Geoscience Reference
In-Depth Information
the flow of pore water. These factors generate the source
current density,
J
S
, of electrokinetic origin. These cou-
plings were first investigated by Marshall and Madden
(1959) and imply the existence of low-frequency polar-
ization mechanisms in the porous material. It is not our
goal to develop a complete theory of polarization in this
book, but rather to provide a practical view of the prob-
lem that can be used to analyze seismoelectric effects.
One of the most effective mechanisms of polarization is
the coupling of the flux densities with the electrochemi-
cal potential gradients as discussed by Marshall and
Madden (1959). The polarization implies a phase lag
between the current and the electrical field and defines
the frequency dependence of the conductivity of the
material. Despite the fact that the seismoelectric theory
contains an electroosmotic polarization effect (which is
the one used by Pride (1994)), it has been known since
Marshall and Madden (1959) that this mechanism cannot
explain the low-frequency dependence of the conductiv-
ity of the material. While this assumption is clearly stated
in Pride (1994), it seems to have been lost in translation in
all the following works. In those works, the model of Pride
is used to explain the low-frequency polarization of
porous rocks, and as such, those authors have considered
the mathematical expression of Pride (1994) as valid to
describe the complex conductivity of porous materials.
This is unfortunately not correct since the model of Pride
does not account for low-frequency polarization mechan-
isms known to control the quadrature conductivity.
Continuing from the preceding text, the total current
density entering, for instance, Ampère
and discussed in detail by Vinegar and Waxman (1984)
and more recently by Revil (2012, 2013a, b). This
dependency will be therefore neglected in the following.
The quadrature conductivity of clean sands and sand-
stones shows a clear frequency peak, but the magnitude
of the quadrature conductivity is usually low. The only
case of a strong and highly frequency-dependent induced
polarization effect is the case of disseminated ores (e.g.,
sulfides like pyrite and oxides like magnetite). In this
case, there is the possibility (still unexplored) to use the
seismoelectric method to detect and image ore bodies.
1.3.1 Effective conductivity
The displacement field is related to the electrical field by
D
=
ε
E
where
ε
denotes the permittivity or dielectric con-
stant (in F m
−
1
) of the material. We consider a harmonic
external electrical field:
E
=
E
0
exp
−
i
ω
t
1 85
where
f
is the frequency in Hz,
f
denotes the angu-
lar frequency (pulsation in rad s
−
1
), and
E
0
represents the
amplitude of the alternating electrical field. Equation
(1.83) can be written as
ω
=2
π
σ
∗
−
J
=
i
ωε
E+J
S
1 86
This total current density can be written as an apparent
Ohm
'
s law:
J
=
σ
eff
∗
E+J
S
1 87
'
s law is
where
σ
eff
∗
=
σ
eff
−
i
ωε
eff
is the effective complex conduc-
J
=
σ
∗
E
+
J
S
+
∂
D
∂
1 83
tivity and
ε
eff
are real positive frequency-
dependent scalars (at least in isotropic media) defined by
σ
eff
and
t
where the first term on the right side of Equation (1.83)
corresponds to a frequency-dependent electrical conduc-
tivity,
σ
eff
=
σ
1 88
∗
, characterized by a real (inphase) component
σ
σ
ε
eff
=
ε
−
σ ω
1 89
and a quadrature (out-of-phase) component
σ
:
Equations (1.88) and (1.89) are a direct consequence of
Ampère
σ
∗
=
σ
+
i
σ
1 84
s law in which the conductivity is considered com-
plex (ion drift is coupled to diffusion), the permittivity is
real, and the Maxwell
'
where
i
denotes the pure imaginary number
1 .
The second term of Equation (1.83) corresponds to the
source current density of electrokinetic origin, and the
third term corresponds to the displacement current den-
sity. Note that in clayey materials, while
i
=
−
Wagner polarization and the polar-
ization of the water molecules at few GHz are neglected.
The effective properties measured in the laboratory or in
the field contain both dielectric and conduction compo-
nents. It is clear fromEquation (1.89) that the effective per-
mittivity is expected to be very strong at low frequencies
-
both
depend on frequency, this dependence is weak as shown
σ
and
σ
