Mathematical Modeling of Electrokinetic Transport and Enhanced Oil Recovery In Porous Geo-Media - Electrokinetics for Petroleum and Environmental Engineers - page 190

be applied. If the voltage difference between the anode and cathode is kept
constant, the following boundary conditions can be used:

(5.36)

ΦΦ =

max ;

Φ = 0

S 1 S 2

Where, S 1 is the boundary surface at the anode and S 2 is the bound-
ary surface at the cathode. However, if the current density is maintained
constant the following boundary conditions can be used (Acar, et al., 1997;
Cao, 1997):

⎣

∑

N

⎦

Φ| S2 =0 (5.37)

* *

∇+∇ F ZDC s Φ = I

j j j S 1

j = 1

Boundary conditions for the species transport equations are evaluated
based on the electrolysis reactions at the electrodes. As a result of elec-
trolysis reactions, the chemistry at the electrodes will continuously change.
These changes either enhance the EK process (e.g., generation of acid at
the anode) or retard the process (e.g., generation of the base at the cath-
ode). In some cases, chemical reagents are introduced at the electrodes to
enhance the EK process. Upon identification of the electrolysis reactions
type, the boundary conditions can be evaluated from mass equilibrium in
electrodes compartments. The rate of concentration change of a specific
species in the electrode compartment will be equal to the net mass flow
rate of the species (Acar et al., 1997; Cao, 1997).

5.4.2 Preservation of Electrical Neutrality

For a unit volume of the soil, the rate of change in the electric charge equals
the total rate of change of chemical species concentrations multiplied by
their charge and the Faraday's constant expressed as:

∂ nC

t

∑

N

j

Rate of change in electrical charge (5.38) = C Z F

p j

∂

j = 1

The charge transport equation preserves the electrical neutrality of the
porous medium (Acar et al., 1997; Cao, 1997) so that:

∂ nC

t

N

∑

j

(5.39)

ZF = 0

j

j = 1

∂