Experimental analysis and numerical simulation of bed elevation change in mountain rivers
 Truong An Dang^{1}Email author and
 Sang Deog Park^{2}
Received: 28 March 2016
Accepted: 29 June 2016
Published: 13 July 2016
The Erratum to this article has been published in SpringerPlus 2016 5:1245
Abstract
Studies of sediment transport problems in mountainous rivers with steep slopes are difficult due to rapid variations in flow regimes, abrupt changes in topography, etc. Sediment transport in mountainous rivers with steep slopes is a complicated subject because bed materials in mountainous rivers are often heterogeneous and contain a wide range of bed material sizes, such as gravel, cobbles, boulders, etc. This paper presents a numerical model that was developed to simulate the river morphology in mountainous rivers where the maximum bed material size is in the range of cobbles. The governing equations were discretized using a finite difference method. In addition, an empirical bed load formula was established to calculate the bed load transport rate. The flow and sediment transport modules were constructed in a decoupled manner. The developed model was tested to simulate the river morphology in an artificial channel and in the Asungjun River section of the mountainous Yangyang Namdae River (South Korea). The simulation results exhibited good agreement with field data.
Keywords
Background
Sediment transport in mountain rivers is a complex phenomenon because it is characterized by steep slopes, water depths based on the order of the height, a wide range of bed material sizes, and distinct bed structures. However, our current knowledge of mountain river flow is still improving and progressing due to the lack of understanding of the interrelationship between flow and sediment. In particular, mountainous catchments with the riverbed gradients larger than 0.05 and bed load transport containing a high portion of gravel, cobbles, boulders, and transport capacities during flood events can reach very high values (Chiari 2008; Rickenmann 1990).
In recent decades, numerical models have become useful tools for studying sediment transport problems in mountain rivers. Li and Fullerton (1987) developed a model for simulating channel aggradation and degradation in gravel and cobblebed rivers. Silvio and Peviani (1989) constructed a numerical model to study the shortand longterm evolution of mountainrivers. Pianese and Rossi (2005) developed a mathematical model to study the longterm scale changes of a riverbed. In 2004, Papanicolaou developed a new 1D numerical model to calculate flow and sediment transport in steep mountain rivers. According to Mosconi (1988), failure to predict bed loads in mountain river flows may be due to most common equations not considering the morphological peculiarities of study areas in conjunction with their limited capability to cover a wide distribution range of bed material sizes. Bed load equations obtained by several authors on low slopes are rarely applicable to mountain rivers, where the river beds contain wide ranges of bed material sizes, large roughness elements, etc. Therefore, these features largely affect the research results (D’Agostino and Lenzi 1999).
In this paper, a 2D numerical model has been developed to simulate the flow and morphological changes in steep channels where bed materials have large size distributions. The model system consists of a flow module and bed load transport module. The flow module is based on the mass and momentum conservation equations in the Cartesian coordinate system. The sediment transport module only comprises empirical bed load formulas. The river morphology module is based on the sediment continuity equation, and a grain material distribution is applied for individual size fractions. The solution method was implemented in a computer source code and written in structured Fortran 90.
Numerical method
Governing equations
Sediment transport equations
In the past, most bed load formulae have been developed and widely used based on laboratory investigations with uniform particle size. Unfortunately, when applied to natural rivers with nonuniform particle size, the calculated bed load transport rates often differ by orders of magnitude and do not exhibit high confidence levels. One of the major causes is often the influence of the local conditions, which are very different from the laboratory conditions where the bed load formulae were constructed. As noted above, mountain rivers is a dominant and fundamental process in the hydrodynamic rivers, and often have nonuniform particle size distributions, the particle sizes are divided into several fractions.
Bed level variation
The transverse bed slope was small and had little effect on the flow calculations. Therefore, in the research, the effect of transverse bed slope is ignored.
Discretization of governing equations
 Step 1:

Initializing all the variables. This step usually corresponds to time T_{0}. In this step, the values of the water depth and flow field within the computational domain and at the boundaries are specifically established. It is assumed that the velocity components and water depth are known at time T_{0} and that the boundary conditions of the velocity components and water depth are given.
 Step 2:

