Page 317 - Standard Handbook Petroleum Natural Gas Engineering VOLUME2

484 Reservoir Engineering

where E,, is the volumetric sweep efficiency in a linear displacement, V, is the
displaceable pore volumes injected, and the other terms are as already defined.
Vertical or invasion Sweep Efficiency (E,)

For well-ordered sandstone reservoirs, the permeability measured parallel to
the bedding planes of stratified rocks is generally larger than the vertical
permeability. For carbonate reservoirs, permeability (and porosity) may have
developed after the deposition and consolidation of the formation; thus the
concept of a stratified reservoir may not be valid. However, in stratified rocks,
vertical sweep efficiency takes into account the inherent vertical permeability
variations in the reservoir. Vertical sweep efficiency of a waterflood depends
primarily upon the vertical distribution of permeabilities within the reservoir,
on the mobility of fluids involved, and on the density differences between
flowing fluids. As a result of nonuniformity of permeabilities in the vertical
direction, fluid injected into an oil-bearing formation will seek the paths of least
resistance and will move through the reservoir as an irregular front. Con-
sequently, the injected fluid will travel more rapidly in the more permeable zones
and will travel less rapidly in the tighter zones. With continued injection and
displacement of some of the resident fluids, the saturation of the injected fluid
will become greater in the more permeable areas than in the low-permeability
strata. This can cause early breakthrough of injected fluid into the producing
wells before the bulk of the reservoir has been contacted. In addition, as the
saturation of the injected fluid increases in the highly permeable zones, the
relative permeability to that fluid also increases. All of these effects can lead
to channeling of the injected fluid, which is aggravated by the unfavorable
viscosity ratio common in waterflooding. In many cases, permeability stratifica-
tion has a dominant effect on behavior of the waterflood.
Permeability Variation

Two methods of quantitatively defining the variation in vertical permeabilities
in reservoirs are commonly used. The extent of permeability stratification is
sometimes described with the Lorenz coefficient [293] and is often described
with the Dykstra-Parsons [294] coefficient of permeability variation.
Lorenz Coefficient. Schmalz and Rahme [293] suggested arranging the vertical
distribution of permeabilities from highest to lowest, and plotting the fraction
of total flow capacity (kh) versus the fraction of total volume (h9). To obtain
the Lorenz coefficient (see Figure 5-162), the area ABCA is divided by the area
ADCA. Values of the Lorenz coefficient can range from zero for a uniform
reservoir to a theoretical maximum value of one. However, the Lorenz coefficient
is not a unique measure of stratification, and several different permeability
distributions can give the same Lorenz coefficient [133].
Dykstra-Parsons Coefficient of Permeability Variation. The coefficient of
permeability variation described by Dykstra and Parsons [294] is also referred
to as the permeability variation or permeability variance. This method assumes
that vertical permeabilities in a reservoir will have a log-normal distribution.
The procedure outlined by Dykstra and Parsons was to: (1) divide permeabilities
(usually from core analysis) so that all samples are of equal thickness (often 1 ft),
(2) arrange the permeabilities in descending order from highest to lowest, (3) cal-
culate for each sample the percent of samples that have a higher permeability