Identifiability and physical interpretability of hybrid, gray-box models -- a case study

Model identifiability concerns the uniqueness of uncertain model parameters to be estimated from available process data and is often thought of as a prerequisite for the physical interpretability of a model. Nevertheless, model identifiability may be challenging to obtain in practice due to both stochastic and deterministic uncertainties, e.g. low data variability, noisy measurements, erroneous model structure, and stochasticity and locality of the optimization algorithm. For gray-box, hybrid models, model identifiability is rarely obtainable due to a high number of parameters. We illustrate through an industrial case study - modeling of a production choke valve in a petroleum well - that physical interpretability may be preserved even for non-identifiable models with adequate parameter regularization in the estimation problem. To this end, in a real industrial scenario, it may be beneficial for the model's predictive performance to develop hybrid over mechanistic models, as the model flexibility is higher. Modeling of six petroleum wells on the asset Edvard Grieg using historical production data show a 35\% reduction in the median prediction error across the wells comparing a hybrid to a mechanistic model. On the other hand, both the predictive performance and physical interpretability of the developed models are influenced by the available data. The findings encourage research into online learning and other hybrid model variants to improve the results.Comment: 6 pages, 4 figure

Bayesian Neural Networks for Virtual Flow Metering: An Empirical Study

Recent works have presented promising results from the application of machine learning (ML) to the modeling of flow rates in oil and gas wells. Encouraging results and advantageous properties of ML models, such as computationally cheap evaluation and ease of calibration to new data, have sparked optimism for the development of data-driven virtual flow meters (VFMs). Data-driven VFMs are developed in the small data regime, where it is important to question the uncertainty and robustness of models. The modeling of uncertainty may help to build trust in models, which is a prerequisite for industrial applications. The contribution of this paper is the introduction of a probabilistic VFM based on Bayesian neural networks. Uncertainty in the model and measurements is described, and the paper shows how to perform approximate Bayesian inference using variational inference. The method is studied by modeling on a large and heterogeneous dataset, consisting of 60 wells across five different oil and gas assets. The predictive performance is analyzed on historical and future test data, where an average error of 4-6% and 8-13% is achieved for the 50% best performing models, respectively. Variational inference appears to provide more robust predictions than the reference approach on future data. Prediction performance and uncertainty calibration is explored in detail and discussed in light of four data challenges. The findings motivate the development of alternative strategies to improve the robustness of data-driven VFMs.Comment: 34 pages, 11 figure

On a hybrid approach to model learning applied to virtual flow metering

Process modeling using first-principle equations has existed for centuries as a methodology to represent and analyze real-world processes. In time with increasing computing power and sensor data availability, data-driven modeling for processes has gained attention. Even though data-driven modeling, or machine learning, has shown remarkable results in fields such as image classification and speech recognition, it has yet to be adopted as the preferred approach for process modeling. Arguably, this is due to the long history of first-principles modeling, along with the inherent black-box nature of datadriven models. The latter causes a lack of model explainability, which, in turn, can result in distrusting the predictions originating from data-driven models. Furthermore, disregarding physical laws that have been acknowledged for centuries to model processes can seem irrational. Hybrid, or gray-box, modeling is a methodology with a vision to utilize all available knowledge, both physics and data, to model processes. It combines firstprinciple equations with data-driven techniques and is especially intriguing for inherent complex processes where the physical behavior is partly unknown or challenging to model with first principles. One such process is the petroleum production system. The multiphase flow rate through the production system is challenging to model with required precision using first principles due to uncertain subsurface properties and complex dynamic behavior. Furthermore, available sensor data is often limited or of low quality. Therefore, a hybrid modeling approach seems of significant importance to models predicting the multiphase flow rates as it attempts to exploit all available information to its full extent. The work leading to this thesis has explored hybrid solutions for virtual flow metering. A virtual flow meter (VFM) is a soft-sensor that utilizes process models and already existing sensor measurements, such as pressures and temperatures, to compute the multiphase flow rate at strategic locations in a petroleum asset. The main part of this thesis is a collection of six peer-reviewed papers, three journal publications, and three conference publications. In addition to the paper collection, this thesis introduces the topic of hybrid modeling for virtual flow metering to provide context to the publications. The main contributions of the six publications can be summarized as follows: firstly, a framework for simultaneous estimation of all parameters in a model with varying degrees of hybridity has been proposed. Secondly, six hybrid VFM model types were developed from real and historical production data from a petroleum asset. Thirdly, several hybrid model properties such as explainability, scientific consistency, flexibility, and accuracy have been examined. Lastly, two methods, one to include uncertainty in the modeling, and one to address the inherent nonstationarity of the underlying process to sustain the long-term VFM performance, have been proposed. The key takeaway of the work leading to this thesis is that hybrid modeling is challenging, yet, also essential for obtaining high accuracy VFMs in certain scenarios. The contributions have shown that the task of balancing learning from physics and learning from data is nontrivial, and if incautious, the hybrid model can exploit the disadvantages of both the mechanistic and data-driven modeling domain instead of the advantages. On the other hand, the results also showed that for processes with unknown or unmodeled physics, a hybrid model can offer improved performance over a mechanistic model, and with little available process data, a hybrid model can obtain a higher performance than a data-driven model. Moreover, in the presence of nonstationarity and little data, frequent updating of a hybrid VFM has shown essential to sustain the prediction accuracy over time. From the results, it is believed that hybrid modeling can be generalized to other applications and can offer improved performance over a mechanistic and datadriven approach. Furthermore, the solution for hybrid modeling presented in this thesis can be conveniently integrated with existing mechanistic process models in the industry. Naturally, the domain of hybrid modeling for virtual flow metering has not been fully explored. The most promising future research direction is combining hybrid modeling with methods that enable learning from more than one petroleum well at a time

