Electrokinetic Effect in Water-Saturated Rock - Ultra and Extremely Low Frequency Electromagnetic Fields - page 322

conducting the electrokinetic current. It should be emphasized that a variety of the
crack sizes can be described only in the framework of rather complicated percolation
theory. Below we restrict our analysis to a simple percolation theory (e.g., see Snante
and Kirkpatrick 1971 ; Staüffer 1979 ) in which the percolation cluster is able to
appear under the requirement that the average rock porosity n is greater than the
percolation threshold n c . In this approximation the average rock conductivity is
described by Eq. ( 8.9 ).
In reality the dry rock conductivity is never equal to zero because of the presence
of ion conductivity in the solid matrix. Actually, as n<n c then the average rock
conductivity falls off abruptly but it is not equal to zero. However, the parameter
n c can serve as a percolation threshold for both the fluid flow and the electrokinetic
current.
The fractal properties near the threshold are determined by the correlation length

0
.p p c /

0
.n n c / ;

(8.17)

where is the correlation length critical exponent, p is the probability that an infinite
cluster will appear in porous rocks while p c and n c denote the critical probability
and porosity related to the percolation threshold, and 0 is a constant of dimension
of length. The infinite cluster has a fractal structure above the percolation threshold
within spatial scale, which does not exceed the correlation length (Feder 1988 ;
Staüffer 1979 ).
The upper crust contains a great deal of fluid-filled cracks, fracture zones, the
sealed underground compartments with high pore pressure and etc. Some of such
formations with high pore pressure may become unstable due to the variations of
tectonic stresses (Bernard 1992 ; Fenoglio et al. 1995 ). The focal zone of a forthcom-
ing earthquake is frequently associated with such unstable inhomogeneities which
are capable of sustaining both the outward fluid migration and the electrokinetic
current. In our model the pore space of the inhomogeneity forms the infinite cluster
which has the fractal structure. This implies that the characteristic scale L of this
inhomogeneity is of the order of the correlation length ( 8.17 ) whence it follows that
n n c . 0 =L/ 1= . Combining this relationship with Eqs. ( 8.7 ) and ( 8.9 ) one can
find the rough estimation of the electrokinetic current density in fractal pore space
(Surkov et al. 2002 ; Surkov and Tanaka 2005 )
j ek 0 C 0
L

=

r P: (8.18)

Using the critical exponents D 1:6 and D 0:88 obtained by numerical
simulation on 3D (tree dimensional) grids (Staüffer 1979 ; Feder 1988 )gives=
1:82. We also suppose that jr P j P=L, where P is the pore fluid pressure
difference between the inhomogeneity and surrounding rock. In the case of large-
scaled inhomogeneities such as earthquake hypocenter, P is supposed to be
proportional to the shear stress drop caused by rock fracture before the main shock.
The shear stress drop is of the order of crushing/shear strength which is independent