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UBC Theses and Dissertations

Runoff concentration in steep channel networks Kellerhals, Rolf 1969

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RUNOFF CONCENTRATION IN STEEP CHANNEL NETWORKS by R o l f K e l l e r h a l s D i p l . I n g . , Swiss F e d e r a l I n s t i t u t e o f Technology, Z u r i c h , I 9 6 0 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f G e o g r a p h y ( I n t e r d i s c i p l i n a r y P r o g r a m i n H y d r o l o g y ) We ac c e p t t h i s t h e s i s as co n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In present ing th is thesis in pa r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and Study. I fur ther agree that permission for extensive copying of th is thesis for s cho l a r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t i on of th is thesis for f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT The o b j e c t i v e o f t h i s s tudy i s the development o f a r u n o f f r o u t i n g p r o c e d u r e , a p p l i c a b l e t o steep c h a n n e l networks i n the t u m b l i n g f l o w r e g i m e , and s u i t a b l e f o r i n c o r p o r a t i o n i n t o more comprehensive m a t h e m a t i c a l r e p r e s e n t a t i o n s o f the r u n o f f p r o c e s s . "Steep" i s meant i n the sense t h a t d e g r a d a t i o n i n t o e x i s t i n g , c o a r s e d e p o s i t s (e.g. P l e i s t o c e n e m a t e r i a l s , s l i d e d e b r i s , s c r e e ) i s assumed t o be the major c h a n n e l - f o r m i n g p r o c e s s . S i m i l a r i t y c o n s i d e r a t i o n s show t h a t under t h e s e c i r c u m s t a n c e s two r e l a t i v e l y e a s i l y a v a i l a b l e p a r a m e t e r s , such as c h a n n e l s l o p e and d r a i n a g e a r e a , or c h a n n e l s l o p e and w i d t h are adequate t o d e f i n e the geometry and h y d r a u l i c p e r - formance o f the c h a n n e l s . The h y d r o l o g i c a l l y s i g n i f i c a n t a s p e c t s o f c h a n n e l f l o w are s t o r a g e p e r u n i t l e n g t h ( a r e a ) and d i s c h a r g e , w i t h the r e l a t i o n between the two d e f i n i n g the c h a n n e l performance under stea d y c o n d i t i o n s . T h i s f u n c t i o n , A = f ( Q ) , can be o b t a i n e d i n the f i e l d by o b s e r v i n g the d i s p e r s i o n o f s l u g - i n j e c t e d t r a c e r s t h r o u g h f i x e d t e s t reaches over a range o f d i s c h a r g e s . Measure- ments o f t h i s t y pe were made on t h i r t e e n t e s t r e a c h e s , c o v e r i n g a wide range o f c h a n n e l s i z e and s l o p e . The d a t a from a l l t e s t reaches can be c l o s e l y a p p r o x i - PA mated by e x p o n e n t i a l r e l a t i o n s o f the form A = a^ Q"' . As • i n d i c a t e d by the s i m i l a r i t y c o n s i d e r a t i o n s , the c o n s t a n t s a A and b A o f t h i s s t e a d y f l o w e q u a t i o n are p r e d i c t a b l e from b a s i n parameters." The d e t a i l s o f the s t a t i s t i c a l l i n k between v a r i o u s i i i b a s i n parameters and the above, constants are. d i s c u s s e d i n Day ( 1 9 6 9 ) on the basis, of the steady flow, data of. t h i s study supplemented by e x t e n s i v e . a d d i t i o n a l measurements. Runoff c o n c e n t r a t i o n i s an unsteady flow p r o c e s s , which can only be d e f i n e d with a s i n g l e flow equation i f the flow system i s t r u l y kinematic. In order to i n v e s t i g a t e whether t h i s holds f o r steep channels, a l l t e s t reaches were l o c a t e d below lakes with o u t l e t s s u i t a b l e f o r minor discharge m o d i f i - c a t i o n s . Small, s t e p - l i k e surges ( p o s i t i v e and negative) were cr e a t e d at the lake o u t l e t s and t h e i r p r o p a g a t i o n through the t e s t reaches was observed with accurate water l e v e l gauges. These surge t e s t s i n d i c a t e c o n s i s t e n t l y t h a t the channels act as kinematic flow systems but with c e r t a i n d i s p e r s i v e e f f e c t s added and with a markedly higher-than-kinematic wave c e l e r i t y at very low stage, which i s probably the r e s u l t of dynamic waves i n p o o l s . Due to the frequent occurrence of s u p e r - c r i t i c a l flow, d i s p e r s i o n can only be the r e s u l t of storage i n p o o l s . The d i f f e r e n t i a l equation f o r a kinematic channel with storage i n a l a r g e number of i d e n t i c a l storage elements i s d e r i v e d and s o l v e d i n l i n e a r i z e d form f o r s t e p - l i k e i n put corresponding to the surge t e s t s . The d i s p e r s i o n c o e f f i c i e n t , which has dimen- sions L, i s the only f r e e parameter of the s o l u t i o n . Comparison with the f i e l d data shows, that mean water s u r f a c e width p r o v i d e s a good estimate of t h i s parameter.. i v As a computationally- simpler a l t e r n a t i v e , a r o u t i n g model which r e p l a c e s the a c t u a l channels by a sequence of t r u l y kinematic channels and deep pools with weir o u t l e t s , both obeying the same steady-flow e q u a t i o n , i s a l s o c o n s i d e r e d . Rules f o r determining the two f r e e parameters of t h i s s o l u t i o n are developed on the b a s i s of the f i e l d data. Both r o u t i n g methods pr o v i d e approximately equal f i t to the surge t e s t data and they both appear to be s u i t a b l e compon- ents f o r an o p e r a t i o n a l channel r u n o f f model. Being based mainly on the above steady flow equation, both methods are n o n - l i n e a r . This i s supported by the f i e l d data, which show no tendency towards l i n e a r i t y , except p o s s i b l y at very low stage. V TABLE OF CONTENTS Page A b s t r a c t i i Table of Contents v L i s t of Tables i x L i s t of F i g u r e s x L i s t of Photographs x i i Appendix, L i s t . ofeContents x i i i Acknowledgements x i v 1. N o t a t i o n and A b b r e v i a t i o n s 1 1.1 N o t a t i o n 1 1 . 2 A b b r e v i a t i o n s 6 2 . I n t r o d u c t i o n 8 2 . 1 Past and present approaches to the r u n o f f problem 8 2 . 2 S e p a r a t i o n of the r u n o f f process i n t o land phase and channel phase 11 2 . 3 The o b j e c t i v e of the study 15 2 . 4 Assumptions r e g a r d i n g r e a d i l y a v a i l a b l e data. 16 3 . F i e l d Methods ' 23 3 . 1 S e l e c t i o n of t e s t reaches 23 3 . 2 Survey measurements 29 3 . 3 T r a c e r methods 30 3 . 3 . 1 O b j e c t i v e 30 3 . 3 . 2 P r i n c i p l e s of discharge and v e l o c i t y measurements with s l u g i n j e c t i o n methods 30 . v i TABLE OF CONTENTS (cont'd.) Page 3 . 3 . 3 V e r t i c a l and l a t e r a l d i s p e r s i o n requirements . . 32 3 . 3 . 4 L o n g i t u d i n a l d i s p e r s i o n models . . . 33 3 . 3 . 5 A gamma-distribution model f o r the f i n a l d e c l i n e of C ( t ) 37 3 - 3 . 6 Equipment and procedures f o r s l u g i n j e c t i o n measurements 43 3 . 3 . 7 T r a c e r l o s s e s ^8 3 . 4 Surge t e s t s ^9 3 . 4 . 1 O b j e c t i v e vfc . 49 3 . 4 . 2 Discharge m o d i f i c a t i o n s 50 3 . 4 . 3 Stage measuring equipment 51 3 . 4 . 4 S t i l l i n g w e l l response 54 3 . 4 . 5 Stage-discharge r a t i n g curves . . . 56 4 . F i e l d R e s u l t s 59 4 . 1 Survey r e s u l t s 59 4 . 2 V e l o c i t y and discharge measurements 60 4 . 2 . 1 Conversion of f i e l d data to time- c o n c e n t r a t i o n curves' 60 4 . 2 . 2 Numerical i n t e g r a t i o n 6 l 4 . 2 . 3 R e s u l t s 64 4 . 2 . 4 Accuracy • 65 4 . 3 Surge t e s t s 77 5 . Channel Geometry and Steady ;Flow Equations . . . . . 85 5 . 1 S i m i l i t u d e c o n s i d e r a t i o n s f o r steep, degrading.channel networks . . . 85 v i i TABLE' OF .-CONTENTS (cont'd.) Page 5 . 1 . 1 Assumptions 85 5 . 1 . 2 C o n d i t i o n s f o r s i m i l a r i t y 88 5 . 2 B a s i c equations f o r steady, uniform flow . . 91 5 . 2 . 1 T h e o r e t i c a l c o n s i d e r a t i o n s 91 5 . 2 . 2 Flow equations of the t e s t reaches .• 93 5 . 3 Determining the parameters of the steady flow equation . . . 100 5 . 4 The f r i c t i o n concept a p p l i e d to tumbling flow 103 5 . 4 . 1 Open channel flow formulas 104 5 . 4 . 2 Comparison with the data 106 6 . Unsteady Flow i n Steep Channels 109 6 . 1 Kinematic waves and the surge t e s t r e s u l t s . . 109 6 . 1 . 1 Some f e a t u r e s of kinematic wayes . . 109 6 . 1 . 2 I n d i c a t i o n s from r e s e a r c h t e s t r e s u l t s 112 6 . 2 Kinematic waves with storage d i s p e r s i o n . . . 114 6 . 2 . 1 D i s p e r s i o n through dynamic e f f e c t s . 114 6 . 2 . 2 The d i f f e r e n t i a l equations of kinematic waves with storage d i s p e r s i o n 116 6 . 2 . 3 A s o l u t i o n f o r s t e p - l i k e i n put . . . 118 6 . 2 . 4 Comparison with f i e l d data 124 6 . 3 A p r a c t i c a l approach to- unsteady, tumbling flow . 127 6 . 3 . I Unsteady flow through a n o n - l i n e a r r e s e r v o i r 128 v i i i TABLE OF CONTENTS ( c a n t ' d. ) Page 6 . 3 . 2 A r o u t i n g model based on a cascade of pools and channels . . 129 6 . 3 . 3 E v a l u a t i o n of the f r e e parameters from f i e l d data 131 7 . C o n c l u s i o n 142 7 . 1 The H y d r a u l i c s of tumbling'.:flow I 1 * 2 7 . 2 B a s i n l i n e a r i t y 144 7 . 3 Towards an o p e r a t i v e channel r u n o f f model . . 145 8 . B i b l i o g r a p h y 147 • Photographs 152 Appendix Computer programs with o p e r a t i n g i n s t r u c t i o n s , p r i n t o u t , and p l o t s 160 i x LIST' OF TABLES Page 1. Comparison of morphometry based-on three map s c a l e s . 19 2 . Test reaches -below, lakes 24 3 . A d d i t i o n a l t e s t reaches (Day, 1 9 6 9 ) 26 4. Summary of t r a c e r measurements: A. Brockton Creek 66 B. P l a c i d Creek 68 C. Blaney Creek 70 D. P h y l l i s Creek 73 5 . Summary of surge t e s t s : A. Brockton Creek 79 B. P l a c i d Creek 80 C. Blaney Creek . 81 D. P h y l l i s Creek. 82 6 . Regression parameters of steady flow 95 7 . Regression parameters of steady flow (Day, 1 9 6 9 ) • • 96 X LIST OF' FIGURES Page- 1. The ; t w o ' r u n o f f phas.es . - 13 2 . Morphometry of t h r e e b a s i n s a t d i f f e r e n t map s c a l e s 18 3 . Comparison between c h a n n e l p r o f i l e s measured o f f maps and s u r v e y e d i n the f i e l d 21 4 . Channel p r o f i l e s 28 5 . L o n g i t u d i n a l d i s p e r s i o n o f s l u g - i n j e c t e d t r a c e r . . 36 6 . The s t o r a g e model a p p r o x i m a t i o n t o l o n g i t u d i n a l d i s p e r s i o n 39 7 . G r a p h i c a l f i t t i n g o f s t o r a g e model 42 8 . C i r c u i t diagram o f r e c o r d i n g c o n d u c t i v i t y b r i d g e . . 45 9 . Response of r e c o r d i n g c o n d u c t i v i t y b r i d g e 47 1 0 . Schematic s e c t i o n of manual gauge 52 1 1 . Schematic view o f stage r e c o r d e r i n s t a l l a t i o n f o r mountain.streams 53 1 2 . Gauge response c u r v e s 55 1 3 . Two t y p i c a l stage - d i s c h a r g e r a t i n g c u r v e s . . . . 57 14. D e f i n i t i o n s k e t c h f o r n u m e r i c a l i n t e g r a t i o n . . . . 62 1 5 . Surge t e s t o f October 1 3 , 1 9 6 8 , on B l a n e y Creek . . 78 1 6 . H y d r a u l i c measurements on the r e a c h , B r o c k t o n Gauge 1 - Gauge 2 98 1 7 . H y d r a u l i c measurements on the r e a c h , B l a n e y Gauge 3 ' - Gauge 5 99 1 8 . Valuesi-Of c 2 f o r b e s t f i t t o - E q u a t i o n 5 . 2 1 105 1 9 . Exponents of E q u a t i o n s 5 . 2 3 and 5.24 v s . . . . . 107 2 0 . D e f i n i t i o n s k e t c h f o r E q u a t i o n 6 . 8 117 2 1 . E f f e c t of /& on the s o l u t i o n o f the k i n e m a t i c wave e q u a t i o n w i t h s t o r a g e d i s p e r s i o n 1 2 3 x i LIST OF FIGURES (Cont'd) Page 22a. Comparison between f i e l d o b s e r v a t i o n s and k i n e m a t i c waves w i t h s t o r a g e d i s p e r s i o n ± 2 5 22b. Comparison between f i e l d o b s e r v a t i o n s and . k i n e m a t i c waves w i t h s t o r a g e d i s p e r s i o n 126 2 3 . D e f i n i t i o n s k e t c h f o r the cascade of c h a n n e l s and r e s e r v o i r s 130 24a. V a r i a b l e number o f r e s e r v o i r s a t f = 0,1 133 24b. V a r i a b l e number o f r e s e r v o i r s a t cf = 0.28 134 24c. V a r i a b l e number o f r e s e r v o i r s at C = 0.7 135 2 5 . The r o u t i n g parameter o " 137 26a. Computed and observed s u r g e s , B r o c k t o n Creek . . . . 138 2 6 b . Computed and obse r v e d s u r g e s , B l a n e y Creek 139 26c. Computed and observed s u r g e s , P h y l l i s Creek 1^0 x i i LIST OF PHOTOGRAPHS Page 1. Brockton Creek, along Reach Br 1 - 2 , l o o k i n g upstream 152 2 . P l a c i d Creek, along Reach P l 3 - 4 , l o o k i n g downstream. T y p i c a l l o g jam i n foreground . . . . 152 3 . Blaney Creek, at B l Gauge 3, l o o k i n g downstream . . . 153 4 . Blaney Creek, at B l Gauge 4 , l o o k i n g upstream from b r i d g e 153 5 . P h y l l i s Creek, at Ph Gauge 2 , l o o k i n g downstream. Stage r e c o r d e r at r i g h t 154 6 . P h y l l i s Creek, at Ph Gauge 4 , l o o k i n g upstream . . . 154 7. Barnstead c o n d u c t i v i t y b r i d g e 154 8 . V o l u m e t r i c g l a s s ware f o r s a l t d i l u t i o n t e s t s . . . . 155 9 . Vats, p a i l , and s t i r r i n g rod f o r s a l t d i l u t i o n t e s t s 155 1 0 . Equipment f o r Rhodamine WTslug i n j e c t i o n t e s t s . . . 156 1 1 . Recording c o n d u c t i v i t y b r i d g e , with e l e c t r o n i c i n t e r v a l timer 156 1 2 . C o n t r o l s t r u c t u r e at the o u t l e t of Blaney Lake. Three f l a s h b o a r d s i n pl a c e 157 1 3 . Timber c r i b dam at o u t l e t of Marion Lake, with two a d d i t i o n s i n pl a c e f o r a down-surge 157 14. Pump at P l a c i d Lake 158 1 5 . I n v e r t e d syphon at p o o l o u t l e t above Brockton Gauge 1 158 1 6 . P l e x i g l a s s tube for. stage readings on r i g h t , constant r a t e i n j e c t i o n apparatus on l e f t 159 1 7 . Recorder i n s t a l l a t i o n w i t h i n v e r t e d syphon at Blaney Gauge 5 159 x i i i APPENDIX COMPUTER PROGRAMS WITH OPERATING INSTRUCTIONS PRINTOUT, AND PLOTS LIST OP CONTENTS NACL Source l i s t i n g 161 Sample p l o t of r a t i n g curve Sample p r i n t o u t DQV Source l i s t i n g 1 6 8 . Sample p r i n t o u t TAILEX Source l i s t i n g 170 Sample p r i n t o u t Sample p l o t s with example f o r d e t e r m i n a t i o n of A, B, and D QVEL Source l i s t i n g , i n c l u d i n g subroutines f o r 176 numerical i n t e g r a t i o n - Three sample outputs' - PL0TGA Source l i s t i n g ' l85\ Sample p l o t s , with and without F - e x t e n s i o n L0GRE Source l i s t i n g , i n c l u d i n g three subroutines 1^9 P r i n t o u t and p l o t s f o r a l l 13 t e s t reaches PD Source l i s t i n g , i n c l u d i n g two subroutines 249 Sample p r i n t o u t SNLR Source l i s t i n g , i n c l u d i n g one subroutine 253- Sample p r i n t o u t x i v ACKNOWLEDGEMENTS The o r i g i n a l support f o r t h i s study, came from the American S o c i e t y of C i v i l Engineers through the award of the 1966 Waldo E. Smith F e l l o w s h i p to the w r i t e r . The N a t i o n a l Research C o u n c i l of Canada and the K i l l a m Foundation gave support l a t e r on In the form of r e s e a r c h funds and f e l l o w - ships r e s p e c t i v e l y . T h i s help i s g r a t e f u l l y acknowledged; I t p e r m i t t e d completion of the study without s e r i o u s f i n a n - c i a l c o n s t r a i n t s . By s u p p o r t i n g the work of Mr. Day, whose r e s u l t s are e s s e n t i a l f o r the p o s i t i v e c o n c l u s i o n s of t h i s study, the Department of C i v i l E n g i n e e r i n g , U.B.C., made a v a l u a b l e c o n t r i b u t i o n . C o l l e c t i o n of the f i e l d data r e q u i r e d f a i r l y l a r g e p a r t i e s and,since much of the work had to be done on r a i n y days, i t was o f t e n l e s s than enjoyable. For t h e i r help as o c c a s i o n a l f i e l d a s s i s t a n t s , whenever s u i t a b l e weather c o n d i t i o n s occurred, the w r i t e r i s indebted to Mr. M. K. Woo, to Mr. Roy P u r s s e l l , a n d to many graduate students of the Department of Geography, U.B.C., a l s o to h i s wife Heather, who d i d much of the l a b o r a t o r y work, data h a n d l i n g , and t y p i n g . The w r i t e r ' s two s u c c e s s i v e s u p e r v i s o r s , Drs. M. A. Melton and G. R. Gates and other members of the i n t e r d e p a r t - mental Ph.D. committee supported the study with t h e i r experience i n conducting r e s e a r c h p r o j e c t s on r e l a t e d t o p i c s . The i n t e r - departmental arrangements, which pr o v i d e d a very s u c c e s s f u l academic and a d m i n i s t r a t i v e framework, were made p o s s i b l e through the e f f o r t s of Dr. Ian McTaggart Cowan, Dean of Graduate S t u d i e s . Dr. 0 . Slaymaker and Mr. M. Church k i n d l y reviewed the t h e s i s manuscript. 