Dynamic Real-Time Optimisation of an Amine-Based Post-Combustion CO2 Capture Facility using Single-Level Nonlinear Model Predictive Control

A complete model of a CO2 capture facility has been optimised with the aid of Dynamic Real-Time Optimisation (DRTO) utilising single-level, Nonlinear Model Predictive Control to merge regulatory and economic objectives. The goal has been to, during 24 hours, minimise the cost related to the energy consumption in the reboiler by variable solvent regeneration, whilst achieving a specified accumulated, or overall, capture ratio of CO2 at the end of the simulation horizon. An hourly varying price of energy with a period of 24 hours have been included in the optimisation problem. The complete model is based on a previous model from Cybernetica AS, the original model, with model reductions suggested by Hotvedt (2017) for the absorber, desorber and heat exchanger. The suggestions included modelling using molar amounts as state variables for each substance in the facility and discretising the unit models in space using control volumes. The complete reduced model has been validated against the original model in addition to instrumental measurements from an existing test facility at Tiller in Trondheim. It was found the reduced model yielded adequate behaviour although with deviations from both the original model responses and instrumental measurements. Introduction of simple estimator; bias updating, removed the deviations significantly. Eigenvalue analysis of the original and the reduced model were performed, and results show that the reduced model yielded only minor reductions in stiffness. On the other hand, the reductions decreased the dimension of the state space with 225 states, resulting in a simulation time reduction of 73%. The DRTO was designed using the infeasible soft-constraint method where constraints on the energy costs have been set infeasible. Results from simulation show that the DRTO is able to achieve the reference accumulated capture ratio after 24 hours in addition to utilise the time varying price of energy to minimise cost. The performance was compared to a basic case where the accumulated capture ratio of CO2 was forced constant during the prediction horizon, obtaining a constant solvent regeneration, and a cost reduction of 13.0% and 10.9% was found using a reference value for the accumulated capture ratio of 85% and 91% respectively. The DRTO was further tested for robustness by firstly inducing a step change in inlet conditions of the flue gas, secondly by abruptly increasing the price of energy and lastly by applying stricter constraints on the reboiler duty. The DRTO accomplished the capture goal in all cases, except for a step in inlet conditions close to the end of the simulation horizon. Lastly, the optimal solution resulted in unnecessary use of reboiler duty analysing a simulated plant replacement model, and consequently was bias updating introduced to enhance cost minimisation

Developing a Hybrid Data-Driven, Mechanistic Virtual Flow Meter - a Case Study

Virtual flow meters, mathematical models predicting production flow rates in petroleum assets, are useful aids in production monitoring and optimization. Mechanistic models based on first-principles are most common, however, data-driven models exploiting patterns in measurements are gaining popularity. This research investigates a hybrid modeling approach, utilizing techniques from both the aforementioned areas of expertise, to model a well production choke. The choke is represented with a simplified set of first-principle equations and a neural network to estimate the valve flow coefficient. Historical production data from the petroleum platform Edvard Grieg is used for model validation. Additionally, a mechanistic and a data-driven model are constructed for comparison of performance. A practical framework for development of models with varying degree of hybridity and stochastic optimization of its parameters is established. Results of the hybrid model performance are promising albeit with considerable room for improvements

Dynamic Real-Time Optimisation of a CO2 Capture Facility

This work investigates economic optimisation of an energy-intensive amine regeneration process in a post-combustion CO2 capture plant, subject to a minimum CO2 capture ratio over 24 hours. A Dynamic Real-Time Optimisation algorithm is implemented as a single-level Nonlinear Model Predictive Control scheme by utilising the infeasible soft-constraint method to include economic objectives in an industrial tracking NMPC package. A time-varying price of electricity is exploited to enhance cost minimisation by adjusting the regeneration according to the peaks of the price curve. This flexible mode of operation is compared to a fixed mode of operation with constant amine regeneration. Simulation results indicate a cost reduction of 10.9% for a reference accumulated capture ratio of 91%. Robustness of the optimisation to abrupt changes in CO2 feed composition and electricity price is also investigated in simulations and results are promising. The NMPC controller uses a reduced, control-oriented model of the capture plant developed from first principle conservation laws with control volumes to discretize the model equations in space

On gray-box modeling for virtual flow metering

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. Therefore, providing general recommendations towards the suitability of different hybrid models is challenging. Nevertheless, the results are promising and indicate that gray-box VFMs can 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