Abstract
Arrivals of sediment particles into open channel flows are typically simulated deterministically. However, there are random quantities of sediment particles brought into the open channel by natural phenomena such as rainfalls, landslides, mudflows, and dam breaches. As such, sediment particles may arrive at channel flow at random times in probabilistic quantities. Such arrival processes of sediment particles cannot be fully described without using probability. A stochastic framework that can account for the arrivals of random-sized batches of sediment particles into receiving waters is proposed. In this work, the random-sized batch arrival process of sediment particles is introduced to evaluate comprehensively the effects of particle arrival patterns, including random occurrences and random quantities of incoming sediment particles, on particle transport rates and sediment concentrations. Random arrivals are simulated as a Poisson process and the number of sediment particles in each arriving batch is described by a stochastic process that is specified in terms of binomially distributed random variables herein. The stochastic diffusion particle tracking model is used to simulate random trajectories of moving particles. Particle deposition and resuspension processes are considered. A probabilistic description of discrete sediment transport based on ensemble statistics of sediment concentrations and transport rates is presented. Simulation results are validated with experimental observations. Finally, time-dependent risks (i.e., probability of exceeding a pre-established turbidity standard with respect to time) are provided for decision makers to have a more comprehensive assessment for water quality management.
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Abbreviations
- B(n, p):
-
The binomial distribution with parameters n ∈ N and p ∈ [0, 1]. p is the probability of success in one trial
- D :
-
\(\varvec{D} = \{ D_{x} , D_{y} ,D_{z} \}\), the turbulence diffusivity
- d :
-
Particle diameter (m)
- d B t :
-
Wiener process at time t in three-dimensional vector form
- D k :
-
The representative matrix of a BMAP with batch size k
- E :
-
The state space
- g :
-
The gravity acceleration (m/s2)
- H :
-
The flow height (m)
- J(t):
-
The phase process
- κ :
-
The Von Karman constant
- L :
-
The length of the control volume (m)
- N :
-
The concentration of all particles in the control volume (bead/m)
- N(t):
-
The counting process
- N d :
-
The concentration of deposition particles in the control volume (bead/m)
- N m :
-
The concentration of moving particles in the control volume (bead/m)
- N p :
-
The total number of incoming particles up to time t
- p ij (k):
-
The transition probability from sate i to state j with batch size k
- \(P_{{v^{\prime } }}\) :
-
The probability of vertical velocity fluctuation v′
- Q :
-
The infinitesimal generator matrix
- q ii :
-
The ith diagonal element of \(\varvec{Q}\)
- SG :
-
The specific gravity
- T :
-
The simulation time (s)
- T j :
-
The interarrivals
- \(\bar{\varvec{U}}\) :
-
\(\bar{\varvec{U}} = \left( {\bar{U},\bar{V},\bar{W}} \right)\), the mean drift flow velocity (m/s)
- \(\bar{\varvec{u}}\left( {t,\varvec{X}_{t} } \right)\) :
-
The drift velocity vector (m/s)
- u * :
-
The shear velocity (m/s)
- w′:
-
The velocity fluctuations in vertical direction
- \(\hat{w}\) :
-
\(\hat{w} = w^{\prime } /\sigma_{2}\), σ2 is the root-mean square of w′
- w s :
-
The settling velocity (m/s)
- X t :
-
The sediment particle position, X t = {X(t), Y(t), Z(t)}T
- Δt :
-
The time step (s)
- λ :
-
The mean rate of a Poisson process
- λ a :
-
The arrival rate of sediment particles (bead/s)
- λ s :
-
The resuspension rate of sediment particles (bead/s)
- μ :
-
The depart rate of sediment particles (bead/s)
- μ d :
-
The deposition rate of sediment particles (bead/s)
- σ (t, X t ):
-
The diffusion coefficient tensor (m/s1/2)
References
Abiodun BJ, Adegoke J, Abatan AA, Ibe CA, Egbebiyi TS, Engelbrecht F, Pinto I (2017) Potential impacts of climate change on extreme precipitation over four African coastal cities. Clim Change 143(3–4):399–413
