In the strong-constraint formulation of the history-matching problem, we assume that all the model errors relate to a selection of uncertain model input parameters. One does not account for additional model errors that could result from, e.g., excluded uncertain parameters, neglected physics in the model formulation, the use of an approximate model forcing, or discretization errors resulting from numerical approximations. If parameters with significant uncertainties are unaccounted for, there is a risk for an unphysical update, of some uncertain parameters, that compensates for errors in the omitted parameters. This paper gives the theoretical foundation for introducing model errors in ensemble methods for history matching. In particular, we explain procedures for practically including model errors in iterative ensemble smoothers like ESMDA and IES, and we demonstrate the impact of adding (or neglecting) model errors in the parameter-estimation problem. Also, we present a new result regarding the consistency of using the sample covariance of the predicted nonlinear measurements in the update schemes.
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The work has benefited from the interaction and collaborations with the Nordforsk Nordic center of excellence in data assimilation, EMBLA. In-depth discussions with Patrick Raanes, regarding the linear-regression derivation, have helped to improve the manuscript, and the author is also grateful for comments by Geir Nævdal and Alberto Carrassi on early versions of the manuscript. Constructive comments by three anonymous reviewers lead to the inclusion of a section on how to practically account for model errors in reservoir history matching and the section on the specification of model errors.
This work was supported by the Research Council of Norway and the companies AkerBP, DEA, ENI, Petrobras, Equinor, Lundin, and Neptune Energy, through the Petromaks–2 project DIGIRES.
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Evensen, G. Accounting for model errors in iterative ensemble smoothers. Comput Geosci 23, 761–775 (2019). https://doi.org/10.1007/s10596-019-9819-z