Model help:DeltaBW

DeltaBW

This is used to Calculate evolution of long profile of a river ending in a 1D migrating delta, using a backwater formulation.

Model introduction

This program calculates bed surface evolution for a narrowly channelized1D fan-delta prograding into standing water using a backwater formulation, as well as calculating the final water surface of the system and the mass balance of sediment in the system.

Model parameters

Input Files and Directories

Run options

Run Parameters

About

Parameter	Description	Unit
Input directory	path to input files
Site prefix	Site prefix for Input/Output files
Case prefix	Case prefix for Input/Output files

Parameter	Description	Unit
Chezy Or Manning, Chezy-1 or Manning-2

Parameter	Description	Unit
Flood discharge (q)		m2 / s
Intermittency (I)	flood intermittency	-
upstream bed material sediment feed rate during flood (Q)		m2 / m
Grain size of bed material (D)		mm
Chezy resistance coefficient (C)	cofficient in the Chezy relation	-
Exponent in load relation (n)		-
Critical Shields stress in load relation (T)		-
Elevation of top of forest (E)		m
Initial elevation of forest bottom (e)		m
Water surface elevation of lake (x)		m
Initial fluvial bed slope (f)		-
Subaqueous basement slope (b)		-
initial length of fluvial zone (s)		m
Maximum length of fluvial zone (m)		m
Slope of forest face (Sa)		-
Submerged specific gravity of sediment (R)		-
Bed porosity (L)		-
Manning-Strickler coefficient (k)	coefficient in the Manning-Strickler relation	-
Manning-Strickler coefficient (r)	coefficient in the Manning-Strickler relation	-
Coefficient in total bed material relation (a)		-
Number of fluvial nodes (M)		-
Time step (t)		days
Number of printouts after initial one		-
Iterations per each printout		-

Parameter	Description	Unit
Model name	name of the model	-
Author name	name of the model author	-

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

• Water surface elevation

[math]\displaystyle{ \eta = \eta_{f}[s_{s} \left (t\right ), t] - S_{a}[x - s_{s}\left ( t \right )] }[/math]

(1)

• Exner equation for shock condition

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) \int _{s_{s}\left (t\right )} ^ \left ( s_{b} \left (t\right ) \right ){\frac{\partial \eta}{\partial t}} d x = I_{f} \{q_{t}[s_{s}\left (t \right ), t] - q_{t} [s_{b}\left (t\right ),t] \} }[/math]

(2)

[math]\displaystyle{ \dot{s_{s}} = {\frac{1}{\left (S_{a} - S_{s} \right )}}[{\frac{I_{f} q_{ts}}{\left ( 1 - \lambda_{p}\right ) \left (s_{b} - s_{s} \right )}} - {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}}] }[/math]

(3)

[math]\displaystyle{ \left (S_{a} - S_{b} \right ) \dot{s}_{b} = \left (S_{a} - S_{s}\right ) \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}} }[/math]

(4)

• Moving boundary coordinate

[math]\displaystyle{ \hat{x} = {\frac{x}{S_{s}\left (t\right )}} }[/math]

(5)

[math]\displaystyle{ \hat{t} = t }[/math]

(6)

• Exner equation for moving-boundary coordinate

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) [\left ({\frac{\partial \eta_{f}}{\partial \hat{t}}} - {\frac{\dot{s}_{s}}{s_{s}}} \hat{x} {\frac{\partial \eta_{f}}{\partial \dot{x}}}\right )] = - {\frac{1}{s_{s}}} I_{f} {\frac{\partial q_{t}}{\partial \dot{x}}} }[/math]

(7)

• Shock condition for moving-boundary coordinate

[math]\displaystyle{ \left (s_{b} - s_{s} \right )[{\frac{\partial \eta_{f}}{\partial \hat{t}}}|_{\hat{x} = 1} + S_{a} \dot{s}_{s}] = {\frac{I_{f} q_{t} \left (1, \hat{t}\right )}{\left ( 1 - \lambda_{p}\right )}} }[/math]

(8)

• Continuity condition for moving-boundary coordinate

[math]\displaystyle{ \dot{s}_{b} = {\frac{S_{a} \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial \hat{t}}}|_{\hat{x} = 1}}{\left ( S_{a} - S_{b}\right )}} }[/math]

(9)

• Sediment transport relation

1) Total bed material transport

[math]\displaystyle{ q_{t} = \sqrt{R g D} D q_{t} ^* }[/math]

(10)

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

(11)

• Normal flow approximation

[math]\displaystyle{ \tau^* = \left ( {\frac{C_{f} q_{w}^2}{g}}\right )^ \left ({\frac{1}{3}}\right ) {\frac{S^ \left ({\frac{2}{3}}\right )}{R D}} }[/math]

(12)

[math]\displaystyle{ C_{f} = Cz^ \left (-2\right ) }[/math]

(13)

[math]\displaystyle{ S = - {\frac{1}{s_{s}}} {\frac{\partial \eta _{f}}{\partial \dot{x}}} }[/math]

