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and where the streaming potential coupling coefficient at
saturation is given by
C
r
=
s
−
n
+1
w
s
3+2
λ
e
3 218
as proposed by Revil et al. (2007). In the next section, we
will see how this equation needs to be modified if we use
thevanGenuchtenmodel, insteadof theBrooks andCorey
(1964) model, to represent the capillary pressure curve.
C
S
=
Q
V
k
0
η
w
σ
3 215
ρ
w
g
(
g
being the
acceleration of the gravity in m s
−
2
), the coupling coeffi-
cient units transform to V m
−
1
. The relative streaming
potential coupling coefficient is given by
If Equation (3.215) is multiplied by
3.5.4 Relative coupling coefficient with
the Van Genuchten model
An alternative to the Brooks and Corey model is the van
Genuchten model (1980) (see discussion in Linde et al.,
2007; Revil et al., 2007). This model can be written as
C
r
≈
s
e
Model A
3 216
C
r
=
s
2
n
−
1
e
Model B
3 217
−
m
v
n
v
s
e
=1+
p
c
p
e
1
n
v
,
p
c
≥
with
m
v
≈
1
−
p
e
3 219
In Figure 3.10, we compare the two models to the
existing data, replacing the water saturation by the irre-
ducible saturation, to satisfy to the additional constrain
that there is no flow at irreducible water saturation. Very
clearly, Model B is unable to explain these data.
Note that the general form of the relative coupling
coefficient with the Brooks and Corey model is
m
v
2
s
1
m
v
e
k
r
s
w
≈
s
e
1
−
1
−
3 220
where
n
v
and
m
v
are the van Genuchten exponents.
Therefore, the relative coupling coefficient is given by
m
v
2
C
r
=
s
−
n
+1
w
s
1
m
v
e
s
e
1
−
1
−
3 221
+
Revil et al. (2011) (clayrock)
Mboh et al. (2012) measured the relative streaming
potential coupling coefficient of a clean sand character-
ized by the following properties: 99.3% silica, porosity
ϕ
Vinogradov and Jackson (2012) (St Bees sandstone)
Guichet et al. (2003) (Fontainebleau sand, Argon)
1.0
= 0.41, hydraulic conductivity
K
= 8.25 × 10
−
5
ms
−
1
,
and mean grain diameter
d
= 160
μ
m (Table 3.2). Their
0.8
+
Table 3.2
Petrophysical properties of the samples discussed in the
main text.
Sr
0.6
Property
Symbol M
E3
E39
+
0.4
Saturation
exponent
n
(
—
)
1.87
2.7
3.5
Model B
Model A
Cementation
exponent
m
(—)
-
1.93
2.49
0.2
k
(m
2
)
8.4 × 10
−
12
48.4 × 10
−
15
23.8 × 10
−
15
Permeability
0
0
Porosity
ϕ
(
—
)
0.41
0.203
0.159
0.2
0.4
0.6
0.8
1.0
Residual
saturation
s
r
(—)
0.09
0.31
0.34
Water saturation,
s
w
(—)
Grain size
d
(μm)
160
-
-
Figure 3.10
Comparison between Model A and Model B to
predict the value of the relative streaming potential coupling
coefficient as a function of the water saturation. We use an
irreducible water saturation
s
r
= 0.2. Data from Revil et al
.
(2011), Vinogradov and Jackson (2011), and Guichet et al
.
(2003). The data seem to favor Model A over Model B.
Pore size
r
(
μ
m)
-
1.18
0.17
M corresponds to the sand sample investigated by Mboh et al. (2012),
while Samples E3 and E39 are dolomitic samples investigated by
Revil et al. (2007).
