Summary
The paper presents a novel method for rapid quantification of uncertainty in
history matching reservoir models using a two-stage Markov Chain Monte Carlo
(MCMC) method. Our approach is based on a combination of fast linearized
approximation to the dynamic data and the MCMC algorithm. In the first stage,
we use streamline-derived sensitivities to obtain an analytical approximation
in a small neighborhood of the previously computed dynamic data. The
sensitivities can be conveniently obtained using either a finite-difference or
streamline simulator. The approximation of the dynamic data is then used to
modify the instrumental proposal distribution during MCMC. In the second stage,
those proposals that pass the first stage are assessed by running full flow
simulations to assure rigorousness in sampling. The uncertainty analysis is
carried out by analyzing multiple models sampled from the posterior
distribution in the Bayesian formulation for history matching. We demonstrate
that the two-stage approach increases the acceptance rate, and significantly
reduces the computational cost compared to conventional MCMC sampling without
sacrificing accuracy. Finally, both 2D synthetic and 3D field examples
demonstrate the power and utility of the two-stage MCMC method for history
matching and uncertainty analysis.
Introduction
Uncertainty exists inherently in dynamic reservoir modeling because of
several factors, the primary ones being the modeling error, data noise, and the
nonuniqueness of the inverse problems that causes several models to fit the
dynamic data. Under a Bayesian framework, the uncertainty in the reservoir
models can be evaluated by a posterior probability distribution, which is
proportional to the product of a likelihood function and a prior probability
distribution of the reservoir model. To quantify the uncertainty, it is
necessary to generate a sequence of model realizations that are sampled
appropriately from the posterior distribution. Rigorous sampling methods, such
as Markov Chain Monte Carlo (MCMC) (Oliver et al. 1997; Robert and Casella
1999), provide the accurate sampling albeit at a high cost because of their
high rejection rates and the need to run a full flow simulation for every
proposed candidate. There is also additional cost associated with a burn-in
time needed for the MCMC to assure that the starting state does not bias
sampling. Approximate sampling methods, such as randomized maximum likelihood
(RML) (Oliver et al. 1996; Kitanidis 1995), are commonly used to avoid the high
cost associated with the MCMC methods. For linear problems (Gaussian posterior
distributions), RML has an acceptance probability of unity; however, the
assumptions made in RML may be too restrictive for nonlinear problems, which is
typically the case for reservoir history matching. The main appeal of RML is
its computational efficiency and ease of implementation within the framework of
traditional automatic history matching via minimization. There are also some
examples in the literature that the RML has favorable sampling properties for
nonlinear problems (Liu et al. 2001), although it is likely to be
problem-specific. There is a need for an efficient and rigorous approach to
uncertainty quantification for general nonlinear problems related to history
matching.
We propose a two-stage MCMC approach for quantifying uncertainty in history
matching geological models. Our proposed sampling approach is computationally
efficient with a significantly higher acceptance rate compared to traditional
MCMC algorithms. In the first stage, we compute the acceptance probability for
a proposed change in reservoir parameters based on a fast linearized
approximation to flow simulation in a small neighborhood of the previously
computed dynamic data. In this stage, no reservoir simulations are needed to
explore the model parameter space. In the second stage, those proposals that
passed a selected criterion of the first stage are assessed by running full
flow simulations to assure the rigorousness in sampling. Then, these samples
are either rejected or accepted using the MCMC selection criterion. It can be
shown that the modified Markov chain converges to a stationary state
corresponding to the posterior distribution. Moreover, the two-stage approach
increases the acceptance rate, and reduces the computational cost required for
the MCMC sampling. To propose MCMC samples, we consider two instrumental
probability distributions, the random walk sampler and the Langevin sampler
(Robert and Casella 1999). Both 2D synthetic and 3D field examples demonstrate
that the two-stage MCMC method is computationally more efficient than the
conventional MCMC methods, but does not sacrifice their accuracy. The proposed
method has been successfully used in conjunction with single-phase upscaling
methods (Efendiev et al. 2005). All examples in the paper are based on the
fine-scale geological models.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
28 June 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
20 July 2007
- Manuscript approved:
7 August 2007
- Version of record:
20 March 2008
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