# On gray-box modeling for virtual flow metering

arXiv: https://arxiv.org/abs/2103.12513 · HTML: https://mlsanity.com/paper/2103.12513/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 44 references that are in the map.
- First posted on arXiv 23 Mar 2021.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [Developing a Hybrid Data-Driven, Mechanistic Virtual Flow Meter -- a Case Study](https://mlsanity.com/paper/2002.02737/) (2020)
- [Bayesian Neural Networks for Virtual Flow Metering: An Empirical Study](https://mlsanity.com/paper/2102.01391/) (2021)
- [A Predictive Model for Steady-State Multiphase Pipe Flow: Machine Learning on Lab Data](https://mlsanity.com/paper/1905.09746/) (2019)

## Built on by (most cited first, 4 of 4)

- [Passive learning to address nonstationarity in virtual flow metering applications](https://mlsanity.com/paper/2202.03236/) (2022)
- [Learning characteristic parameters and dynamics of centrifugal pumps under multiphase flow using physics-informed neural networks](https://mlsanity.com/paper/2310.03001/) (2023)
- [When is gray-box modeling advantageous for virtual flow metering?](https://mlsanity.com/paper/2110.05034/) (2021)
- [Sequential Monte Carlo applied to virtual flow meter calibration](https://mlsanity.com/paper/2304.06310/) (2023)

## Abstract (arXiv)

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.