2012), increases with decreasing the pore size,
and is also affected by the pore geometry (Beskok and Karniada-
kis 1999; Civan et al. 2011; Veltzke and Tho¨ming 2012). Many
experimental or molecular-simulation data are needed to deter-
mine the rarefaction coefficients under different Knudsen num-
bers. According to molecular-simulation results, the relationship
of the rarefaction coefficient to the Knudsen number can be
expressed as
a¼ao
2
ptan1ða1KnbÞ;ð4Þ
where a
o
is the rarefaction coefficient at Kn !1and is dimen-
sionless and a
1
and bare fitting constants that are dimensionless
and obtained through molecular-simulation and experimental data.
According to molecular-simulation results, the relationship of
the rarefaction coefficient to the Knudsen number is shown in
Fig. 1 (Karniadakis et al. 2005).
Eq. 3 also shows that the slip effect and rarefaction effect are
obvious with an increase in the Knudsen number, and the gas-
transport flux continually increases. However, even if the Knud-
sen number is very large, Eq. 3 cannot be degraded to a Knudsen-
diffusion equation and cannot describe Knudsen diffusion.
Knudsen Diffusion. When the Knudsen number Kn 1, the
wall/molecule collisions dominate, the transfer mechanism is
Knudsen diffusion, and the gas-diffusion flux in a circular tube
with a radius of rcan be expressed with the Knudsen equation
(Choi et al. 2001):
Jk¼2
3fmbr8
pRTM
0:5dp
dl;ð5Þ
where J
k
is the Knudsen-diffusion flux in mol/(m
2
s) and Mis gas
molar mass in kg/mol.
The Knudsen-diffusion flux increases with smaller gas-mole-
cule weight. In other words, under the same pressure gradient,
lighter gas molecules can move farther than heavy molecules.
Compared with continuum flow and slip flow, Knudsen diffusion
is independent of pressure or viscosity and it is inversely propor-
tional to the 0.5 power of temperature.
After the unit conversion, the Knudsen-diffusion coefficient
obtained from Eq. 5 is
Dk¼2
3fmbr8RT
pM
0:5
;ð6Þ
where D
k
is the Knudsen-diffusion coefficient in m
2
/s.
The effect of wall roughness on the Knudsen diffusion is sig-
nificant. The Knudsen-diffusion efficiency decreases under severe
wall roughness because of long gas retention in the vicinity of a
pore wall. The Knudsen diffusion can be described with wall
roughness as (Darabi et al. 2012)
Deffk¼dDf2Dk;ð7Þ
where D
effk
is the Knudsen-diffusion coefficient considering wall
roughness in m
2
/s; D
f
is the fractal dimension of the pore wall and
is dimensionless; dis the ratio of the gas-molecule diameter d
M
to
the local average pore diameter 2rand is dimensionless.
The pore-wall roughness can be evaluated by the fractal
dimension of the pore wall, which is a quantitative measurement
of wall roughness varying from 2 to 3, representing a smooth wall
and a space-filling wall, respectively (Coppens 1999; Coppens
and Dammers 2006).
By combining Eqs. 5 through 7, the Knudsen diffusion consid-
ering pore/wall roughness yields
Jk¼2
3fmbrdDf28
pRTM
0:5dp
dl:ð8Þ
Coupling of Bulk-Gas-Transfer Mechanisms. In the actual
shale-gas production, the Knudsen number ranges from 0.0002 to 6,
leading to the intermolecular collision and wall/molecule collision
both becoming important. The gas-transfer mechanisms of the bulk-
gas phase include continuum flow, slip flow, and Knudsen diffusion.
Hence, the key point is how to determine reasonable weighting coef-
ficients of all gas-transfer mechanisms in shale-gas reservoirs
(SGRs). The ratios of molecule-collision frequency to total collision
frequency and nanopore-wall/molecule-collision frequency to the
total collision frequency are used to calculate the weighting coeffi-
cients of slip flow and Knudsen diffusion, respectively.
In nanopores, the number of collisions occurring within the
unit time period can be calculated as
1
tT
¼1
tM
þ1
tS
;ð9Þ
where t
T
is the average time consumed for one collision of overall
gas molecules in seconds; t
M
is the average time required for one
collision between the molecules in seconds; and t
S
is the averaged
time required for one collision between molecule and nanopore
wall in seconds.
The number of collisions can also be expressed as (Reif 1965)
1
tT
¼hvi
kT
;ð10Þ
1
tM
¼hvi
k;ð11Þ
......
.......................
..................
.......................
..........................
..............
.............................
...............................
................................
1.4
Analytic soluion
Tison (1993)
1.2
1
0.8
0.6
α (dimensionless)
0.4
0.01 0.10 1 10 100
Kn (dimensionless)
Fig. 1—The curves for a-vs.-Kn of the bulk gas in nanopore.
Note that experimental data are performed by Tison (1993).
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Methane molecule
Surface diffusion
High pLow p
Desorption
Slip flow or
Knudsen diffusion
(a)
(b)
(c)
(d)
Adsorption
Shale matrix
Slip flow
Knudsen diffusion
Slip flow + Knudsen diffusion
Total flux
Surface diffusion
Surface diffusion
Vacancies
Fig. 2—Schematic of gas-transport mechanisms in nanopores of SGRs.
φ
Table 2—Summary of modeling parameters used in the calculation.
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2016 SPE Journal 5
1
tS
¼hvi
2r;ð12Þ
where k
T
is the average free path of the overall gas molecules in
m and <v>is the average velocity of molecule movement in m/s.
In addition, the gas-molecule free path can be calculated by
(Civan et al. 2011)
k¼g
pffiffiffiffiffiffiffiffiffi
pRT
2M
r:ð13Þ
Substituting Eqs. 10 through 12 into Eq. 9 yields
1
kT
¼1
kþ1
2r:ð14Þ
The contributions of slip flow and Knudsen diffusion to the
total gas transfer in the bulk-gas phase can be determined by the
ratios of molecule-collision frequency to overall collision fre-
quency and nanopore-wall/molecule-collision frequency to over-
all collision frequency, respectively. Therefore, the bulk-gas-
transfer flux in nanopores is
Jb¼kT
kJvs þkT
2rJk;ð15Þ
where J
b
is the total bulk-gas flux in mol/(m
2
s).
