Geoscience Reference

In-Depth Information

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