Uncertainty Quantification for Damping in Transient Pressure Oscillations
Publication: Journal of Water Resources Planning and Management
Volume 145, Issue 9
Abstract
The uncertainty of a model to determine the damping of pressure head oscillations following a sudden valve closure in a simple piping system in a pressurized closed conduit is quantified using Bayesian inference. The joint probability density of the model parameter is estimated based on experimental results published in the literature as well as experiments performed at the University of South Carolina. A Markov chain of the posterior joint distribution of the model parameters is calculated and used to predict the pressure head oscillations. The prediction is performed in a probabilistic fashion, estimating an interval of pressures as a function of time rather than estimating a single point by a traditional regression analysis. The 95% high posterior density of the damping ratio ranges from 1% to 6%. The probabilistic model also correctly predicts the experimental damping ratios when compared with experimental data.
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Data Availability Statement
The following data, models, or code generated or used during the study are available from the corresponding author by request: data for Test 1 (Fig. 2) and codes for the Bayesian inference process.
Acknowledgments
Many thanks and appreciations to the Ministry of Higher Education and Scientific Research (MOHESR) for providing a scholarship to the first author in coordination with the Northern Technical University and the Iraqi Cultural Office in Washington DC. Also, the authors thank the Higher Committee for Education Development (HCED) in Iraq for providing scholarships to the third author in coordination with the University of Basrah to conduct this research.
References
Adamkowski, A., and M. Lewandowski. 2006. “Experimental examination of unsteady friction models for transient pipe flow simulation.” J. Fluids Eng. 128 (6): 1351–1363. https://doi.org/10.1115/1.2354521.
Bayarri, M. J., and J. O. Berger. 2004. “The interplay of Bayesian and frequentist analysis.” In Statistical science, 58–80. Bethesda, MD: Institute of Mathematical Statistics.
Bayes, M., and M. Price. 1763. “An essay towards solving a problem in the doctrine of chances. By the late Rev. Mr. Bayes, FRS communicated by Mr. Price, in a letter to John Canton, AMFRS.” Philos. Trans. 53: 370–418. https://doi.org/10.1098/rstl.1763.0053.
Beck, J. L., and L. S. Katafygiotis. 1998. “Updating models and their uncertainties. I: Bayesian statistical framework.” J. Eng. Mech. 124 (4): 455–461. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(455).
Behmanesh, I., B. Moaveni, and C. Papadimitriou. 2017. “Probabilistic damage identification of a designed 9-story building using modal data in the presence of modeling errors.” Eng. Struct. 131 (Jan): 542–552. https://doi.org/10.1016/j.engstruct.2016.10.033.
Bonin, C. 1960. “Water-hammer damage to Oigawa Power Station.” J. Eng. Power 82 (2): 111–116. https://doi.org/10.1115/1.3672721.
Bughazem, M., and A. Anderson. 1996. “Problems with simple models for damping in unsteady flow.” In Vol. 19 of Proc., BHR Group Conf. Series Publication, 537–548. Mechanical Engineering Publications.
Carlsson, J. 2016. “Water hammer phenomenon analysis using the method of characteristics and direct measurements using a ‘stripped’ electromagnetic flow meter.” M.S. thesis, Dept. of Engineering Physics, Royal Institute of Technology.
Chaudhry, M. H. 2014. Applied hydraulic transients, 3rd edn. New York: Springer.
Chib, S., and E. Greenberg. 1995. “Understanding the Metropolis-Hastings algorithm.” Am. Stat. 49 (4): 327–335. https://doi.org/10.1080/00031305.1995.10476177.
Ching, J., and J. Beck. 2004. “Bayesian analysis of the phase. II: IASC-ASCE structural health monitoring experimental benchmark data.” J. Eng. Mech. 130 (10): 1233–1244. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:10(1233).