Substituting Eqs. 3 and 8 into Eq. 15 yields
Jb¼ 1
ð1þKnÞfmb
r2p
8gRT ð1þaKnÞ1þ4Kn
1bKn
dp
dl
1
ð1þ1=KnÞ
2
3fmbrdDf28
pRTM
0:5dp
dl:
ð16Þ
After the unit conversion, the apparent permeability of the
bulk-gas phase in nanopores obtained from Eq. 16 is
kb¼1
ð1þKnÞfmb
r2ð1þaKnÞ1þ4Kn
1bKn
8
þ1
ð1þ1=KnÞ
2
3fmbrdDf28RT
pM
0:5g
p;
ð17Þ
where k
b
is the apparent permeability for the bulk gas through
nanopores in m
2
.
...............................
.............................
.............................
........................
4
Analytic solution
Tison (1993)
Loyalka & Hamoodi (1990)
Analytic solution
Tison (1993)
Loyalka & Hamoodi (1990)
3.2
2.4
1.6
0.8
0.0
4
3.2
2.4
1.6
0.8
0.0
0.0 0.1 0.2 0.3 0.4
Kn (dimensionless) Kn (dimensionless)
Jb /Jk (dimensionless)
Jb /Jk (dimensionless)
0.5 0.0 0.1 1 10 100
Fig. 3—Comparison of analytic solution and experimental and molecular-simulation data for the bulk gas. Note that the square and
triangle symbols are experimental data of Tison (1993) and linearized Boltzmann data of Loyalka and Hamoodi (1990), respectively.
10 Analytical solution 1
Analytical solution 2
Analytical solution 3
Experiment data 1
Experiment data 2
Experiment data 3
8
6
4
2
0
0.0 0.2 0.4 0.6 0.8
θ (dimensionless)
1
Ds
/Ds (dimensionless)
0
Fig. 4—The curves for D0
s-vs.-hof adsorbed gas.
Table 3—Modeling parameters and experimental conditions for model verification with adsorption-gas
coverage.
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Table 4—Summary of surface-diffusion coefficients in published literature.
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2016 SPE Journal 7
Table 4 (continued)—Summary of surface-diffusion coefficients in published literature.
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8 2016 SPE Journal
Table 4 (continued)—Summary of surface-diffusion coefficients in published literature.
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2016 SPE Journal 9
–
–
–
–
–
–
–
Table 4 (continued)—Summary of surface-diffusion coefficients in published literature.
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10 2016 SPE Journal
Poromechancial Response. Organic matter in shale has weak
strength and strong sensitivity to stress change. From experimen-
tal results, Tinni et al. (2012) found that permeability in Ordovi-
cian shale decreases by one order of magnitude when the ambient
pressure increases from 1,000 to 5,000 psi. For Devonian shale,
the permeability decreases by three orders of magnitude. The
decrease in permeability is explained by a decrease in nanopore
diameter. The variation in permeability because of poromechan-
cial response is correlated by Wang et al. (2012)
xm¼1þðafasÞ
ð1þKn=EsÞ
sðppintÞ
bintEs
3
;ð18Þ
............
Table 4 (continued)—Summary of surface-diffusion coefficients in published literature.
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2016 SPE Journal 11
where x
m
is the poromechancial-response coefficient of shale ma-
trix and is dimensionless; a
f
is the Biot’s coefficient for Kn micro-
fracture and is dimensionless (Biot and Willis 1957); a
s
is the
Biot’s coefficient for shale matrix and is dimensionless (Biot and
Willis 1957); Kn is average microfracture normal stiffness in Pa/
m; sis microfracture spacing in m; E
s
is the shale-matrix Young’s
modulus in Pa; b
int
is the initial microfracture aperture in m; and
p
int
is the initial pressure in Pa.
Sorption-Induced Swelling Response. The gas transmissibility
decreases because adsorbed gas occupies a portion of the nano-
pore volume. During the development of SGRs, the gas desorp-
tion results in an increased effective hydraulic diameter and
transmissibility in nanopores (Alnoaimi and Kovscek 2013). The
variation in permeability caused by sorption-induced-swelling
response is correlated as (Wang et al. 2012)
xs¼13
/
eLpLðppintÞ
ðpþpLÞðpint þpLÞ
3
;ð19Þ
where x
s
is a sorption-induced-swelling-response coefficient of
shale matrix and is dimensionless; e
L
is the Langmuir strain and is
dimensionless; and p
L
is the Langmuir pressure in Pa.
Apparent Permeability of Bulk Gas. According to Eqs. 17
through 19, with the combination of slip flow and Knudsen diffu-
sion, the bulk-gas apparent permeability can be expressed as
kbt ¼kbxmxs;ð20Þ
where k
bt
is the apparent permeability for bulk gas through shale
nanopores in m
2
.
Eqs. 17 through 20 show that the apparent permeability for the
bulk gas considers the effects of rarefaction, nanopore structure
(nanopore size, porosity, tortuosity, and wall roughness), porome-
chancial response, and sorption-induced swelling response on the
bulk-gas transfer.
.............
............................
Table 5—Effects of gas types on surface diffusion for different surface/gas systems.
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12 2016 SPE Journal
Adsorbed-Gas Surface Diffusion in Shale
Nanopores
Langmuir Isotherm Equation. Under the initial condition of
shale-gas reservoirs (SGRs), there is adsorption equilibrium
between the adsorbed gas on the nanopore surface and the bulk-
gas phase, and the amount of adsorbed gas can be expressed with
the Langmuir isotherm (Cui et al. 2009).
During the development of SGRs, although surface diffusion
of adsorbed gas occurs, the assumption of equilibrium-sorption
isotherm still holds valid because the exchange rate between the
gaseous and adsorbed phases is much higher than the surface-
diffusion rate (Lancaster and Hill 1993; Ross and Bustin 2007;
Xiong et al. 2012):
qa¼qLp
pLþp;ð21Þ
where q
a
is a standard volume of adsorbed gas per unit mass in
shale rock in m
3
/kg and q
L
is a Langmuir volume in m
3
/kg.
Although the Langmuir adsorption is monolayer adsorption,
the coverage can be defined as the ratio of the adsorption volume
to the Langmuir volume (Cui et al. 2003),
h¼p
pLþp;ð22Þ
where his the gas coverage on the pore wall at equilibrium state
and is dimensionless.
According to Eq. 22, the concentration of Langmuir-mono-
layer adsorbed gas is shown as
Cs¼4hM
pd3
MNA
;ð23Þ
where C
s
is the adsorbed-gas concentration in kg/m
3
;d
M
is the
gas-molecule diameter in m; and N
A
is Avogadro’s constant,
6.0221415 10
23
/mol.
Adsorbed-Gas Surface Diffusion. According to the Maxwell-
Stefan method, the driving force of surface diffusion is a chemi-
cal-potential gradient (Krishna and Wesselingh 1997):
Js¼Lm
fmsCs
M
@u
@l;ð24Þ
............................
