Geoscience Reference
In-Depth Information
Table 3.2
Statistics for a simple example of estimation
of the sand volume fraction, V
s
, in a reservoir unit at
different stages of well data support
With 2 wells With 5 wells With 30 wells
in using the available data. To put it simply,
variance
refers to the spread of the data you
have (in front of you), while
uncertainty
refers
to some unknown variability beyond the infor-
mation at hand. From probability theory we can
establish that 'most' values lie close to the mean.
What we want to know is 'how close' - or how
sure we are about the mean value. The funda-
mental difficulty here is that the true (population)
mean is unknown and we have to employ the
theory of confidence intervals to give us an esti-
mate. Confidence limit theory is treated well in
most topics on statistics; Size (
1987
) has a good
introduction.
Chebyshev's inequality
gives us the theoretical
basis (and mathematical proof) for quantifying
how many values lie within certain limits. For
example, for a Gaussian distribution 75 % of the
values are within the range of
Mean
38.5
36.2
37.4
˃
4.9
6.6
7.7
SE
3.5
3.0
1.4
C
v
-
0.18
0.21
N
0
-
3
4
N
2
5
30
data is some limited subset of the true subsurface,
and the model should extend from the data in
order to make estimates of the true subsurface. In
terms of set theory:
Data
Model
Truth
∈
∈
Our models should be consistent with that data
(in that they encompass it) but should aim to cap-
ture a wider range, approaching reality, using both
geological concepts and statistical methods. In
fact, as we shall see later (in this section and in
Sect.
3.4
) bias in the data sample and upscaling
transforms further complicate this picture whereby
the data itself can be misleading.
Table
3.2
illustrates this principle using the
simple case of estimating the sand volume frac-
tion, V
s
(or N/G
sand
), at different stages in a field
development. We might quickly infer that the 30
well case gives us the most correct estimate and
that the earlier 2 and 5 well cases are in error due to
limited sample size. In fact, by applying the N-zero
statistic (explained below) we can conclude that
the 5-well estimate is accurate to within 20 % of
the truemean, and that by the 30-well stage we still
lie within the range estimated at the 5-well stage.
In other words, it is better to proceed with a realis-
tic estimate of the range in V
s
from the available
data than to assume that the data you have gives the
“correct” value. In this case, V
s
¼
two standard
deviations
from the mean. Stated
simply
Chebyshev's theory gives:
1
ʺ
Px
ð
j
ʼ
j ʺ˃
Þ
ð
3
:
13
Þ
2
where
is the number of standard deviations.
The
standard error
provides a simple measure
of uncertainty. If we have a sample from a popu-
lation (assuming a normal distribution and statis-
tically independent values), then the standard
error of the mean value,
x
, is the standard devia-
tion of the sample divided by the square root of
the sample size:
ʺ
SE
x
¼
˃
s
p
n
ð
3
:
14
Þ
˃
s
is the standard deviation of the sample
and n is the sample size.
The standard error can also be used to calculate
confidence intervals. For example, the 95 % con-
fidence interval is given by (
x
where
96).
The
Coefficient of Variation
,C
v
,sa
normalized measure of the dispersion of a proba-
bility distribution, or put simply a normalised
standard deviation:
SE
x
1
:
7%
constitutes a good model at the 5-well stage in this
field development.
36 %
p
Var p
3.3.2 Variance and Uncertainty
ðÞ
C
V
¼
ð
3
:
15
Þ
Ep
ðÞ
There are a number of useful measures that can
guide the reservoir model practitioner in gaining
a realistic impression of the uncertainty involved
where Var(p) and E(p) are the variance and
expectation of the variable, p.
