Environmental Engineering Reference
In-Depth Information
than water (by volume) under the same pressure gradient.
Equation 9.29 shows that the mass flow rate of water will
be higher than the mass flow rate of air under the same
pressure gradients.
The volume flow ratio of air to water under a soil tem-
perature of 10
◦
C can be written as follows:
K
w
. It has been noted that the intrinsic permeability measured
when using water is still smaller than the intrinsic permeabil-
ity measured when using air even after the air permeability
results have been corrected for gas slippage.
The Klinkenberger effect becomes important when the
measured intrinsic permeability is lower than 10
−
18
m
2
and the applied differential pressures are low during the
measurement of permeability. When the rate of air flow is
high and the applied pressures are high, the two intrinsic
measurements come closer to one another. The correction
of air intrinsic permeability for the Klinkenberger effect
becomes relevant as the permeability measurements become
extremely low. In general, the application of a correction for
gas slippage is not necessary because soils have a sufficiently
high coefficient of permeability.
The air coefficient of permeability under dry conditions,
K
a
, is often considered to be a constant for simplicity.
However, the intrinsic permeability varies with the absolute
pore-air pressure, and this is also part of the Klinkenberg
effect (Klinkenberg, 1941). The Klinkenberg effect can be
approximated using the following relationship:
Q
a
Q
w
=
υ
w
υ
a
=
1
.
310
0
.
0176
=
74
.
3
(9.30)
The mass flow ratio of air to water under the same soil
temperature can be written as follows:
k
a
k
w
=
dM
a
/
dt
dM
w
/
dt
=
1
9
.
18
(9.31)
The volume flow rate of air is 74.3 times more than the
volume flow rate of water. However, the mass flow rate of
water will be 9.18 times more than the mass flow rate of air.
9.3.5 Klinkenberg Effect
The intrinsic permeability measured by passing air through
a dry porous medium,
K
a
, should be similar to the intrinsic
permeability measured by passing water through the same
saturated porous medium,
K
w
. However, air passage does
not behave quite the same as water flow because the air
velocity at the solid boundaries of a pore is not zero as it
is for the case when water is used. The intrinsic permeabil-
ity measured using air,
K
a
, will appear to be larger than
the intrinsic permeability measured when using water, as
shown in Fig. 9.7. This difference in flow behavior is known
as “gas slippage” or the “Klinkenberger effect” (Klinken-
berger, 1941). Klinkenberger demonstrated that the perme-
ability measured using gases was approximately a linear
function of the reciprocal of the applied gas pressure.
Klinkenberger (1941) showed that the effect of gas slippage
can be evaluated by measuring the air intrinsic permeability
K
a
for a range of pressures and extrapolating the results to an
infinite mean pressure. The extrapolated value will be simi-
lar to the intrinsic permeability measured when using water,
1
b
u
a
K
a
=
K
∞
+
(9.32)
where:
K
a
=
intrinsic permeability measured using air,
K
∞
=
air phase intrinsic permeability when using a high
air pressure, and
=
b
parameter representing the porous medium.
Heid et al. (1950) proposed an empirical relationship
between
K
and
b
based on experimental data:
∞
0
.
147
K
−
0
.
39
∞
b
=
(9.33)
where
b
is in kilopascals and
K
is in meters squared.
Equation 9.33 has been adopted by the American Petroleum
Institute for correcting for the Klinkenberg effect (Baehr and
Hult, 1991). The maximum percent error in estimating the
air permeability by ignoring the Klinkenberg effect was stud-
ied by Baehr and Hult (1991). The Klinkenberg effect was
shown to produce an error greater than 10% when measur-
ing the air permeability of a porous media where the intrinsic
permeability was 10
−
13
m
2
(or 10
−
7
mm
2
) or less.
∞
Liquid flow and gas flow at high mean flowing pressure is
laminar
- Darcy's Law is valid
- Flow velocity at walls is zero
At low mean flowing pressure gas slippage occurs
- Non-Darcy flow is observed
- Flow at walls is not zero
Klinkenberg developed a method to correct gas permeability
measures at low mean flowing pressure to equivalent liquid
permeability
9.4 DIFFUSION OF AIR THROUGH WATER
The diffusion process occurs in response to a concentration
gradient. Ionic or molecular movement will take place
from regions of higher concentration to regions of lower
Figure 9.7
Difference in intrinsic permeability observed when
using water and a gas such as nitrogen.
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