### “Parameter Estimation for a Gas Lifting Oil Well Model Using Bayes' Rule and the Metropolis–Hastings Algorithm”

**Authors:**Zhe Ban, Ali Ghaderi, Nima Janatian and Carlos F. Pfeiffer,

**Affiliation:**University of South-Eastern Norway

**Reference:**2022, Vol 43, No 2, pp. 39-53.

**Keywords:**Gas Lifting Oil Well, Parameter Estimation, Markov Chain Monte Carlo

**Abstract:**Oil well models are frequently used in the oil production process. Estimation of unknown parameters of these models has long been a question of great interest in the oil industry field. Data collected from an oil well system can be useful for identifying and characterizing the parameters in the corresponding model. In this article, we present a solution to estimate the parameters and uncertainty of a gas lifting oil well model by designing Bayesian inference and using the Metropolis-Hastings algorithm. To present and evaluate the estimation, the performance of the chains and the distributions of the parameters were shown, followed by posterior predictive distributions and sensitivity analysis. Compared with the conventional maximum likelihood estimation methods that tried to identify one optimum value for each parameter, more information of the parameters is obtained by using the proposed model. The insights gained from this study can benefit the optimization and advanced control for the oil well operation.

PDF (1883 Kb) DOI: 10.4173/mic.2022.2.1

**DOI forward links to this article:**

[1] Kushila Jayamanne and Bernt Lie (2023), doi:10.4173/mic.2023.1.2 |

**References:**

[1] Brown, K.E. (1977). Technology of artificial lift methods, volume 1. inflow performance, multiphase flow in pipes, the flowing well. 1977.

[2] Busby, D., Farmer, C.L., and Iske, A. (2007). Uncertainty evaluation in reservoir forecasting by bayes linear methodology, In Algorithms for Approximation, pages 187--196. Springer, 2007. doi:10.1007/978-3-540-46551-5_14

[3] Costa, E., deAbreu, O., Silva, T. d.O., Ribeiro, M., and Schnitman, L. (2021). A bayesian approach to the dynamic modeling of esp-lifted oil well systems: An experimental validation on an esp prototype, Journal of Petroleum Science and Engineering. 205:108880. doi:10.1016/j.petrol.2021.108880

[4] Craig, P., Goldstein, M., Seheult, A., and Smith, J. (1996). Bayes linear strategies for matching hydrocarbon reservoir history, Bayesian statistics. 5:69--95.

[5] Craig, P.S., Goldstein, M., Seheult, A.H., and Smith, J.A. (1997). Pressure matching for hydrocarbon reservoirs: a case study in the use of bayes linear strategies for large computer experiments, In Case studies in Bayesian statistics, pages 37--93. Springer. doi:10.1007/978-1-4612-2290-3_2

[6] Cumming, J.A. and Goldstein, M. (2010). Bayes linear uncertainty analysis for oil reservoirs based on multiscale computer experiments, The Oxford handbook of applied Bayesian analysis. pages 241--270.

[7] Gyamfi, K.S., Brusey, J., Hunt, A., and Gaura, E. (2018). Linear dimensionality reduction for classification via a sequential bayes error minimisation with an application to flow meter diagnostics, Expert Systems with Applications. 91:252--262. doi:10.1016/j.eswa.2017.09.010

[8] Hastings, W.K. (1970). Monte carlo sampling methods using markov chains and their applications, 1970. doi:10.1093/biomet/57.1.97

[9] Holder, M.T., Lewis, P.O., Swofford, D.L., and Larget, B. (2005). Hastings ratio of the local proposal used in bayesian phylogenetics, Systematic biology. 54(6):961--965. doi:10.1080/10635150500354670

[10] Hu, C., Zhang, C., Zhang, Z., and Xie, S. (2021). Comparative study on defects and faults detection of main transformer based on logistic regression and naive bayes algorithm, In Journal of Physics: Conference Series, volume 1732. IOP Publishing, page 012075. doi:10.1088/1742-6596/1732/1/012075

[11] JCGM. (2008). Evaluation of measurement data—guide to the expression of uncertainty in measurement, JCGM. 100(2008):1--116.

[12] Jeffreys, H. (1935). Some tests of significance, treated by the theory of probability, In Mathematical Proceedings of the Cambridge Philosophical Society, volume31. Cambridge University Press, pages 203--222. doi:10.1017/S030500410001330X

[13] Khaliullin, F.K., Matyashin, A.V., Akhmetzyanov, R.R., Medvedev, V.M., and Lushnov, M.A. (2019). Prospects for using the bayes algorithm for assessing the technical condition of internal combustion engines, In IOP Conference Series: Materials Science and Engineering, volume 635. IOP Publishing, page 012016. doi:10.1088/1757-899X/635/1/012016

[14] Kruschke, J. (2014). Doing bayesian data analysis: A tutorial with r, jags, and stan, 2014.

[15] Lambert, B. (2018). A student’s guide to Bayesian statistics, Sage.

[16] Lindley, D.V. (1961). The use of prior probability distributions in statistical inference and decision, In Proc. 4th Berkeley Symp. on Math. Stat. and Prob. pages 453--468.

[17] Maraggi, L. M.R., Lake, L.W., and Walsh, M.P. (2020). A bayesian framework for addressing the uncertainty in production forecasts of tight oil reservoirs using a physics-based two-phase flow model, In SPE/AAPG/SEG Latin America Unconventional Resources Technology Conference. OnePetro. doi:10.15530/urtec-2020-10480

[18] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E. (1953). Equation of state calculations by fast computing machines, The journal of chemical physics. 21(6):1087--1092. doi:10.1063/1.1699114

[19] Narasimhan, S. and Jordache, C. (1999). Data reconciliation and gross error detection: An intelligent use of process data, Elsevier.

[20] Sharma, R., Fjalestad, K., and Glemmestad, B. (2011). Modeling and control of gas lifted oil field with five oil wells, In 52nd International Conference of Scandinavian Simulation Society, SIMS. pages 29--30.

[21] Sivia, D. and Skilling, J. (2006). Data analysis: a Bayesian tutorial, OUP Oxfor.

**BibTeX:**

@article{MIC-2022-2-1,

title={{Parameter Estimation for a Gas Lifting Oil Well Model Using Bayes' Rule and the Metropolis–Hastings Algorithm}},

author={Ban, Zhe and Ghaderi, Ali and Janatian, Nima and Pfeiffer, Carlos F.},

journal={Modeling, Identification and Control},

volume={43},

number={2},

pages={39--53},

year={2022},

doi={10.4173/mic.2022.2.1},

publisher={Norwegian Society of Automatic Control}

};