UNCERTAINTY APPROACH TO THE INFILL OF NAVIGATION CHANNELS AND TRENCHES
WITH TELEMAC2D
University of Twente
Faculty of Engineering Technology Civil Engineering
BSc Thesis
Cynthia Wertwijn 26 – July - 2013
Uncertainty approach to the infill of navigation channels and trenches
BSc Thesis Cynthia Wertwijn
University of Twente Faculty of Engineering Technology Bachelor Civil Engineering
Supervisors:
Dr. Ir. P. C. Roos
Dr. Ir. M.A.F. Knaapen
PREFACE
Preface
This document is for the thesis of the Bachelor Civil Engineering at the University of Twente. In this bachelor thesis, a study of the uncertainty on sediment infill is shown. The research conducted for this thesis report is done at HR Wallingford under supervision of Michiel Knaapen, from HR Wallingford, and Pieter Roos, from the University.
First of all I want to thank my supervisors. I would like to thank Michiel Knaapen for his help with the problems in the programme and support during my placement period. I want to thank Pieter Roos for the effort and help with my progression in the research and report. Furthermore, I want to thank Pieter for his support during my placement period.
Secondly I want to thank the HR Wallingford for giving me the opportunity to do my research. I especially want to thank the Coast and Estuaries department of HR Wallingford for making me feel at home at the company. I also want to thank David Wyncoll for helping with special problems that arose in the program. During my placement period I met some amazing peers, whom I would like to thank for the pleasant time at the office, diners, lunches and sightseeing.
At last I would like to thank my family and friends for their help, love and support during this research period. Even when I was too far away to see them, they made me feel close to home and loved.
I hope you enjoy reading this report.
Cynthia Wertwijn
ABSTRACT
Abstract
Trenches and navigation channels are designed according to precise research in each area where they are placed. In this research the quantification of the uncertainties in the infill of these channels and trenches is studied. For studying this area the following goal is set:
Quantifying the uncertainties of a deterministic morphodynamic model for sandy beds by a sensitivity and probabilistic analysis with taken into account the input parameters and their underlying relationships.
The infill used to find the uncertainties needed to accomplish the goal is the amount of sediment that is placed in the trench after a time period. This infill is found by using the program SISYPHE and the by HR Wallingford recommended formulae for a flume. This goal can be split into 3 analyses, the sensitivity analysis, research of underlying relationships and probabilistic analysis. The sensitivity analysis will investigate the effect of each input parameter on the infill prediction. The settling velocity has the most influence on the infill prediction and the number of sediment classes does not have any influence on the infill prediction on its own. The influence of each input parameter on the infill prediction is shown in the table below.
Input parameter Range Infill rate uncertainty Effect
Settling velocity ( ) Positive effect
Median diameter( ) Negative effect
Porosity( ) Positive effect
90% grain diameter( ) Positive effect
Sediment density( ) Negative effect
Nikuradse parameter( ) Negative effect
Number of sediment classes( ) No effect
The research of underlying relationships between the input parameters resulted in simplified formulations. To create the simplified formulae a range is found by conducting a range for each found formula en combining these ranges to make the range that fits all the ranges for each formula.
With these relations and the sensitivity analysis the probabilistic analysis is done. In this research, the Monte Carlo method is used. In the Monte Carlo approach, a model prediction will be computed multiple times, each time with different values for the input parameters. For every run, the Monte Carlo analysis chooses a value randomly for each input parameter within their ranges and uses these values to predict the sand bed infill. The different results together give a normal distribution for the sediment infill. With this normal distribution the average infill value and the uncertainty of the infill can be found.
The average channel and trench infill is 0.1 . Approximately 98% of the channels and trenches with sandy beds will have a sand infill lower than 0.35m. This infill can be taken into account with designing a channel to keep the trench depth below a needed depth, which can save time and effort in researching the sediment characteristics of the site.
For the trench and channel width on trench depth 0.75m the average narrowing of the width for sandy beds is 11 . 98% of the trenches and channel will lie between 14 and 34m. This result can also be taken into account with designing a channel to keep the trench width above a certain wide.
CONTENTS
Contents
Preface ... 1
Abstract ... 2
Contents ... 3
Notation ... 5
1 Introduction ... 7
1.1. External organisation ... 7
1.2. Project description ... 7
1.3. Problem definition... 8
1.4. Objective ... 8
1.5. Researched questions ... 8
1.6. Methodology ... 8
1.7. Limitations ... 10
1.8. Thesis outline ... 10
2 Description of the deterministic model ... 11
2.1 Model geometry ... 11
2.2 Morphological change ... 12
2.3 Bed load transport ... 12
2.4 Suspended load transport ... 14
2.5 Hydrodynamic model ... 15
2.6 Example run ... 16
3 Sensitivity analysis ... 17
3.1 Effect of input parameters on the deterministic model ... 17
3.2 Range of input parameters ... 17
3.3 Effect of input parameters on the bed infill prediction ... 18
4 Relations ... 19
4.1 Different relations ... 19
4.2 Relations between input parameters ... 19
4.3 Additional ranges ... 20
5 Probabilistic analysis ... 21
5.1 Generating input for the probabilistic infill ... 21
5.2 Probabilistic infill ... 22
5.3 Conclusion of the infill quantities and uncertainties ... 23
6 Discussion ... 24
CONTENTS
6.1 Discussion on input ... 24
6.2 Discussion on methods ... 24
6.3 Discussion on results ... 24
7 Conclusion ... 25
8 Further research ... 26
9 References ... 27
Appendix ... 31
List of Figures Figure 1.1: Sequence of analysis with the in- and output according to Bakker, et al. (2009) ... 9
Figure 2.1: Top view of flume for deterministic model [m] ... 11
Figure 2.2: Side view of flume for deterministic model [m] ... 12
Figure 2.3: Comparison of modelled and experimented infill ... 16
Figure 5.1: Infill predictions of the 400 runs ... 22
Figure 5.2: Normal distribution of trench depth ... 22
Figure 5.3: Normal distribution of trench width ... 23
Figure A.1: Positive correlation (left), Negative correlation (middle), No correlation (right) ... 32
Figure A.2: Scatterplot of relation between settling velocity and 90% grain diameter ... 35
List of Tables Table 2.1: Input parameters for running deterministic model with their reference values ... 16
Table 3.1: Input parameters with influence on the bed infill prediction multiplying with 0.5 and 2 ... 17
Table 3.2: Input parameters with influence on the bed infill prediction within the ranges ... 18
Table 5.1: Order for generating the input parameters and the ranges for each input parameters. ... 21
Table 5.2: Mean and standard deviation for trench depth at end of run ... 23
Table 5.3: Mean and standard deviation for trench width at end of run ... 23
Table 7.1: Input parameters with influence on the bed infill prediction within the ranges ... 25
NOTATION
Notation
Symbol Definition Dimension
empirical bed transport coefficient in the flume [s2.4 m-1.4]
empirical bed transport coefficient for each sediment class [s2.4 m-1.4]
empirical suspended transport coefficient at in the flume [s2.4 m-1.4]
empirical suspended transport coefficient for each sediment class [s2.4 m-1.4]
flume width [m]
concentration of sediment in the flume [m3m-3]
drag coefficient at point x,y in the flume []
equilibrium concentration at point x,y in the flume [m3m-3]
concentration at time -1 [m3m-3]
deposition of suspended load [ms-1]
median grain diameter for each sediment class [m]
median grain diameter for each sediment class [m]
grain diameter for which 90% of the grains by mass is finer [m]
erosion of suspended load [ms-1]
gravity acceleration [ms-2]
water depth at point x,y in the flume [m]
significant wave height [m]
wave number [m-1]
Nikuradse roughness parameter [m]
Von Karman constant []
Number of sediment classes []
porosity of the bed []
volumetric bed-load flux at point x,y in the flume [m2s-1]
suspended load flux at point x,y in the flume [m2s-1]
sediment transport rate [m2s-1]
relative density of sediment []
NOTATION
slope friction in direction [ms-2]
slope of the waterway in direction [mm-1]
time [s]
depth average flow velocity in direction x [ms-1]
depth average flow velocity [ms-1]
critical velocity for each sediment class [ms-1]
total critical velocity at point x,y in the flume [ms-1]
wave velocity at point x,y in the flume [ms-1]
depth average flow velocity in y direction [ms-1]
average settling velocity [ms-1]
horizontal coordinate in the along-flume direction [m]
horizontal coordinate orthogonal to x [m]
vertical coordinate [m]
trench/channel depth (bottom elevation) [m]
reference bottom elevation [m-1]
coefficient for sloping bed effect []
dispersion coefficient in direction for suspended load [m2s-1]
dispersion coefficient in direction for suspended load [m2s-1]
change in bed level [m]
fluid density [kgm-3]
sediment density [kgm-3]
dry sediment density [kgm-3]
shear stress in direction [kgms-2]
kinematic viscosity [m2s-1]
orbital frequency [s-1]
INTRODUCTION
1 Introduction
Trenches and channels are chosen according to precise research in each area where they are placed. In this research the quantification of the uncertainties in the infill of the channels and trenches is researched. A general explanation of this research and the external organisation is given in this chapter.
