Rain-induced flash floods are common events in regions near mountain ranges. In peri-urban areas near the Andes the combined effects of the changing climate and El Niño–Southern Oscillation (ENSO) have resulted in an alarming proximity of populated areas to flood-prone streams, increasing the risk for cities and infrastructure. Simulations of rapid floods in these watersheds are particularly challenging, due to the complex morphology, the insufficient hydrometeorological data, and the uncertainty posed by the variability of sediment concentration. High concentrations produced by hillslope erosion and rilling by the overland flow in areas with steep slopes and low vegetational covering can significantly change the dynamics of the flow as the flood propagates in the channel.
In this investigation, we develop a two-dimensional finite-volume numerical model of the nonlinear shallow water equations coupled with the mass conservation of sediment to study the effects of different densities, which include a modified version of the quadratic stress model to quantify the changes in the flow rheology.
We carry out simulations to evaluate the effects of the sediment concentration on the floods in the Quebrada de Ramón watershed, a peri-urban Andean basin in central Chile. We simulate a confluence and a total length of the channel of 10.4 km, with the same water hydrographs and different combinations of sediment concentrations in the tributaries.
Our results show that the sediment concentration has strong impacts on flow velocities and water depths. Compared to clear-water flow, the wave-front velocity slows down more than

Flash floods with high sediment concentrations are common natural events in mountain rivers, which generate hazards in cities and other smaller human communities located near river channels

The spatial and temporal distribution of precipitation, the morphology of the basin, soil properties, and vegetation characteristics naturally influence the magnitude and frequency of floods and sediment transport. Anthropogenic factors also affect the volume and peak discharges of floods in mountain rivers. Climate models predict a larger frequency of intense precipitation events and cyclonic weather systems that will increase the vulnerability in many mountainous regions in the future

The effectiveness of assessing flood hazards and designing strategies aimed at reducing potential damage caused by flooding is closely related to understanding the dynamics of the flow in real conditions. Recent physical models and experiments have provided relevant insights into the flow physics of flash floods in extreme conditions

In most hydrodynamic models to simulate flood propagation, the nonlinear shallow water equations (NSWEs) or Saint-Venant equations are employed to describe the dynamics of the flow in homogeneous and incompressible fluids.
They are obtained by vertically averaging the three-dimensional Navier–Stokes equations, assuming a hydrostatic pressure distribution, resulting in a set of horizontal two-dimensional (2-D) hyperbolic conservation laws that describe the evolution of the water depth and depth-averaged velocities in space and time.
In flows where discontinuities and rapid wet–dry interfaces develop, numerical models employ Godunov-type formulations, solving a Riemann problem at the interfaces of the elements of the discretization

The development of efficient and accurate numerical models to simulate flash floods, however, is far from trivial, since multiple factors control the dynamics of the flow. This is particularly true in mountainous regions, where rivers are characterized by three important features that complicate their representation: (1) complex bathymetries and steep slopes produce rapid changes on velocities and water depths, formation of bores, and wet–dry interfaces; (2) large sediment concentrations directly affect the flow hydrodynamics by introducing additional stresses that alter the momentum balance of the instantaneous flow; and (3) there is a lack of accurate field data, used for validation, due to the difficulties on measuring hydrometeorological variables in high-altitude environments, with difficult access, and during episodes of severe weather.

The Andes mountains in South America incorporate all these characteristics, and they have been the site of many recent catastrophic events, leaving a significant human toll and economic losses

High sediment concentrations during floods cause additional stresses produced by the increase in the density and viscosity of the water–sediment mixture.
Models need to account for the internal stresses that emerge from the particle–flow and particle–particle interactions in the sediment-laden flow. These stresses transform the rheological behavior of the mixture, represented by additional terms of momentum transfer in the governing equations.
A wide variety of rheological models have been proposed depending on the sediment properties and concentration

The main objective of this investigation is to gain fundamental insights into the effects of high sediment concentrations on the propagation of floods in an Andean watershed.
We develop a 2-D finite-volume numerical model of the NSWEs, building on the work of

The paper is organized as follows.
The governing equations of the model and its implementation in the study area are presented in Sects.

Rapid floods over the complex topography of mountainous regions are commonly affected by high sediment concentrations, which change the rheology of the flow. By assuming that the mixture preserves the Newtonian constitutive relation between stress and rate of strain, the NSWEs can be modified to account for the heterogeneous density distribution in space and time

The NSWE model implemented in this investigation has the following assumptions: (i) hydrostatic pressure distribution, (ii) negligible vertical velocities, (iii) vertically averaged horizontal velocities, (iv) horizontal heterogeneous fluid density, (v) homogeneous density in the vertical direction, and (vi) a fixed bed (neither erosion nor deposition).
The momentum sources and sinks consider the gravity term, the bed resistance, and the rheology of the mixture, including the yield stress, Mohr–Coulomb stress, viscous stresses, and turbulent and dispersive stresses, as discussed in Sect.

If we denote the dimensional variables of the flow with a hat (

Here we follow the same procedure outlined in

To adapt the computational domain to the complex arbitrary topography in mountainous watersheds, we use a boundary fitted curvilinear coordinate system, denoted by the coordinates

The Jacobian of the coordinate transformation

The model considers three source terms:

Since the objective of this investigation is to exclusively study the impacts of sediment transport on the hydrodynamics of rapid floods in mountain rivers, where the geomorphic features of the channel play a significant role in flood propagation, we are not considering the erosion or deposition of the bed. The channel of the Quebrada de Ramón stream has a long bedrock section, and the urban area is completely paved. No significant erosion of the channel was reported in the most recent flood, but these conditions cannot apply to other similar cases

It is important to note that mountain rivers with steep slopes in peri-urban watersheds exhibit a wide variety of bed conditions, i.e., bedrock channels, boulders, coarse gravel surfaces, armoring, sand and gravel mixtures, and the concrete surface of the urban setting. The complexity of these environments and the unknown effects of the high sediment concentrations that originate in the high-altitude sections of the mountains prompted the development of this model that couples the transport equation to the flow in mass and momentum with high resolution.
The system of governing Eq. (

The classification and rheology of gravity-driven flows with higher concentrations usually depend on the particle size distribution and sediment composition. Depending on these characteristics, the flows can vary from nearly dry landslides to water flow, with intermediate conditions such as debris flows, mudflows, and mud floods

In the present numerical model we modify the quadratic model of

The yield and Mohr–Coulomb stresses

The viscous term

The value of

To account for the dispersive effects,

To obtain the terms

Appendix

As has been previously mentioned, one of the cases where the sediment concentration plays a significant role is in the flood propagation in mountain rivers. We select as a study case the Quebrada de Ramón watershed in the Andes of central Chile to evaluate how the hydrodynamics of the flow are altered by the magnitude of the sediment concentrations. We simulate the same scenario, but we modify the volume sediment concentration in a range of 0 %–60 %. We locally characterize the depths and velocities of the flow as the flood propagates, but we also evaluate the extension of flooded area and the momentum of the flow in the lower section of the watershed, within the city of Santiago.

Satellite image of the Quebrada de Ramón watershed and the computational domain. The area enclosed by the black line is defined from the lidar topography and incorporates the section of the city around the river channel. The main channel is depicted in blue and the tributary in turquoise. Background image of the terrain from © Google Earth.

Three photos along the Quebrada de Ramón stream from downstream to upstream:

In this watershed there are two major tributaries draining the north and south sections of the watershed. We define the computational domain shown in Fig.

The bed roughness is represented by a mean sediment grain diameter

Distribution of the mean sediment size in millimeters. White circles denote the measurement sites where we report the propagation of the flood.

The hydrographs of the event studied in this investigation for the two main rivers that correspond to the tributaries of the confluence are shown in Fig.

The 50-year return period hydrographs in both streams obtained from a hydrological model of the catchment

The actual sediment concentration during flash floods in Andean watersheds is unknown in most cases. In addition to the lack of information in these rivers, landslides due to erosion produced by soil saturation in the steep slopes of the mountains are common. These conditions can considerably increase the sediment supply to the streams during flood events, with material that does not come from the channel but mostly from hillslope erosion. We study the dynamics of the flood for different scenarios by carrying out a series of simulations to compare and understand the flood hazards and effects of hyperconcentration on the two main streams of the Quebrada de Ramón watershed.

We simulate four different scenarios considering different concentrations of

The river bed is considered dry at the beginning of the simulation, to avoid the additional effects of different sediment concentrations of the initial flow.

We perform the simulations for a total physical time of 1 d, using a simulation time step defined by the Courant–Fiedrichs–Lewy (CFL) stability criterion, defined as

The inflow boundary condition in Fig.

To evaluate the impacts of different sediment concentrations on the flood dynamics, in the following subsections we analyze the flow hydrodynamics including (1) the position and velocity of the flood wave front, (2) the peak flow and arrival time, (3) the flooded areas, (4) the effect of the sediment concentration on the depth and flow velocity, and (5) the momentum of the flow in the urban zone.

