Geoscience Reference
In-Depth Information
Table 4.5
Material properties for the numerical simulation
corresponding to the case study #2 for which the inclusion
U2 is used to simulate a porous formation with oil.
measurements can be better than one nanovolt per meter
for offshore applications (classically for CSEM) (Butler &
Russell, 1993; Mikhailov et al., 1997; Dupuis et al., 2007).
Therefore, the signal-to-noise ratio adopted in the pres-
ent paper is rather pessimistic.
The third point is the use of compactness in the
inversion of the electrical potential data. The source
inversion problem identifies the position of the source
between the two wells. This principle is different from
the inverse problem corresponding to the imaging of
thematerial properties using a smoothness regularization
approach. A complete analogous application of the
method we used previously has also been performed in
the medical community using electroencephalographic
(EEG) data (measuring time-varying electrical potentials
on the scalp) and combining this information with
resistivity information to spatially and temporally locate
electrical sources within the brain. Portniaguine et al.
(2001) proposed the compactness approach as a method
to focus the position of the source responsible for the
observed EEG anomalies. This approach has been very
successful in EEG. We think that despite the drawback
associated with the choice to let the user to stop the
focusing of the tomogram, this method provides images
that are more reliable than having only an unfocused
image. The electrical signature collected at the boreholes
due to seismoelectric conversion is due the occurrence of
local electrical source current densities like in EEG. Also,
we point out that the use of compactness does not
remove the nonuniqueness of the inverse problem. As
for other deterministic methods, this approach provides
a set of solutions on the location of the sources. The
choice of the best solution is dependent on the availabil-
ity of independent information.
Parameter Description
Unit U1
Unit U2
0.01 S m
−
1
0.001 S m
−
1
σ
Conductivity of the
medium
0
V
0.203 C m
−
3
1585 C m
−
3
Q
Excess of charge per
unit pore volume
2650 kg m
−
3
2650 kg m
−
3
ρ
s
Bulk density of the
solid phase
1000 kg m
−
3
983 kg m
−
3
ρ
f
Bulk density of the
fluid phase
ϕ
Porosity
0.25
0.33
36.5 × 10
9
Pa 37 × 10
9
Pa
K
s
Bulk modulus of the
solid phase
0.25 × 10
9
Pa 2.40 × 10
9
Pa
K
f
Bulk modulus of the
fluid phase
4.00 × 10
9
Pa 5 × 10
9
Pa
G
Shear modulus of the
frame
2.22 × 10
9
Pa 9.60 × 10
9
Pa
K
fr
Bulk modulus of the
frame
10
−
12
m
2
10
−
11
m
2
k
0
Low-frequency
permeability
10
−
3
Pa s
10
−
1
Pa s
η
f
Dynamic viscosity of the
pore fluid
Following Linde et al. (2007) and Revil et al. (2007), the charge density
of a partially water-saturated reservoir,
0
V
Q
, should be replaced by
0
V
s
w
where
Q
s
w
(unitless) represents the partial saturation in water.
4.4.7 Discussion
In the present section, we address a few questions that
may arise regarding the applicability of the present
methods. The first is related to the sensitivity of the
approach to the choice of the source parameters: What
happens indeed to the model outputs if we change the
source characteristics? Our choice of the source para-
meters was to make the source as impulsive as possible.
That said, the choice of the source is totally arbitrary, and
our approach is totally independent of the choice of the
source. We can choose any type of source (e.g., a sweep),
and we can always performa deconvolution of the result-
ing seismoelectric signals with the (known) source in
order to retrieve the impulse function of the system.
The second question we want to address is the noise
level in the electrical data. Typically on land, it is easy
to record the electrical field with a precision of at least
one microvolt per meter, and the precision on the
4.5 Conclusions
We have described the implementation of the seismic
and seismoelectric equations in finite elements using
the
u
p
formulation. We used the softwares COMSOL
and MATLAB to perform the forward modeling and
MATLAB to implement routines to invert seismoelectric
signals in terms of material properties and in terms
of boundaries. We have introduced PML boundary
conditions for the seismoelectric problem. Then, several
synthetic case studies were discussed, and we demon-
strated how the forward modeling is implemented and
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