Response Surface Methodology Approach for History Matching and Uncertainty Assessment of Reservoir Simulation Models
- Per Arne Slotte (Statoil) | Eivind Smorgrav (Statoil)
- Document ID
- Society of Petroleum Engineers
- Europec/EAGE Conference and Exhibition, 9-12 June 2008, Rome, Italy
- Publication Date
- Document Type
- Conference Paper
- 2008. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 5.6.4 Drillstem/Well Testing, 5.1.2 Faults and Fracture Characterisation, 5.4.2 Gas Injection Methods, 5.5.8 History Matching, 5.5.11 Formation Testing (e.g., Wireline, LWD), 4.1.2 Separation and Treating, 5.1.5 Geologic Modeling, 5.6.9 Production Forecasting
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We present theory and application of a new approach for assisted history matching and uncertainty assessment. In a Bayesian framework the a priori geological information is conditioned to the production history to give the posterior probability distribution function (pdf). The full posterior pdf is explored to assess the uncertainty through ensembles of reservoir models sampled by a Markov chain Monte Carlo algorithm. To achieve this we construct proxy functions for the output of the flow simulator for all measurements that enter a global objective function. The proxy functions are constructed using polynomials and multi dimensional kriging. An iterative loop, in which ensembles of reservoir models are sampled from the posterior pdf, is run to improve the quality of the proxy functions.
The power of the application is demonstrated on two reservoir models. First we apply the method on a synthetic case. A modified reservoir simulation model from a small StatoilHydro operated oil field is investigated with a synthetic production history and 20 tuning parameters. Finally we apply the method to the StatoilHydro operated Heidrun field. A model which covers the upper formations of the field with 26 production wells, 11 injector wells, and 56 tuning parameters is conditioned to 11 years of production history. We show that it is possible to construct proxy functions accurate enough to describe the full posterior pdf and thereby assess the uncertainty associated with these reservoir models.
Computer assisted history matching has been a topic for decades, and several strategies for minimizing a global objective function with as little computational effort as possible have been developed. Recently more focus has been put on risk management and uncertainty assessment, and the dangers of basing decisions on a single "base case?? reservoir simulation model are widely recognized.
The history matching and the uncertainty assessment challenges may be united within a Bayesian framework. Geological knowledge is used as prior information, which is conditioned to the historic production data to give a posterior probability distribution function (pdf). The reservoir models sampled from this posterior pdf will be history matched in the sense that the simulated responses are focused around the observed responses within a prescribed error tolerance. A popular approach has been to perform a global search for the reservoir model that maximizes the posterior pdf. However, the most probable model does not need to be representative, and to assess the uncertainty one needs to sample from the full posterior pdf.
Several strategies of exploring the full posterior pdf exist, and the most prominent example is perhaps the Ensemble Kalman Filter (EnKF) (Evensen 2006; Evensen et al. 2007; Gaoming Li and Reynolds 2007; Haugen et al. 2006). The EnKF method starts out with an ensemble sampled from the a priori pdf. The ensemble is then approximately conditioned to the measurements sequentially under the assumption that the underlying fields are Gaussian. Randomized Maximum Likelihood (RML) methods (Kitanidis 1995; Ning et al. 2001), is another class of methods which samples approximately from a full posterior pdf using a Gaussian assumption. In the RML methods each member of an ensemble sampled from the a priori pdf are "history matched?? to form a posterior ensemble. Characterizing the full posterior pdf by running the flow simulator in a
Monte Carlo loop is prohibitively computationally demanding, and only a few attempts on synthetic models (Floris et al. 2001; Barker et al. 2001; Hegstad and Omre 2001; Oliver et al. 1996) have been reported. Monte Carlo sampling schemes require an overwhelming number of steps as they only sample asymptotically correct. Reduced physics models and stream line simulators have been proposed as means to lower the computational cost (Ma et al. 2006; Mau??ec et al. 2007), but even these fast simulators are too slow to enable reliable sampling of the posterior pdf for real field cases.
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