Model help:AgDegBW

AgDegBW

It is the Calculator for aggradation and degradation of a river reach using a backwater formulation. This program computes 1D bed variation in rivers due to differential sediment transport. The sediment is assumed to be uniform with size D. All sediment transport is assumed to occur in a specified fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler relation is used for bed resistance. A generic Meyer-Peter Muller relation is used for sediment transport. The flow is computed using a backwater formulation for gradually varied flow.

Model introduction

The model computes variation in river bed level η(x, t), where x denotes a streamwise coordinate and t denotes time, in a river with constant width B. The bed sediment is characterized in terms of a single grain size D and submerged specific gravity R. The reach under consideration has length L. Water surface elevation at the downstream end is prescribed. The model is based on a calculation of total bed material load. The model is 1D, assumes a rectangular channel and neglects wall or bank effects.

By modifying the upstream sediment feed rate G_tf and/or the downstream water surface elevation ξ_d, the river can be forced to aggrade or degrade to a new equilibrium. The program computes this evolution.

Model parameters

First tab header

Second tab header

Etc. tab header

Parameter	Description	Unit
First parameter	Description parameter	[Units]

Parameter	Description	Unit
First parameter	Description parameter	[Units]

Uses ports

This will be something that the CSDMS facility will add

Provides ports

This will be something that the CSDMS facility will add

Main equations

• Computation of the flow

The backwater equation

[math]\displaystyle{ {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - F_{r} ^2}} }[/math]

(1)

[math]\displaystyle{ S_{f} = C_{f} F_{r} ^2 }[/math]

(2)

[math]\displaystyle{ F_{r} ^2 = {\frac{U^2}{g H}} = {\frac{q_{w} ^2}{g H^3}} }[/math]

(3)

[math]\displaystyle{ U = {\frac{q_{w}}{H}} }[/math]

(4)

• Manninbg-Strickler resistance

[math]\displaystyle{ C_f ^ \left ( {\frac{-1}{2}} \right ) = C_{z} = \alpha_{r} \left ( {\frac{H}{k_{c}}} \right ) ^{\frac{1}{6}} }[/math]

(5)

[math]\displaystyle{ k_{s} = n_{k} D }[/math]

(6)

[math]\displaystyle{ S = -{\frac{\eta}{x}} }[/math]

(7)

[math]\displaystyle{ \epsilon = \eta + H }[/math]

(8)

• Shields number

[math]\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} = {\frac{C_{f} U^2}{R g D}} = {\frac{C_{f} {\frac{q_{w} ^2}{H^2}}}{R g D}} }[/math]

(9)

[math]\displaystyle{ \tau_{b} = \rho C_{f} U^2 }[/math]

(10)

[math]\displaystyle{ R = {\frac{\rho_{s}}{\rho}} - 1 }[/math]

(11)

• Computation of the sediment transport

equation of the type of Meyer-Peter and Muller

• τ^* > τ_c ^*

[math]\displaystyle{ q_{t} ^* = \alpha_{t} \left ( \phi_{s} \tau ^2 - \tau_{c} ^* \right ) ^ \left ( n_{t} \right ) }[/math]

(12)

• τ^* <= τ_c ^*

[math]\displaystyle{ q_{t} ^* = 0 }[/math]

(13)

[math]\displaystyle{ G_{t} = \rho_{s} q_{t} Bl_{f} t_{a} }[/math]

(14)

[math]\displaystyle{ q_{t} = {\frac{G_{tf}}{\rho_{s}Bl_{f} t_{a}}} }[/math]

(15)

• Computation of bed variation

Exner equation

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\eta}{t}} = - {\frac{q_{t}}{x}} }[/math]

(16)

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\eta}{t}}= - {\frac{l_{f} q_{t}}{x}} }[/math]

(17)

Nomenclature

Symbol	Description	Unit
Q	flood discharge	m 3 / s
x	imposed water surface elevation	m
η	river bed level	-
t	time step	year
B	channel width	m
D	grain size	mm
l	bed porosity	-
R	submerged specific gravity	-
ξd	downstream water surface elevation	m
qw	water discharge per unit width	m2 / s
H	depth	m
kc	the composite roughness height	m
S	ambient bed slope	-
G	imposed annual sediment transfer rate from upstream	tons / annum
L	length of reach	m
i	number of time steps per printout	-
p	number of printouts desired	-
M	intervals	-
u	upwinding coefficient (1 = full upwind, 0.5 = central difference)	-
r	coefficient in Manning-Strickler Resistance relation	-
a	coefficient in sediment transport relation	-
n	exponent in sediment transport relation	-
P	fraction of bed shear stress that is Skin Friction	-
R	submerged specific gravity of sediment	-
Sf	friction slope	-
Fr	Froude number	-
U	flow velocity	m / s
Cf	bed friction coefficient	-
g	acceleration of gravity	m/ s^2
k	roughness height	mm
αr	dimensionless coefficient between 8 and 9	-
kc	absent roughness height	m
ks	grain roughness	-
nk	dimensionless coefficient typically between 2 and 5	-
τ*	Shield number	-
τb	bed shear stress	kg / (s^2 m)
ρ	fluid density	kg / m3
ρs	sediment density	kg / m3
τc	critical Shields number for the onset of sediment motion	-
ϕs	the fraction of bed shear stress	-
qt *	Einstein number	-
qt	volume sediment transport rate per unit width	-
lf	intermittency	-
tf	cumulative time the river has been in flood	s
Gt	the annual sediment yield	tone/yr
ta	the number of seconds in a year	-
λp	the porosity of the bed deposit	-
Qf	sediment transport rate during flood discharge

Output

Symbol	Description	Unit
x	downstream coordinate	m
η	bed surface elevatioon	m
Sl	slope of the bed surfacew	-
H	water depth	m
τ	shear stress on bed surface	N / m2
qb	bed material load	tons / year

Notes

Actual rivers tend to be morphologically active only during floods. That is, most of the time they are not doing much to modify their morphology. The simplest way to take this into account is to assume an intermittency If such that the river is in flood a fraction I of the time, during which Q = Qf and qt is the sediment transport rate at this discharge (Paola et al., 1992). For the other (1 – If) fraction of time the river is assumed not to be moving sediment.

Output is controlled by the parameters Ntoprint and Nprint. The code will implement Ntoprint time steps. In addition to output pertaining to the initial state, the code implements Nprint outputs, to that the total number of time steps executed is equal to Nprint x Ntoprint.

Examples

An example run with input parameters, BLD files, as well as a figure / movie of the output

Follow the next steps to include images / movies of simulations:

• Upload file: http://csdms.colorado.edu/wiki/Special:Upload
• Create link to the file on your page: [[Image:<file name>]].

See also: Help:Images or Help:Movies

Developer(s)

Gary Parker

References

• Paola, C., Heller, P. L. & Angevine, C. L. 1992 The large-scale dynamics of grain-size variation in alluvial basins. I: Theory. Basin Research, 4, 73-90.

• Meyer-Peter, E., and Müller, R. 1948 Formulas for bed-load transport. Proceedings, 2nd Congress International Association for Hydraulic Research, Rotterdam, the Netherlands, 39-64.

Links

• [Model:AgDegBW]