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
Chap. 4 ) , that the scale of measurement is a key
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
Þ
e
e ʼ Y .
Log-normal distributions are appealing and
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
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