.............................
............................
........................
Table 6—Effects of surface types on surface diffusion for different surface/gas systems.
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2016 SPE Journal 13
where L
m
is the gas mobility in mols/kg; f
ms
is a correction factor
of surface diffusion of adsorbed gas in the nanopores of SGRs and
is dimensionless, with the detailed formulas in Appendix B; and u
is the chemical potential in J/mol.
The gas is assumed as ideal gas and the chemical potential can
be expressed as (Krishna and Wesselingh 1997; Do et al. 2001a)
u¼uoþRTlnp;ð25Þ
where u
o
is the chemical potential in a reference state in J/mol.
If the surface-diffusion rate is far less than the gas-adsorption
and -desorption rate, it is assumed that dynamic equilibrium is
achieved between the bulk-phase gas and the adsorption-phase
gas in the nanopores (Hwang and Kammermeyer 1966; Medved
and Cerny 2011). Combining with Eq. 25, Eq. 24 can be stated as
Js¼D0
s
fmsCs
Mp
@p
@l;ð26Þ
where
D0
s¼LmRT ð27Þ
and D0
sis the surface-diffusion coefficient when the gas coverage
is zero (Guo et al. 2008; Sheng et al. 2014), in m
2
/s.
The surface-diffusion coefficient can also be expressed as
(Hwang and Kammermeyer 1966; Guo et al. 2008; Sheng et al.
2014)
D0
s¼XTmexp E
RT
;ð28Þ
where Xis a constant that relates to gas-molecule weight in m
2
/
(sK
0.5
); mis a constant that is dimensionless; and Eis gas-activa-
tion energy in J/mol.
The gas-activation energy is the function of gas-isosteric-
adsorption heat, and is less than isosteric-adsorption heat (Hwang
..........................
........................
.............................
......................
Table 7—Effects of temperature on surface diffusion for different surface/gas systems.
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14 2016 SPE Journal
Δ
Table 8 —Summary of isosteric-sorption heats published in the literature.
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2016 SPE Journal 15
and Kammermeyer 1966; Kapoor and Yang 1989). According to
experimental results, it can be expressed as (Guo et al. 2008)
E¼DH0:8:ð29Þ
Isosteric-adsorption heat is independent of temperature (Jagiełło
et al. 1992; Nodzen´ski 1998) and dependent on adsorbent and ad-
sorbate (Nodzen´ ski 1998). Isosteric-adsorption heat decreases with
an increase in gas coverage on surface. However, even if the cover-
age is high, the isosteric-adsorption heat of methane is always
higher than the condensation heat of 8.8 kJ/mol with the capillary-
condensation phenomenon (Nodzen´ski 1998; Guo et al. 2008).
In combination with Eqs. 28 and 29, experimental data for
methane/activated-carbon adsorption are fitted to obtain the sur-
face-diffusion coefficient (Guo et al. 2008):
D0
s¼8:29 107T0:5exp DH0:8
RT
:ð30Þ
The surface-diffusion coefficient in Eqs. 28 and 30 is obtained
in a low-pressure condition by theory and experiments, respec-
tively, which is the function of gas molecular weight, temperature,
and gas-activation energy/isosteric-adsorption heat, and independ-
ent of pressure (Hwang and Kammermeyer 1966; Guo et al.
2008). Hence, it is only suitable to surface diffusion of gas in a
low-pressure condition. Although the physical and chemical prop-
erties of a methane/activated-carbon system are similar to those of
a methane/shale-rock system, the calculated results in Eq. 30 can
be considered as the surface-diffusion coefficient when the cover-
age of shale gas is zero in nanopores of SGRs.
The reservoir depth is generally 500–3000 m and the reservoir
pressure is typically 5–30 MPa in SGRs in North America (Curtis
.............................
............
Table 8 (continued)—Summary of isosteric-sorption heats published in the literature.
100
Do et al. 2000
Guo et al. 2008
Calculation value
10–2
10–4
10–6
D0
s (cm2/s)
10–8
0153045
ΔH (kJ/mol)
60
Fig. 5—The curves for D0
s-vs.-DHfor CH
4
in a porous medium.
×10–17
×10–18
kvs (m2)
kvs (%)
×10–19
×10–20
×10–21
0102030
2 nm
25 nm
5 nm
50 nm
10 nm
100 nm
2 nm
25 nm
5 nm
50 nm
10 nm
100 nm
40
p (MPa)
50 010
100
80
60
40
20
0
20 30 40
p (MPa)
(a) kvs ~ p (b) kvs×100%/(kvs+kk+ks) ~ p
50
Fig. 6—Comparison of slip flow with pressure under different nanopore radius (DH514 kJ/mol).
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16 2016 SPE Journal
2002). Hence, to describe the gas-surface diffusion in high pres-
sure, the influence of gas coverage on surface diffusion is consid-
ered. Chen and Yang (1991) used a kinetic method to derive a
surface-diffusion coefficient:
Ds¼D0
s
ð1hÞþj
2hð2hÞþ½Hð1jÞð1jÞj
2h2
ð1hþj
2hÞ2;
ð31Þ
Hð1jÞ¼ 0;j1
1;0j1
ð32Þ
j¼jb
jm
;ð33Þ
where D
s
is a gas-surface-diffusion coefficient in m
2
/s; H(1–j)is
the Heaviside function and is dimensionless; j
m
is a forward-ve-
locity coefficient of surface-gas molecule in m/s; j
b
is a blocking-
velocity coefficient of surface-gas molecule; and jis the ratio of
the blocking-velocity coefficient j
b
to the forward-velocity coeffi-
cient j
m
and is dimensionless.
It can be demonstrated from Eq. 32 that when j
m
>j
b
, surface
diffusion will occur no matter whether the forward position is
occupied by another gas molecule; when j
b
>j
m
, surface diffu-
sion will stop because the gas molecule is blocked, and the block-
ing phenomenon will not cause inverse diffusion of gas molecules.
When gas molecules diffuse on an infinite surface, the block-
ing phenomenon will not happen, the blocking coefficient j¼0,
and Eq. 31 can be simplified to
Ds¼D0
s
1
1h:ð34Þ
Eq. 34 shows that Eq. 31 can be simplified into the Higashi-
Ito-Oishi model (Higashi et al. 1963).
Apparent Permeability of Adsorbed Gas. Through combining
Eqs. 26 and 31 with the unit conversion, the apparent permeability
of adsorbed-gas surface diffusion can be expressed as
ks¼JsVg
dp=dl¼fmsDs
CsRTg
p2M;ð35Þ
.................
................................
...........................
