Page 407 - Fundamentals of Reservoir Engineering

IMMISCIBLE DISPLACEMENT 342

By static it is meant that the downdip water injection is stopped when the maximum
water saturation plane, S w = 1 − S or, at which P c = 0, has just reached point X in the
linear displacement path. If the imbibition capillary pressure curve has a distinct
capillary transition zone, as shown on the right hand side of the diagram, then above
point X the water saturation will be distributed in accordance with the capillary pressure
(capillary rise)-saturation relationship. In particular, the capillary pressure at point A, a
distance y above the base of the reservoir, in the dip-normal direction (normal to the
flow direction), can be calculated as

γ
∆
P (S ) = p − p = ρ g cosθ (10.4)
o
w
c
w
from which the saturation at point A can be determined from the capillary pressure
curve, as shown in fig. 10.5.
Equation (10.4) is referred to, in this text, as the capillary pressure equation expressed,
in this case, in absolute units. In Darcy units, which are used in this chapter to develop
theoretical arguments, the general equation becomes

∆ ρ gy cosθ
P(S ) = 1.0133 10 6 (atm) (10.5)
c
w
×
while in the field units defined in table 4.1, which are employed in the exercises
P c (S w ) = 0.4335 γ ycosθ (psi) (10.6)
∆

where ∆γ is the difference between the water and oil specific gravities in the reservoir.

Allowing the S w = 1 − S or plane to rise incrementally in fig. 10.5 will result in a different
water saturation distribution, in the dip-normal direction, at point X in the displacement
path. This concept is applied in sec. 10.7 in which dynamic displacement is viewed as
a series of static positions of the S w = 1 − S or plane as the flood moves through the
reservoir, each position leading to a new water saturation distribution dictated by the
capillary pressure-saturation relationship.

b) Displacement generally occurs under conditions of vertical equilibrium
5
Coats has qualitatively explained the concept of vertical equilibrium by drawing an
analogy with a simple problem in heat conduction. If one were trying to mathematically
describe heat conduction in a thin metal plate, say, 1/8th of an inch thick and with an
area of several square feet, no allowance would be made for the heat distribution
across the thickness of the metal, in which direction thermal equilibrium would be
assumed. Since reservoirs, typically, have dimensions in proportion to those described
for the metal plate, displacement problems can be frequently tackled in a similar
manner. In this case, however, the assumption made is that of fluid potential
equilibrium across the thickness of the reservoir.

The condition for fluid potential equilibrium is simply that of hydrostatic equilibrium,
discussed previously, for which the saturation distribution can be determined as a
function of capillary pressure and, therefore, height, as