# A stochastic framework for modeling random-sized batch arrivals of sediment particles into open channel flows

• Christina W. Tsai
• Serena Y. Hung
• Jungsun Oh
Original Paper

## 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.

## Keywords

Sediment transport Random arrival processes Stochastic method Particle tracking model Risk assessment

## List of symbols

B(np)

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)

dBt

Wiener process at time t in three-dimensional vector form

Dk

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

Nd

The concentration of deposition particles in the control volume (bead/m)

Nm

The concentration of moving particles in the control volume (bead/m)

Np

The total number of incoming particles up to time t

pij(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

qii

The ith diagonal element of $$\varvec{Q}$$

SG

The specific gravity

T

The simulation time (s)

Tj

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

ws

The settling velocity (m/s)

Xt

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

λa

The arrival rate of sediment particles (bead/s)

λs

The resuspension rate of sediment particles (bead/s)

μ

The depart rate of sediment particles (bead/s)

μd

The deposition rate of sediment particles (bead/s)

σ (t,Xt)

The diffusion coefficient tensor (m/s1/2)

## Notes

### 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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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Christina W. Tsai
• 1
• Serena Y. Hung
• 1
• Jungsun Oh
• 2
1. 1.Department of Civil EngineeringNational Taiwan UniversityTaipeiTaiwan
2. 2.School of Civil and Environmental EngineeringYonsei UniversitySeoulSouth Korea