.................
×10–17
2 nm
25 nm
5 nm
50 nm
10 nm
100 nm
2 nm
25 nm
5 nm
50 nm
10 nm
100 nm
×10–18
×10–19
×10–20
×10–21
×10–22
×10–23
0102030
p (MPa)
ks (m2)
ks (%)
40 50
100
80
60
40
20
0
0102030 40
p (MPa)
50
(a) ks ~ p (b) ks×100%/(kvs+kk+ks) ~ p
Fig. 8—Comparison of surface diffusion with pressure under different nanopore radius (DH514 kJ/mol).
×10–17
×10–18
×10–19
×10–20
×10–21
×10–22
×10–23
×10–24
010
kk (m2)
kk (%)
20 30
2 nm 5 nm
50 nm
10 nm
10
1
0.1
0.01
0.001
0.0001
0.00001
100 nm
25 nm
2 nm 5 nm
50 nm
10 nm
100 nm
25 nm
p (MPa)
40 50 010203040
p (MPa)
50
(a) kk ~ p (b) kk×100%/(kvs+kk+ks) ~ p
Fig. 7—Comparison of Knudsen diffusion with pressure under different nanopore radius (DH514 kJ/mol).
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2016 SPE Journal 17
where k
s
is the apparent permeability of adsorbed-gas surface dif-
fusion in m
2
and Vis the mole volume of gas at the subsurface
conditions of SGRs in m
3
/mol.
A Model For Gas Transfer in Shale Nanopores
Model Description. Gas-transfer mechanisms include slip flow,
Knudsen diffusion, and surface diffusion in nanopores of shale-gas
reservoirs (SGRs). The total gas flux is a sum of each flux from dif-
ferent mechanisms (Hwang and Kammermeyer 1966; Cunningham
and Williams 1980; Rao and Sircar 1993; Do et al. 2001b; Medved
and Cerny 2011), as shown in Fig. 2. Although the bulk-gas flux
and adsorbed-gas flux are a simple linear sum (Cunningham and
Williams 1980), the slip-flow flux and Knudsen-diffusion flux are a
weighted sum. Therefore, the total gas flux can be expressed as
Jt¼JbþJs¼1
1þKn Jvs þ1
1þ1=Kn JkþJs:ð36Þ
Note that gas adsorption/desorption is the only phase transition
between free gas in the bulk-gas phase and adsorbed gas on the sur-
face in nanopores of SGRs, which affects bulk-gas transfer and sur-
face diffusion but has no contribution to gas-mass transfer. Therefore,
gas adsorption/desorption is not considered when analyzing the con-
tributions of gas-mass transfer from different mechanisms.
The apparent permeability can successfully describe the con-
tributions of each transfer mechanism during the development of
SGRs. By combining Eqs. 17 and 20, the apparent permeability of
slip flow and Knudsen diffusion are expressed as, respectively,
kvs ¼1
ð1þKnÞfmbxmxs
r2ð1þaKnÞVstd
81þ4Kn
1bKn
;
ð37Þ
kk¼1
ð1þ1=KnÞfmbxmxs
2
3rdDf28RT
pM
0:5g
p:ð38Þ
By combining Eqs. 35 through 38, the apparent permeability
for the total gas through shale nanopores is expressed as
kt¼kvs þkkþks;ð39Þ
where k
t
is the apparent permeability for the total gas through
shale nanopores in m
2
.
The physical meaning of Eq. 39 can be described as
•It considers the effects of slippage, rarefaction, nanopore
structure (pore size, porosity, tortuosity, and surface rough-
ness), poromechancial response, and sorption-induced-swel-
ling response on the bulk-gas transfer.
•It characterizes the changed rules of contributions of slip
flow and Knudsen diffusion to bulk-gas transfer during the
development of SGRs.
•It considers the contribution of surface diffusion to gas-mass
transfer in nanopores.
...
.....
.........................
100
80
60
40
20
0
100
80
60
40
20
0
01020
30
kvs kkkskvs kkks
kvs kkkskvs kkks
40 50
p (MPa)
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
01020
30 40 50
p (MPa)
ks (%)
ks (%)
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50
p (MPa)
01020
30 40 50
p (MPa)
ks (%)
ks (%)
Fig. 9—Comparison of transfer capacities of different transport mechanisms with pressure (DH58 kJ/mol).
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18 2016 SPE Journal
Model Validation. Bulk-Gas-Transfer Model. To validate the
bulk-gas-transfer model, the calculation results are compared with
the molecular-simulation and experimental data published. To
facilitate comparison on the basis of the continuum-flow flux and
Knudsen-diffusion flux, the dimensionless bulk-gas fluxes are
expressed as, respectively,
Jb
Jv
¼ð1þaKnÞ
ð1þKnÞ1þ4Kn
1bKn
þdDf2Kn
ð1þ1=KnÞ
128
3p;
ð40Þ
Jb
Jk
¼ð1þaKnÞ
ð1þKnÞ1þ4Kn
1bKn
3p
128dDf2Kn þ1
ð1þ1=KnÞ;
ð41Þ
where J
b
/J
v
is the dimensionless bulk-gas flux with the basis of
the continuous-flow flux and is dimensionless, and J
b
/J
k
is the
dimensionless bulk-gas flux with the basis of the Knudsen-diffu-
sion flux and is dimensionless.
The modeling parameters are shown in Table 2, and the calcu-
lation results are shown in Fig. 3.
Fig. 3 shows that analytical solutions are consistent with mo-
lecular-simulation and experimental data published, and that the
new model of the bulk-gas transfer is reliable.
Adsorbed-Gas Surface-Diffusion Model. The surface-diffu-
sion coefficient D0
swith gas coverage of zero is derived from Eq.
28 and fit to Eq. 30 with experimental data. Therefore, the D0
scal-
culated by Eq. 30 is reliable.
To verify the model that considers the influence of adsorbed-
gas coverage on the surface-diffusion coefficient at high pressure,
Eq. 31 is transformed into
Ds=D0
s¼
ð1hÞþj
2hð2hÞþ½Hð1jÞð1jÞj
2h2
ð1hþj
2hÞ2:
ð42Þ
The analytical solution can be calculated by Eq. 42, and is con-
sistent with the experimental data (as shown in Fig. 4), indicating
that the model is reliable and can be applied at high pressure.
Modeling parameters and experimental conditions are shown in
Table 3.
Results and Discussion
Gas-transfer mechanisms in shale nanopores include slip flow,
Knudsen diffusion, and surface diffusion. Different transfer mecha-
nisms have different sensitivities regarding pore size and pressure.
