AN EFFICIENT METHOD OF ESTIMATING SEDIMENT DISCHARGE IN RIVERS
by
Hadi Emdad
B.S. in Irrigation Engineering, Shiraz University, Iran, 1975
M. Engineering in Engineering Science, Pennsylvania State University, 1989
Submitted to the Graduate Faculty of
The School of Engineering in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
University of Pittsburgh
2004
UNIVERSITY OF PITTSBURGH
SCHOOL OF ENGINEERING
This dissertation was presented
by
Hadi Emdad
It was defended on
December 3, 2004
and approved by
Rafael G. Quimpo, Professor, Department of Civil and Environmental Engineering
Tin-Kan Hung, Professor, Department of Civil and Environmental Engineering
William Matlack, Professor, Graduate School of Public and International Affairs
Allan Bullen, Emeritus Professor, Department of Civil and Environmental Engineering
Dissertation Director: Chao-Lin Chiu, Professor, Department of Civil and Environmental Engineering
ii AN EFFICIENT METHOD OF ESTIMATING SEDIMENT DISCHARGE IN RIVERS
Hadi Emdad, PhD
University of Pittsburgh, 2004
Estimation of sediment discharge by conventional measurement methods is expensive and
labor intensive. An efficient method that can estimate the average cross-sectional sediment
concentration and discharge in a river channel was developed. This method relies on the
Chiu’s velocity and sediment distribution models, which are based on the probability concept.
The advantage of this approach compared to conventional methods is that the velocity and sediment concentration can be accurately estimated over the entire water depth, from the channel bed to the water surface. This method requires determining the location of a vertical (y-axis), where the maximum velocity of the entire cross section occurs. The correlation between the mean sediment concentration on each vertical and the cross-sectional mean concentration was analyzed. It led to the conclusion that sediment sampling should be
conducted on y-axis at a point where sediment concentration is equal to the mean concentration on y-axis. A family of plots for selection of the sampling location was developed.
Finally, a method based on Chiu’s sediment transport model and data analysis was developed to estimate the mean sediment concentration in a channel section. This cross-sectional mean sediment concentration can be used to calculate the sediment discharge.
iii
TABLE OF CONTENTS
1.0 INTRODUCTION...... 1 1.1 PROBLEM AND BACKGROUND ...... 1
1.2 OBJECTIVE AND SCOPES OF STUDY ...... 3
2.0 REVIEW OF LITERATURE...... 5 2.1 DISCHARGE MEASUREMENT BY CONVENTIONAL METHODS...... 5
2.1.1 Mid-Section Method ...... 6
2.2 MEASUREMENT OF SUSPENDED SEDIMENT DISCHARGE...... 8
2.2.1 The Equal-Discharge–Increment Method (EDI) ...... 9
2.2.2 Equal-Width-Increment Method (EWI)...... 10
2.2.3 Suspended Sediment Sampling Methods...... 12
2.2.4 Location of Sampling Points on the Vertical...... 13
2.3 ESTIMATION OF SUSPENDED SEDIMENT DISCHARGE...... 14
2.3.1 Calculation of Cross-sectional Concentration (Single Vertical Method) ...... 15
2.3.2 Rating Curve Method...... 16
3.0 DESCRIPTION OF DATA...... 18 4.0 MATHEMATICAL MODELS OF VELOCITY DISTRIBUTION ...... 21 4.1 SIMPLE POWER LAW VELOCITY DISTRIBUTION ...... 21
4.2 PRANDTL-VON KARMAN UNIVERSAL VELOCITY EQUATION...... 22
4.2.1 Parameter Estimation of the Logarithmic Velocity Distribution...... 23
4.3 CHIU’S VELOCITY DISTRIBUTION...... 25
4.3.1 Model Formulation ...... 25
iv 4.3.2 Location of y-axis ...... 28
4.3.3 Relation Between Mean and Maximum Velocities ...... 37
4.3.4 Parameter Estimation of Chiu’s Velocity Distribution...... 43
5.0 ANALYSIS OF SEDIMENT SIZE DISTRIBUTION...... 52 6.0 MATHEMATICAL MODELS OF SEDIMENT DISTRIBUTION ...... 64 6.1 CONVENTIONAL MODLE-ROUSE EQUATION ...... 64
6.1.1 Parameter Estimation...... 66
6.2 CHIU’S MATHEMATICAL MODEL OF DISTRIBUTION OF SEDIMENT
CONCENTRATION...... 67
6.2.1 Parameter Estimation...... 72
7.0 CALCULATION OF THE CROSS-SECTIONAL MEAN CONCENTRATION BY THE EFFICIENT METHOD ...... 83 7.1 CORRELATION BETWEEN THE CROSS-SECTIONAL MEAN
CONCENTRATION AND THE MEAN CONCENTRATION ON EACH VERTICAL ...... 83
7.2 CORRELATION BETWEEN CROSS-SECTIONAL MEAN CONCENTRATION
AND MEAN CONCENTRATION ON Y-AXIS...... 90
7.2.1 Relation of r2 to Sediment Diameter and Location of Verticals ...... 100
7.2.2 Relation of ψ to Q and d50...... 109
7.3 RELATION BETWEEN D50 AND WATER DISCHARGE...... 116
7.4 MEAN SEDIMENT CONCENTRATION ON Y-AXIS USING CHIU’S
EQUATION(2)...... 121
7.5 RELATION BETWEEN Θ AND D 50 ...... 123
7.6 COMPUTATION STEPS FOR DETERMINING LOCATION OF MEAN SEDIMENT
CONCENTRATION ON Y-AXIS...... 126
v y y y 7.7 RELATION OF M TO c , v AND ...... 141 D D D
7.8 RELATION BETWEEN M AND Ψ ...... 148
8.0 SUMMARY AND CONCLUSIONS ...... 154 BIBLIOGRAPHY...... 156
vi
LIST OF TABLES
Table 4-1 h/D for Missouri River at Nebraska City, M=2.56...... 46
Table 5-1 Sediment Size Distribution, Mississippi River...... 53
Table 5-2 d50 along y-axis, Missouri River at Ponca Station...... 60
Table 5-3 d50 along y-axis, Missouri River at Nebraska City...... 62
Table 6-1 Hydraulic Data, Missouri River ...... 74
Table 6-2 Hydraulic Data, Missouri River at Omaha...... 76
Table 7-1 Relation between ψ and M ...... 152
Table 7-2 Location of Sampling Point on y-axis...... 153
vii
LIST OF FIGURES
Figure 2-1 Sketch of Mid-section Method of Computing Cross-sectional Area for Discharge Measurement...... 8
Figure 3-1 Sampling Schematic Missouri River...... 20
Figure 4-1 Prndlt-von Karman Velocity Distribution near Channel Bed...... 24
Figure 4-2 Isovels Showing Location of y-axis Mississippi River at Tarbert (2/1/1996) ...... 29
Figure 4-3 Isovels Showing Location of y-axis Missouri River at Omaha (4/26/1978) ...... 29
Figure 4-4 Location of y-axis, Mississippi River at Tarbert...... 30
Figure 4-5 Location of y-axis, Mississippi River at Union Point...... 31
Figure 4-6 Location of y-axis, Mississippi River at Range 362.2 ...... 32
Figure 4-7 Location of y-axis, Missouri River at Omaha...... 33
Figure 4-8 Location of y-axis, Missouri River at Sioux City ...... 34
Figure 4-9 Channel Cross Section at Measured Verticals for Different Discharges, Missouri River at Sioux City...... 35
Figure 4-10 Channel Cross Section at Measured Verticals for Different Discharges, Missouri River at Nebraska City...... 36
Figure 4-11 Relation between u and umax Mississippi River at Union Point, 1994, 95 and 96 ... 38
Figure 4-12 Relation between u and umax Mississippi River at Tarbert,1995 and 1996 ...... 39
Figure 4-13 Relation between Mean and Maximum Velocities, Missouri River at Ponca, 10/26/1978 to 10/18/1979...... 40
Figure 4-14 Relation between Mean and Maximum Velocities Missouri River at Gayville 4/22/1980 to 7/21/1981...... 41
viii Figure 4-15 Relation between Mean and Maximum Velocities Missouri River at Omaha, 10/6/1976 to 7/15/1981...... 42
Figure 4-16 Relation between h/D and umax Missouri River at Nebraska City ...... 47
Figure 4-17 Relation between h and umax Missouri River at Nebraska City ...... 48
Figure 4-18 Velocity Distribution (Chiu’s Equations) ...... 49
Figure 4-19 Comparison of Velocity Distributions ...... 50
Figure 4-20 Applicability of Chiu’s Velocity Distribution Near Bed and umax below Water ...... 51
Figure 5-1 Determining Sample d50 on y-axis, from Grain Size Distribution,...... 57
Figure 5-2 Distribution of Sample d50 on y-axis, Omaha Station, D=18.4 ft, 10/16/1976...... 58
Figure 5-3 Relation between d50 and y/D, 8-foot Wide Flume...... 59
Figure 6-1 Comparison of Sediment Distribution Models; Missouri River at Nebraska City, 6/14/1977 ...... 69
Figure 6-2 Comparison of Sediment Distribution Models ...... 70
Figure 6-3 Distribution of Sediment Concentration for two Sediment Diameters, Missouri River at Nebraska City, 9/13/1977 ...... 71
Figure 6-4 Chiu’s Distribution of Sediment Concentration, Missouri River at Nebraska City, D=11.0 ft, 9/13/1977...... 77
Figure 6-5 Chiu’s Distribution of Sediment Concentration, Missouri River at Omaha, D=18.4 ft, 10/6/1976 ...... 78
Figure 6-6 Chiu’s Distribution of Sediment Concentration, Missouri River at Omaha, D=18.5, 4/26/1978 ...... 79
Figure 6-7 Comparing the Computed and Measured Mean Concentration on y-axis, Missouri River at Nebraska City...... 80
Figure 6-8 Transverse Distribution of Average Sediment Concentration and Velocity, Missouri River at Nebraska City...... 81
Figure 6-9 Distribution of Average Concentration and Velocity (Measured) across the Channel of Missouri River at Nebraska City ...... 82
ix Figure 7-1 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration (All Sizes), Mississippi River at Tarbert, 1995 and 96 ...... 84
Figure 7-2 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Mississippi River at Union Point, 1994, 95 and 96...... 85
Figure 7-3 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Missouri River at Ponca, 1979-81...... 86
Figure 7-4 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Missouri River at Nebraska City 1977-81...... 87
Figure 7-5 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Mississippi River at Range 362.2...... 88
Figure 7-6 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Missouri River at Omaha, 1976-81 ...... 89
Figure 7-7 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for various sediment diameters, Mississippi River at Range 362.2, 1991 (M=2.67) ...... 91
Figure 7-8 Relation between Mean Concentration on y-axis and Cross- sectional Mean Concentration for various sediment diameters, Mississippi River at Tarbert 1995-96 (M=2.65) ...... 92
Figure 7-9 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for various Sediment Diameters, Missouri River at Omaha (M=3.22)...... 93
Figure 7-10 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for All Sizes Missouri River at Omaha (M=3.22)...... 94
Figure 7-11 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for various Sediment Diameters, Mississippi River at Union Point, 1994, 95 and 96 (M=2.07) ...... 95
Figure 7-12 Relation between Mean Concentration on y-axis and Cross –sectional Mean Concentration for various Sediment Diameters, Missouri River at Ponca (M=2.89)...... 96
Figure 7-13 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for All Sizes Missouri River at Ponca (M=2.89) ...... 97
Figure 7-14 Relation between Mean Concentration on y-axis and Cross –sectional Mean Concentration for All Sizes Sacramento River at Station 37.85 (M=1.6) ...... 98
x Figure 7-15 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for Various Sediment Diameters, Niobrara River Near Cody at Gaging Station (M=1.73)...... 99
Figure 7-16 Location of Measured Verticals Relative to y-axis, Mississippi River at Tarbert.. 101
Figure 7-17 Relation of r2 (of ψ) to Sediment Diameter on y-axis Mississippi River at Union Point, 1994, 95 and 96 ...... 102
Figure 7-18 Relation of r2 (of ψ) to the Location of Verticals (Relative to y-axis) Mississippi River at Union Point, 1994, 95 and 96...... 103
Figure 7-19 Relation of r2 (of Ψ) to Sediment Diameter on y-axis Missouri River at Omaha 1976-81 ...... 104
Figure 7-20 Relation of r2 (of ψ) to the Location of Verticals Missouri River at Omaha 1976-81 ...... 105
Figure 7-21 Relation of r2 (of ψ) to Sediment Diameter on y-axis Mississippi River at Tarbert, 1995-96 ...... 106
Figure 7-22 Relation of r2 (of ψ) to Location of Verticals, Mississippi River at Tarbert, 1995-96 ...... 107
Figure 7-23 Relation of r2 (of ψ) to Sediment Diameter on y-axis, Mississippi River at Range 362.2, 1991...... 108
Figure 7-24 Relation of r2 (of ψ) to Location of Verticals (Relative to y-axis) Mississippi River at Range 362.2, 1991 ...... 109
Figure 7-25 Relation between ψ and Discharge, Mississippi River at Union Point, 1995-96... 110
Figure 7-26 Relation between ψ and d50 on y-axis, Mississippi River at Union Point, 1995-96 111
Figure 7-27 Relation of ψ to d50 and Discharge, Missouri River at Nebraska City...... 112
Figure 7-28 Relation of ψ to d50 and Discharge, Missouri River at Sioux City ...... 113
Figure 7-29 Relation of ψ to d50 and Discharge, Missouri River at Gayville...... 114
Figure 7-30 Relation of ψ to Sediment Diameter (d) and r2, Mississippi River at Union Point, 1994-96 ...... 115
Figure 7-31 Relation between Discharge and d50 on y-axis, Missouri River at Ponca, Nebraska City, Omaha, Gayville and Sioux City ...... 117
xi Figure 7-32 Relation between Discharge and d50 at y-axis, Niobrara River at Gaging Station (1950-52) and Middle Loup River (1948-60)...... 118
Figure 7-33 Relation between Discharge and d50 on y-axis and Discharge, Mississippi River at Union Point, 1995-96...... 119
Figure 7-34 Relation between d50 and y/D, Niobrara River near Cody 3/3/1950-5/8/52 (Q=302– 405 cfs); and Middle Loup River Section B at Dunning, 6/20/1950 (Q= 403 cfs)...... 120
Figure 7-35 Relation between Discharge and d50 along y-axis, Missouri River at Ponca, 1977-81 ...... 121
Figure 7-36 Comparison of Depth- and Velocity-weighted concentration, Missouri River ...... 123
Figure 7-37 Relation between θ and d50 on y-axis ...... 125
Figure 7-38 Relation between λ and d50 ...... 127
Figure 7-39 Effect of Sediment Size on Sediment Concentration...... 130
Figure 7-40 Effect of λ and h/D on Sediment Distribution, Chiu (2) ...... 131
Figure 7-41 Relation between G’(M) and M ...... 132
Figure 7-42 Effect of M on Sediment Distribution for λ = 2...... 133
Figure 7-43 Effect of M on Location of Mean Concentration (on y-axis) for Constant d50..... 134
Figure 7-44 Location of Mean Concentration on y-axis by Simulation and Observed Data, Missouri River at Omaha, D=16.6 ft, 9/14/1977 ...... 136
Figure 7-45 Location of Mean Concentration on y-axis, Missouri River at Omaha, 4/26/1978 137
Figure 7-46 Location of Mean Concentration on y-axis (Simulation with M and d50 as input), Omaha Station 4/26/1978 ...... 138
Figure 7-47 Location of Mean Concentration on y-axis, Rio Grande Channel, Section 2249, 12/21/1965 ...... 139
Figure 7-48 Location of Mean Concentration on y-axis, Missouri River at Nebraska City...... 140
Figure 7-49 Location of Mean Concentration on y-axis, Mississippi River at Line 13, 4/17/1998 ...... 141
Figure 7-50 Relations among Locations of Mean Concentration on y-axis, Cross-sectional Mean Velocity and Mean Velocity on y-axis ...... 143
xii
Figure 7-51 Locations of Mean Concentration and Velocity on y-axis and Cross-sectional Mean Velocity for M=1 ...... 144
Figure 7-52 Locations of Mean Concentration on y- axis (by two methods), Mean Velocity on y- axis and Cross-sectional Mean Velocities for M=3...... 145
Figure 7-53 Locating Mean Concentration on y-axis by Depth- and Velocity-weighted Methods for M=3 ...... 146
Figure 7-54 A comparison of Normalized Chiu’s Sediment Distribution Applied to Data and two Simulations Normalized by Depth-and Velocity-weighted Averages, Missouri River..... 147
Figure 7-55 Sediment Concentration Distribution Normalized by Depth-and Velocity-weighted mean Concentration ...... 148
Figure 7-56 Relation between M and ψ...... 150
Figure 7-57 Relation between M and ψ for Different Sediment Diameters, Mississippi, Missouri, and Middle Loup Rivers, and Rio Grande Channel...... 151
xiii ACKNOWLEDGEMENT
I would like to express my sincere appreciation to my advisor Dr. Chao-Lin Chiu for his
advice and patient guidance throughout my graduate program at the University of Pittsburgh.
Without his encouragement this dissertation would not be possible. I would like to thank Dr.
Allan Bullen, Dr. William Matlack, Dr. Rafael Quimpo and Dr. Tin-Kan Hung for serving on my
doctoral committee.
The Missouri River sediment reports and field data were obtained from Mr. Larry J.
Morong with Department of the Army, Corps of Engineers, Omaha District. The field data on
Mississippi River was obtained from Dr. Albert Molinas. The efforts of Mr. Morong and Dr.
Molinas in providing the data are appreciated.
xiv
1.0 INTRODUCTION
1.1 PROBLEM AND BACKGROUND
Suspended sediment is a major carrier of many contaminants including nutrients and heavy minerals. Control and mitigation of these problems require understanding of the sediment transport processes. Monitoring of sediment transport of a stream provides useful information concerning changes in the watershed conditions and will determine the quantity of sediment discharge. Quantifying the amount of sediment transport within a watershed is important due to the fact that:
1. Expected capacities for reservoirs may be underestimated.
2. Sedimentation of rivers can cause increase in drinking water treatment costs and may
impact the public health.
3. Ecologically sensitive areas can be adversely impacted by excessive sedimentation.
Without a monitoring system, the watershed problems and changes in environmental conditions cannot be measured and addressed adequately. The quantity of sediment in water must be estimated so that the sediment may be removed before the water enters a distribution system.
Pesticides and organic materials are adsorbed by sediment, causing potential health hazards in some streams, estuaries, and water reservoirs.
There is not a single sediment transport equation, which is efficient and provides reliable results
1 without requiring a large number of sampling and complex computations. Many sediment
transport equations have been developed: HEC-6 model offers the user a choice of 12 different
equations. The Corps of Engineers Sediment Transport Package (ODSET) uses 11 equations.
Unlike hydraulic equations such as Chezy’s and Manning’s, which give approximately same
results, the existing sediment transport equations applied to the same data set can generate
estimates of sediment transport rates ranging over more than 2 orders of magnitude (34).
