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
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} = \bar{t}_{s} = 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)
4) 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] (17)
5) 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] (18)
6) 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] (19)
- 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] (20)
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] (21)
3) boundary condition at x = L
[math]\displaystyle{ \left ( \eta_{s} + H_{sand} \right )|_{\bar{X}_{s} = 1} = \xi_{do} + \xi_{d} }[/math] (22)
[math]\displaystyle{ H_{grav}|_{\bar{x}_{g} = 1} = H_{sand}|_{\bar{X}_{s} = 0} }[/math] (23)
- 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] (24)
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] (25)
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] (26)
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] (27)
Symbol | Description | Unit |
---|---|---|
Q_{bf} | bankfull water discharge at flood | L^{3} / T |
I_{fg} | flood intermittency, gravel-bed reach | - |
I_{fs} | flood intermittency, sand-bed reach | - |
Q_{g,feed} | feed rate of gravel (volume feed rate per unit width) at x = 0 | - |
Q_{sand,feed} | feed rate of sand (volume feed rate per unit width) at x = 0 | - |
B_{c,grav} | gravel-bed channel width | L |
B_{c,sand} | sand-bed channel width | L |
B_{d,grav} | depositional or floodplain width, gravel-bed reach | L |
B_{d,sand} | depositional or floodplain width, sand-bed reach | L |
Ω_{g} | sinuosity of gravel-bed reach | - |
Ω_{s} | sinuosity of sand-bed reach | - |
Λ_{sg} | fraction of sand deposited per unit gravel in depositional zone of gravel-bed reach | - |
Λ_{ms} | fraction of mud deposited per unit sand in depositional zone of sand-bed reach | - |
D_{g} | grain size of gravel | L |
D_{s} | grain size of sand | L |
Cz_{g} | dimensionless Chezy resistance coefficient gravel-bed reach | - |
Cz_{s} | dimensionless Chezy resistance coefficient sand-bed reach | - |
L | reach length | L |
S_{gsl} | initial position of gravel-sand transition (must be < L) | L |
S_{gl} | initial slope of gravel-bed reach | - |
S_{sl} | initial slope of sand-bed reach | - |
σ | subsidence rate | L / T |
ξ_{do } | initial water surface base level | L |
ξ_{d} | rate of base level rise | L / T |
Δt | time step | T |
λ_{pg} | bed porosity, gravel-bed reach | - |
λ_{ps} | bed porosity, sand-bed reach | - |
R | submerged specific gravity of sediment | - |
k | initial water surface base level | L |
y | year to start sea level rise | T |
Y | year to end sea level rise | T |
p | number of prints | - |
i | number of iterations per print | - |
d | depositional flood plain width | L |
M | number of spatial steps | - |
x | downstream coordinate | L |
η | bed elevation | L |
Sl | bed slope | - |
q_{b} | volume bedload transport per unit width | L^{2} / T |
τ | shear stress | - |
sgs | location of the gravel-sand transition | L |
L_{max} | maximum reach length | L |
ε_{min} | minimum water elevation | L |
η_{g} | bed elevation on gravel-bed reach | L |
η_{s} | bed elevation on sand-bed reach | L |
q_{g} | total volume gravel load per unit width | L^{2} / T |
q_{s} | total volume sand load per unit width | L^{2} / T |
r_{Bg} | ratio of channel width Bc to depositional width Bd (basin or floodplain width) in gravel-bed reach | - |
r_{Bs} | ratio of channel width Bc to depositional width Bd (basin or floodplain width) in sand-bed reach | - |
S_{ggs} | gravel bed slope | - |
S_{sgs} | sand bed slope | - |
dot{s}_{sg} | migration speed of the gravel-sand transition | - |
v_{v} | equals to δ for the case of constant subsidence without base level rise; equals to ξ_{d} for the case of base level rise at a constant rate without subsidence | - |
S_{gs,ss} | gravel-sand transition position where the gravel 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) | - |
H_{grav} | flow depth on the gravel-bed reach | L |
H_{sand} | flow depth on the sand-bed reach | L |
C_{fg} | friction coefficients on the gravel-bed reaches | - |
C_{fs} | friction coefficients on the sand-bed reaches | - |
q_{w} | water discharge per unit width (during floods) | L^{2} / T |
τ_{b} | boundary shear stress | - |
τ^{*} | shields number | - |
H | water depth | L |
ρ | water density | M / L^{3} |
D | appropriate grain size | 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 |
---|---|---|
S_{s,feed,n} | normal slope associated with sand feed rate | - |
H_{s,feed,n} | normal depth associated with sand feed rate | L |
S_{g,feed,n} | normal slope associated with gravel feed rate | |
H_{g,feed,n} | normal depth associated with gravel feed rate | L |
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 I_{fg} for the gravel-bed reach and fraction I_{fs} for the sand-bed reach. All sediment transport is assumed to take place when the river is in flood.
