Geoscience Reference
In-Depth Information
log k
(
L1)
l
o
g k(L
2
)
lo
g
k(R)
ϕ
(L1)
ϕ
(L
2
)
-14
-12
-10
-18
-16
-14
-12
-10
-12
-10
-8
-6
0.2
0.3
0.4
0.5
0.1
0.2
0.3
log K
s
(L1)
ϕ
(R)
log σ(L1)
log K
s
(L2)
log K
s
(R)
0.2
0.4
0.6
0.8
8
10
12
14
8
10
12
14
8
10
12
14
-3
-2
-1
log σ(L2)
log
σ
(R)
lo
g
K
f
(L1)
log
K
f
(L2)
l
o
g K
f
(R)
-1.5
-1
-0.5
-3.5
-3
-2.5
6
8
10
12
6
8
10
12
6
8
10
12
lo
g
G(L
1
)
lo
g
G(L2
)
l
o
g G(R
)
log
K
fr
(L1)
log
K
fr
(L2)
8
10
12
14
8
10
12
14
8
10
12
14
8
10
12
14
8
10
12
14
log K
fr
(R)
8
10
12
14
Figure 4.13
Posterior probability density functions of the material properties for the three geological units (the two layers
L1 and L2 and the reservoir R). The vertical bars indicate the real value of the material properties (see Table 4.3).
the domain of interest (see Jardani et al., 2010, for further
details on the implementation).
The first problem is to simulate the seismic wave prop-
agation associated with the source in order to get the dis-
tribution of the two components of the displacement
along the coordinates,
x
and
z
, and the mean pore fluid
pressure,
p
, at each time,
t
. Since we are solving station-
ary partial differential equations in the frequency
domain, we use the stationary parametric solver PAR-
DISO (
http://www.pardiso-project.org/
). This solver is
used to determine the distribution of the following para-
meters: first, the fields
u
x
,
u
z
,
p
are determined as a func-
tion of space and time using an IFFT of the solution to the
spatial and temporal domain from the frequency domain.
The quasistatic scalar potential
water-saturated poroelastic material. Then, we use the
components of the solid displacement and the fluid
pressure to determine the electrical potential distribution
at each time step. In our modeling, we also neglect the
Stoneley waves propagating along the boreholes in
which the seismic sources are located. These Stoneley
waves can generate seismoelectric signals (Mikhailov
et al., 2000; Hunt & Worthington, 2000) and should be
considered in the case of a real field experiment.
However, we believe that they can be easily filtered
out of the recorded signals by performing forward mod-
eling of this contribution (see Ardjmandpour et al., 2011,
for an example) using downhole measurements of the
slowness and resistivity.
The geometry of the two synthetic case studies investi-
gated in the succeeding text is shown in Figure 4.15. The
domainconsists of a600m×600mregionand is infinite in
the strike direction (2.5D assumption). In the first case
study, the geometry is made of two half-spaces (U1 and
U2) separated by a vertical interface located at
x
= 300
m. The seismic sources are located in borehole #1 at
is computed by solving
the Poisson equation coupled to the solution of
u
x
,
u
z
,
p
through its source term. In the frequency domain, we
solve the partial differential equations from 1 to 100 Hz
with a step of 1 Hz. The seismic forward modeling code
we used was benchmarked by Jardani et al. (2010)
using the analytical solution of Dai et al. (1995) for a
ψ
