Debris flow material properties change during the initiation, transportation and deposition processes, which influences the runout characteristics of the debris flow. A quasi-three-dimensional depth-integrated numerical model, EDDA (Erosion–Deposition Debris flow Analysis), is presented in this paper to simulate debris flow erosion, deposition and induced material property changes. The model considers changes in debris flow density, yield stress and dynamic viscosity during the flow process. The yield stress of the debris flow mixture determined at limit equilibrium using the Mohr–Coulomb equation is applicable to clear water flow, hyper-concentrated flow and fully developed debris flow. To assure numerical stability and computational efficiency at the same time, an adaptive time stepping algorithm is developed to solve the governing differential equations. Four numerical tests are conducted to validate the model. The first two tests involve a one-dimensional debris flow with constant properties and a two-dimensional dam-break water flow. The last two tests involve erosion and deposition, and the movement of multi-directional debris flows. The changes in debris flow mass and properties due to either erosion or deposition are shown to affect the runout characteristics significantly. The model is also applied to simulate a large-scale debris flow in Xiaojiagou Ravine to test the performance of the model in catchment-scale simulations. The results suggest that the model estimates well the volume, inundated area, and runout distance of the debris flow. The model is intended for use as a module in a real-time debris flow warning system.

Debris flow is a flow of a sediment–water mixture driven by gravity. The mechanical triggers of debris flows can be classified into three types, namely erosion by surface runoff, transformation from landslides, and collapse of debris dams (Takahashi, 2007). Basal erosion, side erosion, and any other surficial material entrainment during the marching process entrain additional material into the flow; the final volume can be several or dozens of times of the initial volume (e.g. Hungr et al., 2005; Chen et al., 2006, 2012, 2014; Berger et al., 2010). When the debris flow moves to a flatter area, the coarse materials can deposit gradually. During the entire movement process, not only the debris flow volume, flow velocity and flow depth change significantly, the properties of the debris flow mixture also change substantially, which in turn influence the runout characteristics.

The mechanisms of changes in debris flow mass are important and have attracted the attention of many researchers (e.g. Cannon and Savage, 1988; Takahashi et al., 1992; Hungr, 1995; Egashira et al., 2001; Iverson, 2012). Cannon and Savage (1988) and Hungr (1995) proposed one-dimensional lumped-mass models based on momentum conservation to describe the entrainment or loss of material during the movement of a debris flow. Takahashi et al. (1992) proposed a model to describe erosion and deposition based on volumetric sediment concentration and flow velocity. Researchers have also described the erosion process from a stress point of view (e.g. Medina et al., 2008; Iverson, 2012; Quan Luna et al., 2012): erosion occurs when the basal shear stress exceeds the critical erosive shear stress of the bedding material.

During the entire process of a debris flow, the debris flow properties can change significantly. The volumetric sediment concentration (i.e. ratio of the solid volume to the total volume of the debris flow mixture) can increase substantially due to entrainment of solid materials (e.g. Takahashi et al., 1992; Egashira et al., 2001) or decrease due to deposition (e.g. Takahashi et al., 1992) and dilution (e.g. Pierson and Scott, 1985). Accordingly, the rheological characteristics of debris flows (e.g. yield stress and dynamic viscosity) will change with the volumetric sediment concentration, which has been observed in a large number of experiments (e.g. O'Brien and Julien, 1988; Rickenmann, 1991; Major and Pierson, 1992; Sosio and Crosta, 2009; Bisantino et al., 2010). Various rheological models have been adopted to describe debris flows, such as the laminar flow model (e.g. Takahashi, 2007), the Bingham fluid model (e.g. Fraccarollo and Papa, 2000), the Voellmy model (e.g. Medina et al., 2008), and the quadratic rheological model (Julien and Lan, 1991).

