Reservoir Model Types - Reservoir Model Design - page 181

Observations

x y

?

?

Realisation 1 Realisation 2

x y y

x

Fig. 6.12 Simple illustration of the sand connectivity problem

Percolation theory, widely used in many
branches of applied physics, describes connectiv-
ity in a statistical network using probability the-
ory. To summarise the concept, it has been found
that by adding conducting elements randomly in
a non-conductive network or lattice, connectivity
occurs (statistically) when a predictable number
of nodes or sites are filled. This point is the
percolation threshold, p c . The value for p c
depends on the dimensions and geometry of the
system being considered. The theory is applied to
a wide range of physical phenomena (de Gennes
1976 ) and has been widely applied in subsurface
flow studies (e.g. Stauffer and Ahorony 1994 ).
King ( 1990 ) showed how the theory can be
applied to overlapping sand bodies in reservoir
characterisation studies and Table 6.1 shows
some example percolation thresholds.
When the theory is applied to permeability
(e.g. Deutsch 1989 ; King 1990 ; Renard and de
Marsily 1997 ) we find that the effective perme-
ability, k eff , in such a system follows a power law
defined by p c :

Table 6.1 Some example percolation thresholds

Percolation
threshold System References

Square Lattice (bond
percolation)

0.5000 Stauffer and

Ahorony
( 1994 )

Simple cubic lattice
(site percolation)

0.3116 Stauffer and

Ahorony
( 1994 )

Simple cubic lattice
(bond percolation)

0.2488 Stauffer and

Ahorony
( 1994 )

Overlapping sandstone
objects (rectangles in
2D)

~0.667 King ( 1990 )

Overlapping sandstone
objects (boxes in 3D)

~0.25 King ( 1990 )

Multiple stochastic
models of intersecting
sinuous channels

~0.2 to ~0.6 Larue and

Hovadik
( 2006 )

The simple case of 2D overlapping sand bod-
ies is illustrated in Fig. 6.13 (based on results
from King 1990 ). For more realistic systems,
the problem is how the constants are to be
estimated. However, all object-based geological
reservoir models will tend to exhibit
characteristics related to percolation phenomena,
and it

For p < p c k eff ¼ 0

e

For p > p c k eff ¼ Ap ð p c Þ

where A and e are characteristic constants. is useful to establish the expected