# 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)

## Notes

The program generates a plot of bed and water surface elevations η and ξ versus streamwise distance, as well as a plot of depth H and depth due to skin friction Hs versus streamwise distance.

In the calculation of φs, a Newton-Raphson iterative scheme solution is implemented. The solution is initiated with some guess φs,1. The calculation is continued until the relative error is under some acceptable limit. There always seems to be a first guess of φs for which the Newton-Raphson scheme converges. When τnom * is only slightly greater than τmin * (in which case φs is only slightly less than 1), however, the right initial guess is sometimes hard to find. For example, the scheme may bounce back and forth between two values of φs without converging, or may yield at some point a negative value of φs. The following technique was adopted to overcome these difficulties:

a) The initial guess for φs is set equal to 0.9.

b) Whenever the iterative scheme yields a negative value of φs, φs is reset to 1.02 and the iterative calculation recommenced.

c) Whenever the calculation does not converge, it is assumed that φs is so close to 1 that it can be set equal to 1.

• Note on input parameters:

If the minimum shear stress due to skin friction τs,min, calculation bombs at any point the program will end.

If the height due to skin friction Hs, calculation bombs, the program will assign the last value in the calculations to Hs.

If the Hnorm calculation bombs, the value for Hnorm is not outputted, but this does not affect the other values that are calculated.

This program requires a given downstream water water elevation, ξd, such that Frd < 1, because the flow is assumed subcritical, and the program will alert the user and quit if the condition is not met.

Downstream bed elevation is set equal to 0, so that at normal conditions the downstream water surface elevation ξd = Hn. The user may then specify a value of ξd that differs from Hn (as long as the corresponding downstream Froude number is less than unity), and compute the resulting backwater curve.

## 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: