van der scheer et al

Transport Formulas for Graded Sediment

Behaviour of Transport Formulas and Verification with Data

P. van der Scheer J.S. Ribberink

A. Blom

Civil Engineering University of Twente

The Netherlands

ISSN: 1568-4652 Research report 2002R-002

Enschede, April 2002

Abstract

University of Twente III

Abstract

This study is aimed at the behaviour analysis and verification of sediment transport formulas for graded sediment in equilibrium conditions. Graded sediment is a mixture of grains with different sizes. This study is a joint project of Rijkswaterstaat RIZA and the University of Twente. The predictive ability of transport formulas for graded sediment is investigated since these predictions still have many uncertainties. Some processes important for transport of graded sediment are given below:

• Incipient motion: Larger grains require a higher shear stress at the point of incipient motion. This means that finer grains are transported more easily than coarser grains.

• Hiding/exposure: Coarse grains protrude more from the bed and, as such, they are more exposed to the flow than in a situation in which only these large grains would be present. Small grains can hide behind coarser grains making them less mobile compared to a situation in which only these small grains would be present.

In this study, ten transport formulas are analysed for their validity ranges, their behaviour and their transport predictions. The predicted transport rates and compositions are verified with measurements from experimental series. Uniform and fractional versions of these transport formulas result in 17 different formulas:

formula transport type uniform or fractional hiding/exposure

correction Ackers & White (1973) total uniform Ackers & White (1973) total fractional Day Ackers & White (1973) total fractional Proffitt & Sutherland Parker (1990) bed fractional Parker Engelund & Hansen total uniform Engelund & Hansen total fractional Meyer-Peter & Müller bed uniform Meyer-Peter & Müller bed fractional Egiazaroff Meyer-Peter & Müller bed fractional Ashida & Michiue Van Rijn bed uniform Van Rijn bed fractional Hunziker/Meyer-Peter & Müller bed fractional Hunziker Gladkow & Söhngen bed fractional Wu et al. bed fractional Wu et al. Wilcock & Crowe bed fractional Wilcock & Crowe Ribberink bed uniform Ribberink bed fractional Ashida & Michiue

The validity ranges of the fractional formulas were determined by analysing the data used for the original development (calibration and verification) of the transport formula. Three different parameters were chosen to represent the validity of the formula: the median grain size of the bed material (D50), the geometric standard deviation (σg) of the bed material and the Shields parameter (θ). The analysis showed that the fractional formulas of Wu et al., Ackers & White with the hiding/exposure correction of Day and Ackers & White with the hiding/exposure correction of Proffitt

Abstract

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& Sutherland have the largest validity ranges. This is caused by the large data sets that were used for deriving these formulas.

Four series of calculations have been done with hypothetical hydraulic and sedimentological conditions to investigate the behaviour of the sediment transport formulas under different circumstances. Both the median grain size (D50) and the geometric standard deviation (σg) of the bed material were varied. The grain size was set at either 2 or 10 mm and the geometric standard deviation was set at either 2 or 5. Combining these sediment properties gives four different sediment mixtures. The Shields parameter in each series was varied between 0 and 0.6.

The transport rates predicted by the several formulas differ much in conditions with Shields parameters below 0.3. This is mainly caused by the different ways of modelling incipient motion since it is difficult to predict sediment transport at these conditions. The behaviour of the transport rate versus the Shields parameter shows no changes when the median grain size (D50) of the bed material is varied. Varying the geometric standard deviation (σg) of the bed material however, does result in changes of the behaviour of some transport formulas. The predicted transport rates of the uniform formulas do not differ much from their fractional counterparts when the conditions are far from incipient motion.

At low Shields parameters, all formulas predict a transport composition finer than the bed composition. At higher Shields parameters, the transport composition predicted by most formulas approximates the composition of the bed material. The largest differences in predictions of the transport composition occur at low Shields parameters due to different ways of modelling incipient motion. The behaviour of the transport composition versus the Shields parameter of the formulas is not very sensitive to variations in D50 of the bed material, but does show differences when σg is varied.

The transport formulas were verified by comparing transport predictions with measurements originating from ten sets of laboratory experiments. These experiments were carried out in flumes with sediment recirculation systems. In this study, only data from equilibrium phases of the tests have been used. The initial bed material has been used as input for the transport predictions.

The fractional formula of Wu et al. [2000] gives the best transport rate predictions, but the formula should be used carefully in conditions with low Shields parameters and relatively coarse material. The fractional formula of Wilcock & Crowe [2001] is a good second option. For all formulas, the transport rates are difficult to predict in Shields parameter ranges below 0.1. The fractional formulas give better results than their uniform counterparts. Many formulas give good transport rate predictions for experiments with a median grain size (D50) smaller than their validity range.

The formula of Wu et al. gives the best transport composition predictions. All formulas have problems predicting a good transport composition at low Shields parameters. At higher Shields parameters and a median grain size (D50) even smaller than their validity range, the composition predictions are good.

Abstract

University of Twente V

Using the surface layer composition instead of the composition of the initial material as input for the transport formulas was only possible for six of the ten experimental series. For most formulas, using the surface layer composition leads to better predictions. The formula of Wilcock & Crowe gives now the best predictions of the transport rate followed by the formula of Wu et al. The formula of Wu et al. remains the best formula to predict the transport composition, but has a much better score than when using the initial bed composition.

Overall, the formula of Wu et al. [2000] is the best formula to predict transport of non-uniform sediment. The formula has large validity ranges due to the large amount of data that was used for the original calibration of the formula. The formula shows stable behaviour concerning the transport rate and the transport composition. Even outside the validity ranges of the formula, it gives good predictions. Based on the present study, it can be concluded that the formula of Wu et al. is the best formula for sediment transport predictions.

The formula of Meyer-Peter & Müller with the hiding/exposure correction of Ashida & Michiue [1973] shows good results for the sediment transport predictions if the Shields parameter is high enough (>0.15). The validity range of this formula is much smaller than the formula of Wu et al., but the formula of Meyer-Peter & Müller with Ashida & Michiue shows good predictions outside its validity range as well. The formula is not suitable for conditions with Shields parameters below 0.1. The formula predicts no sediment transport in this Shields parameter range while experiments show that sediment transport occurs at these conditions.

Preface

University of Twente VII

Preface

This report is the result of a continuation of the project “Transport formulas for graded sediment - Behaviour of transport formulas and verification with experimental data” and is a joint research project between Rijkswaterstaat RIZA and the University of Twente. Rijkswaterstaat RIZA will bring forward this report as a discussion paper in the study group “Morphological Models” of the CHR commission (Commission for the Hydrology of the Rhine basin).

The aim of this project is to gain more insight in predicting the transport of graded sediment. This was done by a literature study on different sediment transport formulas. The data on which the different formulas are originally calibrated have been analysed to determine the validity ranges of the formulas. The behaviour of the transport formulas has been analysed by making predictions for hypothetical conditions. Finally, the predictions of the formulas have been verified with the measured transport rates and compositions from ten sets of flume experiments.

Peter van der Scheer 15 April 2002 Enschede The Netherlands

Table of contents

University of Twente IX

Table of contents

1 DESCRIPTION OF RESEARCH PROJECT 13 1.1 INTRODUCTION 13 1.2 RESEARCH OBJECTIVES 13 1.3 ACTIVITIES 14

1.3.1 Overview transport formulas 14 1.3.2 Validity ranges of transport formulas 14 1.3.3 Behaviour of transport formulas 15 1.3.4 Verification sediment transport formulas 16 1.3.5 Surface layer as active layer 16

2 GRADED SEDIMENT 19 2.1 INTRODUCTION 19 2.2 SEDIMENT PROPERTIES 19 2.3 INCIPIENT MOTION 20 2.4 HIDING/EXPOSURE 21

3 SEDIMENT TRANSPORT FORMULAS 23 3.1 INTRODUCTION 23 3.2 ACKERS & WHITE 23

3.2.1 General 23 3.2.2 Uniform transport 24 3.2.3 Fractional transport 25

3.2.3.1 Day's hiding/exposure factor 25 3.2.3.2 Proffitt & Sutherland's hiding/exposure factor 26

3.3 PARKER 27 3.3.1 General 27 3.3.2 Surface-based bed load formula 28

3.4 ENGELUND & HANSEN 30 3.4.1 General 30 3.4.2 Uniform transport 30 3.4.3 Fractional transport 30

3.5 MEYER-PETER & MÜLLER 31 3.5.1 General 31 3.5.2 Uniform transport 31 3.5.3 Fractional transport 32

3.5.3.1 Egiazaroff’s hiding/exposure factor 32 3.5.3.2 Ashida & Michiue’s hiding/exposure factor 32

3.6 VAN RIJN 33 3.6.1 General 33 3.6.2 Uniform bed load formula 33 3.6.3 Fractional bed load formula 35 3.6.4 Suspended load formula 35

3.7 HUNZIKER/MEYER-PETER & MÜLLER 36 3.7.1 General 36 3.7.2 Fractional bed load formula 37

3.8 GLADKOW & SÖHNGEN 38 3.8.1 General 38 3.8.2 Fractional bed load formula 39

3.9 WU ET AL. 40 3.9.1 General 40 3.9.2 Fractional bed load formula 40 3.9.3 Fractional suspended load formula 41

3.10 WILCOCK & CROWE 41 3.10.1 Fractional formula 42

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3.11 RIBBERINK 43 3.11.1 Uniform formula 43 3.11.2 Fractional formula 44

3.12 VALIDITY RANGES OF THE FRACTIONAL FORMULAS 44 4 BEHAVIOUR TRANSPORT FORMULAS 55

4.1 INTRODUCTION 55 4.2 DIFFERENCES BETWEEN FORMULAS 56

4.2.1 Transport rate 56 4.2.2 Transport composition 59

4.3 UNIFORM VERSUS FRACTIONAL 62 4.4 FOCUS ON LOW SHIELDS PARAMETERS 64 4.5 CONCLUSIONS 67


5 EXPERIMENTAL DATA USED FOR VERIFICATION 71 5.1 INTRODUCTION 71 5.2 BLOM & KLEINHANS 72 5.3 BLOM 72 5.4 KLAASSEN 73 5.5 DAY: HRS A 74 5.6 DAY: HRS B 75 5.7 WILCOCK & MCARDELL AND WILCOCK ET AL. 76

6 VERIFICATION OF TRANSPORT RATE 77 6.1 INTRODUCTION 77 6.2 MEASURED TRANSPORT RATE 77 6.3 RANKING OF THE TRANSPORT FORMULAS 78 6.4 ACKERS & WHITE 79

6.4.1 Ackers & White 79 6.4.2 Ackers & White with Day 80 6.4.3 Ackers & White with Proffitt & Sutherland 81

6.5 PARKER 81 6.6 ENGELUND & HANSEN 82

6.6.1 Engelund & Hansen uniform 82 6.6.2 Engelund & Hansen fractional 83

6.7 MEYER-PETER & MÜLLER 84 6.7.1 Meyer-Peter & Müller uniform 84 6.7.2 Meyer-Peter & Müller with Egiazaroff 84 6.7.3 Meyer-Peter & Müller with Ashida & Michiue 85

6.8 VAN RIJN 86 6.8.1 Van Rijn uniform 86 6.8.2 Van Rijn fractional 87

6.9 HUNZIKER/MEYER-PETER & MÜLLER 88 6.10 GLADKOW & SÖHNGEN 89 6.11 WU ET AL. 90 6.12 WILCOCK & CROWE 90 6.13 RIBBERINK 91

6.13.1 Ribberink uniform 91 6.13.2 Ribberink with Ashida & Michiue 92

6.14 CONCLUSIONS 92 7 VERIFICATION OF TRANSPORT COMPOSITION 95

7.1 INTRODUCTION 95 7.2 MEASURED TRANSPORT COMPOSITION 95 7.3 RANKING OF THE TRANSPORT FORMULAS 96 7.4 ACKERS & WHITE 97

7.4.1 Ackers & White with Day 97 7.4.2 Ackers & White with Proffitt & Sutherland 97

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7.5 PARKER 98 7.6 ENGELUND & HANSEN 99 7.7 MEYER-PETER & MÜLLER 100

7.7.1 Meyer-Peter & Müller with Egiazaroff 100 7.7.2 Meyer-Peter & Müller with Ashida & Michiue 101

7.8 VAN RIJN 102 7.9 HUNZIKER/MEYER-PETER & MÜLLER 102 7.10 GLADKOW & SÖHNGEN 103 7.11 WU ET AL. 104 7.12 WILCOCK & CROWE 105 7.13 RIBBERINK WITH ASHIDA & MICHIUE 105 7.14 CONCLUSIONS 106

8 SURFACE LAYER AS ACTIVE LAYER 107 8.1 INTRODUCTION 107 8.2 MEASUREMENTS 107 8.3 TRANSPORT RATE 108 8.4 TRANSPORT COMPOSITION 110 8.5 CONCLUSIONS 111

9 DISCUSSION 113 9.1 MODIFICATIONS TO VAN RIJN 113 9.2 BED FORM FACTORS 113

10 CONCLUSIONS 115 10.1 VALIDITY RANGES OF FRACTIONAL FORMULAS 115 10.2 BEHAVIOUR ANALYSIS 115


10.3 VERIFICATION WITH EXPERIMENTAL DATA 116 10.3.1 Initial bed mixture as active layer 117 10.3.2 Surface layer as active layer 118

10.4 FINAL CONCLUSIONS 119 10.5 RECOMMENDATIONS 119

REFERENCES 121

APPENDIX 1: BED ROUGHNESS ANALYSIS A-1

APPENDIX 2: SPLITTING A SEDIMENT MIXTURE INTO N FRACTIONS A-5

APPENDIX 3: TRANSPORT RATE SERIES 1 AND 2 A-7

APPENDIX 4: TRANSPORT COMPOSITION SERIES 1 AND 2 A-9

APPENDIX 5: UNIFORM VERSUS FRACTIONAL FOR SERIES 1 AND 2 A-11

APPENDIX 6: VANONI AND BROOKS A-13

APPENDIX 7: DATA WILCOCK A-15

Description of research project

University of Twente 13

1 Description of research project

1.1 Introduction

Many formulas have been developed the last five decades to predict the sediment transport. Formulas for both uniform sediment and for graded sediment exist. All formulas were derived in a different way and calibrated with different data. This can lead to large differences in the predictions of the sediment transport between different formulas. In this project, we will give a clearer picture about which formula gives good sediment transport predictions, and which does not. The present project is a continuation of a joint research project between Rijkswaterstaat RIZA and the University of Twente (Van der Scheer et al., 2001). Rijkswaterstaat RIZA will bring forward this report as a discussion paper in the study group "Morphological Models" of the CHR commission (Commission for the Hydrology of the Rhine basin).

The present project deals with the behaviour and the validity of several sediment transport formulas for graded sediment. Research conducted in the framework of the M.Sc. thesis of P. van der Scheer [Van der Scheer, 2000] is an important basis of this project. During this thesis, software written in the software language MATLAB 5.3 was developed for making predictions by several sediment transport formulas for both uniform and graded sediment. These sediment transport formulas were verified with data of flume experiments, which were performed in 1999/2000 for the Ph.D. project of A. Blom [Blom, 2000].

Within this project, the behaviour of the transport formulas is investigated further. All transport formulas are calibrated with experimental and/or field data. The validity range of the formulas is limited to the specific hydraulic and sedimentological ranges of the calibration data. The influence of important parameters in sediment transport processes like the grain size distribution and the Shields parameter will be investigated.

1.2 Research objectives

There are three research objectives in this project. These are: 1. For several fractional sediment transport formulas we determine the validity

ranges. These validity ranges express for which sediment mixtures and which hydraulic conditions the formula is calibrated and should give good transport predictions.

2. We investigate the behaviour of the transport formulas by predicting transport rates and compositions for hypothetical cases. The predicted transport rates and compositions are compared with each other, which indicates how large is the difference between predictions of sediment transport formulas under different conditions. An important aspect is to gain knowledge on the circumstances when selective transport processes start playing a role and fractional sediment transport formulas should be used.


14 University of Twente

3. We study the performance of the transport formulas. How well a transport formula predicts the transport rate and composition is investigated by verifying predictions of these formulas with measurements in ten sets of experiments.

1.3 Activities

1.3.1 Overview transport formulas

Table 1.1 shows a list of all formulas considered within this project. For this study, the sediment transport formulas of Gladkow & Söhngen, Wu et al., Wilcock & Crowe and Ribberink are included in the list.

Table 1.1: List of sediment transport formulas considered in present study. formula transport type uniform or fractional hiding/exposure

correction Ackers & White (1973) total uniform Ackers & White (1973) total fractional Day Ackers & White (1973) total fractional Proffitt & Sutherland Parker (1990) bed fractional Parker Engelund & Hansen total uniform Engelund & Hansen total fractional Meyer-Peter & Müller bed uniform Meyer-Peter & Müller bed fractional Egiazaroff Meyer-Peter & Müller bed fractional Ashida & Michiue Van Rijn bed uniform Van Rijn bed fractional Hunziker/Meyer-Peter & Müller bed fractional Hunziker Gladkow & Söhngen bed fractional Söhngen Wu et al. bed fractional Wu et al. Wilcock & Crowe bed fractional Wilcock & Crowe Ribberink bed uniform Ribberink bed fractional Ashida & Michiue

1.3.2 Validity ranges of transport formulas

Van der Scheer [2000] found that the hydraulic and sedimentological ranges for which the formulas were originally calibrated are often not clearly given in the scientific papers describing these transport formulas. In this study effort will be put into determining the validity ranges of some fractional transport formulas by means of an extensive literature study. The Shields parameter range will be used to show the hydraulic range of the calibration data. The bed material will be analysed for its median grain size and gradation.

Another important aspect in the literature research will be the definition of the “active layer” used to derive a representative grain size for transport predictions. The “active layer” is defined as the layer that determines the rate and the composition of the transported sediment [Van der Scheer, 2000]. Van der Scheer’s research showed that most formulas do not state which part of the bed is considered to be the active layer and how its composition should be determined. Van der Scheer showed that the composition of the active layer has a large influence on the predictions by the transport formulas. Thus, it is important to present a good method for determining the composition of the active layer to determine the correct composition of the bed material, which is used for sediment transport predictions. If, in the specific papers,



no definition is found of the active layer, the initial bed composition will be used as the composition of the active layer.

A third aspect that will be studied is the type of experiments that are used for calibration of the model. Flume experiments can roughly be divided into two groups: experiments with a sediment recirculation system or with a sediment feed system. Erosion experiments are sediment feed experiments without sediment input at the upstream end of the flume. This leads to erosion of the bed and eventually to armouring. Experiments with a sediment recirculation system are often used to maintain equilibrium during the experiment and measure the sediment transport in that period. Both types of experiments have been used to develop sediment transport formulas and since different processes occur in the two types of experiments, it is important to distinguish them.

1.3.3 Behaviour of transport formulas

Here we deal with the changes in the sediment transport rate and composition when varying hydraulic and sedimentological parameters. Calculations with pre-set hydraulic and sedimentological conditions will be done to get insight in these changes.

The following parameters will be varied: Shields parameter (θ50), grain size (D50) and the geometric standard deviation of the grain size (σg). The variation of these parameters will be within realistic ranges occurring in rivers in The Netherlands and in laboratories. The predicted transport rates and compositions will be plotted versus the Shields parameter. The sediment mixture will be divided into ten fractions according to Ribberink’s method [1987]. Ribberink assumes that the grain size distribution is log-normal. The grain size and the probability of each fraction are calculated by giving the values for D50 and σg.

The range of the Shields parameter is within 0 to 0.6. The Shields parameter will be changed by changing the water depth. The water surface slope and the Chézy value will be held constant at the values 0.0005 and 40 m½/s, respectively. Calculations will be done for different Shields parameters and four different bed compositions. The different bed compositions are characterised by their representative grain size (D50) and their geometric standard deviation of the grain size (σg) (see Table 1.2).

Table 1.2: Bed material properties of the four series σg (-) D50 (mm) 2 5

2 Series 1 Series 2 10 Series 3 Series 4

For the four series, we studied the influence of the Shields parameter on both the sediment transport rate, and the geometric mean grain size of the transported material.



1.3.4 Verification sediment transport formulas

The predictions given by the transport formulas listed in Table 1.1 will be verified with the results of laboratory experiments. A data-file will be created of the experimental data. The following experimental series are examined in this study:

• Blom and Kleinhans, [1999], Non-uniform sediment in morphological equilibrium situations - Data report Sand Flume experiments 97/98, Research report CiT 99R-002/MICS-001, Civil Engineering and Management, University of Twente. Experimental Series 1 was conducted in the Sand Flume Facility of WL | delft hydraulics in 1997/1998. Five tests were performed with a sediment mixture from the river Rhine.

• Blom, [2000], Flume experiments with a trimodal sediment mixture – Data report Sand Flume experiments 99/00, Research report CiT: 2000R-004/MICS-013, Civil Engineering & Management, University of Twente. Experimental Series 2 was conducted in the Sand Flume Facility of WL | delft hydraulics in 1999/2000. Four tests were performed with a tri-modal mixture.

• Klaassen, [1991], Experiments on the effect of gradation and vertical sorting on sediment transport phenomena in the dune phase, Proc. Grain Sorting Seminar, Ascona, Switzerland, pp 127-145. Six tests with graded sediment were conducted in the Sand Flume Facility of WL | delft hydraulics in 1990.

• Day, [1980], A study of the transport of graded sediments, HRS Wallingford, Report No. IT 190. HRS A, experiments conducted by Day. 11 tests with graded sediment were conducted in the flume of HR Wallingford. HRS B, experiments conducted by Day. Nine tests with graded sediment were conducted in the flume of HR Wallingford.

• Wilcock and McArdell, [1993], Surface-Based Fractional Transport Rates: Mobilization Thresholds and Partial Transport of a Sand-Gravel Sediment, Water Resources Research, Vol. 29, No. 4, pp. 1297-1312. Bed Of Many Colours (BOMC): Ten tests with a sand gravel mixture. The tests were carried out at Johns Hopkins University.

• Wilcock et al., [2001], Experimental Study of the Transport of Mixed Sand and Gravel, Water Resources Res., Vol. 37, No. 12, pp. 3349-3358. Four experimental series with different sand gravel mixtures were conducted at Johns Hopkins University. 37 runs are used for verification in this report.

1.3.5 Surface layer as active layer

The active layer is defined as that part of the bed material, which determines the transport rate and composition. The bed surface layer (the material present in the surface layer of the moving bed forms or of a flat bed) seems the most logical representation of the active layer. Namely, the water flow directly interacts with the



bed surface and only grains at the bed surface can be directly transported. Measurements of the bed surface composition are only available from Wilcock & McArdell [1993], Wilcock et al. [2001], and Blom [2000]. However, for most experiments only the composition of the initial bed mixture was known. For this reason, initially, the composition of the initial bed mixture is used as input for the sediment transport predictions. Later the composition of the surface layer is used as input to check if the predictions of the formulas become better.

Graded sediment


2 Graded sediment

2.1 Introduction

In experiments, we often use uniform sediment to increase our insight in sediment transport processes. However, river sediment is mostly graded, i.e. the sediment consists of a range of grain sizes. The gradation of the bed material shows great variation among various rivers and it influences the processes that are involved in sediment transport. This chapter describes several sediment properties and the influence of the gradation of mixtures on some important transport processes.

2.2 Sediment properties

Several sediment properties have to be known to describe the sediment transport. These include size, density, gradation and cohesion. In this report, only non-cohesive material, like sand and gravel, is considered. The sediment transport formulas analysed in the present study are not suitable for clay, silt and other cohesive sediments.

The grain sizes of graded sediment are usually given in weight percentages. The grain sizes are noted as Dx, wherein the subscript x denotes the weight percentage of the sediment mixture with a grain size smaller than D, e.g. D50 = 2.5 mm indicates that 50% of the sediment mixture has a grain size smaller than 2.5 mm.

The density of natural sediments (ρs) is approximately equal to 2650 kg/m3, as they mostly consist of small quartz particles. The sediment density is generally used in sediment transport formulas related to the density of water (ρ) in the form of specific gravity:

ρρ ss = (-)

Or in the form of relative density:

1ss −=−

=∆ρρρ (-)

In literature several definitions of the geometric grain size exist. The most commonly used definition is:

∑ ⋅=i

iim DpD (m) (2.1)

in which pi is the probability of fraction i in the active layer, which determines the transport rate and composition. Di is the grain size of fraction i. However, Parker [1990] uses a different method to calculate the geometric mean grain size. It is widely accepted that grain size distributions of sediment mixtures are log-normal.

Graded sediment


Calculating the geometric mean grain size according to this distribution leads to the following expression for Dm:

i2

i Dlog=−ψ grain size on phi-scale (-)

∑ ⋅=i

iip ψψ mean grain size on phi-scale (-)

ψ−= 2Dm geometric mean grain size (m) (2.2)

In this report, definition (2.1) will be used unless mentioned otherwise.

The geometric standard deviation gives a good indication of the gradation of the sediment:

+=

16

50

50

84g D

DDD

50.σ (-)

Engelund & Hansen [1967] stated that some sediment properties, like shape and gradation, received too little attention. Since then, much research has been done on the gradation of the sediment, but knowledge on the influence of shape is still relatively small.

