Page 291 - Fundamentals of Gas Shale Reservoirs

FLUID TRANSPORT 271
The starting point for this generalization is the direct zRT
extension of Beskok and Karniadakis’ work to a bundle of 2 05 . NP d 2 (12.19)
tubes (Michel et al., 2011a, b). Consider a bundle of n A m
straight circular cross‐section tubes with radii r . If the tubes As long as densities are not too high, this should be an ade­
i
are imbedded in a nonporous matrix they form a simple quate extension of the mean‐free path of an ideal gas. In
porous media with total porosity ϕ. The porosity associated terms of Kn, the Beskok and Karniadakis equation for gas
with each radius r is taken as ϕ . For large enough radii fluid flow in a bundle of circular tubes each of radius r is given in
i
i
flow through the bundle satisfies the Hagen–Poiseuille terms of v the macroscopic velocity density as
equation (Bird et al., 2007), that is Darcy’s law with a
constant permeability k . It is then easy to see that 2
G 1 r dP 1 dP
v f (Kn ) kf (Kn ) (12.20)
1 n 1 n 8 dx G dx
k G r i 2 i r Geff , r Geff r i 2 i (12.15)
2
2
8 1 8 1 4 Kn
where k will be referred to as the geometrical permeability f (Kn ) (1 Kn ) 1 (1 Kn ) (12.21)
G
as it is the permeability that describes the flow of a viscous
liquid that has no chemical interaction with the pore walls, 2 tan 1 Kn (12.22)
and no slippage at the wall boundary. For porous media with 0 1
more complex pore geometries that characterize conven­ 2 125
tional reservoirs, it is generally true that 0 , 1 4, and 0 4. (12.23)
15 2
2
k G Ar Geff m (12.16)
In Equation 12.20, μ is the bulk gas viscosity. Civan (2010a,
where m is the Archie cementation exponent (Katz and b, 2011) has proposed an alternative simpler expression for α.
Thompson, 1986; Sigal, 2002). A is a proportionality con­ The term k f(Kn) is the apparent permeability k which
G
stant that depends on rock type, but for a wide range of describes gas flow in small capillary tubes. When Kn goes to
clastic and carbonate systems its dynamic range is only zero at relatively high pressures or in pores associated with
about 10%. This shows that the difference in formulation for larger diameters, the permeability correction goes to unity
rock permeability between a bundle of tubes and a more gen­ and the apparent permeability reduces to k . It is not only a
eral porous media is just the formation factor. The Beskok function of pore size but also a function of pore pressure
G
and Karniadakis modifications to gas flow are a function of and temperature. The correction factor f(Kn) accounts for
the Knudsen number Kn, which essentially quantifies the slippage and the modification to viscosity that occurs in
importance of methane molecule collisions with the pore nanometer‐scale pores. Gouth et al. (2007) has used molec­
wall relative to collisions with other methane molecules. It is ular dynamic simulation to investigate the Beskok and
the ratio of the average distance a molecule goes between Karniadakis equation. They found that for multicomponent
collisions with another molecule to the characteristic size gases viscosity is not correctly treated by the equation. They
scale of the pore. For a circular cross‐section tube of radius were not able to find a modification to the viscosity that
r, Kn is given by
worked for all possible mixtures of the various components.
Figure 12.2 shows a plot of the correction f(Kn) term as a
Kn (12.17)
r function of pore pressure for a 2.5 nm effective radius and
where λ is the mean‐free path. For an ideal gas λ is given by
(Bird et al., 2007) 10.0
9.0
1 (12.18) 8.0
05
.
2 nd m 2 7.0
6.0
In Equation 12.18, d is the diameter of the gas molecule f (Kn) 5.0
4.0
m
and n is the number density. The equation is derived by 3.0
assuming the gas molecules are spheres that move freely 2.0
except when they collide. As such it breaks down when 1.0
molecules cannot be approximated by spheres, and when 0.0
the density is large enough that the molecules always feel 0 1000 2000 3000 4000 5000 6000
the effects of other molecules. Using Equation 12.12, intro­ Pore pressure (psi)
ducing the Avogadro’s number N , and using the real gas FIGURE 12.2 f(Kn) as a function of pore pressure for radius of
A
equation λ can be written as 2.5 nm and 350 K.