Geoscience Reference
In-Depth Information
with seismic waves. The second consequence is crucial to
the understanding of electrical conductivity in porous
materials. Electrical conductivity of porous media has
two contributions, one associated with conduction in
the bulk pore water and one associated with the electrical
double layer (surface conductivity). Contrary to what is
erroneously assumed in a growing number of scientific
papers in hydrogeophysics, the formation factor will
not be defined as the ratio of the conductivity of the pore
water by the conductivity of the porous material. We will
show that surface conductivity is crucial in obtaining an
intrinsic formation factor characterizing the topology of
the pore space of porous materials. The third conse-
quence is important to the understanding of induced
polarization, which translates into the frequency
dependence of the electrical conductivity. Because of this
low-frequency polarization, the conductivity appears,
generally speaking, as a second-order symmetric tensor
with components that are frequency dependent and
complex. The real (or inphase) components are associ-
ated with electromigration, while the imaginary (quadra-
ture) components are associated with polarization (i.e.,
the reversible storage of electrical charges in the porous
material). We will show in the following that the model
of Pride (1994) does not account correctly for the fre-
quency dependence of electrical conductivity and is
incomplete in its description of the electrical double layer
(no speciation and no description of the Stern layer).
Q
V
v
m
x
=
ρ
x v
m
x
1 73
where the brackets denote a pore volume averaging,
=
1
V
p
V
p
τ
d
1 74
and
d
denotes an elementary volume around point
M
(
x
),
and
v
m
(
x
) denotes the mean velocity averaged over the
pore space. Equation (1.73) is valid whatever the size of
the diffuse layer with respect to the size of the pores. In
the case of a thin double layer (the thickness of the diffuse
layer is much smaller than the thickness of the pores), the
charge density
Q
0
τ
V
is substantially smaller than the (total)
charge density associated with the diffuse layer
Q
V
,
which explains why
Q
0
V
cannot be used to estimate the
CEC of the minerals. In other words, there is no direct
relationship between the effective charge density
Q
0
V
and the CEC of the minerals.
As shown in Figure 1.10, the drag of the excess of
charge of the pore water (more precisely the drag of a
fraction of the diffuse layer) is responsible for a macro-
scopic streaming source current density
J
S
at the scale
of a representative elementary volume of the porous
material. This macroscopic source density is related to
the microscopic (pore scale) current density
j
S
associated
with the local advective transfer of electrical charges by
J
S
=
ϕ
j
S
1 75
J
S
=
ϕρ
v
m
1 76
1.2 The streaming current density
J
S
=
Q
V
ϕ
v
m
1 77
J
S
=
Q
0
V
w
1 78
We evaluate in this section the first consequence associated
with the existence of the electrical double layer coating the
surface of themineral grains in a porous material. We have
established in Section 1.1.1 that there is an excess charge
density in pore water. We have defined the macroscopic
charge density (charge per unit pore volume, in Cm
−
3
)
that is dragged by the flow of pore water as
Q
V
(the reason
for the superscript 0 will be explored in Chapter 3). We
showed that pore water, in proximity to the mineral
grain surface, is characterized by a local charge density
ρ
where
w=
Forchheimer equation)
denotes the macroscopic Darcy velocity (in m s
−
1
; see
Darcy, 1856). This Darcy velocity is not a true pore water
velocity. It represents the flux of water through a cross
section of the porous material (volume of water passing
per surface area and per surface time across a cross
section of the porous material). In Figure 1.10, we show
the effect of the charge distribution and the flow regime
on the source current density. At low frequencies, the
flow is dominated by viscous effects, and the regime is
called the viscous laminar flow regime. At high frequen-
cies, the flow is controlled by the inertial term of the
Navier
ϕ
v
m
(Dupuit
-
(
x
) (in Cm
−
3
) at position
x
due to the presence of the
electrical diffuse layer (
x
= 0 in the bulk pore water
that is electroneutral). We note
v
m
(
x
) as the local instan-
taneous velocity of the pore water relative to the solid (in
ms
−
1
). The macroscopic charge density
Q
V
is defined by
ρ
-
Stokes equation, and the flow regime is called
