Summary Efficient management of smart oil fields requires a reservoir model that can provide reliable forecasts of future production and realistic measures of prediction uncertainty. Reliable forecasts depend on an accurate representation of reservoir geology, which is conveyed largely by the permeabilities used in the reservoir simulator. Because these permeabilities cannot be measured directly, they must be inferred from measurements of related variables, using procedures such as history matching or Bayesian estimation. The ensemble Kalman filter (EnKF) is an attractive option for permeability estimation in real-time reservoir-control applications. It is easy to implement, provides considerable flexibility for describing geological heterogeneity, and supplies valuable information about prediction uncertainty. However, it is more suited for geological heterogeneities that are amenable to second-order (covariance-based) descriptions. In this paper, we investigate the performance of the EnKF for estimation of channel permeabilities that usually follow a bimodal distribution. We consider two synthetic waterflooding problems based on true permeability distributions characterized by conductive channels. The permeability ensembles are obtained from a multipoint geostatistical simulation method. If the ensemble replicates are derived from training images that do not describe the channel geometry properly, the Kalman filter has difficulty identifying the correct permeability field. In fact, the permeability estimates tend to diverge from the true values as more measurements are included. However, if the filter-ensemble replicates are generated by a training image that contains features that are consistent with those in the true permeability field, the filter's estimates are much better. These results emphasize the importance of generating realistic permeability replicates when using ensemble methods to estimate reservoir properties. In fact, a realistic permeability ensemble appears to be essential for successful estimation performance. With a proper ensemble design, despite the bimodality in the initial permeability distribution, the filter exhibits good performance in identifying the patterns in the true permeability field. In practical applications where the true permeability distribution is highly uncertain, the prior information used for ensemble generation should properly reflect the full range of possible geological conditions.

Development of subsurface energy and environmental resources can be improved by tuning important decision variables such as well locations and operating rates to optimize a desired performance metric. Optimal well locations in a discretized reservoir model are typically identified by solving an integer programming problem while identification of optimal well settings (controls) is formulated as a continuous optimization problem. In general, however, the decision variables in field development optimization can include many design parameters such as the number, type, location, short-term and long-term operational settings (controls), and drilling schedule of the wells. In addition to the large number of decision variables, field optimization problems are further complicated by the existing technical and physical constraints as well as the uncertainty in describing heterogeneous properties of geologic formations. In this paper, we consider simultaneous optimization of well locations and dynamic rate allocations under geologic uncertainty using a variant of the simultaneous perturbation and stochastic approximation (SPSA). In addition, by taking advantage of the robustness of SPSA against errors in calculating the cost function, we develop an efficient field development optimization under geologic uncertainty, where an ensemble of models are used to describe important flow and transport reservoir properties (e.g., permeability and porosity). We use several numerical experiments, including a channel layer of the SPE10 model and the three-dimensional PUNQ-S3 reservoir, to illustrate the performance improvement that can be achieved by solving a combined well placement and control optimization using the SPSA algorithm under known and uncertain reservoir model assumptions.

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Part 1--Parameterization Summary Inverse estimation (history matching) of permeability fields is commonly performed by replacing the original set of unknown spatially discretized reservoir properties with a smaller (lower-dimensional) group of unknowns that captures the most important features of the field. This makes the inverse problem better posed by reducing redundancy. The Karhunen-Loeve transform (KLT), also known as principal component analysis, is a classical option for deriving low-dimensional parameterizations for history-matching applications. The KLT can provide an accurate characterization of complex-property fields, but it can be computationally demanding. In many respects, this approach provides a benchmark that can be used to evaluate the performance of more-computationally-efficient alternatives. The KLT requires knowledge of the property covariance function and can give poor results when this function does not adequately describe the actual property field. By contrast, the discrete cosine transform (DCT) provides a robust parameterization alternative that does not require specification of covariances or other statistics. It is computationally efficient and, in many cases, is almost as accurate as the KLT. The DCT is also able to accommodate prior information, if desired. Here, we describe the DCT approach and compare its performance to the KLT for a set of geologically relevant examples. Part 2--History Matching Summary The DCT provides a flexible and effective method for describing spatially distributed reservoir properties such as permeability. This method represents uncertain properties as weighted sums of predefined spatially variable basis functions. The basis function weights may be estimated with iterative or sequential history-matching methods. The compression power of the DCT and its advantages over alternative parameterization techniques are discussed in Part 1. In Part 2, the history-matching capabilities of the DCT parameterization are illustrated with waterflooding examples for synthetic channelized reservoirs. Two history-matching options are examined: an iterative least-squares method and a sequential ensemble Kalman filter (EnKF). Prior information is incorporated through an ensemble of permeability replicates derived from a specified training image. These replicates are used to compute sample covariances for the EnKF and to select basis functions for the DCT expansion in both the least-squares algorithm and the Kalman filter. Prior information improves estimation performance when it is consistent with the directional trends of the true permeability field but may degrade performance if it is incorrectly specified. The most robust history-matching results are obtained with an iterative least-squares algorithm that uses a DCT basis with no directional preference. The experiments documented in this paper indicate that the DCT makes the history-matching problem better-posed and improves the realism of reservoir property estimates. It is efficient and versatile and can be used with or without prior information.