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## River hydraulics and sediment transport

In the following, a brief introduction to river hydraulics and sediment transport will be presented. The purpose is the setting up of the background concepts necessary to the understanding of this work.

**Hydraulics**

To summarize, river flows are the result of, on the one hand, gravity forces applied to the water, and, on the other hand, bed friction which tends to slow it down. Therefore, for a considered water discharge the hydrolics are function of the slope (which will impact the intensity of the gravity), the grain size and topography of the bed stream (De Linares, 2007). Rivers are open-channel flows, and more pre-cisely, natural rivers are much than deep: the width-to-depth ratio is usually very high (Yalin, 1992). Consequently, and for the sake of simplicity, great scale flows can be treated as bi-dimensional prob-lems (shallow flow hypothesis). Nevertheless, a three-dimensional consideration is more adequate for small scale studies and for cases with low width-to-depth ratios, in which the secondary currents may be non negligible.

**Sediment transport in rivers**

Many works have been done in order to provide a better understanding and a better quantification of the sediment dynamics in running water. Indeed, sediment movement initiates when bed shear stress exceeds a threshold value. Shields (1936) defined an non-dimensional parameter µ0, known as Shields parameter, used to calculate the initiation of motion of sediments. Critical Shields µc parameter repre-sent the threshold value of shear stress over which the sediments start moving. From here, one should differentiate two major transport modes: bed load and suspended load. One could also add a crossing mode: saltation, in which case the transported particle bounces in an irregular manner (see figure 1.2).

**Bed load transport**

As said before, bed load is one of the two types of sediment load. It’s important, for it has a large influence on the bed forms and bed stream morphology. This explains why several bedload equations were proposed during the last decades.

**Transport formulas**

Many semi-empirical formulas can be found in literature. Yet, the purpose of this section is not a thorough review of developed formulas. Therefore, only the formulas which were implemented and tested in the numerical model (which is the subject of the study) were discussed hereafter. In addition, most sediment transport formulas assume a threshold conditions under which sediments are assumed to stay at rest. This is the case of all formulas considered in this study.

The following formulas give the non-dimensional sediment bedload transport rate ‘b : in which, Qs is the volumetric bedload transport rate, g is the acceleration of the gravity, s is relative density and D⁄3 is the characteristic sediment diameter (in the following D⁄3=D50).

Meyer-Peter-Müller (1948) Meyer-Peter-Müller formula is one of the most commonly used for bed load transport rate calculation. This threshold formula has been validated for sediments in the range 0.4mm ˙ D50 ˙ 29mm and is based on a large data set provided from flume experiment.

‘b ˘ fimpm ⁄(µ0 ¡µc )3/2 (1.2) where fimpm is a coefficient usually taken equal to 8, µ0 is the Shields parameter and µc is the critical Shields parameter (in this work, and when using the Meyer-Peter-Müller formula, it will be taken equal to 0.047).

Ashmore (1988) This formula has been calibrated on data set from braiding flume experiment with varying slopes. The conducted experiments showed that bed load transport is limited to a small section of the dominant channels. In his flume experiments Ashmore (1988) considered, as being active, only the channels with a water depth greater than or equal to 2mm. ‘b ˘ 3.11 ⁄(µ0 ¡µc )1.37 (1.3)

In the following, when using this formulation of the non-dimensional sediment transport rate, the critical Shields parameter will be taken equal to µc ˘ 0.045. This value of µc gave a better fit to the experiments performed by Ashmore (1988). Van Rijn (1984) Following the approach of Bagnold (2008) and assuming that bed load transport, under the influence of hydrodynamic fluid and gravity forces, is dominated by the saltation motion mode. He provided formulation for the bed load transport rate for particles in the rage of 0.2mm ˙

D 50 ˙ 2mm.

‘ 0.053D 0.3 µp ¡µc 2.1

b ˘ ⁄¡ µ µc ¶ (1.4)

With D⁄ ˘ D50 (s¡1)g i 1/3 µp is the bed-shear velocity related to grains and µc the critical Shields

”2 parameter. h

This formula provides a reliable estimation of bed load transport rate for this range of particles and was verified using 580 data, from the field (56 tests) and flume experiments (524 tests) (Van Rijn, 1984).

