We introduce a novel reservoir parameterization approach to mitigate the challenges associated with field-scale history matching. In this approach, the reservoir property field is mapped to and updated in a low-dimensional transform domain using a linear transformation basis. The transformation basis vectors are the eigenvectors of a Laplacian matrix that is constructed using grid connectivity information and the main features in a given prior model. Because the grid connectivity information is computed only within a small multi-point stencil, the Laplacian is always sparse and is amenable to efficient decomposition. The resulting basis functions are ordered from large to small scale and include prior-specific spatial features. Therefore, the variability in reservoir property distribution can be effectively represented by projecting the property field onto subspaces spanned by an increasing number of leading basis vectors, each incorporating additional heterogeneity features into the model description. This property lends itself to a multiscale history matching algorithm where basis elements are sequentially included to refine the heterogeneity characterization to a level of complexity supported by the resolution of data. While the method can benefit from prior information, in the extreme case where reliable prior knowledge is not available the transformation reduces to a discrete Fourier expansion with model-independent parameterization properties.

We present the derivation and theoretical justification of the proposed method and review its important properties for reservoir parameterization including efficient one-time construction of the basis prior to calibration, applicability to any grid geometry and strong compression performance. The multiscale history matching algorithm begins by updating the prior reservoir model using a parameterized multiplier field that is superimposed onto the grid and assigned an initial value of unity at each cell. The multiplier is sequentially refined from the coarse to finer scales during minimization of production data misfit. This method permits selective updating of heterogeneity at locations and levels of detail sensitive to the available data, otherwise leaving the prior model unchanged as desired. We successfully apply the parameterization approach to history match several reservoir models, including a field case, using an adaptive multiscale algorithm.

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