Afan HA, El-shafie A, Mohtar WHMW, Yaseen ZM (2016) Past, present and prospect of an artificial intelligence (AI) based model for sediment transport prediction. J Hydrol 541:902–913
Ancey C, Davison AC, Bohm T, Jodeau M, Frey P (2008) Entrainment and motion of coarse particles in a shallow water stream down a steep slope. J Fluid Mech 595:83–114. https://doi.org/10.1017/S0022112007008774
Apitz SE, Power EA (2002) From risk assessment to sediment management an international perspective. J Soils Sediments 2(2):61–66
Armanini A, Cavedon V, Righetti M (2015) A probabilistic/deterministic approach for the prediction of the sediment transport rate. Adv Water Resour 81:10–18. https://doi.org/10.1016/j.advwatres.2014.09.008
Batchelor GK (1977) The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J Fluid Mech 83(01):97–117. https://doi.org/10.1017/S0022112077001062
Bennett G, Molnar P, McArdell B, Burlando P (2014) A probabilistic sediment cascade model of sediment transfer in the Illgraben. Water Resour Res 50(2):1225–1244
Bohorquez P, Ancey C (2015) Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: linear stability analysis and numerical simulation. Adv Water Resour 83:36–54
Bose SK, Dey S (2013) Sediment entrainment probability and threshold of sediment suspension: exponential-based approach. J Hydraul Eng 139(10):1099–1106
Cheng N-S, Chiew Y-M (1998) Pickup probability for sediment entrainment. J Hydraul Eng 124(2):232–235
Cheng N-S, Chiew Y-M (1999) Analysis of initiation of sediment suspension from bed load. J Hydraul Eng 125(8):855–861
Coleman NL (1986) Effects of suspended sediment on the open-channel velocity distribution. Water Resour Res 22(10):1377–1384
De Vincenzo A, Brancati F, Pannone M (2016) An experimental analysis of bed load transport in gravel-bed braided rivers with high grain Reynolds numbers. Adv Water Resour 94:160–173. https://doi.org/10.1016/j.advwatres.2016.05.007
Dimou KN, Adams EE (1993) A random-walk, particle tracking model for well-mixed estuaries and coastal waters. Estuar Coast Shelf Sci 37(1):99–110. https://doi.org/10.1006/ecss.1993.1044
Einstein HA (1950) The bed-load function for sediment transportation in open channel flows, vol 1026. United States Department of Agriculture, U.S. Government Printing Office, Washington, D.C.
Fan N, Singh A, Guala M, Foufoula-Georgiou E, Wu B (2016) Exploring a semimechanistic episodic Langevin model for bed load transport: emergence of normal and anomalous advection and diffusion regimes. Water Resour Res 52(4):2789–2801
Farmer WH, Vogel RM (2016) On the deterministic and stochastic use of hydrologic models. Water Resour Res. https://doi.org/10.1002/2016WR019129
Kloeden PE, Platen E, Schurz H (1994) Numerical solution of SDE through computer experiments. Springer, New York
Kondolf GM, Gao Y, Annandale GW, Morris GL, Jiang E, Zhang J, Cao Y, Carling P, Fu K, Guo Q (2014) Sustainable sediment management in reservoirs and regulated rivers: experiences from five continents. Earth’s Future 2(5):256–280
Lisle IG, Rose CW, Hogarth WL, Hairsine PB, Sander GC, Parlange JY (1998) Stochastic sediment transport in soil erosion. J Hydrol 204(1):217–230. https://doi.org/10.1016/S0022-1694(97)00123-6
Liu B, Xu M, Henderson M, Qi Y (2005) Observed trends of precipitation amount, frequency, and intensity in China, 1960–2000. J Geophys Res Atmos 110(D8):D08103
López R, Vericat D, Batalla RJ (2014) Evaluation of bed load transport formulae in a large regulated gravel bed river: the lower Ebro (NE Iberian Peninsula). J Hydrol 510:164–181
Malmon DV, Dunne T, Reneau SL (2003) Stochastic theory of particle trajectories through alluvial valley floors. J Geol 111(5):525–542. https://doi.org/10.1086/376764
Man C (2007) Stochastic modeling of suspended sediment transport in regular and extreme flow environments. State University of New York, Buffalo
Man C, Tsai CW (2007) Stochastic partial differential equation-based model for suspended sediment transport in surface water flows. J Eng Mech 133(4):422–430
