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located in borehole #2. The inverse problem involves
locating the distribution of seismoelectric conversions
at the interface. It will be described in the following
sections using the data inside the SC-time windows for
all the seismic shots.
Our main objective in this section is to estimate the
spatial distribution of heterogeneities that generate the
seismoelectric source current, regardless of the values
of the material properties themselves associated with
these heterogeneities. At each time step, the seismoelec-
tric signals recorded in borehole #2 are like a self-
potential profile (i.e., a distribution of voltages recorded
at a set of electrodes with respect to a reference electrode
due to a quasistatic source of current). At each time in the
SC-time window and for each shot, we use the electrical
potential distribution recorded on the array of electrodes
located in the second well in order to find the position of
the current source generated by the seismoelectric con-
versions between the two wells. At each time step, the
inverse seismoelectric problem is therefore similar to a
self-potential inverse problem for which several algo-
rithms have been developed over the past few years using
deterministic (Jardani et al., 2007, 2008) and stochastic
(Jardani & Revil, 2009; Revil & Jardani, 2010)
approaches. In general, we want to invert these voltages
to recover the position,
r
, and amplitude of volumetric
current source, (
r
,
t
). In our test case, at each time step,
the seismic waves impinging on the interface are respon-
sible for the source current density, which has compact
support (i.e., the spatial distribution of sources at any
given time is very sparse). Therefore, the algorithm we
use in the succeeding text is based on compactness as a
regularization tool, as developed for the self-potential
source inversion problem. A mathematical description
of the algorithm is given in Section 4.3.2 (Figure 4.16).
We now use a small test to analytically check the valid-
ity of our numerical computations and to compute the
size of the first Fresnel zone for the seismic and seismo-
electric problems. To estimate the time at which the
seismoelectric conversion occurs, we can estimate the
P-wave velocity based on the material properties
reported in Table 4.4. We obtain
c
p
= 1935.5 m s
−
1
in unit
U1 and 2163.5 m s
−
1
in U2. Taking the geometry of
Figure 4.15 (case study #1), we obtain a travel time of
0.115 s from seismic source #5 to the interface and a
travel time of 0.213 s from seismic source #5 to geophone
#25. Because the source has a time delay of
t
0
= 0.1 s,
the seismoelectric conversion occurs at
t
1
= 0.215, and
Seismic source
1
×10
−3
t
0
= 0.1 s
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Time (s)
0.02
0.015
0.01
BW=19 Hz
0.005
0
0
20
40
60
80
100
Frequency (Hz)
Figure 4.14
Description of the explosive seismic source. We
use a sharp Gaussian pulse simulating an explosion at
t
0
= 0.1 s.
The standard deviation of the source in the frequency domain
is 19 hertz.
x
= 100 m and
z
= 150, 225, 300, 375, and 450m for shots
#1, #2, #3, #4, and #5, respectively. In borehole #2
(located at
x
= 500 m), we simulate an array of 50 sensors
consisting of electrodes and geophones. The seismic
sources are detonated sequentially in borehole #1.
Meanwhile, seismic and electrical data are recorded for
each individual seismic shot. Table 4.1 provides the
material properties ofmediaU1 andU2used in themodel.
Figure 4.17 shows eight snapshots of the seismic wave
propagation, the associated electrical current density
distribution, and the resulting voltage at electrode #25
in borehole #2. Figure 4.18 represents the signal at elec-
trode #5 at location
z
= 140 m deep from the surface,
which shows that seismoelectric signals are generated
at the interface before the coseismic signal arrives at
the receivers. While the coseismic signals show a charac-
teristic hyperbolic shape, the seismoelectric signals arrive
nearly instantaneously at all receivers. In the following,
we call the
“
seismoelectric conversion (SC-)time win-
dow
the time between the shot of the seismic source
and the time the seismic wave arrives at the receivers
”