1 NOTATION AND 'ABBREVIATIONS 1 . 1 ' N o t a t i o n 2 A Cross s e c t i o n a l area of flow i n a channel . Cm ) A I n i t i a l area, o Ap Area at formative d i s c h a r g e . A t hA/et. a Constant i n the f o l l o w i n g r e g r e s s i o n s : a A l o g A = f ( l o g Q) a v l o g v = f ( l o g Q) a T l o g T = f ( l o g Q) a ; l o g ( A g ^ / ^ / B , . ^ l / S / . S / B , . B( ) Riemann F u n c t i o n b C o e f f i c i e n t i n the above r e g r e s s i o n s (same s u b s c r i p t s ) . b Without s u b s c r i p t : Time constant ( s ) . C Co n c e n t r a t i o n of t r a c e r (g cc" 4') . C^ C o n c e n t r a t i o n of t r a c e r i n r e s e r v o i r i . c Wave c e l e r i t y (ms c. Constants, l D Bed m a t e r i a l s i z e (m). D Mixing c o e f f i c i e n t , f o r one. dimensional d i s p e r s i o n X 2 — 1 over x Cm s ). 2 -1 D c F l o o d wave d i s p e r s i o n c o e f f i c i e n t Cm s ) . 2 DA Drainage area (km ). d Depth, of flow In channel (m) d s Depth, measure,, d e f i n e d as- A/W,o . "s E E x c i t i n g v o l t a g e of. c o n d u c t i v i t y - b r i d g e ( v o l t s ) e Base of n a t u r a l l o g a r i t h m s , 2'. 7 1 8 3 . F(. ) F u n c t i o n of A ( x , t ) . f ( ) U n s p e c i f i e d f u n c t i o n . f ( ) P r o b a b i l i t y d e n s i t y f u n c t i o n . X C o n d u c t i v i t y (mhos). - 2 A c c e l e r a t i o n of g r a v i t y (ms ) ( ? s " P^^fv: JS ' S ' w " t s 5W H Stage r e a d i n g ( f t ) ) ' , h E l e v a t i o n d i f f e r e n c e between, stream and gauge s t i l l i n g w e l l ( f t ) ; . I.(u) M o d i f i e d B e s s e l F u n c t i o n of the f i r s t k i n d of order i and argument u. With J\(u) being a B e s s e l F u n c t i o n of the f i r s t k i n d I. (u) = J . (Y7! u ) . 1 1 i I n teger, counter, j Integer. k( ) F u n c t i o n of gCx,t). L T r a c e r los-s- r a t e (_% per min.).- 1 Length, of t e s t reach (m) 1 ^ , 1 = 0 , 1 , 2 Second order i n t e r p o l a t i o n polyriomlmals-.- M Mass of t r a c e r (g). I n t e g e r , number of r e s e r v o i r s or storage elements. Manning's n. Polynominal approximation of C ( t ) , (g cc "'") 1/2 A b b r e v i a t i o n f o r (240c//3 2 wjp 3 -1 Discharge (m s ). I n i t i a l d i s c h a r g e . Formative d i s c h a r g e . Qutflow from r e s e r v o i r i . Measured d i s c h a r g e . Discharge from upstream r e s e r v o i r . R e l a t i v e d i s c h a r g e , Q/Q^, s u b s c r i p t s as f o r Ad j u s t a b l e r e s i s t a n c e (ohms). Input impedance of r e c o r d e r . Constant r e s i s t a n c e . Volume of r e s e r v o i r i . Argument of f - f u n c t i o n , parameter of f - d i s t r i b u t i o n . Slope F r i c t i o n slope V a l l e y slope Mean water t r a v e l time (min.) Lag to t r a c e r peak.'(min. ) . Lag between p o o l s . R e s e r v o i r f i l l i n g time (s) . Lag to the f irs-t a r r i v a l of t r a c e r (min-.) . Mean tracer, travel, time '(.min-.) . t Time', coo r d i n a t e (min. or s ) . t S t a r t i n g time. t End time. e u B e s s e l f u n c t i o n argument. V P o t e n t i a l d i f f e r e n c e , v o l t a g e ( v o l t ) , v V e l o c i t y ( m s - 1 ) . v m Mean v e l o c i t y . Vp V e l o c i t y at channel forming; d i s c h a r g e . W Width (m). Wp Channel width between high water marks. Wg Water s u r f a c e width. w Exponent i n e x p o n e n t i a l r e l a t i o n between A and Q, Q <=x A W X^ Shape f a c t o r s . x Length c o o r d i n a t e along channel (m). x. Length c o o r d i n a t e s of c r i t i c a l s e c t i o n s . 1 Y. i = l , 2 , 3 Parameters of Storage Model based on 1 f - d i s t r i b u t i o n . A b b r e v i a t i o n f o r exponent (w - l)/w. Exponent i n e x p o n e n t i a l r e l a t i o n between z W and Q, W ex. Q s s Relative- change i n d is. charge dur i n g a surge t e s t Q/QD-1. A Non-dimensional d i s p e r s i o n c o e f f i c i e n t r( ) Gamma F u n c t i o n . A F i n i t e step. A b b r e v i a t i o n s f o r terms i n c and A t . . i 0 Slope angle. K Parameter of - d i s t r i b u t i o n . X Length of a r e s e r v o i r (m). /f V i s c o s i t y . 2 -1 V Kinematic v i s c o s i t y (m s ). | Dummy le n g t h v a r i a b l e (m). ir 3.1416. p S p e c i f i c mass (g cc~^~) . p s S p e c i f i c mass of bed m a t e r i a l . S p e c i f i c mass of water. <5" P r o p o r t i o n of channel l e n g t h occupied by r e s e r v o i r . f Dummy time v a r i a b l e (min. or s ) . 6 1.2 A b b r e v i a t i o n s B l Blaney Creek.. Br Brockton Creek. C.I. Constant i n j e c t i o n . . cc Cubic centimeter, d D e r i v a t i v e . 0 P a r t i a l d e r i v a t i v e . D T o t a l d e r i v a t i v e f o r a moving observer. DO Down-surge, downstream of f t Feet. K 1000 ohms. km Kilometer. Lo Longitude Lat L a t i t u d e 1 L i t e r l o g x N a t u r a l l o g a r i t h m of x. log-j^x Logarithm to base 10. mm M i l l i m e t e r , m Meter, min. Minutes NaCl Sodium c h l o r i d e , common s a l t . NTS N a t i o n a l topographic s e r i e s . Ph P h y l l i s Creek.. P l P l a c i d Creek. RhWT Rhodamine WT,. f lucres-cent dye manufactured by Du Pont. RSQ R-square,.. the. f r a c t i o n of t o t a l sample, v a r i a n c e e x p l a i n e d by- a r e g r e s s i o n . SQD Sodium dichromate Na 2O^CR 2 • 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 c o n c e n t r a t i o n denotes the process which t r a n s - forms r a i n f a l l or snowmelt over a b a s i n to stream discharge at the b a s i n o u t l e t . T h i s t r a n s f o r m a t i o n i s an important and complex problem of hydrology, which has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n , but remains without a s a t i s f a c t o r y , g e n e r a l l y accepted s o l u t i o n . H y d r o l o g i s t s are o f t e n i n t e r e s t e d i n peak flows at c e r t a i n l o c a t i o n s along a stream, but i t i s a f o r t u n a t e a c c i d e n t i f adequate stream flow records happen to be a v a i l a b l e f o r the d e s i r e d s i t e . In many cases, meteoro- l o g i c a l records have to be used to estimate peak r a t e s of r a i n f a l l or snowmelt, which are then transformed to stream- flow . Two d i f f e r e n t approaches towards the p r e c i p i t a t i o n - r u n o f f t r a n s f o r m a t i o n appear to be f e a s i b l e , with the added p o s s i b i l i t y of combinations between the two. The s i m u l a t i o n approach avoids the d e t a i l e d p h y s i c s of the r u n o f f process by s i m u l a t i n g i t , or c e r t a i n p a r t s of i t , with systems which may be c l a s s i f i e d i n t o "black box" systems;., conceptual models, or e l e c t r i c analogues. The a l t e r n a t i v e may be c a l l e d the p h y s i c a l approach, as i t i n v o l v e s d i v i d i n g the r u n o f f process i n t o c l e a r l y i d e n t i f i a b l e subprocesses, which are d e a l t with on a p h y s i c a l b a s i s . Amorocho and Hart (1964) 9 d i s c u s s the v a r i o u s p o s s i b i l i t i e s i n d e t a i l . The p o p u l a r i t y of the "black box" systems approach, of which the U n i t Hydrograph i s the prime example, Is a r e s u l t of the widely h e l d b e l i e f (or hope) t h a t l i n e a r systems 1 r e p r e s e n t most b a s i n responses adequately. The theory of l i n e a r systems i s q u i t e advanced and i t lends i t s e l f to an almost u n l i m i t e d number of mathematical e x e r c i s e s . 2 Recently the number of n o n - b e l i e v e r s has been growing as evidence i s accumulating that most b a s i n s have s u f f i c i e n t l y strong n o n - l i n e a r e f f e c t s to render the l i n e a r approximation dangerous ( l e a d i n g to underestimates of peak f l o w s ) . There are, however, f u r t h e r and b e t t e r reasons f o r the use of "black box"systems. Some of the r u n o f f c o n c e n t r a t i n g processes take p l a c e over the e n t i r e b a s i n area and may be extremely v a r i a b l e , r e n d e r i n g d e t a i l e d p h y s i c a l d e s c r i p t i o n i m p r a c t i c a l (at l e a s t at p r e s e n t ) . A pure " b l a c k box" type systems approach to such a p r o c e s s , i g n o r i n g the p h y s i c a l aspects e n t i r e l y , may l e a d to good r e s u l t s . The non- l i n e a r i t i e s of a b a s i n may a l s o be concentrated i n a few "'"To be l i n e a r , a system has to s a t i s f y the f o l l o w i n g c o n d i t i o n s : Assuming f-j_ ( t ) . and f 2 (t) are the responses to inputs. f 3 (t) and fij ( t ) r e s p e c t i v e l y , then ( f i + f 2 ) Is the response due to input (T3 + fi\). The. mathematical f o r m u l a t i o n of l i n e a r systems leads to l i n e a r d i f f e r e n t i a l equations. 2 Numerous papers i n the Proceedings of the I n t e r n a t i o n a l Hydrology Symposium, h e l d at F o r t C o l l i n s , Sept. 6 - 8 , 1 9 6 7 and i n the Proceedings of the Symposium on Analogue and D i g i t a l Computers, Tucson, 1 9 6 8 , can be c i t e d as evidence of t h i s t r e n d . 10 p r o c e s s e s , so that, even l i n e a r systems may g i v e good a p p r o x i - mations to o t h e r s . Much of the most re c e n t work on r u n o f f s i m u l a t i o n i s based on simple conceptual models of the t o t a l r u n o f f p r o c e s s , such as l i n e a r or n o n - l i n e a r r e s e r v o i r s , i n s e r i e s or i n ' p a r a l l e l , systems of uniform permeable s o i l l a y e r s , or systems based on one-dimensional d i s p e r s i o n - (Overton, 1 9 6 7 ; Sugawara, 1 9 6 7 ; D i s k i n , 1 9 6 7 ) - . The major d i f f i c u l t y with a l l s i m u l a t i o n approaches l i e s i n the problem of parameter i d e n t i f i c a t i o n . Most systems, even q u i t e p r i m i t i v e ones, have enough f r e e parameters to permit c l o s e f i t t i n g to a p a r t i c u l a r set of r a i n f a l l - r u n o f f data. The somewhat more severe t e s t of r e p r e s e n t i n g r u n o f f events of d i f f e r e n t magnitude from the same b a s i n without r e q u i r i n g parameter adjustments, i s a l s o met by many of the r e c e n t l y proposed s i m u l a t i o n systems, but to be r e a l l y u s e f u l the parameters should have f i x e d r e l a t i o n s with i d e n t i f i a b l e b a s i n c h a r a c t e r i s t i c s . This c o n d i t i o n i s not met by any p r e s e n t l y a v a i l a b l e s i m u l a t i o n model. The p h y s i c a l approach c o n s i s t s e s s e n t i a l l y of i d e n t i f y - i n g the processes that c o n t r i b u t e s i g n i f i c a n t l y to the p r e c i p i t a t i o n - r u n o f f t r a n s f o r m a t i o n , f o r m u l a t i n g the d i f f e r e n t i a l equations governing them, s e a r c h i n g f o r p r a c t i c a l s o l u t i o n s , and f i n a l l y d e v eloping f i e l d and o f f i c e procedures which supply the f r e e parameters from r e a d i l y a v a i l a b l e data. 11 Such a t r u l y p h y s i c a l treatment .of. the t o t a l r u n o f f process would avoid the i d e n t i f i c a t i o n problem, but i t i s u n f o r t u n a t e l y q u i t e i m p o s s i b l e at p r e s e n t , not so much f o r lack of understanding of the major processes as due to the complexity of most ba s i n s and the d i f f i c u l t y of s e p a r a t i n g the processes from each other. However, some important processes are both separable and f a i r l y w e l l understood, so that the p h y s i c a l approach has become p o s s i b l e and, by a v o i d i n g the i d e n t i f i c a t i o n problem, has superceded s i m u l a t i o n . Examples of such processes are the p r o p a g a t i o n of f l o o d waves i n p r i s m a t i c channels, s u r f a c e r u n o f f from paved s u r f a c e s , e v a p o r a t i o n from l a r g e , deep l a k e s , and i n f i l t r a t i o n i n t o uniform s o i l s . In c o n c l u s i o n i t appears to the w r i t e r that the most p r o f i t a b l e d i r e c t i o n f o r new r e s e a r c h on r u n o f f processes l i e s i n the p h y s i c a l f i e l d , aiming at a g r a d u a l replacement of s i m u l a t i o n models with more c l o s e l y r e p r e s e n t a t i v e p h y s i c a l models. The r e s e a r c h p r o j e c t which forms the b a s i s of t h i s t h e s i s was designed i n accordance with t h i s b e l i e f . 2 . 2 S e p a r a t i o n of the Runoff Process i n t o Land Phase and Channel Phase Larson ( 1 9 6 5 ) suggests the s e p a r a t i o n of the r u n o f f process i n t o two phases; a l a n d phase, which t r a n s f o r m s . r a i n - f a l l or snowmelt to r u n o f f supply (Larson's term f o r channel i n f l o w ) , and a channel phase, t r a n s f o r m i n g r u n o f f supply to 12 b a s i n outflow. F i g u r e 1 i l l u s t r a t e s this, d i v i s i o n . The land phase i s s i m i l a r to the t o t a l r u n o f f process from v e r y s m a l l b a s i n s and i n c l u d e s a l l the complex i n t e r a c t i n g processes which can take p l a c e over the e n t i r e b a s i n area, but should remain constant over regions of s i m i l a r topography, v e g e t a t i o n and s o i l c h a r a c t e r i s t i c s (e.g. e v a p o - t r a n s p i r a t i o n , i n f i l t r a t i o n , . i n t e r f l o w , e t c . ) . In b a s i n s with n e g l i g i b l e water l o s s e s out of the channel system, which i n c l u d e s most ba s i n s i n the humid zone, the channel phase i s dominated by the s i n g l e process "wave propagat i o n In open channels". For r e p r e s e n t a t i o n of the land phase the s i m u l a t i o n approach appears to be best s u i t e d under the present circum- stances. The parameters can be' evaluated on the b a s i s of r a i n f a l l - r u n o f f data from sma l l t e s t b a s i n s ( s m a l l i n the sense that the channel phase i s n e g l i g i b l e ) i n the r e g i o n of i n t e r e s t . Parameter c o n s i s t e n c y i s not an absolute n e c e s s i t y due to the assumption that the land phase i s r e g i o n a l l y con- s t a n t . Under c o n d i t i o n s of heavy r a i n f a l l on f a i r l y imperm- eable or thoroughly wet b a s i n s with high drainage d e n s i t y , the land phase may even be r e d u c i b l e to " R a i n f a l l - O v e r l a n d Flow-Runoff Supply" (Figure 1) with s m a l l l o s s e s and n e g l i - g i b l e time l a g . The e f f e c t of the land phase on the outflow hydrograph i n the v i c i n i t y of the peak may then be n e g l i g i b l e . In most cases one has to assume that the land phase c o n t r o l s the volume of r u n o f f and c o n t r i b u t e s i n a n o n - n e g l i g i b l e 13 I n put : i Precipitation) > Interception Depression Storage • Overland Flow A, • Infiltration Inter flow Groundwater Eva potransp. J Deep Percol. 