Natural Hazards

, Volume 59, Issue 2, pp 1021–1045 | Cite as

Landslide dam failure and flood hydraulics. Part II: coupled mathematical modelling

  • Zhixian Cao
  • Zhiyuan Yue
  • Gareth Pender
Original Paper


A coupled 2D mathematical modelling study of landslide dam failure and flood is presented, complementing our experimental investigation presented in the companion paper. The model is built upon the shallow water hydrodynamic equations. The governing equations are numerically solved using the total-variation-diminishing version of the second-order weighted-average-flux method along with the HLLC (Harten, Lax and van Leer with Contact wave restored) approximate Riemann solver. Two parameters related to bed-load sediment transport and critical slope stability are calibrated using the measured stage hydrographs from two runs of the flume experiments. The calibrated model is then applied to other independent runs of the experiments featuring different inflow discharges, dam geometry, dam composition and initial breach dimensions. It is found to be able to satisfactorily reproduce the measured stage hydrographs and the widening of initial breach. The experimental observation of the prime role of the inflow discharge and initial breach in dictating the dam failure process and flood is unequivocally resolved, along with the impacts of dam geometry as well the content of cohesive clay and gravel in the dam. Interestingly, the downstream peak discharge and stage of the flood are substantially reduced by initial breach, which clearly exemplifies its role in modulating the flooding.


Landslide dam Dam failure Flood Sediment transport Coupled hydrodynamic model 

List of symbols


Limit function [–]


Local dynamic wave velocity [m/s]


Averaged sediment concentration in volume [–]


Bed-load sediment transport capacity [–]


Courant number [–]


Sediment particle diameter [mm]

E, D

Sediment entrainment and deposition fluxes [m/s]


Vectors of flux variables in x-direction defined in Eq. (1)


Left or right numerical fluxes of an intercell


Numerical flux in the x-direction


Riemann solver


HLLC approximate Riemann solver


Gravitational acceleration [m/s2]


Vectors of flux variables in y-direction defined in Eq. (1)

Gi, j+1/2

Numerical flux in the y-direction


Flow depth [cm]


Initial static water depth at downstream of the dam [cm]


Initial static water depth at upstream of the dam [cm]

i, j

Spatial node indexes [–]


Time step index [–]


Manning roughness [–]


Auxiliary time step index [–]


Bed sediment porosity [–]


Inlet discharge [m3/s]


Unit-width bed-load transport rate under transport capacity status [kg/(m·s)]


Submerged specific gravity of sediment [–]


Vector of source terms defined in Eq. (1)

S1, S2

Initial upstream and downstream slopes of the dam [–]

Sbx, Sby

Bed slopes in x- and y-directions, respectively [–]

Sc1, Sc2

Critical slopes [–]

Sfx, Sfy

Friction slopes in x- and y-directions, respectively [–]

SL, SR, S*

Left, right and middle wave speeds [m/s]


Time [s]


Time for the initial breach to grow laterally to the full width of the channel [s]


Depth-averaged velocity in x-direction [m/s]


Frictional velocity [m/s]


Depth-averaged velocity in y-direction [m/s]


Settling velocity of a single sediment particle in tranquil water [m/s]


Horizontal coordinates in x-direction [m]


Horizontal coordinates in y-direction [m]


Bed elevation [cm]


Time step [s]

Δx, Δy

Spatial steps [m]


Empirical coefficient [–]


Empirical coefficient [–]


Angle of repose [–]


Angle between slope surface and flow direction [–]


Angle between slope surface and horizon plane [–]


Proportionality coefficient [–]


Von Kármán coefficient [–]


Shields parameter [–]

θc, \( \tilde{\theta }_{c} \)

Threshold Shields parameters in steeper and mild slopes [–]


Density of water–sediment mixture [kg/m3]

ρw, ρs, ρ0

Densities of water and sediment, and saturated bed, respectively [kg/m3]


Modification coefficient [–]


Coefficient of turbulent viscosity [–]


Scale for sediment diameter [–]


Length scale [–]


Manning roughness scale [–]


Discharge scale [–]


Time scale [–] and


Velocity scale [–]



This investigation is funded by National Key Basic Research and Development Program (973 Program) of China (Grant No. 2007CB714106) and Natural Science Foundation of China (Grant No. 10932012, 10972164).


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.State Key Laboratory of Water Resources and Hydropower Engineering ScienceWuhan UniversityWuhanChina
  2. 2.School of the Built EnvironmentHeriot-Watt UniversityEdinburghUK
  3. 3.Yangtze River Waterway Research InstituteWuhanChina

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