Page 292 - Fundamentals of Gas Shale Reservoirs

272 A REVIEW OF THE CRITICAL ISSUES SURROUNDING

temperature of 350 K for an ideal gas. For small pore throats, Equation 12.25 is the Klinkenberg equation (Klinkenberg,
the correction term increases effective permeability k by a 1941) that is used to correct laboratory gas permeability
factor of 3–5 at abandonment pressures and about 1.5 at initial measurements. For an ideal gas from Equations 12.12 and
reservoir pressures. For larger pore throats, the correction 12.19, the correction term can be written as b/P where b is a
term is somewhat smaller. constant, so that in the limit of large P k goes to K . To first
G
Measurements of effective pore size based on gas storage order in Kn, Equation 12.21 for the ideal gas case can also
measurements and mercury injection measurements suggest be rewritten as the sum of a Darcy term and a Knudsen
that effective pore sizes of 2–3 nm are not unreasonable diffusion term.
(Sigal, 2013a, c; Sigal et al., 2013). dP d
For a single size tube k and f(Kn) both are functions of a v k G 0 589 D k (12.26)
.
G
single‐scale parameter, the tube radius. Michel et al. (2011a, b) dx dx
investigated the extension of the Beskok and Karniadakis for­ In Equation 12.26, ρ is the density and D is the Knudsen
mulation to bundles of tubes with log normal distributions. It is diffusivity (Civan, 2010a, b, 2011; Michel et al., 2011b). The
k
desirable for the more general case to preserve the form of k as viscosity in the second term is eliminated by using the
the product of k times an apparent permeability correction expression for the viscosity of an ideal gas (Guggenheim,
G
term f(Kn). For a bundle of tubes, Equation 12.15 or its integral 1960). The apparent diffusion term enters from the gas slip­
form provides the scale factor r Geff for k . The complex page effect, and does not imply that the transport equation
G
functional form of f(Kn) does not allow for a simple analytical accounts for classical concentration driven diffusion.
expression for r , the effective size parameter that would The formulation so far discussed has not accounted for
Seff
replace the hydraulic radius, r in Kn when there are multiple gas adsorption on organic pore walls. Adsorption in general
tube sizes. For each of the log normal distributions examined a will reduce the space available for free gas transport. In
r was found numerically that when substituted into f(Kn) addition, adsorption introduces a new transport mode, trans­
Seff
matched the ratio of the calculated value of k/k . In general port in the adsorbed layer. The effect of adsorption on ideal
G
r does not equal r Geff , so that a generalized Beskok and gas transport in smooth circular cross‐section capillary tubes
Seff
Karniadakis permeability requires two different functions of has been investigated in Xiong et al. (2012) (also see Sigal,
the pore throat sizes to characterize it. For the case of a general 2013b). The model assumed the density of the adsorbed
porous media, we have proposed that for simple nonadsorbing layer was in equilibrium with the local pore pressure so that
gases the complexities of the pore geometry are captured in a a pore pressure gradient produces a density gradient in the
k term and there exists a r such that the correction term adsorbed layer. The Beskok and Karniadakis equation con­
G
Seff
keeps the same form as for the bundle of tube case. That is
trolled the transport of the free gas in each tube, and the
adsorbed gas transport was calculated by a Fick’s law diffu­
k kf Kn r
G Seff (12.24) sion equation. For this model study, adsorption significantly
reduced k . The effect on f(Kn) was minimal. Transport in
G
For porous media characterized by a pore size distribution, the adsorbed layer in a 2 nm radius tube was only significant
the apparent permeability for a distribution of pore sizes may for values of diffusivity larger than 0.01 cm /s.
2
be quantified through numerical integration of the formula­ For a smooth tube with geometrical radius r and with
0
tion for apparent permeability for a single capillary tube. single layer adsorption, in the limit of zero pressure the tube
Michel et al. (2011a, b) considered three different log‐ radius for free gas transport is r and its cross‐sectional area is
0
normal pore size distributions (Figure 7 in Michel et al., πr . At infinite gas pressure the effective tube radius is (r – d )
2
m
0
0
2011b). The narrowest distribution is characterized by pores and the cross‐sectional area is π(r – d ) . It is assumed for
2
0
m
from 1 to 10 nm, while the intermediate distribution is char­ intermediate pressures that the appropriate tube radius is
acterized by pores from 1 to 30 nm and the broadest distribu­
tion is characterized by pores from 1 to 100 nm. The r 0 d L (12.27)
m
corresponding permeability correction factors indicate that Using Equation 12.27 transport of free methane through a
the distribution characterized by the smallest pores has the bundle of adsorbing tubes can be computed. Figure 12.3
largest permeability correction (Figure 8 in Michel et al., shows the ratio of k , the geometrical permeability corrected
2011b); however, as pore sizes get larger, the extent of this for adsorption to k for a tube radius of 2.5 nm, P of 1800 psi,
Ga
correction remains significant, but diminishes. at 350 K, with d equal to 0.38 nm. L
G
Using Equations 12.21–12.23, Equation 12.24 can be m
expanded to first order in Kn. This gives This model would understate the effects of adsorption if
the adsorbed zone is not monolayer and if the formation
factor is more typical of a conventional reservoir. On the
k k G 14 (12.25) other hand, a rough pore wall surface should act to reduce
r Seff the permeability loss (Hu et al., 2013a, b).