1 Loss T T h e L a n d P h a s e ( exc lud ing snowmelt) Input: Runoff Supply Summation from sub-areas Translation through channels and lakes Storage in channels and lakes • Evap. from lakes Loss * Deep Percol. T h e C h a n n e l P h a s e THE TWO RUNOFF PHASES F ig . I manner to the shape of the hydrograph. Larson's b a s i c assumptions which, i f proven, c e r t a i n l y w i l l j u s t i f y the two-phase approach, are that the channel phase accounts f o r the d i f f e r e n c e s between b a s i n s due to s i z e and shape and that the dominant process i n that phase (wave propagation) may be f u l l y r e p r e s e n t a b l e from r e a d i l y a v a i l - able data. He envisaged t h i s roughly as f o l l o w s : Maps show the channel network i n p l a n ; channel dimensions and roughnes's can be obtained i n the f i e l d or on l a r g e s c a l e a i r photos. A l t e r n a t i v e l y maps can be used to make rough e s t i - mates of channel forming discharge and the c o n s i d e r a b l e body of knowledge on r e l a t i o n s between s i z e , performance and d i s - charge of s e l f - f o r m e d channels w i l l permit estimates of the necessary parameters. Once the dimensions of the channel system are estimated, standard methods of f l o o d r o u t i n g should give good r e p r e s e n t a t i o n s of the channel phase. The two phase approach cannot be con s i d e r e d o p e r a t i v e at p r e s e n t . The u s e f u l n e s s of the concept hinges on the p h y s i c a l r e p r e s e n t a t i o n of the channel phase. I f t h i s can be done, then the concept does re p r e s e n t a step forward by removing one p a r t of the r u n o f f process from the realm of s p e c u l a t i o n and s i m u l a t i o n . Larson does not seem to have r e a l i z e d t h i s c l e a r l y , as h i s channel phase i s based on long d l s p r o v e n concepts such as u s i n g a s i n g l e constant value of Manning's n f o r l a r g e flow ranges i n many n a t u r a l channels or assuming f l o o d wave movements at mean water v e l o c i t y , and 15 he never I n v e s t i g a t e s the c r u c i a l q u e s t i o n whether the parameters of h i s c h a n n e l phase r e p r e s e n t a t i o n , a s . o b t a i n e d by curve f i t t i n g , a re r e l a t e d t o the f i e l d v a l u e s . 2 . 3 The O b j e c t i v e of T h i s Study F o r s e v e r a l r e a s o n s , the two phase approach appears a p r i o r i t o be p a r t i c u l a r l y s u i t a b l e i n mountainous a r e a s . The c h a n n e l system i s g e n e r a l l y w e l l d e v e l o p e d and e a s i l y t r a c e a b l e on maps. I n a d d i t i o n t o a p l a n o f the c h a n n e l network w i t h c o n t r i b u t i n g d r a i n a g e a r e a s , maps a l s o s u p p l y c h a n n e l s l o p e s , one of the most i m p o r t a n t parameters i n any type of f l o w r o u t i n g . S o i l c o v e r i s o f t e n t h i n and l y i n g on s t e e p , impermeable l a y e r s , r e s u l t i n g I n a f a s t - a c t i n g and t h e r e f o r e l e s s i m p o r t a n t l a n d phase. I f r u n o f f o r i g i n a t e s m a i n l y i n g l a c i e r s and s n o w f i e l d s , the l a n d phase can be r e p l a c e d by an i c e - p h a s e , s u p p l y i n g water t o the c h a n n e l system a t d i s c r e t e l o c a t i o n s , but the g e n e r a l approach remains v a l i d . The major o b s t a c l e s t o a p p l y i n g the two-phase concept i n mountainous areas a r e : i ) The p r e s e n t l a c k o f i n f o r m a t i o n on the f o r m a t i v e laws and h y d r a u l i c performance of s t e e p c h a n nels c h a r a c t e r - i z e d by a l t e r n a t i n g s u p e r - and s u b - c r i t i c a l f l o w and by energy d i s s i p a t i o n due t o r a p i d changes i n c r o s s s e c t i o n and s l o p e , ( P e t e r s o n and Mohanty ( i 9 6 0 ) i n t r o d u c e d the v e r y d e s c r i p t i v e term " t u m b l i n g f l o w " f o r t h i s f l o w regime. Photographs 1 t o 6 i l l u s t r a t e t u m b l i n g f l o w . ) and i i ) The l a c k o f a r e a l i s t i c r o u t i n g method, w h i c h does 16 not r e q u i r e v i r t u a l l y unobtainable i n f o r m a t i o n on the channel system. The present t h e s i s represents an attempt to solve these problems by i n v e s t i g a t i n g the p h y s i c a l laws governing steady and unsteady flow i n steep channels and by t r y i n g to st a t e them i n such a form that a l l parameters can be obtained from data r e a d i l y a v a i l a b l e even i n ungauged basins. The d e t a i l s of the l i n k between the flow parameters and map measures are discussed i n Day ( I 9 6 9 ) . His l i n k s are s t a t i s t i c a l but based on considerations of dynamic s i m i l a r i t y . The problems were approached e m p i r i c a l l y , s t a r t i n g with f i e l d measurements and concluding w i t h a n a l y s i s of the data, a sequence which has been maintained i n t h i s write-up. 2.4 Assumptions regarding Readily A v a i l a b l e Data The design of t h i s p r o j e c t i s based on the assumption that ungauged basins have ( i ) map-., coverage, ( i i ) high a l t i t u d e a i r photo coverage and ( i i i ) data on p r e c i p i t a t i o n and run o f f f o r at l e a s t one l o c a t i o n i n the same c l i m a t i c r e g i o n . Some of the channel phase models developed subsequently r e q u i r e a very rough estimate of mean annual peak flow, or a high flow of some other frequency. I t e m ( i i i ) , combined wit h map i n f o r m a t i o n , should permit such an estimate to + 50%. Map coverage s u p p l i e s the f o l l o w i n g i n f o r m a t i o n : ( i ) a pla n of the channel network, ( i i ) c o n t r i b u t i n g drainage areas a l l along each channel, and ( i i i ) channel slopes. The 17 accuracy of t h i s Information depends p r i m a r i l y on the s c a l e and contour i n t e r v a l of the maps, but other f a c t o r s , such as the procedures used i n map making and the height and d e n s i t y of ground cover i n the case of maps made from a i r photos, may a l s o be Important (Morisawa, 1957; Scheidegger, 1966) . The standard Canadian map s c a l e s are 1:50,000 with 50 f t contour i n t e r v a l and 1:250,000 with 500 f t contour i n t e r - v a l . Only a s m a l l f r a c t i o n of the Canadian C o r d i l l e r a has 1:50,000 coverage but t h i s i n c l u d e s most developed areas and highway r o u t e s . To g a i n some i d e a of the degree to which these two map s c a l e s r e p r e s e n t drainage networks i n s o u t h - c o a s t a l B. C., three of the basins used i n t h i s study (Furry Creek, P h y l l i s Creek, Blaney Creek) were analysed morphometrically f o r stream o r d e r s , number of streams and mean stream l e n g t h s . One b a s i n (Blaney Creek) could a l s o be analysed on maps to a s c a l e of 1:2400 which, based on a few spot checks, appear to give a reasonably t r u e r e p r e s e n t a t i o n of the drainage system, i n c l u d - i n g f i r s t order streams which c o n t a i n some flow d u r i n g most of the wet season. (The S l e s s e and Ashnola b a s i n s could not be i n c l u d e d i n t h i s a n a l y s i s because they l i e p a r t l y i n the United S t a t e s , where the accuracy and s c a l e of the map coverage i s s i g n i f i c a n t l y d i f f e r e n t . ) The r e s u l t s of the morphometric a n a l y s i s are shown i n Table 1 and on F i g u r e 2. The channel network measurements are 1 8 ; 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 P h y l l i s 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 . 2 5 1 2.00 3 1 6. 50 1:50,000 1 34 0.785 6 0.79 12 0 . 6 3 NTS 2 10 0.985 2 1.27 3 1.18 3 2 3 . 8 0 0 1 2.05 1 4.10 4 1 4.100 1:2400 1 86 0. 23 UBC Research 2 21 0.44 Fore s t , . 3 4 - 0.56 Topography 4 2 2.38 and Forest 5 1 1.90 Cover. Notes: - Figure 2 shows the same data g r a p h i c a l l y . - The channel network i s based on the blue l i n e s shown as the maps wit h a d d i t i o n s based on the contour p i c t u r e . - P l a c i d Creek i s part of the Blaney Creek basi n . " M 20 based on the blue l i n e s of the maps with some a d d i t i o n s • and extensions based on the contour p i c t u r e . (Morisawa, 1 9 5 7 ) . Lakes are r e p l a c e d by stream segments. Since running water i s not a c h i e f agent i n developing the s u r f a c e geometry of these b a s i n s , i t i s not s u r p r i s i n g that Figure. 2. does not show l o g a r i t h m i c r e l a t i o n s between stream order and stream number or stream l e n g t h , as found by Horton and others i n many b a s i n s . However, with i n c r e a s i n g b a s i n s i z e and i n c r e a s i n g map s c a l e s , the r e l a t i o n s become more c l o s e l y l o g a r i t h m i c . F i g u r e 2 i n d i c a t e s that the 1 : 5 0 , 000 maps miss most f i r s t order and some second order streams, w h i l e * t h e 1 : 2 5 0 , 000 maps miss the f i r s t , second and p a r t of the t h i r d order. These r e l a t i o n s may be d i f f e r e n t i n other b a s i n s . I f p o s s i b l e , a s i m i l a r comparative study should be made bef o r e data obtained on a l a r g e s c a l e map are e x t r a p o l a t e d to f i r s t order channels. F i g u r e 3 i l l u s t r a t e s the extent to which channel slope i s ob t a i n a b l e from maps of v a r i o u s s c a l e s by comparing the survey r e s u l t s (hand l e v e l p r o f i l e s ) of 2 t e s t reaches with p r o f i l e s from maps of s c a l e s 1:2400 and 1 : 1 2 . } 0 0 0 . . A i r photos are con s i d e r e d p a r t of the e s s e n t i a l data f o r ungauged basins because the Canadian maps pro v i d e l i t t l e i n f o r m a t i o n on v e g e t a t i v e cover and on bed-rock exposures. P a r t i c u l a r l y at the lower end of hanging v a l l e y s , some streams flow d i r e c t l y on bed-rock, o f t e n without as much as a minor canyon. This does a f f e c t the channel performance c o n s i d e r a b l y . Dense t r e e cover and the presence of l o g g i n g s l a s h r e s u l t i n 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- ® 900 J UJ 7001 500 J 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 l o g jams, i n s m a l l streams, again a f f e c t i n g channel performance. The present data are not adequate to d e f i n e the c o n d i t i o n s under which l o g jams become s i g n i f i c a n t , but i t appears that at a channel width of approx. 1 2 m to 1 5 m l o g jams cease to be s i g n i f i c a n t , at l e a s t i n south c o a s t a l B. C. Channel slope has a l s o a pronounced i n f l u e n c e on the formation of l o g jams, with f l a t reaches being g e n e r a l l y more d e b r i s choked. 3. FIELD METHODS 3.1 S e l e c t i o n of Test Heaches The c r i t e r i a f o r s e l e c t i o n of t e s t reaches were as f o l l o w s : ( i ) The data were to cover the l a r g e s t .possible range of s i z e (width) and s l o p e . ( i i ) To t e s t the unsteady flow behaviour, upstream lakes with o u t l e t s s u i t a b l e f o r minor flow m o d i f i c a t i o n s were r e q u i r e d . ( i i i ) Since measurements were to be made over as l a r g e a d i scharge range as p o s s i b l e , easy a c c e s s i b i l i t y from Van- couver was a l s o an important c o n s i d e r a t i o n . Four streams, c o v e r i n g a range of mean width from 0.b9 m to 14.0 . m were f i n a l l y s e l e c t e d and 2 to 5. t e s t reaches were e s t a b l i s h e d on each stream, c o v e r i n g a f a i r l y wide range i n s l o p e . (See Table 2 f o r a l i s t of reaches.) None of the 4 streams had previous discharge r e c o r d s . The s m a l l e s t stream (Brockton Creek) has no o f f i c i a l name and does not appear on any map. Due to i t s l o c a t i o n at t r e e l i n e i t i s not a f f e c t e d by f o r e s t d e b r i s (Photograph 1). At flows In the order of the mean annual flow, some water s p i l l s out of the stream channel proper on the upstream t e s t reach and f o l l o w s the stream i n some p a r a l l e l minor d e p r e s s i o n s . P l a c i d Creek i n the UBC Research F o r e s t flows through an area covered by dense second growth f o r e s t approximately TABLE II TEST REACHES BELOW LAKES Creek L o c a t i o n (mid-reach) Reach (going down- stream) Length Drop Slope No.of (m) (m) s i n Q Steps # 3 i n xlO' Width WD Survey (m) C o e f f . o f D r a i n - E s timated V a r i a t i o n age Mean Annual f o r WT D Area Peak (km 2) (m 3s _ 1~) Brockton Creek M.t. Seymour Br 1-2 119. 0 ,.8.„8 74 36 0.89 ' .499 0.0655 0 . 2 0 Park, E l e v . 4 , 0 0 0 ' Lo : 1 2 2 Q 5 6 ' Br 2-3 8 0 . 5 2 8 . 1 349 27 0 .99 . 611 0 . 0 8 8 0 0 . 2 0 Lat:4 9 ° 2 3 ' P l a c i d UBC Research P l 1-2 960 79. 7 83 64 2.75 .387 0 . 6 1 4 1.5 Creek Forest, E l e v . 1 , 4 0 0 ' P l 2-3 610 2 1 . 6 3 5 . 5 41 3.16 ,.373 1.17 2 . 5 Lo : 1 2 2 ° 3 4 ' L a t : 4 9 ° l 8 . 5 ' P l 3-4 1844 6 2 . 4 33 .9 122 7 .02 . 400 2.60 5 . 0 Blaney UBC Research B l 1-3 685 31 .9 4 6 . 6 46 12.76 .435 7 .43 1 2 . 0 Creek F o r e s t > E l e v . 9 5 0 ' B l 3-5 335 17 .5 39. o x ' 23 11 .06 . 4 1 4 7.70 1 2 . 0 Lo: 1 2 2 0 3 4 . 5 ' Lat:4 9 ° 1 7 T B l 5-4 930 8 5 . 3 9 4 . 7 62 12 .92 .292 7.94 1 3 . 0 P h y l l i s N r . B r i t a n n i a Ph 1-2 770 2 3 . 7 3 0 . 5 52 1 1 . 4 8 .314 8.69 15 .0 Creek Beach, B.C. E l e v . 1 , 4 0 0 ' Ph 2-3 716 3 4 . 9 4 8 . 7 48 12 .57 .216 1 0 . 4 1 17 .0 L o : 1 2 3 ° H ' Lat:4 9 ° 3 4 ' Ph 3-4 617 3 9 . 5 64 42 1 2 . 6 4 .190 10.99 17 .0 Ph 4-6 305 3 0 . 2 99 21 12. 28 .226 11. 34 1 8 . 0 Ph Lo 1 4 0 . 5 3 0 . 8 219 10 1 4 . 0 4 .167 11 .81 19 .0 i T h i s reach has a sudden steep drop In the l a s t 40 m. The slope of the long f l a t t e r IY) p a r t i s shown here 25 20 years o l d and Its. flow regime i s a f f e c t e d by. frequent l o g jams (Photograph 2 ) . The lowest P l a c i d r e a c h flows through two b u l l d o z e d f i r e p o o l s , and the slope i s i r r e g u l a r . Blaney Creek flows through an area t h a t was burnt over approximately 80 years ago. I t s flow regime i s a f f e c t e d by l o g jams at approximately 150 m i n t e r v a l s (Photographs 3 and 4 ) . The v a l l e y of P h y l l i s Creek was logged 20 to 30 years ago but the creek seems to be of s u f f i c i e n t s i z e to have c l e a r e d i t s e l f of most d e b r i s . Only the f l a t t e s t top reach has 2 l o g jams (Photographs 5 and 6 ) . 1 In a d d i t i o n to these 13 t e s t reaches, which were, except f o r two, t e s t e d f o r steady and unsteady flow, Day ( 1 9 6 9 ) t e s t e d another 12 reaches f o r steady c o n d i t i o n s only. As he d i d not r e q u i r e upstream l a k e s , the choice was much wider and the reaches are i n g e n e r a l more s a t i s f a c t o r y f o r the purpose of steady flow t e s t s . Some of Day's streams have discharge records and the t e s t reaches are u s u a l l y more u n i - form i n s l o p e . Table 3 l i s t s the 12 reaches t e s t e d by Day. The t e s t reach l e n g t h v a r i e s from 8 0 . 5 m to 1 8 4 4 . 0 m and the length to width r a t i o s vary from 10 and 3 4 9 , but most reaches have r a t i o s between 30 and 80 . The lowest reach, on P h y l l i s . Creek was. t e s t e d by Mr. T.Day. For convenience,, the. r e s u l t s are presented here, together with a l l other P h y l l i s reaches. TABLE I I I ADDITIONAL TEST REACHES (from Day, 1969) Creek L o c a t i o n Reach Length Drop Slope No. of (m) (m) s i n Q s t e p s i n x l 0 3 survey Width WD (m.) C o e f f . of Drainage V a r i a t i o n Area f o r W D (km2) Fury Creek S l e s s e Creek J u n i p e r Creek Ewart Creek Upper Mid Low Near B r i t a n n i a Beach , B . C . L o : 1 2 3 ° 1 2 » L a t : 4 9 ° 3 5 ' South o f C h l l l i w a c k , - B.C. L o : 1 2 1 ° 3 8 ' L a t : 4 9 ° 0 1 ' L o : 1 2 1 0 3 8 ' L a t : 4 9 o , 0 1 ' L o : 1 2 1 ° 3 9 ' L a t : 4 9 ° 0 2 ' South of Keremeos B'. C . L o : 1 2 0 O 0 2 ' L a t : 4 9 ° 0 6 ' South o f Upper Keremeos ,• B.. C . L o : 1 2 0 O 0 2 ' L a t : 4 9 ° 0 6 ' L o : 1 2 0 ° 0 2 ' Low L a t . 