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and where
ρ
2
θ
2
denotes the aver-
aged density of the mixture. The material properties
entering Equations (3.91)
ρ
T
=
ρ
s
1
−
ϕ
+
ρ
1
θ
1
+
In these formulae,
K
s
denotes the bulkmodulus of solid
phase,
K
1
stands for the bulk modulus of the nonwetting
fluid phase,
K
2
is the bulk modulus of the wetting fluid
phase,
-
(3.92) are given by Lo et al.
(2005) where
is the relative
saturation of the nonwetting fluid, and
p
c
=
p
1
ϕ
represents the porosity,
S
1
=
θ
ϕ
1
p
2
refers
to the capillary pressure. Equations (3.104) and (3.105)
are the classical formulations used to infer the viscous
coefficients of the two fluids that are governed by Darcy
−
a
11
=
K
s
1
−
ϕ
−
δ
s
, with
3 98
a
12
=
a
21
=
−
K
s
δ
1
3 99
'
s
law. The variable
η
i
is the dynamic viscosity of the pore
fluid i,
k
s
is the permeability of the porous medium,
and
k
r
i
is the relative permeability of fluid phase
i
that
can be deduced from the degree of saturation in the
media. In this, tortuosity is represented by
a
s
(
a
s
/
a
13
=
a
31
=
−
K
s
δ
2
3 100
1
M
1
K
1
K
2
dS
1
d
p
c
+
K
1
K
2
S
1
dS
1
d
p
c
+
K
1
S
1
a
22
=
−
δ
1
1
−
S
1
corre-
sponds to the formation factor). The elastic parameters
are mainly dependent on changes in saturation; hence,
we use the van Genuchten (1980) approach to determine
(
dp
c
/
dS
1
) and the relative permeabilities (Lo et al., 2005):
ϕ
+
K
1
K
2
ϕ
S
1
dS
1
d
p
c
+
K
1
S
1
ϕ
3 101
1
−
S
1
δ
1
δ
2
δ
s
K
s
+
K
1
K
2
ϕ
M
1
dS
1
dp
c
a
23
=
a
32
=
−
3 102
d
p
c
d
S
1
=
ρ
2
g
m
v
n
v
χ
1
−
n
v
n
v
n
v
n
v
−
2
n
v
−
1
S
1
−
S
1
−
1
−
−
1
1
−
3 113
K
1
K
2
dS
1
dp
c
+
K
1
K
2
1
−
S
1
dS
1
dp
c
+
K
2
1
−
S
1
1
n
v
−
1
δ
2
1
M
1
S
1
a
33
=
−
+
K
1
K
2
ϕ
1
−
S
1
S
1
dS
1
dp
c
2
m
v
1
m
v
S
2
λ
1
+
K
1
1
−
S
1
ϕ
k
r1
S
2
=
1
−
−
S
2
3 114
3 103
m
v
2
m
v
S
2
λ
1
1
m
v
k
r2
S
2
=
−
1
−
S
2
3 115
−
θ
1
η
1
k
s
k
r1
R
11
=
3 104
where
m
v
are the van Genuchten
parameters (
m
v
and
n
v
should not be confused with the
m
and
n
of Archie
=
1
−
1
n
v
,
n
v
,
χ
, and
λ
R
22
=
−
θ
η
2
k
s
k
r2
2
3 105
'
s laws).
A
11
=
1
−
α
s
ρ
1
θ
1
3 106
3.2.2 The u
-
p formulation for two-phase
flow problems
The classical formulation described previously, using
Equations (3.95) to (3.96), is based on solving partial
differential equations for three unknown fields,
u
(solid
displacement field) and
w
1
and
w
2
, corresponding to
the two filtration displacements for the two fluids.
A two-dimensional discretized problem results in six
degrees of freedom per node that must be solved
to compute the wave propagation problem. We seek to
simplify the problem by reducing the number of degrees
of freedom that need to be solved; therefore, we intro-
duce an alternate formulation that is simpler to solve
numerically. In this alternate formulation, we eliminate
the unknowns
w
1
and
w
2
and only use
u
and
p
i
as
the unknown parameters we solve for. This will reduce
the problem to three unknown parameters (
u
1
,
u
2
, and
p
i
) that we have to solve at each node. To get to this
A
22
=
1
−
α
s
ρ
2
θ
2
3 107
K
b
K
s
K
s
K
s
+
M
1
M
2
K
b
K
s
+
ϕ
−
1
1
−
ϕ
−
δ
s
=
3 108
K
1
S
1
+
K
2
dS
1
dp
c
+
K
2
S
1
1
−
S
1
dS
1
dp
c
1
−
ϕ
−
K
b
K
s
δ
1
=
−
K
s
M
1
+
M
2
K
b
K
s
+
ϕ
−
1
3 109
K
2
1
−
S
1
+
K
1
S
1
dS
1
dp
c
1
−
ϕ
−
K
b
K
s
δ
2
=
K
s
M
1
+
M
2
K
b
K
s
+
ϕ
−
1
3 110
K
1
S
1
dS
1
dp
c
+
K
2
dS
1
dp
c
+1
M
1
=
−
3 111
1
−
S
1
K
1
K
2
dS
1
d
p
c
+
K
1
S
1
+
K
2
1
−
S
1
ϕ
M
2
=
3 112
ϕ
S
1
1
−
S
1
ϕ
