Model help:GravelSandTransition

GravelSandTransition

This model is used to calculate evolution of long profile of river with a migrating gravel-sand transition and subject to subsidence or base level rise.

Model introduction

This program calculates the bed surface evolution at predefined nodes for a river with a gravel to sand transition, as well as calculating the relative location of the transition point and the slopes, bedload transport rate, shear stress, and water depth for plotting.

Model parameters

Input Files and Directories

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
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

• Exner equation (Conservation relations for gravel and sand on the gravel-bed reach )

[math]\displaystyle{ {\frac{\partial \eta_{g}}{\partial t}} + \delta = - {\frac{I_{fg} \left ( 1 + \Lambda _{sg} \right ) \Omega _{g}}{\left ( 1 - \lambda _{pg} \right ) r_{Bg}}} {\frac{\partial q_{g}}{\partial x}} }[/math]

(1)

• Exner equation (Conservation relations for gravel and sand on the sand-bed reach)

[math]\displaystyle{ {\frac{\partial \eta_{s}}{\partial t}} + \delta = - {\frac{I_{fs} \left ( 1 + \Lambda _{ms} \right ) \Omega _{s}}{\left ( 1 - \lambda _{ps} \right ) r_{Bs}}} {\frac{\partial q_{s}}{\partial x}} }[/math]

(2)

[math]\displaystyle{ {\frac{\partial q_{s}}{\partial x}} = \Lambda _{sg} {\frac{\partial q_{g}}{\partial x}} }[/math]

(3)

• gravel bed slope

[math]\displaystyle{ S_{ggs} = - {\frac{\partial \eta_{g}}{\partial x}}|_{x = S_{gs}} }[/math]

(4)

• sand bed slope

[math]\displaystyle{ S_{sgs} = - {\frac{\partial \eta_{s}}{\partial x}}|_{x = S_{gs}} }[/math]

(5)

• bed elevation continuity at the gravel-sand transition

[math]\displaystyle{ \eta_{g} \left (x,t \right ) = \eta_{g} \left (x,t \right ) }[/math]

(6)

• Migration speed of the gravel-sand transition

[math]\displaystyle{ \dot{s}_{sg} = {\frac{{\frac{\partial \eta_{g}}{\partial t}}|_{x=s_{gs}} - {\frac{\partial \eta _{s}}{\partial t}}|_{x=s_{gs}}}{S_{ggs} - S_{sgs}}} = {\frac{1}{\left (S_{ggs} - S_{sgs}\right )}}[{\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) r_{Bs}}} {\frac{\partial q_{s}}{\partial x}}|_{x=S_{gs}} - {\frac{I_{fg} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) r_{Bg}}} {\frac{\partial q_{g}}{\partial x}}|_{x=S_{gs}}] }[/math]

(7)

• Steady state

[math]\displaystyle{ {\frac{\partial \eta_{g}}{\partial t}} = {\frac{\partial \eta_{s}}{\partial t}} = \dot{S}_{sg} = 0 }[/math]

(8)

• Total volume gravel load per unit width created by subsidence or sea level rise in steady state

[math]\displaystyle{ q_{g} = q_{g,feed} - {\frac{\left ( 1 - \lambda _{pg}\right ) r_{Bg}}{I_{fg} \left ( 1 + \Lambda _{sg} \right ) \Omega_{g}}} v_{v}x }[/math]

(9)

• Total volume sand load per unit width created by subsidence or sea level rise in steady state

[math]\displaystyle{ q_{s} = q_{s,feed} - \Lambda_{sg} \left ( q_{g,feed} - q \right ) = q_{s,feed} - \Lambda_{sg}{\frac{\left ( 1 - \lambda _{pg}\right ) r_{Bg}}{I_{fg} \left ( 1 + \Lambda _{sg} \right ) \Omega_{g}}} v_{v}x }[/math]

(10)

• Gravel-sand transition where the gravel transport rate drops to zero at the steady-state condition

[math]\displaystyle{ S_{gs,ss} = {\frac{I_{fg}\left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) r_{Bg}V_{v}}} q_{g,feed} }[/math]

(11)

• Total volume sand load per unit width at the point where the gravel runs out

[math]\displaystyle{ q_{s} |_{x = S_{sg,ss}} = q_{s,feed} - \Lambda_{sg} {\frac{\left ( 1 - \lambda_{pg} \right ) r_{Bg}}{I_{fg} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}} V_{v} S_{sg,ss} = q_{s,feed} - \Lambda_{sg} q_{g,feed} }[/math]

(12)

• ravel-sand transition where the sand transport rate drops to zero at the steady-state condition

[math]\displaystyle{ L_{max} = S_{gs,ss} + {\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) r_{Bs} V_{v}}} \left ( q_{s,feed} - \Lambda_{sg} q_{g,feed} \right ) = {\frac{1}{V_{v}}}[{\frac{I_{fg}\left ( 1 + \Lambda_{sg}\right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) r_{Bg}}} q_{g,feed} + {\frac{I_{fs}\left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left (1 - \lambda_{ps}\right )r_{Bs}}}\left (q_{s,feed} - \Lambda_{sg} q_{g,feed}\right )] }[/math]

(13)

• Moving boundary coordinate

1) downstream coordinate for gravel

[math]\displaystyle{ \bar{X}_{g} = {\frac{x}{S_{gs}\left (t\right )}} }[/math]

(14)

2) time

[math]\displaystyle{ \bar{t}_{g} = t }[/math]

(15)

3) downstream coordinate for sand

[math]\displaystyle{ \bar{X}_{s} = {\frac{x - S_{gs} \left ( t \right )}{L - S_{gs} \left ( t \right )}} }[/math]

(16)

2) time

[math]\displaystyle{ \bar{t}_{s} = t }[/math]

(17)

3) Exner equation for gravel conservation on the gravel-bed reach

[math]\displaystyle{ {\frac{\partial \eta_{g}}{\partial \bar{t}_{g}}} = - \delta + {\frac{\dot{S}_{gs} \bar{X}_{g}}{S_{gs}}} {\frac{\partial \eta_{g}}{\partial \bar{X}_{g}}} - {\frac{I_{f} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) S_{gs} r_{Bg}}} {\frac{\partial q_{g}}{\partial \bar{X}_{g}}} }[/math]

(18)

3) Exner equation for sand conservation on the sand-bed reach

[math]\displaystyle{ {\frac{\partial \eta_{s}}{\partial \bar{t}_{s}}} = - \delta + {\frac{\dot{S}_{gs}\left ( 1 - \bar{X}_{s} \right )}{ L - S_{gs}}} {\frac{\partial \eta_{s}}{\partial \bar{X}_{s}}} - {\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) \left ( L - S_{gs} \right ) r_{Bg}}} {\frac{\partial q_{s}}{\partial \bar{X}_{s}}} }[/math]

(19)

4) * Migration speed of the gravel-sand transition

[math]\displaystyle{ \dot{S}_{gs} = {\frac{1}{\left ( S_{ggs} - S_{sgs}\right )}} [{\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) r_{Bs} \left ( L - S_{gs} \right )}}{\frac{\partial q_{s}}{\partial \bar{X}_{s}}}|_{\bar{X}_{s} = 0} - {\frac{I_{fg} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg}\right ) r_{Bg} S_{gs}}}{\frac{\partial q_{g}}{\partial \bar{X}_{g}}}|_{\bar{X}_{g} = 1}] }[/math]

(20)

• Calculation of flow

1) Backwater formula for the gravel-bed reach

[math]\displaystyle{ {\frac{\partial H_{grav}}{\partial x}} = {\frac{1}{s_{gs}}}{\frac{\partial H_{grav}}{\partial \bar{X}_{g}}} = {\frac{S_{ggs} - C_{fg}{\frac{q_{w}^2}{g H_{grav}^3}}}{1 - {\frac{q_{w}^2}{g H_{grav}^3}}}} }[/math]

(21)

