Page 270 - Fundamentals of Gas Shale Reservoirs

250 GAS TRANSPORT PROCESSES IN SHALE

As shown in Figure 11.6, researchers need an equation Equation 11.6 provides an apparent Darcy permeability
that describes flow beyond the limit of slip flow. Molecular relationship written in the Klinkenberg form as
dynamic (MD) models (Karniadakis et al., 2005) are pow-
erful models that are capable of modeling interaction of b
individual gas molecules and molecules at the pore walls. k app k D 1 (11.7)
MD models are valid for any range of K , but they are com- p avg
n
putationally inefficient and therefore of limited use for 05 . 05 .
describing large units such as shale systems. Javadpour b 16 8 RT 8 RT 2 1 (11.8)
(2009) proposed a model that includes the two major mech- 310 3 r M M r
anisms of Knudsen diffusion and slip‐flow contribution to
the gas flow in a single, straight, cylindrical nanotube. where k is Darcy permeability.
D
Javadpour also asserted that these two processes exist at any Azom and Javadpour (2012) showed how Equation 11.6
K , but their individual contributions to total flux varies.
n can be corrected for a real gas flowing in a porous medium.
The final equation still has the form of Equation 11.7, but
2 rM 8 RT 05 . r 2 p p with b given a
J F avg 2 1 (11.4)
310 3 RT M 8 L
16 cp 8 zRT 05 . 8 RT 05 . 2
b g avg 1 , (11.9)
The first and second terms in the right‐hand‐side bracket in 310 3 r M M r r
Equation 11.4 refer to Knudsen diffusion and slip flow, respec-
tively. The term F is the slip coefficient and is defined as:
where c is gas compressibility and z is compressibility
g
factor. Notice that as the real gas becomes ideal, (Eq. 11.9)
.
8 RT 05 2 becomes (Eq. 11.8), because, for an ideal gas, the compress-
F 1 1 (11.5) ibility c = 1/p and the compressibility factor z = 1.
M rp avg Darabi et al. (2012) later applied several modifications to
avg
g
adapt the model developed by Javadpour (2009) from being
where M is molar mass, r is the pore radius, R is the universal applicable to a single, straight, cylindrical nanotube to being
gas constant, T is the temperature, ρ is the average gas applicable to ultra‐tight, natural porous media characterized
avg
density, μ is the gas viscosity, and p and p are the upstream by a network of inter‐connected tortuous micropores and
1 2
and downstream pressures, respectively. p is the average nanopores.
avg
pressure of the system p = (p + p )/2. The term α is the
avg 1 2
tangential momentum accommodation coefficient or, simply, M 2 b
the fraction of gas molecules reflected diffusely from the k app RT D f D k k D 1 p . (11.10)
pore wall relative to specular reflection. The value of α var- avg avg
ies theoretically in a range from 0 (representing specular
accommodation) to 1 (representing diffuse accommodation), In Equation 11.10, φ is porosity, τ is tortuosity, and δ is
depending on wall‐surface smoothness, gas type, tempera- normalized molecular radius size (r ) with respect to local
m
ture, and pressure (Agrawal and Prabhu, 2008; Arkilic et al., average pore radius (r ), yielding δ = r /r . Knudsen diffu-
m
avg
avg
2001). Experimental measurements are needed to determine sion (D ) is defined as:
k
α for specific shale systems.
.
Javadpour (2009) showed that this model matches data by r 2 avg 8 RT 05
Roy et al. (2003), from flow through an Anodisc membrane D k 3 M , (11.11)
(Whatman Ltd.) with pore sizes of 200 nm, at an average
error of 4.5%. By comparing Equation 11.4 to Darcy’s law where r is the average pore radius of the porous system,
avg
for a single nanotube (Hagen–Poiseuille equation), apparent approximated by r = (8k ) . The average pore radius can
0.5
permeability (k ) for a porous medium containing of avg D
app also be determined by laboratory experiments employing
straight cylindrical nanotubes can be defined as:
such as processes as mercury injection and nitrogen
adsorption tests and pore imaging using SEM and AFM.
r 2 8 RT 05 . Darabi et al. (2012) also included the fractal dimension
k
app 3
310 p M of the pore surface (D ) to consider the effect of pore‐
f
avg surface roughness on the Knudsen diffusion coefficient
r 2 8 RT 05 . 2
1 1 (11.6) (Coppens, 1999; Coppens and Dammers, 2006). Surface
8 M rp avg roughness is one example of local heterogeneity. Increasing