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