Page 305 - Standard Handbook Petroleum Natural Gas Engineering VOLUME2

272 Reservoir Engineering

If L is the distance from injector to producer, the time of water breakthrough,
kt, is given by:
tbt = -- p,,

q
A@ as, (5-2 10)

Equation 5-209 can be used to calculate the saturation distribution in a linear
waterflood as a function of time. According to Equation 5-209, the distance
moved by a given saturation in a given time interval is proportional to the slope
of the fractional flow curve at the saturation of interest. If the slope of the
fractional flow curve is graphically obtained at a number of saturations, the
saturation distribution in the reservoir can be calculated as a function of time.
The saturation distribution can then be used to predict oil recovery and required
water injection on a time basis. A typical plot of dfw/dSw vs. S, will have a
maximum as shown in Figure 5-154. However, a problem is that equal values
of the slope, dfw/dSw, can occur at two different saturations which is not
possible. To overcome this difficulty, Buckley and Leverett [ 1521 suggested that
a portion of the saturation distribution curve is imaginary, and that the real
curve contains a saturation discontinuity at the front. Since the Buckley-Leverett
procedure neglects capillary pressure, the flood front in a practical situation
will not exist as a discontinuity, but will exist as a stabilized zone of finite length
with a large saturation gradient.
Welge Graphical Technique. A more simplified graphical technique was pro-
posed by Welge [153] which involves integrating the saturation distribution from
the injection point to the front. The graphical interpretation of this equation
is that a line drawn tangent to the fractional flow curve from the initial water
saturation (Swi) will have a point of tangency equal to water saturation at the
front (Sd). Additionally, if the tangent line is extrapolated to f, = 1, the water
eturation will correspond to the average water saturation in the water bank,
S,. Construction of a Welge plot is shown in Figure 5-155. The tangent line
should be drawn from the initial water saturation even if that saturation is
greater than the irreducible water saturation.
Welge derived an equation that relates the average displacing fluid saturation
to the saturation at the producing end of the system:
-
S, - S, = Q fO4 (5-211)
where s, = average water saturation, fraction of PV
S,, = water saturation at the producing end of the system, fraction of PV
Q = pore volumes of cumulative injected fluid, dimensionless
fo2 = fraction of oil flowing at the outflow end of the system
Equation 5-211 is important because it relates to three factors of prime impor-
tance in waterflooding [133]: (1) the average water saturation and thus the total
oil recovery, (2) the cumulative injected water volume, and (3) the water cut
and hence the oil cut.
Welge also related the cumulative water injected and the water saturation at
the producing end: