Page 135 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 5

In terms of the oil-field distribution problem we have just considered, Y is the
number of discovery wells in a tract, p is the probability that a given drilling site
contains a discovery well, and k is a measure of the degree of clustering of the
discoveries. If k is large, clustering is less pronounced and the spatial distribution
approaches the Poisson, or randomness. As k approaches zero, the pattern of
clustering becomes more pronounced. The density, A, is equal to
h = kp (5.20)

If k is not an integer (and in general it will not be), this combinatorial equation
cannot be solved. Then, the following approximation must be used:

(5.21)

As with the Poisson distribution, h is estimated by the average density of discoveries
per tract, m/T. The clustering parameter, k, is estimated by

(5.22)

where s2 is the variance in number of discovery wells per tract. Then, the probability
p can be estimated as
(5.23)

We can apply the negative binomial model to the data on discovery wells in
the eastern part of the Permian Basin (Fig. 5-6) to see if this distribution can ade-
quately describe their spatial distribution. The mean and variance of the number
of discovery wells per tract have already been found: m/T = .1.05 and s2 = 1.46.
The clustering effect can be estimated using Equation (5.22)
1.05*
k= = 2.69
1.46 - 1.05
In turn, the probability of a discovery well occurring in a tract is

p=-- "05 - 0.390
2.69
Using the approximation equations, the probability that a given tract will con-
tain no discovery wells is
1
P(0) = = 0.4124
(1 + 0.390)2.69
The probability that a tract will contain exactly one discovery well is

308