Der Kiureghian, A., and O. Ditlevsen. 2009. “Aleatory or epistemic? Does it matter?” Struct. Saf. 31 (2): 105–112. https://doi.org/10.1016/j.strusafe.2008.06.020.
Duan, H. 2016. “Sensitivity analysis of a transient-based frequency domain method for extended blockage detection in water pipeline systems.” J. Water Resour. Plann. Manage. 142 (4): 04015073. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000625.
Duan, H.-F. 2015. “Uncertainty analysis of transient flow modeling and transient-based leak detection in elastic water pipeline systems.” Water Resour. Manage. 29 (14): 5413–5427. https://doi.org/10.1007/s11269-015-1126-4.
Duan, H.-F., M. S. Ghidaoui, P. J. Lee, and Y.-K. Tung. 2012. “Relevance of unsteady friction to pipe size and length in pipe fluid transients.” J. Hydr. Eng. 138 (2): 154–166. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000497.
Duan, H.-F., Y.-K. Tung, and M. S. Ghidaoui. 2010. “Probabilistic analysis of transient design for water supply systems.” J. Water Resour. Plann. Manage. 136 (6): 678–687. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000074.
EL-Turki, A. 2013. “Modeling of hydraulic transients in closed conduits.” Ph.D. thesis, Dept. of Civil and Environmental Engineering, Colorado State Univ.
Garthwaite, P. H., J. B. Kadane, and A. O’Hagan. 2005. “Statistical methods for eliciting probability distributions.” J. Am. Stat. Assoc. 100 (470): 680–701. https://doi.org/10.1198/016214505000000105.
Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin. 2014. Vol. 2 of Bayesian data analysis. Boca Raton, FL: CRC Press.
Geweke, J. 1991. Vol. 196 of Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Minneapolis: Federal Reserve Bank of Minneapolis, Research Dept.
Ghidaoui, M. S., M. Zhao, D. A. McInnis, and D. H. Axworthy. 2005. “A review of water hammer theory and practice.” Appl. Mech. Rev. 58 (1): 49–76. https://doi.org/10.1115/1.1828050.
Hernandez, E., and P. Nestor. 2016. “A lower bound for the variance of frequency and damping ratio identified from noisy vibration measurements.” J. Struct. Control Health Monit. 23 (1): 5–19. https://doi.org/10.1002/stc.1757.
Hoffman, M. D., and A. Gelman. 2014. “The no-u-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo.” J. Mach. Learn. Res. 15 (1): 1593–1623.
Holmboe, E., and W. Rouleau. 1967. “The effect of viscous shear on transients in liquid lines.” J. Basic Eng. 89 (1): 174–180. https://doi.org/10.1115/1.3609549.
Jaynes, E. T. 2003. Probability theory: The logic of science. Cambridge, UK: Cambridge University Press.
Kapelan, Z. S., D. A. Savic, and G. A. Walters. 2007. “Calibration of water distribution hydraulic models using a Bayesian-type procedure.” J. Hydr. Eng. 133 (8): 927–936. https://doi.org/10.1061/(ASCE)0733-9429(2007)133:8(927).
Katafygiotis, L. S., and J. L. Beck. 1998. “Updating models and their uncertainties. II: Model identifiability.” J. Eng. Mech. 124 (4): 463–467. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(463).
Khilqa, S., M. Elkholy, M. Al-Tofan, J. Caicedo, and M. Chaudhry. 2017. “Damping rate of transient oscillations in pressurized closed conduits.” In Proc., 37th IAHR World Congress on Transients in Pipes, 5870–5875. Kuala Lumpur.
Kim, S., P. Parhi, H. Jun, and J. Lee. 2018. “Evaluation of drought severity with a Bayesian network analysis of multiple drought indices.” J. Water Resour. Plann. Manage. 144 (1): 05017016. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000804.