Based on understanding of erosion, deposition and rheology of debris flow materials, great efforts have been made to simulate the movement of debris flows (e.g. Cannon and Savage, 1988; Takahashi et al., 1992; Hungr, 1995; Denlinger and Iverson, 2001; Ghilardi et al., 2001; Chen et al., 2006, 2013; Pastor et al., 2009; Li et al., 2012; van Asch et al., 2014). The numerical methods include the finite difference method (e.g. Takahashi et al., 1992), the finite volume method (e.g. Medina et al., 2008), the finite element method (e.g. Crosta et al., 2003), the distinct element method (e.g. Li et al., 2012), the smoothed particle hydrodynamics method (e.g. Pastor et al., 2009) and others. Several computer programs have been written for debris flow analysis, such as DAMBRK (Boss Corporation, 1989), FLO-2D (O'Brien et al., 1993), DAN (Hungr, 1995), TOCHNOG (Crosta et al., 2003), 3dDMM (Kwan and Sun, 2006), FLATModel (Medina et al., 2008), DAN3D (Hungr and McDougall, 2009), MassMov2D (Beguería et al., 2009), PASTOR (Pastor et al., 2009), and RAMMS (Bartelt et al., 2013).

Depth-integrated models have been widely adopted to describe erosion and deposition (e.g. Takahashi et al., 1992; McDougall and Hungr, 2005; Armanini et al., 2009; Hungr and McDougall, 2009; Iverson et al., 2011; Quan Luna et al., 2012; Ouyang et al., 2015). The Mohr–Coulomb failure process was adopted to simulate bed erosion (e.g. Medina et al., 2008; Quan Luna et al., 2012). Ouyang et al. (2014) further combined the Mohr–Coulomb model and the Voellmy model to overcome the flaws of each of these two models. The changes in flow depth, flow velocity and debris mass have been accounted for in the literature. Limited attempts have also been made to consider the evolution of volumetric sediment concentration (Takahashi et al., 1992; Denlinger and Iverson, 2001; Ghilardi et al., 2001). Several key problems, however, still remain. How can one describe the various phases of a debris flow (e.g. clear water flow, hyper-concentrated flow, and fully developed debris flow) using a general rheological model? How do the properties of debris flows (e.g. volumetric sediment concentration, yield stress, viscosity) change in the erosion and deposition processes? How do these changes affect the runout characteristics of the debris flow? These problems are very important for the risk assessment of debris flows.

The objective of this paper is to develop a numerical model to consider the erosion and deposition processes and debris flow property changes during these processes. The paper is organized as follows. The methodology is introduced in Sect. 2, including the problem description, governing differential equations, constitutive models, initiation of erosion and deposition, numerical solution algorithm, time stepping and numerical stability. The model is tested and verified in Sect. 3 using an analytical solution, numerical solutions, and experimental tests. A large-scale debris flow event in the Wenchuan earthquake zone is simulated as a field application in Sect. 4. The limitations of the model are indicated in Sect. 5.

The volume of a debris flow can increase due to erosion or entrainment and
decrease due to deposition. Due to changes in sediment concentration, a
debris flow triggered by surface runoff may experience several flow regimes.
The debris flow can evolve from a clear water flow to a hyper-concentrated
flow, a fully developed debris flow, and finally a deposit on the debris
fan. The erosion and deposition processes and property changes in debris
flow are illustrated in Fig. 1. The debris flow entrains and incorporates
materials from the channel bed if the volumetric sediment concentration,

Erosion, deposition and property changes in debris flow.

Changes of volumetric sediment concentration of debris flow:

In this study, an integrated numerical model is developed to simulate debris
flow erosion, deposition, and the induced property changes. The model is
named EDDA, which stands for Erosion–Deposition Debris flow
Analysis. The reference frame is defined in Fig. 1. Depth-integrated
mass conservation equations (Eqs. 1, 2) and momentum conservation
equations (Eqs. 3, 4) are adopted to describe the movement of a debris
flow:

Changes in dynamic viscosity and yield stress with
volumetric sediment concentration:

