Page 271 - Well Logging and Formation Evaluation

Additional Mathematics Theory 261

Also

( and C)+ ( and C¢)+ ( and C¢)+ ( and C¢) = 1 (A4.13)
P S 1 P S 2 P S 1 P S 2
Now P(C and S 1 ), the probability of both C and S 1 occurring, is given by:

(
()
P C and S 1 )+ ( P S 1 = R* P S 1
()
P C S 1 )*
and so on for the other combinations. Replacing P(C and S 1) and similar
terms in equation A4.12 yields back the same result as equation A4.7. It
is often useful to make a plot of DEMV as a function of R. In this way it
is possible to determine the value of R for which a particular data acqui-
sition campaign becomes viable.
The above concepts can obviously be extended to cover more than two
states, and specialized software is available that will allow the EMV to be
calculated relatively simply. Note that the reliability of the tool has been
defined as P(C/S 1) etc. One is also interested to know P(S 1/C), i.e., the
probability of the field being in state 1 given that the tool yields a result
C. To make this conversion it is necessary to use Bayes’ theorem. This
uses the fact that:

(
P C S 1 )*
P S C)*
()
P C and S 1 ) = ( P S 1 = ( and C) = ( 1 P C (A4.14)
()
P S 1
P C = ( 1 P S + ( 2 P S 2 (A4.15)
P C S )*
P C S )*
().
()
()
1
Combining these equations:
(
(
(
(
P SC) = [ P S 1 * P C S 1 )] [ P S 1 * P C S 1 )+ ()* P C S 2 )] (A4.16)
()
()
1
P S 2
P C S 1 )
P S 1 * R [ P S 1 * R+ () ( 1- R)], since R = ( (A4.17)
()
()
P S 2 *
Likewise:
P SC¢) = [ P S 1 * P C S 1 )] [ P S 1 * P C S 1 )+ ()* P C S 2 )] (A4.18)
(
(
(
¢
¢
¢
(
()
()
P S 2
1
() (
P S 1 * 1- R) [ P S 1 * 1- R)+ () R] (A4.19)
P S 2 *
() (
(
P SC) = [ P S 2 * P C S 2 )] [ P S 2 * P C S 2 )+ ()* P C S 1 )] (A4.20)
(
(
(
()
()
2
P S 1
() (
P S 2 *
() (
P S 1 * 1- R) [ P S 1 * 1- R)+ () R] (A4.21)