Page 171 - Percolation Models for Transport in Porous Media With

168 CHAPTER 10 PRODUCTION CHANGE: ELECTRIC ACTION

Figure 63: Hook-up diagrams for the electric treatment of solitary wells: 1 -
electric energy source; 2 - electric cable; 3 - grounding electrode; 4 - open shaft;
5 - electrode inside the well; 6 - guiding load; 7- filtering station

The boundary and initial conditions for the considered problem are

'I I
J r--+oo ::::::: 211'r2 ' (10.4)

where r = lrl = rw is the radius of the well;

<Pel = U, <Pel r--+oo = 0, (10.5)
ru,
~(E(r,O),O) = ~o (10.6)

Assume that the electric current is switched on instantly. In the equations
above, U is the voltage supplied to the casing column, I = U / Ro is the current in
the circuit, Ro is the apparent resistance of the medium. It is taken into account
in (10.4) that in the interval Hw ~ r ~ rw the problem has cylindrical symmetry,
and in the interval r ~ Hw, spherical symmetry. The component of the electric
current vector perpendicular to the surface of the ground vanishes.
Taking account of the time dependencies of ~ and E does not cause any change
to the equations (10.1)- (10.3) and the conditions (10.4)- (10.6), since the char-
acteristic times of variation for ~ and E are no less than several tens of minutes,
and therefore the contribution of the non-steady state terms is negligible. There-
fore time can be treated as a parameter in this problem, and the problem itself
becomes quasi-stationary. Using the asymptotics (10.4), we can propose a uniform
relationship for the current density

j(r, t)::::::: I(t)JS (r) (10.7)
0