SPE Journal
Volume 14, Number 1, March 2009, pp. 182-201

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.

© 2009. Society of Petroleum Engineers

View full textPDF ( 1,540 KB )

History

• Original manuscript received: 5 December 2006
• Meeting paper published: 26 February 2007
• Revised manuscript received: 7 August 2008
• Manuscript approved: 24 August 2008
• Published online: 16 March 2009
• Version of record: 1 March 2009