Subsurface Flow Model Calibration with a Spectral-Domain Parameterization Adaptive to Grid Connectivity and Prior Model Information
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A heterogeneity parameterization is introduced to mitigate the challenges associated with field-scale subsurface flow model calibration. The estimated geologic parameter field is mapped to and updated in a low-dimensional transform domain using a linear transformation basis. The basis vectors are the eigenvectors of a Laplacian matrix that is constructed using grid connectivity information and the main features in the prior geologic model. Because the grid connectivity information is computed within a small multipoint stencil, the Laplacian is always sparse and amenable to efficient decomposition. The resulting basis vectors 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 calibration algorithm where basis elements are sequentially included to refine the heterogeneity characterization to a level of complexity supported by the resolution of available data. While the method can benefit from prior information, when the prior is unavailable or unreliable the transformation can reduce to a discrete Fourier expansion with robust model-independent parameterization properties. We present the derivation and theoretical justification of the transformation basis and review its advantageous properties for heterogeneity parameterization including efficient one-time construction, applicability to any grid geometry, and strong prior model compression performance. The multiscale calibration workflow begins by updating the prior 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 well production data misfit. This method permits selective updating of heterogeneity at locations and levels of detail sensitive to the data, otherwise leaving the prior unchanged as desired. The parameterization approach is applied to calibrate several petroleum reservoir models using an adaptive multiscale algorithm. 2012 International Association for Mathematical Geosciences.
