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s
w
Model B
where
is called the pore size distribution index. There-
fore, we identify the following equality:
λ
Λ Λ
3 199
where
n
denotes the second Archie
s exponent (Archie,
1942). The left side of Equations (3.197)
'
2
(3.199) indi-
cates the parameters used to compute the electrical con-
ductivity in fully saturated conditions, and on the right
side, we have the scaling of the same parameters with
saturation for unsaturated materials.
The permeability,
k
, is related to the formation factor,
F
, and the dynamic pore radius,
-
λ
=
−
1
,
r
=
n
+ 2 Model A
3 208
n
2
3
n
−
1
,
r
=3
n
Model B
λ
=
3 209
This result is very important because it provides an
explicit relationship between a hydraulic parameter
and an electrical parameter. To our knowledge, this is
the first time that these relationships are proposed,
despite some attempts by others to connect the resistivity
index and the capillary pressure curves (see, for instance,
Li & Horne, 2005).
Figure 3.8 shows that Model A seems to agree better
than Model B with experimental data but more data
Λ
, by (see Johnson, 1986,
for the saturated case)
s
w
2
8
Fs
w
k
0
s
w
=
Λ
3 200
Therefore, the permeability should scale with the
water saturation as
2
8
F
s
2+
n
Shannon
sandstone
k
0
s
w
=
Λ
Brooks and Corey (1964)
Model A
3 201
w
+
Jougnot et al. (2010)
Clayrock
2
8
F
s
3
w
Model B
k
0
s
w
=
Λ
Revil et al. (2007)
Limestone
3 202
Brooks and Corey (1964)
Jun-Zhi and Lile (1990)
Berea
sandstone
Therefore, according to this scaling process, the perme-
ability can be computed as the product of the permeabil-
ity at saturation,
k
S
=
4
2
8
F
, and a relative permeability
that depends only on the relative water saturation:
Λ
3.5
3
k
0
s
w
=
k
S
k
r
s
w
3 203
+
Model A
2.5
with
k
r
s
w
=
s
n
+2
e
Model A
3 204
2
k
r
s
w
=
s
3
n
e
Model B
3 205
1.5
Model B
Here, we used the effective water saturation,
s
e
, rather
than the water saturation, to enforce the fact that the rel-
ative permeability is null for the irreducible water satura-
tion case. These equations can be compared to the one
proposed by Li and Horne (2005; equations 5 and 6)
k
r
s
w
=
s
e
s
w
. In the Brooks and Corey (1964) model,
the relative permeability is also given by a power law
relationship (see also Purcell, 1949):
1
0.5
0
1.5
2
2.5
3
Saturation exponent
n
(—)
Figure 3.8
Comparison between Model A and Model B used to
predict the value of the Topics and Corey exponent,
, from the
saturation exponent,
n
. Data from Jougnot et al
.
(2010), Revil
et al. (2007), Brooks and Corey (1964), and Jun-Zhi and Lile
(1990). The Shannon sandstone is also known under the term
“
λ
−
2
−
3
λ
=
s
e
p
c
p
e
k
r
s
w
=
3 206
in the literature. The data seem to favor
Model A indicating therefore a saturation dependence of
surface conductivity.
Hygiene sandstone
”
r
=
2+3
λ
3 207
λ