Abstract
A nonhomogeneous discrete-time three-state Markov chain model is developed in this study to quantify the bedload and suspended load discharge under unsteady flow for mixed size sediment particles. When flow is subject to sudden changes, the particle holding time, defined as the amount of time for a sediment particle staying on the bed or in the moving state, needs to be carefully evaluated. The time step used in this study for single-step motion in the discrete-time Markov chain is represented by a characteristic timescale for particle motion. The transition probabilities are functions of flow conditions and particle properties. Specifically, the likelihood of particle movement between the bedload layer and the bed surface is evaluated by the entrainment probability. Exchange of sediment particles between the bedload layer and suspended load layer is quantified by the suspension probability. A nonhomogeneous Markov chain ensures the transition probabilities are time dependent as they are a function of local conditions. The proposed sediment transport model can be used to calculate both bedload and suspended load as a function of time for any size fraction of mixed size sediment particles. The proposed sediment transport relation is compared with a classic sediment transport formula. The comparison shows that the proposed model performs better for particles of smaller grain size. It is also demonstrated that the quasi-steady assumption normally adopted in flow-sediment modeling appears to be a valid approximation when the time for the sediment transport rate to reach an equilibrium state is sufficiently small. The proposed model is also validated against unsteady bedload transport flume data, as well as flume data containing both bedload and suspended load.
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Acknowledgments
The writers gratefully acknowledge the financial support from the National Science Foundation under grant number #0748787.
References
Ancey, C. (2010). “Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions.” J. Geophys. Res. Earth Surface, 115(F2), F00A11.
Ancey, C., Böhm, T., Jodeau, M., and Frey, P. (2006). “Statistical description of sediment transport experiments.” Phys. Rev. E, 74(1), 011302.
Ancey, C. A., Davison, C., Böhm, T., Jodeau, M., and Frey, P. (2008). “Entrainment and motion of coarse particles in a shallow water stream down a steep slope.” J. Fluid Mech., 595, 83–114.
Cheng, N. S. (2005). “Analysis of bedload transport in laminar flows.” Adv. Water Resour., 27(9), 937–942.
Cheng, N. S., and Chiew, Y. M. (1998). “Pick-up probability for sediment entrainment.” J. Hydraul. Eng., 232–235.
Cheng, N. S., and Chiew, Y. M. (1999). “Analysis of initiation of sediment suspension from bed load.” J. Hydraul. Eng., 855–861.
Einstein, H. A. (1937). Bedload transport as probability problem sedimentation, Water Resources Publications, CO, 105–108.
Einstein, H. A. (1950). “The bed load function for sediment transportation in open channel flow.” U.S. Dept. of Agriculture, Washington, DC.
Fang, H. W., and Rodi, W. (2003). “Three-dimensional calculations of flow and suspended sediment transport in the neighborhood of the dam for the Three Gorges Project (TGP) reservoir in the Yangtze River.” J. Hydraul. Res., 41(4), 379–394.
Gomez, B., and Church, M. (1989). “An assessment of bedload sediment transport formulae for gravel bed rivers.” Water Resour. Res., 25(6), 1161–1186.
Jaffe, P. R., and Ferrara, R. A. (1984). “Modeling sediment and water column interactions for hydrophobic pollutants. Parameter discrimination and model response to input uncertainty.” Water Res., 18(9), 1169–1174.
Kirkup, L., and Frenkel, R. B. (2006). “An introduction to uncertainty in measurement: Using the GUM (guide to the expression of uncertainty in measurement).” Cambridge University Press, Cambridge, U.K.
Krishnan, V. (2006). Probability and random processes, Wiley, Hoboken, NJ.
Kuai, K. Z., and Tsai, C. (2012). “Identification of varying time scales in sediment transport using the Hilbert Huang transform method.” J. Hydrol., 420, 245–254.
Malmon, D. V., Dunne, T. S., and Reneau, S. L. (2003). “Stochastic theory of particle trajectories through alluvial valley floors.” J. Geol., 111(5), 525–542.
Niño, Y., and García, M. (1998). “Using Lagrangian particle saltation observations for bedload sediment transport modelling.” Hydrol. Processes, 12(8), 1197–1218.
Papanicolaou, A. N., Diplas, P., Evaggelopoulos, N., and Fotopoulos, S. (2002). “Stochastic incipient motion criterion for spheres under various bed packing conditions.” J. Hydraul. Eng., 369–380.
Ross, S. M. (2000). Introduction to probability models, Academic Press, San Diego.
Samaga, B. R., Raju, K. G. R., and Garde, R. J. (1986a). “Bed load transport of sediment mixtures.” J. Hydraul. Eng., 1003–1017.
Samaga, B. R., Raju, K. G. R., and Garde, R. J. (1986b). “Suspended load transport of sediment mixtures.” J. Hydraul. Eng., 1019–1034.
Sandmann, W. (2008). “Discrete-time stochastic modeling and simulation of biochemical networks.” Comput. Biol. Chem., 32(4), 292–297.
Song, T., and Graf, W. H. (1997). “Experimental study of bedload transport in unsteady open-channel flow.” Int. J. Sediment Res., 12(3), 63–71.
Sun, Z., and Donahue, J. (2000). “Statistically derived bedload formula for any fraction of nonuniform sediment.” J. Hydraul. Eng., 105–111.
Tsai, C., Kuai, K. Z., Bursik, M., and Hess, D. (2005). “Applicability and experimental verification of a dam-break flood and sediment transport model.” Proc., World Water and Environmental Resources Congress, EWRI, Anchorage, AK, 10.
Tsai, C. W., and Franceschini, S. (2005). “Evaluation of probabilistic point estimate methods in uncertainty analysis for environmental engineering applications.” J. Environ. Eng., 131(3), 387–395.
Tsai, C. W., and Lai, K. C. (2014). “Three-state continuous-time Markov chain model for mixed size sediment particle transport.” J. Hydraul. Eng., 04014047.
Tsai, C. W., and Yang, F.-N. (2013). “Modeling bedload transport using a three-state continuous-time Markov chain model.” J. Hydraul. Eng., 1265–1276.
Turowski, J. M. (2009). “Stochastic modeling of the cover effect and bedrock erosion.” Water Resour. Res., 45(3), W03422.
Vanoni, V. A., and Brooks, N. H. (1957). “Laboratory studies of the roughness and suspended load of alluvial streams.”, California Institute of Technology, Pasadena, CA.
Van Rijn, L. C. (1984a). “Sediment transport. Part II: Suspended load transport.” J. Hydraul. Eng., 1613–1641.
Van Rijn, L. C. (1984b). “Sediment transport. Part I: Bed load transport.” J. Hydraul. Eng., 1431–1456.
Wu, F. C., and Chen, C. C. (2009). “Beyesian updating of parameters for a sediment entrainment model via Markov chain Monte Carlo.” J. Hydraul. Eng., 22–37.
Wu, F.-C., and Chou, Y.-J. (2003). “Rolling and lifting probabilities for sediment entrainment.” J. Hydraul. Eng., 110–119.
Wu, F.-C., and Yang, K.-H. (2004). “A stochastic partial transport model for mixed-size sediment: Application to assessment of fractional mobility.” Water Resour. Res., 40(4), w04501.
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© 2016 American Society of Civil Engineers.
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Received: Dec 21, 2014
Accepted: Feb 2, 2016
Published online: Jun 30, 2016
Published in print: Nov 1, 2016
Discussion open until: Nov 30, 2016
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