2.3 Incipient motion

The flow of water over the riverbed causes a stress known as the bed shear stress (τb). The bed shear stress can be divided into a part related to the grains and a part related to the bed forms:

τττ ′′+′=b

in which τ’ notes the shear stress caused by the grains and τ’’ notes the shear stress caused by the bed forms.

lower regime

dunes

ripples plane bed

upper regime

anti - dunes

trans

ition

plan

e be

d an

d st

andi

ng w

aves

τ’

τ’’

τb

u

Figure 2.1: Bed shear stress versus flow velocity (from Engelund & Hansen [1967]).

Graded sediment


Figure 2.1 shows the bed shear stress versus the flow velocity. It can be seen that, in the lower flow regime, the bed form roughness mainly determines the bed shear stress, while in the upper regime, the grain roughness is the most important factor for the bed shear stress.

In an open channel the bed shear stress can be calculated with: wb ihg ⋅⋅⋅= ρτ bed shear stress (N/m2)

in which g is gravity, h is mean depth, and iw is the water surface slope. The Shields parameter (θ) gives a dimensionless bed shear stress related to the grain diameter (D) and is given by:

Dih

Dgb

⋅∆⋅

=⋅∆⋅⋅

=ρτ

θ Shields parameter (-)

Each grain needs a certain shear stress to start moving. This stress is called the critical bed shear stress (τb,cr). The grain is moved when the bed shear stress exceeds the critical bed shear stress. The critical Shields parameter (θcr) represents the dimensionless critical bed shear stress. Shields [1936] determined the relation between the Reynolds number and the critical Shields parameter. Van Rijn [1984a] gives a mathematical approximation of this relation. In Section 3.6, this approximation is given as part of the sediment transport formula of Van Rijn.

Both θ and θcr are related to the grain diameter. This means that grains with different sizes start moving at different shear stresses. This can be simply explained in the case of spherical particles. The weight of a particle is the main force that withholds it from moving. The drag force, which is proportional to the bed shear stress, only applies to the surface area of the particle. The weight is a function of the volume of the particle. When the diameter of a particle is increased, its volume increases faster than its surface area. This means that a higher bed shear stress is needed to move larger particles.

2.4 Hiding/exposure

In sediment mixtures, there is a difference in exposure to the shear stress for different grain sizes. Larger grains protrude more out of the bed and thus are more exposed to the flow. This is called ‘exposure’. Smaller grains can lie in the shade of the larger grains. This ‘hides’ them from the flow. The combined effect is called hiding/exposure. Figure 2.2 illustrates the hiding/exposure effect.

Exposure Hiding

Figure 2.2: Hiding/exposure effect.

Graded sediment


The hiding/exposure effects result in a smaller critical bed shear stress for larger grains and a higher critical bed shear stress for smaller grains. Egiazaroff [1965] was the first who estimated a hiding/exposure factor for the critical bed shear stress. Most hiding /exposure factors are derived for a specific sediment transport formula. They are mostly based on experimental results. This means that these empirical hiding/exposure factors do not only correct for the critical bed shear stress but may take other effects of gradation into account as well.

Sediment transport formulas


3 Sediment transport formulas

3.1 Introduction

Throughout the last fifty years, many researchers have proposed formulas to predict sediment transport rates and compositions. Most of these formulas have an empirical basis. They are mostly based on experiments with uniform bed material. However, natural sediment mostly consists of non-uniform material. A certain grain size is present in the bed material in a certain percentage, its probability. A simple method to determine the transport per size fraction is to multiply the transport calculated for uniform sediment of this size fraction by the probability of this size fraction in the bed:

uiini qpq ,, ⋅=

In which: qi,n volumetric sediment transport of fraction i in non-uniform sediment;

pi probability (volume fraction) of size fraction i being present in the transport layer;

qi,u volumetric sediment transport of fraction i in uniform sediment with similar hydraulic conditions.

In general, this method is considered to be too simple because it does not take hiding and exposure effects into account. The following sections give an overview of the ten transport formulas, listed in Table 1.1, and of available hiding/exposure corrections. The data on which the several fractional transport formulas were originally calibrated are analysed to determine their validity ranges.

3.2 Ackers & White

3.2.1 General

Ackers & White [1973] proposed a formula to estimate the total load transport. No distinction was made between bed load and suspended load. This empirical formula is based on 925 sets of data of flume experiments with a grain size ranging from 0.04 mm to 4 mm. The water depth was mainly below 0.4 m. The analysis of the data showed that the transport of fine material (smaller than 0.04 mm) could be best determined using the shear velocity (u*). The mean velocity (u ) appeared more suitable for coarser grains (larger than 2.5 mm). The method cannot be used for grain sizes smaller than 0.04 mm because of their cohesive properties. Ackers & White excluded data with Froude numbers exceeding 0.8. The method proved not to be sensitive to bed forms. The following sections present the formulas for uniform sediment and sediment mixtures.



3.2.2 Uniform transport

Ackers & White define the dimensionless transport rate as: n

gr uu

DshXG

⋅

⋅⋅

= * dimensionless sediment transport rate (-) (3.1)

m

crgr

grgr 1

FF

KG

−⋅=

, dimensionless sediment transport rate (-) (3.2)

in which:

ρρ ss =

ρρ⋅⋅

=q

qX st mass flux per unit mass flow rate (-)

Combining Equations 3.1 and 3.2 gives the following equation, with which the sediment transport volume without pore volume can be calculated:

m

crgr

grn

t 1FF

uuDuKq

−⋅

⋅⋅⋅=

,* total sediment transport (m2/s) (3.3)

in which: u depth-averaged velocity (m/s)

wihgu ⋅⋅=* shear velocity (m/s)

( )n1

n

gr

Dh1032

uDg

uF

−

⋅⋅⋅

⋅∆⋅=

log* sediment mobility number (-)

ρρρ −

=∆ s relative density of sediment (-)

ρs sediment density (kg/m3)

ρ water density (kg/m3)

D diameter of bed material (m) 31

2gDD

/

*

⋅∆⋅=

ν dimensionless grain size (-)

62 1015Te00068015Te0310141 −⋅−+−−= ])(.)(..[ν kinematic viscosity (m2/s)

Te temperature (°C)

According to Ackers & White [1973], the best representation of the bed material is the D35 grain size. The coefficients n, m, K and Fgr,cr (critical sediment mobility number) are dimensionless. These coefficients depend on the dimensionless particle size. Ackers & White make a distinction between particles with 1 < D* < 60 and particles with D* ≥ 60. Later revisions were made for the K and m coefficients [HR



Wallingford, 1990], because there were uncertainties in the original formula in the sediment transport for relatively fine and coarse sediments. Van der Scheer [2000] showed that the results of the Ackers & White formula with the modified parameters, described in HR Wallingford [1990], are slightly worse than the predictions of the original formula. This is the reason that the modified formula will not be analysed in this report. Table 3.1 shows the original coefficients for the Ackers & White formula.

Table 3.1: Coefficients Ackers & White formula

coefficients 1 < D* < 60 D* ≥ 60

n )log(. *D5601 ⋅− 0

m 341

D669 ..

*+

1.5

K 2DD86253310 )(log)log(.. ** −+− 0.025

Fgr,cr 140

D230 ..

*

+ 0.17

3.2.3 Fractional transport

In this section the hiding/exposure corrections for the Ackers & White transport formula by Day [1980] and Proffitt & Sutherland [1983] are explained. Both correction factors (ξi) given in these sections are applied to the critical sediment mobility number (Fgr,cr). The fractional sediment transport for fraction i yields:

ii m

iicrgr

igrn

iiiit 1F

FuuDuKpq

−

⋅⋅

⋅⋅⋅⋅=

ξ,,

,

*, fractional transport formula (m2/s) (3.4)

3.2.3.1 Day's hiding/exposure factor

Day [1980] proposes a hiding/exposure factor based on a large number of experiments with sediment mixtures (Table 3.2). Unfortunately, Day does not explain which definition he used for the active layer of for which part of the bed the D50 was representative.

Table 3.2: Validity ranges of experimental data used by Day Experiments D50

(mm) σg (-)

θ50 range (-)

Remarks

USWES 1 [1935] 0.42 1.82 0.043 - 0.191 - USWES 2 [1935] 0.44 1.52 0.062 - 0.260 - USWES 9 [1935] 4.10 1.46 0.033 - 0.072 - Gibbs & Neill [1972, 1973] 4.75 2.34 0.042 - 0.108 constant flow, tilting flume Cecen & Bayazit [1973] 13.9 1.73 0.043 - 0.061 removal of armour layer Day: HRS A [1980] 1.75 4.21 0.034 - 0.125 sediment recirculating system Day: HRS B [1980] 1.55 3.50 0.028 - 0.123 sediment recirculating system

Day's correction is a correction factor for the critical sediment mobility number. The hiding/exposure factor is given by:



60DD

40i

Ai .. +⋅=ξ hiding/exposure factor (-) (3.5)

in which DA denotes the grain size in the sediment mixture that does not experience any hiding/exposure effects:

280

84

16

50

A

DD

61DD

.

.

⋅= (-)

3.2.3.2 Proffitt & Sutherland's hiding/exposure factor

Proffitt & Sutherland [1983] also propose a hiding/exposure factor for the Ackers & White formula. This factor can be seen as an extension of Day’s work, since the latter was more aimed at threshold conditions, whereas Proffitt & Sutherland include also the higher transport rates. The hiding/exposure factor is based on laboratory experiments done by Proffitt [1980] and verified with a large number of experiments (Table 3.3). Also, Proffitt & Sutherland do not explain what they defined as the active layer.

Originally, Proffitt & Sutherland's hiding/exposure factor was applied to the sediment mobility number, which is in contrast to Day’s factor. The latter is applied to the critical sediment mobility number. The inverse function of the original Proffitt & Sutherland factor is given to allow the use of the two hiding/exposure factors in the same form in formula 3.4:

Table 3.3: Validity ranges of experimental data used by Proffitt & Sutherland Experiments D50 range

(mm) σg (-)

θ50 range (-)

Remarks

Proffitt: run 1 [1980] 2.90 - 6.85 2.26 0.037 - 0.090 armouring experiments Proffitt: run 2 [1980] 3.25 - 11.7 3.24 0.030 - 0.099 armouring experiments Proffitt: run 3 [1980] 3.07 - 11.7 2.78 0.027 - 0.098 armouring experiments Proffitt: run 4 [1980] 4.20 - 5.45 1.95 0.041 - 0.072 armouring experiments Davies [1974] 2.8 - 4.2 ca. 3 0.050 - 0.087 armouring experiments Day: HRS A [1980] 1.75 4.21 0.038 - 0.125 sediment recirculating system Day: HRS B [1980] 1.55 3.50 0.040 - 0.123 sediment recirculating system Gessler: 1-5 [1967] 1.0 2.78 0.074 Little and Mayer [1972] 1.0 2.05 0.057 East Fork river [1980] 1.3 6.71 0.150 - 0.326 Tanana river [1978] 13 18.9 N.A.1 Snake river [1980] 30 6.87 0.050 - 0.094 river with paved bed Clearwater river [1980] 18 10.9 N.A. river with paved bed

≤

<<

+

⋅

≤

=−

0750DDfor52

73DD0750for1D

D530

DD73for7690

ui

ui

1

ui

ui

i

..

..log.

..

ξ hiding/exposure factor (3.6)

1 N.A.: Not Available



Du denotes the grain size that requires no correction. Du is related to the effective Shields parameter, which denotes:

50

b50 Dg ⋅⋅∆⋅

=ρ

τθ (-)

Proffitt & Sutherland give a relation between Du and θ50 that is shown in Figure 3.1.

(a) ACKERS & WHITE

1.5

1.0

00.02 0.06 0.10 0.14

θθθθ50

du /d50

Figure 3.1: Relation between θ50 and Du given by Proffitt & Sutherland [1983]

As seen in Figure 3.1, the function consists of four straight lines. The authors of this report made the following approximation of the function:

<≤<+⋅−≤<+⋅−

≤

=

50

5050

5050

50

50

u

0970for456009700460for40173904600400for9441621

040for081

DD

θθθθθ

θ

..........

..

3.3 Parker

3.3.1 General

Parker et al. [1982] proposed a bed load transport formula for paved gravel bed streams. In paved rivers, the surface layer (pavement) is significantly coarser than the substrate. Parker et al. distinguish a paved bed from an armoured bed in that a paved bed is transported during peak discharges, in contradiction to armoured beds, which never move. The formula uses the grain distribution of the substrate and is further referred to as "substrate-based formula". Parker [1990] gives a final version of this formula. In the same article, Parker transforms the substrate-based formula into a surface-based formula that uses the grain distribution of the surface layer instead of the grain distribution of the substrate. A description of the surface-based formula is given in the next section. In this project, the substrate-based formula of



Parker is not taken into account since Van der Scheer [2000] showed that the formula gives poor results for the sediment transport predictions.

3.3.2 Surface-based bed load formula

Parker [1990] transformed the substrate-based formula into a surface-based formula to predict pavement as well as selective transport. The surface-based formula includes a hiding/exposure factor. He defined the active layer as the surface layer of the plane bed, with a thickness equal to D90. The transport formula is based solely on field data (Table 3.4).

Table 3.4: Validity ranges of field data used by Parker Stream D50

(mm) σg (-)

θ50 range (-)

Remarks

Oak Creek 20 5.52 0.091 - 0.147 river with paved bed Elbow River [1968] 28 2.79 0.103 - 0.139 river with paved bed Snake River [1974/76] 27 7.09 0.097 - 0.160 river with paved bed Clearwater River [1974/76] 18 9.19 0.058 - 0.134 river with paved bed Vedder River [1974] 19 5.93 0.083 - 0.103 river with paved bed

The dimensionless surface-based transport formula denotes: )]([**

i00sgrsi gGWW δφω ⋅⋅⋅= transport parameter (-) (3.7)

The fractional bed load without pore volume relates to the dimensionless transport parameter as follows:

i2

3b

ibsi

F

gqW

⋅

∆⋅⋅=

ρτ

,* (3.8)

The variables in the equations above are described by:

<

≤≤−⋅−−⋅

<

−

=

1for

5911for12891214

591for853015474

G

214

2

54

φφ

φφφ

φφ

φ

.

.

.])(.)(.exp[

..

)( (-)

sg

ii D

D=δ (-)

∑ ⋅= )lnexp( iisg DFD surface layer geometric mean diameter (m)

*rsg

sg0sg

τ

θφ = (-)

sg

bsg Dg ⋅⋅∆⋅

=ρ

τθ Shields parameter Dsg of surface layer (-)

The hiding/exposure factor for the surface-based formula is given by:



β

δ−

=

sg

ii0 D

Dg )( reduced hiding function (-)

The surface-based formula uses a generalised straining factor to better agree with the equal mobility concept. The straining function yields:

)( 11 00

−⋅+= ωσ

σω

φ

φ straining function (-)

∑ ⋅

= i

2sgi2 F

2DD)ln(

)ln(φσ arithmetic standard deviation (-)

Fi denotes the volume fraction of fraction i in the surface layer. The parameter β equals 0.0951. The reference values *

rsW and *rsgτ equal to 0.0025 and 0.0386,

respectively. The values ω0 and σφ0 depend on φsg0 , as shown in Figure 3.2. The relations in Figure 3.2 are based on the relations given by Parker. Parker [1990] does not give a mathematical relation between ω0 and φsg0 and between σφ0 and φsg0, so two sets of linear relations between the parameters have been estimated by the authors of this report:

≤<≤+⋅−

<≤+⋅−<

=

0sg

0sg0sg

0sg0sg

0sg

0

5for4530552for74800590

521for285127401for0111

φφφ

φφφ

ω

....

....

(-)

≤<≤+⋅<≤+⋅

<

=

0sg

0sg0sg

0sg0sg

0sg

0

5for501152for0001100021for43203840

1for8160

φφφφφ

φ

σφ

.....

.

(-)

σφ0

ω0

1.501

0.453

0 1 2 3 4φsg0

0.816

1.011

0

1

2

σφ0 .ω0

Figure 3.2 Relation between φsg0 and ω0, and the relation betweenφsg0 and σφ0 , after Parker [1990]



3.4 Engelund & Hansen

3.4.1 General

The formula proposed by Engelund & Hansen [1967] is semi-empirical. Although the formula was originally derived for bed load transport, it proved to be very suitable to the total sediment transport of small grain sizes. Grain sizes, ranging from 0.19 to 0.93 mm, were used in the experiments on which the formula is based. Engelund & Hansen note that application of the formula should be done with care for grain sizes smaller than 0.15 mm.


The dimensionless sediment transport parameter for uniform sediment transport is given by:

25

50EH050 )(. θµ ⋅⋅=Φ transport parameter (-) (3.9)

The total sediment transport in volume omitting pores can be determined using:

350

t

Dg

q

⋅⋅∆=Φ

in which: 5

22

EH gC

=µ bed form factor (-)

50

b50 Dg ⋅⋅∆⋅

=ρ

τθ Shields parameter (-)

wihuC⋅

= Chézy value (m1/2/s)


No hiding/exposure correction has been developed for this formula. Laguzzi [1994] suggested that Egiazaroff [1965], modified by Ashida & Michiue [1973], could be used to correct for the hiding/exposure of the fractions present in bed material. This method has not been tested and will not be used here. A fractional transport formula only accounting for the probability of size fractions in the bed is used. The dimensionless transport parameter yields:

25

iEHii 050p )(. θµ ⋅⋅⋅=Φ fractional transport parameter (-) (3.10)

The transported volume is determined by:

3i

iti

Dg

q

⋅⋅∆=Φ ,

in which:



i

bi Dg ⋅⋅∆⋅

=ρ

τθ fractional Shields parameter (-)

3.5 Meyer-Peter & Müller

3.5.1 General

Meyer-Peter & Müller [1948] proposed an empirical sediment transport formula for bed load based on extensive experimental research. Both uniform and non-uniform sediment was used with a grain size ranging from 0.4 to 29 mm.

Egiazaroff [1965] developed a general hiding/exposure factor, which has been used extensively in combination with the Meyer-Peter & Müller transport formula. Ashida & Michiue [1973] modified Egiazaroff's hiding/exposure factor.


The original formula of Meyer-Peter & Müller uses the geometric mean grain size (Dm) as the representative grain size. The dimensionless transport parameter is given by:

23

mMPMb 04708 ).( −⋅⋅=Φ θµ transport parameter (-) (3.11)

The relation between the dimensionless transport parameter and the volumetric sediment transport without pore volume is given by:

3m

bb

Dg

q

⋅⋅∆=Φ

in which:

m

bm Dg ⋅⋅⋅∆

=ρτ

θ Shields parameter for Dm (-)

23

MPM CC

′=µ bed form factor (-)

The bed form factor should always be smaller or equal to 1, since the grain related Chézy value (C90) can never be smaller than the total Chézy value (C). This restriction is applied in the present study.

⋅⋅=′

90Dh1218C log grain related Chézy value (White-Colebrook) (m½/s)

∑ ⋅=i

iim DpD geometric mean grain size (m)

It should be noted that the bed form factor of the Meyer-Peter & Müller formula used here is not the original bed form factor given by Meyer-Peter & Müller [1948]. In Appendix 1, the different bed form factors are analysed in detail.




3.5.3.1 Egiazaroff’s hiding/exposure factor

The hiding/exposure factor developed by Egiazaroff [1965] was calibrated with data from several experiments. Unfortunately, we did not succeed in collecting all information on the experimental data used by Egiazaroff. This means that the validity ranges given in Table 3.5 are not complete. Egiazaroff does not explain his definition of the active layer and his method to determine the composition of the active layer.

Table 3.5: Validity ranges of experimental data used by Egiazaroff Experiments D50

mm σg -

θ50 range -

Remarks

Nizery & Braudeau [1953] 5.50 4.84 0.666 - 0.917 armouring experiments Pantélopulos A [1957] 2.55 2.08 0.042 - 0.101 feed experiments Pantélopulos B [1957] 1.80 2.38 N.A. feed experiments Pantélopulos C [1957] 2.75 2.54 N.A. feed experiments Oumarov [1961] N.A. N.A. N.A. - Ramette [1962] N.A. N.A. N.A. -

The hiding/exposure factor of Egiazaroff [1965] is applied to the critical Shields parameter. This results in the dimensionless transport parameter per fraction:

23

iiMPMiib 04708p ).(, ⋅−⋅⋅⋅=Φ ξθµ fractional transport parameter (-) (3.12)

This parameter is related to the fractional sediment transport (without pore volume):

3i

ibib

Dg

q

⋅⋅∆=Φ ,

,

using:

i

bi Dg ⋅⋅⋅∆

=ρτ

θ fractional Shields parameter (-)

The hiding/exposure factor of Egiazaroff is semi-theoretically determined. It considers the balance of forces working on a spherical grain on the threshold of movement. The correction factor is applied to the critical Shields parameter, which has a value of 0.047 in the formula of Meyer-Peter & Müller.

( )

2

mii DD19

19

⋅

=log

)log(ξ hiding/exposure factor Egiazaroff (-) (3.13)

3.5.3.2 Ashida & Michiue’s hiding/exposure factor

For the range Di/Dm < 0.4, Ashida & Michiue [1973] did not find consistency between their experiments (Table 3.6) and Egiazaroff's correction factor. They present a correction to Egiazaroff for this range, but this correction must be used with care since it is based on only one measurement in this range [Ribberink, 1981].



Table 3.6: Validity ranges of the experimental data used by Ashida & Michiue Experiments D50

(mm) σg (-)

θ50 range (-)

Remarks

Ashida & Michiue [1971] 1.7 3.73 0.043 - 0.185 armouring experiments Hirano A [1970] 0.83 1.24 0.039 armouring experiments Hirano B [1970] 0.98 2.31 0.038 armouring experiments

The corrected hiding/exposure factor yields:

( )

<⋅

≥

⋅=

40DD

forDD

850

40DD

forDD19

19

m

i

i

m

m

i2

mii

..

.log

)log(

ξ hiding/exposure factor A&M (-) (3.14)

3.6 Van Rijn

3.6.1 General

Van Rijn [1984a,b] divided his transport formula in a bed load and a suspended load formula. The total sediment transport is the sum of the bed load and the suspended load.

sbt qqq += total transport (m2/s) (3.15)

The suspended load formula is not used in the present study, but presented for completeness.

3.6.2 Uniform bed load formula

Like Bagnold [1954], Van Rijn assumed in his bed load formula that the bed load particles primarily move because of saltations or jumps. The bed load formula is suitable for sediments with a grain diameter ranging from 0.2 – 2 mm. The dimensionless bed load parameter is given below:

≥⋅⋅<⋅⋅=Φ −

−

3TforTD103TforTD0530

5130

1230

b ..*

..*

.

. bed load parameter (-) (3.16)

The relation between the sediment transport volume without pore volume and the dimensionless transport parameter yields:

350

bb

Dg

q

⋅∆⋅=Φ

in which: 31

250gDD

⋅∆⋅=

ν* dimensionless grain size (-)

crb

crbbT,

,'

τττ −

= bed shear parameter (-)

bb τµτ ⋅=' effective bed shear stress (N/m2)



2

CC

=

'µ bed form factor (-)

⋅⋅

⋅=90D3h1218C log' grain related Chézy value (White-Colebrook) (m1/2/s)

cr50scrb Dg θρρτ ⋅⋅⋅−= )(, critical bed shear stress (N/m2)

The bed form factor can physically not be larger than one because the total Chézy (C) value must always be smaller than or equal to the grain related Chézy value (C’). However, in this formula the Chézy value cancels out of the equation of the effective bed shear stress when the following expressions are used:

wihCu ⋅⋅= 2

wb Cugihg

⋅⋅=⋅⋅⋅= ρρτ

This may cause the bed form factor in this formula to become larger than one, although it is physically not justifiable.

The critical Shields parameter θcr can be determined from Figure 3.3.

100

10-1

10-2

100 101 102 103

θcr

dimensionless particle size, D*

NO MOTION

Figure 3.3: Relation between D* and θcr given by Van Rijn [1984a]

The Shields curve given in Figure 3.3 can be approximated by [Van Rijn, 1984a]:

>≤<⋅≤<⋅≤<⋅

≤⋅

= −

−

−

150D0550150D20D013020D10D040

10D4D1404DD240

290

100

640

1

cr

*

*.

*

*.

*

*.

*

**

.

.

.

.

.

θ critical Shields parameter (-) (3.17)



3.6.3 Fractional bed load formula

No specific hiding/exposure correction has been developed for Van Rijn’s formula. Although Laguzzi [1994] suggests some options, these have not been verified and will not be used here. Nevertheless, a fractional bed load formula is given accounting for the probability of the fractions as described in Section 3.1. This results in the following dimensionless bed load parameter:

>⋅⋅⋅<⋅⋅⋅=Φ −

−

3TforTD10p3TforTD0530p

i51

i30

i

i12

i30

iib ..

*

..*

, .. fractional bed load parameter (3.18)

The relation between sediment transport volume and the dimensionless transport parameter now yields:

3i

ibib

Dg

q

⋅∆⋅=Φ ,

,

in which: 31

2igDD

⋅∆⋅=


icrb

icrbbiT

,,

,,'

τττ −

= fractional bed shear parameter (-)

icrisicrb Dg ,,, )( θρρτ ⋅⋅⋅−= critical fractional bed shear stress (N/m2)

The critical Shields parameter θcr,i can be determined from Figure 3.3 or (3.17). The other variables in the fractional bed load formula are the same as in the uniform bed load formula.

3.6.4 Suspended load formula

The suspended load formula determines the volume of the sediment transported as suspended lead, without pores. Van Rijn [1993] does not give a dimensionless transport formula. Van Rijn’s formula for suspended sediment transport yields:

as chuFq ⋅⋅⋅= suspended sediment transport (m2/s) (3.19)

in which F can only be approximated, as the original differential equation (not given in this report) cannot be integrated analytically. The following equation gives an estimation of F [Van Rijn, 1984]:

( )'.'

.'

Z21ha1

ha

ha

FZ

21Z

−⋅

−

−

= F-factor (-)

40

0

a80

s2

s

s

cc

uw

52

uuw21

wZ

..

**

*

.'

⋅

⋅+

⋅⋅

+

=

κ

suspension number (-)



30

5150

aDT

aD

0150c.

*

.. ⋅⋅= reference sediment concentration (-)

c0 = 0.65 maximum concentration (-)

κ = 0.4 constant of Von Karman (-)

The fall velocity ws (in m/s) is given by Van Rijn [1993]:

>⋅⋅∆⋅

≤≤

−

⋅⋅∆⋅+

⋅

<⋅⋅⋅∆

=

mm1DforDg11

mm1D10for1Dg010

1D

10

mm10D18

Dg

w

ss

s2

3s

s

2s

s

.

..

.

ννν

in which Ds denotes the representative particle size (m) of suspended sediment.

≥<⋅−⋅−⋅+

=25TforD25TforD25T101101

D50

50ss

)]()(.[ σ

crb

crbbT,

,'

τττ −

= bed shear parameter (-)

+=

16

50

50

84s D

DDD

50.σ geometric standard deviation of bed material (-)

The reference level (a) is equal to 0.5 times the bed form height (H) or equal to the overall roughness height (ks).

3.7 Hunziker/Meyer-Peter & Müller

3.7.1 General

Hunziker [1995] proposed a fractional formula for bed load. This formula can be seen as a modified version of the sediment transport formula of Meyer-Peter & Müller [1948]. Hunziker found that the fractional bed load formula of Meyer-Peter & Müller with the hiding/exposure factor of Ashida & Michiue [1973] does not lead to satisfactory predictions of the rotational erosion process of the experiments of Günter [1971]. Hunziker states that all fractions in a uniform mixture experience initial motion at the same critical Shields parameter, but the fine particles are more mobile than the coarse ones. The formula is calibrated with the experiments of Günter [1971] and Suzuki [1992] (Table 3.7).

Table 3.7: Validity ranges of the experimental data used by Hunziker Experiments D50

mm σg -

θ50 range -

Remarks

Günter: mixture 1 [1971] 1.73 3.08 0.077 - 0.085 armouring experiments Günter: mixture 2 [1971] 0.90 3.50 0.127 - 0.148 armouring experiments Günter: mixture 3 [1971] 2.54 3.87 0.065 - 0.071 armouring experiments Suzuki & Hano [1992] 2.10 2.93 0.175 - 0.326 sand feed system



Meyer-Peter & Müller Run 1 [1948] 1.43 2.74 0.062 - 0.197 equilibrium conditions Meyer-Peter & Müller Run 2 [1948] 3.93 2.62 0.078 - 0.127 equilibrium conditions Meyer-Peter & Müller Run 3 [1948] 2.62 2.45 0.061 - 0.115 equilibrium conditions Zarn [1997] N.A. N.A. N.A. equilibrium conditions Gessler i/5 [1965] 1 4.1 0.081 armouring experiment


The fractional dimensionless transport parameter of Hunziker/Meyer-Peter & Müller is given by:

23

cmsmiiib 5p ))(( *,, θθξ −⋅⋅=Φ transport parameter (-) (3.20)

The relation between the fractional dimensionless transport parameter and the volumetric sediment transport without pore volume is given by:

3sm

ibib

Dg

q

,

,,

⋅∆⋅=Φ

in which: α

ξ−

=

sm

ii D

D

, hiding/exposure factor (-) (3.21)

30011051

sm ...*

, −⋅=−

θα (-)

smHsm ,*

, θµθ ⋅= corrected Shields parameter (-) 2

H CC

′=µ bed form factor (-)

fA

gC =′ grain related Chézy value (Yalin & Scheuerlein [1988]) (m1/2/s)

2

90f D2

hb1A−

⋅

= lnκ

grain resistance of bed (Yalin & Scheuerlein) (-)

).( 1B40 seb −⋅= (-)

( )( ) ( )( )22170s e35258B *Reln.

*Reln.. −⋅−+= roughness function (Schlichting, [1968]) (-)

ν90D2u ⋅⋅

= **Re Reynolds number (-)



Hunziker found that using the formula of Strickler [1923] leads to too small values of the bed form factor at the beginning of motion and uses therefore the formula of Yalin & Scheuerlein [1988]. This results in a bed form factor closer to one and thus a larger sediment transport. Hunziker compensated this by applying a smaller value of



the calibration factor in his own formula (‘5’ instead of ‘8’ at Meyer-Peter & Müller), see (3.20). The bed form factor is restricted to values smaller than 1.

sm

bms Dg ,⋅⋅⋅∆

=ρτ

θ Shields parameter for Dms (-) (3.22)

330

sm

omcrcm D

D.

,

,

⋅= θθ critical Shields parameter corrected for layers in bed (-)

∑ ⋅=i

oioiom DpD ,,, geometric mean grain size of the under-layer (m)

∑ ⋅=i

sisism DpD ,,, geometric mean grain size of the surface layer (m)

The critical Shields parameter is determined with the expression given by Iwagaki [1956]. The expression of Iwagaki is rewritten such that the critical Shields parameter can be calculated using the dimensionless grain size.

31

2smgDD

⋅∆⋅=

ν,* dimensionless grain size (-)

≥

<≤⋅⋅∆

⋅⋅∆<≤

<≤⋅⋅∆

⋅⋅∆<

=−

−

676D050

676D829Dg

Dg015050829D3140340

314D661Dg

Dg12350661D140

sm

sm

sm

sm

cr22

1111

322

25

3211

163

3225

..

..).(

...

..).(

..

*

*,

,

*

*,

,

*

ν

ν

θ critical Shields stress (-) (3.23)

Hunziker has made some restrictions for his hiding/exposure correction to counteract instability in certain parameter ranges. These restrictions are: α ≤ 2.3

250DD

sm

i .,≥ in equation 3.21

3.8 Gladkow & Söhngen

3.8.1 General

Gladkow & Söhngen [2000] developed a fractional bed load formula. Gladkow & Söhngen used several formulas and their modifications as basis for their new bed load formula. The critical Shields parameter used in the formula is an approximation of the formula of Knoroz [1958]. Knoroz found a decreasing critical Shields parameter, which is not in concurrence with Shields [1936], who found a decreasing critical Shields parameter after which the parameter increases. The formula was calibrated with field and experimental data. The data used for calibration could not be acquired, so the validity ranges could not be determined.




The fractional bed load formula of Gladkow & Söhngen calculates the bed load transport rate without pores as:

42

iS

icri

42iib

1Frq00140pq

.,

., .

∆

⋅

⋅−

⋅⋅⋅⋅=θµθξ

fractional volumetric transport (m2/s)

(3.24)

in which:

hguFr⋅

= Froude number (-)

2

i

iicr 720D

31D02660

−

+⋅=

..

.*,

*,,θ critical Shields parameter for fraction i (-)

31

2iigDD

⋅∆⋅=

ν*, dimensionless grain size of fraction i (-)

ii D

ih⋅∆⋅

=θ Shields parameter (-)

41

s

ss

′=

κκ

µ bed form factor of Söhngen [1996] (-)

sss κκκ ′′+′= total bed roughness according to Söhngen [1996] (m)

mDms S61D ⋅+=′ .κ grain roughness (m)

∑ ⋅= iim DpD mean grain size (m)

∑ −⋅= 2miiD DDpS

m)( standard deviation grain size (m)

hh2 d

s⋅

=′′κ bed form roughness (-)

( )2

m

mcrd Fr11

61hh −⋅

−⋅⋅=θθ , dune height (m)

2

m

mmcr 720D

31D02660

−

+⋅=

..

.*,

*,,θ critical Shields parameter for Dm (-)

31

2mmgDD

⋅∆⋅=

ν*, dimensionless grain size of Dm (-)

mm D

ih⋅∆⋅

=θ Shields parameter for Dm (-)



( )

≥

<≤

⋅

<⋅

=

01DD

forDD

01DD

40forDD19

19

40DD

forDD

850

m

i90

i

m

m

i2

mi

m

i

i

m

i

.

..log

)log(

..

.

ξ hiding/exposure factor Söhngen (-) (3.25)

3.9 Wu et al.

3.9.1 General

Wu et al. [2000] developed a fractional formula for bed load transport and a fractional formula for suspended load. The total sediment transport is calculated as in (eq. 3.15).


The formula of Wu et al. has been calibrated and tested for a wide range of experimental and field data. The key element of this formula is the development of a new hiding/exposure factor. In this factor the grain size of a fraction is compared with the grain sizes of the other fractions. It is assumed that particles are distributed randomly on the bed. This leads to the assumption that the exposure height of a particle is normally distributed. The bed load formula shows large similarities with the formula of Meyer-Peter & Müller.

Table 3.8: Validity ranges of the experimental data used by Wu et al. Experimental / Field data D50

mm σg -

θ50 range -

Remarks

Samaga [1986a] 0.2 - 0.35 1.91 - 3.79 0.317 - 0.823 Recirculation flume Kuhnle [1993] 0.44 - 5.58 1.28 - 5.27 - Recirculation flume Wilcock [1993] 4.53 9.91 0.009 - 0.196 Recirculation flume Liu [1986] - - - Not available Sustina river 34.8 - 50.2 1.66 - 2.13 0.059 - 0.127 Gravel bed river Chultina river 10.6 - 23.6 2.23 - 2.40 0.032 - 0.074 Gravel bed river Black river 0.38 - 0.53 1.68 - 1.99 0.141 - 0.544 Sand bed river Toutle river 1.82 - 24.8 1.29 - 3.52 0.051 - 1.387 Gravel bed river Yampa river 0.47 - 0.70 1.67 - 2.17 0.444 - 4.076 Sand bed river

The dimensionless transport parameter for fractional bed load yields: 22

ic

b23

iib 1nn00530p

.

,,

'.

−

⋅⋅=

ττ

φ fractional transport parameter (-) (3.27)

The volumetric bed load transport without pore volume can be determined with:

3i

ibib

Dg

q

⋅⋅∆= ,

,φ

in which: ( ) icisic Dg ξθρρτ ⋅⋅⋅⋅−=, critical shear stress (N/m2)



030c .=θ critical Shields parameter (-) 60

ih

iei p

p.

,

,−

=ξ hiding/exposure factor (-) (3.28)

∑= +

=N

1j ji

ijie dd

dpp , exposure probability (-)

∑= +

=N

1j ji

jjih dd

dpp , hiding probability (-)

ihgb ⋅⋅⋅= ρτ bed shear stress (N/m2)

20D

n6

50=' Manning’s roughness related to grains (s/m1/3)

uihn

21

32⋅

= Manning’s roughness (s/m1/3)

3.9.3 Fractional suspended load formula

Wu et al. [2000] also proposed a fractional suspended load formula. This formula is given here for completeness, but it will not be taken into account in the remaining part of this study. The formula was analysed with one set of experimental data and two sets of field data.

The fractional transport parameter for suspended sediment transport is: 741

iiciis

u100002620p.

,, .

−⋅⋅=

ωττφ fractional transport parameter (-) (3.29)

The transport parameter relates to the volumetric sediment transport in the following way:

3i

isis

Dg

q

⋅⋅∆= ,

,φ

in which: ihg ⋅⋅⋅= ρτ shear stress (N/m2)

ii

2

ii D

9513Dg091D

9513 ννω ... −⋅⋅∆⋅+

⋅= settling velocity (m/s)


3.10 Wilcock & Crowe

Wilcock & McArdell [1993] and Wilcock et al. [2001] did extensive research in experimental flumes after the fractional transport of sand/gravel mixtures. Based on these experiments, Wilcock & Crowe [2001] developed a fractional transport formula for mixtures of sand and gravel, using the bed surface as their active layer (the bed



layer determining the transport rate and composition). The experiments showed a relation between the sand content of the surface composition and the sediment transport rate.

3.10.1 Fractional formula

The formula is calibrated on the results of five experimental series with 48 runs (Table 3.9). The bed surface compositions of the experiments were determined using a point count method. The initial bed material was sieved and different fractions were painted in different colours. After a run photos were taken of the bed surface and later the photos were analysed to determine the probability of each fraction in the bed surface.

Table 3.9: Validity ranges of the experimental data used by Wilcock & Crowe [2001] Experiments D50

mm σg -

θ50 range -

Remarks

Wilcock et al: J06 10.22 2.93 0.028 - 0.134 Recirculation flume Wilcock et al: J14 8.38 3.69 0.048 - 0.136 Recirculation flume Wilcock et al: J21 7.06 4.66 0.028 - 0.165 Recirculation flume Wilcock et al: J27 5.76 5.42 0.012 - 0.201 Recirculation flume Wilcock & McArdell: BOMC 4.53 9.91 0.009 - 0.196 Recirculation flume

The dimensionless transport parameter of Wilcock & Crowe is given by:

≥

−

<⋅

=351for8940114

351for0020W 54

50

57

i ..

...

.

.

*

φφ

φφ dimensionless transport parameter (-) (3.30)

The dimensionless transport parameter is related to the volumetric sediment transport (without pore volume) according to:

3i

bii

upqg

W*

*

⋅

⋅⋅∆=

in which:

riττφ =

wihg ⋅⋅⋅= ρτ bed shear stress (N/m2) b

50s

i

50rs

ri

DD

=

ττ hiding/exposure function (-) (3.31)

−+

=

50s

i

DD

511

690b.exp

. (-)

50s

50rs50rs Dg ⋅⋅⋅∆

=ρτ

τ * Shields parameter of median surface grain size (-)

( )s50rs F1401300210 ⋅−⋅+= exp..*τ




The parameter Fs denotes the sand fraction in the active layer. Sand is considered to have a grain size between 0.062 and 2 mm.

3.11 Ribberink

Ribberink [1998] developed a uniform bed load formula for both steady flow and oscillatory flow conditions. This formula is largely based on the formula by Meyer-Peter & Müller [1948], but has some small modifications. Ribberink used laboratory experiments and field data for the calibration of his formula. Most of these experiments were conducted in the sheet flow regime with a flat bed. The grain sizes ranged from 0.19 to 3.8 mm. The effective Shields parameter (part of the bed shear stress responsible for bed load transport) lies within the range of 0.03 - 7.7. The uniform formula was modified to a fractional formula. The hiding/exposure factor of Ashida & Michiue was used.

3.11.1 Uniform formula

The formula of Ribberink uses the median grain size (D50) as the representative grain size. This differs from Meyer-Peter & Müller’s formula that uses the mean grain size (Dm) as its representative grain size. The dimensionless transport parameter can be determined by:

671cr50b 410 .)(. θθ −′⋅=Φ transport parameter (-) (3.32)

The relation between the dimensionless transport parameter and the volumetric sediment transport without pore volume is given by:

350

bb

Dg

q

⋅⋅∆=Φ

in which:

( ) 50s50 Dg ⋅⋅−

′=′

ρρτθ effective Shields parameter (-)

2

2

Cug′

⋅=′ ρτ bed shear stress caused by grains (N/m2)

⋅⋅=′

skh1218C log grain related Chézy value (m½/s)

( )( )

−+⋅⋅

=161D

D3k

5050

90s θ

max roughness height (m)

>≤<⋅≤<⋅≤<⋅

≤⋅

= −

−

−

150D0550150D20D013020D10D040

10D4D1404DD240

290

100

640

1

cr

*

*.

*

*.

*

*.

*

**

.

.

.

.

.

θ critical Shields parameter: Van Rijn [1984] (-)

(3.33)



31

250gDD

⋅∆⋅=


3.11.2 Fractional formula

The hiding/exposure factor of Ashida & Michiue [1973] was used. This is possible because the formula of Ribberink is based on the formula of Meyer-Peter & Müller [1948]. The hiding/exposure factor of Ashida & Michiue is widely used in the fractional formula of Meyer-Peter & Müller. The dimensionless transport parameter now yields:

671icriiiib 410p .

,, )(. θξθ ⋅−′⋅⋅=Φ fractional transport parameter (-) (3.34)

This parameter is related to the fractional sediment transport (without pore volume):

3i

ibib

Dg

q

⋅⋅∆=Φ ,

,

in which:

( ) isi Dg ⋅⋅−

′=′

ρρτθ effective Shields parameter (-)

( )

<⋅

≥

⋅=

40DD

forDD

850

40DD

forDD19

19

m

i

i

m

m

i2

mii

..

.log

)log(

ξ hiding/exposure factor A&M (-)

31

2igDD

⋅∆⋅=

ν* dimensionless fractional grain size (-) (3.35)

The critical Shields parameter per fraction (θcr,i) is determined using (3.33) and (3.35). The remaining parameters remain the same as for uniform formula.

3.12 Validity ranges of the fractional formulas

The validity ranges of the fractional formulas were presented in the accompanying tables in the previous sections. Here, we will give a clearer view on how these validity ranges compare with each other. Figures 3.4-a,b and 3.5-a,b show that there are large differences in the validity ranges of the different fractional formulas and hiding/exposure corrections.

The hiding/exposure correction by Day [1980] and the correction by Proffitt & Sutherland [1983] were derived especially to transform the sediment transport formula of Ackers & White [1973] in a fractional formula. These new fractional formulas with the hiding/exposure corrections were calibrated and verified with new data. Proffitt & Sutherland divide the data they used in two parts. The data from Proffitt [1980] were used to derive the hiding/exposure correction and the rest of the data were used to verify it. The validity ranges of Day and Proffitt & Sutherland can



be seen as validity ranges of the fractional formula of Ackers & White with the hiding/exposure corrections of Day and Proffitt & Sutherland, respectively.

Both Day and Proffitt & Sutherland have large validity ranges for their formulas. The fact that Proffitt & Sutherland used field data of gravel bed rivers for verification of their formula makes it more suitable for predictions with large grain sizes and large geometric standard deviations. Day, on the other hand, used data of fine material, which makes it more suitable to material with median grain sizes smaller than 1 mm.

The hiding/exposure corrections of Egiazaroff [1965] and Ashida & Michiue [1973] were not developed for the sediment transport formula of Meyer-Peter & Müller [1948], especially. Both corrections were initially derived for other sediment transport formulas. Ribberink [1987] was the first who applied these correction factors to the Meyer-Peter & Müller formula in a modified form. Ribberink calibrated the fractional formula with experimental data using a bimodal sediment mixture. These experiments were conducted in a narrow flume and thus the sidewall effects probably influenced the calibration. This means that the validity ranges given in Figures 3.4 and 3.5 should not be seen as the validity ranges of the fractional formula of Meyer-Peter & Müller corrected with Egiazaroff or Ashida & Michiue. Instead, they should be seen as the validity range of the original fractional formula of Egiazaroff and the original formula given by Ashida & Michiue. It should also be noted that we did not manage to collect all the data used by Egiazaroff and thus the validity ranges are not complete.

The Shields parameter range for the Egiazaroff correction factor is large due to the experimental data of Nizery and Braudeau [1953] that is taken from a small canal with a slope of 0.1. Other data that Egiazaroff used are from Pantélopulos [1957], who did experiments with both carbon grains and sand grains.

Parker [1990] and Hunziker [1995] both developed a fractional formula. This means that the data they used for calibration and verification directly show the validity range of their formulas. Parker used data only from rivers with coarse bed material and a large geometric standard deviation. Hunziker used the data sets of Günter [1971] and Suzuki [1992] for calibration of his model and the other data of Table 3.7 for verification.

The data used by Parker et al. [1982] have a small Shields parameter range. Hunziker uses data from armouring tests, which have a different composition for the surface layer and the substrate. The original material of the substrate was used for some experimental series since data on the armour layer were not available.

The formula of Wu et al. has the largest validity ranges of all the formulas analysed. A large amount of flume and field data has been used for calibration of the formula. Data with relatively fine material, e.g. experiments of Samaga et al. [2002], were used as well as data of coarse gravel rivers, e.g. Sustina River. The validity ranges only cover the data used to calibrate the bed load formula. Some of the data used for the bed load formula has also been used for the suspended load formula, but we were not able to acquire all calibration data of the suspended load formula. In the present study only the bed load formula is tested and analysed.



Wilcock & Crowe have used experimental data of Wilcock & McArdell [1993] and Wilcock et al. [2002] in the development of their formula. In all these experiments a sand gravel mixture was used. The experiments were done at relatively low Shields parameters and with coarse material. Although a lot of experiments were used, the validity ranges are small since the data covers a narrow range.

It must be noted that the formula of Wilcock & Crowe used the bed surface composition as the active layer composition, yet in figures 3.4-b and 3.5-b the initial bed material is used. This was done to allow for better comparison with the other formulas for which only the initial material composition of the data is known.



Figure 3.4-a: Validity ranges for several fractional formulas and hiding/exposure corrections given by the median grain size (D50) of the bed material versus the Shields parameters (θ50).



Figure 3.4-b: Validity ranges for several fractional formulas and hiding/exposure corrections given by the median grain size (D50) of the bed material versus the Shields parameters (θ50).



Figure 3.5-a: Validity ranges for several fractional formulas and hiding/exposure corrections given by the median grain size (D50) of the bed material versus geometric standard deviation (σg).



Figure 3.5-b: Validity ranges for several fractional formulas and hiding/exposure corrections given the median grain size (D50) of the bed material versus geometric standard deviation (σg).

Behaviour transport formulas


4 Behaviour transport formulas

4.1 Introduction

In this chapter, the sediment transport formulas described in Chapter 3 are analysed regarding their behaviour under different hydraulic conditions and for different sediment mixtures. The behaviour of the transport formulas is analysed by carrying out four series of calculations. The predicted sediment transport rates and the composition of the transported sediment are analysed.

In Section 4.2 the predicted sediment transport rates and sediment transport compositions are analysed for the 12 fractional formulas to see the differences between these formulas. The transport rate predictions of some fractional formulas are compared with their uniform counterparts (Section 4.3) to find out at which conditions fractional formulas give significant differences compared to their uniform counterparts. Section 4.4 is focussed on the behaviour at low Shields parameters with special attention for incipient motion.

Each of the four series has a different sediment composition. Both the median grain size (D50) and the geometric standard deviation of the bed material (σg) have been varied (Table 4.1).

Table 4.1: Bed material properties of the four series σg (-) D50 (mm) 2 5

2 Series 1 Series 2 10 Series 3 Series 4

The sediment mixtures are divided into n fractions of which the grain sizes (Di) and the probabilities (pi) are calculated. To do this, an assumption must be made for the grain size distribution. After Ribberink [1987], it is assumed that the grain size distribution is log-normal. Ribberink’s method of dividing a sediment mixture with a log-normal grain size distribution into size fractions is given in Appendix 2.

We divided the mixtures into ten grain size fractions. For each series, calculations were made for the Shields parameter range [0 - 0.6]. The following hydraulic parameters were constant in the calculations:

iw = 0.0005 water surface slope

w = 500 m width of hypothetical river

ρs = 2650 kg/m3 sediment density

ν = 0.0000012 m2/s viscosity

Cb = 40 m½/s Chézy value

The width of the hypothetical river was assumed 500 m, so that side wall effects could be neglected, and the hydraulic radius almost equals the water depth. The



Chézy value is supposed constant at 40 m½/s. As the Shields parameter and the median grain size are known, it is now possible to calculate the water depth as function of a certain Shields parameter:

iD

h 5050 ∆⋅⋅=θ water depth (m)

Now the discharge equals:

wihhCBQ ⋅⋅⋅⋅= discharge (m3/s)

Now, all input parameters for the calculations are known. The results are shown in two ways. The transport rate is shown as a function of the Shields parameter. The predicted composition is shown as the ratio between the geometric mean grain size of the transported material and the geometric mean grain size of the bed material (DmT/DmB), as a function of the Shields parameter.

Not every transport formula is valid for the sediment mixtures used in the four series. The median grain size and the geometric standard deviation of the bed material are allowed to have a difference of 20% to increase the number of formulas that are valid for a sediment mixture used in a series. The 80% - 120% validity ranges of the two sedimentological parameters for each series are given in Table 4.2, together with the formulas valid for this series.

Table 4.2: Input parameter ranges of the four model behaviour series and the formulas valid in that series D50 range σg range Valid formulas (mm) (-) Series 1 1.6 - 2.4 1.6 - 2.4 A&W+D, A&W+P&S, MP&M+Eg, Hunziker, Wu et al. Series 2 1.6 - 2.4 4.0 - 6.0 A&W+D, A&W+P&S, Hunziker, Wu et al. Series 3 8.0 - 12 1.6 - 2.4 A&W+D, A&W+P&S, Wu et al. Series 4 8.0 - 12 4.0 - 6.0 A&W+D, A&W+P&S, Wu et al., Wilcock.

