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(called the filtration displacement);
T
is the stress tensor;
I
is the identity matrix;
F
is the body force on the elastic
solid phase;
S
is a pressure source acting on the pore fluid;
ρ
bulk modulus of the solid phase, and
Willis
coefficient. Equation (2.187) is the Gassmann equation,
η
f
is the dynamic viscosity of the pore fluid,
k
0
is the per-
meability of the medium,
α
is the Biot
-
represents the mass density of the saturated medium;
ρ
f
and
ϕ
is the porosity, and
a
is the
ρ
s
are the mass density of the fluid and the solid,
respectively;
tortuosity. The ratio
a
/
corresponds to the electrical for-
mation factor
F
also defined by Archie
ϕ
ϕ
−
m
where
m
is called the cementation exponent. In the fol-
lowing, we consider the tortuosity equal to
ρ
f
is an apparent density of the pore fluid;
p
is the fluid pressure;
'
s law to be
F
=
2 3
G
is the undrained
Lamé modulus of the porous material;
b
is the mobility
of the fluid;
G
is the shear modulus of the porous frame;
and
C
and
M
are elastic moduli. Equation (2.180) corre-
sponds toNewton
λ
U
=
K
U
−
ϕ
−
1 2
, which
is equivalent to a cementation exponent of 1.5, typical of
a pack of spherical grains.
s law, while Equation (2.181) represents
a constitutive expression for the total stress tensor as a
function of the displacement (Hooke
'
2.2.2 The u
-
p formulation
The
s law). This constitu-
tive equation comprises the classical termof linear elasticity
plus an additional term related to the expansion/contrac-
tion of the porous body to accommodate the flow of the
pore fluid relative to a Lagrangian framework attached
to the solid phase. Equation (2.110) is the Darcy constitu-
tive equation in which the bulk force acting on the fluid
phase has been neglected and Equation (2.183) is one of
the classical Biot
'
classical
formulation described in Equations
(2.180)
(2.183) is based on solving partial differential
equations for two unknown fields,
u
and
w
. For a 2D
discretized problem, four degrees of freedom per node
are therefore present. All the papers dealing with the
modeling of the seismoelectric problem use this type of
formulation (e.g., Haarsten & Pride, 1997; Haartsen
et al., 1998; Garambois & Dietrich, 2002). Atalla et al.
(1998) introduced an alternative approach using
u
and
p
as unknowns (see also Karpfinger et al., 2009).
In 2D, this implies three unknown parameters (
u
1
,
u
2
,
and
p
) to be solved at each node.
We start with the Darcy equation where the electro-
osmotic coupling term neglected:
-
Frenkel constitutive equations of poroe-
lasticity. The mass density of the porous material is given
by
-
−
ϕ ρ
s
. These Equations (2.180) through
(2.183) can also be derived from equations presented
in Section 2.1, assuming that the pore fluid cannot bear
shear stresses.
The material
ρ
=
ϕρ
f
+1
properties
entering
Equations
(2.180)
(2.183) are given by Pride (1994) and Rañada
Shaw et al. (2000):
-
η
f
w
=
−∇
p
−
ρ
f
u
+
F
f
,
2 190
k
ω
b
=
η
f
k
0
,
2 184
where
F
f
is the body force acting on the pore fluid phase.
The fact that the electroosmotic term can be safely
neglected in this type of formulation has been discussed
by a number of authors including Revil et al. (1999b).
The dynamic permeability is written as (e.g., Morency
& Tromp, 2008)
ρ
f
=
ρ
f
a
,
2 185
K
fr
K
s
α
=1
−
2 186
K
U
=
K
f
K
s
−
K
fr
+
ϕ
K
fr
K
s
−
K
f
,
2 187
K
f
1
−
ϕ
−
K
fr
K
s
+
ϕ
K
s
1
1
−
i
ω ω
c
k
0
ω
≡
,
2 191
k
K
f
K
s
−
K
fr
C
=
K
s
,
2 188
K
f
1
−
ϕ
−
K
fr
K
s
+
ϕ
which is based on the low-frequency approximation to
the dynamic permeability given by Pride (1994). The crit-
ical frequency is
M
=
C
α
K
f
K
s
=
K
s
,
2 189
K
f
1
−
ϕ
−
K
fr
K
s
+
ϕ
ω
c
=
η
f
k
0
ρ
f
F
=
b
ρ
f
where
b
=
η
f
k
0
and
ρ
f
=
ρ
f
F
. Neglecting the body force acting on the fluid
phase, Equation (2.190) can be used to express the filtra-
tion displacement,
w
, as a function of the pore fluid pres-
sure,
p
, and the displacement of the solid phase,
u
:
where
K
U
(in Pa) is the bulk modulus of the porous
medium,
K
fr
is the bulk modulus of the dry porous frame
(skeleton),
K
f
is the bulk modulus of the fluid,
K
s
is the
