Geoscience Reference
In-Depth Information
E
-
+
+
d
(+)
J
+
-
-
-
+
-
-
+
-
+
+
-
-
+
-
-
+
-
+
-
+
Sorption
+
+
+
-
-
Desorption
+
-
-
-
>SiO
-
+ Na
+
>SiO
-
Na
+
>SiO
-
Na
+
>SiO
-
+ Na
+
Silica grain
-
+
+
-
+
+
-
+
-
-
+
+
+
-
-
+
-
-
-
+
+
+
-
-
-
-
-
-
+
+
+
+
+
d
(+)
+
Decrease in salt concentration
J
Increase in salt concentration
-
Back diffusion
Figure 1.12
The presence of an applied electrical field
E
creates a dipole moment associated with the transfer of the counterions in
both the Stern and the diffuse layers around a silica grain. This dipole moment points in the direction that is opposite to the applied field.
The charge attached to the mineral framework remains fixed. The movement of the counterions in the Stern layer is mainly tangential
along the surface of the grain. However, sorption and desorption of the counterions are in principle possible. Back diffusion of the
counterions can occur both in the Stern and diffuse layers, and diffusion of the salt occurs in the pore space. In both cases, the diffusion
of the counterions occurs over a distance that is equal to the diameter of the grain.
σ
w
(in S m
−
1
)
due to the quadrature conductivity-related term
σ
/
ω
.
as a function of the pore water conductivity
A discussion of the frequency dependence of
upon
the effective permittivity can be found in Revil (2013a, b).
The polarization of the electrical double layer (called
the
σ
by the following expression:
=
1
F
−
1
S
+
f
CEC
σ
F
σ
w
+
ρ
S
β
+
1
−
f
+
β
1 90
F
ϕ
-polarization in electrochemistry) plays a dominant
role at low frequencies through the apparent permittivity
of the material (see Figure 1.12). This is in contrast with
ideas expressed in the geophysical literature since Poley
et al. (1978). In the prior geophysical literature, low-
frequency polarization is envisioned to be dominated
by the Maxwell
α
where
F
is the so-called formation factor,
is the poros-
ity,
f
denotes the fraction of counterions in the Stern
layer,
ϕ
ρ
S
denotes the mass density of the solid phase (typ-
ically 2650 kg m
−
3
),
β
+
corresponds to the mobility of
the counterions in the diffuse layer, and
S
+
denotes
the mobility of the counterions in the Stern layer (both
in m
2
s
−
1
V
−
1
). The partition coefficient,
f
, is salinity
dependent as discussed in Sections 1.1.1 and 1.1.2. For
clay minerals (and for silica as well), the mobility of
the counterions in the diffuse layer is equal to the mobil-
ity of the same counterions in the bulk pore water (e.g.,
β
(+)
(Na
+
,25C) = 5.2 × 10
−
8
m
2
s
−
1
V
−
1
). The mobility of
the counterions in the Stern layer is substantially smaller
and equal to
β
-
Wagner polarization (also called
“
space
charge
polarization) due to the disconti-
nuity of the displacement current at the interfaces
between the different phases of a porous composite.
”
or
“
interfacial
”
1.3.2 Saturated clayey media
Assuming that clayey materials exhibit a fractal or self-
affine behavior through a broad range of scales (e.g.,
Hunt et al., 2012), the inphase and quadrature conduc-
tivities are expected to be weakly dependent on fre-
quency as discussed in detail by Vinegar and Waxman
(1984) and Revil (2012). This has been shown for a range
of frequencies typically used in laboratory measurements
(0.1 mHz to 0.1MHz). Revil (2013a) recently developed
a model to describe the complex conductivity of clayey
materials using a volume-averaging approach. According
to this model, the inphase conductivity
S
+
25 C,Na
+
=15×10
−
10
m
2
s
−
1
V
−
1
for
clay minerals (Revil, 2012, 2013a, b), therefore about
350 times less mobile than in bulk solution. We can
rewrite the inphase conductivity equation as
β
1
F
σ
w
+
1
F
S
+
f
CEC
σ
≈
ρ
S
β
+
1
−
f
+
β
1 91
ϕ
The surface conductivity corresponds to the last termof
Equation (1.91). Equation (1.91) represents the full
(S m
−
1
) is given
σ
