# Model help:DredgeSlotBW: Difference between revisions

## DredgeSlotBW

This model is a calculator for aggradation and degradation of sediment mixtures in gravel-bed streams subject to cyclic hydrographs.

## Model introduction

This program calculates the 1D bed evolution of a sand-bed river after installation of a dredge slot.

## 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
Flow discharge (Q) m3 / s
Flood Intermittency (I) flood intermittency -
Channel width (B) m
Median grain size (sand) (d) median grain diameter of sand mm
Grain size such that 90% is finer (sand) (D) mm
Submerged specific gravity of sediment (R) -
Bed slope (S) -
Reach slope
Reach length (L) m
Depth of dredge slot (H) m
Fraction of reach length defining upstream end of dredge slot (u) -
Fraction of reach length defining downstream end of dredge slot (r) -
Bed porosity (l) -
Upwinding coefficient (1 = full upwinding, 0.5 for central difference) -
Number of spatial steps desired (no more than 2000) (M) -
Time step (t) days
Number of iterations per print statement (i) -
Iterations of prints desired (p) -
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 of sediment continuity
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) = - I_{f} {\frac{\partial q_{t}}{\partial x}} = - I_{f}{\frac{\partial q_{b}}{\partial x}} - I_{f} {\frac{\partial q_{s}}{\partial x}} }$ (1)
• Bedload transport in sand-bed streams (Ashida and Michiue, 1972)
 $\displaystyle{ \tau_{s}^* = {\frac{\tau_{bs}}{\rho R g D_{s50}}} }$ (2)
 $\displaystyle{ q_{b}^* = {\frac{q_{b}}{\sqrt{R g D_{s50} D_{s50}}}} = 17 \left ( \tau_{s}^* - \tau_{c}^* \right ) \left ( \sqrt{\tau_{s}^*} - \sqrt{\tau_{c}^*} \right ) }$ (3)
• Entrainment of sand into suspension (Wright and Parker, 2004)
 $\displaystyle{ E = {\frac{A Z_{u}^5}{1 + {\frac{A}{0.3}} Z_{u}^5}} }$ (4)
 $\displaystyle{ Z_{u} = {\frac{u_{*s}}{v_{s}}} Re_{p}^\left (0.6\right ) S_{f}^ \left (0.07\right ) }$ (5)
 $\displaystyle{ u_{*s} = \sqrt{{\frac{\tau_{bs}}{\rho}}} }$ (6)
 $\displaystyle{ Re_{p} = {\frac{\sqrt{R g D_{s50}} D_{s50}}{\nu}} }$ (7)
• Suspended sediment transport rate (Wright-Parker formulation)
 $\displaystyle{ u_{*} = \left ( g H S_{f} \right )^ \left ({\frac{1}{2}}\right ) }$ (8)
 $\displaystyle{ u_{*s} = \left ( g H_{s} S_{f} \right )^ \left ({\frac{1}{2}}\right ) }$ (9)
 $\displaystyle{ C_{z} = {\frac{U}{u_{*}}} }$ (10)
 $\displaystyle{ k_{c} = 11 {\frac{H}{exp \left ( \kappa C_{z} \right )}} }$ (11)
 $\displaystyle{ q_{s} = {\frac{E u_{*} H}{\kappa}} I }$ (12)
 $\displaystyle{ I \left ( {\frac{u_{*}}{v_{s}}}, {\frac{H}{k_{c}}}, \zeta_{b} \right ) = \int_{\zeta_{b}}^1 [{\frac{\left (1 - \zeta \right ) / \zeta}{\left ( 1 - \zeta_{b}\right ) / \zeta_{b}}}]^ \left ({\frac{v_{s}}{\kappa u_{*}}} \right ) ln \left ( 30 {\frac{H}{k_{c}}} \zeta \right) d \zeta }$ (13)
• Gradually varied flow in sand-bed rivers including the effect of bedforms

1) Backwater equation

 $\displaystyle{ {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - Fr^2}} }$ (14)

2) Froude number

 $\displaystyle{ Fr = {\frac{q_{w}}{g^ \left ({\frac{1}{2}}\right ) H^ \left ( {\frac{3}{2}}\right )}} }$ (15)

3) Friction slope

 $\displaystyle{ S_{f} = C_{f} {\frac{U^2}{g H}} = {\frac{\tau_{b}}{\rho g H}} = \phi_{s}^ \left ({\frac{-4}{3}}\right ) S_{nom} }$ (16)

4) boundary shear stress in a sand-bed river

 $\displaystyle{ \tau_{b} = \tau_{bs} + \tau_{bf} = \rho \left ( C_{fs} + C_{ff} \right ) U^2 }$ (17)

5) boundary depth in a sand-bed river

 $\displaystyle{ H = H_{s} + H_{f} }$ (18)

6) friction coefficient due to skin friction

 $\displaystyle{ C_{fs}^ \left ({\frac{-1}{2}}\right ) = {\frac{q_{w}}{H \sqrt{g H_{s} S_{f}}}} = 8.32 \left ({\frac{H_{s}}{3D_{s90}}}\right )^ \left ({\frac{1}{6}}\right ) }$ (19)

