  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 computationally feasible VFM for use in real-time applications, with high
prediction accuracy and scientifically consistent behavior. This article
investigates five different gray-box model types in an industrial case study
using real, historical production data from 10 petroleum wells, spanning at
most four years of production. The results are diverse with an oil flow rate
prediction error in the range of 1.8%-40.6%. Further, the study casts light
upon the nontrivial task of balancing learning from both physics and data.
Consequently, providing general recommendations towards the suitability of
different hybrid models is challenging. Nevertheless, the results are promising
and indicate that gray-box VFMs may reduce the prediction error of a
mechanistic VFM while remaining scientifically consistent. The findings
motivate further experimentation with gray-box VFM models and suggest several
future research directions to improve upon the performance and scientific
consistency.
