# Model help:GravelSandTransition

## GravelSandTransition

This model is used to calculate evolution of long profile of river with a migrating gravel-sand transition and subject to subsidence or base level rise.

## Model introduction

This program calculates the bed surface evolution at predefined nodes for a river with a gravel to sand transition, as well as calculating the relative location of the transition point and the slopes, bedload transport rate, shear stress, and water depth for plotting.

## 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
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 (Conservation relations for gravel and sand on the gravel-bed reach )
 $\displaystyle{ {\frac{\partial \eta_{g}}{\partial t}} + \delta = - {\frac{I_{fg} \left ( 1 + \Lambda _{sg} \right ) \Omega _{g}}{\left ( 1 - \lambda _{pg} \right ) r_{Bg}}} {\frac{\partial q_{g}}{\partial x}} }$ (1)
• Exner equation (Conservation relations for gravel and sand on the sand-bed reach)
 $\displaystyle{ {\frac{\partial \eta_{s}}{\partial t}} + \delta = - {\frac{I_{fs} \left ( 1 + \Lambda _{ms} \right ) \Omega _{s}}{\left ( 1 - \lambda _{ps} \right ) r_{Bs}}} {\frac{\partial q_{s}}{\partial x}} }$ (2)
 $\displaystyle{ {\frac{\partial q_{s}}{\partial x}} = \Lambda _{sg} {\frac{\partial q_{g}}{\partial x}} }$ (3)
• gravel bed slope
 $\displaystyle{ S_{ggs} = - {\frac{\partial \eta_{g}}{\partial x}}|_{x = S_{gs}} }$ (4)
• sand bed slope
 $\displaystyle{ S_{sgs} = - {\frac{\partial \eta_{s}}{\partial x}}|_{x = S_{gs}} }$ (5)
• bed elevation continuity at the gravel-sand transition
 $\displaystyle{ \eta_{g} \left (x,t \right ) = \eta_{g} \left (x,t \right ) }$ (6)
• Migration speed of the gravel-sand transition
 $\displaystyle{ \dot{s}_{sg} = {\frac{{\frac{\partial \eta_{g}}{\partial t}}|_{x=s_{gs}} - {\frac{\partial \eta _{s}}{\partial t}}|_{x=s_{gs}}}{S_{ggs} - S_{sgs}}} = {\frac{1}{\left (S_{ggs} - S_{sgs}\right )}}[{\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) r_{Bs}}} {\frac{\partial q_{s}}{\partial x}}|_{x=S_{gs}} - {\frac{I_{fg} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) r_{Bg}}} {\frac{\partial q_{g}}{\partial x}}|_{x=S_{gs}}] }$ (7)
 $\displaystyle{ {\frac{\partial \eta_{g}}{\partial t}} = {\frac{\partial \eta_{s}}{\partial t}} = \dot{S}_{sg} = 0 }$ (8)
• Total volume gravel load per unit width created by subsidence or sea level rise in steady state
 $\displaystyle{ q_{g} = q_{g,feed} - {\frac{\left ( 1 - \lambda _{pg}\right ) r_{Bg}}{I_{fg} \left ( 1 + \Lambda _{sg} \right ) \Omega_{g}}} v_{v}x }$ (9)
• Total volume sand load per unit width created by subsidence or sea level rise in steady state
 $\displaystyle{ q_{s} = q_{s,feed} - \Lambda_{sg} \left ( q_{g,feed} - q \right ) = q_{s,feed} - \Lambda_{sg}{\frac{\left ( 1 - \lambda _{pg}\right ) r_{Bg}}{I_{fg} \left ( 1 + \Lambda _{sg} \right ) \Omega_{g}}} v_{v}x }$ (10)
• Gravel-sand transition where the gravel transport rate drops to zero at the steady-state condition
 $\displaystyle{ S_{gs,ss} = {\frac{I_{fg}\left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) r_{Bg}V_{v}}} q_{g,feed} }$ (11)
• Total volume sand load per unit width at the point where the gravel runs out
 $\displaystyle{ q_{s} |_{x = S_{sg,ss}} = q_{s,feed} - \Lambda_{sg} {\frac{\left ( 1 - \lambda_{pg} \right ) r_{Bg}}{I_{fg} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}} V_{v} S_{sg,ss} = q_{s,feed} - \Lambda_{sg} q_{g,feed} }$ (12)
• ravel-sand transition where the sand transport rate drops to zero at the steady-state condition
 $\displaystyle{ L_{max} = S_{gs,ss} + {\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) r_{Bs} V_{v}}} \left ( q_{s,feed} - \Lambda_{sg} q_{g,feed} \right ) = {\frac{1}{V_{v}}}[{\frac{I_{fg}\left ( 1 + \Lambda_{sg}\right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) r_{Bg}}} q_{g,feed} + {\frac{I_{fs}\left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left (1 - \lambda_{ps}\right )r_{Bs}}}\left (q_{s,feed} - \Lambda_{sg} q_{g,feed}\right )] }$ (13)
• Moving boundary coordinate