4 9 ° 0 8 ' 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 4 l 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 I I I (cont'd.) Creek L o c a t i o n Reach Length Drop Slope No. of Width C o e f f . o f Drainage (m) (m) s i n Q steps i n WD V a r i a t i o n Area x l 0 3 survey f o r Ŵ  (km 2) Ashnola South of Upper 490 5 - 0 9 1 0 . 4 17 2 1 . 1 . 1 7 0 2 2 1 . 5 R i v e r Keremeos, B.C. L o : 1 2 0 ° 1 1 * Lat:4 9 ° 0 9 ' L o : 1 2 0 ? 1 0 ' Mid 1 0 0 3 2 6 . 0 2 5 . 9 33 2 8 . 2 . 2 1 8 4 0 8 . 5 L a t . 4 9 0 1 0 - ' L o : 1 2 0 o i 0 ' L o w 747 2 6 . 0 3 5 . 1 25 2 2 - l . 2 2 5 4 0 9 . 5 L a t : 4 9 ° 1 0 ' ro 5 0 0 Chainage (meters) 1000 1500 28 2 0 0 0 t—-Marion Lake Gauge 1 6 0 0 1900 Gauge 2 (m et er s)  1800 Gauge 3 \ Ele va tio n cn O •1700 Gauge 4 X500 1600 Phyllis Creek , Handlevel profile Gauge 6 in E c o a > U J 1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 Chainage (feet) Chainage (meters) 8 0 0 0 4 0 2 0 Z 4000-1 0) CO E c 3 9 6 0 o o > UJ 3 9 4 0 3 9 2 0 3 9 0 0 0 5 0 100 1 150 < f-Pool in Gauge 1 ^ ^ ^ ^ (m ete rs  Gauge Z V c O \ > \ <l> V ^ Brockton Creek profile 1 ! 1 Gauge 3-4 1 —i 100 2 0 0 3 0 0 4 0 0 Chainage (feet) CHANNEL PROFILES 5 0 0 6 0 0 -1320 -1310 1300 1290 H 2 8 0 Fia^4_ 29 With the p o s s i b l e e x c e p t i o n of the most upstream reach on Blaney Creek, none of the t e s t reaches have f l o o d p l a i n s . 3 . 2 Survey Measurements of the channel phase on the b a s i s of map and a i r photo data, i t was n e v e r t h e l e s s c o n s i d e r e d necessary to measure l e n g t h and slope i n the f i e l d , mainly because the reaches could not be l o c a t e d adequately on maps, but a l s o to compare map p r o f i l e s with f i e l d data ( S e c t i o n 2 . 5 ) . by c h a i n i n g and h a n d - l e v e l l i n g i n 50 f t . or 100 f t . steps. The p r o f i l e p o i n t s are the deepest p a r t s of the channel where p o s s i b l e , otherwise water s u r f a c e . The step s i z e i s s u f f i c - i e n t l y long to e l i m i n a t e most of the c h a r a c t e r i s t i c p o o l - r i f f l e sequence. Aneroid measurements were used as a check a g a i n s t l a r g e l e v e l l i n g e r r o r s . F i g u r e s 3 and 4 show the p r o f i l e s of the 13 t e s t reaches of t h i s study. O r i g i n a l l y i t had been planned to measure two h y d r a u l i c parameters, width and roughness, but no s a t i s f a c t o r y method f o r measuring roughness could be designed. When i t became apparent from c o n s i d e r a t i o n s of dynamic s i m i l i t u d e ( S e c t i o n 5 . 1 ) that roughness i s a redundant parameter i n a h y d r o l o g i c a l flow model, attempts to measure i t were abandoned. The channel width, from high-water mark to high-water mark, was measured at each p r o f i l e p o i n t ( 5 0 f t or 100 f t While the o b j e c t i v e of the t h e s i s c a l l s f o r d e f i n i t i o n Length and slope were u s u a l l y o b tained s imultaneously 30 i n t e r v a l s ) . As i n d i c a t e d by Photographs 1 to 6, channel width i s f a i r l y w e l l d e f i n e d by a l i n e of permanent v e g e t a t i o n . The high-water mark f o l l o w s t h i s l i n e c l o s e l y . I t g e n e r a l l y c o n s i s t s of an abrupt change from f o r e s t f l o o r or meadow to channel bed (exposed g r a v e l ) . Tables 2 and 3 show the mean values and c o e f f i c i e n t s of v a r i a t i o n f o r each reach. Most measurements were made with a survey tape, but towards the end of the f i e l d work a r a n g e f i n d e r became a v a i l a b l e , which proved to be very s u i t a b l e f o r t h i s type of survey. 3.3 T r a c e r Methods 3.3.1 Obj e c t i v e The main o b j e c t i v e of the t r a c e r measurements was to e s t a b l i s h the r e l a t i o n s between d i s c h a r g e , v e l o c i t y , and channel area under c o n d i t i o n s of uniform flow and c o v e r i n g the l a r g e s t p o s s i b l e range of d i s c h a r g e . At times t r a c e r methods were a l s o used f o r simple l o c a l d i scharge measurements, which were needed to d e f i n e the s t age-discharge r a t i n g curves as mentioned i n S e c t i o n 3 .4.5. 3-3-2 P r i n c i p l e s of Discharge and V e l o c i t y Measurements with Slug I n j e c t i o n Methods Tr a c e r methods are i d e a l l y s u i t e d f o r measuring the channel flow v a r i a b l e s which are s i g n i f i c a n t i n r u n o f f s t u d i e s , namely d i s c h a r g e , mean v e l o c i t y , and channel storage ( a r e a ) . I f a s l u g of t r a c e r of mass M i s i n j e c t e d i n t o a stream -of d i s c h a r g e Q and the t r a c e r c o n c e n t r a t i o n C (mass per u n i t 31 volume) i s measured at a l o c a t i o n s u f f i c i e n t l y f a r downstream to permit the assumption of complete l a t e r a l and v e r t i c a l mixing (see S e c t i o n 3 . 3 - 2 ) , the p r i n c i p l e c-f c o n s e r v a t i o n of mass takes the f o l l o w i n g form (Replogle e t a l . , 1 9 6 6 ) t M = j Q C dt ... 3 . 1 t i n which t i s the a r r i v a l time of the t r a c e r , and t i s the s 3 e time at which a l l the t r a c e r has passed the sampling s i t e . I f Q i s steady d u r i n g the i n t e r v a l t - t , Equation 3 . 1 g i v e s Q = M / ( C.Vdt ... 3 . 2 t which shows that Q can be measured by i n j e c t i n g a known volume of t r a c e r i n t o a stream and observing or sampling the time- c o n c e n t r a t i o n curve at a downstream l o c a t i o n . The mean t r a v e l time T^ of a t r a c e r cloud between the p o i n t of i n j e c t i o n and the sampling l o c a t i o n i s (Thackston et a l . , 1967) t t r e e T t = j C t dt / j C dt t t s s 3 . 3 and t h i s i s only i d e n t i c a l to the mean t r a v e l time of the stream water,.T, i f instantaneous v e r t i c a l and l a t e r a l mixing at the p o i n t of i n j e c t i o n can be assumed w i t h no t r a c e r b e i n g d i s p e r s e d upstream- or i f the. t r a c e r i s i n j e c t e d above the t e s t reach and T i s obtained as the d i f f e r e n c e between the T^- values f o r the upstream and downstream end p o i n t s of the reach. The mean water v e l o c i t y along.the reach of l e n g t h 1 Is v m 1 T 3 . 4 and from c o n t i n u i t y , the channel area of the t e s t reach becomes A = Q QT 1 3 . 5 v m 3 - 3 . 3 V e r t i c a l and L a t e r a l D i s p e r s i o n Requirements The mechanics of v e r t i c a l and l a t e r a l d i s p e r s i o n i n s t r a i g h t , uniform open channels i s w e l l developed ( D i a c h i s h i n , 1 9 6 3 ; F i s c h e r , 1 9 6 6 ). C r i t e r i a f o r the time or d i s t a n c e which 2 assure adequate mixing have been developed but t h e i r a p p l i - c a t i o n to tumbling flow i s not reasonable. E m p i r i c a l methods, such as sampling across the channel and v i s u a l i n s p e c t i o n of the d i s p e r s i o n of dyes were t h e r e f o r e used to determine whether the l a t e r a l mixing requirements were being met. I f p o s s i b l e , the t e s t reaches were l o c a t e d so that they s t a r t e d at severe channel d i s c o n t i n u i t i e s ( w a t e r f a l l s , c o n s t r i c t i o n s ) , which assure f a s t mixing. On some of the short reaches i t was necessary, however, to i n j e c t the t r a c e r above the reach and Complete d i s p e r s i o n , i s not p o s s i b l e i n f i n i t e time due to the continuous nature of the process (see Equations 3 . 7 and 3 . 1 3 ) - . 33 to observe, two t i m e - c o n c e n t r a t i o n curves for. one measurement of v . m On s e v e r a l occasions the v e l o c i t y of long reaches was measured twice at c l o s e l y s i m i l a r flows by i n j e c t i n g the t r a c e r at the s t a r t i n g p o i n t of a reach and by i n j e c t i n g f u r t h e r upstream. There are no s i g n i f i c a n t d i s c r e p a n c i e s between these r e s u l t s . 3 . 3 . 4 L o n g i t u d i n a l D i s p e r s i o n Models The t r a c e r techniques o u t l i n e d i n S e c t i o n 3 - 3 . 2 r e q u i r e the e v a l u a t i o n of i n t e g r a l s over observed time- c o n c e n t r a t i o n curves. L o n g i t u d i n a l d i s p e r s i o n i s p r i m a r i l y ' r e s p o n s i b l e f o r the shape of these curves. I f the observed C (t) - curves cover the i n t e r v a l t - t adequately f o r numerical I n t e g r a t i o n , the mechanics of d i s p e r s i o n can be ignored. However, i n the course of t h i s study i t happened f r e q u e n t l y t h at f i e l d o b s e r v a t i o n s had to be terminated b e f o r e C (t) had d e c l i n e d to n e g l i g i b l e values and i t became neces- sary , t h e r e f o r e ,to develop a d i s p e r s i o n model which would permit e x t r a p o l a t i o n . P a r t i c u l a r l y i n the equation f o r T̂_ (Equation 3 . 3 ) , the d e c l i n e of C at l a r g e values of t c a r r i e s c o n s i d e r - able weight. The three main processes causing l o n g i t u d i n a l d i s p e r s i o n are: l o n g i t u d i n a l t u r b u l e n c e , t u r b u l e n t mass exchange between stream l i n e s of d i f f e r i n g v e l o c i t i e s , and storage of t r a c e r i n pools and dead : zones. M o l e c u l a r d i f f u s i o n i s only important 34 at extremely small s c a l e s . Taylor ( 1 9 5 4 ) showed that the one-dimensional d i f f u s i o n equation gives a f a i r l y good r e p r e s e n t a t i o n of l o n g i t u d i n a l d i s p e r s i o n i n uniform, t u r b u l e n t pipe flow. E l d e r ( 1 9 5 9 ) extended the a n a l y s i s to i n f i n i t e l y wide open channels and F i s c h e r (1966), Church (1967), Thackston and Krenkel ( 1 9 6 7 ) , and many others have examined i t s a p p l i c a b i l i t y to n a t u r a l channels. The one-dimensional d i f f u s i o n equation i s r r + V T— = D ?r . . . 3-D 6 t ID h x ^ 2 J i n which x i s the l o n g i t u d i n a l coordinate, and D i s the d i s - X p e r s i o n c o e f f i c i e n t . For sl u g i n j e c t i o n of a t r a c e r of mass M at t = o, x = o, the s o l u t i o n takes the f o l l o w i n g form a f t e r v e r t i c a l and l a t e r a l mixing are almost complete M (x - v t ) 2 C ( X J T ) = A V 2 ? D 1 e xp ( i n r T — } ••• 3 - 7 x X I t shows that the t r a c e r i s d i s t r i b u t e d normally over x, with the centre moving downstream at v e l o c i t y v and the m variance i n c r e a s i n g as 2D t . Most d i s p e r s i o n data are based on observation of C at constant x, x = 1 . Under these con- d i t i o n s , Equation 3 . 7 i s skewed to the r i g h t , which agrees with f i e l d data. 35 S u b s t i t u t i n g 3-7 i n t o 3-3 g i v e s . , ;.2D' 1 Y T = - + v -2.. m v : m i n which the second term on the r i g h t accounts f o r the f a c t t h a t i n i t i a l l y some t r a c e r may be d i s p e r s e d upstream-. (Thackston et a l . , 1 9 6 7 ) . Although 2D / v 2 < < 1/v through- out t h i s study, Equation 3 - 8 i s f u r t h e r p r o o f of the need f o r f a s t i n i t i a l mixing. Equation 3 - 7 was f i t t e d to s e v e r a l observed time- c o n c e n t r a t i o n curves, u s i n g the l e a s t squares f i t t i n g method o proposed by Thackston et a l . ( 1 9 6 7 ) . F i g u r e 5 shows a t y p i - c a l f i t . The agreement between Equation 3 . 7 and the f i e l d data i s g e n e r a l l y c l o s e over most of the C (t)--curves but the p r e d i c t e d f i n a l d e c l i n e of C i s always much f a s t e r than observed d e c l i n e s , i n d i c a t i n g that the one-dimensional d i f - f u s i o n equation ( 3 . 6 ) does not r e a l l y r e p r e s e n t d i s p e r s i o n i n n a t u r a l channels. Hays ( 1 9 6 6 ) developed a new model, which i n c l u d e s dead zone storage e f f e c t s besides one-dimensional d i f f u s i o n . I t appears to represent the slow d e c l i n e of C very w e l l , but u n f o r t u n a t e l y , i t i s r a t h e r d i f f i c u l t t o-handle, r e q u i r i n g a F o u r i e r t r a n s f o r m a t i o n of the f i e l d data and subsequent curve f i t t i n g i n frequency space. 3 T h i s method i s based on the IBM Share l i b r a r y prog- ramme NLIN2, d e s c r i b e d by Marquard ( 1 9 6 4 ) . 36 60 Time from injection (min) LONGITUDINAL DISPERSION OF SLUG & INJECTED TRACER Fig. 5 37 3 - 3 - 5 A Gamma-Distribution Model f o r the F i n a l D e c l i n e of C(t) Since the f i e l d data of the present study d e f i n e the main p a r t of the. G ( t ) - c u r v e s adequately, an attempt was made to develop a simple model f o r the f i n a l d e c l i n e , c o n s i d e r i n g only t r a c e r storage i n p o o l s . I t i s based on the assumptions: (I) The stream acts as a cascade o f " r e s e r v o i r s with steady flow Q. ( i i ) A l l r e s e r v o i r volumes FL are equal to T R Q, with T R, the f i l l i n g time, being an a r b i t r a r y time constant. ( i i i ) Mixing i s instantaneous i n each r e s e r v o i r . (Iv) The d i s p e r s i o n process i s i n i t i a t e d by i n j e c t i n g a q u a n t i t y M of t r a c e r i n t o the f i r s t r e s e r v o i r (R Q) at time t = 0 . The i n i t i a l c o n c e n t r a t i o n i n R i s t h e r e f o r e s o C o (t = 0 ) = T^Q ... 3 - 9 (v) The t r a v e l time between two r e s e r v o i r s i s constant f o r a l l water or t r a c e r p a r t i c l e s and can t h e r e f o r e be ignored i n the f o l l o w i n g . These assumptions l e a d to a system of l i n e a r , non- homogeneous d i f f e r e n t i a l equations of f i r s t order f o r C ^ ( t ) . The g e n e r a l form i s dC. C. , C. — i = 1 ~ 1 _1 ^ 1 0 dt T T ... j-xu Through a r e f e r e n c e i n Water Resources Research, V o l . 5 , No. 4 , p. 9 2 7 , August 1 9 6 9 , a paper by MacMullin and Weber (Trans. Am. I n s t . Chem. Engrs., V o l . 3 1 , pp. 4 0 9 - ^ 5 8 , 1 9 3 5 ) has r e c e n t l y come to the w r i t e r ' s a t t e n t i o n . I t contains an i d e n t i c a l - d e r i v a t i o n , b a s e d on c o n s i d e r i n g the outflow from a s e r i e s of well-mixed v e s s e l s . 38 The s o l u t i o n s are T R e s e r v o i r R C (t) = =—•̂  e o o T RQ t • T R e s e r v o i r R, C-. ( t ) = — ^ — t e T R Q t • T R e s e r v o i r R. C. ( t ) = M t 1 e R ... 3 . 1 1 1 1 i ! T R 1 + 1 Q The s u c c e s s i v e peaks occur at t = i T _ and the mean t r a c e r R t r a v e l time i s T f c = ( i + 1) T R . Equation 3 . 1 1 can be compared to the gamma d i s t r i b u t i o n fx <*> = r f ? T *~Kt ... 3.12 r > o K > 0 x > 0 Keeping i n mind t h a t f(r + 1) = r ! f o r r = 1 , 2 , 3 • . . E q u a t i o n 3 . 1 1 can be r e w r i t t e n as t C j ( t ) Q _ i i " T R N _ M ." T c r ( i + l ) T„ T e n n which shows that the g e n e r a l s o l u t i o n i s p r o p o r t i o n a l to a f- d i s t r i b u t i o n with parameters (I + 1 ) , ( 1 / T R ) , and t . I n i t i a l l y the q u e s t i o n of whether t h i s storage model could r e p r e s e n t the t o t a l t i m e - c o n c e n t r a t i o n curyes was explored by f i t t i n g i t to s e v e r a l s e t s of f i e l d data; ( F i g u r e 6),. In comparing I t with the d i f f u s i o n model one may 39 30 Storage model (Least squares fit to all points) Test. Ph R 10, G2 - 3 X - 4 - 6 140 160 Time from injection (min.) THE STORAGE MODEL APPROXIMATION TO LONGITUDINAL DISPERSION Fig. 6 40 say t h a t : ( i ) Both models, can be. f i t t e d almost e q u a l l y w e l l to the observed t i m e - c o n c e n t r a t i o n data, but both f a i l to rep- resent the slow d e c l i n e of C at l a r g e values of t . ( i i ) The storage model can account f o r the f i n i t e time l a g between t r a c e r i n j e c t i o n and the f i r s t a r r i v a l at the sampling l o c a t i o n . In the d i f f u s i o n model the t r a c e r covers the whole reach immediately. ( i i i ) The d i f f u s i o n model giv e s b e t t e r parameter s t a b i l i t y . The number of r e s e r v o i r s i n the storage model does not n e c e s s a r i l y i n c r e a s e with channel l e n g t h , n e i t h e r does T R, the r e s e r v o i r f i l l i n g time, decrease with flow Q. ( i v ) The t h i r d moment r a t i o of the P-distribution-is 2/ V~r" , i n d i c a t i n g t h a t with i n c r e a s i n g channel l e n g t h the skewness of C (t)=curves should decrease, which does not appear to be c o n s i s t e n t with f i e l d r e s u l t s . In s p i t e of these d e f i c i e n c i e s , the storage model can r e p r e s e n t the f i n a l d e c l i n e of C ( t ) , i f the f i t over the main par t of the C ( t ) - c u r v e s i s i g n o r e d , which amounts to s p l i t t i n g the d i s p e r s i o n phase i n t o two p a r a l l e l phases; a d i s p e r s i o n phase, r e s p o n s i b l e f o r moving most of the t r a c e r and a storage phase, which dominates the f i n a l d e c l i n e . Equation 3.11 i s e s s e n t i a l l y of the form Y _Y. t C (t) = Y 1 t ^ e J i n which the Y. are c o n s t a n t s . 