Application of Electrokinetics for Enhanced Oil Recovery - Electrokinetics for Petroleum and Environmental Engineers - page 129

where: p xyz is the local power loss, at the point of interest.

E is the local electric field vector, at the point of interest.

Ø is the local electrical potential field gradient, at the point of
interest.
σ is the local electrical conductivity tensor, at the point of interest.
ρ is the local electrical resistivity tensor (inverse of σ), at the point
of interest.
J is the local current density vector, at the point of interest.
I is a constant, in Equations 3.5 to 3.7, dependent only on the total volt-
age drop across the entire circuit and the sum of the circuit element resis-
tances in the circuit. For a heterogeneous Earth model, however, all of the
parameters in Equations 3.8 to 3.10 are functions of position, with the vec-
tor and second rank tensor properties, directional, in nature.
Electrical current densities follow paths of least resistance, such that the
total Joule power loss;

∇

P T = p xyz dxdydz , (3.11)

∫∫∫

for the entire (earth) system , is minimized. This means that in regions of
high resistivity, | J | is much lower than in regions of low resistivity. Since p xyz
depends upon the local | J | 2 , Joule power loss is much less in regions of high
resistivity than in regions of low resistivity. The exceptions to this rule-of-
thumb are in the vicinity of the power electrodes, where the current den-
sity is controlled by the geometry of the power electrodes. Even in these
situations, however, DCEOR electrode arrays are designed to maximize
Joule heating where it is desired and minimize Joule energy loss, where this
result is the desired outcome.

3.9.2 Electric Field Mapping

The primary objective of DCEOR is to pass electrical current through the oil
reservoirs in an optimum manner to achieve the above mechanisms. This is
achieved by electrode array design. Figure 3.17 shows a simple three-layer
earth model, used to design electrode arrays. Figures 3.18 and 3.19 illustrate
in cross section and plan view, respectively, the equipotential surfaces about
parallel sets of three anodes and cathodes, in the model of figure 3.17.

3.9.3 Joule Heating and Energy Loss

Electrode arrays are specifically designed, in DCEOR systems, to produce
high current densities, where high Joule heating is desired, and low current