Geoscience Reference
In-Depth Information
2.1.3.4 for a complete derivation) to the following
equation (which will be solved numerically in the follow-
ing text):
∇
P
p
f
1
B
BB
u
s
=
−
K
U
w
,
2 101
α
∇
P
K
K
U
=
−
=
,
2 102
∇ Υ
=
F
,
2 109
∇
u
s
∇
w
=0
1
−
B
α
where
B
=
p
f
T
xx
T
xz
P
∇
w
=0
,
2 103
T
zx
T
zz
Υ
=
,
2 110
−
p
f
0
0
−
p
f
α
=
1
B
K
K
U
1
−
,
2 104
and
−
F
x
+
ρ
u
x
+
ρ
f
w
x
and
C
=
K
U
B
and
α
is the bulk Biot coefficient defined
−
F
z
+
ρ
f
u
z
+
ρ
f
w
z
F
=
2 111
ρ
f
u
x
+
ρ
f
w
x
ρ
f
u
z
+
ρ
f
w
z
previously.
2.1.3.2 The field equations
We start with the Darcy equation, Equation (2.39), writ-
ten in the following form and neglecting the electroos-
motic term:
In the frequency domain, the stress tensor is written as
2
3
G
U
2
3
C
G
T
=
K
U
−
∇
u
s
I
+
C
−
∇
w I
2 112
u
s
+
C
G
∇
w
T
+
G
U
∇
u
s
+
∇
w
+
∇
η
f
w
=
−∇
p
f
−
ρ
f
u
s
+
F
f
2 105
k
ω
For a two-dimensional (2D) problem, the components
of the stress tensor are
The dynamic permeability is written as (see Eq. 2.40)
T
xx
T
xz
T
zx
T
zz
m
c
T
≡
2 113
1
1
−
i
ω ω
c
1
−
i
ωτ
ω
≡
2 106
ωτ
m
c
k
k
0
1
−
i
which are given by
Using the fact that
w
=
−
i
ω
w
, we can easily rewrite
Equation (2.105) as
2
3
G
U
∂
u
x
∂
x
+
∂
u
z
∂
2
3
C
G
∂
w
x
∂
+
∂
w
z
∂
T
xx
=
K
U
−
+
C
−
z
x
z
bt
w
+
ρ
f
w
=
− ∇
p
f
+
ρ
f
u
s
−
F
f
,
2 107
+2
G
U
∂
u
x
∂
+2
C
G
∂
w
x
∂
,
2 114
where the effective fluid density is given by
ρ
f
F
,
where
F
is the formation factor defined by Chapter 1
(see Eq. 1.90). This is the same result than for a viscous
Newtonian fluid. Indeed, for a viscous Newtonian fluid,
the frequency-dependent permeability is given by
ρ
f
=
x
x
2
3
G
U
∂
u
x
∂
x
+
∂
u
z
∂
2
3
C
G
∂
w
x
∂
+
∂
w
z
∂
T
zz
=
K
U
−
+
C
−
z
x
z
+2
G
U
∂
u
z
∂
+2
C
G
∂
w
z
∂
,
2 115
z
z
k
0
ω
=
k
ω ω
c
,
2 108
1
−
i
T
zx
=
T
xz
=
G
U
∂
u
x
∂
z
+
∂
u
z
∂
+
C
G
∂
w
x
∂
+
∂
w
z
∂
2 116
x
z
x
where
ρ
f
F
where
F
is the formation factor.
Equation (2.108) is a simplified version of Equation
(236) of Pride (1994). Inserting Equation (2.108) into
Equation
ω
c
=
η
f
k
0
With
these
notations,
Equation
(2.107)
and
(2.105)
results
in
Equation
(2.107)
ρ
T
=
F
yield directly Equation (2.109). In
the special case for which the fluid does not bear any
shear stress (Newtonian case), we have
G
U
=
G
fr
and
C
G
= 0 and we recover the classical poroelastic equations.
u
s
+
ρ
f
w
−∇
with
ρ
f
F
.
The second point studied in this section is to go from
Equation (2.107) and
ρ
f
=
ρ
u
s
+
ρ
f
w
−∇
T
=
F
(see Section
