Electromagnetic Effects Resulted from Natural Disasters - Ultra and Extremely Low Frequency Electromagnetic Fields - page 382

˃ 1 , C 1 ˃ 2 , C 2

z x

- x 0

Fig. 10.6 A model of conductive medium containing two half-spaces with different conductivity
and streaming potential C coefficients. A pore pressure inside the fill area differs significantly
from that in the surrounding media

Below we do not follow the cited papers in any detail since they consider only
a plane medium model. For simplicity, we assume that the pore pressure is a given
function and the pressure gradient is not equal to zero at the boundary between
two half-spaces. The pore pressure P has an axially symmetric distribution around
x-axis which is directed perpendicular to the boundary surface. The center of
this distribution is situated inside the focal region on the x-axis at the point with
coordinate x D x 0 .
For the axially symmetric problem, Eq. ( 8.12 ) in cylindrical coordinates x and r
reads

@ x ‰ 1;2 C r 1 @ r .r@ r ‰ 1;2 / D 0;

(10.29)

where ‰ 1;2 D dž 1;2 C C 1;2 P and dž stands for the electric potential. The subscripts
1 and 2 are related to the first and second regions, respectively. We seek for the
solution of Eq. ( 10.29 ) in the form of Bessel transform

Z

‰ 1;2 D

1;2 .x;k/J 0 .kr/kdk; (10.30)

0

where J 0 denotes Bessel function of the first kind and zero order, 1;2 D
' 1;2 .x;k/ C C 1;2 p.x;k/ while ' and p are Bessel transforms of the potential and
pore pressure, respectively. Substituting Eq. ( 10.30 )for‰ 1;2 into Eq. ( 10.29 ) yields

0 1;2 k 2 1;2 D 0;

(10.31)