Estimating Mature Plays
31
where the β value ranges from negative to positive. The larger the β
value, the greater the exploration effi ciency will be. When β = 0, the
discovery process can be considered as a random sampling process.
Therefore, the probability of observing (x 1 , … , x n ), given Y i , i = 1, … , N,
is expressed as
1
1
1
1
P
,...,
,...,
···
n
j
j
N
N
j
j
n
N
X
X
X X
X
b Y
Y
=
+
=
+
+ +
∏
b
b
b
(3.4)
where b j = x j + · · · + x n (discovered pool sizes) and Y is equal to the
undiscovered pool sizes.
The probability that the j th pool is deposited and discovered is the
product of the following two probabilities: the probability of the deposition of a pool, j, with size, x j , in the lognormal pool-size distribution,
f (x j ); and the probability of the pool j being discovered at a certain
point in the sequence. Thus, the joint density function of all discovered
pools can be shown as follows:
( ) (
)
( )
1
1
1
!
E
!
···
n
n
j
j
j
j
j
n
N
X
N
L
f X
N n
b Y
Y
=
=
+
=
−
+
++
∏
∏
b
u
u
b
b
u
(3.5)
where θ represents the distribution parameters (µ, σ
2
), the factorial
operation N!/(N – n)! is the number of ordered samples of size n without
replacement from a population of N pools, b j is equal to x j + · · · + x n (discovered pools), and y n+1 , … , y N is equal to the undiscovered pool sizes.
Quantity L(θ), which is the likelihood function of LDSCV, indicates
the likelihood of a discovery sequence. What we attempt to do here is
to reenact the exploration history. By doing so, we maximize the likelihood function by searching those values of µ, σ
2 , and N for which the
function L(θ) is maximized. The resultant L(θ) value is the maximized
log-likelihood value. This procedure is called the maximum-likelihood
method in statistics. The pool-size distribution f θ (y) can be any probability distribution, but the lognormal family is applied here. In addition, the pool size variable can be replaced by any variable, such as pool
area or net pay.
Equation 3.5 consists of two parts, f θ and E[•]. The term f θ represents
the pool-size distribution, which results from tectonics, sedimentation,
generation, migration, and accumulation of hydrocarbons, whereas
E[•] represents the manner in which pools are discovered (Fig. 3.4).
31
where the β value ranges from negative to positive. The larger the β
value, the greater the exploration effi ciency will be. When β = 0, the
discovery process can be considered as a random sampling process.
Therefore, the probability of observing (x 1 , … , x n ), given Y i , i = 1, … , N,
is expressed as
1
1
1
1
P
,...,
,...,
···
n
j
j
N
N
j
j
n
N
X
X
X X
X
b Y
Y
=
+
=
+
+ +
∏
b
b
b
(3.4)
where b j = x j + · · · + x n (discovered pool sizes) and Y is equal to the
undiscovered pool sizes.
The probability that the j th pool is deposited and discovered is the
product of the following two probabilities: the probability of the deposition of a pool, j, with size, x j , in the lognormal pool-size distribution,
f (x j ); and the probability of the pool j being discovered at a certain
point in the sequence. Thus, the joint density function of all discovered
pools can be shown as follows:
( ) (
)
( )
1
1
1
!
E
!
···
n
n
j
j
j
j
j
n
N
X
N
L
f X
N n
b Y
Y
=
=
+
=
−
+
++
∏
∏
b
u
u
b
b
u
(3.5)
where θ represents the distribution parameters (µ, σ
2
), the factorial
operation N!/(N – n)! is the number of ordered samples of size n without
replacement from a population of N pools, b j is equal to x j + · · · + x n (discovered pools), and y n+1 , … , y N is equal to the undiscovered pool sizes.
Quantity L(θ), which is the likelihood function of LDSCV, indicates
the likelihood of a discovery sequence. What we attempt to do here is
to reenact the exploration history. By doing so, we maximize the likelihood function by searching those values of µ, σ
2 , and N for which the
function L(θ) is maximized. The resultant L(θ) value is the maximized
log-likelihood value. This procedure is called the maximum-likelihood
method in statistics. The pool-size distribution f θ (y) can be any probability distribution, but the lognormal family is applied here. In addition, the pool size variable can be replaced by any variable, such as pool
area or net pay.
Equation 3.5 consists of two parts, f θ and E[•]. The term f θ represents
the pool-size distribution, which results from tectonics, sedimentation,
generation, migration, and accumulation of hydrocarbons, whereas
E[•] represents the manner in which pools are discovered (Fig. 3.4).