Partial differential Eqs. (18), (19), and (20) for the flow and Eq. (21) for sediment continuity are solved with the finite difference code. Discretized equations are obtained for the shallow water and sediment continuity equations using the staggered numerical grid. The initial and boundary conditions used to solve the momentum Eqs. (19) and (20) and the continuity Eq. (18), i.e., the values of u, v, and h at time T + 1, are determined at every interior node (I = 2,…, N). The values of the dependent variables u and v at the boundary nodes 1 and N + 1 are determined using the boundary conditions. The values of the dependent variables that are not specified through boundary conditions can be determined by extrapolation of the interior points or equivalently by approximation of the derivatives at fictitious boundary points. We then obtain the corrected water depth and (u, v) velocity components at every interior node in the computational domain.
 Step 3:

The velocity components are calculated at time step T = T_{0} + ΔT until a converged solution is obtained. In this step, convergence criteria must be checked because the scheme used in this research is an iterative scheme. Then, the velocity components and water depth are updated with their corresponding values.
 Step 4:

The water depth and velocity components are used to calculate the dimensionless particle diameter, dimensionless Shields stress, dimensionless critical Shields stress, critical shear stress, boundary shear stress, etc. Finally, the dimensionless particle diameter is calculated.
 Step 5:

The parameters calculated in Step 4 are used to calculate the bed load transport rate.
 Step 6:

Erosion and deposition are calculated using the sediment continuity Eq. (21) to determine the bed level variation and update the new water depth if the channel bed has changed.
 Step 7:

Return to Step 2 and repeat the preceding calculation until the specified final time. If a steady state solution is required, a specified convergence criterion must be satisfied.
 Step 8:

The last step in the calculation process involves storing and updating variables at each time step, moving to the next time step, and repeating Step 2 through Step 7.
Stability conditions
Application of model verification and discussion
To investigate the applicability of the developed model, the present model has been tested in two experimental cases. The first case was obtained from the Large Scale Hydraulic Models of the University of Calabria, Italy (Bellos and Hrissanthou 2003; Bor 2008; Miglio et al. 2009). The second was obtained from a flood event in the Asungjun River.
Numerical models can be calibrated by comparing measured and computed results and adjusting the empirical coefficients in the associated empirical relationships. By a trial and error procedure, the agreement between calculations and measurements can be satisfied. However, this procedure is difficult to apply because of the lack of input data, especially for simulating flow and sediment transport in natural rivers. Several researchers have determined the goodness of fit of hydrodynamic models by computing the root mean square differences (RMSD) and mean absolute errors (MAE) between observed and simulated results.
Experimental data from a seal aggradation test
Model setup

Width of the artificial channel: 0.194 m;

Length of the artificial channel: 5.0 m;

Water depth: 4.3 cm;

Flow discharge: 0.0242 m^{3}/s;

Bed slope: 1.0 %;

Grid spacing: Dx = Dy = 0.01 m;

Median diameter: D_{50} = 3.0 mm;

Porosity: p = 0.35;

Manning’s coefficient: 0.015.
The experimental flume data were measured over a period of 30 min with a time step of 1.0 s.
Results and discussion of the seal aggradation test case
Model test of the Asungjun River section
Model setup

Width of the artificial channel: 250 m;

Length of the artificial channel: 600 m;

Grid spacing: Dx = Dy = 1.0 m;

Porosity: p = 0.40.
Distribution of bed material diameter fractions
Fraction  Material diameter (mm)  d_{im} (mm)  d_{i} (mm) 

No.1  <0.01  0.005  0.005 
No.2  0.125–0.5  0.31  0.3 
No.3  0.5–2.0  1.25  1.25 
No.4  2.0–8.0  5.0  5.0 
No.5  8.0–16  12.0  10.0 
No.6  16–32  24.0  25.0 
No.7  32–64  48.0  50.0 
No.8  64–128  95.0  100.0 
No.9  128–265  196  150.0 
No.10  >265  250  250.0 
In Table 1, d_{im} is the material diameter and d_{i} is the mean material diameter.
Manning’s coefficients in the roughness zones shown in Fig. 13
Zone  Zone 1  Zone 2  Zone 3  Zone 4 