The most important results are discussed in the following sections. The rest of the results can be found in Appendices 3, 4, and 5.

4.2 Differences between formulas

In this section, the behaviour of the different transport formulas is discussed based on the calculations of the four series. The formulas are compared for both the transport rate and the composition of the transported material. The validity ranges are taken into account in this comparison.

It should be noted that the formulas of Van Rijn and Engelund & Hansen have never been officially derived for fractional sediment transport. This leads to exceptionally high transport rates of the finest fractions because no hiding/exposure correction was introduced.

4.2.1 Transport rate

In this section not all results of all series are given. The median grain size (D50) of the bed material appeared to have little influence on the behaviour of the transport



formulas. A variation of the geometric standard deviation of the bed material does influence the transport rate. This is why Series 3 and 4 are chosen to be representative for the behaviour of the transport formulas under different conditions. The results for Series 1 and 2 are shown in Appendix 3.

0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2

Serie 3: D50=10mm, σσσσg=2

θ50 (-)

qs (m

3 /s/m

)

Ackers & White + DayAckers & White + Proffitt & SutherlandParker surface basedHunziker/Meyer-Peter & MüllerWilcock & CroweWu et al.

0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)

Engelund & Hansen fractionalGladkow & SöhngenMeyer-Peter & Müller + EgiazaroffMeyer-Peter & Müller + Ashida & MichiueVan Rijn fractionalRibberink + Ashida & Michiue

Figure 4.1: Series 3 (D50 = 10 mm, σg = 2): Sediment transport rate versus the Shields parameter for the fractional transport formulas.

From Figure 4.1 it can be seen that the transport formulas of Parker, Wilcock & Crowe and Engelund & Hansen predict the highest transport rates. The differences between these three formulas are small for low Shields parameters. Parker predicts a much higher transport rate for Shields parameters higher than 0.1. It must be noted that Parker’s formula is not valid for all conditions in Series 1 through 4, since the sediment mixtures are too fine. Nevertheless, Parker’s results are comparable to other formulas at Shields parameters smaller than 0.1.

The formulas of Ackers & White with Day, Ackers & White with Proffitt & Sutherland, Meyer-Peter & Müller with Ashida & Michiue, Ackers & White with Egiazaroff and Wu et al. show medium transport rates. There are no noticeable differences between the two fractional formulas of Meyer-Peter & Müller, which means that the modification of Ashida & Michiue on the hiding exposure correction of Egiazaroff has little effect on mixtures with a small gradation. For lower Shields parameters, Ackers & White with Day predicts a higher transport rate than Meyer-Peter & Müller with Ashida & Michiue. For higher Shields parameters, the predictions for the transport rate become almost equal. Note that the correction factor of Ashida & Michiue was developed for sediment mixtures finer than in Series 1 through 4.

The formulas of Hunziker/Meyer-Peter & Müller, Gladkow & Söhngen and the fractional formula of Van Rijn predict the lowest transport rates. Only at low Shields parameters predicts the formula of Van Rijn a larger transport rate than most

qs (m

3 /s/m

)

qs (m

3 /s/m

)



formulas. This is mainly transport of fine fractions due to the lack of a hiding/exposure correction in this formula. Hunziker’s formula starts predicting sediment transport at rather high Shields parameters. This is partly caused by its bed form factor, which is smaller than one and partly because the formula uses the equal mobility concept based on the Shields parameter of the geometric mean grain size (Dm). The geometric mean grain size (eq. 2.1) is larger than the median grain size (D50). This results in lower Shields parameters.

0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)


0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)


Figure 4.2: Series 4 (D50 = 10 mm, σg = 5): Sediment transport rate versus the Shields parameter for the fractional transport formulas.

Figure 4.2 shows the transport rate predictions for Series 4 (D50 = 10 mm, σg = 5). The two fractional formulas of Ackers & White show practically similar transport rate predictions. These predictions are relatively high compared with the other formulas. Wilcock & Crowe and Engelund & Hansen predict the highest transport rates. Parker remarkably gives fluctuating transport rate predictions around the Shields parameter of 0.1. The formula predicts small transport rates at low Shields parameters and large rates at high Shields parameters, relative to other formulas. The fractional formula of Van Rijn shows the opposite behaviour: large transport rates at low Shields parameters and small transport rates at high Shields parameters.

The formula of Meyer-Peter & Müller with Ashida & Michiue predicts no sediment transport at all for almost its whole validity range. The formula differs from Egiazaroff’s formula especially when sediment mixtures with a strong gradation are used. When the hiding/exposure factor of Egiazaroff is used the formula does not have problems with predicting sediment transport at low Shields parameters. For conditions with higher Shields parameters, the transport rate is somewhat larger for Ashida & Michiue than for Egiazaroff.

qs (m

3 /s/m

)

qs (m

3 /s/m

)



The formulas of Hunziker/Meyer-Peter & Müller and Gladkow & Söhngen predict the smallest transport rates. The formula of Hunziker/Meyer-Peter & Müller predicts no sediment transport until a Shields parameter of 0.19, which is partly caused by the fact that the geometric mean grain size (eq. 2.1) is much larger than the median grain size (D50). In the formula of Hunziker, the Shields parameter based is on Dm, which, in its turn, is much smaller than the Shields parameter based on D50. Together with a relatively small bed form factor, this causes incipient motion at high Shields parameters. After the critical Shields parameter is reached, the formula of Hunziker shows a large increase in the predicted transport rate, but remains the formula that predicts the smallest amount of sediment transport.

Although there are large differences between the formulas at high Shields parameters, up to a factor 10, the biggest differences occur at the low Shields parameter range. There are large differences in the transport rate and in predicting incipient motion. Some formulas only start predicting sediment transport at high Shields parameters, while other formulas predict sediment transport at all conditions.

4.2.2 Transport composition

The composition of the transported material is represented by the ratio between the geometric mean grain size of the transported material (DmT) and the geometric mean grain size of the bed material (DmB):

∑

∑⋅

⋅=

iii

iii

m

m

DBp

DTp

BDTD

piT probability of fraction i in transported material

piB probability of fraction i in bed material

In this section, again the results of Series 3 and 4 are shown to represent the behaviour of the several transport formulas under different conditions. The results of Series 1 and 2 are shown in Appendix 4. Only the fractional formulas of Ackers & White show sensitivity towards the grain size of the bed material. The DmT/DmB ratio becomes larger when the grain size of the bed material increases.



0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


Figure 4.3: Series 3 (D50 = 10 mm, σg = 2): Ratio of geometric mean grain size of transported material and geometric mean grain size of bed material versus the Shields parameters for the fractional transport formulas.

Most formulas show an asymptotic behaviour of the predicted composition of the transported material for Shields parameters of about 0.6 (Figure 4.3). Only Van Rijn still shows a coarsening transport composition at these Shields parameters.

The formulas of Hunziker/Meyer-Peter & Müller, Wu et al. and the two versions of Ackers & White predict a transport coarser than the bed material at high Shields parameters.

The formulas of Parker, Wilcock & Crowe, Gladkow & Söhngen and both formulas of Meyer-Peter & Müller predict a transport composition equal to the bed material composition. This means that the formulas predict equal mobility at conditions with high enough Shields parameters. There is no difference between the different formulas of Meyer-Peter & Müller.

The formulas of Engelund & Hansen and Van Rijn show the finest transport composition. The formula of Engelund & Hansen does not show differences in the transport composition at different hydraulic conditions. This is caused by the fact that there is no form of critical shear stress or hiding/exposure factor incorporated within the fractional formula. Van Rijn’s formula also predicts very fine sediment transport due to a lack of a hiding/exposure factor. Both formulas of Engelund & Hansen and Van Rijn were not officially derived for fractional sediment transport.

Dm

T/D

mB (-

)

Dm

T/D

mB (-

)



0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


Figure 4.4: Series 4 (D50 = 10 mm, σg = 5): Ratio of geometric mean grain size of transported material and geometric mean grain size of bed material versus the Shields parameters for the fractional transport formulas.

The series with the larger geometric standard deviation (Series 2 and 4, σg=5) show a larger variation in the predicted transport composition than Series 1 and 3 (see Appendix 4 for Series 1 and 2). The asymptotic behaviour for σg=2 (Series 1 and 3) is not shown for σg=5 (Series 2 and 4). The asymptotic behaviour occurs at higher Shields parameters since only then the coarsest fractions will start being transported.

Hunziker/Meyer-Peter & Müller predicts a composition of the transported material as coarse as or coarser than the bed material (Figure 4.4). This is caused by Hunziker’s assumption of equal mobility, combined with his hiding/exposure factor. The equal mobility concept enables the grain sizes much larger than Dm to be transported, while other formulas predict no transport of these large grains.

The formula of Ackers & White with Day predicts the second coarsest transport composition. The formula of Gladkow & Söhngen shows a fast coarsening for low Shields parameters and asymptotic behaviour for Shields parameters larger than 0.3. The formula of Parker shows strange behaviour again at Shields parameters around 0.1. The formulas of Engelund & Hansen and Van Rijn show the finest sediment transport. Engelund & Hansen shows again the same composition for all hydrological conditions.

The two hiding/exposure factors used for the Meyer-Peter & Müller (Egiazaroff and Ashida & Michiue) give different results for sediment mixtures with a large gradation. In these mixtures, the small fractions are sufficiently smaller than the mean grain size, so that the modification by Ashida & Michiue has effect. The formula with Egiazaroff shows a very fine transport at the lowest Shields parameters. This is caused by the fact that if the ratio Di/Dm is smaller than 1/(192) (see eq.

Dm

T/D

mB (-

)

Dm

T/D

mB (-

)



3.13), fraction i suffers “exposure” instead of “hiding”. This is caused by the asymptotic behaviour of the correction factor at grain sizes 19 times smaller than the mean grain size.

4.3 Uniform versus fractional

In this section, the transport rates predicted by the uniform formulas are compared with those by their fractional counterparts. The goal of this comparison is to determine for which hydraulic and sedimentological conditions a fractional formula shows different predictions compared to the uniform formula, and hence it would be preferred to use the fractional formula. Only the uniform formulas of Ackers & White and Meyer-Peter & Müller are compared with their fractional versions. The uniform formula of Parker is not taken into account in this report, as mentioned in Section 3.3. Therefore, a comparison cannot be made. Engelund & Hansen and Van Rijn did not derive a fractional version of their formulas. The fractional formulas of Engelund & Hansen and Van Rijn given in this report have not officially been tested and, as such, they are not taken into account in this section.

An analysis shows that the differences between uniform and fractional formulas are not sensitive to a change in the median grain size (D50) of the bed material, besides a higher transport rate for predictions with a larger median grain size. This is because the transport rate is shown as a function of the Shields parameter in which the grain size is incorporated. A variation in the geometric standard deviation in the bed material does show significant differences in the transport predictions. This is the reason why again only Series 3 and 4 are shown whereas, Series 1 and 2 are shown in Appendix 5. The results are shown as the ratio between the fractional formula and its uniform counterpart versus the Shields parameter.

Figure 4.5 shows the ratio of fractional transport rates and uniform transport rates versus the Shields parameter in Series 3 (σg = 2) for four selected transport formulas. The formula of Ackers & White with Day predicts a much smaller transport rate than the uniform formula at low Shields parameters, but the transport rate predictions approach each other at high Shields parameters. The ratio of Proffitt & Sutherland with the uniform formula increases and then somewhat decreases for Shields parameters up to 0.14. At higher Shields parameters, the formula of Proffitt & Sutherland predicts a transport rate, which is a factor 1.5 higher than the uniform formula of Ackers & White.

The uniform and fractional formulas of Meyer-Peter & Müller predict the same sediment transport rate at Shields parameters higher than 0.15. The uniform formula predicts a lower transport rate than the fractional formula at Shields parameters smaller than 0.15.

There are only large differences between the uniform formulas and the fractional formulas at the lower Shields parameter range. Most fractional formulas converge with their uniform counterparts at Shields parameters higher than 0.2.



Figure 4.5: Series 3 (D50 = 10 mm, σg = 2): Ratio of fractional transport rates with uniform transport rates versus the Shields parameters for four transport formulas.

Figure 4.6 shows the predicted transport rate ratios versus the Shields parameter in Series 4 for four combinations of uniform and fractional formulas. At low Shields parameters, the uniform formula of Ackers & White predicts a higher transport rate than the fractional formulas. At Shields parameters higher than 0.2 the different uniform and fractional formulas converge (ratio ≈ 1).

For Shields parameters smaller than 0.25, the uniform formula of Meyer-Peter & Müller predicts a much smaller transport rate than the fractional formulas. At higher Shields parameters, the uniform formula shows more agreement with the fractional formula of Ashida & Michiue than the one of Egiazaroff.

It can be concluded that the use of fractional formulas is especially important at low Shields parameters. The higher the gradation of a sediment mixture, the more a prediction of a fractional formula differs from its uniform counterpart.

Series 3

0.1

1.0

10.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Shields D50 (-)

qs_f

r/qs_

u (-)

Ackers & White + Day

Ackers & White + Proffitt & Sutherland

Meyer-Peter & Muller + Egiazaroff

Meyer-Peter & Muller + Ashida & Michiue



Figure 4.6: Series 4 (D50 = 10 mm, σg = 5): Ratio of fractional transport rates with uniform transport rates versus the Shields parameters for four transport formulas.

4.4 Focus on low Shields parameters

In lower Shields parameter ranges (0 - 0.2), the transport formulas show very different behaviour. The problem is that incipient motion is difficult to define and to predict. This is why different ways of modelling incipient motion have been developed, which causes the formulas to differ a lot from each other. In this section, a short analysis will be given concerning the differences between the uniform formulas of Ackers & White and Meyer-Peter & Müller. For the formula of Meyer-Peter & Müller, incipient motion occurs at high Shields parameters. Calculations with increasing Shields parameters (steps of 0.01) have been done to determine the ‘actual’ critical Shields parameter in different sediment mixtures. The ‘actual’ critical Shields parameter is the lowest Shields parameter based on the median grain size (D50) at which sediment is transported. Table 4.3 shows the three formulas and their ‘actual’ critical Shields parameters (based on D50) for the four Series.

Table 4.3: ‘Actual’ critical Shields parameter (D50) of different formulas in the four series. Name formula ‘Actual’ critical Shields parameter (-) Series 1 Series 2 Series 3 Series 4 Ackers & White 0.05 0.04 0.05 0.04 Meyer-Peter & Müller 0.10 0.19 0.10 0.19 Meyer-Peter & Müller (µµµµMPM = 1) 0.06 0.12 0.06 0.12

The uniform formula of Meyer-Peter & Müller has large critical Shields parameters, especially for Series 2 and 4 (σg = 5). An important reason for this is the fact that the bed form factor is already smaller than one at incipient motion. Theoretically, this is

Series 4

0.1

1.0

10.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Shields D50 (-)

qs_f

r/qs_

u (-)



Meyer-Peter & Muller + Egiazaroff

Meyer-Peter & Muller + Ashida & Michiue



not possible since at conditions of incipient motion no bed forms are formed, so the bed form factor should equal one. It can be that bed forms were formed during previous flow conditions and are still present. These situations cannot be predicted by a transport formula since these cannot deal with historical effects. A morphological model is needed if sediment transport predictions are needed in these conditions. The effect of a bed form factor below one can be seen by comparing the predictions of the original Meyer-Peter & Müller formula with the Meyer-Peter & Müller with a constant bed form factor of one (Figure 4.7 and 4.8).

The second reason why the ‘actual’ critical Shields parameter differs from 0.047 is that these transport formulas use a Shields parameter based on the geometric mean grain size (Dm) instead of the median grain size (D50), which is used in Table 4.3. Since in the behaviour calculations grain size of the bed material is assumed to be log-normal distributed, the Dm can be expressed in terms of D50 and σg:

).exp( 2yym 50D σµ += geometric mean grain size (m) (4.1)

In which: )ln( 50y D=µ

9950g

y .)ln(σ

σ =

Equation 4.1 can be written as: ).exp( 2

y50m 50DD σ⋅=

The geometric standard deviation (σg) cannot be smaller than one. This means that the geometric mean grain size (Dm) cannot be smaller than the median grain size (D50). It must be noted that due to splitting the log-normal distribution into fractions the geometric mean grain size according to (4.1) does not equal the geometric mean grain size according to (4.2). For more information see Appendix 2.

∑ ⋅=i

iim DpD geometric mean grain size (m) (4.2)

In Table 4.4 the ratio between Dm and D50 are given, where Dm is calculated with different formulas.

Table 4.4: Ratio between geometric mean grain size (Dm) and median grain size (D50) for different geometric standard deviations (σg) according to (4.1) and (4.2).

σg Dm/D50 based on Eq. (4.1)

Dm/D50 based on Eq. (4.2)

2 1.275 1.202 5 3.699 2.527

In this report, (4.2) is used to determine the geometric mean grain size (Dm). The uniform formula of Meyer-Peter & Müller uses Shields parameters based on the geometric mean grain size instead of the median grain size as used in the plots in this report. Since the Shields parameter is inversely proportional to the grain size,



the transport formulas show high Shields parameters for conditions of incipient motion.

The transport rates of three formulas: Ackers & White, Meyer-Peter & Müller and Meyer-Peter & Müller with bed form factor equal to one for Series 3 and 4 are shown in Figure 4.7 and 4.8 respectively. The effect of the bed form factor appears from the differences between the two different Meyer-Peter & Müller formulas. The differences caused by the difference in geometric standard deviation can be seen by comparing Figure 4.7 with Figure 4.8.

The transport rates of the Meyer-Peter & Müller formula with the bed form factor equal to one is only shown at its conditions of incipient motion, because after sediment is transported, the bed form factor will theoretically be smaller than one because bed forms will be formed.

Series 3

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Shields_D50 (-)

qs (m

2/s)

Ackers & White uniformMeyer-Peter & Muller uniformMeyer-Peter & Muller uniform (bed form factor = 1)

Figure 4.7: Series 3 (D50 = 10 mm, σg = 2): Sediment transport rate versus the Shields parameter for the three selected transport formulas at low Shields parameters.



Series 4

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Shields_D50 (-)

qs (m

2/s)

Ackers & White uniformMeyer-Peter & Muller uniformMeyer-Peter & Muller uniform (bed form factor = 1)

Figure 4.8: Series 4 (D50 = 10 mm, σg = 5): Sediment transport rate versus the Shields parameter for the three selected transport formulas at low Shields parameters.

The problems with modelling incipient motion also occur in the formula of Hunziker/Meyer-Peter & Müller. The bed form factor of this formula is also not equal to one at incipient motion. According to Hunziker [1995], the bed form factor of Yalin and Scheuerlein [1988] is closer to one at conditions of incipient motion. The use of Yalin and Scheuerlein leads to better predictions than when the bed form factor of Stickler is used, but even if the bed form factor is equal to one at conditions of incipient motion, the formula predicts incipient motion at high Shields parameters. This is an important problem in the formulas of Meyer-Peter & Müller and Hunziker/ Meyer-Peter & Müller at these conditions. The formula of Hunziker also uses the geometric mean grain size (Dm) for its Shields parameter, which leads to problems in predicting sediment transport with strongly graded sediment mixtures.

4.5 Conclusions


The transport formulas of Parker [1990] and Wilcock & Crowe [2001] and the fractional formula of Engelund & Hansen [1967] predict the highest transport rates. The formulas of Hunziker/Meyer-Peter & Müller [1995], Gladkow & Söhngen [2000] predict low transport rates. The formula of Hunziker predicts lower transport rates than the different formulas of Meyer-Peter & Müller (Egiazaroff [1965] and Ashida & Michiue [1973]). The formulas of Ackers & White [1973] with the hiding/exposure correction of Proffitt & Sutherland [1983] and of Van Rijn [1984a] show relatively high transport rates for low Shields parameters. Both formulas predict a slowly



increasing transport rate for higher Shields parameters. Unlike the formulas of Meyer-Peter & Müller, these show small sediment transport rates at low Shields parameters and a fast increasing transport rates for higher Shields parameters.

Defining and predicting incipient motion is difficult, which leaded to differences in modelling incipient motion between the formulas. That is why the differences between the formulas are larger for small Shields parameters (θ50 < 0.3). Another reason is the use of different hiding/exposure factors. The differences in the transport rate predictions become even larger when the geometric standard deviation of the bed material is larger. This is due to larger differences in incipient motion when dealing with a widely graded bed mixture instead of dealing with a less graded mixture.

The influence of the grain size on the behaviour of the different sediment transport formulas is small. The transport rate increases for calculations with a larger median grain size of the bed material. This is because the transport rate is shown as a function of the Shields parameter. The Shields parameter in itself is a function of the grain size.

Variations in the geometric standard deviation of the grain size result in different behaviour for the different transport formulas. Engelund & Hansen as well as Van Rijn show an increase in the transport rate when the standard deviation is increased, whereas the formulas of Parker and Hunziker/Meyer-Peter & Müller show a decrease in the transport rate. The transport rate predictions of the other formulas show little sensitivity concerning the geometric standard deviation of the bed material.

The largest differences between the uniform and the fractional versions of the formulas occur in the Shields parameter range up to 0.25. Most uniform and fractional formulas have converged at this point. The differences between the uniform and fractional formulas become larger when the geometric standard deviation of the bed material increases.

The bed form factors in the formulas of Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller are an important reason why these formulas show incipient motion at high Shields parameters. The problem is that the bed form factor is not equal to one at conditions of incipient motion. Another reason for incipient motion at high Shields parameters is the fact that the formulas use Shields parameters based on the geometric mean grain size (Dm, eq. 2.1) instead of median grain size (D50). The geometric mean grain size is larger than the median grain size when the geometric standard deviation of a mixture is larger (eq. 4.1).


All formulas show a transport composition finer than the bed composition (DmT < DmB) for small Shields parameters. This is caused by the fact that the critical Shields parameter of the larger grain sizes is not reached so that only the fine material is transported. The composition of the transported sediment coarsens gradually when the Shields parameter is increased. For some formulas the composition of the transported material approximates the composition of the bed for high Shields parameters. The formulas of Hunziker and Wu et al. show different behaviour: they predict a sediment transport coarser than the bed material at high Shields



parameters. This effect is strongest for Hunziker’s formula for sediment mixtures with a strong gradation. At which Shields parameter the formulas predict a transport composition coarser than the bed material depends on the geometric standard deviation of the material.

The transport composition predicted by the formulas differs most at low Shields parameters (θ50 < 0.3). This is caused by the differences in modelling incipient motion by the different formulas.

Both fractional Ackers & White formulas show sensitivity towards the grain size of the bed material. The DmT/DmB ratio becomes larger when the grain size of the bed material increases. Other formulas do not show changes in the ratio when this grain size is varied.

The predictions of the transport composition by all formulas are sensitive to changes in the geometric standard deviation of the bed material. All formulas predict a finer composition when the geometric standard deviation of the bed material increases. The asymptotic behaviour of the DmT/DmB ratio is also smaller in most cases. The formula of Hunziker/Meyer-Peter & Müller predicts a much coarser sediment transport when the Shields parameter exceeds 0.3.

Experimental data used for verification


5 Experimental data used for verification

5.1 Introduction

In this chapter, ten sets of experiments will be described. In Chapter 6 and 7, predictions of transport rate and transport composition will be verified with, respectively, the measured transport rate and transport composition of these experiments. The experimental data that are analysed in this chapter are:

• Blom and Kleinhans, [1999], Non-uniform sediment in morphological equilibrium situations - Data report Sand Flume experiments 97/98, Research report CiT 99R-002/MICS-001, Civil Engineering and Management, University of Twente. Experimental Series 1 was conducted in the Sand Flume Facility of WL | delft hydraulics in 1997/1998. Five tests were performed with a sediment mixture from the river Rhine.

• Blom, [2000], Flume experiments with a trimodal sediment mixture – Data report Sand Flume experiments 99/00, Research report CiT: 2000R-004/MICS-013, Civil Engineering & Management, University of Twente. Experimental Series 2 was conducted in the Sand Flume Facility of WL | delft hydraulics in 1999/2000. Four tests were performed with a tri-modal mixture.

• Klaassen, [1991], Experiments on the effect of gradation and vertical sorting on sediment transport phenomena in the dune phase, Proc. Grain Sorting Seminar, Ascona, Switzerland, pp 127-145. Six tests with graded sediment were conducted in the Sand Flume Facility of WL | delft hydraulics in 1990.

• Day, [1980], A study of the transport of graded sediments, HRS Wallingford, Report No. IT 190. HRS A, experiments conducted by Day. 11 tests with graded sediment were conducted in the flume of HR Wallingford. HRS B, experiments conducted by Day. Nine tests with graded sediment were conducted in the flume of HR Wallingford.

• Wilcock and McArdell, [1993], Surface-Based Fractional Transport Rates: Mobilization Thresholds and Partial Transport of a Sand-Gravel Sediment, Water Resources Research, Vol. 29, No. 4, pp. 1297-1312. Bed Of Many Colours (BOMC): Ten tests with a sand gravel mixture. The tests were carried out at Johns Hopkins University.