Meyer-Peter E, Müller R (1948) Formulas for bed-load transport. In: IAHSR 2nd meeting, Stockholm, Sweden. IAHR, pp 39–64
Muste M, Yu K, Fujita I, Ettema R (2009) Two-phase flow insights into open-channel flows with suspended particles of different densities. Environ Fluid Mech 9(2):161–186. https://doi.org/10.1007/s10652-008-9102-7
Nardin W, Edmonds DA, Fagherazzi S (2016) Influence of vegetation on spatial patterns of sediment deposition in deltaic islands during flood. Adv Water Resour 93:236–248
Oh J, Tsai CW (2010) A stochastic jump diffusion particle-tracking model (SJD-PTM) for sediment transport in open channel flows. Water Resour Res. https://doi.org/10.1029/2009wr008443
Oh J, Tsai CW (2018) A stochastic multivariate framework for modeling movement of discrete sediment particles in open channel flows. Stoch Env Res Risk Assess 32(2):385–399
Parker G, Andrews ED (1985) Sorting of bed load sediment by flow in meander bends. Water Resour Res 21(9):1361–1373. https://doi.org/10.1029/WR021i009p01361
Pendergrass AG, Hartmann DL (2014) Changes in the distribution of rain frequency and intensity in response to global warming. J Clim 27(22):8372–8383
Priya KL, Jegathambal P, James EJ (2016) Salinity and suspended sediment transport in a shallow estuary on the east coast of India. Reg Stud Mar Sci 7:88–99. https://doi.org/10.1016/j.rsma.2016.05.015
Ramesh NI, Onof C, Xie D (2012) Doubly stochastic Poisson process models for precipitation at fine time-scales. Adv Water Resour 45:58–64
Ross SM (2007) Introduction to probability models. Academic Press, Burlington
Safari M-J-S, Aksoy H, Mohammadi M (2016) Artificial neural network and regression models for flow velocity at sediment incipient deposition. J Hydrol 541:1420–1429
Schumer R, Meerschaert MM, Baeumer B (2009) Fractional advection-dispersion equations for modeling transport at the Earth surface. J Geophys Res Earth Surf. https://doi.org/10.1029/2008jf001246
Tregnaghi M, Bottacin-Busolin A, Marion A, Tait S (2012a) Stochastic determination of entrainment risk in uniformly sized sediment beds at low transport stages: 1. Theory. J Geophys Res Earth Surf 117(F4):F04004
Tregnaghi M, Bottacin-Busolin A, Tait S, Marion A (2012b) Stochastic determination of entrainment risk in uniformly sized sediment beds at low transport stages: 2. Experiments. J Geophys Res Earth Surf 117(F4):F04005
Tsai CW, Hsu Y, Lai K-C, Wu N-K (2014) Application of gambler’s ruin model to sediment transport problems. J Hydrol 510:197–207
Visser AW (1997) Using random walk models to simulate the vertical distribution of particles in a turbulent water column. Mar Ecol Prog Ser 158:275–281
Wu F-C, Jiang M-R (2007) Numerical investigation of the role of turbulent bursting in sediment entrainment. J Hydraul Eng 133(3):329–334
Wu F-C, Lin Y-C (2002) Pickup probability of sediment under log-normal velocity distribution. J Hydraul Eng 128(4):438–442. https://doi.org/10.1061/(ASCE)0733-9429(2002)128:4(438)
Yen BC (2002) Stochastic inference to sediment and fluvial hydraulics. J Hydraul Eng 128(4):365–367
Yettella V, Kay JE (2017) How will precipitation change in extratropical cyclones as the planet warms? Insights from a large initial condition climate model ensemble. Clim Dyn 49(5–6):1765–1781
Yoo C, Kim D, Kim T-W, Hwang K-N (2008) Quantification of drought using a rectangular pulses Poisson process model. J Hydrol 355(1–4):34–48
Acknowledgements
This research is supported by Taiwan Ministry of Science and Technology (MOST) under Grant Contract No. 104-2628-E-002-011-MY3. Financial support from US National Science Foundation under Grant Contract No. EAR-0748787 to the first author is greatly appreciated. The authors acknowledge that the data used in this manuscript are listed in the tables.
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Tsai, C.W., Hung, S.Y. & Oh, J. A stochastic framework for modeling random-sized batch arrivals of sediment particles into open channel flows. Stoch Environ Res Risk Assess 32, 1939–1954 (2018). https://doi.org/10.1007/s00477-018-1529-x
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DOI: https://doi.org/10.1007/s00477-018-1529-x