Manning’s n  0.033  0.042  0.052  0.048 
Wet/dry treatment
In shallow water regions of natural rivers where the water depth is small and the channel bed exhibits irregular geometry, the water edges change with time. In those cases, wet and dry treatments in numerical simulations are often used to determine the wet and dry cells. The water depth defined by the user will often depend on the scale of the simulation. In numerical models, the process of drying and wetting is represented by a flow domain that becomes dry when the water depth decreases and wet when the water depth increases.
Existing 2D models have taken a number of approaches to solve the problem associated with some areas being wet and others dry, or fluctuations between the two (Bates and Hervouet 1999; Begnudelli and Sanders 2006; DHI 2003). Several models turn cells on and off based on the minimum depth criteria (Delft 2002; King and Roig 1988; Leclerc et al. 1990). Other models change the fluid properties at very small depths so that a very thin layer of fluid is always present. Most approaches attempt to reformulate the flow equations over partially wet elements by introducing a scaling coefficient, representing the true volume of water at each element. This coefficient varies from zero to one as the cells tend from fully dry to fully wet (Bates and Hervouet 1999; Defina 2000).
In this study, a threshold value of the water depth (0.03 m) based on the river bed material size is used to establish drying and wetting. If the water depth in a cell is larger than this threshold value, this cell is considered wet, and if the water depth is lower than this threshold value, the cell is dry.
Results and discussion of the Asungjun River section case
Comparison of results calculated between the numerical models
Model  Root mean square difference  Absolute difference mean 

Developed model  0.0032  0.030 
Mike21C model  0.0039  0.043 
Conclusions
A depthaveraged 2D numerical model was developed for simulating river morphology in mountain rivers. FDM is used to solve the momentum equations and sediment continuity equations. The model system consists of flow and river morphology modules. Both the flow and river morphology modules are solved using an iteration method that constitutes a coupling procedure.
The bed material size distribution is separated using a fractional approach. This approach is more complex than the classical method, which only uses a value of particle size diameter (D_{50}).
The simulation results of the river morphology of the flood event in the natural river using the developed model are more accurate than those produced by the Mike21C model. Generally, the simulation results were in good agreement with the measured data compared to the results of the Mike21C model.

The robustness of the developed model under the various cases studied, such as lateral water and abrupt cross section variations, division of bed material into a number of size fractions, and division of Manning’s roughness coefficient into different values in study zones to fit the real bed topography conditions.

The simple structure of the developed model allows users to easily control the calculation procedure, and it has a relatively fast computational speed.

The complicated procedure of constructing the numerical grid;

The sensitivity of the coefficients to the river morphology;

The need to calibrate multiple parameters when constructing the bed load formula.
More testing of the model may be necessary to improve its predictive ability. It is expected that the model will become a useful predictive tool for mountainous river studies.
Notes
Declarations
Authors’ contributions
In this study, SDP proposed the research project and outlined the project, designed the research proposal, and wrote the manuscript. He also has participated in field surveys, analyzed output data, and prepared and edited the manuscript. TAD participated in the field surveys, analyzed measurement data, analyzed and established input–output data, wrote source code, implemented the simulation model, analyzed the simulated results, edited the manuscript, and wrote the main paper. Both authors contributed equally to this work. TAD jointly conceived the study with SDP. We participated in field surveys, collected and analyzed input–output data, and administered the experiments. Both authors discussed the results and commented on the manuscript in all stages. Both authors read and approved the final manuscript.
Acknowledgements
The model presented in this study is part of a research project sponsored by the Institute for Disaster Prevention, GangneungWonju National University, South Korea. An important part of this study was performed during the first author’s 3year tenure at the GangneungWonju National University, South Korea. The study was financially supported by the Institute for Disaster Prevention, GangneungWonju National University, South Korea.
Competing interests
To carry out this study, we received the following support. We received support from the Institute for Disaster Prevention, GangneungWonju National University, South Korea. T.A. Dang had full access to all the study data and take full responsibility for the accuracy of the data analysis. T.A. Dang authorized manuscript preparation and the decision to submit the manuscript for publication.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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