Existing methods of predicting bed material load in sand bed channel flow is unreliable. In field data collection sedimentation quantities have to be separately sampled and measured.
Measurements involve collection and transportation of large volumes and numbers of samples to laboratories and methods of analysis are slow and cumbersome. Sediment transport in natural channels is highly variable in space and time, so the accuracy of measurement depends on exposure time and density of sampling. In view of this situation, it is only realistic to expect substantial level of errors in sedimentation data (12).
Because of the costs, difficulties and uncertainties associated with the traditional methods of sediment load measurements and estimate, many environmental groups and government agencies have not adopted a sediment measurement program.
The common method used by U.S. Geological Survey (USGS) is to measure the suspended sediment concentration by depth-and point-integrating samplers. By conducting depth-averaged sampling the vertical mean concentration is determined directly. However, the method is impractical under strong currents in high flows. In addition, a distance of 0.3 to 0.5 foot from the channel bed is left unmeasured due to the sampler’s restrictions (14).
The US Geological Survey uses US PS-69 automatic pumping samplers at gaging stations in
Pennsylvania to measure the suspended sediment concentration of streams and some small
2 rivers. Data collected by pumping samplers are taken from a single point in the flow, which
represents neither the cross-sectional nor vertical mean sediment concentrations. Without a
workable mathematical model of distribution of sediment concentration, it is difficult to establish
a stable relation between the concentration at the sample point and the mean sediment
concentration in the vertical and the cross-section.
Sediment rating curves are often used to estimate suspended sediment loads where the sampling
program is insufficient to define the continuous record of sediment concentration. Use of this
technique will involve errors in the values of sediment load produced. Values of annual sediment
load estimated by using a rating curve could involve errors of up to 280%. Careful consideration
should be given to possible error terms before rating curve estimates of sediment load are used in statistical and other analyses (16, 20).
The widely used Rouse equation- a suspended sediment concentration model is based on the
Prandtl-von Karman velocity distribution, which is incapable of predicting maximum velocities near water surface and the channel bed. Rouse equation can not predict sediment concentration at or near the channel bed, where the highest sediment concentration occur (2).
1.2 OBJECTIVE AND SCOPES OF STUDY
The main objective of this research is to develop an efficient method of estimating sediment
discharge in rivers based on Chiu’s velocity and sediment-distribution models. To achieve this
goal, the following objectives have been established:
3 • Develop a method to estimate a coefficient for converting sediment concentration at the
optimal point on y-axis to the cross-sectional mean concentration. This sediment
concentration can be used to calculate the sediment discharge.
• Determine the location of the optimal sampling station (hereinafter is referred to as the y-
axis) across a river cross section at a monitoring station.
• Determine the optimal depth of sampling on y-axis. Sediment concentration at this depth
should represent the mean concentration on y-axis.
• Apply Chiu’s sediment transport model to sediment and velocity data obtained from
Missouri River Special Point-Integrated Sediment Sampling Program initiated in 1976 by
the USGS and the U.S. Corps of Engineers (USACE).
• Apply Chiu’s equations to sediment and velocity data on Mississippi River obtained from
Lower Mississippi River Sediment Study by USACE and others. Data analysis on other
rivers.
4
2.0 REVIEW OF LITERATURE
2.1 DISCHARGE MEASUREMENT BY CONVENTIONAL METHODS
Determination of water discharge is needed for estimation of sediment discharge. At many
locations water discharge can be accurately determined from a stage-discharge relation. If,
however, the relation of discharge to stage is not stable, as for most sand bed streams, or an accurate relation is not available, as for a new station, a discharge measurement is necessary at the time sediment concentration is sampled in the cross section (25).
Average velocity is required for calculating water discharge. Direct measurement of the average velocity of an entire cross section is impossible, so the conventional method uses incremental method, which is time consuming.
The discharge of a stream is calculated from measurement of velocity and depth. In the field, a marked line is stretched across the stream. At regular intervals along the line, the depth of the water is measured with a graduated rod or by lowering a weighted line from the surface to the streambed, and the velocity is measured using a current meter (36). The discharge at a cross section is found by summing the incremental discharges from each measurement i, i = 1, 2, 3,…,
n of velocity ui and depth Di .
n Q = ∑ u i D i ∆ w i (2-1) i =1
∆wi =width increment; Di=water depth; ui = average velocity at a vertical
5 Each measurement represents the conditions up to halfway between this measurement and the
adjacent measurements on each side.
1 1 ∆w = (b − b ) + (b − b ) (2-2) i 2 i+1 i 2 i i−1
th bi = horizontal distance in feet of i velocity vertical from the left water edge; b1=0; bn = channel width.
Discharge increment is
qi = Diui ∆wi (2-3)
The average vertical velocity is computed by averaging point velocities at two points on each vertical. These point velocities are measured by current-meter and located at 0.2 and 0.8 of the total depth below the water surface.
A comprehensive study by the USGS in 1950 making a comparison of mid-section versus the mean-section methods of computing cross-sectional area and/or discharge resulted in USGS adopting the mid-section method as the standard or recommended procedure. The main reason for adopting the mid-section method was time and money savings over the mean-section method(35).
2.1.1 Mid-Section Method
The mid-section method of measurement assumes that the velocity samples at each depth
sampling point, represents the mean velocity in partial rectangular cross-sectional area. The
partial area extends laterally from half the distance from the preceding meter location to half the
distance to the next, and vertically from the water surface to the measured depth (Figure 1). The
channel cross section is defined by depths at locations 1,2,3,…n. At each location, the velocities
6 are measured by current-meter to obtain the mean of vertical distribution of velocities. The
partial discharge is now computed for any subsection i as
⎡ ⎤ (bi − b(i−1) ) (b(i +1) − bi ) ⎡b(i+1) − b(i−1) ⎤ qi = ui ⎢ + ⎥Di = ui ⎢ ⎥Di (2-4) 2 2 2 ⎣⎢ ⎦⎥ ⎣ ⎦ where qi = discharge through partial section i,
ui = mean velocity at subsection i, bi = distance from initial point to location i,
b(i−1) = distance from initial point to preceding location,
b(i +1) = distance from initial point to next location
Di = depth of water at location i.
Thus, for example, the discharge through partial section 5 is
⎡b6 − b4 ⎤ q5 = u5 ⎢ ⎥D5 (2-5) ⎣ 2 ⎦
The preceding location at the beginning of the cross section is considered coincident with
location 1; the “next location “ at the end of the cross section is considered coincident with
location n. Thus,
⎡b2 − b1 ⎤ ⎡bi − b(n−1) ⎤ q1 = u1 ⎢ ⎥D1 and qn = un ⎢ ⎥Dn (2-6) ⎣ 2 ⎦ ⎣ 2 ⎦
For the example shown in the figure, q1 is zero because the depth at observation point 1 is zero.
The summation of the discharges for all partial sections is the total discharge of t he stream (35).
7
Figure 2-1 Sketch of Mid-section Method of Computing Cross-sectional Area for Discharge Measurement
2.2 MEASUREMENT OF SUSPENDED SEDIMENT DISCHARGE
Sediment concentration may be determined as the ratio of the weight of sediment to the weight
of water-sediment and is the method referred in USGS manual (25). The purpose of collecting sediment samples is to determine the instantaneous sediment concentration at a cross section.
USGS uses two basic methods to define the location or spacing of the verticals. One is based on equal increments of water discharge (EDI); the second is based on equal increments of stream or channel width (EWI).
8 Units of measurement of sediment concentration are reported in the U.S.G.S. Computation of
Fluvial-Sediment Discharge (25) as follows: Milligrams of sediment per liter of water-sediment mixture is a concentration unit. However, as a matter of convenience, it is determined in the laboratory in parts per million, which is the dry weight of suspended material per million equal weights of water-sediment mixture, or milligram per kilogram, and is found by the formula:
parts per million = (weight of sediment X 1000000)/(weight of water-sediment mixture) or
0.001g 1g 1ppm= = 1000g 1000000g
Concentration in milligram per liter is the weight in milligrams (mg) of sediment per thousand
milliliters (ml) of mixture and is the ratio of dry weight of sediment to the volume of mixture, or
0.001g 1mg Milligrams per liter= = 1000ml 1000ml
The numerical values of part per million and milligrams per liter are equal when the density of
the mixture is equal to 1.00, and for all practical purposes, 1 liter weighs 1000 g.
2.2.1 The Equal-Discharge–Increment Method (EDI)
In this method, the location of verticals can only be done after the discharge distribution across
the width is known. It is then a simple matter to locate the verticals at the middle of strips of
equal discharge. Obviously the average of the mean concentrations measured at these verticals
gives the average concentration for entire cross section. This value can then be multiplied by the
discharge obtained with the help of a stage-discharge curve (30).
9 In EDI method, the cross-sectional area is divided laterally into a series of subsections, each of
which conveys the same water discharge. Depth integration is then carried out at vertical in each
subsection where half of the subsection discharge is on one side and half is on the other side. The
method requires that we have some knowledge of stream flow distribution in the cross section before sampling verticals can be selected. EDI method can save and labor over the ETR method, especially on the larger streams, because fewer verticals are required. From a discharge measurement, a graph is drawn of cumulative discharge in percent of total discharge versus distance from left or right bank or the station numbers on the cableways or bridges. A decision is made as to the number of verticals required to adequately define the suspended sediment concentration across the stream (13).
The average sediment concentration for the whole cross section in EDI method is calculated as
1 n C X = ∑Ci (2-7) n i=1
Ci = average concentration at vertical, i =1.. n
If the partial discharge in the subsections is not equal, the discharge distribution weight is not
1 equal to . n
2.2.2 Equal-Width-Increment Method (EWI)
In this method the measured sediment concentration in a vertical has to be multiplied the
discharge through the strip it represents to compute the load passing the strip. The discharge
through the strip can be obtained by current meter measurements in the vertical or with the help
of a relation between the discharge in the strip and the total discharge in the cross section. The
10 total discharge could be obtained from the stage-discharge curve. The relation between the
discharge through a strip and the total discharge will have to be obtained earlier from extensive
current-meter investigations (30).
This method also is called Equal-width-increment method. The EWI method requires that all verticals be traversed using the same transit rate (14). The number of verticals required for an
EWI sediment discharge measurement depends on the distribution of concentration and flow in
the cross section at the time of sampling and accuracy of the results. Water discharge
measurement is not required preceding use of the EWI method.
If the sampled verticals do not represent centroids of equal discharge (EDI method), the mean
concentration is not the average of all measured verticals. In this case (EWI) the mean
concentration in the cross section is calculated as:
1 1 n n u A n C = uCdA = C q = C ( i i ) = C W (2-8) x ∫ ∑ i i ∑i ∑i i Q A Q i=1 i==11Q i
qi Wi = discharge distribution weight = Q
Ai = Di ∆wi (2-9)
∆wi = width of vertical
11
2.2.3 Suspended Sediment Sampling Methods
The usual purpose of sediment sampling is to determine the instantaneous mean discharge-
weighted suspended sediment concentration at a cross section. Such concentrations are combined
with water discharge to compute the measured suspended sediment discharge.
In the depth-integration method the sampler traverses the depth of the stream at a uniform speed.
At every point in the vertical, a volume of sediment-water mixture proportional to the stream
velocity is collected by the sampler. The concentration of the sample thus gives the mean
concentration in the vertical. The product of this concentration and discharge corresponding to
the strip within which the vertical lies is the suspended sediment load for the strip (14, 30).
Depth-integrated samplers normally collect water and sediment mixture only from the surface to
about 0.3 foot from the streambed (29).
In the point-integrating method sediment samples are taken at a number of points along the
vertical to obtain the concentration distribution in the vertical. If the velocity distribution in the
vertical is known or measured, the concentration and velocity profiles could be combined to
prepare the sediment distribution curve for the vertical and the suspended load is calculated.
Point-integrating sampler will have to be used on streams, which are too deep or too swift for the
depth-integrating sampler (14, 30).
Point-integrated sample is a sample of sediment that is accumulated continuously in a sampler that is held at a relatively fixed point and that admits water and sediment mixture at a velocity about equal to the instantaneous stream velocity at that point (29). Because of the cost of sampling, this method is used mostly for research programs.
12
2.2.4 Location of Sampling Points on the Vertical
A precise method of point-integration involves the determination of the velocity and
concentration distribution curves in the vertical measurement at a large number of points. But this is too time-consuming to be practicable for routine sediment investigations. Therefore, such detailed profiles are taken for research purposes (14, 30). In practice point-integration method follows one of the following schemes: One-point method, two-point method, and. three-point method.
In the one-point method the concentration is measured at the surface or at 0.6 times the depth below the surface. An empirical coefficient has to be used to get the mean sediment concentration from known surface concentration. This method is undependable. Sampling at 0.6 depths has been used in the hope that it gives the mean concentration, presumably because the mean velocity occurs approximately at this level. This method is not reliable because the mean sediment concentration varies with the flow and sediment size.
Straub (30, 33) showed that the mean concentration of suspended sediment in a vertical is given by
3 5 Ci = C0.8D + C0.2D 8 8 (2-10)
Where C0.8 and C0.2 are sample concentration taken at 0.8 and 0.2 depth below water surface, respectively.
The three-point method involves the measurement of the concentration at surface, bottom and mid-depth. The two-point and three-point methods do not give a correct idea about the size distribution of the suspended load. Such information can only be obtained using a depth-
13 integrating sampler or from records of a point-integrating sampler at a number of points in the
vertical.
The current method of sediment measurement, which is used widely by USGS is automatic pump sampling (23). However, when the sampler intake is fixed, without a sediment distribution
model, the point sampling does not represent the vertical or cross-sectional mean concentration.
It is evident from the vertical distribution of sediment concentration that relative depth of the sampler intake may have a relatively small effect on the measured concentrations of silt and clay it will have a very large impact on all sands, and even coarse silt. The sample intake should be located at a point within the cross-section that approximates the mean concentration across the full range of sampled flows or have an adjustment factor.
Single-point sampling can accurately represent the mean concentration of fine sediment (silt and clay) that exhibits uniform concentration distribution. However, the measurement of sand load presents a difficult problem and can produce very poor correlation between the point samples and the mean concentration.
2.3 ESTIMATION OF SUSPENDED SEDIMENT DISCHARGE
Generally, the mean discharge-weighted concentration (Cx ) of a stream can be used directly to compute the rate of sediment discharge moving in the stream (14, 25).
If water discharge is in cubic feet per second
Qs = 0.0027QC x (2-11)
14 Qs = tons per day (English short tons)
If water discharge is in cubic meters per second
Qs = 0.0864QC x (2-12)
Qs = metric tons per day where
C x = discharge-weighted mean concentration (whole cross-section), in mg/L
Q = stream flow rate, in cubic feet per second, or cubic meters per second.
2.3.1 Calculation of Cross-sectional Concentration (Single Vertical Method)
Ideally, one must measure the suspended sediment concentration at all points in a cross section.
Because of the cost constraints, and difficulties of sampling during floods, measuring along a
fixed vertical or at a single point is a more practical option.
USGS uses a coefficient, which is computed by dividing the mean concentration of the cross
section by the concentration obtained from the point or single-vertical sample. This coefficient is
called cross-section or box coefficient. The average cross-sectional concentration is measured by
means of depth/width integrating technique, which is known as EWI or EDI methods. The
coefficient is multiplied by a concentration obtained from a point or single- vertical in order to
obtain an estimate of the mean sediment concentration in the stream cross section (24, 25).
The USGS method does not use a sediment transport model to convert the single-sample to the mean vertical concentration. Moreover, the locations of single-vertical and the single- point sampling should be investigated to determine the correlations between their concentrations and cross-sectional concentrations especially for streams with sand size (coarser than 0.062 mm)
15 suspended sediment. This method may cause errors when used in rivers with coarse suspended
sediment, because the sediment size distribution is non-uniform across the channel and along the
water depth. According to Morris, G. and Fan (34), one possible approach is to separate samples
into fine and coarse fractions, and to establish relationships for each individually.
L.Yuqian (22) suggested that for a sediment size of less than 0.075 mm, the vertical distribution is rather uniform and sampling at any point in the vertical can be considered representative of the average sediment concentration in the vertical, however, it does not apply to certain rivers.
2.3.2 Rating Curve Method
After collecting sufficient sediment and water data, average sediment and average water
discharge are computed for each cross section. Then, they are plotted against each other. By
using least square method, the best line is fitted through the data on a log-log paper.
The relation between suspended sediment and water discharge is defined by a power function,
C = a Qb and referred to rating curve(16, 20). Normally this function is formulated as linear model to find the solution of the rating curve parameters (a and b). Formulation of the power function as a linear model requires a logarithmic transformation to linearize the function and a subsequent correction for transformation bias (by the method of least square).
In order to use the rating curve method for estimating the suspended sediment loads and sediment discharge of streams, parameters a and b of the power function must be determined at
high and low-flow periods.
Even when rating curves differentiated by season and stage, errors can be expected in estimates
of annual suspended sediment loads. If rating curve approach produces unacceptable errors, then
16 at present the only viable alternative is some form of field monitoring enabling the direct calculation of loads as there is no viable prediction model for stream sediment transport (16, 20).
17
3.0 DESCRIPTION OF DATA
Sediment data on Missouri (10), Mississippi (27), Atchafalaya (27), Sacramento (17), Niobrara (29) and
Middle Loup Rivers (37) as well as Rio Grande Channel (38), 8-ft McQuivery Flume (19) and
Coleman Flume (21) were analyzed.
The Missouri River sediment data was provided by USACE, Omaha District (10).
The U.S. Geological Survey, under a cooperative stream gaging program with the U.S. Corps of
Engineers, has collected bed material and suspended sediment samples since 1976 at two
locations on Missouri River: Omaha, Nebraska, and Nebraska City, Nebraska. They have
collected data at Sioux City, Iowa since 1979. The data collected from 1979 to 1986 used by
USACE to estimate the bed material loads in order to study the degradation problems and to
define the balance of sediment load between different stations of Missouri River.
The sediment data includes the size distribution of the suspended sediment and the laboratory
results such as total weight of the sample, volume of the sample, plus cumulative weight retained
on the various sieves. Also, the actual concentration of sediment retained on each sieve was
included.
Data collected include five to seven point-integrated samples per stream vertical at five locations
in the cross-section (Figure 3.1). Each station was sampled by boat at about six week intervals
during the open water season.
In addition to the sediment data, velocity measurements, discharge measurements, water surface
elevation and slope, sonic soundings and temperature data were collected at each station.
Laboratory analyses of the sediment were conducted and recorded.
18 Suspended sediment samples were collected with a US P-61 point-integrating suspended
sediment sampler.
Discharge distribution data across the channel was included with the data. This is used in this
study to compute weight factors for different sample verticals. The factors are then used to
calculate sediment load across the section. This data is needed for calculation of sediment load
by the new sediment transport model as well as the traditional method.
The velocity and sediment data collected by USGS at Nebraska City, Omaha, Sioux City,
Gayville, and Ponca stations were selected for analysis in this research. These stations are
briefly described below (8, 9, 11).
The Nebraska City Station (River Mile 561.8) is at the Waubonsie Highway Bridge (Nebraska-
Iowa State Highway 2). The station was established in 1957.
