Streamline-Assisted Ensemble Kalman Filter for Rapid and Continuous Reservoir Model Updating
- Elkin Arroyo (Texas A&M University) | Deepak Devegowda (U. of Oklahoma) | Akhil Datta-Gupta (Texas A&M University) | Jonggeun Choe (Seoul Natl. U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- December 2008
- Document Type
- Journal Paper
- 1,046 - 1,060
- 2008. Society of Petroleum Engineers
- 4.3.4 Scale, 3.3 Well & Reservoir Surveillance and Monitoring, 5.1.2 Faults and Fracture Characterisation, 5.4.1 Waterflooding, 5.5.7 Streamline Simulation, 7.2.2 Risk Management Systems, 5.1 Reservoir Characterisation, 7.6.2 Data Integration, 5.1.5 Geologic Modeling, 5.5 Reservoir Simulation, 2.3 Completion Monitoring Systems/Intelligent Wells, 5.5.8 History Matching, 5.6.4 Drillstem/Well Testing, 5.3.1 Flow in Porous Media, 5.3.2 Multiphase Flow, 5.1.9 Four-Dimensional and Four-Component Seismic, 5.6.9 Production Forecasting
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The use of the ensemble Kalman filter (EnKF) is a promising approach for data assimilation and assessment of uncertainties during reservoir characterization and performance forecasting. It provides a relatively straightforward approach to incorporating diverse data types, including production and/or time-lapse seismic data. Unlike traditional sensitivity-based history matching methods, the EnKF relies on a cross-covariance matrix computed from an ensemble of reservoir models to relate reservoir properties to production data. For practical field applications, we need to keep the ensemble size small for computational efficiency. However, this leads to poor approximations of the cross-covariance and, often, loss of geologic realism through parameter overshoots, in particular by introducing localized patches of low and high permeabilities. Because the EnKF estimates are "optimal" only for Gaussian variables and linear dynamics, these difficulties are compounded by the strong nonlinearity of the multiphase history matching problems and for non-Gaussian prior models. Specifically, the updated parameter distribution tends to become multi-Gaussian with loss of connectivities of extreme values, such as high permeability channels and low permeability barriers, which are of special significance during reservoir characterization.
We propose a novel approach to overcome some of these limitations by conditioning the cross-covariance matrix using information gleaned from streamline trajectories. Our streamline-assisted EnKF is analogous to the conventional assisted history matching, whereby the streamline trajectories are used to identify gridblocks contributing to the production response of a specific well. We then use these gridblocks only to compute the cross-covariance matrix and eliminate the influence of unrelated or distant observations and spurious correlations. We show that the streamline-assisted EnKF is an efficient and robust approach for history matching and continuous reservoir model updating. We illustrate the power and utility of our approach using both synthetic and field applications.
Proper characterization of the reservoir and the assessment of uncertainty are crucial aspects of any optimal reservoir development plan and management strategy. To achieve this goal, it is necessary to reconcile geological models to the dynamic response of the reservoir through history matching. The topic of history matching has been of great interest and an area of active research in the oil industry (Datta-Gupta and King 2007; Emanuel and Milliken 1998; Oliver et al. 2001). The past decade has seen some significant developments in assisted and automatic history matching of high-resolution reservoir models and associated uncertainty quantification. Many of these techniques involve computation of sensitivities that relate changes in production response at a well to a change in reservoir parameters. Techniques of automatic history matching that typically do not use parameter sensitivities or gradient of the misfit function are stochastic algorithms such as Markov Chain Monte Carlo (MCMC), simulated annealing and genetic algorithms (Ma et al. 2008; Sen et al. 2005). A relatively recent and promising addition to this class of techniques is the use of ensemble Kalman Filters (EnKF) for data assimilation (Gu and Oliver 2005, 2006; Naevdal et al. 2005; Gao et al. 2006; Skjervheim et al. 2007; Dong et al. 2006). It is a Monte-Carlo approach that works with an ensemble of reservoir models. Specifically, the method utilizes cross-covariances between measurements and model parameters computed directly from the ensemble members to sequentially update the reservoir models.
A major advantage of the EnKF is that it can be readily linked to any existing reservoir simulator. The ability to assimilate diverse data types and the ease of implementation have resulted in considerable interest in the approach. Moreover, EnKF uses a sequential updating technique; that is, the reservoir data is assimilated as and when it becomes available. The EnKF can assimilate the latest production data without re-running the simulator from the initial conditions. These characteristics make it particularly well-suited for continuous model updating. The increased application of downhole monitors, intelligent well systems, and permanent sensors to continuously record pressure, well rates, and temperature has provided a further boost to the sequential model updating through EnKF.
In spite of all its favorable properties, the current implementation of EnKF approach comes with its own share of challenges. A key requirement in history matching is that the final model should honor the available geological information and retain geologic realism. It has been shown that the EnKF works well when the prior distribution of parameters is Gaussian; however, the estimates are suboptimal for non-Gaussian distributions. Over a sequence of many updates, multimodal permeability distributions tend to transform to Gaussian distribution. During geologic model updating, this can lead to a loss of structure and connectivity of the extremes in the permeability field. This has serious implications in the fluid flow because of the influence of high-permeability channels and low-permeability barriers. Although there are some variants of the Kalman filter that work with non-Gaussian distributions, such as the Gaussian summation approximation, the implementation on an ensemble framework tend to be very expensive (Anderson and Moore 1979).
In the past few years, we have seen several applications of the EnKF for field-scale history matching, including some recent papers that attempt to deal with some of the challenges pertaining to its use (Gu and Oliver 2005, 2006; Naevdal et al. 2005; Gao et al. 2006; Skjervheim et al. 2007; Dong et al. 2006). In particular, localized overshooting of permeabilities has been reported, resulting in loss of geologic continuity. This is aggravated by the strong non-linearity inherent in multiphase flow simulations.
Another common difficulty experienced when using the EnKF is filter divergence. The effect of filter divergence is such that the distribution produced by the filter drifts away from the truth. Filter divergence normally occurs because the prior probability distribution becomes too narrow (loss of variance) and the observations have progressively less impact on the model updates.
One common approach to deal with filter divergence is to add some (white) noise to the prior ensemble to "inflate" its distribution and enhance the impact of new observations. Other problems and limitations of the EnKF, particularly for nonlinear problems and non-Gaussian parameter distributions, can be partly controlled using a large ensemble. However, for practical field applications, the ensemble size needs to be kept relatively small for computational efficiency.
This paper describes an approach to address many of the currently reported difficulties in the use of the EnKF applied to reservoir history matching. The unique feature of our proposed approach is that the final models that constitute the ensemble tend to retain the geological information that went into building them initially. Over a sequence of many updates, our approach tends to preserve the shape of the initial permeability distribution and consequently retains key geological features. Our approach greatly decreases the severity of the overshooting problem reported in earlier implementations of the EnKF. Moreover, it allows the use of smaller ensemble size, while providing results comparable or better than the standard EnKF.
The paper is organized as follows. First, we briefly review the major steps of the EnKF and the additional streamline-based conditioning of the cross-covariance proposed here. We also illustrate these steps using a synthetic example. Next, we discuss the underlying mathematical formulation in detail. We then demonstrate the power and practical utility of the approach using the benchmark PUNQ-S3 synthetic example (Gu and Oliver 2005) and a field example. Finally, an analysis of the scalability and speed-up factor for the parallel implementation of our code is given.
|File Size||19 MB||Number of Pages||15|
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