Page 303 - Standard Handbook Petroleum Natural Gas Engineering VOLUME2

470 Reservoir Engineering

g = acceleration due to gravity
Ap = water-oil density differences = p, - po
ud = angle of the formation dip to the horizontal.

This equation is derived in an appendix in the monograph by Craig [133].
Because relative permeabilities and capillary pressure are functions of only fluid
saturation, the fractional flow of water is a function of water saturation alone.
In field units, Equation 5-204 becomes [133]:

--
1+0.001127J- :(z 0.433Apsinad
f, =
1+-- Pw ko (5-205)
Po kw

where permeability is in md, viscosities are in cp, area is in sq ft, flow rate is
in B/D, pressure is in psi, distance is in ft, and densities are in g/cc.
In practical usage, the capillary pressure term in Equation 5-204 is neglected
[133]:

I--- km (g Apsinu,)
f, = v Po
1+-- Pw ko (5-206)
Po kw

and for a horizontal displacement of oil by water, the simplified form of this
equation is [13S]:

1
f, =
I+-- Pw km (5-207)
Po k,

Examples of idealized fractional flow curves, f, vs. S,, are given in Figure 5-153
for strongly water-wet and strongly oil-wet conditions [133].
Based on the initial work of Leverett [loo], Buckley and Leverett [152]
presented equations to describe an immiscible displacement in one-dimensional
flow. For incompressible displacement, the velocity of a plane of constant water
saturation traveling through a linear system was given by:

(5-208)

where q is the flow rate in cc/sec (or ft3/D), A is the cross-sectional area in
cm* (or ft'), t) is the fractional porosity, v is the velocity or rate of advance in
cm/sec (or ft/ D), and (af,/aS,,) is the slope of the curve of f,, vs. S,. This
equation states that the rate of advance or velocity of a plane of constant water
saturation is directly proportional to the derivative of the water cut at that water