Omaha Station is in Douglas County, Nebraska. The station is on the left side of the concrete floodwall on the right bank, 275 feet downstream from the Interstate 480 highway bridge in
Omaha, at River Mile 615.9. The drainage area upstream of this station is approximately
322,800 mi2 . The station has been in operation since 1939 (8,9).
Sioux City Station is located in Dakota County, Nebraska, on the right bank on the upstream side
of the bridge on US Highway 77 at South Sioux City. At River Mile 732.3 mi downstream of Big
Sioux River, the Missouri River drains an area of 314.6 mi2. The Sioux City Station was established in 1954.
The basic measurements in the point-integrated sampling on Missouri River comprise the depth of flow at sampling vertical; the number of sampling points; the velocity and distance from the
local bed for each sampling point and for each sample, the measured point-integrated sample
volume, weight of sediment in the point-integrated sample and its particle size distribution.
19 These data are used to compute the suspended bed material load from water surface up to a
specified distance from the bed. The distance is the outer limit of the bed layer for computing the
suspended bed material load and is 0.5 ft (10,11,12).
The Mississippi River (27) data included data on Old River Complex (ORCC). The discharge and sediment data were collected at seven ranges along the Mississippi River and ORCC (Union
Point, Hydropower Station, Line 13, Low Sill, Auxiliary, Line 6 and Tarbert).
Figure 3-1 Sampling Schematic Missouri River
20
4.0 MATHEMATICAL MODELS OF VELOCITY DISTRIBUTION
A velocity distribution model is required for calculating sediment discharge in rivers. Many
researchers have studied the velocity distribution in open channel flow. The most widely used
conventional equation for velocity distribution is Prandtl-von Karman Universal velocity
equation. USACE used this logarithmic velocity profile for calculating sediment discharge of
Missouri River (8, 9). Toffaleti (1969) suggested the velocity distribution in the form of power law(9). The existing equations cannot predict the maximum velocity below the water surface.
Also, they are deficient near the channel bed and in sediment-laden water. Chiu developed a velocity distribution by using entropy maximization, which can describe the vertical velocity profile from the water surface to the channel bed and is applicable in steady and unsteady flows with or without sediment (1).
4.1 SIMPLE POWER LAW VELOCITY DISTRIBUTION
The simplest form of power law is (4)
u = ay b (4-1)
du ab = (4-2) dy y1−b
21 This velocity distribution goes to infinity at y = 0 which is considered the weakness of
power law, considering the fact that the velocity is zero at the bed. Toffalete (1969) suggested that
u y = n ( ) n2 (4-3) u 1 D
Many attempts have been made to divide flow velocity into several regions, but it is difficult
with only five or six data measurements at different levels (8).
4.2 PRANDTL-VON KARMAN UNIVERSAL VELOCITY EQUATION
The well-known von Karman equation is (3, 4)
u y u = * ln (4-4) k y0
then
du u = * (4-5) dy ky
u* = shear velocity; k = von Karman’s constant, k ≈ 0.4 for clear water.
For sediment-laden water, k is less than 0.4, and k decreases with sediment concentration. At the
channel bed, the velocity gradient goes to infinity, which is the weakness of this velocity
distribution. The velocity is zero at a distance y0 above the channel bed
22 4.2.1 Parameter Estimation of the Logarithmic Velocity Distribution
This research also investigates the applicability of velocity distributions near channel beds. The
logarithmic equation was applied to velocity data of Missouri River. As it was shown in Figure
4.1, the velocity at a distance equal to y0 becomes zero.
The parameters of the logarithmic velocity profile are estimated as follows:
The Prandtl-von Karman equation can be written in the form of
u = A1 ln(y) + A2 (4-6)
where
u u A = ∗ and A = − ∗ ln(y ) (4-7) 1 k 2 k 0
The two parameters are estimated from regression. u∗ (Shear Velocity) is obtained from
u∗ = gRS (4-8) where g = gravity acceleration, R= hydraulic Radius, and S = water surface slope (in the case of
Missouri River). Therefore, k can be calculated.
To illustrate applicability of the logarithmic velocity profile, the 6-point velocity data at a vertical with the depth of 15.7 feet, was analyzed (Missouri River, Omaha Station, dated
4/16/80). As the result, the following logarithmic equation was calculated
u = 0.599 ln( y) + 3.819 (4-9)
From A1 and A2, the parameters, k=0.389 and y0 = 0.0017 ft were calculated (u∗ =0.233 ft/s).
23 1.0 Observed Data, Missouri River at Omaha 0.9 4/16/80, D=15.7 ft Prandtl-von Karman equation
0.8 Calculated k=0.389 and y0=0.002 ft
0.7
0.6 10
0.5 1 y/d )
0.4 t f 0.1 y ( 0.3 0.01 0.2 0.001 0.1 01234 u (ft/s) 0.0 0123456
u (ft/s)
Figure 4-1 Prndlt-von Karman Velocity Distribution near Channel Bed
24
4.3 CHIU’S VELOCITY DISTRIBUTION
4.3.1 Model Formulation
Chiu pioneered the probability-based mathematical models of velocity distribution, which was
published in several papers in conference proceedings and journal papers. Chiu’s velocity
distribution equation was presented as (4, 5, 6)
umax ⎡ M ξ −ξ0 ⎤ u = ln⎢1+ (e −1) ⎥ (4-10) M ⎣ ξmax −ξ0 ⎦ in which
y y ξ = exp(1− ) (4-11) D − h D − h
D = water depth at the y-axis; h = a parameter
Chiu derived the velocity distribution equation (4-10) as below
ξ − ξ u u 0 = ∫ p(u)du = ∫ e(a1 +a2u) du (4-12) ξ max − ξ 0 0 0
Where a1 and a2 = parameters, 0 ≤ ξ ≤ 1; ξ = is a value attached to an isovel on which the
velocity is u; ξ 0 is the value of ξ when u = 0; and ξ max = the value of ξ when u is the maximum velocity.
The probability of velocity less than or equal to u, is the area between the isovels of ξ and ξ 0 ,
y divided by the total area of the channel section. In wide channels this probability is equal to . D
25 Chiu’s probability-based velocity distribution equation was determined by maximizing
Shannon’s information entropy (3,5). Shannon’s information entropy, which is the average
information content in a massage, or the measure of uncertainty can be expressed in discrete
form as
H = −∑ p(x j )ln p(x j ) (4-13) j
where H is the mean value of − ln p(x j ) ; x j is the value of discrete random variable X; and
p(x j ) is the prior probability of X being equal to x j . For a continuous random variable, the information entropy can be expressed as
umax H = − ∫ p ln p(u)d(u) (4-14) 0
where p(u) is the probability density function of continuous random variable U.
A system has the tendency to move toward chaos or uncertainty rather than toward order. p(u)
can be determined so that H is maximized subject to a set of constraints (3):
umax ∫ p(u)d(u) = 1 (4-15) 0 and
umax Q ∫ up(u)du = u = (4-16) 0 A
A=Cross-sectional Area
The entropy can be maximized by the method of Lagrange multipliers, and the probability
density function can be determined as
p(u) = e(λ1 −1)eλ2u (4-17)
in which λ1 and λ2 are the Lagrange multipliers.
26 Substitution of (4-17) in to (4-12) yields:
umax ∫ e(λ1 −1)eλ2u du = 1 (4-18) 0
or
λ eλ1 −1 = 2 (4-19) eλ2umax −1
By defining M = λ2umax and substituting (4-19) in to (4-17), the probability density function becomes
u M Me umax p(u) = M (4-20) umax (e −1)
u e M −1 u u p(u)d(u) = max (4-21) ∫ M 0 e −1
Through mathematical manipulation of these equations, Chiu (1989) derived a velocity distribution as:
u 1 ⎡ M ξ −ξ 0 ⎤ = ln⎢1+ (e −1) ⎥ (4-22) umax M ⎣ ξ max −ξ 0 ⎦
Equation (4-20) gives
u M u max Me umax u du = u (4-23) ∫ M 0 umax (e −1) which, when integrated, can be expressed as
u e M 1 = M − = φ (4-24) umax e −1 M
27
4.3.2 Location of y-axis
y-axis is a vertical on which the maximum velocity of the cross section occurs. In order to illustrate the procedure of determining the location of y-axis, this research used velocity data of
Missouri River at Nebraska City, Sioux City, Gayville, Ponca and Omaha stations, and
Mississippi River at Union Point, Range 362.2 and Tarbert. Data collected on Missouri River by
USGS include five to seven point velocities on each of the five verticals in the channel section.
The velocities were measured by Price current-meter. Because of the operational restriction of current-meters, velocity measurements were conducted approximately between 0.07 and 0.77 of the total water depth. The measured velocities were used to plot the isovels in the cross sections.
Figures 4.2 and 4.3 show the isovel pattern and indicate the location of the y-axis.
The maximum velocities were estimated at the average location of the y-axis. Figures 4.4 to 4.8 show the locations of the y-axis on different dates. The figures demonstrate that even for a sand bed channels such as Missouri River, the location of y-axis was stable. Figures 4.9 and 4.10 show the channel cross sections of Missouri River at Sioux City and Nebraska City at different discharges. The location of y-axis is shown on theses channel sections.
28 50 4.95 4.54 4.43 5.88 45 5.6 3.9 4.4 40 4.78 4.29 5.65 4.4 35 5.3 4.22 3.5 4.56 ) t 30 4.10 5.34 4.2 (f 4.9 3.97 h 25 4.24
Dept 20 3.33 3.83 4.82 15 3.63
10 3.3 3.89 5 0 1000 1500 2000 2500 3000 3500 4000 4500
Distance from Reference Point (ft)
Figure 4-2 Isovels Showing Location of y-axis Mississippi River at Tarbert (2/1/1996)
y-axis
18 2.85 4.15 4.8 16 5.7 5.67 4.59 3.28 14 4.1 3.94 3.07 4.15 12 5.9 5.89 2.63 3.72 3.9 4.15 10 2.2 3.94 3.07 4.6 8 3.28 Depth (ft) 5.3 2.42 3.72 3.5 6 5.24 3.72 5.0 3.85 4 5.02 3.28 4.37 2 3.07 4.15 0 100 200 300 400 500 600 700
Distance from Reference Point (ft)
Figure 4-3 Isovels Showing Location of y-axis Missouri River at Omaha (4/26/1978)
29 2202
y-axis is on vertical 2 ) t (f y Z
ence, 2201 fer e R om r f axis - y f 2200 o ance t s i D
2199 5 5 9 96 9 v c n b 96 No Ja De Fe Dates of Measurements
Figure 4-4 Location of y-axis, Mississippi River at Tarbert
30 3300
3000 ft) ( y 2700 Z , int
o 2400 e P c
n 2100 e r fe e 1800 R
1500 axis to - y 1200 of ce
n 900 sta i
D 600
300 6 4 6 5 9 94 9 9 9 / / /95 7 0/ 0 6/ 9/ 7/96 4 2 2 1 / 2/ /1 /1 2/ 2 1/ 2 1/2 1 12 1 Dates of Measurements (1994,95 and 96)
Maximum Water Surfce Width =3200 feet _
Zy=1883 ft (Mean Location of y-axis) _
Zy+σ (σ=296 ft) _
Zy-σ
Figure 4-5 Location of y-axis, Mississippi River at Union Point
31 3500 ) y Z (
t 3000 t, f in Po
e 2500 c n e r fe e 2000 R e th to
is 1500 x a - y f
o 1000 ce an st i D 500 7 0 7 12 9 1 26 21 23 16 24 6/ 8/ 9/ 7/ 5/ 4/ 6/ 7/ 8/ 5/
Dates of Measurement (April 26,1991- September 12, 1991)
Maximum Channel Width = 3200 ft _
Zy=2010 _ Zy+σ=2320 ft (σ=310 ft) _
Zy-σ=1700 ft
Figure 4-6 Location of y-axis, Mississippi River at Range 362.2
32 800 (ft) y 700 , Z ge 600 r Ed te a 500 ft W
400 from Le
s 300 i x
Y-a 200 f o e c
n 100 a t s i
D 0 6 1 0 /7 /8 78 79 /8 77 /6 14 3/ 26 21/ 18/ 10 8/ 8/ 6/ 7/ 10/ Dates of Measurement
Maximum Water Surface Width=700 ft _
Zy=472 ft _
Zy+σ=539 ft _
Zy- σ =404 ft
Figure 4-7 Location of y-axis, Missouri River at Omaha
33 900 850 t
in 800 750 Po
e 700 c
n 650 re 600 fe
e 550 500
(ft) 450 y from R Z 400 350 300 250 of y-axis e
c 200 n
a 150 t s i 100 D 50 0 Oct-79 Apr-80 May-80 Jul-80 Oct-80 Apr-81 Jun-81 Jul-81 Aug-81 Oct-81 Date of Measured Data
Observed Data, Maximum Water Surface Width=700 ft _
Zy=444 ft, Mean Location of Y-axis _ Zy+ SD=496ft _ Zy-SD=392 ft
Figure 4-8 Location of y-axis, Missouri River at Sioux City
34 y-axis
0
-5 e rfac u -10 r S te a
-15 w W lo Be -20 (ft) h t p e
D -25
-30 0 100 200 300 400 500 600 700 800 Distance from Reference Point (ft)
10/11/79, Q=39600 cfs
4/17/80, Q=31000 cfs 5/29/80, Q=31900 cfs 7/10/80, Q=33500 cfs 4/16/81, Q=32700 cfs
Figure 4-9 Channel Cross Section at Measured Verticals for Different Discharges, Missouri River at Sioux City
35 y-axis
0
-5 ace r Surf e -10 low Wat e -15 B ) t h (f
-20 Dept
-25 0 100 200 300 400 500 600 700
Distance from Reference Point (ft)
4/26/77, Q=41000 cfs 6/14/77, Q=37500 cfs 7/25/78, Q=69200 cfs 10/18/77, Q=36800 cfs 9/12/78, Q=57000 cfs 8/28/79, Q=49700 cfs
Figure 4-10 Channel Cross Section at Measured Verticals for Different Discharges, Missouri River at Nebraska City
36
4.3.3 Relation Between Mean and Maximum Velocities
The entropy parameter M of a channel section can be determined from the relation between the mean and maximum velocities. Chiu(1) developed a technique to determine the discharge from velocity profile on the y-axis in a channel section. The technique is an efficient way to estimate the discharge in streams and rivers during unsteady, high flow conditions.
Figures 4.11-4.15 show the relation between the mean and maximum velocities at a channel section. The slope of the regression line isφ , and M was computed using (4-24). M is the channel constant. For a specific channel section M is invariant with the discharge or water depth.
To establish an equilibrium state and the corresponding M value, an erodible channel section adjusts the form and material, roughness, geometrical shape, slope, and alignment, under various values of discharge and water depth. A nonerodible or well-established channel section maintains the equilibrium state and the corresponding M and entropy by adjusting the velocity distribution through modifying the maximum velocity and h when the flow condition changes.
The location of y-axis at a channel section is stable and independent of the discharge and water
(1) depth . Equation (4-24) is the slope of the u -umax line.
It is desirable that the velocity data to be used include those collected as close to the water surface as possible so that maximum velocity and h determined may be accurate. However these data are often missing due to the difficulties in measurements near the water surface, or due to ignorance about the value of information contained in the maximum velocity. Without measurements near the water surface, the velocity data tend to indicate a continuously increasing trend, and often lead to overestimating the maximum and mean velocities.
37 7 _ u=Q/A
6 umax Obtained from Regression M=2.07, φ = 0.66, r=0.98
5
4 (ft/s)
_ u 3
2
1
0 01234567891011 u (ft/s) max
Figure 4-11 Relation between u and umax Mississippi River at Union Point, 1994, 95 and 96
38 7 _ u=Q/A 6 umax Obtained from Regression M=2.66, φ = 0.70, r=0.96 5
4 s) t/
_ u (f 3
2
1
0 012345678910 u (ft/s) max
Figure 4-12 Relation between u and umax Mississippi River at Tarbert,1995 and 1996
39 7 _ u=Q/A, Observed Data 6 φ=0.714, Μ=2.89
5
4 s) t/
_ u (f 3
2
1
0 01234567 u (ft/s) max
Figure 4-13 Relation between Mean and Maximum Velocities, Missouri River at Ponca, 10/26/1978 to 10/18/1979
40 5 _
u=Q/A, Calculated umax φ = 0.715, Μ=2.92 4
3 ) c e
_ u (ft/s 2
1
0 0123456 u (ft/s) max
Figure 4-14 Relation between Mean and Maximum Velocities Missouri River at Gayville 4/22/1980 to 7/21/1981
41
9 _ 8 u=Q/A, Observed Data φ=0.73, Μ=3.22 7
6
5 s) t/
_ u (f 4
3
2
1
0 0123456789 u (ft/s) max
Figure 4-15 Relation between Mean and Maximum Velocities Missouri River at Omaha, 10/6/1976 to 7/15/1981
42
4.3.4 Parameter Estimation of Chiu’s Velocity Distribution
(1, 2) The three parameters in Chiu’s velocity distribution equation , M, h, and umax , can be determined by regression analysis on a set of velocity data at the verticals with maximum velocity in the cross section. The procedure is:
1. At the cross section of interest, determine the location of y-axis, which is the vertical with
the maximum cross-sectional velocity, by plotting the isovel distribution. This research
and prior studies have shown that the location of y-axis is stable over long period of time.
Since a slight shift in the y-axis location has minimal effect on maximum velocity, the
mean location of the y-axis can be used to estimate the maximum velocity.
2. Use the velocity distribution data along y-axis and apply regression analysis to solve for
the M, h, and umax values in Chiu’s velocity distribution equation.
3. If the maximum velocity occurs on the water surface, h ≤ 0, y = D, and
D D (1− ) ξ = e D−h (4-25) max D − h then by substitution, the following velocity distribution is obtained, which is used in regression analysis
D− y ξ y ( ) = e D−h (4-26) ξ max D
D− y ( ) u ⎡ M y ⎤ u = max ln⎢1+ (e −1) e D−h ⎥ (4-27) M ⎣ D ⎦
4. When the maximum velocity is below the water surface, h f 0 , ξ 0 = 0 , and ξ max = 1
43 u u = max ln[1+ (e M −1)ξ ] (4-28) M
The following equation is used in regression analysis
y (1− ) u ⎡ M y ⎤ u = max ln⎢1+ (e −1) e D−h ⎥ (4-29) M ⎣ D − h ⎦
5. Compute the values of φ by linear regression for each of the velocity data sets using the
relationship between u andumax . M is found from the Equation (4-24) in which
u = mean velocity of the cross section and is calculated as
Q u = (4-30) A
The values of umax are obtained by regression analysis in step 2.
6. Use the cross-sectional constant M in the Chiu’s velocity distribution equation from step
4 and by regression compute umax and h for the second time for the 5 or 6- point velocity
data at each y-axis corresponding to a discharge.
7. For each cross section, plot the h/D versus umax for each velocity dataset to estimate the
mean of h/D, which is the cross-sectional constant. The values of h and umax , that are
computed in step 6 are used to generate these plots (Figures 4.16 and 4.17).
