Page 408 - Fundamentals of Reservoir Engineering

IMMISCIBLE DISPLACEMENT 343

∆ ρ gy cosθ
PS w ) = (10.5)
(
c
1.0133 10 6
×
that is, the fluids are distributed in accordance with capillary-gravity equilibrium. The
vertical equilibrium condition can therefore be interpreted in the following manner.
When, during the displacement of oil by water, the water saturation at any point in the
reservoir increases by a small amount, the new water saturation is instantaneously
redistributed as indicated by equ. (10.5). This means that the vertical velocities of oil
and water, as the two are redistributed in accordance with capillary-gravity equilibrium,
appear to be infinite in comparison with the velocity of fluid movement parallel to the
reservoir bedding planes resulting from the Darcy or viscous forces.

The condition of vertical equilibrium will be promoted by

- a large vertical permeability (k v)

- small reservoir thickness (h)

- large density difference between the fluids (∆ρ)

- high capillary forces (large capillary transition zone H)

- low fluid viscosities

- low injection rates.

5,6
Coats has presented two dimensionless groups, relating the above terms, the
magnitudes of which can be used as "rough rules of thumb" for deciding whether
vertical equilibrium conditions prevail in the reservoir. The two cases considered can
be applied when the capillary transition zone is large and also when it is negligible.
These dimensionless groups are not presented in this text since, irrespective of their
magnitude, the only way to check the validity of the vertical equilibrium is by using the
numerical simulation techniques described in sec. 10.10. In any case, when applying
simple, analytical techniques to describe the displacement process one is obliged to
assume that either vertical equilibrium is valid or else the complete opposite, that there
is a total lack of vertical equilibrium. The latter case will apply when, for instance, the
injection rate is so high that the water and oil velocities, parallel to the bedding planes
are much greater than their velocity components in the dip-normal direction. Under
these circumstances the water saturation will be uniformly distributed with respect to
thickness. These two extremes both represent conditions under which the saturation
profile in the dip-normal direction is definable and this facilitates the application of the
analytical techniques described in this chapter. For in-between cases the engineer
must resort to numerical simulation techniques (refer sec. 10.10). It has been found
that the vertical equilibrium condition is approximately satisfied in a great many
reservoirs. Nevertheless, it will be repeatedly stated throughout the remainder of the
chapter precisely when this condition is being assumed, and when not.