Page 148 - Percolation Models for Transport in Porous Media With

144 CHAPTER 8. CONDUCTIVITY AND ELECTRIC CURRENT

tive to the average in the medium. Since the fluid flow in cement is hardly ever
observed, temperature can fall primarily due to heat conductance of the skele-
ton of the medium, the fluid, and the cement. When certain threshold values of
temperature Tc ("temperature mechanism") or temperature gradient ("gradient
mechanism") are reached, the mechanical tensions developing in the thin capillar-
ies of the cement destroy it. The cement is then ejected entirely or partially into
larger cavities. This effect results in a notable increase of the electric conductivity
of the medium {by up to several tens of per cent) and to substantial increase of
the permeability (by several times).
We shall determine the values of these thresholds for electric treatment with
impulse current as functions of the impulse parameters, of the pore space structure
and the fraction of the non-conducting capillaries in the medium. Since the electric
conductivity of the skeleton is usually negligible, we can suppose that the current
flows only through the inter-grain space filled with fluid or cement. Let the average
field intensity in the medium E(t) = Eou(t), where Eo is the amplitude and u(t)
is a time function that sets the form of the impulse, be given. If the contribution
of the high-frequency harmonics {i.e., those with frequencies 100kHz and more)
to the Fourier spectrum of the impulse is small {this is valid for the duration of
impulse T > w-s - w- 6 s), then the deviations of the form of the current in the
medium from u(t) due to the reactive components of the electric conductivity can
be neglected.
If the radii of the capillaries in the medium are sufficiently larger > >..' fu' :=::$
w- 6 m, then the surface conductivity of the capillaries can also be neglected.
For the majority of rocks, the coefficients of temperature conductivity of the
fluid, cement, and skeleton differ by no more than 3 to 5 times. Therefore, for
clarity, without significant loss of accuracy, the coefficients of temperature con-
ductivity of the fluid, cement, and skeleton can be considered the same, and equal
to "'t ("'t :::;$ w- m /s).
7
2
Since the characteristic size of a grain l :=::$ w- 3 m and theradii of the capillaries
filled with cement r :=::$ w- 6 - w- 4 m, it follows that the characteristic periods of
1
temperature exchange for a capillary Tk = 1/4r 2 K"t :::;$ 2 ·10- 6 - 2 ·10- 2 sis much
2 1
less than those for a grain Tl = 1/4l K"t :=::$ 2 s. Hence for impulse current with
the impulse period T < Tj, the overlap of the thermal fields of adjacent capillaries
can be neglected.
For r ;: w- 6 m, the current density in a capillary can be considered constant
across the cross-section, and the boundary effects at the capillary (or the capillary
- pore) junctions can be neglected.
Suppose the electric current I(t) = Iou(t), where Io is the amplitude, flows
through two successive capillaries oflength leach and with radii r1 and r2. Assume
that l ~ r1, r2 and r2 2: r1 and address the determination of the temperature
distribution in two semi-infinite capillaries.