Application of Electrokinetics for Enhanced Oil Recovery - Electrokinetics for Petroleum and Environmental Engineers - page 128

anode(s) and cathode(s) in the subsurface. Successful field pilots have uti-
lized anode-cathode separations of as much as 400 m, or 0.25 mi (Wittle
and Hill, 2006a, 2006b; Wittle et al, 2008a, 2008b, and 2008c; Wittle et al,
1011).

3.9.1 Three-Dimensional Current Flow Ramifications

A common misconception is that three-dimensional electrical cur-
rent flow in the subsurface can be represented by one-dimensional lin-
ear (Kirchhoff) ) circuit theory. For one-dimensional linear circuit theory ,
Ohm's law is:

Δ V

R

(3.5)

I =

or:

(3.6)

VIR

Δ =

Joule heat loss across a given circuit element , of resistance R , is given by:

2

, (3.7)

PIVIR = Δ =

where: P is the power loss, over the individual circuit element, of resis-
tance, R .
ΔV is the voltage drop across the individual circuit element.
R is the individual circuit element resistance.
I is the current through the entire circuit, controlled by the total
voltage drop across the entire circuit and the sum of all of the
resistances in the circuit.
In three-dimensions , however, Ohm's law (Equation 3.5) becomes:

−
ssfrf

1

J ==∇=∇ E

, (3.8)

or:

∇ f J .

(3.9)

In three-dimensions, the local Joule heating power loss is given by:

2

(3.10) p xyz =• =•∇= JE J f J r

,,