Mathematical Modeling of Electrokinetic Transport and Enhanced Oil Recovery In Porous Geo-Media - Electrokinetics for Petroleum and Environmental Engineers - page 193

Cao (1997) developed a one-dimensional model to investigate mul-
tispecies transport under transient electric field as an extension to the
Alshawabkeh and Acar model (1996). The governing equation of mass
transport in their model is defined as:

2 2

∂ nC

t

∂
∂

C
x

− ∂
∂

Φ

vk C
x

] ∂ ] ∂

Φ

[ [

(5.45)

*

i

= D

i

+

i

− Cv + k + nR

i meo i m eo i

∂

2

x ∂ ∂ x

2

Where, D* is effective diffusion coefficient, k eo is effective EO mobility,
v m is effective electro-migration mobility, E is electric potential, and R is
the term of chemical reaction rate.
Concentration distributions of charged ions change over time while
transporting under applied electric field resulting in change to the local
electrical conductivity, which in turn alters the value of the electric gra-
dient. Therefore, the changing electric conductivity and electrical field
describe the transport process of the species implicitly. One of the impor-
tant features of the Cao (1997) model is that the model updates the
current density due to ionic motion under applied electric field and dif-
fusion motion under concentration gradient as expressed in the following
equation:

∂
∂

Φ

FZD C
x

∂
∂

∑ ∑

= ⎣

⎦

*

j FZ C x +

i

(5.46)

b i i i i i

i i

The model uses the updated concentration [ C i ( x,t )], mobility [ u i
( x,t )], and current density [ j b ( t )] to update the electric field distribution
expressed as:

( )

⎛
⎜

∂ Cxt

x

, ⎞

⎟

( ) +

∑ s

jt F ZD

* i

b i i

i

∂

( ) =

Ext ,

(5.47)

( )

xt ,

b

∑

where, s b is the bulk conductivity [ s b

=

FZuC

].
Figures 5.2 and 5.3 present the spatial distribution of the total lead and
the pH for 1-day and 35-day runs of this model compared with the results
from the laboratory tests of Pb(NO 3 ) contaminated kaolinite clay subjected
to long-term EK treatments (Pamukcu, 2009). Although the initial concen-
trations of the lead used in the numerical simulator and the experiments
were different (0.05 M in the numerical and 0.15 M in the experiment), as

i i i