Page 270 - Intelligent Digital Oil And Gas Fields

220 Intelligent Digital Oil and Gas Fields

where C D is the data covariance matrix. The relationship between the data
and the model parameters is expressed as a nonlinear function that maps the
model parameters into the data space, d5g[m], where d is the data vector
with N observations representing the output of the model, m is a vector of
size M whose elements are the model parameters, and g is the forward model
operator, a function that relates the model parameters to the output. For
history-matching problems, g represents the reservoir simulator.
Using Bayes’ inference (Eq. 6.8), the posterior pdf can be defined as
follows:
p m|d mj dÞ∝ exp O m (6.11)
ð
½
ðފ
where O(m) represents the Bayesian OF in multi-Gaussian notation:
1 T 1 1 0 T 1 0

O mðÞ ¼ ð d gmðÞÞ C ð d gmðÞÞ + m m C M m m
d
2 2
(6.12)
The first right-hand term of Eq. 6.12 represents the data misfit term (mis-
match between observed data and simulated response), while the second
right-hand term corresponds to a regularization term, usually represented
by the prior/known geomodel. The result of minimizing the O(m) (Eq.
6.12) is called the Maximum-A-Posteriori (MAP) estimate because it rep-
resents the most likely posterior model. The set of parameters m that min-
imizes O(m) is the most probable estimate.
The goal of the Bayesian approach is to derive a statistical distribution
for the model parameters via posterior distribution, constrained through
the prior distribution. Because the reservoir history-matching problem is
an inverse problem, its solution renders multiple plausible models (i.e.,
multiple realizations), and the consequence of nonlinearity is that one
must resort to an iterative solution. The MAP estimate is often insufficient
because it does not provide the uncertainty quantification in the posterior
model. As a solution, Kitanidis (1995) and Oliver et al. (1996) introduced
the randomized maximum likelihood (RML) method which provides a
theoretically rigorous approach to sample from the posterior distribution
but holds only for linear Gaussian problems, which are seldom the case
for reservoir simulation model inversion. Several alternative approaches
for a rigorous sampling from posterior distribution for nonlinear problems
have been proposed, such as traditional Markov chain Monte Carlo
(McMC) (Neal, 1993), its multistage implementation with enhancements