This chapter starts with a description of the external organisation. It is followed by the description of the project itself in section 1.2. With this general description of the project, the problem is defined. This is followed by the main objective, section 1.4, for solving the problem described in section 1.3. Section 1.3 and 1.4 are used to create the questions which will be studied in this research, shown in section 1.5.
The objective and questions studied in this research are used to create a plan for this research. The used methods in used order are described in section 1.6. This research works with some limitations which are described in section 1.7. The last section of this chapter gives an outline of this report.
1.1. External organisation
About 66 years ago HR Wallingford was founded by the British government (Members of FLOODsite, 2009). Nowadays HR Wallingford is an independent non-profit organisation that works in the field of civil engineering and environmental hydraulics (About HR Wallingford, 2013). In this field, particularly the areas of rivers catchments, estuaries, coast and offshore, HR Wallingford is seen as a European leader with their knowledge and research (Members of FLOODsite, 2009). The company’s reputation is kept by re-investing their profit into research and development. Through this research and development HR Wallingford creates solutions for complicated problems and obstacles. The organisation has a global network of clients and partners from governmental organisations to universities (About HR Wallingford, 2013). The organisation invests in creating more knowledge and improving the knowledge they and others already have. HR Wallingford also helps others, by handing out information to universities and hosting students or young researchers on-site for collaborative projects (Members of FLOODsite, 2009).
The company consists of more than 280 employees all over the world (Members of FLOODsite, 2009).
HR Wallingford has offices in among others India, China and United Arab Emirates. The head-quarter is situated at Wallingford in the United Kingdom. The park is 76 acres and placed on the banks of the Thames in the region South Oxfordshire (British Council for Offices, 2011). On these 76 acres there are facilities from physical modelling to ship simulation centre and flood product testing. At the moment HR Wallingford is working on a project that is creating a cleaner, healthier tideway tunnel for the Thames by means of physical modelling the river and its engineering work (HR Wallingford, 2013).
This research project has been performed at the Coasts and Estuaries Group. The Coasts and Estuaries Group offers knowledge about environments and processes in coasts and estuaries. Research in this field provides information about managing water and sediment from erosion, sediment transport to flooding and the protection of the coastal environments.
1.2. Project description
This research is a part of a bigger project within HR Wallingford, called Application of uncertainty analysis. This research is about analysing the uncertainty, range in which the parameters can vary, of the infill of channels and trenches. Because of the uncertainties in the infill prediction, channel design is a risk in projects about for example building harbours or installing pipelines. To decrease the uncertainty for each input parameter the ranges need to be narrowed or confirmed. As a result the calculated trench and channel infill are among others more precise or certain for the use in harbour and pipeline installation design (DPW, 2012) (Wilkens & Chesher, 2011).
INTRODUCTION
The infill prediction can be modelled with SISYPHE by using HR Wallingford’s standard deterministic model for calculating trench infill in sandy beds (Wilkens & Chesher, 2011). SISYPHE is a numerical transport model that belongs with the TELEMAC flow model. This model calculates the morphological changes with various flow and sediment conditions. The calculations are functions of time-varying flow conditions, like wave height and flow velocity, and sediment characteristics. SISYPHE uses among others the formulae of Soulsby and Chezy to compute the change bed levels over time. The variables taken into account by the SISYPHE are among others the effect of waves and currents and cohesive and non- cohesive sediment characteristic. It can be used for a large variety of hydrodynamic situations and sediment mixtures from rivers to harbours. In this research SISYPHE will be run on an idealised geometry to keep the simulation time short allowing for the number of runs required to assess the uncertainties in the predictions (Villaret & El Kadi, 2010).
1.3. Problem definition
The previous section explained that channel design is a risk in projects about for example building harbours or installing pipelines. In recent research there are models created to calculate the morphological changes of channels and trenches over time with the use of SISYPHE. However, these deterministic models still have uncertainties in the calculation of migration and infill. By generating a tool that quantifies the uncertainties, the best channel or trench design can be modelled more accurately. This will optimise the design for each trench of channel on maintenance and dredging. This will save costs in these kinds of projects and supply more knowledge about uncertainties in the whole coastal area (Wilkens & Chesher, 2011).
1.4. Objective
The main objective of this research is:
Quantifying uncertainties of the deterministic model for sandy sea beds by a sensitivity and probabilistic analysis with taken into account the input parameters and their underlying relationships.
1.5. Researched questions
The main question put up for this study can be deduced from the problem definition in section 1.3 and objective in section 1.4. The mean question is as followed:
What is the magnitude of uncertainties in channel and trench infill?