The code locates the gravel-sand transition at a point determined by the continuity condition. At this point the gravel transport rate is only a small fraction of the feed value, but it is not precisely zero. In rivers, the small residual gravel load at gravel-sand transitions is either buried or consists of grains that easily break down to sand. In the code, the residual gravel load at the gravel-sand transition is added to the sand load.
The position of the gravel-sand transition x = s_{gs}(t) may change in time.
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.
It is assumed that there are no significant tributaries along the entire reach from x = 0 to x = L, so that water discharge during floods is constant downstream.
The model allows for depositional widths B_{dgrav} and B_{dsand} that are wider than the corresponding bankfull channel widths B_{grav} and B_{sand} 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 initial downstream bed elevation is taken to be zero. As a result, the initial downstream water surface elevation ξ_{do} also equals the initial downstream depth. In order to ensure subcritical flow (and thus keep the calculation from crashing), ξ_{do} must be exceed the critical flow depth H_{c} = [(Q_{bf}/B_{c,sand})2/g]-1/3.
- 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 should be chosen to be less than the maximum value L_{max}, in order to ensure that there is enough sediment supply to fill the accomodation space created by subsidence or sea level rise.
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}.
Depending on the input values, there may be no steady-state solution allowing a gravel-sand transition to equilibrate at a position between 0 and L. For example, if ξ_{d} = 0 and δ = 0, the only steady-state solution is one for which the sand is all driven into the sea. In such cases, the code will fail. The code can be run, however, to a time at which the gravel-sand transition is nearly driven out of the domain of interest.
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: https://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)
References
- Aalto, R., 2002, Geomorphic form and Process of Sediment Flux within an Active Orogen: Denudation of the Bolivian Andes and Sediment Conveyance across the Beni Foreland, PhD thesis, University of Washington, USA, 365 p.
- Cui, Y. and Parker, G., 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 2: General numerical solution, Journal of Hydraulic Research, 36(2): 159-182.
- Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.
- Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams, Technisk Vorlag, Copenhagen, Denmark.
- Ferguson, R. I., 2003, Emergence of abrupt gravel-sand transitions along rivers through sorting processes, Geology 31, 159-162.
- Paola, C., Heller, P. L., and Angevine, C. L., 1992, The large-scale dynamics of grain-size variation in alluvial basins, 1: Theory, Basin Research, 4, 73-90.
- Parker, G., 1979, Hydraulic geometry of active gravel rivers, Journal of Hydraulic Engineering,105(9), 1185‑1201.
- Parker, G., 1991a, Selective sorting and abrasion of river gravel: theory, Journal of Hydraulic Engineering, 117(2), 131-149.
- Parker, G., 1991b, Selective sorting and abrasion of river gravel: applications, Journal of Hydraulic Engineering, 117(2), 150-171.
- Parker, G., and Y. Cui, 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 1: Simplified analytical solution, Journal of Hydraulic Research, 36(1): 75-100.