Electrokinetic Effect in Water-Saturated Rock - Ultra and Extremely Low Frequency Electromagnetic Fields - page 316

en
"" 0

@ x dž C @ y dž D ; (8.3)

where n denotes the ions excess in the solution, and e is their charge. The
dielectric permittivity " of the fluid is assumed to be constant. Near the channel
surface the potential changes rapidly along the normal to the surface that is along
the y-axis so that we can neglect the derivative with respect to x in Eq. ( 8.3 ).
In order to calculate the extrinsic/fluid-driven current I e caused by convective
transfer of the ions one should integrate the extrinsic current density j e D enV x
over the channel cross section. Taking into account of Eq. ( 8.2 )forV x and Eq. ( 8.3 )
for n, we get

Z

r 0

Z

r 0

r 0 "" 0 d 2 dž

dy 2 ydy:

dP
dx

I e 2r 0 e nV x dy D (8.4)

0 0

Performing integration by parts, we arrive at

r 0 "" 0 &

dP
dx ;

I e (8.5)

where & D dž.0/ dž.r 0 /. Here we have taken into account that the potential dž
is a rapidly decreasing function whose typical y-scale is much smaller than r 0 .The
parameter & in Eq. ( 8.5 ) is, in fact, the potential jump across the DEL that is the
&-potential/zeta potential which we have introduced above.
The compensated conduction current is I D r 0 f E x where f denotes the
fluid conductivity and E x is the projection of electric field on x-axis. At the state
of equilibrium this current is equal to the electrokinetic current given by Eq. ( 8.5 ).
Equating these currents we can find the effective field of extrinsic force E eff D E x :

E eff D C dP

"" 0 &
f : dx ;C D

(8.6)

Here C is the so-called streaming potential coefficient. This formula was first
derived by Helmholtz and then by Smoluchowski on the basis of more rigorous
treatments. In practice, Eq. ( 8.6 ) serves as a basic equation of the theory of
electrokinetic effect.
The value of parameter f is controlled by both intrinsic and extrinsic/impurity
conductivity of the water depending on the contents of mineral salts in the solution.
For the case of high mineralized groundwater the conductivity f can reach a
value of several unities or tens S/m, whereas for the sweet water the groundwater
conductivity does not exceed several thousandth S/m (e.g., Semenov 1974 ).
Assuming for the moment that the uniaxial stress is applied to the porous rock
sample with the length l, cross section S, and mean rock conductivity , then
the electromotive force E eff l arises in the sample. According to Ohm's law, it