To reveal the contribution of each mechanism in a different produc-
tion process of shale-gas reservoirs (SGRs),the transfer capacities of
differenttransfer mechanisms are compared and analyzed.
100
80
60
40
20
0
100
80
60
40
20
0
01020
30
kvs kkkskvs kkks
kvs kkkskvs kkks
40 50
p (MPa)
01020
30 40 50
p (MPa)
ks (%)
ks (%)
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
p (MPa)
ks (%)
ks (%)
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
p (MPa)
Fig. 10—Comparison of transfer capacities of different transport mechanisms with pressure (DH14 kJ/mol).
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2016 SPE Journal 19
Surface diffusion for the adsorbed gas is affected by gas type,
surface type, and operation parameters. Generally, there is a great
difference between the surface-diffusion coefficients for different
surface/gas systems. For example, the surface-diffusion coefficients
of a methane (CH
4
)/shale system are significantly different from
those of a CH
4
/zeolite system. However, because the chemical and
physical properties of the CH
4
/shale system are similar to those of a
CH
4
/activated-carbon system, the surface-diffusion coefficients of
the CH
4
/shale system are comparable with those of the CH
4
/acti-
vated-carbon system. They range from 1.0 10
4
to 1.0 10
1
cm
2
/s at pressure less than 22 MPa and temperature less than 365
K, as shown in Table 4. A surface-diffusion coefficient increases
with decreasing gas molar mass. The relationships of different
gas surface-diffusion coefficients can be observed with H
2
>
CH
4
>CO
2
>i-C
4
H
10
,NH
3
>CO
2
,C
2
H
6
>C
3
H
8
>n-C
4
H
10
>
CH
3
OH >C
6
H
5
CH
3
,C
2
H
4
>C
3
H
6
>i-C
4
H
10
>SO
2
>CF
2
Cl
2
,and
CO
2
>CF
2
Cl
2
, as shown in Table 5. The surface-diffusion coeffi-
cient of the same gas varies with the solid type, as shown in Table
6. The surface-diffusion coefficient increases with an increasing
temperature, as shown in Table 7. The chemical and physical
properties of a CH
4
/shale system are similar to those of a CH
4
/
activated-carbon system and a CH
4
/coal system. The isosteric-
sorption heats of the CH
4
/shale system are comparable with those
of the CH
4
/activated-carbon system and the CH
4
/coal system, as
shown in Table 8. Therefore, it is reasonable that the isosteric-
sorption heats of the CH
4
/shale system range from 7.8 to 54 kJ/
mol. The surface-diffusion coefficient D0
sfor CH
4
decreases with
an increasing isosteric sorption heat in SGRs, and calculation val-
ues of Eq. 30 are comparable with the experimental results pub-
lished, as shown in Fig. 5.
Slip Flow. Under the condition of large pores or high pressure,
the gas-molecule free path is much smaller than a pore diameter,
the collision frequency among gas molecules is far greater than
that between gas molecule and solid surface, and the gas transfer
is dominated by slip flow. The gas apparent permeability of slip
flow first decreases and then increases with increasing pressure;
this is because the slip effect significantly weakens with an
increasing pressure at low pressure and the stress dependence of
gas-transport capacity significantly strengthens with an increasing
pressure at high pressure. In addition, the gas apparent permeabil-
ity of slip flow increases with an increasing pore radius, as shown
in Fig. 6a; under the condition of meso/macropores (pore radius
more than 2 nm) and high pressure, slip flow makes a great contri-
bution to gas transfer, which is up to 99%; under the condition of
micropores (pore radius 2 nm) and pressure less than 1 MPa, the
contribution of slip flow is lower than 3.10%, which can be
ignored, as shown in Fig. 6b.
Knudsen Diffusion. Under the condition of small pores and low
pressure, the gas-molecule free path is greater than a pore
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50
p (MPa)
01020
30 40 50
p (MPa)
ks (%)
ks (%)
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50
p (MPa)
01020
30 40 50
p (MPa)
ks (%)
ks (%)
kvs kkkskvs kkks
kvs kkkskvs kkks
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
Fig. 11—Comparison of transfer capacities of different transport mechanisms with pressure (DH520 kJ/mol).
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20 2016 SPE Journal
diameter, the collisions are mainly between gas molecules and
solid surface, and the gas transfer in the bulk phase is dominated by
Knudsen diffusion. The gas apparent permeability of the Knudsen
diffusion increases with a decrease in pressure and decreases with a
decrease in the pore radius, as shown in Fig. 7a; this is because the
gas-molecule free path and Knudsen diffusion both increase with a
decrease in pressure, and transport capacity for Knudsen diffusion
increases with an increase in the pore radius. Under the condition
of macropores (pore radius more than 50 nm) and pressure less
than 1 MPa, Knudsen diffusion makes an important contribution to
gas transfer, which is up to 7.64%; at pressure greater than 1 MPa,
the contribution of Knudsen diffusion is lower than 0.53%, which
can be ignored, as shown in Fig. 7b.
Surface Diffusion. As seen from Fig. 8a, the gas apparent per-
meability of surface diffusion decreases with an increase in pres-
sure; this is because the apparent permeability of surface diffusion
relates not only to the surface-diffusion coefficient but also to the
ratio of the adsorbed-gas concentration to the square of pressure,
C
s
/p
2
. Compared with the surface-diffusion-coefficient degree of
increase, the C
s
/p
2
degree of increase is greater with the increase
of pressure. In addition, the apparent permeability of surface dif-
fusion increases with a decrease in the pore radius. This is
because, under the condition of the same porosity, the smaller the
pore radius, the larger the pore number, the greater the ratio of the
adsorbed-gas-section area to the total pore-section area, and the
greater the transfer volume of surface diffusion. As seen from Fig.
8b, under the condition of micropores (pore radius 2 nm), gas
transfer is always dominated by surface diffusion over all the
ranges of pressure, and the contribution of surface diffusion to gas
transfer ranges from 48.34 to 99.97%; under the condition of a
pore radius larger than 25 nm and pressure greater than 1 MPa,
the contribution of surface diffusion is lower than 3.28%, which
can be ignored; under the condition of a pore radius ranging from
2 to 25 nm and the pressure less than 5 MPa, the contribution of
surface diffusion is greater than 10%, which cannot be ignored.
