Model help:SubsidingFan
SubsidingFan
This model is an calculator for evolution of profiles of fans in subsiding basins.
Model introduction
This model is the calculation of Sediment Deposition in a Fan-Shaped Basin, undergoing Piston-Style Subsidence.
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
- Subsidence rate
<math>\delta = - {\frac{\partial \eta_{base}}{\partial t }} </math> (1)
- Exner equation
<math> \left ( 1 - \lambda_{p} \right ) \left ( {\frac{\partial \eta}{\partial t}} + \delta \right ) = -{\frac{I_{f} \left ( 1 + \Lambda \right )}{B_{f}}} {\frac{\partial Q_{tbf}}{\partial x_{b}}} </math> (2)
- Relation for sediment transport (sand-bed)
1) Dimensionless bankfull width
<math> \hat{B} = {\frac{C_{f}}{\alpha_{EH} \left ( \tau_{form}^* \right ) ^ \left (2.5 \right )}} \hat{Q}_{t} </math> (3)
2) Down-channel bed slope
<math> S = {\frac{R^ \left ({\frac{3}{2}}\right ) C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{\hat{Q}_{t}}{\hat{Q}}} </math> (4)
3) Dimensionless bankfull depth
<math> \hat{H} = {\frac{\alpha_{EH} \left ( \tau_{form} \right )^2}{\left ( R C_{f} \right ) ^\left ({\frac{1}{2}}\right )}}{\frac{\hat{Q}}{\hat{Q}_{t}}} </math> (5)
- Relation for sediment transport (gravel-bed)
1) Dimensionless bankfull width
<math> \hat{B} = {\frac{1}{\alpha_{P} \left ( \tau_{form}^* \right ) ^ \left ({\frac{3}{2}}\right ) r_{form}}} \hat{Q}_{t} </math> (6)
2) Down-channel bed slope
<math> S = {\frac{R^ \left ({\frac{3}{2}}\right )}{\alpha_{P} r_{form} C_{f}^ \left ({\frac{1}{2}}\right )}}{\frac{\hat{Q}_{t}}{\hat{Q}}} </math> (7)
3) Dimensionless bankfull depth
<math> \hat{H} = {\frac{\alpha_{P} \tau_{form}^* r_{form} C_{f}^ \left ({\frac{1}{2}}\right )}{\sqrt{R}}} {\frac{\hat{Q}}{\hat{Q}_{t}}} </math> (8)
4)
<math> r_{form} = \left ( 1 - {\frac{\tau_{c}^*}{\tau_{form}^*}}\right ) ^ \left (4.5\right ) = 0.0135 </math> (9)
- basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)
<math> \dot{\eta}_{d} = {\frac{\left ( 1 + \Lambda \right ) I_{f} Q_{tbf,feed} - \left ( 1 - \lambda_{p} \right ) \int _{0}^ \left ( L_{b} \right ) B_{b} \delta d x_{b}}{\left ( 1 - \lambda_{p} \right ) \int _{0} ^ \left (L_{b} \right ) B_{b} d x_{b}}} </math> (10)
- A bajada formed by sand-bed rivers equations
1) Water surface elevation
<math> \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x\right ) </math> (11)
2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)
<math> \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) B_{b} L_{b}}} - \delta </math> (12)
3)
<math> \hat{x} = {\frac{x_{b}}{L_{b}}} </math> (13)
4)
<math> \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} </math> (14)
5)
<math> {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x} </math> (15)
6)
<math> {\frac{S}{S_{u}}} = 1 - \hat{x} </math> (16)
7)
<math> S_{u} = {\frac{R C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{Q_{tbf,feed}}{Q_{bf}}} </math> (17)
8)
<math> \hat{\eta} = \Omega S_{u} \left ( {\frac{1}{2}} - \hat{x} + {\frac{1}{2}} \hat{x} ^2 \right ) </math> (18)
9)
<math> {\frac{B_{bf}}{D}} = {\frac{C_{f}}{\alpha_{EH} \left (\tau_{form}^* \right ) ^ \left (2.5\right ) R^ \left ({\frac{1}{2}}\right )}} {\frac{Q_{tbf}}{\sqrt{gD} D^2}} </math> (19)
10)
<math> {\frac{H_{bf}}{D}} = {\frac{\alpha_{EH} \left ( \tau_{form}^* \right ) ^2}{C_{f}^ \left ({\frac{1}{2}}\right )}}{\frac{Q_{bf}}{Q_{tbf}}} </math> (20)
- A bajada formed by gravel-bed rivers equations
1) Water surface elevation
<math> \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x\right ) </math> (21)
2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)
<math> \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) B_{b} L_{b}}} - \delta </math> (22)
3)
<math> \hat{x} = {\frac{x_{b}}{L_{b}}} </math> (23)
4)
<math> \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} </math> (24)
5)
<math> {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x} </math> (25)
6)
<math> {\frac{S}{S_{u}}} = 1 - \hat{x} </math> (26)
7)
<math> S_{u} = {\frac{R}{\alpha _{P} r_{form} C_{f}^ \left ( {\frac{1}{2}}\right )}} {\frac{Q_{tbf,feed}}{Q_{bf}}} </math> (27)
8)
