Page 322 - Standard Handbook Petroleum Natural Gas Engineering VOLUME2

Fluid Movement in Waterflooded Reservoirs 489

A recent analytical extension [299] of the Dykstra-Parsons method allows
calculations of total flow rates and flow rates in each layer for both a constant
injection rate and for a constant pressure drop. The ability to calculate cumula-
tive injection into a layer allows the incorporation of sweep efficiency of each
layer as a function of mobility ratio and displaceable pore volumes injected for
the pattern used in the waterflood.
Crossflow. In the usual cases where there is vertical communication between
the different layers of varying permeabilities, the effect of vertical crossflow must
be considered [300,301]. Goddin et al. [301] performed a numerical simulation
in a 2-D, 2-layer, water-wet system. For mobility ratios ranging from 0.21 to 0.95,
oil recovery with crossflow was between that computed for a uniform reservoir
and that for a layered reservoir with no crossflow. Goddin et al. [301] defined
a crossflow index, which is a measure of the extent the performance varies from
that of a uniform permeability system:

N,, - N,,
crossflow index = (5-225)
N, - Npd
where NPU = oil recovery from uniform system with the average permeability
N, = oil recovery from layered system with crossflow
Npnd = oil recovery from stratified system with no crossflow
Of the variables investigated, mobility ratio and the permeability ratio of the
two layers had the largest effect on crossflow (see Figures 5165 and 5-166,
respectively). Crossflow was more pronounced at lower mobility ratios or at high
ratios of layer permeabilities. The crossflow index of one means that the
performance of the layered system with crossflow is identical to the performance
of the system with uniform permeability.
Still at issue is the relative importance of mobility ratio and gravity in
waterflooding stratified reservoirs [302-3061. For wetting conditions that are
not strongly water-wet, additional complications will arise.

Estimates of Volumetrlc Sweep Efficiency. Volumetric sweep efficiency ranges
from about 0.1 for very heterogeneous reservoirs to greater than 0.7 for
homogeneous reservoirs with good flooding characteristics [278]. For a liquid-
filled, 5spot pattern, Craig [298] found that the volumetric sweep efficiency (EJ
at breakthrough decreases sharply as the permeability variation increases (see
Figure 5-167). Similar trends were observed for initial gas saturations of 10%
and 20%. These data indicated that the major effect of mobility ratio on E, at
breakthrough occurs for mobility ratios ranging fmm 0.1 to 10.
More recent simulations [307] of 5-spot patterns with a streamtube model
yielded the volumetric sweep efficiencies shown in Figures 5-168 and 5-169 for
WORs of 25 and 50, respectively. Mobility ratios of 0.1, 1, 10, 30, and 100 were
used. The permeabilities in the 100-layer model were assumed to have a log-
normal distribution, and pseudo-relative permeability expressions were used. In
a companion paper [308] the streamtube model (no crossflow) was compared
to the Dykstra-Parsons method (no crossflow) and with a model having the
assumption of equal pressure gradient in each layer (with crossflow). The
streamtube model was more closely described by the model with vertical
communication for unfavorable (high) mobility ratios and by the Dykstra-Parsons
model for favorable (low) mobility ratios.
(tat continued on page 292)