Comparison of Transfer Capacities of Different Transfer
Mechanisms. The contributions of different transfer mechanisms
to the total gas flux are controlled by pore size and pressure in the
nanopores of SGRs. Because of the uncertainty of isosteric-sorp-
tion heats in CH
4
/shale systems, ranging from 7.8 to 54 kJ/mol,
isosteric-sorption heats significantly affect the surface-diffusion
contribution. The contributions of different transfer mechanisms
must be determined and discussed with the isosteric-sorption heats
of 8, 14, 20, 26, 32, 38, and 44 kJ/mol, respectively, as shown in
Figs. 9 through 15. Results show that there is a great difference in
different transfer-mechanism contributions with different isosteric-
sorption heats. Under the condition of pore radius of 2 nm and
isosteric-sorption heat less than 32 kJ/mol, surface diffusion is im-
portant, even dominating the gas-mass transfer over the range of
pressure. The surface-diffusion contribution decreases with an
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50
p (MPa)
01020
30 40 50
p (MPa)
ks (%)
ks (%)
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50
p (MPa)
01020
30 40 50
p (MPa)
ks (%)
ks (%)
kvs kkkskvs kkks
kvs kkkskvs kkks
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
Fig. 12—Comparison of transfer capacities of different transport mechanisms with pressure (DH526 kJ/mol).
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2016 SPE Journal 21
increase in pore size, isosteric-sorption heat, and pressure. For the
slip flow, under the condition of pore radius of 2 nm and isosteric-
sorption heat greater than 32 kJ/mol, it is important and even dom-
inates the gas-mass transfer over the range of pressure. Further-
more, when the pore radius is larger than 5 nm, it dominates the
gas-mass transfer over the range of pressure with the isosteric-
sorption heats ranging from 8 to 44 kJ/mol. In general, the contri-
bution of Knudsen diffusion is always very weak over the range of
pressure and can be ignored. Only under the condition of macro-
pores (pore radius larger than 50 nm) and with the pressure less
than 1 MPa does the Knudsen diffusion make an important contri-
bution to gas transfer, which is up to 7.64%.
There are abundant nanopores in North American SGRs, and
for the majority, the nanopore radius is smaller than 20 nm with a
surface area of 2–18 m
2
/g and with a large amount of adsorbed
gas (Clarkson et al. 2013). A surface-diffusion coefficient is com-
parable with a pore-diffusion coefficient. Hence, surface diffusion
is important for gas-mass transfer in SGRs. Kang et al. (2011)
estimated methane-transport parameters for organic-rich-shale
samples by use of isothermal pulse-decay measurements at vary-
ing pore pressures, and their results showed that a surface-diffu-
sion coefficient ranges from 1.55 10
3
to 8.80 10
2
cm
2
/s.
Xiong et al. (2012) calculated that the surface-diffusion coeffi-
cient is larger than 1.0 10
3
cm
2
/s under the condition of pres-
sure of 6.895–34.474 MPa and temperature of 363 K. The
experimental results of Akkutlu and Fathi (2012) showed that a
surface-diffusion coefficient ranges from 5.1 10
4
to 8.8 10
2
cm
2
/s, and that surface diffusion plays an important role in gas
transport, especially when the kerogen pore network is not fully
developed and thus not interconnected, and the surface-diffusion
coefficient is significantly high compared with a pore-diffusion
coefficient. Fathi and Akkutlu (2014) indicated that even if the ra-
tio of the surface-diffusion coefficient to the pore-diffusion coeffi-
cient is 0.01, their simulation results showed the importance of
the surface diffusion, which leads to an additional recovery of
35%, and the surface diffusion is likely to play an important and
positive role during production of SGRs.
Some similar results are obtained in other material/gas sys-
tems. Barrer and Barrie (1952) indicated that surface diffusion is
comparable with pure-volume diffusion of the same gases in
channels of a mean radius of 3 nm in a porous glass, and the ratio
of surface flow to volume flow ranges from 0.38 to 1.05. Gilliland
et al. (1958) showed that permeabilities considerably larger than
the values predicted from a nonadsorbed-gas correlation, some-
times more than 17 times larger, were observed for hydrocarbon/
Vycor systems. Weaver and Metzner (1966) concluded that the
ratio of the surface-transport rate to the total transport rate ranges
from 0.227 to 0.315 for isobutane on porous Vycor at a tempera-
ture of 298.15 K.
In addition to isosteric-sorption heats, surface diffusion and its
contribution are affected by pore size, pressure, and temperature.
Sheng et al. (2014) indicated that the contribution of surface
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
p (MPa) p (MPa)
ks (%)
ks (%)
p (MPa) p (MPa)
ks (%)
ks (%)
kvs kkkskvs kkks
kvs kkkskvs kkks
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
Fig. 13—Comparison of transfer capacities of different transport mechanisms with pressure (DH532 kJ/mol).
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22 2016 SPE Journal
diffusion to gas-mass transfer decreases from 0.78 to 0.20 when
the pore radius increases from 2.5 to 10nm under the condition of
pressure of 2.5 MPa and temperature of 366.5 K in SGRs. A sur-
face-diffusion coefficient increases, but its contribution decreases
when the pressure increases. Medved and Cerny (2011) showed
that the surface-diffusion coefficient increases 20 times when the
gas coverage increases from 0.2 to 0.9. Do et al. (2000) concluded
that surface diffusion is important and even dominates the gas-
mass transfer, especially at low pressure. Guo et al. (2008) demon-
strated that the ratio of the surface-diffusion flux to the total flux
decreases from 0.9189 to 0.8115 when pressure increases from 0.1
to 2.5 MPa at a temperature of 273 K. Temperature dependence of
surface diffusion is weaker compared with pore flow. Hence, the
surface-diffusion contribution decreases with an increasing tem-
perature. Barrer and Strachan (1955) concluded that the ratio of
the surface-diffusion flux to the bulk-gas flux decreases from 1.51
to 1.28 when the temperature increases from 273.15 to 298.15 K
for nitrogen gas in a carbolic-carbon plug. Barrer and Gabor
(1959) also indicated that the ratio of the surface-diffusion flux to
the bulk-gas flux decreases from 0.21 to 0.14 when the temperature
increases from 273.15 to 323.15 K for nitrogen gas in catalyst
plug. Schneider and Smith (1968a, b) studied the monolayer
adsorption and surface diffusion of ethane, propane, and n-butane
on silica gel, and their results showed that the surface contribution
ranges from 0.72 to 0.61, depending on temperature.
Conclusions
Gas-transfer mechanisms include pore diffusion for the bulk gas
and surface diffusion for adsorbed gas in nanopores of shale-gas
reservoirs (SGRs). A unified model for gas transfer coupling dif-
ferent gas-transfer mechanisms in nanopores of SGRs has been
established, and some key conclusions can be drawn.
