# Model help:BackwaterWrightParker

## BackwaterWrightParker

This is used to calculate backwater curves in sand-bed streams, including the effects of both skin friction and form drag due to skin friction.

## Model introduction

This program calculates backwater curves over a sand-bed stream with a specified spatially constant bed slope S. The calculation uses the hydraulic resistance formulation of Wright and Parker (2004) (without the flow stratification correction), as well as calculating the normal depth.

## Model parameters

Parameter Description Unit
Input directory Path to input file
Site prefix site prefix for Input/Output files -
Case prefix Case prefix for Input/Output files -
Parameter Description Unit
bed slope -
Submerged specific gravity of sediment -
Median grain size (D50) -
Grain size such that 90% passes (D90) grain diameter such that 90% of the distribution is finer mm
channel width (B) m
flow discharge (Q) m3 / s
downstream water surface elevation (k) m
reach length (L) m
number of spatial nodes (max of 99) -
Parameter Description Unit
Model name name of the model -
Author name Name of the model author -
Median grain size (D50) -

## 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)
 $\displaystyle{ q_{s} = {\frac{E u_{*} H}{\kappa}} I \left ( {\frac{u_{*}}{v_{s}}}, {\frac{H}{k_{c}}}, \zeta_{b} \right ) }$ (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)