Another method for finding h, was developed by Chiu and Tung (7). The following equations show that by knowing M, the location of maximum velocity (h) below the surface at y-axis can be determined.
h j(M ) = −0.2ln (4-31) D 58.3
e M −1 j(M ) = (4-32) Mφ
44 For wide channels
y ξ = (4-33) D
u = mean water velocity in the cross section; φ = section constant; A = cross-sectional area.
Table 4.1 is the summary of computed h/D and umax related to Missouri River at Nebraska City.
Figure 4.16 gives the average h/D and suggests that h/D does not change with umax in a channel section. Figure 4.17 shows that h/D will increase when umax increases.
Figure 4.18 is the Chiu’s velocity distribution applied to the observed data on Missouri River at
Omaha (4/26/1978). The figure depicts the location of maximum velocity below water surface.
Figure 4.19 compares logarithmic velocity distribution with Chiu’s velocity distribution. It shows that the logarithmic equation can not compute the maximum velocity below water surface.
Figure 4.20 depicts the applicability of Chiu’s velocity distribution near the channel bed and below water surface. It shows that Chiu’s distribution becomes zero at the channel bed, but
Prandtl-von Karman’s distribution becomes zero at y=0.002 ft.
45
Table 4-1 h/D for Missouri River at Nebraska City, M=2.56
Date Depth-D h h/D umax
4/26/1977 13.5 8.19 0.61 6.64
6/14/1977 12.9 7.72 0.6 6.09
8/2/1977 12.6 6.4 0.51 5.8
9/13/1977 11 6.09 0.55 5.85
10/18/1977 12.5 6.74 0.54 6.02
4/25/1978 15.8 6.71 0.42 7.54
7/25/1978 17.6 9.7 0.55 7.12
9/12/1978 17.1 10.55 0.62 6.72
10/24/1978 18.3 -0.69 -0.04 7.07
10/9/1979 13.7 8.42 0.61 6.12
8/28/1979 16.2 7.48 0.46 6.15
46 1.0
0.9
0.8
0.7
0.6
D 0.5 h/
0.4
0.3 M=2.56 (φ = 0.69) 0.2 Mean of h/D=0.55 0.1 h/D+σ (σ = 0.07) h/D-σ 0.0 5.56.06.57.07.5
umax (ft/s)
Figure 4-16 Relation between h/D and umax Missouri River at Nebraska City
47 14 M=2.56 (φ = 0.69) r2=0.75
12
10 t) h (f
8
6
4 5.0 5.5 6.0 6.5 7.0 7.5 8.0 u (ft/s) max
Figure 4-17 Relation between h and umax Missouri River at Nebraska City
48 1.0 Observed Data 0.9 Missouri River at Omaha, 4/26/78 D=18.5 ft 0.8 Chiu's Equation umax=5.75 ft/s, M=3.22 0.7 h=5.78 ft, h/D=0.312
0.6
0.5 y/D
0.4
0.3
0.2
0.1
0.0 0.00.10.20.30.40.50.60.70.80.91.01.1 u/u max
Figure 4-18 Velocity Distribution (Chiu’s Equations)
49 1.0 Observed Data, Missouri River at Omaha 0.9 4/16/80, D=15.7 ft Logarithmic Velocity Distribution 0.8 k=0.389 Chui's Equation 0.7 M=3.22, h=8.6 ft
0.6
0.5 y/d
0.4
0.3
0.2
0.1
0.0 0123456
u (ft/s)
Figure 4-19 Comparison of Velocity Distributions
50 20 Obseved Data, Missouri River at Omaha 18 4/16/80, D=15.7 ft Prandtl-von Karmn's Velocity Distribution Estimated Parameters: k=0.389, y =0.002 ft 16 0 Chiu's Equation 14 M=3.22, h=8.6 ft
12 10
10 1 y (ft) 8 ) t
f 0.1 y ( 6 0.01 4 0.001 2 01234 u (ft/s) 0 0123456
u (ft)
Figure 4-20 Applicability of Chiu’s Velocity Distribution Near Bed and umax below Water
51
5.0 ANALYSIS OF SEDIMENT SIZE DISTRIBUTION
Information on sediment movement and particle size distribution is needed in the design of dams, canals and other hydraulic structures. It is also required in determining the average sediment concentration.
In this thesis research, the median grain diameter (d50) was calculated by using the sediment data at 5 or 6 points along the y-axis. Figure 5.1 shows the grain size distribution for the sediment sample taken on y- axis at a point 1.1 ft above the channel bed of the Missouri River at Omaha.
The sample d50 is obtained from Figure 5.1. d50 at other five points along y-axis were determined by first plotting the figures similar to Figure 5.1 at each sample point. Figure 5.2 shows the distribution of d50 along the y-axis. Figure 5.3 shows the relation between d50 and y/D on y-axis in an 8-foot flume (McQuivey (19), 1973).
Table 5.1 shows the sediment size distribution and the calculated d50 of the Mississippi and
(27) Atchafalaya Rivers (Lower Mississippi River Study). Tables 5.2 and 5.3 show d50 at 5 or 6 points along y-axis of the Missouri River at Ponca and Nebraska City. d50 of each sample along y-axis was calculated by plotting sediment size distribution data similar to Figures 5.1 and 5.2.
52 Table 5-1 Sediment Size Distribution, Mississippi River
Sta.
Percent Finer Than Indicated Sizes in mm d50 d50
1 0.5 0.25 0.125 0.0625 0.0442 0.0312 0.0221 0.0156 0.011 0.0075 mm mm
Union Point
2/6/1996 100 99.9 96.9 71 61.6 44.6 41.5 35 32 0.051
2/27/1996 100 100 98.3 73.4 64.6 51.3 49.5 42 38 0.041
3/12/1996 100 99.9 97.4 74.6 67.5 54.7 51 43 39 0.031
4/2/1996 100 100 97.1 82.6 76.5 62.5 58 49.9 45 0.022
5/7/1996 100 99.5 96 88.5 83.6 67 61 51 46 0.02
5/21/1996 100 99.9 97 75.9 67.1 53.4 50.8 45 41 0.029
6/11/1996 100 99.8 96.1 83 79.2 70.4 69.2 64 61 0.01
6/25/1996 100 99.9 96.8 76.5 71.1 57.2 55 50 47 0.022
8/6/1996 100 100 99.7 94.7 92.6 86.9 83.8
10/8/1996 100 100 98.6 92.2 88.3 80.5 74
12/3/1996 100 99.9 96.3 83.9 77 59.5 55.4
12/17/1996 100 99.8 96.3 74.8 66.8 47.7 44.9
Union Point
1/24/1995 100 99.9 95.7 82.7 69.7 51.8 48 40.5 35.7 32.2 0.037
2/7/1995 100 100 96.9 82.3 71.3 58.3 56.2 52.2 52.3 38.1 0.015
3/7/1995 100 100 98.8 89.8 84.9 72.1 69.9 61 56 51 0.008
3/28/1995 100 100 98.5 87 77.9 65.9 62.1 55 51 47.7 0.014
4/11/1995 100 99.9 99.1 94.5 92.3 81.2 73.9 56 46 39 0.02
4/25/1995 100 100 99.1 95.8 91.4 79.2 72.2 58.5 50 0.015
5/9/1995 100 98.5 92.6 82.2 74.4 68.2 55.6 49 0.023
53 Table 5.1 (continued)
6/6/1995 100 100 98.5 88.5 64.8 52.9 49.9 44.1 0.031
6/20/1995 100 100 98.7 90.8 76 67.5 65.5 61.6 59.2 56 0.008 0.022
Range 362.2
4/26/1991 100 99.7 97.4 85.6 82.8 72.8 70.4 63.7 0.008
5/10/1991 100 100 99.9 95.5 93.4 87.7 85.5 80.7 76.9 0.004
5/24/1991 100 100 99 88.9 83.7 73.9 71.6 65.5 61 57.1 0.007
6/7/1991 100 99.8 97.5 90 87.5 81.4 78.9 71.8 67.5 63.4 0.005
6/21/1991 100 99.9 99.3 96.5 94.9 88.9 85.2 77.7 72.7 68.5 0.004
7/9/1991 100 99.7 97.9 96.9 94.6 93 90.9 83.3 78.2 75 0.003
7/23/1991 100 100 96.8 94.3 93
8/7/1991 100 99.1 98.5 96.6 94.6
8/16/1991 100 98.5 98.2 97.1 96.2
9/12/1991 100 100 99.2 94.1 92.9 73.3 65.5 49.3 40 0.025
10/1/1991 100 100 100 100 99.5 96.5 91.2 74.1 63 0.01
10/10/1991 100 99.9 99.6 99.2 98.7 90.5 83.9 70.3 0.009
10/25/1991 100 100 99.9 99.7 98.3 0.007
11/6/1991 100 100 99.9 98.9 96.6
11/22/1991 100 98.7 98 95.1 91.4
12/6/1991 100 99.5 98.6 92.7 86.7
12/18/1991 100 99.8 96.5 84 74.9
Tarbert
1/11/1995 100 100 99.6 95 91.1 81.8 75.5 61.4 53.8 48 0.012
1/26/1995 100 100 97 88 74 56 52 45 0.03
2/9/1995 100 100 99 86.9 77.5 67 64 56 52 48 0.012
2/23/1995 100 100 99.5 93.7 85.6 78.7 73.8 62.7 56 51 0.01
54 Table 5.1 (continued)
3/9/1995 100 100 99.5 95 89.1 74.6 67 54 46 40 0.017
3/22/2009 100 99.9 97.7 87.4 76.4 62.7 58 51 48 45 40 0.02
4/6/2009 100 100 99.2 95.9 89.3 78.6 74 66 60 56.9 51.6 0.007
5/4/1995 100 99.9 99.3 95.9 88.5 69 63 52 47 43 0.019
5/17/1995 100 99.9 99.2 93.5 83.6 71.9 66 54.9 49 45 0.04
6/1/1995 100 99.7 98 88.3 74.2 51 47 39 35 32 0.007
6/15/1995 100 100 98.7 92.1 86.5 81 78 71 66 62 0.004
6/29/1995 100 100 97.9 87.7 75.5 64.5 61.9 56 52 50 0.011
7/12/1995 100 100 100 99.2 96.1 93.2 88.9 78.8 73 68 62 0.004
8/3/1995 100 100 99.9 98.7 94.2 87 82 68 61 56 50 0.008
8/31/1995 100 100 99.5 98.3 94.8 88 87.4 74 68 62 51 0.007
9/14/1995 100 100 99.7 98.5 97.2 93.6 86 71 62 55 46 0.008
9/26/1995 100 100 100 99.3 98 96 93 78 71 65 57 0.007
2/1/1996 100 100 98.3 84.6 75.1 48 45 38 34 31 0.05
2/15/1996 100 100 98.3 81 71.4 63 59 51 47 44 0.02
3/14/1996 100 99.9 97.8 80.9 76.2 65 62 56 52 48 0.01
3/28/1996 100 100 98.6 83.7 78.8 70 66 56 50 45 0.015
4/11/1996 100 99.9 96.6 78.9 73 55.9 53 46 41 0.026
4/25/1996 100 100 99.8 95.3 93.5 81 75.7 65 59 54 49 0.008
5/22/1996 100 99.8 96.7 75.5 67 56 53 47 44 41 0.03
6/6/2009 100 99.9 97.1 74.9 69.5 57 55 48.7 45 42 0.023
6/19/2009 100 100 97.2 79.9 76.7 64 63 57 54 52 46 0.01
7/18/1996 100 100 99.8 97.9 95 92 88 75 67 62 54 0.007
8/1/1996 100 99.6 98.3 93 87.1 75.7 69 59 49 44 0.016
10/10/1996 100 100 99.8 96.3 91.7 84 77 61 54 0.01
55 Table 5.1 (continued)
11/21/1996 100 99.9 98.7 88.8 82.4 65 60 50 44 0.022
12/5/1996 100 100 98.1 88.8 82.9 66 62 53 48 0.017
12/19/1996 100 100 97.9 85 78.2 65 61 52 47 0.018
12/30/1996 100 100 95.9 78.6 71.7 57 54 47 42 0.026 0.012
Atchafalaya
River
1/5/1994 100 100 99.9 90.9 83.6 64 59.7 51.1 48 0.02
1/19/1994 100 100 99.4 90.9 81.1 68.8 63.1 50 45 0.022
2/2/1994 100 100 99.3 95.7 62.5 54.8 43.8 26.8 0.035
2/16/1994 100 99.5 95 71.2 44.2 33.7 32.2 28.9 0.07
3/2/1994 100 100 96.3 81.3 66 57 53 46 41 0.03
3/16/1994 100 99.8 94 75.1 54.2 34.7 32 27 25 0.05
3/30/1994 100 99.8 91.2 64.2 51.5 47 44.7 40 37.7 34 0.059
4/13/1994 100 100 93.6 72.3 57.2 50.7 48 43 42 0.043
5/2/1994 100 99.9 93.1 62.1 44.8 30.4 29 25.9 24.9 0.08
5/18/1994 100 100 96.3 78.4 72.5 65.7 63 57.9 54.9 51 0.01
6/15/1994 100 100 100 98.5 96.3 91 86 77 72 66 0.008
6/27/1994 100 100 99.3 97.1 95.1 93 90 83.7 79 75 0.007
7/13/1994 100 100 99.9 99.6 97.7
7/28/1994 100 100 99.8 98.4 93 96 91 76 70 67 53 0.007
8/17/1994 100 100 100 99.4 94.6 96 94.4 79.7 73 71 57 0.006
10/5/1994 100 100 100 99.8 96 99 95 81 73 72 55 0.007
11/9/1994 100 100 99.8 98.3 88.8 78 74 64 59 55 50 0.008
12/21/1994 100 100 99.4 96.5 75.3 58 53 43 37 0.03
12/28/1994 100 100 98.7 90.2 63.5 50.8 46.7 39 0.044 0.026
56 100 Sample at 1.1 ft from the Channel Bed 90 Fitted Line
80
70 r
e 60 n i
t F 50 en
rc 40 Pe 30
20
10 d50=0.19 mm 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Sediment Diameter (mm)
Figure 5-1 Determining Sample d50 on y-axis, from Grain Size Distribution, Missouri River at Omaha, D=18.4 ft, 0/16/1976
57 1.0
0.9
0.8
0.7
0.6
0.5 y/D
0.4
0.3
0.2 Point d50 0.1 Fitted Line
0.0 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
d50 (mm)
Figure 5-2 Distribution of Sample d50 on y-axis, Omaha Station, D=18.4 ft, 10/16/1976
58 1.0
0.9
0.8
0.7
0.6
0.5 y/D
0.4
0.3
0.2 d50 obtained on the y-axis or 0.1 Centerline of the Flume d50=0.117-0.24ln(y/D) 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 d (mm) 50
Figure 5-3 Relation between d50 and y/D, 8-foot Wide Flume (Data from R.S. McQuivey, 1973)
59 Table 5-2 d50 along y-axis, Missouri River at Ponca Station
Q=53100 cfs Q=52200 cfs Q=54000 cfs Q=41700 cfs
10/26/1978 9/20/1978 7/27/1978 6/22/1978
D=21.9 ft D=20.9 ft D=17.1 ft D=18.2 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.191 0.041 0.21 0.053 0.308 0.058 0.158 0.769
0.18 0.05 0.185 0.105 0.305 0.099 0.126 0.5
0.187 0.091 0.19 0.172 0.219 0.17 0.128 0.286
0.188 0.16 0.17 0.287 0.207 0.287 0.138 0.165
0.178 0.279 0.159 0.502 0.139 0.497 0.16 0.099
0.17 0.498 0.153 0.77 0.122 0.766 0.16 0.06
0.154 0.767
Q=35000 cfs Q=31000 cfs Q=33500 cfs Q=32700 cfs
8/4/1977 4/28/1977 6/16/1977 4/23/1981
D=18.1 ft D=15.5 ft D=12.9 ft D=11 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.197 0.061 0.183 0.065 0.192 0.062 0.151 0.745
0.188 0.099 0.176 0.103 0.21 0.101 0.179 0.445
0.166 0.166 0.181 0.174 0.178 0.163 0.213 0.209
0.156 0.282 0.163 0.29 0.168 0.287 0.219 0.073
0.134 0.503 0.172 0.503 0.136 0.535 0.327 0
0.179 0.768 0.161 0.768 0.123 0.767
Q=32500 cfs Q=31800 cfs Q=32500 cfs Q=36300 cfs
10/20/1977 6/11/1981 8/21/1980 7/17/1980
60 Table 5.2 (continued)
D= 12.2 ft D=11.7 ft D=12.2 ft D=8.2 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.204 0.098 0.148 0.769 0.137 0.77 0.139 0.768
0.2 0.164 0.147 0.504 0.122 0.451 0.183 0.5
0.19 0.287 0.158 0.291 0.139 0.287 0.223 0.28
0.176 0.5 0.158 0.171 0.155 0.164 0.235 0.171
0.163 0.77 0.205 0.111 0.15 0.098 0.246 0.098
Q=29800 cfs Q=29800 cfs Q=38400 cfs Q=31000 cfs
4/27/1978 4/24/1980 10/18/1979 9/15/1977
D=9.9 ft D=14.8 ft D=19.4 ft D=14.8 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.149 0.768 0.164 0.77 0.151 0.768 0.203 0.061
0.179 0.495 0.149 0.5 0.173 0.5 0.198 0.101
0.21 0.293 0.18 0.284 0.202 0.284 0.193 0.169
0.248 0.172 0.194 0.169 0.228 0.165 0.164 0.284
0.277 0.101 0.284 0.101 0.221 0.098 0.159 0.5
0.125 0.77
61 Table 5-3 d50 along y-axis, Missouri River at Nebraska City
Q=42000 cfs Q=41000 cfs Q=37500 cfs Q=38500 cfs
10/5/1976 4/26/1977 6/14/1977 8/2/1977
D=13.4 ft D=13.5 ft D=12.9 ft D=12.6 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.153 0.067 0.164 0.067 0.182 0.07 0.159 0.103
0.154 0.097 0.161 0.104 0.176 0.101 0.158 0.167
0.157 0.164 0.172 0.17 0.169 0.163 0.146 0.286
0.159 0.284 0.174 0.289 0.165 0.287 0.149 0.5
0.162 0.5 0.163 0.504 0.163 0.504 0.115 0.77
0.165 0.769 0.153 0.77 0.168 0.767
Q=38000 cfs Q=36800 cfs Q=51000 cfs Q=69200 cfs
9/13/1977 10/18/1977 4/25/1978 7/25/1978
D=11.0 ft D=12.5 ft D=15.8 ft D=17.6 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.178 0.1 0.18 0.096 0.183 0.057 0.219 0.057
0.174 0.164 0.179 0.168 0.177 0.101 0.218 0.102
0.173 0.282 0.176 0.28 0.169 0.165 0.185 0.165
0.159 0.5 0.164 0.496 0.164 0.285 0.194 0.284
0.137 0.773 0.147 0.768 0.145 0.5 0.183 0.5
0.142 0.766 0.177 0.767
Q=57000 cfs Q=60500 cfs Q=44900 cfs 49700 cfs
62 Table 5.3 (continued)
9/12/1978 10/24/1978 10/9/1979 8/28/1979
D= 17.1 ft D=18.3 ft D=13.7 D=16.2 ft d50 y/D d50 y/D d50 y/D d50 y/D
0.224 0.041 0.237 0.06 0.158 0.051 0.143 0.062
0.223 0.099 0.214 0.098 0.163 0.095 0.159 0.099
0.203 0.17 0.212 0.164 0.174 0.161 0.167 0.167
0.196 0.292 0.201 0.284 0.188 0.277 0.175 0.284
0.173 0.772 0.186 0.497 0.198 0.496 0.172 0.5
0.206 0.765 0.194 0.766 0.18 0.772
63
6.0 MATHEMATICAL MODELS OF SEDIMENT DISTRIBUTION
6.1 CONVENTIONAL MODLE-ROUSE EQUATION
The concentration profile C(y) is calculated by knowing the diffusion concept for suspended sediment (31, 32). In steady uniform flow, this leads to a balance between the downward settling of sediment due to gravity and upward diffusion associated with turbulent fluctuations
dC v C + ε = 0 (6-1) s s dy
dC v = − s dy (6-2) C ε s
Where vs = settling velocity; and ε s = diffusion coefficient for suspended sediment, which is normally taken to be proportional to the eddy viscosity of the flowε .