**Morphodynamics**

To talk about morphodynamics, one should first introduce the theoretical principle of dynamic equi- librium of water streams. Indeed, Rivers tend to achieve “dynamically stable” equilibrium conciliating between two types of variables, control and response variables. This concept is schematically repre-sented by the lane balance presented in figure 1.3. On the one hand, the control variables, as the stream flow and the sediment discharge are imposed to the system and are the result of the watershed’s con-figuration, its hydrology, the climate and so on. These control variables are imposed to the system, which will adjust its response variables in order to reach the state of equilibrium. These response vari-ables are therefore the local slope, the sinuosity, and the channel width among others (Malavoi and Bravard, 2010). For instance, as one can see in the Lane’s balance, a decrease of the stream flow will cause an aggradation phase. Subsequently, the equilibrium will be reached by the adjustment of the slope. A higher slope will induce a higher sediment transport in order to achieve a new equilibrium state in agreement with the new control variables.

Nevertheless, this schematic representation (Lane balance) of water streams dynamics remains sim-plistic, for it does not account of other response variables such as the channel width, or more specifi-cally its transverse geometry, nor its sinuosity, etc.

### Morphologic changes due to anthropogenic influences

Needless to say how much the human activities have a significant impact on the environment. In particular, anthropogenic actions affecting the hydrology and sediment yield of a catchment will have inevitably an effect on the alluvial style of its watercourse.

For instance, braiding rivers (which are the subject of this work) are highly sensitive to the changes affecting sediment supply and/or flood patterns (Ferguson, 1993). On the one hand, activities that will tend to increase the sediment influx can make a different river pattern shift to a braiding scheme and conversely a drastic decrease of the sediment discharge, due to construction of a dam or major climate change for example will lead to the gradual fading of the braiding pattern to a meandering like configuration (Malavoi and Bravard, 2010; Piégay and Grant, 2009).

**Slope and width**

The slope is an important morphological parameter. It is a “response variable” (the increase of the slope is due to the system’s need to ensure a higher dynamic required by the high sediment influx rate, see figure 1.3) which differentiates the resulting rivers morphology. Indeed, ceteris paribus, a meandering system will have a lower slope than a braiding one (Ashmore, 2009). Also, the width helps segregating different systems from straight, meandering, braiding river channel pattern, the latter will have the highest width to depth ratio Schumm and Khan (1972).

**Channel types**

Among the different alluvial rivers, and depending on their slope and morphology, one can differenti-ate several alluvial types. Starting from the upstream part of a water stream, the waterfalls are present when the slopes are high (more than 10%) and usually in confined valleys. The waterfalls can be de-scribed as an entanglement of rocks and boulders with no apparent organization. They are usually stable, even at the occurrence of floods. At lower slopes (from 3% to 10%), step-pools are formed with a regular succession of steps, wich can be assimilated to staircase in the bed of stream, formed with coarsest fraction of bed material interlaced with fines. And pools, which provide a storage for finer bed material.

Then, when the slope is lower (from 1% to 3%), one can encounter two alluvial styles: alternate bars and braiding rivers. Alternate bars are typically a succession of emerged sediments deposits alternat-ing between the river’s sides. The braiding rivers are formed, as their designation might indicate, (see figure 1.1) with large gravel bed in which a network of channels which dived and merge around usually temporary islands called bars.

However, one should differentiate braiding rivers from other comparable alluvial styles: anabranch and anastomosis. The first can be assimilated to a system of multiple channels characterized by allu-vial islands which are vegetated and stable (Nanson and Knighton, 1996). The term anabranch refers to a stream which diverts from the main channel and rejoins it downstream. The anastomosis designates multiple channels, which are more stable than braiding rivers’, more sinuous, narrower and deeper with a lower slope. This channels delimitate relatively steady and large islands (Malavoi and Bravard, 2010).

Finally, for even lower slopes and finer sediments, one encounters meandering and straight channel patterns.

The braiding rivers are the main interest of this work and are presented in more details in the next chapter.

