Page 61 - Principles of Catalyst Development

42 Principles of Applied Reservoir Simulation

dx\ -'
j f l ,<,K ^c I (5,12)
* i

The derivative (dxidi) Sw is the velocity of the moving plane S w, and (dfJdS w) Sw
is the slope of the fractional flow curve. The integral of the frontal advance
equation gives

rs - W < ( dL \ (5J3)
-~7* \7s~l
\ ' $„
where
distance traveled by a particular S w contour [ft]
x Sw
Wj cumulative water injected [cu ft]
(df w/dsj\ slope of fractional flow curve

Water Saturation Profile
A plot of S w versus distance using Eq. (5.13) and typical fractional flow
curves leads to the physically impossible situation of multiple values of S w at
a given location. A discontinuity in S w at a cutoff location x c is needed to make
the water saturation distribution single valued and to provide a material balance
for wetting fluids. The procedure is described by Collins [ 1961 ] and summarized
below.

5.2 Welge's Method

In 1952, Welge published an approach that is widely used to perform the
Buckley-Leverett frontal advance calculation. Welge's approach is best
demonstrated using a plot off w vs S w (Figure 5-2).
A line is drawn from the water saturation S w before the waterflood -
irreducible water saturation S wirr - and tangent to a point on thef w curve. The
resulting tangent line is called the breakthrough tangent, or slope. It is illustrated
in Figure 5-2. Water saturation at the flood front S^is the point of tangency on
curve. Fractional flow of water at the flood front is/^and occurs at the
thef w
point of tangency S^ on the/ w curve. In Figure 5-2, S wf\s 65% and/ w/is 95%