Geoscience Reference
In-Depth Information
following the so-called BET isotherm. Such types of
experiment have usually two parts. In the first, known
volumes of nitrogen are allowed to expand into the appa-
ratus and the resulting pressure is recorded. From this
information, the volume of the chamber can be com-
puted. Then, any gas adsorbed on the surface of the solid
grains is removed by heating a sample of the porous
material in a vacuum. Once the surface has been stripped
of adsorbed gas molecules, known volumes of nitrogen
are fed in and allowed to reach equilibrium with the sur-
face of the mineral grains. By measuring the resulting
pressure in the apparatus, the amount of nitrogen that
remains unadsorbed can be found, and therefore, the
amount adsorbed on the solid can also be computed.
Knowing the size of a nitrogen molecule, it is possible
to determine the surface area per unit mass of grains
(expressed therefore in m
2
and the quadrature conductivity depend on the conduc-
tivity of the pore water. The formation factor and the sur-
face conductivity can be found by using the in-phase
conductivity data performed at the different salinities
by fitting the core sample conductivity versus pore water
conductivity using an electrical conductivity equation
such as Equation (1.91) (the surface conductivity corre-
sponding to the last term of this equation). In this case,
we have two unknowns to fit the formation factor and
the surface conductivity, and such fitting (using a least-
square approach) is shown in Figure 2.8.
2.3.1.3 Measuring the streaming potential
coupling coefficient
The quasistatic streaming potential coupling coefficient is
defined by
kg
−
1
in the International
∂
ψ
∂
System of Units).
C
0
= lim
ω
2 216
p
J
=0
0
2.3.1.2 Measuring the complex conductivity
The experimental setup used to measure the complex
conductivity of a rock sample is shown in Figure 2.7.It
consists of an impedance meter able to measure the com-
plex conductance or impedance in a broad range of fre-
quencies (1 mHz to 45 kHz for the equipment described
in Figure 2.7b). A harmonic current is imposed on the
sample (through the current electrodes A and B; see
Figure 2.7a), and the harmonic electrical field is meas-
ured with the pairs of electrodes M and N. The phase
angle between the current and the voltage is generally
very small (less than 30 mrad), and measuring it ade-
quately used to be quite challenging. Nowadays, how-
ever, we can easily reach a phase accuracy of 0.1 mrad
from 1 mHz to 10 kHz by using a high sampling rate.
The amplitude of the measured conductance can be
transformed into the amplitude of the electrical conduc-
tivity using a geometrical factor, which depends on the
geometry of the sample measurement system. This
includes the position of the electrodes and the boundary
conditions for the electrical current and the electric
potential. The geometrical factor can be determined
using two approaches: (1) using a benchmark with a
material of same shape as the test sample but of known
conductivity and (2) by solving numerically the Laplace
equation (see Jougnot et al., 2010, for additional details).
One example of an experimental data set performed at
different salinities but at a single frequency is shown in
Figure 2.8. We see that both the in-phase conductivity
Measuring this fundamental coefficient, used in the
whole electrokinetic theory, is actually pretty simple.
Figure 2.9a shows a very simple experimental setup in
which a gradient in hydraulic head
h
=
p g
ρ
f
(g denotes
the acceleration of the gravity, 9.81 m
2
s
−
1
) is applied to
the core sample. Figure 2.9b shows that the measured
potential drops
, at the end faces of the cylindrical core
sample, are proportional to the imposed head differences.
This result is confirmed in Figure 2.10 by directly plotting
the potential differences as a function of the hydraulic
head differences. The quasistatic coupling coefficient is
determined from the slope of the linear trends shown
in Figure 2.10. The permeability can be measured from
the same setup by measuring the flow rate of water
through the core sample and applying Darcy
∂
ψ
s law to
determine either the hydraulic conductivity or the per-
meability. If the electrical conductivity of the core sample
has also been determined, we can use the expression of
the streaming potential coupling coefficient to determine
the effective charge density
Q
0
'
V
. An example is provided
in the next section.
2.3.2 Streaming potential dependence
on salinity
First, we investigate the effect of salinity upon the
streaming potential coupling coefficient
C
. At low fre-
quencies (
k
1), the streaming potential coupling
coefficient is given by
ωτ