Martins, N. M., A. K. Soares, H. M. Ramos, and D. I. Covas. 2016. “CFD modeling of transient flow in pressurized pipes.” Comput. Fluids 126 (Mar): 129–140. https://doi.org/10.1016/j.compfluid.2015.12.002.
Meniconi, S., H. Duan, B. Brunone, M. S. Ghidaoui, P. Lee, and M. Ferrante. 2014. “Further developments in rapidly decelerating turbulent pipe flow modeling.” J. Hydr. Eng. 140 (7): 04014028. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000880.
Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. 1953. “Equation of state calculations by fast computing machines.” J. Chem. Phys. 21 (6): 1087–1092. https://doi.org/10.1063/1.1699114.
Neal, R. M. 2011. “MCMC using Hamiltonian dynamics.” In Vol. 2 of Handbook of Markov chain Monte Carlo. Boca Raton, FL: CRC Press.
Nerella, R., and E. V. Rathnam. 2015. “Fluid transients and wave propagation in pressurized conduits due to valve closure.” Procedia Eng. 127: 1158–1164. https://doi.org/10.1016/j.proeng.2015.11.454.
O’Hagan, A. 1998. “Eliciting expert beliefs in substantial practical applications.” J. R. Stat. Soc. Series D Stat. 47 (1): 21–35. https://doi.org/10.1111/1467-9884.00114.
Pezzinga, G., and P. Scandura. 1995. “Unsteady flow in installations with polymeric additional pipe.” J. Hydr. Eng. 121 (11): 802–811. https://doi.org/10.1061/(ASCE)0733-9429(1995)121:11(802).
Poulakis, Z., D. Valougeorgis, and C. Papadimitriou. 2003. “Leakage detection in water pipe networks using a Bayesian probabilistic framework.” Probab. Eng. Mech. 18 (4): 315–327. https://doi.org/10.1016/S0266-8920(03)00045-6.
Puust, R., Z. Kapelan, D. Savic, and T. Koppel. 2008. “Probabilistic leak detection in pipe networks using the SCEM-UA algorithm.” In Proc., Water Distribution Systems Analysis Symp. 2006, 1–12. Reston, VA: ASCE.
Reddy, P., W. F. Silva-Araya, and M. Hanif Chaudhry. 2012. “Estimation of decay coefficients for unsteady friction for instantaneous, acceleration-based models.” J. Hydr. Eng. 138 (3): 260–271. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000508.
Sattar, A. M. 2016. “A probabilistic projection of the transient flow equations with random system parameters and internal boundary conditions.” J. Hydraul. Res. 54 (3): 342–359. https://doi.org/10.1080/00221686.2016.1140682.
Sattar, A. M., and M. El-Beltagy. 2016. “Stochastic solution to the water hammer equations using polynomial chaos expansion with random boundary and initial conditions.” J. Hydr. Eng. 143 (2): 04016078. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001227.
Tanner, M. A., and W. H. Wong. 1987. “The calculation of posterior distributions by data augmentation.” J. Am. Stat. Assoc. 82 (398): 528–540. https://doi.org/10.1080/01621459.1987.10478458.
Thyer, M., B. Renard, D. Kavetski, G. Kuczera, S. W. Franks, and S. Srikanthan. 2009. “Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: A case study using Bayesian total error analysis.” Water Resour. Res. 45 (12): W00B14. https://doi.org/10.1029/2008WR006825.
Trenkle, C. J. 1979. “Failure of riveted and forge-welded penstock.” J. Energy Div. 105 (1): 93–102.
Wahba, E. 2016. “On the propagation and attenuation of turbulent fluid transients in circular pipes.” J. Fluids Eng. 138 (3): 031106. https://doi.org/10.1115/1.4031557.
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Published In
Journal of Water Resources Planning and Management
Volume 145 • Issue 9 • September 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Jul 17, 2018
Accepted: Jan 18, 2019
Published online: Jul 12, 2019
Published in print: Sep 1, 2019
Discussion open until: Dec 12, 2019
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