An adaptively scaled frequency-domain parameterization for history matching
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History matching problems are typically underdetermined and are at once confronted with the problems of solution non-uniqueness, instability, and the ability to both reproduce field observations and provide a reliable forecast. We address these challenges with a new adaptive multiscale history matching formulation that parameterizes the reservoir properties in the frequency domain. The geologic model updating is carried out by successively increasing the level of detail up to a spatial scale sufficient to match the observed data. Our method begins by constructing a coarse representation of the field using the lowest-frequency components of its discrete cosine parameterization. This substantially reduces the number of unknown parameters to be resolved during history matching, leading to a better posed inverse problem. A gradient-based minimization is then performed to match the production data. Next, the updated model is incrementally refined in the frequency domain and the minimization is repeated until the data misfit is reduced below a pre-specified criterion or until no further improvements are observed. During minimization, components of the gradient insensitive to production information are removed by a truncated singular value decomposition (TSVD), facilitating iterative convergence and providing additional regularization. In this manner a balance is achieved between parameter reduction which is required for stability, and the spatial resolution of heterogeneity required for reproduction of the production history. The low-frequency approximation of the permeability field helps to honor geologic continuity and is particularly suited for resolving the large-scale heterogeneity that has a dominant influence on the field-scale flow regime and production response. Applications of the approach are demonstrated using the SPE10 and PUNQ-S3 models and involve waterflood history matching with water-cut and bottom-hole pressure data. Our results show that the principle geologic features of the reference field are adequately resolved only if we begin at low resolution. As the resolution is iteratively increased, the history match is improved while the TSVD step removes insensitive parameter combinations thereby decreasing the likelihood of convergence to less plausible local minima. Notably, in all our applications an adequate history match is achieved using less than 1% of the original parameter dimension, which leads to increased solution stability and computational savings in history matching large geologic models. 2010 Elsevier B.V.
