Geoscience Reference

In-Depth Information

the mean of a given sample dataset can be made

using confidence interval theory (e.g. Isaaks and

Srivastava
1989
; Jensen et al.
2000
).

This analysis gives a useful framework for

judging how variable your reservoir data really

is. Note that more than half the datasets included

in Fig.
3.15
are heterogeneous or very heteroge-

nous. Carbonate reservoirs and highly laminated

or inter-bedded formations show the highest C
v

values. This plot should in no way be considered

as definitive for reservoirs for any particular

depositional environment. We shall see later (in

factor within essentially multi-scale geological

reservoir systems. Also keep in mind that your

dataset may be too limited to make a good assess-

ment of the true variability - the C
v
from a

sample dataset is an estimate. Jensen et al.

(
2000
) give a fuller discussion of the application

of the C
v
measure to petrophysical reservoir data.

The function is also fundamental to a wide range

of statistical methods and the basis for most

geostatistical modelling tools. It is also important

to say that many natural phenomena do not con-

form to the Gaussian distribution - they may, for

example, be better approximated by a another

function such as the Poisson distribution and in

geology have a strong tendency to be more com-

plex and multimodal.

Permeability data is often found to be

approximated by a log-normal distribution. A

variable X is
log-normally distributed
if its

natural logarithmic transform Y is normally

distributed with mean

ʼ
Y
and standard deviation

˃
Y
2
. The probability density function for X is

given by:

2

½

ln
ðÞʼ
Y

1

2

˃
Y
x
e

Y

fx

ðÞ
¼

p

2

˃

if x

>

0

ð

3

:

19

Þ

ˀ

˃
X
2
are related

to the log transform parameters

The variable statistics,

ʼ
X
and

˃
Y
2

ʼ
Y
and

as follows:

2

Y

e
ʼ
Y
þ
0
:
5
˃

ʼ
X
¼

ð

3

:

20a

Þ

3.3.3 The Normal Distribution

and Its Transforms

2

Y

2

2

X

e
˃

˃

X
¼
ʼ

1

ð

3

:

20b

Þ

Probability theory is founded in the properties of

the Normal (or Gaussian) Distribution. A vari-

able X is a
normal random variable
when the

probability density function
is given by:

This can lead to some confusion, and it is

important that the reservoir modelling practi-

tioner keeps close track of which distributions

relate to which statistics. For
˃
¼ 0 the mean

obeys the simple law of the log transform,

ʼ
x
=
e
ʼ
Y
, but generally

1

2

ð

xʼ

Þ

e

e
ʼ
Y
.

Log-normal distributions are appealing and

useful because (a) they capture a broad spread

of observations in one statistic, and (b) they are

easily manipulated using log transforms. How-

ever, they also present some difficulties in reser-

voir modelling:

They tend to generate excessive distribution

tails

It is tempting to apply them to multi-modal

data

They can cause confusion (e.g. what is the

average?)

Note that the correct 'average' for a log-

normal distribution of permeability is the geo-

metric average - equivalent to a simple average

of ln (k) - but this does not necessarily mean that

gx

ðÞ
¼

p

2

ð

:

Þ

ð

˃ >

0

Þ

,

ʼ
x
>

3

18

2

2
˃

˃

ˀ

2
is the variance.

This bell shaped function is completely deter-

mined by the mean and the variance. Carl

Friedrich Gauss became associated with the

function following his analysis of astronomical

data (atmospheric scatter from point light

sources), but the function was originally pro-

posed by Abraham de Moivre in 1733 and devel-

oped by one of the founders of mathematics,

Pierre-Simon de Laplace in his topic
Analytical

Theory of Probabilities
in 1812. Since that time,

a wide range of natural phenomena in the

biological and physical sciences have been

found to be closely approximated by this distri-

bution - not least measurements in rock media.

where

ʼ

is the mean and

˃