Geoscience Reference
In-Depth Information
bottom interfaces of the reservoir. These coseismic signals
are labeled RCS1 and RCS2, respectively, and occur
when a seismic wave travels through a porous material,
creating a relative displacement between the pore water
and the solid phase. The associated current density is bal-
anced by a conduction current density that results an
electrical field traveling at the same speed as the seismic
wave. The second type of seismoelectric signals corre-
spond to the converted seismoelectric signals associated
with the arrival of the P-waves at the top and bottom
interface of the oil reservoir. These converted seismoelec-
tric signals are labeled IR1 and IR2, respectively. When
crossing an interface between two domains characterized
by different properties, a seismic wave generates a time-
varying charge separation, which acts as a dipole radiat-
ing EMenergy. These dipoles oscillate with the waveform
of the seismic waves. Because the EM diffusion of the
electrical disturbance is very fast (instantaneous in our
simulations), the seismoelectric conversions are observed
nearly at the same time by all the electrodes but with dif-
ferent amplitudes. The seismoelectric conversions appear
therefore as flat lines in the electrograms shown in
Figure 2.4a.
Experiment #2 uses the same geometry as Experiment
#1, but the fluid in the reservoir is now a viscoelastic
wetting oil. The dynamic permeability response of the
reservoir filled with the viscoelastic fluid is shown in
Figure 2.6a. The time function of the source is a Dirac
and we investigate the electrical field response in the
frequency band (1
expected. In the next section, we are dealing with the
classical Biot
Frenkel theory for which the pore fluid is
a Newtonian fluid that cannot sustain shear stresses. In
this case, we expect two P-waves and one type of S-wave.
-
2.2 Poroelastic medium filled with a
Newtonian fluid
We consider now that the pore fluid is a Newtonian fluid
(in the previous section, we were dealing with a more
general fluid that was Maxwellian in nature). Water is
the best example of a linear Newtonian viscous fluid.
We can eventually use the previous theory in the low-
frequency behavior of the Maxwell fluid neglecting shear
stresses in the fluid. The corresponding theory is the one
developed by Biot and is generally called the classical
dynamic Biot (or Biot
Frenkel) theory of linear poroelas-
ticity. We present in the following text the fundamental
equations of this theory and their application to the seis-
moelectric problem.
-
2.2.1 Classical Biot theory
The Biot theory (Biot, 1956a, b, 1962a, b) provides a start-
ing framework to model the propagation of seismic waves
in linear poroelastic media saturated by a Newtonian
viscous pore fluid like water. The theory predicts the exist-
ence of an additional compressional wave by comparison
with the P- and S-waves found for purely (nonporous)
elastic materials. The existence of this slow P-wave was
first confirmed by Plona (1980). The physical interpreta-
tions of the elastic constants in the Biot theory were given
by Biot and Willis (1957) and Geertsma and Smit (1961).
According to the Biot theory, the equations of motion in a
2D statistically isotropic, fully saturated, heterogeneous,
porous elastic medium are given, in the frequency
domain, by (e.g., Haartsen et al., 1998)
1000 Hz). A plot of the vertical com-
ponent of the electrical field at an observation point at
depth is shown in Figure 2.6b. The maximum at the
electrical field occurs at the same frequency as that of
the resonance of the viscoelastic fluid. The amplification
of the electrical field reaches several orders of magnitude
with respect to the DC value. This very strong amplification
of the signal could be the basis of a newdetection andmap-
pingmethod of heavy oils andDNAPL in the ground. How-
ever, we feel that such theory should be checked first
through experimental investigations in the laboratory.
-
2
−ω
ρ
u
+
ρ
f
w
=
∇
T
+
F
,
2 180
u
T
T
=
λ
u
∇
∇
∇
u
+
∇
u
+
C
w I
+
G
2 181
2.1.7 Conclusions
We have developed previously a general extension of the
Biot
2
−ω
ρ
f
u
+
ρ
f
w
−
ib
ω
w
=
−∇
p
,
2 182
Frenkel dynamic theory in the case of a porous mate-
rial saturated by a generalized Maxwell-type fluid that can
sustain shear stresses. In this case, the whole Biot
-
−
p
=
C
∇
u
+
M
∇
w
+
S
,
2 183
Frenkel
theory becomes symmetric in its constitutive equations,
and two types of P-waves and two types of S-waves are
-
where
i
2
=
1;
u
is the displacement vector of the solid;
w
is the displacement vector of the fluid relative to the solid
−