Similar to the two-dimensional model proposed by O'Brien et al. (1993), the
governing equations above use a global coordinate system, which has been
proven to simulate well flows in channels and alluvial fans (Akan and Yen,
1981; O'Brien et al., 1993). The difference is that EDDA considers changes
in debris flow properties due to material entrainment and the induced
momentum exchange. In Eqs. (3) and (4), the flow velocity gradient in the
orthogonal direction is neglected, since very little accuracy is sacrificed
by neglecting this term (Akan and Yen, 1981; O'Brien et al., 1993). In this
study, erosion and deposition are investigated while surficial material
entrainment is not. The bed elevation changes in the erosion and deposition
processes and can be expressed as

Various forms of rheological models can be implemented in the momentum
conservation equation, which allows for the simulation of various types of
flows. Several of the most widely used rheological models are introduced below to
compute

The laminar flow model is useful to describe the movement of a fully
liquefied flow, which is governed by viscous behaviour. The flow resistance
slope is expressed as

Turbulent flows with low volumetric sediment concentration are often
analysed using the Manning equation:

The Bingham fluid model considers both plastic and viscous behaviours. A
Bingham fluid does not move if the shear stress is smaller than a threshold
yield stress, but behaves as a viscous material when the shear stress
exceeds the threshold. The model is expressed as

The Voellmy model (Voellmy, 1955) combines the effects of frictional and
turbulent behaviours:

The quadratic rheological model proposed by Julien and Lan (1991) considers
the effects of frictional behaviour, viscous behaviour, and turbulent
behaviour plus the resistance arising from solid-particle contacts, which
are represented by three terms as follows:

Since the quadratic rheological model accounts for the most comprehensive flow behaviour, it is adopted into the governing differential equations in this paper.

O'Brien and Julien (1988) proposed the following empirical relationships to
estimate the yield stress,

The effective cohesion of the debris flow material is taken as zero in
the above equation. If only the particles in contact are considered, the
yield stress can be calculated by incorporating a coefficient of suspension
of solid particles as follows:

Three typical suspension scenarios are shown in Fig. 4: partial suspension,
0 <

Equation (17) is suitable for calculating the changing yield stress especially at low solid concentrations. Equation (13) is suitable for calculating the changing yield stress especially at high solid concentrations and performs well on alluvial fans (O'Brien et al., 1993). Therefore, Eqs. (17) and (13) are adopted to calculate the yield stress in a confined channel with erodible materials and an unconfined flat area, respectively. The combination of the two equations overcomes the drawbacks of each of the two equations.

The values of

Erosion occurs when the bed shear stress is sufficiently large and the
volumetric sediment concentration is smaller than an equilibrium value. The
equilibrium value proposed by Takahashi et al. (1992) is adopted in this
study:

The erosion rate can be described approximately by the following equation:

Three typical suspension scenarios:

Medina et al. (2008) and Quan Luna et al. (2012) consider bed erosion as a
Mohr–Coulomb failure process. In this study, the critical erosive shear
stress can be calculated by considering the partly suspended particles at
limit equilibrium using the Mohr–Coulomb equation:

When the debris flow moves to a flatter place, deposition occurs if

The analysis domain is discretized into a grid first, with properties of
each cell assigned, including the initial flow depth, the thickness and
properties of the erodible soil layer, the elevation of the non-erodible
layer, Manning's coefficient and so on. As shown in Eqs. (1) and (2), the
changes in

As shown in Fig. 5, each cell has eight flow directions, namely four
compass directions (i.e. north, east, south and west) and four diagonal
directions (i.e. northeast, southeast, southwest and northwest). In each
time step, the changes in

Eight flow directions and flow boundaries of each cell.

At the beginning of each time step, the erosion rate or deposition rate
is computed for each cell. The flow depth and volumetric sediment concentration of each cell are updated using Eqs. (1) and (2) as follows:

At each flow boundary, the average flow depth, flow density, volumetric sediment concentration and roughness of the two cells bounded at the boundary are computed. The bed slope between the two cells is defined using the gradient between the centres of the cells.

