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We focus on specific parameterization techniques developed in inverse stochastic modeling for determining permeability fields from dynamic data using a reduced number of parameters. Two major contributions are the pilot point method and the gradual deformation method. They were designed to reduce the number of parameters and to respect the inferred spatial structure. Weaknesses have been revealed for the pilot point method: pilot points can be assigned unreasonably extreme values and possible correlations among the pilot points are neglected. To bypass these limitations, a new approach, called the gradual pilot point method, is suggested. It follows the basic workflow of the pilot point method, but the pilot point values are not driven by the optimization procedure. Intermediate gradual deformation parameters are introduced which govern the pilot point values. Compared to the original pilot point method, the gradual pilot point method does not produce extreme variations. Moreover, when the whole set of pilot points is modified simultaneously from a single deformation parameter, the correlations among the pilot points are accounted for. Thus, many pilot points can be placed on the permeability field, whatever their locations. They can produce local and global deformation. The performed numerical experiments show that a two-step approach for calibrating permeability fields is useful. First, the gradual deformation method is used to globally deform the permeability fields. Once the permeability fields have been globally improved, they can be locally refined using the gradual pilot point method.

Combining the Pilot Point and Gradual Deformation Methods for Calibrating Permeability Models to Dynamic Data