41 Taking logarithms on both s i d e s g i v e s l o g 1 Q C = l o g 1 Q Y 1 + Y 2 l o g 1 Q t - Y 3 t which shows that (3-11) can be t e s t e d by p l o t t i n g the f i e l d data i n the form ( l o g ^ C - Y^og^Qt) vs. ( t ) , f o r s e l e c t e d values of • Equation 3.11 i s a good f i t i f the data p o i n t s f a l l on a s t r a i g h t l i n e . With very few e x c e p t i o n s , which are a t t r i b u t a b l e to the d i f f i c u l t i e s of determining low C v a l u e s , the f i e l d data p l o t as shown by the two examples of F i g u r e 7- Some s i m i l a r computer-made p l o t s are shown i n the Appendix under Subroutine "TAILEX". The f i n a l d e c l i n e of C (t) appears to be s i m i l a r to a r - d i s t r i b u t i o n with 1< r ^ 3• A good f i t was g e n e r a l l y achieved by s e t t i n g r = 2 ( i = 1), but r = 1 (negative e x p o n e n t i a l d e c l i n e ) could have been s e l e c t e d with almost equal j u s t i f i c a t i o n . F i g u r e 7A shows a set of data which covers almost the complete d e c l i n e of C (t) and F i g u r e 7B i l l u s t r a t e s the g r a p h i c a l f i t t i n g of a P - d i s t r i b u t i o n exten- s i o n to an incomplete set of C ( t ) — d a t a . On F i g u r e 5 the r e s u l t i n g curve i s p l o t t e d i n C - t c o o r d i n a t e s . S i m i l a r computer-made p l o t s are shown i n the Appendix under Subrout- ine "PL0TGA". The above storage model r e p r e s e n t s only a s m a l l f i r s t step towards an understanding of l o n g i t u d i n a l d i s p e r s i o n i n 4 The parameters Y,, Y ?, and Y_ appear as A, B, and C i n the Appendix. -> 42 \0] Fig. 7A Test covering C(t) decline almost completely 5̂ Rg. 7B Incomplete test with r~ extension Test BR R2, GIUP-|~2X GRAPHICAL FITTING OF STORAGE MODEL Fig.7 43 tumbling, flow, channels. The l a r g e numher of C .(t)-curves measured In the course o f t h i s .study- should permit a more com- p l e t e i n v e s t i g a t i o n , c o n c e n t r a t i n g on the p r e d i c t i v e q u a l i t i e s of the storage model, but this- i s not p a r t of the present obj e c t i v e . 3 . 3 . 6 Equipment and Procedure f o r Slug I n j e c t i o n Measurements A l l the C (t) curves i n c l u d e d i n t h i s study were mea- sured with e i t h e r one of the f o l l o w i n g methods: ( i ) the r e l a t i v e s a l t d i l u t i o n method, based on e l e c t r i c a l d e t e c t i o n of a Na C l - s o l u t i o n ; ( i i ) the dye d i l u t i o n method, based on f l u o r o m e t r i c d e t e c t i o n of a f l u o r e s c e n t t r a c e r (Rhodamine WT). A d e t a i l e d d e s c r i p t i o n of both methods, based p a r t l y on the experience gained i n the course of t h i s study, i s In press. • (Church and K e l l e r h a l s , 1 9 6 9 ) . Only a b r i e f summary • 5 w i l l be given here. The r e l a t i v e s a l t d i l u t i o n method (Aastad and Sognen, 1 9 5 4 ; 0strem, 1 9 6 4 ) uses the l i n e a r r e l a t i o n between concen- t r a t i o n of s a l t and c o n d u c t i v i t y . A known volume ( g e n e r a l l y 5 I n i t i a l l y a few d i s c h a r g e s were measured by i n j e c t i n g the t r a c e r (Sodium Dichromate or Rhodamine WT) at. a constant r a t e and then determining the d i l u t i o n r a t i o between i n j e c t e d s o l u t i o n and stream water. This "constant r a t e i n f e c t i o n method" i s d e s c r i b e d i n Church and K e l l e r h a l s .(1969,) . The equipment i s , shown on Photograph. 1 6 . The method, was- not s u i t a b l e f o r t h i s study, because i t cannot g i v e v e l o c i t y and o f f e r s no advantages over s l u g i n j e c t i o n methods f o r simple discharge measurements. 44 10 to 100. l i t e r s ) of a s a l t s o l u t i o n , whose c o n c e n t r a t i o n need not be known, i s s l u g - i n j e c t e d i n t o the stream and the passage of the s a l t wave i s observed downs-tream w i t h a p o r t a b l e con- d u c t i v i t y meter and e l e c t r o d e . A s m a l l sample of the i n i t i a l s o l u t i o n i s r e t a i n e d f o r the c o n s t r u c t i o n of a c o n d u c t i v i t y - c o n c e n t r a t i o n r a t i n g curve by s u c c e s s i v e d i l u t i o n . The main advantages of the method are the p o s s i b i l i t i e s of computing discharge i n the f i e l d and avoidance of l a b o r a t o r y work. A disadvantage i s the r e l a t i v e l y bulky f i e l d equipment, con- s i s t i n g of 2 vats with needle gauges, p a i l s , approximately 3 - 1 1 kg of NaCl per m s to be measured, p i p e t s , 2 v o l u m e t r i c f l a s k s , c o n d u c t i v i t y meter, e l e c t r o d e , and stop watch. Photographs 7 , 8 , and 9 show the main items. The dye d i l u t i o n method used here i s p a r t i c u l a r l y s u i t a b l e under d i f f i c u l t f i e l d c o n d i t i o n s as during severe r a i n s t o r m s . A c c u r a t e l y measured amounts of the l i q u i d t r a c e r (Rhodamine WT-dye) are i n j e c t e d from the p i p e t d i r e c t l y i n t o the stream and the C ( t ) - c u r v e i s d e f i n e d by t a k i n g 10 - 20 s m a l l water samples at the downstream l o c a t i o n f o r l a t e r a n a l y s i s on a fluorometer. Photograph 10 shows the equipment. Not d i s c u s s e d i n Church & K e l l e r h a l s ( 1 9 6 9 ) i s the r e c o r d i n g c o n d u c t i v i t y b r i d g e (Photograph 11) b u i l t to a v o i d the tedium of measuring c o n d u c t i v i t y - t i m e curves at extremely low flows. A b r i e f d e s c r i p t i o n of t h i s b r i d g e may be i n order as no s i m i l a r instrument appears to be a v a i l a b l e 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 V A C Isolation or stepdown transformers 40V £-V II5V, Variable transformer CIRCUIT DIAGRAM OF RECORDING CONDUCTIVITY BRIDGE Fig.8 46 The p r i n c i p l e of o p e r a t i o n is. to r e c o r d the o f f - b a l a n c e p o t e n t i a l V of an AC-bridge on a 0 - 1 v o l t Rustrak r e c o r d e r with high input impedance, Rn = 10Q K. The c i r c u i t diagram of the b r i d g e i s shown on F i g u r e 8. The response V i s ERQ. (RG - .1) . . V = (RG+1) (R C+2R Q)+2R 3 ' 1 4 i n which E i s the e x c i t i n g v o l t a g e , which i s a d j u s t a b l e between zero and 40 v o l t s , R Is the a d j u s t a b l e r e s i s t a n c e which can take any value between 0 K and 85 K, R c i s f i x e d at 5.11 K and G i s the c o n d u c t i v i t y measured by the probe. Computed and measured'responses are p l o t t e d on F i g u r e 9. The d i f f e r e n c e between the two i s caused by the s i g n i f i c a n t but n e g l e c t e d t h r e s h o l d v o l t a g e and v o l t a g e l o s s of the i n v e r t e r . With c a r e f u l s e l e c t i o n of e x c i t i n g v o l t a g e , amount of s a l t to be i n j e c t e d , and i n i t i a l background adjustments, the b r i d g e can be operated i n the l i n e a r range between responses of 0.2 and 0.9 v o l t s . To conserve the e x c i t i n g v o l t a g e E d u r i n g long p e r i o d s of continuous o p e r a t i o n , an e l e c t r o n i c i n t e r v a l timer was b u i l t ^ and i n s e r t e d between the b r i d g e and i t s power supply. The r e c o r d e r runs o f f an independent power source. The design of. the timer was developed r e c e n t l y by S. O u t c a l t of the Dept.. of Geography, UBC, and W. Schmitt, Dept. of C i v i l E n g i n e e r i n g , UBC. 47 Recorded potential (Volts) RESPONSE OF RECORDING CONDUCTIVITY BRIDGE Fig. 9 48 3 . 3 - 7 ' T r a c e r Losses The t r a c e r methods f o r discharge measurements assume c o n s e r v a t i o n of t r a c e r mass. Prom Equation 3 . 2 one can see that t r a c e r l o s s e s due to a b s o r p t i o n . o r chemical r e a c t i o n s r e s u l t i n overestimated d i s c h a r g e , w h i l e t r a c e r l o s s due to seepage of water out of the channel r e s u l t s i n underestimates. The t r a v e l time, Equation 3 . 3 3 however, i s independent of the t r a c e r mass as long as the l o s s e s do not a f f e c t the shape of the C ( t ) - c u r v e . To permit c o r r e c t i o n f o r l o s s e s , most t e s t s with long mean r e s i d e n c e times T were only i n t e r p r e t e d f o r T^ a c c o r d i n g to Equation 3 . 3 , discharge being measured with a separate t e s t over the s h o r t e s t p e r m i s s i b l e reach. Almost a l l Rhodamine WT t e s t r e s u l t s show a c e r t a i n amount of t r a c e r l o s s due to a b s o r p t i o n or chemical d i s - i n t e g r a t i o n of the t r a c e r . The l o s s r a t e L, i n percent per minute can be estimated from two simultaneous t e s t s , one over a long reach and the other over a short reach, both ending at the same sampling p o s i t i o n . Assuming that the measured discharge and the true discharge Q are r e l a t e d as " - . . . . 3 - 1 5 100 t leads to " 1 0 0 ( 1 — Q S/Q] L) L = ... 3 . 1 6 m " ̂  ° m t , l 'Q1: t,s i n which the s u b s c r i p t s 1 and s r e f e r to the long and the short reach r e s p e c t i v e l y , and Q and are computed a c c o r d i n g 49 to Equations 3 . 2 and 3 . 3 r e s p e c t i v e l y . L i s commonly i n the order of 0 . 1 to 0 . 3 percent per minute. No reason c o u l d be found f o r the observed v a r i a t i o n i n L. Most di s c h a r g e s based on Rhodamine WT t e s t s were c o r r e c t e d a c c o r d i n g to Equation 3 . 1 5 , with L-values estimated from double t e s t s . No c o n s i s t e n t evidence of t r a c e r l o s s appears i n the s a l t d i l u t i o n data, but long t r a v e l times d e f i n i t e l y tend to r e s u l t i n u n r e l i a b l e d i s c h a r g e s . The cause i s probably a combination of t r a c e r l o s s e s and changes i n the background c o n d u c t i v i t y of the stream. The c o n d u c t i v i t y changes observed d u r i n g the passage of a s a l t wave can only be converted to s a l t c o n c e n t r a t i o n i f the background c o n d u c t i v i t y remains constant or changes i n a p r e d i c t a b l e manner, n e i t h e r of which was true i n very extended t e s t s . 3 . 4 Surge Tests 3 . 4 . 1 Ob,]* e c t i v e I f a channel r o u t i n g method i s capable of r e p r o d u c i n g the discharge Q ( t ) at the downstream end of a t e s t r each, r e s u l t i n g from s m a l l , s t e p - l i k e i n c r e a s e s or decreases i n Q (t) at the upstream end, and i f t h i s holds over the complete range of Q, then one may assume t h a t the method should a l s o be adequate to route complex storm hydrographs through the channel reach, s i n c e they can be decompos.ed i n t o a sequence of s m a l l steps. T h i s g e n e r a l statement i s a b s o l u t e l y c o r r e c t i f the channel response i s l i n e a r , but. f o r a l l p r a c t i c a l purposes i t 50 w i l l a l s o h o l d as long aa the n o n - l i n e a r i t i e s , are not s t r o n g enough to l e a d to severe, di's-continuities sucfi as bores. The main o b j e c t i v e of. the surge t e s t s was. t h e r e f o r e to impose s m a l l , s t e p l i k e discharge m o d i f i c a t i o n s , A Q, at the upstream end of t e s t reaches and to observe the p r o p a g a t i o n of these p o s i t i v e and negative surges. The t e s t s were to cover as l a r g e a range of discharge as p o s s i b l e . A few t e s t s on the e f f e c t of the r e l a t i v e s i z e of A Q were a l s o run by imposing s m a l l and l a r g e A Q's at constant Q. 3.4 . 2 Discharge M o d i f i c a t i o n s D i f f e r e n t methods f o r modifying Q were used at each of the f o u r l a k e o u t l e t s on the t e s t streams of t h i s study. A s m a l l dam was b u i l t at the o u t l e t of Blaney Lake. Discharge could be i n c r e a s e d or decreased by adding or remov- i n g f l a s h b o a r d s (Photograph 1 2 ) . " An o l d t i m b e r - c r i b dam at the o u t l e t of Marion Lake gave e x c e l l e n t c o n t r o l over the P h y l l i s Creek reaches. Photograph 13 shows the dam, with two f l a s h b o a r d - l i k e a d d i t i o n s i n p l a c e . The o u t l e t of P l a c i d Lake was so marshy that no c o n t r o l s t r u c t u r e could be b u i l t . Surges were produced by pumping water across the swamp i n t o the creek (Photograph 14). The p o o l above the Brockton reaches was so s m a l l that i t was d i f f i c u l t to m a intain steady d i s - charges d i f f e r e n t from the p o o l i n f l o w . The i n i t i a l surge was produced by adding or removing a few rocks at the p o o l o u t l e t . A g r a v i t y - o p e r a t e d i n v e r t e d siphon was then used to' m a i n t a i n more or l e s s steady f l o w f o r 5 to 15 mi.nutes v (Photograph 1 5 ) . 51 3.4 . 3 ' 'Stage Measuring Equipment Discharge changes at the end p o i n t s of the. t e s t reaches were monitored by observing or r e c o r d i n g stage, changes and e s t a b l i s h i n g stage discharge r a t i n g curves f o r co n v e r s i o n to di s c h a r g e . Even on the steepest and most t u r b u l e n t reaches i t was g e n e r a l l y p o s s i b l e to f i n d s t a b l e p o o l s , e i t h e r on bed- rock or between l a r g e boulders (Photograph 1 8 ). The t u r b u l e n t l e v e l f l u c t u a t i o n s and a i r entrainment made d i r e c t l e v e l measurements i m p o s s i b l e , but the p l e x i g l a s s s t i l l i n g w e l l s i l l u s t r a t e d on F i g u r e 10 and Photographs 16 and 18 p e r m i t t e d 7 the r e a d i n g of water l e v e l s to + 0.001 f t . This gave s a t i s - f a c t o r y r e s o l u t i o n , s i n c e most surge t e s t s caused l e v e l changes i n the order of 0.02 to 0 . 0 5 f t . To g a i n some i n f o r m a t i o n about the dis c h a r g e range of the t e s t reaches, automatic stage r e c o r d e r s were i n s t a l l e d on a l l but one of the t e s t creeks. The instruments were Stevens A - 3 5 r e c o r d e r s , with the f a s t e s t a v a i l a b l e c l o c k g e a r i n g (9-6 in/24 h o u r s ) , and a 12:10 l e v e l s c a l e . These l a r g e s c a l e s made i t p o s s i b l e to r e l y on the r e c o r d e r s f o r the surge t e s t s , thereby saving one f i e l d a s s i s t a n t . With c a r e f u l pro- cedures the time s c a l e could be i n t e r p r e t e d to' + 1/2 min. The r e c o r d e r i n s t a l l a t i o n s were somewhat, un c o n v e n t i o n a l , due to the i n v e r t e d siphon connecting the stream to the s t i l l i n g ' A l l t h e r e o f due stage measurements are i n f e e t to a l a c k of r e a d i l y a v a i l a b l e and decimals m e t r i c 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 Kellerhalŝ  1969) 54 w e l l . F i g u r e 11 and Photographs 17 and 18 I l l u s t r a t e the method. The. experience gained w.ith a l l the above stage measuring equipment (.tubes and r e c o r d e r s ) i s d i s c u s s e d at len g t h i n Church & K e l l e r h a l s ( 1 9 6 9 ) . 