Multiresolution Wavelet Analysis for Improved Reservoir Description

Abstract

Summary It is well documented that history matching is a problem with possibly nonunique solutions. In the past few years, several automated or semiautomated history-matching algorithms have been proposed. Depending on the algorithm used, it is possible that the final estimated reservoir-property distribution that allows for a good history match may not be geologically realistic. Therefore, there is a need to include other constraints to generate multiple, geologically realistic history-matched realizations. These constraints might, for example, include the variogram, a training image, the distribution of net-to-gross, pore volume, or other geostatistical information about the reservoir. This inclusion is particularly useful because it introduces uncertainty information in the reservoir description when we have limited history from existing wells in the field and intend to drill infill wells or implement a secondary-recovery process. The algorithm proposed in this paper uses multiresolution wavelet analysis to integrate history data with the geostatistical information contained in the variogram proposed for the reservoir. Wavelets allow the representation and manipulation of property distributions at various resolutions at the same time. Using wavelets, information from different sources such as production history and seismic surveys (that would be at different resolutions) can be incorporated directly at the appropriate resolution level. In the first step, we fix the wavelet coefficients sensitive to the history-match data. This has the effect of fixing the field history without fixing individual gridblock properties. In the second step, the remaining free wavelet coefficients are modified to integrate variogram information into the reservoir description. Generating multiple realizations of only the second set of wavelets coefficients results in multiple history-matched, variogram-constrained descriptions of the reservoir. The computational investment is very modest because the history match is done only once. In a number of example cases, different areal Gaussian fields with varying amounts of available production-history data were studied to test the algorithm. It was found that the wavelet coefficients constraining the history can be decoupled from those constraining the variogram. The implication of this observation is that the history data and variogram can be integrated sequentially into the reservoir model—that is, after the initial history match, new information can be added to the model without disturbing the original match to yield multiple history-matched and geostatistically constrained realizations.