**Numerical modeling and morphodynamic models’ state of the art**

In order to understand, quantify and predict rivers behavior and changes, several numerical models were developed. Depending on the scale, in time and space, of the study, different approaches can be considered, and in that sense, each model will reveal its strengths and weakness. In what follows, a general review of some hydro-sedimentary models will be presented. The purpose of this section is to give a global, yet not exhaustive, appreciation of existing models and the work conducted for the case of braiding systems.

Table 1.1 presents seven examples of braiding river’ numerical models. The work of Jagers (2003) gave a substantial review of the different existing models. He compared different modeling approaches: neu-ral network, branches model, cellular model and general 2/3D models (see table 1.1) and concluded that although each of these models has its strength and weaknesses, the branches and general 2/3D models seem to be the most promising.

The work of Schuurman et al. (2013) showed representative results of braiding emergence and forma-tion on a physics-based 2-D model of a sand-bed river. Schuurman et al. (2013) also highlighted the importance of the bed slope and spiral flow effects (see paragraph 3.1.6.3 page 26). The 2-D depth averaged module of the morphodynamic model Delft3D succeeded on reproducing braiding rivers’ characteristics such as the wave length, the bars shape and scour holes among others. However, past the near-equilibrium stage the model started showing rather unrealistic morphology: the bars become static and develop exaggerated length and height.

Rüedlinger and Molnar (2010) performed both 1-D and 2-D modeling of a braiding segment of Rhone river. They concluded that even if the modeling of the hydrodynamics might give satisfactory results, simulating the mophodynamics with 2-D physics based model remains a difficult task for its accu-racy may rely a lot on data availability. However, 1-D models do not succeed on predicting short term processes and, therefore, the result obtained by this models must be considered with additional care.

Leduc (2010) underlined several limitations of the 2-D physics based model RUBAR 20. From this work one can site the inadequacy of rectangular mesh, the uncoupling of the hydrodynamics and morpho-dynamics increases the instabilities and the inadequacy of the available sediment transport formulas for low depth-width ratio streams.

The two last models presented in table 1.1 are Caesar RCM , which is a Reduced complexity model, and a Cellular model. Both showed relatively good computational performances. However, Ziliani et al. (2013) and Murray and Paola (1994) observed that in the case of long-term simulations the cited models tend to lack on reproducing braiding channels dynamic. A further analysis of this “alternative” models is presented in the work of (Jagers, 2003).

To summarize, one can state that a 2/3D approach presents numerous advantages from which we can cite:

• The modeling is based on fundamental physics laws and empirical relations of sediment trans-port;

• The results can be as detailed as the computation performances can allow (in terms of process time and available memory space);

• The possibility to refine the mesh in areas where more precision is needed;

• Depending on one’s interest, the model can be complemented with various physical processes (i.e. accounting of the slope effect, hiding-exposure factor, . . . );

Despite this advantages, a conventional 2/3D model remains highly time consuming and requires more data than a branches or neural model for instance. In addition, long term simulations are subject to great incertitude in this kind of models. This latter issue can be addressed by a Monte Carlo simula-tion Jagers (2003). But here again, the coupling of a probabilistic approach will increase, say 100 times at least, the computation time.

Nonetheless, for the sake of this work, a 2D model proved to be adequate. On the one hand, the inves-tigation of the braiding system response to various configurations and the effect of different transport formulas for instance can accurately be examined with a 2D model. On the other hand, the choice of a 2D rather than a 3D consideration is justified by both the shallowness of braiding pattern and the scale in which the problem is discussed (a global appreciation of the braiding pattern formation and evolution).

Table 1.1 presented several numerical models that have been used in the case of braided rivers. To the best of our knowledge, no work was conducted with TELEMAC-Mascaret modeling system (see paragraph below). The following modeling work will be conducted using this modeling system. The next section of this chapter will therefore be dedicated to its presentation.

#### Why TELEMAC?

Among other sets of solvers, the TELEMAC-Mascaret system presents many interesting features. First, TELEMAC-Mascaret suit of solvers is open source and therefore free to use. Secondly, it can be de-scribed as flexible, for it offers a rather easy access to all the subroutines that intervene throughout the problem resolution. Finally, it can be used in parallel form; a huge advantage in terms of reduction of computation time. A detailed description of the TELEMAC system is presented in the following.