• Wilcock et al., [2001], Experimental Study of the Transport of Mixed Sand and Gravel, Water Resources Res., Vol. 37, No. 12, pp. 3349-3358. Four experimental series with different sand gravel mixtures were conducted at Johns Hopkins University. 37 runs are used for verification in this report.



5.2 Blom & Kleinhans

A set of flume experiments was carried out in the Sand Flume Facility of WL | delft hydraulics from November 1997 until January 1998. Blom & Kleinhans [1999] give a description of the experimental set-up, the measurements and the resulting data. Sediment was recirculated and uniform flow conditions were maintained in five tests until a morphological equilibrium was reached. Morphological equilibrium was defined as: water surface slope equal to the bed slope, constant bed roughness, constant sediment transport rate and composition, and constant and uniform bed form dimensions [Blom & Kleinhans, 1999]. Only the equilibrium conditions are analysed in the present study.

The Sand Flume of WL | delft hydraulics has a width of 1.5 m and a sediment recirculation system. The D50 of the initial bed mixture was 1.28 mm and σg was 4.28. The geometric mean grain size of the initial bed mixture is 1.68 mm. The sediment density is 2620 kg/m3. All tests were done in the dune phase, except in T1 where longitudinal stripes moved over an armour layer. The conditions measured in the equilibrium phases of the tests are shown in Table 5.1. The data are corrected for side wall roughness by the method of Vanoni and Brooks [1956]. The method of Vanoni and Brooks is given in Appendix 6.

Table 5.1: Equilibrium conditions in experimental series of Blom & Kleinhans Test Q h Rb Cb visc. θ50 qs DmT

m3/s m m m1/2/s *10-6 m2/s - m2/s mm T1 0.130 0.201 0.177 48.60 1.20 0.038 4.73 * 10-7 1.21 T5 0.254 0.245 0.224 37.18 1.20 0.166 3.15 * 10-5 1.23 T7 0.419 0.354 0.316 35.84 1.20 0.233 5.46 * 10-5 1.69 T9 0.272 0.260 0.240 34.06 1.20 0.202 3.77 * 10-5 1.52

T10 0.170 0.193 0.176 41.43 1.20 0.097 1.15 * 10-5 1.21

The sediment mixture was divided into 12 fractions. The size (Di) and the probability (pi) of these fractions are given in Table 5.2.

Table 5.2: Composition of the initial mixture in experimental series of Blom & Kleinhans fraction 1 2 3 4 5 6 7 8 9 10 11 12 Di (mm) 0.27 0.33 0.39 0.46 0.55 0.65 0.84 1.18 1.67 2.83 5.66 11.3 pi (%) 3 9 13 15 13 10 7 6 5 6 6 6

5.3 Blom

Blom [2000] conducted a second set of experiments in the Sand Flume facility of WL | delft hydraulics. The experimental series can be divided into two test series (A and B) of each two tests. In Test series A, the initial bed material consisted of a trimodal sediment mixture with a probability ratio of 1:1:1 of the three size fractions in the bed. The initial bed of test series B consisted of two layers: a coarse surface layer and a fine substratum. The surface layer of 3 cm had the same composition as the initial bed material of test series A and the substratum consisted of only the finest of the three fractions. Only the data that were gathered in the equilibrium phases of the



tests were used in this study. The tests were maintained until an equilibrium was reached in: average bed level, water level, sediment transport rate, sediment transport composition, bed form dimensions and a uniform patterns of bed forms [Blom, 2000] (Table 5.3). The width of the Sand Flume was reduced to 1 m. A sediment recirculation system was used.

Table 5.3: Equilibrium conditions in experimental series of Blom Test Q h Rb Cb visc. θ50 qs DmT

m3/s m m m1/2/s *10-6 m2/s - m2/s mm A1 0.098 0.154 0.140 37.8 1.20 0.082 1.13 * 10-5 2.25 A2 0.267 0.321 0.271 38.3 1.20 0.137 4.33 * 10-5 2.41 B1 0.098 0.155 0.140 38.7 1.21 0.077 1.17 * 10-5 2.06 B2 0.267 0.389 0.351 25.0 1.21 0.626 4.45 * 10-5 1.01

The initial bed mixture of test series A had a D50 of 2.1 mm and a σg of 2.93. The geometric mean grain size of the initial mixture was 2.83 mm. The properties of the sediment mixture used as surface layer in test series B were the same as those of the initial mixture in test series A. The substratum in test series B had a D50 of 0.68 mm and a σg of 1.26. The geometric mean grain size of the substratum was 0.71 mm (Table 5.4). The composition of the bed surface layer is also given in this table.

Table 5.4: Composition of the initial mixture and surface layer in experimental series of Blom fraction 1 2 3 Di (mm) 0.68 2.1 5.7 pi (%) initial material A1, A2 and B1 33.3 33.3 33.3 pi (%) initial material B2 80 10 10 pi (%) surface material A1 49 44 7 pi (%) surface material A2 36 38 26 pi (%) surface material B1 52 39 9 pi (%) surface material B2 89 6 5

5.4 Klaassen

Klaassen [1990] conducted experiments with graded sediment in the Sand Flume facility of WL | delft hydraulics. The experiments consisted of six tests. In this study, only the equilibrium periods of the tests are analysed.

The width of the Sand Flume was reduced to 1.125 m and a sediment recirculation system was used. The D50 of the initial bed mixture was 0.66 mm and σg was 2.34. The geometric mean grain size of the initial mixture was 0.93 mm. The sediment density was assumed to be 2650 kg/m3. All tests were done in the dune phase and no armour layers were formed. The conditions in the equilibrium periods of the tests are given in Table 5.5. The side wall roughness correction of Vanoni and Brooks [1956] has been applied to the data.

Table 5.5: Equilibrium conditions in experimental series of Klaassen Test Q h Rb Cb visc. θ50 qs DmT

m3/s m m m1/2/s *10-6 m2/s - m2/s mm T49 0.110 0.178 0.164 33.8 1.07 0.242 7.92 * 10-6 0.849 T50 0.240 0.337 0.305 28.6 1.06 0.451 1.59 * 10-5 0.710 T51 0.300 0.402 0.361 27.4 1.06 0.537 1.82 * 10-5 0.738



T52 0.241 0.349 0.317 27.2 1.06 0.469 1.36 * 10-5 0.861 T53 0.111 0.189 0.175 31.6 1.06 0.251 6.45 * 10-6 0.761 T54 0.050 0.091 0.085 41.9 1.05 0.125 2.80 * 10-6 0.709

The composition of the initial material has been used as input for the transport formulas. The initial bed material has been divided into 14 fractions (Table 5.6).

Table 5.6: Composition of the initial mixture in experimental series of Klaassen fraction 1 2 3 4 5 6 7 Di (mm) 0.07 0.1 0.15 0.19 0.25 0.36 0.52 pi (%) 0.15 0.11 0.54 2.7 8.5 18 16

fraction 8 9 10 11 12 13 14 Di (mm) 0.74 0.95 1.22 1.6 2.4 3.4 4.3 pi (%) 15 8.0 13 6.0 7.5 3.4 1.1

5.5 Day: HRS A

Day [1980] conducted his experiments in a 2.46 m wide tilting flume with a sediment recirculation system. These experiments were conducted at the HR Wallingford and consisted of 11 tests. After each test, the top few centimetres of the bed were replaced by initial bed material and the bed was levelled. The standard deviations of the measured parameters were sufficiently small, so that it can be assumed that the tests were in equilibrium. No data is available concerning the presence of bed forms.

The experiments were aimed at studying incipient motion of different grain sizes, so the tests were done in low Shields parameter ranges. The D50 of the initial mixture was 1.75 mm and σg was 4.21. The geometric mean grain size of the mixture was 2.72 mm. Day did not give a sediment density, and we assume it to be 2650 kg/m3. The conditions in the tests are given in Table 5.7. No side wall roughness correction was needed for these experiments due to the fact that the width/depth ratio in these experiments was larger than five.

Table 5.7: Equilibrium conditions in experimental series of Day: HRS A Test Q h Rb Cb visc. θ50 qs DmT

m3/s m m m1/2/s * 10-6 m2/s - m2/s mm 1 0.199 0.166 0.146 48.9 1.24 0.034 4.36 * 10-7 0.575 2 0.199 0.159 0.141 48.9 1.24 0.038 5.91 * 10-7 0.586 3 0.199 0.145 0.130 55.1 1.24 0.036 5.43 * 10-7 0.763 4 0.197 0.169 0.149 47.9 1.24 0.034 2.68 * 10-7 0.740 5 0.196 0.147 0.131 50.1 1.24 0.040 7.42 * 10-7 0.927 6 0.197 0.133 0.120 52.6 1.24 0.045 9.07 * 10-7 1.128 7 0.197 0.123 0.112 47.2 1.24 0.066 2.95 * 10-6 1.622 8 0.197 0.131 0.118 45.4 1.24 0.063 2.52 * 10-6 1.659 9 0.196 0.121 0.110 39.8 1.24 0.095 4.71 * 10-6 2.083

10 0.198 0.112 0.103 41.7 1.24 0.103 1.32 * 10-5 2.374 11 0.193 0.107 0.098 38.6 1.24 0.124 2.47 * 10-5 2.626



The composition of the active layer was not determined. Instead, the initial bed composition was used as input for the bed material for the sediment transport formulas. Day divided the initial mixture into 19 fractions (Table 5.8).

Table 5.8: Composition of the initial mixture in experimental series of Day: HRS A fraction 1 2 3 4 5 6 7 8 9 10 Di (mm) 0.15 0.21 0.3 0.39 0.46 0.55 0.65 0.78 0.93 1.2 pi (%) 1 4 11 6 6 4 3 3 2 5

fraction 11 12 13 14 15 16 17 18 19 Di (mm) 1.55 2.03 2.86 4.06 5.56 7.18 8.73 11.1 14.2 pi (%) 3 7 10 10 11 7 3 2 0

5.6 Day: HRS B

Day [1980] conducted a second set of experiments (HRS B), which were performed in the same flume facility as series HRS A. Experiments HRS B consisted of nine tests. After each test, the top few centimetres of the bed were replaced and the bed was levelled, like in series HRS A. The conditions at the end of the tests are again assumed to be in equilibrium.

The experiments were aimed at studying incipient motion. In HRS B the initial bed mixture was finer and less graded than the mixture used in HRS A. The D50 of the sediment mixture was 1.55 mm and σg was 3.5. The geometric mean grain size was 1.74 mm. The density of the mixture was again assumed to be 2650 kg/m3. The conditions of the nine tests are given in Table 5.9.

Table 5.9: Equilibrium conditions in experimental series of Day: HRS A Test Q h Rb Cb visc. θ50 qs DmT

m3/s m m m1/2/s *10-6 m2/s - m2/s mm 1 0.203 0.189 0.164 51.1 1.24 0.029 5.04 * 10-8 1.34 2 0.210 0.184 0.160 54.9 1.24 0.028 9.60 * 10-8 1.08 3 0.202 0.162 0.143 48.2 1.22 0.043 6.08 * 10-7 1.17 4 0.202 0.154 0.137 35.9 1.24 0.087 9.53 * 10-7 1.20 5 0.200 0.145 0.130 37.0 1.24 0.090 2.31 * 10-6 1.41 6 0.202 0.119 0.109 44.0 1.24 0.096 2.32 * 10-5 1.62 7 0.204 0.124 0.113 42.6 1.24 0.097 2.11 * 10-5 1.77 8 0.200 0.117 0.107 41.6 1.21 0.109 2.56 * 10-5 1.81 9 0.201 0.115 0.105 40.1 1.24 0.123 3.36 * 10-5 1.83

Too little data is available to determine the composition of the active layer for the tests. Instead, the composition of the initial bed material is used for the sediment transport predictions. Day divided the mixture into 16 fractions (Table 5.10).

Table 5.10: Composition of the initial mixture in experimental series of Day: HRS B fraction 1 2 3 4 5 6 7 8 Di (mm) 0.15 0.21 0.28 0.33 0.39 0.46 0.55 0.65 pi (%) 2 5 6 6 5 4 3 2 fraction 9 10 11 12 13 14 15 16 Di (mm) 0.78 0.93 1.2 1.55 2.03 2.86 4.06 5.56 pi (%) 3 3 7 5 17 14 15 1



5.7 Wilcock & McArdell and Wilcock et al.

Wilcock & McArdell [1993] and Wilcock et al. [2001] conducted five sets of experiments with graded sediments. The 48 tests were done in a tilting flume with different recirculating systems for water and sediment. Gravel coarser than 16 mm was recirculated manually. The flume had a width of 60 cm and a working length of 7.9 m. In this report the data from Wilcock et al. has been used for verification purposes. For each of the five sets of experiments a different sediment mixture of sand and gravel has been used. The mixtures in the five sets of experiments (J06, J14, J21, J27 and BOMC) contained 6, 15, 21, 27 and 34 percent sand, respectively. For the five mixtures, the same gravel has been used. Only the sand in the series with the 34% sand mixture had a different composition than the sand of the other series. The bed was remixed and screeded after two or three tests. In case the bed was not remixed the test was followed by a test with a higher transport rate, so that “history” effects of the bed were negligible. After each test the bed surface composition was determined. Appendix 7 lists the hydraulic and sedimentology data.

Verification of transport rate


6 Verification of transport rate

6.1 Introduction

The selected sediment transport formulas have been verified by comparing predicted transport rates to measured transport rates. Chapter 5 describes the ten sets of experiments that are used for this verification. In this chapter, the composition of the initial bed material was used as input for the sediment transport predictions, since for most experiments only the composition of the initial mixture was known. However, Chapter 8 does consider transport predictions based on the composition of the bed surface layer. In Section 6.2, we consider the variations in the measured transport rates. In Section 6.3, a classification method is given together with the score table of all the transport formulas. In the Sections 6.4 to 6.13, the results of the formulas are discussed separately. In Section 6.14, the conclusions concerning the transport rate predictions are given. Chapter 7 considers the verification of the transport formulas by comparing the predicted transport composition with the measured one.

6.2 Measured transport rate

In this section the measured transport is discussed in relation to the Shields parameter, to show the variation in the measurements. Figure 6.1 shows the measured transport rates as a function of the Shields parameter. The relation qsmeas = a θ503/2 is shown as well. This power relation is used by Meyer-Peter & Müller and other researchers. The experiments of Blom & Kleinhans, Klaassen, and Blom (except test B2) show good agreement with this relation. Tests performed at lower Shields parameters show a relation with a larger power. This power is higher if the initial bed material is coarser. E.g., the tests of J06 show agreement with a power of ca. 8, the tests of BOMC with a power of ca. 3.



10−2

10−1

100

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

Measured transport rate

θ50

(−)

qsm

eas (

m3 /s

/m) ← θ

503/2

Blom & Kleinhans Blom Klaassen Day: HRS A Day: HRS B Wilcock et al: J06 Wilcock et al: J14 Wilcock et al: J21 Wilcock et al: J27 Wilcock & McArdell: BOMC

Figure 6.1: Measured transport rates versus the Shields parameters for the experiments.

6.3 Ranking of the transport formulas

A classification method was developed to be able to compare the performance of the different formulas. The classification of the formulas is based on the ratio between the predicted sediment transport and the measured sediment transport. For the classification of the transport rate predictions, the ratio is defined as:

)()(

)()( ,

measuredqpredictedq

measuredqpredictedq

ratios

is

s

s ∑==

In total, 82 tests were used for the verification. Table 6.1 is the score table of the predictions for the transport rate. The score is calculated with:

n

jfactorScore

n

1j∑==

)(

>≤

= 1jratioifjratio1

1jratioifjratiojfactor )()(

)()()(

in which n is the total number of tests, and j the specific test. The maximum score is one.

Table 6.1 gives the scores and the rankings of the different transport formulas. In the following sections, the results of each transport formula will be explained in detail.

Table 6.1: Score table for the predictions of the transport rate. Name Score Ranking Ackers & White uniform 0.34 5 Ackers & White + Day 0.36 3



Ackers & White + Proffitt & Sutherland 0.32 7 Parker 0.21 13 Engelund & Hansen uniform 0.37 2 Engelund & Hansen fractional 0.33 6 Meyer-Peter & Müller 0.24 10 Meyer-Peter & Müller + Egiazaroff 0.27 8 Meyer-Peter & Müller + Ashida & Michiue 0.26 9 Van Rijn uniform 0.22 12 Van Rijn fractional 0.17 15 Hunziker/Meyer-Peter & Müller 0.22 11 Gladkow & Söhngen 0.11 17 Wu et al. 0.41 1 Wilcock & Crowe 0.36 3 Ribberink uniform 0.14 16 Ribberink + Ashida & Michiue 0.18 14

6.4 Ackers & White

6.4.1 Ackers & White

The uniform formula of Ackers & White is the 5th best formula with a score of 0.34. Figure 6.2 shows the transport rate ratio versus the Shields parameter of the tests used for verification. The ratios with values larger than 1000 or smaller than 0.001, are shown with values of, respectively, 1000 or 0.001. In general, the uniform formula of Ackers & White has the tendency to overestimate the sediment transport rate. Particularly at low Shields parameter ranges, the overestimation can be large. Underestimations are only significant in the experiments of Blom and Wilcock & McArdell. Especially the latter shows large underestimations at the lowest Shields parameters. The experiments of Wilcock et al. with the coarsest sediment (J06, J14 and J21) show large overestimations at the lower Shields parameters. The difference between the measured and predicted transport rate becomes smaller with increasing Shields parameters. At Shields parameters higher than 0.1, the predicted and measured transport rates are of the same order of magnitude. The formula predicts the transport rates for the experiments of Blom & Kleinhans nearly perfectly. Bad results are there for Day HRS A. The results for the other experimental series are average.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Ackers & White uniform

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.34


Figure 6.2: Transport rate ratio versus the Shields parameter for the uniform formula of Ackers & White.

6.4.2 Ackers & White with Day

The fractional formula of Ackers & White with the hiding/exposure correction of Day shows the 3rd best results of all formulas (Figure 6.3). It has a score of 0.36.

The fractional formula of Ackers & White with Day has a large scatter at low Shields parameters, for which it has a strong tendency to overestimate the transport rate. The formula overestimates for relatively coarse sediment mixtures (J06, J14 and J21) and underestimates when fine material is used (HRS B, BOMC). The predictions show more stable and better results at higher Shields parameters. The formula shows good results for most experimental series when the Shields parameter is higher than 0.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103


θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.36


Figure 6.3: Transport rate ratio versus the Shields parameter for the fractional formula of Ackers & White with Day.



6.4.3 Ackers & White with Proffitt & Sutherland

With a score of 0.32 the fractional formula of Ackers & White with the hiding/exposure factor of Proffitt & Sutherland is the 7th best formula (Figure 6.4).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103


θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.32


Figure 6.4: Transport rate ratio versus the Shields parameter for the fractional formula of Ackers & White with Proffitt & Sutherland.

This formula has problems predicting the transport rate at Shields parameters lower than 0.05 where it underestimates very strongly. This especially occurs for the BOMC experiments. The transport rate is largely overestimated at low Shields parameters for experiments J06 and J14. At higher Shields parameters, the predictions become much better. Only for the experiments of Klaassen a systematic overestimation is shown. The formula is not valid for fine sediment mixtures as used by Klaassen, which is confirmed by the relatively bad predictions.

6.5 Parker

With a total score of 0.21, Parker’s surface based formula has a 13th ranking for the transport rate predictions (Figure 6.5). Parker’s formula shows a systematic underestimation of the transport rate for low Shields parameters and a systematic overestimation for high Shields parameters. Only the transport rates of the coarse material (J06 and J14) are overestimated at low Shields parameters. There are a few good predictions in the Shields parameter range 0.08 - 0.23. This is similar to the Shields validity range of the formula (0.06 - 0.16). It must be noted that the formula is not valid for any of the experimental series used for verification because the sediment mixtures of the verification data were too fine. The formula shows good results for the experiments of Blom & Kleinhans only. The results for the BOMC experiments are very poor. It should be noted that the formula of Parker is based on the surface layer composition but the initial bed composition is used here as input of the formula due to lack of surface layer composition data.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Parker surface

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.21


Figure 6.5: Transport rate ratio versus the Shields parameter for the fractional formula of Parker.


6.6.1 Engelund & Hansen uniform

The predictions of the uniform Engelund & Hansen formula are the 2nd best of all formulas (Figure 6.6). It has a score of 0.37. Engelund & Hansen’s uniform formula overestimates the transport rate in many cases. This is due to the lack of a critical shear stress, which results in high predicted transport rates at low Shields parameters, while the measured transport rates are much lower. The formula shows some underestimations, most of them in the Shields parameter range 0.09 - 0.23. The overestimations at low Shields parameters are high, especially for the experiments of Wilcock et al. with the coarse sediment (J06, J14, and J21).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Engelund & Hansen uniform

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.37


Figure 6.6: Transport rate ratio versus the Shields parameter for the uniform formula of Engelund & Hansen.

6.6.2 Engelund & Hansen fractional

It is noted again that the fractional formula has not been officially developed and tested. Figure 6.7 shows the transport rate ratios of the fractional formula of Engelund & Hansen. The score of the formula is 0.33, which gives a 6th ranking.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Engelund & Hansen fractional

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.33


Figure 6.7: Transport rate ratio versus the Shields parameter for the fractional formula of Engelund & Hansen.

The transport rates predicted by the fractional formula are larger than by the uniform formula due to the lack of a critical shear stress and a hiding/exposure factor. This leads to a relatively higher transport rates of smaller fractions, thus the total transport rate of the fractional formula is higher than the one of the uniform



formula. The fractional formula overestimates the transport in almost 75% of the cases. Only the predictions for the experiments by Blom & Kleinhans are good. The transport rates of the experiments of Wilcock et al. are again strongly overestimated, especially at the lower Shields parameter range. If the sediment becomes coarser, the overestimation becomes larger.


6.7.1 Meyer-Peter & Müller uniform

The uniform formula of Meyer-Peter & Müller gives the 10th best predictions for the transport rate with a total score of 0.24 (Figure 6.8). This formula has a lot of problems with predicting the sediment transport rate at Shields parameters smaller than 0.1. In these conditions, the formula predicts no sediment transport at all. In Chapter 4, we found that the bed form factors were too small at conditions of incipient motion, which results here in problems with predicting sediment transport at the same conditions. The formula shows rather good results when the Shields parameter exceeds 0.1. In this Shields parameter range, most predictions deviate little from the measured transport rates.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Meyer−Peter & Müller uniform

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.24


Figure 6.8: Transport rate ratio versus the Shields parameter for the uniform formula of Meyer-Peter & Müller.

6.7.2 Meyer-Peter & Müller with Egiazaroff

With a score of 0.27, the formula is the 8th best formula of all formulas; this formula gives the best predictions of the Meyer-Peter & Müller formulas (Figure 6.9). Like the uniform formula of Meyer-Peter & Müller, this formula has the tendency to predict no sediment transport at Shields parameters smaller than 0.1, although the validity range for the Shields parameter for this formula is 0.04 - 0.9. This is due to the same problems with the bed form factor as the uniform formula. The predictions for



the experiments of Wilcock et al. differ much between each other. For the experiments with coarse material there is a rather high overestimation, while there is a strong underestimation for the experiments with fine sediment. The formula is not valid for the experimental series of Blom & Kleinhans, test B2 by Blom, and Klaassen, because the sediment mixtures were too fine. Yet, the transport rate predictions for these experiments are rather good.

Analysing the transport rate predictions show that that the asymptotic behaviour of the hiding/exposure factor (Section 4.2.2) becomes important when dealing with strongly graded mixtures. Fractions more than 19 times smaller than the geometric mean grain size have a smaller hiding coefficient, while fractions of ca. 19 times smaller than the geometric grain size experience a very strong hiding coefficient. This is also the reason why transport rates are overestimated in some tests of J06 and J14 with a low Shields parameter. In these tests only the smallest fraction is transported.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Meyer−Peter & Müller + Egiazaroff

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.27


Figure 6.9: Transport rate ratio versus the Shields parameter for the fractional formula of Meyer-Peter & Müller with Egiazaroff.

6.7.3 Meyer-Peter & Müller with Ashida & Michiue

The fractional transport formula of Meyer-Peter & Müller with the hiding/exposure correction of Ashida & Michiue gives the 9th transport rate predictions of all formulas with a score of 0.26 (Figure 6.10). It is slightly worse than Meyer-Peter & Müller with Egiazaroff. The fractional formula with Ashida & Michiue has the same tendency as the other formulas of Meyer-Peter & Müller to predict no sediment transport at Shields parameters below 0.1. At higher Shields parameters, the formula shows good results. The formula has the tendency to underestimate the transport rate except for the experiments of Wilcock et al. with relatively coarse material: J06, J14 and J21. The formula is not valid for the experiments of Klaassen, because the sediment mixture was too fine, but the formula’s results can be considered very good for these experiments.



The correction made by Ashida & Michiue on the hiding/exposure of Egiazaroff has lead to slightly worse verification results. This is due to the fact that the correction eliminates the asymptotic effect for fractions 19 times smaller than the geometric grain size. This also means that the smallest fraction is not transported in all the tests of J06 and J14. It can be concluded that although the correction by Ashida & Michiue has a better theoretic basis, the original hiding/exposure factor of Egiazaroff gives better transport rate results due to its asymptotic behaviour.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Meyer−Peter & Müller + Ashida & Michiue

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.26


Figure 6.10: Transport rate ratio versus the Shields parameter for the fractional formula of Meyer-Peter & Müller with Ashida & Michiue.