ε s = βε m (6-3)
Where β = momentum correction factor and ε m = diffusion coefficient for momentum transfer.
du ⎛ y ⎞ τ = ρε m = τ 0 ⎜1− ⎟ (6-4) dy ⎝ D ⎠
τ y = 1− (6-5) τ 0 D
τ 0 y du −1 2 τ du −1 ε m = (1− )( ) = u∗ ( ) (6-6) ρ D dy τ 0 dy
64 du where ρ = fluid density; = velocity gradient; u =shear velocity; and τ and τ = shear stress dy ∗ 0 at y and y = 0, respectively. Therefore, solution of (6-1) gives
Y C ⎡ v τ −1 du ⎤ = exp⎢− s ( ) dy⎥ (6-7) C 2 ∫ τ dy 0 ⎣ βu∗ 0 0 ⎦
where C0 = C at y = 0.
Mathematical models of distribution of sediment concentration may be derived by substituting velocity distributions in Equation (6-7). The well-known Rouse equation is derived when
Prandtl-von Karman logarithmic equation is used. The shear stress distribution, equation (6-5) was used in deriving Rouse equation.
z C ⎡ a D − y ⎤ = ⎢ ⎥ (6-8) Ca ⎣ D − a y ⎦
v z = s (6-9) kU* β
C = concentration of suspended sediment; Ca = a reference concentration at a distance a above the bed; y = vertical coordinate, measured upward from the bottom; D = water depth, and z=
Rouse number.
The conventional method of computation of sediment load is to substitute the logarithmic or power velocity distribution and the Rouse concentration profile in the following integral (8,11, 32):
D q = Cudy = C q (6-10) s ∫ i a
qs = sediment discharge per unit width; q = water discharge per unit width; a = two times the
(32) sediment diameter at the bed according to Einstein ; Ci = average concentration per unit width.
65 When Prandtl-von Karman universal velocity equation is used in derivation of Rouse equation, sediment concentration cannot be determined within a distance “a” from the channel bed.
6.1.1 Parameter Estimation
In order to estimate parameters of Rouse equation, A and B in the following equation must be calculated first by regression (10, 11, 18).
B ⎡ D − y ⎤ C = A⎢ ⎥ (6-11) ⎣ y ⎦ in which
B ⎡ a ⎤ A = Ca (6-12) ⎣⎢ D − a ⎦⎥
v B = s (6-13) ku∗
To calculate the Rouse parameters z and C0 , the following assumptions are made: β = 1, and a = 0.05D.
66
6.2 CHIU’S MATHEMATICAL MODEL OF DISTRIBUTION OF SEDIMENT
CONCENTRATION
The velocity and sediment concentration profiles for the y-axis are defined by Chiu’s velocity and sediment concentration equations (2), which estimate sediment concentration from the channel bed (y = 0) to the water surface (y= D).
1. If maximum velocity occurs on the water surface, h ≤ 0, y = D, then, Equations
(4-25), (4-26), and (4- 27) are used. In this case, the shear stress and velocity distribution are
y represented by Equations (6-5) and (4-10) with ξ = respectively, therefore, (6-7) gives the D following equation:
λ′ ⎡ y ⎤ 1− C ⎢ D ⎥ = ⎢ ⎥ (6-14) C M y 0 ⎢1+ (e −1) ⎥ ⎣⎢ D ⎦⎥
in which C0 is C at y = 0 and y = D at C = 0
v u (1− e −M ) v u (1− e −M ) ′ s max s λ = 2 = 2 = λG (6-15) βu∗ M βu∗ Mφ where
1− e −M G = (6-16) Mφ and
M vsu φM (2 + e ) λ = 2 = θ M (6-17) βu∗ 6(e −1)
67 Equation (6-14) is applicable at the channel bed but Rouse equation is not although both equations appear to be similar (2).
2. If maximum velocity occur below water surface, h>0, ξ max = 1, ξ 0 = 0 , then, Equations
(4-28) and (4-29) are used.
Shear stress distribution compatible with the velocity distribution with ξ given by (4-11) is expressed in the following power series form:
2 τ h ⎛ y ⎞ ⎛ h ⎞⎛ y ⎞ = ⎜1− ⎟ + ⎜1− ⎟⎜1− ⎟ (6-18) τ 0 D ⎝ D − h ⎠ ⎝ D ⎠⎝ D − h ⎠
which satisfies the boundary conditions that τ = τ 0 at y = 0; τ = 0 at y = h-D where u = umax and
h also at y = D (water surface). As approaches negative infinity, Equation (6-18) becomes (6-5). D
du Equation (4-29) with given by (6-19) dy
du u M ⎡ M ξ ⎤ −1 = (e −1){hξ ⎢1+ (e −1) ⎥ } (6-19) dy φMξ max ⎣ ξ max ⎦ gives
C ⎡ ' ⎛ y h ⎞⎤ = exp⎢− λ I⎜ , , M ⎟⎥ (6-20) C0 ⎣ ⎝ D D ⎠⎦
⎡ ' y h ⎤ C ⎢−λ I ( , ,M )⎥ = e ⎣ D D ⎦ (6-21) C0
−1 e M Y ⎡ ξ ⎤ τ I = h −1 1+ (e M −1) ( ) −1 dy (6-22) ∫ ξ ⎢ ⎥ ξ max 0 ⎣ ξ max ⎦ τ 0
y −1 ⎡ (1− ) ⎤ ⎛ y ⎞ D−h hξ = ()D − h ⎢⎜1− ⎟e ⎥ (6-23) ⎣⎝ D − h ⎠ ⎦
68 −1 M Y y D− y −1 e ⎡ D − h − y (1− ) ⎤⎡ y ( ) ⎤ ⎡ h y h y ⎤ I = e D−h 1+ (e M −1) e D−h (1− ) + (1− )(1− ) 2 dy ∫ ⎢ 2 ⎥⎢ ⎥ ⎢ ⎥ ξ max 0 ⎣ (D − h) ⎦⎣ D ⎦ ⎣ D D − h D D − h ⎦ (6-24)
Figures 6.1 and 6.2 compare Chiu’s sediment distribution with Rouse equation near the channel bed. They show that Chiu’s sediment distribution estimates the sediment concentration from the water surface to the channel bed, while Rouse equation is not applicable within a distance “a” from the channel bed. Figure 6.3 depicts Chiu’s sediment distribution model applied to a sediment data with the grain diameters (d=0.062 and 0.254 mm). It shows that for d=0.062 mm, the distribution of sediment concentration is uniform along y-axis.
1
0.1
0.05 y/D
0.01 Observed Data, D=12.9 ft Chiu's Equation, M=2.56, h=7.7
C0=2.25 g/L, λ=0.71 Rouse Equation
a/D=0.05, z=0.36, Ca=1.2 g/L 0.002 0.01 0.1 1 10 C (g/L)
Figure 6-1 Comparison of Sediment Distribution Models; Missouri River at Nebraska City, 6/14/1977
69 15 14 13 12 Rouse Equation Observed Data, Nebraska City Sta. 11 4/26/77, D=13.5 ft 10 Chiu's Equation M=2.56 9 8
y (ft) 7 6 5 4 3 2 1 0 0.00.20.40.60.81.01.21.41.61.82.0
C (g/L)
Figure 6-2 Comparison of Sediment Distribution Models
70 12
11
10 d=0.062 mm Observed Data; d=0.062mm 9 Chiu's Equation
M=2.56, h=6, C0=0.032, λ=0.12 8 Observed Data, d=0.254 mm 7 d=0.254 mm Chiu's Equation C0=1.026, λ=1.44 6 y (ft) 5
4
3
2
1
0 0.00.10.20.30.40.5 C (g/L)
Figure 6-3 Distribution of Sediment Concentration for two Sediment Diameters, Missouri River at Nebraska City, 9/13/1977
71
6.2.1 Parameter Estimation
To use the sediment distribution equations the parameters C0 and λ must be calculated by regression method as explained in the following steps:
Apply the Equations (6-14) or (6-21) to sediment data at 5 or 6 points on y-axis, and find the
parameters C0 and λ’
1. Find λ from the equation
λ′ λ = (6-15) G
2. θ and β are calculated from the following equations, having M and λ .
M M vs D vsu e −1 6(e −1) θ = = 6 2 M = λ M (6-17) ε s βu∗ φM (2 + e ) φM (2 + e )
A mathematical equation (2) (Cheng 1997) for settling velocity is used to calculate λ .
v d s = ( 25 +1.2d 2 − 5)1.5 (6-25) ν ∗ where d = particle diameter; ν = kinematics viscosity of the fluid; and
1 ⎛ ()ρ − ρ g ⎞ 3 d = ⎜ s ⎟ d (6-26) ∗ ⎜ 2 ⎟ ⎝ ρν ⎠
where ρ s = density of sediment.
The coefficient β that relates ε s to ε m is about unity, varying little with sediment concentration, and is constant along a vertical. It also tends to vary with sediment size and it is impossible to be calculated accurately. Therefore, by probabilistic approach Chiu presented the method of calculation as follows:
72 M M vs D vsu e −1 6(e −1) θ = = 6 2 M = λ M (6-17) ε s βu∗ φM (2 + e ) φM (2 + e )
θ is the only unknown parameter in the above equation that can be calculated.
θ is fairly invariant with sediment concentration but increases with sediment size. A practical procedure of determining β , is to obtain θ first, then substitute it into (6-17). Since M at a channel section is constant, the value of λ and λ′ can be obtained by (6-17) and (6-15) respectively.
Tables 6.1 and 6.2 present the summary of discharge, d50, λ and θ of Missouri River at four monitoring stations. The values of λ and θ were computed using Equations (6-15) and (6-17) respectively.
Figures 6.4, 6.5, and 6.6 show Chiu’s distribution of sediment concentration applied to sediment data on y-axis of Missouri River at Nebraska City and Omaha. Parameters C0 and λ’ were estimated by regression method and λ from Equation (6-15).
Figure 6.7 shows there is a good agreement between the measured and computed mean concentration using Chiu’s equations on y-axis.
Figures 6.8 and 6.9 present the transverse distribution of mean sediment concentration and mean velocity (measured) on each vertical in the Missouri River at Nebraska City. These figures show that the maximum sediment concentration and maximum velocity occur on y-axis.
73
Table 6-1 Hydraulic Data, Missouri River
Missouri River at Date of Water Total Vertical
Sediment discharge Depth d50 λ θ
Measurement D
M-D-Y cfs ft mm
Sioux City 10/7/1976 36500 11.4 0.177 0.86 1.53
10/11/1979 39600 15 0.181 1.76 3.03
4/17/1980 31000 18.9 0.197 1.53 2.62
5/29/1980 31900 20.8 0.162 1.54 2.64
7/10/1980 33500 24.2 0.184 2.03 3.49
8/28/1980 33700 12 0.144 0.85 1.45
10/16/1980 36600 17.2 0.17 0.54 0.92
4/16/1981 32700 18.8 0.188 1.55 2.65
6/4/1981 32100 18.8 0.168 1.19 2.04
7/16/1981 33400 17.2 0.227 1.69 2.9
8/27/1981 33400 16.6 0.246 3.57 6.13
10/15/1981 35100 13.6 0.224 1.04 1.78
Gayville 7/21/1981 35300 16.4 0.187 1.24 3.05
6/9/1981 32200 15.7 0.174 1.35 3.31
4/21/1981 33900 12.9 0.258 1.56 3.83
10/7/1980 37600 20 0.21 1.39 3.41
8/19/1980 32300 15.7 0.143 1.96 4.81
7/15/1980 36600 13.7 0.199 1.65 4.04
6/3/1980 28500 12.8 0.177 1.24 3.03
74 Table 6.1 (continued)
4/22/1980 31400 15.4 0.166 1.73 4.25
Nebraska City 10/5/1976 42000 13.4 0.158 0.27 0.72
4/26/1977 41000 13.5 0.164 0.61 1.64
6/14/1977 37500 12.9 0.169 0.89 2.41
8/2/1977 38500 12.6 0.149 0.96 2.6
9/13/1977 38000 11 0.173 0.87 2.35
10/18/1977 36800 12.5 0.176 1 2.69
4/25/1978 51000 15.8 0.167 1.43 3.87
7/25/1978 69200 17.6 0.19 0.85 2.31
9/12/1978 57000 17.1 0.203 1.6 4.33
10/24/1978 60500 18.3 0.209 2.01 5.41
10/9/1979 44900 13.7 0.181 1.72 4.66
8/28/1979 49700 16.2 0.169 0.96 2.6
75 Table 6-2 Hydraulic Data, Missouri River at Omaha
Missouri River at Date of Water Total Vertical
Sediment discharge Depth d50 λ θ
Measurement D
M-D-Y cfs ft mm
Omaha 10/6/1976 38000 18.4 0.164 1.59 3.59
4/27/1977 31500 13.2 0.149 0.72 1.63
6/15/1977 34500 15.5 0.15 0.47 1.07
7/26/1978 63300 13.1 0.187 1.28 2.88
8/3/1977 34500 20 0.167 1.17 2.64
9/14/1977 34500 16.6 0.15 1.46 3.38
10/19/1977 33000 15.1 0.171 1.34 3.03
4/26/1978 34500 18.5 0.145 1.07 2.24
6/21/1978 47200 21.4 0.155 1.89 4.28
10/25/1978 58100 19 0.173 1.76 3.99
9/12/1978 54800 19.8 0.172 1.04 2.35
4/30/1979 43100 12.8 0.186 1.43 3.53
6/6/1979 46400 20.2 0.164 1.34 3.03
7/18/1979 43000 18.1 0.19 1.38 3.12
8/29/1979 43900 16 0.177 1.56 3.53
4/16/1980 34700 15.7 0.182 0.73 1.66
10/10/1979 41200 17.5 0.163 0.87 1.97
5/30/1980 32900 17.4 0.181 1.90 2.65
8/26/1980 34100 17.4 0.181 1.85 4.29
7/9/1980 32800 19 0.153 1.57 3.56
76 Table 6.2 (continued)
10/15/1980 37100 16.2 0.175 1.48 3.34
4/15/1981 35700 13.6 0.179 0.99 2.23
6/3/1981 31700 17.2 0.167 0.99 2.24
10/14/1981 32900 14.2 0.176 1.51 3.37
8/26/1981 33800 16.6 0.187 1.16 2.69
7/15/1981 33300 16.2 0.187 1.58 3.66
12
Observed Data Chiu's Equation, M=2.56, 10 h=6 ft, C0=1.763 g/L, λ=0.82
8
ft) 6 y (
4
2
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
C (g/L)
Figure 6-4 Chiu’s Distribution of Sediment Concentration, Missouri River at Nebraska City, D=11.0 ft, 9/13/1977
77 20 Observed Data, Missouri River at Omaha, 10/6/76 18 Chiu's Sediment Distribution Equation, M=3.22, h=11.0 ft, C =8.61 g/L, λ=1.5 16 0
14
12
10 y (ft) 8
6
4
2
0 012345678 C (g/L)
Figure 6-5 Chiu’s Distribution of Sediment Concentration, Missouri River at Omaha, D=18.4 ft, 10/6/1976
78 20 Observed Data
Chiu's Equation
M=3.22, h=5.8 ft, C0=1.879 g/L, λ =1.07 15
ft) 10 y (
5
0 0.0 0.5 1.0 1.5 2.0
C (g/L)
Figure 6-6 Chiu’s Distribution of Sediment Concentration, Missouri River at Omaha, D=18.5, 4/26/1978
79 0.8 Concentrations at y-axis Line of Perfect Agreement 0.7 's Method
0.6 ions by Chiu
t 0.5 a r t n
0.4 Conce d e
ut 0.3 (g/L) y Comp _ C 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Measured Concentrations by Two-point Method _ C (g/L) y
Figure 6-7 Comparing the Computed and Measured Mean Concentration on y-axis, Missouri River at Nebraska City
80 y-axis
7 0.8
6 0.7
0.6 5
0.5 )
4 ) L
t/s 0.4 (f _ C (g/ _ u 3 0.3
2 0.2
1 0.1
0 0.0 0 100 200 300 400 500 600 Distance Between the Reference Point and Each Vertical (ft)
Average Vertical Velocity, Measured, 6/14/1977, Q=37500 cfs Average Vertical Concentration, Measured, 6/14/1977 Average Vertical Concentration, Measured, 4/26/1977 Average Vertical Velocity, Measured, 4/26/1977, Q=42000 cfs
Figure 6-8 Transverse Distribution of Average Sediment Concentration and Velocity, Missouri River at Nebraska City
81 Mean Velocity on Verticals y-axis 5 0.5 Cross-sectional Mean Velocity 4 0.4
) 3 0.3 /L) g ft/s Cross-sectional Mean Conc. (
2 ( y y
0.2 _ C _ u 1 0.1 0 Mean Concentration on Verticals 0.0 0
-5 (ft) h t p e -10 l D ne an Ch
-15
0 200 400 600
Distance (ft) from the reference point
Figure 6-9 Distribution of Average Concentration and Velocity (Measured) across the Channel of Missouri River at Nebraska City
82 7.0 CALCULATION OF THE CROSS-SECTIONAL MEAN CONCENTRATION BY
THE EFFICIENT METHOD
7.1 CORRELATION BETWEEN THE CROSS-SECTIONAL MEAN
CONCENTRATION AND THE MEAN CONCENTRATION ON EACH
VERTICAL
Figures 7.1 to 7.6 show the correlation between cross-sectional mean concentration and the mean concentration on each vertical across the channel sections of different rivers. These figures demonstrate that the correlation is highest on y-axis and decreases toward the channel banks.
Therefore, it provides a basis for collecting the data on y-axis.
