Geoscience Reference
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charge density on the mineral surface is exactly counter-
balanced by the charge density in the Stern layer and the
charge density in the diffuse layer. In order to get an ana-
lytical solution for the zeta potential, we are going to omit
the charge density in the Stern layer (a fair approxima-
tion for silica but not for clays). It follows that the total
electroneutrality condition can be written as
the hydrodynamic shear plane, which is defined as the
position of zero relative velocity between the solid and liq-
uid phases. The exact position of the zeta potential is
unknown but likely pretty close to the mineral surface.
If we assume that the zeta potential represents the poten-
tial on the OHP (see Figure 1.1 for the position of this
plane), it follows from Equation (1.30) that we can write
the zeta potential as (Revil et al., 1999a, b)
Q S + Q 0 =0
1 23
ζ = b log 10 C f + c
1 31
Using Equations (1.7), (1.10), (1.14), and (1.22) into
Equation (1.23), the potential of the Stern layer φ d is
the solution of the following equation:
where
b = k b T
3 e
ln10
1 32
1
X
X 2
α
X
1+
β
1=0
1 24
ε f k b TN 1 2
2 eK Γ
c = 2 k b T
3 e
8×10 3
10 pH
ln
1 33
0
S
where
This equation shows how the zeta potential depends
on the salinity C f for simple supporting 1:1 electrolytes.
Note that Pride and Morgan (1991, their Figure 4) came
to Equation (1.31) on purely empirical grounds, fitting
experimental data with such an equation and getting
empirically the values of b and c . Typically, the seismo-
electric community has been using Equation (1.31) only
as an empirical equation while it can derived from
physical grounds as demonstrated by Revil et al.
(1999a). The previous model yields b = 20mV per tenfold
change in concentration (salinity) for a 1:1 electrolyte.
A comparison between the prediction of Equation (1.31)
and a broad dataset of experimental data is shown in
Figure 1.3. The slope b of the experimentally determined
zeta potential is actually closer to 24mV per tenfold
change in concentration, therefore fairly close to the pre-
dicted value.
Equation (1.31) is not valid at very high salinities (10 1
mol l 1 and above). Jaafar et al. (2009) presented mea-
surements of the streaming potential coupling coefficient
in sandstone core samples saturated with NaCl solutions
at concentrations up to 5.5 mol l 1 (Figure 1.3). Using
measurement of the streaming potential coupling coeffi-
cient, they were able to determine the zeta potential up to
the saturated concentration limit in salinity. They found
that the magnitude of the zeta potential also decreases
with increasing salinity, as discussed previously and as
predicted by Equation (1.31), but approaches a constant
value at high salinity around
aC f
2 e
α
=
1 25
0
S
Γ
= 10 pH + K M C f
K
β
1 26
and where X is defined by Equation (1.15) and a by
Equation (1.21). At low salinities, we have X (1/ X ).
With this assumption, Equation (1.24) simplified to the
following cubic equation:
X 3 + pX + q =0
1 27
with p =1
β
and q =
1
αβ
. The real root of this cubic
equation is given by
q
2 +
1 3
q
2 − Δ
1 3
X =
Δ
+
1 28
where
q
2
2
p
3
3
1
Δ =
+
=
2 >0
1 29
4
α
2
β
2
(assuming 4
1, which can be easily checked).
Using Equation (1.15), the solution is simply given by
α
27
β
φ d = 2 k b T
3 e
ln
αβ
1 30
In electrokinetic properties, the zeta potential represents
the electrical potential of the diffuse layer at the position of
20 mV. This value is, so far,
not captured by exiting models.
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