# 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)

## Notes

This model calculates the long profile of a river with a gravel-sand transition. The model uses two grain sizes: size Dg for gravel and size Ds for sand. The river is assumed to be in flood for the fraction of time Ifg for the gravel-bed reach and fraction Ifs for the sand-bed reach. All sediment transport is assumed to take place when the river is in flood.

The code locates the gravel-sand transition at a point determined by the continuity condition. At this point the gravel transport rate is only a small fraction of the feed value, but it is not precisely zero. In rivers, the small residual gravel load at gravel-sand transitions is either buried or consists of grains that easily break down to sand. In the code, the residual gravel load at the gravel-sand transition is added to the sand load.

The position of the gravel-sand transition x = sgs(t) may change in time.

Gravel transport is computed using the Parker (1979) approximation of the Einstein (1950) bedload transport relation. Sand transport is computed using the total bed material transport relation of Engelund and Hansen (1967).

In this simple model the gravel is not allowed to abrade. Both the gravel-bed and sand-bed reaches carry the same flood discharge Qbf.

Gravel is transported as bed material in, and deposits only in the gravel-bed reach. A small residual of gravel load is incorporated into the sand at the gravel-sand transition. Sand is transported as washload in the gravel-bed reach, and as bed material load in the sand-bed reach.

It is assumed that there are no significant tributaries along the entire reach from x = 0 to x = L, so that water discharge during floods is constant downstream.

The model allows for depositional widths Bdgrav and Bdsand that are wider than the corresponding bankfull channel widths Bgrav and Bsand of the gravel-bed and sand-bed channels. As the channel aggrades, it is assumed to migrate and avulse to deposit sediment across the entire depositional width. For each unit of gravel deposited in the gravel-bed reach, it is assumed that Lamsg units of sand are deposited. For each unit of sand deposited on the sand-bed reach, it is assumed that Lamms units of mud are deposited.

The initial downstream bed elevation is taken to be zero. As a result, the initial downstream water surface elevation ξdo also equals the initial downstream depth. In order to ensure subcritical flow (and thus keep the calculation from crashing), ξdo must be exceed the critical flow depth Hc = [(Qbf/Bc,sand)2/g]-1/3.

• Note on model running

The values for the spatial step of both gravel and sand max out at 100, allowing 200 spatial steps total (the number of sand spatial nodes, Ms and gravel spatial nodes, Mg must be ≤ 100).

When the slopes are unequal at gravel sand transition the GetData function and the plot data output a slope of 0 at that point.

The reach length L should be chosen to be less than the maximum value Lmax, in order to ensure that there is enough sediment supply to fill the accomodation space created by subsidence or sea level rise.

The sand and gravel Froude numbers must be less than 1 (i.e. the flow is assumed subcritical in the sand and gravel portions of the bed); if they are not, the program will quit and alert the user what their values were.

The user inputted downstream water elevation, ξd, must be greater than the program calculated minimum water elevation, ξmin; if this is not the case, the program will quit and the user will be alerted what the value is for ξmin.

Depending on the input values, there may be no steady-state solution allowing a gravel-sand transition to equilibrate at a position between 0 and L. For example, if ξd = 0 and δ = 0, the only steady-state solution is one for which the sand is all driven into the sea. In such cases, the code will fail. The code can be run, however, to a time at which the gravel-sand transition is nearly driven out of the domain of interest.

## 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:

## References

• Aalto, R., 2002, Geomorphic form and Process of Sediment Flux within an Active Orogen: Denudation of the Bolivian Andes and Sediment Conveyance across the Beni Foreland, PhD thesis, University of Washington, USA, 365 p.
• Cui, Y. and Parker, G., 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 2: General numerical solution, Journal of Hydraulic Research, 36(2): 159-182.
• Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.
• Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams, Technisk Vorlag, Copenhagen, Denmark.
• Ferguson, R. I., 2003, Emergence of abrupt gravel-sand transitions along rivers through sorting processes, Geology 31, 159-162.
• Paola, C., Heller, P. L., and Angevine, C. L., 1992, The large-scale dynamics of grain-size variation in alluvial basins, 1: Theory, Basin Research, 4, 73-90.
• Parker, G., 1979, Hydraulic geometry of active gravel rivers, Journal of Hydraulic Engineering,105(9), 1185‑1201.
• Parker, G., 1991a, Selective sorting and abrasion of river gravel: theory, Journal of Hydraulic Engineering, 117(2), 131-149.
• Parker, G., 1991b, Selective sorting and abrasion of river gravel: applications, Journal of Hydraulic Engineering, 117(2), 150-171.
• Parker, G., and Y. Cui, 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 1: Simplified analytical solution, Journal of Hydraulic Research, 36(1): 75-100.