Environmental Engineering Reference
In-Depth Information
(typical for the range of EO velocities in geological media) the inertial
force is zero. Therefore, the continuity equations can be simplified as:
∇=
−∇
uv
P
0
⎧
⎪
⎪
(5.71)
a
(
)
+∇=
+ ∇
m
∇
v
()
r
E
0
a
a
a
e
a
The effect of the electrical gradient on the flow can be evaluated by set-
ting to zero the pressure gradients in both phases. Hence, when the electri-
cal gradient is applied to the water phase only, equation 5.71 reduces to:
v
∇=
()
∇
⎧
2
m
v
r
E
⎪
⎪
(5.72)
w
w
e
w
2
m
∇=
0
o
o
The charge distribution of each phase is required to calculate the phase
velocities. The Poisson equation describes the electric potential distribu-
tion in cylindrical coordinates (Hunter, 2001):
1
∂
∂
∂
∂
yyr
e
⎟
+
∂
2
−
⎛
⎜
⎞
(5.73)
rr
r
=
e
2
r
∂
z
Where,
y
is the surface potential and
e
is the permittivity of the fluid.
Equation 5.73 can be further simplified based on the assumption of
thermodynamic equilibrium and the Debye-Huckel approximation for a
low surface potential (Hunter, 2001), as:
1
d
dr
∂
∂
y
⎛
⎜
⎞
2
y
(5.74)
r
⎟
=
k
r
r
Where,
k
is the reciprocal of EDL (Electric Double Layer) thickness
(Debye length) defined as (Hunter, 2001):
12
/
=
⎛
2
2
⎞
⎟
2
ne z
K
B
(5.75)
k
0
⎜
e
Where,
e
is the elementary charge;
n
is the ionic concentration in equilib-
rium solution,
K
B
is the Boltzman constant; and T is absolute temperature.
The boundary conditions imposed on the channel wall and at the oil-
water interface are (see figure 5.7):
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