# 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

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
Upstream bed material sediment feed rate during floods m3 / s
Basin length m
Flood discharge m3 / s
Grain size of bed grain diameter mm
Units of wash load deposited in the fan per unit bed material load -
Fan angle degrees
Bed porosity -
Intermittency flood intermittency -
Chezy Resistance Coefficient coefficient in the Chezy relation -
Channel sinuosity -
Subsidence rate must equal mean aggradation for perfect filling of hole, else under or over filling mm / yr
Engelund-Hansen coefficient (sand) -
Engelund-Hansen exponent (sand) -
Channel-forming Shields number (sand) -
Parker coefficient (Gravel) -
Parker exponent (Gravel) -
Critical Shields number (Gravel) -
Channel-forming Shields number (Gravel) -
Submerged specific gravity of sediment -
Number of spatial intervals (< 200) -
Time step yr
Number of iterations desired -
Number of printouts desired -
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

• Subsidence rate
 $\displaystyle{ \delta = - {\frac{\partial \eta_{base}}{\partial t }} }$ (1)
• Exner equation
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) \left ( {\frac{\partial \eta}{\partial t}} + \delta \right ) = -{\frac{I_{f} \left ( 1 + \Lambda \right )}{B_{f}}} {\frac{\partial Q_{tbf}}{\partial x_{b}}} }$ (2)
• Relation for sediment transport (sand-bed)

1) Dimensionless bankfull width

 $\displaystyle{ \hat{B} = {\frac{C_{f}}{\alpha_{EH} \left ( \tau_{form}^* \right ) ^ \left (2.5 \right )}} \hat{Q}_{t} }$ (3)

2) Down-channel bed slope

 $\displaystyle{ S = {\frac{R^ \left ({\frac{3}{2}}\right ) C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{\hat{Q}_{t}}{\hat{Q}}} }$ (4)

3) Dimensionless bankfull depth

 $\displaystyle{ \hat{H} = {\frac{\alpha_{EH} \left ( \tau_{form} \right )^2}{\left ( R C_{f} \right ) ^\left ({\frac{1}{2}}\right )}}{\frac{\hat{Q}}{\hat{Q}_{t}}} }$ (5)
• Relation for sediment transport (gravel-bed)

1) Dimensionless bankfull width

 $\displaystyle{ \hat{B} = {\frac{1}{\alpha_{P} \left ( \tau_{form}^* \right ) ^ \left ({\frac{3}{2}}\right ) r_{form}}} \hat{Q}_{t} }$ (6)

2) Down-channel bed slope

 $\displaystyle{ S = {\frac{R^ \left ({\frac{3}{2}}\right )}{\alpha_{P} r_{form} C_{f}^ \left ({\frac{1}{2}}\right )}}{\frac{\hat{Q}_{t}}{\hat{Q}}} }$ (7)

3) Dimensionless bankfull depth

 $\displaystyle{ \hat{H} = {\frac{\alpha_{P} \tau_{form}^* r_{form} C_{f}^ \left ({\frac{1}{2}}\right )}{\sqrt{R}}} {\frac{\hat{Q}}{\hat{Q}_{t}}} }$ (8)

4)

 $\displaystyle{ r_{form} = \left ( 1 - {\frac{\tau_{c}^*}{\tau_{form}^*}}\right ) ^ \left (4.5\right ) = 0.0135 }$ (9)
• basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)
 $\displaystyle{ \dot{\eta}_{d} = {\frac{\left ( 1 + \Lambda \right ) I_{f} Q_{tbf,feed} - \left ( 1 - \lambda_{p} \right ) \int _{0}^ \left ( L_{b} \right ) B_{b} \delta d x_{b}}{\left ( 1 - \lambda_{p} \right ) \int _{0} ^ \left (L_{b} \right ) B_{b} d x_{b}}} }$ (10)
• A bajada formed by sand-bed rivers equations

1) Water surface elevation

 $\displaystyle{ \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x\right ) }$ (11)

2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)

 $\displaystyle{ \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) B_{b} L_{b}}} - \delta }$ (12)

3)

 $\displaystyle{ \hat{x} = {\frac{x_{b}}{L_{b}}} }$ (13)

4)

 $\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} }$ (14)

5)

 $\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x} }$ (15)

6)

 $\displaystyle{ {\frac{S}{S_{u}}} = 1 - \hat{x} }$ (16)

7)

 $\displaystyle{ S_{u} = {\frac{R C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }$ (17)

8)

 $\displaystyle{ \hat{\eta} = \Omega S_{u} \left ( {\frac{1}{2}} - \hat{x} + {\frac{1}{2}} \hat{x} ^2 \right ) }$ (18)

9)

 $\displaystyle{ {\frac{B_{bf}}{D}} = {\frac{C_{f}}{\alpha_{EH} \left (\tau_{form}^* \right ) ^ \left (2.5\right ) R^ \left ({\frac{1}{2}}\right )}} {\frac{Q_{tbf}}{\sqrt{gD} D^2}} }$ (19)

10)

 $\displaystyle{ {\frac{H_{bf}}{D}} = {\frac{\alpha_{EH} \left ( \tau_{form}^* \right ) ^2}{C_{f}^ \left ({\frac{1}{2}}\right )}}{\frac{Q_{bf}}{Q_{tbf}}} }$ (20)
• A bajada formed by gravel-bed rivers equations