The new flow velocity across each flow boundary is obtained by solving the momentum conservation equation:

The discharge,

To make the solution more robust, the average values of

On the one hand, the time step should be sufficiently small to ensure the numerical stability. On the other hand, the time step should be large enough to attain reasonable computational efficiency. An adaptive time stepping scheme is adopted in this research to ensure both the numerical stability and the computational efficiency, especially for cases which involve a large number of cells so that the simulation time is likely long. The algorithm for the adaptive time stepping scheme is shown in Fig. 6.

Three convergence criteria are adopted in this study. The first criterion is
the Courant–Friedrichs–Lewy (CFL) condition; namely, a particle of fluid
should not travel more than the cell size in one time step,

Algorithm for adaptive time stepping.

In this section, four numerical tests are conducted to verify the performance of the proposed model. In Test 1, an analytical solution to one-dimensional debris flow is adopted to validate the performance of the model in simulating the movement of a debris flow with constant material properties. In Test 2, a two-dimensional dam-break flow problem is adopted to validate the performance of the model in simulating two-dimensional problems. In Test 3, a flume test is adopted to validate the performance of the model in describing the erosion process and material property changes. In Test 4, another flume test is adopted to validate the performance of the model in describing the movement of a debris flow considering the material property changes due to both erosion and deposition.

The problem described by Liu and Mei (1989) is adopted in this test. The
materials are initially retained as a triangular pile by a board and the
initial profile is shown in Fig. 7. The materials start moving after the
board is removed and cease moving finally due to the presence of yield
stress. The final profile for the one-dimensional flow is (Liu and Mei,
1989)

Comparison of the final debris flow depth profiles from the analytical solution and the numerical solution in Test 1.

A two-dimensional partial dam-breach problem reported by Fennema and Hanif
Chaudhry (1987) is adopted. A sketch of the problem is shown in Fig. 8a. The
computation domain is a channel 200 m in length and 200 m in width. The
depth of the reservoir water is 10 m, and the depth of the tail water is 5 m. The boundary is assumed to be frictionless. The dam is assumed to fail
instantaneously and the breach width is 75 m. The computation domain is
discretized into a grid with cell dimensions of 2.5

Numerical solutions in Test 2:

Experiment setup in Test 3:

A series of flume experiments conducted by Takahashi et al. (1992) is
simulated in this numerical test. The width of the flume was 10 cm. Four
experiments with different lengths of erodible bed layer (

Soil properties in Test 3, Test 4 and field application.

Hydrological parameters for simulating the erosion and deposition processes in Test 3, Test 4 and field application.

In the simulation, the flume is discretized into a grid with cell dimensions
of 0.02

A sensitivity analysis is conducted to investigate the influence of

Results of erosion sensitivity analysis in Test 3.

Another series of flume tests conducted by Takahashi et al. (1992) is
simulated in Test 4. The experiment setup is shown in Fig. 10. In the test
series, the flume width was also 10 cm. The bed layer had a length of 3.0 m
and a thickness of 10 cm, which was located 5.5 m from the outlet of the
flume. A partition with a height of 10 cm was used to retain the sediment in
the experiment, and the partition is assumed to have no influence on the
flow. A board inclined at 5

Experiment setup in Test 4:

In the simulation, the flume is discretized into a grid with cell dimensions
of 0.02

The computed discharges at the outlet and the measured results from the last
four experiments are compared in Fig. 11. Time

When the debris flow moves to the flood board (Fig. 10), it decelerates and
deposits gradually. The flow depth, deposit thickness, volumetric sediment
concentration, and flow velocity can be monitored for all cells. If
deposition occurs somewhere, the deposit thickness there will be larger than
zero. The thickness of the debris fan is the sum of the flow depth and the
deposit thickness. The debris fans in the experimental tests and numerical
solution are compared in Fig. 12, with contours of the thickness of the debris
fans. At

Comparison of discharge hydrographs at the downstream end of the flume in Test 4.