<math> \hat{\eta} = \Omega S_{u} \left ( {\frac{1}{2}} - \hat{x} + {\frac{1}{2}} \hat{x} ^2 \right ) </math> (28)
9)
<math> {\frac{B_{bf}}{D}} = {\frac{1}{\alpha_{P} \left (\tau_{form}^* \right ) ^ \left ({\frac{3}{2}}\right ) R^\left ({\frac{1}{2}}\right ) r_{form}}} {\frac{Q_{tbf}}{\sqrt{gD} D^2}} </math> (29)
10)
<math> {\frac{H_{bf}}{D}} = \alpha_{P} \tau_{form}^* r_{form} C_{f}^\left ({\frac{1}{2}}\right ) {\frac{Q_{bf}}{Q_{tbf}}} </math> (30)
- An axisymmetric fan formed by sand-bed rivers equations
1) Water surface elevation
<math> \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x,t\right ) </math> (31)
2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)
<math> \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) \theta_{f} {\frac{L_{b}^2}{2}}}} - \delta </math> (32)
3)
<math> \hat{x} = {\frac{x_{b}}{L_{b}}} </math> (33)
4)
<math> \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} </math> (34)
5)
<math> {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x}^2 </math> (35)
6)
<math> {\frac{S}{S_{u}}} = 1 - \hat{x}^2 </math> (36)
7)
<math> S_{u} = {\frac{R C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha _{EH} r_{form}^ *}} {\frac{Q_{tbf,feed}}{Q_{bf}}} </math> (37)
8)
<math> \hat{\eta} = \Omega S_{u} \left ( {\frac{2}{3}} - \hat{x} + {\frac{1}{3}} \hat{x} ^3 \right ) </math> (38)
9)
<math> {\frac{B_{bf}}{D}} = {\frac{C_{f}}{\alpha_{EH} \left (\tau_{form}^* \right ) ^ \left ({\frac{5}{2}}\right ) R^\left ({\frac{1}{2}}\right )}} {\frac{Q_{tbf}}{\sqrt{gD} D^2}} </math> (39)
10)
<math> {\frac{H_{bf}}{D}} = {\frac{\alpha_{EH} \left ( \tau_{form}^* \right )^2 }{C_{f}\left ({\frac{1}{2}}\right )}} {\frac{Q_{bf}}{Q_{tbf}}} </math> (40)
- An axisymmetric fan formed by gravel-bed rivers equations
1) Water surface elevation
<math> \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x,t\right ) </math> (41)
2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)
<math> \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) \theta_{f} {\frac{L_{b}^2}{2}}}} - \delta </math> (42)
3)
<math> \hat{x} = {\frac{x_{b}}{L_{b}}} </math> (43)
4)
<math> \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} </math> (44)
5)
<math> {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x} </math> (45)
6)
<math> {\frac{S}{S_{u}}} = 1 - \hat{x}^2 </math> (46)
7)
<math> S_{u} = {\frac{R}{\alpha _{P} r_{form} C_{f}^ \left ({\frac{1}{2}}\right )}} {\frac{Q_{tbf,feed}}{Q_{bf}}} </math> (47)
8)
<math> \hat{\eta} = \Omega S_{u} \left ( {\frac{2}{3}} - \hat{x} + {\frac{1}{3}} \hat{x} ^2 \right ) </math> (48)
9)
<math> {\frac{B_{bf}}{D}} = {\frac{1}{\alpha_{P} \left ( \tau_{form}^* \right )^ \left ({\frac{3}{2}}\right ) R^ \left ({\frac{1}{2}}\right ) r_{form}}}{\frac{Q_{tbf}}{\sqrt{gD} D^2}} </math> (49)
10)
<math> {\frac{H_{bf}}{D}} = \alpha_{P} \tau_{form}^* r_{form} C_{f}^ \left ({\frac{1}{2}} \right ) {\frac{Q_{bf}}{Q_{tbf}}}</math> (50)
Symbol | Description | Unit |
---|---|---|
Qbf | flood discharge | L3 / T |
Qtbf,feed | upstream bed material sediment feed rate during floods, equals to Qtbf when x_{b} = 0 | L3 / T |
Λ | units of wash load deposited in the fan per unit material load deposited | - |
If | flood intermittency | - |
D | grain size of bed material | L |
R | submerged specific gravity of sediment (e.g. 1.65 for quartz) | - |
Lb | basin length | L |
θf | fan angle in degrees | deg |
xb | radially outward coordinate from the fan vertex (for fan subsidence) | L |
Ω | channel sinuosity | - |
λp | bed porosity | - |
Cz | Chezy resistance coefficient | - |
σ | subsidence rate (must equal mean aggradation for perfect filling of hole; otherwise overfilled or underfilled ) | L / T |
M | number of intervals | - |
Δt | time step | T |
Mtoprint | number of time steps to printout | - |
Mprint | number of printouts | - |
η | water surface elevation | L |
Bb | basin width | L |
nt | exponent in the Engelund-Hansen relation, equals to 2.5 | - |
B^ | dimensionless bankfull width | L |
H^ | dimensionless bankfull depth | L |
Q^t | dimensionless total volume bed material load | - |
Q^ | dimensionless flow discharge | - |
S | down-channel bed slope | - |
Cf | bed friction coefficient | - |
