Page 225 - Fundamentals of Enhanced Oil and Gas Recovery

Waterflooding 213

with flow area A 5 2πrh, where h is the reservoir thickness; hence, Eq. (7.30)
becomes
1 2 2πhkk ro =q t μ r@P c =@r 1 P c

f w 5 o (7.31)
1 1 k ro μ =k rw μ o
w
Water saturation is a function of time, t, and position, r; we can express
@S w @S w
dS w 5 dt 1 dr (7.32)
@t @r
At the displacement front, the water saturation is constant, thus provides a bound-
ary condition for us.

@S w @S w @S w @S w dt
dS w 5 dt 1 dr 5 0- 52 (7.33)
@t @r @t @r dr
Recall that water fraction is function of water saturation, f w ðS w Þ, and partial differ-
ential equation result for change of fluid density governing equation:

df w @S w ð 2r e 2 2rÞπhφ @S w
2 5 (7.34)
dS w @r q t @t
Substituting Eq. (7.33) into Eq. (7.34)

df w @S w dt ð 2r e 2 2rÞπhφ @S w df w ð 2r e 2 2rÞπhφ
2 2 5 - dt 5 dr (7.35)
dS w @r dr q t @t dS w q t
Integrating Eq. (7.35) yields an equation for displacement front position, r f .

2
r 2 2r e r f 1 tq t df w 5 0 (7.36)
f
πhφ dS w f
where r f is the displacement front position in radial system; there are two solutions to
Eq. (7.36), which are

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tq t df w
2
r f 5 r e 6 r 2 (7.37)
e
πhφ dS w f
For Eq. (7.37), only one solution is correct to match the physical phenomenon.
Considering at the beginning of the displacement as t - 0, we have r f - 0; there-
fore, we can ignore the solution
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tq t df w
2
r f 5 r e 1 r 2 (7.38)
e
πhφ dS w f