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where
i
denotes the pure imaginary number and
is
the angular frequency. Inserting Equation (1.106) into
Equation (1.104) shows that the amplitudes obey the
scalar Helmholtz equation:
ω
1
3
Trace
T
P
=
−
1 111
Still, Equation (1.111) is not good enough to be com-
patible with the velocity of the P-wave in a porous mate-
rial. First, the bulk modulus that should be considered is
generally the undrained bulk modulus (the fluid in the
pores resists to the deformation). The modulus
K
u
(in
Pa) is defined by
r
+
ω
2
2
p
ω
∇
p
ω
r
=0
1 107
c
f
s law, Equation (1.101),we
obtain the following equation for the displacement of
the fluid:
If we consider Newton
'
K
u
=
K
f
K
s
−
K
+
ϕ
KK
s
−
K
f
1 112
K
f
1
−
ϕ
−
K K
s
+
ϕ
K
s
1
ρ
f
ω
ur
,
t
=
−
∇
p
ω
r
exp
−
i
ω
t
1 108
2
In addition, the porous material, at the opposite of a
viscous fluid, can sustain shear stresses. This means that
Equation (1.110) needs to be replaced by
Because the curl of a gradient is always equal to zero,
we have the property
2
P
∂
K
u
+
4
1
ρ
∇
t
2
−
3
G
∇
P
=
f
r
,
t
1 113
∇
×
ur
,
t
=0
1 109
∂
where
G
is described as the shear modulus of the skeleton
(frame) of the porous material (the reason for the term
K
u
+43
G
can be obtained from elastic theory).
Equation (1.113) can be used to solve the poroacoustic
problem for the P-wave propagation in a porous material.
Assuming that the viscous coupling between the pore
water and the solid phase can be neglected, the velocity
of the P-waves is approximated by (Gassmann, 1951)
The displacement is irrotational, which means that the
pressure wave is purely longitudinal and corresponds to a
P-wave with a seismic velocity given by Equation (1.105).
1.4.2.2 Extension to porous media
Usually, the seismoelectric problem is formulated in
terms of a coupling between the Maxwell equations
and the Biot
-
Frenkel theory (e.g., Pride, 1994; Revil &
Jardani, 2010), and the Biot
-
Frenkel theory will be
discussed in Chapter 2. In the present section, we adapt
the acoustic wave developed in the previous section to
a porous body, and we simplify the seismoelectric theory
to make it compatible with this acoustic approximation.
We need an acoustic approximation solving now for
the macroscopic pressure perturbation
P
of a porous
material containing a fluid that cannot support shear
stress (for instance, water). This formulation is obtained
by adapting Equation (1.104) to a porous material:
1
2
K
u
+
4
3
G
c
p
=
1 114
ρ
Now,weneed todescribehowthemacroscopicperturba-
tion
P
ontheelastic skeletonaffects thepore fluidpressure
p
,
at least in an approximate way. In the undrained regime of
poroelasticity, the pressure,
P
, is related to the so-called
undrained pore fluid pressure
p
by (see Section 1.5.3)
p
=
BP
1 115
2
P
∂
∂
1
ρ
∇
t
2
−
∇
K
P
=
f
r
,
t
1 110
where 0
1 is called the Skempton coefficient (see
Section 1.5). It is given by
≤
B
≤
where
P
denotes the confining pressure (in Pa),
is the
mass density of the material (in kg m
−
3
),
K
is the bulk
modulus of the porous material (in Pa), and
f
(
r
,
t
) denotes
the source function (seismic source) at position
r
and
time
t
. The acoustic pressure corresponds to the hydro-
static part of the stress tensor
T
:
ρ
B
=
1
−
K K
u
1 116
1
−
K K
S
where
K
is the bulk modulus (in Pa),
K
u
is the undrained
bulk modulus (in Pa), and
K
S
is the bulk modulus of the