The mean concentration on each vertical (Ci , i =1..5) was obtained by averaging the sediment concentration measured at two-tenths and eight-tenths of water depth on each vertical using
Equation (2-10). The cross-sectional mean concentration (Cx ) was calculated using the Equation
(2-8). A study of Figures 7.1 to 7.6 revealed that the river sections with coarser d50 such as
2 Missouri River have lower r than the river sections with finer d50 such as Mississippi River. It is obvious that in channel sections with coarse particles, the transverse distribution is not uniform.
According to L.Yuqian (22) appreciable errors would be expected in the concentration of coarse sediment if only a limited number of verticals were adopted.
83 700 700 Averrage Vertical by y-axis is on vertical 2 600 Two-Point Method 600 2 r =0.969 r2=0.954 _ _ 500 _ _ 500 C =0.81C +19.9 x y Cx=0.95C1+9.5 )
) 400 400 (ppm x (ppm 300 x 300 _ C _ C
200 200
100 100
0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 _ _ C1 (ppm) Cy (ppm)
700 700 Average Vertical by Two-Point Average Vertical by Two-Point 600 r2=0.96 600 r2=0.959 _ _ _ _ 500 500 C =1.12C -2.1 Cx=0.96C3+1.5 x 4 ) 400 400 (ppm (ppp) x 300 x 300 _ C _ C
200 200
100 100
0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600
_ _
C3 (ppm) C4 (ppm)
Figure 7-1 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration (All Sizes), Mississippi River at Tarbert, 1995 and 96
84 800 800 Measured Averaged Vertical Measured Averaged Vertical r2=0.937 600 r2=0.852, _ _ 600 _ _ ) Cx=0.89 Cy+19 Cx=0.90C1+14 ) pm 400 (p ppm
x 400 ( x _ C _ C 200 200
0 0 100 200 300 400 500 600 700 800 0 0 200 400 600 800 _ _ C y (ppm) C1 (ppm) 800 Measured Averaged Vertical 800 Measured Vertical Concentration r2=0.937 600 700 2 _ _ r =0.919 600 _ _ ) Cx=0.87C2+21.8 ) Cx=0.88C3+27 pm 400 m 500 (p x (pp 400 x _ C _ C 200 300 200
0 100 0 100 200 300 400 500 600 700 800 0 _ 0 100 200 300 400 500 600 700 800 C (ppm) 2 _ C3 (ppm) 800 Measured Vertical Concentration 700 r2=0.900 600 _ _ Cx=1.04C4+36 500 400 (ppm) x
_ C 300 200 100 0 0 200 400 600 800 _ C (ppm) 4
Figure 7-2 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Mississippi River at Union Point, 1994, 95 and 96
85 0.6 0.7 Vertical Average Concentration 0.5 Average Concentration on y-axis 0.6 by two-point 2 2 r =0.78 0.5 r =0.20 0.4 _ _ C =0.59C +0.06 x y 0.4 0.3 /L) (g/L) (g x x 0.3 _ C _ C 0.2 0.2
0.1 0.1
0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 _ _ Cy (g/L) C1 (g/L)
0.6 0.6
0.5 Vertical Line #2 0.5 Vertical Line #3 2 r =0.60 r2=0.51 0.4 0.4 ) /L /L) g
0.3 ( 0.3 (g x x _ C _ C 0.2 0.2
0.1 0.1
0.0 0.0 0.00.10.20.30.40.50.6 0.00.10.20.30.40.50.6 _ _ C (g/L) C (g/L) 2 3
0.6 0.6 Vertical Line #4 Vertical Line #5 0.5 2 0.5 r =0.78 r2=0.20 0.4 0.4 /L) /L) g
0.3 ( 0.3 (g x x _ C _ C 0.2 0.2
0.1 0.1
0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 _ _ C (g/L) C4 (g/L) 5
Figure 7-3 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Missouri River at Ponca, 1979-81
86 0.7 0.50 Measured Concentrations 0.6 By 2-point method 0.45 r2=0.05 0.5 0.40 )
0.4 ) /L /l g ( (g
x 0.35
0.3 x _ C _ C 0.2 0.30 Vertical No. 2 0.1 0.25 Measured Concentrations Vertical No.1 2 0.0 r =0.65 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.20 _ 0.2 0.3 0.4 0.5 0.6 0.7 C1 (g/L) _ C2 (g/L)
0.50 0.50 Measured Concentrations Measured Concentrations 2 0.45 r2=0.04 0.45 r =0.035
0.40 0.40 ) ) /L /L
(g 0.35 (g
x 0.35 x _ C _ C 0.30 0.30
0.25 0.25 Vertical No.4 Vertical No.3 0.20 0.20 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.0 0.1 0.2 0.3 0.4 0.5 0.6 _ _ C (g/L) C3 (g/L) 4 0.60 0.40 Measured Concentrations Measured Concentrations 0.55 r2=0.16 r2=0.77 0.50 0.35 ) L
0.45 / ) (g /L 0.40 x 0.30 (g _ C x _ C 0.35
0.30 0.25
0.25 Vertical No.5 y-axis 0.20 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 _ _ C (g/L) C5 (g/L) y
Figure 7-4 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Missouri River at Nebraska City 1977-81
87 350 Vertical Concentration 300 Vertical Concentration 2=0.978 300 r 2 _ _ r =0.917 250 _ _ Cx=0.89Cy+9.9 250 Cx=C1-28.3
) 200 pm)
200 pm (p (p x x _ C 150 150 _ C
100 100
50 50 50 100 150 _ 200 250 300 350 50 100 150 _ 200 250 300 350 Cy (ppm) _ C1 (ppm) Distance of y-axis distance from the reference Zy=1950 ft Distance of Vertical 1 from reference Z1=1100 ft
300 350 Vertical Concentration Vertical Concentration 2 2 300 r =0.965 250 r =0.978 _ _ _ _ C =0.89C +7.1 Cx=0.86C3+24.9 250 x 2 ) 200 ) pm pm 200 (p (p x
x 150 _ C _ C 150
100 100
50 50 50 100 150 200 250 300 350 50 100 150 _ 200 250 300 350 _ C2 (ppm) C3 (ppm) Distance of Vertical 3 from reference Z =2250 ft Distance of Vertical 2 from reference Z2=1650 ft 3
Vertical Concentration 300 r2=0.988 _ _ Cx=1.01C4+15.1 ) 200 pm (p x _ C
100
0 0 100 200 300 Distance of Vertical 4 from reference Z4=2950 ft _ C4 (ppm)
Figure 7-5 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Mississippi River at Range 362.2
88
0.8 0.8 y-axis 2 Vertical No.1 r = 0.63 2 0.6 _ _ 0.6 r = 0.23 Cx=0.64Cy+0.04 ) L / /L) g (g 0.4 0.4 ( x x _ C _ C
0.2 0.2
0.0 0.0 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.0 0.1 0.2 0.3 0.4 0.5 _ _ C (g/L) Cy (g/L) 1 0.8 0.8 Vertical No.2 Vertical No.3 2 2 = 0.42 0.6 r = 0.31 0.6 r /L) /L) 0.4 0.4 (g (g x x _ C _ C
0.2 0.2
0.0 0.0 0.10.20.30.40.50.60.7 0.1 0.2 0.3 0.4 0.5 0.6 _ _ C (g/L) C2 (g/L) 3 0.8 0.8 Vertical No.5 Vertical No.4 r2 = 0.2 0.6 r2 = 0.50 0.6 ) ) L / 0.4
0.4 (g/L (g x x _ C _ C 0.2 0.2
0.0 0.0 0.00.10.20.30.40.50.6 0.0 0.1 0.2 0.3 0.4 _ _ C4 (g/L) C5 (g/L)
Figure 7-6 Correlation between Mean Concentration on each Vertical and Cross-sectional Mean Concentration, Missouri River at Omaha, 1976-81
89 7.2 CORRELATION BETWEEN CROSS-SECTIONAL MEAN CONCENTRATION
AND MEAN CONCENTRATION ON Y-AXIS
The ratio of cross-sectional mean concentration (Cx ) to mean concentration on y-axis (Cy ) is
C x defined ψ = . This coefficient was calculated for each sediment size (d) and median size (d50) C y on y-axis. The ψ value representative of each station was determined by regression.
The ψ values obtained from the plots of Cx against Cy were used in developing the relation between M and ψ . Figures 7.7 to 7.15 show the correlation between cross-sectional mean concentration and the concentration on y-axis. These figures show ψ and r2 decrease as sediment size (d=average of each size class) increases. Also, ψ found to be higher for channel sections with finer d50.
The suspended sediment of Mississippi River consists predominantly of particles finer than 0.06 mm, as opposed to the Missouri and Niobrara Rivers, which contain coarser particles (0.13-0.19 mm). It was found that the Mississippi River (Figures 7.7, 7.8, and 7.11) has higher r2 between
Cx and Cy than Missouri River.
Figure 7.14 shows strong correlation between the cross-sectional mean concentration and mean concentration on y-axis of Sacramento River at station 37.85. The suspended sediment of the
(17) Sacramento River consists of predominantly fine sediment with d50 of smaller than 0.06 mm.
90 400 All Sizes (d=0.004-0.25 mm) 60 d=0.125-0.25 mm (0.19 mm) _ _ Ψ=0.62 C =0.93C 300 x y r2=0.87 2
Ψ=0.93, r =0.98 ) ) 40 pm pm 200 (p (p x x _ C _ C 20 100
0 0 0 102030405060 0 100 200 300 400 _ _ Cy (ppm) Cy (ppm)
50 20 d=0.062-0.25 mm (0.16 mm) d=0.06-0.125mm Ψ=0.68 Ψ=0.9 40 2 2 r =0.85 15 r =0.87
) 30 )
10 (ppm x (ppm 20 _ C x _ C
5 10
0 0 0 10203040506070 0 5 10 15 20 _ _ Cy (ppm) Cy (ppm)
300 d=0.004-0.06 mm (0.03 mm)
250 Ψ=0.97 r2=0.97
) 200 (ppm
x 150 _ C
100
50 50 100 150 200 250 300 350
_ Cy (ppm)
Figure 7-7 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for various sediment diameters, Mississippi River at Range 362.2, 1991 (M=2.67)
91
700 200 All Sizes (d=0.004-0.25 mm, d50=0.012 mm) 600 _ _ d=0.125-0.25 mm (d=0.19 mm) C = 0.87C Ψ=0.87, r2=0.97 Ψ= 0.56 x y 2 500 150 r =0.87
) 400
(ppm 100
x 300 (ppm) _ C x _ C 200 50 100
0 0 100 200 300 400 500 600 700 0 0 50 100 150 200 _ _ C (ppm) y Cy (ppm) 250 140
d=0.062-0.25 mm (d=0.16 mm) d=0.062-0.125 mm (d=0.09 mm) 200 120 Ψ=0.61 Ψ=0.62 2 2 r =0.91 100 r =0.91
) 150 80 ppm pm) x ( (p _ C 100 x 60 _ C
40 50
20
0 0 50 100 150 200 250 0 0 20 40 60 80 100 120 140 _ _ C (ppm) y Cy (ppm)
500 d=0.004-0.062 mm (d=0.03 mm) Ψ=0.98 400 r2=0.99
) 300 (ppm x
_ C 200
100
0 0 100 200 300 400 500 _
Cy (ppm)
Figure 7-8 Relation between Mean Concentration on y-axis and Cross- sectional Mean Concentration for various sediment diameters, Mississippi River at Tarbert 1995-96 (M=2.65)
92 20 120 d=0.359 mm d=0.254 mm ψ=0.53 Ψ=0.76 r2=0.29 100 15 r2=0.21
) 80 pm
10 ppm) (p ( x x _ C _ C 60
5 40
0 20 0 5 10 15 20 0 20406080100120 _ _ Cy (ppm) Cy (ppm)
300 160 d=0.18 mm 140 d=0.127 mm 250 Ψ=0.76 Ψ=0.7 2 r =0.72 120 r2=0.84 200
) 100 pm) pm
150 (p 80 x (p x _ C
_ C 60 100 40 50 20
0 0 0 50 100 150 200 250 300 0 20 40 6_0 80 100 120 140 160 _ C (ppm) Cy (ppm) y 40 50 d=0.064 mm d=0.09 mm ψ=0.9 40 ψ=0.76 30 r2=0.89 r2=0.73 pm) ) 30 m (p
x 20 _ C (pp x
_ C 20
10 10
0 0 0 5 10 15 20 25 30 35 40 0 1020304050 _ _ Cy (ppm) Cy (ppm)
Figure 7-9 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for various Sediment Diameters, Missouri River at Omaha (M=3.22)
93
600 Whole Sample d=0.053-0.42 mm 500 Ψ=0.75 r2=0.63
400 )
ppm 300 ( x _ C
200
100
0 0 100 200 300 400 500 600 _ C (ppm) y
Figure 7-10 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for All Sizes Missouri River at Omaha (M=3.22)
94 1000 200 All sizes (d =0.022 mm) 50 d=0.125-0.25 mm (d=0.19 mm) 800 Ψ=0.89 Ψ=0.60 2 150 2 r =0.95 r =0.72
600 ) ) pm 100 pm (p x (p x
400 _ C _ C
50 200
0 0 0 200 400 600 800 1000 0 50 100 150 200 _ _ Cy (ppm) Cy (ppm)
120 400 d=0.062-0.125 mm (d=0.09 mm) d=0.062-.25 mm (d=0.16 mm) 100 Ψ=0.76 Ψ=0.65 r2=0.81 300 r2=0.76 80 ) ) pm m 60 200 (p x (pp _ C x
_ C 40
100 20
0 0 0 20406080100120 0 100 200 300 400 _ _ Cy (ppm) Cy (ppm)
900
800 d=0.004-0.062 mm (d=0.03 mm)
700 Ψ=0.99 r2=0.0.99 600 ) 500 pm (p
x 400 _ C 300
200
100
0 0 100 200 300 400 500 600 700 800 900 _ Cy (ppm)
Figure 7-11 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for various Sediment Diameters, Mississippi River at Union Point, 1994, 95 and 96 (M=2.07)
95 160 50 d=0.254 mm 140 d=0.359 mm Ψ=0.64 40 Ψ=0.60 120
) 100 ) 30 80 (ppm (ppm x x 20 _ C 60 _ C 40 10 20
0 0 0 20 40 60 80 100 120 140 160 0 1020304050 _ _ Cy (ppm) Cy (ppm)
140 100 d=0.127 mm d=0.18 mm 120 Ψ=0.77 Ψ=0.65 80 100 ) ) 60 80 (ppm x (ppm x 60 _ C 40 _ C 40 20 20
0 0 0 20 40 60 80 100 120 140 160 180 200 220 0 20406080100120 _ _ Cy (ppm) Cy (ppm) 50 30 d=0.09 mm d=0.064 mm Ψ=0.95 25 Ψ=1.00 40 20 ) )
30 m p 15 (p (ppm x x _ C
_ C 20 10
10 5
0 0 0 1020304050 0 5 10 15 20 25 30 _ _ C (ppm) y Cy (ppm)
Figure 7-12 Relation between Mean Concentration on y-axis and Cross –sectional Mean Concentration for various Sediment Diameters, Missouri River at Ponca (M=2.89)
96 600
Whole Sample Concentration d=0.053-0.42 mm 500
Ψ=0.75 r2=0.76 _
400 Cx=0.75Cy )
300 (ppm x _ C
200
100
0 0 100 200 300 400 500 600 _ C (ppm) y
Figure 7-13 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for All Sizes Missouri River at Ponca (M=2.89)
97 300
d50=0.06 mm 250 ψ=0.87, r2=0.99
200 )
150 (ppm x _ C
100
50
0 0 50 100 150 200 250 300 _ C (ppm) y
Figure 7-14 Relation between Mean Concentration on y-axis and Cross –sectional Mean Concentration for All Sizes Sacramento River at Station 37.85 (M=1.6)
98 3500 300
3000 All Sizes 250 Ψ=0.51 d=0.064 mm 2500 r2=0.56 Ψ=0.97 200 2 ) r =0.91 2000 ) pm 150 (p (ppm x 1500 x _ C _ C 100 1000
500 50
0 0 0 500 1000 1500 2000 2500 3000 3500 0 50 100 150 200 250 300 350 _ _
Cy (ppm) Cy (ppm) 700 2000 600 d=0.09 mm d=0.18 mm Ψ=0.65 Ψ=0.39 2 ) 2 500 r =0.51 1500 r =0.83 (ppm) x (ppm _ C
x 400 _ C 1000 300
200 500 100
0 0 0 100 200 300 400 500 600 700 0 500 1000 1500 2000 2500 _ _ C (ppm) y Cy (ppm) 500
400 d=0.37 mm Ψ=0.31 r2=0.64 300 (ppm) x
_ C 200
100
0 0 100 200 300 400 500 _
Cy (ppm)
Figure 7-15 Relation between Mean Concentration on y-axis and Cross-sectional Mean Concentration for Various Sediment Diameters, Niobrara River Near Cody at Gaging Station (M=1.73)
99
7.2.1 Relation of r2 to Sediment Diameter and Location of Verticals
2 The coefficient of determination (r ) of Cx -C y relation is higher for fine sediment particles
(d50<0.062 mm). It means that in these streams, the sediment particles are uniformly distributed,
and C y is a good predictor of Cx . The distribution of sediment concentration in channel
2 sections, with coarse sediment particles (d50>0.062 mm) is not uniform and r is lower.
Figure 7.16 depicts the location of measured verticals relative to y-axis on Mississippi River at
Tarbert. Vertical 2 is coincident with y-axis (Figure 4.4). Bj (j=1, 2) = the distance between y- axis and the channel banks, and z3/B2 = the ratio of the distance between vertical 3 and y-axis, to the distance between y- axis to the channel bank.
Figures 7.17, 7.19, 7.21, and 7.23 show the relation of r2 (of ψ ) to the sediment diameter on y- axis. These figures show that when sediment diameter increases, r2 (of ψ ) decreases. Figures
7.18, 7.20, 7.22 and 7.24 show the relation of r2 (of ψ ) to the location of each vertical relative to
y-axis. These figures show that the correlation between Cx and C y is higher on y-axis than other verticals. Also, the verticals closer to y-axis have higher values of r2.