To quantify the propagation of the flood along the channels and the arrival time of the flood to the city, we compute the mean velocity of the wave front by tracking its position in time.
Table

Mean velocity of the front for different sediment concentrations. In parentheses, the percentage change from clear water is shown.

Location of the wave front in time for both streams, considering four different sediment concentrations, to characterize the advance of the flood in the river:

Figure

The numerical results show that the sediment concentration produces a significant change in the evolution of the flood, as it is the only factor that we modify in these simulations. The local variations in these velocities are produced by the gradual change of bed roughness and the slope of the river channels, which is approximately constant in large portions of the reaches. The Quillayes stream exhibits higher propagation velocities, which are also consistent with the steeper slopes and finer sediment diameters of the bed.

Figure

Downstream of the confluence, we also observe the effects of the interaction between the morphology and the sediment concentration. The wave-front speed is affected by the different arrival times of flows from both tributaries. Figure

Location of the wave front in time for the Quebrada de Ramón stream downstream of the confluence considering four sediment concentrations in both streams.

By comparing the hydrographs computed using different sediment concentrations at the four points monitored upstream of the confluence, we observe that the most important difference is the magnitude of the peak flow for different concentrations. The relative difference between the peak discharge simulated with clear-water flow compared to a sediment concentration of

Normalized hydrographs at the gauged points upstream of the confluence. In panels

Normalized hydrographs at the gauged points downstream of the confluence. In panel

In both streams, the time to the peak of the hydrograph, however, is not significantly affected by the different concentrations. This seems to be related to the shape of the inflow hydrograph and to the location of the gauged points. The time to reach the peak discharge is around

When we analyze the hydrographs downstream of the confluence we observe similar results, as shown in Fig.

The total area in the watershed that is inundated for different sediment concentrations is depicted in Fig.

Contours of the flooded area for different sediment concentrations along the channel, and in the city of Santiago. Image of the terrain from © Google Earth.

Major differences, however, appear in regions with milder slopes, around the confluence and in the city, near the outlet of the watershed.
At the confluence, simulations with higher concentrations of

Figure

Maximum depth computed at every gauged point depending on the sediment concentration.

By comparing the flow depths in the simulations, we note that the deepest flow is always located downstream of the confluence (QR-C). At this location, a difference of 0.80 m is measured between the clear water and the flow with the maximum concentration of

Additionally, in Fig.

Mean velocity computed at each gauged point depending on the sediment concentration.

Stage–velocity relation between the computed flow depths and velocities for different sediment concentrations. The data at the point QR-U are shown in panel

In Fig.

Even though a direct relation is not observed between mean velocity and sediment concentration at points QR-U and Qui-U (Fig.

To evaluate the potential damage to the infrastructure generated by floods, we can compute the flow momentum at each cross section of the flooded area.
In this case we compare the maximum force produced by the flow in the urban area of the watershed, considering flows with different sediment concentrations coming from the Quebrada de Ramón and the Quillayes streams.
Figure

Maximum momentum of the flow in the urban area. Panel

The approaching flow has an approximate force of 700 kN in both simulations. For these two cases, the areas with the highest momentum correspond to (1) the confined zone in the right of the image and (2) at the outlet of the basin in the urban area. However, the force is on average 14.5 % higher in the second case, which could be related to the higher flow density of the flow that is obtained downstream of the confluence.

Since the density in these simulations is different in both streams upstream of the confluence, the density of the fluid in the main channel varies in time and space, both along and across the flow. The mean concentration in the main channel, downstream of the confluence, is around

The results evidence the competition between two main factors that control the dynamics of the flow in mountain rivers at different spatial and temporal scales: (1) the geomorphological features of the river represented by the bathymetry, the slope, and the channel width and (2) the flow resistance due to the internal sediment dynamics that changes the rheology of the mixture.

At timescales of seconds or minutes, flow velocities and depths along the channel are significantly affected by both factors, having a great impact on global variables such as the wave-front velocity, the total inundated area, and the cross-section momentum of the flow.
As reported in Sect.

Global bulk variables, on the other hand, such as the normalized hydrograph shape and the time to the peak discharge, show a geomorphic control at the scales of the duration of the entire event. Except in areas where there is a change in the flow regime, the effects of the sediment concentration are not observed for the time-averaged velocities along the channel.
As shown in the normalized plots in Fig.

It is important to point out that the sensitivity of the flow physics affected by the sediment concentration, such as the mean velocity of the wave front, flow depth, instantaneous velocity, flooded area, and flow momentum, decreases for higher sediment concentrations. We show that as the sediment concentration increases, the changes are more significant in the range between 0 % and 20 %, compared to the flood propagation for increments over 40 %. These new insights are relevant to determine flood hazard in mountain rivers and define a reduced number of possible scenarios for different concentrations in these rivers.

