The Bayesian framework is the standard approach for data assimilation in
reservoir modeling. This framework involves characterizing the
posterior distribution of geological parameters in terms of a given
prior distribution and data from the reservoir dynamics, together with a
forward model connecting the space of geological parameters to the data
space. Since the posterior distribution quantifies the uncertainty in
the geologic parameters of the reservoir, the characterization of the
posterior is fundamental for the optimal management of reservoirs.
Unfortunately, due to the large-scale highly nonlinear properties of
standard reservoir models, characterizing the posterior is
computationally prohibitive. Instead, more affordable ad hoc techniques,
based on Gaussian approximations, are often used for characterizing the
posterior distribution. Evaluating the performance of those Gaussian
approximations is typically conducted by assessing their ability at
reproducing the truth within the confidence interval provided by the ad
hoc technique under consideration. This has the disadvantage of mixing
up the approximation properties of the history matching algorithm
employed with the information content of the particular observations
used, making it hard to evaluate the effect of the ad hoc approximations
alone. In this paper, we avoid this disadvantage by comparing the ad
hoc techniques with a fully resolved state-of-the-art probing of the
Bayesian posterior distribution. The ad hoc techniques whose performance
we assess are based on (1) linearization around the maximum a
posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble
Kalman filter-type methods. In order to fully resolve the posterior
distribution, we implement a state-of-the art Markov chain Monte Carlo
(MCMC) method that scales well with respect to the dimension of the
parameter space, enabling us to study realistic forward models, in two
space dimensions, at a high level of grid refinement. Our implementation
of the MCMC method provides the gold standard against which the
aforementioned Gaussian approximations are assessed. We present
numerical synthetic experiments where we quantify the capability of each
of the ad hoc Gaussian approximation in reproducing the mean and the
variance of the posterior distribution (characterized via MCMC)
associated to a data assimilation problem. Both single-phase and
two-phase (oil–water) reservoir models are considered so that
fundamental differences in the resulting forward operators are
highlighted. The main objective of our controlled experiments was to
exhibit the substantial discrepancies of the approximation properties of
standard ad hoc Gaussian approximations. Numerical investigations of
the type we present here will lead to the greater understanding of the
cost-efficient, but ad hoc, Bayesian techniques used for data
assimilation in petroleum reservoirs and hence ultimately to improved
techniques with more accurate uncertainty quantification.