•Slip flow makes a great contribution to gas transfer under the
condition of meso/macropores (pore radius larger than 2 nm)
and high pressure, but it can be ignored under the condition of
micropores (pore radius 2 nm) and pressure less than 1 MPa
in SGRs.
•Knudsen diffusion makes an important contribution to gas
transfer under the condition of macropores (pore radius larger
than 50 nm) and pressure less than 1 MPa, whereas it can be
ignored in other cases.
•A surface-diffusion coefficient is comparable with a pore-diffu-
sion coefficient, and gas transfer is always dominated by surface
diffusion over all the ranges of pressure in micropores (pore
radius 2nm). Surface diffusion cannot be ignored under the con-
dition of a pore radius ranging from 2 to 25nm and pressure less
than 5 MPa, whereas it can be ignored under the condition of the
pore radius greater than 25 nm andpressure higher than 1 MPa.
•Temperature and pressure dependence of surface diffusion is
weaker compared with that of pore flow. Surface-diffusion
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
p (MPa) p (MPa)
ks (%)
ks (%)
p (MPa) p (MPa)
ks (%)
ks (%)
kvs kkkskvs kkks
kvs kkkskvs kkks
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
Fig. 14—Comparison of transfer capacities of different transport mechanisms with pressure (DH538 kJ/mol).
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2016 SPE Journal 23
contribution decreases with an increase in the pore size, isos-
teric sorption heat, pressure, and temperature in SGRs.
•There is a great difference in different transfer-mechanism con-
tributions with different isosteric-sorption heats. Surface-diffu-
sion contribution decreases, whereas slip-flow and Knudsen-
diffusion contributions increase with an increasing isosteric-
sorption heat.
Nomenclature
A
a
¼bulk gas flowing a sectional area in shale, m
2
A
b
¼sectional area of shale, m
2
A
t
¼sectional area of a capillary in shale, m
2
b¼gas slip constant, dimensionless
b
int
¼initial microfracture aperture, m
C
s
¼adsorbed gas concentration, kg/m
3
d
M
¼gas molecular diameter, m
D
eff-k
¼Knudsen diffusion coefficient considering wall rough-
ness, m
2
/s
D
f
¼fractal dimension of the pore wall, dimensionless
D
k
¼Knudsen diffusion coefficient, m
2
/s
D
s
¼gas surface diffusion coefficient, m
2
/s
D0
s¼surface diffusion coefficient when the gas coverage is
zero, m
2
/s
E¼gas activation energy, J/mol
E
s
¼shale matrix Young’s modulus, Pa
H(1–j)¼Heaviside function, dimensionless
J
b
¼mass flux of the bulk phase transfer, kg/(m
2
s)
J
b
/J
k
¼dimensionless bulk gas flux with the basis of the Knud-
sen diffusion flux, dimensionless
J
b
/J
v
¼dimensionless bulk gas flux with the basis of the con-
tinuous flow flux, dimensionless
J
k
¼Knudsen diffusion flux, mol/ (m
2
s)
J
s
¼mass flux of the adsorbed gas surface diffusion, kg/
(m
2
s)
J
tot
¼mass flux of the total fluid transfer, kg/(m
2
s)
J
v
¼continuum flow flux, mol/ (m
2
s)
J
vs
¼slip flow flux, mol/ (m
2
s)
k
a
¼shale permeability, m
2
k
b
¼apparent permeability for the bulk gas through nano-
pores, m
2
k
bt
¼apparent permeability for bulk gas through shale nano-
pores, m
2
k
k
¼apparent permeability of Knudsen diffusion, m
2
k
s
¼apparent permeability of adsorbed gas surface diffu-
sion, m
2
k
t
¼apparent permeability for the total gas through shale
nanopores, m
2
k
tt
¼permeability of a capillary in shale, m
2
k
vs
¼apparent permeability of slip flow, m
2
Kn¼average microfracture normal stiffness, Pa/m
l¼gas-transport distance, m
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
100
80
60
40
20
0
100
80
60
40
20
0
01020
30 40 50 01020
30 40 50
p (MPa) p (MPa)
ks (%)
ks (%)
p (MPa) p (MPa)
ks (%)
ks (%)
kvs kkkskvs kkks
kvs kkkskvs kkks
(a) r = 2 nm (b) r = 5 nm
(c) r = 10 nm (d) r = 25 nm
Fig. 15—Comparison of transfer capacities of different transport mechanisms with pressure (DH544 kJ/mol).
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24 2016 SPE Journal
l
b
¼shale length, m
l
t
¼length of a capillary in shale, m
L
m
¼gas mobility, mols/kg
m¼constant, dimensionless
M¼gas molar mass, kg/mol
N
A
¼Avogadro’s constant, 6.022141510
23
/mol
p¼pressure, Pa
p
int
¼initial pressure, Pa
p
L
¼Langmuir pressure, Pa
q
a
¼standard volume of adsorbed gas per unit mass in shale
rock, m
3
/kg
q
L
¼Langmuir volume, m
3
/kg
r
t
¼radius of a capillary in shale, m
R¼universal gas constant, J/(molk)
s¼microfracture spacing, m
t
M
¼average time required for one collision between the
molecules, seconds
t
S
¼averaged time required for one collision between the
molecule and the nanopore wall, seconds
t
T
¼average time consumed for one collision of overall gas
molecules, seconds
T¼temperature, K
u¼chemical potential, J/mol
u
O
¼chemical potential in a reference state, J/mol
V¼mole volume of gas at the subsurface conditions of
SGRs, m
3
/mol
a
o
¼rarefaction coefficient at Kn!1, dimensionless
a
f
¼Biot coefficient for microfracture, dimensionless
a
s
¼Biot coefficient for shale matrix, dimensionless
a
1
¼fitting constant, dimensionless
b¼fitting constant, dimensionless
DH¼gas isosteric adsorption heat, kJ/mol
e
L
¼Langmuir strain, dimensionless
f
mb
¼a correction factor of apparent permeability in nano-
pores of SGRs, dimensionless
f
ms
¼correction factor of surface diffusion of adsorbed gas
in the nanopores of SGRs, dimensionless
g¼gas viscosity, Pas
h¼gas coverage on the pore wall at equilibrium state,
dimensionless
j¼ratio of the blocking velocity coefficient j
b
to the for-
ward velocity coefficient j
m
,dimensionless
j
b
¼blocking velocity coefficient of surface gas molecule, m/s
j
m
¼forward velocity coefficient of surface gas molecule,
m/s
k
T
¼average free path of the overall gas molecules, m
<v>¼average velocity of molecule movement, m/s
r¼ratio of the gas molecule diameter to the local average
pore diameter, dimensionless
/¼shale porosity, dimensionless
x
m
¼poromechancial response coefficient of shale matrix,
dimensionless
x
s
¼sorption-induced swelling response coefficient of shale
matrix, dimensionless
X¼a constant that relates to gas molecular weight, m
2
/
(sK
0.5
)
Acknowledgments
The authors would like to acknowledge the Natural Sciences and
Engineering Research Council of Canada, Alberta Innovates
Energy and Environment Solutions, Foundation CMG, and Alberta
Innovates Technology Futures for providing research funding.
