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DEPARTMENT OF LAND AND AGROFOREST ENVIRONMENTS
UNIVERSITY OF PADOVA

Muskingum One-Dimensional Debris flow Simualtion (MODDS)

by Mario A. Lenzi

The main goal of the 1-D simulation is the location of critical cross-sections along the channel, where the debris flow spreading on the fan begin for a given scenario of event. In alpine areas fan slope of catchments where debris flow may occur (basin area < 10 km2) are rarely lower than 3%. Considering  the unsteady flow for a gradually varied open channel, the inertial terms are negligible in the momentum equation in comparison with the bed slope. In this case the Saint-Venant equations reduce to:

  (1)

where: t is the time; x the spatial coordinate; Q  is the flow rate; c=dQ/dA=(1/B) dQ/dh= kinematic wave speed  (B= channel top width; h= flow depth; A=flow area); D=Q/(2BS)=diffusion coefficient (S=channel bed slope).
Cunge proved that the conventional Muskingum equations are assimilable to a convective-diffusive equation like (1), if the numerical diffusion is equated to the physical diffusion (D, eq.1). Developing this hypothesis in a computational space-time grid, it follows:

  (2)

where the variable parameters gi are expressed by the relations:

  (3)

with P=2D/(c Dx) and  N=c Dt/Dx..P and N parameters are computed by averaging the three respective values coming from point 1, 2 and 3.
For the evaluation of the kinematic speed (c) of a debris flow it is necessary an assumption on the joint between the surge depth (h) and its velocity (v). This assumption may be supported by the analysis of the observed maximum velocity field data in monitored stream. Even if different rheologic models may interpret the debris flow movement, the analysis of precise measurements of several granular surges moving on a mobile bed shows the capability of the Newtonian flow to  give acceptable results. Some authors estimated the flow resistance in terms of the dimensionless Chezy coefficient (C*). They obtained that most data are included in the region:

  (4)

where S =sin q; q= inclination angle of the channel and h  is the surge depth (m).
The adoption of eq.(4) linked with the hypothesis to equate the energy slope to the bed slope, allows both the computation of the kinematic speed (c=1.5 v) and the surge depth associated to the flow rate . 
Following the above described procedure, the model MODDS (Muskingum One-Dimensional Debris flow Model) has been developed for a channel geometry described by eight points. Each of  these contains three absolute coordinate (X and Z in the cross-section plan and X and Y for a plan view). In addition 3 additional points are automatically generated by the program: point 9 represents the channel axis, points 10 and 11 are fictitious points, located vertically 50 m above point 7 and 8 respectively. Points 10 and 11 are necessary if the bank overflow has been prevented by the user (the setting of this option may be useful in a design perspective) or if  the overflow occur (above point 7 and/or 8) and temporary cross-section parameters have to be evaluated for a given depth (h).
The described computational approach has been developed using the Borland Delphi 5.0 language.
Furthermore, the program estimates surface superelevations in bends, the presence of bridges, the diversion of the flow due right and/or left bank overflowing.
A summary of  input parameters and output results in presented in the appendix. 
In figures 1-2 the results of an application are shown considering a channel having a  rectangular  cross-section and a constant slope equal to 19% (the same of the Rio Lenzi, Trento – Italy, along the fan). The channel is 5 m wide for the firsts 210 m, it follows an abrupt narrowing to 3 m for a 30 m long reach, after which the channel enlarged again to 5 m for the remaining 210 m. The input debris-flow hydrograph is triangular shaped with a peak value equal to 100 m3 s-1 and the assumed roughness C* is 3 for the whole channel. 
The computation has been conducted choosing a  Dx value ranging from 20 m  - in the narrowing - to 50 m - in the remaining reach - and a  Dt value of  5 s.
The model responds satisfactorily to the sharp input hydrograph and to the narrowing. The pattern of flow rates with time (fig.1) at different locations shows a certain lamination of the peak discharge and a progressive smoothing of the initial triangular input hydrograph. The depth pattern for the cross-sections inside the narrowing increases markedly (fig. 2) and returns downstream of the contraction to values comparable with those upstream of  the narrowing. The volume conservation is still acceptable for a debris flow wave, because the entering sediment volume of 33000 m3 of the initial cross-section becomes equal to 33650 m3 in the terminal one, with a volume increase equal to 2%. 

  Figure 1 - Debris flow routing trough a rectangular channel 450 m long (C*  = 3; S = 19%; width B = 5 m; abrupt narrowing to B =3 m fromx  = 210 m to x  = 240 m):

low rates hydrographs

 

  

Figure 2 - Debris flow routing trough a rectangular channel (same conditions of figure 1):

surge depths hydrographs

APPENDIX

INPUT PARAMETERS

General simulation data:

  • Input discharge time interval (s)

  • Simulation duration (hours) 

  • Temporal  integration step (s)

  • Bend superelevation coefficient α

General boundary conditions:

  • Intial discharge (m3s-1)

  • Peak discharge (m3s-1)

  • Final discharge (m3s-1)

For each section of the channel:

  • Mean reach gradient

  • Non-dimensional Chèzy coefficient 

  • Longitudinal distance from the initial point of the simulation (m) 

  • Cross-section coordinates (8 points) 

  • Bridge characteristics (thickness and position)

  OUTPUT DATA 

At each cross-section for each time step:

  • Debris flow discharge (m3s-1)

  • Oveflow discharge at left and right banks  (m3s-1)

  • Mean velocity (m/s)

  • Flow depth  (m)