Geoscience Reference
In-Depth Information
a
Good approximately log-normal dataset
Well data
Model
Assume a log-normal
distribution and apply
ln(x) transform during
modelling
0.1
0.1
F
F
Use Normal Score
or Box-Cox transform
to ensure data is
honoured during
modelling
0
0
Permeability (md)
Permeability (md)
b
Poor non-Guassian dataset
Well data
Model
Fit a Gausian
distribution using a
transform or user
judgement
0.1
0.1
F
F
Sub-divide data into
several Gaussian
distributions based on
geological knowledge
0
0
Permeability (md)
Permeability (md)
Fig. 3.16
Illustration of data-to-model transforms for (
a
) a well-sampled dataset, and (
b
) a poorly-sampled datasets
this is the “correct” average for flow calculations.
In fact, for a layered model with layer values d-
rawn from a log-normal distribution the layer-p-
arallel effective permeability is given by the
arithmetic average (see previous section).
There are several useful transforms other than
the log-normal transform. The Box-Cox trans-
form (Box and Cox
1964
), also known as a power
transform, is one of the most versatile and is
essentially a generalisation of the log normal
transform. It is given by:
form of ranking (Deutsch and Journel
1992
;
Journel and Deutsch
1997
). This is done using a
cumulative distribution function (cdf) where
each point on the cdf is mapped into a standard
normal distribution using a transform (the score).
There are several ways of doing this but the most
common (and simple) is the linear method in
which a linear factor is applied to each step
(bin) of the cumulative distribution (for a fuller
explanation see Soares
2001
). This allows any
arbitrary distribution to be represented and
modelled in a geostatistical process (e.g. Sequen-
tial Gaussian Simulation). Following simulation,
the results must be back transformed to the origi-
nal distribution.
These transforms are illustrated graphically in
Fig.
3.16
. We should add an important note of
caution when selecting appropriate transforms in
any reservoir modelling exercise. It may be
tempting to allow default transforms in a given
modelling package (notably the NST) to auto-
matically handle a series of non-Gaussian input
data (e.g. Fig.
3.16b
). This can be very misleading
and essentially assumes that your data cdf's are
x
ʻ
1
x
ðÞ
¼
if
ʻ 6
¼
0
ð
3
:
21
Þ
ʻ
x
ðÞ
¼
ln
x
ðÞ
if
ʻ
¼
0
where the power
determines the transformed
distribution x
(
ʻ
)
. The square-root
ʻ
transform is
given by
ʻ
¼
½ and for
ʻ
¼
0 the transform is
the log-normal transform.
Another transform widely used in reservoir
property modelling is the
normal score transform
(NST)
in which an arbitrary distribution is
transformed into a normal distribution, using a
