Geoscience Reference
In-Depth Information
The transversal resistance TR
i
when the electrical
current is applied from the top and the bottom of the
layers' package is:
TR
i
¼
R
i
¼
q
i
h
i
:
ð
2
:
2
:
11
Þ
From Eqs. (
2.2.10
) and (
2.2.11
) it can be deter-
mined that:
h
i
¼
p
C
i
TR
i
r
TR
i
C
i
ð
2
:
2
:
12
Þ
q
i
¼
:
For the total thickness H
¼
P
N
h
i
the total parallel
i
¼
1
conductivity C and the total transversal resistance TR
are:
C
¼
X
N
h
i
q
i
i
¼
1
ð
2
:
2
:
13
Þ
TR
¼
X
N
Fig. 2.12
Model of a sequence of horizontal layers
h
i
q
i
:
i
¼
1
The apparent resistivity is a function of the specific
electrical resistivity of the rocks, the thickness and the
depth of the layers and other rock inhomogeneities as
fractures and the type of their filling substance. Thus,
the electrical field should be not symmetric in the
space. Its configuration will depend on the superpo-
sition of the electrical conductivity of the studied
''package'' of layers and structures. The same is valid
for the electrical resistivity.
Let us consider the simplest case, representing a
package of horizontal layers with electrical resistivity
of every layer q
1
, q
2
, q
3
, … q
N
(Fig.
2.12
).
In this model L = 1. The thickness of the layers is
h
1
, h
2
, h
3
, … h
N
. From Eq. (
2.2.1
) is seen that the
resistance R is:
The average parallel resistivity q
l
and the average
transversal resistivity q
t
will be:
q
l
¼
H
C
q
t
¼
TR
H
ð
2
:
2
:
14
Þ
:
It is proven that always the transversal resistivity q
t
is higher than the parallel resistivity q
l
(Hmelevskoy
1984
).
Let us now consider a more complicated and close
to the practice case, when the layers are dipping from
the surface at angle a and the measurement of the
electrical resistivity is executed using the symmetric
four electrodes array Schlumberger (Fig.
2.13
). The
array
can
be
oriented
randomly
on
the
surface,
R
¼
q
L
S
:
ð
2
:
2
:
9
Þ
crossing the parallel layers at angle b.
Without entering in the theoretical details, it can be
shown (Yakubovskiy
1973
) that:
The parallel conductivity C
i
when the electrical
current is applied on the transversal walls of every
layer i is:
p
q
l
ð
cos
2
b
þ
sin
2
b
cos
2
a
Þþ
q
t
sin
2
b
sin
2
a
q
t
q
l
q
a
¼
q
:
C
i
¼
1
R
i
¼
h
i
q
i
:
ð
2
:
2
:
10
Þ
ð
2
:
2
:
15
Þ
