Geoscience Reference
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following expression for the bulk force acting on the
solid phase:
∇
σ
∇
ψ
=
∇
J
S
,
4 37
J
S
=
j
x
1
s
x
2
;
j
x
1
s
x
1
,
4 38
s
x
2
s
x
1
F
x
,
y
,
ω
=
F
ω
∇
δ
x
−
x
0
δ
y
−
y
0
,
4 45
σ
0
σ
=
4 39
2
s
x
1
s
x
2
F
ω
=FT
t
−
t
0
exp
−
π
f
0
t
−
t
0
,
4 46
0
σ
where FT[
f
(
t
)] is the Fourier transformof the function
f
(
t
),
t
0
is the time delay of the source, and
f
0
is its dominant
frequency. In this chapter, we will neglect the pore fluid
pressure source term,
S
, which is equivalent to neglecting
the electromagnetic effects of the seismic source itself.
A complete analysis of the electromagnetic effects gener-
ated by elementary sources can be found in Pride and
Haartsen (1996) and will be developed in Chapter 5.
This novel formulation of the electrical problem
includes the perfect matching layers. For the boundary
values, we applied the Neumann boundary condition
at
=0
(Dirichlet boundary condition) at the other boundaries.
the insulating air
-
ground interface and
ψ
4.1.3 Boundary conditions at an interface
If we consider an interface between two poroelastic
media 1 and 2, the boundary conditions at this interface
are given by (Pride & Haartsen, 1996)
4.1.5 Lateral resolution of cross-hole
seismoelectric data
In the seismicwavedomain, the first Fresnel zone isdefined
as the areaof the reflector that contributes energyconstruc-
tively to the total reflectionenergy reaching anobservation
point P. The same definition can be used for the transmis-
sion part of problem as well. In our case, the seismoelectric
Fresnel zones may be defined similarly as the area of an
interface that contributes constructively to the total
transmitted energy reaching an observation point P. If we
consider amonochromatic seismic source,
S
, located above
a horizontal interface betweenmedia of different electrical
properties, the spherically spreading seismic wave inter-
sects the interface and causes fluid flowacross the interface.
The resulting electrical field is due to the streaming current
imbalance at the interface. This is equivalent to having
electrical dipoles oscillating in phase with the seismic wave
along the interface. As a consequence, the electromagnetic
disturbances are radiatedaway fromthe dipole sources and
are recordedremotelyat theobservationpoint P.Assuming
that the shot point S and the observation point P are
collocated, the first Fresnel zones correspond to two circles
of radii
r
S
and
r
SE
, respectively. Fourie (2003)provideda
complete analysis of the reflection problem and found that
the seismic and seismoelectric first Fresnel zone radii are
given by
u
1
=
u
2
,
4 40
p
1
=
p
2
,
4 41
n w
1
−
w
2
=0,
4 42
n T
1
−
T
2
=0,
4 43
n
×
E
1
−
E
2
=0,
4 44
where
n
denotes the unit vector normal to the interface
between the media 1 and 2. These boundary conditions
express the continuity in the solid displacement, the pore
fluid pressure, the fluid displacement, the momentum
flux, and the tangential components of the electrical field
across the interface.
4.1.4 Description of the seismic source
In the following example, we use a source generating
P-waves only. This force creates a net force on the solid
phase of the porous rock. Because the source generates
a displacement of pore water relative to the grain frame-
work, it creates a source current density and therefore an
electromagnetic disturbance. This disturbance diffuses
nearly instantaneously to all receivers with an amplitude
that can be pretty strong in the vicinity of the seismic
source. Because this contribution can be easily removed
from the electrograms (using its temporal and spatial
characteristics), we will not model it in the succeeding
text. We will focus on this contribution in Chapter 5.
Using the Fourier transform of the first time derivative
1 2
2
d
+
λ
d
λ
S
4
S
2
,
d
2
−
≈
r
S
=
4 47
1 2
2
d
+
λ
S
2
d
2
r
SE
=
−
≈
d
λ
S
,
4 48
of
the Gaussian function for the source yields the
