Electrokinetic Effect in Water-Saturated Rock - Ultra and Extremely Low Frequency Electromagnetic Fields - page 317

produces the electrokinetic current I ek D E eff l=Rwhere R D l=.S/ is the sample
resistance. Combining these formulas and taking the notice of j ek D I ek =S,we
obtain the electric current density j ek due to the electrokinetic effect. The result can
be written in the vector form as follows (Frenkel 1944 ; Mizutani and Ishido 1976 ;
Mizutani et al. 1976 ;Pride 1994 )

j ek D C r P: (8.7)

The total current density which includes both the conduction and electrokinetic
currents can thus be written as

j D E C r P: (8.8)

In the case of narrow cracks and capillaries .r 0 r D / the DEL can contribute
to the total conductivity due to the presence of ion excess in the DEL. The electric
current in the surface layer with thickness r D can be estimated as I s D 2r 0 r D f E x

while the surface current density is j s D I s = r 0 . In order to take into account
the surface current in the capillaries one should replace f in Eq. ( 8.6 )by eff D
f C 2 f r D =r 0 (Watillon and de Backer 1970 ; Dukhin 1975 ). If we take the values
r 0 D 10 7 5 10 6 m which is typical for granites Westerly and Sherman (Brace

et al. 1968 ; Brace 1977 ) and take into account the parameters f D 0:025 S/m and
2 f r D D 10 8 S reported by Watillon and de Backer ( 1970 ), then we come to the
following numerical estimate for surface conductivity: 2 f r D =r 0 D 0:002 0:1
S/m. The typical values of the streaming potential coefficient C are as follows:
0:8 V/Pa for granite Westerly, 4:2 V/Pa for sandstones, and 4:7 V/Pa for
porous rocks (r 0 >10 5 m).
Since the solid matrix/dry rock conductivity is much smaller than that for the
fluid, the average rock conductivity in Eq. ( 8.8 ) is mainly determined by the fluid
content. In the simple model the rock conductivity and permeability are proportional
to the rock porosity n, which is equal to the volume fraction of fluid-filled pores and
cracks (Frenkel 1944 ; Mizutani and Ishido 1976 ). As the non-conductive matrix
approximation is assumed, the single pores and cracks cannot conduct the electric
current and thus only those cracks and channels, which create a connected system
or cluster are able to contribute to the conductivity . In the percolation theory the
effective conductivity of the medium near the percolation threshold depends on the
porosity by a power law (Snante and Kirkpatrick 1971 ; Staüffer 1979 ; Feder 1988 ):

D 0 .n n c / t ; (8.9)

where n c is the percolation threshold, 0 is a constant with dimension of conduc-
tivity, and t is the transport critical exponent. The numerical modeling based on 3D
grid has shown that t D 1:6 (Snante and Kirkpatrick 1971 ; Staüffer 1979 ).
Actually the dry rock conductivity is never equal to zero because there is
ion conductivity of the solid matrix, i.e., the percolation threshold of the rock
conductivity is absent. However, the nonlinear character of the .n/ dependence,