6.8 Van Rijn

6.8.1 Van Rijn uniform

The uniform formula has a score of 0.22, which means a 12th ranking (Figure 6.11). The uniform formula of Van Rijn overestimates the transport rate for all tests except for the experiments of Blom and some tests of Wilcock with very low Shields parameters (<0.03). The transport rates are strongly overestimated at low Shields parameters (0.03 – 0.15). The formula shows reasonable results at Shields parameters higher than 0.15. The formula seems to overestimate the transport rate stronger if the median grain size increases.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Van Rijn uniform

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.22


Figure 6.11: Transport rate ratio versus the Shields parameter for the uniform formula of Van Rijn.

6.8.2 Van Rijn fractional

The fractional formula of Van Rijn has not been officially published and tested. It is noted that Kleinhans & Van Rijn [2001] have developed a fractional bed load formula, however, it could not be taken into account within the present study. The results of the fractional formula of Van Rijn are shown in Figure 6.12 (score of 0.17, 15th ranking). The predicted transport rates by the fractional formula of Van Rijn are even larger than the ones of the uniform formula. This is caused by the lack of a hiding/exposure correction, which could have decreased the large transport of the fine fractions. The transport rate predictions are especially high at low Shields parameters. Many experiments have an overestimation of more than 1000 times the measured transport rate. The formula shows good results for the experiments of Blom and Klaassen. The results for the other series are very poor.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Van Rijn fractional

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.17


Figure 6.12: Transport rate ratio versus the Shields parameter for the fractional formula of Van Rijn


The formula of Hunziker/Meyer-Peter & Müller has a score of 0.22, which means the 11th best formula (Figure 6.14). This formula has the same problems with predicting sediment transport at low Shields parameters as the formulas of Meyer-Peter & Müller and in most cases shows the same underestimation of the transport rate. The formula predicts no sediment transport at Shields parameters below 0.1, where the formula is sensitive. The two runs of Blom (A1 and B1) show this. These tests had basically the same conditions, but the predicted transport rates vary strongly. At conditions with Shields parameters higher than 0.1, the predictions are reasonably well. The formula gives good results for the experiments of Klaassen, although it is not valid for the median grain size used in these experiments. Rather good results are there for Blom & Kleinhans as well, but the formula shows poor results for the other experimental series.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Hunziker/Meyer−Peter & Müller

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.22


Figure 6.14: Transport rate ratio versus the Shields parameter for the fractional formula of Hunziker/Meyer-Peter & Müller.


The formula of Gladkow & Söhngen gives the worst predictions of all formulas. The formula has a score of 0.11 and Figure 6.15 shows the transport ratios versus the Shields parameters.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Gladkow & Söhngen

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.11


Figure 6.15: Transport rate ratio versus the Shields parameter for the formula of Gladkow & Söhngen.

The formula has a strong tendency to underestimate the sediment transport. This formula often predicts no sediment transport at conditions with Shields parameters smaller than 0.1. At higher Shields parameters, the formula underestimates the transport rate as well. Only for a couple of runs of Wilcock et al., the transport rate is



overestimated. The formula does not give good predictions for any of the experimental series. The transport rate predictions agree better with the measurements if the sediment mixture is coarser (J06 and J14).

6.11 Wu et al.

The formula of Wu et al. is the best formula with a score of 0.41. Figure 6.16 shows the transport ratios versus the Shields parameters. The predictions of the formula of Wu et al. are reasonably good. It shows a large overestimation at low Shields parameters, but still gives the best results in this Shields parameter range. This overestimation is the largest for experiments with coarse material, i.e. Wilcock et al: J06 and J14. The conditions of all tests are within the validity ranges of the formula, which may be the reason for the good results of the formula. At higher Shields parameters, the formula slightly underestimates the transport rates.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Wu et al.

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.41


Figure 6.16: Transport rate ratio versus the Shields parameter for the formula of Wu et al.


The formula of Wilcock & Crowe gives the third best predictions (Figure 6.17). It has a score of 0.36. The formula of Wilcock & Crowe shows good results. However, this is not surprising since 57% of the data used for the verification was also used by Wilcock & Crowe to calibrate the formula, although they used the surface layer as active layer and in this section the initial bed composition is used. Nevertheless, this formula shows a large variation at low Shields parameters. The transport rates of the runs of BOMC (fine material) are strongly underestimated, while the rates of the experiments J06 and J14 (coarse material) are strongly overestimated. At higher Shields parameters, the formula overestimates the sediment transport rates.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Wilcock & Crowe

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.36


Figure 6.17: Transport rate ratio versus the Shields parameter for the formula of Wilcock & Crowe.

6.13 Ribberink

6.13.1 Ribberink uniform

The uniform formula of Ribberink gives the second worst predictions of all formulas. The formula has a score of 0.14. In Figure 6.18 the results of this formula are shown.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Ribberink uniform

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.14


Figure 6.18: Transport rate ratio versus the Shields parameter for the uniform formula of Ribberink.

The formula strongly overestimates the sediment transport rates. For some runs the formula predicts no transport, but this occurs less frequently than with the other Meyer-Peter & Müller type formulas. At higher Shields parameters, the predictions



are good. At low Shields parameters, the transport rates for the experiments of Wilcock are strongly overestimated. The overestimation is greater when the sediment is coarser.

6.13.2 Ribberink with Ashida & Michiue

The formula of Ribberink with Ashida & Michiue is the 14th best formula with a score of 0.18. Figure 6.19 shows the transport ratios versus the Shields parameters.

This formula also strongly overestimates the transport rate, although less than its uniform counterpart. The formula shows good predictions at higher Shields parameters and strong overestimations at the low Shields parameters (< 0.2).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Ribberink + Ashida & Michiue

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.18


Figure 6.19: Transport rate ratio versus the Shields parameter for the fractional formula of Ribberink with Ashida & Michiue.

6.14 Conclusions

The best formula for predicting the sediment transport rate is the fractional formula of Wu et al. The validity ranges indicate that the formula can be used in a wide range of conditions. However, the formula should be used cautiously at low Shields parameters, since it has the tendency to strongly overestimate the transport rate.

The formulas of Wilcock & Crowe and Ackers & White with the hiding/exposure correction of Day are good alternatives, but also with these formulas caution is required for low Shields parameters.

The sediment transport rate is extremely difficult to predict at Shields parameters up to 0.1 with the selected formulas. Some formulas show a strong overestimation of the transport rate, others predict no sediment transport at all, while again other formulas show both strong overestimations as underestimations of the transport rate. In this study, many tests with low Shields parameters have been used for verification. We deliberately put strong emphasis on these conditions because of the known problems with predicting the sediment transport in this range.



In this range, the formulas of Meyer-Peter & Müller, Hunziker/Meyer-Peter & Müller and Gladkow & Söhngen often predict no sediment transport. In the Meyer-Peter & Müller type of formulas, this is because the applied bed form factors are too small at conditions with low Shields parameters. But the formulas of Ribberink, which can be seen as a type of Meyer-Peter & Müller formula, show a strong overestimation. This shows that Meyer-Peter & Müller type of formulas are very sensitive at conditions with low Shields parameters and are thus recommended not to be used at these conditions.

Other formulas, like Ackers & White with Day, Engelund & Hansen, Van Rijn and Wu et al. strongly overestimates the transport rate at low Shields parameters. This overestimation can be more than a factor 1000.

Although only the formulas of Ackers & White with Day and Wu et al. are valid for the experiments of Klaassen also the formulas of Meyer-Peter & Müller and of Hunziker/Meyer-Peter & Müller give good results for these experiments. This means that the formulas of Meyer-Peter & Müller and of Hunziker/Meyer-Peter & Müller are suitable for transport rate predictions with finer sediment mixtures and higher Shields parameters.

The fractional formulas of Ackers & White with Day, Meyer-Peter & Müller with Egiazaroff, Meyer-Peter & Müller with Ashida & Michiue and Ribberink with Ashida & Michiue give better predictions than their uniform counterparts.

The fractional formulas of Engelund & Hansen and Van Rijn show worse results than their uniform formulas. A hiding/exposure factor could solve some of the problems of the fractional formulas, since these show a relatively large transport of small fractions.

Verification of transport composition


7 Verification of transport composition

7.1 Introduction

In this chapter, the fractional formulas are verified by comparing predicted transport compositions with the measured ones in the experiments discussed in Chapter 5. In this comparison, like in Chapter 6, the predictions are done using the composition of the initial bed mixture as input for the formulas. In Chapter 8, the predicted composition using the bed surface layer as input for the formulas will be discussed. The composition is represented by its geometric mean grain size. Section 7.2 shows the measured transport composition related to the initial material composition. A classification method with the score table of the fractional formulas for the composition predictions is given in Section 7.3. In Sections 7.4 to 7.13, the results of the formulas will be discussed separately. The conclusions are given in Section 7.14.

7.2 Measured transport composition

Figure 7.1 shows the ratio of the geometric mean grain sizes of the measured transport composition and the composition of the initial bed mixture as a function of the Shields parameter. It can be seen that at low Shields parameters (up to 0.1) the transport composition is much finer than the initial bed material, which we call selective transport. At higher Shields parameters, the transport composition approaches the initial material composition. The transported sediment does not become systematically coarser than the initial bed material, not even at high Shields parameters.

10−2

10−1

100

10−1

100

Measured DmT

/DmB

θ50

(−)

Dm

T/D

mB (

−)


Figure 7.1: Ratio of measured geometric mean grain size of transported material and geometric mean grain size of the initial bed mixture versus the Shields parameters.



It must be noted that the initial mixture composition is not the best definition for the active layer. But due to lack of active layer definitions and measurements in some experimental series, the composition of the initial bed mixture is used. Chapter 8 will address this problem.

7.3 Ranking of the transport formulas

The ratio between the predicted geometric mean grain size and the measured geometric mean grain size will be used for this verification. This ratio is calculated as follows:

∑

∑

⋅

⋅

==

iii

ii

totals

is

measmT

predmT

measuredDp

predictedDq

q

DD

ratio))((

)(,

,

,

,

In which: DmT geometric mean grain size (m)

qs,i sediment transport rate for fraction i (m2/s)

∑=i

istotals qq ,, total sediment transport rate (m2/s)

pi measured probability of fraction i in sediment transport (-)

Di grain size of fraction i (m)

The scores of the formulas for the composition predictions are calculated in a similar way as described in Section 6.3. Table 7.1 lists the scores for the verification of the composition of the transported material and the ranking of all the fractional formulas.

Table 7.1: Score table for the transport composition of all fractional formulas Name Score Ranking Ackers & White + Day 0.59 4 Ackers & White + Proffitt & Sutherland 0.54 6 Parker 0.58 5 Engelund & Hansen fractional 0.54 6 Meyer-Peter & Müller + Egiazaroff 0.34 11 Meyer-Peter & Müller + Ashida & Michiue 0.27 12 Van Rijn fractional 0.45 9 Hunziker/Meyer-Peter & Müller fractional 0.35 10 Gladkow & Söhngen 0.53 8 Wu et al. 0.79 1 Wilcock & Crowe 0.67 2 Ribberink + Ashida & Michiue 0.62 3



7.4 Ackers & White

7.4.1 Ackers & White with Day

The fractional formula of Ackers & White with the hiding/exposure factor of Day shows the 4th best results concerning the transport composition with a score of 0.59 (Figure 7.2). The ratios shown in these figures are within the range 0.1 – 10. If the ratio is smaller than 0.1 (0 in case of no sediment transport), or if the ratio exceeds 10, it is set on 0.1 or 10, respectively.

At low Shields parameters, the predicted sediment transport by this formula is often coarser than measured. In this range, there is a large variation in the predicted composition. This is especially the case in the BOMC experiments. At high Shields parameters, the variation is smaller and the predicted compositions approximate the measured composition better, although the predicted geometric mean grain size is finer than the one measured. The formula shows reasonable results for most experimental series, and has the largest problems with tests BOMC and J27.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101


θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.59


Figure 7.2: Transport composition ratio versus the Shields parameter for the fractional formula of Ackers & White with Day.

7.4.2 Ackers & White with Proffitt & Sutherland

The fractional formula of Ackers & White with Proffitt & Sutherland (score = 0.54) shows the 6th result of all fractional formulas (Figure 7.3). The results are worse than the formula of Ackers & White with the hiding/exposure of Day.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101


θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.54


Figure 7.3: Transport composition ratio versus the Shields parameter for the fractional formula of Ackers & White with Proffitt & Sutherland.

For about half of the tests the fractional formula of Proffitt & Sutherland overestimates the geometric mean grain size of the transported material, provided that the sediment transport is not equal to zero. The formula especially overestimates the grain size for Shields parameters smaller than 0.1. In this range, the formula shows large variations in the predicted transport composition. In many tests, the formula does not predict any sediment transport, so the predicted geometric mean grain size is automatically zero (in figure set on 0.1). At higher Shields parameters, the formula underestimates the geometric mean grain size, and the variations become smaller. The formula shows reasonable results for the experiments of Klaassen, although the formula is not valid for such a fine sediment mixture. The predictions for BOMC and Wilcock et al. are very poor.

7.5 Parker

The geometric mean grain size of the transported material, predicted by the surface based formula of Parker, finds good agreement with the measurements. With a score of 0.58 the formula is the 5th best formula (Figure 7.4).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Parker surface

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.58


Figure 7.4: Transport composition ratio versus the Shields parameter for the fractional formula of Parker.

The formula of Parker underestimates the geometric mean transported grain size for Shields parameters smaller than 0.2. At Shields parameters larger than 0.2, the predicted transported mean grain size is similar to the measured one. This behaviour is in concurrence with the behaviour of the formula for strongly graded sediments (Figure 4.4). Although, due to their fine sediment mixtures, the formula of Parker is not valid for the experiments by Blom & Kleinhans, Blom, and Klaassen, the predictions of the geometric mean grain size are good for these experiments. The results for the experimental series Day HRS A and HRS B are average. The formula has problems predicting the composition for the experiments of Wilcock et al. The BOMC experiments are predicted well at very low Shields parameters, but the results are poor when the Shields parameter increases.


The fractional formula of Engelund & Hansen shows the 6th best results for the transport composition predictions with a score of 0.54 (Figure 7.5).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Engelund & Hansen fractional

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.54


Figure 7.5: Transport composition ratio versus the Shields parameter for the fractional formula of Engelund & Hansen.

For most tests, the fractional formula of Engelund & Hansen underestimates the geometric mean grain size of the transported material. As mentioned before, this formula has never been officially derived, and thus, it has not been calibrated for fractional transport predictions. It should be noted that, under different hydraulic conditions, the formula predicts the same transport composition for identical bed composition (see Section 4.2.2). Therefore, the ratio of the calculated and measured geometric mean grain sizes of the transported material for experiments with the same initial bed material only differs because of differences in the measured transport composition. Since the measured composition becomes coarser if the Shields parameter increases, the ratio decreases. The predictions are not good for any of the experimental series, but on average the formula predicts the composition rather well.


7.7.1 Meyer-Peter & Müller with Egiazaroff

The fractional sediment transport formula of Meyer-Peter & Müller with the hiding/exposure correction of Egiazaroff is the 11th best formula to predict the transport composition with a score of 0.34 (Figure 7.6).

In all cases, the formula of Meyer-Peter & Müller with Egiazaroff predicts a finer geometric mean grain size than measured. At low Shields parameters, the underestimation tends to be larger than at high Shields parameters. The formula has large problems predicting the sediment transport composition at Shields parameters up to 0.1: it often predicts no transport at all (Section 6.7.2). This is the most important factor for the poor result of this formula. At higher Shields parameters the composition predictions are quite good.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Meyer−Peter & Müller + Egiazaroff

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.34


Figure 7.6: Transport composition ratio versus the Shields parameter for the fractional formula of Meyer-Peter & Müller with Egiazaroff.

7.7.2 Meyer-Peter & Müller with Ashida & Michiue

With a score of 0.27, the fractional formula of Meyer-Peter & Müller with the hiding/exposure correction of Ashida & Michiue is the worst formula for predicting the transport composition (Figure 7.7).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Meyer−Peter & Müller & Ashida & Michiue

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.27


Figure 7.7: Transport composition ratio versus the Shields parameter for the fractional formula of Meyer-Peter & Müller with Ashida & Michiue.

The predicted geometric mean grain size of the transported sediment is smaller than the measured one. The geometric mean grain size predicted by the combination of Meyer-Peter & Müller and Ashida & Michiue is mostly slightly smaller than by the combination of Meyer-Peter & Müller and Egiazaroff. This leads to larger



underestimations and thus worse predictions. The difference between the two formulas is largest at low Shields parameters.

The differences between the predictions using the two different hiding/exposure factors are caused by the asymptotic behaviour of the hiding/exposure factor of Egiazaroff. This behaviour makes the hiding factor of some small fractions so high that these fractions are never transported according to the formula. This means that the composition of the transported material becomes coarser.

7.8 Van Rijn

The fractional version of Van Rijn’s formula gives the 9th best result concerning the sediment transport composition. The formula has a score of 0.45 and its results are shown in Figure 7.8. It should be noted that this formula has not been officially derived and tested by Van Rijn [1984].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Van Rijn fractional

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.45


Figure 7.8: Transport composition ratio versus the Shields parameter for the fractional formula of Van Rijn.

The fractional formula of Van Rijn always predicts a finer sediment transport composition than measured, except in some runs of BOMC at very low Shields parameters. In many tests, the formula predicts that the geometric mean grain size of the transported material is less than half of the measured one. The results for the BOMC tests are good, whereas the results for the experiments of Klaassen and Day HRS A are average. The results for the other experiments are poor. The predictions of the transport composition are somewhat better for higher Shields parameters.


The predictions of the transport composition of the formula of Hunziker/Meyer-Peter & Müller are the 10th best with a score of 0.35 (Figure 7.9). For small Shields



parameters, the formula of Hunziker predicts a finer sediment transport composition than measured. For Shields parameters higher than about 0.15, the formula predicts a coarser transport composition than measured. This concurs with the results of Section 4.2.2. Like the fractional formulas of Meyer-Peter & Müller, Hunziker’s formula has problems predicting sediment transport for small Shields parameters (smaller than 0.1), for which the formula often predicts no sediment transport at all. For tests where the formula does predict sediment transport to occur, the composition ratio is mostly within the 0.5 - 2 range, which is quite good.

The hiding/exposure factor of Hunziker (eq. 3.21) predicts an inverse hiding/exposure effect (i.e. small grains are assumed to experience a higher shear stress, and large grains a smaller shear stress than they would experience in a bed with uniform material of the same grain size) at grain roughness related Shields parameters ( *

msθ ) smaller than 0.11. In combination with the equal mobility concept, this can be a reason for the predicted transport composition being too fine at low Shields parameters, although the predictions are still reasonable.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Hunziker/Meyer−Peter & Müller

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.35


Figure 7.9: Transport composition ratio versus the Shields parameter for the formula of Hunziker/Meyer-Peter & Müller.


The score of the formula of Gladkow & Söhngen is 0.53, which means an 8th ranking. Figure 7.10 shows the results of this formula.

This formula overestimates as well as underestimates the geometric mean grain size of the transported material. For many tests, it does not predict any sediment transport to occur, which results in a less than average ranking. For the tests for which sediment transport does occur, the composition is predicted rather well or slightly too coarse (BOMC experiments).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Gladkow & Söhngen

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.53


Figure 7.10: Transport composition ratio versus the Shields parameter for the formula of Gladkow & Söhngen.

7.11 Wu et al.

The formula of Wu et al. gives the best predictions of all formulas. With a score of 0.79 it is much better than the other formulas (Figure 7.11).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Wu et al.

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.79


Figure 7.11: Transport composition ratio versus the Shields parameter for the formula of Wu et al.

The formula of Wu et al. gives very good predictions for the composition of the transported material. Only for Shields parameters smaller than 0.08, the formula has a tendency to predict a finer transport than measured. At higher Shields parameters, the composition is predicted very well with both small overestimations and small underestimations of the geometric grain size.




The formula of Wilcock & Crowe gives the 2nd best predictions of all formulas. It is has a score of 0.67 and its results are shown in Figure 7.12. The formula has the tendency to predict the sediment transport being finer than measured. For most tests, the composition ratio lies within the 0.5 – 1 range. This stable pattern results in the good classification of this formula. For many of the BOMC experiments, the composition is predicted coarser than measured, but for the most other tests the predicted composition is mostly somewhat finer.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Wilcock & Crowe

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.67


Figure 7.12: Transport composition ratio versus the Shields parameter for the formula of Wilcock & Crowe.

7.13 Ribberink with Ashida & Michiue

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Ribberink + Ashida & Michiue

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.62




Figure 7.13: Transport composition ratio versus the Shields parameter for the fractional formula of Ribberink with Ashida & Michiue.

The fractional formula of Ribberink with the hiding/exposure correction of Ashida & Michiue gives the 3rd best predictions for the transport composition, with a score of 0.62 (Figure 7.13). For most tests, the predicted composition is finer than the measured composition. Only in a couple of tests of HRS A and BOMC the predicted composition is coarser. The formula gives the best composition predictions of the Meyer-Peter & Müller type formulas. This is because the formula nearly always predicts sediment transport to occur.

7.14 Conclusions

The formula of Wu et al. gives the best transport composition predictions. For conditions with low Shields parameters, the formula shows in general small underestimations of the geometric mean grain size.

The formula of Wilcock & Crowe gives the second best composition predictions. This is not really surprising since 57% of the verification data used in this report was used for calibrating the formula. But the formula shows good results for the other experimental data as well.

All formulas have problems predicting the sediment transport composition for low Shields parameters. For this range, some formulas do not predict any sediment transport at all, and others show a large variation in the predicted compositions. Only the formula of Wu et al. shows good and stable predictions for the transport composition in this range.

All fractional formulas predict the transport composition of the experiments of Klaassen well. Only the formulas of Ackers & White with Day and Wu et al. are valid for these experiments, but the other formulas seem also suitable for these fine sediment mixtures and high Shields parameters.

The formulas of Meyer-Peter & Müller, Hunziker/Meyer-Peter & Müller and Gladkow & Söhngen have problems predicting any sediment transport for low Shields parameters. This is the main reason why the scores of the transport composition of the formulas are poor.

Also the formula of Hunziker/Meyer-Peter & Müller has problems predicting sediment transport for low Shields parameters. The formula overestimates the geometric mean grain size for Shields parameters larger than 0.15. Hunziker found that, at these high Shields parameters, the transported material was coarser than the bed material. This assumption was based on armouring tests of Suzuki.

Surface layer as active layer


8 Surface layer as active layer

8.1 Introduction

An important input parameter for fractional sediment transport formulas is the composition of the part of the bed layer that determines the rate and composition of the transported sediment, the so-called ‘active layer’. Wilcock et al. [2001] state that the bed surface layer is the bed material that determines the transport rate and composition. However, for some of the experiments described and used in this study, only the composition of the initial bed mixture was known. That is why we chose to base our verification in Chapter 6 and 7 on the composition of the initial bed mixture. In this chapter we will use the composition of the surface layer as input for the transport predictions, for the tests for which we do know this surface layer composition.

The next paragraphs are based on Wilcock [2001] and the accompanying discussion by Kleinhans and Blom [2001]. A definition of the ‘active layer’ is relatively straightforward when one is speaking of flat bed situations: the surface layer. When bed forms are present, different definitions are given in literature. One definition states that the ‘active layer’ can be considered as the layer over which the bed forms migrate. This is usually a relatively coarse layer, which is only exposed to the flow in the trough areas.

Another definition of the ‘active layer’ can be considered the actual bed surface: the top layer of bed forms and trough areas. The second definition gives the best basis for sediment transport predictions since the shear stress acting by the flow interacts only with the bed surface.

In the following sections the results will be discussed. The differences between the transport rate using the initial bed material composition or the surface composition are discussed in Section 8.3. The differences in the transport composition will be discussed in Section 8.4. This is followed by conclusions in Section 8.5.

8.2 Measurements

Wilcock & McArdell [1993] (BOMC) and Wilcock et al. [2001] (J06, J14, J21, and J27) directly measured their surface layer compositions. In all of these tests the bed was essentially planar [Wilcock et al., 2001]. Blom [2000] took box core samples and measured the bed elevation distributions. By comparing the vertical profiles of bed composition and the bed elevation probabilities, the surface layer can be estimated. The surface compositions of Blom [2000] are shown in Section 5.3. Appendix 7 gives the bed surface compositions for the tests of Wilcock & McArdell [1993] and Wilcock et al. [2001].

The surface layer composition measured in experimental series J06 and J14 is much coarser than the composition of their initial bed material. In test series J21 and



J27, it became on average slightly coarser than the initial material. The surface compositions in the BOMC experiments are much finer than its initial composition, and the experiments of Blom show a surface composition slightly finer than the initial material. Table 8.1 gives the scores of the formulas using as input the initial bed or surface composition for both the transport rate and the transport composition.

