RUNOFF CONCENTRATION IN STEEP CHANNEL NETWORKS by Rolf Kellerhals Dipl. Ing., Swiss Federal Institute of Technology, Zurich, I960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Geography (Interdisciplinary Program in Hydrology) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada ii ABSTRACT The objective of this study is the development of a runoff routing procedure, applicable to steep channel networks in the tumbling flow regime, and suitable for incorporation into more comprehensive mathematical representations of the runoff process. "Steep" is meant in the sense that degradation into existing, coarse deposits (e.g. Pleistocene materials, slide debris, scree) is assumed to be the major channel-forming process. Similarity considerations show that under these circumstances two relatively easily available parameters, such as channel slope and drainage area, or channel slope and width are adequate to define the geometry and hydraulic per formance of the channels. The hydrologically significant aspects of channel flow are storage per unit length (area) and discharge, with the relation between the two defining the channel performance under steady conditions. This function, A = f(Q), can be obtained in the field by observing the dispersion of slug-injected tracers through fixed test reaches over a range of discharges. Measure ments of this type were made on thirteen test reaches, covering a wide range of channel size and slope. The data from all test reaches can be closely approxi-PA mated by exponential relations of the form A = a^ Q"' . As • indicated by the similarity considerations, the constants aA and bA of this steady flow equation are predictable from basin parameters." The details of the statistical link between various iii basin parameters and the above, constants are. discussed in Day (1969) on the basis, of the steady flow, data of. this study supplemented by extensive.additional measurements. Runoff concentration is an unsteady flow process, which can only be defined with a single flow equation if the flow system is truly kinematic. In order to investigate whether this holds for steep channels, all test reaches were located below lakes with outlets suitable for minor discharge modifi cations. Small, step-like surges (positive and negative) were created at the lake outlets and their propagation through the test reaches was observed with accurate water level gauges. These surge tests indicate consistently that the channels act as kinematic flow systems but with certain dispersive effects added and with a markedly higher-than-kinematic wave celerity at very low stage, which is probably the result of dynamic waves in pools. Due to the frequent occurrence of super-critical flow, dispersion can only be the result of storage in pools. The differential equation for a kinematic channel with storage in a large number of identical storage elements is derived and solved in linearized form for step-like input corresponding to the surge tests. The dispersion coefficient, which has dimen sions L, is the only free parameter of the solution. Comparison with the field data shows, that mean water surface width provides a good estimate of this parameter.. iv As a computationally- simpler alternative, a routing model which replaces the actual channels by a sequence of truly kinematic channels and deep pools with weir outlets, both obeying the same steady-flow equation,is also considered. Rules for determining the two free parameters of this solution are developed on the basis of the field data. Both routing methods provide approximately equal fit to the surge test data and they both appear to be suitable compon ents for an operational channel runoff model. Being based mainly on the above steady flow equation, both methods are non-linear. This is supported by the field data, which show no tendency towards linearity, except possibly at very low stage. V TABLE OF CONTENTS Page Abstract ii Table of Contents v List of Tables x List of Figures x List of Photographs xii Appendix, List. ofeContents xiiAcknowledgements xiv 1. Notation and Abbreviations 1 1.1 Notation 1 1.2 Abbreviations 6 2. Introduction 8 2.1 Past and present approaches to the runoff problem2.2 Separation of the runoff process into land phase and channel phase 11 2.3 The objective of the study 5 2.4 Assumptions regarding readily available data. 16 3. Field Methods ' 23 3.1 Selection of test reaches 23.2 Survey measurements 29 3.3 Tracer methods 30 3.3.1 Objective3.3.2 Principles of discharge and velocity measurements with slug injection methods 30 . vi TABLE OF CONTENTS (cont'd.) Page 3.3.3 Vertical and lateral dispersion requirements . . 32 3.3.4 Longitudinal dispersion models ... 33 3.3.5 A gamma-distribution model for the final decline of C(t) 37 3-3.6 Equipment and procedures for slug injection measurements 43 3.3.7 Tracer losses ^8 3.4 Surge tests ^9 3.4.1 Objective vfc . 49 3.4.2 Discharge modifications 50 3.4.3 Stage measuring equipment 51 3.4.4 Stilling well response 4 3.4.5 Stage-discharge rating curves . . . 56 4. Field Results 59 4.1 Survey results 54.2 Velocity and discharge measurements 60 4.2.1 Conversion of field data to time-concentration curves' 60 4.2.2 Numerical integration l 4.2.3 Results 64 4.2.4 Accuracy • 65 4.3 Surge tests 77 5. Channel Geometry and Steady ;Flow Equations . . . . . 85 5.1 Similitude considerations for steep, degrading.channel networks . . . 85 vii TABLE' OF .-CONTENTS (cont'd.) Page 5.1.1 Assumptions 85 5.1.2 Conditions for similarity 88 5.2 Basic equations for steady, uniform flow . . 91 5.2.1 Theoretical considerations 91 5.2.2 Flow equations of the test reaches .• 93 5.3 Determining the parameters of the steady flow equation . . . 100 5.4 The friction concept applied to tumbling flow 103 5.4.1 Open channel flow formulas 104 5.4.2 Comparison with the data 106 6. Unsteady Flow in Steep Channels 109 6.1 Kinematic waves and the surge test results. . 109 6.1.1 Some features of kinematic wayes . . 109 6.1.2 Indications from research test results 112 6.2 Kinematic waves with storage dispersion . . . 114 6.2.1 Dispersion through dynamic effects . 114 6.2.2 The differential equations of kinematic waves with storage dispersion 116 6.2.3 A solution for step-like input . . . 118 6.2.4 Comparison with field data 124 6.3 A practical approach to- unsteady, tumbling flow . 127 6.3.I Unsteady flow through a non-linear reservoir 128 viii TABLE OF CONTENTS (cant' d. ) Page 6.3.2 A routing model based on a cascade of pools and channels . . 129 6.3.3 Evaluation of the free parameters from field data 131 7. Conclusion 142 7.1 The Hydraulics of tumbling'.:flow I1*7.2 Basin linearity 144 7.3 Towards an operative channel runoff model . . 145 8. Bibliography 147 • Photographs 152 Appendix Computer programs with operating instructions, printout, and plots 160 ix LIST' OF TABLES Page 1. Comparison of morphometry based-on three map scales. 19 2. Test reaches -below, lakes 24 3. Additional test reaches (Day, 1969) 26 4. Summary of tracer measurements: A. Brockton Creek 66 B. Placid Creek 8 C. Blaney Creek 70 D. Phyllis Creek 3 5. Summary of surge tests: A. Brockton Creek 79 B. Placid Creek 80 C. Blaney Creek . 81 D. Phyllis Creek. 82 6. Regression parameters of steady flow 95 7. Regression parameters of steady flow (Day, 1969) • • 96 X LIST OF' FIGURES Page-1. The; two'runoff phas.es . - 13 2. Morphometry of three basins at different map scales 18 3. Comparison between channel profiles measured off maps and surveyed in the field 21 4. Channel profiles 28 5. Longitudinal dispersion of slug-injected tracer . . 36 6. The storage model approximation to longitudinal dispersion 39 7. Graphical fitting of storage model 42 8. Circuit diagram of recording conductivity bridge. . 45 9. Response of recording conductivity bridge 47 10. Schematic section of manual gauge 52 11. Schematic view of stage recorder installation for mountain.streams 53 12. Gauge response curves 5 13. Two typical stage - discharge rating curves .... 57 14. Definition sketch for numerical integration .... 62 15. Surge test of October 13, 1968, on Blaney Creek . . 78 16. Hydraulic measurements on the reach, Brockton Gauge 1 - Gauge 2 917. Hydraulic measurements on the reach, Blaney Gauge 3 ' - Gauge 5 99 18. Valuesi-Of c2 for best fit to-Equation 5.21 105 19. Exponents of Equations 5.23 and 5.24 vs. .... 107 20. Definition sketch for Equation 6.8 1121. Effect of /& on the solution of the kinematic wave equation with storage dispersion 123 xi LIST OF FIGURES (Cont'd) Page 22a. Comparison between field observations and kinematic waves with storage dispersion ±25 22b. Comparison between field observations and . kinematic waves with storage dispersion 126 23. Definition sketch for the cascade of channels and reservoirs 130 24a. Variable number of reservoirs at f = 0,1 133 24b. Variable number of reservoirs at cf = 0.28 134 24c. Variable number of reservoirs at C = 0.7 135 25. The routing parameter o" 137 26a. Computed and observed surges, Brockton Creek .... 138 26b. Computed and observed surges, Blaney Creek 139 26c. Computed and observed surges, Phyllis Creek 1^0 xii LIST OF PHOTOGRAPHS Page 1. Brockton Creek, along Reach Br 1-2, looking upstream 152 2. Placid Creek, along Reach Pl 3-4, looking downstream. Typical log jam in foreground .... 152 3. Blaney Creek, at Bl Gauge 3, looking downstream . . . 153 4. Blaney Creek, at Bl Gauge 4, looking upstream from bridge 155. Phyllis Creek, at Ph Gauge 2, looking downstream. Stage recorder at right 154 6. Phyllis Creek, at Ph Gauge 4, looking upstream . . . 154 7. Barnstead conductivity bridge 158. Volumetric glass ware for salt dilution tests .... 155 9. Vats, pail, and stirring rod for salt dilution tests 155 10. Equipment for Rhodamine WTslug injection tests . . . 156 11. Recording conductivity bridge, with electronic interval timer 1512. Control structure at the outlet of Blaney Lake. Three flashboards in place 157 13. Timber crib dam at outlet of Marion Lake, with two additions in place for a down-surge 157 14. Pump at Placid Lake 158 15. Inverted syphon at pool outlet above Brockton Gauge 1 158 16. Plexiglass tube for. stage readings on right, constant rate injection apparatus on left 159 17. Recorder installation with inverted syphon at Blaney Gauge 5 15xiii APPENDIX COMPUTER PROGRAMS WITH OPERATING INSTRUCTIONS PRINTOUT, AND PLOTS LIST OP CONTENTS NACL Source listing 161 Sample plot of rating curve Sample printout DQV Source listing 168. Sample printout TAILEX Source listing 170 Sample printout Sample plots with example for determination of A, B, and D QVEL Source listing, including subroutines for 176 numerical integration -Three sample outputs' -PL0TGA Source listing ' l85\ Sample plots, with and without F -extension L0GRE Source listing, including three subroutines 1^9 Printout and plots for all 13 test reaches PD Source listing, including two subroutines 249 Sample printout SNLR Source listing, including one subroutine 253-Sample printout xiv ACKNOWLEDGEMENTS The original support for this study, came from the American Society of Civil Engineers through the award of the 1966 Waldo E. Smith Fellowship to the writer. The National Research Council of Canada and the Killam Foundation gave support later on In the form of research funds and fellow ships respectively. This help is gratefully acknowledged; It permitted completion of the study without serious finan cial constraints. By supporting the work of Mr. Day, whose results are essential for the positive conclusions of this study, the Department of Civil Engineering, U.B.C., made a valuable contribution. Collection of the field data required fairly large parties and,since much of the work had to be done on rainy days, it was often less than enjoyable. For their help as occasional field assistants, whenever suitable weather conditions occurred, the writer is indebted to Mr. M. K. Woo, to Mr. Roy Purssell,and to many graduate students of the Department of Geography, U.B.C., also to his wife Heather, who did much of the laboratory work, data handling, and typing. The writer's two successive supervisors, Drs. M. A. Melton and G. R. Gates and other members of the interdepart mental Ph.D. committee supported the study with their experience in conducting research projects on related topics. The inter departmental arrangements, which provided a very successful academic and administrative framework, were made possible through the efforts of Dr. Ian McTaggart Cowan, Dean of Graduate Studies. Dr. 0. Slaymaker and Mr. M. Church kindly reviewed the thesis manuscript. 1 NOTATION AND 'ABBREVIATIONS 1.1' Notation 2 A Cross sectional area of flow in a channel . Cm ) A Initial area, o Ap Area at formative discharge. At hA/et. a Constant in the following regressions: aA log A = f(log Q) av log v = f(log Q) aT log T = f(log Q) a; log (Ag^/^/B, . ^l/S/.S/B,. B( ) Riemann Function b Coefficient in the above regressions (same subscripts) . b Without subscript: Time constant (s). C Concentration of tracer (g cc"4') . C^ Concentration of tracer in reservoir i. c Wave celerity (ms c. Constants, l D Bed material size (m). D Mixing coefficient, for one. dimensional dispersion X 2—1 over x Cm s ). 2 -1 Dc Flood wave dispersion coefficient Cm s ) . 2 DA Drainage area (km ). d Depth, of flow In channel (m) ds Depth, measure,, defined as- A/W,o . "s E Exciting voltage of. conductivity- bridge (volts) e Base of natural logarithms, 2'. 7183. F(. ) Function of A(x,t). f( ) Unspecified function. f ( ) Probability density function. X Conductivity (mhos). -2 Acceleration of gravity (ms ) (?s " P^^fv: JS ' S ' w " t s 5W H Stage reading (ft))', h Elevation difference between, stream and gauge stilling well (ft);. I.(u) Modified Bessel Function of the first kind of order i and argument u. With J\(u) being a Bessel Function of the first kind I. (u) = J. (Y7! u). 1 1 i Integer, counter, j Integer. k( ) Function of gCx,t). L Tracer los-s- rate (_% per min.).-1 Length, of test reach (m) 1^,1 = 0,1,2 Second order interpolation polyriomlmals-.-M Mass of tracer (g). Integer, number of reservoirs or storage elements. Manning's n. Polynominal approximation of C(t), (g cc "'") 1/2 Abbreviation for (240c//32wjp 3 -1 Discharge (m s ). Initial discharge. Formative discharge. Qutflow from reservoir i. Measured discharge. Discharge from upstream reservoir. Relative discharge, Q/Q^, subscripts as for Adjustable resistance (ohms). Input impedance of recorder. Constant resistance. Volume of reservoir i. Argument of f -function, parameter of f -distribution. Slope Friction slope Valley slope Mean water travel time (min.) Lag to tracer peak.'(min. ) . Lag between pools. Reservoir filling time (s) . Lag to the f irs-t arrival of tracer (min-.) . Mean tracer, travel, time '(.min-.) . t Time', coordinate (min. or s). t Starting time. t End time. e u Bessel function argument. V Potential difference, voltage (volt), v Velocity (ms-1). vm Mean velocity. Vp Velocity at channel forming; discharge. W Width (m). Wp Channel width between high water marks. Wg Water surface width. w Exponent in exponential relation between A and Q, Q <=x AW X^ Shape factors . x Length coordinate along channel (m). x. Length coordinates of critical sections. 1 Y. i=l,2,3 Parameters of Storage Model based on 1 f -distribution. Abbreviation for exponent (w - l)/w. Exponent in exponential relation between z W and Q, W ex. Q s s Relative- change in d is. charge during a surge test Q/QD-1. A Non-dimensional dispersion coefficient r( ) Gamma Function. A Finite step. Abbreviations for terms in c and At.. i 0 Slope angle. K Parameter of -distribution. X Length of a reservoir (m). /f Viscosity. 2 -1 V Kinematic viscosity (m s ). | Dummy length variable (m). ir 3.1416. p Specific mass (g cc~^~) . ps Specific mass of bed material. Specific mass of water. <5" Proportion of channel length occupied by reservoir. f Dummy time variable (min. or s). 6 1.2 Abbreviations Bl Blaney Creek.. Br Brockton Creek. C.I. Constant injection. . cc Cubic centimeter, d Derivative. 0 Partial derivative. D Total derivative for a moving observer. DO Down-surge, downstream of ft Feet. K 1000 ohms. km Kilometer. Lo Longitude Lat Latitude 1 Liter log x Natural logarithm of x. log-j^x Logarithm to base 10. mm Millimeter, m Meter, min. Minutes NaCl Sodium chloride, common salt. NTS National topographic series. Ph Phyllis Creek.. Pl Placid Creek. RhWT Rhodamine WT,. f lucres-cent dye manufactured by Du Pont. RSQ R-square,.. the. fraction of total sample, variance explained by- a regression. SQD Sodium dichromate Na2O^CR2 • 2R^Q . s Seconds. UP Up-surge, upstream of X Time-concentration curve measured here. 8 2. • INTRODUCTION 2.1. Past and Present Approaches to the Runoff Problem Runoff concentration denotes the process which trans forms rainfall or snowmelt over a basin to stream discharge at the basin outlet. This transformation is an important and complex problem of hydrology, which has received considerable attention, but remains without a satisfactory, generally accepted solution. Hydrologists are often interested in peak flows at certain locations along a stream, but it is a fortunate accident if adequate stream flow records happen to be available for the desired site. In many cases, meteoro logical records have to be used to estimate peak rates of rainfall or snowmelt, which are then transformed to stream-flow . Two different approaches towards the precipitation-runoff transformation appear to be feasible, with the added possibility of combinations between the two. The simulation approach avoids the detailed physics of the runoff process by simulating it, or certain parts of it, with systems which may be classified into "black box" systems;., conceptual models, or electric analogues. The alternative may be called the physical approach, as it involves dividing the runoff process into clearly identifiable subprocesses, which are dealt with on a physical basis. Amorocho and Hart (1964) 9 discuss the various possibilities in detail. The popularity of the "black box" systems approach, of which the Unit Hydrograph is the prime example, Is a result of the widely held belief (or hope) that linear systems1 represent most basin responses adequately. The theory of linear systems is quite advanced and it lends itself to an almost unlimited number of mathematical exercises. 2 Recently the number of non-believers has been growing as evidence is accumulating that most basins have sufficiently strong non-linear effects to render the linear approximation dangerous (leading to underestimates of peak flows). There are, however, further and better reasons for the use of "black box"systems. Some of the runoff concentrating processes take place over the entire basin area and may be extremely variable, rendering detailed physical description impractical (at least at present). A pure "black box" type systems approach to such a process, ignoring the physical aspects entirely, may lead to good results. The non-linearities of a basin may also be concentrated in a few "'"To be linear, a system has to satisfy the following conditions: Assuming f-j_ (t). and f2 (t) are the responses to inputs. f3 (t) and fij (t) respectively, then (fi + f2) Is the response due to input (T3 + fi\). The. mathematical formulation of linear systems leads to linear differential equations. 2 Numerous papers in the Proceedings of the International Hydrology Symposium, held at Fort Collins, Sept. 6-8, 1967 and in the Proceedings of the Symposium on Analogue and Digital Computers, Tucson, 1968, can be cited as evidence of this trend. 10 processes, so that, even linear systems may give good approxi mations to others. Much of the most recent work on runoff simulation is based on simple conceptual models of the total runoff process, such as linear or non-linear reservoirs, in series or in 'parallel, systems of uniform permeable soil layers, or systems based on one-dimensional dispersion- (Overton, 1967; Sugawara, 1967; Diskin, 1967)-. The major difficulty with all simulation approaches lies in the problem of parameter identification. Most systems, even quite primitive ones, have enough free parameters to permit close fitting to a particular set of rainfall-runoff data. The somewhat more severe test of representing runoff events of different magnitude from the same basin without requiring parameter adjustments, is also met by many of the recently proposed simulation systems, but to be really useful the parameters should have fixed relations with identifiable basin characteristics. This condition is not met by any presently available simulation model. The physical approach consists essentially of identify ing the processes that contribute significantly to the precipitation-runoff transformation, formulating the differential equations governing them, searching for practical solutions, and finally developing field and office procedures which supply the free parameters from readily available data. 11 Such a truly physical treatment .of. the total runoff process would avoid the identification problem, but it is unfortunately quite impossible at present, not so much for lack of understanding of the major processes as due to the complexity of most basins and the difficulty of separating the processes from each other. However, some important processes are both separable and fairly well understood, so that the physical approach has become possible and, by avoiding the identification problem, has superceded simulation. Examples of such processes are the propagation of flood waves in prismatic channels, surface runoff from paved surfaces, evaporation from large, deep lakes, and infiltration into uniform soils. In conclusion it appears to the writer that the most profitable direction for new research on runoff processes lies in the physical field, aiming at a gradual replacement of simulation models with more closely representative physical models. The research project which forms the basis of this thesis was designed in accordance with this belief. 2.2 Separation of the Runoff Process into Land Phase and Channel Phase Larson (1965) suggests the separation of the runoff process into two phases; a land phase, which transforms.rain fall or snowmelt to runoff supply (Larson's term for channel inflow), and a channel phase, transforming runoff supply to 12 basin outflow. Figure 1 illustrates this, division. The land phase is similar to the total runoff process from very small basins and includes all the complex interacting processes which can take place over the entire basin area, but should remain constant over regions of similar topography, vegetation and soil characteristics (e.g. evapo-transpiration, infiltration, . interflow, etc.). In basins with negligible water losses out of the channel system, which includes most basins in the humid zone, the channel phase is dominated by the single process "wave propagation In open channels". For representation of the land phase the simulation approach appears to be best suited under the present circum stances. The parameters can be' evaluated on the basis of rainfall-runoff data from small test basins (small in the sense that the channel phase is negligible) in the region of interest. Parameter consistency is not an absolute necessity due to the assumption that the land phase is regionally con stant . Under conditions of heavy rainfall on fairly imperm eable or thoroughly wet basins with high drainage density, the land phase may even be reducible to "Rainfall-Overland Flow-Runoff Supply" (Figure 1) with small losses and negli gible time lag. The effect of the land phase on the outflow hydrograph in the vicinity of the peak may then be negligible. In most cases one has to assume that the land phase controls the volume of runoff and contributes in a non-negligible 13 I n put: i Precipitation) > Interception Depression Storage • Overland Flow A, • Infiltration Inter flow Groundwater Eva potransp. J Deep Percol. 1 Loss T The Land Phase (excluding snowmelt) Input: Runoff Supply Summation from sub-areas Translation through channels and lakes Storage in channels and lakes • Evap. from lakes Loss * Deep Percol. The Channel Phase THE TWO RUNOFF PHASES Fig. I manner to the shape of the hydrograph. Larson's basic assumptions which, if proven, certainly will justify the two-phase approach, are that the channel phase accounts for the differences between basins due to size and shape and that the dominant process in that phase (wave propagation) may be fully representable from readily avail able data. He envisaged this roughly as follows: Maps show the channel network in plan; channel dimensions and roughnes's can be obtained in the field or on large scale air photos. Alternatively maps can be used to make rough esti mates of channel forming discharge and the considerable body of knowledge on relations between size, performance and dis charge of self-formed channels will permit estimates of the necessary parameters. Once the dimensions of the channel system are estimated, standard methods of flood routing should give good representations of the channel phase. The two phase approach cannot be considered operative at present. The usefulness of the concept hinges on the physical representation of the channel phase. If this can be done, then the concept does represent a step forward by removing one part of the runoff process from the realm of speculation and simulation. Larson does not seem to have realized this clearly, as his channel phase is based on long dlsproven concepts such as using a single constant value of Manning's n for large flow ranges in many natural channels or assuming flood wave movements at mean water velocity, and 15 he never Investigates the crucial question whether the parameters of his channel phase representation, as.obtained by curve fitting, are related to the field values. 2.3 The Objective of This Study For several reasons, the two phase approach appears a priori to be particularly suitable in mountainous areas. The channel system is generally well developed and easily traceable on maps. In addition to a plan of the channel network with contributing drainage areas, maps also supply channel slopes, one of the most important parameters in any type of flow routing. Soil cover is often thin and lying on steep, impermeable layers, resulting In a fast-acting and therefore less important land phase. If runoff originates mainly in glaciers and snowfields, the land phase can be replaced by an ice-phase, supplying water to the channel system at discrete locations, but the general approach remains valid. The major obstacles to applying the two-phase concept in mountainous areas are: i) The present lack of information on the formative laws and hydraulic performance of steep channels character ized by alternating super- and sub-critical flow and by energy dissipation due to rapid changes in cross section and slope, (Peterson and Mohanty (i960) introduced the very descriptive term "tumbling flow" for this flow regime. Photographs 1 to 6 illustrate tumbling flow.) and ii) The lack of a realistic routing method, which does 16 not require virtually unobtainable information on the channel system. The present thesis represents an attempt to solve these problems by investigating the physical laws governing steady and unsteady flow in steep channels and by trying to state them in such a form that all parameters can be obtained from data readily available even in ungauged basins. The details of the link between the flow parameters and map measures are discussed in Day (I969). His links are statistical but based on considerations of dynamic similarity. The problems were approached empirically, starting with field measurements and concluding with analysis of the data, a sequence which has been maintained in this write-up. 2.4 Assumptions regarding Readily Available Data The design of this project is based on the assumption that ungauged basins have (i) map-., coverage, (ii) high altitude air photo coverage and (iii) data on precipitation and runoff for at least one location in the same climatic region. Some of the channel phase models developed subsequently require a very rough estimate of mean annual peak flow, or a high flow of some other frequency. Item(iii), combined with map information, should permit such an estimate to + 50%. Map coverage supplies the following information: (i) a plan of the channel network, (ii) contributing drainage areas all along each channel, and (iii) channel slopes. The 17 accuracy of this Information depends primarily on the scale and contour interval of the maps, but other factors, such as the procedures used in map making and the height and density of ground cover in the case of maps made from air photos, may also be Important (Morisawa, 1957; Scheidegger, 1966) . The standard Canadian map scales are 1:50,000 with 50 ft contour interval and 1:250,000 with 500 ft contour inter val. Only a small fraction of the Canadian Cordillera has 1:50,000 coverage but this includes most developed areas and highway routes. To gain some idea of the degree to which these two map scales represent drainage networks in south-coastal B. C., three of the basins used in this study (Furry Creek, Phyllis Creek, Blaney Creek) were analysed morphometrically for stream orders, number of streams and mean stream lengths. One basin (Blaney Creek) could also be analysed on maps to a scale of 1:2400 which, based on a few spot checks, appear to give a reasonably true representation of the drainage system, includ ing first order streams which contain some flow during most of the wet season. (The Slesse and Ashnola basins could not be included in this analysis because they lie partly in the United States, where the accuracy and scale of the map coverage is significantly different.) The results of the morphometric analysis are shown in Table 1 and on Figure 2. The channel network measurements are 18; STREAM ORDER MORPHOMETRY OF THREE BASINS AT DIFFERENT MAP SCALES TABLE I COMPARISON OF MORPHOMETRY BASED ON THREE MAP SCALES Map Scale and Type • Stream Order Furry No. of Streams Creek Mean (km) Length Phyllis No. of Streams Creek Mean (km) Length Blaney No. of Streams Creek Mean (km) Length 1:250,000 1 14 1. 62 3 1.35 2 3.75 NTS 2 3 1.75 1 3.25 1 2.00 3 1 6. 50 1:50,000 1 34 0.785 6 0.79 12 0.63 NTS 2 10 0.985 2 1.27 3 1.18 3 2 3.800 1 2.05 1 4.10 4 1 4.100 1:2400 1 86 0. 23 UBC Research 2 21 0.44 Forest, . 3 4 - 0.56 Topography 4 2 2.38 and Forest 5 1 1.90 Cover. Notes: - Figure 2 shows the same data graphically. - The channel network is based on the blue lines shown as the maps with additions based on the contour picture. - Placid Creek is part of the Blaney Creek basin. " M 20 based on the blue lines of the maps with some additions • and extensions based on the contour picture. (Morisawa, 1957). Lakes are replaced by stream segments. Since running water is not a chief agent in developing the surface geometry of these basins, it is not surprising that Figure. 2. does not show logarithmic relations between stream order and stream number or stream length, as found by Horton and others in many basins. However, with increasing basin size and increasing map scales, the relations become more closely logarithmic. Figure 2 indicates that the 1:50, 000 maps miss most first order and some second order streams, while*the 1:250, 000 maps miss the first, second and part of the third order. These relations may be different in other basins. If possible, a similar comparative study should be made before data obtained on a large scale map are extrapolated to first order channels. Figure 3 illustrates the extent to which channel slope is obtainable from maps of various scales by comparing the survey results (hand level profiles) of 2 test reaches with profiles from maps of scales 1:2400 and 1:12.}000. . Air photos are considered part of the essential data for ungauged basins because the Canadian maps provide little information on vegetative cover and on bed-rock exposures. Particularly at the lower end of hanging valleys, some streams flow directly on bed-rock, often without as much as a minor canyon. This does affect the channel performance considerably. Dense tree cover and the presence of logging slash result in 1,000 Chainage (meters) 2p00 3Q00 4.000 5000 6,000 I700i 1500 •Placid Lake Gauge I \ Placid Creek CD CO £ o > CD 13001 IIOO-® 900J UJ 7001 500J 3] (O OJ Gauge 3 Blaney Lake Gauge 4 Gauge I Blaney Creek I" = 200' map Handlevel profile I" = 1000' map Gauge 3 Gauge 5 5,000 10,000 Chainage (feet) COMPARISON BETWEEN CHANNEL OFF MAPS AND SURVEYED Gauge 4 500 400 CD E c o 300 5 ; UJ 200 15000 20,000 PROFILES MEASURED IN THE FIELD frequent log jams, in small streams, again affecting channel performance. The present data are not adequate to define the conditions under which log jams become significant, but it appears that at a channel width of approx. 12 m to 15 m log jams cease to be significant, at least in south coastal B. C. Channel slope has also a pronounced influence on the formation of log jams, with flat reaches being generally more debris choked. 3. FIELD METHODS 3.1 Selection of Test Heaches The criteria for selection of test reaches were as follows: (i) The data were to cover the largest .possible range of size (width) and slope. (ii) To test the unsteady flow behaviour, upstream lakes with outlets suitable for minor flow modifications were required. (iii) Since measurements were to be made over as large a discharge range as possible, easy accessibility from Van couver was also an important consideration. Four streams, covering a range of mean width from 0.b9 m to 14.0 . m were finally selected and 2 to 5. test reaches were established on each stream, covering a fairly wide range in slope. (See Table 2 for a list of reaches.) None of the 4 streams had previous discharge records. The smallest stream (Brockton Creek) has no official name and does not appear on any map. Due to its location at tree line it is not affected by forest debris (Photograph 1). At flows In the order of the mean annual flow, some water spills out of the stream channel proper on the upstream test reach and follows the stream in some parallel minor depressions. Placid Creek in the UBC Research Forest flows through an area covered by dense second growth forest approximately TABLE II TEST REACHES BELOW LAKES Creek Location (mid-reach) Reach (going down stream) Length Drop Slope No.of (m) (m) sin Q Steps #3 in xlO' Width WD Survey (m) Coeff.of Drain- Estimated Variation age Mean Annual for WT D Area Peak (km2) (m3s_1~) Brockton Creek M.t. Seymour Br 1-2 119. 0 ,.8.„8 74 36 0.89 ' .499 0.0655 0.20 Park, Elev. 4,000' Lo:122Q56' Br 2-3 80. 5 28.1 349 27 0.99 . 611 0.0880 0.20 Lat:49°23' Placid UBC Research Pl 1-2 960 79. 7 83 64 2.75 .387 0.614 1.5 Creek Forest, Elev. 1,400' Pl 2-3 610 21.6 35.5 41 3.16 ,.373 1.17 2.5 Lo:122°34' Lat:49°l8.5' Pl 3-4 1844 62.4 33.9 122 7.02 . 400 2.60 5.0 Blaney UBC Research Bl 1-3 685 31.9 46.6 46 12.76 .435 7.43 12.0 Creek Forest > Elev.950' Bl 3-5 335 17.5 39. ox' 23 11.06 .414 7.70 12.0 Lo:122034.5' Lat:49°17T Bl 5-4 930 85.3 94.7 62 12.92 .292 7.94 13.0 Phyllis Nr.Britannia Ph 1-2 770 23.7 30.5 52 11.48 .314 8.69 15.0 Creek Beach, B.C. Elev.1,400' Ph 2-3 716 34.9 48.7 48 12.57 .216 10.41 17.0 Lo:123°H' Lat:49°34' Ph 3-4 617 39.5 64 42 12.64 .190 10.99 17.0 Ph 4-6 305 30. 2 99 21 12. 28 .226 11. 34 18.0 Ph Lo 140. 5 30.8 219 10 14.04 .167 11.81 19.0 iThis reach has a sudden steep drop In the last 40 m. The slope of the long flatter IY) part is shown here 25 20 years old and Its. flow regime is affected by. frequent log jams (Photograph 2). The lowest Placid reach flows through two bulldozed fire pools, and the slope is irregular. Blaney Creek flows through an area that was burnt over approximately 80 years ago. Its flow regime is affected by log jams at approximately 150 m intervals (Photographs 3 and 4). The valley of Phyllis Creek was logged 20 to 30 years ago but the creek seems to be of sufficient size to have cleared itself of most debris. Only the flattest top reach has 2 log jams (Photographs 5 and 6).1 In addition to these 13 test reaches, which were, except for two, tested for steady and unsteady flow, Day (1969) tested another 12 reaches for steady conditions only. As he did not require upstream lakes, the choice was much wider and the reaches are in general more satisfactory for the purpose of steady flow tests. Some of Day's streams have discharge records and the test reaches are usually more uni form in slope. Table 3 lists the 12 reaches tested by Day. The test reach length varies from 80.5 m to 1844.0 m and the length to width ratios vary from 10 and 349, but most reaches have ratios between 30 and 80 . The lowest reach, on Phyllis. Creek was. tested by Mr. T.Day. For convenience,, the. results are presented here, together with all other Phyllis reaches. TABLE III ADDITIONAL TEST REACHES (from Day, 1969) Creek Location Reach Length Drop Slope No. of (m) (m) sin Q steps in xl03 survey Width WD (m.) Coeff. of Drainage Variation Area for W D (km2) Fury Creek Sless e Creek Juniper Creek Ewart Creek Upper Mid Low Near Britannia Beach , B.C. Lo:123°12» Lat:49°35' South of Chllliwack,- B.C. Lo:121°38' Lat:49°01' Lo:121038' Lat: 49o,01' Lo:121°39' Lat:49°02' South of Keremeos B'. C . Lo:120O02' Lat:49°06' South of Upper Keremeos ,• B.. C . Lo:120O02' Lat:49°06' Lo:120°02' Low Lat.49°08' 229 29.5 129:- 16 584 20.0 34.3 21 541 23.5 43.3 18 1402 47.0 33.5 47 610 81.5 134 4l 579 20.2 34.8 19 1125 45.9 40.8 36 20.2 27.0 22.8 19-5 6.03 16.36 14. 2 .304 .222 • 239 .285 .407 . 267 .302 39.0 105.1 110. 4 126.0 21. 8 80. 8 95.8 TABLE III (cont'd.) Creek Location Reach Length Drop Slope No. of Width Coeff.of Drainage (m) (m) sin Q steps in WD Variation Area xl03 survey for W^ (km2) Ashnola South of Upper 490 5-09 10.4 17 21.1 .170 221.5 River Keremeos, B.C. Lo:120°11* Lat:49°09 ' Lo:120?10' Mid 1003 26.0 25.9 33 28.2 .218 408.5 Lat .49010-' Lo:120oi0' Low 747 26.0 35.1 25 22-l .225 409.5 Lat:49°10' ro 500 Chainage (meters) 1000 1500 28 2000 t—-Marion Lake Gauge 1 600 1900 Gauge 2 (meters) 1800 Gauge 3\ Elevation cn O •1700 Gauge 4 X500 1600 Phyllis Creek , Handlevel profile Gauge 6 in E c o a > UJ 1000 2000 3000 4000 5000 6000 7000 Chainage (feet) Chainage (meters) 8000 4020 Z 4000-1 0) CO E c 3960 o o > UJ 3940 3920 3900 0 50 100 1 150 < f-Pool in Gauge 1 ^^^^ (meters Gauge ZV c O \ > \ <l> V ^ Brockton Creek profile 1 ! 1 Gauge 3-4 1 —i 100 200 300 400 Chainage (feet) CHANNEL PROFILES 500 600 -1320 -1310 1300 1290 H280 Fia^4_ 29 With the possible exception of the most upstream reach on Blaney Creek, none of the test reaches have flood plains. 3.2 Survey Measurements of the channel phase on the basis of map and air photo data, it was nevertheless considered necessary to measure length and slope in the field, mainly because the reaches could not be located adequately on maps, but also to compare map profiles with field data (Section 2.5). by chaining and hand-levelling in 50 ft. or 100 ft. steps. The profile points are the deepest parts of the channel where possible, otherwise water surface. The step size is suffic iently long to eliminate most of the characteristic pool-riffle sequence. Aneroid measurements were used as a check against large levelling errors. Figures 3 and 4 show the profiles of the 13 test reaches of this study. Originally it had been planned to measure two hydraulic parameters, width and roughness, but no satisfactory method for measuring roughness could be designed. When it became apparent from considerations of dynamic similitude (Section 5.1) that roughness is a redundant parameter in a hydrological flow model, attempts to measure it were abandoned. The channel width, from high-water mark to high-water mark, was measured at each profile point (50 ft or 100 ft While the objective of the thesis calls for definition Length and slope were usually obtained s imultaneously 30 intervals). As indicated by Photographs 1 to 6, channel width is fairly well defined by a line of permanent vegetation. The high-water mark follows this line closely. It generally consists of an abrupt change from forest floor or meadow to channel bed (exposed gravel). Tables 2 and 3 show the mean values and coefficients of variation for each reach. Most measurements were made with a survey tape, but towards the end of the field work a rangefinder became available, which proved to be very suitable for this type of survey. 3.3 Tracer Methods 3.3.1 Obj ective The main objective of the tracer measurements was to establish the relations between discharge, velocity, and channel area under conditions of uniform flow and covering the largest possible range of discharge. At times tracer methods were also used for simple local discharge measurements, which were needed to define the stage-discharge rating curves as mentioned in Section 3.4.5. 3-3-2 Principles of Discharge and Velocity Measurements with Slug Injection Methods Tracer methods are ideally suited for measuring the channel flow variables which are significant in runoff studies, namely discharge, mean velocity, and channel storage (area). If a slug of tracer of mass M is injected into a stream -of discharge Q and the tracer concentration C (mass per unit 31 volume) is measured at a location sufficiently far downstream to permit the assumption of complete lateral and vertical mixing (see Section 3.3-2), the principle c-f conservation of mass takes the following form (Replogle et al., 1966) t M = j Q C dt ... 3.1 t in which t is the arrival time of the tracer, and t is the s 3 e time at which all the tracer has passed the sampling site. If Q is steady during the interval t - t , Equation 3.1 gives Q = M / ( C.Vdt ... 3.2 t which shows that Q can be measured by injecting a known volume of tracer into a stream and observing or sampling the time-concentration curve at a downstream location. The mean travel time T^ of a tracer cloud between the point of injection and the sampling location is (Thackston et al., 1967) t t r e e Tt = j C t dt / j C dt t t s s 3.3 and this is only identical to the mean travel time of the stream water,.T, if instantaneous vertical and lateral mixing at the point of injection can be assumed with no tracer being dispersed upstream- or if the. tracer is injected above the test reach and T is obtained as the difference between the T^- values for the upstream and downstream end points of the reach. The mean water velocity along.the reach of length 1 Is v m 1 T 3.4 and from continuity, the channel area of the test reach becomes A = Q QT 1 3.5 v m 3-3.3 Vertical and Lateral Dispersion Requirements The mechanics of vertical and lateral dispersion in straight, uniform open channels is well developed (Diachishin, 1963; Fischer, 1966). Criteria for the time or distance which 2 assure adequate mixing have been developed but their appli cation to tumbling flow is not reasonable. Empirical methods, such as sampling across the channel and visual inspection of the dispersion of dyes were therefore used to determine whether the lateral mixing requirements were being met. If possible, the test reaches were located so that they started at severe channel discontinuities (waterfalls, constrictions), which assure fast mixing. On some of the short reaches it was necessary, however, to inject the tracer above the reach and Complete dispersion, is not possible in finite time due to the continuous nature of the process (see Equations 3.7 and 3.13)-. 33 to observe, two time-concentration curves for. one measurement of v . m On several occasions the velocity of long reaches was measured twice at closely similar flows by injecting the tracer at the starting point of a reach and by injecting further upstream. There are no significant discrepancies between these results. 3.3.4 Longitudinal Dispersion Models The tracer techniques outlined in Section 3-3.2 require the evaluation of integrals over observed time-concentration curves. Longitudinal dispersion is primarily' responsible for the shape of these curves. If the observed C (t) - curves cover the interval t - t adequately for numerical Integration, the mechanics of dispersion can be ignored. However, in the course of this study it happened frequently that field observations had to be terminated before C (t) had declined to negligible values and it became neces sary ,therefore ,to develop a dispersion model which would permit extrapolation. Particularly in the equation for T^_ (Equation 3.3), the decline of C at large values of t carries consider able weight. The three main processes causing longitudinal dispersion are: longitudinal turbulence, turbulent mass exchange between stream lines of differing velocities, and storage of tracer in pools and dead: zones. Molecular diffusion is only important 34 at extremely small scales. Taylor (1954) showed that the one-dimensional diffusion equation gives a fairly good representation of longitudinal dispersion in uniform, turbulent pipe flow. Elder (1959) extended the analysis to infinitely wide open channels and Fischer (1966), Church (1967), Thackston and Krenkel (1967), and many others have examined its applicability to natural channels. The one-dimensional diffusion equation is rr + V T— = D ?r ... 3-D 6t ID h x ^2 J in which x is the longitudinal coordinate, and D is the dis-X persion coefficient. For slug injection of a tracer of mass M at t = o, x = o, the solution takes the following form after vertical and lateral mixing are almost complete M (x - v t)2 C(XJT) = AV2?D1 exp ( inrT—} ••• 3-7 x X It shows that the tracer is distributed normally over x, with the centre moving downstream at velocity v and the m variance increasing as 2D t. Most dispersion data are based on observation of C at constant x, x = 1. Under these con ditions, Equation 3.7 is skewed to the right, which agrees with field data. 35 Substituting 3-7 into 3-3 gives . , ;.2D' 1 Y T = - + v -2.. m v : m in which the second term on the right accounts for the fact that initially some tracer may be dispersed upstream-. (Thackston et al., 1967). Although 2D /v2 < < 1/v through-out this study, Equation 3-8 is further proof of the need for fast initial mixing. Equation 3-7 was fitted to several observed time-concentration curves, using the least squares fitting method o proposed by Thackston et al. (1967). Figure 5 shows a typi cal fit. The agreement between Equation 3.7 and the field data is generally close over most of the C (t)--curves but the predicted final decline of C is always much faster than observed declines, indicating that the one-dimensional dif fusion equation (3.6) does not really represent dispersion in natural channels. Hays (1966) developed a new model, which includes dead zone storage effects besides one-dimensional diffusion. It appears to represent the slow decline of C very well, but unfortunately, it is rather difficult to-handle, requiring a Fourier transformation of the field data and subsequent curve fitting in frequency space. 3 This method is based on the IBM Share library prog ramme NLIN2, described by Marquard (1964). 36 60 Time from injection (min) LONGITUDINAL DISPERSION OF SLUG & INJECTED TRACER Fig. 5 37 3-3-5 A Gamma-Distribution Model for the Final Decline of C(t) Since the field data of the present study define the main part of the. G(t)-curves adequately, an attempt was made to develop a simple model for the final decline, considering only tracer storage in pools. It is based on the assumptions: (I) The stream acts as a cascade of"reservoirs with steady flow Q. (ii) All reservoir volumes FL are equal to TR Q, with TR, the filling time, being an arbitrary time constant. (iii) Mixing is instantaneous in each reservoir. (Iv) The dispersion process is initiated by injecting a quantity M of tracer into the first reservoir (RQ) at time t =0. The initial concentration in R is therefore s o Co (t = 0) = T^Q ... 3-9 (v) The travel time between two reservoirs is constant for all water or tracer particles and can therefore be ignored in the following. These assumptions lead to a system of linear, non-homogeneous differential equations of first order for C^(t). The general form is dC. C. , C. —i = 1~1 _1 ^10 dt T T ... j-xu Through a reference in Water Resources Research, Vol. 5, No. 4, p. 927, August 1969, a paper by MacMullin and Weber (Trans. Am. Inst. Chem. Engrs., Vol. 31, pp. 409-^58, 1935) has recently come to the writer's attention. It contains an identical-derivation,based on considering the outflow from a series of well-mixed vessels. 38 The solutions are T Reservoir R C (t) = =—•^ e o o TRQ t • T Reservoir R, C-. (t) = —^— t e TR Q t • T Reservoir R. C. (t) = M t 1 e R ... 3.11 1 1 i!TR1+1Q The successive peaks occur at t = iT_ and the mean tracer R travel time is Tfc = (i + 1) TR. Equation 3.11 can be compared to the gamma distribution fx <*> =rf?T *~Kt ... 3.12 r > o K > 0 x > 0 Keeping in mind that f(r + 1) = r! for r = 1,2,3 • . . Equation 3.11 can be rewritten as t Cj(t)Q _ i i " TR N_ M ." Tc r (i+l) T„ T e n n which shows that the general solution is proportional to a f-distribution with parameters (I + 1), (1/TR), and t. Initially the question of whether this storage model could represent the total time-concentration curyes was explored by fitting it to several sets of field data; (Figure 6),. In comparing It with the diffusion model one may 39 30 Storage model (Least squares fit to all points) Test. Ph R 10, G2 - 3X-4-6 140 160 Time from injection (min.) THE STORAGE MODEL APPROXIMATION TO LONGITUDINAL DISPERSION Fig. 6 40 say that: (i) Both models, can be. fitted almost equally well to the observed time-concentration data, but both fail to rep resent the slow decline of C at large values of t. (ii) The storage model can account for the finite time lag between tracer injection and the first arrival at the sampling location. In the diffusion model the tracer covers the whole reach immediately. (iii) The diffusion model gives better parameter stability. The number of reservoirs in the storage model does not necessarily increase with channel length, neither does TR, the reservoir filling time, decrease with flow Q. (iv) The third moment ratio of the P-distribution-is 2/ V~r" , indicating that with increasing channel length the skewness of C (t)=curves should decrease, which does not appear to be consistent with field results. In spite of these deficiencies, the storage model can represent the final decline of C (t), if the fit over the main part of the C (t)-curves is ignored, which amounts to splitting the dispersion phase into two parallel phases; a dispersion phase, responsible for moving most of the tracer and a storage phase, which dominates the final decline. Equation 3.11 is essentially of the form Y _Y. t C (t) = Y1 t ^ e J in which the Y. are constants. 41 Taking logarithms on both sides gives log1Q C = log1QY1 + Y2 log1Q t - Y3t which shows that (3-11) can be tested by plotting the field data in the form (log^C - Y^og^Qt) vs. (t), for selected values of • Equation 3.11 is a good fit if the data points fall on a straight line. With very few exceptions, which are attributable to the difficulties of determining low C values, the field data plot as shown by the two examples of Figure 7- Some similar computer-made plots are shown in the Appendix under Subroutine "TAILEX". The final decline of C (t) appears to be similar to a r -distribution with 1< r ^ 3• A good fit was generally achieved by setting r = 2 (i = 1), but r = 1 (negative exponential decline) could have been selected with almost equal justification. Figure 7A shows a set of data which covers almost the complete decline of C (t) and Figure 7B illustrates the graphical fitting of a P -distribution exten sion to an incomplete set of C (t)—data. On Figure 5 the resulting curve is plotted in C - t coordinates. Similar computer-made plots are shown in the Appendix under Subrout ine "PL0TGA". The above storage model represents only a small first step towards an understanding of longitudinal dispersion in 4 The parameters Y,, Y?, and Y_ appear as A, B, and C in the Appendix. -> 42 \0] Fig. 7A Test covering C(t) decline almost completely 5^ Rg. 7B Incomplete test with r~ extension Test BR R2, GIUP-|~2X GRAPHICAL FITTING OF STORAGE MODEL Fig.7 43 tumbling, flow, channels. The large numher of C .(t)-curves measured In the course of this .study- should permit a more com plete investigation, concentrating on the predictive qualities of the storage model, but this- is not part of the present obj ective. 3.3.6 Equipment and Procedure for Slug Injection Measurements All the C (t) curves included in this study were mea sured with either one of the following methods: (i) the relative salt dilution method, based on electrical detection of a Na Cl-solution; (ii) the dye dilution method, based on fluorometric detection of a fluorescent tracer (Rhodamine WT). A detailed description of both methods, based partly on the experience gained in the course of this study, is In press. • (Church and Kellerhals, 1969). Only a brief summary • 5 will be given here. The relative salt dilution method (Aastad and Sognen, 1954; 0strem, 1964) uses the linear relation between concen tration of salt and conductivity. A known volume (generally 5 Initially a few discharges were measured by injecting the tracer (Sodium Dichromate or Rhodamine WT) at. a constant rate and then determining the dilution ratio between injected solution and stream water. This "constant rate infection method" is described in Church and Kellerhals .(1969,) . The equipment is, shown on Photograph. 16. The method, was- not suitable for this study, because it cannot give velocity and offers no advantages over slug injection methods for simple discharge measurements. 44 10 to 100. liters) of a salt solution, whose concentration need not be known, is slug-injected into the stream and the passage of the salt wave is observed downs-tream with a portable con ductivity meter and electrode. A small sample of the initial solution is retained for the construction of a conductivity-concentration rating curve by successive dilution. The main advantages of the method are the possibilities of computing discharge in the field and avoidance of laboratory work. A disadvantage is the relatively bulky field equipment, con sisting of 2 vats with needle gauges, pails , approximately 3 -1 1 kg of NaCl per m s to be measured, pipets, 2 volumetric flasks, conductivity meter, electrode, and stop watch. Photographs 7, 8, and 9 show the main items. The dye dilution method used here is particularly suitable under difficult field conditions as during severe rainstorms. Accurately measured amounts of the liquid tracer (Rhodamine WT-dye) are injected from the pipet directly into the stream and the C (t)-curve is defined by taking 10 - 20 small water samples at the downstream location for later analysis on a fluorometer. Photograph 10 shows the equipment. Not discussed in Church & Kellerhals (1969) is the recording conductivity bridge (Photograph 11) built to avoid the tedium of measuring conductivity-time curves at extremely low flows. A brief description of this bridge may be in order as no similar instrument appears to be available commercially. 6 V Power pack 45 -Interval timer •I'h-T 'IH 6 V. Batteries Recorder (O.V-I.V) Galvo 100 K. 6 V. D.C. Inverter -115 VAC Isolation or stepdown transformers 40V £-V II5V, Variable transformer CIRCUIT DIAGRAM OF RECORDING CONDUCTIVITY BRIDGE Fig.8 46 The principle of operation is. to record the off-balance potential V of an AC-bridge on a 0 - 1 volt Rustrak recorder with high input impedance, Rn = 10Q K. The circuit diagram of the bridge is shown on Figure 8. The response V is ERQ. (RG - .1) . . V = (RG+1) (RC+2RQ)+2R 3'14 in which E is the exciting voltage, which is adjustable between zero and 40 volts, R Is the adjustable resistance which can take any value between 0 K and 85 K, Rc is fixed at 5.11 K and G is the conductivity measured by the probe. Computed and measured'responses are plotted on Figure 9. The difference between the two is caused by the significant but neglected threshold voltage and voltage loss of the inverter. With careful selection of exciting voltage, amount of salt to be injected, and initial background adjustments, the bridge can be operated in the linear range between responses of 0.2 and 0.9 volts. To conserve the exciting voltage E during long periods of continuous operation, an electronic interval timer was built^ and inserted between the bridge and its power supply. The recorder runs off an independent power source. The design of. the timer was developed recently by S. Outcalt of the Dept.. of Geography, UBC, and W. Schmitt, Dept. of Civil Engineering, UBC. 47 Recorded potential (Volts) RESPONSE OF RECORDING CONDUCTIVITY BRIDGE Fig. 9 48 3.3-7 ' Tracer Losses The tracer methods for discharge measurements assume conservation of tracer mass. Prom Equation 3.2 one can see that tracer losses due to absorption.or chemical reactions result in overestimated discharge,while tracer loss due to seepage of water out of the channel results in underestimates. The travel time, Equation 3.33 however, is independent of the tracer mass as long as the losses do not affect the shape of the C (t)-curve. To permit correction for losses, most tests with long mean residence times T were only interpreted for T^ according to Equation 3.3, discharge being measured with a separate test over the shortest permissible reach. Almost all Rhodamine WT test results show a certain amount of tracer loss due to absorption or chemical dis integration of the tracer. The loss rate L, in percent per minute can be estimated from two simultaneous tests, one over a long reach and the other over a short reach, both ending at the same sampling position. Assuming that the measured discharge and the true discharge Q are related as " - .... 3-15 100 t leads to " 100(1— QS/Q]L) L = ... 3.16 m " ^ ° m t,l 'Q1: t,s in which the subscripts 1 and s refer to the long and the short reach respectively, and Q and are computed according 49 to Equations 3.2 and 3.3 respectively. L is commonly in the order of 0.1 to 0.3 percent per minute. No reason could be found for the observed variation in L. Most discharges based on Rhodamine WT tests were corrected according to Equation 3.15, with L-values estimated from double tests. No consistent evidence of tracer loss appears in the salt dilution data, but long travel times definitely tend to result in unreliable discharges. The cause is probably a combination of tracer losses and changes in the background conductivity of the stream. The conductivity changes observed during the passage of a salt wave can only be converted to salt concentration if the background conductivity remains constant or changes in a predictable manner, neither of which was true in very extended tests. 3.4 Surge Tests 3.4.1 Ob,]* ective If a channel routing method is capable of reproducing the discharge Q (t) at the downstream end of a test reach, resulting from small, step-like increases or decreases in Q (t) at the upstream end, and if this holds over the complete range of Q, then one may assume that the method should also be adequate to route complex storm hydrographs through the channel reach, since they can be decompos.ed into a sequence of small steps. This general statement is absolutely correct if the channel response is linear, but. for all practical purposes it 50 will also hold as long aa the non-linearities, are not strong enough to lead to severe, di's-continuities sucfi as bores. The main objective of. the surge tests was. therefore to impose small,steplike discharge modifications, A Q, at the upstream end of test reaches and to observe the propagation of these positive and negative surges. The tests were to cover as large a range of discharge as possible. A few tests on the effect of the relative size of A Q were also run by imposing small and large A Q's at constant Q. 3.4.2 Discharge Modifications Different methods for modifying Q were used at each of the four lake outlets on the test streams of this study. A small dam was built at the outlet of Blaney Lake. Discharge could be increased or decreased by adding or remov ing flashboards (Photograph 12). " An old timber-crib dam at the outlet of Marion Lake gave excellent control over the Phyllis Creek reaches. Photograph 13 shows the dam, with two flashboard-like additions in place. The outlet of Placid Lake was so marshy that no control structure could be built. Surges were produced by pumping water across the swamp into the creek (Photograph 14). The pool above the Brockton reaches was so small that it was difficult to maintain steady dis charges different from the pool inflow. The initial surge was produced by adding or removing a few rocks at the pool outlet. A gravity-operated inverted siphon was then used to' maintain more or less steady flow for 5 to 15 mi.nutesv (Photograph 15). 51 3.4.3 ' 'Stage Measuring Equipment Discharge changes at the end points of the. test reaches were monitored by observing or recording stage, changes and establishing stage discharge rating curves for conversion to discharge. Even on the steepest and most turbulent reaches it was generally possible to find stable pools, either on bed rock or between large boulders (Photograph 18). The turbulent level fluctuations and air entrainment made direct level measurements impossible, but the plexiglass stilling wells illustrated on Figure 10 and Photographs 16 and 18 permitted 7 the reading of water levels to + 0.001 ft. This gave satis factory resolution,since most surge tests caused level changes in the order of 0.02 to 0.05 ft. To gain some information about the discharge range of the test reaches, automatic stage recorders were installed on all but one of the test creeks. The instruments were Stevens A-35 recorders, with the fastest available clock gearing (9-6 in/24 hours), and a 12:10 level scale. These large scales made it possible to rely on the recorders for the surge tests, thereby saving one field assistant. With careful pro cedures the time scale could be interpreted to' + 1/2 min. The recorder installations were somewhat, unconventional, due to the inverted siphon connecting the stream to the stilling 'All thereof due stage measurements are in feet to a lack of readily available and decimals metric equipment. 52 Tube cover Plexiglass tube, 2"O.D., 1/8" wall Wooden post Strip of rod cloth glued to plexiglass tube "/^-Pocket mirror in position j/ for water level reading 1/4 Hose connector Nail tied into nearby BM SCHEMATIC SECTION OF MANUAL GAUGE (from Church and Kellerhals, 1969) Fig. 10 -n u5 Air bleed plug,(rubber stop peri-Glass or plastic container 1/2 to 2 gal. Possible locations for taps (not essential) Automatic stage recorder Recorder stand and stilling well (plywood) •%%S£HEMAft IC VIEW .,0F; STAGE "RECORJDER" INSTALL ATlbN oo '-FOR*-MOUNTAIN STREAMS" (from Church and Kellerhals^ 1969) 54 well. Figure 11 and Photographs 17 and 18 Illustrate the method. The. experience gained w.ith all the above stage measuring equipment (.tubes and recorders) is discussed at length in Church & Kellerhals (1969). 3.4.4 Stilling Well Response The hose connections between plexiglass tube wells or recorder wells and the streams cause considerable damping (in the sense that the gauge level cannot follow high fre quency fluctuations of the stream level). This damping is essential for accurate level readings but the question arises as to how much it affects the surge test data. Under normal circumstances the flow in the connecting g hoses is laminar and the inertia of the flowing water is negligible. The well response Is then governed by the follow ing equation: dh h = - b || ... 3.17 in which h is the elevation difference between the stream and the stilling well and b is a time 'constant. If the stream level is constant and the well level is off by h at time t , the solution is ^o~^ h = hQe . ... 3-18 Assuming a steady rise of 0.02 ft/min. in the well, which is approximately the maximum observed during surge tests, gives Reynolds Numbers of 24. for. the connecting hose of plexi glass tubes and 500 for recorder well connections. 31 ro CD CB . "*:.v,:... • • • GAUGE^:^RE^feo>JSE 'CURVES ^ ';rA:.' 56 which indicates that a plot of (t - t ). vs. (h/h,) should, give a straight line of slope (-b) on semi-log paper. Some devia tions should be expected since Equation 3.17 only, considers pipe friction and there are certain other losses present. Figure 12 shows some typical gauge response curves. Instead of plotting a response curve, one can measure the time,At, it takes the stilling well to drop from an arbitrary h^ to an arbitrary h^ and compute b as follows: b = logth*/h2) 3,19 All gauge and recorder setups were tested in this manner and, where necessary, the surge test records were cor rected for lag according to Equation 3-17. About one third of the gauges needed lag corrections. 3.4.5 Stage-Discharge Rating Curves At the time when most of the test reaches of this study were installed, the destructive force of the streams under severe flood conditions was not properly appreciated. Many of the gauges were located at pools that proved subsequently to be unstable. In hindsight, it appears that there was no lack of stable pools; only a lack of experience in locating Q them. As the gauges could not be moved without altering the 9 Pools formed by large, preferably angular rocks, arranged in such a way that they do not easily catch drift wood, are best. Locations below well established log jams are excellent, as the jams tend to catch most debris and coarse bed load. 57 TWO TYPICAL STAGE - DISCHARGE RATING CURVES 58 test reach length, some parts of the stage-discharge curves had to be re-deflned two or three times, to permit conversion of the surge data from stage to discharge. The stage recorders were installed at the most stable gauging sites. Figure 13 shows two rating curves, one for the stable pool of Blaney Gauge 5 (Photograph 18) and the other for the more troublesome Blaney Gauge 1. 59 4. FIELD RESULTS 4 .1 Survey Results Tables 2 and 3 and Figures 3 and 4 summarize the survey results which consist of profiles of the test reaches and width measurements (Section 3-2)' The width data were pro cessed with a set of programs developed by Day (1969). Tables 2 and 3 are summaries of the program output. There is undoubtedly a considerable operator effect in the width data, since the high water mark was often, particularly on the bushier streams, rather ill defined. The large number of width measurements tends to. compensate for this, but dis crepancies of 10% to 15% could, still occur between different field parties. On most test reaches there is no significant difference between actual length and length in plan (map length), but In the case of the few very steep reaches it is worth noting that the surveyed length is the actual length on the slope. The relative accuracy of the chaining and hand-levelling is estimated at + 3 percent. Slope is defined as drop divided by length, sin Q, if Q is the slope angle.. The drainage areas were measured on the best available maps and refer to the middle point of a test reach. The Brockton Creek basin .does not appear on any map, as the 60 1:50,000, coverage, of. this area happens, to be exceptionally poor. The drainage area was. therefore measured off air photos. 4.2 Velocity and Discharge Measurements 4.2.1 Conversion of Field Data to Time-Concentration Curves The field data resulting from a relative salt dilution test consist of . the following: (i) location and time of injection, (ii) volume of brine injected, (iii) sampling location, list of times and corres ponding conductivity readings, (iv) rating curve, covering range of observed con ductivity readings (it consists of dilution rates and corres ponding conductivity readings), (v) water temperature in stream and in rating tank. The computational procedure for converting the time-conductivity data to time-concentration is described in detail in Church and Kellerhals (1969). The procedure proposed in earlier publications on this method (0strem, 1964) should not be used, as the correction for different background readings in the stream and in the rating tank is in error. A Fortran IVG program "NACL" was developed for this conversion.' it prints the time-conductivity and time-concentration data and plots the rating curve. The program is listed in the Appendix, together with operating instructions and sample, output. 61 The field data resulting from a Rhodamine WT test are: (i) location and time of injection, (ii) volume of injected dye, (iii) sampling location, list of sampling times,and 10 to 30 samples. The samples are subsequently analysed on a Tluorometer, and the instrument reading is converted to concentration with a rating curve based on standard dilutions of the tracer. Wilson (1968) describes the laboratory procedures in. great detail. The necessary computations were done manually, but the final time-concentration data were put on cards for pro cessing by an input program "DQV" analogous to "NACL",which is also listed in the Appendix. 4.2.2 Numerical Integration The Equations 3.2, for discharge Q, and 3.3, for tracer travel time T, are evaluated by a Fortran IV.G subroutine "QVEL" (see Appendix). The integral over C(t) and the first moment of C(t) are computed twice, first on the basis of the trapezoidal rule and then with a second order method similar to Simpson's rule but capable of handling unevenly spaced points. The procedure is discussed briefly below, because the formulas do not appear to be readily available in texts on numerical analysis. The "Turner Model 110".fluorometer. of the B.C. Research Council was used here. 62 Assume that C(t) is. defined at t =. t , t^, and t2 as shown on Figure 14. FIGURE 14. 0 "I 2 DEFINITION SKETCH FOR NUMERICAL INTEGRATION The function C(t) can be approximated by a second order polynomial P (t) of the form P (t) = CQ 1q (t) + c1 i1 (t) + c2 i2 (t) in which the 1^ (t) are 'second order polynomials in terms of t^jtpand t2 (Herrio, 1963), e.g. yt) t - tt1 - tt2 •+-- t1t2 . <v- V (to - V The integral C(t)dt can be estimated as t„ t„ t„ J P(t)dt = Co J 1q (t.).dt + C1 / 1 (-t.).dt + C / l2(t)dt Without loss of generality, one. can substitute t = 0 o tx = Atx t2 = At2 The integrals over 1 (t) are then: At f 2 ! At?2At1 At 0J v*'* - <-^^ i-) At2 2 A t0 (t)dt = 2 6(A tx2 -A tx A.t2) A t, At 3 At0 At. 1? (t)dt = -r-' 1 = ( i i 1 ) A\t2 -A t1At2 3 At which reduces to Simpson's Rule if.At-, = 1 2 An approximation to the first moment with respect to t = o is At2 At2 At2 At2 Jp(t)'tdt = cQ fio(t)tdt+c1 Ji1(t)tdt+c2 Ji2(t)tdt = (<Ti+r2+r3) -T- - rri(AvAV At 3 At 3 + r2At2+r3Ati] -f- +riAti-^ 64 in which C. r _ 9_ & 1 At-jAt r - 1 2 2 At^ -At-^At^ C2 ^3 = 3 At2-At1At2 In those cases where the concentration decline has not been defined adequately in the field, a subroutine "TAILEX" (see Appendix) is first called from "NACL" or DQV". It plots (log CY.'2 log t) vs. (t) to permit the fitting of a f-distribution extension, as discussed in Section 3.3-5 and shown on Figure 7. The parameters of the extrapolation are read off the "TAILEX" plot and punched onto the control card of the CXt.)-.;.datac.che.Gk. If the subroutine "QVEL" is then called, it will use the /~"-distribution to extend the integral over C(t) to infinite time. Finally, the C(t)-curves, with or without r -extension, can be plotted by calling subroutine "PL0TGA" from the input programs "DQV" or "NACL". "PL0TGA" is also listed In the Appendix, together with operating instructions and sample plots 4.2.3 Results The tracer data consist of 111 slug injections, for which 146 time-concentration curves were determined. In other words, approximately two thirds of. the slug injection tests contain one downstream sample only; the other third consists 65 mainly of runs "with two samples. Only three runs extend over three sampling locations and none over four. In addition, there are eight discharge measurements with the constant rate injection method. Presentation of the complete data, which consist of approximately 7000 time and concentration values did not appear justified. Tables 4A to 4D show the results in summarized form. There is one table per stream, with the test runs arranged in historical sequence. The identification code is explained in a footnote. With the exception of the simple discharge measurements, the same data appear again in the Appendix, as printout of the program "L0GRE " (see Section 5.2.2). The arrangement there is by test reach. For cross reference from Table 4 to the Appendix it is best to use the test number. Note that the discharges are slightly different because Table 4 refers to the tracer data and therefore gives the discharge at the sampling point, whereas the "L0GRE" printout refers to test reaches, and the discharge is the estimated mean between inflow and outflow. 4.2.4 Accuracy The relative accuracy, or internal consistency of the data appears to be satisfactory. As can be seen from Table 4, tests on the same day give similar flows at all stations of a stream (after correction for tracer loss in long dye-dilution runs) and tests made at different times but with apparently similar flows, show a consistent time distribution of the TABLE IVA SUMMARY OF TRACER MEASUREMENTS: BROCKTON CREEK Test Date No. of T " • Peak Mean Discharge Identification1 Pointsr. J\ (min) Lag /m3a-l\ forf Method (mm) extension BrRl,2UP-2X-3 Aug.15,67 17 0 . 4 1. 5 2. 61 0 .00564 RhWT BrRl,2UP-2-3X Aug. 15,6.7 28 14. 0 33- 0 37. 6,'- 0 .00770 + RhWT BrR2,lUP-IX-2 Aug.15,67 16 0. 5 1. 5 2. 28 ' "0 .00856 .' RhWT BrR2,lUP-l-2X Aug.15,67 16 17. 5 27. 0 32. 4\ 0 .0100 + RhWT BrR3 , 2UP-2X Aug.17,67 0 .00336 RhWT r T BrR4,lUP-lX-2-3 Aug.17,67 19 1. 0 2. 2 3. 43 0 . 00504 0. j-. RhWT BrR4,1UP-1-2X-3 Aug.17,67 18 25. 0 '39. 5 47. 9 - 0 .00617 + RhWT BrR4 ,1UP-1-2-3X Aug.17,67 13 52. 0 74. 0 82. 1 0 .00738 RhWT BrR5,2UP-2X Aug.17,67 0 . 0108 RhWT P T BrR6 ,1UP-1X Aug.17,67 15 1. 0 2. 2 2. 99 0 .00706 0 . I . RhWT BrR7 ,1UP-IX Sept.15,67 15 4. 2 9. 0 14. 50 0 ,00067 RhWT BrR8,2-2DOX Sept.15,67 13 1. 0 5. 0 8. 71 0 .000703 RhWT BrR9,lUP-lD0X Sept.24,67 32 4. 0 19. 0 39. 9 0 .000154 RhWT 1 The test identification code is as follows: the first two letters identify the stream (Br=Brockton, Pl=Placid, Bl=Blaney, Ph=Phyllis), then comes the test run number (R1,R2,... in a more or less historical sequence, next is the location (gauge number) of injections; (UP or DO meaning shortly upstream of ... or shortly downstream from ...), finally the sample locations (gauge numbers), with an X indicating the particular time concentration curve one is dealing with. Examples: PhR5,2UP-2X is a simple discharge determination at Gauge #2 on Phyllis Creek, with injection shortly above Gauge #2 and sampling at the gauge. B1R10 1-3-5X would indicate that the test run #10 on Blaney Creek covers two reaches '1-3, and 3-5'). 2 C.I. means "constant rate injection test." TABLE IVA (Cont'd.) SUMMARY OF TRACER MEASUREMENTS: BROCKTON CREEK Test No. of "T" "'• Peak Mean Discharge : + Identification! Date Points',3. y Lag Lag (m3s-1) for ^ Method' ^Tnln^ (min) (min) extension BrR10,l-2X Aug.18,68 18 70. 117, 160. 0 .00124 + RhWT BrRll,2-3X. Aug.18,6 8 16 53. 97. 120 . 0 .00126 + RhWT BrR12,lUP.-IDOX Aug.18,68 9 2. 5 5. 7 8. 50 0 .00122 RhWT BrR13,2UP-2Dp"X Aug.18,68 9 0 .00114 RhWT BrRl4,1-IDOX Aug.26,68 9 0 .00453 RhWT BrR15,2U,P-2X , Aug.26,68 9 . 0 .00555 RhWT BrRl6,l-2X Sept.14,68 13 4. 9 8. 1 9. 08 0 .0648 RhWT BrR17 ,2-3X Sept.14,68 12 3. 0 7. 2 8. 50 0 .0556 + RhWT BrRl8,l-2X-3 Sept.14,68 19. 7. 2 11. 6 12. 6 0 .0550 RhWT BrRl8,l-2-3X Sept.14,68 13 14. 5 20. 0 21. 2 9 + RhWT BrR193l-2X Sept.14,68 10 2. 5 5. 0 5. 40 0 .175 RhWT BrR20,2-3X Sept.22,68 13. 2. 2 4. 7 5. 39 0 .0889 RhWT BrR21,l-2X Sept.22,68 11 4. 2 7. 3 8. 33 0 . 102 + RhWT BrR22,2-3X Sept.22,68 12 1. 8 4. 0 4. 53 0 .109 RhWT BrR23,l-2X Sept.22,68 11 3. 2 5. 5 6. 97 0 . 152 + RhWT 0s —3 TABLE IVB SUMMARY OF TRACER MEASUREMENTS: PLACID CREEK Test No. of m;Ts Peak Mean Discharge + Identification Date Points (min) Lag Lag , 3 -1\ for/-' Method < (min) (min) extension P1R1,2UP-2X . June 3,68 18 1. 8-- 3. 5 4. 57 0. 0687 RhWT P1R2,1-2X June 3,68 14 122. 164. 210. 0. 064 + RhWT P1R4,3UP-3X June 9,68 17 3. 2 6. 4 7- 8 0. 0183 + .RhWT P1R532-3X June 9,68 17 160. 236. 351. 0. 018 + RhWT P1R6,4-4DOX June 9,68 46 4. 5 9. 3 14. 8 0. 0408 NaCl P1R7,2UP-2X June 9,68 16 5. 8 11. 0 13. 8 0. 0117 + RhWT P1R9,2UP-2X June 18,68 18 1. 9 4. 8 8. 29 0. 00358 RhWT P1R10,2-3X June : 2Yy68 17 62. 0 94. 121. 0. 085 + RhWT P1R11,3-3DOX June 27,68 18 11. 0 20 . 26. 9 0. 144^ RhWT P1R11,3-3DO-4X June 27,68 6 211. 261. 9 RhWT P1R12,4UP-4X June -27/68 27 119. 170. 206. 0. 185 RhWT P1R13,1-2X June 28,68 17 165. 208. 281. 0. 045 + RhWT PlRl4,2UP-2X June 28,67 18 24. 0 28. 3 29. 7 0. 0418 RhWT PlRl6,4-4DOZ Aug. 20,68 40 8. 8 11. 6 2 0. 035 NaCl P1R17,3-4X Aug. 20-1,68 674. 100. 0 1200. 0. 020 NaCl Below confluence of Gauge 3. OA 00 TABLE IVB (Cont'd.) SUMMARY OP TRACER MEASUREMENTS: PLACID CREEK Test No. of T.Ts Peak Mean Discharge + Identification Date Points (min) Lag Lag , 3 -1\ for f1 Method (min) (min) extensions P1R17A,2UP-2X Aug.28,68 9 8. 11. 5 12. 5 0. 0824 RhWT PlRl8,2-3X-4 Aug.28,68 60 48. 71. 83. 3 0. 157 NaCl PlRl8,2-3-4X Aug.28,68 68 236. 294. 339. 0. 245 + •NaCl P1R19,3-4X Aug.28,68 36 162. 199. 239. 0 . 268 + NaCl P1R20,4-4D0 Aug.28,68 13 1. 15 3. 8 4. 82 0 . 249 NaCl P1R21,3-4X Aug.30,68 39 302. 386. 482. 0 . 097 RhWT P1R22,4UP-D0X Sept.21,68 12 8. 5 14. 5 19. 2 0. 122 + RhWT P1R23,2UP-2X Sept.21,68 12 4. 0 8. 0 9. 4 0. 0363 RhWT PlR24,2-3X-4 Oct.12,68 73 47. 5 74. 92. 1 0. 142 + NaCl PlR24,2-3-4X Oct.12,68 52 205. 274. 346. 0. 368 + NaCl P1R2531-2X Oct.12,68 92 115. 159. 181. 0. 0822 + NaCl P1R26,2UP-2X Oct.12,68 12 4. 2 7. 3 8. 68 0. 0822 RhWT P1R27,2-3X-4 Oct.22,68 80 36. 1 54. 8 62. 1 0. 212 NaCl PlR27,2-3-4X Oct.22,68 38 157. 224. 271. 0. 560 + NaCl P1R28,1-2X Oct.22,68 75 94. 130. 144. 0. 122 + NaCl P1R28A,2UP-2X Oct.22,68 11 3. 2 5. 7 7. 02 0. 122 RhWT P1R29,4UP-4DOX Oct.22,68 12 3. 0 6. 0 7. 45 0. 474 RhWT TABLE IVC SUMMARY OP TRACER MEASUREMENTS: BLANEY CREEK Test Identification Date No. of ~Jis-' Peak Points (min) Mean Lag & (min) (min) Discharge + for r extension (mV1) Method B1R1,1-2X Mayl5,67 0 .190 B1R1,1-3X May 15,67 0 . 260 B1R2,1-2X May 18,67 . 0 .159 B1R3,4UP-4DOX May 19,67 0 . 123 B1R4,4UP-4DOX May 19,67 0 .168 B1R5,1-2X June 9,67 0 .064 BIR6,4UP-4DOX June 9,67 10 4. 2 9. 0 10. 5 0 .054 B1R7,1-2X June 9,67 11 17. 5 27. 0 36. 0 .060 B1R7A,5UP-5X Sept.30,67 18 2. 8 5. 5 12. 2 0 .0317 B1R8,3-5X Oct.6,67 14 8. 5 16. 2 21. 2 0 .873 BIR9 ,4UP-4DO:X Nov.19,67 9 0. 9 1. 82 2. 19 0 .595 B1R10,3-5X-4 Nov.19,67 22 12. 0 20. 5 26. 3 0 .538 B1R10,3-5-4X Nov. 19,6.7 14 58. 77. 5 89. 4 0 .588 BlRll,l-3X-5 Nov.19,67 15 30. 44. 5 54. 7 0 .517 BlRll,l-3-5X Nov.19,67 11 51. 67. 5 77. 2 0 . 520 SoD •CI. SoD C.I.. SoD .C.I. SoD ,.C_,I. SoD C.I. RhWT •C.I. RhWT + RhWT RhWT + RhWT RhWT RhWT + RhWT + RhWT + RhWT TABLE IVC (Cont'd.) SUMMARY OF TRACER MEASUREMENTS: BLANEY CREEK Test No. of j:T:s: Peak Mean Discharge + Identification Date Points (min) Lag Lag , 3 -1\ for F Method (min) (min) extension BlR12,l-3X-5-4 Dec.26,67 15 17. 23. 5 27. 5 1. 76 + RhWT BlR12,l-3-5X-4 Dec.26,67 19 28. 37. 5 43- 2 1. 86 + RhWT BlR12,l-3-5-4X Dec.26,67 9 53. 67. 7 74. 0 2. 06 + RhWT B1R13,3-5X-4 Jan.20,68 18 3. 3 5. 6 7. 1 10. 4 + RhWT BlR13,3-5-4X Jan.20,68 12 14. 5 19. 5 21. 6 12. 0 RhWT B1R14,3-5X Jan.20,68 14 3. 2 • 5. 5. 6. 24 11. 7 RhWT BlR15,l-3X-5 Jan.20,68 13 7. 5 10. 6 12. 10 11. 8 RhWT BlR15,l-3-5X Jan.20,68 18 11. 5 16. 5 18. 6 11. 8 + RhWT BlRl6,l-3X-5 Feb.3,68 16 19. 5 26. 4 30. 2 1. 64 RhWT BlRl6,l-3-5X Feb.3,68 23 31. 5 40. 2 47. 8 1. 66 + RhWT B1R17,1-3X Feb.3,68 39 18. 3 25. 5 29. 6 1. 64 NaCl B1R18,3-5X March 5,68 25 6. 2 22. 4 0. 862 RhWT B1R19,l-3X-5-4 March 5,68 16 15. 0 22. 5 25. 4 1. 95 RhWT BlR19,l-3-5X-4 March 5,6 8 16 26. 0 33. 7 38. 0 2. 00 RhWT BlR19,l-3-5-4x March 5,68 13 50. 0 61. 5 68. 3 2. 33 + RhWT B1R20,3-5X March 20,68 62 10.. 0 18. 2 24. 8 5. 34 + NaCl B1R21,5-4X March 31,68 77 38. 5 51. 4 56. 6 0. 804 NaCl B1R22,3-5X May 25,68 17 20. 0 36. 0 45. 5 0. 162 + RhWT TABLE IVC (Cont'd.) SUMMARY OP TRACER MEASUREMENTS: BLANEY CREEK Test No. of .'i'-sl Peak Mean Discharge + Identification Date Points (min) Lag Lag , 3 -1\ for/7 Method (min) (min) extension B1R22A,4UP-4DOX May 27,68 18 1.2 2. 0 3. 70 0. 131 RhWT B1R23,5-4X May 27,68 '.. 9 98CV0 137. 0 165. 0. 130 RhWT B1R24,1-3X May 27,68 7 71.0 101. 0 123. 0. 120 RhWT B1R25,3-5X June 3,68 18 9.5 16. 6 20. 8 0. 682 RhWT B1R26,3-5X June 6,68 17 16. 0 26. 0 34. 2 0. 262 RhWT B1R27,4-4D©X June 6,68 18 0. 264 RhWT B1R28,5-4X June 6,68 67 56. 0 81. 0 93. 9 0. 280 NaCl BlR29,3-5X-4 June 13, 68 45 21. 2 38. 5 50. 5 0 . 140 NaCl B1R29,3-5-4X June 13,68 84 115. 172. 202. 1 0. 140 NaCl B1R30,4UP-4X June 13,68 18 4.5 9. 6 11. 41 0. 139 RhWT B1R31,1-3X June 13,68 18 57-0 90. 0 117. 0 0. 146 + RhWT B1R32,3-5X June 18,68 56 27.5 53. 0 71. 4 0 . 0903 NaCl B1R33,3-5X June 18,68 18 29.0 54. 5 71. 4 0. 083 + RhWT B1R34,3-5X Sept.21,68 12 8.9 15. 0 19. 5 0 . 748 + RhWT B1R35,1-3X Oct.1,68 69 40. 5 61. 0 71. 2 0 . 285 NaCl B1R3631-3X Oct.12,68 16 24. 0 34. 0 40. 4 0. 741 + RhWT BlR37,3-5X-4 • Oct.13,68 31 7.5 12. 7 15. 5 1. 35 NaCl BlR37,3-5-4x Oct.13,68 41 36. 0 45. 4 51. 4 1. 30 + NaCl TABLE IVD SUMMARY OF TRACER MEASUREMENTS: PHYLLIS CREEK Test No. of 3^ Peak Mean Discharge + Identification Date Points Lag- Lag , 3„-l\ for P Method (mm) i . \ i • \ (m s ; ) . (mm) (mm) . . > extension PhRl,1-1D0X June 22,67 1. 980 SoDi C T PhR3,2-3X-4 July 21,67 22 25. 0 38. 0 •43. 5 0. 748 + RhWT PhR3,2-3-4X July 21,67 30 56. 75. 83. 1 0. 817 + RhWT PhR4,l-2X July 27,67 18 38. 5 56. 69. 92 0. 369 + RhWT PhR5,3-4X-6 July 28,67 14 32. 0 45. 52. 9 0. 339 + RhWT PhR5,3-4-6x July 28,67 21 52. 0 73. 82. 3 0. 385 + RhWT PhR6,2-3X-4 July 28,67 18 37. 5 57. 64. 4 0. 352 + RhWT PhR6,2-3-4x July 28,67 12 77. 0 104. 118. 1 0. 338 + RhWT PhR7,1-2X July 29,67 15 43. 5 61. 0 76. 0 0. 312 + RhWT PhR8,4-6X July 29,67 17 15. 0 26. 31. 5 0. 366 + RhWT PhR9,4UP-4DOX Aug. 8,67 0. 232 RhWT P T PhRlO,2-3X-4-6 Aug. 8,67 19 43. 0 70. 87. 6 0. 228 + L . ± . RhWT PhR10,2-3-4X-6 Aug. 8,67 14 96. 0 134. 152. 3 0. 239 + RhWT PhR10,2-3-4-6x Aug. 8,67 12 127. 169. 187. 0. 240 + RhWT PhRll,l-2X-3 May 17,68 17 21. 25 28. 5 36. 2 1. 47 + RhWT PhRll,l-2-3X May 17,68 11 40. 0 58. 0 64. 2 1. 59 + RhWT PhR12,4-6X May 19,68 14 5. 1 8. 4 9. 54 2. 37 RhWT PhR13,3-4x May 19,68 16 11. 5 16. 8 18. 8 2. 49 RhWT TABLE IVD (Cont'd.) SUMMARY OP TRACER MEASUREMENTS: PHYLLIS CREEK Test Identification Date No. of Points rip, ,. Jl-s-(min) Peak Lag (min) Mean Lag (min) Dis.charg (m3s-1) je + for r extension Method PhRl4,2-3X May 19,68 12 15. 21. 23.4 2.55 + RhWT PhR15,1-2X May 19,68 16 1 16. 5 22. 5 27.8 2. 40 + RhWT PhRl6,l-2X May 17,68 37 19. 28. 5 36.7 1. 40 NaCl PhR17,2-3X May 19,68 32 14. 25 21. 2 . 24.2 2. 42 NaCl PhRl8,4-6X May 24,68 13 8.00 12.8 14.4 1.10 RhWT PhR19,3-4X May 24,68 15 19. 26.5 29.9 1.07 RhWT PhR20,2UP-2X-3 May 30,68 16 2.6 5.0 6.31 0.945 RhWT PhR20,2UP-2-3X May 30,68 16 25. 38.5 49.8 0.985 + RhWT PhR21,l-2X May 30,68 18 24. 33. 39.8 1.05 RhWT PhR22,l-2X May 30,68 70 22.5 33. 38.8 1. 02 NaCl PhR23,2-3X June 1,68 13 17. 24. 26.8 1.88 RhWT PhR24,2-3X June 22,68' 15 23.5 35.5 42.1 0.955 + RhWT PhR25,l-2X June 22,68 17 23. 35.4 46.4 0. 826 + RhWT PhR26 ,2-3X July 3,68 18 20. 0 31. 2 35.5 1.19 + RhWT PhR27,3-4X-6 July 3,68 35 17. 24.5 28.5 1, 26 NaCl PhR27,3-4-6X July 3,68 66 26. 38.5 44.8 1.25 ^NaCl PhR28,3-4X-6 Sept .17,68 28 10. 3 14.5 17.33 3.10 + NaCl PhR28,3-4-6X Sept .17,68 64 15. 0 21.8 24.9 3.10 NaCl -<1 4=" TABLE IVD (Cont'd.) SUMMARY OF TRACER MEASUREMENTS: PHYLLIS CREEK Test No. of Identification Date Points rT£. • Peak Mean (min) Lag Lag (min) (min) Dis charge (mV1) + for r extension Method PhR29,3-4X-6 PhR29,3-4-6x PhR30,1-2X-3 PhR30,1-2-3X Oct.17,68 31 8.5? 13.5 15.4 3.69 Oct.17,68 31 14.5 20.4 23.2 3.72 Oct.17,68 35 14.5 20. 25.1 3.48 Oct.17,68 36 27. 37.2 44.5 3.61 + NaCl NaCl NaCl NaCl —3 ui 76 tracer. The relative salt dilution method also agrees with the dye dilution method on the few. occasions when simultaneous tests were run,. (Ph Rll-Ph R12; Ph R21-Ph R22; Bl Rl6 -Bl R17; Bl R32 - Bl R33)• Absolute accuracy is more.difficult to estimate, par ticularly with regard to discharge, because none of the four testestreams is gauged by the Water Survey of Canada. During the tests Bl R32 and Bl R33 over the reach 3 - 5 of Blaney Creek, the Water Survey of Canada measured discharge at Gauge 1. With mean tracer travel times of over one hour, the tracer methods cannot be expected to give very reliable discharges. The salt dilution method, Run 32, indicated a discharge of 90.3 Is-1, the dye dilution method, Run 33, gave 96.8 Is-1, or 83 Is ^ with the customary tracer loss adjustment (L = 0.2% per minute). The current meter measurement was made at a poor section with depth of less than 1 ft throughout; it indicated 75 Is-1. Coverage of the discharge range varies from stream to stream. The low flows are reasonably well defined on all four, but Brockton and Placid Creeks go dry regularly so that one could conceivably observe tracer travel times of several days at extremely low flows. Run 9, on Brockton Creek gives the lowest velocity, 3-5 mmsT1. The high flow range is reasonably well defined on only two of the four streams, Blaney and Brockton. Runs 13, l4,and 15 on. Blaney Creek coincide with the largest observed flow at a stream guage in the neighbouring valley, for which there 77 are 3 years of records. Several major log jams were moved at this flow. The control structure (Photograph 12) was com pletely submerged. Run 19 on Brockton Creek was observed at the runoff' peak during a very severe rain storm. On Phyllis and Placid Creeks,- the discharge range of the tracer tests extends to approximately 20% of the highest flows that have occurred during the last 3 years. 4.3 Surge Tests The result of a surge test consists of a graph showing water levels vs. time for all the gauges on one stream. Figure 15 is.a typical example. The curves are well defined because of the inherently high accuracy of time and stage measurements. For convenience the data are plotted as t vs. H rather than the more significant t vs. Q. This does, however, not affect the conclusions, because the gauge rating curves are practically linear in the small discharge range encoun tered during any one surge test. A total of 22 surge tests were made; 7 on Phyllis Creek, 6 on Blaney and Brockton Creeks,and 3 on Placid. Data from 1 or 2 gauges are missing in approximately 50% of the tests, due to either lack of field assistants or difficulties with the tube gauges. Only the Brockton Creek tests cover the complete range of flows. The control structure on Blaney Lake was submerged and inoperative during the highest flows : (Photograph 12). Gauge I Gauge 3 15.26 Gauge 4 4.80 1400 SURGE TEST OF OCTOBER 13,1968 ON BLANEY CREEK Fig. 15 TABLE VA SUMMARY OP SURGE TESTS, BROCKTON CREEK Date Element Used to Determine Lag Q at G. 1 (mV1 1) 2) AQi tL) ) Lag 1-2 (min Q at G. 2 (m3s_1) A Q 2 Lag 2-3 Q at G.3 (m3s_1) AQ3 Aug. 15,67 Start of UP3' .0094 -'+.0013 6.25 .010 .0010 6. .0084 + . 0002 sharp peak . 0107 5.50 .011 5. .0086 start of DS .0101 -.0015 5.00 . 011 — .0012 5. . 0091 - .0012 Aug. 17,67 Start of UP . 0027 +.0015 11. 0 .0035 + .0011 8. 5 .0036 - + .0008 Start of DS . 0040 -.0006 13.0 .0045 - .0008 9. 0 .0042 - .0006 Start of UP . 00675 +.00055 7.8 .0066 + .00045 Start of DS . 0070 -.00145 8.2 .0070 -.0011 Start of UP . 0068 -+.0008 8.2 .0072 + . 00095 7.. 0 . < : .0062 + .00075 Aug. 26,68 Start of UP .0043 -.0012 11.5 .0057 -.0013 5. 8 .0056 + . 00105 Start of DS 1.0 .0068 -.0011 7. 2 — .00085 Sept .14,68 Start of UP . 047 +.0050 4.2 .038 + .0055 3. 2 .041 • + .0025 Start of DS . 052 -.0130 4.5 .045 — .014 3. 0 .044 - .0080 Sept .22,68 Start of UP .082 + .009 3.0 .084 •4-.008 3. 0 .081 . 0040 Start of DS .086 3.0 .088 - .014 2. 8 .085 -.0045 1) Q is the flow immediately prior to the test. 2) AQis the change in flow as observed at a particular gauge. 3) UP = Up-surge; DS = Down-surge. TABLE VB SUMMARY OF SURGE TESTS, PLACID CREEK Date Element Used to Determine Lag Lag 1-2 (min) Q at G. 2 (mV1) AQ2 Lag 2-3 (min) Q at G.3 (mV1) AQ3 Lag 3-4 (min) Q at G.4 (mV1) AQ4 June 28, 1968 Start Mid-of UP • UP 92. 110. . 042 +. 0.0 4 56. 56. .065 + .004 Start Mid-of DS DS 113. 96. .044 -.004 56. 49. .067 -.004 Aug.28, 1968 Start Mid-of UP-. UP 69.5 77.5 .082 + .005 Start Mid-of DS DS .086 -.006 40. 43. .156 -.010 Aug.30, 1968 Start Mid-of UP-. UP , 89. 104. . 042 + .007 51. 57. .058 + .004 Start Mid-of DS DS 89. 106. .048 -.007 49. 56. .061 -.008 142. 142. .109 -.006 co o TABLE VC SUMMARY OF SURGE TESTS, BLANEY CREEK n , Element Used Q at A Q-, Lag Q at AQ.-, Lag Q at AQ,- Lag Q at A Qu uaze to Determine G.l 1 1-3 G.3 3-5 G.5 5-4 G.4 La^ (mV1) (mln) (mV1) (min)(m3s-l)(min) ^-l. 3-4 May 19, Mid UP-; •' a 118 + . 052 45. .119 + .047 16. 5 66.5 123 045 1967 Start of UP ' 37-5 15. 62. Mid DS 162 042 47. .158 — .038 17. 70. 165 —. 040 Start of DS . 38. 12. 52. r— t 1 June 9, Mid UP-" 063 018 59. .061 + .019 23. .055 016 5-4 72. 066 + . 019 1967 Start of UP 49. 21. 72. Nov.19, Mid UP # 505 + . 0 20 31.5 .495 + .020 9. .435 + . 020 19. 555 + . 015 1967 Start of Up 24.5 21. March 5, Mid DS 2. 100 — . 140 13.5 2 .180 — . 140 7. 2.120 —. 120 1968 Start of DS 9.5 8. Mid UP,' ' 2. 100 + . 160 13. 2 .180 + .120 6. 2.080 + . 120 Start of UP- 10 . 5. Oct.13, Mid DS 1. 370 — , 180 18.5 1 .090 — .140 7. 1.180 —. 130 21. 5 1. 170 —. 130 1968 Start of DS 13; 8. 20. Mid UP- ' 1-. 240 + . 200 18.5 .970 + .180 7. 1,050 +. 180 21. 1. 050 + . 170 Start of UP . 12. 7. 22. Nov.30, Mid UP., (i small) 1. 240 + . 020 19.5 • 990 + .030 9. 5 1.120 + . 020 14.5 1. 060 + . 030 1968 Start of UP " 16. 8. 18, Mid DS(small) 1. 260 — . 030 19.5 1 .020 040 9. 5 1.140 —. 020 15. 1. 090 —. 030 Start of DS 17. 7. 15.5 Mid UP 1 (large) 1. 240 + . 210 19 .980 240 8. 5 1.120 210 16 1. 060 + . 2'10 Start" of UP ' -• 13...5 9- .16 TABLE VD SUMMARY OF SURGE TESTS, PHYLLIS CREEK Date Element Used to Determine Lag Lag 1-2 (min) Q at G.2 OmV1) A Q2 ( Lag Q at 2-3 G.3 1,1111 > (mV1) AQ3 Lag 3-4 (min) Q at G.4 (.mV1) A Qa Lag 4 4-6 (min) Q at G.6 mV1) A Q6 July 28, Mid UP-:' 36. .338 .+ . 00 8 25. .342 + .010 18.2 .340 + . 013 14. .370 + . 010 1967 Start of UP. 29- 22 . 20 . 12. Mid DS 32.5 . 345 Oil 22.8 .350 -. 010 19. .352 • 014 12. .377 + . 017 Start of DS 24. . 24. 22. 11. May 19, Mid DS 16. 2. 340 —. 040 11. 2.450 -.040 9.5 2.380 - . 040 1968 Start of DS 9.5 11.5 8. Mid UP 17. 2. 300 + . 050 10. 2.'440 + .040 10. 2. 430 + . 060 Start DS 11. 10 . 8. June 1, Mid DS 17. 2.550 + . 070 9. 2.500 -.070 8. 2.570 - . 050 4. 2.550 050 1968 Start DS 11. 11. 8. 5. Mid UP 1-6. 5 2.530 + . 20 10.5>2. 500 + .20 6. 2. 540 + . 17 5. 2.520 + . 150 Start UP . 12. 9. 25? 6 . June 22, Mid DS 25. .815 - . 095 17. ..790 -.090 14. .345 - , 090 8. . 880' — . 105 1968 Start DS 17. 14. 12. 7.5 Mid UP' 25. .720 + . 10 16. .700 + .10 14. .755 095 8. • 775 + . 105 Start UP 17. 13. 14. 6.5 Sept •17, Mid DS 16. 2.750 - . 33 11.5 2.770 -.27 1968 Start DS 7. 13.5 Mid UP 17. + . 25 10. 2.450 + .27 Start UP 10. 10. 5 00 TABLE VD (Cont'd.) SUMMARY OF SURGE TESTS, PHYLLIS CREEK Date Element Used to Determine Lag Lag 1-2 (min) AQ G^ at AQ ! Lag Q at d 2-3 G.3 (min)(m3s-l) AQ3 Lag 3-4 (min) Q at G.4 (m3s-l-0 AQ,. Lag ^ 4-6 (min)( Q G. m3 at 6 s-1) AQg 2-4 Nov.18, Mid DS 1.8. 5 2. 740 —. 26 36.5 2/-890 -.240 1968 Start DS 11.5 25.5 Mid UP 18. 2. 450 + . 16 35.5 2.580 + .170 Start . .. UP' 11.5 25.5 3-6 Nov.29, Mid DS 19. 2. 770 -. 32 8. 3.200 -.35 9.5 4. 200 -.30 1968 Start DS 11.5 9. 6. Mid UP-.-(small) 21. 2. 450 + . 030 11.5 2. 820 + .035 6.5 3. 900 + .030 Start UP- ' (small) 17. 14. 6. Mid DS 18. 5 2. 480 _ 17 9. 2.850 -.13 9.5 3. 900 -.20 Start DS 10.5-12.5 10. 10. Mid UP--. (large) 20. 2 . 310 31 12. 2.720 + .28 5. 3. 700 + .30 Start UP' (large) 13.5 10. 5 7. or LO 84 The test results are summarized in Tables. 5A to 5D in which the gauge readings have been converted to discharges, on the basis of the tube rating curves (Section 3.4.5S Figure 13). The data appear to be consistent, insofar as the observed change in discharge remains constant from gauge to gauge along a test stream. The surge lags from station to station- as shown in Table 53 have relatively low accuracy, particularly if they are short because the gauges could only be read at 30 to 60 second intervals. With lags in the order of 5 to 10 minutes, this introduces an immediate uncertainty of 10% to 20%. The lags based on the mid-points of the surges are substantially more reliable than the lags based on the starting points because the mid-points were derived by smooth ing the level-time curve. 85 5. CHANNEL GEOMETRY AND STEADY FLOW EQUATIONS 5.1 Similitude Considerations for Steep, Degrading Channel Networks The conditions under which readily available information, as defined in Section 2.5, may be adequate for evaluation of the channel network hydraulics will be discussed here. 5.1.1 Assumptions Dynamic similitude between related physical systems can only be examined on the basis of a complete list of the forces affecting the system. Similarity between channel networks is possible if the major processes of formation are similar (Barr, 1968). Obviously there are a large number of pro cesses which could conceivably affect the channel network of mountainous basins; the problem lies in identifying the dominant ones . The formative processes assumed here may be biased towards the present problem in the sense that they describe a system that can be defined adequately with the readily avail able information. However, the field data supply evidence indicating that the system is reasonable and explains a major portion of the variation in and between channel networks. The assumptions are listed and discussed below:;. 86 '(i) The drainage network occupies, valleys whose longi tudinal slopesjSy,are remnants of the Pleistocene period. The streams form their channels by degrading into glacial debris.-, which contains sufficient coarse material to prevent the stream from reaching bed-rock or from degrading enough to achieve a channel slope ,S,significantly different from S^. In other words, S is imposed on the channel network, but the size of the material lining the channel is one of the results of the channel-forming process. This is the very opposite of the common regime-type assumption with respect to slope and bed material, which states that a regime canal will adjust its slope by erosion or deposi tion, until it is adequate to handle the upstream supply of water and sediment. Most rivers fall between the two extremes with meandering and braiding playing an important role in reaching adjustment between water, sediment load, and valley slope. The main support for the present assumption lies in the consistent downstream steepening of the test- streams (hanging valleys, Figures 2 and 3), lack of flood plains, absence of braiding or meandering, and the apparent close correlation between slope and size of the bed material. (ii) The coarse material lining the degraded channels can only be moved at extreme flows, which are therefore solely responsible for the channel, form. A single discharge value is adequate to represent these formative high flows. 87 This assumption is supported by the work of Miller (1958), who found high correlation between discharges of a given frequency and hydraulic parameters such as width, depth and velocity. Day (1969) presents similar correlations for the test reaches of this study. The theory on channel performance developed here does not depend on this assumption. It is only used in stating the similitude criteria. (iii) The transport rates of material finer than the bed material are low and do not affect the performance of the channel. Supply of coarse material to the channel through slides, rockfalls, bank erosion, etc., is low and in balance with the transporting capacity of the channel. Up to flows in the order of the mean annual peak, this assumption is well supported by field observation, but it may break down under extreme flood conditions, such as the event described by Stewart and LaMarche (1967). A sufficient amount of fine gravel and sand may then be in motion to lower the channel resistance significantly, thereby starting a chain reaction of higher velocity - more bank and bed erosion -lower resistance. None of the reaches showed much evidence of active bed or bank erosion except in a few isolated locations, mainly associated with damaged vegetation cover of the stream banks due to recent logging. (iv) The channel forming process is repeatable. If the same flow regime, Q(t), is diverted down identical valleys, 88 containing debris, of identical gradation,, the mean properties of the resulting channels will also be identical. This assumption is well supported in regime-type situations, where an identical supply of water and sediment to a straight channel segment eventually produces identical channel dimensions. The application of this concept to the present situation is speculative, but Section 5-3 will show that the significant channel parameters can be derived from the imposed or independent effects such as Q(t) and S, with out requiring a knowledge of any dependent parameters, such as width or roughness. This eliminates the possibility of a large random effect in the channel forming process. 5.1.2 Conditions for Similarity Considering a short, straight channel reach, one can identify the following forces, (Barr, 1967; Barr and Herbertson, 1968):gravity g acting on the water (waves); gravity acting along the valley slope, g'S; ; gravity acting on the submerged grains ((ps - pw)/ys) g = gg,^ and viscosity, V. is imposed from upstream. Some of the resultant measures ar'e W^, the water surface width at flow Herbertson and Barr like, to consider 4 gravitational forces g, gS, as above and gs as. the net. gravitational force on the submerged grains as: it affects, the surrounding grains, and gw = ((ps - $>w)/rj>w) g as the net gravitational force on the submerged grains as it affects the displaced, water. Since gw can be computed from gs and g , it need not be specified. 89 Q^; AD/WD, a depth measure based on the. flow area at. dis charge QDJ. the. velocity, v^, and D,. the size parameter of. the material lining the channel. These, terms can be arranged in dimensionally homogeneous functional forms e.g. using dimensions of length fi < 2/5 4D .1/5 'D QDS 2/3 2/3 1/3 &s 0 . . .5.1 D VD in which the terms between vertical bars can be used alter natively. Note that Equation 5.1 has only 5 terms as it con sists of ratios of active forces and boundary actions. The dimensions 1 and t are reduced to 1 only. A functional form containing n + 1 dimensionally homogeneous variables can be reduced to a n-term non-dimensional form without reducing the generality. Of the numerous non-dimensional groupings pos sible with Equation 5.1, the following is most suitable for the experimental set-up at hand; 1/3 f ( 4D (»g) 1/3 5/3 v D ) = 5.2 Equation 5-2 shows that in the present situation, where g/g , )? , and g are constant,, correlations between Q ,S, and s D any one resultant measure such as WQ or v^ should be complete It also shows that complete kinematic similarity is only 90 possible if the resultant measures and \) vary with accord ing to the following proportionalities (Barr and Herbertson, 1968), AD 4 vD «x %+'2 5-3 S r-2 Since i? Is not variable In the field and S is independent of Q, the field data do not have to match the above proportional ities, but it is reasonable to expect fairly close corres pondence. Equation 2 can also be derived from dimensional con siderations (Yalim, 1966) . The basic variables are o , V , ps, g, S, QD, and one resultant measure, say D. There are 7 variables containing 3 dimensions so that a non-dimensional form in 4 terms, such as Equation 5-2 is adequate for a description of the problem. The system described by Equation 5.2 neglects many signi ficant processes. In the field area of this study vegetation affects the smaller streams (WD< 15 m) considerably through the formation of frequent log jams. Some minor reaches appear to be alluvial and may be aggrading (which would introduce the transport rate as a. variable); others are- occasionally cn bed rock. The transport rates and sediment supply rates are unknown. 91 5.2 ' Basic' Equations Tor Steady, Uniform Flow 5.2.1 Theoretical Considerations The uniform flow parameters,' mean velocity, v , surface width,W , and flow area,A, of a given straight open s channel segment are fully defined by three equations: (i) the Equation of Continuity which may be written as Q = v A (=• v'rW. d#) ... 5.4 m A • m sr * where d» = A/W , and s (ii) a geometrical equation linking A and W (e.g. s Wg <x VA" in the case of a triangular channel)} and (iii) a flow equation linking Q and one or several of the parameters v , A, W , and dw in a form which is linearly f m5 5 s' * independent of Equation 5.4. The constants of this equation will depend on the boundary and cross sectional shape of the channel segment. In the case of natural "tumbling flow" channels, the relations between W and A are virtually unobtainable but this s J is not a serious drawback as long as the aim is hydrological. The main parameters are then only mean velocity, vm?and channel storage per unit length, A, so that the Wg vs. A relation becomes redundant. The flow equation can take many forms. Some of the more relevant possibilities are listed below, all assuming broad 92 rectangular channels (dx = depth = hydraulic radius) of width 3W, and depth.^d. For flow governed by friction over a hydraulically rough boundary with roughness ratios between 7 and 130 (Ackers, 1958) 5/3 Q <x A (Manning's Equation) and for roughness ratios between 1.5 and 11 7/4 Q cx A (Lacey's Equation) For horizontalj:frictionless channel segments controlled by a step-like drop at the downstream end, an approximate flow equation is Q - A3/2 If the control is a triangular weir with apex on the channel floor Q <~ A5/2 and in the case of a parabolic weir Q - A7/2 If the channel segment is obstructed'by a dam with outflow below the watersurface Q - A1/2 Note that since A = Q/v , a relation of the form m' Q °< Aw ' ...5.5 can also be stated as w Q -x-v/-1 ... 5.6 93 or as w-1 1 w Q <* - T# 5.7 in which T refers to unit length. In-natural channels W is usually related to Q by an Equation of the form W o< Qz, s s The observed exponents z cover a range from 0.05 to 0.5 (Miller, 1958), but the most common values fall between 0.15 to 0.25. If this effect is included, Q <x Aw becomes w Q « Azw+1 ... 5.8 Considering tumbling flow as a randomly arranged sequence of short channel segments governed by flow equations similar to the ones listed above, and' considering further that the travel times through channel segments in series are additive, one can express the relation between Q and T for long reaches ( I > > W ) as s lim n G Y Q = n-~ [ £ _A T i ] ... 5-9 I,.- 0 i = i L *+-Wj_-1 The constants c. and the exponents y.(= ) are random 1 1 w. l variables with unknown distributions. Equation 5.9 can therefore not be solved to predict the form of the Q = f(T) relation. 5.2.2 Flow Equations of the Test Reaches With the form of the tumbling flow equation not being predictable, a number of possibilities were tried by plotting 94 various transformations, of .the' Q - A data of several reaches against each other. Pure, exponentials of the form bA A = aA Q ... 5-10 or = ( — ) . . .5.10a aA give consistently good fits, although in a few cases the fit can be slightly improved by assuming a small remnant flow area at zero discharge. Table 6 lists the coefficients aA and b^ for all 13 test reaches of this study. The coefficients are based on linear regressions of log^A on log-^Q of the form l°g10A = log10 aA +j bA log1QQ ... 5.11 RSQ is percentage of the variance of the logarithms explained by the regression equation. The standard error of estimate is only meaningful for the reaches with fairly high number of degrees of freedom. Additional data collected by Day (1969) are listed on Table 1. Figures 16 and 17 show data points and fitting lines for 2 sample reaches. A Fortran IVG program"L0GRE"was used to compute the Q - A, Q - vm, and Q - T regressions. Since v = 1/T and Q = Avm,and since these facts were used to obtain the regression data, any one of the regressions is adequate, to establish all 95 TABLE VI REGRESSION PARAMETERS OF STEADY FLOW Degrees "h Correla- Approx. St.Error of Reach of aA bA tlon . Estimate {%) Freedom Coefficient Brockton 1-2 :8 8104 .3376 0.98 12.5 Brockton 2-3 "'5 • 7183 . 2830. 0.99 4.0 Placid 1-2 <2 1. 879 .3403 0.96 3-2 Placid 2-3 ••3 1. 930 .3065 0.95 6.2 Placid 3--4 ::'4 3. 943 .4656 0.99 5-3 Blaney 1-3 -.6 3. 375 . 4784 0.99 7.8 Blaney 3-5 • 16 3. 460 .5339 0.99 9-0 Blaney 5-4 7 3. 110 .4389 0.98 8.9 Phyllis 1-2 J 3. 262 • 5413 0.98 9.0 Phyllis 2-3 .9 3. 202 .4577 0.99 3-6 Phyllis 3-4 .7 3. 021 .4787 0.99 3.9 Phyllis 4-6 "6 3. 199 .4023 0.97 7-8 Phyllis Lower 2 3. 207 .3557 0.99 7.0 96 TABLE VII REGRESSION PARAMETERS OP STEADY FLOW (from Day, 1969) Reach Degrees Gorrelation of: aA 13 Coefficient Freedom Furry 5 5.788 .5009 .95 Slesse U 3 2.922 .5993 .99 Sles se M 4 4.179 .5213 .97 Slesse L 4 2.926 .5765 1.0 Juniper 3 3-377 .4075 .99 Ewart U 3 4.091 .4184 .99 Ewart L 4 3.253 . .4365 .97 Ashnola U 3 3. 092 " .5806 1.0 Ashnola . • M 3 4.638 .4647 1.0 Ashnola L 3 •3.967 .4208 1.0 97 three and any significant deviation of the regression coefficients from their theoretical relations indicates com putational errors. With ' T = aTQ ' ... 5.12 and b vm= avQ v ... 5.13 one obtains b = - b„ ... 5.14A v T av ~&0~a~T ... 5.14B b = 1 - b. ... 5.14C v A a _1 v aA ... 5.14D "L0GRE"also computes two sets of Q - T (T = arrival s s time of tracer) and Q - Tp (Tp = peak time) regressions, one using all data, and one using only those tests for which the dye has been injected at the upstream end-point of the test reach. The above regressions constitute one of the main results of the field work, therefore all the"L0GRE"printouts and plots with lists of the data are included in the Appendix. Inspection of. the plots (Figures 16 and 17, and Appen dix) shows that, over the range of discharges covered here, there appear to be no significant deviations from Equation 98 1 1 1 r— • 1 1 1 1— • • y : 1 1 1 1— y /*• y y 1 .1 _ 1 e • s / - / >r\\ 4- y ff>fo ' 1. © 1 .Ol J> 0.001 .1 CM E c 0 0) < .01 I III 1 III • ••• .0001 .001 .01 Discharge in m 3 s-l + mean velocity t Vm 0 mean cross - sectional area , A • surge celerity observations, c HYDRAULIC MEASUREMENTS ON THE REACH BROCKTON GAUGE I - GAUGE 2 Fig.16 0.1 1.0 Discharge in m3 s-1 10.0 4 mean velocity , Vm © mean cross - sectional area , A • surge celerity observations for mid - surge , c HYDRAULIC MEASUREMENTS ON THE REACH BLANEY GAUGE 3 - GAUGE 5 Fig. 17 100 5.10. In particular, the Q - plots of Blaney Creek,which has the best coverage of the discharge range, show no tendency towards linear basin response (v '.independent of Q) at high flows as observed by Pilgrim (1966). With the almost total absence of flood plains along the test reaches this is not surprising. 5.3 Determining the Parameters of the Steady Flow Equation The basic flow equation (5.10) can be re-written in various non-dimensional forms; one possibility, using terms similar to Equation 5.2 is " •»* ^Vr' ... 5.10b or, if the concept of a formative discharge is retained bA f- ( §—:) ... 5.10C D V' Either version is suitable for comparison with the basic similitude criterion, Equation 5-2. The parameters of the steady flow equation (a^' , A^, b^) can be considered "resultant measures" of the channel forming process, similar to channel width,W^, depth,and roughness, D. Two possible forms of the similitude criterion 5.2 are therefore QD -S ' K ,f ( '5/3 , S , -|- , b ) = 0 ... 5.2a 101 1/3 f ( > s > ' gf , aA' ) = 0 ... 5 • 2b which shows that a^ and b^ should be determined by QD and S, as all other variables can be assumed constant. Exploring this possibility in detail is the main objective of Day (1969). Some of his findings will be summarized briefly here. QD is obviously not a "readily available" parameter, but in a region with reasonably homogeneous climate the con tributing drainage area of a channel segment, DA, can be used instead. As there is virtually nothing known about the physics of the process expressed by Equation 5.2, multiple regressions of aA and bA on the independent variables drainage area (in km ), and slope were tried, including various combin ations of transformed data. For the test reaches in the mountainous areas surround ing the lower Fraser Valley the following two equations give best fit aA = 1.738 , DA0:2922 " ... 5.15 bA = 0.28:88 S-°-10a6 DA0*0756' ... 5.16 They explain 96.1% and 69.7% of the variance in the data. With 11 degrees of freedom, both equations are significant well beyond the 1% level. The' 6 test reaches in the dry interior of'B. C. (Juniper ,Creek, Ewart Creek and the Ashnola River, Tables 3 and 7) do not cover a wide enough range of the independent variables to justify a meaningful 102 relationship for that region. In generally applicable relations for a^ and b^ drainage area can obviously not take the place of Q^'. If it is to be used in the analysis, some correction factor for regional variations in the drainage area-runoff relation has to be added. An alternative to such a factor is the use of a consistent discharge value, such as the estimated mean annual flood. A further possibility which may prove interesting in areas with sparse hydro-meteorological records, is to use width,WD,as an independent variable. The data of this study give good fit to regionally constant relations between width and drainage area, and these relations are easily established by measuring a few channel widths on streams of various sizes in the region of interest. Slope appears to have no effect on width. With W^ and S as independent variables and including the data from all areas one obtains the following relations for a^ and bA 0 ii 7 p aA = 0.943 8 WD ^' : ... 5.17 0.1368 _ Or: 0,82 2 bA = 0,2519 WD • S ' ... 5.18 They explain &6:% and 6;3% of the data variance and are statistically significant at the 1% level, having 18 degrees of freedom. It is not surprising that width and slope deter mine b.A to a lesser degree than aA> The factor aA, which is 3 -1 identical to the flow area at Q = 1 m s can. vary over -a •. 103 large range and needs to be predicted closely. The range of b^ is limited to approximately 0.25 < b^ < 0.65, so that accurate prediction of b^ is not essential as long as the flows of interest are of the order of 1 m^ s-1. More general pre dictive equations should be possible on the basis of Equation 2 5.10c by substituting measured values of DA or for the uncertain QD (the relation between DA and WQ found by Day are 2 consistently close to DA <*" ) . 5.4 The Friction Concept Applied to Tumbling Flow Although the present study does not reply on the friction-concept and the data are less than ideally suited for appli cation of generally accepted open channel friction formulas, a> brief comparison between friction formulas and the general flow equation (5-10) may be interesting and may facilitate comparison with other studies. To•the writer's knowledge, 2 all previous work on very rough channels or on tumbling flow is based on the friction concept, which, by requiring virtually unobtainable roughness data, tends to yield results that cannot be applied to hydrological problems. ^Utah State University appears to have a continuing research program on tumbling flow. Some of the results are published in Peterson and Mohanty, I960 and in an extensive-number of M.Sc. and Ph.D. theses, such as Al Kafaji, 196l; Judd, 1963; Abdelsalam, 1956. Other studies on rough, nat ural channels are: Leopold, Bagnold e_t al/, I960 ; Mirajgaoker and Charlu, 1963; Johnson, 1964; Herbich, 1964; Argyropoulos, 1965; Kellerhals, 1967; Hartung and Scheuerlein, 1967; Scheuerlein, 1968. 104 5.4.1 Open Channel Flow Formulas The problem to be solved by an open channel flow formula is of the form vm = f (D, ds, S, g, ?w, /i , \) 5.19 in which d% = A/W replaces the more commonly used hydraulic radius (they are almost indistinguishable in most stream channels),^ is viscosity and the X^ are non-dimensional cor rection factors which vanish in the case of a broad rectangular channel section and a particular shape of the roughness ele ments of diameter D. Assuming this to be the case,one obtains the commonly used non-dimensional form of Equation 5.19 2 v v d« ~ "PUTS- - f (S> — > d- ) 5.20 It is well established that S on the right side of (5.20) can be neglected as long as steady flow does not lead to the formation of surface waves and either one of the two remaining parameters on the right is often negligible also, depending on their relative size. A formulation of Equation 5.20^ for the case of steep, rough channels is (Keulegan, 1938) |AW = 6.25 + 5.75 log10(^) ... 5.21 which can be fitted approximately with exponential functions of the form 105 . 5.22 .3 .6 .8 1.0 2 4 6 8 10 20 FIGURE 18. VALUES OF c2 FOR BEST FIT TO EQUATION 5.21 CA I, -The commonly used Manning Equation assumes a c„ of 1/6, which a, provides good fit to Equation 5-21 over the range 7< — < 130. It is important to note that Equation 5-22 neglects the terms vmd%/u> and S of Equation 5.20. While the theory of turbulent boundary layers justifies the former, there is no a priori justification for the latter in cases where the 106 roughness elements affect the free surface, as in tumbling flow. 5.4.2 Comparison with the Data The steady-flow data of this study consist of exponen tial relations between discharge and velocity for conditions of constant, but unknown roughness, constant known slope, and constant, but unknown cross-sectional shape. To transform Equation 5-22 into comparable form requires some assumptions regarding cross-sectional shape. Two assumptions will be used which should bracket the true situation (Section 5.2.1). (i) W = c3 v s -v(ii) Wa Q° ' 2 s Equation 5.22 can be written as 1 -, c„ + 0.5 d„ <x v 2 * m and with Assumption (1) this leads to e2+0.5 c?+1.5 vm <* Q ... 5.23 and with Assumption(ii) it leads to 0.8(c2+0.5) cp+1.5 v «r Q ... 5.24 m • The Q-exponents of the last two equations correspond to b of Equation 5.13, and b is related to b, of Tables 6 and v ^ ' v A 107 7 by b. = 1 - b (Equation 5-14C). The observed b cover V Jr\. \f the range 0.4 < bv < 0.72. On Figure 19, the exponents of Equations 5.23 and 5.24 have been plotted against c0. FIGURE 19. EXPONENTS OF EQUATIONS 5-23 AND 5-24 vs. c2 Using Figure 19 to convert the observed by to c2, and Figure 18 to convert c2 to apparent roughness ratios (assuming Equation 5.2 is valid) one can see that the data cover the approximate range of roughness ratios from 0.4 to 8, which is compatible with the appearance of the channels. From Equation 5.18 one can obtain an explicit equation for b^ by = 1 - 0.25 WD14S"-08 ... 5.25 which, if used in conjunction with Figures 18 and 19 indicates that at a fixed slope, large channels are relatively smoother than small channels and, for a given channel size, steep channels are rougher than flat channels. This also agrees with field observations. 108 In conclusion, the data of this study appear to be compatible with the commonly accepted logarithmic law for rough channels, Equation 5.21, but without information on roughness size and on shape of the flow sections, It is not possible to decide whether this equation gives a meaningful representation of flow in extremely rough channels. 109 6. _ UNSTEADY FLOW IN STEEP CHANNELS 6.1 Kinematic Waves and the Surge Test Results 6.1.1 Some Features of Kinematic Waves Lighthill and Whitham (1955) introduced the term "kinematic wave" for a class of waves which arise In one-dimensional flow systems if there is a unique functional relation between: (i) the flow Q, (ii) the position x, and (iii) the quantity per unit distance (A in the case of a stream). The wave motions are then governed by the equation of con tinuity alone. It has long been recognized that the movement of a flood down a long river can be approximated by this type of wave (Seddon, 1900; Masse', 1935). The equation of continuity for unsteady flow in a long channel is AR + AA = o 6i which can also be written as o t d A d x b Q/d A has dimensions of a velocity and can only' depend on Q and on the position x if the assumptions for'kinematic waves are satisfied. An observer moving along x at speed 6Q/6 A 110 will then observe no change In area or discharge (DA/Dt = 0), which shows that Equation 6.2 defines a wave motion with dQ = c(x,Q) ... 6.3 6 A being the celerity of these kinematic waves. In Chapters 8 and 9 of his book on open channel flow, Henderson (1966) examines the conditions under which kine matic waves can approximate the movement of flood waves. The equation of motion for a prismatic channel can be written as (Henderson's Equation 8.5) '• v - dv <Sv , ,Q _ -Q dd m .-.--m 1 m c u •• 1 2 3. 'h 5 2 2 4/3 in which S„ is the friction slope (e.g. n v /d« if the f ^ & m m * Manning Equation is applicable). Terms 1 and 2 define steady, uniform flow,' terms 1 to 4 define steady, non-uniform flow, and the complete equation applies to un-steady, non-uniform conditions. If S is much larger than the 3 other terms on the right side of Equation 6.4, the wave motion is approximately kinematic. Henderson shows that this condition is satisfied in a relatively steep, alluvial river, even during a very rapid flood rise. In tumbling flow channels the 5th term is always one or more orders of magnitude smaller than the averaged S but may be comparable with the local S in a few places (big pools). Terms 3 and 4 however, may be of order S or more and, like S, Ill they are highly and unpredictably variable with x. Equation 6.4 does not,therefore,permit any definite conclusions regard ing the applicability of kinematic wave theory to tumbling flow. Only experimental evidence can do this. The equation for the kinematic wave celerity at a fixed location, c = dQ/dA, can also be stated as d(vA) . .dv C = ~dA = v dA which shows that c Increases with discharge in natural river channels. In a truly kinematic channel, c(x) , is therefore a unique and increasing function of A or Q. As a consequence, a kinematic wave cannot disperse but the higher parts of the wave will tend to overrun the lower parts, resulting in a gradually steepening wave front of positive waves. In Section 5.2 it was shown that channel reaches in the tumbling flow regime obey equations of the form bA A = a^Q (5.10). Kinematic waves in such channels should therefore have the celerity dq i ' ,.ybAr-c = iiS. = ± A-- • " 6 5 c dA 1/b A . R ...DO aA A bA or, in terms of discharge n 1-bA ° ~ aA°A ...6.6 Substituting b = 1 - b and a = 1/a. into Equation 6.6 gives v m bA ...6.7 112 which shows that the kinematic wave celerity is proportional to v • m 6.1.2 Indications from the Surge Test Results The three features of kinematic waves which are suit able for immediate comparison with the field data are: (i) the steepening of positive wave fronts, (ii) the non-dispersive nature of kinematic waves, and (iii) the wave celerity c = v/b . All surge tests show a consistent downstream flattening of positive and negative wave fronts and a tendency towards increasingly smooth Q(t)--curves in the downstream direction. Both facts are clear evidence for dispersive effects. According to Equation 6.7, the kinematic surge celerity plots as a straight line on logarithmic paper, parallel to the Q - v line. Figures 16 and 17 show these lines for two m to test reaches, together with the observed surge celerities based on the mid-points (over Q) of the observed rise or fall. The agreement between the theoretical line and the observed surge celerities is consistently similar to the situation shown In Figures 16 and 17. At intermediate to high flows the agreement between observed and kinematic celerities is always close but at low flows, the observed celerities tend to be significantly higher than kinematic. Symmetrical tests, consisting of an up-surge followed 113 by a similar down-surge (or vice versa) are particularly instructive on the mechanism of wave propagation (Figure 15). The time lag between the mid-points of the up and down-surges remains very closely constant and identical to the original lag at the lake outlet, with the celerity of these mid-points being close to kinematic at all but the lowest flows and with the sharp changes in discharge, becoming gradually smoother as noted above. From these comparisons one can conclude that the mechanism-' of wave propagation at intermediate to high flows through channels in the tumbling flow regime is essentially kinematic with a certain dispersive effect added. The digression from kinematic conditions at low flows can be explained as follows: at low stage large parts of any tumbling flow channel are occupied by;.relatively deep and slow-moving pools, which are not kinematic according to the assumptions stated In Section 6.1.1. Changes in discharge propagate through pools at the dynamic wave celerity \/g d , which will generally be much larger than the corres ponding kinematic wave celerity. For example, at a flow of 3 — 1 0.01 m s Brockton Creek (Figure 16) contained a few pools with depth of more than 0.3m and a large number of pools with depths between 0.1 and 0.3 m. The kinematic wave celerity at this flow is 0.18 ms and the dynamic celerities for 0.1 m -1 -1 and 0.3 m depth are 1 ms and 1.7 ms . The effects of pools will be examined in greater detail in Section 6.3. 114 6.2 Kinematic Waves with Storage Dispersion 6.2.1 Dispersion through Secondary Dynamic Effects In the case of long rivers with relatively prismatic channels and sub-critical flow throughout, it has long been recognized that the propagation of flood waves is mainly kinematic with some dispersive effects added. Hayami (1951) introduced the equation d A ^jn b_k _ <^2A r „ 4t 2 dx c £x2 ... o.r for this type of kinematic wave. This corresponds to Equation 6.2, with 3vm/2 being the kinematic wave celerity according to the Che'zy friction formula and D being an c undetermined dispersion coefficient. Hayami arrived at Equation 6.7 by adding the effect of the changed water surface slope, which occurs during the passage of a flood wave, to the basic flood wave equation (6.2). Without detailed argument he claims further that the dispersion coefficient consists of a sum of two terms, one accounting for the slope effect and the other accounting for wave dispersion in storage elements, such as pools or permeable stream banks. He gives an explicit solution for Equation 6.7, based on the linearizing assump tions of constant v and constant D , and found good agreement m c' . between computed and observed propagation of an artificially produced symmetrical flood wave. The main difficulties with 115 Hayami's solutions are the normally unpredictable size of D and v . c m Lighthill and Witham discuss several different forms of the dispersion term in the kinematic wave equation. They claim that, since the dispersive term is probably small, when compared with the terms on the left of Equation 6.7 b A ^ _b_k "ST c b x and the dispersion term can therefore be stated in any one of the three forms: b2k/bx2, 1/c2 b 2A/Jt2, or 1/c b A/b x b t . The dispersion coefficient, however, may be more easily established for one form than for the others, depending on circumstances. Two methods for determining the dispersion coefficient are given, one based on observed flood profiles at fixed times (which is rarely possible), and the other based on the well-known (but difficult to measure) hysteresis effect, which occurs in stage-discharge rating curves during the passage of a flood wave. In tumbling flow channels the dispersion term may not be small because of the extensive pool storage, so that the above transformation of the term containing second derivation is not justified. Henderson (1966) also discusses the dispersion of kinematic waves and shows that his form of the equation of motion (6.4) can be transformed into the kinematic wave equation of Hayami, Equation 6.7, if terms 4 and 5 are neglected. 116 In conclusion it appears that there is a considerable body of knowledge on dispersive kinematic waves, but it is based on, and applicable to long, relatively prismatic channels, in which the dispersion is mainly the result of differences between steady . and unsteady slopes and the resulting hysteresis in the stage-discharge relations. Tumbling flow is characterized by frequent transitions from sub-critical to supercritical depths, which means that, at least at the critical section, the stage-discharge relation is unique. The dispersion is therefore attributable solely to storage. The consequences of this do not appear to have been investigated before. 6.2.2 The Differential Equation of Kinematic Waves with Storage Dispersion A channel, In which the relation between Q and A; is only unique at regularly spaced discrete locations , has the following Q - A relation at intermediate points x. , <x<x. (see Figure 20) l-l I & Q = f (A) I + ,<?WD -|-| ... 6.8 1 x. I since, at rising stage, some of the discharge at x will go into storage between x and x.. 117 FIGURE 20. DEFINITION SKETCH FOR EQUATION 6.8 The coefficient of b A/b t has dimension L, and with Wp being the only length readily available in the present problem (Section 5.3), it is convenient to use it in Equation 6.8, together with a non-dimensional coefficient fi , whose value will have to be determined later on. Physically, the factor ft WD is a length measure in direction x, related to the average size of pools. In some alluvial rivers, pool-riffle sequences scale approximately with width (Leopold et al., 1963). By using WD in (6.8) one assumes that a similar rela tion holds in tumbling flow channels. Assuming a long channel with densely spaced control sections x^ , one may substitute Equation 6.8 into the equation of continuity (6.1), at least as an approximation, giving AA + AR AA + Q b2A _ n it 6 A bx *At <*xdt " u 118 in which stands for & A/ 6t. Noting that d Q/d A = c and, according to Equation 6.8, b Q/d A^ = W^, one obtains o d t d x "Do xdt This is the basic equation for kinematic waves with storage dispersion. Lighthill and Whitham (1955) arrive at the same equation by considering the effect of hysteresis in a stage-discharge rating curve, applicable to the whole reach. Equation 6.8 also defines a hysteresis effect in the Q - A relation, but in a natural tumbling flow channel this is probably highly variable and could certainly not lead to a practical method for . estimating /? , the one remaining free parameter. An alternative is to obtain an explicit solution for Equation 6.9, with the relatively simple Initial con ditions of the surge tests, and then to obtain/^ by fitting the solution to the observed surges. This will be done in the following two sections. 6.2.3 A Solution for Step-like Input An explicit solution of Equation 6.9 is only obtainable with the linearizing assumptions of constant c and constant A , which is justifiable for the surge tests, since AQ <<Q. With these assumptions- Equation 6.9 becomes a linear, homo geneous, partial differential equation of second order. The substitution 119 /S w F = A e D ct /3 W D 6. 10 transforms the equation into the compact, first canonical form a2F dxd.t cF ^2wr2 = 0 6.11 D which has the characteristics x = const, and t = const. The initial conditions and boundary conditions of a surge test can be approximated as: A(x k 0,0) = A . 6.12 A(0,0) = A o A(0, t > 0) = (1 + <* ) A After the above transformation they become x <* « 1.0 F(x^0,0) F(0,0) = A e o = A ct F(0,t > 0) = (1 + °<)Aoe D 6.13 6.12a 6.13a oc « 1-0 Since the initial conditions define F on two of its characteristics, the problem to be solved is a so-called Goursat problem. Under the above conditions it has a unique solution (Mikhlin, 1966), which can be found by Riemann's method. 12.0 The Riemann Function B is B = I [-v/4V xt ] in which I (u) is the "Modified Bessel Function of the First o Kind of Order Zero". The solution of (6.9) is of the form x c F(x,t) = F(0,0)B(0,0) + j B dF^| >0) dj + J ' dF(0,r) d r dr 6.14 in which J and T are dummy variables in length and time coordinates respectively. The last term of"(6.14) cannot be evaluated in the above form since the derivative dF(05T)/dT" is undefined at Y'= 0. Partial integration of this term gives x r=t. F(x,t) = F(O,O)B(O,O) + CB dF^?0) d/ + B(o,r)P(o,r) j Q' r=o t - / f| P(.o,r) dr ... 6.15 Since dIQ(u)/du = I#1 (u) and (-u) is the "Modified Bessel Function of the First Kind of Order One", substituting for B. and the starting conditions of F in 6.15 gives 121 0 D t cr 2W& J 1 MM V , 1cl(t-rt7-1/2 6.16 With A(l,t) = F exp(-l//?WD - ct^Wp) this is the explicit solution of Equation 6.9 for a reach of length 1. Under normal circumstances it can be simplified considerably. For large -arguments u the Bessel functions l0(!u ) and I^( u) tend asymptotically towards the function \/l/2Tu exp(u) According to Jahnke and Emde (1945)- the agreement is within 5% at u = 9. With the 1-values and time lags of the present surge data, the arguments will always be much larger than 10, so that the Bessel functions can be replaced by their more easily computed asymptote. The middle term on the left of (6.16) is negligible, when compared with the two integral terms. Time lag is generally given in minutes, while all other data are in meters and seconds. With these assumptions, Equation 6.16 becomes A 122 1 ^^ir h/——-^=r^EXP (PV^ sf? + *f 0 120(1 +<*) A CI ^ f— : • ) dx + 5 p — / -i / = exp (p v/l Vt-/9^IAL/ J / „_ /TT;—ZTT" 60ct ^ 0J V2TPV^TT 60cT 60ct 1_ x , ^ ,„ + /?W /?W j ar •' • D / 2 2*" in which p = ^/2k0o./fi . Equation 6.17 poses.no computa tional problems. The Fortran IVG program "PD" computes A(l,t) for given values of QQ,o<, fi, W^, 1, a^, and b^ using Equation 5.10a to convert discharge to area and vice versa and Equation 6.6 to compute c. On output, the program lists several parts of (6.17). to permit an assessment of the contribution of the two terms on the right. The program, with operating instruc tions/ and sample output, is listed in the Appendix. During the period of rapidly changing A(l,t), both terms of (6.17) are of similar magnitude. The first term dominates before that,when A(1,t) ~ A ,and the second term dominates the period when A(l,t)~ (1+ <xr ) A . Away from t = 1/c, the dominant terms become time independent. Close to t = 0, the solution fails as a result of substituting an asymptotic function for the two Bessel functions. 123 0.8 10 12 14 16 18 20 22 24 26 28 30 Time in Minutes EFFECT OF J9 ON THE SOLUTION OF THE KINEMATIC WAVE EQUATION WITH STORAGE DISPERSION Fig. 21 124 6.2.4 Comparison with Field Data To evaluate the probable range of ft , which is the only-free parameter in Equation 6.17, a few surge tests were com pared with computed Q(l,t) curves covering a wide range of . Figure 21 shows a typical comparison. Obviously the computed c of that test is somewhat too small as .pointed out in Section 6.1.2. The best fitting values of fall consistently between 0.5 and 1, with the greater values occurring at the larger discharges. Fit was determined by inspection. A least squares fit of Equation 6.17 to evaluate optimal values of /3 , or /? and c is feasible, using the programs "NLIN2" (Section 3.3.1) and "PD", but it would involve excessive com puter time. As noted in Section 5.2.1 the actual water surface width of natural channels, W , is generally related to Wn by functions of the form with the factor z probably falling into the range 0.1''" to 0.2 in the case of tumbling flow channels (Miller, 1958). This suggests that W might be a good estimate of^Wn or A - ... 6.19 Figures 22a and 22b show a few typical results obtained on the basis of Equations 6.17 and 6.19, with z taken as 0.2 and QD taken as the estimated mean annual peak flow (Table 2). 125 J300 15 30 45 14OO 15 30 45 ^00 Time Phyllis Creek, June 22, 1968. /? = 0.54 -2.8 \n -2.7 T= -2.6 •— -2.5 8. o -2.4 1 Q , , , , ! , , 1500 15 30 45 |g00 15 30 Phyllis Creek, Sept. 17, 1968. jS = 0.69 COMPARISON BETWEEN FIELD OBSERVATIONS AND KINEMATIC WAVES WITH STORAGE DISPERSION Fig. 22a 'w 0.08" 10 E ro CD TJ c o (5 o 0.07-0.06 126 -GI G3, observed-^/ L*-G3, computed fi =0.25 n 1 1 r 0 10 20 30 40 50 60 70 80 90 100 110 Time in minutes Blaney Creek, Upsurge GI-G3, June 9,1967 1.0 W IO E a O 0.8 / / / G 3,observed-^ A-*-G3, computed / / fi =0.6 / / 1.2 CO ro j 1.1 ^ CD +-o O 1.0 10 20 30 Time in minutes 40 Blaney Creek , Upsurge GI-G3, Nov. 30,1968 ro E CD ° 2.0 o -GI G 3 .computed fi =0.725 G3, observed h2.2 _ "v> ro E -21 10 o O 0 2 4 6 8 10 12 14 16 18 20 Time in minutes Blaney Creek, Downsurge GI-G3, March 5,1968 COMPARISON BETWEEN FIELD OBSERVATIONS AND KINEMATIC WAVES WITH STORAGE DISPERSION Fjg.22b 127 Only reaches immediately below lakes can be used for com parison, because the surges on lower reaches do not fit the initial conditions, as stated in (6.12) and (6.13). The writer knows of no other routing method which could give com parable fit without having to evaluate some free parameters from other- unsteady flow data beforehand. Equation 6.17 is not a practical routing equation for routine hydrological work. It is a means of obtaining the dispersion coefficient of the basic wave equation (6.9) if circumstances permit the creation of a small, step-like surge To obtain an operational flow forecasting system, Equation 6. would have to be considered non-linear and programmed for numerical solution, possibly using the methods discussed by Lighthill and Whitham (1955) or Henderson (1966). 6.3 A Practical Approach to Unsteady, Tumbling Flow As an alternative to the routing method of the last section, which considers the tumbling flow channel as a large sequence of storage elements with unique Q-A relations at their outlets-, it appears worth investigating whether a sequence of a few reservoirs and channels could represent tumbling flow. The basic steady flow Equation 5.10 can be satisfied physically either by an inclined rough channel with the appropriate roughness elements or by a smooth, almost horizontal reservoir-like channel with a weir-like outlet. The following 3 sections will explore the consequences of 128 assuming that a channel reach in the tumbling regime can be represented by a relatively small number of alternating reservoirs and channels, both meeting the steady flow equation bA A = aAQ ...5.10 6.3.1'' Unsteady Flow through a Non-linear Reservoir Unsteady flow through a prismatic reservoir of length A and area A has to satisfy the continuity'relation Qu(t) - Q(t) = -|| ... 6.20 in which Qu(t) is the inflow, and Q(t)the outflow. Evaluating the derivative dA/dt with the dimensionally homogeneous form of the steady flow equation ' 5•10b and representing all dis charges as fractions of QD (Q = q Q^), leads to da = _!5 ( l-b 2-b } dt HcbA \q - Q } ... 6.21 which is a separable but non-linear differential equation. The introduction of into (6.21) is purely for ease in converting the formulas to other systems of units; it does not limit Q to values less than To obtain an explicit solution for a constant q , one can write (6.21) as follows v1 D f ,. q-J dt = — + constant AADbA J qu- q 129 Substituting (q - q)/q = k gives t V1 r D A / dk , . = -q j —^— + constant J ki AVA U J k(l+k)-BA The integral on the right has explicit solutions for rational values of bA (b^ = i/j), but the form of the solution can vary widely, depending on i and j (Edwards, 1921). This approach is therefore not suitable for applications in which bA may take a fairly wide range of values. In general it is most efficient to solve Equation 6.21 by a standard numerical method, such as Runge-Kutta. 6.3-2 A Routing Model Based on a Cascade of Channels and Pools If a tumbling flow channel is represented as a cascade of kinematic channels and non-linear reservoirs, as shown in Figure 23,. Equation 6.21 becomes dq.(t) nQ 2-b -at- = *i A° bA <*i-i (t-V ^ ~ ^ A> ••• 6-21a The time lag between pools, T^, is evaluated by neg lecting the time lag between a pool and the following channel, by assuming that the channels are kinematic flow systems,and by assuming further that the pool water-surface is horizontal at all times.1 With this , T^ becomes "'"This is identical with the assumption of an infinite dynamic wave celerity in the pools. The wave celerities quoted in•Section.6.1.1 would appear to justify this. 130 m = (l - oO -i L n c.(t) l and with a substitution for c, according to Equation 6.6 °0 Channel i Pool i FIGURE 23. DEFINITION SKETCH FOR THE CASCADE OF CHANNELS AND RESERVOIRS = (1 -dQ 1 Ap bA 'D .. 6.22 The two equations (6.21a) and (6.22) define a channel routing system suitable for numerical evaluation. A Fortran IV G program "SNLR'-' -written for this purpose is listed in the Appendix. - - -In '-"SNLR"; the parameter is called o< . 131 Suitable assumptions for the free parameters n and d will be discussed in the following section. However, con siderations of computing economics and stability of the computations impose fairly narrow limits on both parameters. Stability is assured as long as the following two con ditions are met.. R± >> Qj_ At ... 6.23 TL(J + 1) < TL(J) + At ••• 6'24 in which A t Is the finite time step in the numerical inte gration of (6.21a), and the subscript j refers to these time steps. The exact formulation of Inequality 6.23 depends on the integrating method, but by-requiring either large reservoirs or small time intervals, it certainly narrows the range of possible cf and n values. Violation of Inequality 6.24 would indicate a tendency towards the formation of a bore. This may be a rather remote possibility,since even the largest surges of.this study do not come close to violating the inequality. 6.3.3 Evaluation of the Free Parameters from Field Data Figure 23 shows that the parameter n is a scale para meter, which should express whether a reach is relatively "long" or "short". With the ratio 1/W^ being the most reason able and practical measure of relative length, n will be 132 assumed a priori to be a function of 1/W^ alone. It is important that n be independent of Q because it would be dif ficult to change n in the course of a computation. The parameter indicates how much of a given channel reach is acting like a reservoir. Since pools are prominent at low flows and tend to disappear during floods, it seems reasonable to expect d to be an decreasing function of Q/Q^. Prom a practical point of view this is a feasible assumption. The boundary between a channel segment and the adjoining reservoir can be shifted during a surge computation as they have identical A-Q relations. The combined effect of Inequality 6.23 and the above assumption on the relation between 6 and Q is somewhat unfortunate, since (6.23) indicates a need for larger reser voirs with increasing flow while the proposed decrease of d with Q has the opposite effect. The two conditions can be met simultaneously by decreasing the time step of the numerical integration as -Q increases. To gain a clearer picture of the effects of changes in n and in d on the computed downstream flow, the surge test shown in Figure 15 was routed through the three Blaney Creek reaches with various assumed combinations of n and d . The results are shown on Figures 24 a, b, and c. They indicate that good fit can be achieved with a wide range of n- d combinations. If the reach is divided into many 1200 13°° Gauge I (ft) 1.25-1.2 15.3 Gauge 3 (ft) 15.25 15.2 • r observed NR = L/(40 WD) oooo NR = L/(IO WD) I 0 Ip If 1/ jj © V o \ / 1 o 1° 133 |400 time 12 00 1300 1400 Blaney Creek, Oct. 13,1968. Q approx. I m3 s-' VARIABLE NUMBER OF RESERVOIRS AT <J = 0.1 Fig. 24a 1200 Gauge I (ft) 1.25 1.2 15.3 Gauge 3 (ft) 15.25 15.2 4.9 Gauge 4 (ft) 4.8 1200 I300 134 1400 -r observed n = L/(40W ) + n = L/(20 W ) o9oo n = L/(IOW ) \ 0 \\ \\ \ ' 7° /o 9 time — —-"\Y n /+/© */ © r o A, V u o f ° v h \\° /A «° I300 I400 Blaney Creek, Oct. 13,1968. Q approx. Im3s-' VARIABLE NUMBER OF RESERVOIRS AT (J = 0.28 Fig. 24b 135 1.3 Gauge (ft) 1.25 1.2 15.3 Gauge 3 (ft) 15.25 15.2 1 r observed - —- n = L/(40WD) OGGO n = L/(|0 W[)) —vOO© \ \ ' \ b / s T ^ v. / 7 time Gauge 4 (ft) j2oo l3oo I400 Blaney Creek, Oct. 13,1968. Q approx. I m3 s-1 VARIABLE NUMBER OF RESERVOIRS AT (p = 0.7 Fig. 24c 136 reservoirs (n large) ' they have to cover a large part of the length (cf large) and vice versa. Since computing time increases more than, linearly with n due to (6.23) and the increase in computing time with decreasing d is relatively small, it is obviously advantageous to keep n as small as possible. The surge test data give no clear indication of a lower limit. The practical solution appears to be to select a "1/Wp" criterion for n that results in a single reservoir for the shortest reaches of interest, which will depend on the scale at which one is working. As can be seen from Table 2 n = 1/40WD ... 6.25 is a suitable assumption for the present set of data. The second parameter,6 , was estimated by trial and error, using all the surge tests of Blaney Creek. The resulting d - q relation is naturally only valid in combin ation with '(6.25). As in the case of the kinematic waves with storage dispersion, the free parameter d could again be obtained with a non-linear least squares fit (Section 6.2.4),. but the amount of computing time required would be even greater here. A simultaneous fit of n and d would probably not lead to significant results, since the minimum, of the squared residuals appears to be an elongated valley, lying more or less in a diagonal direction across the d -n plane. 137 .3 .2-.1-.001 .01 .1 1.0 q FIGURE 25. THE ROUTING PARAMETER FOR n = 1/40WD Figures 26 a, b, and c show several typical comparisons between computed and observed surges for Blaney Creek and for other streams, which were not used for deriving the 6 -q relation of Figure 25. direct comparison of observed and computed H - T curves. Since several of the gauge rating curves are not too well defined, some of the differences between observed and com puted surges are the result of this, rather than of any defi ciency in the routing method. The closeness of fit is com parable to the result obtained with Equation 6.17. At very low stage the surge celerity is again underestimated. With an increase in 6 it is possible to achieve a correct surge arrival time3but this leads to excessive damping. As pointed out in Section 6.1.2, this effect is probably the result of neglecting dynamic waves. There is, however, some slight evidence, particularly in the case of Brockton Creek (Figure 16) The program "SNLR" converts discharge to stage for 13.8 9.80 9.78 9.76 9.74 9.72 Gauge I (ft) 4.06 4.04 4.02 4.00 3.98 3.96 Gauge 2 (ft) 4.8 4.6 4.4 4.2 A/1 30 40 50 15OO 10 20 30 40 40 50 £00 10 Gauge 3 (ft) August 15, 1967 Q^O.OI m3 s"1, A Q-0.001 Sept. 14, 1968 0.05, AQ—0.01 COMPUTED AND OBSERVED SURGES, BROCKTON CREEK Fig. 26a .55 .50] .45 .40 12.33 12.31 12.29 12.27 12.25 observed /^computed _j Gauge I (ft) 15.30 15.28 15.26 15.24 15.22-1 4.60 4.58 4.56 4.54 4.52-1 Gauge 3 (ft) 4.9 4.8 139 1.4 1.3 -T r IO00 II00 j2oo I300 -T r-May 19, 1967 Q~0.I2 m3 s"1, A Q-0.05 20 30 40 50 j^oo 10 20 30 Gauge 4 (ft) October 13, 1968 Q ~ 1.2, A Q -0.15 COMPUTED AND OBSERVED SURGES, BLANEY CREEK Fig. 26b .34 .33 K32 -| 1 1 1 1 r ~i r-mo 2.4 2.2 2.0 -i—.—•—•—i—-—r Discharge at outlet of Marion Lake (m3s_l) .30 .29 .28 " I 1 computed^ Gauge 2 (ft) .38-.37-l4oo 1.90 '' 1.854 1 1 1 1 1 1 1 1 1 1 1 1 r-I500 ,600 1.80-1 n 1 • ' ' ' 1 50 ||00 10 20 30 40 50 |200 Gauge 6 (ft) July 28, 1967 Q~0.33m3s-', AQ-O.OI Nov. 29, 1968 Q ~ 2.2 , AQ -0.3 COMPUTED AND OBSERVED SURGES, PHYLLIS CREEK Fig 26 c 141 that the true Q - A relation is somewhat curved on double log paper, probably due to a residual area at zero flow,and this would result in higher values for the kinematic celerity c at small discharges. The major difference between, the routing method based, cn channels and reservoirs and routing based on Equation 6.17 is that the former makes no assumption about the shape of the surge input to the reach and proceeds with finite time steps,, which,In the case of a runoff model,permits the addition of local inflow. 142 7 CONCLUSIONS 7.1 The Hydraulics of Tumbling Flow The objectives of this investigation are stated in the Introduction as: (i) finding the laws that govern steady and unsteady tumbling flow, and (ii) expressing these laws in terms of parameters which can be related to readily available basin data. Within certain limitations, to be discussed below, this has been accomplished. The aspects of steady channel flow,which are signi ficant in a runoff model,are completely described by the functions A = f (Q) for all segments.of the channel network. The field data indicate that these functions can be approxi-bA mated by simple exponentials of the form A = aA Q . On the basis of similarity considerations It is shown that the two parameters aA and bA depend on basin parameters which can be considered "readily available". This is confirmed by the highly significant correlations appearing in Day's (1969) regression models. The physics of the channel forming process, which is implicit in Day's regression equations, remains unknown. The two proposed methods for extending the steady flow equations to unsteady flow routing are shown to be capable of reproducing the downstream propagation of step-like surges. Since more complicated channel inflows can be approximated by a series of steps, both methods are capable of routing all hydrograph shapes except in those cases where the non-linearity 143 of the channel response leads to the formation of a bore. The data impose a number of limitations on the. conclusions. None of the test reaches is a first order channel and only three reaches represent small streams; Brockton 1-2 and 2-3, and Placid 1-2. Their steady flow behaviour differs markedly because Brockton Creek lies at tree-line and is essentially debris free while Placid Creek is severely choked by logging slash. The results of the study are therefore not suitable for application to almost complete channel networks as they appear on large scale maps such as the 1:2400 map used in Figure 2. The channel networks appearing on 1:50,000 NTS maps represent the approximate lower limit to which the results of this study may be applied. The first order and most second order channels have to be included in the land phase. The conclusions are further limited by the lack of unsteady flow tests on the larger streams. The flow regime in large streams is rarely "tumbling" over long reaches and it is questionable whether the unsteady flow models, parti cularly the rules for determining the parameters and & , apply to ordinary, rough turbulent channel flow. There is a twofold regional limitation on the data: (i) The relations between channel parameters and drain age area depend on climatic factors. This can be overcome by establishing relations between DA and W-^, and then predicting the channel performance from and S. This requires some field work and may therefore not always be feasible. 144 (ii) Climate and elevation are the main factors determin ing vegetation and can therefore have a considerable effect on the performance of the smaller streams. . It follows that logging and clearing operations are of similar importance. 7 • 2 Basin Linearity Both routing models developed in Section 6 show that the non-linear response of a channel segment is largely a consequence of the dependency between c and Q, which, for the kinematic approximation,is positive exponential since b c = dQ/dA «x v and v •= a Q . The surge tests do however m m v & indicate a consistent tendency towards c values larger than dQ/dA (Figures 26 a, b and c) at low flows, which may be interpreted as a trend towards linear response of the channel system at low flows. That the high flows show no tendency towards linearity is not surprising since the only reason which is generally advanced for such a trend, the rapid increase in flow area as the stream channels overflow onto the flood plain, is rarely applicable in steep mountainous basins. One may even argue that, since the bed material of degrading tumbling flow channels moves o.nly under extreme flood conditions and since a mobile bed would probably offer lower resistance to the flowing water (Kellerhals, 1967), there could be stronger non-linear trends during extreme runoff events. 145 7.3 Towards an Operative Channel Runoff Model Neither of the two routing methods of Section 6 represents an operative channel runoff model, but both can be considered as building blocks out of which a channel runoff model can now be assembled. Since the writer plans to pursue this after completion of the present study, a brief note on what remains to be done may be in order. (i) If the routing method based on Equation 6.9 is to be used, a considerable programming problem remains to be solved. One can either adopt the present solution for step like input (6.17) to unspecified input shapes or one might choose a purely numerical solution based on successive approxi mations to Q (x,t) in a finite grid on the x - t planes of all channel segments. (ii) The routing method based on non-linear reservoirs is closer to being operational. The gradual change in d with Q remains to be incorporated in the program "SNLR". (iii) A mathematical formulation for the drainage network will have to be devised to permit proper representation of the channel parameters in a computer. The model recently pro posed by Surkan (1969) appears to be adaptable to this purpose. (iv) A suitable representation of the land-phase input to the channel system will have to be found. Hydrographs of a few small source areas may be acceptable, possibly in com bination with meteorological records. 146 (v) With increasing basin size, the channel phase becomes dominant over the land phase. It is important, therefore, to extend the routing methods to larger streams than those con sidered in the present study. It will probably become necessary to obtain the basic A(Q)-equation from extensive river surveys rather than with tracer methods. Dispersion coefficients and wave celerities will have to be established through controlled releases from several major dams. Only after completion of all this will it be possible to pass a final judgement on the usefulness of the two-phase approach to runoff. However, the results presented here leave little room for doubt that it will be positive. 147 8. BIBLIOGRAPHY Aastadj J. and Sdgnen, R. 1954. "Discharge Measurements by-Means of a Salt Solution., 'the Relative Salt Dilution Method'," International Association of Scientific Hydrology. Publication No. 38. Assemble Internationale d'Hydrologle de Rome. Tome III. Abdelsalam, W. W. , 1965. Flume Study of the Effect of Con centration and Size of Roughness Elements on Flow in High Gradient Natural Channels. Ph.D. Thesis, Utah State University. Ackers, Peter, 1958. "Resistance of Fluids Flowing in Channels and Pipes," Department of Scientific and In dustrial Research, Hydr. Research Station, Hydr. Research Paper No. 1, London. Al Khafaji, Abbas Nasser, 1961. The Dynamics of Twc-Dlmensional Flow in Steep Natural Streams. Ph.D. Thesis, Utah State University. Amorocho, J., and Hart, W. E. 1964. "A Critique of Current Methods in Hydrologic Systems Investigations," American Geophysical Union, Transactions? Vol. 45, No. 2, pp. 307-321. Argyrcpoulos, Praxitelis A., 1965- "High Velocity Flow in Irregular Natural Streams," Inter. Assoc. for Hydraulic . Research, 11th Meeting, Leningrad, Paper 1-26. Barr, David I. H. - 1968. "Discriminating Formulation of n-term Non-Dimensional Functional Equations from (n+1)-term Dimensionally Homogeneous Equations with Particular Reference to Incompressible Viscous Flow," Institution of Civil Engineers,Proc., Vol. 39, pp. 305-312. Barr, David 1. H., and Herbertson, John G. 1968. "Similitude Theory Applied to Correlation of Flume Sediment Transport Data," Water Resources Research, Vol. 4, No.2,p.307. Church,Michael.1967. "Observations of Turbulent Diffusion in a Natural Channel," Canadian Journal of Earth Sciences, Vol. 14. Church,: M. . and Kellerhals, R. 1969. "Stream Gauging in Isolated Areas Using Portable Equipment," Technical Bulletin No. , Inland Waters Branch, Canada Dept. of Energy, Mines, and Resources, (in press). 148 Day, T., 1969- The Channel Geometry of Mountain Streams. -•.M. A. Thesis, University,'Of British Columbia (in press). Diachishin, A. N., 1963. "Dye Dispersion Studies," Journal of the Sanitary Engineering Division, American Society of Civil Engineers,Proceedings. Vol. 89, No. SA1, pp. 29-49. .Diskin, M. H., 196-7. "A. Dispersion Analog Model for Watershed Systems," Proceedings of the International Hydrology - ' Symposium. Fort Collins. Vol. 1. Edwards, Joseph, 1921. A Treatise on the Integral Calculus. Macmillan and Co., London. Vol. 1. Elder, J. W. , 195-9. "The Dispersion of Marked Fluid in Turbulent Shear Flow," Journal of Fluid Mechanics, Vol. 5. Fischer, H. B., 1966. "Longitudinal Dispersion in Laboratory and Natural Streams," Report No..KH-12-12, W. M. Kech Hydrologlc Laboratory, California Institute of Technology, Pasadena, Cal. Hartung, Fritz, and Scheuerlein, Helmut, 1967. "Macrotur-bulent Flow in Steep Open.v'Channels with High Natural Roughness," Proceedings of the International Hydrology Symposium, Fort Collins. Vol. 1. Hayami, Shoitiro, 1951. "On the Propagation of Flood Waves," Disaster Prevention Research Institute, Kyoto University, Bulletdns , No . 1. Hays, J. R., 1966. Mass Transport Mechanism in Open Channel Flow. Ph.D. Thesis, Vanderbilt University. Henderson, F. M., 1966. Open Channel Flow. Macmillan Co., New York Herbich, John B.., and Shullts, Sam, Nov. 1964. "Large-Scale Roughness in Open;Channel Flow," Journal of the Hydraulic Division, ASCE, Vol. 90, N0.HY6, pp. 203-236~7 ; ~~' Herrio, John G..., 1963.. Methods of Mathematical Analysis and Computation.. John Wiley & Sons, Inc. New York. Jahnke, Eugene, and Emde, Fritz, 19 45. Tables of Functions. Dover Publications, New York. Johnson, Martin L., 1964. "Channel Roughness in Steep Moun tain Streams," American Geophysical Union, Ann. Meeting. Judd, Harl E., 1963- A Study of Bed- Characteristics in Relation to Flow in Rough, High-gradient, Natural Channels. Ph.D. Thesis, Utah State University. 149 Kellerhals, Rolf, 1967. "Stable Channels With Gravel-Paved Beds," Journal of Waterways and Harbors Division, ASCE, Vol. 93, No. WW1, Proc. Paper 5091. pp. 63-84. Larson, C. L., 1965. "A Two-Phase Approach to the Prediction of Peak Rates and Frequencies of Runoff for Small Ungauged Watersheds," Technical Report No. 53, Dept. of Civil Engineers, Stanford. Leopold, L. B., Wolman, M. Gordon, and Miller, John P., 1963. 'Fluvial Processes in Geomorphology. W. H. Freeman and Co., San Francisco and London. Leopold, L. B., Bag^old, R. A., and Wolman, M. Gordon, and Bush,, L. M. , i960. "Flow Resistance in Sinuous or Irregular; Channels," United States Geological Survey, Professional Paper, -No . 2 82-D . Lighthill, M. J. and Whitham, G. B., 1955- On Kinematic Waves: I. Flood Movement in Long Rivers, Proc. Royal Soc. of London (A) , Vol. 229, pp.. ' 281-31.6. Marquardt, D. W., 1964. "Least Squares Estimation of Non-Linear Parameters," A Computer Program in Fortran IV Language,•IBM Share Library, Distribution,No. 3094. Masse', Pierre, 1935. • Hydrodynamlque fluvlale, regimes variables.' • Hermann et Cie, Paris. Mikhlin, S. G., 1967.' Linear Equations of Mathematical Physics. Holt, Rhinehart, and Winston, Inc. Miller, J. P., 1958. "High Mountain .Streams ^Effects of Geo logy on Channel Characteristics and Bed Material," New Mexico, State Bureau of Mines and Mineral Resources, Memoir No. 4. Mirajgaoker, A. G., and Charlu, K.L.N., Sept. 1963. "Natural Roughness Effects in Rigid Open Channels," Journal of the Hydraulics Division, ASCE, Vol. 89, No. HY5, Proc. Paper 3630~ pp. 29-44. Morisawa, M. 1957- "Accuracy of Determination of Stream Lengths from Topographic Maps," American Geophysical Union Transactions, Vol. 38, No. 1. 0strem, G., 1964. "A Method of Measuring Water Discharge in Turbulent Streams," Geographical Bulletin No. 21, Geographical Branch, Dept. of Mines and Technical Surveys, Ottawa. 150 Overton, D. E., 1967. "Analytical Simulation of Watershed Hydrographs from Rainfall," Proceedings of the Inter national Hydrology Symposium, Fort Collins, Vol. 1. Peterson, D. F. and Mohanty, P. K., i960. "Flume Studies in Steep, Rough Channels," Journal of the Hydraulics Division, Proceedings of the American Society of Civil Engineers, Vol. 86, No. HY9, pp. 55-76. Pilgrim, D. H., 1966. "Radioactive Tracing of Storm Runoff on a Small Catchment," Journal of Hydrology. Vol. 4, pp. 289-326. Replogle, J. A. et al., 1966. "Flow Measurements with Fluorescent Tracers," Journal of the Hydraulics Division, American Society of Civil Engineers, Vol. 92, No. HY5, Proc. Paper 4895, pp. 1-15. Scheidegger, A. E., 1966. "Effect of Map Scale on Stream Orders," Bulletin of the International Association of Scientific Hydrology, Vol. 11, No. 3. Scheuerlein, Helmut, 1968. "Der Rauhgerinneabfluss," Bericht No. 14 der Versuchsanstalt fur Wasserbau der Technischen Hochschule Munchen, Oskar V. Miller Institut. Seddon, J. A., 1900. "River Hydraulics," Transactions, American Society of Civil Engineers, Vol. 43, pp. 179-Stewart, John H., and LaMarche, Valmore, C., 1967. "Erosion and Deposition Produced by the Flood of December 1964 on Coffee Creek, Trinity County, California," United States Geological Survey, Professional Paper, No. 422-K. Sugaware, M., 1967. "Runoff Analysis and Water Balance Analysis By a Series Storage Type Model," Proceedings of the International Hydrology Symposium, Fort Collins, Vol. 1. Surkan, A. J. 1969. "Synthetic Hydrographs: Effects of Network Geometry." Water Resources Research, Vol. 5, No. 1, pp. 112-128. . Taylor, G. I., 1954. "The Dispersion of Matter in Turbulent Flow Through a Pipe," Proceedings of the Royal Society of London, Vol. 223-Thackston, E. L. et_ al. , 1967. "Least Squares Estimation of Mixing Coefficients," Journal of the Sanitary Engineering Division, American Society of Civil Engineers, Proceedings, Vol. 93, No. SA 3, PP- 47-58. 151 Thackston, E. L. and Krenkel, P. A., 1967. "Longitudinal Mixing in Natural Streams," Journal of the Sanitary Engineering Division, American Society of Civil Engineers, Vol. 93, No. SA 5, pp. 67-90. Wilson, J. P., 1968. "Fluorometric Procedures for Dye Tracing, in Techniques of Water Resources Investigations of the U. S. Geological Survey, Chapter A 12, Book 3. Yalin, M. S., Jan. 1966. "A Theoretical Study of Stable Alluvial Systems," Golden Jubilee Symposium, Central Water and Power Research Station, Poona. PHOTOGRAPHS 152 PHOTOGRAPH 2. PLACID CREEK, ALONG REACH PI 3 - 4, LOOKING DOWNSTREAM. TYPICAL LOG JAM IN FOREGROUND. PHOTOGRAPH 4. BLANEY CREEK, AT Bl GAUGE 4, LOOKING UPSTREAM FROM BRIDGE, PHOTOGRAPH 5. PHYLLIS CREEK, AT PH GAUGE STAGE RECORDER AT RIGHT. 2, LOOKING DOWNSTREAM. PHOTOGRAPH 6. PHYLLIS CREEK, AT PH GAUGE 4, LOOKING UPSTREAM. PHOTOGRAPH 7. BARNSTEAD CONDUCTIVITY BRIDGE. 155 PHOTOGRAPH 8. VOLUMETRIC GLASS WARE FOR SALT DILUTION TESTS. PHOTOGRAPH 9. VATS, PAIL, AND STIRRING ROD FOR SALT DILUTION TESTS. 156 PHOTOGRAPH 10 EQUIPMENT FOR RHODAMINE WT SLUG INJECTION TEST PHOTOGRAPH 11. RECORDING CONDUCTIVITY BRIDGE, WITH ELECTRONIC INTERVAL TIMER. PHOTOGRAPH 12. CONTROL STRUCTURE AT THE OUTLET OF BLANEY LAKE. THREE FLASHBOARDS IN PLACE. PHOTOGRAPH 13. TIMBER CRIB DAM AT OUTLET OF MARION LAKE, WITH TWO ADDITIONS IN PLACE FOR A DOWN-SURGE. 158 159 i -f 160 APPENDIX COMPUTER PROGRAMS WITH OPERATING INDSTRUCTIONS PRINTOUT, AND PLOTS. LIST OP CONTENTS NACL Source listing Sample plot of rating curve Sample printout DQV Source listing Sample printout TAILEX Source listing Sample printout Sample plots with example for determination of A, B, and D QVEL . Source listing, including subroutines for numerical integration Three sample outputs PLOTGA Source listing Sample plots, with and without -extension LOGRE Source listing, including three subroutines Printout and plots for all 13 test reaches PD Source listing, including two subroutines Sample printout SNLR Source listing, including one subroutine Sample printout 161 168 170 176 185 189 249 253 C C FORTRAN /360 MAIN PROGRAM CALLED NACL ', WHICH CONVERTS C TIME-CONDUCTIVITY DATA TO TIME-CONCENTRATION DATA. TIME IN C MINUTES AND SECONDS IS CONVERTED TO MINUTES AND DECIMALS. C C INPUT C CONTRQL . CARDS _ „ . C I ONE CARD PER RUN, C NO. OF DATA SETS, KTOT, (12) _£ 2 ONE PER DATA SFT . : C NO OF DATA POINTS, K, 113J C ARRIVAL TIME OF TRACER WAVE, TST, (F7.2) C . TITLE_OR RUN I DENRIFICAT I ON NO. (7A4 L _ C ,„. PARAMETERS OF GAMMA EXTENSION, IF DESIRED, C LOG A, B, D,(2X,3E10.5). LOG A IS CONVERTED TO A. C 3 ONE PER $TET OF DATA. ; C NO. OF POINTS ON THE RATING CURVE, N, (13) C DILUTION RESULTING FROM 10 CC OF SECONDARY SOLUTION C , IN RATING ...TANK , DlL10, ( F10.0) C RATING TANK TEMPERATURE, TEMP, (F10.0) C BACKGROUD READING AT START OF TEST, BACKST, (F10.0) _C BACKGROUND READING AT END OF TEST OR BLANCK IF IT IS C EQUAL TO BACKST, BACKND, (F10.0) C PARAMETER NPLO, WHICH SHOULD BE .GT. 0, IF NO PLOT C . . DESIRED, j 13) .' .... C PARAMETER NPUNCH, TO BE SET .GT. 0, IF NO PUNCHED C OUTPUT DESIRED, (13) _C DATA CARDS OF STREAM MEASUREMENTS, K CARDS PER DATA SET. C CONDUCTIVITY READINGS YC,TIME IN MINUTES AND SECONDS C FROM INJECTION, XT, (2F9.3) C DATA CARDS OF RATING CURVE, N CARDS PER SET C AMOUNT OF SECONDARY SOLUTION IN RATING TANK IN CC, CC, C CONDUCTIVITY READING, READ, (2F9.3) C OUTPUT C PRINTOUT OF INPUT DATA c PRINTOUT.OF CONVERTED DATA, WITH CONCENTRATION. IN ..PPM c OF THE PRIMARY SOLUTION. c OPTIONAL, c PUNCHED DATA CAROSCONTAINING CONCENTRATION YC, AND c TIME XT, AS REQUIRED FOR PROGRAM DQV, (2F9.3), c PLOT OF RATING CURVE. CALL PLOTS DIMENSION XT (200) , YC (200), .TITLE (7) , CC(20) ,R FAD (20) 1 FORMAT (13, F7.2,7A4,2X,3E10.5 ) 2 ' FORMAT (2F 9.3) '; > "-3 FORMAT (13, 4F10.0,213 ) 5 FORMAT!. 8H0RATING,4X, 4HSTEP, 9X ,2HCC, 8X,. THREADING ,/ K13X , 12 , 7X, F6.0 ,4 X , F9.4 ) ) 7 FORMAT (12) 16 FORMAT (18H1C0NTR0L CARD 1 = /IX, 13, 2X, F7.2, 7A4, 1 2X, 3E12.6 / 18H0C0NTR0L CARD 2 = /.IX, 12, 2X, 4F12.2, 2 3X, 12, 3X, 12 ) 19 FORMAT ( 22H0CONVERTED FINAL DATA . .. ). 32 c FORMAT(5H0DATA , 4X,2HN0,8X,2HXT , 7X , 7HREA0ING • / 1( 9X , 12 ,5X, F7.2 , 2X ,F10.4 )) V, C LOOP FOR SETS c READ (5,7) KTOT DO 8 KSET = 1, KTOT C READ DATA READ(5,1) K , TST, TITLE, A, B, D IF ( A .NE. 0.0 ) A = EXP (A) r READ(5,3) N » OIL 10 , TEMP , BACKST, BACKND, NPLOT, NPUNCH IF ( BACKND .LE. 0.0 ) BACKND = BACKST READ (5,2) ( YC ( I ) , XT (I), I = I , K ) . . .. . READ (5,2) (CC (I), READd),I = 1,N) c PRINT DATA f WRITE (6,16) K, TST, TITLE ,A,B,D, N, DIL10, TEMP, 1 BACKST, BACKND, NPLOT, NPUNCH WRITE (6,32) (I , XT ( I) , YC ( I ), I = 1, K ) WRITE (6, 5) (I , CC(I) , READd), 1=1, N ) c CONVERT SECONDS TO MINUTES 9 DO 9 I =1 , K IXT =XTd) . TMIN .... =IXT . L... XT(I) =TMIN + (XT(I)-TMIN) / 0.6 C CONCENTRATION RATING CURVE c c. 10 CONVERT CC(I) TO CONCENTRATION IN PPM RI = READd) . DO 10 I = 1 , N READ (I) = READ(I) - RI CCd) =(CCd) /(10.0 * OIL 10))* 10.0E+5 c c COMPUTE B OF REGRESSION LINE SYX = o.o . ;. : .. SXX = 0.0 00 12 I = It N SYX = CCd) * REAO(I) + SYX 12 C C SXX =READ(I)**2 + SXX BB = SYX / SXX . ' TION ADJUST YCd) TO A ZERO BACKGROUND AND CONVERT TO CONCEMTRA-DBACK =(BACKST - BACKND )/ (XT(K) -TST) OD 11 1=1, K 11 IF (XTU) .LE. TST ) YCd) =0.0 * DBACK IF (XT(I) .GT. TST ) YCd) = YCd) - BACKST +((XTd)-TST) YCd ) = BB* YCd ) ,. WRITE (6, 19 ) WRITE (6,32) (I XT(I).3, YC ( I ) , I = 1, K ) IF ( NPLOT .NE. 0 ) GO TO 17 C c PLOT RATING CURVE ; . CCMAX =8B* READ(N) 1 .... 163 CALL SCALE ( READ ,N, 7.0 ,RXMIN ,RDX, 1) CALL SCALE { CC ,N, 9.0 ,CYMIN ,CDY, 1) CCMAX = (CCMAX - CYMIN) / CDY CALL AXIS ( 0.0 , 0.0, 7HREADING ,-7, 7.0 , 0.0, RXMIN,RDX ) CALL AXIS I 0.0 , 0.0,20HCONCENTRATION IN PPM, +20,9.0, 90.0, 1 CYMIN, CDY) CALL .SYMBOL..:.(. 1.0, 9.5 , 0.21 ,TITLE. , 0.0 ,..28) CALL SYMBOL ( 1.0, 9.0 , 0.21, 7HTEMP. =,0.0, 7) CALL NUMBER ( 2.4, 9.0 , 0.21, TEMP ,0.0, 1 ) CALL SYMBOL ( 3.5, 9.0 , 0.21, 4HB8 =, 0.0 , 3 ) . CALL NUMBER ( 4.6, 9.0 , 0.21 ,BB , 0.0 , 4 ) DO 13 I = 1 t" N " • CALL...SYMBOL... i READ( I ) , CCC I) ,0. 14 ..,4 , 0.0 v -1 ) CALL PLOT (0.0 « 0.0 ,+3 ) CALL PLOT {READ{N) , CCMAX , +2 ) CALL PLOT t 9.6 , 0.0 , -3 ) CONTINUE PUNCH'. DATA .CARDS .. ... _ IF .< NPUNCH <, NE. 0 ) GO TO 18 WRITE «7»1J K » 1ST, TITLE, A,, B, D WRITE < '7,2) (YCU), XT(I), 1= 1,K ) CONTINUE CALLS _Tn_ FITTING. AND PLOTTING. ROUT INES . ... _ THE CALL CARDS FOR NUMERICAL INTEGRATION OF THE C-T CURVE AND FOfl PITTING A GAMMA-DISTRIBUTION EXTENSION GO IN HERE, f.A\L.JJSJ. LEX IK; TST, XT, YC , TITLE ) CONTINUE C A L ( PLOT NO _ ...... STOP END CONCENTRATION IN PPM 24. Q 32.Q 40.0 46. Q _J 56.Q 64. Q 72. Q _J > -o r m o > o m TQ CO 4^ CO •J-LU LH cu CD LU I 4^ I CD X LO m * CD CONTROL CARD 1 = .• . 63 15.00PH R 28, 3-4-6X, SEPT. 17, 6 0.0 0.0 CONTROL CARD 2 ' = 6 > 83333.25 11.40 1.62 1.62 10 10 r DATA NO XT READING 1 14.30 1.6150 —_ 2 _ 15.30 . 1.6190 „ 3 16.00 1.6240 4 16.30 1.6360 5 17.00 1.6610 6 17.30 . 1.7090 7 18. 10 1.7900 ' 8 ' .18.30^ 1.8640.* 9 19.00 1.9800 10 19.30 2.0610 11 20.00 2.1610 'NI API V 12 20.35 2.2510 13 21.00 2.3000 . . _._ ... 14 21.30 2.3290 . SAMPLE PRINTOUT 15 22.00 2.3340 -16 22.30 2.3210 17 23.00 2.2900 18 23.30 2.2550 19 24.00 2.2020 - 20__- .24. 30... 2. 1440 21 25.00 2.0930 22 25.30 2.0390 23 26.05 1.9970 24 26.30 1.9640 25 27.00 1.9250 ... ......26 27.30 1.8840 27 28.30 1.8210 " 28 29.00 1.7900 29 29.30 1.7730 30 30.00 1.7540 31 30.30 1.7420 _ 32 .31...00_ 1.7270 . 33 31.30 1.7150 34 32.00 1.7060 35 32.30 1.6950 36 33.00 1.6880 37 33.35 1.6800 .38. .34.00_ 1.6760. . 39 34.30 1.6730 40 35.00 1.6690 41 35.30 1.6650 42 36.00 1.6610 43 36.40 1.6590 44 37.00 1.6580 . _ 45 37.30 1.6560 46 38.05 1.6530 47 38.30 1.6510 48 39.10 1.6490 49 39.30 1.6480 50 40.00 1.6470 _ ,._ 51 41.00 1.6450 52 42.00 1.6420 53 43.00 1*6490 ( 54 44.00 1.6390 55 45.00 1.6380 166 56 46.00 1.6370 57 47.00 1.6360 58 48.00 1.6350 L 59 49.00 1.6340 ' 60 51.00 1.6320 61 53.00 1.6300 62 .55.00 1. 6300, 63 60. 15 1.6280 RATING STEP CC READING 1 0. 1.5970 2 10. 1.8790 3 .20. 2.1460 4 30. 2.4170 5 40. . 2.7000 6 50. 2.9780 CONVERTED FINAL DATA DATA NO XT ;" READING .... 1 14.50 0.0 2 15.50 0.0402 3 16.00 0.2545 4 16.50 0.7737 . 5 17.0.0.. . 1.8589 6 17.50 3.9456 7 18. 17 7.4681 8 18.50 10.6881 9 19.00 15.7357 10 19.50 19.2594 11 20.00 23.6103 12 20.58 27.5252 13 21.00 29.6560 14 21.50 30.9154 15 22.00 31.1298 16 22.50 30.5603 17 _23.00 29.2072 . 18 23.50 27.6797 19 24.00 25.3686 20 24.50 22.8397 21 25.00 20.6157 22 25.50 18.2610 23. 26.08 16.4282 24 26.50 14.9885 25 27.00 13.2869 26 27.50 11.4983 27 28.50 8.7483 28 29.00 7.3952 29 29.50 6.6515 30 30.00 5.8208 31 30.50 5.2950 32 31.00 4.6384 33 31.50 4.1125 34 32.00 3.7173 . 35 . 32.50... 3.2350 .. 36 33.00 2.9268 37 33.58 2.5745 t 38 34.00 2.3975 39 34.50 2.2635 167 40 35.00 2.0860. 41 35.50 1.9085 42 36.00 1.7309 43 36.67 1.6393 44 37.00 1.5936 45 37.50 1.5031 46 38.08 1.3685 ... 47. 38.50 1.2786 48 39.17 1.1870 49 39.50 1.1413 50 40.00 1.0943 51 41.00 1.0005 52 42.00 0.8631 53 _ 43.00 ... 0. 7693 54 44.00 0.7191 55 45.00 0.6688 56 46.00 0.6185 57 47.00 0.5682 58 48.00 ,. 0.5179 59 .._ _ 49.00 „ ... . 0.4677 60 51.00 0.3671 61 53.00 0.2665 62 55.00 0.2530 63 60.25 0.1306 . D Q V ;-c c c c FORTRAN /360 MAIN PROGRAM, CALLED DQV, FOR READING TIME -CONCENTRATION DATA INTO ARRAYS SUITABLE FOR FURTHER PROCESSING BY SUBROUTINES 9V, PLOTGA, AND TAILEX. c c c INPUT CONTROL CARDS c c c 1 ONE CARD PER RUN* NO. OF DATA SETS, KTOT, (12) 2 ONE PER DATA SET. c c c NO OF DATA POINTS, K, (13) ARRIVAL TIME OF TRACER WAVE, TST, (F7.2) TITLE OR RUN IDENRIFICATION NO. (7A4) c c c PARAMETERS OF GAMMA EXTENSION, IF DESIRED, LOG A, B, D,(2X,3E10.5). LOG A IS CONVERTED TO A. DATA CARDS c c c TRACER CONCENTRATION, YC,IN PPB,TIME FROM INJECTION, IN MINUTES AND DECIMAL FRACTIONS, (2F9.3) XT, c c c OUTPUT PRINTOUT OF DATA c CALL PLOTS DIMENSION XT(50) ,YC(50) , TITLE (7) c 1 2 FORMAT (13 , F7.2,7A4,2X, 3E10.5) FORMAT (2F9.3) 7 16 FORMAT (12) FORMAT (13H1CONTROL CARD ,5X , 13 , F7.2, 7A4./15H A, B, 1 , 3E10.5) AND> 32 C FORMAT (5H0DATA ,4X , 2HN0 , 8X, 2HXT ,9X , 2HYC ,/ 1 (9X, 12, 5X, F7.2, 2X, F9.3) ) C LOOP FOR SETS READ (5,7) KTOT DO 8 KSET = 1 , KTOT c c READ AND WRITE DATA READ (5,1) K , TST ,TITLE , A, B , D IF ( A .NE. 0.0 ) A = EXP (A) READ (5,2)(YC (I), XTU), I = 1 ,K ) WRITE (6,16) K , TST , TITLE , A, B, D r WRITE (6, 32) ( I, XTU) , YCd) , I = 1, K) c CALLS TO SUBROUTINES GO HERE c CALL TAILEX (K, TST, XT, YC, TITLE ) CALL TAILEX (K, TST, XT, YC, TITLE ) 8 CONTINUE CALL PLOTND WRITE (6,100) 100 FORMAT (1H1) STOP 169 r CONTROL CARD 17 14. OOBR R2 G1UP- i-2Xt AUG 15,67 A» B, AND D = .16323E 01. 10000E 01.86000E-01 DATA NO XT YC 1 12.00 0.0 \ 2 15.00 0.100 I 3 18.00 0.600 4 21.00 12.100 5 .... 24.00 . 38.000 _L 6 27.00 50.500 7 29.00 47.000 8 31.00 38.000 n Q V 9 33.00 27.800 10 35.00 20.200 . 11 37.00 14.100 . S AMP.LE. .PRINTOUT 12 41.00 7.300 13 45.00 4.500 14 49.00 3.100 15 54.00 2.400 16 59.00 1. 800 17 .69.00 . . _ - 1.000 170 TAILEX c c c SUBROUTINE TAILEX ( K , TST , XTIN.YC , TITLE ) THIS FORTRAN /360 SUBROUTINE IS CALLED BY DQV OR BY NACL, IF A GAMMA EXTENSION IS TO BE FITTED. IT PLOTS THE C C C C-T DARA IN THE FORM (LOG C -B*LOG T ) VS. (T), FOR B-VALUES OF -1,0, 1, 2, 3, AND4. DIMENSION XT1200), YC(200), 1 XTIN(200), XTPL01200) DO 1 J = 1, K BXL(200), YL(200), T ITLEl7 f , XT(J) = XTIN(J) - TST XTPLO IJ) = XT (J ) IF ( XTPLO <J) .LT. 0.0 ) XTPLO(J) * 0.0 IF (YC(J> .LE. 0.0) YL(J) =-10.0 IF ( YC(J> .GT. 0.0) YL(J) = ALOG ( YC(J)) CONTINUE UUIMI inut CALL SCALE (XTPLO , K , 14.0, XTMIN, DXT , 1 ) <XTIV CALL AXIS ( 0.0, 7.0 ,15HXT - TST, (MIN),-13 ,14.0 ,0^ 1 DXT ) .XTMIN, N = 0.0 DO 2 J = 1, K YDIFF = YC (J+l) - YC(J) The parameters, A, B, and D, of 'TAILEX' correspond 2 .... 4 IF (YDIFF .LT. 0.0)N = N+l IF (N .EQ. 3) GO TO 3 CONTINUE to YI.YZ.and Y3 of text. WRITE (6, 4 ) TITLE , TST , J , XT(J) FORMAT ( 18H1SUBROUTINE TAILEX ,/9H RUN NO. , 7A4, 2X, 1 18H STARTING TIME , F7.2 , / "2 28H TAILEX STARTS AT POINT NO ,I 2,10X,7HXTIME = ,F8.2) LOOP FOR DIFFERENT B VALUES DO 6 J5 = 0,5 __ J2 = J5 - ] B = J2 DO 7 J3 = J K AX= XT(J3) IF ( AX .GT.0.01 ) BXL (J3) = B * ALOG (AX) IF ( AX .LE. 0.01 ) BXL (J3) = - 10.0 Y(J3) =-BXL(J3) IF ( IWRITE (6,8) TST + YL(J3) J2 .EQ. 1 .OR. J2 .EQ. 4 ) NXT(M) , B , ( M ,YC(M) , YUM) , BXL (M) , Y(MT> 2 , M = J , K ) FORMAT (16H0INITIAL TIME = , F7.2 ,/ 1 54H NO CONC. LOG C B*LOG T 5H B = , F6.2, / Y XT - TIN ,/ 2 (IX, 12 , F8.2 ,F9.4 , F10.4 , F10.4 , F 9.2 )) JNC.) ELIMINATION OF DATA POINTS TO SECOND POINT BEYOND PEAK(NOT"> IF (J2 .NE.(-l) )G0 TO 10 •__ YMAX = 4.* ALOG (AX) IF (YL(K) .LE. 0.0 .AND. YL(K) .GT.(-9.))YMAX = YMAX + YL(K) DY = 0.5 IF (YMAX .GT. 3.5) DY = 1.0 IF (YMAX .GT. 7.0) DY = 2.0 IF (YMAX .GT. 14.) DY = 5.0 IF (YMAX .GT. 35.0) DY = 10.0 YLMAX = Y (J) IF ( Y(K) .GT. Y(J)) YLMAX = Y(K) 171 DX =0.5 IF ( YLMAX -GT. 1.5) DX = 1.0 IF ( YLMAX .GT. 3.0 ) DX = 2.0 IF ( YLMAX .GT. 6.0 ) DX = 5.0 IF ( DX .GT. DY ) DY = DX YMIN =-7.0* DY YMAX =*3.0* DY . , ; ; CALL AXIS ( 0.0,0.0,15HL0G C - B.LOG T , +15 , 10.0, 90.0, 1 YMIN, OY ) CALL SYMBOL (1.0, 1.0, 0.21, TITLE, 0.0, 28 ) CALL SYMBOL (1.0, 0.5, 0.21, 15HSTARTING TIME = , 0.0, 15 ) CALL NUMBER( 4.0, 0.5, 0.21, TST, 0.0, 2 ) 10 00 12 J4 = J , K 12 ~Y(J4) = Y(J4)/ DY + 7.0 CALL SYMBOL(XTPL0(J)»Y(J), 0.14 , 3 , 0.0, -1 ) KS= J +1 DO 11 M = KS , K IF (YL(M) .LT. (-9.0V) GOTO 11 IF ( Y(M).GT.10.0 .OR. Y(M).LT. 0.0 )G0 TO 13 CALL SYMBOL(XTPL0(M) ,Y(M) ,0.14 , 3 ,0.0 , -2 ) GO TO 11 (_2) 13 IF ( Y(M).GT.IO.O) CALL SYMBOL (XTPLO (M), 10. , 0. 14 , 7 , 0.0 ,) • IF ( Y(M).LT. 0.0 )CALLSYMBOL(XTPLO(M),0.0,0.14 , 5 . 0.0 ,) 11 CONTINUE ^26^ _ CONTINUE . . CALL PLOT ( 16.0 , 0.0 , -3 ) RETURN END SUBROUTINE TAILEX RUN NO. BR R2 G1UP-1-2X, AUG 15,67 STARTING TIME 14.00 TAILEX STARTS AT POINT NO 8 XTIME = 17.00 INITIAL TIME = 14.00 V R = i .no NO CONC. LOG C B*LOG T Y XT - TIN 8 38.00 3.6376 2.8332 0.8044 17.00 9._ _27-80__ 3.3250 2.9444 .. .0.3806. 19.00 10 20.20 3.0057 3.0445 -0.0388 21.00 11 14.10 2.6462 3.1355 -0.4893 23.00 1? 7.^0 1-9879 3.7958 -1.3080 27.00 13 4.50 1.5041 3.4340 -1.9299 31.00 14 3.10 1.1314 3.5553 -2.4239 35.00 15 2.40 0.8755 3.6889 -2.8134 _ 40.00 „ 16 1.80 0.5878 3.8067 -3.2189 45.00 17 1.00 0.0 4.0073 -4.0073 55.00 INITIAL TIME 14.00 B = 4.00 NO '..CONC. - LOG. C _B$LOG_T__ y_ XT„.-...TIN • 8 38.00 3.6376 11.3329 -7.6953 17.00 9 . 27.80 3.3250 11.7778 -8.4527 19.00 10 20.20 3.0057 12.1781 -9.1724 21.00 11 14. 10 2.6462 12.5420 -9.8958 23.00 12 7.30 1.9879 13.1833 -11.1955 27.00 13- 50 _1.5C41 13.7359_._ -12.2319. 31.00 14 3. 10 1.1314 14.2214 -13.0900 35.00 15 2.40 0.8755 14.7555 -13.8800 40.00 16 1 .80 0.5878 15.2267 -14.6 389 45.00 17 1.00 0.0 16.0293 -16.0293 55.00 ... ... • -1 TAIL EX "... . --• -SAM PI F PRINT-OUT 174 'TAILEX' o in SAMPLE PLOT I sL BL R 29. 3-5-4X. JUNE 13. 68 STARTING TIME = 115.00 CD I o a m CM-SAMPLE PLOT BR R2 G1UP-1-2X- AUG 15.67 STARTING TIME = 14-00 176 QVEl SUBROUTINE Q VEL ( KK, TST , X , Y , TITLE, A, B, 0 ,NRWT, NIGA) C C SUBROUTINE FOR NUMERICAL INTEGRATION AND NUMERICAL EVALUATION C OF FIRST MOMENTS OF TIME-CONCENTRATION CURVES« EXTENSI ON TO C INFINITE TIME, BASED ON A DECLINE OF C SIMILAR TO A GAMMA C DISTRIBUTION,IS OPTIONAL. C .. THIS PROGRAM REQUIRES .4.. SUBROUTINES, GAUSS1, GAUSS2, AUX1, C AND AUX2 C £ INPUT [ C KK IS THE NUMBER OF DATA POINTS, CALLED K IN NACL AND DQV, C TST IS STARTING TIME, AS BEFORE, C X...AND „Y. ARE. THE T-C DATA, ..CALLED .XT AND..YC. IN NACL AND DQV , C TITLE IS AS BEFORE, C A, B, AND D ARE THE PARAMETERS OF THE GAMMA EXTENSION, C NRWT " C 0 IF RHWT TEST C 1 IF NA CL TEST, 50 LITER TANK C 2 IF NA CL.JPST,. 16 LLTER TANK J. _ C NIGA, C 0 IF A, B, D ARE NOT GIVEN C 1 IF A, B, D ARE AVAILABLE FOR EXTENSION TO INF. C C OUTPUT C..„ INTEGRALS..AND. FIRST MOM. OVER THE DATA. POI NTS, USING C FIRST AND SECOND ORDER METHODS. MEAN TIME IS (FIRST MOMENT C S STARTING TIME, TST ). C OPTIONAL, C WITH NIGA = 1, INTEGRALS AND MOMENTS WITH EXTENSION TO C INFINITE TIME. * TNT TO XT(K)«IS THE NEGLECTED PART OF C . _ THE.. INTEGRAL OVER THE GAMMA DISTRIBUTION, UP TO _.. C TIME XT(K). 'FACTOR FAM• IS THE ADJUSTMENT TO A, TO C ACHIEVE CLOSEST FIT TO THE LAST 3 DATA POIMTS. C * FACTOR A' IS THE CORRECTED VALUE OF A. A*FAM. C WITH NRWT =1 OR 2, THE PROGRAM COMPUTES THE DISCHARGE. C DIMENSI0N_XT(260I.,_„YC.(200),„X(2a) ,.Y(20) ..,_II.TLE(7.),YC0(3), 1 FACT0R(3) DOUBLE PRECISION DGAMMA , GX, G Y REAL MT, MEAN TI, MEAN T2 ,MEAN T 3 ; ._ C C ELIMINATION OF SUPERFLUOS DATA POINTS DO .9...J_=_1_^__K_K IF <X(J) .GT. TST) GO TO 10 9 CONTINUE 10 K = KK-J + 2 DO 11 Jl = 1 , K I =J1 + J -1 XT (J 1-H) = X ( I) .-..TST. 11 YC (Jl + 1) = Y "(I) XT (1) =0.0 YC (1) = 0.0 C C. INTEGRAL OF DATA POINTS , FIRST AND SECOND ORDER METHODS 177 C FIRST ORDER 1 CT MT 1 = 0.0 CT INT1 = 0.0 DO 4 J = 2 T K TRAPEZ = UYC(J-1) + YC(J))* (XT(J)- XTU-UJI/ 2.0 CT INT1 = CT INT1+ TRAPEZ - CT~MT_l....=._.C.T-MT-_.l„+__T.RA.P..E2._*_(-_XT.(.J^.l.) +„.„(.(.2 .* Y.C (J) + YC(J-l)) / 1 (3.0 *(YC(J-1)+ YC(J)))) * ( XT(J) -XT(J-i))) 4 CONTINUE FIRST M = CT MT 1 / CT I NT 1 MEAN Tl = FIRST M + TST C C SECOND-ORDER 1 .._ _ ^ ... KTEST =(K / 2)* 2 KL IM = K - 1 CT INT2 a 0.0 ; CT MT 2 = 0.0 IF (KTEST .NE. K ) KLIM = K „ DO 5 J_=_3_,_.KLI M » 2 .. DTI = XT (J-l) - XT(J-2) DT2 = XT (J) - XTU-2) FO =((((DT2**?)*0Tl)/2.) - (PT2**3)/6.0) / (DT1*DT2) Fl=1(DT2**3)/(~6.0) )/ ((DT1**2) - (DT1*DT2)) t-pTI F2 = (UDT2**3)/ 3.0) -(((DT2**2)*DTI) / 2.0)) /((DT2**2V ... * . DT2 ) )... D INT = FO * YC (J-2) + FI * YC (J-l) + F2 * YC(J) CT INT2 = CT INT 2 + D INT U =. YC( J-2) / (DTI * DT2) V = YC(J-l) / (DT1**2 - DT1*DT2) W = YC(J) / (DT2**2 - DT1*0T2) CT. .MT_2—=_CT_MT_.2- .+..D INT.* (±DT_2)) 1 ( XT (J-2) +( ((U*V+W)*(DT2**4))/ 4.0 - ((U* (DTl) ? +(V * DT2) +(W * DTD) * (OT2**3) / 3.0 +(U* DTI* 3 (DT2«* 3) ) / 2.0 ) / DINT ) , 5 CONTINUE IF (KTEST .NE. K ) GO TO 6 TRAPEZ =_(.{y.C-<K-l.J*„yC.(K)..l*UX.T..-.(.Kl==_XT-(K-=l)).).../-_2.0 CT INT 2 = TRAPEZ + CT INT 2 CT MT 2 = CT MT 2 + TRAPEZ * (XT(K-l) + (2.0*YC(K) +YC(K-1))/ 1 (3.0* (YC(K) + YC(K-H)) * (XT(K) - XT(K-I))) 6 FIRST N = CT MT2 / CT INT 2 MEAN T2 = TST + FIRST N C WRITE RESULTS (MEANT2 WRITE (6, 7) CT INT 1 • FIRST M » MEAN Tl , CTINT2,FIRSTN,) V 7 FORMAT ( 32H11NTEGRATION OF MEASURED POINTS . / 1 19H0INTEGRAL CT1 t F15.5 t 10HPPB * MIN t / 2 19H FIRST MOMENT (1)= t F15.5 * 4HMIN T / . ..... 3 .. ... 19 H _M EAN_t.IJM.E_.' (1 ) =— -•?- -F15.5 4HMIN „ ... f / 4 19H INTEGRAL CT2 t F15.5 » 10HPPB * MIN f / 5 19H FIRST MOMENT (2)= t F15.5 • 4HMIN t / 6 19H MF AN TIMF (?) t. F15.5 t 4HMTN 1 (INF. C INTEGRATION OF DATA POINTS COMBINED WITH FITTED EXTENSION TO) IF (NIGA .EQ. 0 ) GO TO 3 178 C CORRECTION FACTOR K3 = K - 3 DO 8 J = 1 , 3 I = K3 + J YCO (J) = A*(Xt(I)**B) * EXP< -D*XT(I)) 8 FACTOR(J) = YCIII / YCO(J) - F A M_=JJj? A C.TO.R_. . ).„_+.„ 2...0„JL., J£A CT OR (2 ) _t_3.. 0_*F A CT OR (3 I.) / 6.0 C C INTEGRATION OF GAMMA DISTRIBUTION R = B + 1.0 GX = R GY = DGAMMA(GX) .. G = GY _I __ GAMMA = A * G / (D**R) XMOl = R / D CTINTF = (D **".* ) / G FIRST3 = 0.0 XINT= XT IK) / 20.0 CTINT3 =0.0. 1 ._ ; DO 12 J = 1 ,20 COUNT = J XUL = XT(K) - XINT * (COUNT -1.0) , XLL = XUL - XINT CALL GAUSSK XLL , XUL , DB, A, B , D) CAL L. .GAUSS 2 (._ XL.L_i_X UL__t_D A.t..A B, D.) CTINT3 = CTINT3 + DB FIRST3 = FIRST3 + DA 12 CONTINUE ; DEBCT = ( CTINT3 * A ) / GAMMA D INT3 =(GAMMA - CTINT 3* A ) * FAM DFM03 = ((XMOl * GAMMA) - (FIRST3 * A)) * FAM CT INT 4 = CTINT1 + DINT3 FIR m =(CTMT1 + DFM03 ) / CT INT .4 CT I NT 5 = CTINT2 + DINT3 FIR M5 =(CTMT2 + DFM03 ) / CT INT 5 MEAN T3 = FIR M5 + TST C WRITE RESULTS " WRITE(6,13) DINT 3 , DFMO 3 , CTINT4 , FIRM4 , CTINT5, FIRM5 1 , MEAN T 3 , DEBCT, FAM 13 FORMAT ( 45H0INTEGRATION OF DATA POINT WITH FITTED EXT. , / I 18H0AREA CORR. , F15.5, 7HPPB*MIN , / 2 18H_.FIR.SX_.MaM.-.C0RR.=_ ,. F15.5, . 15H MIN**.2^*„PPB ,// 3 18H AREA BY TRAPEZ = , F15.5, 7HPPB*MIN , / 4 18H FIRST MOM. (TR) = , F15.5, 7HMIN , / 5 18H AREA BY PARAB. = , F15.5, 7HPPB*MIN , / 6 18H FIRST MO. BY PA.= , F15.5, 7HMIN / 7 18H MEAN TIME 3 , F15.5 , 7HMIN // 8 ... 18H INT. TO XT(K) =.. ,F15.5 __../_ _.. 9 18H FACTOR FAM , F15. 5 ) A = FAM * A WRITE (6,16) A 16 FORMAT { 18H FACTOR A = , F15.5 ) 3 CONTINUE C _ 179 COMPUTE DISCHARGE OF SALT TESTS. IF ( NRWT .EQ. 0 ) RETURN DISCH = 0.0 IF ( NRWT . F Q . 1 .AND. NIGA .EQ. 0 10ISCH = 833.3 / CTINT2 IF { NRWT .EQ. 2 .AND. NIGA .EQ. 0 JDISCH = 833.3 /(3. * CTINT2) IF ( NRWT .EQ. 1 .AND. NIGA .GT.O ) DISCH = 833.3 / CTINT 5 ~.IF_J N R W.T__. E Q. ..1.2—. AN D. __N J G A„... GT... 0....J _DJ.S.C H _.= „8 3 3 . 3_ 11 3 .*CTINT5 ) IF ( DISCH .GT. 0.0 ) WRITE (6,15) DISCH FORMAT ( 18H0DISCHARGE = , F15.5 , 17H CUBIC M PER SEC.) RFTURN ; ; ; END .' .. ' 180 SUBROUTINE GAUSS1 (A, B, AREA, XA, XB, XD) C C SUBROUTINE IN FORTRAN /360 CALLED BY SUBROUTINE QVEL. C DIMENSION AXI4), H(4) DOUBLE PRECISION AX, H -AXi.l)_.^Jl..a6Q2B9_8.56.49753^ AX{2) = 0.796666477413627 AX(3) = 0.525532409916329 AXt4) •= 0.183434642495650 HQ) = 0.101228536290376 H(2) =0.222381034453374 . H(3) =„0.313706645877887 H(4) = 0.362683783378362 P = (B+A)*0.5 Q = (B-A)*0.5 SUM = 0.0 DO 30 J = 1,4 R = AX(J)*Q ___ _ _ X = P+R CALL AUXl (X, Y, XA, XB, XD) Z ' = Y X = P-R CALL AUXl (X, Y, XA, XB, XD) 30 _SUM_j=.„.SUM_+_H ( J ) * ( Z+Y ) J J :_. AREA = 0*SUM RETURN END " SUBROUTINE AUXl ( X, Y,A, B, D) C C CALLED BY GAUSS 1. C : Y = ( X ** Bl' * EXP( (- D)* X ) RETURN END -• '• " - " • '•• 181 SUBROUTINE GAUSS2 (A, B, AREA, XA, XB, XO) C C SUBROUTINE IN FORTRAN /360 CALLED BY SUBROUTINE QVEL. _c 1 ; : DTMENSIGN AXC4), H(4) DOUBLE PRECISION AX, H A XXI .)._=_0..9 60.285.8 56 4973 3.6 _ AX(2) = 0.796666477413627 AX(3) = 0.525532409916329 AX(4) = 0.183434642495650 HID = 0. 101228536290376 H(2) =' 0.222381034453374 _ _ H.C3.)_S--Q^313706.645.877 887. : H(4) =0.362683783378362 P = (B+A)*0.5 0 = IB-A)*0.5 ; SUM = 0.0 DO 30 J = 1,4 R_ = . AX( j)#o. ;j . . X = P+R CALL AUX2 (X, Y, XA, XB, XD) i = Y : ; X = P-R CALL AUX2 (X, Y, XA, XB, XD) .30 ... - SUW_=_SUM._+_.H.U.)-*XZ*Y.)._, AREA = Q*SUM RETURN EMU 1 SUBROUTINE AUX2 ( X , Y , A , B , D ) C C CALLED BY GAUSS2. Y = (X **(B+1.0)) * EXP <(-D)* X ) RETURN .... END _ — _1 X 182 INTEGRATION OF MEASURED POINTS INTEGRAL CT1 FIRST MOMENT (11= MEAN TIME (1) = INTEGRAL CT2 678.49805PPB * MIN 17.41537MIN 31.41537MIN 674.12207PPB * MIN FIRST MOMENT (2)= MEAN TIME (2) 17.38902MIN 31.38902MIN INTEGRATION OF DATA POINT WITH FITTED EXT. AREA CORR. 13.54708PPB*MIN FIRST MOM. CORR.= 930.09424 MIN#*2.* PPB AREA BY.__TRAP.EZ i FIRST MOM. (TR) AREA BY PARAB. FIRST MO. BY PA.-.6.9 2 . 04492 P_PB*_M I N. 18.41844MIN 687.66895PPB*MIN 18.39899MIN MEAN TIME 3 32.39899MIN I NT JO JKTtK). FACTOR FAM FACTOR A .0.94942. 1.21365 1.98106 1 Q V E L1 SAMPLE PRINTOUT FOR 'B R R 2 , GI UP I - 2X' INTEGRATION OF MEASURED POINTS INTEGRAL CT1 FIRST MOMENT (1) = MEAN TIME (1) INTEGRAL CT2 6153.19922PPB * MIN 96.25470MIN 211.25470MIN 6149.38672PPB * MIN J83 FIRST MOMENT (2)= MEAN TIME (2) 96.21269MIN 211.21269MIN INTEGRATION OF DATA POINT WITH FITTED EXT. AREA CORR. 459.59253PPB*MIN FIRST MOM. CORR.= 176856.87500 MIN**2 * PPB AREA. BY. TRAPEZ FIRST MOM. (TR) AREA BY PARAB. FIRST MO. BY PA. .6612.78906PPB*MIN . 116.30965MIN 6608.97656PPB*MIN 116.28214MIN MEAN TIME 3 231.28214MIN INT TO XT(K) FACTOR FAM FACTOR A .0.85202. 0.99531 0.44722 DISCHARGE 0.12609 CUBIC M PER SEC, Q V E L' SAMPLE PRINTOUT FOR J BL. R 2 9, G 3 - 5 - 4X' INTEGRATION OF MEASURED POINTS 184 INTEGRAL CT1 FIRST MOMENT < 1): MEAN TIME (1) INTEGRAL GT2 268.61548PPB * MIN 9.89905MIN 24.89903MIN 268.43823PPB * MIN FIRST MOMENT (2); MEAN TIME (2) 9.90452MIN 24.90451MIN DISCHARGE 3.10425 CUBIC M PER SEC 'Q V E L* _.. SAMPI F PRINTOUT FOR 'PH R2 8, G3- 4 - 6X' 185 PI 0 T fi A SUBROUTINE PLOTGA (Kt TST,XT IN,YCIN,A, B, D, TITLE ) C C THIS SUBROUTINE IN FORTRAN /360 IS CALLED BY THE MAIN C PROGRAMS NACL AND DQV TO PLOT THE C-T CURVES . OPTIONALLY C IT WILL ALSO PLOT THE GAMMA EXTENSIONS. IN THIS CASE IT SHOULD C BE CALLED AFTER THE SUBROUTINE QVEL HAS BEEN CALLED, AS QVEL C IMPROVE.S__THE..ES_TIMAXE_OF_A. C DIMENSION XT(200),YC(200), TITLE(7) , XTINI200), YCIN(200) DO 11 I =1 , 200 ' XT(I) = XTIN(I) 11 YC(I) = YCINU) CALL. SCAL E (XT, K, 10.0,.. XTMIN, DXT, 1 ) CALL SCALE ( YC, K, 9.0, YCMIN, DYC, 1 ) (DXT) CALL AXIS (0.0, 0.0, 15HTI ME IN MINUTES, -15, 13.0,0.0, XTMIN,) CALL AXIS (0.0, 0.0, 20HC0NCENTRATION IN PPB, »20 , 9.0,) 1 YCMIN , DYC ) (90.0, DO 1 J = 1, K 1 CALL . SYMBOL .(XT.(J), _YC( JL.„,.„0..1A. , 2. ,. „0. 0,-1 .) CALL SYMBOL (6.0, 9.0, 0.28, TITLE, 0.0, 30 ) CALL SYMBOL (6.0, 8.5 , 0.14 , 5HTST = , 0.0 , 5 ) CALL NUMBER (7.5 , 8.5 , 0.14 , TST , 0.0 , 2) IF (A .EQ. 0.0 ) GO TO 6 CALL PLOT (0.0 ,0.0, +3) ...YMAX__=.LY.CM.IN_£OYXJ_•__.9...0. : : 00 2 I = 1, 131 F = I- 1 X . = F / 10.0 T = XTMIN + X * DXT - TST IF { T .LT. 0.0 ) T = 0.0 Y__=( {... A*(_t_**.. Bl)*. EXP(-D* T ) -YCMIN ) / DYC IF ( Y .LE. YMAX ) GO TO 5 CALL SYMBOL ( X ,9.0, 0.07 , 13 ,0.0, -2 ) GO TO 2 5 CALL PLOT ( X ,Y , + 2 ) 2 CONTINUE CALL SYM80L._ (...6 ,0..,_ 8.0...,__0. 14. .. 24HGAMMA ..PARAM. A, B, D = 1 0.0 , 24 ) CALL NUMBER ( 9.0 , 8.0 , 0.14 , A , 0.0 , 6 ) CALL NUMBER ( 10.3 , 8.0 , 0.14 , B , 0.0 , 2 ) CALL NUMBER ( 11.6 , 8.0 , 0.14 , 0 , 0.0 , 6 ) 6 CALL PLOT (15.0, 0.0, -3 ) RETURN • END PH R 28. 3-4-6X. SEPT. 17. TST = 15.00 A A A A A A A A A A A 'PLOT G A* SAMPLE PLOT A A 18.0 ,A A A—,_A ,— SO.O 58.0 66.0 TIME IN MINUTES 10.0 26.0 I 34.0 42.0 -1 74.0 B2.0 90.0 I 9B.0 -r 106 A A A A A A A A A A A BL R 29. 3-5-4X. JUNE 13. 68 TST = 115.00 GAMMA PARAM. A. B. D 0.447722 1.00 0.012499 'PL 0 T G A' SAMPLE PLOT A 80.0 120.0 160.0 280.0 TIME 320.0 360.0 IN MINUTES BR R2 G1UP-1-2X. RUG 15.67 J TST = 14.00 GAMMA PARAM. A. B. D = 1.981554 1.00 0.086499 A A ' P L 0 T G A* SAMPLE PLOT A 189 L0GRE ' r c C THIS FORTRAN /360 PROGRAM COMPUTES THE LINEAR REGRESSIONS C ON LOG Q (DISCHARGE) OF THE FOLLOWING VARI ABLE S= C LOG TM (MEAN TRACER TRAVEL TIME) C LOG A ( CROSSECTIONAL AREA) C LOG V (VELOCITY) C LOG TS _( STARTI N.G.TIME) . C LOG TP (PEAK TIME) C LOG TSS (STARTING TIME, BUT OMITTING RUNS WITH TRACER C INJECTION ABOVE THE REACH ) C LOG TPP (PEAK TIME, OMITTING RUNS AS FOR TSS) C THE ACTUAL REGRESSION ANALYSIS IS DONE BY A SUBR. •REGR* . C TWO PLOTTING.SUBROUTINES ..CAN. ALSO. BE. CALLED_FROM THIS PROGRAM. C C INPUT C FIRST CONTROL CARD, ONE PER JOB SUBMISSION, = C NO. OF SETS, (12). C SECOND CONTROL CARD, ONE PER DATA SET, = C TITLE, (6X, ?A4) C DATA CARDS, ONE PER TEST RUN, = C RUN NO. (12) COL 1 £ 2 C IDENTIFICATION (ID COL 5 J C THIS IS I FOR RUNS WITH INJECTION OF TRACER AT UPSTREAM C END OF TEST REACH, 0 FOR OTHER RUNS. C DATA .._ . (6F6.0) COL 7 . £ ON, (Q, TS , TP , TM , A , V ) C Q IN L/S, TIMES IN MIN.,A IN SO M, C V IN M/S, 0 IS CONVERTED TO CU M/S. C C OUTPUT C PRINTOUT OF DATA, C LINEAR REGRESSION EQUATIONS, STANDARD ERROR OF ESTIMATE, C CORRELATION COEFF., DEGREES OF FREEDOM, F-RATIO. C CALL PLOTS DIMENSION NOI30), ID (30) , Q(30), TS(30), TP (30), TM(30), 1 A(30), V (30), TSSI30) , TPP (30), TIT (7 ) ,QQ(30) 2,Q1 (30) .. C C. LOOP FOR NUMBER OF SETS , READ (5, 1) KTOT 1 FORMAT (12) DO 2 KS=1, KTOT C READING AND PRTNTING OF ~DATA~ READ (5,3) TIT 3 FORMAT (6X,7A4) DO 4 K - 1, 30 ~~ (V(k) READ (5, 5) NO(K), ID(K) , QIK), TS(K), TP(K), TM (K) , A (K) ,) 5 FORMAT (12, 2XT. I I, IX, 12F. 6.0 ) i IF (NO.(K).LE. 0 ) GO TO 6 4 CONTINUE 6 K = K -1 • WRITE (6, 7) TIT 7 FORMAT (1H1,7A4,/ 1 58H0NO..I0. Q.J L/S) TS TP_ TM. A v/) 190 WRITE (6, 8) <N0(I1), 10(11), QUI), TSUI), TPU1), TMQ 1) » 1 ACID , V(I1), II = 1, K ) 8 FORMAT (IH , 1 12, 13, F10.2 , 3F9.2 , 2F9.4 ) r KID =0 c TRANSFORMATION TO. LOGS...... ..EVALUATION..OF. _.N0.._D£ SIMPLE. RUNS DO 9 12 = 1, K Q( 12) = ALOG10 (QU2J/1000.) Q1U2) = Q(I2) TS(I 2) = AL0G10 (TS(12)) TP(12) = AL0G10 (TP(I 2)) TM ( 12) = ALOG10 (TM (1 2 ) ) .. A(12) = ALOG10 (A(I 2)> VU2) = AL0G10 (V(I 2)) 9 KID = KID + ID (12) C C REGRESSIONS QN ORIGINAL DATA 10 ..FORMAT „( 1H0 /._1_8H LOG TM.VS. LOG Q ) . WRITE (6,10) CALL REGR ( Q , TM , K ,ATM , BTM ) 11 FORMAT (1H0 / 18H LOG A VS. LOG Q ) WRITE (6, 11) CALL REGR ( Q , A , K ,AA , BA ) 12 FORMAT J1H0 / 18H LOG V VS. LOG Q ) WRITE (6, 12) CALL REGR ( Q , V , K ,AV , BV ) IF (KID .EQ. K ) GO TO 18 13 FORMAT (1H0 / 18H LOG TS VS. LOG Q ) WRITE (6, 13) CALL REGR ( Q , TS , K ,ATS , BTS ) 14 FORMAT (1H0 / 18H LOG TP VS. LOG Q ) WRITE (6, 14) CALL REGR ( Q ", TP , K ,ATP , BTP ) r IF (KID .EQ. 0 ) GO TO 20 L C TP AND TS REGRESSIONS __ 18 NID =0 DO 15 15= 1 , K IF (ID (15 ) .LE. 0 ) GO TO 15 NID = NID + 1 TSS(NID) = TS 115 ) TPP(NID) = TP (15 ) .„ _ . QQ(NIO) = Q(15) 15 CONTINUE IF(NID .NE. KID ) WRITE (6, 16) KS 16 FORMAT (1H1, 17H NID ERROR IN SET , 12 ) 17 FORMAT (1H0 / 18H LOG TSS VS. LOG Q ) WRITE (6, 17) '_ : CALL REGR ( QQ, TSS, KID , ATSS , 8TSS : 'k 19 FORMAT (1H0 / 18H LOG TPP VS. LOG Q ) WRITE (6, 19) CALL REGR ( QQ, TPP, KID , ATPP , BTPP ) GO TO 21 191 C CALLS TO PLOTTING SUBROUTINES C 20 CONTINUE (BTM, CALL TPLO I Q , TS , TP, TH , K , K , ATS, ATP, ATM, BTS, BTPT) 1 TIT,Q) GO TO 22 21 CONTINUE. _ ... . _ ... . (BTSS, CALL TPLO ( Q , TSS ,TPP, TM , K , KID , ATSS, ATPP , ATM ,) 1 BTPP , BTM , TIT , QQ ) 22 CONTINUE ' CALL HYPLO IQ1, A , V , K ,AA , AV , BA , BV , TIT ) C 2 CONTINUE.• _ CALL PLOTNO STOP END SUBROUTINE REGR (X,Y,N , A » B ) 192 A SUBROUTINE IN FORTRAN /360 CALLED BY THE MAIN ROUTINE LOGRE IT COMPUTES THE REGRESSION OF Y ON X AND PRINTS THE RESULT DIMENSION X(100),Y(100) . . SUMX=O.O _._ ._ SUMY=0.0 SUMX2=0.0 SUMY2=0.0 SUMP=0.0 DO 1 J=1,N SUMX = SUMX+X( J) ... _. SUMY=SUMY+Y(J) SUMX2=SUMX2+X(J)**2 SUMY2=SUMY2+Y(J )**2 1 SUMP=SUMP+X(J)*Y(J) AN=N . SSX=SUMX2-{.SUMX**2)/AN •.. _ SSY=SUMY2—(SUMY**2)/AM SP=SUMP-(SUMX*SUMY)/AN B=SP/SSX A=(SUMY/AN)-(B*(SUMX/AN)) R=SP/(SQRT(SSX*SSY)) WRITE (6,4) SSX,SSY,SP 4 FORMAT (// 4H SSX,F12.4,4X,4H SSY,F12.4,4X,3H SP,F12.4 //) WRITE (6,5)A,B 5 FORMAT (26H REGRESSION EQUATION Y= ,F10.4,2H +, FI0.4,2H X) S = SQRT ( (SSY - SP*SP/SSX) / (AN - 2.0 ) ) WRITE (6, 9) S 9 FORMAT (27H STANDARD ERROR.0F_ EST I MATE ,.F10.4 ) WRITE (6,6) R 6 FORMAT (24H CORRELATION COEFFICIENT , F10.4 ) NDF = N - 1 WRITE (6, 7) NDF 7 FORMAT ( 24H DEGREE OF FREEDOM ,110) F = ( R*R*( AN . - 2.0) ) / ( 1.0 R*R) _._ WRITE (6, 8) F 8 FORMAT ( 4H F = , 20X, F10.4) RETURN END 193 SUBROUTINE TPLO ( Q, TSS, TPP, TM , K ,KID , ATSS , ATPP ,ATM, 1 BTSS , BTPP , BTM , TIT , QQ ) (LOGRE THIS FORTRAN /360 SUBROUTIN IS CALLED FROM THE MAIN PROGRAM) IT PLOTS THE REGRESSIONS OF TSS, TPP, TM ON Q , INCLUDING DATA POINTS. DI MENS I ON ol 30 ) , TSS ( 30 F, TPP ( 30)", TM (30),T ( 90 ) , TI T (7) I , QQ(30) SCALE OATA DO 1 I - 1, K T ( I ) ._ = TM .( I ) DO ION = 1, KID K5 = N + K T( K5) = TPP(N) Nl = K + KID K6 = N + Nl T( K6_) = TSS(N)_ _ __. N2 =N1 + KID CALL SCALE ( t, N2 , 5.0 , TMIN , DT , 1 ) K7 = K + 2 * KID DO 100 J5 = 1 , K7 T(J5) = T( J5) + 3.0 DO 2 J_= 1, K TM(J) = T (J) DO 11 Jl = 1, KID K7 = Jl + Nl TSS (Jl) = T ( K7 ) K8 = Jl + K TPP (Jl) = T(. K8 ) CALL SCALE ( Q, K , 5.0 , QMIN , OQ , 1 ) DO 200 JJ7 = 1 , KID QQ(JJ7) ( QQ(JJ7) - QMIN ) / DQ DRAW AXIS (DO) CALL AXIS ( 0.0, 3.0,16HQ IN CU M / SEC , -16, 5.0 , 0 . , QMll^) CALL AXIS ( 0.0, 3.0,12HTIME IN MIN. , +12 , 5.0 ,90^ PLOT POINTS (TMIN , DT) DO 311 = 1 , K . CALL SYMBOL ( Q (II) , TM(I1), 0.07, 2 , 0.0, -I ) DO 4 12 = 1 , KID CALL SYMBOL ( QQ(I2) , TSS(I2), 0.07_, 3 , 0.0 , -1 ) CALL SYMBOL I QQ(12) , TPP(I2), 0.07 , 4 , 0.0 , -1 ) PLOT REGRESSION LINES TM VS Q LINE YB = (((ATM + BTM* QMIN). -. TMIN).. / DT) +J3.0 XB = 0.0 IF < YB .LE. 8.0 ) GO TO 101 YB = 8.0 : . - . XB = ( 5.0 * DT + TMIN -• ATM - QMIN* BTM ) / ( DQ * BTM) YE = (((ATM + BTM* (QMIN + 5.0* DQ)) - TMIN) / DT ) + 3.0 XE = 5.0 194 IF ( YE . GE .3.0 ) GO TO 102 YE = 3.0 XE = ( TMIN - ATM - QMIN * BTM J / ( DQ * BTM ) 102 CALL PLOT < XB , YB , +3 ) CALL PLOT (XE , YE , +2 ) C c TS VS Q LINE . YB = (((ATSS+BTSS* QMIN) - TMIN) / DT) + "3.0" XB = 0.0 IF I YB .LE. 8.0 ) GO TO 103 YB = 8.0 XB = ( 5.0 * DT + TMIN -r ATSS - QMIN* BTSS ) / ( DQ * BTSS ) 103 YE = ( ( ( ATSS+BTSS* .(QMIN ..+ 5.0*. DQ) ) - TMIN) / DT. » +3.0 XE = 5.0 IF ( YE . GE . 3.0 ) GO TO 104 YE = 3.0 XE = ( TMIN - ATSS - QMIN * BTSS ) / { DQ * BTSS ) CALL PLOT (XB , YB , +3) CALL PLOT (XE ... , YE , +2).. . C TP VS Q LINE YB = (UATPP + BTPP* QMIN) - TMIN ) / DT ) +3.0 XB = 0.0 IF ( YB .LE. 8.0 ) GO TO 105 YB _„=.._8.0. __ .. .. ... XB = ( 5.0 * DT + TMIN - ATPP - QMIN* BTPP ) / ( DQ * BTPP ) 105 YE = IUATPP + BTPP* (QMIN + 5.0* DQ)) - TMIN ) / DT ) +3.0 XE = 5.0 IF { YE . GE . 3.0 ) GO TO 106 YE = 3.0 XE = ( TMIN. -..ATPP - QMIN .*... BTPP..).../ ( DQ * BTPP ) 106 CALL PLOT (XB , YB , +3 ) CALL PLOT (XE , YE , +2 ) C C WRITE TITLE AND LEGEND CALL SYMBOL ( 0 .5 , 9 . 0 , 0 . 2 1, TIT , 0 . 0 , 28 ) CALL SYMBOL ( .1 .0 ., 1.5 , .0.07 , 2.., .0.0. , -1). CALL SYMBOL ( 1.0 ,1.0 , 0.07 , 4 , 0.0 , -I ) CALL SYMBOL ( 1.0 ,0.5 , 0.07 , 3 , 0.0 , -1 ) CALL SYMBOL ( 1.5 ,1.5 , 0.14 , 9HMEAN TIME , 0.0 , 9 ) CALL SYMBOL ( 1.5 ,1.0 , 0.14 , 9HPEAK TIME , 0.0 , 9 ) CALL SYMBOL ( 1.5 ,0.5 , 0.14 , 13HSTARTING TIME , 0.0 , 13) C : C COMPLETE OUTLINE , MOVE ON CALL PLOT ( 0.0, 8.0, +3 ) CALL PLOT ( 5.0, 8.0, +2 ) CALL PLOT ( 5.0, 3.0, +1 ) CALL PLOT (11.0, 0.0, -3 ) RETURN .„.;„ END 195 SUBROUTINE HYPLO (Q , A , V , K , AA , AV, BA , BV , TIT ) C (LOGRE C THIS FORTRAN /360 SUBROUTINE IS CALLED FROM THE MAIN PROGRAM ) C IT PLOTS THE REGRESSIONS OF A, AND V t ON Q . C DIMENSION Q(30), A(30) , V(30) , TIT (7) C _ .._ C SCALE DATA CALL SCALE I Q , K , 5.0, QMIN , OQ ,1 ) CALL SCALE ( V , K , 5.0, VMIN , DV ,1 ) CALL SCALE ( A , K , 5.0, AMIN , DA ,1 ) C C .. DRAW.AXIS. _' (QMIN, CALL AXIS ( 0.0 , 3.0 , 16HQ IN CU M / SEC ,-16 ,5.0, 0.0, I 1 DQ ) (AMIN. CALL AXIS (0.0, 3.0 , 14HAREA IN SQ M. , H4 ,5.0,90.0,; 1 DA ) (VMIN. CALL AXIS (5.0, 3.0 , 17HVEL0CITY IN M/SEC , — 17,5.0,9 U. U,) 1 0V..1 „_ C C PLOT POINTS DO 1 I = 1 , K A(I) = A(I) + 3.0 V( I) - V ( I) + 3.0 CALL SYMBOL ("_Q( I )..,.. A(I)_,0.07 , 2 0. 0_,_-1 ) 1 CALL SYMBOL ( Q(I) , V(I) ,0.07 , 3 , 0.0 , -1 ) C C PLOT REGRESSION LINE C C Q VS. A LINE YB = (((AA.+ BA*„QMIN) -AMIN ) /OA )+3.0 _.. XB = 0.0 IF ( YB .GE. 3.0 ) GO TO 101 YB = 3.0 XB = (AM IN - AA - QMIN * BA) / (DQ * BA) 101 XE = 5.0 YE = ((( AA + BA*( QMIN +5.0 * DQ)) -AMIN) A OA ) + 3.0 IF ( YE .LE . 8.0 ) GO TO 102 YE = 8. 0 XE = (5.0 * DA + AMIN - AA - QMIN* BA) / (OQ * BA ) 102 CALL PLOT ( XB , YB , + 3 ) CALL PLOT ( XE , YE , + 2 ) c ..;„ C Q VS V LINE XB = 0.0 YB = (<( AV + BV*QMIN ) -VMIN) /DV ) + 3.0 IF ( YB .GE. 3.0 ) GO TO 103 YB = 3.0 XB = ( VMIN - AV - QMIN * BV. ) / (DQ * BV) 103 XE = 5.0 YE = ((( AV + BV *(QMIN +5.0 * DQ)) -VMIN )/ DV ) + 3.0 IF ( YE .LE. 8.0 ) GO TO 104 YE = 8.0 XE = ( 5.0 * DV + VMIN - AV -QMIN *BV ) / ( DQ * BV) 104 CALL PLOT ( XB:.,__YB ., +3.J > 3r9-6 r CALL PLOT ( XE , YE , +2 ) c TITLE AND LEGEND CALL SYMBOL ( 0.5, 9.0, 0.21, TIT , 0.0 , 28 ) CALL SYMBOL ( 1.0, 1.6, 0.14, 2 , 0.0 , -1 ) CALL SYMBOL (1.0, 1.1, 0.14, 3 , 0.0 , -1 ) ... CALL SYMBOL. I 1.5, 1.5, .0.14, . ..9HFL0W AREA. ., . .. 0.0. , .9 .) r CALL SYMBOL ( 1.5, 1.0, 0.14 , 9HVEL0CITY , , 9 ) C COMPLETE OUTLINE , MOVE ON CALL PLOT i 0.0 , 8.0 , +3 ) CALL PLOT I 5.0 , 8.0 , +2 ) . CALL PLOT.. ( 8.0 , 0.0 , -3.) ... RETURN END BROCKTON CK, REACH 1-2 197 > NO ID 0 (L/S) TS TP TM A V 2 1 9.20 17.00 25.50 30.10 0.1400 0.0660 4 1 5.60 24.00 37.30 44.50 0.1260 0.0450 7 0 0.67 110.00 180.00 193.00 0.0650 0.0103 9 0 0.16 350.00 540.00 560.00 0.0450 0.0035 ... .10. _1 __1...2.3 .7.0.00 l.l.7.._00_..'. 1159... 5.0. -.0..Q99Q ... _ 0.0125 16 1 58.00 4.90 8.10 9.08 0.2660 0.2180 18 1 50.00 7.20 11.60 12.58 0.3170 0. 1580 19 1 158.00 2.50 5.00 5.40 0.4300 0.3680 21 1 92.00 4.20 7.30 8.33 0.3860 0.23 80 23 1 137.00 3.20 5.50 6.97 0.4810 0.2850 LOG TM VS. LOG Q SSX 10.1903 SSY 4.5008 SP ^6.7536 REGRESSION EQUATION Y= 0.2056 + -0.6628 X STANDARD ERROR OF ESTIMATE 0.0558 CORRELATION COEFFICIENT -0.9972 DEGREE OF FREEDOM 9 .£,_.= . 1439.3.098— _ . :L.0 G R E COMPLETE PRINTOUT LOG A VS. LOG Q SSX 10* 15LQ.3 SSY. 1. 1856 SP ..... 3. 4399. REGRESSION EQUATION Y= -0.0913 •+ 0.3376 X STANDARD ERROR OF ESTIMATE 0.0553 CORRELATION COEFFICIENT 0.9896 DEGRE E_ OF„FR E E DOM . .9 F = 379.7510 LOG V VS. LOG Q SSX 10.1903 SSY 4.5095 SP 6.7599 REGRESSION EQUATION Y= 0.0934 + 0.6634 X STANDARD ERROR OF ESTIMATE 0.0561 CORRELATION COEFF.ICIENT. 0.9972 _„ DEGREE OF FREEDOM 9 F = 1423.0298 LOG TS VS. LOG Q SSX 10.1903 SSY 4.7987 SP -6.9778 * , 198 r REGRESSION EQUATION Y= -0.1173 * STANDARD ERROR OF ESTIMATE 0.0509 CORRELATION COEFFICIENT -0.9978 DEGREE OF FREEDOM 9 ^ F = 1841.9011 -0.6847 X ? LOG TP VS. LOG Q . SSX 10.1903 SSY 4.5419 SP -6.7840 REGRESSION EQUATION Y= 0.1390 + STANDARD ERROR OF ESTIMATE 0.0566 CORRELATION COEFFICIENT -0.9972 DEGREE OF FREEDOM 9 -0.6657 .X F = 1412.1279 LO6"YSS^S7"LOG~Q" SSX 4.0349 SSY 1.7498 SP -2.6472 RE GR E S SI g^YoljATl ON V-T"" -076787 + STANDARD ERROR OF ESTIMATE 0.0466 CORRELATION COEFFICIENT -0.9963 -0.6561 X DEGREE OF FREEDOM 7 F = 798.5081 LOG TPP VS. LOG Q SSX 4.0349 SSY 1.6039 SP -2.5322 REGRESSION EQUATION Y= 0.1896 + -0.6276 X STANDARD ERROR OF ESTIMATE 0.0496 CORRELATION COEFFICIENT -0.9954 DEGREE OF FREEDOM 7 F = __ . „ 646.0571...7 199 BROCKTON CK. REACH 1-2 A MERN TIME' PEAK TIME + STflRTING TIME" 200 BROCKTON CK. REACH 1-2 Q IN CU M / SEC A + FLOW AREA VELOCITY BROCKTON CK, REACH 2-3 NO 10 Q (L/S) TS TP 1 1 6.70 13.60 33.00 4 0 6.80 27.00 34.50 8 0 0.73 76.00 160.00 11 1 1.20 53.00 97.00 1.7. _l. 34.00 „,L 3.00 7.20 20 1 89.00 2.20 4.70 22 1 110.00 1.80 4.00 TM 35.00 34. 17 0.1740 0.1720 0.0390 0.0400 172.00 119.80 8.5.0 5.39 4.53 0.0935 0.1065 0.340.0. 0.3560 0.3690 0.0078 0.0113 0.1590 0.2500 0.2980 LOG TM VS. LOG Q SSX 4.7054 SSY 2.4174 SP -3.3716 REGRESSION EQUATION Y= -0.0129 • ST ANDARD ERROR OF. EST I MATE 0.0179. CORRELATION COEFFICIENT -0.9997 DEGREE OF FREEDOM 6 F = 7506.9609 -0.7165 X LOG...A VS. L.QJS_Q_ SSX 4.7054 SSY 0.3784 SP 1.3316 REGRESSION .EQUATION.. . Y= -0.1437 + STANDARD ERROR OF ESTIMATE 0.0179 CORRELATION COEFFICIENT 0.9979 DEGREE OF FREEDOM 6 .0.2 83.0 X F = 1182.4143 LOG V VS. LOG Q SSX 4.7054 SSY 2.4167 SP 3.3710 REGRESSION EQUATION Y= 0.1442 + STANDARD ERROR OF ESTIMATE 0.0185 CORRELATION COEFFICIENT 0.9996 0.7164 X DEGREE OF FREEDOM F = 7072.7969 LOG TS VS. LOG Q SSX 4.7054 SSY 2.7562 SP -3.5577 REGRESSION EQUATION Y= -0.4422 + -0.7561 X STANDARD ERROR OF ESTIMATE 0.1151 202 CORRELATION COEFFICIENT -0.9879 DEGREE OF FREEDOM 6 F = 203.1117 LOG TP VS. LOG Q SSX 4.7054 SSY 2.4596 SP -3.3980 REGRESSION EQUATION Y= -0.0710 + -0.7221 X STANDARD ERROR OF ESTIMATE 0.0342 CORRELATION COEFFICIENT -0.9988 DEGREE OF FREEDOM 6 F = 2102.5312 LOG TSS VS. LOG Q SSX 2.8700 SSY 1.5750 SP -2.1253 REGRESSION EQUATION Y= -0.4533 + STANDARD ERROR OF ESTIMATE 0.0194 CORRELATION COEFFICIENT -0.9996 -0.7405 X DEGREE OF FREEDOM 4 F = 4198.3047 LOG TPP VS. LOG Q SSX 2.8700 SSY 1.4477 SP -2.0364 REGRESSION EQUATION Y= -0.0606 + STANDARD ERROR OF ESTIMATE 0.0305 . CORRELATION COEFFICIENT -0.9990 DEGREE OF FREEDOM 4 F = 1557.8159 -0.7095 X - - - • — - -BROCKTON CK.'.REACH 2-3 Q IN CU M / SEC A MEAN TIME . PEAK TIME + STARTING TIME 204 PLACID CREEK, REACH 1-2 20 5 NO ID Q (L/S) TS TP TM A V 2 1 50.30 122.00 ^ 13 1 35.40 165.00 164.00 210.00 0.6600 0. 208.00 280.60 0.6180 0. 0763 0573 ? 25 1 64.60 115.00 28 1 95.90 94.00 159.00 180.60 0.7280 0. 130.00 144.20 0.8630 0. 0890 1110 ICG TM VS. LOG Q SSX 0.0997 SSY 0.0441 SP -0.0660 REGRESSION EQUATION Y= STANDARD ERROR OF ESTIMATE 1.4758 + 0.0147 -0.6622 X CORRELATION COEFFICIENT DEGREE OF FREEDOM F = -0.9951 3 203.3226 LOG A VS. LOG Q SSX 0.0997 SSY 0.0119 SP 0.0339 REGRESSION EQUATION Y= 0.2739 + 0.3403 X STANDARD ERROR OF ESTIMATE CORRELATION COEFFICIENT DEGREE OF FREEDOM 0.0141 0.9833 3 F = 58.3338 LOG V VS. LOG Q SSX 0.0997 SSY 0.0437 SP 0.0656 REGRESSION EQUATION Y= STANDARD ERROR OF ESTIMATE CORRELATION COEFFICIENT -0.2748 + 0.0147 0.9950 0.6587 X DEGREE OF FREEDOM F = 3 199.5172 LOG TSS VS. LOG Q SSX 0.0997 SSY 0.0307 SP -0.0541 REGRESSION EQUATION •Y= STANDARD ERROR OF ESTIMATE 1.4121 + 0.0257 -0.5425 X CORRELATION COEFFICIENT DEGREE OF FREEDOM I F = -0.9782 3 44.3011 206 LOG TPP VS. LOG Q SSX 0.0997 SSY 0.0210 SP -0.0448 > REGRESSION EQUATION Y- 1.6546 + -0.4498 X STANDARD ERROR OF ESTIMATE _ 0.J3202 CORRELATION COEFFICIENT -6.9T04 "~ ~ ~ ~~ ~~ ~" DEGREE OF FREEDOM 3 F = 49.3988 PLACID CREEK. REACH 1-2 .035 .04 Q IN CU M / SEC MEAN TIME PEAK TIME STARTING TIME 208 A FLOW AREA + VELOCITY PLACID CREEK, REACH 2-3 NO ID Q (L/S) TS 209 5 10 15.40 72.60 160.00 62.00 TP 236.00 94.00 TM 351.30 121.40 0^5340 0.8690 V 0.0289 0.0834 r 18 1 114.50 48.00 71.00 24 1 121.20 47.50 74.00 27 1 181.00 36.10 54.80 83.30 0. 92.10 I. 62.10 1. 9390 0. 1000 0. 1100 0. 1220 1100 1630 LOG TM VS. LOG 0 SSX 0.6955 SSY 0.3363 SP -0.48 21 REGRESSION EQUATION Y= 1.2919 + -0.6932 X STANDARD ERROR OF ESTIMATE 0.0265 CORRELATION COEFFICIENT -0.9969 DEGREE OF FREEDOM 4 F = 477.4194 LOG A VS. LOG Q SSX 0.6955 SSY 0.0674 SP 0.2131 REGRESSION EQUATION Y= 0.2855 STANDARD ERROR OF ESTIMATE 0.0266 CORRELATION COEFFICIENT 0.9842 + 0.3065 X DEGREE OF FREEDOM 4 F = 92.410LOG V VS. LOG Q SSX 0.6955 SSY 0.3356 SP 0.4815 REGRESSION EQUATION Y= -0;2868 STANDARD ERROR OF ESTIMATE 0.0270 + 0.6924 CORRELATION COEFFICIENT 0.9967 DEGREE OF FREEDOM 4 F = 456.2595 LOG TSS VS. LOG Q SSX 0.6955 SSY 0.2499 SP -0.4168 REGRESSION EQUATION Y= 1.1170 STANDARD ERROR OF ESTIMATE 0.0074 CORRELATION COEFFICIENT -0.9997 , DEGREE OF FREEDOM 4 + -Q.5993 X y 210 F = 4588.8828 LOG TPP VS. LOG Q SSX 0.6955 SSY 0.2397 SP~ -0.4077 REGRESSION EQUATION Y= 1.3101 + -0.5862 X STANDARD ERROR OF ESTIMATE 0.0146 CORRELATION COEFFICIENT -0.9987 DEGREE OF FREEDOM 4 F = 1118.6213 211 PLACID CREEK. REACH .2-3 \ MEAN TIME K PEAK TIME STARTING TIME PLACID CREEK. REACH 2-3 A FLOW AREA + VELOCITY PLACID CREEK, REACH 3-4 NO ID Q (L/S) TS TP TM A V < 17 1 14.50 674.00 18 1 245.00 188.00 1000.00 1200.00 223.00 255.30 0.5650 0. 2.0300 0. 0256 1210 ? 19 1 268.00 162.00 21 1 70.00 302.00 _24 1. 2.67.00 157.50 199.00 238.90 386.00 481.60 200.00 254.00 2.0000 0.1340 1.0950 0.0638 2.2000 0.1210 27 1 404.00 121.00 169.00 209.00 2.7400 0. 1480 LOG TM VS. LOG Q SSX 1.4894 SSY 0.4213 SP -0.7908 REGRESSION EQUATION Y= 2.0902 + STANDARD ERROR OF ESTIMATE 0.0187 CORRELATION COEFFICIENT -0.9983 -0. 5310 X DEGREE OF FREEDOM F = 5 1203.0806 LOG A VS. LOG Q SSX 1.4894 SSY 0.3250 SP 0.6934 REGRESSION EQUATION Y= 0.5958+ STANDARD ERROR OF ESTIMATE 0.0232 0. 4656 X CORRELATION COEFFICIENT DEGREE OF FREEDOM F = 0.9967 5 : -'• 599.9749 LOG V VS. LOG Q SSX 1.4894 SSY 0.4296 SP 0.7980 REGRESSION EQUATION Y= -0.5947 + 0. 5358 X STANDARD ERROR OF ESTIMATE 0.0230 CORRELATION COEFFICIENT 0.9975 DEGREE OF FREEDOM 5 F = 810.3389 LOG TSS VS. LOG Q SSX 1.4894 SSY 0.3661 SP -0.7345 \— REGRESSION EQUATION Y= 1.9227 + STANDARD ERROR OF ESTIMATE 0.0312 CORRELATION COEFFICIENT -0.9947 -0. 4931 X DEGREE OF FREEDOM F = 5 372.5400 2T? LOG TPP VS. LOG Q SSX i.4894 SSY 0.4252 SP -0.7939 REGRESSION EQUATION Y= 2.0035 + -0.5330 X STANDARD ERROR OF ESTIMATE 0.0226 . CORRELATION COEFFICIENT -0.9976 DEGREE OF FREEDOM 5 F = _826.400?_ _____ PLOT TAPE SUCCESSFULLY WRITTEN DONE STOP 0 EXECUTION TERMINATED $SIG PLACID" GREEK.REACH 3-4 4 MEAN TIME PEAK TIME STARTING TIME PLACID CREEK. REACH 3-4 o Q IN CU M / SEC ~ A FLOW AREA VELOCITY BLANEY CREEK, REACH 1-3 217 NO ID Q (L/S) TS TP TM A v 11 1 500.00 30.00 '> 44.50 V 12 1 1750.00 17.00 23.50 54.70 27.48 2.3900 4.2200 0.2090 0.4160 R 15 1 11500.00 7.50 10.60 16 1 1630.00 19.50 26.40 17 1 1600.00 18.30 25-50 12.10 30.16 29.63 12.1500 4.3200 4.2500 0.9470 0.3800 0.3870 19 1 1950.00 15.00 22.50 24 1 120.00 71.00 101.00 31 1 146.00 57.00 90.00 2 5.37 123.00 117.00 4.3200 1.2900 1.4900 0.4510 0.0960 0.0980 35 1 285.00 40.50 61.00 36 1 741.00 24.00 34.00 71.20 40.40 1.7700 2.6200 0.1550 0.2830 LOG TM VS. LOG Q SSX 3.2537 SSY 0.8994 SP -1.7018 REGRESSION EQUATION Y= 1.5854 STANDARD ERROR OF ESTIMATE 0.0340 + -0. 5230 X CORRELATION COEFFICIENT -0.9948 DEGREE OF FREEDOM 9 F = 770.1775 LOG A VS. LOG Q SSX 3.2537 SSY . . . 0.7537 SP 1.5566 REGRESSION EQUATION Y= 0.5283 + 0.4784 X STANDARD ERROR OF ESTIMATE 0.0335 CORRELATION COEFFICIENT 0.9940 DFGRFF OF FREEDOM 9 F = 661.6221 LOG V VS. LOG Q SSX 3.2537 SSY 0.8945 SP 1.6981 REGRESSION EQUATION Y= -0.5270 STANDARD ERROR OF ESTIMATE 0.0321 CORRELATION .COEFFICIENT 0.9554 • 0.5219 X DEGREE OF FREEDOM 9 F = 858.608LOG TSS VS. LOG Q SSX 3.2537 SSY 0.7642 SP. -1.5678 218 REGRESSION EQUATION Y= 1.3558 + -0.4819 X STANDARD ERROR OF ESTIMATE 0.0331 CORRELATION COEFFICIENT -0.9943 DEGREE OF FREEDOM 9 F = 690.2686 LOG TPP VS. LOG Q SSX 3.2537 SSY 0.8187 SP -1.6255 REGRESSION EQUATION Y= . . 1 .5139 + -0.4996 X STANDARD ERROR OF ESTIMATE 0.0287 CORRELATION COEFFICIENT -0.9960 DEGREE OF FREEDOM 9 • F = 985.6753 BLANEY CREEK. REACH 1-3 220 BLANEY CREEK. REACH 1-3 Q IN CU M 7 SEC A FLOW AREA + VELOCITY BLANEY CREEK, REACH 3-5 221 NO ID 0 (L/S) TP TM V 8 10 870.00 530.00 8.50 12.00 1.6.20 20.5 0 21.15 26. 31 3.3000 2.500 0 0.26 40 0.2120 11 12 13. 14 15 16 0 0 JL 1 0 0 520.00 1820.00 J.XL4..0.Q....Q.Q. 11700.00 11700.00 1650.00 21.00 11.00 3.30. 3.20 4.00 12.00 23.00 15.00 5.60 5.50 5.00 14.00 22.5 5 15.67 6. 24 6.45 17.62 2.1400 5.0300 13.0000 13.500 0 5.2100 0.24 30 0. 35 3C _0_._7LQJQ-0.9000 0.8700 0.3170 19 0 20 1 22 1_ 25 1 26 1 29 1 2050.00 530.00 1 60.. 0.0. 682.00 262.00 140.00 11.00 10.00 2 COO. 9.50 16.00 21.20 12.00 18.20 3A..,.Q_0_. 16.60 26.00 38.50 12.62 24.78 45.5.0... 20. 85 34.20 47. 85 4.630 0 2.3500 .:1_.3JQ0JL. 2.5500 1.6000 1.200 0 0.44 30 0.22 60 .0. 1 ?3 0 6.2680 0.1640 0.1160 32 1 33 1 34 JL 37 1 80.00 80.00 .. 748...00. 1350.00 27.50 29.00 _8.90 53.0 0 54.50 15.00 7.50 12.70 71.43 71.45 15.46 1 .0200 1 .0200 2.6200 3.7300 0.0780 0.0780 .Q..28 5C... 0.3620 LOG TM VS. LOG Q SSX 7.8009 SSY 1 .7186 SP •3.633? REGRESSION EQUATION Y=. 1.2*62 + STANDARD ERROR OF ESTIMATE 0.0407 COR R..E.L AT I 0 N_ _C 0 JE F FJ CJ E_N T -0.9923 DEGREE OF FREEDOM 17 F •= 1023.4202 -0.4657 X LOG A VS. LOG Q SSX 7.8009 SSY 2.248 7 SP 4.1651 REGRESSION EQUATION Y= S T A N D A RD...E R R.OR__0 F__ ESTIMATE CORRELATION COEFFICIENT 0.9945 DEGRFE OF FREEDOM ' 17 F = 1433.6624 0.5391 + 0.0394 0.5339 X 1.0G V VS ....LOG. 0 SSX 7.8009 SSY 1 .6867 SP 3.5944 REGRESSION E.Q.UATI ON. v = -0.5423 + 0.4608 X STANDARD ERROR OF ESTIMATE 0.0437 CORRELATION COEFFICIENT 0.9909 OEGREF OF FREEDOM ' 17 222 F = 687.9802 LOG TS VS. LOG Q SSX 7.5619 SSY 1.4003 SP -3.0375 REGRESSION EQUATION Y= 1.0047 + STANDARD ERROR OF ESTIMATE 0.1061 CORRELATION COEFFICIENT -0.9334 -0.4017 X DEGREE OF FREEDOM F = 17 108.3378 LOG TP VS. LOG Q SSX 7.5619 SSY 1.5980 SP -3.4337 REGRESSION EQUATION Y= 1.2003 + STANDARD ERROR OF ESTIMATE 0.0493 -0.4541 X CORRELATION COEFFICIENT -0.9878 DEGREE OF FREEDOM 17 F = . _ 64.1.9316. LOG TSS VS. LOG Q SSX. .5.55.11 SSY. .1.1274 _S.P_ -2.4827 REGRESSION EQUATION Y= 0.9407 + -0.4472 X STANDARD ERROR OF ESTIMATE 0.0394 CORRELATION COEFFICIENT -0.9924 0EGR.EE CF.. FREEDOM _1.2 1_ F = 716.6177 LOG TPP VS. LOG Q SSX 5.5511 SSY 1.2177 SP -2.5748 REGRESSION EQUATION Y= 1.1846 + -0.4638 X STANDARD ERROR OF ESTIMATE 0.0461 C 0 RR.E LA T ION.COEFF ICIENJ ±0.. 9.904 , . DEGREE OF FREEDOM 12 F = 561.8801 BLANEY CREEK. REACH 3-5 A X + MEAN TIME PEAK TIME STARTING TIME BLANEY CREEK. REACH 3-5 2 24 T 1 1—i—r T 1—i—i r -i r O o o CM o J C3 CO CE cr o cvi .04 .06 .1 Q IN CU M / SEC A FLOW AREA + VELOCITY -A. BLANEY CREEK, REACH 5-4 225 NO ID Q (L/S) TS TP TM 10 12 0 0 570.00 1960.00 46.00 25.00 57.50 29.00 63. 10 30.80 2.3200 3.9000 0.2460 0.5030 13 0 11000.00 19 0 22C0.00 .21 ... 1 .JULQ.JOXL 23 1 28 1 29 0 125.00 280.00 140.00 13.50 24.00 38.50 98.00 56.OC 93 . 80 13.90 26.00 51.40 137.00 81.00 133.50 14.40 30.30 ...J5iu.6iL 165.00 93.90 158.00 10.2000 4.3000 _2....850.Q„ 1.3300 1.7000 1.4300 1.0800 0.5120 .0.2740 0.0940 0. 1650 0.0980 37 0 1350.00 28.50 32.70 35.90 3.1300 0.4320 LOG TM VS. LOG Q SSX 3.1444 SSY 0.9987 SP -1.7627 REGRESSION EQUATION Y= 1.6828 + -0.5606 X STANOARD ERROR OF ESTIMATE 0.0389 CORRELATION COEFFICIENT -0.9947 DEGREE OF FREEDOM 8 F = 654.3979 LOG A VS. LOG Q SSX 3.1444 SSY 0.6162 SP 1.3802 REGRESSION EQUATION Y= 0.4927 + 0.4389 X STANDARD ERROR OF ESTIMATE 0.0386 CORRELATION COEFFICIENT 0.9915 DEGREE OF FREEDOM 8 _F_= : 406.31.18... LOG V VS. LOG Q . SSX 3_, 1444 SSY 1 ...OjOi)7_ . S_P_ 1.7646. REGRESSION EQUATION Y= -0.4922 + 0.5612 X STANDARD ERROR OF ESTIMATE 0.0387 CORRELATION COEFFICIENT 0.9948 DEGREE OF FREEDOM JL_ F = 662.0354 LOG TS VS. LOG Q SSX 3.1444 SSY 676478" SP -1.4164 v. ( 869 .0930 226 LOG TS VS. LOG Q >- SSX 7.8009 SSY 1.4003 SP •3.0879 REGRESSION EQUATION Y= 1.0024 + STANDARD ERROR OF ESTIMATE 0.1055 CORRELATION COEFFICIENT -0.9343 -0.3958 X DEGREE OF FREEDOM F = 17 109.8279 LOG TP VS. LOG Q SSX 7.8009 SSY 1.5980 SP -3.4937 REGRESSION EQUATION Y= 1.1977 +• STANDARD ERROR OF ESTIMATE 0.0457 -0.4479 X CORRELATION COEFFICIENT -0.9895 DEGREE OF FREEDOM 17 F = 750.7678 LOG TSS VS. LOG Q SSX 5.7530 SSY 1.1274 SP •2.5322 REGRESSION EQUATION Y= 0.9382 + -0.4401 X STANDARD ERROR OF ESTIMATE 0.0343 CORRELATION COEFFICIENT -0.9943 DEGRE.E OF FREEDOM' . 12 F = 948.2959 LOG TPP VS. LOG Q SSX 5.7530 SSY 1.2177 SP •2.6281 REGRESSION EQUATION Y= 1.1819 + STANDARD ERROR OF ESTIMATE 0.0394 .CORR EL AT. I 0 N JC.QE F FIX J IRI - - 0. 9.13.0 DEGREE OF FREEDOM 12 F = 772. 1868' PLOT TAPE SUCCESSFULLY WRITTEN •0.4568 X DONE STOP 0 EXFCUTTON TERMINATED BLANEY CREEK. REACH 5-4 1 T 1 i r 1 1 1 r Q IN CU M / SEC A MEAN TIME ' PERK TIME + STARTING TIME 228 BLANEY CREEK. REACH 5-4 d Q IN CU "M / SEC A FLOW AREA + VELOCITY PHYLLIS CREEK, REACH 1-2 NO ID Q <L/S) TS TP TM 369.00 38.50 56.00 69.90 312.00 43.50 61.00 76.40 1.9900 1.8300 V 0.1850 0.1710 1470.00 2400.00 J.398.00. 945.00 945.00 826.00 21.25 16.50 OSL-.0JCL 24.00 22.50 23.00 28.50 22.50 28.50 33.00 33.00 35.40 36.19 27.79 36.66 39.80 38.75 46.39 4.1000 5.1500 .3...9600 2.9000 2.8300 2.9600 0.3580 0.4670 .0. 3540 0.3260 0.3340 0.2790 3480.00 14.50 20.00 25.81 6.9400 0.5020 LOG TM VS. LOG Q SSX 0.9355 SSY 0.2044 SP -0.4306 REGRESSION EQUATION Y= 1.6262 + STANDARD ERROR OF ESTIMATE 0.0296 CORRELATION COEFFICIENT -0.9849 -0.4603 X DEGREE OF FREEDOM F = 8 225.9623 LOG A VS. LOG Q SSX 0.9355 SSY 0.2802 SP 0.5064 REGRESSION EQUATION Y= 0.5135 + STANDARD ERROR OF ESTIMATE 0.0294 0.5413 X CORRELATION COEFFICIENT 0.9891 DEGREE OF FREEDOM 8 F = _ 317. 0220. LOG V VS. LOG Q SSX 0...93.55 SSY., Q.. 20.32. S.P .0.4295 REGRESSION EQUATION Y= -0.5133 + 0.4591 X STANDARD ERROR OF ESTIMATE 0.0294 CORRELATION COEFFICIENT 0.9850 DEGREE OF FREEDOM _._ 8 F = 228.4597 LOG TSS VS. LOG Q SSX 0.9355 SSY 0.1958 SP -0.4191 REGRESSION EQUATION Y= 1.3751 + -0.4480 X STANDARD ERROR OF ESTIMATE 0.0339 CORRELATION COEFFICIENT -0.9792 DEGREE OF FREEDOM 8 F = 162.962f LOG TPP VS. LOG Q • ) SSX 0.9355 SSY 0.2079 SP -0.4382 REGRESSION EQUATION Y= 1.5288 + STANDARD ERROR OF EST I.MAT E 0.0195 CORRELATION COEFFICIENT -0.9936 DEGREE OF FREEDOM 8 F = 538.0989 -0.4684 X — —~ — •. — — - - — - - — — — • — _ „ — PHYLLIS CREEK. REACH 1-2 Q IN CU M / SEC A MEAN TIME-PERK TIME STARTING TIME 232 PHYLLIS CREEK. REACH 1-2 Q IN CU M / SEC A + FLOW flREft VELOCITY PHYLLIS CREEKt REACH 2-3 NO ID Q (L/S) TS TP TM 748.00 25.00 38.00 43.52 352.00 37.50 57.00 64.76 233 2.7200 1.9100 0.2780 0.1840 228.00 1590.00 .255.Q..JD.GL 2415.00 985.00 1880.00 43.00 18.75 15.00 14.25 22.40 17.00 70.00 29.50 .21.00 21.20 33.50 24.00 87.62 28.05 23,43 24.15 40.00 26.78 1.6720 3.9300 5..O000. 4.8700 3.3000 4.2200 0.1362 0.40 50 .0.5100 0.4960 0.2980 0.4450 840.00 1194.00 3610.00 23.50 20.00 12.50 35.50 31.20 .17.. 20. 40.84 35.48 18.65 2.9600 3.5500 5.6300 0.2840 0.3360 0.6410 LOG TM VS. LOG Q SSX. .1.3517 JSSAL .0.4005 SP —0.7340 REGRESSION EQUATION Y= 1.5798 + -0.5430 X STANDARD ERROR OF ESTIMATE 0.0145 CORRELATION COEFFICIENT -0.9976 D E.GR EE_.OF__ERJE£.O.OM 10 F = 1902.6912 LOG A VS. LOG Q SSX 1.3517 SSY 0.2844 SP 0.6187 REGRESSION EQUATION Y= 0.5057 + STANDARD ERROR OF ESTIMATE 0.0115 CO R RE L AT I ON. CQEFF IC IE NT 0 .9979... DEGREE OF FREEDOM 10 F = 2144.6553 0.4577 X LOG V VS. LOG Q SSX 1.3517 SSY 0.3986 SP 0.7327 REGRESSION EQUATION Y= -0.5055 + 0.5421 X STANDARD .ERRO R OF. EST IM A T E 0 .012 2 CORRELATION COEFFICIENT 6.9983 DEGREE OF FREEDOM 10 F = 2648.800LOG TS VS. LOG Q f" f REGRESSION EQUATION Y- 1.3481 • STANDARD ERROR OF ESTIMATE 0.0135 CORRELATION COEFFICIENT -0.9971 L DEGREE OF FREEDOM 10 -0.4546 X 234 1-^ F = 1522.280r LOG TP VS. LOG Q SSX 1.3517 SSY 0.3411 SP -0.6764 f" REGRESSION EQUATION Y= 1.5257 • STANDARD ERROR OF ESTIMATE 0.0170 CORRELATION COEFFICIENT -0.9962 -0.5004 X ;•" DEGREE OF FREEDOM 10 F = 1166.0486 f- LOG TSS VS. LOG Q ( SSX 1.0294 SSY 0.2185 SP •0.4729 ( REGRESSION EQUATION Y= 1.3457 + STANDARD ERROR OF ESTIMATE 0.0148 -0.4593 X r CORRELATION COEFFICIENT -0.9970 DEGREE OF FREEDOM 7 F = 985.6196 r LOG TPP VS. LOG Q (".. SSX 1.0294 _ _SSY 0.2615 . _SP - 0,5186 REGRESSION EQUATION Y= 1.5219 + -0.5037 X STANDARD ERROR OF ESTIMATE 0.0071 CORRELATION COEFFICIENT -0.9994 DEGREE OF FREEDOM 7 F = 5139.066. c, (•• . —-1:: --- - — -r PHYLLIS CREEK. REACH 2-3 MERN TIME-PERK TIME STARTING TIME X PHYLLIS CREEK. REACH 2-3 Q IN CU M / SEC A FLOW AREA . + VELOCITY PHYLLIS CREEK, REACH 3-4 237 NO ID Q (L/S) TS TP TM V 3 5 0 1 750.00 339.00 18.00 32.00 37.00 45.00 39.56 52.46 2.8800 1.7400 0.2600 0. 1950 6 10 .13. 19 27 28 0 0 1 1 338.00 228.00 24.9il.00.. 1070.00 1200.00 3100.00 39.50 53.00 11.50 19.00 17.00 10.30 47.00 64.00 -1J&-.JB.0. 26.50 24.50 14.50 53.34 64.70 .18.83 29.88 28.52 17.33 1.7500 1.4870 Jt.5600. 3.1200 3.3300 5.2200 0.1930 0.1608 0.546Q. 6.3440 0.3610 0.5950 29 1 3690.00 8.90 13.50 15.41 5.5300 0.6550 LOG TM VS. LOG Q SSX 1.5952 SSY 0.4250 SP -0.8211 REGRESSION EQUATION Y= 1.4903 + STANDARD ERROR OF ESTIMATE 0.0184 CORRELATION COEFFICIENT -0.9972 -0.5147 X DEGREE OF FREEDOM F = 8 1248.8062 LOG A VS. LOG Q SSX 1.5952 SSY 0.3676 SP 0.7636 REGRESSION EQUATION Y= 0.4801 + STANDARD ERROR OF ESTIMATE 0.0173 0.4787 X CORRELATION COEFFICIENT 0.9971 DEGREE OF FREEDOM 8 F..= 1219.084 5 LOG V VS. LOG Q SSX. .1,5952 „__S5Y_. 0,4183. SP 0.8146 REGRESSION EQUATION Y= -0.4786 + 0.5106 X STANDARD ERROR OF ESTIMATE 0.0182 CORRELATION COEFFICIENT 0.9972 D E G R EE OF FREE DOM 81 F = 1260.845LOG TS VS. LOG Q SSX 1.5952 SSY 0.5792 SP -0.9452 238 r REGRESSION EQUATION Y= 1.2809 + -0.5925 X STANDARD ERROR OF ESTIMATE 0.0523 CORRELATION COEFFICIENT -0.9833 DEGREE OF FREEDOM 8 F = 204.5374 > : ; ; : LOG TP VS. LOG Q SSX 1.5952 SSY 0.4718 SP -0. 8616 REGRESSION EQUATION Y= STANDARD ERROR OF ESTIMATE 1.4390 + 0.0303 -0.5401 X CORRELATION COEFFICIENT DEGREE OF FREEDOM F = -0.9932 8 506.9019 LOG TSS VS. LOG Q SSX 0.7485 SSY 0.2131 SP -0. 3982 REGRESSION EQUATION Y= STANDARD ERROR OF ESTIMATE CORRELATION COEFFICIENT DEGREE OF FREEDOM 0 1.2698 + 0.0173 .9972 5 .-0..532O. _x_ :~- -F = 706 .3044 LOG TPP VS. LOG Q SSX 0.7485 SSY 0.1959 SP -o. 3825 REGRESSION EQUATION Y= STANDARD ERROR OF ESTIMATE CORRELATION COEFFICIENT 0 "1.4235 + 0.0113 .9987 -0.5109 ~x DEGREE CF FREEDOM F = 1521 5 .2705 PLGT_J AP_E_SUCCESS.F UL L Y WR ITTEN_ DONE STOP 0 EXECUTION TERMINATED PHYLLIS CREEK. REACH 3-4 A + MEAN TIME PERK TIME STARTING TIME PHYLLIS CREEK. REACH 3-4 PHYLLIS CREEK, REACH 4-6 2i|1 NO ID Q (L/S) TS TP TM A V 5 0 385.00 20.00 28.00 29.44 2.2300 0. 1730 8 1 366.00 15.00 26.00 31.49 2.2700 0.1610 10 0 239.00 31.00 35.00 35.00 1.6580 0.1450 12 1 2370.00 5.10 8.40 9,54 4.4500 0.5330 . is. ...1 ...1.1 CO.00 .8,00 12...8D 14.43 3. 1300 0.3520 27 0 1200.00 9.00 14.00 16,23 3.8400 0.3130 28 0 3100.00 4.70 7.30 7.57 4.6200 0.6710 29 0 3720.00 5.60 6.90 7,79 5.7100 0.6510 LOG _T.M._V.S.,_L.O.G_Q SSX 1.4633 SSY 0.5280 SP -0.8730 REGRESSION EQUATI.ON._Ys_ 1.2102 + _-0.5966 X STANDARD ERROR OF ESTIMATE 0.0345 CORRELATION COEFFICIENT -0.9932 DEGREE OF FREEDOM 7__ F = 437.0291 LOG A VS. LOG Q SSX 1.4633 SSY 0.2438 SP 0.5887 REGRESSION EQUATION Y= 0.5049 + 0.4023 X STANDARD ERROR OF ESTIMATE 0.0341 CORRELATION COEFFICIENT 0.9856 DEGREE OF FREEDOM 7 F = 203.3331 LOG V VS. LOG Q SSX 1.4633 SSY 0.5279 SP 0.8729 REGRESSION EQUATION Y= -0.5044 + 0.5965 X STANDARD ERROR OF ESTIMATE 0.0346 CORRELATION COEFFICIENT 0.9932 DEGREE OF FREEDOM 7 F = '_ . 435.2229 1 LOG TS VS. LOG Q SSX 1.46 33 SSY „ REGRESSION EQUATION Y= 0.6307 SP _ -0. 9286 1.0009 + -0.6346 X STANDARD ERROR OF ESTIMATE 0.0831 CORRELATION COEFFICIENT -0.9666 DEGREE OF FREEDOM 7 F = 85.3634 242 LOG TP VS. LOG Q SSX 1.4633 SSY 0.5447 SP -0.8898 REGRESSION EQUATION Y= 1.1672 + STANDARD ERROR OF ESTIMATE 0.0246 CORRELATION_COEFFICIENT -0.9967 __ DEGREE OF FREEDOM 7 F = 893.3013 -0.6081 X LOG TSS VS. LOG Q SSX 0.3326 SSY 0.1108 SP -0.1919 REGRESSION EQUATION Y= 0.9250 + STANDARD ERROR_OF ESTIMATE^. _0.0028_ CORRELATION COEFFICIENT -1.0000 DEGREE OF FREEDOM 2 F = 14613.2969 -0.5771 X LOG.TPP VS. LOG. SSX 0.3326 SSY 0.1230 SP -0.2020 REGRESS 10N_ E.QUATION Y=. STANDARD ERROR OF ESTIMATE CORRELATION COEFFICIENT -0.9991 DEGREE OF FREEOOM 2 1.1447 + 0.0152 -0.6076 X F = 529.8909 PHYLLIS CREEK. REACH 4-6 MEAN TIME PEAK TIME STARTING TIME 244 PHYLLIS CREEK. REACH 4-6 ft IN CU M / SEC A FLOW AREA + VELOCITY PHYLIS LOWER 245 NO ID Q CL/S) TS TP TM 10 20 365.00 269.00 4.58 4.75 10.33 12.00 13.79 18.18 2.3300 2.3190 0.1565 0.1159 30 40 4358.00 3415.00 1.56 1.75 2.55 3.00 2.78 3.60 5.6140 5.6910 0.7760 0.6000 LOG TM VS. LOG Q SSX 1.2032 SSY 0.5027 SP -0.7765 REGRESSION EQUATION Y= 0.8764 + -0.6454 X STANDARD ERROR OF ESTIMATE 0.0280 CORRELATION COEFFICIENT -0.9984 DEGREE OF FREEDOM 3 F = 639.7966 LOG A VS. LOG SSX 1.203 2 SSY 0.1489 S_P Q .4206 REGRESSION EQUATION Y= 0.5449 + 0.3496 X STANDARD ERROR OF ESTIMATE 0.0310 CORRELATION COEFFICIENT 0.9935 DEGREE^OF FREEJDOM„ 3__I_ F = 152.6978 LOG V VS. LOG Q 'SS X""1.2032 SSY 0.5114 SP0.7829 REGRESSION EQUATION Y= -0.5451 + 0.6507 X STANDARD ERROR OF ESTIMATE 0.0309 CORR E L AT I ON „C0 EFF I CI ENT 0 . 9981 __„._ . DEGREE OF FREEDOM 3 F = 532.2039 LOG TSS VS. LOG Q SSX 1.2032 SSY 0.2045 SP -0.4951 REGRESSION EQUATION Y= 0.4604 + -0.4115 X STANDARD ERROR 0F_ ESTIMATE 0.0197 _ CORRELATION "COEFFICIENT -0.9981 DEGREE OF FREEDOM 3 F = 525.7346 -2-46-LGG TPP VS. LOG Q SSX 1.2032 SSY 0.3704 SP -0.6675 > REGRESSION EQUATION Y= 0.7671 + -0.5548 X STANDARD ERROR OF ESTIMATE 0.0073 __• CORRELATION COEFFICIENT -0.9999 DEGREE OF FREEDOM 3 F = 6927.8672 r-PHYL IS'. LOWER STARTING TIME PHYLIS LOWER 248 249 p. D: C FORTRAN /360. PROGRAM *PD* FOR SOLUTION OF THE DIFFERENTIAL C EQUATION.OF UNSTEADY FLOW THROUGH A CASCADE OF'.RESERVOIRS. C CONTROL CARDS . v./, . -C NO 1, Q0> Q AT START ,., AL..= ALPHA, .. BE = BETA (F6.0) C • NO 2, STARTING T , TE = END T ; DT = TIME I NT C TO. = START OF TIME. COUNT , v, y (F6.Q) C NO 3, W = WIDTH , L '= LENGTH ' ' ' AA £ BA (F6.0) C NO 4 TIT= TITLE (20A4) 0 • • C EXPLANATION OF TERMS: C . .: ''/••••'' C UNITS"ARE METERS ANO SECONDS EXCEPT'AS NOTED BELOW. _C BETA IS THE DISPERSION COEFFICIENT, WHICH CAN BE C ESTIMATED AS (QO / QD1**0.2 . C ALPHA IS THE RELATIVE CHANGE IN DISCHARGE, C .. ( 0 ( END.) / Q( ST AR T) ) - 1.0.. .... C THE COMPUTATIONS ARE PERFORMED FOR THE PERIOD TS - TE. C TS AND TE ARE IN MINUTES FROM THE START OF THE TEST, _C TO IS THE STARTING TIME IN HOURS AND MINUTES(E.G. 1420. C FOR »20 MINUTES PAST 2PM' ) C WIDTH IS THE CHANNEL WIDTH, WD. C . LENG.NTH IS THE TOTAL LENGTH OF THE TEST REACH. c .-. •' • -: ':• c ' C . OUTPUT:: V r "C THE PROGRAM PRINTS THE.SOLUTION OF THE CASCADE EQUATION C A(T) , THE CORRES PONDING Q ( T ) , AND SOME OF THE TERMS C IN THE EQUATION FOR A ( T) . c • DIMENSION TIT(20) COMMON L, P, BE, W, C, T, EX1 , EX2, K EXTERNAL AUX1, AUX2 REAL L , LS , LE c. .. ..J. ... . C READ INITIAL DATA READ (5,1) QO, AL, BE READ (5,1 ) TS, TE, DT , TO READ (5,1) W, L , AA, BA 1 FORMAT (12F6.0) READ (5,2) JIT. ; 2 FORMAT (20A4) r. C INITIALIZE AO = A A *(Q0 .** BA) .: ' AEND = AA*((QO*(1. + AL)) *#BA ) • \ AL = .( AEND - AO )/ AO Si: . ' •." C = ( QO ** (l.-BA)) / (AA*BA )••',• P ' .= SQRTU240.* C) / ((BE *'W)**2M COEFl = AO / (BE *W * SQRT (6.2 83 * P )) PI8=A0* ( 1. + AL)/ SQRT( 8. * 3.1416 ) C0EF2 = PI8 * ((( 240. * C * L ) / HPv B E * W ) **2 ) ) **0 . 2 5 DL . = L/20. . c c N = (TE - TS) /DT 250 WRITE INITIAL DATA WRITE (6, 3) TIT FORMAT( 1H 1, 20A4 ) I TO =TO WR.I TE 1.6 , .4 1......Q0.. ,.,...AL_BE . , I TO , TS , TE ,, ' DT 1 W , L , A A, B A , 2 AO , C y" "'" CGEF1 ,COEF2 ,P 4 FORMAT ( * INITIAL CONDITIONS' / l» INITIAL 0 (QOI , ALPHA (AL), BETA (BE) = 3F10.3 / 2' START OF TIME COUNT (ITO), TS, TE, DT = », 16, 3F8. 1 / 3». WIDTH ( W) .. , LENGTH (i. ) , AA, BA 4F10.3 / 4« INITIAL AREA (AO), CELERITY (C) 2F10.3 // 5« COEF1 , COEF2, P 3E14.5 ) WRITE (6,5) 5 FORMAT ( 10 TIM E 1ST EXPO. 1ST INTEG. 1ST TERM 2ND EXPU c c 12ND I NT E• . 2ND TERM A(T) Q(T) '/) DO LOOP FOR N VALUES OF AIT) DO 6 I = I, N T = TS+ (1-1)* DT C c FIRST INTEGRAL OVER X A I NT I = 0.0,... L E = 0.0 DO 7 J= 1, 20 LS = LE IE = LS + DL K =1 All =. _FGAU16(LS, LE, AUXl ) A I NT 1 = AINT1 + All CONTINUE FT1 = C0EF1 * AlNT 1 / (T **0.25) C C SECOND INTEGRAL OVER T Al NT 2. = 0.0 '.; TTE = 0.0 TO IFF = T / 20.0 DO 8 J = 1 ,20 C C TT S = TTE TTE = TTS + TDIFF K _ = 1 AT ? = FGAU16( TTS, AINT2 = AINT2 + AI2 CONTINUE TTE AUX2 ) FT2 = C0EF2 * AINT2 A = FT 1 + FT2 0 . = ( A. / AA)**U.O /BA) . ;._ WRITE RESULTS WRITE (6,9) T, EX1, A INT 1 , FT 1 , EX2 , AINT2, FT2 , A 9 FORMAT (IX, F4.0 , 8E12.4 ) 6 CONTINUE STOP 251 FUNCTION • AUXl { X ) C C FUNCTION CALLED BY THE LIBRARY PROGRAM FGAU16 C REAL L COMMON L» P» BE, H,. C, T, EX1,EX2, K . EX1 = P. *SQRT_( (L-X)*T) + (X -L -(60.* ..C. * T) ) /.(BE* W ) AUX1= EXP (EX1) /(( L-X}*# 0.25) K = 0 RETURN •-' •- END FUNCTION AUX2 { Z) c C FUNCTION CALLED BY THE LIBRARY PROGRAM FGAU16 r __- L COMMON L, PT BE, W, C, T, EX1, EX 2 ,K E X 2 = P * SORT (L*(T - I) ) + (60.*C*Z - 60.*C*T - L ) I, AU X 2 = EXP (EX?) / (( T - Z)**0.75) ^ BE*W) K =0 RE TURN END •p. D: SAMPLE OUTPUT PHYLLIS CREEK, JUNE 22, 1968. DOWNSURGE INITIAL CONDITIONS INITIAL Q IQO) , ALPHA IAD, 8ETA (BE) = START OF TIME COUNT UTO», TS, TE, OT = WIDTH (W) , LENGTH (LI, AA. BA INITIAL*AREA (AO), CELERITY tC) ; 0.815 -0.065 0.540 1300 12.0 60.0 2.0 11.500 777.000 3.260 2.918 0.516 0.541 C0EF1 , C0EF2, P 0.14005E 00 0.38484E 01 0.17919F 01 TlMF 1ST EXPO. 1ST INTEG. 1ST TERM 2ND EXPO. 2ND INTE. 2ND TERM Am Q(T) 12. -0.3461E 02 C.3874E 02 0.2915E 01 -0.9813E 02 0.4538E-06 0.1746E-05 0.2915E 01 0.8134E 00 14. -0.4229E 02 0.4027F 02 0.2915E 01 -0.9611E 02 0.2572E- 04 0.9899E- 04 0.2916E 01 0.8136E 00 16. -0.5012E 02 0.4161E 02 0.2914E 01 -0.9425E 02 0.5727E- 03 0.2204E- 02 0.2916E 01 0.8137E 00 18. _-_0...5m9.E_ _Q2L_ 0 .4.2.5.3 E _Q2_ —.0*23320. _0_1_ -0.9251E _02_ C.60Q9E- 0? 0_._23.12Er 01 0.2915E 01 0.8131E 00 20. -0.6616E 02 0.4192E 02 0.2776E 01 -0.9089E 02 0.34C8E- 01 0.I312E 00 0.2908E 01 0.8095E 00 22. -0.7433E 02 0.3768E 02 0.2437F 01 -0.8936E 02 0.1168E 00 0.4496E 00 0.2887E 01 0.7987E 00 24.. -0.8257E 02 0.2881E. 02 0.1823E 01 -0.8791F 02 C.2664E 00 0.1025E 01 0.2848E 01 0.7792E 00 26. -0.9089E 02 0.1775F 02 0.I101E 01 -0.8653E 02 0.4423E 00 0.1702E 01 0.2803E 01 0.7565E 00 28. -0.9927E 02 0.8570F 01 0.5218E 00 -0.8521E 02 0.5833E 00 0.2245E 01 0.2767E 01 0.7384E 00 30. -0.1077E 03 0.3222E 01 0.1928E 00 -0.8395E 02 0.6634E 00 0.2553E 01 0.2746E 01 0.7283E 00 32. -0.1162E 03 0.9474E 00 0.5579E-01 -0.8274E 02 0.6968E 00 0.2681E 01 0.2737E 01 0.7240E 00 34. -0.1247E 03 0.2202E 00 0.1277E-01 -0.8157E 02 C.7073E 00 0.2722E 01 0.2735E oi 0.7228E 00 36. -0.1333E 03 0.4098E-01 0.2343E-02 -0.8045E 02 0.7098E 00 0.2732E 01 0.2734E 01 0.7224E 00 38. -0.1419E 03 0.6187E-02 0.3490E- 03 -0.7936E 02 0.7103E 00 0.2733E 01 0.2734E 01 0.7224E 00 40. -0.1505E 03 0.7677E-03 0.4275E- 04 -0.7831E 02 0.7103E 00 0.2734E 01 0.27 34E 01 0.7224E 00 42. -0.1592E 03 0.7926E-04 Q.4360E- 03 -0. 77.29 E _0.2_ __Q..J_LOAE_ _.0.Q_ 0_.2_7_i_tE_ 01 _0_^2J_34E_ 01 0.7223E 00 44. -0.1679E 03 0.6885E-05 0.3744E- 06 -0.7631E 02 0.7104E 00 0.2734E 01 0.2734E 01 0.7223E 00 46. -0.1767E 03 0.5086E-06 0.2735E- 07 -0.7534E 02 0.7104E 00 0.2734E 01 0.2734E 01 0.7223E 00 48. -0.1854E 03 0.3224E-C7 0.1716E- 08 -0.7441E 02 0.7104E 00 0.2734E 01 0.2734E 01 0.7224E 00 50. -0 52. -0 54. -0 56. -0 58. -0 .1942F 03 .2030E 03 ...2.L12.E_03_ .2207E 03 •2296E 03 0. 0. JQ. 0. 0. 1770E-08 8486F-10 3576F-11 1334E-12 4435E-14 0.9325E-10 0.4426E-11 J3_..1_8A8.E__LL2_ 0.6832E-14 0.2251E-15 -0.7351E 02 -0.7262S 02 _i0_..J_l_76E_Q2. -0.7092E 02 -0.7010E 02 0.7104E 0.7104E JD.7_L04E_ 0.7104E 0.7104E 00 0. 00 0. JD_0 Q__. 00 0. 00 0. 2734E 01 0. 2734E 01 0. 2734E 01 0. 2734E 01 0. 2734E 01 2734E 01 2734_E__QJL. 27346 01 2734E 01 0.7224E 00 0.7224E 00 JL._I22_t_E._0.Q... 0.7224E 00 0.7223E 00 ro ro TSN L R * 253 C PROGRAM FOR FLOOD ROUTING THROUGH SEQUENCES OF NONLINEAR C RESERVOIRS AND KINEMATIC CHANNELS, WRITTEN IN FORTRAN /360. _C C 1 CONTROL CARD PER CHANNEL (CONSISTING OF SEVERAL REACHES): C NO OF REACHES, KR, (12) fi TITLE OF CHANNEL, TITR, (10A4). C C 3 CONTROL CARDS PER REACH; C _C NO 1 CONTAINS: C NO OF RESERVOIRS, N, (12); fi FACTOR ALPHA , AL, (F6.0); C LOCAL INFLOW ALONG REACH, QINC, (F6.0). C _ _ •. c NO 2 CONTAINS: Parameter "Sigma" of text C TITLE FOR REACH, TIT, ( 10AA) . . uA1 . II . !_..„ Di f is Alpha in SNLR . __ NO 3 CONTAINS: C LENGTH OF REACH, L, (F6.0); STEADY FLOW PARAMETERS, AA AND C BA, (F6.0); FORMATIVE DISCHARGE, QD, (F6.0) „_ C RATING CURVE PARAMETERS, AH1, BH1, HOI, AH2, C BH2, H02, (F6.0); STARTING TIME, TS, (F6.0); TIME INT. _C OF H-DATA, PELT, (F6.0); 2X ; NO OF INITIAL H-DATA, IN, C (12); NO OF INTERVALS TO BE COMPUTED, K, (F6.0). C R C DATA CARDS: C C INITIAL H-DATA , (12F6.0) _C C EXPLANATIONS: C C ALPHA = L(RESERVOIR) / L(TOTAL REACH) ._ _ C GAUGE RATING CURVES ARE DEFINED AS: C Q = AH* (H-HO)**BH , INDEX 1 REFERS TO THE UPSTREAM _C GAUGE, INDEX 2 TO THE DOWNSTREAM ONE. C TS IS THE STARTING TIME IN MINUTES OF THE INPUT DATA. C DATA : THE PROGRAM READS »IN' H-DATA, AND ASSUMES THAT C ALL FURTHER (K-IN) DATA POINTS ARE_EQUAL_TO.THE LAST C INPUT VALUE C C OUTPUT: C C THE PROGRAM PRINTS THE CONTROL CARD DATA AND INITIAL C H-DATA; IT THEN ROUTES THE FLOW THROUGH SUCCESSIVE REACHFS, C WITH THE OUTPUT OF REACH (1-1) + QINC BECOMING INPUT OF C REACH (I). C . . •• • C THE OUTFLOW OF EACH REACH IS PRINTED, AND, IF THE RATING C CURVE PARAMETERS ARE GIVEN, THE GAUGE READINGS. DIMENSION Q(100), Y(2) ,F(2) ,TEMP(2),H(IOO),T(100),Q1(100), 1 TIC 100) , TIT(IO) , TITR (10) COMMON FQ, QIM1, BA REAL L 254 C LOOP FOR REACHES READ (5,40) KR, TITR 40 FORMAT ( 12, 10A4) DO 41 KK = I ,KR C C READ CONSTANTS READ ( 5 ,30). ..N.. , AL ,_.Q.I.NC_ _._ 30 FORMAT (I2,2F6.0 ) READ ( 5,20) TIT 20 FORMAT ( 10A4) ; ; (H02, READ (5 , 1) L , AA , BA, QD , AH1 , BH1, HOI, AH2, BH? ,) 1 TS , DELT , IN , K 1 FORMAT! 12F6.0, 2X, 2 12 .) ... . .._ :.. KL= K K= K+1 IF ( KK .GT. 1 ) GO TO 43 C C READ INITIAL HIGHT DATA READ (5,2) ( H ( I), 1= ..1,IN) 2 FORMAT(12F6.0) IF ( IN .EQ. K ) GO TO 4 C ; C COMPLETE INITIAL ARRAY DO 3 I = IN, K 3 H( I) = H( IN)... ._ _ . C C CONVERT HIGHT TO DISCHARGE, COMPUTE TIMES 4 DO 5 I = 1 , K T( I) = TS + (I-l) * DELT Q ( I )•=( (H( I )-H01 )/AHl )** (1. /BHD 5 CONTINUE ..... .. ._ _i GO TO 45 C C ADJUST FOR KK .GT. 1 AND KLA .NE. K 43 IN = K IF < KLA .EQ. K ) GO TO 45 DO 90 NI. = KLA,. K . 90 Q(N1) = Q(KLA) C C INITIAL CONSTANTS 45 AD = AA* QD **BA FTT= ( L*AD* BA) / (N*QD*60.0 ) FT = FTT. * j .1 .0 .-„ AL .) _ FQ = ( 1. - AL )i / (FT * AL ) TC = ( FTT * AL ) / ( Q(l) ** ( 1.0- BA)) QFAC = Q(l) / QD IDT = 2.0 * DELT / TC IF ( IDT .LT. 1 ) IDT =1 IF ( OFAC .GT.0.15 ).IOT = 2*IDT _ C C WRITE INITIAL CONDITIONS WRITE (6, 6) TITR , KK 6 FORMAT < * 1NON-LI NEAR RESERVOIR ROUTING' / 'INITIAL CONDITIONS' 1 / 10A4, 'ROUTING OVER ',12,'. REACH •) WRITE (6,21) TIT _ 255 21 FORMAT(1H0, 10A4 ) WRITE ( 6, 31> N • AL , QINC 31 FORMAT CON = NO OF RES. = • , 15, • AL(PHA) = L(RES)/L = \ 1 F10.5 / ' QINCREMENT « ', F10.5 ) ITS = TS WRITE (6,7) KL,IN,AA,BA,DELT,IDT,ITS *t » QD, 1 AH1, BH1,H01, AH2, BH2, H02, AO, FQ, FT 7 FORMAT POK , IN • , 217 , / • AA , BA • , 2F12.5 , / 1 * DT OF QO IN MIN , NO. STEPS PER DT • F10.5, 18 / 2 'STARTING TIME , LENGTH (M) , QO ', 3X,I4, 2F 12.3 / 3 'AH1, BH1,H01,« , 3F15.6 / 4 «AH2, BH2, H02, « , 3F15.6 / / 5 'AD, FQ, FT, • , 5X, 3E15.6 // ) IF ( KK .GT. 1 ) GO TO 65 WRITE(6,8) ( I , T(I) , H (I) , Q(I ), I = 1, IN) 8 FORMAT ( 'NO TIME LEVEL DISCHARGE ' // 1 < IX, 12,IX, F7.1,2X, F7.3, 2X, F10.5 >) C C CONVERT Q TO NONDI MENS IONAL Q 65 CONTINUE DO 46 1= 1, K 46 Qll) =(Q(I) + QINC)/ QD C C ADVANCE SOLUTION BY 1 RESERVOIR DO 9 I =1,N Y(l) = T(l) + FT / ( Q(1)** (l.-BAJ) Y(2) = Q(l) Ql(l) = Y(2) TKll = Yd) AIN = IDT D = DELT /AIN OO 10 J= 2,K C C COMPUTE Q(I-l) : DT = FT / (((Q(J-l) + Ql(J-l))/2.) ^*(1. - BA)) TO = TKJ-l) + DELT / 2. - DT DO 82 M = 1 , K IF (T(M) .GT. TO ) GO TO 85 82 CONTINUE 85 IF (M .EQ.l) GO T0B3 QIM1 = Q(M-i) +((Q(M) - Q (M-1)) / DELT) #(T0 -T(M-l) ) GO TO 89 83 QIM1 = Q (1) • __ _ _ __ 89 CALL RK ( Y , F ,TEMP,D , 2 ,IDT ) 12 FORMAT (6E14.4, 215 ) QKJ) = Y(2) T1IJI = Yd) 10 CONTINUE ! DO 13 II = 1,K ." _ . Q (II) = Q1(11) 13 T (II) = TKII) 9 CONTINUE ' C C PRINT OUT RESULTS AFTER 1 REACH DO 50 I = 1,K . _ _ 256 Q( I) = Q( I) * QD IF { BH2 .LF. 0.0 ) GO TO 50 HU) = AH2 *(Q(I) **BH2 )+ H02 50 CONTINUE WRITE (6,11) N 11 FORMAT(•ODISCHARGE AFTER «,I2,» RESERVOIRS * //) IF (. BH2 .LE. _Q..Q ) WRITE ..(..6,61 ...I (_J_ ,. T ( I ) , . Q( I ) ,1= l,K) 61 FORMAT ( » NO TIME LEVEL DISCHARGE « // 1 ( IX, I2,1X, F7.W2X, 7X, 2X, F10.5 )) IF ( BH2 .GT. 0.0. WRITE ( 6, 8 ) ( I, T(I), H(I), Q(I),I = I,K) KLA = K C 41 CONTINUE WRITE ( 6 , 70) 70 FORMAT ( 1H1 ) STOP C END SUBROUTINE AUXRK (Y,F) C DIMENSION Y(2), F(2) COMMON FO, Q1M1, BA . IF ( Y(2) .LE. 0.0 ) STOP 6 4 F(2)= FQ* (QIM1*(Y(2)**(1. - BA)) - Y(2)**(2. - BA )) 11 RETURN A END NON-LINEAR RESERVOIR ROUT IMG INITIAL CONDITIONS 8LANFY CREEK, OCT. 13,1968, 1.-3-5-4 BL. CK. REACH 1-3, DAM IN 12H15, OUT 45 257 ROUTING OVER 1. REACH N = NO OF RES. = 9 AL(PHA) = L<RES)/L = QINCREMENT = 0.02000 0.70000 K , IN 50 36 AA , BA 3,37500 0.47800 DT.OF QO IN MIN , NO. STEPS PER DT 1.00000 STARTING TIME , LENGTH (M5 , QD AH 1, BHUHOl, 1. 320000 AH2» 8H2, H02, 1. „230000 AD 14 686.000 12.000 0.445000 -0.110000 0.354000 14.000000 FQ, FT, 0, 11069 3F 02 0.255037E 01 0.168043E CO NO TIME LEVEL 01 SCHARGE 1 14.0 1.265 1.09607 2 15.0 1. 265 1.09 607 3 16.0 1.185 0.95794 4 5 17.0 18.0 1. 180 1. 180 0.94965 0.94965 'SNLR i 6 19.0 1. 180 0 .94965 7 20.0 1.181 0.95130 Printout for one of the 8 9 21.0 22.0 1.182 1.183 0.95296 0.95462 computations of Figure 24 10 23.0 1. 184 0. 95628 11 24.0 1.185 0.95794 12 25.0 1.186 0.95960 13 26.0 1. 186 0.95960 14 27.0 1.187 0.96 127 15 28.0 1.188 0.96 29 3 16 29.0 1. 188 0.96293 17 30.0 1. 189 0.96460 18 31.0 1.189 0.96460 19 32.C 1. 190 0.96627 20 33.0 1. 190 0.96627 21 34.0 1. 191 0.96794 22 35.C 1.192 0.9696 2 23 36.0 1. 193 0.97129 24 37.0 1.193 0.97 129 25 38.0 1. 194 0.97297 26 39.0 1. 195 0.97464 27 40.0 1. 195 0.97464 28 41.0 1. 196 0.97632 29 42.0 1.197 0.97 800 ' 30 43.0 1.197 0.97800-31 44.0 1.198 0.97969 32 45. C 1. 199 0.98 137 33 46.0 1.285 1. 13 223 34 47 .0 I. 290 1. 14137 35 48.0 1.293 1.14687 36 49.0 1.295 1.15054 DISCHARGE AFTER 9 RESERVOIRS NO TIME LEVEL 258 DISCHARGE 1 19.2 15.279 1.11607 2 20.2 15.279 1.11607 V 3 21.2 15.279 1.11605 ( 4 22.2 15.279 1.11603 5 23.2 15.279 1.11596 6 24.2 .. 15.279 1.11568 7 25.2 15.278 1.11488 8 26.2 15.278 1.11302 9 27.2 15.276 1.10951 10 28.2 15. 2 74 1. 10382 11 29.2 15.270 1.09577 12 30.2 15.266 1.08554 13 31.2 15.261 1.07 365 14 32.2 15.256 1.06033 15 33.2 15.251 1.04785 16 34.2 15.245 1.03539 17 35.2 15.240 1.02 39 6 18 36.2 15.236 1.01392 19 37.2 15.232 1.00 543 20 38.2 15.229 0.99853 ' 21 39.2 15.227 0.99313 22 40.2 15.225 0.98908 23 41.2 15.224 0.98619 24 42.2 15.223 C.98 42 7 25 43.2 15.223 G.98311 26 44.2 15.222 0.^8255 27 45.2 15.222 0.93244 28 46.2 15.222 0.98266 29 47.2 .15.223 C.98314 30 48.2 15.223 G.98379 31 49.2 15.223 0.98459 32 50.2 15.224 C.98548 33 51.2 15.224 0.98 64 5 34 52.2 15.225 0.98748 35 53.2 15.225 0.98858 36 54.2 15.226 0.9893? 37 55.2 15.226 0.99 144 38 56.2 15.227 C.99390 . 39 '57.? 15.229 0.99737 40 58.2 15.232 1.C0415 41 59 . 2 15.236 1.01337 42 60.? 15.241 1.02574 4 3 61.2 15.24 8 1 .04094 44 6 2.2 15.255 1.05812 4 5 63.2 15.2 62 1.07612 46 64.2 15.270 1.09 37 3 47 65.2 15.276 1.10993 48 66.2 15.282 1.12 406 49 67.2 15.287 1 . 1353 2 50 68. 2 15.290 1.14 521 51 69.2 15.293 1.15 245 f NON -LINEAR RESERVOIR ROUTING 259 INITIAL CONDITIONS BLANEY CREEK. OCT. 1.3, 1968, 1-3-5-4 ROUTING OVER 2. REACH s BL . CK. REACH 3-5, DAM IN 12H15, OUT 45 ? N = NC OF PES. 6 AL(PHA) - L.RES./L = 0.700 00 QINCREMENT = 0.0800 0 K , IN 60 61 AA , BA 3.48300 0 . 52700 DT OF QO IN MIN , NO. STEPS PER DT 1.00000 4 STARTING TIME , LENGTH (M) , QD 14 335 .000 12.000 AH 1 , BH1,H01, 1. 230000 0 .354000 1.4.000000 AH 2 , BH2, H02 _ 1. 459999 0 . 331000 9.50 0000 AD, FQ, FT, 0.129027E 02 0.270925E 01 0.158188F 00 DISCHARGE AFTER 6 RESERVOIRS NO TIME LEVEL 0 I SCHARGE 1 22.0 1 1.0 49 1. 1.9 60 7 2 23.0 1 1.049 1. 19606 3 24.0 1 1.049 1 . iQ605 4 25.0 1 1.049 1 . 19 604 5 26.0 1 1.C49 1. 19 603 6 27.C 1 1.049 L. 19 602 7 28.0 1 1.049 1. 19 601 8 29.0 1 1.049 1. 1959 3 9 30.0 1 1.049 1.19 59 2 10 31.0 1 1.049 1. 19576 11 32.0 1 1.049 1 . 1953 9 12 33.0 1 1.049 1. 1 9 46 6 13 34.0 1 1.048 1. 19332 14 35.0 1 1. 047 1.19110 15 36.0 1 1.046 1.18775 16 37.0. 1 1.044 1.18 306 17 38.0 1 1.041 1.17693 18 39.0 1 1.038 1.16 942 19 40.0 1 1.034 1.16 071 20 Al .0 1 1.03C 1.1510 9 2L 42.0 1 1.025 1.1409 I 22 43.0 1 1.021 1. 13055 23 44 .C I 1.016 1 . 12033 24 4 5.0 1 1.012 1.11071 25 46.C 1 1.008 1.10179 26 47.0 1 1.00 4 1.09379 2 7 48. C 1 1.00 I 1.08 63 3 28 49.0 1 0.998 1. 0 8 09 4 29 50.0 1 0.996 1.07609 3C 51.0 1 0.994 1.07224 31 52.0 1 0.99.3 1.069 2 9 32 53.0 1 0. 992 1.06713 3 3 54.0 1 0.991 1 .0656 5 34 55.0 1 0.991 1.06474 35 56.0 1 0.990 1.06429 260 ( 36 57.G 10.990 • 1 .06 42 2 3 7 58 .0 10.990 1.C6445' 3B 59.0 10.99 1 1.06 4 9 1 39 60 .0 10.991 1.06 557 40 6 1 .0 10.991 1.0664 2 V 41 62.0 10.992 1.06749 f 42 6 3 . 0 10.99 3 1. 06 88 9 4 3 64.0 10.99 3 1 .0708 1 44 65.0 10.995 1.0735 3 45 66.0 10.996 1.07740 46 67.0 10.99 9 1.08 28 2 47 68.0 1 1.00 2 1.09009 48 69. C 1 1.007 1.G9 94? 49 70.0 11.012 1.110 7 9 5 0 71.0 11.018 1.12 394 51 72.0 1 1.024 1 . ] 3 H 3 7 52 7 3.0 1 1 .031 1.15345 53 74.0 1 1.0 37 1.1 ', * '* 3 54 75.0 11.043 1 . 18 258 55 76.0 11.049 1.19 5 21 5 6 77.0 11.05 3 1.20 58 2 5 7 78 .0 1. 1. 0 5 7 1.2142 0 5 8 79. C 11.0 60 1.22"4 -\ 59 HO . 0 1 1.061 1.22481 60 B 1 .0 1 1.0 63 1.22 7 74 61 8 2.0 11.063 1.22^62 ' NGN-LI NEAR RESERVOIR ROUTING INITIAL CONDITIONS BLANEY CREEK, OCT. 13,1968, 1-3-5-4 261 ROUTING OVER 3. .PEACH BL . CK. REACH 5-4, DAM IN 12H15, OUT 45 N = NO OF RES. = 12 AL(PHA) = L ( R F S . / L = C.70000 QINCREMENT = 0.0 K , IN • 70 71 A A » 3. 1 1000 0.4 3 900 OT OF QO IN MIN t NO. STEPS PFR DT 1.00 00 0 STARTING Tl ME , LENGTH ( '") , 00 14 9 30.' )00 13.000 AH 1 RH1,H0 L , .1. . 4 59999 ( .331000- 9.500000 AH? BH2 , HG? 0.7 8 30 00 0 .63000 0 4.00 00 0 0 • AD, FO, FT, 0.95 89 17E 0 1 0.341546E 0 0 . 1. ? 5 4 3 0 0 0 0 D T SCHARGE AFTER 12 RESERVOIRS NO TIME LEVEL 01 SCHARGE 1 2 7.8 4.37 3 1 . 19 607 2 28 . 3 4.873 1.. 1.9 60 6 3 29.3 4.373 1 . 19 605 4 30 . 8 4. 8 7.3 1. 1960 4 5 31.8 4. 87 3 1 . 1 960 3 6 32.8 4.87 3 1 . 19 60? 7 3 3.6 4.87 3 1.19 691 8 34 .8 4. 8 7.3 1 . 19 600 Q 3 5 . P 4. 3 7 3 1.19 600 10 36.8 4. P. 7 3 1.195°9 11 37.8 4.373 1.19 593 12 38.8 4.873 1 . 1 9 59 7 13 39.8 4.873 1.. 19 00 6 14 4 0.8 4.87 3 1.1959 3 15 4 1.8 4.373 1 . 19 5'"! 3 16 42 . 8 4 . P 7 3 1 . 19 59 1 17 43.3 4.873 1.19 5 3 3 18 44.8 4.873 1.19 5 3 3 19 45 .8 4.873 1.19 5 7 5 20 46 .8 4.873 1 . 1 956? 21 47 . 8 4.873 1.1 •') 5 39 2? 48 .P 4.873 1. 1950 1 2 3 49.6 4. 37? 1. 19 4 4 1. 24 50.8 4.87? 1.19 3 5 0 25 51.8 4.371 1.19 217 2 6 52.8 4. 370 1. 19 0 3 4 2 7 53.8 4.869 1. 1. R79 0 23 54 .8 4. 86 8 1. 18 476 ?9 53 . « 4.866 1 . 1 •<• }3 9 30 56.8 4. 8 64 1 . 1 7 62 6 31 57.8 4. 86 2 1.17Q39 3? 58.3 4:. 35 9 1.1648 5 3 3 5 9.8 4.856 1 .15 8 2 2 34 60. 8 4.85? 1.13 11? 3 5 6 1 .3 4. 849 1 .14 3 70 262 ( 36 62.8 4. 845 1 . 1 3 610 37 63. S 4. 842 1.12843 3 8 64.8 4.838 1 . 1.2 09 9 39 65.8 4.8 35 1.11374 40 66. 8 4.832 1.10 686 67 . 8 4.82 8 1. 10 044 1 42 68.8 4.826 1.09 45 5 43 69.8 4. 82 3 1.0892 5 44 70.8 4.821 1.08456 45 71.8 4.819 1.08G49 46 72.8 4.81 7 1.07704 47 73 . 8 4.816 1.07419 4 8 74 . 8 4.815 1.07190 49 75.8 4.814 1.07014 50 76. P 4.813 1.06 88 7 51 77 .8 4.813 1.C6807 52 78 . 8 4.813 1.06 771 53 79.6 4.813 1.06 73 0 54 00 .8 4.813 1.06837 55 81.8 4.814 1.06947 56 82.8 4.815 1.0 7 120 57 8 3.8 4.816 1.07^68 5 0 84.8 4.817 1.07706 59 8 5.8 4.819 1.08143 60 86.8 4.822 1.08 709 61 87.8 4.825. 1 .09 39 7 62 88. 8 4.829 1.10 216 63 89. 8 4.834 1. 11 1 5 9 64 90.8 4.83 9 1.12 211 6 5 91.8 4. 644 1.13346 66 92.8 4.85 0 1.14 527 6 7 v 3 . 8 4.855 1.15 715 6 8 94 . 8 4. 8 60 1 . 1 6 8 6 9 6 9 95.8 4. 866 1. 17952 70 96.8 4.8 70 1. 18 9 3 5 71 97 . R 4.8 74 1.19 797
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Runoff concentration in steep channel networks Kellerhals, Rolf 1969-05-19
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Title | Runoff concentration in steep channel networks |
Creator |
Kellerhals, Rolf |
Publisher | University of British Columbia |
Date Issued | 1969 |
Description | The objective of this study is the development of a runoff routing procedure, applicable to steep channel networks in the tumbling flow regime, and suitable for incorporation into more comprehensive mathematical representations of the runoff process. "Steep" is meant in the sense that degradation into existing, coarse deposits (e.g. Pleistocene materials, slide debris, scree) is assumed to be the major channel-forming process. Similarity considerations show that under these circumstances two relatively easily available parameters, such as channel slope and drainage area, or channel slope and width are adequate to define the geometry and hydraulic performance of the channels. The hydrologically significant aspects of channel flow are storage per unit length (area) and discharge, with the relation between the two defining the channel performance under steady conditions. This function, A = f(Q), can be obtained in the field by observing the dispersion of slug-injected tracers through fixed test reaches over a range of discharges. Measurements of this type were made on thirteen test reaches, covering a wide range of channel size and slope. The data from all test reaches can be closely approximated by exponential relations of the form [formula omitted]. As indicated by the similarity considerations, the constants aA and bA of this steady flow equation are predictable from basin parameters. The details of the statistical link between various basin parameters and the above constants are discussed in Day (1969) on the basis of the steady flow data of this study supplemented by extensive.additional measurements. Runoff concentration is an unsteady flow process, which can only be defined with a single flow equation if the flow system is truly kinematic. In order to investigate whether this holds for steep channels, all test reaches were located below lakes with outlets suitable for minor discharge modifications. Small, step-like surges (positive and negative) were created at the lake outlets and their propagation through the test reaches was observed with accurate water level gauges. These surge tests indicate consistently that the channels act as kinematic flow systems but with certain dispersive effects added and with a markedly higher-than-kinematic wave celerity at very low stage, which is probably the result of dynamic waves in pools. Due to the frequent occurrence of super-critical flow, dispersion can only be the result of storage in pools. The differential equation for a kinematic channel with storage in a large number of identical storage elements is derived and solved in linearized form for step-like input corresponding to the surge tests. The dispersion coefficient, which has dimensions L, is the only free parameter of the solution. Comparison with the field data shows that mean water surface width provides a good estimate of this parameter. As a computationally simpler alternative, a routing model which replaces the actual channels by a sequence of truly kinematic channels and deep pools with weir outlets, both obeying the same steady-flow equation, is also considered. Rules for determining the two free parameters of this solution are developed on the basis of the field data. Both routing methods provide approximately equal fit to the surge test data and they both appear to be suitable components for an operational channel runoff model. Being based mainly on the above steady flow equation, both methods are non-linear. This is supported by the field data, which show no tendency towards linearity, except possibly at very low stage. |
Subject |
Runoff |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0093330 |
URI | http://hdl.handle.net/2429/34727 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geography |
Affiliation |
Arts, Faculty of Geography, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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