This main question can be divided into 4 sub-questions listed below:
How works the deterministic model used by HR Wallingford for trench and channel infill calculations?
What is the influence of the input parameters on the infill prediction model?
What are the underlying relationships that will have influence on the values of the input parameters?
How should uncertainty in channel and trench infill be quantified?
1.6. Methodology
Figure 1.1 shows the sequence of analysis according to Bakker, Uijttewaal, Winterwerp, Jonkman and Veale (2009) that will be followed. The starting values of the input parameters are the values used in the flume experiment, shown in model geometry in section 2.1 and table 2.1.
INTRODUCTION
Figure 1.1: Sequence of analysis with the in- and output according to Bakker, et al. (2009)
This research starts with identifying the deterministic model and its input parameters. The input parameters are the parameters that hold the input for the deterministic model. The other parameters, that describe for example the relation between input parameters, will be called parameters. With these input parameters the infill rate can be found. The infill rate is the percentage ratio between the sediment infill at time in the trench and the initial trench depth.
The uncertainty analysis starts with investigating the effect is of each input parameter on the deterministic model. For each input parameter the value is multiplied by a half and two. With these values the model is run multiple times. The variations in results show the effect of an input parameter on the model. Afterwards the range for each input parameter is found. This is just done by doing some literature research about how much they can vary within the limitations of this research. A comparison between these ranges and the temporary ranges given by multiplying by and 2 is conducted to find the effect of each input parameter on the infill rate. The conclusion that is drawn from this comparison is checked by running the deterministic model for each input parameter’s range edges and a few values within the range. The amount in which the infill predictions varies by changing an input parameter within its range, shows how the sensitivity of the deterministic model is on each input parameter.
Afterwards the change in infill rate for each input parameter is compared with each other. This shows the order of influence of input parameters on the infill (Booij, 2012).
Before the probabilistic analysis starts, the relationship between the input parameters needs to be identified. By identifying the relation and the causal connections, the sequence of important and influential input parameters can change. An input parameter that is not influential on its own can have an immense impact on the infill in combination with a change of another input parameter. This would make this input parameter due to its relation to the other input parameter more important in the infill prediction than expected from the sensitivity analysis (Van Gelder, 2000). Furthermore, relation is researched to narrow the ranges for each input parameter. To find out the relations, formulae which link the input parameters will be searched. For each found formula a range is conducted by changing all the parameters in the formula and the input parameters within given ranges. These ranges can be combined to make a range for each input parameter that fits all the ranges for each formula.
Bakker, et al. used the Monte Carlo analysis for analysing the mud infill prediction, because it is an easy applicable approach that works with ranges. In this research, which is similar, the Monte Carlo would suit the best as well. In the Monte Carlo approach, a model prediction will be computed multiple times, each time with different values for the input parameters. For every run, the Monte Carlo analysis randomly chooses a value to each input parameter within their ranges and uses these values to predict the bed infill. The different results together can be put in a probabilistic distribution to find the most likely value and the uncertainty of the infill.
Input Sequence of analysis Output
Deterministic calculation
Sensitivity analysis Probabilistic
analysis Start values of the
input parameters Range within the input
parameters can vary Relations between the
input parameters Probability density
function of the infill rate Order of influence of input parameters on
infill Infill rate
INTRODUCTION
1.7. Limitations
For this research some variables are fixed, which are explained in section 1.7.1. Furthermore, this research is only done for sandy beds. The explanation of this limitation is explained in section 1.7.2. In the last section the researched conditions will be defined.
1.7.1. Fixed variables
The wave effects are also fixed variables. At the moment these variables cannot be changed in the Monte Carlo tool for SISYPHE. When these variables can be changed the uncertainty, analysis should be done with these input parameters, because for example storm and wind will change the wave effects which can have a big effect on the infill prediction (Landry & Garcia, 2011).
Furthermore, the empirical slope factor, which is a factor that included the diffusion in the bed evolution to make the bed evolution more in line with the actual bed load transport, will be seen as a fixed variable (Villaret & El Kadi, 2010). This empirical factor can vary between 0 and infinity. To get the best possible value for this factor, real-life measurements need to be done (Wilkens & Chesher, 2011). Next to this, studying the empirical factor can be investigated on its own due to the big range and especially the little amount of knowledge about reasons for choosing a value. In this research the empirical slope factor will be kept at the recommended rate for TELEMAC, which is 1.3 (Villaret & El Kadi, 2010).
