{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T07:01:15Z","timestamp":1768806075476,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,2,28]],"date-time":"2020-02-28T00:00:00Z","timestamp":1582848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005416","name":"Norges Forskningsr\u00e5d","doi-asserted-by":"publisher","award":["267620"],"award-info":[{"award-number":["267620"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Uncertainty propagation is used to quantify the uncertainty in model predictions in the presence of uncertain input variables. In this study, we analyze a steady-state point-model for two-phase gas-liquid flow. We present prediction intervals for holdup and pressure drop that are obtained from knowledge of the measurement error in the variables provided to the model. The analysis also uncovers which variables the predictions are most sensitive to. Sensitivity indices and prediction intervals are calculated by two different methods, Monte Carlo and polynomial chaos. The methods give similar prediction intervals, and they agree that the predictions are most sensitive to the pipe diameter and the liquid viscosity. However, the Monte Carlo simulations require fewer model evaluations and less computational time. The model predictions are also compared to experiments while accounting for uncertainty, and the holdup predictions are accurate, but there is bias in the pressure drop estimates.<\/jats:p>","DOI":"10.3390\/a13030053","type":"journal-article","created":{"date-parts":[[2020,2,28]],"date-time":"2020-02-28T09:30:36Z","timestamp":1582882236000},"page":"53","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Uncertainty Propagation through a Point Model for Steady-State Two-Phase Pipe Flow"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7679-7816","authenticated-orcid":false,"given":"Andreas","family":"Strand","sequence":"first","affiliation":[{"name":"Department of Structural Engineering, Faculty of Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3454-1409","authenticated-orcid":false,"given":"Ivar Eskerud","family":"Smith","sequence":"additional","affiliation":[{"name":"SINTEF Multiphase Flow Laboratory, 7491 Trondheim, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6867-0287","authenticated-orcid":false,"given":"Tor Erling","family":"Unander","sequence":"additional","affiliation":[{"name":"SINTEF Multiphase Flow Laboratory, 7491 Trondheim, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ingelin","family":"Steinsland","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Information Technology and Electrical Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leif Rune","family":"Hellevik","sequence":"additional","affiliation":[{"name":"Department of Structural Engineering, Faculty of Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s00158-008-0234-7","article-title":"A comparative study of uncertainty propagation methods for black-box-type problems","volume":"37","author":"Lee","year":"2009","journal-title":"Struct. Multidiscipl. Optim."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4034","DOI":"10.1021\/ef300190p","article-title":"Characterization of confidence in multiphase flow predictions","volume":"26","author":"Cremaschi","year":"2012","journal-title":"Energ. Fuel."},{"key":"ref_3","unstructured":"Holm, H., Saha, P., Suleymanov, V., Vanvik, T., and Hoyer, N. (2011, January 15\u201317). Shtokman flow assurance challenges\u2014A systematic approach to analyze uncertainties\u2013Part 1. Proceedings of the 15th International Conference on Multiphase Production Technology, Cannes, France."},{"key":"ref_4","unstructured":"Holm, H., Saha, P., Suleymanov, V., Vanvik, T., and Hoyer, N. (2011, January 15\u201317). Shtokman flow assurance challenges\u2014A systematic approach to analyze uncertainties\u2013Part 2. Proceedings of the 15th International Conference on Multiphase Production Technology, Cannes, France."},{"key":"ref_5","unstructured":"Hoyer, N., Kirkedelen, M., Biberg, D., Johnson, G., Valle, A., Johansson, P., and Nossen, J. (2013, January 12\u201314). A structured approach for the evaluation of uncertainties in flow assurance systems. Proceedings of the 16th International Conference on Multiphase Production Technology, Cannes, France."},{"key":"ref_6","unstructured":"Klavetter, K., Posluszny, D., Warr, J., Cremaschi, S., Sarica, C., and Subramani, H. (2012, January 20\u201322). Uncertainty analysis of multiphase flow models: A comparison of three propagation approaches. Proceedings of the 8th North American Conference on Multiphase Technology, Banff, AB, Canada."},{"key":"ref_7","unstructured":"Keinath, B. (2017, January 7\u20139). Decision analysis in multiphase flow: Uncertainty, risk and application. Proceedings of the 18th International Conference on Multiphase Production Technology, Cannes, France."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1016\/j.ijheatmasstransfer.2019.04.075","article-title":"Uncertainty quantification of two-phase flow and boiling heat transfer simulations through a data-driven modular Bayesian approach","volume":"138","author":"Liu","year":"2019","journal-title":"Int. J. Heat. Mass. Tran."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.ijmultiphaseflow.2016.12.004","article-title":"Uncertainty quantification and global sensitivity analysis of mechanistic one-dimensional models and flow pattern transition boundaries predictions for two-phase pipe flows","volume":"90","author":"Picchi","year":"2017","journal-title":"Int. J. Multiphas. Flow"},{"key":"ref_10","unstructured":"Klinkert, J. (2018). The characterization of Uncertainty for Steady State Multiphase Flow Models in Pipelines. [Master\u2019s Thesis, Delft University of Technology]."},{"key":"ref_11","unstructured":"Lataniotis, C., Marelli, S., and Sudret, B. (2020, February 26). Uqlab User Manual\u2014The Input Module. Available online: https:\/\/www.researchgate.net\/profile\/Bruno_Sudret\/publication\/281232688_UQLab_user_manual_-_the_INPUT_module\/links\/55dc1df708ae9d6594937608.pdf."},{"key":"ref_12","unstructured":"Smith, I.E., Nossen, J., and Unander, T.E. (2013, January 12\u201314). Improved holdup and pressure drop predictions for multiphase flow with gas and high viscosity oil. Proceedings of the 16th International Conference on Multiphase Production Technology, Cannes, France."