Introduction to Electrokinetics - Electrokinetics for Petroleum and Environmental Engineers - page 8

If the total flow rate of water, q , in a porous medium consists of (1) the
flow rate where there is no electrical potential effect, q n , and (2) an osmotic,
countercurrent flow, q os , then:

kdp
dx

d
dx

y

− ⎛

⎞
⎟

,

(1.5)

qq q =− =

n

k

⎜

n os os

m

where k n is the permeability in the absence of electrical phenomena; p
is the pressure; k' os is the transport coefficient resulting from the streaming
potential, ψ ; and μ is the viscosity.
The coefficient, k' os is obtained from the Helmholtz equation for the
velocity of electroosmotic flow in a tortuous capillary (Adamson, 1960;
Scheidegger, 1974):

e fz

pmt

d
dx

y

(1.6) U =

os

2

4

where ϕ is the porosity and τ is the tortuosity.
Combining Eqs. 1.4 and 1.6 into Eq. 1.5 yields the fluid flow equation
that includes the effect of electroosmotic flow (Donaldson and Alam,
2008):

⎛
⎜

⎞
⎟

22
0

k e fz

pmt

R dP
dx

⎛
⎜

kkdp
dx

⎞
⎟

(1.7)

q =−

(

n

=−

n os

)

2

m

2

mm

4

1.3 Coehn's Rule

A general rule for the potential difference of the double layer was given by
Coehn in 1909 as follows:
Substances of higher dielectric constants are positively charged in
contact with substances of lower dielectric constants. The corresponding
potential difference is proportional to the difference of the dielectric con-
stants of the touching substances.
Later, researchers (Smoluchowski, 1921; and Adamson et al., 1963)
investigated this qualitative rule.
They found that this rule does not apply to pure organic liquids with low
dielectric constants, such as benzene and carbon tetrachloride. However,
Coehn's rule is still used to indicate the sign of the zeta potential and,
hence, the direction of movement of phases past each other.