1.7.2. Sediment limitation
This research only uses bed material that is sand for two reasons. First of all estuaries are dominated by sand (McNally & Mehta, 2002). Secondly most of the harbours are placed in sandy sites (Van Rijn, Harbour siltation and control measures, 2012) (Leys & Mulligan, 2012).
1.7.3. Researched conditions
Sediment conditions changes within each place on the bed and changes over time in water systems. This means that the precise sediment characteristics are uncertain even after expensive testing. To find and narrow the uncertainty ranges it is important to see this group as unknown input parameters. Because of these reasons the sediment conditions will be researched in this research (Ongley, 1996).
The fluid conditions contains among others the wave forces, the water depth and flow velocity. The average values needed for these input parameters can be obtained with SeaZone (SeaZone Solutions division of HR Wallingford Ltd.) According to Soulsby (1997) these input parameters except the water depth have an uncertainty of 10% and the water depth has an uncertainty of 5%. By researching the fluid conditions, the wave forces will change the velocities and other fluid conditions. Because of this the uncertainty cannot be narrowed that much till none, so these conditions are not researched.
1.8. Thesis outline
Model description In chapter 2, the main physical processes underlying the deterministic model used in this research.
Sensitivity analysis In chapter 3, the input parameters for sandy beds will be quantified by a sensitivity analysis.
Input relationships Chapter 4 will explain the relations between the input parameters Probabilistic analysis Chapter 5 will show the uncertainty by doing a probabilistic analysis
Discussion Chapter 6 discusses in which extent the objectives are achieved and gives an evaluation on the methodology and results.
Conclusion Chapter 7 gives the conclusion form this research
Further research Chapter 8 gives recommendations and a set-up for further research.
DESCRIPTION OF THE DETERMINISTIC MODEL
2 Description of the deterministic model
In this chapter the main processes of the deterministic model are described to answer ‘How works the deterministic model used by HR Wallingford for trench and channel infill calculations?’ which is the first sub-question. The deterministic model, HR-SISYPHE, can run a variety of sediment transport formulae.
The formulae used in this research and described in this part, are the by HR-Wallingford recommended settings (Wilkens & Chesher, 2011) (Villaret & El Kadi, 2010). The model geometry is defined in section 2.1. The calculations are divided into processes which will be explained in different sections. Section 2.2 explains how the morphological changes in the flume are calculated. To calculate morphological changes the bed load (section 2.3) and suspended load transport (section 2.4) need to be calculated. These transport processes are driven by a hydrodynamic model, which is explained in section 2.5. The chapter ends with a section, in which the settings and results of an example run are described.
2.1 Model geometry
This research works with a flume, the top view of which is shown in figure 2.1. This figure shows the 4 boundaries of the model which are red and blue. At the left blue boundary a uniform discharge of water and sediment flows into the flume. The water with sediment will flow out of the flume at the right blue boundary with a uniform water depth. This water flow direction is subject to the currents. The two red boundaries are closed-off, this means that no water or sediment flows in or out through these boundaries. The figure shows that the geometry of this model is smaller than 1 meter. Because the Soulsby-Van Rijn formulae are not valid for water depths smaller than 1 meter, the flume will be scaled up by multiplying the laboratory dimension by 10 and the time by to obtain the researched dimensions (Van Rijn, 1984) (Wilkens & Chesher, 2011).
Figure 2.1: Top view of flume for deterministic model [m]
Furthermore figure 2.2 gives a side impression of the flume. In this figure the velocity in the direction x at height is shown. In this research the depth averaged shallow water equations are applied. These equations assume that a logarithmic velocity profile at the start of the flume stays in the same shape, at every point in the flume, which will be stretched out when the water depth increases. The velocity profile changes due to vegetation, which is not included in the research, and a high slope gradient. When the slope gradient is high the velocity profile in the trench will differ from the velocity profile outside the trench, and there might even be flow reversal in the trench. The trench is in figure 2.2 the isosceles trapezium with the maximum depth which is 0.125m in this research. In this research the slope gradient , formula is shown in equation 2.1, is 0.125 depth change from 5 to 6.25m and 8.25 to 9.5 m in direction . This is much smaller than which still gives the logarithmic velocity profile according to Berger, Teeter and Pankow (1993).
(2.1)
DESCRIPTION OF THE DETERMINISTIC MODEL
Figure 2.2: Side view of flume for deterministic model [m]
This justifies the use of a depth averaged velocity in this deterministic model. So the model will be run according to a 2D horizontal approach.
2.2 Morphological change
When water flows over a granular bed, it causes a shear stress on that bed. If the flow velocity is high enough, this bed shear stress can exceed the threshold value. The threshold value marks the point from which the sediment will start moving. When the bed shear stress exceeds the threshold value, the sediment will start to move. This movement is called transport, because the sediment is transported to another place (Fox, McDonald, & Prichard, 2011).