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2550","DOI":"10.2514\/1.G004346","article-title":"Frequency Responses Identification from Multi-Axis Maneuver with Simultaneous Multisine Inputs","volume":"42","author":"Lichota","year":"2019","journal-title":"J. Guid. Contr. Dynam."},{"key":"ref_14","unstructured":"Lichota, P., and Nore\u00f1a, D.A. (June, January 29). A priori model inclusion in the multisine maneuver design. Proceedings of the 17th International Carpathian Control Conference (ICCC), Tatranska Lomnica, Slovakia."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1109\/TCST.2007.899728","article-title":"NARX-based nonlinear system identification using orthogonal least squares basis hunting","volume":"16","author":"Chen","year":"2007","journal-title":"IEEE Trans. Contr. Syst. Tech."},{"key":"ref_16","unstructured":"Efremov, A. (2010, January 29). System Identification Based on Stepwise Regression for Dynamic Market Representation. Proceedings of the International Conference on Data Mining and Knowledge Engineering, Rome, Italy."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1021\/i160056a011","article-title":"A model for gas-liquid slug flow in horizontal and near horizontal tubes","volume":"14","author":"Dukler","year":"1975","journal-title":"Ind. Eng. Chem. Fund."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/0301-9322(78)90023-X","article-title":"Correlation of the liquid volume fraction in the slug for horizontal gas-liquid slug flow","volume":"4","author":"Gregory","year":"1978","journal-title":"Int. J. Multiphas. Flow"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kora, C., Sarica, C., Zhang, H.q., Al-Sarkhi, A., and Al-Safran, E. (2011, January 15\u201317). Effects of high oil viscosity on slug liquid holdup in horizontal pipes. Proceedings of the Canadian Unconventional Resources Conference, Calgary, AB, Canada.","DOI":"10.2118\/146954-MS"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/0301-9322(84)90057-0","article-title":"An experimental investigation of the motion of long bubbles in inclined tubes","volume":"10","author":"Bendiksen","year":"1984","journal-title":"Int. J. Multiphas. Flow"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"593","DOI":"10.2118\/151616-PA","article-title":"Drift-velocity closure relationships for slug two-phase high-viscosity oil flow in pipes","volume":"17","author":"Jeyachandra","year":"2012","journal-title":"SPE J."},{"key":"ref_22","unstructured":"Nuland, S. (1998, January 8\u201312). Bubble front velocity in horizontal slug flow with viscous Newtonian, shear thinning and Bingham fluids. Proceedings of the 3rd International Conference on Multiphase Flow (ICMF\u201998), Lyon, France."},{"key":"ref_23","unstructured":"Hinze, J.O. (1975). Turbulence, McGraw-Hill."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/0301-9322(87)90022-X","article-title":"Stratified gas-liquid flow in downwardly inclined pipes","volume":"13","author":"Andreussi","year":"1987","journal-title":"Int. J. Multiphas. Flow"},{"key":"ref_25","unstructured":"White, F.M. (2011). Fluid Mechanics, McGraw-Hill."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Brki\u0107, D., and Praks, P. (2019). Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright \u03c9-function. Mathematics, 7.","DOI":"10.3390\/math7050410"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.ijmultiphaseflow.2014.07.006","article-title":"Investigation of two-phase flow pattern, liquid holdup and pressure drop in viscous oil\u2013gas flow","volume":"67","author":"Khaledi","year":"2014","journal-title":"Int. J. Multiphas. Flow"},{"key":"ref_28","unstructured":"ISO, I., and OIML, B. (2020, February 26). Guide to the Expression of Uncertainty in Measurement. Available online: https:\/\/www.bipm.org\/utils\/common\/documents\/jcgm\/JCGM_100_2008_E.pdf."},{"key":"ref_29","first-page":"407","article-title":"Sensitivity estimates for nonlinear mathematical models","volume":"1","author":"Sobol","year":"1993","journal-title":"Math. Model. Comput. Exp."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.cpc.2009.09.018","article-title":"Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index","volume":"181","author":"Saltelli","year":"2010","journal-title":"Comput. Phys. Comm."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/S0010-4655(02)00280-1","article-title":"Making best use of model evaluations to compute sensitivity indices","volume":"145","author":"Saltelli","year":"2002","journal-title":"Comput. Phys. Comm."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1016\/j.ress.2006.07.001","article-title":"Estimating the approximation error when fixing unessential factors in global sensitivity analysis","volume":"92","author":"Sobol","year":"2007","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0010-4655(98)00154-4","article-title":"Analysis of variance designs for model output","volume":"117","author":"Jansen","year":"1999","journal-title":"Comput. Phys. Comm."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Xiu, D. (2010). Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton university press.","DOI":"10.1515\/9781400835348"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.jocs.2015.08.008","article-title":"Chaospy: An open source tool for designing methods of uncertainty quantification","volume":"11","author":"Feinberg","year":"2015","journal-title":"J. Comput. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1080\/01932691.2014.989568","article-title":"Development of a Steady-State Point Model for Prediction of Gas\/Oil and Water\/Oil Pipe Flow","volume":"36","author":"Smith","year":"2015","journal-title":"J. Dispers. Sci. Technol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1175\/MWR2904.1","article-title":"Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation","volume":"133","author":"Gneiting","year":"2005","journal-title":"Mon. Weather Rev."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/3\/53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:02:30Z","timestamp":1760173350000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/3\/53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,28]]},"references-count":37,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["a13030053"],"URL":"https:\/\/doi.org\/10.3390\/a13030053","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,28]]}}}