Mathematical Modeling of Electrokinetic Transport and Enhanced Oil Recovery In Porous Geo-Media - Electrokinetics for Petroleum and Environmental Engineers - page 188

5.4 Mathematical Model and Solution of Ek

Tr a n s p or t

A mathematical model consisting of system of equations describing the
transient reactive multi-component species transport under applied
hydraulic, electric, and chemical gradients can be developed to model the
transport phenomena in EK remediation of contaminated soils. The sys-
tem consists of partial differential equations (PDEs) for transport and alge-
braic equations for geochemical reactions. The transport PDEs are divided
into three types. The first type consists of one equation that describes tran-
sient fluid flow. The second type consists of N number of equations that
describe reactive transport of N species. The third type is described by one
equation for charge transport. In the following section the transport PDEs
are discussed.
Substituting fluid flux into volume change equation results in the EO
consolidation equation expressed as:

∂
∂

h
t

Ch K
m

(5.33) =∇+

2 e

∇

2

Φ

v

g

Vw

This equation is necessary to describe the changes in the hydraulic head
across the soil, which affects the advective component of mass transport
(Acar et al., 1997). The transient reactive PDEs for mass transport are
derived by substituting mass flux equation into mass conservation equa-
tion given as:

∂ nC

t

( ) ∇+∇

⎣ ⎦

i

DC

* 2

Cuk

*

kh R

i =1,2,..., N (5.34)

=∇+∇ ( + Φ ) + i i i i eo h i

∂

Changes in the electric potential distribution across the soil as a result of
changes in the geochemistry can be found by substituting the charge flux
equation in the charge conservation equation as:

∂
∂

Φ

( )

N

∑

* 2 *

(5.35) C = F ZDC ∇ + ∇ s ∇ Φ

p j j j

t

j = 1

The total number of differential equations described for this system is
N+2 including N equations for mass transport, one equation for charge