Geoscience Reference
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compression of the solid phase on the pore fluid pressure
and the hydraulic pressure exercised mutually between
the two fluids are taken in the account.
The parameters used into Equations (3.131)
3.2.3 Seismoelectric conversion
in two-phase flow
To describe the electromagnetic response in the quasi-
static limit, we consider that the target is relatively close
(less than 1000 m) to the sensors (antennas, nonpolariz-
ing electrodes, and magnetometers). In this case, we can
neglect the time required for the electromagnetic distur-
bances to diffuse between the reservoir and the electro-
magnetic sensors. In this case, we can model the
problem by solving only the quasistatic electromagnetic
problem using the Poisson equation. We note that this
seismoelectric signal is a result of the relative displace-
ment of the fluids generated by seismic source. Hence,
the conversion of the mechanical energy into an electri-
cal signal is due to the electrokinetic coupling where the
drag of the charge density contained in the pore fluids is
responsible for the polarization of the medium (Revil
et al., 1999a; Leroy & Revil, 2004). The hydroelectric
problem is coupled to hydromechanical via the current
density term that is colinear with velocity of the relative
displacement of the two fluid phases. The electric prob-
lem can be written in frequency domain as
-
(3.134)
are given by
s
=
a
1s
+
a
12
s
1
+
a
13
2
3
G
λ
θ
1
α
θ
2
α
−
3 135
s2
ρ
i
=
ρ
i
θ
A
ii
θ
i
−
3 136
2
i
1
k
i
ω
=
−
3 137
R
ii
θ
i
ωρ
i
+
2
i
k
i
ω
i
1
k
i
=
=
3 138
R
ii
θ
ω
ω
2
ρ
i
−
i
ω
2
i
2
2
1
k
1
−
ω
2
2
2
k
2
ρ
T
=
ρ
T
−
ω
ρ
ρ
3 139
2
θ
1
=
−α
s1
−
ω
ρ
1
k
1
3 140
2
θ
2
=
−
α
s
2
−
ω
ρ
2
k
2
3 141
a
2s
a
33
−
a
3s
a
23
θ
1
α
s1
=
3 142
a
23
−
a
33
a
22
∇
σ
∇
ψ
=
∇
J
S
3 150
a
3s
a
22
−
a
2s
a
23
θ
2
Q
0
1
S
1
w
1
+
Q
0
α
s2
=
3 143
a
23
−
a
33
a
22
2
S
2
w
2
J
S
=
−
i
ω
3 151
2
1
a
33
θ
M
11
=
3 144
Q
1
S
1
k
1
∇
Q
2
S
2
k
2
∇
a
23
−
a
33
a
22
2
2
J
S
=
−
i
ω
p
1
−
ω
ρ
1
u
−
i
ω
p
2
−
ω
ρ
2
u
3 152
a
23
θ
1
θ
2
a
23
−
M
12
=
3 145
a
33
a
22
2
2
where
in the
quasistatic limit of the Maxwell equations). The electrical
conductivity of the medium is
ψ
is the electrostatic potential (
E
=
−∇
ψ
a
22
θ
M
22
=
3 146
a
23
−
a
33
a
22
, and it can be inferred
from electrical resistivity tomography or other electro-
magnetic techniques. The medium conductivity can also
be expressed as a function of the pore water conductivity,
σ
f
,as
σ
a
1s
=
a
11
+
a
12
+
a
13
3 147
a
2s
=
a
12
+
a
22
+
a
23
3 148
a
3s
=
a
13
+
a
23
+
a
33
3 149
where
θ
2
are the volumetric hydromechanical
coupling coefficients of the each fluid,
θ
1
and
=
1
F
S
2
σ
f
+
S
2
n
−
1
σ
σ
S
3 153
λ
s
is the Lamé coef-
ficient of the solid phase,
k
1
and
k
2
are the dynamic per-
meability of the each fluid,
ρ
T
is an apparent mass density
for the solid phase at a given frequency
where
F
denotes the formation factor, introduced previ-
ously (ratio of the pore space tortuosity by the connected
porosity),
n
(
, and M
11
and
M
22
are the storativity coefficients for the nonwetting
and wetting fluids,
ω
s exponent,
σ
S
denotes the surface conductivity associated with elec-
trical conduction in the electrical double layer, and
J
S
is
the source current density of an electrokinetic nature.
≥
1) denotes the second Archie
'
ρ
i
denotes the inertial drag interactions between the solid
and fluid phase
i.
respectively. The parameter