3 . 4 . 4 S t i l l i n g W ell Response The hose connections between p l e x i g l a s s tube w e l l s or r e c o r d e r w e l l s and the streams cause c o n s i d e r a b l e damping ( i n the sense that the gauge l e v e l cannot f o l l o w high f r e - quency f l u c t u a t i o n s of the stream l e v e l ) . T h i s damping i s e s s e n t i a l f o r accurate l e v e l readings but the q u e s t i o n a r i s e s as to how much i t a f f e c t s the surge t e s t data. Under normal circumstances the flow i n the connecting g hoses i s laminar and the i n e r t i a of the f l o w i n g water i s n e g l i g i b l e . The w e l l response Is then governed by the f o l l o w - i n g equation: dh h = - b | | ... 3 . 1 7 i n which h i s the e l e v a t i o n d i f f e r e n c e between the stream and the s t i l l i n g w e l l and b i s a time 'constant. I f the stream l e v e l i s constant and the w e l l l e v e l i s o f f by h at time t , the s o l u t i o n i s ^o~^ h = h Q e . ... 3 - 1 8 Assuming a steady r i s e of 0.02 f t / m i n . i n the w e l l , which i s approximately the maximum observed d u r i n g surge t e s t s , gives Reynolds Numbers of 24. for. the connecting hose of p l e x i - g l a s s tubes and 500 f o r r e c o r d e r w e l l connections. 31 r o CD CB . "*:.v,:... • • • GAUGÊ :̂ RÊ feo>JSE 'CURVES ̂  ' ; r A : . ' 56 which i n d i c a t e s that a p l o t of (t - t ). vs. (h/h,) should, g i v e a s t r a i g h t l i n e of slope (-b) on semi-log paper. Some d e v i a - t i o n s should be expected s i n c e Equation 3 . 1 7 only, c o n s i d e r s pipe f r i c t i o n and there are c e r t a i n other l o s s e s p r e s e n t . F i g u r e 12 shows some t y p i c a l gauge response curves. Instead of p l o t t i n g a response curve, one can measure the t i m e , A t , i t takes the s t i l l i n g w e l l to drop from an a r b i t r a r y h^ to an a r b i t r a r y h^ and compute b as f o l l o w s : b = l o g t h * / h 2 ) 3 , 1 9 A l l gauge and r e c o r d e r setups were t e s t e d i n t h i s manner and, where necessary, the surge t e s t records were cor- r e c t e d f o r l a g a c c o r d i n g to E q u a t i o n 3 - 1 7 . About one t h i r d of the gauges needed l a g c o r r e c t i o n s . 3 . 4 . 5 Stage-Discharge Rating Curves At the time when most of the t e s t reaches of t h i s study were i n s t a l l e d , the d e s t r u c t i v e f o r c e of the streams under severe f l o o d c o n d i t i o n s was not p r o p e r l y a p p r e c i a t e d . Many of the gauges were l o c a t e d at pools that proved subsequently to be u n s t a b l e . In h i n d s i g h t , i t appears that there was no l a c k of s t a b l e p o o l s ; only a l a c k of experience i n l o c a t i n g Q them. As the gauges could not be moved without a l t e r i n g the 9 Pools formed by l a r g e , p r e f e r a b l y angular r o c k s , arranged i n such a way that they do not e a s i l y catch d r i f t - wood, are b e s t . L o c a t i o n s below w e l l e s t a b l i s h e d l o g jams are e x c e l l e n t , as the jams tend to catch most d e b r i s and coarse bed l o a d . 57 TWO TYPICAL STAGE - DISCHARGE RATING CURVES 58 t e s t reach l e n g t h , some p a r t s of the stage - d i s c h a r g e curves had to be r e - d e f l n e d two or three times, to permit c o n v e r s i o n of the surge data from stage to d i s c h a r g e . The stage r e c o r d e r s were i n s t a l l e d at the most s t a b l e gauging s i t e s . F i g u r e 13 shows two r a t i n g curves, one f o r the s t a b l e p o o l of Blaney Gauge 5 (Photograph 18) and the other f o r the more troublesome Blaney Gauge 1 . 59 4 . FIELD RESULTS 4 . 1 Survey R e s u l t s Tables 2 and 3 and F i g u r e s 3 and 4 summarize the survey r e s u l t s which c o n s i s t of p r o f i l e s of the t e s t reaches and width measurements ( S e c t i o n 3 - 2 ) ' The width data were pro- cessed with a set of programs developed by Day ( 1 9 6 9 ) . Tables 2 and 3 are summaries of the program output. There i s undoubtedly a c o n s i d e r a b l e operator e f f e c t i n the width data, s i n c e the high water mark was o f t e n , p a r t i c u l a r l y on the b u s h i e r streams, r a t h e r i l l d e f i n e d . The l a r g e number of width measurements tends to. compensate f o r t h i s , but d i s - crepancies of 10% to 15% could, s t i l l occur between d i f f e r e n t f i e l d p a r t i e s . On most t e s t reaches there i s no s i g n i f i c a n t d i f f e r e n c e between a c t u a l l e n g t h and l e n g t h i n p l a n (map l e n g t h ) , but In the case of the few very steep reaches i t i s worth n o t i n g that the surveyed l e n g t h i s the a c t u a l length on the s l o p e . The r e l a t i v e accuracy of the c h a i n i n g and h a n d - l e v e l l i n g i s estimated at + 3 percent. Slope i s d e f i n e d as drop d i v i d e d by l e n g t h , s i n Q, i f Q i s the slope angle.. The drainage areas were measured on the best a v a i l a b l e maps and r e f e r to the middle p o i n t of a t e s t reach. The Brockton Creek b a s i n .does not appear on any map, as the 60 1 : 5 0 , 0 0 0 , coverage, of. t h i s area happens, to be e x c e p t i o n a l l y poor. The drainage area was. t h e r e f o r e measured o f f a i r photos. 4 . 2 V e l o c i t y and Discharge Measurements 4 . 2 . 1 Conversion of F i e l d Data to Time-Concentration Curves The f i e l d data r e s u l t i n g from a r e l a t i v e s a l t d i l u t i o n t e s t c o n s i s t of . the f o l l o w i n g : ( i ) l o c a t i o n and time of i n j e c t i o n , ( i i ) volume of b r i n e i n j e c t e d , ( i i i ) sampling l o c a t i o n , l i s t of times and c o r r e s - ponding c o n d u c t i v i t y r e a d i n g s , ( i v ) r a t i n g curve, c o v e r i n g range of observed con- d u c t i v i t y readings ( i t c o n s i s t s of d i l u t i o n r a t e s and c o r r e s - ponding c o n d u c t i v i t y r e a d i n g s ) , (v) water temperature i n stream and i n r a t i n g tank. The computational procedure f o r c o n v e r t i n g the time- c o n d u c t i v i t y data to t i m e - c o n c e n t r a t i o n i s d e s c r i b e d i n d e t a i l i n Church and K e l l e r h a l s ( 1 9 6 9 ) . The procedure proposed i n e a r l i e r p u b l i c a t i o n s on t h i s method ( 0 s t r e m , 1 9 6 4 ) should not be used, as the c o r r e c t i o n f o r d i f f e r e n t background readings i n the stream and i n the r a t i n g tank i s i n e r r o r . A F o r t r a n IVG program "NACL" was developed f o r t h i s c o nversion.' i t p r i n t s the t i m e - c o n d u c t i v i t y and t i m e - c o n c e n t r a t i o n data and p l o t s the r a t i n g curve. The program i s l i s t e d i n the Appendix, together with o p e r a t i n g i n s t r u c t i o n s and sample, output. 61 The f i e l d data r e s u l t i n g from a Rhodamine WT t e s t are: ( i ) l o c a t i o n and time of i n j e c t i o n , ( i i ) volume of i n j e c t e d dye, ( i i i ) sampling l o c a t i o n , l i s t of sampling times,and 10 to 30 samples. The samples are subsequently analysed on a Tluorometer, and the instrument r e a d i n g i s converted to c o n c e n t r a t i o n with a r a t i n g curve based on standard d i l u t i o n s of the t r a c e r . Wilson (1968) d e s c r i b e s the l a b o r a t o r y procedures in. great d e t a i l . The necessary computations were done manually, but the f i n a l t i m e - c o n c e n t r a t i o n data were put on cards f o r pro- c e s s i n g by an input program "DQV" analogous to "NACL",which i s a l s o l i s t e d i n the Appendix. 4.2.2 Numerical I n t e g r a t i o n The Equations 3.2, f o r discharge Q, and 3.3, f o r t r a c e r t r a v e l time T, are e v a l u a t e d by a F o r t r a n IV.G subroutine "QVEL" (see Appendix). The i n t e g r a l over C(t) and the f i r s t moment of C(t) are computed twice, f i r s t on the b a s i s of the t r a p e z o i d a l r u l e and then with a second order method s i m i l a r to Simpson's r u l e but capable of h a n d l i n g unevenly spaced p o i n t s . The procedure i s d i s c u s s e d b r i e f l y below, because the formulas do not appear to be r e a d i l y a v a i l a b l e i n t e x t s on numerical a n a l y s i s . The "Turner Model 110".fluorometer. of the B.C. Research C o u n c i l was used here. 62 Assume that C(t) is. d e f i n e d at t =. t , t ^ , and t 2 as shown on F i g u r e 14. FIGURE 14. 0 "I 2 DEFINITION SKETCH FOR NUMERICAL INTEGRATION The f u n c t i o n C(t) can be approximated by a second order p olynomial P ( t ) of the form P (t) = C Q 1 q ( t ) + c 1 i 1 ( t ) + c 2 i 2 ( t ) i n which the 1^ (t) are 'second order polynomials i n terms of t ^ j t p a n d t 2 ( H e r r i o , 1 9 6 3 ) , e.g. y t ) t - tt1 - t t 2 •+-- t 1 t 2 . <v- V ( t o - V The i n t e g r a l C ( t ) d t can be estimated as t„ t„ t„ J P ( t ) d t = C o J 1 q (t.).dt + C 1 / 1 (-t.).dt + C / l 2 ( t ) d t Without l o s s of g e n e r a l i t y , one. can s u b s t i t u t e t = 0 o t x = A t x t 2 = At2 The i n t e g r a l s over 1 ( t ) are then: A t f 2 ! A t ? 2 A t 1 A t 0 J v * ' * - < - ^ ^ i - ) A t 2 2 A t 0 ( t ) d t = 2 6(A t x 2 - A t x A.t 2) A t, At 3 A t 0 A t . 1 ? ( t ) d t = - r - ' 1 = ( i i 1 ) A \ t 2 -A t 1 A t 2 3 A t which reduces to Simpson's Rule if . A t - , = 1 2 An approximation to the f i r s t moment with r e s p e c t to t = o i s A t 2 A t 2 A t 2 A t 2 Jp(t)'tdt = c Q fio(t)tdt+c1 J i 1 ( t ) t d t + c 2 Ji 2(t)tdt = ( < T i + r 2 + r 3 ) - T - - r r i ( A v A V At 3 A t 3 + r 2 A t 2 + r 3 A t i ] - f - + r i A t i - ^ 64 i n which C. r _ 9 _ & 1 At-jAt r - 1 2 2 A t ^ -At-^At^ C2 ^ 3 = 3 A t 2 - A t 1 A t 2 In those cases where the c o n c e n t r a t i o n d e c l i n e has not been d e f i n e d adequately i n the f i e l d , a subroutine "TAILEX" (see Appendix) i s f i r s t c a l l e d from "NACL" or DQV". I t p l o t s ( l o g CY.'2 l o g t ) vs. ( t ) to permit the f i t t i n g of a f-distribution e x t e n s i o n , as d i s c u s s e d i n S e c t i o n 3 . 3 - 5 and shown on F i g u r e 7 . The parameters of the e x t r a p o l a t i o n are read o f f the "TAILEX" p l o t and punched onto the c o n t r o l card of the CXt.)-.;.datac.che.Gk. I f the subroutine "QVEL" i s then c a l l e d , i t w i l l use the / ~ " - d i s t r i b u t i o n to extend the i n t e g r a l over C(t) to i n f i n i t e time. F i n a l l y , the C ( t ) - c u r v e s , with or without r - e x t e n s i o n , can be p l o t t e d by c a l l i n g subroutine "PL0TGA" from the input programs "DQV" or "NACL". "PL0TGA" i s a l s o l i s t e d In the Appendix, together with o p e r a t i n g i n s t r u c t i o n s and sample p l o t s 4 . 2 . 3 R e s u l t s The t r a c e r data c o n s i s t of 111 s l u g i n j e c t i o n s , f o r which 146 t i m e - c o n c e n t r a t i o n curves were determined. In other words, approximately two t h i r d s of. the s l u g i n j e c t i o n t e s t s c o n t a i n one downstream sample only; the other t h i r d c o n s i s t s 65 mainly of runs "with two samples. Only three runs extend over three sampling l o c a t i o n s and none over f o u r . In a d d i t i o n , there are e i g h t discharge measurements with the constant r a t e i n j e c t i o n method. P r e s e n t a t i o n of the complete data, which c o n s i s t of approximately 7000 time and c o n c e n t r a t i o n values d i d not appear j u s t i f i e d . Tables 4A to 4D show the r e s u l t s i n summarized form. There i s one t a b l e per stream, with the t e s t runs arranged i n h i s t o r i c a l sequence. The i d e n t i f i c a t i o n code i s e x p l a i n e d i n a f o o t n o t e . With the e x c e p t i o n of the simple discharge measurements, the same data appear again i n the Appendix, as p r i n t o u t of the program "L0GRE " (see S e c t i o n 5 . 2 . 2 ) . The arrangement there i s by t e s t reach. For cross r e f e r e n c e from Table 4 to the Appendix i t i s best to use the t e s t number. Note that the d i s c h a r g e s are s l i g h t l y d i f f e r e n t because Table 4 r e f e r s to the t r a c e r data and t h e r e f o r e g i v e s the discharge at the sampling p o i n t , whereas the "L0GRE" p r i n t o u t r e f e r s to t e s t reaches, and the discharge i s the estimated mean between i n f l o w and outflow. 4 . 2 . 4 Accuracy The r e l a t i v e accuracy, or i n t e r n a l c o n s i s t e n c y of the data appears to be s a t i s f a c t o r y . As can be seen from Table 4 , t e s t s on the same day give s i m i l a r flows at a l l s t a t i o n s of a stream ( a f t e r c o r r e c t i o n f o r t r a c e r l o s s i n long d y e - d i l u t i o n runs) and t e s t s made at d i f f e r e n t times but with apparently s i m i l a r flows, show a c o n s i s t e n t time d i s t r i b u t i o n of the TABLE IVA SUMMARY OF TRACER MEASUREMENTS: BROCKTON CREEK Test Date No. of T " • Peak Mean Discharge I d e n t i f i c a t i o n 1 P o i n t s r . J \ (min) Lag /m3a-l\ forf Method (mm) extension BrRl,2UP-2X - 3 Aug. 1 5 , 6 7 17 0 . 4 1. 5 2. 61 0 . 0 0 5 6 4 RhWT BrRl,2UP-2-3X Aug. 15,6.7 28 14. 0 3 3 - 0 3 7 . 6,'- 0 . 0 0 7 7 0 + RhWT BrR2,lUP-IX-2 Aug. 1 5 , 6 7 16 0 . 5 1. 5 2. 28 ' "0 . 0 0 8 5 6 . ' RhWT BrR2,lUP-l-2X Aug. 1 5 , 6 7 16 1 7 . 5 2 7 . 0 3 2 . 4\ 0 . 0 1 0 0 + RhWT BrR3 , 2UP-2X Aug. 1 7 , 6 7 0 . 0 0 3 3 6 RhWT r T B r R 4 , l U P - l X - 2 - 3 Aug. 1 7 , 6 7 19 1. 0 2. 2 3 . 43 0 . 0 0 5 0 4 0 . j-. RhWT BrR4,1UP-1-2X -3 Aug. 1 7 , 6 7 18 2 5 . 0 '39. 5 4 7 . 9 - 0 . 0 0 6 1 7 + RhWT B r R 4 ,1UP-1-2-3X Aug. 1 7 , 6 7 13 5 2 . 0 7 4 . 0 8 2 . 1 0 . 0 0 7 3 8 RhWT BrR5,2UP-2X Aug. 1 7 , 6 7 0 . 0 1 0 8 RhWT P T BrR6 ,1UP-1X Aug. 1 7 , 6 7 15 1. 0 2. 2 2. 99 0 . 0 0 7 0 6 0 . I . RhWT BrR7 ,1UP-IX Sept. 1 5 , 6 7 15 4 . 2 9 . 0 14. 50 0 , 0 0 0 6 7 RhWT BrR8,2-2DOX Sept. 1 5 , 6 7 13 1. 0 5 . 0 8 . 71 0 . 0 0 0 7 0 3 RhWT BrR9,lUP-lD0X Sept.24 , 6 7 32 4 . 0 1 9 . 0 3 9 . 9 0 . 0 0 0 1 5 4 RhWT 1 The t e s t i d e n t i f i c a t i o n code i s as f o l l o w s : the f i r s t two l e t t e r s i d e n t i f y the stream (Br=Brockton, P l = P l a c i d , Bl=Blaney, P h = P h y l l i s ) , then comes the t e s t run number ( R 1 , R 2 , . . . i n a more or l e s s h i s t o r i c a l sequence, next i s the l o c a t i o n (gauge number) of i n j e c t i o n s ; (UP or DO meaning s h o r t l y upstream of ... or s h o r t l y downstream from . . . ) , f i n a l l y the sample l o c a t i o n s (gauge numbers), with an X i n d i c a t i n g the p a r t i c u l a r time c o n c e n t r a t i o n curve one i s d e a l i n g with. Examples: PhR5 , 2 U P - 2 X i s a simple discharge determination at Gauge #2 on P h y l l i s Creek, with i n j e c t i o n s h o r t l y above Gauge #2 and sampling at the gauge. B 1 R 1 0 1-3-5X would i n d i c a t e that the t e s t run #10 on Blaney Creek covers two reaches ' 1 - 3 , and 3 - 5 ' ) . 2 C.I. means "constant r a t e i n j e c t i o n t e s t . " TABLE IVA (Cont'd.) SUMMARY OF TRACER MEASUREMENTS: BROCKTON CREEK Test No. o f "T" "'• Peak Mean D i s c h a r g e : + I d e n t i f i c a t i o n ! Date P o i n t s ' , 3 . y Lag Lag ( m 3 s - 1 ) f o r ̂  Method' ^ T n l n ^ (min) (min) e x t e n s i o n BrR10,l-2X Aug. 1 8 , 6 8 18 70. 117, 1 6 0 . 0 .00124 + RhWT B r R l l , 2 - 3 X . Aug.18,6 8 16 53. 9 7 . 120 . 0 . 0 0 1 2 6 + RhWT BrR12,lUP.-IDOX Aug. 1 8 , 6 8 9 2 . 5 5. 7 8 . 50 0 . 0 0 1 2 2 RhWT BrR13,2UP-2Dp"X Aug. 1 8 , 6 8 9 0 .00114 RhWT BrR l 4 , 1 - I D O X Aug.2 6 , 6 8 9 0 .00453 RhWT BrR15,2U,P-2X , Aug.2 6 , 6 8 9 . 0 .00555 RhWT B r R l 6 , l - 2 X S e p t . 1 4 , 6 8 13 4. 9 8 . 1 9 . 08 0 . 0 6 4 8 RhWT BrR17 ,2 -3X S e p t . 1 4 , 6 8 12 3 . 0 7. 2 8 . 50 0 . 0 5 5 6 + RhWT BrRl8,l-2X-3 S e p t . 1 4 , 6 8 19. 7. 2 1 1 . 6 1 2 . 6 0 .0550 RhWT BrRl8,l - 2-3X S e p t . 1 4 , 6 8 13 14. 5 2 0 . 0 2 1 . 2 9 + RhWT B r R 1 9 3 l - 2 X S e p t . 1 4 , 6 8 10 2 . 5 5. 