# Model help:DeltaBW

## DeltaBW

This is used to Calculate evolution of long profile of a river ending in a 1D migrating delta, using a backwater formulation.

## Model introduction

This program calculates bed surface evolution for a narrowly channelized1D fan-delta prograding into standing water using a backwater formulation, as well as calculating the final water surface of the system and the mass balance of sediment in the system.

## 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
Chezy Or Manning, Chezy-1 or Manning-2
Parameter Description Unit
Flood discharge (q) m2 / s
Intermittency (I) flood intermittency -
upstream bed material sediment feed rate during flood (Q) m2 / m
Grain size of bed material (D) mm
Chezy resistance coefficient (C) cofficient in the Chezy relation -
Exponent in load relation (n) -
Critical Shields stress in load relation (T) -
Elevation of top of forest (E) m
Initial elevation of forest bottom (e) m
Water surface elevation of lake (x) m
Initial fluvial bed slope (f) -
Subaqueous basement slope (b) -
initial length of fluvial zone (s) m
Maximum length of fluvial zone (m) m
Slope of forest face (Sa) -
Submerged specific gravity of sediment (R) -
Bed porosity (L) -
Manning-Strickler coefficient (k) coefficient in the Manning-Strickler relation -
Manning-Strickler coefficient (r) coefficient in the Manning-Strickler relation -
Coefficient in total bed material relation (a) -
Number of fluvial nodes (M) -
Time step (t) days
Number of printouts after initial one -
Iterations per each printout -
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

• Water surface elevation
 $\displaystyle{ \eta = \eta_{f}[s_{s} \left (t\right ), t] - S_{a}[x - s_{s}\left ( t \right )] }$ (1)
• Exner equation for shock condition
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) \int _{s_{s}\left (t\right )} ^ \left ( s_{b} \left (t\right ) \right ){\frac{\partial \eta}{\partial t}} d x = I_{f} \{q_{t}[s_{s}\left (t \right ), t] - q_{t} [s_{b}\left (t\right ),t] \} }$ (2)
 $\displaystyle{ \dot{s_{s}} = {\frac{1}{\left (S_{a} - S_{s} \right )}}[{\frac{I_{f} q_{ts}}{\left ( 1 - \lambda_{p}\right ) \left (s_{b} - s_{s} \right )}} - {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}}] }$ (3)
 $\displaystyle{ \left (S_{a} - S_{b} \right ) \dot{s}_{b} = \left (S_{a} - S_{s}\right ) \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}} }$ (4)
• Moving boundary coordinate
 $\displaystyle{ \hat{x} = {\frac{x}{S_{s}\left (t\right )}} }$ (5)
 $\displaystyle{ \hat{t} = t }$ (6)
• Exner equation for moving-boundary coordinate
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) [\left ({\frac{\partial \eta_{f}}{\partial \hat{t}}} - {\frac{\dot{s}_{s}}{s_{s}}} \hat{x} {\frac{\partial \eta_{f}}{\partial \dot{x}}}\right )] = - {\frac{1}{s_{s}}} I_{f} {\frac{\partial q_{t}}{\partial \dot{x}}} }$ (7)
• Shock condition for moving-boundary coordinate
 $\displaystyle{ \left (s_{b} - s_{s} \right )[{\frac{\partial \eta_{f}}{\partial \hat{t}}}|_{\hat{x} = 1} + S_{a} \dot{s}_{s}] = {\frac{I_{f} q_{t} \left (1, \hat{t}\right )}{\left ( 1 - \lambda_{p}\right )}} }$ (8)
• Continuity condition for moving-boundary coordinate
 $\displaystyle{ \dot{s}_{b} = {\frac{S_{a} \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial \hat{t}}}|_{\hat{x} = 1}}{\left ( S_{a} - S_{b}\right )}} }$ (9)
• Sediment transport relation

