ABSTRACT
This is a multinominal model that is a generalization of the binomial model. If we assume a prior distribution of the form
g p p p p p p B p p p p( , , ) ( )1 2 3 0 1 2 3 0 1 2 1 2 3 1 2 3 1 2 1= = − −B α α α α α α (21.2)
where α1, α2, and α3 > −1; p1 ∈ [0, 1], p2 ∈ [0, 1 − p1], and
B0
3 1 1 1
=
+ + +
+ +
Γ Γ Γ Γ
[ ] ] [ ] [ ]
α α α
α α[α +
Then, since data are available in discrete form, the posterior conditional probability can be represented as
w p p p x x x g p p p f x x x p p p
g p ( , , , , )
( , , ) ( , , , , )
| |
= ∑ p p f x x x p p p2 3 1 2 3 1 2 3, ) ( , , , , )| (21.3) and for the continuous case, we have
W p p p x x x
g p p p f x x x p p p p x x
( , , , , ) ( , , ) ( , , , , )
1 |
| =
2 3, )x (21.4)
where
P x x x g p p p f x x x p p p dp dp
( , , ) ( , , ) ( , , , , )1 2 3 0
= ∫ ∫− | = − −∫ ∫− + + +A B p p p p dp dpp x x x0 0
P x x x
x x x x x x
1 1 1 =
+ + + + + +
+ + +
A B [ ] [ ] [ ] [
Γ Γ Γ Γ
α α α
α1 2 3 3+ + +α α ] (21.5)
Then, to estimate the probabilities, Pi’, i = 1, 2, and 3 of a well discovering yi barrels of reserves, we have
ˆ [ , , ]p E p x x xi i= | 1 2 3
which implies that
ˆ ( , , , , )p p W p p p x x x dp dp p
ˆ ( , , ) ( , , , , ) ( ,
p p g p p p f x x x p p p dp dp
p x x
= ∫ ∫− |2 3, )x
pˆ x
x x x 1
1 3
=
+ +
+ + + + + +
α
α α α (21.6)
Similarly
ˆ [ , , ]p E p x x x2 2 1 2 3= |
pˆ
x x x x
1 3
=
+ +
+ + + + + +
α
α α α (21.7)
pˆ
x x x x
1 3
=
+ +
+ + + + + +
α
α α α (21.8)
Q x yi i= ∑ (21.9)
E Q W P X Qi i( ) ( )= ∑ | (21.10)
21.5 The k-category case The k-category case is a generalization of the three-category case. Pore and Dennis (1980) obtained their k-category pixel expression using Bayesian estimation procedure. Their result is, therefore, extended here to multicategory reservoir systems. Then, the probabilities (Pi), i = 1, 2, 3, …, k of a well discovering yi barrels of reserves, can be written as
ˆ ( )
p x
x i
=
+ +
+ +∑ α
α
(21.11)
Table 21.1 is the result obtained by McCray (1975) for various values of x1, x2, and x3 that can appear in a sample. Each line in the table represents one possible sample, and the probability of that particular sample is calculated using Equation 21.1. McCray (1975) assumed p1 = 0.5, p2 = 0.3, and p3 = 0.2 and their corresponding reserves are, respectively, y1 = 0, y2 = 15, and y3 = 60.