The primary emphasis of this work is to examine the effects of the sediment concentration on the flood dynamics in an Andean watershed.
To simulate different scenarios, we developed a finite-volume numerical model that solves the hydrodynamics of hyperconcentrated fluids in complex natural topographies.
The model is based on the work of

To investigate the effects of the sediment concentration in floods that occur in mountain rivers, we perform simulations in the Quebrada de Ramón watershed, an Andean catchment located in central Chile. We analyze the changes in hydrodynamic variables such as peak discharge, arrival time of the flood wave, cross-section momentum, flow depth, mean velocity, and total flooded area. Most of the these results are compared and analyzed in seven points along the channel.

The most important effects on the flood propagation are observed for the increments of sediment concentration just above the clear-water flow, in the range of concentrations from

Some of the hydrodynamic variables analyzed were more sensitive to changes in sediment concentration.
We observed significant effects on the total flooded area and momentum of the flow as the flood arrives at the urban area.
While the extent of the 2-D flooded area in the entire basin remains more or less constant for different concentrations, the largest difference is observed in the city, where the slopes are milder. The simulations show a difference of

The numerical solution of the system of Eq. (

The initial hyperbolic step consists of numerically solving the following equation:

To compute the numerical fluxes we implement the VFRoe-ncv method

The bed-slope source term

The concentration gradient term

In the second step of the numerical solution, we incorporate the momentum source terms in vector

Finally, the temporal integration of Eq. (

The boundary conditions are handled by creating two rows of “ghost cells” outside of the computational domain

This benchmark test is developed to demonstrate the capacity of the model to preserve the hydrostatic state with density differences. An analytical solution is obtained from the procedure developed by

The dimensions and initial conditions of this test are presented in Fig.

Dimensions and initial conditions of the rectangular tank used for the quiescent equilibrium test

Quiescent equilibrium test. Comparison between theoretical and numerical profiles of hydrodynamic variables.

Results show that there is an excellent agreement between the analytical and numerical solutions for the free surface, as shown in Fig.

To test the model in unsteady conditions, we simulate a density-driven dam break to evaluate the evolution of the hydrodynamic variables in space and time. The numerical experiment is based on the work developed by

Initial state of the density-driven dam break

Two different simulations are performed for

In Fig.

Note that when

To test the rheological model, we simulate the large-scale dam-break experiment with high sediment concentration performed by

The experiment consists of the sudden release of a large volume of a sediment–water mixture in a 95 m long rectangular channel, with a cross section that is 2 m wide by 1.2 m deep. The channel is very steep, with an inclination of 31

The unsteady inflow condition is the debris flow at a distance of 2 m downstream from the gate, which is shown in Fig.

In Fig.

To compare the numerical results directly with data provided by the experiments, we apply the same moving-average filter used to smooth the experimental data (black line in Fig.

The simulated and observed wave-front positions in time are very similar (Fig.

Overall, the validation study shows that the numerical model in these extreme cases is very robust and it is able to reproduce many of the phenomena of interest that appear in hyperconcentrated flash floods.

Density-driven dam break. Case

Density-driven dam break. Case

Density driven dam break. Case

Unsteady inflow boundary condition corresponds to the cross-section flow measured at a location of 2 m downstream of the gate

Comparison of the flow thickness measured in the experiment of

Comparison of the position of the flow front as a function of time:

The code and data are available at

MTC and CE conceived the study, analyzed the data, interpreted the results, and wrote the paper. MTC developed the numerical code and performed the simulations for the cases presented in the paper, under the supervision of CE.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Advances in computational modelling of natural hazards and geohazards”. It is a result of the Geoprocesses, geohazards – CSDMS 2018, Boulder, USA, 22–24 May 2018.

This work has been supported by Conicyt/Fondap grant 15110017. This research has been partially supported by the supercomputing infrastructure of the NLHPC (ECM-02). Additional funding from VRI of the Pontificia Universidad Católica de Chile, internationalization of research, project PUC1566, MINEDUC. We thank the Ministry of Public Works for providing the lidar topography. Verónica Ríos and Jorge Gironás provided the flood hydrographs.

This research has been supported by the CONICYT/FONDAP (grant no. 15110017) and the VRI internationalization of research, MINEDUC (grant no. PUC1566).

This paper was edited by Albert J. Kettner and reviewed by two anonymous referees.