34
Statistical Methods for Estimating Petroleum Resources
Unfortunately, the n pools are not a random sample, but a biased sample from the play. Therefore, the statistical estimation of p i requires use
of LDSCV, as described earlier. On the other hand, with the discovery
process model and the underlying empirical superpopulation distribution, p i can also be estimated without making any assumptions about
its shape, such as lognormal distribution.
As we have discussed for LDSCV, the likelihood function can adopt
any probability distribution, such as a Weibull or Pareto distribution.
Each distribution, however, would require a specifi c numerical algorithm to solve the likelihood function. Consequently, as a logical extension of the lognormal model, the birth of the nonparametric discovery
process (NDSCV) ensued. It is used in the following ways:
To estimate the empirical pool-size distribution and
•
N nonparametrically
To provide estimates of
•
p i to validate distributional assumptions
To act as a validation tool for LDSCV
•
The statistical treatment of NDSCV is explained in Appendix B.
Estimating Pool-Size Distribution for the Beaverhill Lake Play
The in-place oil volumes and their discovery dates for the Beaverhill
Lake play data set (shown in Fig. 2.9) were entered into LDSCV and
NDSCV. The number of discoveries (sample size) equals 92. This data
set includes commercial as well as noncommercial pools, with the
smallest pool size equal to 0.001 MMbbls (1000 bbls).
Table 3.1, column 1, lists all the N values. For each N value, the
values of µ, σ
2 , β, and the log likelihood were estimated by LDSCV
(columns 2 to 5) and by NDSCV (columns 6 to 9). The curve of log L
versus N derived by both models increases rapidly (Fig. 3.5), but when
N > 400, both curves increase slowly. On the other hand, if we examine
the estimates from N = 400 to 500, we can visualize that by increasing
the value of N, the number of small pools increases rapidly, whereas
the number of pools for the midsize classes increases slowly. The point
estimates for µ and σ
2 derived from both models when N = 400 (Table 3.1)
are used in the matching process. From the estimated µ and σ
2 , the
corresponding lognormal and/or empirical pool-size probability distribution can be generated.
Figure 3.6 displays the estimated pool-size distributions derived
from LDSCV (Fig. 3.6, line A, β ˆ = 0.4 ) and NDSCV (Fig. 3.6, line B,
Statistical Methods for Estimating Petroleum Resources
Unfortunately, the n pools are not a random sample, but a biased sample from the play. Therefore, the statistical estimation of p i requires use
of LDSCV, as described earlier. On the other hand, with the discovery
process model and the underlying empirical superpopulation distribution, p i can also be estimated without making any assumptions about
its shape, such as lognormal distribution.
As we have discussed for LDSCV, the likelihood function can adopt
any probability distribution, such as a Weibull or Pareto distribution.
Each distribution, however, would require a specifi c numerical algorithm to solve the likelihood function. Consequently, as a logical extension of the lognormal model, the birth of the nonparametric discovery
process (NDSCV) ensued. It is used in the following ways:
To estimate the empirical pool-size distribution and
•
N nonparametrically
To provide estimates of
•
p i to validate distributional assumptions
To act as a validation tool for LDSCV
•
The statistical treatment of NDSCV is explained in Appendix B.
Estimating Pool-Size Distribution for the Beaverhill Lake Play
The in-place oil volumes and their discovery dates for the Beaverhill
Lake play data set (shown in Fig. 2.9) were entered into LDSCV and
NDSCV. The number of discoveries (sample size) equals 92. This data
set includes commercial as well as noncommercial pools, with the
smallest pool size equal to 0.001 MMbbls (1000 bbls).
Table 3.1, column 1, lists all the N values. For each N value, the
values of µ, σ
2 , β, and the log likelihood were estimated by LDSCV
(columns 2 to 5) and by NDSCV (columns 6 to 9). The curve of log L
versus N derived by both models increases rapidly (Fig. 3.5), but when
N > 400, both curves increase slowly. On the other hand, if we examine
the estimates from N = 400 to 500, we can visualize that by increasing
the value of N, the number of small pools increases rapidly, whereas
the number of pools for the midsize classes increases slowly. The point
estimates for µ and σ
2 derived from both models when N = 400 (Table 3.1)
are used in the matching process. From the estimated µ and σ
2 , the
corresponding lognormal and/or empirical pool-size probability distribution can be generated.
Figure 3.6 displays the estimated pool-size distributions derived
from LDSCV (Fig. 3.6, line A, β ˆ = 0.4 ) and NDSCV (Fig. 3.6, line B,