Page 221 - Fundamentals of Enhanced Oil and Gas Recovery

Waterflooding 209

For a given rock, the fraction of flow for water f w is a function only of the water
saturation S w , as indicated by Eq. (7.4), assuming constant oil and water viscosities.
The water saturation, however, is a function of both time and position, which may be
express as f w 5 F(S w ) and S w 5 G(t, x). Then,

@S w @S w
dS w 5 dt 1 dx (7.5)
@t @x
x t

dS w @S w @S w dx
5 1 (7.6)
dt @t @x dt
x t
Now, there is interest in determining the rate of advance of a constant saturation

plane or front @x=@t , where S w is constant and dS w 5 0. So, from Eq. (7.5),
S w

dx @S w =@t
x
5 (7.7)
dt @S w =@x
t
Substituting Eqs. (7.5) and (7.6) into Eq. (7.7) gives the Buckley Leverett frontal
advance equation:

dx 2 q t df w
5 (7.8)
dt Aφ dS w Sw
S w

is the slope of the fractional flow curve and derivative
The derivative df w =dS w
S w

dx=dt is the velocity of the moving plane with water saturation S w . Because the
S w
porosity, area, and flow rate are constant and because for any value of S w , the deriva-

tive df w =dS w is a constant, then the rate dx=dt is constant.
S w
This means that the distance a plane of constant saturation, S w , advances is propor-

tional to time and the value of the derivative df w =dS w at that saturation, or
S w

2 q t df w
5
Aφ dS w S w
X S w (7.9)
is the distance traveled by a particular S w contour and q t is the cumulative
where X S w
water injection at reservoir conditions.
In field units,

5:615q t df w
52 (7.10)
Aφ dS w Sw
X S w
Fig. 7.2 shows the linear flow through a body of constant cross-section as well as
series and parallel flow in linear bed. Consider displacement of oil by water in a sys-
tem with dip angle α.