100
y-axis Reference Point B1 B2
z4 z 1 z2=0 z 50 3 ) t
h (f 25 pt De
0 1000 1500 2000 2500 3000 3500 4000 4500
Distance from Reference Point (ft)
Figure 7-16 Location of Measured Verticals Relative to y-axis, Mississippi River at Tarbert
101
1.0
0.9
0.8 r2
0.7 Average of upper and lower of each size class 0.6 r2=0.93
d50 of All Sizes 0.5 0.00 0.05 0.10 0.15 0.20 0.25
d, mm (Sediment Size)
Figure 7-17 Relation of r2 (of ψ) to Sediment Diameter on y-axis Mississippi River at Union Point, 1994, 95 and 96
102
1.00
0.95
0.90 r2
0.85
z = Distance from y-axis 0.80 i BJ=Distance of y-axis from channel banks r2=0.64 0.75 0.0 0.2 0.4 0.6
zi/BJ
Figure 7-18 Relation of r2 (of ψ) to the Location of Verticals (Relative to y-axis) Mississippi River at Union Point, 1994, 95 and 96
103
1.0
0.8
0.6 r2
0.4
0.2 Average of each size class r2=0.78
d50 of all sizes 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
d, mm (sediment Size)
Figure 7-19 Relation of r2 (of ψ) to Sediment Diameter on y-axis Missouri River at Omaha 1976-81
104
y-axis
zi = Distance from y-axis 0.8 BJ=Distance of y-axis from channel banks r2=0.95
0.6 r2 0.4
0.2
0.0 0.0 0.2 0.4 0.6 0.8
zi/BJ
Figure 7-20 Relation of r2 (of ψ) to the Location of Verticals Missouri River at Omaha 1976-81
105
1.00
0.98
0.96
0.94 r2
0.92
0.90 d=Average of each size class 0.88 r2=0.89
d50 of All Sizes 0.86 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
d, mm (Sediment Size)
Figure 7-21 Relation of r2 (of ψ) to Sediment Diameter on y-axis Mississippi River at Tarbert, 1995-96
106
y-axis 0.99
0.98
0.97
0.96 r2 0.95
0.94
0.93 z1= Distance from y-axis
BJ= Distance of y-axis from channel banks 0.92 r2=0.84
0.91 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
z /B i J
Figure 7-22 Relation of r2 (of ψ) to Location of Verticals, Mississippi River at Tarbert, 1995-96
107
1.02 Average of each size class 1.00 r2=0.86 d of all sizes 0.98 50
0.96 r2 0.94 0.92
0.90
0.88
0.86
0.84 0.00 0.05 0.10 0.15 0.20
d,mm (Sediment Size)
Figure 7-23 Relation of r2 (of ψ) to Sediment Diameter on y-axis, Mississippi River at Range 362.2, 1991
108
y-axis 1.00
0.98
0.96 r2
0.94
zi = Distance from y-axis 0.92 BJ =Distance of y-axis from channel banks r2=0.91
0.90 0.0 0.1 0.2 0.3 0.4 0.5 0.6
z /B i J
Figure 7-24 Relation of r2 (of ψ) to Location of Verticals (Relative to y-axis) Mississippi River at Range 362.2, 1991
7.2.2 Relation of ψ to Q and d50
Data analysis on different rivers in this research indicate that ψ is not sensitive to changes in water discharge and d50, but it decreases as sediment diameter (average of each size class) increases at a station. Figures 7.25 to7.29 show that ψ has poor relation with Q and
d50. Figure 7.30 shows as sediment diameter (d, not d50) increases for a channel section, ψ and r2 will decrease. r2 is defined as the coefficient of determination between cross-sectional mean concentration and mean concentration on y-axis as indicated on Figures 7.7 to 7.15.
109 1.0
0.9
0.8
0.7
0.6
ψ 0.5
0.4
0.3 ψ = C /C 0.2 x y _ =0.9 0.1 ψ
0.0 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 1.4e+6 Water Discharge (cfs)
Figure 7-25 Relation between ψ and Discharge, Mississippi River at Union Point, 1995-96
110 1.0
0.9
0.8
0.7
0.6
ψ 0.5
0.4
0.3
Measured ψ=Cx/Cy 0.2 _ ψ=0.90 0.1
0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06
d50 (mm)
Figure 7-26 Relation between ψ and d50 on y-axis, Mississippi River at Union Point, 1995-96
111 700
600 Measured Concentration _ 500 Ψ=0.68
400 (ppm) x 300 C
200
100
0 0 100 200 300 400 500 600 700
Cy (ppm) a) Relation Between Mean Cross-sectional Concentration and and Mean Concentration on y-axis
1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6
Ψ 0.5 0.5 Ψ 0.4 0.4 0.3 0.3 Measured Ψ=Cx/Cy Measured Ψ=Cx/Cy 0.2 0.2 0.1 _ 0.1 _ Ψ=0.68 Ψ =0.68 0.0 0.0 0.15 0.16 0.17 0.18 0.19 0.20 35000 40000 45000 50000 55000 60000 65000 70000 75000 d (mm) Q (cfs) 50
c) Relation Between Ψ and d b) Relation Between Ψ and Q 50
Figure 7-27 Relation of ψ to d50 and Discharge, Missouri River at Nebraska City
112 500
Measured Concentration 400 Ψ=0.78
) 300 (ppm x
_ C 200
100
0 0 100 200 300 400 500 _ Cy (ppm)
a) Relation Between Cross-Sectional Mean Concentration and Mean Concentrations on y-axis
1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6
Ψ 0.5
Ψ 0.5 0.4 0.4
0.3 Measured Ψ=C /C 0.3 x y Measured Ψ=Cx/Cy 0.2 0.2 _ _ 0.1 Ψ=0.78 0.1 Ψ=0.78 0.0 0.0 30000 32000 34000 36000 38000 40000 42000 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26
Q (cfs) d50 (mm)
b) Relation Between Ψ and Q c) Relation Between Ψ and d50
Figure 7-28 Relation of ψ to d50 and Discharge, Missouri River at Sioux City
113 100 Measured Concentration 80 Ψ=0.72
60 Ψ 40
20
0 0 20406080100 _ Cy (ppm) a) Relation Between Cross-sectional Mean Concentration and Mean Concentration on y-axis
1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6
0.5 Ψ 0.5 Ψ 0.4 0.4 Measured Ψ=C /C 0.3 Μeasured Ψ=Cx/Cy 0.3 x y 0.2 _ 0.2 _ Ψ=0.72 0.1 Ψ=0.72 0.1 0.0 0.0 28000 30000 32000 34000 36000 38000 40000 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28
Q (cfs) d50 (mm) b) Relation Between Ψ and Q c) Relation Between Ψ and d50
Figure 7-29 Relation of ψ to d50 and Discharge, Missouri River at Gayville
114
1.0 1.0
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6 r2 on y-axis
ψ 0.5 0.5
0.4 d= Mean of each Size Class, related to ψ 0.4
d50 of the Station (on y-axis) related to ψ 0.3 d= Mean of each Size Class, related to r2 0.3 2 d50 of the Station (on y-axis), related to r 0.2 0.2 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
d (mm) sediment diameter
Figure 7-30 Relation of ψ to Sediment Diameter (d) and r2, Mississippi River at Union Point, 1994-96
115
7.3 RELATION BETWEEN d50 AND WATER DISCHARGE
Hubbell and Matejka(37) found that “On streams such as the Middle Loup River, where sediment is relatively coarse and unlimited in supply and where the size distribution is evidently unrelated to water discharge and water temperature, the most representative size distribution can probably be determined from an average of all available samples.” Also, they showed that d50 and water discharge do not have direct relation.
(26) Kircher investigated the relation between d50 and water discharge for Platte River in Nebraska and found that the median diameter decreases somewhat with discharge at Overton, but no trend is detected at Grand Island.
Colby and Hembree(29) indicated in their report on Niobrara River, that no relationships have been clearly defined for Niobrara River between water discharge and bed-material sizes. They also reported that at this station suspended sediment tend to become smaller at high flows.
To study the relationship between discharge (Q) and d50, data sets on different rivers were analyzed as part of this research. Consequently, plots of Q against d50 on y- axis for different rivers did not reveal any trend, although poor relations for some datasets were found. Figures
7.31 to 7.35 show poor relations between Q and d50 on y-axis for various rivers.
116
0.40
0.35
0.30 d50 on y-axis (Calculated) 0.25 ) m 0.20 (m 50 d 0.15
0.10
Cross-sectional d50 at Nebraska City and Omaha (11) 0.05
0.00 20000 30000 40000 50000 60000 70000 80000 River Discharge (cfs)
Figure 7-31 Relation between Discharge and d50 on y-axis, Missouri River at Ponca, Nebraska City, Omaha, Gayville and Sioux City
117 0.50 Niobrara River 0.45 Middle Loup River Sections B, C, and D 0.40
0.35
0.30 )
0.25 (mm 50 d 0.20
0.15
0.10
0.05
0.00 250 275 300 325 350 375 400 425 450 475 500 Q (cfs)
Figure 7-32 Relation between Discharge and d50 at y-axis, Niobrara River at Gaging Station (1950-52) and Middle Loup River (1948-60)
118 0.14
0.12
0.10 )
m 0.08 (m 50 d 0.06
0.04
0.02
0.00 4.0e+5 6.0e+5 8.0e+5 1.0e+6 1.2e+6 1.4e+6
Discharge (cfs)
Figure 7-33 Relation between Discharge and d50 on y-axis and Discharge, Mississippi River at Union Point, 1995-96
119 1.0
Niobrara River, d50 obtained from 0.9 Size Distribution on y-axis Middle Loup River d =0.135-0.04ln(y/D) 0.8 50
0.7
0.6
/D 0.5 y
0.4
0.3
0.2
0.1
0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 d (mm) 50
Figure 7-34 Relation between d50 and y/D, Niobrara River near Cody 3/3/1950-5/8/52 (Q=302– 405 cfs); and Middle Loup River Section B at Dunning, 6/20/1950 (Q= 403 cfs)
120 1.0
0.8
Q= 54000 cfs 0.6 53100 cfs 52200 cfs 39800 cfs y/D 36500 cfs 0.4 33500 cfs 32700 cfs 32500 cfs 31000 cfs 0.2
0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 d (mm) on y-axis 50
Figure 7-35 Relation between Discharge and d50 along y-axis, Missouri River at Ponca, 1977-81
7.4 MEAN SEDIMENT CONCENTRATION ON Y-AXIS USING CHIU’S EQUATION(2)
Method 1
Depth-averaged concentration (mean spatial concentration (35) ) on y-axis is defined by:
1 D C = C C(y)dy (7-1) y 0 ∫ D 0
121 Chiu’s (2) equations are used in Equation (7-1). The depth-weighted mean concentration is generally higher than the velocity-weighted concentrations for the rivers analized in this research.
Method 2
The velocity-weighted, mean sediment concentration at y-axis for h>0 (maximum velocity below water surface):
⎛ y ⎞ D ⎡ 1− ⎤ u ⎡ y ⎜ ⎟ ⎤ ' ⎢ max ln⎢1+ ()e M −1 e⎝ D−h ⎠ ⎥⎥C e −λ I ( y) dy ∫ M D − h 0 0 ⎣⎢ ⎣⎢ ⎦⎥⎦⎥ C y = (7-2) D ⎛ y ⎞ u ⎡ y ⎜1− ⎟ ⎤ max ln⎢1+ ()e M −1 e⎝ D−h ⎠ ⎥dy ∫ M D − h 0 ⎣⎢ ⎦⎥
Figure 7.36 shows calculated depth-averaged concentration is higher than the velocity-weighted concentration for Missouri River at Nebraska City and Sioux City. The same results were obtained for other river sections.
122 0.8
Nebraska City
ion 0.7 Line of perfect agreement at r Sioux City 0.6
) 0.5 /L g (
erage Concent 0.4 v axis
-
ed A 0.3 on y ght
0.2 -wei y t ci o
l 0.1 e V 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Depth-averaged Concentration on y-axis axis (g/L)
Figure 7-36 Comparison of Depth- and Velocity-weighted concentration, Missouri River
7.5 RELATION BETWEEN θ AND d 50
This research has improved the regression relationship between θ and d 50 by analyzing a large number of sediment data on various rivers during different flow events. To establish the relation, first λ' is estimated by applying Chiu’s sediment equations to multi-point sediment data on y- axis.
123 Then λ is calculated from the equation
1− e−M λ' = λ (6-15) Mφ
For each dataset, θ is calculated from
6(e M −1) θ = λ (6-17) φM (2 + e M )
As part of this research, different d50 on y-axis of various rivers were calculated from particle size distribution related to different flow events. Also, different θ were calculated using above equations and the following regression relationship was determined Fig 7.37.
1.26 θ = 22.54(d 50 + 0.023) (7-3)
124 20
10 7 5 3 2
1
θ 0.5 0.3 0.2
0.1
0.01 0.01 0.02 0.03 0.05 0.1 0.15 0.2 0.3
d50 (mm)
Missouri River at Omaha 10/6/1976 to 10/14/81, Q=31500 to 63300 cfs Missouri River at Ponca 4/28/1977 to 7/23/81, Q=29800 to 54000 cfs Missouri River Nebraska city 10/5/1976 to 10/9/79, Q=36800 to 69200 cfs Missouri River at Sioux City 10/7/1976 to 10/15/1981, Q=31000 to 39600 cfs Sacramento River, River Mile 35.64 3/19/1979, 2/29/1980; and RM 37.85, 1/19/79, Q=16800 to 76200 cfs Rio Grand Conveyance Channel Sections 2243+62, and 2249+93, 12/21/1965, Q=1860 cfs Mississippi River at Union Point 3/28/1995, Q=806,000 cfs, and 6/6/1995 Q=1,191,000 cfs Mississippi River at Tarbert 2/9/1995, Q=651,000 cfs and 12/30/96, Q=829,000 cfs Missouri River at Gayville 4/22/1980 to 7/21/1981, Q=28500 to 37600 cfs 2 1.26 r =0.8, θ=22.54(d50+0.023)
Figure 7-37 Relation between θ and d50 on y-axis
125
7.6 COMPUTATION STEPS FOR DETERMINING LOCATION OF MEAN
SEDIMENT CONCENTRATION ON Y-AXIS
For a channel section with known M and median sediment size (d50) on y-axis the location of
mean concentration on y-axis ( yc ) is computed as follows: h/D is a function of M and is calculated by using the following equations (Chiu and Tung) (7):
h j(M ) = −0.2ln (4.31) D 58.3
eM −1 j(M ) = (4.32) Mφ
e M 1 φ = − (4-24) e M −1 M
φM (2 + e M ) λ = G ' (M )θ , and G ' (M ) = (6-17) 6(e M −1)
Figure 7.37 yields
1.26 θ = 22.54(d 50 + 0.023) (7-3)
λ can be obtained from either Equations (4-24), (6-17) and (7-3) or Figure 7.38.
Figure 7.38 gives the relation between λ and d50 for different M values. It can be used to obtain
λ in a channel section with known values of M and d50.
126 8
7 M=4
6
3 5
λ 4 2 1 3
2
1
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 d (mm) 50
Figure 7-38 Relation between λ and d50
For M=1.0 -5.6 and h/D>0, maximum velocity (28) is below water surface, Chiu’s sediment transport equation is:
C ⎡ ' y h ⎤ = exp⎢− λ I( , , M )⎥ (6-21) C0 ⎣ D D ⎦ where λ’ = Chiu’s sediment distribution parameter and is a function of M and λ
1− e −M λ' = λ (6-15) Mφ
Equations (4-11), (6-18), (6-22) and (6-23) can be written for y/D
127 −1 y e M 1 ⎡ ξ ⎤ τ y I( ) = h −1 1+ (e M −1) ( ) −1 d( ) (7-4) ∫ ξ ⎢ ⎥ D ξ max 0 ⎣ ξ max ⎦ τ 0 D
2 ⎛ y ⎞ ⎛ y ⎞ ⎜ ⎟ ⎜ ⎟ τ h ⎛ h ⎞ = ⎜1− D ⎟ + ⎜1− ⎟⎜1− D ⎟ (7-5) τ D ⎜ h ⎟ ⎝ D ⎠⎜ h ⎟ 0 ⎜ 1− ⎟ ⎜ 1− ⎟ ⎝ D ⎠ ⎝ D ⎠
y y (1− D ) h y 1− ξ( ) = D e D (7-6) D h 1− D
ξ max =1
−1 ⎡ y ⎛ y ⎞⎤ ξ ( ) y ⎢ ⎜ ⎟⎥ h ( ) = ⎢ D ⎜1− D ⎟⎥ (7-7) ξ D y ⎜ h ⎟ ⎢ ⎜ 1− ⎟⎥ ⎣⎢ D ⎝ D ⎠⎦⎥
yc = Location of the point on y-axis, where the concentration is equal to C y .
y y if = c is substituted in the concentration equation, C can be calculated from: D D y
⎡ ' yc ⎤ C y = C0 exp⎢− λ I( )⎥ (7-8) ⎣ D ⎦
On the other hand the depth-averaged concentration on y-axis
1 y y C = C C( )dy( ) (7-9) y 0 ∫ 0 D D
And the velocity-weighted average concentration
128 ⎛ y ⎞ ⎡ ⎡ ⎜ ⎟ ⎤⎤ y ⎜1− D ⎟ 1 ⎢ ⎢ ⎜ h ⎟ ⎥⎥ y 1− −λ'I ( ) ⎢u M ⎜ ⎟ ⎥ y max ln⎢1+ ()e −1 D e⎝ D ⎠ ⎥ C e D d( ) ∫ ⎢ M ⎢ h ⎥⎥ 0 D 0 ⎢ ⎢ 1− ⎥⎥ D ⎣⎢ ⎣⎢ ⎦⎥⎦⎥ C y = (7-10) ⎛ y ⎞ ⎡ ⎡ ⎜ ⎟ ⎤⎤ y ⎜1− D ⎟ 1 ⎢ ⎢ ⎜ h ⎟ ⎥⎥ 1− ⎢u M ⎜ ⎟ ⎥ y max ln⎢1+ ()e −1 D e⎝ D ⎠ ⎥ d( ) ∫ ⎢ M ⎢ h ⎥⎥ D 0 ⎢ ⎢ 1− ⎥⎥ D ⎣⎢ ⎣⎢ ⎦⎥⎦⎥
Therefore,
⎡ ' yc ⎤ 1 ⎢−λ I ( )⎥ y y C e ⎣ D ⎦ =C C( )d( ) (7-11) 0 0 ∫ 0 D D
yc Consequently, can be computed for each d50 D
Figure 7.39 is the plots of sediment concentration normalized by depth-averaged concentration against their relative depths. In this figure, λ was calculated from Equations (6-17) and (7-3) or
Figure 7.38; and h/D from Equation (4-31). It is evident that the location of mean concentration on y-axis is lower for larger sediment diameters.
A comparison of Figure 7.40 by Chiu (2) and the simulated plots in Figure 7.39 indicates that
y Figures 7.39 and 7.40 both give the same c for C=C . h/D is a function of M, therefore in D y
Figure 7.40, for M =3, the plots corresponding to h/D = 0.4 is to be used.
Figure 7.41 presents the relation between M and G’(M). It can be used to calculate λ.
In Figure 7.43, concentration is normalized by depth-averaged concentration. It suggests that when d50 is kept constant in different channel sections, M will affect slightly the distribution of sediment concentration and the location of sampling. On the other hand, when λ is kept constant,
129 yc as M increases, will increase, because, d50 will decrease, as it is evident from Equation (6- D
17). Both Figure 7.41 and Equation (6-17) show that G’(M) will increase as M increases. Figure
7.42 indicates when λ is kept constant, yc is located closer to the channel bed for lower M values.