1) downstream coordinate for gravel

 $\displaystyle{ \bar{X}_{g} = {\frac{x}{S_{gs}\left (t\right )}} }$ (14)

2) time

 $\displaystyle{ \bar{t}_{g} = \bar{t}_{s} = t }$ (15)

3) downstream coordinate for sand

 $\displaystyle{ \bar{X}_{s} = {\frac{x - S_{gs} \left ( t \right )}{L - S_{gs} \left ( t \right )}} }$ (16)

4) Exner equation for gravel conservation on the gravel-bed reach

 $\displaystyle{ {\frac{\partial \eta_{g}}{\partial \bar{t}_{g}}} = - \delta + {\frac{\dot{S}_{gs} \bar{X}_{g}}{S_{gs}}} {\frac{\partial \eta_{g}}{\partial \bar{X}_{g}}} - {\frac{I_{f} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg} \right ) S_{gs} r_{Bg}}} {\frac{\partial q_{g}}{\partial \bar{X}_{g}}} }$ (17)

5) Exner equation for sand conservation on the sand-bed reach

 $\displaystyle{ {\frac{\partial \eta_{s}}{\partial \bar{t}_{s}}} = - \delta + {\frac{\dot{S}_{gs}\left ( 1 - \bar{X}_{s} \right )}{ L - S_{gs}}} {\frac{\partial \eta_{s}}{\partial \bar{X}_{s}}} - {\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) \left ( L - S_{gs} \right ) r_{Bg}}} {\frac{\partial q_{s}}{\partial \bar{X}_{s}}} }$ (18)

6) Migration speed of the gravel-sand transition

 $\displaystyle{ \dot{S}_{gs} = {\frac{1}{\left ( S_{ggs} - S_{sgs}\right )}} [{\frac{I_{fs} \left ( 1 + \Lambda_{ms} \right ) \Omega_{s}}{\left ( 1 - \lambda_{ps} \right ) r_{Bs} \left ( L - S_{gs} \right )}}{\frac{\partial q_{s}}{\partial \bar{X}_{s}}}|_{\bar{X}_{s} = 0} - {\frac{I_{fg} \left ( 1 + \Lambda_{sg} \right ) \Omega_{g}}{\left ( 1 - \lambda_{pg}\right ) r_{Bg} S_{gs}}}{\frac{\partial q_{g}}{\partial \bar{X}_{g}}}|_{\bar{X}_{g} = 1}] }$ (19)
• Calculation of flow

1) Backwater formula for the gravel-bed reach

 $\displaystyle{ {\frac{\partial H_{grav}}{\partial x}} = {\frac{1}{s_{gs}}}{\frac{\partial H_{grav}}{\partial \bar{X}_{g}}} = {\frac{S_{ggs} - C_{fg}{\frac{q_{w}^2}{g H_{grav}^3}}}{1 - {\frac{q_{w}^2}{g H_{grav}^3}}}} }$ (20)

2) Backwater formula for the sand-bed reach

 $\displaystyle{ {\frac{\partial H_{sand}}{\partial x}} = {\frac{1}{L - s_{gs}}} {\frac{\partial H_{sand}}{\partial \bar{x}_{s}}} = {\frac{S_{sgs} - C_{fs}{\frac{q_{w}^2}{g H_{sand}^3}}}{1 - {\frac{q_{w}^2}{g H_{sand}^3}}}} }$ (21)

3) boundary condition at x = L

 $\displaystyle{ \left ( \eta_{s} + H_{sand} \right )|_{\bar{X}_{s} = 1} = \xi_{do} + \xi_{d} }$ (22)
 $\displaystyle{ H_{grav}|_{\bar{x}_{g} = 1} = H_{sand}|_{\bar{X}_{s} = 0} }$ (23)
• Calculation of Shields numbers

1) Boundary shear stress

 $\displaystyle{ \tau_{b} = \rho C_{f} U^2 = \rho C_{f} {\frac{q_{w}^2}{H^2}} }$ (24)

2) Shield number

 $\displaystyle{ \tau^* = {\frac{C_{f} U^2}{R g D}} = C_{f} {\frac{q_{w}^2}{R g D H^2}} }$ (25)

3) Volume gravel transport per unit width for the ith node

 $\displaystyle{ q_{g,i} = \sqrt{R g D_{g}} D_{g} 11.2 \left ( \tau_{grav,i}^* \right ) ^ \left (1.5 \right ) \left ( 1 - {\frac{0.03}{\tau_{grav,i}^*}}\right ) ^ \left ( 4.5 \right ) }$ (26)

4) Volume sand transport per unit width for the ith node

 $\displaystyle{ q_{s,i} = \sqrt{R g D_{s}} D_{s} {\frac{0.05}{C_{fs}}} \left (\tau_{sand,i}^* \right ) ^ \left (2.5 \right ) }$ (27)