0 5 . 40 0 . 1 7 5 RhWT BrR20 , 2-3X S e p t . 2 2 , 6 8 13. 2 . 2 4. 7 5. 39 0 . 0 8 8 9 RhWT BrR21,l-2X S e p t . 2 2 , 6 8 11 4 . 2 7. 3 8 . 33 0 . 102 + RhWT BrR22,2-3X S e p t . 2 2 , 6 8 12 1 . 8 4. 0 4. 53 0 .109 RhWT B r R 2 3 , l - 2 X S e p t . 2 2 , 6 8 11 3 . 2 5 . 5 6. 97 0 . 152 + RhWT 0 s —3 TABLE IVB SUMMARY OF TRACER MEASUREMENTS: PLACID CREEK Test No. of m ; T s Peak Mean Discharge + I d e n t i f i c a t i o n Date P o i n t s (min) Lag Lag , 3 -1\ f o r / - ' Method < (min) (min) extension P1R1,2UP-2X . June 3 , 6 8 18 1. 8-- 3 . 5 4. 57 0. 0 6 8 7 RhWT P1R2,1-2X June 3 , 6 8 14 122. 164. 210. 0. 064 + RhWT P1R4,3UP-3X June 9 , 6 8 17 3 . 2 6 . 4 7- 8 0. 0 1 8 3 + .RhWT P1R5 32-3X June 9 , 6 8 17 160. 236. 3 5 1 . 0. 018 + RhWT P1R6,4-4DOX June 9 , 6 8 46 4 . 5 9 . 3 14. 8 0. 0 4 0 8 NaCl P1R7,2UP-2X June 9 , 6 8 16 5 . 8 11. 0 1 3 . 8 0. 0117 + RhWT P1R9,2UP-2X June 1 8 , 6 8 18 1. 9 4 . 8 8 . 29 0. 0 0 3 5 8 RhWT P1R10,2-3X June : 2 Y y 6 8 17 62. 0 9 4 . 121. 0. 085 + RhWT P1R11,3-3DOX June 2 7 , 6 8 18 11. 0 20 . 26. 9 0. 144^ RhWT P1R11,3-3DO-4X June 2 7 , 6 8 6 211. 261. 9 RhWT P1R12,4UP-4X June - 2 7 / 6 8 27 1 1 9 . 170. 2 0 6 . 0. 185 RhWT P1R13,1-2X June 28 , 6 8 17 1 6 5 . 208. 2 8 1 . 0. 045 + RhWT PlRl4,2UP-2X June 28 , 6 7 18 24. 0 28. 3 2 9 . 7 0. 0418 RhWT PlRl6 , 4-4DOZ Aug. 20 , 6 8 40 8 . 8 11. 6 2 0. 035 NaCl P 1 R 1 7 , 3-4X Aug. 20-1 , 6 8 6 7 4 . 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 + I d e n t i f i c a t i o n Date P o i n t s (min) Lag Lag , 3 -1\ f o r f1 Method (min) (min) extensions P1R17A,2UP-2X Aug.28 , 6 8 9 8 . 11. 5 12. 5 0. 0824 RhWT PlRl8,2-3X - 4 Aug.28 , 6 8 60 48. 71. 8 3 . 3 0. 157 NaCl PlRl8,2 - 3-4X Aug.28 , 6 8 68 236. 2 9 4 . 3 3 9 . 0. 245 + •NaCl P1R19 ,3-4X Aug.28 , 6 8 36 1 6 2 . 1 9 9 . 2 3 9 . 0 . 268 + NaCl P1R20,4-4D0 Aug.28 , 6 8 13 1. 15 3 . 8 4. 82 0 . 249 NaCl P1R21 ,3-4X Aug.30 , 6 8 39 302. 3 8 6 . 482. 0 . 097 RhWT P1R22,4UP-D0X Sept.21 , 6 8 12 8 . 5 14. 5 1 9 . 2 0. 122 + RhWT P1R23,2UP-2X Sept.21 , 6 8 12 4 . 0 8 . 0 9 . 4 0. 0 3 6 3 RhWT PlR24,2-3X-4 Oct.12 , 6 8 73 4 7 . 5 7 4 . 92. 1 0. 142 + NaCl PlR24,2 -3-4X Oct.12 , 6 8 52 2 0 5 . 2 7 4 . 346. 0. 368 + NaCl P1R25 31-2X Oct.12 , 6 8 92 115. 159. 1 8 1 . 0. 0 8 2 2 + NaCl P1R26,2UP-2X Oct.12 , 6 8 12 4 . 2 7. 3 8 . 68 0. 0 8 2 2 RhWT P1R27,2-3X-4 Oct.22 , 6 8 80 3 6 . 1 5 4 . 8 62. 1 0. 212 NaCl P l R 2 7 , 2 - 3 - 4 X Oct.22 , 6 8 38 1 5 7 . 224. 271. 0. 560 + NaCl P1R28,1-2X Oct.22 , 6 8 75 9 4 . 130. 144. 0. 122 + NaCl P1R28A,2UP-2X Oct.22 , 6 8 11 3 . 2 5. 7 7 . 02 0. 122 RhWT P1R29,4UP-4DOX Oct.22 , 6 8 12 3 . 0 6 . 0 7 . 45 0. 474 RhWT TABLE IVC SUMMARY OP TRACER MEASUREMENTS: BLANEY CREEK Test I d e n t i f i c a t i o n Date No. of ~Jis-' Peak P o i n t s (min) Mean Lag & (min) (min) Discharge + fo r r extension (mV1) Method B 1 R 1 , 1 - 2 X M a y l 5 , 6 7 0 . 1 9 0 B 1 R 1 , 1 - 3 X May 1 5 , 6 7 0 . 260 B 1 R 2 , 1 - 2 X May 1 8 , 6 7 . 0 . 1 5 9 B 1 R 3 , 4 U P - 4 D O X May 1 9 , 6 7 0 . 123 B 1 R 4 , 4 U P - 4 D O X May 1 9 , 6 7 0 . 1 6 8 B 1 R 5 , 1 - 2 X June 9 , 6 7 0 . 0 6 4 B I R 6 , 4 U P - 4 D O X June 9 , 6 7 1 0 4 . 2 9 . 0 1 0 . 5 0 . 0 5 4 B 1 R 7 , 1 - 2 X June 9 , 6 7 1 1 1 7 . 5 2 7 . 0 3 6 . 0 . 0 6 0 B 1 R 7 A , 5 U P - 5 X S e p t . 3 0 , 6 7 18 2 . 8 5 . 5 1 2 . 2 0 . 0 3 1 7 B 1 R 8 , 3 - 5 X O c t . 6 , 6 7 1 4 8 . 5 1 6 . 2 2 1 . 2 0 . 8 7 3 B I R 9 , 4 U P - 4 D O : X N o v . 1 9 , 6 7 9 0 . 9 1 . 82 2 . 19 0 . 5 9 5 B 1 R 1 0 , 3 - 5 X - 4 N o v . 1 9 , 6 7 2 2 1 2 . 0 2 0 . 5 2 6 . 3 0 . 5 3 8 B 1 R 1 0 , 3 - 5 - 4 X Nov. 19 ,6.7 1 4 5 8 . 7 7 . 5 8 9 . 4 0 . 5 8 8 B l R l l , l - 3 X - 5 N o v . 1 9 , 6 7 1 5 3 0 . 4 4 . 5 5 4 . 7 0 . 5 1 7 B l R l l , l - 3 - 5 X Nov. 1 9 , 6 7 1 1 5 1 . 6 7 . 5 7 7 . 2 0 . 5 2 0 SoD • C I . 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 + I d e n t i f i c a t i o n Date P o i n t s (min) Lag Lag , 3 -1\ f o r F Method (min) (min) extension B l R 1 2 , l - 3 X - 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 1 0 . 4 + RhWT B l R 1 3 , 3 - 5 - 4 X 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 1 0 . 6 12. 1 0 1 1 . 8 RhWT BlR15,l - 3 - 5 X Jan.20 ,68 18 11. 5 16. 5 18. 6 1 1 . 8 + RhWT B l R l 6 , l - 3 X - 5 Feb.3,68 16 19. 5 26. 4 30. 2 1. 6 4 RhWT B l R l 6 , l - 3 - 5 X Feb.3,68 23 31. 5 40. 2 47. 8 1. 66 + RhWT B 1 R 1 7 , 1 - 3 X Feb.3,68 39 18. 3 25. 5 29. 6 1. 6 4 NaCl B1R18,3-5X March 5,68 25 6. 2 22. 4 0 . 862 RhWT B 1 R 1 9 , l - 3 X - 5 - 4 March 5,68 16 15. 0 22. 5 25. 4 1. 95 RhWT B l R 1 9 , l - 3 - 5 X - 4 March 5,6 8 16 26. 0 33. 7 38. 0 2. 0 0 RhWT BlR19,l - 3 - 5 - 4 x 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 + I d e n t i f i c a t i o n Date P o i n t s (min) Lag Lag , 3 -1\ f o r / 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 B 1 R 2 5 , 3 - 5 X June 3,68 18 9.5 16. 6 20. 8 0. 682 RhWT B1R26 , 3 - 5 X 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 B l R 2 9 , 3 - 5 X - 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 - 3 X June 13,68 18 57-0 90. 0 117. 0 0. 146 + RhWT B1R32,3-5X June 18,68 56 27.5 5 3 . 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 B 1 R 3 5 , 1 - 3 X Oct.1 ,68 69 40. 5 61. 0 71. 2 0 . 285 NaCl B1R36 31 - 3 X Oct.12 ,68 16 24. 0 34. 0 40. 4 0. 741 + RhWT BlR37 , 3 - 5 X - 4 • Oct.13,68 31 7.5 12. 7 15. 5 1. 3 5 NaCl BlR37 , 3 - 5 - 4 x 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 + I d e n t i f i c a t i o n Date P o i n t s Lag- Lag , 3„-l\ f o r 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 J u l y 21,67 22 25. 0 38. 0 •43. 5 0. 7 4 8 + RhWT PhR3,2-3-4X J u l y 21,67 30 56. 75. 83. 1 0. 817 + RhWT PhR4,l-2X J u l y 27,67 18 38. 5 56. 69. 92 0. 369 + RhWT PhR5,3-4X-6 J u l y 28,67 14 32. 0 45. 52. 9 0. 339 + RhWT PhR5 ,3-4-6x J u l y 28,67 21 52. 0 73. 82. 3 0. 385 + RhWT PhR6,2-3X-4 J u l y 28,67 18 37. 5 57. 6 4 . 4 0. 352 + RhWT PhR6,2-3-4x J u l y 28,67 12 77. 0 104. 118. 1 0. 338 + RhWT PhR7,1-2X J u l y 29,67 15 43. 5 61. 0 76. 0 0. 312 + RhWT PhR8,4-6X J u l y 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 P h R l l , l - 2 X - 3 May 17,68 17 21. 25 28. 5 36. 2 1. 47 + RhWT Ph R l l , l - 2 - 3 X May 17,68 11 40. 0 58. 0 6 4 . 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 I d e n t i f i c a t i o n Date No. of Poin t s rip, ,. Jl-s-(min) Peak Lag (min) Mean Lag (min) Dis.charg ( m 3 s - 1 ) je + f o r r extension Method PhRl4,2-3X May 1 9 , 6 8 12 1 5 . 21. 2 3 . 4 2 . 5 5 + RhWT P h R 1 5 ,1-2X May 19 , 6 8 16 1 1 6 . 5 22. 5 2 7 . 8 2 . 40 + RhWT P h R l 6 , l - 2 X May 17 , 6 8 37 1 9 . 28. 5 3 6 . 7 1. 40 NaCl PhR17,2-3X May 19 , 6 8 32 14. 25 21. 2 . 24.2 2 . 42 NaCl PhRl8 , 4-6X May 24 , 6 8 13 8.00 12 . 8 14 . 4 1.10 RhWT PhR19,3-4X May 24 , 6 8 15 1 9 . 26 . 5 2 9 . 9 1.07 RhWT PhR20,2UP-2X-3 May 30 , 6 8 16 2 . 6 5.0 6.31 0 . 9 4 5 RhWT PhR20,2UP-2-3X May 3 0 , 6 8 16 25. 3 8 . 5 4 9 . 8 0 . 9 8 5 + RhWT PhR21,l-2X May 3 0 , 6 8 18 24. 3 3 . 3 9 . 8 1.05 RhWT PhR22,l-2X May 3 0 , 6 8 70 22.5 3 3 . 3 8 . 8 1. 02 NaCl P h R 2 3 , 2 - 3 X June 1 , 6 8 13 1 7 . 24. 26 . 8 1 . 8 8 RhWT PhR24,2-3X June 22 , 6 8 ' 15 2 3 . 5 3 5 . 5 42.1 0 . 9 5 5 + RhWT P h R 2 5,l-2X June 22 , 6 8 17 2 3 . 3 5 . 4 46 . 4 0 . 826 + RhWT P h R 2 6 ,2-3X J u l y 3 , 6 8 18 20. 0 31. 2 3 5 . 5 1.19 + RhWT PhR27 ,3-4X-6 J u l y 3 , 6 8 35 17. 24 . 5 28 . 5 1, 26 NaCl PhR27,3-4-6X J u l y 3 , 6 8 66 26. 3 8 . 5 4 4 . 8 1 . 2 5 ^NaCl PhR28,3-4X-6 Sept .17 , 6 8 28 10. 3 1 4 . 5 1 7 . 3 3 3.10 + NaCl PhR28,3-4-6X Sept .17 , 6 8 64 1 5 . 0 21 . 8 24 . 9 3 . 1 0 NaCl -<1 4=" TABLE IVD (Cont'd.) SUMMARY OF TRACER MEASUREMENTS: PHYLLIS CREEK Test No. of I d e n t i f i c a t i o n Date Po i n t s rT£. • Peak Mean (min) Lag Lag (min) (min) Dis charge (mV1) + f o r r extension Method P h R 2 9 ,3 - 4 X - 6 P h R 2 9 , 3 - 4 - 6 x P h R 3 0 , 1-2X - 3 P h R 3 0 , 1 - 2 - 3 X 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 u i 7 6 t r a c e r . The r e l a t i v e s a l t d i l u t i o n method a l s o agrees with the dye d i l u t i o n method on the few. occasions when simultaneous t e s t s were run,. (Ph R l l - P h R12; Ph R21-Ph R22; B l R l 6 - B l R 1 7 ; B l R 3 2 - B l R33)• Absolute accuracy i s m o r e . d i f f i c u l t to estimate, par- t i c u l a r l y with regard to d i s c h a r g e , because none of the fou r t e s t e s t r e a m s i s gauged by the Water Survey of Canada. During the t e s t s B l R32 and B l R 3 3 over the reach 3 - 5 of Blaney Creek, the Water Survey of Canada measured discharge at Gauge 1 . With mean t r a c e r t r a v e l times of over one hour, the t r a c e r methods cannot be expected to give very r e l i a b l e d i s c h a r g e s . The s a l t d i l u t i o n method, Run 3 2 , i n d i c a t e d a discharge of 9 0 . 3 I s - 1 , the dye d i l u t i o n method, Run 3 3 , gave 9 6 . 8 I s - 1 , or 83 Is ^ with the customary t r a c e r l o s s adjustment (L = 0 . 2 % per minute). The cu r r e n t meter measurement was made at a poor s e c t i o n with depth of l e s s than 1 f t throughout; i t i n d i c a t e d 75 I s - 1 . Coverage of the discharge range v a r i e s from stream to stream. The low flows are reasonably w e l l d e f i n e d on a l l f o u r , but Brockton and P l a c i d Creeks go dry r e g u l a r l y so that one could c o n c e i v a b l y observe t r a c e r t r a v e l times of s e v e r a l days at extremely low flows. Run 9, on Brockton Creek g i v e s the lowest v e l o c i t y , 3 - 5 mmsT1. The high flow range i s reasonably w e l l d e f i n e d on only two of the f o u r streams, Blaney and Brockton. Runs 1 3 , l4,and 15 on. Blaney Creek c o i n c i d e with the l a r g e s t observed flow at a stream guage i n the neighbouring v a l l e y , f o r which there 77 are 3 years of r e c o r d s . S e v e r a l major l o g jams were moved at t h i s flow. The c o n t r o l s t r u c t u r e (Photograph 12) was com- p l e t e l y submerged. Run 19 on Brockton Creek was observed at the runoff' peak dur i n g a very severe r a i n storm. On P h y l l i s and P l a c i d Creeks,- the discharge range of the t r a c e r t e s t s extends to approximately 20% of the h i g h e s t flows that have occurred d u r i n g the l a s t 3 years. 4 . 3 Surge Tests The r e s u l t of a surge t e s t c o n s i s t s of a graph showing water l e v e l s vs. time f o r a l l the gauges on one stream. F i g u r e 15 i s . a t y p i c a l example. The curves are w e l l d e f i n e d because of the i n h e r e n t l y high accuracy of time and stage measurements. For convenience the data are p l o t t e d as t vs. H r a t h e r than the more s i g n i f i c a n t t vs. Q. This does, however, not a f f e c t the c o n c l u s i o n s , because the gauge r a t i n g curves are p r a c t i c a l l y l i n e a r i n the s m a l l discharge range encoun- t e r e d d u r i n g any one surge t e s t . A t o t a l of 22 surge t e s t s were made; 7 on P h y l l i s Creek, 6 on Blaney and Brockton Creeks,and 3 on P l a c i d . Data from 1 or 2 gauges are m i s s i n g i n approximately 50% of the t e s t s , due to e i t h e r l a c k of f i e l d a s s i s t a n t s or d i f f i c u l t i e s with the tube gauges. Only the Brockton Creek t e s t s cover the complete range of flows. The c o n t r o l s t r u c t u r e on Blaney Lake was submerged and i n o p e r a t i v e d u r i n g the h i g h e s t flows : (Photograph 1 2 ) . 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 (mV 1 1) 2 ) A Q i t L ) ) Lag 1-2 (min Q at G. 2 ( m 3 s _ 1 ) A Q 2 Lag 2-3 Q at G.3 ( m 3 s _ 1 ) A Q 3 Aug. 15,67 S t a r t of UP 3' .0094 -'+.0013 6 . 2 5 .010 .0010 6. .0084 + . 0002 sharp peak . 0 1 0 7 5.50 .011 5. . 0 0 8 6 s t a r t of DS .0101 -.0015 5.00 . 011 — .0012 5. . 0091 - .0012 Aug. 17 , 6 7 S t a r t of UP . 0 0 2 7 + . 0 0 1 5 11. 0 . 0 0 3 5 + .0011 8. 5 .0036 - + .0008 S t a r t of DS . 0040 - . 0 0 0 6 13.0 .0045 - . 0 0 0 8 9. 0 .0042 - . 0 0 0 6 S t a r t of UP . 0 0 6 7 5 +.00055 7.8 . 0 0 6 6 + .00045 S t a r t of DS . 0 0 7 0 -.00145 8.2 . 0 0 7 0 - .0011 S t a r t of UP . 0 0 6 8 - + . 0 0 0 8 8.2 . 0 0 7 2 + . 0 0 0 9 5 7.. 0 . < : . 0 0 6 2 + . 0 0 0 7 5 Aug. 26 , 6 8 S t a r t of UP .0043 -.0012 11.5 . 0 0 5 7 - . 0 0 1 3 5. 8 . 0 0 5 6 + . 0 0 1 0 5 S t a r t of DS 1.0 . 0 0 6 8 - .0011 7. 2 — .00085 Sept .14 , 6 8 S t a r t of UP . 047 +.0050 4.2 . 0 3 8 + . 0 0 5 5 3. 2 .041 • + . 0 0 2 5 S t a r t of DS . 052 - . 0 1 3 0 4.5 .045 — . 0 1 4 3. 0 .044 - . 0 0 8 0 Sept .22 , 6 8 S t a r t of UP .082 + .009 3.0 .084 •4-.008 3. 0 . 0 8 1 . 0040 S t a r t of DS . 0 8 6 3.0 . 0 8 8 - .014 2. 8 .085 - .0045 1) Q i s the flow immediately p r i o r to the t e s t . 2) AQis the change i n flow as observed at a p a r t i c u l a r 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) A Q 3 Lag 3-4 (min) Q at G .4 (mV1) A Q 4 June 2 8 , 1968 S t a r t Mid- of UP • UP 9 2 . 1 1 0 . . 042 +. 0.0 4 56 . 5 6 . . 0 6 5 + . 0 0 4 S t a r t Mid- of DS DS 1 1 3 . 9 6 . . 0 4 4 - . 0 0 4 5 6 . 49 . . 0 6 7 - . 0 0 4 Aug.28, 1968 S t a r t Mid- of UP-. UP 6 9 . 5 7 7 . 5 . 0 8 2 + . 0 0 5 S t a r t Mid- of DS DS . 0 8 6 - . 0 0 6 4 0 . 4 3 . . 1 5 6 - . 0 1 0 Aug.30, 1 9 6 8 S t a r t Mid- of UP-. UP , 8 9 . 1 0 4 . . 042 + . 0 0 7 5 1 . 5 7 . . 0 5 8 + . 0 0 4 S t a r t Mid- of DS DS 8 9 . 1 0 6 . . 0 4 8 - . 0 0 7 4 9 . 5 6 . . 0 6 1 - . 0 0 8 1 4 2 . 1 4 2 . . 1 0 9 - . 0 0 6 c o 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 u a z e to Determine G .l 1 1-3 G . 3 3 - 5 G .5 5 - 4 G .4 L a ^ (mV1) ( m l n ) (mV1) ( m i n ) ( m 3 s - l ) ( m i n ) ^ - l . 3-4 May 19, Mid UP-; •' a 118 + . 052 45. . 1 1 9 + .047 1 6 . 5 66.5 123 045 1967 S t a r t of UP ' 37-5 1 5 . 62. Mid DS 162 042 47. . 1 5 8 — .038 17 . 7 0 . 165 — . 040 S t a r t of DS . 38. 12. 5 2 . r— t 1 June 9, Mid UP-" 0 6 3 0 1 8 59. .061 + .019 2 3 . .055 016 5-4 7 2 . 066 + . 019 1967 S t a r t of UP 49. 21. 72. Nov.19 , Mid UP # 505 + . 0 20 31.5 .495 + .020 9. .435 + . 020 19. 555 + . 015 1 9 6 7 S t a r t of Up 24.5 21. March 5, Mid DS 2. 100 — . 140 13.5 2 . 1 8 0 — . 140 7. 2.120 — . 120 1968 S t a r t of DS 9.5 8. Mid UP,' ' 2. 100 + . 160 1 3 . 2 . 1 8 0 + .120 6. 2 . 0 8 0 + . 120 S t a r t of UP- 10 . 5. Oct.13, Mid DS 1. 370 — , 180 1 8 . 5 1 .090 — .140 7. 1 . 1 8 0 — . 130 21. 5 1. 170 — . 130 1968 S t a r t of DS 1 3 ; 8. 20. Mid UP- ' 1-. 240 + . 200 1 8 . 5 .970 + . 1 8 0 7. 1,050 + . 180 2 1 . 1. 0 5 0 + . 170 S t a r t of UP . 12. 7. 22. Nov.3 0 , Mid UP., ( i small) 1. 240 + . 020 19.5 • 990 + .030 9. 5 1.120 + . 020 14.5 1. 0 6 0 + . 0 3 0 1968 S t a r t of UP " 1 6 . 8. 1 8 , Mid DS(small) 1. 260 — . 030 1 9 . 5 1 .020 040 9. 5 1.140 — . 020 15. 1. 090 — . 030 S t a r t of DS 1 7 . 7. 15.5 Mid UP 1 ( l a r g e ) 1. 240 + . 210 19 . 9 8 0 240 8. 5 1.120 210 16 1. 060 + . 2'10 Sta r t " 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 Q 2 ( Lag Q at 2-3 G.3 1,1111 > (mV1) A Q 3 Lag 3-4 (min) Q at G.4 (.mV1) A Q a Lag 4 4-6 (min) Q at G.6 mV1) A Q 6 J u l y 28, Mid UP-:' 36. .338 .+ . 00 8 25. .342 + .010 18.2 .340 + . 013 14. .370 + . 010 1967 S t a r t of UP. 29- 22 . 20 . 12. Mid DS 32.5 . 345 O i l 22.8 .350 - . 010 19. .352 • 014 12. .377 + . 017 S t a r t of DS 24. . 24. 22. 11. May 19, Mid DS 16. 2. 340 — . 040 11. 2.450 -.040 9.5 2.380 - . 040 1968 S t a r t of DS 9.5 11.5 8. Mid UP 17. 2. 300 + . 050 10. 2.'440 + .040 10. 2. 430 + . 060 S t a r t 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 S t a r t 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 S t a r t UP . 12. 9. 25? 6 . June 22, Mid DS 25. .815 - . 095 17. ..790 -.090 14. .345 - , 090 8. . 880' — . 105 1968 S t a r t DS 17. 14. 12. 7.5 Mid UP' 25. .720 + . 10 16. .700 + .10 14. .755 095 8. • 775 + . 105 S t a r t UP 17. 13. 14. 6.5 Sept •17, Mid DS 16. 2.750 - . 33 11.5 2.770 -.27 1968 S t a r t DS 7. 13.5 Mid UP 17. + . 25 10. 2.450 + .27 S t a r t 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) A Q Ĝ at AQ ! Lag Q at d 2 - 3 G .3 ( m i n ) ( m 3 s - l ) A Q 3 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 3 6 . 5 2/-890 -.240 1 9 6 8 S t a r t DS 1 1 . 5 2 5 . 5 Mid UP 1 8 . 2 . 450 + . 16 3 5 . 5 2.580 + . 