1) Total bed material transport

 $\displaystyle{ q_{t} = \sqrt{R g D} D q_{t} ^* }$ (10)
 $\displaystyle{ q_{t}^* = \alpha_{t}[\tau^* - \tau_{c}^*]^ \left (n_{t}\right ) }$ (11)
• Backwater formula
 $\displaystyle{ {\frac{dH}{d \hat{x}}} = s_{s} {\frac{S - S_{f}}{1 - Fr^2}} }$ (12)
 $\displaystyle{ S = - {\frac{1}{s_{s}}} {\frac{\partial \eta _{f}}{\partial \hat{x}}} }$ (13)
 $\displaystyle{ Fr^2 = {\frac{q_{w}^2}{g H^3}} }$ (14)
 $\displaystyle{ S_{f} = C_{f} Fr^2 }$ (15)
 $\displaystyle{ H \left (1,t\right ) = \xi_{s} \left (\hat{t} \right ) - \eta_{f} \left (1,\hat{t} \right ) }$ (16)
 $\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} = {\frac{C_{f} U^2}{R g D}} = {\frac{C_{f} {\frac{q_{w}^2}{H^2}}}{R g D}} }$ (17)
• Boundary conditions
 $\displaystyle{ s_{s} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{s} \left (\hat{t}\right ) + \dot{s}_{s} \Delta \hat{t} }$ (18)
 $\displaystyle{ s_{b} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{b} \left (\hat{t}\right ) + \dot{s}_{b} \Delta \hat{t} }$ (19)
 $\displaystyle{ \eta_{b} \equiv \eta [S_{b} \left (\hat{t} \right ), \hat{t}] = \eta_{d} - S_{s} \left ( s_{b} - s_{s}\right ) }$ (20)
• Calculation of derivatives
 $\displaystyle{ {\frac{\partial \eta}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{\eta_{i+1} - \eta_{i}}{\Delta \hat{x}}} & i = 1 \\ {\frac{\eta_{i+1} - \eta_{i-1}}{2 \Delta \hat{x}}} & i = 2...M \\ {\frac{\eta_{i} - \eta_{i-1}}{\Delta \hat{x}}} & i = M+1 \end{matrix}\right. }$ (21)
 $\displaystyle{ {\frac{\partial q_{t}}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{q_{t,i} - q_{tf}}{ \Delta \hat{x}}} & i = 1 \\ {\frac{q_{t,i} - q_{t,i-1}}{ \Delta \hat{x}}} & 1 \lt i \lt = M+1 \end{matrix}\right. }$ (22)

## Notes

This module is a calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width.The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width. The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs a full backwater calculation.

In the normal flow formulation, for any given time t = t^: a) Specify the downstream bed elevation ηd

b) Calculate the backwater curve upstream from x^ = 1.

c) Use this to evaluate qt everywhere, including qts at x^ = 1.

d) Implement the shock condition to find dot{s}s. This shock condition requires knowledge of the term d ηf / d t^ |x^ = 1 . It is sufficient to evaluate this term using the current bed profile and that obtained one step earlier, at t^ = 0, this term can be ignored.

e) Solve Exner everywhere to find new bed elevations at time Δt^ later.

f) Use continuity condition to find dot{s}b.

• Note on model running

A uniform grain size is assumed, and the same choice of Manning Strickler or Chézy is posed to the user.

The fan builds outward by forming a prograding delta front with an assigned foreset slope.

If the flow becomes supercritical, the backwater calculation will fail, so the program automatically alerts the user and exits.

The initial depth at the top of the foreset must be greater than the critical water depth Hcrit; if it isn't the program will alert the user and exit

## 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:

See also: Help:Images or Help:Movies

## References

• Grover, N.C., and Howard, C.L., 1937, The passage of turbid water through Lake Mead, Transactions, American Society of Civil Engineers, 103, 720-732.
• Kostic, S. and Parker, G., 2003a, Progradational sand-mud deltas in lakes and reservoirs. Part 1. Theory and numerical modeling, Journal of Hydraulic Research, 41(2), 127-140.
• Kostic, S. and Parker, G., 2003b, Progradational sand-mud deltas in lakes and reservoirs. Part 2. Experiment and numerical simulation, Journal of Hydraulic Research, 41(2), 141-152
• Swenson, J. B., Voller, V. R., Paola, C., Parker G. and Marr J., 2000, Fluvio-deltaic sedimentation: a generalized Stefan problem, European Journal of Applied Math., 11, 433-452.