Comparison of the time-varying geometry and elevations of the debris fan in Test 4 from the numerical solution and the experimental tests.

Sensitivity analysis is conducted to investigate the influence of deposition
rate on the runout characteristics of debris flows. The debris fans at the
final stage considering different

Debris fans at final stage considering different coefficients of deposition rate:

Rainfall-induced landslides are one of the most catastrophic hazards in
mountainous areas (e.g. Chen et al., 2012; Chen and Zhang, 2014; Raia et al.,
2014; Zhang et al., 2013, 2014). Decisions for effective risk
mitigation require hydrological and landslide analyses at the regional scale
(e.g. O'Brien et al., 1993; Formetta et al., 2011; Archfield et al., 2013;
Chen et al., 2013). The 2008 Wenchuan earthquake triggered numerous
landslides, leaving a large amount of loose materials on the hill slopes or
channels. From 12 to 14 August 2010 a storm swept the
epicentre, Yingxiu, and its vicinity, triggering a catastrophic debris flow
in Xiaojiagou Ravine (Fig. 14). About 1.01

Location of the study area and a satellite image shortly after the Xiaojiagou debris flow.

Interpretation of the satellite images taken before and after the debris
flow reveals that the source material of this debris flow was mainly the
channel colluvium (Chen et al., 2012). The deposits in the main channel
marked “location of the main source material” in Figs. 14 and 15 had a
volume of approximately 0.74

Hydrological parameters for rainfall-runoff and debris flow runout simulations in field application.

Grid system for rainfall runoff simulation and debris flow runout simulation.

The study area is divided into two domains, one for rainfall-runoff
simulation and the other for debris flow runout simulation (Fig. 15). Grid
systems are created within the two domains with grid sizes of 30

Rainfall runoff simulation is conducted first in domain one using FLO-2D
(FLO-2D Software Inc., 2009). The rainfall data at Yingxiu is adopted. The
runoff water would be retained by the colluvium and accumulate behind the
landslide deposits, forming landslide barrier ponds. The cumulative runoff
water at Section 1-1 (Figs. 14, 15) can be computed, which is applied at
Section 1-1 as the inflow hydrograph for debris flow runout simulation in
domain two. Debris flow would occur when the barrier ponds breach. The
source materials are assumed to be saturated before the occurrence of the
debris flow. As water flows over the source material, erosion occurs if the
conditions in Eqs. (18)–(21) are met. Since the debris flow was witnessed to
occur at the ravine mouth about 36 h after the storm started and lasted
about 30 min, the cumulative runoff water at Section 1-1 in Fig. 14 at
36 h after the storm started is adopted to create the inflow hydrograph, and
the surface runoff is determined to be 0.5

In domain two, the source material is distributed into 329 cells with a
thickness of 10 m. The internal friction angle,

Surface water inflow hydrograph for debris flow runout simulation.

The values of the volumetric sediment concentration,

The change of

The volume of the simulated debris flow is about 1.0

Change of volumetric sediment concentration at Section 3-3 in Fig. 14.

The simulated and observed deposition zones are shown in Fig. 18a. The simulated inundated area and runout distance match the observed results reasonably well. It is noted that the debris fan front has very small depths; the fan front is the precursory surge in front of the boulder front in Fig. 1. The distribution of the maximum flow velocity is shown in Fig. 18b, which indicates that the debris flow moves very rapidly, especially in the ravine channel. Taking into account the large volume, the debris flow is very destructive, which has been observed by Chen et al. (2012).

The comparison between the simulation results and the observations suggests that the model evaluates well the debris flow volume considering the erosion process. The inundated area and the runout distance can also be predicted reasonably well.

The mathematical model proposed in this study has limitations due to the simplifying assumptions and approximations in the underlying theory. The main limitations are as follows.

