Estimating Mature Plays
43
Let x j be the value associated with the jth discovery, and the observed
ordered sample be denoted by (x 1 , … , x n ). Then the probability of
observing the ordered sample x 1 , … , x n under the successive sampling
discovery model will be
(
)(
)
( )
( )
( )
1
1
1
1
P
,...,
,...,
···
n
j
n
N
j
j
n
N
w x
x
x
y
y
b w y
w y
=
+
=
+
+ +
∏
(3.8)
where b j = w(x j ) + · · · + w(x n ). In other words, the sample is obtained
by selecting successively, without replacement and with a probability
proportional to w( y), from the fi nite population of N pools.
To obtain the unconditional distribution of the random variable
(X 1 , … , X n ), we have to sum Equation 3.8 over all possible {y 1 , … , y N }
values, multiply by the joint density of random variable Y 1 , … , Y N , and
integrate over the unobserved values (Y n+1 , … , Y N ). This gives the joint
density of X j = x i , i = 1, … , n, as
(
)
( )
( )
( )
( )
1
1
1
!
E
!
···
n
n
j
j
j
j
j
n
N
w x
N
f x
N n
b w Y
w Y
=
=
+
−
+
++
∏
∏
u
u
(3.9)
Equation 3.9 is the multivariate case of Equation 3.5 and was implemented in PETRIMES/W. The following subsections contain two
examples that demonstrate the applications of Equation 3.9.
Bivariate Lognormal Distribution for Oil and Gas Pools
So far, there are three ways to evaluate a trap that contains both oil
and gas:
Assess the oil and gas separately.
1.
Convert the gas into oil-equivalent volume and add it to the oil
2.
volume of the same pool.
Compute the trap volume for oil and gas together.
3.
The fi rst method presents two assessments, one for oil and one for
gas. The second method reports an assessment for oil equivalent. The
third method predicts the trap volume.
Equation 3.9 can also be used to estimate the oil and gas joint distribution (Lee, 1998). The following is such an example. The discovery
sequence of the Leduc isolated reef play (Fig. 3.9) from the Western
43
Let x j be the value associated with the jth discovery, and the observed
ordered sample be denoted by (x 1 , … , x n ). Then the probability of
observing the ordered sample x 1 , … , x n under the successive sampling
discovery model will be
(
)(
)
( )
( )
( )
1
1
1
1
P
,...,
,...,
···
n
j
n
N
j
j
n
N
w x
x
x
y
y
b w y
w y
=
+
=
+
+ +
∏
(3.8)
where b j = w(x j ) + · · · + w(x n ). In other words, the sample is obtained
by selecting successively, without replacement and with a probability
proportional to w( y), from the fi nite population of N pools.
To obtain the unconditional distribution of the random variable
(X 1 , … , X n ), we have to sum Equation 3.8 over all possible {y 1 , … , y N }
values, multiply by the joint density of random variable Y 1 , … , Y N , and
integrate over the unobserved values (Y n+1 , … , Y N ). This gives the joint
density of X j = x i , i = 1, … , n, as
(
)
( )
( )
( )
( )
1
1
1
!
E
!
···
n
n
j
j
j
j
j
n
N
w x
N
f x
N n
b w Y
w Y
=
=
+
−
+
++
∏
∏
u
u
(3.9)
Equation 3.9 is the multivariate case of Equation 3.5 and was implemented in PETRIMES/W. The following subsections contain two
examples that demonstrate the applications of Equation 3.9.
Bivariate Lognormal Distribution for Oil and Gas Pools
So far, there are three ways to evaluate a trap that contains both oil
and gas:
Assess the oil and gas separately.
1.
Convert the gas into oil-equivalent volume and add it to the oil
2.
volume of the same pool.
Compute the trap volume for oil and gas together.
3.
The fi rst method presents two assessments, one for oil and one for
gas. The second method reports an assessment for oil equivalent. The
third method predicts the trap volume.
Equation 3.9 can also be used to estimate the oil and gas joint distribution (Lee, 1998). The following is such an example. The discovery
sequence of the Leduc isolated reef play (Fig. 3.9) from the Western