**TELEMAC-Mascaret modeling system**

TELEMAC-Mascaret modeling system is an open source program developed by the Research and De-velopment division of Electricité de France in the Laboratoire National d’Hydraulique. It’s a powerful tool for free-surface flows problems modeling and is commonly used in river and maritime hydraulics. The geometry which will be considered in the models is meshed into a grid of triangular elements. It has numerous simulation modules for hydrodynamics (1D, 2D or 3D), sediment transport, dispersion and underground flow, in addition to pre and post-processors (http://www.opentelemac.org/). In this work, only the hydrodynamics and sediment transport are of interest. Therefore, only the modules TELEMAC-2D and SISYPHE will be used (see the following paragraphs).

One should also note that the TELEMAC-Mascaret modeling system, from the mathematics, the physics, to the advanced parallelisation, is written in Fortran.

One other major strength of TELEMAC is that it allows the user to implement any functions of a simu-lation module by modifying specific subroutines in Fortran. This makes it easy, in a way, to work with specific formulas for the sake of test and validation for example.

**TELEMAC-2D**

TELEMAC-2D code is a hydrodynamics module of the TELEMAC-MASCARET system. It solves the 2D Saint-Venant equations, or shallow water equations (Hervouet, 2007).

These equations are derived from equations of conservation of mass and conservation of momentum (Navier-Stokes equations).

The results provided from solving these equations are therefore water depths and the depth-averaged velocities for each node of the problem’s grid.

TELEMAC-2D is a polyvalent tool, for it allows to considers several phenomenon : treatment of sin-gularities, friction on the bed, supercritical and subcritical flows, dry areas in the computational field like tidal flats and/or flood-plains, inclusion of porosity… and, as it’s the case in this work, can be also coupled with a 2D sediment transport module (SISYPHE).

**Table of contents :**

**1 Introduction and State of the Art **

1.1 Introduction

1.2 River hydraulics and sediment transport

1.2.1 Hydraulics

1.2.2 Sediment transport in rivers

1.2.3 Bed load transport

1.3 Morphodynamics

1.3.1 Morphologic changes due to anthropogenic influences

1.3.2 Slope and width

1.4 Channel types

1.5 Numerical modeling and morphodynamic models’ state of the art

1.6 Why TELEMAC?

1.7 TELEMAC-Mascaret modeling system

1.7.1 TELEMAC-2D

1.7.2 SISYPHE

1.7.3 Parallel simulation

**2 Dynamics of Braided Streams **

2.1 Braiding water streams

2.1.1 Origin of braiding

2.1.2 Braiding rivers’ particularities

2.1.3 Braided rivers’ components

2.1.4 Morphological changes

2.1.5 Morphometric parameters

2.2 Flume experiments

2.2.1 Material and methods

2.2.2 Results

2.2.3 Purposes and limitations

2.3 Conclusion and recommendations

**3 Numerical Experiment **

3.1 Modeling with TELEMAC-Mascaret system

3.1.1 The geometry

3.1.2 The mesh

3.1.3 Modeling organization

3.1.4 Initial conditions

3.1.5 Boundary conditions

3.1.6 Modeling parameters

3.2 Results presentations’ plan and expected results

3.3 Results

3.3.1 Establishment of the braiding pattern(Run 0 and Run 00)

3.3.2 Erosion (Run 01)

3.3.3 Aggradation (Run 02)

3.4 Additional results

3.4.1 Variation of the water inflow

3.4.2 Ashmore and Van Rijn

3.4.3 Widening of the domain and slope change

3.4.4 Sediment grading effects

**4 Discussion and conclusion **

4.1 Discussion of the results

4.2 Conclusion

**Bibliography **

Appendixes

A DEM extraction with photogrammetry method

A.1 Photogrammetry

A.2 Agisoft Photoscan

A.2.1 Principles

A.2.2 Results and Bias

A.3 Recommendations

B Steering files

B.1 TELEMAC-2D steering file

B.2 SISYPHE steering file

C Simulation results