1) Water surface elevation

 $\displaystyle{ \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x\right ) }$ (21)

2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)

 $\displaystyle{ \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) B_{b} L_{b}}} - \delta }$ (22)

3)

 $\displaystyle{ \hat{x} = {\frac{x_{b}}{L_{b}}} }$ (23)

4)

 $\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} }$ (24)

5)

 $\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x} }$ (25)

6)

 $\displaystyle{ {\frac{S}{S_{u}}} = 1 - \hat{x} }$ (26)

7)

 $\displaystyle{ S_{u} = {\frac{R}{\alpha _{P} r_{form} C_{f}^ \left ( {\frac{1}{2}}\right )}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }$ (27)

8)

 $\displaystyle{ \hat{\eta} = \Omega S_{u} \left ( {\frac{1}{2}} - \hat{x} + {\frac{1}{2}} \hat{x} ^2 \right ) }$ (28)

9)

 $\displaystyle{ {\frac{B_{bf}}{D}} = {\frac{1}{\alpha_{P} \left (\tau_{form}^* \right ) ^ \left ({\frac{3}{2}}\right ) R^\left ({\frac{1}{2}}\right ) r_{form}}} {\frac{Q_{tbf}}{\sqrt{gD} D^2}} }$ (29)

10)

 $\displaystyle{ {\frac{H_{bf}}{D}} = \alpha_{P} \tau_{form}^* r_{form} C_{f}^\left ({\frac{1}{2}}\right ) {\frac{Q_{bf}}{Q_{tbf}}} }$ (30)
• An axisymmetric fan formed by sand-bed rivers equations

1) Water surface elevation

 $\displaystyle{ \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x,t\right ) }$ (31)

2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)

 $\displaystyle{ \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) \theta_{f} {\frac{L_{b}^2}{2}}}} - \delta }$ (32)

3)

 $\displaystyle{ \hat{x} = {\frac{x_{b}}{L_{b}}} }$ (33)

4)

 $\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} }$ (34)

5)

 $\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x}^2 }$ (35)

6)

 $\displaystyle{ {\frac{S}{S_{u}}} = 1 - \hat{x}^2 }$ (36)

7)

 $\displaystyle{ S_{u} = {\frac{R C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha _{EH} r_{form}^ *}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }$ (37)

8)

 $\displaystyle{ \hat{\eta} = \Omega S_{u} \left ( {\frac{2}{3}} - \hat{x} + {\frac{1}{3}} \hat{x} ^3 \right ) }$ (38)

9)

 $\displaystyle{ {\frac{B_{bf}}{D}} = {\frac{C_{f}}{\alpha_{EH} \left (\tau_{form}^* \right ) ^ \left ({\frac{5}{2}}\right ) R^\left ({\frac{1}{2}}\right )}} {\frac{Q_{tbf}}{\sqrt{gD} D^2}} }$ (39)

10)

 $\displaystyle{ {\frac{H_{bf}}{D}} = {\frac{\alpha_{EH} \left ( \tau_{form}^* \right )^2 }{C_{f}\left ({\frac{1}{2}}\right )}} {\frac{Q_{bf}}{Q_{tbf}}} }$ (40)
• An axisymmetric fan formed by gravel-bed rivers equations

1) Water surface elevation

 $\displaystyle{ \eta = \eta_{d} \left (t\right ) + \eta_{dev} \left (x,t\right ) }$ (41)

2) basin aggrade/ subside rate in time uniformly in space (positive for overfilled basin, and negative for underfilled basin)

 $\displaystyle{ \dot{\eta}_{d} = {\frac{I_{f} \left ( 1 + \Lambda \right ) Q_{tbf,feed}}{\left ( 1 - \lambda_{b} \right ) \theta_{f} {\frac{L_{b}^2}{2}}}} - \delta }$ (42)

3)

 $\displaystyle{ \hat{x} = {\frac{x_{b}}{L_{b}}} }$ (43)

4)

 $\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L_{b}}} }$ (44)

5)

 $\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = 1 - \hat{x} }$ (45)

6)

 $\displaystyle{ {\frac{S}{S_{u}}} = 1 - \hat{x}^2 }$ (46)

7)

 $\displaystyle{ S_{u} = {\frac{R}{\alpha _{P} r_{form} C_{f}^ \left ({\frac{1}{2}}\right )}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }$ (47)

8)

 $\displaystyle{ \hat{\eta} = \Omega S_{u} \left ( {\frac{2}{3}} - \hat{x} + {\frac{1}{3}} \hat{x} ^2 \right ) }$ (48)

9)

 $\displaystyle{ {\frac{B_{bf}}{D}} = {\frac{1}{\alpha_{P} \left ( \tau_{form}^* \right )^ \left ({\frac{3}{2}}\right ) R^ \left ({\frac{1}{2}}\right ) r_{form}}}{\frac{Q_{tbf}}{\sqrt{gD} D^2}} }$ (49)

10)

 $\displaystyle{ {\frac{H_{bf}}{D}} = \alpha_{P} \tau_{form}^* r_{form} C_{f}^ \left ({\frac{1}{2}} \right ) {\frac{Q_{bf}}{Q_{tbf}}} }$ (50)

## 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:

## 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.