2) Backwater formula for the sand-bed reach

[math]\displaystyle{ {\frac{\partial H_{sand}}{\partial x}} = {\frac{1}{L - s_{gs}}} {\frac{\partial H_{sand}}{\partial \bar{x}_{s}}} = {\frac{S_{sgs} - C_{fs}{\frac{q_{w}^2}{g H_{sand}^3}}}{1 - {\frac{q_{w}^2}{g H_{sand}^3}}}} }[/math]

(22)

3) boundary condition at x = L

[math]\displaystyle{ \left ( \eta_{s} + H_{sand} \right )|_{\bar{X}_{s} = 1} = \xi_{do} + \xi_{d} }[/math]

(23)

[math]\displaystyle{ H_{grav}|_{\bar{x}_{g} = 1} = H_{sand}|_{\bar{X}_{s} = 0} }[/math]

(24)

• Calculation of Shields numbers

1) Boundary shear stress

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

(25)

2) Shield number

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

(26)

3) Volume gravel transport per unit width for the ith node

[math]\displaystyle{ q_{g,i} = \sqrt{R g D_{g}} D_{g} 11.2 \left ( \tau_{grav,i}^* \right ) ^ \left (1.5 \right ) \left ( 1 - {\frac{0.03}{\tau_{grav,i}^*}}\right ) ^ \left ( 4.5 \right ) }[/math]

(27)

4) Volume sand transport per unit width for the ith node

[math]\displaystyle{ q_{s,i} = \sqrt{R g D_{s}} D_{s} {\frac{0.05}{C_{fs}}} \left (\tau_{sand,i}^* \right ) ^ \left (2.5 \right ) }[/math]

(28)

Nomenclature

Symbol	Description	Unit
Qbf	bankfull water discharge at flood	m3 / s
Ifg	flood intermittency, gravel-bed reach	-
Ifs	flood intermittency, sand-bed reach	-
qg,feed	feed rate of gravel (volume feed rate per unit width) at x = 0	-
Qsand,feed	feed rate of sand (volume feed rate per unit width) at x = 0	-
Bc,grav	gravel-bed channel width	m
Bc,sand	sand-bed channel width	m
Bd,grav	depositional or floodplain width, gravel-bed reach	m
Bd,sand	depositional or floodplain width, sand-bed reach	m
Ωg	sinuosity of gravel-bed reach	-
Ωs	sinuosity of sand-bed reach	-
Λsg	fraction of sand deposited per unit gravel in depositional zone of gravel-bed reach	-
Λms	fraction of mud deposited per unit sand in depositional zone of sand-bed reach	-
Dg	grain size of gravel	mm
Ds	grain size of sand	mm
Czg	dimensionless Chezy resistance coefficient gravel-bed reach	-
Czs	dimensionless Chezy resistance coefficient sand-bed reach	-
L	reach length	m
Sgsl	initial position of gravel-sand transition (must be < L)	m
Sgl	initial slope of gravel-bed reach	-
Ssl	initial slope of sand-bed reach	-
σ	subsidence rate	mm / year
ξdo	initial water surface base level	m
ξd	rate of base level rise	mm / year
Δt	time step	years
λpg	bed porosity, gravel-bed reach	-
λps	bed porosity, sand-bed reach	-
R	submerged specific gravity of sediment	-
k	initial water surface base level	m
y	year to start sea level rise	yr
Y	year to end sea level rise	yr
p	number of prints	-
i	number of iterations per print	-
d	depositional flood plain width	m
M	number of spatial steps	-
x	downstream coordinate	m
η	bed elevation	m
Sl	bed slope	-
qb	volume bedload transport per unit width	m2 / s
H	water depth	m
τ	shear stress	N / m2
sgs	location of the gravel-sand transition	m
Lmax	maximum reach length	m
εmin	minimum water elevation	m
ηg	bed elevation on gravel-bed reach	L
ηs	bed elevation on sand-bed reach	L
qg	total volume gravel load per unit width	L2 / L
qs	total volume sand load per unit width	L2 / L
rBg	ratio of channel width Bc to depositional width Bd (basin or floodplain width) in gravel-bed reach	-
rBs	ratio of channel width Bc to depositional width Bd (basin or floodplain width) in sand-bed reach	-
Sggs	gravel bed slope	-
Ssgs	sand bed slope	-
dot{s}sg	migration speed of the gravel-sand transition	-
vv	equals to δ for the case of constant subsidence without base level rise; equals to ξd for the case of base level rise at a constant rate without subsidence	-
Sgs,ss	gravel-sand transition position where the gravel transport rate drops to zero at the steady-state condition	-
Lmax	gravel-sand transition position where the sand transport rate drops to zero at the steady-state condition	-
bar{X}_g	moving boundary coordinate on the gravel-bed reach (0~1)	-
bar{X}_s	moving boundary coordinate on the sand-bed reach (0~1)	-
Hgrav	flow depth on the gravel-bed reach	L
Hsand	flow depth on the sand-bed reach	L
Cfg	friction coefficients on the gravel-bed reaches	-
Cfs	friction coefficients on the sand-bed reaches	-
qw	water discharge per unit width (during floods)	-
τb	boundary shear stress	-
τ*	shields number	-
H	water depth	L
ρ	water density	M / L3
D	appropriate grain size	L
Dg	appropriate grain size of gravel	L
Ds	appropriate grain size of sand	L
τ* grav,i	shields number for the ith node of the gravel-bed reach	L
τ* sand,i	shields number for the ith node of the sand-bed reach	L