Author K. Wu also acknowledges the National Science and Tech-
nology Major Project of China (No. 2011ZX05030-005-04) and
National Natural Science Foundation of China (Nos. 51490654
and 51374222) for their support of part of this work.
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Appendix A—Derivation of Correction Factor of
Apparent Permeability in Nanopores of Shale-Gas
Reservoirs (SGRs)
SGRs are characterized by myriad nanopores. The SGRs are theo-
retically considered to be the integration of capillaries and matrix,
and the assumption of the nanopores being simplified to be
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28 2016 SPE Journal
capillaries is often given. In this study, this assumption has been
put forward for the following calculation. A shale reservoir is rep-
resented by a cuboid with porosity of /,permeability of k
a
,a sec-
tional area of A
b
,and a length of l
b
,and consisting of n
nanopores. Consequently, it can be further developed from the
assumption that there exist ncapillaries with the average perme-
ability of k
tt
,the average radius of r
t
,the average length of l
t
,and
the average sectional area of A
t
in the cuboid.
The gas flux of SGRs consisting of ncapillaries can be written,
without consideration of absorbed gas, as
qb¼nkttAt
l
dp
dlt
:ðA-1Þ
The gas flux can be also represented by
qb¼kaAb
l
dp
dlb
:ðA-2Þ
The permeability k
a
can be derived from the comparison of
Eqs. A-1 and A-2:
ka¼nAtktt
Ab
dlb
dlt
:ðA-3Þ
In SGRs, the relation between the capillary size and capillary
number can be expressed as
n¼/Ab
At
¼/Ab
pr2
t
:ðA-4Þ
In addition, the tortuosity of the nanopores can be written as
s¼lt
lb
;ðA-5Þ
where the tortuosity of the nanopores can be tested by diffusion
experiments.
Substituting Eqs. A-4 and A-5 into Eq. A-3, we obtain
ka¼/
sktt:ðA-6Þ
Then, a correction factor for the apparent permeability in
nanopores of SGRs can be expressed as
fmb ¼/
s:ðA-7Þ
Eq. A-7 indicates that the porosity and tortuosity have a pro-
nounced influence on the gas-transport efficiency.
Appendix B—Derivation of Correction Factor of
Surface Diffusion of Adsorbed Gas in Nanopores
of Shale-Gas Reservoirs (SGRs)
The assumption made in Appendix A is also suitable in Appendix
B. In addition, gas adsorption on nanopore walls of SGRs is
regarded as Langmuir-monolayer adsorption and the height of the
adsorption monolayer is set to be equivalent to the diameter of a
gas molecule d
M
. Therefore, the bulk gas flowing in a sectional
area A
a
in the nanopores can be described as
Aa¼pðrtdMÞ2:ðB-1Þ
The adsorbed-gas-diffusion sectional-area A
s
in nanopores can
be expressed as
As¼pr2
tpðrtdMÞ2:ðB-2Þ
By combining Eqs. B-1 and B-2, the ratio of A
s
to A
a
,A
s–a
, can
be achieved by
Asa¼As
Aa
¼1dM
r
2
1:ðB-3Þ
The bulk-gas flux of nanopores can be written as
qb¼Aa
ktt
l
dp
dlt
:ðB-4Þ
The surface-diffusion flux in the walls of nanopores can be
expressed as
qs¼As
ks
l
dp
dlt
:ðB-5Þ
Eq. B-5 can be rearranged with the consideration of Eq. B-3 as
qs¼Aa
Asaks
l
dp
dlt
:ðB-6Þ
According to Eqs. B-4 and B-6, the sum of the bulk-gas flux
and surface-diffusion flux in nanopores walls can be derived by
qbs¼qbþqs¼Aa
ðktt þAsaksÞ
l
dp
dlt
:ðB-7Þ
By combining Eqs. A-7, B-3, and B-7, the correction factor
for gas diffusion in the nanopore walls of SGRs, f
ms
, can be
expressed as
fms ¼fmbAsa¼/
s1dM
r
2
1
"#
:ðB-8Þ
Keliu Wu is currently a post-doctoral fellow at the University of
Calgary. His research interests are microscale- and nanoscale-
flow mechanisms in shale-gas reservoirs, upscaling of the fluid
transport in heterogeneous porous media, and numerical
modeling and simulation of coupled flow and geomechanics
processes. Wu also focuses on carbon dioxide sequestration in
shale-gas reservoirs and tight oil reservoirs. He holds a bache-
lor’s degree in petroleum engineering from China University of
Geosciences, Wuhan, and a PhD degree in petroleum engi-
neering from China University of Petroleum, Beijing.
Xiangfang Li is a professor of petroleum engineering at the
China University of Petroleum, Beijing. His research interest is in
integrated studies in enhanced oil recovery, oil-production
technology, and mechanics of multiphase flows in porous
media.
Chaohua Guo is a PhD degree student at Missouri University of
Science and Technology, majoring in petroleum engineering.
His research interests include flow mechanics of gas in nano-
pores, numerical simulation of shale-gas reservoirs, and well
testing of low-permeability reservoirs with multifractured hori-
zontal wells. Guo holds a bachelor’s degree in petroleum en-
gineering from China University of Petroleum.
Chenchen Wang is currently a post-doctoral fellow at the Uni-
versity of Calgary. His research interests are flow behavior and
numerical modeling in shale-gas reservoirs. Wang holds bach-
elor’s and PhD degrees in petroleum engineering from China
University of Petroleum.
Zhangxin Chen is a professor in the Department of Chemical
and Petroleum Engineering and the Director of the Founda-
tion CMG/Frank and Sarah Meyer Collaboration Centre at the
University of Calgary. His research specialty is in reservoir mod-
eling and simulation and scientific computing. Chen has the
distinction of holding two chair awards: the Natural Sciences
and Engineering Research Council of Canada/AIEE (AERI)/
Foundation CMG Senior Research Chair in Reservoir Simulation
and the Alberta Innovates Technology Futures (Alberta Inno-
vates Technology Futures, formerly iCORE) Industrial Chair in
Reservoir Modeling. He holds a PhD degree from Purdue
University.
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2016 SPE Journal 29
