Mathematical Modeling of Electrokinetic Transport and Enhanced Oil Recovery In Porous Geo-Media - Electrokinetics for Petroleum and Environmental Engineers - page 184

The total charge flux (migration and diffusion charge flux) can be
found as:

( ) +∇− ∑

N species

)

(

* *

(5.17)

I = F Z D j ∇ − C s E ]

j j

j = 1

For one dimensional applications, the total charge flux will be given by:

∂
∂

c
x

∂
∂

E
x

N species

∑

j

(5.18) I

*

=− F Z D − s j j

j = 1

5.3.4 Conservation of Mass and Charge

Applying the conservation of fluid, mass and charge equation to fluid flux
in saturated soil medium results in the following expression:

∂
∂

e

m u
t

∂
∂

∂
∂

h
t

(5.19)

V

=−∇ J = = m g

w V V w

t

where, e V is the volumetric strain of the soil mass, m V is the coefficient of
volume compressibility of the soil, u is the pore water pressure, g w is the
unit weight of water, h is hydraulic head, and t is time. Equation 5.19 is
Terzaghi's classical consolidation equation which describes the change in
hydraulic head due to soil volume change. This equation becomes impor-
tant in cases where hydraulic gradients are also used to enhance transport
under applied electrical gradient.
The mass conservation equation describes transient reactive transport
of i chemical species under hydraulic, electric, and chemical concentration
gradients. Applying the conservation equation to mass transport of species
i results in:

∂ nC

t

i

(5.20)

i

=−∇ + JnR

i

∂

where, R i is the production/consumption rate of the i th aqueous chemi-
cal species per unit fluid volume due to geochemical reactions, and n is the
porosity of the soil.
Applying conservation of charge to the charge flux equation, results in:

∂
∂

T
t

IC E
t

∂
∂

(5.21)

e

=−∇ =

p