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Bayes' Theorem and Markov Chain Theory
Also see:
Bayes' theorem and Markov Chain theory are
increasingly used to treat the problems of uncertainty quantification and
optimization in reservoir characterization and simulation (search One Petro
for "Bayes
Theorem" and "Markov
Chain").
The Petroleum Engineering Handbook does not define
Bayes' theorem.
From
Bayes' Theorem Definition | Investopedia:
"The formula is written as P(A|B) = P(B|A) * P(A) /
P(B). P(A) and P(B) are the probabilities of A and B without regard to each
other. P(B|A) is the probability that B will occur given A is true. Finally,
the answer, P(A|B) is the conditional probability of A occurring given B is
true."
The condition applying to the theorem is P(B) > 0.
In reservoir
characterization and simulation and all applications such as reserves
estimation, there is no such thing as an anterior or a posterior
probability and Bayes' theorem does not apply, because P(A) and P(B) are both
0, due to the very large number of unknown variables. That number is
generally equal to many times the numbers of gridblocks and wells.
For any given real input in a simulation model
of a real reservoir, what is the probability
that it is exactly correct? (0) I'm thinking of a real number between 0 and
1. What is the probability that anyone
can guess what it is? (0) What is the probability that any given reservoir
realization is correct? (0). If a given realization is found to better match
history, what is it's probability of being correct? (0) If a given parameter
in a given realization is adjusted to better match history, what is the
probability that the right parameter to adjust is selected, and that
the correct adjustment is applied to it? (0)
Markov chain theory defines the
probability of a modified realization from the probability of an existing
one. Since those probabilities are generally both 0 in reservoir
simulation due to very large numbers of uncertainties, the theory does not apply.
Therefore, to answer any question
by reservoir
characterization and modeling when there is significant uncertainty in the
inputs, some number of equally (im)probable realizations must be used in probabilistic analysis. |