τform * | equals to 0.0487 | - |
rform | equals to 0.0135 | - |
αP | equals to 11.2 | - |
dot{η} d | basin aggrade/ subside rate in time uniformly in space | L / T |
x^ | dimensionless down-channel streamwise coordinate | L |
η^ | dimensionless water surface elevation | L |
Su | upstream channel bed slope | - |
Hbf | bankfull cross sectionally-averaged flow depth | L |
P | coefficient in Parker relation | - |
N | exponent in Parker relation | - |
c | critical Shields number | - |
G | channel-forming Shields number (gravel) | - |
g | acceleration due to gravity | L / T2 |
t | time step | yr |
i | number of intervals per print | - |
p | number of prints | - |
αt | equals to αEH / Cf | - |
αEH | equals to 0.05 | - |
ηbase | basement elevation | L |
Bbf | bankfull channel width | L |
Qtbf | bed material load at the flood flow | M / T |
Bf | floodplain width (excluding channel) | L |
ηdev | deviation of bed elevation from base state | L |
τc * | critical Shields number at the threshold of motion | - |
Notes
The model assumes that the mean annual load of bed material sediment is transported by a bankfull flow continuing for fraction If of any year. The bed material load at the flood (bankfull) flow is Qtbf, and the mean annual bed material load is IfQtbf. The sediment is carried in the channel(s) traversing the basin, but deposited uniformly across the basin width as a result of channel migration, avulsion and overbank deposition. Besides, it is assumed that for every 1 unit of bed material load deposited in the basin Λ units of wash load are deposited.
Assume the sediment flow into a basin of area Ab subsiding at (constant, uniform) rate δ. The rate at which “accomodation space” (volume available to store sediment) is created by subsidence is given as Abδ. Any deposit that formed would have pores, so the actual rate of creation of storage space for sediment is (1-λp)Abδ. Note that even a small subsidence rate δ can create considerable accommodation space if basin area Ab is sufficiently large.
When wash load is included, the annual supply of sediment available for deposition is given as If(1+Λ)Qtbf. If (1+Λ)Qtbf = (1-λp)Abδ the basin is perfectly filled with sediment, resulting in no net vertical movement of the sediment surface, even though the basement continues to subside; If If(1+Λ)Qtbf < (1-λp)Abδ then the basin underfills with sediment, and the sediment surface continues to move down, though at a rate that is less than basement subsidence; If on the other hand If (1+Λ)Qtbf > (1-λp)Abδ then the basin overfills with sediment, and the sediment surface will move upward even though the basement continues to subside.
- Note on model running
For a grain size of < 2 mm the Engelund-Hansen formulation will be used for the bedload, and for a grain size of ≥ 2 mm the Parker bedload formulation will be used, aside from this the diameter is not taken into account for the calculation.
In the case of a gravel bed, the wash load is assumed to be sand, and in the case of a sand bed the wash load is assumed to be silt.
The initial bed is assumed to be horizontal.
The downstream boundary condition is one of vanishing sediment transport, and therefore it is one of vanishing slope.
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
- Allen, P. A. and Allen, J. R., 1990, Basin Analysis Principles and Applications, Blackwell Science, Oxford, U.K., 451 p.
- Heller, P. L., and Paola, C., 1992, The large-scale dynamics of grain-size variation in alluvial basins, 2: Application to syntectonic, Basin Research, 4, 91-102.
- Leeder, M. L., 1999, Sedimentology and Sedimentary Basins From Turbulence to Tectonics, Blackwell Science, Oxford, U.K., 592 p.
- 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., Paola, C., Whipple, W. and Mohrig, D., 1998a, Alluvial fans formed by channelized fluvial and sheet flow: Theory, Journal of Hydraulic Engineering, 123(10), 985-995.
- Parker, G., Paola, C., Whipple, W., Mohrig, D., Toro-Escobar, C., Halverson, M., Skoglund, T., 1998b, Alluvial fans formed by channelized fluvial and sheet flow: Application, Journal of Hydraulic Engineering, 124(10), 996-1004.