Summary
The well known PUNQ-S3 reservoir model represents a synthetic problem which
was formulated to test the ability of various methods and research groups to
quantify the uncertainty in the prediction of cumulative oil production.
Previous results reported on this project suggest that the randomized maximum
likelihood (RML) method gives a biased characterization of the uncertainty. A
major objective of this paper is to show that this is incorrect. With a correct
implementation of the RML method within a Bayesian framework, we show that RML
does an adequate job of sampling the a posteriori distribution for the PUNQ
problem. In particular, the true predicted oil production lies within the band
of predictions generated with the RML method and is not biased. We also apply
the ensemble Kalman Filter (EnKF) method to the PUNQ data set, and show that
this method also gives a reasonable quantification of the uncertainty in
performance predictions with an uncertainty range similar to the one obtained
with RML.
Introduction
We consider conditioning models to production data in a Bayesian framework
and wish to generate a suite (ensemble) of models which represent a correct
sampling of the conditional probability density function (pdf). By predicting
future reservoir performance with each realization, we obtain a
characterization of the uncertainty in predicted performance. Both the
rejection algorithm and Markov chain Monte Carlo (MCMC) are theoretically sound
sampling procedures, but they are too computationally inefficient for practical
applications (Liu and Oliver 2003). Oliver et al. (1996) and Kitanidis (1986)
independently proposed the randomized maximum likelihood (RML) method to
generate an approximate sampling of the a posteriori pdf. Two different proofs
(Oliver 1996; Reynolds et al. 1999) have been presented which show that the RML
method samples the posterior probability density function (pdf) correctly if
data are linearly related to the model; however, no rigorous theoretical
foundation exists for the method when the relation between data and model is
nonlinear, which is the case when the data represent production data.
Computational results indicate that the RML method generates reasonable
characterization of uncertainty for single-phase flow (Oliver et al. 1996;
Reynolds et al. 1999; Liu and Oliver 2003). Our first objective is to show
that, contrary to a previous claim (Floris 2001), RML gives a reasonable
characterization of the uncertainty in predicted performance for the PUNQ-S3
problem; our second objective is to compare the quantification of uncertainty
obtained with RML with the one obtained with the ensemble Kalman filter
(EnKF).
The PUNQ-S3 reservoir represents a synthetic model based on an actual
reservoir (Floris et al. 2001; Barker et al. 2001). The problem was set up as a
test case to allow various research groups to test their own methodology for
the characterization of the uncertainty in reservoir performance predictions
given some geologic information on the reservoir, hard data at well gridblocks
and some scattered production data from the first 8 years of production. Then
participants were asked to predict cumulative oil production for 16.5 years of
total production and characterize the uncertainty in this prediction.
© 2006. Society of Petroleum Engineers
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History
- Original manuscript received:
7 December 2004
- Revised manuscript received:
18 May 2006
- Manuscript approved:
3 July 2006
- Version of record:
20 December 2006
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