Environmental Engineering Reference
In-Depth Information
Ψ
2
Ψ
1
r
1
r
2
Ψ
2
Figure 5.7
Schematic of oil and water filled capillary used in the analysis
to pump non-conductive liquids (e.g., oil) in micro-channels (Santiago,
2001; Brask et al., 2002; Gao et al., 2005, 2007; Wang et al., 2006; Liu et al.,
2009). A simple micro-channel (capillary), similar to EO viscous pump, is
considered to evaluate the coefficient. Figure 5.7 shows a schematic of the
micro-channel filled with the two immiscible fluids. In this depiction, the
inner layer is oil (crude oil, assumed to possess no electrical conductivity)
and the outer layer, near the channel wall is water (the formation water,
with electrical conductivity).
Under an applied electrical gradient, water with EO mobility will
be driven by electro-osmosis, and oil with no or negligible EO mobility
will be dragged by the interfacial viscous force between the water and oil
(Santiago, 2001; Brask et al., 2002; Gao et al., 2005, 2007; Wang et al., 2006;
Liu et al., 2009). To evaluate the
k
er,ow
coefficient, the electric field is applied
only to the water phase through the two end electrodes. The velocity and
volume of flow of each phase can then be calculated and used for evalua-
tion of the
k
er,ow
coefficient as a function of water saturation, or the volume
fraction of water in the capillary.
Several simplifying assumptions are needed for this analytical treatise,
including i) the two fluids are simple Newtonian fluids, ii) the fluid proper-
ties are constant and independent of local electric field, ion concentration,
and temperature, and iii) two-phase flow is steady and fully developed
with no slip-boundary conditions both at the interface and at the wall.
Accordingly, the continuity equations (velocity and momentum) for each
phase can be expressed as:
∇=
∂
∂
v
0
⎧
⎪
⎪
a
(5.70)
v
t
⎛
⎜
⎞
(
)
+∇
r
a
+∇
vv
⎟
=−∇
P
+∇
m
∇
v
()
r
E
a
a
a
a
a
a
e
a
r
e
are the velocity, mass density, and the electric
charge density of the phase α, respectively. For small Reynolds number
where,
v
α
,
r
α
, and
()
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