Table 8.1: Score table for the predictions of the transport rate and transport composition for two active layer compositions: initial bed mixture and surface layer composition. Name transport rate transport composition Initial bed surface Initial bed surface Ackers & White uniform 0.27 0.24 - - Ackers & White + Day 0.23 0.27 0.52 0.51 Ackers & White + Proffitt & Sutherland 0.22 0.26 0.49 0.39 Parker 0.19 0.25 0.52 0.76 Engelund & Hansen uniform 0.29 0.34 - - Engelund & Hansen fractional 0.25 0.29 0.53 0.68 Meyer-Peter & Müller 0.11 0.15 - - Meyer-Peter & Müller + Egiazaroff 0.19 0.19 0.27 0.26 Meyer-Peter & Müller + Ashida & Michiue 0.12 0.19 0.20 0.23 Van Rijn uniform 0.14 0.13 - - Van Rijn fractional 0.08 0.10 0.42 0.56 Hunziker/Meyer-Peter & Müller 0.10 0.13 0.25 0.26 Gladkow & Söhngen 0.16 0.13 0.52 0.51 Wu et al. 0.30 0.36 0.74 0.85 Wilcock & Crowe 0.27 0.53 0.65 0.77 Ribberink uniform 0.06 0.06 - - Ribberink + Ashida & Michiue 0.08 0.08 0.58 0.69

8.3 Transport rate

Most sediment transport formulas show better transport rate predictions when the surface layer composition is used as input for these formulas instead of the initial bed composition. But there are also formulas that give worse predictions. Only the formula of Wilcock & Crowe shows a big improvement when the surface layer is used as input for the formula. The score of the formula is 0.53 for the transport predictions, which is the highest of all formulas. This is not a coincidence since the formula of Wilcock & Crowe was calibrated on the tests of Wilcock & McArdell and Wilcock et al. using the surface layer composition as input for the transport formula.

With a score of 0.36, the formula of Wu et al. gives the 2nd best result. In this section only the results of the two best formulas will be discussed, because the differences are small for most other formulas. The transport rate decreases for the tests for which the ‘active layer’ composition becomes coarser (J06, J14, J21, and J27) and increases for the tests for which the ‘active layer’ becomes finer (Blom and BOMC).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Wilcock & Crowe

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.27

Blom Wilcock et al: J06 Wilcock et al: J14 Wilcock et al: J21 Wilcock et al: J27 Wilcock & McArdell: BOMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−3

10−2

10−1

100

101

102

103

Wilcock & Crowe

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.53


Figure 8.1: Transport rate ratio versus the Shields parameter for the formula of Wilcock & Crowe (left: using initial bed material composition, right: using bed surface composition)

Figure 8.1 shows the transport rates predicted by the formula of Wilcock & Crowe for both active layer definitions: the initial bed composition (left) and the surface layer composition (right). The large variation of the predicted transport rates at small Shields parameters is much smaller when the surface layer composition is used. This is caused by the fact that for the tests with an overestimation of the transport rate using the initial mixture composition, i.e. J06, J14 and J21, the surface layer composition is coarser than the initial material composition. This reduces the transport rate and comes closer to the measured transport rate. For the tests with an underestimation of the transport rate, i.e. BOMC, the surface layer composition is finer than the initial bed mixture composition, which leads to higher transport rates.

The transport rates predicted by the formula of Wu et al. are shown in Figure 8.2. On the left are shown the results when using the initial bed material composition and on the right the results when using the surface layer composition. The formula of Wu et al. is also sensitive to changes in the ‘active layer’ composition for the transport rate predictions. The overestimations at low Shields parameters become smaller, but are still large, yet they do show a significantly smaller scatter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−3

10−2

10−1

100

101

102

103

Wu et al.

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.3


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−3

10−2

10−1

100

101

102

103

Wu et al.

θ50

(−)

qsca

lc/q

s mea

s (−

) Score: 0.36


Figure 8.2: Transport rate ratio versus the Shields parameter for the formula of Wu et al. (left: using initial bed material composition, right: using bed surface composition)



8.4 Transport composition

The predictions for the transport composition are more influenced by changes in the ‘active layer’ than those of the transport rate. Almost all the formulas result in better predictions when the surface layer composition is used instead of the initial mixture composition. All formulas predict a coarser or finer sediment transport when the ‘active layer’ composition becomes coarser or finer, respectively. In this section the results of the three best formulas will be discussed. The best formula is the one of Wu et al. with a score of 0.85. The second best formula is the one of Wilcock & Crowe with a score of 0.77. The surface formula of Parker is the third best formula with a score of 0.76.

The formula of Wu et al. already gave good predictions of the transport composition, but they improve when the surface layer composition is used as input for the predictions (Figure 8.3). If the composition of the active layer becomes finer or coarser, the transport becomes finer or coarser, respectively. This gives better results for the experiments BOMC, where the ‘active layer’ becomes finer and tests J06, J14 and J21, where the ‘active layer’ becomes coarser.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Wu et al.

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.74


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−1

100

101

Wu et al.

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.85


Figure 8.3: Transport composition ratio versus the Shields parameter for the formula of Wu et al. (left: using initial bed material composition, right: using bed surface composition).

The formula of Wilcock & Crowe shows improvements comparable to the formula of Wu et al. (Figure 8.4). The left picture shows the predictions using the initial material composition and the right the predictions using the surface layer composition.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Wilcock & Crowe

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.65


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−1

100

101

Wilcock & Crowe

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.77


Figure 8.4: Transport composition ratio versus the Shields parameter for the formula of Wilcock & Crowe (left: using initial bed material composition, right: using bed surface composition).

The formula of Parker gives much better results when the surface layer composition is used for the ‘active layer’ composition instead of the initial bed material composition (Figure 8.5). The improved results are mostly found in the experimental series J06, J14 and J21. The other three experimental series (Blom, J27, and BOMC) do not show a big improvement. Parker’s formula was calibrated using the compositions of surface layers. The results show that the transport composition of the formula is very sensitive to changes in the ‘active layer’ composition and better results are achieved when the surface layer composition is used instead of the initial bed composition.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−1

100

101

Parker surface

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.52


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−1

100

101

Parker surface

θ50

(−)

Dm

,cal

c/Dm

,mea

s (−

)

Score: 0.76


Figure 8.5: Transport composition ratio versus the Shields parameter for the formula of Parker (left: using initial bed material composition, right: using bed surface composition).

8.5 Conclusions

Most formulas give better sediment transport predictions using the surface layer composition as input for these formulas instead of using the composition of the initial bed material. It is logical that using the surface layer composition leads to better results since the shear stress caused by the water flow is applied directly to



the bed surface, which gives the direct availability of the different grain size fractions. The surface layer composition can differ from the composition of the initial bed mixture in equilibrium situations due to vertical sorting.

The formula of Wilcock & Crowe [2001] shows the best results for the transport rate predictions. The score of the formula is almost twice as good when the surface layer composition is used instead of the initial bed material composition. The formula of Wu et al. [2000] also shows better results, there is less scatter in the predictions and the predictions are better.

The transport composition is very well predicted by the formula of Wu et al. using the surface layer composition. The formulas of Wilcock & Crowe and Parker [1990] give significantly better predictions for the transport composition when the surface composition is used.

Discussion


9 Discussion

9.1 Modifications to Van Rijn

In this study, the formula of Van Rijn adapted to the case of non-uniform sediment did not give good predictions. Kleinhans & Van Rijn [2001] propose a number of modifications to this formula and create a fractional bed load formula based on a stochastic representation of the grain shear stress. The formula was calibrated on the tests of Blom & Kleinhans [1999] and according to Kleinhans & Van Rijn, it gave good predictions for the sediment transport in two rivers. This formula could not be verified within the time frame of this project.

9.2 Bed form factors

The bed form factors in the formulas of Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller result in some difficulties (Sections 4.4, and Appendix 1). Bed form factors denote the part of the bed roughness that is related to grain roughness and not to bed form roughness. This means that at conditions of incipient motion, where no bed forms are present, the bed form factor should be one. However, in the formulas mentioned above, the bed form factor is much smaller than one at conditions of incipient motion.

The bed form factors of Strickler, Yalin & Scheuerlein, and White-Colebrook all have this problem. This means that the formulas of Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller should be used carefully at conditions of incipient motion.

Conclusions


10 Conclusions

In the first part of the study, validity ranges of a number of selected fractional formulas were determined. In the second part, the behaviour of the transport formulas was analysed for different hydraulic and sedimentological conditions. The final part deals with the verification of the transport formulas with measurements of ten sets of experimental data. In this chapter, the conclusions of each separate part will be given, followed by some overall conclusions.

10.1 Validity ranges of fractional formulas

The validity ranges of the original fractional formulas were determined. Data used for the original calibration of the formulas were analysed concerning the ranges of the median grain size (D50), the geometric standard deviation of the bed material (σg), and the Shields parameter (θ50). The formula of Wu et al. [2000] has the largest validity ranges of all analysed formulas. This formula is based on a large set of experimental and field measurements. The fractional formulas of Ackers & White [1973] with Day [1980] and of Ackers & White with Proffitt & Sutherland [1983] have large validity ranges as well. Wilcock & Crowe [2002] and Hunziker [1995] used large data sets for calibration and verification of their formulas, but the conditions in their experiments did not vary much, so that their validity ranges are small.

10.2 Behaviour analysis

The behaviour of the transport formulas was analysed in terms of transport rate and the transport composition as a function of the Shields parameter. This was done by computing the transport rate and composition under different hydraulic and sedimentological conditions. Four series of calculations were done with different bed compositions: the bed mixtures varied in median grain size (2 or 10 mm) and geometric standard deviation (2 or 5). Calculations were done for Shields parameters up to 0.6.


The formulas of Wilcock & Crowe [2001], Parker [1990] and Engelund & Hansen [1967] predict the highest sediment transport rates in their calculations. The formulas of Van Rijn [1984a], Gladkow & Söhngen [2000] and Hunziker/Meyer-Peter & Müller [1995] predict the lowest transport rates.

Defining and predicting incipient motion is difficult, resulting in great differences in modelling transport rates around incipient motion between the formulas, i.e. for small Shields parameters (θ50 < 0.3).

The behaviour of the transport formulas appears not very sensitive to changes in the grain size of the bed material. This is due to the fact that the transport is shown as a function of the Shields parameter, in which the grain size is incorporated.

Conclusions


A variation in the geometric standard deviation of the bed material does influence the predicted transport rates. The formulas of Engelund & Hansen and Van Rijn show an increase in the transport rate when the geometric standard deviation (σg) is increased, while Parker and Hunziker/Meyer-Peter & Müller show a decrease in the transport rate. The formulas of Ackers & White and Meyer-Peter & Müller are not very sensitive to a change in the geometric standard deviation of the bed material.

The formulas of Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller predict incipient motion at high Shields parameters. One reason is the fact that the bed form factors used in these formulas are smaller than one at conditions of incipient motion. However, theoretically, the bed form factor should be equal to one at these conditions: all bed shear stress is related to the grains since there are no bed forms at incipient motion. However, even if the bed form factor is one at conditions of incipient motion, incipient motion still occurs at relatively high Shields parameters for the formulas of Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller. This is an important problem of these formulas. The formulas use the geometric mean grain size (Dm) (eq. 2.1) for its Shields parameter, which leads to problems in predicting sediment transport for strongly graded sediment mixtures.

There are little differences between the uniform formulas and their fractional counterparts. The largest differences occur at Shields parameters below 0.25. Most fractional formulas converge with their uniform formula at higher Shields parameters.


For low Shields parameters, all formulas predict a transport composition that is much finer than the bed composition. The composition of the transported material coarsens gradually for higher Shields parameters. In some cases, the composition of the transported material approximates the composition of the bed material when the Shields parameter increases. For these high Shields parameters, the formulas of Wu et al. and Hunziker/Meyer-Peter & Müller generally predict a transport composition coarser than the bed composition.

The transport compositions predicted by the different formulas differ most for Shields parameters smaller than 0.3, as predicting sediment transport at these conditions of incipient motion is difficult. This has led to many different approaches of modelling incipient motion.

Only the predictions of the transport composition by the two fractional formulas of Ackers & White are sensitive to variations in the median grain size of the bed material. All formulas are sensitive to variations in the geometric standard deviation in the bed material. They all predict a finer transport composition when the geometric standard deviation is increased. Only the formula of Hunziker/Meyer-Peter & Müller shows opposite behaviour.

10.3 Verification with experimental data

Predictions of the sediment transport formulas were verified with measurements from ten sets of experiments. These experiments were all carried out in flumes with sediment recirculation systems. Only data of the equilibrium phases of the

Conclusions


experiments were used for verification. Both the transport rate and transport composition were used for verification.

Since, the surface layer of the bed material determines the rate and composition of the transported material, the composition of the surface layer should be used as input for the transport formulas. However, for some experiments used for verification, only data on the composition of the initial bed mixture was available. For this reason, initially, we used the composition of the initial bed mixture as composition of the active layer (Section 10.3.1). After that, for the experiments for which data on the composition of the surface layer was available, we compared prediction on the basis of i) the initial bed mixture and ii) the surface layer (Section10.3.2).

10.3.1 Initial bed mixture as active layer

Transport rate

The formula of Wu et al. gives the best transport rate predictions. The formula should be used with care at conditions with low Shields parameters and relatively coarse material. At other conditions the formula gives good predictions.

The formula of Wilcock & Crowe is a good second option, yet this formula is not reliable in the low Shields parameter range as well.

The sediment transport is extremely difficult to predict at Shields parameters smaller than 0.1. The formulas of Gladkow & Söhngen, Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller frequently predict no sediment transport at all, whereas formulas like Engelund & Hansen, Wu et al., Ribberink with Ashida & Michiue and Van Rijn strongly overestimate the transport rate for many tests.

The bed form factors in the formulas of Meyer-Peter & Müller and of Hunziker/Meyer-Peter & Müller seem to be too small in conditions with low Shields parameters. This results in predicting no sediment transport, while, according to measurements, sediment transport did occur.

Although only the formulas of Wu et al. and Ackers & White with Day are valid for the conditions of the experiments of Klaassen, also the formulas of Meyer-Peter & Müller and Hunziker/Meyer-Peter & Müller give good results for these experiments. This means that these formulas are also suitable for transport rate predictions with sediment mixtures finer and Shields parameters higher than their validity range.

The fractional formulas of Ackers & White with Day, Ackers & White with Proffitt & Sutherland, Meyer-Peter & Müller with Egiazaroff and Meyer-Peter & Müller with Ashida & Michiue give better predictions than their uniform counterparts.

Conclusions


Transport composition

Also for the transport composition the formula of Wu et al. shows the best predictions. For conditions with low Shields parameters, the formula shows small underestimations of the geometric mean grain size of the transported mixture.

All formulas have problems predicting the transport composition for low Shields parameters. Some formulas predict no sediment transport at all, while other formulas show a large variation in the predicted compositions. Only the formula of Wu et al. shows good and stable predictions for the transport composition.

All fractional formulas predict the transport composition of the experiments of Klaassen well. Only the formulas of Ackers & White with Day, Wu et al. are valid for these experiments, but the other formulas are also suitable for predictions with sediment this fine and Shields parameters this high.

The formulas of Meyer-Peter & Müller, Hunziker/Meyer-Peter & Müller, and Gladkow & Söhngen have problems predicting sediment transport at low Shields parameters. This is the main reason why the scores for the transport compositions of the formulas are poor.

10.3.2 Surface layer as active layer

When the bed surface is used as the ‘active layer’, which is defined as the part of the bed that determines the rate and composition of the transported sediment, the transport predictions of most formulas are better. The surface layer composition was only determined for the experiments of Blom [2000], Wilcock & McArdell [1993], and Wilcock et al. [2001]. Wilcock & Crowe [2001] now lead to the best transport rate predictions of all the formulas. Nonetheless, we need to remark that the measurements of the surface composition in the experiments by Wilcock & McArdell, and Wilcock et al. were used for the development of their formula. When the surface layer composition is used the large variation at the lower Shields parameters has strongly decreased. The formula of Wu et al. [2000] shows better results with the surface layer as ‘active layer’, as well. It still strongly overpredicts the transport rate at low Shields parameters, yet it shows much less scatter.

The transport composition predicted by Wu et al. improves when the bed surface composition is used as input. The predictions agree very well with the measurements, although at low Shields parameters the predicted sediment composition is finer than measured. The formula of Wilcock & Crowe shows good results too, although somewhat worse than Wu et al., because it shows more variation in the Shields parameter range 0.05 - 0.25.

The use of the surface layer as active layer does not only give better results, there is also a theoretical basis why the surface layer should be used as input for the sediment transport predictions. Only the material present directly at the bed surface is exposed to the water flow and can be transported.

Conclusions


10.4 Final conclusions

Among the transport formulas analysed in this study, the formula of Wu et al. is the best formula for sediment transport predictions. The formula has large validity ranges due to the large data set that was used for the original calibration of the formula. The formula is the best formula to predict the transport rate and the best to predict the transport composition when the composition of the initial bed mixture is used as input for the transport formulas. If the bed surface composition is used, the formula of Wu et al. is also the best formula, together with the formula of Wilcock & Crowe.

The formula of Meyer-Peter & Müller with the hiding/exposure correction of Ashida & Michiue [1973] shows good results for the transport rate predictions. However, the formula does not perform well when predicting the composition of the transported sediment. The validity range of this formula is much smaller than the formula of Wu et al., but it shows good predictions outside its validity range as well. The formula is not suitable to conditions with Shields parameters below 0.1. The formula often predicts no sediment transport in this Shields parameter range, whereas experiments of Day [1980], Wilcock & McArdell [1993] and Wilcock et al. [2001] show that sediment transport did occur in these conditions.

Using the bed surface composition instead of the composition of the initial bed mixture as the active layer composition leads to better predictions for most formulas. The use of the surface layer has also a better theoretical basis because the shear stress acts directly on the bed surface.

10.5 Recommendations

In this research, data from only flume experiments was used for the verification of the sediment transport formulas. Although many tests were used with large ranges in hydraulic and sedimentological conditions, it is recommended that the transport formulas be also verified with field measurements. It is important to know how the formulas behave in a natural environment where uncertainties in the measurements are much larger. However, when selecting field data, it is important to know how reliable the measurements are. The hydraulic conditions can be measured reasonably well, but the composition of the bed surface layer introduces a lot of problems. Three measurements are needed for verification: hydraulic conditions, sediment transport, and the ‘active’ layer composition. It is difficult to find combined measurements that were taken at the same time and at the same place.

More research is needed for a definition of the active layer, i.e. the material that determines the transport rate and composition of the transported material. The experiments of Wilcock & McArdell [1993], Wilcock et al. [2001], Blom & Kleinhans [1999], and Blom [2000] give a good basis, but more coupled measurements of the composition of the surface layer, sediment transport (rate and composition), and hydraulic conditions are required.

Conclusions


The influence of the geometric mean grain size of the mixture in the active layer on the predicted sediment transport should be investigated. The commonly used definition of the geometric mean grain size leads to problems predicting the sediment transport of widely graded sediment. Parker (1990) gives a different definition, which seems to solve a part of our problems.

The modified formula of Van Rijn [Kleinhans & Van Rijn, 2001] shows promising results according to the developers of this formula. It is recommended to verify this formula with a larger data set and compare its results with results from other formulas. Introducing the stochastic approach of Kleinhans & Van Rijn [2001] into other transport formulas might lead to better results as well. Especially in conditions of incipient motion, the stochastic approach seems very useful.

It is useful to test the formula of Wu et al. [2000] in a graded morphological model. This will show if the formula is also able to predict morphological changes and if the formula introduces stability problems into morphological models.

References


References

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Ashida, K. and Michiue, M., (1971), An investigation of river bed degradation downstream of a dam, Proc. 14th Congress IAHR, Vol. 13, Paris.

Ashida, K. and Michiue, M., (1973), Studies on bed load transport rate in alluvial streams, Trans. JSCE, Vol. 4.

Blom, A. and Kleinhans, M., (1999), Non-uniform sediment in morphological equilibrium situations - Data report Sand Flume experiments 97/98, Research report CiT 99R-002/MICS-001, Civil Engineering and Management, University of Twente.

Blom, A., (2000), Flume experiments with a trimodal sediment mixture - Data report Sand Flume experiments 99/00, Research report CiT: 2000R-004/MICS-013, Civil Engineering & Management, University of Twente.

Blom, A. and Ribberink, J.S., (1999), Non-uniform sediment in rivers: vertical sediment exchange between bed layers, Proc. IAHR Symposium on River, Coastal and Estuarine Morphodynamics, Genova, Italy 6-10 September 1999, pp. 45-54.

Blom, A., Ribberink, J.S. and Scheer, P. van der, (2000), Sediment transport in flume experiments with a trimodal sediment mixture, CD-rom contribution, Gravel-bed Rivers Workshop, New Zealand, 27 Aug-2 Sept.

Bredius, J., (1998), Verificatie van transportformules voor gegradeerd sediment, (in Dutch), M.Sc. thesis Civil Engineering and Management, University of Twente.

Colebrook C. F., (1939), Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws, Institution of Civ. Eng. journal v11, paper no. 5204.

Day, T.J., (1980), A study of the transport of graded sediments, HRS Wallingford, Report No. IT 190.

Egiazaroff, I.V., (1965), Calculation of non-uniform sediment concentrations, J. of Hydr. Div., ASCE, Vol. 91, No. 4, pp. 225-248.

Engelund, F. and Hansen, E., (1967), A monograph on sediment transport in alluvial streams, Teknisk Forlag, Copenhagen, Denmark.

Gladkow, G.L., and Söhngen, B., (2000), Modellierung des Geschiebetransports mit unterschiedlicher Korngröße in Flüssen, Mitteilungsblatt der Bundesanstalt für Wasserbau, Vol. 82, pp 123-129.

Günter, A., (1971), Die kritische mittlere Sohlenschubspannung bei Geschiebemischungen unter Berücksichtigung der Deckschichtbildung und der turbulenzbedingten Sohlenschub-spannungsschwankungen, Mitteilungen Nr. 3 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, ETH Zürich.

References


Hirano, M., (1970), On the river bed degradation downstream of a dam and the armoring phenomena, Proc. 14th Conference on Hydraulics, (in Japanese).

Hunziker, R.P., (1995), Fraktionsweiser Geschiebetransport, Ph.D. thesis Mitteilungen Nr. 138 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, ETH Zürich, Switserland.

Iwagaki, Y., (1956), Hydrodynamical Study on Critical Tractive Force, Trans. of JSCE, Vol. 41, Tokyo.

Klaassen, G.J., (1991), Experiments on the effect of gradation and vertical sorting on sediment transport phenomena in the dune phase, Proc. Grain Sorting Seminar, Ascona, Switzerland, pp 127-145.

Kleinhans, M.G. and Blom, A., (2001), Discussion of (The Flow, The Bed, and the Transport: Interaction in the Flume and the Field, by Wilcock, 2001), proc. Gravel-Bed Rivers V, edit. Mosley, M.P., The Caxton Press, pp. 212-214, Christchurch.

Kleinhans, M.G. and Rijn, L.C. van, (2002), Stochastic prediction of sediment transport in sand-gravel bed rivers, J. of Hydr. Eng., Vol. 128, No. 4, pp. 412-425.

Kuhnle, R.A., (1993), Fluvial transport of sand and gravel mixtures with bimodal size distributions, Sedimentary Geology, Vol. 85, pp. 17-24.

Laguzzi, M., (1994), Modelling of sediment mixtures, Report No. Q 1660, WL | delft hydraulics.

Meyer-Peter, E. and Müller, R., (1948), Formulas for bed-load transport, Proc. 2nd Congress IAHR, Stockholm, Sweden.

Nizery, A. and Braudeau, G., (1953), Variation de la Granulometrie de Charriage dans une section de Riviere, Proc. 5th Congress IAHR, Minnesota, pp 49-60.

Pantélopulos, J., (1955), Note sur la granulometrie de charriage et la loi du debit solide par charriage de fond d’un melange de materiaux, Proc. 6th Congress IAHR, The Hague, Vol 4, pp D10/1-D10/11.

Pantélopulos, J., (1957), Étude Expérimentale du mouvement par harriage de fond d’un mélange de matériaux recherches sur la similitude du charriage, Proc. 7th Congress IAHR, Lissabon, pp D30/1-D30/24.

Parker, G., Klingeman, P.C. and McLean, D.G., (1982), Bedload and size distribution in paved gravel-bed streams, Proc. ASCE, J. of the Hydr. Div., Vol. 108, HY 4, pp. 544-571.

Parker, G. and Klingeman, P.C., (1982), On why gravel bed streams are paved, Water Resources Res., Vol. 18, No. 5, pp. 1409-1423.

Parker, G., (1990), Surface-based bedload transport relation for gravel rivers, J. of Hydr. Res., Vol. 28, No. 4, pp. 417-436.

Proffitt, G.T. and Sutherland, A.J., (1983), Transport of non-uniform sediments, J. of Hydr. Res., Vol. 21, No. 1, pp. 33-43.

References


Ribberink, J.S., (1981), Bed-load formulae for non-uniform sediment, Delft University of Technology, Department of Civil Engineering, Fluid Mechanics Group, Internal Report no. 4-78, The Netherlands.