1.0 M=3, h/D=0.38 0.9
0.8 d50=0.06 mm λ=0.41 0.7
0.6 d50=0.17 λ=1.2 0.5 y/D
0.4 d50=0.3 0.3 λ=2.26
0.2
d50=0.5 0.1 λ=4.14
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 _ C/C . y
Figure 7-39 Effect of Sediment Size on Sediment Concentration
130 1.0 M=3 h/D=0 0.9 h/D=0.4 λ=0.5 h/D=-1 0.8 λ=1 h/D=- 8 0.7 λ=2
0.6 λ=4
0.5 y/D 0.4
0.3
0.2
0.1
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 _ C/C y
Figure 7-40 Effect of λ and h/D on Sediment Distribution, Chiu (2)
131 1.6
1.4
1.2
1.0 )
M 0.8 G'(
0.6
0.4
0.2
0.0 012345678910
M Figure 7-41 Relation between G’(M) and M
132 1.0 λ=2 0.9 _ M=1, d50=0.4 mm, θ=7.6, yc/D=0.14 0.8 _
M=2, d50=0.34 mm, θ= 6.3, yc/D=0.18 0.7 _
M=3, d50=0.27 mm, θ=4.8, yc/D=0.23 0.6 1.26 θ=22.54(d50+0.023) 0.5 M=3 y/D
0.4
0.3 2
0.2 1 0.1
0.0 0.00.51.01.52.02.53.03.54.0 _
C/Cy
Figure 7-42 Effect of M on Sediment Distribution for λ = 2
133 1.0 M=1, h/D=0.56, λ=0.66, d =0.15 mm 0.9 50 M=2, h/D=0.45, λ=0.80, d50=0.15 mm
0.8 M=3, h/D=0.38, λ=1.03, d50=0.15 mm M=4, h/D=0.24, λ=1.34, d =0.15 mm 0.7 50 M=6, h/D= -0.06, λ=2.08, d50=0.15 mm 0.6 Location of Average Vertical Concentration 0.5 _ y/D ( yc/D ) 0.4
0.3
0.2
0.1
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 _ C/C y
Figure 7-43 Effect of M on Location of Mean Concentration (on y-axis) for Constant d50
Figure 7.44 compares observed data with the simulated sediment distributions when M and d50 are used as input. Both distributions fit the data. The difference between the two distributions is due to the methods of calculations of θ and h/D for each point. Figures 7.45 to 7.49 are also
Chiu’s sediment concentration distributions. The location of mean concentration on y-axis is at a depth where concentration is equal to the average vertical concentration.
Figure 7.45 is the sediment distribution applied to observed data of Omaha Station. This figure was plotted based on the parameters obtained from applying Chiu’s equations to the observed
134 data on y-axis. Figure 7.46 is the simulation using the input of M and d50 related to Figure 7.45.
The simulation uses the regression relation Equation (7-3) for θ and Equation (4-31) for calculating h/D. Figure 7.48 shows Chiu’s Equation (6-20) applied to the sediment data on
Missouri River at Nebraska City (4/25/1978). The location of mean vertical concentration is also shown.
Figure 7.49 shows Chiu’s sediment concentration distribution applied to the observed data at
Line 13, Mississippi River (d50 = 0.02 mm). The mean concentration on y-axis is located at y=0.48D.
135
1.0 Chiu's Equations Applied to Data 0.9 M=3.22, h/D=0.55, λ=0.72 Observed data,Omaha Sta. 4/27/77, D=13.2 ft 0.8 Simulation, M=3.22, h/D=0.35,
d50=0.15 mm, λ=1.09 0.7 1.26 θ=22.54(d50+0.023) 0.6
0.5 y/D _
0.4 yc/D
0.3
0.2
0.1
0.0 0.00.51.01.52.02.53.03.54.0 _ C/C y
Figure 7-44 Location of Mean Concentration on y-axis by Simulation and Observed Data, Missouri River at Omaha, D=16.6 ft, 9/14/1977
136 1.0 Observed Data, y-axis d =0.145 mm 0.9 50 Chiu's Equation applied on sediment data
0.8 M=3.22, h/D=0.48, λ=1.07, C0=1.88 Mean Concentration by Depth-averaged 0.7 _ Cy=0.32 g/L 0.6
0.5 y/D
0.4 _
0.3 yc/D
0.2
0.1
0.0 0.0 0.5 1.0 1.5 2.0
C (g/L)
Figure 7-45 Location of Mean Concentration on y-axis, Missouri River at Omaha, 4/26/1978
137
1.0
0.9 Simulation, Chiu's Eqution M=3.22, h/D=0.48, λ=1.07, d =0.15 mm 0.8 50 Location of Average y-axis Concentration 0.7 _ _ yc/D=0.36 at C/Cy=1 0.6
0.5 y/D
0.4
0.3
0.2
0.1
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 _ C/C y
Figure 7-46 Location of Mean Concentration on y-axis (Simulation with M and d50 as input), Omaha Station 4/26/1978
138 1.0
Observed Data on y-axis, d50=0.095 mm 0.9 Chiu's Eq. applied to observed data M=3.8, h/D=0.26, λ=1.28 0.8 _ _
yc/D at Cy=5.09 g/L 0.7 _
Cy =Mean Vertical Concentration 0.6 by Depth-Averaged
0.5 y/D
0.4
0.3
0.2
0.1
0.0 0 5 10 15 20 25 30 35
C (g/L)
Figure 7-47 Location of Mean Concentration on y-axis, Rio Grande Channel, Section 2249, 12/21/1965
139 1.0
0.9 Observed Data, Nebraska City, 4/25/78
D=15.8 ft, d50=0.167 mm 0.8 Chiu's Equations
M=2.56, h/D=0.42 , C0=3.99, λ=1.25 0.7 Location of Average Vertical Concentration 0.6
0.5 y/D
0.4
0.3
0.2
0.1
0.0 012345678910111213 _ C/ C y
Figure 7-48 Location of Mean Concentration on y-axis, Missouri River at Nebraska City
140 105
90
75
60 y (ft) 45 Observed Data, d50=0.02 mm Chiu's Equations, 30 M=1.23, h=37.8 ft, λ=0.17 Estimated from Data Depth-averaged Concentration 15 _
yc/D=0.48
0 0 50 100 150 200 250 300 350 400 450 500 550 600 650
C (ppm)
Figure 7-49 Location of Mean Concentration on y-axis, Mississippi River at Line 13, 4/17/1998
y y y 7.7 RELATION OF M TO c , v AND D D D
For a channel section with known M and d50, the location of a single sampling point on y-axis
y y ( c ) can be determined from c - M relations. The collected sample at this location represents D D
the average concentration on y-axis (Cy ). For comparison, the locations of average velocities on y-axis and cross-sectional average velocities have been plotted based on their relations with M.
141 Figure 7.50 shows the relation of M to locations of average concentration on y-axis for different
yc λ and d . The figure indicates that is a function of d50. 50 D
yc yc The plots of M against show that when M increases, for a given d50, tends to be D D
yc insensitive to variation of M for fine sediment (d50 ≤ 0.06 mm). varies slightly (5%) with M D
yc for coarse sediment (d50 =0.3 mm). Therefore, it can be said that is not a function of M, but D
depends on d 50 . If d50 (y-axis) of a river is known, location of the single-point sampling can be obtained by referring to related plots presented here (Figure 7.50).
y On the other hand, location of average velocity on y-axis ( v ) and location of cross-sectional D
y average velocity ( ) are functions of M and do not change in a channel section (Figures 7.51 D and 7.52).
y Figures 7.51 and 7.52 give the location of mean concentration on y-axis ( c ) in a channel D
y section with known M. Figure 7.53 compares c obtained from depth-and velocity-weighted D averaging. This figure demonstrates that as d50 reduces, the distributions of sediment concentration determined from depth- and velocity-weighted methods become closer.
Figures 7.54 and 7.55 suggest that data support the results obtained from Figure 7.50.
Ingram (15, 18) studied the sampling depths of several rivers and found that the sampling point in the Middle Loup River and Rio Grande Conveyance Channel were located at between y = 0.19D and to 0.33D.
142
0.60
0.55
0.50 d50=0.03 mm _ y /D 0.45 c 0.06 mm 0.40 _ _ 0.15 mm 0.35 y /D yc/D c 0.17 mm _ 0.30 _ λ= 1 y /D yv/D 0.25 v _ 0.3 mm _ y/D 0.20 yc/D λ= 2 0.15 λ= 3 0.10 _ y/D 0.05
0.00 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
M
Figure 7-50 Relations among Locations of Mean Concentration on y-axis, Cross-sectional Mean Velocity and Mean Velocity on y-axis
143
1.0 M=1, h/D=0.56
0.9 d50=0.06 mm λ=0.26 0.8 _ _ u/u 0.7 u/ uv _ d50=0.15 0.6 λ=0.66 yc/D _ 0.5 _ yv/D yc/D _ 0.4 y/D
0.3 d50=0.3 λ=1.45 d50=0.5 0.2 λ=2.65 _
0.1 yv/D
0.0 0.00.51.01.52.02.53.03.54.0 _ _ _
C/Cy u/uv u/u
Figure 7-51 Locations of Mean Concentration and Velocity on y-axis and Cross-sectional Mean Velocity for M=1
144 1.0
d50=0.06 mm 0.9 λ=0.41
0.8 d50=0.15 mm λ=1.03 0.7
d50=0.3 mm 0.6 λ=2.26
0.5 y/D _ u/u _ 0.4 Depth-averaged Concentration Cy1 _ 0.3 Velocity-weighted Concentration Cy2
0.2 _ 0.1 u/uv
0.0 0.00.51.01.52.02.53.03.54.0 _ _ _ _ C/C C/C u/u u/u y1 y2 v
Figure 7-52 Locations of Mean Concentration on y- axis (by two methods), Mean Velocity on y- axis and Cross-sectional Mean Velocities for M=3
145
1.0
0.9
0.8 d50= 0.7 0.06 mm 0.15 mm 0.30 mm 0.6
0.5 y/D
0.4 _
Depth-averaged Concentration Cy1 0.3 _
Velocity-weighted Concentration Cy2 0.2
0.1
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 _ C/C y
Figure 7-53 Locating Mean Concentration on y-axis by Depth- and Velocity-weighted Methods for M=3
146 1.0 Observed Data, Nebraska City, 4/25/78
0.9 D=15.8 ft, d50=0.167 mm Chiu's Equations 0.8 M=2.56, h/D=0.42 , C0=3.99, λ=1.25 Location of Average Vertical Concentration 0.7 Simulation- Depth-averaged Concentration, M=2.56, h/D=0.43, λ=1.03 0.6 Simulation- Velocity-weighted Concentration 0.5 y/D
0.4 _ 0.3 yc/D
0.2
0.1
0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 _ C/ C y
Figure 7-54 A comparison of Normalized Chiu’s Sediment Distribution Applied to Data and two Simulations Normalized by Depth-and Velocity-weighted Averages, Missouri River
147 1.0 Observed Data, Missouri River at Omaha,
d50=0.15 mm, D=13.2 ft, 4/27/197 Chiu's Equations, M=3.22, C =1.156 g/l, λ=0.722 0.8 0 Simulation, M=3.22, λ=1.09 Depth-averaged Concentration Simulation, Velocity-weighted Concentration 0.6 Location of Depth-averaged Concentration y/D
0.4
0.2
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3_.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 C/C y
Figure 7-55 Sediment Concentration Distribution Normalized by Depth-and Velocity-weighted mean Concentration
7.8 RELATION BETWEEN M AND Ψ
Relations between Ψ and M for different river sections were determined by data analysis. Figure
7.56 shows relations between M and Ψ for river sections with fine size sediment d 50 <0.1 mm,
and coarse size sediment 0.19 ≥ d 50 ≥ 0.13 mm.
148 The scatter of the points indicates two separate relations due to difference in d50 of different river sections. For the sections with larger sediment size d50=0.13-0.19 mm, the values of Ψ were found to be between 0.50 and 0.75, while for the sections having finer sediments d50=0.007-
0.095 mm, the Ψ values were between 0.80 and 0.99 (Figure 7.56).
In order to use these relations, d 50 on y-axis in the channel section must be determined from sediment size distribution either from the long term historic data (for established stations) or by taking several samples on y-axis (for new stations).
Figure 7.57 shows the relations between M and Ψ for a range of particle diameters (0.06-0.36 mm). For the data analyzed in this research, the sediment diameters of 0.15 to 0.18 mm have the highest coefficient of determination r2 and fine sediment with diameter of 0.06 mm has the lowest r2.
Table 7.1 shows the computed values of M and Ψ related to a channel section. It indicates a trend that Ψ will increase with M, but does not vary with discharge.
149 1.0
0.9
0.8 r2=0.56 Ψ=0.86+0.05lnM d = 0.007 to 0.095 mm Ψ 50 0.7
0.6 r2=0.88 Ψ=0.4+0.3lnM d = 0.13 to 0.19 mm 0.5 50
0.4 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 M
Misssouri River at Nebraska City, 12 Sets (d50=0.17mm)
Niobrara River at Gaging Station 6 Sets (d50=0.16mm)
Missouri River at Ponca 14 Sets (d50=0.17mm)
Missouri River at Gayvill 8 Sets (d50=0.18mm)
Missouri River at Maskell 7 Sets (d50=0.15mm)
Missouri River at Sioux City 11 Sets (d50=0.182mm)
Middle Loup River at Dunning Section B 6 Sets (d50=0.16 mm)
Mississippi River at Line 6, RM 307 (d50 =0.02mm)
Mississippi River at Line 13, RM 313 (d50=0.02mm)
Mississippi River at Low Sill Channel (d50=0.02mm)
Mississipi River at ORC Outflow Channel (d50=0.02mm) Rio Grande Channel Sections 245, 255,
2249, 2243 and 1318 (d50=0.09mm) Sacramento River at River Miles
38.45, 37.85, 37.16, and 35.64 (d50 =0.01-0.06 mm) Mississippi River at Union Point RM=265, 32 Sets
and RM 323 (d50=0.02 mm)
Mississippi River at Range 362.2, 9 Sets (d50=0.007mm) Mississippi River at Tarbert RM306, 35 Sets
and RM 304 (d50=0.012mm) Atchafalaya River at Simmesport, 15 Sets (0.026mm)
Flume Data, Coleman 3 Runs (d50=0.05mm)
Missouri River at Omaha 16 Sets (d50=0.17mm)
Mississippi River at Auxiliary Channel (d50=0.02mm)
Mississippi River at Outflow Auxiliary Structure (d50=0.09mm)
Mississippi River at Hydroinflow Channel (d50=0.02 mm)
Middle Loup River at Dunning, Section C (d50=0.13 mm)
Middle Loup River at Dunning, Section D (d50=0.19 mm) Regression Lines
Figure 7-56 Relation between M and ψ
150 1.0
0.9 d=0.06 mm
0.8
0.7 0.09 mm
0.6 0.15 mm ψ 0.5 0.18 mm 0.4
0.3 0.36 mm 0.2
0.1
0.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 M
d=0.36 mm: ψ=0.23M-0.16, r2=0.52 0.18 mm ψ=0.2M+0.11, r2=0.79 0.15 mm ψ=0.18M+0.2, r2=0.67 0.09 mm ψ=0.14Μ+0.41, r2=0.58 2 0.06 mm ψ=0.02M+0.9, r =0.09
Figure 7-57 Relation between M and ψ for Different Sediment Diameters, Mississippi, Missouri, and Middle Loup Rivers, and Rio Grande Channel
151
Table 7-1 Relation between ψ and M
Range Discharge Range d50 ψ M
River Station cfs mm
Missouri River at
Omaha 31,500-63,300 0.149-0.19 0.75 3.22
Ponca 29,800-54,000 0.139-0.223 0.75 2.89
Nebraska City 36,800-69,200 0.115-0.237 0.68 2.56
Sioux City 31,000-39,600 0.162-0.246 0.78 4
Gayville 28,500-37,600 0.143-0.258 0.72 2.9
Niobrara River at
Gaging Station 298-405 0.13-0.24 0.51 1.73
Middle Loup River at
Sect.D 376-403 0.195-0.24 0.73 3.18
Sect.C 376-450 0.1-0.137 0.71 2.7
Sec.B 367-415 0.15-0.2 0.73 3
Mississippi River at
Union Point 51,000-127000 0.022 0.89 2.07
Range 326.2 24,000-105,000 0.007 0.93 3.18
Tarbert 38,000-102,000 0.012 0.87 2.65
Atchfalaya River 12,000-57,000 0.026 0.95 0.96
Coleman Flume
8/1/1990 0.27 0.03 0.95 4.65
8/2/1990 0.21 0.03 0.98 4.69
7/30/1990 0.54 0.03 1 4.89
152 Table 7-2 Location of Sampling Point on y-axis
y c River d50 D
(mm)
Missouri River 0.13-0.18 0.32D-0.37D
Niobrara River
Middle Loup River
Mississippi River at 0.022 0.48D
Union Point
Mississippi River at 0.012 0.48D
Tarbert
Sacramento River at 0.06 0.45D
Four Stations
Rio Grande Channel at 0.095 0.40D
Sections 245 and1318
153
8.0 SUMMARY AND CONCLUSIONS
• The proposed method is compatible with the efficient method of water discharge
measurement based on probability concept pioneered by Chiu. Chiu’s equations were
used to determine the location of single sampling point on y-axis. It was found that the
y sample taken at a relative depth of c could represent the mean concentration on y-axis, D
Cy . This sample concentration in turn can be converted into the cross-sectional mean
concentration (Cx ) using their relations determined in this research. The cross-sectional
mean concentration can be determined asCx =ψC y . The sediment discharge in turn can
be calculated as Qs = C xQ , where Q = Au = φAumax .
• The y-axis is a vertical on which the maximum velocity of the entire river section occurs.
It was determined from the velocity data and verified from the pattern of the isovels. The
location of y-axis is stable and does not change with time and discharge.
• The correlations between cross-sectional mean concentration and the mean concentration
on each vertical across the channel sections of different rivers were determined. It was
shown that the correlation is highest on y-axis and decreases toward the channel banks.
Therefore, it provides a basis for collecting the sediment sample on the y-axis.
154 • The ratio of cross-sectional mean concentration (Cx ) to the mean concentration on y-axis
C x (Cy ) is defined as ψ = . This coefficient was calculated for each grain diameter (d) C y
and median size (d50) on y-axis. The ψ value representative of each river section was
determined by regression.
2 • The coefficient of determination (r ) of Cx -C y relation is higher for fine sediment
particles (d50<0.062 mm). It means that in rivers, that transport predominantly fine
sediment, the sediment particles are uniformly distributed, and C y is a good predictor of
Cx . The distribution of sediment concentration in river sections, with coarse sediment
2 particles (d50>0.062 mm) is not uniform and r was found to have lower values.
• One needs to know d50 on the y-axis in order to choose an appropriate sampling depth.
d50 affects the location of sampling on y-axis, but M affects it slightly.
yc • Table 7.2 summarizes the computed values of and d50 of the suspended sediment for D
several rivers. For example, d50 on y-axis of Missouri, Niobrara and Middle Loup Rivers
at different discharges used in this research are in the range of 0.13-0.18 mm. According
to Figure 7.50 and Table 7.2 the sample point should be between y = 0.32D to 0.37D
(32% to 37% of the depth from the channel bed).
• In this research, only the field data related to suspended sediment of different rivers were
analyzed. Therefore, the estimated Cx determined from the relations of Ψ and M
represents the mean suspended sediment concentration in the overall cross section.
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