28
Statistical Methods for Estimating Petroleum Resources
obtained from exploration. Taking a lognormal distribution as an
example, if the parameters—mean and variance—have been estimated, all the upper percentiles or the density of the distribution can
then be generated.
We shall now discuss the principle of petroleum resource estimation
from a statistical point of view. In cases in which the discovery data for
a play come from a random sample or, alternatively, if all the discoveries have been made, the sample mean and variance adequately represent the population. However, in reality, discovery is infl uenced by
many factors, including exploration techniques, drilling technology,
acreage availability, and company objectives. Furthermore, geologists tend to test what is perceived to be the best or largest prospect,
which might not be the largest pool of the play. Testing fi rst for the
best prospect tends to characterize the discovery process as a sampling
procedure (as was indicated in Figure 2.9, which shows that discovered
pool size gradually decreases with time). However, variations from that
trend, or “waves,” occur during the course of exploration. We are then
faced with the question of how to use these types of biased samples to
estimate the population. For the superpopulation model, a lognormal
pool-size distribution is defi ned as
( )
2
1
l n
1
exp 2
2
x
f x
x
−
=
−
u
m
s
s p
(3.1)
for x > 0, where θ = (µ, σ
2
) is the population parameter to be estimated.
Examples of lognormal distribution shapes are presented in Figure 3.1.
Here, µ is the mean of the population of logarithmic pool sizes and σ
2 is
the variance of the population, n is the sample size (i.e., number of discoveries), and N is the total number of pools (discovered and undiscovered) in a play. The N value is also an unknown value to be estimated.
A fi nite population was created from a random sample of size 300
(N = 300) drawn from the lognormal population with parameters µ = 3.0
and σ
2 = 5.0. The histogram of the lognormal population (Fig. 3.2)
exhibits a J-shaped distribution (the term J-shaped is used to describe
a distribution monotonically increasing toward its left side) if an arithmetic scale is used for the horizontal axis. On the other hand, an almost
symmetrical pattern results when a logarithmic scale is applied.
The estimation is based on the principle that the probability of
discovering a pool is proportional to its size, and that a pool will not
be discovered twice (Barouch and Kaufman, 1977; Kaufman, 1963;
Kaufman et al., 1975). For the sake of simplicity, the concept of the
Statistical Methods for Estimating Petroleum Resources
obtained from exploration. Taking a lognormal distribution as an
example, if the parameters—mean and variance—have been estimated, all the upper percentiles or the density of the distribution can
then be generated.
We shall now discuss the principle of petroleum resource estimation
from a statistical point of view. In cases in which the discovery data for
a play come from a random sample or, alternatively, if all the discoveries have been made, the sample mean and variance adequately represent the population. However, in reality, discovery is infl uenced by
many factors, including exploration techniques, drilling technology,
acreage availability, and company objectives. Furthermore, geologists tend to test what is perceived to be the best or largest prospect,
which might not be the largest pool of the play. Testing fi rst for the
best prospect tends to characterize the discovery process as a sampling
procedure (as was indicated in Figure 2.9, which shows that discovered
pool size gradually decreases with time). However, variations from that
trend, or “waves,” occur during the course of exploration. We are then
faced with the question of how to use these types of biased samples to
estimate the population. For the superpopulation model, a lognormal
pool-size distribution is defi ned as
( )
2
1
l n
1
exp 2
2
x
f x
x
−
=
−
u
m
s
s p
(3.1)
for x > 0, where θ = (µ, σ
2
) is the population parameter to be estimated.
Examples of lognormal distribution shapes are presented in Figure 3.1.
Here, µ is the mean of the population of logarithmic pool sizes and σ
2 is
the variance of the population, n is the sample size (i.e., number of discoveries), and N is the total number of pools (discovered and undiscovered) in a play. The N value is also an unknown value to be estimated.
A fi nite population was created from a random sample of size 300
(N = 300) drawn from the lognormal population with parameters µ = 3.0
and σ
2 = 5.0. The histogram of the lognormal population (Fig. 3.2)
exhibits a J-shaped distribution (the term J-shaped is used to describe
a distribution monotonically increasing toward its left side) if an arithmetic scale is used for the horizontal axis. On the other hand, an almost
symmetrical pattern results when a logarithmic scale is applied.
The estimation is based on the principle that the probability of
discovering a pool is proportional to its size, and that a pool will not
be discovered twice (Barouch and Kaufman, 1977; Kaufman, 1963;
Kaufman et al., 1975). For the sake of simplicity, the concept of the