Geoscience Reference
In-Depth Information
saturation version of a more general model. This model
implies that the surface conductivity is controlled either
by the grain diameter (or from the grain diameter prob-
ability distribution as discussed by Revil & Florsch, 2010;
see Figure 1.13a) or by the CEC (Figure 1.13b). Surface
conductivity could be also expressed as a function of
the specific surface area. Indeed, the CEC and the specific
surface area are related to each other by Equation (1.34):
Q
0
= CEC/
S
sp
where
Q
0
, the surface charge density of the
counterions, is about 0.32 Cm
−
2
for clayminerals. For sil-
ica grains, there is a relationship between the mean grain
diameter and the surface area or the equivalent CEC of
the material. Indeed, the specific surface area
S
sp
was cal-
culated from the median grain diameter,
d
, using
S
sp
=6/
(
ρ
s
= 2650 kg m
-
3
denotes the density of the
silica grains. This also yields an equivalent CEC given
by CEC = 6 CEC = 6
Q
0
ρ
s
d
with
Q
0
= 0.64 Cm
−
2
, and
ρ
s
= 2650 kg m
−
3
. In Figure 1.13b, the surface conductiv-
ity data of silica sands and glass beads and clayey media
are all along a unique trend. This is consistent with the
idea that surface conductivity is dominated by the diffuse
layer. Indeed, the mobility of the counterions in the Stern
layer is much smaller than the mobility of the counter-
ions in the bulk pore water (see discussion in Revil,
2012, 2013a, b).
The quadrature conductivity expression obtained by
Revil (2013a)is
ρ
s
d
) where
Glass beads
5×10
-4
NaCl
4×10
-4
3×10
-4
2×10
-4
1×10
-4
0
0
0.005
0.01
0.015
0.02
Inverse of the grain diameter (μm
-1
)
1
F
(a)
S
σ
≈ −
ρ
S
β
+
f
CEC
1 92
ϕ
Glass beads, silica sands, and clayey materials
0.1
For the reasons explained previously for the surface
conductivity, the quadrature conductivity can be
expressed as a function of the specific surface area or as
a function of the CEC. At saturation, a comparison
between the equation for the quadrature conductivity
and experimental data is shown in Figure 1.14 where
we used the relationship between the CEC and the
specific surface area given by Equation (1.53).
For
Clayey sands (Vinegar and Waxman, 1984)
+
Fontainebleau sand (Lorne et al., 1999)
Glass bead (Bolève et al., 2007)
0.01
0.001
S
+
Na
+
=15×10
−
10
clayey sands,
taking
β
m
2
s
−
1
V
−
1
at 25 C,
f
= 0.90,
Q
0
=032 C m
−
2
, and
ρ
s
=
2650 kg m
−
3
, yields
cS
Sp
with
c
= 7.6 × 10
−
8
Skg
m
−
3
. For the clean sands and sandstones, using
σ
≈ −
+
0.0001
β
(+)
(Na
+
)=
5.2 × 10
−
8
m
2
s
−
1
V
−
1
,
f
=0.50,
Q
0
=064 C m
−
2
, and
ρ
s
= 2650 kg m
−
3
, yields
c
= 2.9 × 10
−
5
Skgm
−
3
. The dif-
ference between the trend for the clean sands and sand-
stones and the trend for the clayey materials illustrates
how different are the mobilities of the counterions in
the Stern layer of silica and clays. In Figure 1.15, we plot
directly the quadrature conductivity as a function of the
CEC for shaly sands. In Figure 1.16, we plot the quadra-
ture conductivity of sands as a function of the mean grain
diameter. Using the transform given previously between
the mean grain diameter and the CEC, we plot in
Figure 1.17 the quadrature conductivity as a function
NaCl/KCl
10
-5
1
10
100
1000
10
4
CEC (C/kg)
(b)
Figure 1.13
Surface conductivity.
a)
For glass beads and silica
sands, the surface conductivity is controlled by the size of the
grains (Data from Bolève et al., 2007).
b)
All the data for glass
beads, silica sands, and shaly sands are on the same trend when
plotted as a function of the (total) CEC. This is consistent with a
surface conductivity model dominated by the contribution of the
diffuse layer (Data from Vinegar & Waxman, 1984 (shaly sands,
NaCl); Bolève et al., 2007 (glass beads, NaCl); and Lorne et al.,
1999a, b (Fontainebleau sand KCl)).
