Environmental Engineering Reference
In-Depth Information
Four generalized relative permeability coefficients are used in pres-
sure-driven flow. These include two diagonal (
k
r,w w
and
k
r,nn
), and the two
off-diagonal (
k
r,w n
and
k
r,nw
) coefficients. The off-diagonal coefficients cap-
ture the contribution of the viscous drag through momentum transfer
(Kalaydjian, 1991; Liang and Lohrenz, 1994; Avraam and Payatakes, 1995;
Bentsen and Manai, 1993; Bentsen 1998; Dullien and Dong 1996; Li et al.,
2005), as given in the following equation:
k
k
⎡
⎤
rww
,
rwn
,
⎢
⎢
⎢
⎢
⎥
⎥
⎥
⎥
v
v
∇
∇
P
P
⎡
⎢
⎤
⎥
=−
m
m
⎡
⎢
⎤
⎥
w
w
w
n
(5.67)
K
k
k
n
rnw
,
rnn
,
n
m
m
⎣
⎦
w
n
Where,
K
is the intrinsic permeability of the porous medium;
v
w
and
v
n
are the flow velocities;
m
w
and
m
n
are the dynamic viscosities; and
∇
P
w
P
n
are the pressure gradients in the wetting and non-wetting phases,
respectively. In this particular case, the non-wetting phase is oil therefore
the subscript identifier “
n
” is changed to “
o
”, representing oil in the follow-
ing presentation.
The two diagonal coefficients can be determined as a function of the
wetting phase saturation using standard test methods of steady state or
un-steady state flow in rock cores. This requires the injection of water and
oil simultaneously at constant rates into the core. Once steady state flow
is reached, the volumetric flow rate is measured. Given that the intrinsic
permeability of the porous medium, pressure gradients, and dynamic vis-
cosity of the fluids are known, the diagonal relative permeability coeffi-
cients can be determined at specific saturation levels of the wetting phase.
Figure 5.4 shows the experimentally evaluated two diagonal coefficients
for a sandstone rock sample (Core C in table 5.2) under applied pressure
gradient.
The experimental evaluation of the off-diagonal coefficients (
k
r,w n
and
k
r,nw
) is more challenging and only a few studies have reported these coef-
ficients from steady state experiments (Kalaydjian, 1991; Bentsen and
Manai, 1993; Liang and Lohrenz, 1994; Bentsen, 1998; Dullien and Dong,
1996). These tests require that upon achieving steady state, the gradient in
one phase is set to zero (i.e., the injection of that phase is cut of ) while the
gradient in the other phase is maintained. Then the volumetric flow rate
of the cut-off phase is measured to compute the corresponding off-diago-
nal coefficient. These coefficients are reported to range between 0 to 24%
of the diagonal coefficients depending on the fluids and porous medium
and
∇
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