7) Shields number due to form drag

 $\displaystyle{ \tau_{s}^* = {\frac{H_{s} S_{f}}{R D_{50}}} = \left\{\begin{matrix} 0.05 + 0.7 \left (\tau^* Fr^ \left (0.7\right ) \right )^ \left (0.8\right ) & \tau^* \gt = \tau_{min}^* \\ \tau^* & \tau^* \lt \tau_{min}^*\end{matrix}\right. }$ (20)

8) Shields number

 $\displaystyle{ \tau^* = {\frac{H S_{f}}{R D_{s50}}} }$ (21)
• Bed shear stress due to skin friction to total bed shear stress
 $\displaystyle{ \phi = \left\{\begin{matrix} {\frac{0.05 + 0.7 \left ( \tau^* Fr ^ \left (0.7\right ) \right ) ^ \left (0.8\right )}{\tau^*}} & \tau^* \gt =\tau_{min}^* \\ 1 & \tau^* \lt \tau_{min}^* \end{matrix}\right. }$ (22)
• Minimum Shields number
 $\displaystyle{ \tau_{min}^* = 0.05 + 0.7 \left ( \tau_{min}^* Fr^ \left (0.7\right ) \right ) ^ \left (0.8\right ) }$ (23)
• Calculation of Hs and Sf from known depth H
 $\displaystyle{ F \left (\phi_{s} \right ) = \left\{\begin{matrix} \phi_{s} - [{\frac{\phi_{s}^ \left ({\frac{-1}{3}}\right ) \tau_{nom}^* - 0.05}{0.7 \left ( \tau_{nom}^* \right ) ^ \left ({\frac{4}{5}}\right ) Fr^ \left ({\frac{14}{25}}\right )}}]^ \left ({\frac{-15}{16}}\right ) & \phi_{s} \lt = \left (\tau_{nom}^* / \tau_{min}^* \right )^ \left ({\frac{3}{4}}\right ) \\ \tau_{s} - 1 & \tau_{s} \gt \left ( \tau_{nom}^* / \tau_{min}^* \right) ^ \left ({\frac{3}{4}}\right )\end{matrix}\right. = 0 }$ (24)
• Calculation of the normal flow condition prevailing in the absence of the dredge slot
 $\displaystyle{ S_{f} = S }$ (25)
 $\displaystyle{ F_{N} \left (H\right ) = \left\{\begin{matrix} H \phi_{s} \left (H\right ) - {\frac{R D_{50}}{S}}[0.05 + 0.7 \left ({\frac{H S}{R D_{s50}}}\right )^ \left ({\frac{4}{5}}\right )\left ({\frac{q_{w}}{\sqrt{g}H^ \left ({\frac{3}{2}}\right )}}\right )^ \left ({\frac{14}{25}}\right )] & H \gt = {\frac{R D_{50} \tau_{min}^*}{S}} \\ H \phi_{s} \left (H\right ) - H & H \lt {\frac{R D_{50} \tau_{min}^*}{S}} \end{matrix}\right. = 0 }$ (26)

1) Concentration of wash load in the ith grain size range

 $\displaystyle{ C_{wi} = C_{uwi} exp [- {\frac{V_{swi}}{q_{w}}} \left (x - L_{su} \right )] }$ (27)

2) Exner equation

 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}} = - I_{f} {\frac{\partial \left ( q_{b} + q_{s} \right )}{\partial x}} + I_{f} \sum V_{swi} C_{wi} }$ (28)

3) Relaxation distance for suspended sediment profile

 $\displaystyle{ C = {\frac{C}{r_{o}}} [1 - exp \left ( - {\frac{r_{o} v_{s}}{q_{w}}} x \right )] }$ (29)

4) Characteristic relaxation distance

 $\displaystyle{ L_{sr} = {\frac{q_{w}}{r_{o} v_{s}}} }$ (30)

## Notes

The model implements BackwaterWrightParker for the case of filling of a dredge slot. It first computes the equilibrium normal flow values of depth H, depth due to skin friction Hs and volume bed load and bed material suspended load transport rates per unit width qb and qs for given values of flood water discharge Qw, flood intermittency If, channel width B, bed sediment sizes D50 and D90 (both assumed constant), sediment submerged specific gravity R and (constant) bed slope S.

A dredge slot is then excavated at time t = 0. The hole has depth Hslot, width B and length (rd - ru)L, where L is reach length, ruL is the upstream end of the dredge slot. and rdL is the downstream end of the dredge slot. Once the slot is excavated, it is allowed to fill without further excavation. Specification of the bed porosity λp, the number of spatial intervals M, the time step Δt, the number of steps to printout Mtoprint, the number of printout after the one corresponding to the initial bed Mprint and the upwinding coefficient au completes the input.

A sufficiently deep dredge slot can capture wash load (e.g. material finer than 62.5 μm) as well as bed material load. As long as the dredge slot is sufficiently deep to prevent re-entrainment of wash load, the rate at which wash load fills the slot can be computed by means of a simple settling model.

• Note on model running

The model is allowed up to 2000 spatial steps, any more than that will cause a memory overwrite and the data cannot be trusted

The Normal flow data is included in the initial output, at the bottom of the file.

The sediment is assumed to be sufficiently uniform such that D50 and D90 do not change in space or time.

The bedload transport calculations use an Ashida-Michiue formulation (1972), the entrainment of suspended sediment uses that of Wright and Parker (2004), as does the formulation for flow resistance.

## 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: