Field Studies of Mixing in Rivers

Transcription

1 Chapter 7 1/213 Field Studies of Mixing in Rivers Light Source Green LED Photodetector excited light 540 nm emitted light 580 nm Optical Filter wavelength Optical Fiber

2 Chapter 7 Field Studies of Mixing in Rivers 2/213 Contents 7.1 Introduction 7.2 Field Survey Methods 7.3 Data Analysis 7.4 Calculation of Dispersion Coefficients Objectives - Classify stages of pollutant mixing in rivers - Introduce methods for field survey and measurements for river mixing study - Introduce data analysis methods - Calculate dispersion coefficients from concentration data

3 7.1 Introduction 3/ Governing Equations of Pollutants in Rivers Mixing stages of pollutants in rivers: Stage I: Three-dimensional mixing (vertical + lateral + longitudinal mixing) Stage II: Two-dimensional mixing (lateral + longitudinal mixing) Stage III: One-dimensional mixing (longitudinal mixing) Stage I: Three-dimensional mixing Two types of contaminant source 1) Effluent discharge through outfall structure 2) Accidental spill of slug of contaminant

4 7.1 Introduction 4/213 c c c c c c c + u + u + u = ( ε ) + ( ε ) + ( ε ) x y z l t v t x y z x x y y z z (7.1) c t u, u, ε where = time-averaged concentration; = time; u = velocity l x y z components; = longitudinal turbulent mixing coefficient; = transverse turbulent mixing coefficient; ε v = vertical turbulent mixing coefficient ε t

5 7.1 Introduction 5/213 a) Continuous Source I.P C C C Near field Intermediate field Far field

6 7.1 Introduction 6/213 b) Instantaneous Source I.P C C C Near field Intermediate field Far field

7 7.1 Introduction 7/213 Stage II: Two-dimensional mixing (longitudinal + lateral mixing) ~ Intermediate field mixing ~ Contaminant is mixed across the channel primarily by turbulent dispersion and spread longitudinally in the receiving stream. apply 2D depth-averaged advection-dispersion equation for mixing in rivers c c c c c u v D + + = + D t x y x x y y L T (7.2)

8 7.1 Introduction 8/213

9 7.1 Introduction 9/213 c where = depth-averaged concentration; = depth-averaged longitudinal velocity; v = depth-averaged transverse velocity; D = 2D longitudinal mixing coefficient; D T = transverse mixing coefficient. u L Stage III: Longitudinal dispersion ~ Far field mixing ~ Process of longitudinal shear flow dispersion erases any longitudinal concentration variations. ~ Apply 1D longitudinal dispersion model proposed by Taylor (Fischer et al., 1979)

10 7.1 Introduction 10/213 C C C + U = K t x x x (7.3) where C = cross-sectional-averaged concentration; U = cross-sectionalaveraged longitudinal velocity; K = 1D longitudinal mixing coefficient.

11 7.1 Introduction 11/ Field Studies on River Mixing Field survey and measurements are needed to validate mixing predictions and to estimate the dispersion coefficients. Field data include - Bathymetric data: needed to make reliable mixing predictions especially when using numerical models - Hydraulic data: local velocity and depth, discharge - Concentration data: tracer concentration at sites below injection point

12 7.1 Introduction 12/213 Field studies should be designed and planned according to the mixing models that are investigated. Field study for1d model: bathymetric and hydraulic data at each section; cross-sectional averaged concentration data ( C( xt, )) Field study for 2D model: bathymetric and hydraulic data at each section; depth averaged concentration data Field study for 3D model: bathymetric and hydraulic data at each section; local concentration data ( cxyzt (,,, )) ( c( xyt,, ))

13 7.1 Introduction 13/213 Estimation of dispersion coefficients can be done by using - change of variance of the tracer distribution - routing method in which a concentration profile is routed from one site to the other site - graphical methods using concentration profile or peak concentration

14 7.2 Field Survey and Experiments 14/ Planning Field Work Good planning is essential to get the most out of a field experiment minimizing the number of field crew and logistics. Data needed to make reliable mixing predictions especially when using 2D numerical models are given below: 1) Topographic and bathymetric data 2) Hydraulic data 3) Concentration data

15 7.2 Field Survey and Experiments 15/213 Good radio communication between field parties is necessary so that sampling procedures can be changed during the course of an experiment. Careful book keeping is essential to label each gaging, echo sounding, dye sampling with site, date, time, location and direction of travel. It is necessary to carefully select sampling sites and to optimize sampling procedures.

16 7.2 Field Survey and Experiments 16/213 Field experimental procedure is given below: 1) Selecting sampling sites 2) Installing tag lines 3) Surveying channel bathymetry 4) Measuring velocity and discharge 5) Measuring dye concentration

17 7.2 Field Survey and Experiments 17/ Selection of Sampling Sites Sampling sections should be decided considering distances for vertical and transverse mixing. 1) 1D mixing test The first sampling section should be located below L t downstream of the injection point. L t L t 2 UW = 0.1, D T 2 UW 0.4, D T centerline injection = side injection (7.4) (7.5)

18 7.2 Field Survey and Experiments 18/213 2) 2D mixing test The first sampling section should be located below L v downstream of the injection point, and last sampling section located inside of L t downstream of the injection point. L v = UH 0.35 ε v 2 (7.6) [Ex] For slowly meandering streams, D t T ( 0.3 ~ 0.9) W U 50; 10 * H U L = ( 50 ~ 150) W HU *

19 7.2 Field Survey and Experiments 19/213 When ten or less sampling sections can be selected, the interval between sections can be 5~15 times of channel width. Tag lines should be installed for accurate position fixing, which are stretched across the channel, in small rivers. Tag lines are used to measure transverse distances, a tape for longitudinal distance, and a calibrated rod for vertical distance.

20 7.2 Field Survey and Experiments 20/ Surveying River Bathymetry Accurate estimates of river width, depth are needed to make reliable mixing calculations. Channel topography and bathymetry are needed for calculation of dispersion coefficients and application of the 2D numerical model.

21 7.2 Field Survey and Experiments 21/213 Position fixing in rivers can be done by given methods: i) Satellite position fixing system: use GPS; accuracy is only of the order of meters ii) Remote shore-based systems: use Lidar iii) Land-based surveying: gives excellent results but is very labor intensive

22 7.2 Field Survey and Experiments 22/213 Typical procedure of the land-based surveying can be as follow: 1) Install tag line at each sampling section - Install GHP using wooden or steel pile at both banks of the channel - Use GPS to measure position and datum of the GHP - Need benchmarks along the bank whose position (x, y) and height above datum (z) are known 2) At each cross-section, measure water level using echo sounder or ADCP across the channel by boat

23 7.2 Field Survey and Experiments 23/213 3) In case tag lines are not possible, automatic position fixing systems can be installed in the boat which follows a random path thereby require line of sight between the shore beacons and the boat 4) For surveying land in the floodplain area, use Real Time Kinematic-GPS (RTK-GPS) - Set up the base station around the surveying field - Yield absolute positions of 1~2 cm horizontal accuracy in real time with no post-processing - The receiver with unknown position is called Rover

24 7.2 Field Survey and Experiments 24/213

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26 7.2 Field Survey and Experiments 26/ Velocity Measurements Accurate measurements of river discharge and velocity are essential for reliable mixing and mass balance calculations. It is advisable during tracer tests to have one or more hydrological field parties undertake river gauging.

27 7.2 Field Survey and Experiments 27/ ADV (Acoustic Doppler Velocimetry) - Sends out a beam of acoustic waves at a fixed frequency from a transmitter probe. - Waves bounce off of moving particulate matter in the water and three receiving probes listen for the change in frequency of the returned waves. - Calculates the velocity of the water in the x, y, and z directions as following formulas.

28 7.2 Field Survey and Experiments 28/213 F d = V Fs C s (7.7) where F = change in received frequency (Doppler shift); F = frequency of d transmitted sound; V = velocity of source relative to receiver; C = speed of sound. s

29 7.2 Field Survey and Experiments 29/213

30 7.2 Field Survey and Experiments 30/213

31 7.2 Field Survey and Experiments 31/ ADCP (Acoustic Doppler Current Profiler) - Applied the Doppler principle by bouncing an ultrasonic sound pulse off small particles of sediment and other material. - Transmits a pulse into the water column and then listens for the return echo from the acoustic backscatters in the water column. - The sound is shifted one time (as perceived by the backscatter) and a second time (as perceived by the ADCP transducer). Because there are two Doppler shifts, the velocity of the water along the acoustic path can be computed from the following.

32 7.2 Field Survey and Experiments 32/213 V Fd = 2Fs C S (7.8) - Computation of the real time discharge with ADCP deployed on a moving boat Q= QLE + QTop + QMeasured + QBottom + QRE (7.9) where Q LE bank; Q = discharge estimated for the top unmeasured area; Q = Top = discharge estimated for the unmeasured area near the left discharge measured directly by the ADCP; for the bottom unmeasured area; unmeasured area near the right bank. Q RE Q Bottom Measureed = discharge estimated = discharge estimated for the

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39 7.2 Field Survey and Experiments 39/ Other Methods (1) Mechanical velocity meters - The principle of operation is based on the proportionality between the velocity of the water and the resulting angular velocity of the meter rotor. - Velocity is determined by placing meter in stream and counting number of revolutions in a measured amount of time.

40 7.2 Field Survey and Experiments 40/213

41 7.2 Field Survey and Experiments 41/213

42 7.2 Field Survey and Experiments 42/213 (2) Volumetric measurements - The most accurate way of measuring small discharges - Observing the time it takes to fill a container of known capacity

43 7.2 Field Survey and Experiments 43/213

44 7.2 Field Survey and Experiments 44/213 (3) Portable weir plates - Used when depths are too shallow and velocity too low for a conventional velocity meter - Discharge from 90 degree portable weir plates is computed using the formula. Q = 2.49H 2.48 (7.10) where H = height of water behind the notch.

45 7.2 Field Survey and Experiments 45/213

46 7.2 Field Survey and Experiments 46/213 (4) Float measurements - Distribute a number of floats uniformly over the stream width, noting the position of each with respect to the bank. - The velocity of the float is equal to the distance between the cross sections divided by the time of travel.

47 7.2 Field Survey and Experiments 47/213

48 7.2 Field Survey and Experiments 48/ Tracer Tests Tracers ~ common method of measuring velocities and dispersion rates ~ can use two kinds of tracers: i) Artificial tracers ii) Natural (environmental) tracers

49 7.2 Field Survey and Experiments 49/213 1) Artificial tracer Two types of artificial tracers ⅰ) Soluble tracer: ~ not affected by photochemical decay or adsorption ~ tracer input rate is controlled ⅱ) Floating tracer: ~ use surface floats or sub-surface drogues ~ used in the oceans to study advection and dispersion and in early laboratory dispersion studies

50 7.2 Field Survey and Experiments 50/213 ~ disadvantage of only sampling surface currents which tend to be higher than the depth-averaged current ~ affected by the wind especially in wide river channels ~ difficult to use in natural rivers because of snags ~ travel faster than the dye and give advance warning to the sampling teams Commonly used dyes (Smart & Laidlaw, 1977) i) Rhodamine WT ~ conservative ~ easy to handle and cheaper than Rhodamine B ~ readily measurable to very low levels

51 7.2 Field Survey and Experiments 51/213 ⅱ) Rhodamine B ~ absorbed onto sediments ⅲ) Salt ~ detection limit is high, thus large amounts are required ⅳ) Radioactive tracers: ~ require specialist injection and detection equipment ~ arouses public disquiet 2) Conservative natural tracer ~ present in a tributary (Stallard, 1987) ~ STP effluents sometimes contain a suitable tracer such as EC, temperature.

52 7.2 Field Survey and Experiments 52/213 Temperature ~ can be used to quantify mixing rates in rivers downstream from large cooling water discharges of power plants ~ non-conservative behavior in the far-field Sediment ~ especially used at junction of tributary ~ using the concentration of suspended sediment Conductivity ~ nearby inflow of effluents are needed ~ using the conductivity difference by effluent mixing

53 7.2 Field Survey and Experiments 53/ Tracer Injection ~ can use two types of tracer injection: i) Steady source ~ represent effluent discharge from STP or power plant ~ neglect longitudinal dispersion ~ do not give estimates of velocity Use special apparatus for steady introduction of the tracer - Marriott vessel - Floating siphon - Peristaltic pump

54 7.2 Field Survey and Experiments 54/213

55 7.2 Field Survey and Experiments 55/213 ii) Slug input ~ represent instantaneous injection of accidental spills ~ use simple apparatus for point or line sources Instruments ~ can use two types of instruments for measurements of Rhodamine concentration: 1) Spectro-fluorometer ~ expensive, complex, delicate

56 7.2 Field Survey and Experiments 56/213 2) Filter fluorometer ~ moderately expensive, simple to use, robust in the field sensitivity comparable - Turner Aminco Bowman fluoro / colorimeter - YSI

57 7.2 Field Survey and Experiments 57/213 ~ for natural tracer ⅰ) Electrical conductivity meter ~ simple to use, commonly used for conductivity measurement ~ need temperature compensated instrument ⅱ) Thermistor ~ simple to use, commonly used for temperature measurement ~ high precision, typical range -90 to 130 C

58 7.2 Field Survey and Experiments 58/213 Light Source Green LED Photodetector excited light 540 nm emitted light 580 nm Optical Filter wavelength Optical Fiber

59 7.2 Field Survey and Experiments 59/213 Light source low-pressure mercury lamp 1 st filter excited light 555 nm wavelength absorbed by dye sample emitted light 580 nm Sample compartment 2 nd filter Reader amount of light vs. reference light Photomultiplier

60 7.2 Field Survey and Experiments 60/ Sampling of the tracer ~ can use two types of tracer sampling: i) Sampling bottle ~ take many samples with time using sampling bottle ~ can use tubing and pump attached to the tag line in small-to-medium streams or take samples on the boat in large rivers ~ analyze concentration of each bottle at the base camp or in the laboratory

61 7.2 Field Survey and Experiments 61/213 ii) Flow-through type sensor ~ measure tracer concentration using flow-through type sensors in sites ~ Sensors can be fixed to the tag line in small streams ~ in large rivers, sensors are mounted in the boat along with position fixing system, GPS, and dragged in the river

62 7.2 Field Survey and Experiments 62/213 (1) Unsteady source ~ ensure that the maximum tracer concentration is well above the detection limit ~ there is a large enough change in concentration between sites to calculate the velocity and the dispersion coefficient ~ need to conduct a number of identical field tests and average the results to measure an ensemble average ~ when studying longitudinal dispersion, it is necessary to sample for long enough to measure the entire concentration versus time profile at each site and check for tracer loss ~ can be very time consuming because there are often long tails associated with tracer becoming trapped in dead zones

63 7.2 Field Survey and Experiments 63/213 (2) Steady source ~ is possible to collect replicate samples over a reasonably long time period and calculate time averaged concentration ~ when studying transverse dispersion, it is desirable to traverse the plume several times at the same transect and average the results Check for fluorometer drift ~ a number of discrete water samples should be collected from time to time, analyzed later in the laboratory to check the fluorometer drift

64 7.3 Data Analysis 64/ Topographic Data Channel 1) Thalweg line ~ a line that connects the lowest points within a river channel ~ a line of fastest flow along a river s course 2) Sinuosity, S n ~ Sinuosity of river is its tendency to move back and forth across its floodplain, in an S-shaped pattern, over time ~ S n = thalweg line / straight line = actual path length / shortest path length

65 7.3 Data Analysis 65/213 ~ S n > 1.5 : meandering channel ~ S n < 1.05 : straight channel 3) Riffle-pool sequence ~ a stream's hydrological flow structure alternates from areas of relatively shallow to deeper water 4) Hydraulic radius, R ~ the ratio of the water area to its wetted perimeter - R = A / P

66 7.3 Data Analysis 66/213 1) Manning s roughness coefficient, n ~ approaches in the proper determination of n - to understand the factors that affect the value of n and thus to acquire a basic knowledge of the problem and narrow the wide range of guesswork - to consult a table of typical n values for channels of various types - to examine and became acquainted with the appearance of some typical channels whose roughness coefficients are known - to determine the value of n by an analytical procedure based on the data of either velocity distribution in the channel cross section and on the data of either velocity or roughness measurement

67 7.3 Data Analysis 67/213

68 7.3 Data Analysis 68/ Floodplain 1) Merge of Topographic Information for Channel and Floodplain ~ Merge the two data based on a unified reference point to the same coordinate system ~ Adopted Geodetic Reference System 1980 and calculated by converting UTM latitude and longitude coordinates

69 7.3 Data Analysis 69/213 Gangjeong-Goryong RTK-GPS : GRX1(GPS+GLONASS Receiver and Antena, Digital UHF Modem, Blue Tooth, GCS/GPRS Module and Integrated Receiver with Removable Battery 72 Channels for GPS + GLONASS + SBAS Dalseong Total Station RTK-GPS

70 7.3 Data Analysis 70/213 + = Floodplain (RTK-GPS) Channel (Single-beam) Merged

71 7.3 Data Analysis 71/213

72 7.3 Data Analysis 72/ Hydraulic Data Velocity Data 1) Velocity profile ~ Velocity profile is measured by wading with ADV or boatmounted ADCP ~ Lateral distribution of velocity is obtained ~ Spatial averaged velocity and discharge at each section can be calculated from velocity profiles

73 7.3 Data Analysis 73/213

74 7.3 Data Analysis 74/213 2) Reach-averaged velocity 1 Velocity measurements at a number of cross sections ~ large number of gauging is needed ~ more accurate reach-averaged velocity would be obtained according to the number of cross sections

75 7.3 Data Analysis 75/213

76 7.3 Data Analysis 76/213 2 Tracer method ~ cost effective if tracer is made with an instantaneous injection v x = x t x t (7.11) v t x 1 2 where = reach - average velocity; and = times of passage if the centroid if tracer profile past sites 1 and 2 t t 1 0 = 0 (, ) τc x τ dτ 1 (, τ) C x 1 dτ (7.12) where Cxτ (, ) = cross-sectional averaged concentration at Site 1 1

77 7.3 Data Analysis 77/213 In the mid-field, 1 W C( x, t) = h( x, y) c( x, y, t) dy 0 A (7.13) where c = local depth-averaged concentration; h = local depth; A = crosssectional area. The observed tracer versus time profile is often uneven because of experimental errors and turbulent eddies in the plume so that some smoothing may be required.

78 7.3 Data Analysis 78/ River depth 1) Use of wading rod - Depth is shallow enough or measuring from a low footbridge - Wading rods have a small foot on the bottom to allow the rod to be placed firmly on the streambed. 2) Use of sonic sounder - The sonic sounder has been used primarily for measuring depth when making a moving boat measurement and in ADCP discharge measurements. - For moving boat measurements, the sonic sounder records a continuous trace of the streambed on a digital or analog chart. Ex) Depth measurement using sonic sounder in the Mihocheon

79 7.3 Data Analysis 79/213 N W E S Sec. 3 Sec. 2 Sec kmkilometers

80 7.3 Data Analysis 80/213

81 7.3 Data Analysis 81/ Discharge 1) Velocity-Area method - The most practical method of measuring the discharge of a stream. - The total discharge is the summation of the products of the partial areas of the stream cross section and their respective average velocities. Q n = av i i= 1 i (7.14) where a is the cross section area, is the corresponding mean velocity i vi

82 7.3 Data Analysis 82/213 2) Mid-section method - Once the velocity, depth, and distance of the cross section have been determined, the mid-section method can be used for determining the discharge. Q D + D V + V Wi n 1 i i+ 1 i i+ 1 = i= (7.15)

83 7.3 Data Analysis 83/213

84 7.3 Data Analysis 84/213 Ex) Velocity measurements with ADCP and cumulative discharge in the Mihocheon

85 7.3 Data Analysis 85/213 N W E S Sec. 3 Sec. 2 Sec kmkilometers

86 7.3 Data Analysis 86/213

87 7.3 Data Analysis 87/ Concentration Data C-t profile 1) Central tendency ~ Mean of concentration ~ Median of concentration 2) Dispersion tendency ~ Range of concentration and time ~ Variance of concentration, σ 2

88 7.3 Data Analysis 88/213 3) Symmetry ~ Skewness of concentration 4) Peakedness ~ peak of concentration, C p ~ time of concentration peak, t p

89 7.3 Data Analysis 89/213 a) Sec Conc. (ppb) ,100 1,400 1,700 2,000 2,300 2,

90 7.3 Data Analysis 90/213 b) Sec Conc. (ppb) ,100 1,400 1,700 2,000 2,300 2,600

91 7.3 Data Analysis 91/213 c) Sec Conc. (ppb) ,100 1,400 1,700 2,000 2,300 2,600

92 7.3 Data Analysis 92/ Tracer Mass Flow It is advisable to check for tracer loss due to either photochemical decay, adsorption or poor sampling. Equations for estimating dispersion coefficients require knowledge of the tracer mass inflow rate. In the mid-and far-field the mass flux is W mx ( ) = hxyuxycxydy (, ) (, ) (, ) 0 W mx ( ) hxyuxycxytdtdy (, ) (, ) (,, ) = 0 0 for continuous source for instantaneous source (7.16a) (7.16b)

93 7.3 Data Analysis 93/213 [Re] Mass flux = mass / time = conc vol / time = C Q = C V A Concentration field 1) Compare the behavior of the tracer cloud ~ plot the contours of the tracer concentration at certain time intervals ~ analysis the spatial distribution of tracer

94 7.3 Data Analysis 94/213 y (m) 299, , , , ,800 (a) 21 min. 130, , , , , , ,400 x (m) Conc. (ppb)

95 7.3 Data Analysis 95/213 y (m) 299, , , , ,800 (b) 26 min. 130, , , , , , ,400 x (m)

96 7.3 Data Analysis 96/213 y (m) 299, , , , ,800 (c) 36 min. 130, , , , , , , x (m)

97 7.3 Data Analysis 97/ C-y profile - At steady state, C-y profile is used to analyze transverse mixing. - As cloud moving further downstream, the variance of transverse concentration distribution gets larger.

98 7.3 Data Analysis 98/213 EC(µS/cm) Sec.1 Sec.2 Sec.3 Sec y/w

99 7.3 Data Analysis 99/ Uncertainty in Field Measurements The issue of accuracy in data sampling and measurements is very important for the validity of the model output Methods of Measurements The physical property of a compound or an element can be applied as the basis for an analytical measurement.

100 7.3 Data Analysis 100/213 i) Definite methods These methods are based on the laws that govern the physical and chemical parameters. ~ include volumetric analysis, gravimetry and potentiometry measurements, activation analysis, isotope dilution mass spectrometry, and voltammetry and polarography

101 7.3 Data Analysis 101/213 ⅱ) Relative methods These methods consist of comparing the sample with a set of calibration samples of known content. The sample value is determined by interpolation of the measured quantity with respect to the response curves of standard samples. ⅲ) Comparative methods ~ use a detection system sensitive to the content of the molecules or elements to be determined ~ X-ray fluorescence spectrometry

102 7.3 Data Analysis 102/213 ⅳ) On-line monitoring methods The on-line monitoring consists of the sampling system and sensors. The collected sample from a direct intake of surface water is transferred to the measuring systems. The sensors: ~ is able to respond rapidly and continuously using the active microzone where biological and chemical reactions take place ~ optical sensors / biosensors / chemical sensors are connected to the micro-zone ~ need frequent calibration, growth of biofilm

103 7.3 Data Analysis 103/213 Traceability: ISOC (1994) defined traceability as the ability to verify the history, location, or application of an item by means of recorded identification. The traceability of measurement depends on proper functioning of measuring equipment, which can be assured by calibration, using reference standards. The measurement accuracy also depends on the type of sample and the analytical procedure.

104 7.3 Data Analysis 104/ Uncertainty (1) Uncertainty of Sampling A surface water sample belongs to a population, a statistical term that includes all the possible available measurements of a variable. The sampling process has to collect data as accurate as possible. In order to be accurate, the data must be both unbiased and precise. Random sampling eliminates the sampling bias.

105 7.3 Data Analysis 105/213 New bias can be generated due to errors of the analytical measurement. In case that a sample is representative of the population, this sample may be used for extracting general conclusions concerning the entire population. Statistical inference Factors responsible of uncertainty during sampling analysis includes i) inappropriate methodology of measurement ii) improper calibration of the instrument iii) scarce representativity of the collected sample

106 7.3 Data Analysis 106/213 (2) Uncertainty of Measurement Uncertainty of measurement is a non-negative parameter characterizing the dispersion of the values that could be attributed to a variable, based on the used information. i) Definitional uncertainty ~ a component of measurement uncertainty which result from a finite amount of detail in the definition of the variable ii) Standard uncertainty ~ a result of a measurement is expressed as a standard deviation

107 7.3 Data Analysis 107/213 [Cf] Measurement error ~ the difference between the expected and determined values Uncertainty ~ a range into which the expected value may fall within a certain probability [Cf] Bias of measurement ~ gives an estimate of how for the result is from the true value Precision ~ gives information on the dispersion of the results

108 7.3 Data Analysis 108/213 Estimation of total uncertainty total = ( sampling + measure + population ) u u u u where u sampling samples u population u measure = is determined by repeated collection of at least seven identical = is determined by repeated analyses of homogeneous samples = is related to the sample population

109 7.3 Data Analysis 109/213

110 7.4 Calculation of Dispersion Coefficients 110/213 Observation: calculating the observed values from field concentration data Prediction: estimating the dispersion coefficient using theoretical or empirical equations Longitudinal Dispersion Coefficient of 1D ADE Approximate Model for 1D Advection-Dispersion Approximate model for 1D advection-dispersion in the far-field + U = K t x x 2 C C C 2 (7.17)

111 7.4 Calculation of Dispersion Coefficients 111/213 ~ neglects the advective zone ~ assume that the mean velocity and longitudinal dispersion coefficient are constant Taylor solution of Eq. (7.17) for a slug injection is M ( x Ut) Cxt (, ) = exp A 4π Kt 4Kt 2 (7.18) Concentration versus distance profile at t n ~ symmetrical about the peak concentration

112 7.4 Calculation of Dispersion Coefficients 112/213 C max M M = = A 4π Kt x n A 4π K U max (7.19) x max = Ut n (7.20) x ( t ) = x = Ut n max n (7.21) Concentration versus time profile at x n ~ not symmetrical The peak concentration occurs at t 2 2 K x n max 4 2 K = + U U U (7.22)

113 7.4 Calculation of Dispersion Coefficients 113/213 The centroid of the concentration-time curve is t xn K = + 2 U U 2 (7.23) ( x Ut) 2 M C( xt, ) = exp A 4π Kt 4Kt (7.24) 2 2 ( 5, 1, 2 /, 20 / ) M = kg A = m U = m s K = m s

114 7.4 Calculation of Dispersion Coefficients 114/213

115 7.4 Calculation of Dispersion Coefficients 115/213 C b) C - t profile x = x 1 (100 m) x = x 2 (200 m) x x = x 3 (400 m) t max3 = 196 sec t max2 = 96 sec t max1 = 46 sec t

116 7.4 Calculation of Dispersion Coefficients 116/213 The mean travel time is t t = max x n U t < t < t t (7.25) (7.26) The Frozen Cloud Approximation In many applications, advection dominates dispersion x K >> U U 2 xu >> K (7.27) (7.28)

117 7.4 Calculation of Dispersion Coefficients 117/213 Then, the maximum tracer concentration occurs at t max t x n U (7.29) The frozen cloud approximation exploits Eq. (7.29) to estimate C-x curves at a fixed time from C-t curves at a fixed location and vice versa. Frozen cloud concept assumes that no longitudinal dispersion occurs during the time taken for the tracer to pass a sample site. This concept is not strictly valid in most natural channels because although advection is more rapid than dispersion, the latter does have a discernible effect on tracer concentrations.

118 7.4 Calculation of Dispersion Coefficients 118/213 Under the frozen cloud assumption x = Ut 1 1 [ ] Cx (, t) = Cx+ Ut ( t), t (7.30) (7.31) C( x, t ) = C x, t x 1 U x (7.32) Eq. (7.31) is used to estimate C-t data from C-x data, and Eq. (7.32) to go the other way. Note that inversion of the axes in Fig concentration measured at negatives values of x x 1 of. t t 1 (i.e. in the leading edge) have positive values

119 7.4 Calculation of Dispersion Coefficients 119/213 [Re] In Eq. (14), t = t + ( t t ) 1 1 (A) Multiplying Eq. (A) by U results in x = Ut = Ut + U ( t t ) = x+ Ut ( t)

120 7.4 Calculation of Dispersion Coefficients 120/ Change of Moment Method For C-x profiles, the change of moment method is written as K K = = dσ dt σ 2 x ( t ) σ ( t ) 2 2 x 2 x 1 t t 2 1 (7.33) (7.34) where 2 σ x = longitudinal variance It is rare for experiments to yield concentration versus distance profiles. It is more common to obtain the concentration data as a function of time at a fixed sampling site downstream.

121 7.4 Calculation of Dispersion Coefficients 121/213 For C-t profiles, Fischer (1966) revised Eq. (7.34) as K = 1 U σ ( x ) σ ( x ) 2 t 2 t 1 t t 2 1 (7.35) where 2 σ ( x ) t i = ( t t ) C( x,) t dt i C( x,) t dt i i (7.36) t i = tc( x,) t dt i C( x,) t dt i (7.37)

122 7.4 Calculation of Dispersion Coefficients 122/213 Practical difficulties - Tracer profiles exhibit long tails and sampling must continue for a long time after the bulk if the tracer has passed. - Small measurement errors in the low concentrations of the leading and trailing edge of the tracer profile are inevitable and these greatly influence the variance and hence the dispersion coefficient. - Thus, it is difficult to compute a meaningful value of variance when concentration distributions are skewed because of long tails on the observed distributions.

123 7.4 Calculation of Dispersion Coefficients 123/ Routing Method (1) Routing a spatial concentration profile Use analytical solution, Eq. (7.18) at a specified time superposition principle t = t 1 and apply i) Divide the channel into a number of short segments each x long and centered on x i

124 7.4 Calculation of Dispersion Coefficients 124/213 ii) Then, the mass of tracer in segment i is behave like an instantaneous source at Cxt (, ) x i 1 and is assumed to x= x i released t = t1 iii) This source results in concentrations at a later time which are predicted using Eq. (7.18), and the separate concentration profiles are summed. t = t 2 C( ξ, t ) = 4 π Kt ( t) 1 (, ) exp 2 Cxt 2 1 [ x ξ Ut ( t) ] Kt ( t) dξ (7.38)

125 7.4 Calculation of Dispersion Coefficients 125/213 where t ; 1 2 Cxt (, 1) = observed concentration as a function of distance at time Cxt (, ) = predicted concentration at time t ; ξ 2 = dummy distance variable of integration. Restrictions: Since the analytical solution, Eq. (7.18), is an asymptotic solution which is accurate only a long distance below an instantaneous point source, t1 t2 t1 should be sufficiently large after the injection, and ( ) should also be large.

126 7.4 Calculation of Dispersion Coefficients 126/213 (2) Routing a temporal concentration profile 1) Frozen cloud method i) Use the frozen cloud assumption to transform the C t profile at Site 1 (Profile 1) into a C x profile (Profile 1A) at the mean time of passage using Eq. (7.32). ii) Profile 1A is then routed downstream using Eq. (7.38) to give a C x profile (Profile 2A) at the mean time of passage past Site 2.

127 7.4 Calculation of Dispersion Coefficients 127/213 iii) Profile 2A is then transformed into a C t profile (Profile 2) at Site 2 using Eq. (7.31). iv) K can be estimated by comparing the Cx ( 2,) t with the observed C t profile at Site 2. The best fit value is regarded as the observed dispersion coefficient C( x, τ) U U ( t t t + τ) C( x, t) = exp 2 d 4 π K( t t) 4 K( t t) τ (7.39)

128 7.4 Calculation of Dispersion Coefficients 128/213 where Cx (,) t = observed concentration as a function of time at Site 1; 1 Cx ( 2,) t t 1 t 2 = predicted concentration as a function of time at Site 2; and = mean time of passages at Sites 1 and 2, respectively. t = t x x 2 n U (7.40) Restrictions: 1 Site 1 must be outside the advective zone ( x' > 0.4). 2 The entire C t profile must be measured at Site 1. 3 Tracer loss must be negligible so it is necessary to correct for tracer losses caused by adsorption, decay or dilution by tributaries.

129 7.4 Calculation of Dispersion Coefficients 129/213 Calculate the mass flux If necessary scale the downstream concentrations so that the mass fluxes are identical. 2) Hayami solution Barnett (1983) suggested an alternative method for routing a C t profile using Hayami solution to Eq. (7.17). Mx ( x Ut) Cxt (, ) = exp AUt 4π Kt 4Kt 2 (7.41)

130 7.4 Calculation of Dispersion Coefficients 130/213 This solution is simply Eq. (7.18) multiplied by x Ut. It satisfies the boundary conditions that at x = 0, C = 0 except t = 0 when a tracer slug of mass M is introduced over an infinitesimally small time interval. for all values of t It can be used to route downstream an observed needing to invoke the frozen cloud approximation. C t profile without

131 7.4 Calculation of Dispersion Coefficients 131/213 Comparison between Taylor solution and Hayami solution x t t For large and, two solutions give similar results. For small, the Taylor solution predicts positive concentrations upstream from the source, where the Hayami solution predicts small negative concentrations upstream from the source. Apply the superposition principle to Eq. (7.41) C( x, τ)( x x ) [ x x U( t τ)] C( x, t) = exp d 2 ( t τ) 4 πk(( t τ)) 4 Kt ( τ ) τ (7.42)

132 7.4 Calculation of Dispersion Coefficients 132/213

133 7.4 Calculation of Dispersion Coefficients 133/213

134 7.4 Calculation of Dispersion Coefficients 134/ Graphical Methods (1) Chatwin s method Rearrange Eq. (7.18) based on Chatwin s transformation (1980) a x Vt t log ( ) = e (7.43) C t 2 K 2 K where a = A M 4π K (7.44)

135 7.4 Calculation of Dispersion Coefficients 135/213 a t log ( ) e C t Plot versus t (7.45) Then, slope is 2 V K (7.46) And intercept is 2 x K (7.47) This method is only valid in the equilibrium zone (at asymptotically long times).

136 7.4 Calculation of Dispersion Coefficients 136/213 (2) Thackston s method Thackston et al (1967) developed an estimation technique similar to Chatwin s by transforming Eq. (7.18) to give log ( C t) = log a e e ( x Ut) 4Kt 2 (7.48) Plot log ( C t) e versus ( x Ut) 4Kt 2 (7.49)

137 7.4 Calculation of Dispersion Coefficients 137/213 Then, slope is 1 4K (7.50) (3) Cmax method For a conservative tracer, the peak concentration the square root of distance below the source. C max decreases inversely as C max M M = = (7.51) A 4πKtn 4πKxmax A U

138 7.4 Calculation of Dispersion Coefficients 138/213 tn c x where = time where profile is measured; = location of the maximum concentration. xmax Plot C max versus 1 x (7.52) Then, the slope is M A 4 π K / U (7.53) An advantage of this method is that it requires only the peak concentration and not the entire tracer profile.

139 7.4 Calculation of Dispersion Coefficients 139/ Use of Velocity Measurements In large rivers tracer tests may not be viable because the advective zone is very long and prohibitively large amounts of dye are required. More cost effective to measure the flow distribution at several sites along the channel Calculate LDC from the velocity and depth distribution within the channel. Flow gauging is made at water level recording stations. Such data can be used to estimate K.

140 7.4 Calculation of Dispersion Coefficients 140/213 Fischer (1967) extended the Taylor and Elder method to natural channels where transverse velocity shear dominates longitudinal dispersion. 1 W y 1 y K q '( y) dy dy q '( y) dy A D( yhy ) ( ) = (7.54) T where h( y) q '( y) = u '( y, z) dz 0 u'( y, z) = u( y, z) u( y) (7.55) (7.56) where u( y) = local depth - averaged velocity

141 7.4 Calculation of Dispersion Coefficients 141/213 Disadvantage: D T varies significantly between channels and along a given channel. The uncertainty in estimating is of the order of and this leads to a comparable uncertainty in estimates of K. DT ± 50% The reach-averaged dispersion coefficient is 1 n K = K x L i = 1 i i (7.57)

142 7.4 Calculation of Dispersion Coefficients 142/213 Homework Assignment Due: Two weeks from today Concentration-time data listed in Table are obtained from dispersion study by Godfrey and Fredrick (1970). 1) Plot concentration vs. time 2) Calculate time to centroid, variance, skew coefficient. 3) Calculate dispersion coefficient using the change of moment method and routing procedure.

143 7.4 Calculation of Dispersion Coefficients 143/213 4) Compare and discuss the results. Test reach of the stream is almost straight and necessary data for the calculation of dispersion coefficient are U = 0.52 m/s; W = 18.3 m; H = = * 0.84 m; U 0.10 m/s

144 7.4 Calculation of Dispersion Coefficients 144/213 Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 x = 192 m x =1,009 m x = 1,728 m x = 2,399 m x = 3,353 m x = 4,130 m T (hr) C/C 0 T (hr) C/C 0 T (hr) C/C 0 T (hr) C/C 0 T (hr) C/C 0 T (hr) C/C

145 7.4 Calculation of Dispersion Coefficients 145/

146 7.4 Calculation of Dispersion Coefficients 146/ Longitudinal and Transverse Dispersion Coefficients for 2D ADE Introduction For mixing predictions in the intermediate field, the depth-averaged 2D advection-dispersion equation can be used. c c c + ( huc) + ( hvc) = ( hd ) + ( hd ) L T t x y x x y y (7.58) where over-bars for depth-averaged concentrations are eliminated.

147 7.4 Calculation of Dispersion Coefficients 147/213 Field study for 2D model is conducted to obtain bathymetric and hydraulic data at each section and depth averaged concentration data ( cxyt (,, )). Then, longitudinal and transverse dispersion coefficients are calculated from the field data. Two types of tracer injection are used: i) Continuous injection: steady source ii) Instantaneous injection: non-steady source

148 7.4 Calculation of Dispersion Coefficients 148/213 A single instantaneous tracer injection gives information about only one of the ensemble of possible outcomes. In practice, results from a single instantaneous injection are usually assumed to describe the typical plume behavior. A steady source is preferred when studying transverse dispersion because the plume can be traversed several times and the results time averaged.

149 7.4 Calculation of Dispersion Coefficients 149/213 For steady source, only transverse dispersion coefficient can be obtained by i) Moment method ii) Routing method iii) Graphical method For non-steady source, both longitudinal and transverse dispersion coefficients can be obtained by the 2D routing methods whereas only the transverse dispersion coefficient can be obtained by the moment methods.

150 7.4 Calculation of Dispersion Coefficients 150/213 Calculation of LDC & TDC No Instantaneous tracer injection? Yes Convert into steady-state concentration data No Consider Irregularities of depth & width? No Consider transverse velocity? Need LDC? No Consider transverse velocity? No Consider Irregularities of depth & width? No Yes Yes Yes Yes Yes High quality variance for distribution? No No Consider Irregularity of depth & width? No High quality variance for distribution? Yes Yes Yes SMM STMM STRM GMM 2D RM 2D STRM GMM-θ STRM-θ STMM-θ SMM-θ

151 7.4 Calculation of Dispersion Coefficients 151/ Moment-based Methods for Steady Source For steady injection of tracer, only transverse dispersion coefficient (TDC) is calculated by the moment method. (1) Simple Moment Method (SMM) TDC is determined by the rate of change of spatial variance of the tracer profile. D T = 1 2 dσ dt y 2 (7.59)

152 7.4 Calculation of Dispersion Coefficients 152/213 where t = time of travel; and σ y w 2 ( ) (, ) 2 0 ( x) = y y c x y dy 0 w c( x, y) dy (7.60) y( x) w 0 = 0 w yc( x, y) dy c( x, y) dy (7.61) Finite difference form is D T = 1 U 2 σ y ( x ) σ ( x ) y 1 x x 2 1 (7.62)

153 7.4 Calculation of Dispersion Coefficients 153/213 Graphically, the slope of plot of σ 2 y ( x) versus x is 2D T U (7.63) Restrictions: Tracer dispersion must obey Fick s law with a constant dispersion coefficient. Tracer must mot impinge on either bank. The channel must be uniform so that the plume does not expand and contract. The upstream transect must be placed far enough downstream from the source for tracer to be well mixed vertically.

154 7.4 Calculation of Dispersion Coefficients 154/213 Experimental errors (uncertainties): The sampling error is ±10% plus random error by sensor limit. The entire transverse tracer profile must be measured because the variance estimates are greatly affected by experimental errors in concentration measurements at the edges of the tracer profile where s is small but ( y y) 2 is large. It is advisable to plot concentration profile and either drop or smooth out any suspicious concentration data at the edges of the plume.

155 7.4 Calculation of Dispersion Coefficients 155/213 (2) Generalized Moment Method (GMM) When the channel bathymetry varies longitudinally then the plume undergoes cyclical expansions and contractions, SMM cannot give accurate results. Hollley et al (1972) developed the GMM for a steady tracer input which takes into account transverse and longitudinal variations of depth and velocity, non-zero transverse velocities, and allows for tracer impinging on the bank.

156 7.4 Calculation of Dispersion Coefficients 156/213 The depth-averaged 2D advection-dispersion equation for channels with irregular depth is c c c + ( huc) + ( hvc) = ( hd ) + ( hd ) L T t x y x x y y (7.64) For steady-state, Eq. (6) becomes c ( huc) + ( hvc) = ( hd ) T x y y y (7.65) where the longitudinal dispersion term is neglected

157 7.4 Calculation of Dispersion Coefficients 157/213 Multiply each term in Eq. (7) by Here y is given as ( y y) 2 y w 0 = 0 w yhucdy hucdy (7.66) Then, integrate each term in Eq. (7.65) across the channel to have d dx 2 w ( y y ) hucdy 2 w ( y y ) hvcdy 2 w ( y y ) D dy = 0 0 T (7.67) c y

158 7.4 Calculation of Dispersion Coefficients 158/213 Normalize each term in Eq. (7.67) by dividing the total longitudinal tracer flux w hucdy 0 (7.68) Then, in case of constant h, u, and identical. D T, Eq. (7.67) and Eq. (7.59) are 2 dσ ( x ) y1 g( x) + 2 D f ( x) = 0 (7.69) y dx

159 7.4 Calculation of Dispersion Coefficients 159/213 where σ y1 ( x) 2 0 w = ( ) 2 y y hucdy 0 w hucdy (7.70) g( x) w 2 ( y y ) hvcdy 0 = w 0 hucdy (7.71) f ( x) = 0 w c ( y y ) hϕ dy y 0 w hucdy (7.72) D = D ϕ( y) T y (7.73)

160 7.4 Calculation of Dispersion Coefficients 160/213 where 0 w ϕ ( y) dy = 1 (7.74) Practical difficulties 1 The transverse velocity is rarely measured in the field. 2 In an irregular channel, the variance of the tracer flux linearly with distance. σ 2 ( x) 3 does not vary In a non-uniform channel both g(x) and f(x) also vary non-linearly. Holley suggests that this problem can be overcome by integrating Eq. (7.69) along the channel to give

161 7.4 Calculation of Dispersion Coefficients 161/213 σ x x ( x ) σ ( x ) g( x) dx 2 D f ( x) dx 0 y1 2 y1 1 + = x y (7.75) x 1 1 Eqs. (7.69) and (7.75) require accurate estimation of the variance of the tracer flux and measurement errors in concentrations at the edges of the tracer cloud affect the variance. GMM does not produce robust estimates of the dispersion coefficient for field measurements in irregular natural channels (Rutherford, 1994). The stream-tube method is much to be preferred in natural streams.

162 7.4 Calculation of Dispersion Coefficients 162/213 (3) Stream-tube Moment Method (STMM) Yotsukura and Sayre (1976) developed a more popular tool than the GMM in a non-uniform channel. This transformation is of merit because a fixed value of q is attached to a fixed streamline so that the coordinate system shifts back and forth within the cross section along with the flow. Assuming that the factor of diffusivity is constant, the solution for a steady vertical line source is

163 7.4 Calculation of Dispersion Coefficients 163/213 m cxq (, ) = exp 4π Dx ( q q ) 0 4Dx 2 (7.76) where m = tracer injection rate; q y = 0 hudy 2 D = ψ H UD T q 0 = cumulative discharge at the source, and (7.77) (7.78) where H = mean depth; U = mean velocity; and ψ = 1 W h 2 u ( ) dy 0 W H U (7.79)

164 7.4 Calculation of Dispersion Coefficients 164/213 where ψ = a dimensionless shape factor. Rewrite Eq. (7.76) as cxq (, ) = c ( x) exp max ( q q ) 0 4Dx 2 (7.80) If tracer does not impinge on either bank, Plot log e c ( ) max x cxq (, ) versus q Then the slope is 1 4Dx (7.81)

165 7.4 Calculation of Dispersion Coefficients 165/213 [Re] Stream-tube GMM dσ q dx 2 = 2 D f ( x) q (7.82) where σ q Q 2 ( ) (, ) 2 0 ( x) = q q c x q dq 0 Q c( x, q) dq (7.83) q( x) Q 0 = 0 Q qc( x, q) dq c( x, q) dq (7.84)

166 7.4 Calculation of Dispersion Coefficients 166/213 f ( x) = 0 Q cxq (, ) qψ ( q) dq q 0 Q c( x, q) dq (7.85) Dq ( ) = Dψ ( q) q (7.86) D q = 1 2 d ( ) q dx σ 2 (7.87)

167 7.4 Calculation of Dispersion Coefficients 167/213 (4) Graphical estimation method 1) Nokes method (1986) This method assumes that the tracer concentration distributions are Gaussian. Use a normalized cumulative concentration c c ( y) = o o y w c( y) dy c( y) dy (7.88)

168 7.4 Calculation of Dispersion Coefficients 168/213 Plot c ( ) c y versus y on arithmetic probability paper. Then, a straight line can be fitted to a plot of experimental data Gaussian distributions plot as a straight line A robust estimate of the variance of the tracer distribution is 1 σ = ( y y y ) (7.89) where and y = transverse locations at which c ( y) = 0.84 and y respectively. c

169 7.4 Calculation of Dispersion Coefficients 169/213 2) Constant - coefficient model m u( y y ) 0 cxy (, ) = exp h π D xu 4Dx ( ) m c x = max h 4π D xu T 4 T T 2 (7.90) (7.91) Plot log e c ( ) max x cxy (, ) versus y Then, the slope of this plot is u Dx 4 T (7.92)

170 7.4 Calculation of Dispersion Coefficients 170/ Moment-based Methods for Non-Steady Source For non-steady injection of tracer, the transverse dispersion coefficient can be calculated by the moment method after converting into steady-state concentration data. Beltaos (1975) presented a procedure for converting the governing equation of the transient condition to the steady-state condition. Integrate Eq. (7.2) with respect to time assuming that u, v, h, D, L and DT are independent of time to yield

171 7.4 Calculation of Dispersion Coefficients 171/213 ( huθ ) + ( hvθ ) x y = hd x L θ + x hd y T θ y (7.93) where θ is the dosage of the tracer, which is defined as, θ ( xy, ) cxytdt (,, ) 0 (7.94) Eq. (7.93) is the same differential equation as Eq. (7.2), with the dosage θ used instead of the concentration without the time variation term. c in a continuous injection tracer test,

172 7.4 Calculation of Dispersion Coefficients 172/213 (1) Generalized Moment Method for θ (GMM-θ) Simplify Eq. (7.2) by dropping the longitudinal dispersion term under a steady concentration condition, and assuming that the depth and the transverse dispersion coefficient are constant in the transverse direction ( uθ ) + x y ( vθ ) = D 2 θ 2 y T (7.95) Multiplying each side of Eq. (7.95) by 2 y and integrating it with respect to y using the Leibnitz s rule and integral by parts to yield the generalized moment equation as

173 7.4 Calculation of Dispersion Coefficients 173/213 D T 2 1 σ 2 1 g = θ 2 x f f (7.96) where σ f W 2 uθ y dy 2 0 θ 2 W = W 0 uθ dy [ θy] θdy 0 = W uθdy 0 W 0 (7.97a) (7.97b) g W 0 = W 0 vθydy uθdy (7.97c)

174 7.4 Calculation of Dispersion Coefficients 174/213 Plot 2 σθ 2 against x Then, the slope of the straight line is 2 f DT g + f (7.98) (2) Simple Moment Method for θ (SMM- θ) The simple moment method (Sayre and Chang, 1968) can be deduced from the generalized moment method. Assuming that the longitudinal velocity is constant, and the tracer impinging on banks and transverse velocity are negligible, Eq. (7.97) becomes

175 7.4 Calculation of Dispersion Coefficients 175/213 f 1 =, g U = 0 (7.99) where U is the reach-averaged velocity. Incorporating Eq. (7.99) into Eq. (7.96) yields D T = U 2 2 σ θ x (7.100) where 2 σ θ = the variance of the transverse distribution of the tracer dosage W θ 0 = W 0 2 y dy θdy (7.101)

176 7.4 Calculation of Dispersion Coefficients 176/213 (3) Stream-tube Moment Method for θ (STMM- θ) Beltaos (1980) derived the stream-tube moment method which can be applied to meandering streams that have a skewed concentration distribution due to irregular depths and velocities. Apply the stream-tube concept neglecting the transverse velocity into Eq. (7.95) to get θ = x q E T θ q (7.102)

177 7.4 Calculation of Dispersion Coefficients 177/213 where E T = the diffusion factor defined as E T = ΨD UH T 2 (7.103) where Ψ is a dimensionless shape-velocity factor that has a range of (Sayre, 1979; Beltaos, 1980) and is defined by Ψ = 1 Q Q 0 h 2 udq (7.104) Normalizing Eq. (7.102) using η and S S x = E Q T 2 2 S 2 η (7.105)

178 7.4 Calculation of Dispersion Coefficients 178/213 where η q /Q S θ / Θ 1 Θ= θdη. 0 Conducting a similar procedure as the simple moment method for Eq. (7.105) gives D T = 2 2 Q dσ η 1 2 2ΨUH dx 0 1 0S0 ( 1 (1 η ) S η ) (7.106)

179 7.4 Calculation of Dispersion Coefficients 179/213 2 where and η are the variance and the centroid of the S η distribution, ση 0 respectively, and 0 and S are the normalized dosages at the left and right banks, respectively. S 1 Ex) Case study of streamtube moment method Field tracer experiment in Nakdong River Pollutant inflow from left side of tributaries Using conductivity as tracer Using Beltaos(1980) s streamtube moment method

180 7.4 Calculation of Dispersion Coefficients 180/213 i) Hydraulic data measurement Discharge, velocity, water depth

181 7.4 Calculation of Dispersion Coefficients 181/213 Velocity (m/s) Sec W (m) Depth (m)

182 7.4 Calculation of Dispersion Coefficients 182/213 Date Section Q H U u* W W/H U/u* (m 3 /s) (m) (m/s) (m/s) (m)

183 7.4 Calculation of Dispersion Coefficients 183/213 ii) Concentration data measurement Conductivity, temperature

184 7.4 Calculation of Dispersion Coefficients 184/213 EC(µS/cm) Sec.1 Sec.2 Sec y/w

185 7.4 Calculation of Dispersion Coefficients 185/213 iii) calculation of ψ ψ = 1 Q 2 h udq 0 Q iv) plot C η graph q C 1 η, C, C Cdη Q C = 0

186 7.4 Calculation of Dispersion Coefficients 186/ Sec.1 Sec.2 Sec.3 C' ƞ

187 7.4 Calculation of Dispersion Coefficients 187/213 ( ) 2 v) plot σ F x graph ( η ) f( x) = 1 1 C η C x ( ) ( ) F x = f x dx Slope = 2E y 2 Q from diffusion factor calculated. E y =ψ D UH T 2, transverse dispersion coefficient can be

188 7.4 Calculation of Dispersion Coefficients 188/ σ F(x) (m)

189 7.4 Calculation of Dispersion Coefficients 189/ Routing Method for Non-Steady Source (1) Stream-tube Moment Method for θ (STRM- θ) The stream-tube routing method can be derived from the analytical solution of the stream-tube model of 2D ADE for tracer dosage transport in irregular channels (Seo et al., 2006). The analytical solution to Eq. (7.102) for a mass being instantaneously injected at x = 0 and η = ω can be obtained as

190 190/ Calculation of Dispersion Coefficients Θ = x B x B x S C C 4 ) ( exp 4 ), ( 2 ω η π η (7.107) 2 /. C T B E Q = where Using the frozen cloud assumption in which no dispersion occurs during the passage of the tracer cloud past the measuring station (Fischer, 1968; Rutherford, 1994), the routing equation of the tracer dosage can be written as ω ω η π ω η d x x B x x B x S x S C C = ) ( 4 ) ( exp ) ( 4 ), ( ), ( (7.108)

191 7.4 Calculation of Dispersion Coefficients 191/213 where S( x 1, ω) = the observed dosage distribution as a function of the dimensionless discharge at an upstream site; S( x 2, η) = the predicted dosage distribution as a function of dimensionless discharge at a downstream site; and ω = a normalized dummy transverse distance variable of integration. Eq. (19) cannot obtain the longitudinal dispersion coefficient because the concentration profile is converted into the dosage profile.

192 192/ Calculation of Dispersion Coefficients ξ ψ (2) Two-dimensional Routing Method (2D RM) The two-dimensional routing method can be derived from the analytical solution of the 2D advection-dispersion equation for the initial distribution of the mass which is injected instantaneously at and (Baek et al., 2006). ψ ξ ψ ξ π ψ ξ d d t D y t D x D D t f t y x C T L W T L = 4 ) ( exp 4 ) ( exp 4 ), ( ),, ( (7.109)

193 193/ Calculation of Dispersion Coefficients where is a dummy function. Using the frozen cloud assumption in which no dispersion occurs during the passage of the tracer cloud past the measuring station (Fischer, 1968; Rutherford, 1994), the routing equation of the temporal tracer concentration can be written as ( ) f ψ τ ψ τ π τ ψ d d t t D y t t D t t t U D D t t U x C t y x C T L W T L + = ) ( 4 ) ( exp ) ( 4 ) ( exp ) ( 4 ),, ( ),, ( (7.110)

194 7.4 Calculation of Dispersion Coefficients 194/213 where C( x 2, y, t) = the predicted depth-averaged concentration as a function of time and lateral distance, y, at x ; (,, ) 2 C x1 ψ τ = the observed depth-averaged concentration as a function of time and lateral distance at x 1.

195 7.4 Calculation of Dispersion Coefficients 195/213 I.P. η η Dosage S Discharge η Sobs ( x, η) 1 Sobs Dosage S ( x, η) 3 Discharge η η Input Best Fitting S ( x, η) f( D ) dη = S ( x, η) obs 1 T pre 3 Find D T