Output

Symbol	Description	Unit
Ss,feed,n	normal slope associated with sand feed rate	-
Hs,feed,n	normal depth associated with sand feed rate	m
Ss,feed,n	normal slope associated with gravel feed rate
Hs,feed,n	normal depth associated with gravel feed rate	m

Notes

This model calculates the long profile of a river with a gravel-sand transition. The model uses two grain sizes: size D_g for gravel and size D_s for sand. The river is assumed to be in flood for the fraction of time Ifg for the gravel-bed reach and fraction Ifs for the sand-bed reach. All sediment transport is assumed to take place when the river is in flood.

Gravel transport is computed using the Parker (1979) approximation of the Einstein (1950) bedload transport relation. Sand transport is computed using the total bed material transport relation of Engelund and Hansen (1967).

In this simple model the gravel is not allowed to abrade. Both the gravel-bed and sand-bed reaches carry the same flood discharge Q_bf.

Gravel is transported as bed material in, and deposits only in the gravel-bed reach. A small residual of gravel load is incorporated into the sand at the gravel-sand transition. Sand is transported as washload in the gravel-bed reach, and as bed material load in the sand-bed reach.

The model allows for depositional widths Bdgrav and Bdsand that are wider than the corresponding bankfull channel widths Bgrav and Bsand of the gravel-bed and sand-bed channels. As the channel aggrades, it is assumed to migrate and avulse to deposit sediment across the entire depositional width. For each unit of gravel deposited in the gravel-bed reach, it is assumed that Lamsg units of sand are deposited. For each unit of sand deposited on the sand-bed reach, it is assumed that Lamms units of mud are deposited.

The gravel-bed reach has sinuosity Omegag and the sand-bed reach has sinuosity Omegas.

Bed resistance is computed through the use of two specified constant Chezy resistance coefficients; C_zg for the gravel-bed reach and C_zs for the sand-bed reach.

• Note on model running

The values for the spatial step of both gravel and sand max out at 100, allowing 200 spatial steps total (the number of sand spatial nodes, Ms and gravel spatial nodes, Mg must be ≤ 100).

When the slopes are unequal at gravel sand transition the GetData function and the plot data output a slope of 0 at that point.

The reach length, L, must be less than the maximum reach length, L_max; if it is not the program will quit and alert the user what the value was for L_max.

The sand and gravel Froude numbers must be less than 1 (i.e. the flow is assumed subcritical in the sand and gravel portions of the bed); if they are not, the program will quit and alert the user what their values were.

The user inputted downstream water elevation, ξ_d, must be greater than the program calculated minimum water elevation, ξ_min; if this is not the case, the program will quit and the user will be alerted what the value is for ξ_min.

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)

Name of the module developer(s)

References

Key papers

Links

[Model:GravelSandTransitio]