The model is suitable for describing the initiation and movement of debris flows originated from runoff-driven channel bed failure or breaching of landslide dams by overtopping erosion, which has been tested in this study. The model is also able to consider surficial material entrainment from collapses of bank material or detached landslide material as shown in the governing differential equations. But the latter capability has not been tested and further work is needed.

The new model is suitable for channels and flat alluvial fans, but may not be ideal for steep terrains.

The flow velocity in each of the eight flow directions is computed independently without considering the flow velocity gradient in the orthogonal direction (Eqs. 3, 4). Such influence is not significant in a confined channel since the orthogonal gradient is small. In an unconfined flat area, the eight flow directions account for the influence to certain extent, but further work is needed to test the performance of the model.

Further work is needed to test the performance of some empirical equations adopted in the model.

The governing equations are in a depth-integrated form; hence, the particle segregation in the vertical direction cannot be considered.

A hydrostatic pressure distribution is assumed along the vertical direction, which affects the computed yield stress.

The suspension coefficient,

Simulation results of the Xiaojiagou debris flow:

A new depth-integrated numerical model for simulating debris flow erosion, deposition, and property changes (EDDA) is developed in this study. The model considers the changes in debris flow density, yield stress, and dynamic viscosity, as well as the influences of such changes on the runout characteristics of the debris flow.

The model is unique in that it considers erosion and deposition processes, changes in debris flow mass, debris flow properties and topography due to erosion and deposition. Considering the partly suspended solid particles at limit equilibrium, the yield stress of the debris flow mixture estimated using the Mohr–Coulomb equation is suitable for water flows, hyper-concentrated flows, and fully developed debris flows. An adaptive time stepping algorithm is developed to assure both numerical stability and computational efficiency.

Four numerical tests have been conducted to verify the performance of the model. In Test 1, an analytical solution to one-dimensional debris flow with constant properties is adopted. Comparison between the numerical solution and the analytical solution indicates that the model simulates exceptionally well the one-dimensional movement of debris flow with constant properties. In Test 2, a two-dimensional dam-break water flow problem is adopted. The model simulates very well the two-dimensional dam-break water flow. Flume tests are simulated in Tests 3 and 4. The calculated volumetric sediment concentration at the debris flow front agrees with the experimental results reasonably well in Test 3. In Test 4, the model simulates reasonably well the erosion and deposition processes, and the movement of the multi-directional debris flows in the confined channel and the unconfined flat area in terms of the discharge hydrographs at the outlet and the time-varying geometry and elevations of the debris fan. Sensitivity analyses in Tests 3 and 4 indicate that erosion and deposition processes influence the property changes and runout characteristics significantly.

The model is also applied to simulate a large-scale debris flow in Xiaojiagou Ravine to test the performance of the model in catchment-scale simulations. The model describes well the changes in debris flow properties and estimates the volume of debris flow. Considering the deposition process, the inundated area and the runout distance are predicted properly. The model is shown to be a powerful tool for debris flow risk assessment in a large area and intended for use as a module in a real-time warning system for debris flows.

EDDA is written in FORTRAN, which can be compiled by Intel FORTRAN Compilers. The source code is enclosed as supplement files. The main subroutine is dfs.F90, which contains the numerical solution algorithm for solving the governing equations. Two input files are needed. One is edda_in.txt, which is the file for inputting material properties and hydrological parameters and setting controlling options. EDDA is designed as the debris flow simulation part of a cell-based model for analysing regional slope failures and debris flows, so the edda_in.txt file also includes the material properties and controlling options for slope stability analysis. The other is inflow.txt, which is the inflow hydrograph file. Digital terrain data (e.g. surface elevation, slope gradient, erodible layer thickness) are included in separate ASCII grid files and enclosed in the data folder. Output files are stored in the results folder. Investigated variables at selected points are stored in EDDALog.txt.

This research was substantially supported by the Sichuan Department of Transportation and Communications, the Natural Science Foundation of China (grant no. 51129902), and the Research Grants Council of the Hong Kong SAR (no. 16212514). Edited by: H. McMillan