Sediment can be transported through suspension or by rolling, hopping and sliding alongside the bed (Snellink & de Korte, 2010). The transport of suspended sediment particles is known as suspended load.
The transport of sediment by rolling, hopping and sliding is called bed load transport. The bed load flux and the suspended load flux together is the total sediment load. This total load at place on the bed for a time period is needed for calculating the change in bed shape. The change in bed-level is calculated with the porosity , which is a value for the percentage of space in the soil (Soulsby, 1997).
(2.3)
2.3 Bed load transport
The bed load transport can occur over a flat bed by the fluid velocity and over a slope by gravity and fluid velocity. The velocity differs at different heights above the bed. The velocity near the bed will be lower than the velocity near the surface. This change in velocity is due to the bed friction. The friction rate changes by changes in the bed roughness. The bed roughness impacts, as explained in section 2.3.2, the velocity near the bed and the bed load transport. Next to this, the velocity depends on currents and waves. The effects that currents and waves have on the bed load transport, is explained in section 2.3.3.
All these effects together give the bed load transport, defined in section 2.3.1.
2.3.1 General bed-load transport formulae
The bed load transport is calculated according to Soulsby and Van Rijn (Soulsby, 1997) :
(2.4)
(2.5)
The bed load transport is calculated with a correction method. This correction method takes in account the bottom elevation over length in the fluid flow direction and an empirical factor . The (2.2)
DESCRIPTION OF THE DETERMINISTIC MODEL
empirical factor ensures that a diffusion factor is included in the bed evolution (Villaret & El Kadi, 2010). With this correction method the bed evolution is more realistic.
The critical velocity also needs to be specified for the bed-load transport. This parameter shows from which velocity the sediment start moving on the bed and is explained in section 2.3.4. The depth- averaged velocity is a vector with -component and -component . This velocity expresses the current effects in the bed transport formula by changing its sign. The wave effects are included in the equation 2.4 by the wave velocity . This wave effect is explained in section 2.3.3. Furthermore, the bed roughness affects the bed evolution. This effect is implemented by the drag coefficient , to be explained in section 2.3.2. Next to this, the bed evolution is calculated with an empirical bed transport factor with each median diameter for sediment class by Soulsby and Van Rijn (Soulsby, 1997):
(2.6)
(2.7)
These equations need the median grain diameter for each sediment class , the water depth , acceleration due to gravity and the relative density . The gravitational acceleration is (Fox, McDonald, & Prichard, 2011). The relative density of sediment is given by dividing the sediment by the fluid density (Soulsby, 1997):
(2.8)
2.3.2 Bed roughness effects
The drag coefficient, also known als bottom friction is calculated according to Nikuradse’s law (1933):
(2.9)
The value of the bottom friction is determined by the water depth, Van Karman constant , which is 0.4 according to Fox, McDonald and Prichard (2011), and the Nikuradse roughness parameter .The Nikuradse parameter is a parameter that introduces the bed roughness to the transport processes and is related to the different bottom types.
The fixed variables are among others the gravity acceleration and Von Karman constant ( are by definition the shown value.
2.3.3 Wave effects
The wave effects can be expressed in terms of the orbital velocity. The orbital velocity can be calculated in different ways. In this research orbital velocity is generated from the linear wave theory with the orbital (wave) frequency and significant wave height (Villaret & El Kadi, 2010):
(2.10)
This formula depends on the wave frequency and wave number . The orbital frequency is the frequency in which waves arrive. The wave number k is a function of the orbital frequency , water depth and the gravity acceleration , shown in equation 2.11 (Soulsby, 1997):
(2.11)
DESCRIPTION OF THE DETERMINISTIC MODEL
2.3.4 Critical velocity
The critical velocity is the depth-averaged speed required to move the sediment on the bed and is according to Van Rijn (1984)1 given by:
for (2.12)
for (2.13)
(2.14)
In these formulae the median diameter has a range within which formula is the most suitable. Next to the median diameter, the grain diameter for which 90% of the grains by mass are finer and the water depth are key elements to compute the critical current velocity.
2.4 Suspended load transport
The suspended load transport is the sediment that is carried into suspension by currents, waves and flow velocities. The general suspension discharge formulae, defined in section 2.4.1, use the net sediment flux. The net sediment flux is the net difference between the sand picked up and positioned at the same place (section 2.4.2).
