Upscaling Flow Properties - Reservoir Model Design - page 125

Fig. 4.12 Summary of a
waterflood experiment
across a laminated water-
wet rock sample (Redrawn
from Huang et al. 1995 ,
# 1995, Society of
Petroleum Engineers Inc.,
reproduced with
permission of SPE. Further
reproduction prohibited
without permission)

0.6 10000

Capillary trapping of oil
upstream of low-k layer

0.4

1000

0.2

Flow direction

100

0.0

0 10 20

Length along sample (cm)

4.2.3 Heterogeneity and Fluid Forces

To treat this issue more formally, we use scal-
ing group theory (Rapoport 1955 ; Li and Lake
1995 ;Lietal. 1996 ; Dengen et al. 1997 )tounder-
stand the balance of forces. The viscous/capillary
ratio and the gravity/capillary ratio are two of a
number of dimensionless scaling group ratios that
can be determined to represent the balance of fluid
forces. For example, for an oil-water system we
can define the following force ratios:

It is important to relate these multi-phase fluid
flow processes to the heterogeneity being
modelled. This is a fairly complex issue, and
fundamental to what reservoir model design is
all about. As a way in to this topic we use the
balance of forces concept to give us a framework
for understanding which scales most affect a
particular flow process. For example, we know
that capillary forces are likely to be important for
rocks with strong permeability variations at the
small scale (less than 20 cm scale is a good rule
of thumb).
Figure 4.13 shows a simple sketch of the end-
members of the fluid force system. We have three
end members: gravity-, viscous- and capillary-
dominated. Reality will lie somewhere within the
triangle, but appreciation of the end-member
systems is useful to understand the expected
flow-heterogeneity interactions. Note, that for
the same rock system the flow behaviour will
be completely different for a gravity-dominated,
viscous-dominated or capillary-dominated flow
regime. The least intuitive is the capillary-
dominated case where water (for a water-wet
system) imbibes preferentially into the lower
permeability layers.

Viscous
Capillary ¼

μ o
k x dP c =

u x ʔ x

ð 4 : 11 Þ

ð dS Þ

Capillary ¼ ʔˁ

Gravity g ʔ z

ð 4 : 12 Þ

ð dP c = dS Þ

where,
ʔ

z are system dimensions,
u x is fluid velocity,
μ o is the oil viscosity,
k x is the permeability in the x direction,
(dP c /dS) is the slope of the capillary pressure
function,
ʔˁ is the fluid density difference and g is the
constant due to gravity.
The viscous/capillary ratio is essentially a
ratio of Darcy's law with a capillary pressure

x, ʔ