Ribberink, J.S., (1987), Mathematical modelling of one-dimensional morphological changes in rivers with non-uniform sediment, Ph.D. thesis, Report No. 87-2, Comm. On Geot. And Hydr. Eng., Delft Univ. of Technology, The Netherlands.

Ribberink, J.S., (1998), Bed-load transport for steady flows and unsteady oscillatory flows, Coastal Eng., Vol. 34, pp. 59-82.

Rijn, L.C. van, (1993), Principles of sediment transport in rivers, estuaries and coastal seas, Aqua Publications, Amsterdam, The Netherlands.

Rijn, L.C. van, (1984a), Sediment transport, Part I: Bed Load Transport, J. of Hydr. Eng., ASCE, Vol. 110, No. 10, pp. 1431-1456.

Rijn, L.C. van, (1984b), Sediment transport, Part II: Suspended Load Transport, J. of Hydr. Eng., ASCE, Vol. 110, No. 11, pp. 1613-1641.

Samaga, B.R., Ranga Raju, K.G. and Garde, R.J., (1986a), Bed Load Transport of Sediment Mixtures, J. Hydr. Eng., Vol. 112, No. 11, pp. 1003-1018.

Samaga, B.R., Ranga Raju, K.G. and Garde, R.J., (1986b), Suspended Load Transport of Sediment Mixtures, J. Hydr. Eng., Vol. 112, No. 11, pp. 1019-1035.

Scheer, P. van der, (2000), Transport Formulae for Graded Sediment - Verification with Flume Data 1999-2000, M.Sc. thesis, University of Twente, Enschede.

Strickler, A., (1923), Beiträge zur Frage der Geschwindigkeitsformel und der Rauhigkeitszahlen für Ströme, Kanäle und geschlossene Leitungen, Mitt. No. 16 des Amtes für Wasserwirtschaft, Eidgenössisches Departement des Innern, Bern.

Suzuki, K. and Hano A., (1992), Grain Size change of bed surface layer and sediment discharge of an equilibrium river bed, Proc. Grain Sorting Seminar, Ascona, Switzerland, pp 151-156.

Wallingford, (1990), Sediment transport: the Ackers & White theory revised, Report SR 237, HR Wallingford.

Wilcock, P.R. and McArdell, B.W., (1993), Surface-Based Fractional Transport Rates: Mobilization Thresholds and Partial Transport of a Sand-Gravel Sediment, Water Resources Res., Vol. 29, No. 4, pp. 1297-1312.

Wilcock, P.R., (2001), The Flow, The Bed, and the Transport: Interaction in the Flume and the Field, proc. Gravel-Bed Rivers V, edit. Mosley, M.P., The Caxton Press, pp. 183-219, Christchurch.

Wilcock, P.R., Kenworthy, S.T. and Crowe, J.C., (2001), Experimental study of the transport of mixed sand and gravel, Water Resources Res., Vol. 37, No. 12, pp. 3349-3358.

Wilcock, P.R. and Crowe, J.C. (2001), A surface-based transport model for sand and gravel, submitted to J. Hydr. Eng.

References


Wilcock, P.R. and Kenworthy, S.T., (2001), A two fraction model for the transport of sand/gravel mixtures, submitted to Water Resources Res.

Williams, G.P. and Rosgen, D.L., (1989), Measured total sediment loads (suspended loads and bedloads) for 93 United States streams, USGS Open_File Report 89-67, Denver, CO.

Wu, W., Wang, S.S.Y. and Jia, Y, (2000), Nonuniform sediment transport in alluvial rivers, J. of Hydr. Res., Vol. 38, No. 6, pp 427-434.

Yalin, M.S., Scheuerlein, H., (1988), Friction Factors in alluvial rivers, Bericht Nr. 59, Instituts für Wasserbau und Wassermengenwirtschaft, Oskar van Miller Institut, Obernach.

Zarn, B., (1997), Einfluss des Flussbettbreite auf die Wechselwirkung zwischen Abfluss, Morphologie und Geschiebetransportkapazität, Mitteilung Nr. 154 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, ETH Zürich, Switserland.

Appendix 1

University of Twente A-1

Appendix 1: Bed roughness analysis

In general, three expressions for the bed roughness are used. These are given by:

21

21

ihfg8

u ⋅⋅= Darcy-Weisbach

21

32

ihn1u ⋅⋅= Manning

21

21

ihCu ⋅⋅= Chézy

The relations between the different roughness coefficients is:

fg8

nhC

61

==

C Chézy value [m1/2/s]

n Manning coefficient [s/m1/3]

f Darcy-Weisbach coefficient [-]

The formulas of Meyer-Peter & Müller [1948] use the following expression for the bed form factor:

23

kk

=

'µ bed form factor [-]

The formula of Strickler [1923] is used for the calculation of k and k’:

6 hck = total roughness according to Strickler [m1/3/s]

690D

ck =' grain roughness according to Strickler [m1/3/s]

It should be noted that Strickler’s roughness is the inverse roughness of Manning’s roughness. The constant c is not dimensionless, but has the dimension [m1/2/s]. The value of c varies in expressions of different authors. Meyer-Peter & Müller [1948] use c = 26, while Van Rijn [1993] uses c = 25.

The total bed shear stress (τ) can be divided in a shear stress related to the grains (τ’) and a shear stress related to the bed forms (τ’’):

''' τττ +=

This relation can be rewritten for Chézy values which results in:

222 C1

C1

C1

'''+=

The bed form factor of Meyer-Peter & Müller can now be rewritten as:

Appendix 1

A-2 University of Twente

23

23

23

CC

nn

kk

=

=

=

''

'µ

The grain roughness by Strickler can now be rewritten to the other grain roughness predictors:

690

6 D26

hC

n1k ===

''

'

Which leads to: 6

1

90Dh26C

=' (1)

Strickler’s equation approximates the White-Colebrook [1939] formula. This approximation can only be used in the Chézy value range 40 to 70 m1/2/s. The White-Colebrook relation for the grain related Chézy value yields:

=

90Dh1218C log' grain related Chézy value according to White-Colebrook (2)

The equation above can be seen as the White-Colebrook expression of the original Meyer-Peter & Müller expression for the grain roughness.

Note that the expression for the grain related Chézy value used in the bed load formula of Van Rijn [1984a] differs from the one used in the Meyer-Peter & Müller formula.

⋅⋅

=90D3h1218C log' grain related Chézy value used by Van Rijn [1984a]

In the figure below, the differences between equations 1 and 2 can be seen.

Appendix 1


Comparison grain roughness predictors

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350 400

h/D90 (-)

Gra

in ro

ughn

ess

(m1/

2/s)

StricklerWhite-Colebrook

Appendix 2


Appendix 2: Splitting a sediment mixture into n fractions

In this appendix, a summary is given of the equations used to split a sediment mixture with a log-normal distribution in n size fractions. Only the equations used in the MATLAB model are given in this appendix. Ribberink [1987] gives a more thorough description. The log-normal probability distribution of grain diameter D50 can be written as:

{ }2y

2y50

50y50 D50

2D1Df σµ

πσ)(ln.exp)( −−= log-normal probability distribution

In which: 50y Dln=µ mean value of y

9950g

y .lnσ

σ = standard deviation of y

The standard normal distribution is given by z. A lower and upper boundary of the grain size distribution (Dl and Du) were chosen according to:

%.)()( 282DDPDDP ul =≥=≤

These boundaries coincide with the following values of z: zl = -2 lower boundary z

zu = 2 upper boundary z

The region between these boundaries is divided into n equal parts ∆z:

nzz

z lu −=∆

The z-value of each fraction i (zi) is:

z2

1i2zz li ∆⋅−

+=)(

This can be transformed into Di using: )exp( yyii zD µσ += grain size of fraction i (m)

In order to determine the probabilities of each size fraction, an upper (zui) and lower (zli) boundary of each fraction i is necessary:

z50zz ili ∆−= .

z50zz iui ∆+= .

The probability of fraction i (pi) can be determined with:

Appendix 2


{ }).().(.

.liuii z250erfz250erf

9545050p ⋅−⋅=

Note: the probability of fraction i is divided by a factor 0.9545 to correct for the upper and lower boundaries of the grain size distribution (1 - 2 * 0.0228 ≈ 0.9545).

The sediment transport formulas use a number of representative grain size diameters, which are calculated as follows:

).exp( 9950D

Dy

5016 −⋅−

=σ

).exp( 3850D

Dy

5035 −⋅−

=σ

).exp( 9950D

Dy

5084 ⋅−

=σ

).exp( 2811D

Dy

5090 ⋅−

=σ

Note: Due to the discretisation of the log-normal distribution some errors are introduced when calculating the geometric mean grain size. There is an error between the two ways the geometric mean grain size can be calculated.

).exp( 2y50m 50DD σ⋅= continuous geometric mean grain size (m)

and ∑ ⋅=

iiim DpD discrete geometric mean grain size (m)

The error caused by the discretisation is caused by the lower and upper boundary (zl and zu) and the number of fractions. If these boundaries are set on higher values (e.g. -10 and 10 respectively) and the number of fractions is set very large (e.g. 100), the discrete geometric mean grain size approaches the continuous geometric mean grain size.

Appendix 3


Appendix 3: Transport rate Series 1 and 2

0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)


0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)


0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)


0 0.1 0.2 0.3 0.4 0.5 0.610

-8

10-7

10-6

10-5

10-4

10-3

10-2


θ50 (-)

qs (m

3 /s/m

)


qs (m

3 /s/m

)

qs (m

3 /s/m

)

qs (m

3 /s/m

)

qs (m

3 /s/m

)

Appendix 4


Appendix 4: Transport composition Series 1 and 2

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


θ50 (-)


Dm

T/D

mB (-

)

Dm

T/D

mB (-

)

Dm

T/D

mB (-

)

Dm

T/D

mB (-

)

Appendix 5


Appendix 5: Uniform versus fractional for

Series 1 and 2

Serie 1

0.1

1.0

10.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Shields D50 (-)

qs_f

r/qs_

u (-)

Ackers & White + DayAckers & White + Proffitt & SutherlandMeyer-Peter & Muller + EgiazaroffMeyer-Peter & Muller + Ashida & Michiue

Serie 2

0.1

1.0

10.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Shields D50 (-)

qs_f

r/qs_

u (-)

Ackers & White + DayAckers & White + Proffitt & SutherlandMeyer-Peter & Muller + EgiazaroffMeyer-Peter & Muller + Ashida & Michiue

Appendix 6


Appendix 6: Vanoni and Brooks

Vanoni and Brooks [1957] made a correction for the sidewall roughness for flume experiments with a width/depth ratio smaller than 5. This correction was made because the sidewall roughness has a big influence under these circumstances and since only the bed roughness is relevant for the sediment transport formulas. The correction factor is applied to the hydraulic radius and the Chézy coefficient.

wbb iRgu ⋅⋅=*, shear velocity related to the bed (m/s)

Rff

R bb ⋅= hydraulic radius related to the bed (m)

h2bhbR⋅+

⋅= hydraulic radius (m)

2

uu

8f

⋅= * friction coefficient (-)

wiRgu ⋅⋅=* shear velocity (m/s)

hbQu⋅

= mean flow velocity (m/s)

)( bb ffb

h2ff −⋅⋅

+= friction coefficient related to the bed (-)

18840f

04280f

00260f2

w .Relog.Relog. +

⋅−

⋅= friction coefficient

related to smooth sidewalls (105 ≤ Re/f < 108) (-)

νRu4 ⋅⋅

=Re Reynolds’ number (-)

h water depth (m)

iw water surface slope (-)

b width (-)

Q discharge (m3/s)

ν kinematic viscosity coefficient (m2/s)

Appendix 7


Appendix 7: Data Wilcock

Hydraulic data Wilcock & McArdell [1993] and Wilcock et al. [2001] Test Q h Rb Cb visc θ50 qs DmT

m3/s m m m1/2/s m2/s - m2/s mm J06-1 0.047 0.104 0.095 26.42 1.11E-06 0.028 8.51E-11 5.40J06-2 0.052 0.108 0.102 27.75 1.11E-06 0.032 9.69E-10 6.14J06-3 0.058 0.104 0.094 26.53 1.11E-06 0.059 3.55E-08 10.29J06-4 0.062 0.102 0.086 28.61 1.11E-06 0.082 4.37E-07 16.23J06-5 0.054 0.103 0.094 27.9 1.11E-06 0.042 7.70E-09 9.12J06-6 0.063 0.103 0.092 30.28 1.11E-06 0.058 1.65E-07 10.59J06-7 0.073 0.106 0.103 25.28 1.11E-06 0.102 5.56E-06 17.19J06-8 0.047 0.105 0.099 31.49 1.11E-06 0.036 1.27E-09 6.73J06-9 0.077 0.109 0.101 27.71 1.11E-06 0.117 1.13E-05 16.48J06-10 0.08 0.108 0.118 25.11 1.11E-06 0.134 7.82E-05 17.60J14-1 0.076 0.117 0.102 26.34 1.11E-06 0.143 1.97E-05 14.18J14-2 0.075 0.109 0.112 25.87 1.11E-06 0.140 2.85E-05 14.50J14-3 0.05 0.107 0.107 30.87 1.11E-06 0.048 1.15E-08 6.61J14-4 0.061 0.104 0.094 30.79 1.11E-06 0.082 6.82E-07 8.84J14-5 0.066 0.106 0.114 25.71 1.11E-06 0.113 4.79E-06 12.15J14-6 0.047 0.102 0.101 36.45 1.11E-06 0.033 7.36E-09 7.27J14-7 0.057 0.106 0.106 30.69 1.11E-06 0.065 5.90E-07 7.94J14-8 0.055 0.106 0.097 31.23 1.11E-06 0.060 2.10E-07 8.10J14-9 0.08 0.117 0.131 25.19 1.11E-06 0.136 4.41E-05 14.83J21-1 0.076 0.118 0.105 26.58 1.11E-06 0.161 5.17E-05 12.71J21-2 0.047 0.108 0.096 35.85 1.11E-06 0.041 1.87E-07 4.84J21-3 0.053 0.102 0.095 33.45 1.11E-06 0.064 2.34E-06 5.39J21-4 0.06 0.105 0.097 28.64 1.11E-06 0.105 4.56E-06 7.38J21-5 0.044 0.109 0.1 36.27 1.11E-06 0.033 5.10E-08 4.06J21-6 0.054 0.104 0.094 31.73 1.11E-06 0.071 4.90E-06 5.32J21-7 0.039 0.099 0.09 38.94 1.11E-06 0.028 6.40E-09 4.43J21-8 0.067 0.102 0.096 27.16 1.11E-06 0.153 5.82E-05 10.81J27-1 0.039 0.102 0.098 38.03 1.11E-06 0.032 1.85E-07 2.81J27-2 0.054 0.101 0.101 33.54 1.11E-06 0.076 1.40E-05 4.12J27-3 0.03 0.11 0.112 42.51 1.11E-06 0.012 1.12E-09 1.76J27-4 0.034 0.101 0.098 35.71 1.11E-06 0.028 7.85E-08 2.35J27-5 0.049 0.093 0.09 34.01 1.11E-06 0.074 1.06E-05 5.00J27-6 0.045 0.098 0.094 38.33 1.11E-06 0.045 2.15E-06 3.45J27-7 0.062 0.106 0.101 34.17 1.11E-06 0.091 2.56E-05 5.35J27-8 0.068 0.106 0.098 34.14 1.11E-06 0.112 4.02E-05 6.17J27-9 0.075 0.106 0.082 34.77 1.11E-06 0.163 1.30E-04 9.03J27-10 0.078 0.111 0.106 27.79 1.11E-06 0.201 2.98E-04 10.85BOMC-14c 0.017 0.111 0.098 33.86 1.11E-06 0.009 8.85E-10 0.60BOMC-7a 0.021 0.11 0.098 33.42 1.11E-06 0.013 1.26E-08 0.55BOMC-14b 0.022 0.109 0.096 35.57 1.11E-06 0.014 1.48E-08 0.54BOMC-7b 0.024 0.111 0.098 34.38 1.11E-06 0.017 3.65E-08 0.54BOMC-7c 0.029 0.105 0.094 35.79 1.11E-06 0.024 1.64E-07 0.57BOMC-1 0.04 0.12 0.104 41.01 1.11E-06 0.030 2.22E-06 0.56BOMC-2 0.04 0.112 0.102 33.32 1.11E-06 0.049 2.72E-06 0.83BOMC-6 0.047 0.096 0.088 33.29 1.11E-06 0.091 4.79E-05 1.90BOMC-4 0.049 0.094 0.086 33.81 1.11E-06 0.099 6.02E-05 2.07BOMC-5 0.057 0.088 0.083 29.53 1.11E-06 0.195 2.19E-04 5.97Initial bed and surface layer composition of the tests of Wilcock & McArdell [1993] and Wilcock et al. [2001].

Appendix 7


fraction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Size (mm) 0.355 0.75 1.2 1.7 2.4 3.4 4.85 6.85 9.65 13.65 19.3 27.3 38.65 54.65 test p_1 p_2 p_3 p_4 p_5 p_6 p_7 p_8 p_9 p_10 p_11 p_12 p_13 p_14 J06 initial 0.001 0.034 0.016 0.012 0.037 0.084 0.120 0.097 0.083 0.091 0.143 0.139 0.084 0.060 J06-1 0.000 0.001 0.000 0.002 0.004 0.038 0.103 0.125 0.118 0.132 0.199 0.178 0.065 0.035 J06-2 0.000 0.001 0.000 0.002 0.004 0.038 0.103 0.125 0.118 0.132 0.199 0.178 0.065 0.035 J06-3 0.000 0.000 0.000 0.001 0.003 0.068 0.069 0.111 0.117 0.133 0.195 0.186 0.074 0.043 J06-4 0.000 0.000 0.000 0.001 0.004 0.077 0.052 0.111 0.104 0.131 0.211 0.192 0.078 0.041 J06-5 0.000 0.000 0.000 0.001 0.003 0.038 0.075 0.104 0.106 0.128 0.193 0.174 0.104 0.073 J06-6 0.000 0.000 0.000 0.001 0.005 0.038 0.088 0.098 0.102 0.120 0.177 0.187 0.109 0.075 J06-7 0.000 0.000 0.000 0.001 0.006 0.040 0.103 0.117 0.095 0.103 0.179 0.183 0.094 0.080 J06-8 0.000 0.000 0.000 0.000 0.005 0.037 0.105 0.133 0.108 0.118 0.175 0.172 0.090 0.057 J06-9 0.000 0.000 0.000 0.002 0.008 0.041 0.101 0.094 0.089 0.103 0.172 0.217 0.102 0.074 J06-10 0.000 0.000 0.000 0.001 0.005 0.055 0.128 0.117 0.101 0.095 0.148 0.183 0.101 0.067 J14 initial 0.001 0.077 0.037 0.034 0.033 0.076 0.109 0.088 0.076 0.083 0.130 0.126 0.076 0.054 J14-1 0.000 0.001 0.001 0.015 0.016 0.085 0.054 0.110 0.091 0.105 0.166 0.203 0.102 0.050 J14-2 0.000 0.001 0.003 0.007 0.012 0.076 0.067 0.105 0.112 0.096 0.163 0.197 0.105 0.056 J14-3 0.000 0.000 0.000 0.006 0.012 0.118 0.068 0.129 0.117 0.118 0.152 0.154 0.077 0.049 J14-4 0.000 0.002 0.002 0.013 0.010 0.092 0.051 0.118 0.099 0.122 0.163 0.169 0.095 0.066 J14-5 0.000 0.001 0.001 0.007 0.013 0.113 0.054 0.108 0.089 0.102 0.178 0.175 0.088 0.072 J14-6 0.000 0.002 0.003 0.012 0.015 0.124 0.070 0.134 0.108 0.103 0.132 0.151 0.089 0.058 J14-7 0.000 0.002 0.004 0.012 0.021 0.121 0.067 0.114 0.107 0.099 0.131 0.165 0.096 0.063 J14-8 0.000 0.001 0.001 0.011 0.015 0.110 0.062 0.130 0.111 0.111 0.152 0.165 0.085 0.046 J14-9 0.000 0.000 0.001 0.010 0.006 0.086 0.054 0.126 0.091 0.105 0.160 0.201 0.099 0.063 J21 initial 0.001 0.106 0.051 0.048 0.031 0.071 0.102 0.082 0.071 0.077 0.121 0.117 0.071 0.051 J21-1 0.000 0.003 0.001 0.030 0.027 0.080 0.188 0.124 0.079 0.082 0.129 0.149 0.068 0.040 J21-2 0.000 0.021 0.004 0.046 0.031 0.116 0.130 0.115 0.097 0.088 0.123 0.127 0.060 0.041 J21-3 0.000 0.021 0.010 0.027 0.025 0.091 0.117 0.127 0.103 0.099 0.119 0.138 0.075 0.047 J21-4 0.000 0.005 0.003 0.026 0.026 0.120 0.113 0.114 0.097 0.092 0.122 0.142 0.085 0.056 J21-5 0.000 0.018 0.012 0.044 0.031 0.155 0.114 0.140 0.094 0.092 0.109 0.097 0.061 0.034 J21-6 0.000 0.016 0.016 0.040 0.028 0.154 0.091 0.132 0.099 0.085 0.111 0.107 0.074 0.046 J21-7 0.000 0.015 0.015 0.045 0.030 0.176 0.069 0.120 0.102 0.090 0.125 0.135 0.051 0.029 J21-8 0.000 0.057 0.041 0.067 0.038 0.148 0.034 0.068 0.063 0.075 0.115 0.168 0.083 0.043 J27 initial 0.001 0.140 0.068 0.065 0.029 0.065 0.094 0.076 0.065 0.071 0.112 0.109 0.066 0.038 J27-1 0.000 0.074 0.038 0.069 0.029 0.155 0.073 0.103 0.074 0.079 0.101 0.113 0.054 0.039 J27-2 0.000 0.036 0.039 0.080 0.038 0.153 0.063 0.107 0.079 0.072 0.106 0.120 0.059 0.047 J27-3 0.000 0.078 0.051 0.082 0.041 0.158 0.087 0.107 0.068 0.079 0.087 0.091 0.044 0.028 J27-4 0.000 0.079 0.035 0.062 0.028 0.160 0.078 0.128 0.091 0.080 0.087 0.098 0.046 0.028 J27-5 0.000 0.041 0.041 0.098 0.039 0.148 0.063 0.111 0.083 0.079 0.097 0.108 0.057 0.036 J27-6 0.000 0.078 0.040 0.076 0.035 0.141 0.058 0.110 0.076 0.080 0.108 0.117 0.055 0.026 J27-7 0.000 0.100 0.059 0.068 0.026 0.113 0.051 0.091 0.067 0.074 0.119 0.139 0.063 0.031 J27-8 0.000 0.085 0.050 0.077 0.031 0.133 0.051 0.084 0.070 0.080 0.119 0.129 0.062 0.030 J27-9 0.000 0.119 0.066 0.092 0.032 0.140 0.040 0.068 0.062 0.059 0.085 0.127 0.073 0.039 J27-10 0.000 0.065 0.045 0.069 0.032 0.157 0.055 0.100 0.074 0.078 0.102 0.117 0.067 0.040 fraction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Size (mm) 0.355 0.75 1.2 1.7 2.4 3.4 4.85 6.85 9.65 13.65 19.3 27.3 38.65 54.65 test p_1 p_2 p_3 p_4 p_5 p_6 p_7 p_8 p_9 p_10 p_11 p_12 p_13 p_14 BOMC initial 0.177 0.104 0.033 0.029 0.026 0.060 0.086 0.069 0.059 0.065 0.102 0.099 0.053 0.038 BOMC-14c 0.198 0.130 0.059 0.051 0.029 0.085 0.114 0.088 0.069 0.055 0.062 0.048 0.010 0.003 BOMC-7a 0.246 0.174 0.037 0.043 0.028 0.067 0.110 0.077 0.055 0.052 0.055 0.038 0.014 0.005

Appendix 7


BOMC-14b 0.210 0.119 0.048 0.063 0.035 0.076 0.114 0.088 0.069 0.055 0.062 0.048 0.010 0.003 BOMC-7b 0.242 0.114 0.040 0.045 0.041 0.102 0.094 0.080 0.059 0.063 0.054 0.043 0.018 0.007 BOMC-7c 0.150 0.148 0.032 0.049 0.036 0.089 0.141 0.091 0.069 0.061 0.057 0.053 0.019 0.007 BOMC-1 0.325 0.158 0.034 0.020 0.014 0.052 0.065 0.061 0.051 0.047 0.068 0.061 0.027 0.018 BOMC-2 0.216 0.180 0.044 0.026 0.014 0.047 0.066 0.071 0.058 0.057 0.082 0.071 0.046 0.020 BOMC-6 0.205 0.194 0.056 0.033 0.019 0.064 0.056 0.056 0.046 0.056 0.081 0.082 0.027 0.025 BOMC-4 0.217 0.197 0.053 0.039 0.022 0.057 0.070 0.064 0.048 0.047 0.076 0.073 0.027 0.009 BOMC-5 0.332 0.171 0.054 0.039 0.023 0.068 0.072 0.048 0.031 0.028 0.042 0.047 0.032 0.014

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