2.4.1 General suspended transport formulae
The suspend transport can be described as the depth integrated concentration of sediment in the water multiplied by the flow velocity (Fox, McDonald, & Prichard, 2011):
(2.15)
The concentration of suspended sediment in a place is the amount in dispersion with the net sediment flux minus the concentration that is moved away from this place by the velocities. Suspended sediment concentration is calculated with the following formula (Huybrechts, Villaret, & Hervouet, 2010):
(2.16)
The formula works with the dispersion coefficient , depth averaged velocities and , which are described with the hydrodynamic model in section 2.5, and net sediment flux .
2.4.2 Bed evolution due to suspension
The difference in erosion and deposition will be calculated for the length above the bed equal to which is zero (Knaapen & Kelly, 2011). The bed evolution can be put down according to Miles’
equation 2.17 (1981) with the use of the settling velocity and the two concentrations.
(2.17)
The concentration is the concentration in the previous time step. In the first time step the value is set to zero. The other concentration, equilibrium concentration, is calculated with Soulsby-Van Rijn (Soulsby, 1997):
(2.18)
The sediment transport rate can be calculated in the same way as the bed evolution at equation 2.4 and 2.5 with an empirical suspended transport factor (Soulsby, 1997):
1 Keep attention, because the dimensions do not match
DESCRIPTION OF THE DETERMINISTIC MODEL
(2.19)
(2.20)
The depth-averaged current velocity , threshold current velocity , orbital velocity , drag coefficient and bed slop factor are already defined in section 2.3 Bed load transport. This sediment transport rate is calculated with empirical suspended transport factor for each median diameter by Soulsby and Van Rijn (Soulsby, 1997):
(2.21)
(2.22)
2.5 Hydrodynamic model
The 2D hydrodynamic model solves the basic mass and momentum conservation of the horizontal components and . In this section among others the mass conservation equation and the momentum conservation equations are given in section 2.5.1. These equations need the stresses on the fluid, shown in section 2.5.2 and the slope effects, shown in section 2.5.3.
2.5.1 General equations
The mass conservation law states that the amount of change in volume the fluid is the same as the amount fluid and sediment added and removed to the volume (Fox, McDonald, & Prichard, 2011). With this law the mass conservation equation has been composed with the scalars for the depth-averaged velocity in the and direction, and , in equation 2.23.
(2.23)
The momentum law states that the change in momentum in a time period is the same as the change in rate of momentum added to the sum of forces acting on the volume. With this law the momentum conservation equations have been composed.
(2.24)
(2.25)
2.5.2 Stresses on the fluid
The stresses on the fluid are the bed shear stress and the transverse shear stresses. According to Villaret and El Kadi (2010) in and direction is:
(2.26)
(2.27)
The transverse shear stresses are described with the kinematic viscosity (Steffler & Blackburn, 2002):
(2.28)
DESCRIPTION OF THE DETERMINISTIC MODEL
2.5.3 Slope friction
According to Steffler and Blackburn (2002) and Villaret and El Kadi (2010) the friction slope in direction x and in direction y can be defined as:
(2.29)
(2.30)
2.6 Example run
In the previous section is shown that the model needs a lot of data. The geometry shown in section 2.1 gives the slope, bed shape and width of the model. The width and other needed parameters for the run are shown in table 2.1.
Table 2.1: Input parameters for running deterministic model with their reference values
The run is done with time steps of 1 second for a time period of 113841 seconds which is in real-life 36000 seconds which is 10 hours. The result of this run, shown in figure Figure 2.3, with the initial bed-level and the measured bed level after 10 hours in real-life have been put together to present the difference between the modelled infill and the measured infill after 10 hours in real-life. The figure that shows the results is taken in the middle width of the flume over the length from 6 till 14 meter. It appears that the model has a slight difference in the infill to the measured bed level. The infill rate of this run is 52% of the trench infill at point 84,5.5.
Figure 2.3: Comparison of modelled and experimented infill
Parameter Value Parameter Value
Median diameter for sediment
class 1 ( ) Sediment discharge at start of flume
( ) 0.528
Median diameter for sediment
class 2 ( ) Depth averaged velocity ( )
Coefficient for sloping bed effect ( )
90% grain diameter ( ) Settling velocity ( ) Water depth ( ) 2.55 m Dispersion coefficient ( and ) Significant wave height ( ) Fluid density ( ) Nikuradse parameter ( ) 0.015 m Sediment density ( ) Number of sediment classes ( ) 2 Kinematic viscosity ( )
Porosity of bed ( ) 0.4 Orbital frequency ( )
Von Karman constant ( ) 0.4 Gravity acceleration ( )
SENSITIVITY ANALYSIS
3 Sensitivity analysis
In this section the second sub-question ‘What is the influence of the input parameters on the infill prediction model?’ by conducting a sensitivity analysis with each input parameter. This analysis starts with finding out the effect of the input parameter on the deterministic model in section 3.1. This is followed by section 3.2 which determines the range for each input parameter can vary. With this range and the effect on the model, the effect on the bed prediction can be defined in section 3.3.
