Page 269 - Fundamentals of Gas Shale Reservoirs

GAS FLOW IN MICROPORES AND NANOPORES 249
recently for tight gas systems (with pores of 1–10 µm in where
size). Brown et al. (1946) added to our understanding of
these processes and articulated the concept of slip flow, kT , (11.3)
B
which provided an explanation for the observed relationship 2 2 p
between gas‐flow rate and mean pressure.
As explained earlier, the pores in producing shale gas in which k is the Boltzmann constant (1.3805 × 10 J/K), T
−23
reservoirs are in the range of 1–100s nm; the gas molecules is temperature (K), p is pressure (pa), and δ is the collision
B
contained in the pores are of comparable size (~0.5 nm), diameter of the gas molecule (m). Table 11.1 presents flow
and under certain pressure and temperature conditions the regimes corresponding to Knudsen number ranges.
distance between gas molecules (mean free path) exceeds Continuum no‐slip flow or Darcy equation is valid for
the size of the pores. In such conditions, the gas molecules K < 10 . Continuum flow with slip correction (Klinkenberg)
−3
n
might move singly through the pores and the concept of is valid for K < 10 , which covers most conventional gas
−1
n
continuum and bulk flow may not be applicable. Knudsen reservoirs and many tight gas reservoir conditions as well.
number (K ) is the ratio of mean free path (λ) to pore diam-
n However, because of the existence of nanopores in a shale
eter (d), and can be used to identify different flow regimes.
system, K could be larger than 0.1, and hence new forms of
n
gas‐flow equations are needed. Figure 11.6 presents Knudsen
K n , (11.2) number as a function of pore size and mean reservoir
d pressure, and shows the validity of different flow equations.
TAbLE 11.1 Validity of different flow equations as functions of Knudsen number
Knudsen number (K )
n
Lower bound Upper bound Flow regime
Shale gas reservoirs 0 10 −3 Continuum/Darcy flow Navier–Stokes equation
(no‐slip flow)
10 −3 10 −2 Slip flow (Klinkenberg
10 −2 10 −1 model)
10 −1 10 0 Transition flow
10 0 10 1
10 1 ∞ Free‐molecule flow

10 1

10 0

10 –1

10 –2
K R
Shale gas 10 –3

10 –4 d = 1 nm Klinkenberg equation
d = 10 nm
d = 100 nm Darcy equation
10 –5 d = 1 m
d = 10 m
10 –6
10 –1 10 0 10 1 10 2
p m (MPa)
FIGURE 11.6 Knudsen number (K ) as function of mean reservoir pressure for porous system with different mean pore size. Validity of
n
different gas‐flow equations and condition corresponding to shale gas reservoirs are marked.