Geoscience Reference
In-Depth Information
Given that we have a set of recipes for differ-
ent property modelling methods, how do we
combine them to make a good property model?
Remember that the mark of a good model is that
it is geologically-based and fit for purpose. To
illustrate the different approaches to property
model design we describe two approaches
based on case studies:
￿ An object-based model of channelized facies
based on detailed outcrop data;
￿ A seismic-based facies model, exploiting
good 3D seismic data.
P(B|A) is the likelihood (of what was actually
observed),
P(B) is the underlying evidence (also termed the
marginal likelihood).
This comparison of prior and posterior
probabilities may at first appear confusing, but
is easily explained using a simple example (Exer-
cise 3.5), and fuller discussion can found else-
where (e.g. Howson and Urbach 1991 ). The
essence of Bayesian estimation is that a probabi-
listic variable (the posterior) can be estimated
given some constraints (the prior). This allows
probabilistic models to be constrained by data
and observations, even when those data are
incomplete or uncertain. This is exactly what
probabilistic reservoir models need - a depen-
dence on, or conditioning to, observations.
Bayesian methods are used to condition reservoir
models to seismic, well data and dynamic data,
and are especially valuable for integrating seis-
mic and well data.
3.4.4 Bayesian Statistics
We have now reviewed the main statistical tools
employed in property modelling; however, one
important concept is missing. We argued in
Chap. 2 that reservoir modelling must find a
balance between determinism and probability,
and that more determinism is generally desirable.
Using Gaussian simulation methods without firm
control from the known data is generally unhelp-
ful and dissatisfying. We ideally want
geostatistical property models rooted in geologi-
cal concepts and conditioned to observations
(well, seismic and dynamic data), and this is
where Bayes comes in. Thomas Bayes
(1701-1761) developed a theorem for updating
beliefs about the natural world and then later his
ideas were developed and formalised by Laplace
(in Th ´orie analytique des probabilit ´s , 1812).
Subsequently, over the last 50 years Bayesian
theory has revolutionized most fields of statisti-
cal analysis, not least reservoir modelling.
Bayesian inference derives one uncertain
parameter (the posterior probability) from
another (the prior probability) via a likelihood
function. Bayes' rule states that:
PA B ¼
Exercise 3.5
Bayes and the cookie jar.
A simple example to illustrate Bayes
theory is the “cookie jar” example. There
are two cookie jars. One jar has 10 choco-
late chip cookies and 30 plain cookies,
while the second jar has 20 of each. Fred
picks a jar at random, and then picks a
cookie at random - he gets a plain one.
We all know intuitively he could have
picked from either jar, but most likely
picked from Jar 1. Use Bayes theory
Eq. ( 3.27 ) to find probability that Fred
picked the cookie from Jar 1.
The answer is 0.6 - but why?
3.4.5 Property Modelling:
Object-Based Workflow
PB A PA
ðÞ
ð
3
:
27
Þ
PB
ðÞ
where:
P(A|
Geological modelling using object-based methods
was explained in Chap. 2 . The geological objects
(i.e. model elements such as channels, bar
forms, or sheet deposits) need petrophysical
properties to be defined. This could be done in a
) is the posterior - the probability of A
assuming B is observed,
P(A) is the prior - the probability of A before B
was observed,
Β
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