3.1 Effect of input parameters on the deterministic model
The effect of each input parameter on the deterministic model is done by multiplying the start value and dividing the start value by two. These two values define a range in which each input parameter varies.
The model is run for values within these ranges. Afterwards, the differences between the infill rates for the start values and the boundaries are calculated. This difference is measured in the middle of the flume width (5.5 m in y direction in the deterministic model) and on the whole of the flume length. The place where the measurement is taken, is shown by the purple line in Figure 2.1. The list of input parameters is shown in Table 3.1 with the infill rate. The infill rate is calculated at the deepest point of the modelled reference bed level from chapter 2 at the end of the run period (84 m in the x direction in the deterministic model). This point is chosen to keep into account that the trench will migrate over time. A positive effect implies that increasing the value for the input parameter gives an increased infill rate. A negative effect of an input parameter on the infill rate gives a decreased infill rate with an increased value for the input parameter.
Table 3.1: Input parameters with influence on the bed infill prediction multiplying with 0.5 and 2
3.2 Range of input parameters
Sediment conditions change within each place on the bed and change over time in water systems. This means that the precise sediment characteristics are uncertain even after extensive testing. To find and narrow the range of uncertainty it is important to see the input parameters in this group as unknown.
This means that the input parameters will not vary around a measured value (Ongley, 1996). The sediment density actually varies around with a variation percentage of 2% according to Soulsby (1997). The ranges for the density of sandy beds given by Hillel (1980b) and Atkinson (Atkinson)are approximately the same as the range given by Soulsby (1997). This means that the range for the sediment density will be from to .
As this research only covers sandy beds, the sediment characteristics have to work for a grain size diameter between 0.06 and 4.76 mm to cover the ranges given for sand by among others The Dutch Normalisation Institution (Commissie Geotechnics, 1989), American Society for Testing and Materials (ASTM, 2000) and Wentworth’s grain size scale (Soulsby, 1997). These edges in grain sizes are the edges for the median diameter and the 90% grain diameter. Within these grain sizes the porosity of the bed can vary from 0.25 to 0.46 (Environmental Science Division, 2013) (Soulsby, 1997).
Input
Parameter For 0.5 value Infill rate For 2 value Effect
Negative effect
Positive effect
Negative effect
52% No effect
Positive effect
Positive effect
No effect
SENSITIVITY ANALYSIS
The restrictions in sediment diameter also create a range for the settling velocity. The settling velocity can be calculated in multiple ways. Sadat-Helbar, Amiri-Tokaldany, Darby and Shafaie (2009) defined the settling velocity for each diameter according to, among others, Stokes (Graf, 1971) and Van Rijn (Van Rijn, 1989). The settling velocity for sand lies approximately between and (Sadat-Helbar, Amiri-Tokaldany, Darby, & Shafaie, 2009). This range is bigger than the range, from till , Soulsby (1997) would give for sediment particles within these grain size edges and sediment densities.
To get the largest uncertainty the range for the settling velocity will be from and .
Furthermore, the ranges for the Nikuradse roughness coefficient and the number of sediment classes need to be specified. The number of sediment classes is minimal 1. For the upper boundary 7 classes are chosen, because sand can be divided into 7 groups according to Folk (1954). And the Nikuradse roughness coefficient will vary between and . This range results from the range for the bed roughness length , shown in equation 3.1 (Nikuradse, 1933).
(3.1)
The bed roughness length varies between 0.3 mm and 6.0 mm for sandy beds. This will give a range from 0.009 till 0.18 m for the Nikuradse roughness parameter (Soulsby, 1997).
3.3 Effect of input parameters on the bed infill prediction
With the ranges given in the previous section, the infill prediction is calculated by varying each input parameter separately within its ranges. The differences between the infill for the start values and the edges are calculated on the same place as section 3.1. The order of influence with percentage of change on the infill at point (84, 5.5) for each input parameter is shown in Table 3.2. In the order of influence number 1 is the most influential and number 7 is the least influential. A positive difference in infill rate between the upper and bottom edge shows that the input parameter has a positive effect, explained in section 3.1.
Table 3.2: Input parameters with influence on the bed infill prediction within the ranges Order of
influence Input
Parameter Range
Infill rate Infill
uncertainty percentage*
For lowest
parameter value For highest parameter value
1
2
3
4
5
6
7
