Geology Reference
In-Depth Information
i
V
(3.113)
V
A
B
h
i
i
mz
i
(3.101)
m
3
2
2
h
i
i
Again, in equation (3.110) the unknown parameter
mx
appears on both sides of the equation. The equivalent
quadratic equation is
where
(3.102)
A
2
4
ii
hi
2
12
V
0 (3.114)
mx
i
h
i x
h i
2
2
(3.103)
B
5
4
i
i
h
h
a. Solution for the Case
V
i
≤ 0.1
By rationalizing
3.113,
b. Solution for the Case
V
i
> 0.1
The solution repre-
sented by equations (3.94)-(3.96) can be applied by using
the following coefficients of the quadratic equation
derived from equation (3.99):
2
(3.115)
1
i
h
i
h
i
mx
h
2
2
i
h
i
aa ja
2
30
j
(3.104)
1
2
4
(3.116)
ii h
mx
2
bb jb
h
3
5
V
i
h
i
1
2
i
h
i
i
h
(3.105)
j
3
5
V
h
i
i
i
b. Solution for the Case
V
i
> 0.1
The quadratic
equation (3.114) can then be solved as shown in equa-
tions (3.95) and (3.96) after replacing
m
with
mx
and
using the following coefficients of the quadratic equa-
tion derived from equation (3.110):
cc jc
1
2
(3.106)
where
2
2
cV
i
3
(3.107)
aa ja
1
j
(3.117)
1
i
h
i
h
i
i
i
h
i
h
1
2
bb jb
12
V
j
12
V
(3.118)
1
2
i
i
h
i
i
c
i
(3.108)
2
3
2
i
i
i
h
i
h
i
h
i
h
cc jc
j
hi hi
(3.119)
1
2
3. Oriented Needle Inclusion
By assuming the needle
axis in the
Z
direction, then
A
x
=
A
y
= 0.5 and
A
z
= 0. The
dielectric constant of the mixture becomes anisotropic.
Nevertheless
mx
=
my
since
A
x
=
A
y
. In this case equation
(3.84) can be rewritten as follows:
Results from the above model, based on inputs from
the physical model of brine volume and salinity (sec-
tion 3.4.), are presented in
Shokr
[1998]. Figures 3.29
and 3.30 show the results from using the assumptions of
random-needle and oriented‐needle inclusions, respec-
tively. The permittivity from using the random‐needle
assumption is almost constant, about 3.4 for tempera-
tures below −8 °C or salinities less than 4‰ (for constant
density). Beyond these limits, the permittivity increases
nonlinearly at a relatively fast rate as temperature or
salinity increases.
Arcone et al.
[1986] shows similar
results. The loss factor is more sensitive to temperature
and salinity, as indicated by the monotonic nonlinear
increase throughout the examined range. Results from
using the oriented needle assumption (Figure 3.30)
reflect the anisotropic character of the complex dielectric
constant in this case. Both permittivity and loss compo-
nents calculated for the needle axis being parallel to the
electric field are higher than those calculated for the axis
12
V
i
h
for
V
01
.
(3.109)
mx
h
i
i
i
h
2
V
i
h
for
V
0 1
.
(3.110)
m
x
h
im
i
x
i x
V
(3.111)
mz
h
i
i
h
Note that in deriving equation (3.111) the term * disap-
pears since
A
z
=0. Consequently, the equation is valid for any
value of
V
i
. By rationalizing equation (3.111),
V
(3.112)
mz
h
i
i
h
