Geoscience Reference
In-Depth Information
2.1.3.3 Note regarding the material properties
We look for an expression of bt =
1 ϕ T s + ϕ T f + F =1 ϕ ρ s u s + ϕρ f u f ,
2 123
η f
t k 0 , where
bt =FT 1
η ω
k 0 , and
T + F =
ρ
u s +
ρ f w ,
2 124
ω η f
k 0
1
where the bulk density and the bulk force are defined by
bt =FT 1
b
,
2 117
m c
1
i
ωτ
ρ = ϕρ f +1 ϕ ρ s ,
2 125
where FT 1 stands for the inverse Fourier transform of a
given frequency-dependent function. Let us start with
the simpler case where the distribution of the relaxation
time obeys a Dirac distribution (Debye relaxation, c = 1).
In this case, we have
F =1
ϕ
F s +
ϕ
F f ,
2 126
respectively.Anexpression inthe timedomainfor the force
in terms of sourcemechanismis given in the following text.
The total stress tensor and the stress tensor on the pore
fluid are given by
η f
k 0
1
= η f
k 0
t
τ m
FT 1
1
exp
2 118
1
i
ωτ m
T = K U
u s I + C
w I +2 G U d s +2 C G d w ,
2 127
The more general case c
1 can be found in Revil et al .
(2006). It yields
T f = C
u s I + M
w I +2 C G d s +2 M G d w ,
2 128
1 n
τ m nc
bt = η f
k 0
t
where the circled cross stands for the Stieltjes convolution
product, K U and G U are the undrained bulk and shearmod-
uli of the porous medium, C and M are two Biot moduli,
and d s and d w denote the deviatoric components of the
deformation tensors for the solid and fluid, respectively.
1
,
2 119
Γ
1+ nc
n =0
where
Γ
( ) is the gamma function defined by
u x 1 e u du
Γ
x
2 120
2.1.4 The Maxwell equations
As shown by Pride (1994), the local Maxwell equations
can be volume averaged to obtain the macroscopic
Maxwell equations. With the Donnan model developed
by Revil and Linde (2006), the Maxwell equations are
0
We can easily check that with the case where c = 1, and
Equation (2.119) becomes equal to (2.118) (Revil et al . ,
2006). We can apply the same approach to the properties
of the fluid discussed in Section 2.1.1. For example, we
can determine the inverse Fourier transform of the fre-
quency-dependent shear modulus of the fluid defined
by Eq. (2.51):
× E =
B ,
2 129
× H = J + D ,
2 130
ω ω m c
i
G f ω
=
ω ω m c G f ,
2 121
B =0,
2 131
1
i
D =
ϕ
Q V ,
2 132
and G f t =FT 1 G f ω . If the Cole - Cole exponent is equal
to 1, we have G f
t = G f exp t
τ m . The more general
where H is the magnetic field, B is the magnetic induc-
tion, and D is the displacement vector. These equations
are completed by two electromagnetic (EM) constitutive
equations: D =
case c
1 can be found easily as
t = G f
n =0
1 n
τ m nc
t
G f
2 122
Γ
1+ nc
ε
E and B =
μ
H where
ε
is the permittivity
of the medium and
is the magnetic permeability. If the
porous material does not contain magnetized grains,
these two material properties are given by (Pride, 1994)
μ
2.1.3.4 Force balance equations
Performing a volume average of the force balance equa-
tions in each phase, Equations (2.17) and (2.25) yield a
total force balance equation in the time domain
ε = 1
F ε f + F 1 ε s ,
2 133
 
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