Mathematical Modeling of Electrokinetic Transport and Enhanced Oil Recovery In Porous Geo-Media - Electrokinetics for Petroleum and Environmental Engineers - page 218

5.8.5 Two-phase Immiscible Flow Under Applied Pressure and
Electrical Gradient

Superimposing electrical gradient to the pressure and elevation gradients,
the resulting fluid flow expressions for each phase can be expressed as:

k k

⎡ ⎤ rww , rwo ,

⎢
⎢
⎢
⎢

⎥
⎥
⎥
⎥

( k ) ( k )

v
v

∇− ∇

∇− ∇

P gz

Pg

r w ⎡ ⎤ ∇

∇

E
E

⎡
⎢

⎤
⎥ =−

⎡
⎢

⎤
⎥ −

⎡
⎢

⎤
⎥

m m

er ww , er wo ,

w w w

k

w o

k e

⎢ ⎥

( )

k k r z

( k ) k

⎢

⎣ ⎦

o row , roo , o o o

er ow , er oo ,

m m

⎣ ⎦

(5.85)

w o

As discussed in section 5.9.4, setting k er,ow and k er,oo coefficients to zero,
and

∇=∇ E E

o , Eq. 5.85 reduces to: w

k k

⎡ ⎤ rww , rwo ,

⎢
⎢
⎢
⎢

⎥
⎥
⎥
⎥

v
v

∇− ∇

∇− ∇

P gz

Pg

r w k

k

∇
∇

E
E

⎡
⎢

⎤
⎥ =−

m m ⎡

⎢

⎤
⎥ −

⎡
⎢

⎤
⎥

⎡
⎢

⎤
⎥

(5.86)

w w er ww , w

w o

k k e

k k r z

o row , roo , 0 o er ow , w

m m

⎣ ⎦

w o

Having the velocities of each phase under the combined pressure and
electrical gradients, the system of equations to be solved is then:

∂
∂

( ) =−∇ [ ] +

(5.87) Φ S r r v q

ww ww w

t

k k

( ) − ( ) − ( )

rww , rwo ,

v =− k ∇ P − r g ∇ z k ∇ Pg − r ∇ z k e k ∇ E

(5.88) w w w o o er ww , w

m m

w o

∂

∂ [

] =−∇ [ ] +

(5.89) Φ S r . r v q

oo oo o

t

k k

( ) − ( ) − ( )

row , roo ,

v =− k ∇ P − r g ∇ z k ∇ Pg − r ∇ z k e k ∇ E

(5.90)

o w w o o er ow , w

m m

w o

1 (5.91)

SS
w

+=
0

PPP
C

=−

(5.92)

o w