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.
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