# 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)