(14)

• Boundary conditions

[math]\displaystyle{ s_{s} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{s} \left (\hat{t}\right ) + \dot{s}_{s} \Delta \hat{t} }[/math]

(15)

[math]\displaystyle{ s_{b} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{b} \left (\hat{t}\right ) + \dot{s}_{b} \Delta \hat{t} }[/math]

(16)

[math]\displaystyle{ \eta_{b} \equiv \eta [S_{b} \left (\hat{t} \right ), \hat{t}] = \eta_{d} - S_{s} \left ( s_{b} - s_{s}\right ) }[/math]

(17)

• Calculation of derivatives

[math]\displaystyle{ {\frac{\partial \eta}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{\eta_{i+1} - \eta_{i}}{\Delta \hat{x}}} & i = 1 \\ {\frac{\eta_{i+1} - \eta_{i-1}}{2 \Delta \hat{x}}} & i = 2...M \\ {\frac{\eta_{i} - \eta_{i-1}}{\Delta \hat{x}}} & i = M+1 \end{matrix}\right. }[/math]

(18)

[math]\displaystyle{ {\frac{\partial q_{t}}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{q_{t,j+1} - q_{tf}}{2 \Delta \hat{x}}} & i = 1 \\ {\frac{q_{t,i} - q_{t,i-1}}{2 \Delta \hat{x}}} & 2 \lt = i \lt = M \end{matrix}\right. }[/math]

(19)

Nomenclature

Symbol	Description	Unit
qw	water discharge / width	m2 / s
If	intermittency	-
Cz	dimensionless Chezy resistance coefficient	-
D	grain size of sediment	mm
R	submerged specific gravity of sediment	-
λp	bed porosity	-
qtf	sediment input rate	-
at	coeff in total bed material load relation	-
nt	exponent in load relation	-
τsc *	critical Shields stress in load relation
εd	water surface elevation of the lake	m
ηsl	initial elevation of top of the foreset	m
ηbl	initial elevation of bottom of the foreset	m
Sfl	initial fluvial bed slope	-
Sb	subaqueous basement slope	-
sfl	initial length of fluvial zone	m
sfmax	maximum length of fluvial zone	m
Sa	slope of foreset face	-
Δt	time step	days
M	number of fluvial nodes	-
Mtoprint	number of steps until a printout is made	-
Mprint	number of printouts after the initial one	-
Hi	initial depth at top of foreset deposit	m
Hcrit	depth for Froude-critical flow	m
Ui	initial flow velocity at top of foreset deposit	m / s
τsi	initial Shields stress at top of foreset	m / s
Hni	normal depth associated with initial slope	m
Cf	friction coefficient	-
dt	time step	s
dxbar	dimensionless spatial step	s
τsn	normal Shields stress associated with qw and qtf
Sn	normal slope associated with qw and qtf
Hn	normal depth associated with qw and qtf	m
Frn	normal Froude number associated with qw and qtf
x	downstream coordinate	m
η	bed surface elevation	m
Sl	bed slope	-
qb	volume bedload transport per unit width	m2 / s
H	water depth	m
τ	shear stress	N / m2
sbb	reach of the alluvium bottom	m
sss	reach of the alluvium top	m
etaup	upstream bed surface elevation	m
etatop	bed surface elevation of the top of the forest	m
etabot	bed surface elevation of the bottom of the forest	m
q	flood discharge	m2 / s
I	flood intermittency	-
Q	upstream bed material sediment feed rate during floods	m2 / s
D	grain diameter	mm
Cz	coefficient in the Chezy rlation, Cf	-
n	exponent in the load relation	-
T	critical shields stress in load relation	-
E	elevation of the top of the forest	m
e	initial elevation of the bottom of the forest	m
f	initial fluvial bedslope	-
b	subaqueous basement slope	-
s	initial length of the fluvial zone	m
m	maximum length of the fluvial zone	m
S	slope of the forest face, Sa	-
R	submerged specific gravity	-
L	bed porosity	-
k	coefficient in the Manning-Strickler relation	-
a	coefficient in the total bed material load relation	-
r	coefficient in the Manning-Strickler relation	-
M	number of fluvial nodes	-
t	time step	days
p	number of prints	-
i	number of iterations per print	-

Output

Symbol	Description	Unit
qbT	total volume gravel bedload transport rate per unit width summed over all sizes	-
v	flow velocity	m / s
τsg	shield stress	kg / (m s)

Notes

This module is a calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width.The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width. The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs a full backwater calculation.

• Note on model running

A uniform grain size is assumed, and the same choice of Manning Strickler or Chézy is posed to the user.

The fan builds outward by forming a prograding delta front with an assigned foreset slope.

If the flow becomes supercritical, the backwater calculation will fail, so the program automatically alerts the user and exits.

The initial depth at the top of the foreset must be greater than the critical water depth H_crit; if it isn't the program will alert the user and exit

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

Key papers

Links

• Model:DeltaBW