  A virtual flow meter (VFM) enables continuous prediction of flow rates in
petroleum production systems. The predicted flow rates may aid the daily
control and optimization of a petroleum asset. Gray-box modeling is an approach
that combines mechanistic and data-driven modeling. The objective is to create
a VFM with higher accuracy than a mechanistic VFM, and with a higher scientific
consistency than a data-driven VFM. This article investigates five different
gray-box model types in an industrial case study on 10 petroleum wells. The
study casts light upon the nontrivial task of balancing learning from physics
and data. The results indicate that the inclusion of data-driven elements in a
mechanistic model improves the predictive performance of the model while
insignificantly influencing the scientific consistency. However, the results
are influenced by the available data. The findings encourage future research
into online learning and the utilization of methods that incorporate data from
several wells.
