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a slow rotational S-wave. The fast rotational wave repre-
sents the classical shear wave, while the slow rotational
wave is related to the fact that the fluid can sustain shear
stresses above its resonance frequency. This wave does
not exist at low frequencies
F ρ f
ρ
Ψ ρ f
ρ
2
2
2
ρ f Ψ = C G
Ω + M G
G U
Ω
2 164
ρ f
ρ
2
Ψ
Ψ
C G
bt
ω ω m , because the pore
fluid behaves as a Newtonian fluid and cannot sustain
shear stresses. The phase speeds of these two waves are
given by 1 c S =Re z S 1
We assume plane wave propagation in the x -direction:
Ω
= B 1 exp ilx
ω
t
,
2 165
and 1 c I S =Re z I S 1
, and
2
2
Ψ
= B 2 exp ilx
ω
t
,
2 166
the inverse quality factors are 1 Q S =Im z S
Re z S
2
2
and 1 Q I S =Im z I S
Re z I S
, respectively. To the best
of our knowledge, this second type of fast shear wave
has been introduced first by Revil and Jardani (2010).
To check the consistency of the model, we can look for
the case where G f ω = 0 and G U = G fr , C G ω = 0, and
M G ω = 0. In this case, we obtain
where ω is the angular frequency and l is the complex
wave number. Using these equations, we obtain
ρ ρ f
F
2 B 1 = G U
B 1 l 2
C G B 2 l 2
ω
ω
+ 1
F
b
F
C G B 1 l 2 + M G B 2 l 2
i
ω
B 2 ,
2 167
ρ ρ f
F + i b ω
ρ
ρ f
a =
,
2 174
ω
ρ f
ρ
2 B 2 =
C G l 2 B 1
M G l 2 B 2
F
ρ f ω
b = G fr 1+ i b
ω
ωρ f
,
2 175
+ ρ f
ρ
G U l 2 B 1 + C G l 2 B 2 + i
ω
B 2 b
ω
2 168
c =0,
2 176
and therefore
Eliminating the constants B 1 and B 2 between these two
equations yields the following quadratic equation for the
speed: a z S + b z S + c = 0, where z p = ω l is the complex
speed and
2
z S
=0,
2 177
2
f
+ b 2
2
= G fr ρ f
ρ
ρ f ρ
ρ
ω ω
ρ f
ib
ω ω ρ f F
ρ
2
z I S
2
2
2
f
ρ f ρ
ρ
+ b ω ω
ρ ρ f
ω
ω
ρ
ρ f
F + i b
a =
,
2 169
2 178
In the absence of dissipation, this second wave corre-
sponds to the classical shear wave with phase speed
given by
M G
ω ρ
ρ f
2 C G
ω
+ i b
ω
ωρ f G U ω
b = 1
+ G U ω
,
F
2 170
c = 1
ρ f
G fr
ρ ρ f F
G fr
ρ
2
G U ω
M G ω
C G ω
2 171
c S =
,
2 179
The two roots of this equation are
when F >> 1.
The next section will be concerned with a simplifica-
tion of the present equations to the case of a poroelastic
material filled with a Newtonian fluid like liquid water.
b + b 2
=
4 a c
2
z S
,
2 172
2 a
b 2
2 = b
4 a c
z I S
,
2 173
2.1.6 Synthetic case studies
To support our analysis, we introduce two 2D numerical
experiments (Experiment #1 and Experiment #2) to
demonstrate the occurrence of the seismoelectric signals
2 a
with z S > z I S . These two waves correspond to the complex
wave speeds associated with a fast rotational S-wave and
 
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