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uniform conducting half-space with constant conductivity
g
while '
p
stands for the
perturbations caused by the appearance of the fractured region with conductivity
p
.
In the atmosphere .
z
0/ the potential '
0
reads
8
<
q
.
z
C
h
0
/
2
C
2
z
h
0
I
2
g
.h
h
0
/
'
0
D
q
.
z
C
h/
2
ln
:
C
2
z
h
9
=
q
.
z
C
h
0
/
2
1
ln
C
2
C
z
C
h
0
h
h
0
C
p
z
2
;
(11.27)
;
C
2
C
z
where denotes the polar radius shown in Fig.
11.8
. The approximate relationship
for the perturbations caused the rock fracture is given by (Surkov
1989
)
I
g
p
2
g
.h
h
0
/
R
r
kC1
X
ƒ
k
P
k
.cos /
k
p
C
.k
C
1/
g
'
p
D
;
(11.28)
k
D
1
where
1
R
h
k
hR
k
h
0
.h
h
0
/
k
2k
C
1
k
C
1
h
h
0
ƒ
k
D
:
(11.29)
n
.
z
C
h/
2
C
2
o
1=2
Here P
k
.cos /are Legendre's polynomials, r
D
, and the angle
is shown in Fig.
11.8
.
If R
h and R
h
h
0
, then Eq. (
11.28
) is simplified because only first term
of the series can be taken into account. If, in addition,
h,s then Eq. (
11.27
)
is also simplified. For example, on the plane
z
D
0 the solution of the problem is
reduced to
3I
g
p
hR
2
4 .h
h
0
/
g
p
C
2
g
3
:
Ih.h
C
h
0
/
4
g
3
'
0
D
;
p
D
(11.30)
It follows from Eq. (
11.30
) that both the casing pipe and fractured zone produce
only a local effect since the electric field
E
Dr
' falls off faster with distance,
that is, as
4
. The numerical estimates of this effect seem to be accident-sensitive
because of the lack of information about actual values of the parameters I and h
0
.
The magnetic perturbation in the ground has only the azimuthal component B
,
owing to the symmetry of the given problem, though B
D
0 in the atmosphere.
Actually the magnetic perturbation is equal to a finite value near the ground surface
because the symmetry of the problem gets broken due to asperity of the ground
surface. Notice that the magnetic field associated with this effect and the field due to
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