1 7 0 S t a r t . .. UP' 1 1 . 5 2 5 . 5 3 - 6 Nov.2 9 , Mid DS 1 9 . 2 . 770 - . 32 8 . 3.200 - . 3 5 9 . 5 4 . 200 - . 3 0 1968 S t a r t DS 1 1 . 5 9 . 6 . Mid UP-.- (small) 2 1 . 2 . 450 + . 030 11 .5 2 . 820 + . 0 3 5 6 . 5 3 . 900 + . 0 3 0 S t a r t UP- ' (small) 1 7 . 14. 6 . Mid DS 1 8 . 5 2 . 480 _ 17 9. 2.850 - . 1 3 9 . 5 3 . 900 -.20 S t a r t DS 1 0 . 5 - 1 2 . 5 1 0 . 1 0 . Mid UP--. ( l a r g e ) 20. 2 . 310 31 1 2 . 2.720 + . 2 8 5 . 3 . 700 + . 3 0 S t a r t UP' ( l a r g e ) 1 3 . 5 1 0 . 5 7. or LO 84 The t e s t r e s u l t s are summarized i n Tables. 5A to 5 D i n which the gauge readings have been converted to d i s c h a r g e s , on the b a s i s of the tube r a t i n g curves ( S e c t i o n 3 . 4 . 5 S F i g u r e 1 3 ) . The data appear to be c o n s i s t e n t , i n s o f a r as the observed change i n discharge remains constant from gauge to gauge along a t e s t stream. The surge lags from s t a t i o n to s t a t i o n - as shown i n Table 5 3 have r e l a t i v e l y low accuracy, p a r t i c u l a r l y i f they are short because the gauges could only be read at 30 to 60 second i n t e r v a l s . With lags i n the order of 5 to 10 minutes, t h i s i n t r o d u c e s an immediate u n c e r t a i n t y of 10% to 20%. The lags based on the mid-points of the surges are s u b s t a n t i a l l y more r e l i a b l e than the lags based on the s t a r t i n g p o i n t s because the mid-points were d e r i v e d by smooth- i n g the l e v e l - t i m e curve. 85 5 . CHANNEL GEOMETRY AND STEADY FLOW EQUATIONS 5 . 1 S i m i l i t u d e C o n s i d e r a t i o n s f o r Steep, Degrading Channel Networks The c o n d i t i o n s under which r e a d i l y a v a i l a b l e i n f o r m a t i o n , as d e f i n e d i n S e c t i o n 2 . 5 , may be adequate f o r e v a l u a t i o n of the channel network h y d r a u l i c s w i l l be d i s c u s s e d here. 5 . 1 . 1 Assumptions Dynamic s i m i l i t u d e between r e l a t e d p h y s i c a l systems can only be examined on the b a s i s of a complete l i s t of the f o r c e s a f f e c t i n g the system. S i m i l a r i t y between channel networks i s p o s s i b l e i f the major processes of formation are s i m i l a r (Barr, 1 9 6 8 ) . Obviously there are a l a r g e number of pro- cesses which c o u l d c o n c e i v a b l y a f f e c t the channel network of mountainous b a s i n s ; the problem l i e s i n i d e n t i f y i n g the dominant ones . The formative processes assumed here may be b i a s e d towards the present problem i n the sense that they d e s c r i b e a system that can be d e f i n e d adequately with the r e a d i l y a v a i l - able i n f o r m a t i o n . However, the f i e l d data supply evidence i n d i c a t i n g t h at the system i s reasonable and e x p l a i n s a major p o r t i o n of the v a r i a t i o n i n and between channel networks. The assumptions are l i s t e d and d i s c u s s e d below:;. 8 6 '(i) The drainage network occupies, v a l l e y s whose l o n g i - t u d i n a l s l o p e s j S y , a r e remnants of the P l e i s t o c e n e p e r i o d . The streams form t h e i r channels by degrading i n t o g l a c i a l debris.-, which contains s u f f i c i e n t coarse m a t e r i a l to prevent the stream from r e a c h i n g bed-rock or from degrading enough t o achieve a channel slope , S , s i g n i f i c a n t l y d i f f e r e n t from S^. In other words, S i s imposed on the channel network, but the s i z e of the m a t e r i a l l i n i n g the channel i s one of the r e s u l t s of the channel-forming p r o c e s s . T h i s i s the very opposite of the common regime-type assumption with r e s p e c t to slope and bed m a t e r i a l , which s t a t e s that a regime c a n a l w i l l a d j u s t i t s slope by e r o s i o n or d e p o s i - t i o n , u n t i l i t i s adequate to handle the upstream supply of water and sediment. Most r i v e r s f a l l between the two extremes with meandering and b r a i d i n g p l a y i n g an important r o l e i n r e a c h i n g adjustment between water, sediment l o a d , and v a l l e y s l o p e . The main support f o r the present assumption l i e s i n the c o n s i s t e n t downstream steepening of the test- streams (hanging v a l l e y s , F i g u r e s 2 and 3 ) , l a c k of f l o o d p l a i n s , absence of b r a i d i n g or meandering, and the apparent c l o s e c o r r e l a t i o n between slope and s i z e of the bed m a t e r i a l . ( i i ) The coarse m a t e r i a l l i n i n g the degraded channels can only be moved at extreme flows, which are t h e r e f o r e s o l e l y r e s p o n s i b l e f o r the channel, form. A s i n g l e discharge value i s adequate to rep r e s e n t these formative high flows. 87 T h i s assumption i s supported by the work of M i l l e r ( 1 9 5 8 ) , who found high c o r r e l a t i o n between di s c h a r g e s of a g i v e n frequency and h y d r a u l i c parameters such as width, depth and v e l o c i t y . Day ( 1 9 6 9 ) presents s i m i l a r c o r r e l a t i o n s f o r the t e s t reaches of t h i s study. The theory on channel performance developed here does not depend on t h i s assumption. I t i s only used i n s t a t i n g the s i m i l i t u d e c r i t e r i a . ( i i i ) The t r a n s p o r t r a t e s of m a t e r i a l f i n e r than the bed m a t e r i a l are low and do not a f f e c t the performance of the channel. Supply of coarse m a t e r i a l to the channel through s l i d e s , r o c k f a l l s , bank e r o s i o n , e t c . , i s low and i n balance with the t r a n s p o r t i n g c a p a c i t y of the channel. Up to flows i n the order of the mean annual peak, t h i s assumption i s w e l l supported by f i e l d o b s e r v a t i o n , but i t may break down under extreme f l o o d c o n d i t i o n s , such as the event d e s c r i b e d by Stewart and LaMarche ( 1 9 6 7 ) . A s u f f i c i e n t amount of f i n e g r a v e l and sand may then be i n motion to lower the channel r e s i s t a n c e s i g n i f i c a n t l y , thereby s t a r t i n g a c h a i n r e a c t i o n of h i g h e r v e l o c i t y - more bank and bed e r o s i o n - lower r e s i s t a n c e . None of the reaches showed much evidence of a c t i v e bed or bank e r o s i o n except i n a few i s o l a t e d l o c a t i o n s , mainly a s s o c i a t e d with damaged v e g e t a t i o n cover of the stream banks due to recent l o g g i n g . ( i v ) The channel forming process i s r e p e a t a b l e . I f the same flow regime, Q ( t ) , i s d i v e r t e d down i d e n t i c a l v a l l e y s , 88 c o n t a i n i n g debris, of i d e n t i c a l gradation,, the mean p r o p e r t i e s of the r e s u l t i n g channels w i l l a l s o be i d e n t i c a l . T h i s assumption i s w e l l supported i n regime-type s i t u a t i o n s , where an i d e n t i c a l supply of water and sediment to a s t r a i g h t channel segment e v e n t u a l l y produces i d e n t i c a l channel dimensions. The a p p l i c a t i o n of t h i s concept to the present s i t u a t i o n i s s p e c u l a t i v e , but S e c t i o n 5-3 w i l l show that the s i g n i f i c a n t channel parameters can be d e r i v e d from the imposed or independent e f f e c t s such as Q(t) and S, w i t h - out r e q u i r i n g a knowledge of any dependent parameters, such as width or roughness. This e l i m i n a t e s the p o s s i b i l i t y of a l a r g e random e f f e c t i n the channel forming p r o c e s s . 5.1.2 Cond i t i o n s f o r S i m i l a r i t y C o n s i d e r i n g a s h o r t , s t r a i g h t channel reach, one can i d e n t i f y the f o l l o w i n g f o r c e s , (Barr, 1967; Barr and Herbertson, 1968):gravity g a c t i n g on the water (waves); g r a v i t y a c t i n g along the v a l l e y s l o p e , g'S; ; g r a v i t y a c t i n g on the submerged g r a i n s ( ( p s - p w ) / y s ) g = g g , ^ and v i s c o s i t y , V. i s imposed from upstream. Some of the r e s u l t a n t measures ar'e Ŵ , the water s u r f a c e width at flow Herbertson and B a r r li k e , to c o n s i d e r 4 g r a v i t a t i o n a l f o r c e s g, gS, as above and g s as. the net. g r a v i t a t i o n a l f o r c e on the submerged g r a i n s as: i t a f f e c t s , the surrounding g r a i n s , and g w = ( ( p s - $>w)/rj>w) g as the net g r a v i t a t i o n a l f o r c e on the submerged g r a i n s as i t a f f e c t s the disp l a c e d , water. Since g w can be computed from g s and g , i t need not be s p e c i f i e d . 89 Q^; AD/WD, a depth measure based on the. flow area at. d i s - charge Q D J . the. v e l o c i t y , v^, and D,. the s i z e parameter of. the m a t e r i a l l i n i n g the channel. These, terms can be arranged i n d i m e n s i o n a l l y homogeneous f u n c t i o n a l forms e.g. u s i n g dimensions of l e n g t h f i < 2/5 4 D .1/5 'D Q D S 2/3 2/3 1/3 & s 0 . . . 5 . 1 D V D i n which the terms between v e r t i c a l bars can be used a l t e r - n a t i v e l y . Note that Equation 5 . 1 has only 5 terms as i t con- s i s t s of r a t i o s of a c t i v e f o r c e s and boundary a c t i o n s . The dimensions 1 and t are reduced to 1 only. A f u n c t i o n a l form c o n t a i n i n g n + 1 d i m e n s i o n a l l y homogeneous v a r i a b l e s can be reduced to a n-term non-dimensional form without r e d u c i n g the g e n e r a l i t y . Of the numerous non-dimensional groupings pos- s i b l e with Equation 5 . 1 , the f o l l o w i n g i s most s u i t a b l e f o r the experimental set-up at hand; 1/3 f ( 4 D (»g) 1/3 5 / 3 v D ) = 5 . 2 E q u a t i o n 5 - 2 shows that i n the present s i t u a t i o n , where g/g , )? , and g are constant,, c o r r e l a t i o n s between Q , S , and s D any one r e s u l t a n t measure such as WQ or v^ should be complete I t a l s o shows that complete kinematic s i m i l a r i t y i s only 90 p o s s i b l e i f the r e s u l t a n t measures and \) vary with accord- i n g to the f o l l o w i n g p r o p o r t i o n a l i t i e s (Barr and Herbertson, 1 9 6 8 ) , AD 4 v D «x % + ' 2 5 - 3 S r- 2 Since i? Is not v a r i a b l e In the f i e l d and S i s independent of Q, the f i e l d data do not have to match the above p r o p o r t i o n a l - i t i e s , but i t i s reasonable to expect f a i r l y c l o s e c o r r e s - pondence. Equation 2 can a l s o be d e r i v e d from dimensional con- s i d e r a t i o n s (Yalim, 1 9 6 6 ) . The b a s i c v a r i a b l e s are o , V , p s , g, S, Q D, and one r e s u l t a n t measure, say D. There are 7 v a r i a b l e s c o n t a i n i n g 3 dimensions so t h a t a non- dimensional form i n 4 terms, such as Equ a t i o n 5 - 2 i s adequate f o r a d e s c r i p t i o n of the problem. The system d e s c r i b e d by Equation 5.2 n e g l e c t s many s i g n i - f i c a n t p rocesses. In the f i e l d area of t h i s study v e g e t a t i o n a f f e c t s the s m a l l e r streams (W D< 15 m) c o n s i d e r a b l y through the formation of frequent l o g jams. Some minor reaches appear to be a l l u v i a l and may be aggrading (which would i n t r o d u c e the t r a n s p o r t r a t e as a. v a r i a b l e ) ; others are- o c c a s i o n a l l y cn bed rock. The t r a n s p o r t r a t e s and sediment supply r a t e s are unknown. 91 5.2 ' Basic' Equations T or Steady, Uniform Flow 5.2.1 T h e o r e t i c a l C o n s i d e r a t i o n s The uniform flow parameters,' mean v e l o c i t y , v , su r f a c e width,W , and flow area,A, of a g i v e n s t r a i g h t open s channel segment are f u l l y d e f i n e d by three equations: ( i ) the Equation of C o n t i n u i t y which may be w r i t t e n as Q = v A (=• v'rW. d#) ... 5.4 m A • m s r * where d» = A/W , and s ( i i ) a g e o m e t r i c a l equation l i n k i n g A and W (e.g. s Wg <x VA" i n the case of a t r i a n g u l a r c h a n n e l ) } and ( i i i ) a flow equation l i n k i n g Q and one or s e v e r a l of the parameters v , A, W , and d w i n a form which i s l i n e a r l y f m5 5 s' * independent of Equation 5.4. The constants of t h i s equation w i l l depend on the boundary and cross s e c t i o n a l shape of the channel segment. In the case of n a t u r a l "tumbling flow" channels, the r e l a t i o n s between W and A are v i r t u a l l y unobtainable but t h i s s J i s not a s e r i o u s drawback as long as the aim i s h y d r o l o g i c a l . The main parameters are then only mean v e l o c i t y , v m ? a n d channel storage per u n i t l e n g t h , A, so that the Wg vs. A r e l a t i o n becomes redundant. The flow equation can take many forms. Some of the more r e l e v a n t p o s s i b i l i t i e s are l i s t e d below, a l l assuming broad 92 r e c t a n g u l a r channels ( d x = depth = h y d r a u l i c r a d i u s ) of width 3W, and depth.^d. For flow governed by f r i c t i o n over a h y d r a u l i c a l l y rough boundary with roughness r a t i o s between 7 and 130 (Ackers, 1 9 5 8 ) 5/3 Q <x A (Manning's Equation) and f o r roughness r a t i o s between 1 . 5 and 11 7/4 Q cx A (Lacey's Equation) For h o r i z o n t a l j : f r i c t i o n l e s s channel segments c o n t r o l l e d by a s t e p - l i k e drop at the downstream end, an approximate flow equation i s Q - A 3 / 2 I f the c o n t r o l i s a t r i a n g u l a r weir with apex on the channel f l o o r Q <~ A 5 / 2 and i n the case of a p a r a b o l i c weir Q - A 7 / 2 I f the channel segment i s obstructed'by a dam w i t h outflow below the watersurface Q - A 1 / 2 Note t h a t s i n c e A = Q/v , a r e l a t i o n of the form m' Q °< A w ' . . . 5 . 5 can a l s o be s t a t e d as w Q - x - v / - 1 ... 5 . 6 93 or as w-1 1 w Q <* - T # 5 . 7 i n which T r e f e r s t o u n i t l e n g t h . I n - n a t u r a l channels W i s u s u a l l y r e l a t e d t o Q by an E q u a t i o n o f the form W o< Qz, s s The observed exponents z cov e r a range from 0 . 0 5 t o 0 . 5 ( M i l l e r , 1958), but the most common v a l u e s f a l l between 0 . 1 5 t o 0 . 2 5 . I f t h i s e f f e c t i s i n c l u d e d , Q <x A w becomes w Q « A z w + 1 ... 5 . 8 C o n s i d e r i n g t u m b l i n g f l o w as a randomly a r r a n g e d sequence o f s h o r t c h a n n e l segments governed by f l o w e q u a t i o n s s i m i l a r t o the ones l i s t e d above, and' c o n s i d e r i n g f u r t h e r t h a t the t r a v e l times t h r o u g h c h a n n e l segments i n s e r i e s are a d d i t i v e , one can exp r e s s the r e l a t i o n between Q and T f o r l o n g reaches ( I > > W ) as s l i m n G Y Q = n - ~ [ £ _A T i ] ... 5 - 9 I,.- 0 i = i L *+- Wj_-1 The c o n s t a n t s c. and the exponents y.(= ) are random 1 1 w. l v a r i a b l e s w i t h unknown d i s t r i b u t i o n s . E q u a t i o n 5 .9 can t h e r e f o r e not be s o l v e d t o p r e d i c t the form o f the Q = f ( T ) r e l a t i o n . 5 . 2 . 2 Flow E q u a t i o n s o f the Test Reaches With the form o f the t u m b l i n g f l o w e q u a t i o n not b e i n g p r e d i c t a b l e , a number o f p o s s i b i l i t i e s were t r i e d by p l o t t i n g 94 v a r i o u s transformations, of .the' Q - A data of s e v e r a l reaches a g a i n s t each other. Pure, e x p o n e n t i a l s of the form b A A = a A Q ... 5 - 1 0 or = ( — ) . . . 5 . 1 0 a aA give c o n s i s t e n t l y good f i t s , although i n a few cases the f i t can be s l i g h t l y improved by assuming a s m a l l remnant flow area at zero d i s c h a r g e . Table 6 l i s t s the c o e f f i c i e n t s a A and b^ f o r a l l 13 t e s t reaches of t h i s study. The c o e f f i c i e n t s are based on l i n e a r r e g r e s s i o n s of l o g ^ A on log-^Q of the form l ° g 1 0 A = l o g 1 0 a A +j b A l o g 1 Q Q ... 5 . 1 1 RSQ i s percentage of the v a r i a n c e of the logarithms e x p l a i n e d by the r e g r e s s i o n equation. The standard e r r o r of estimate i s only meaningful f o r the reaches with f a i r l y high number of degrees of freedom. A d d i t i o n a l data c o l l e c t e d by Day ( 1 9 6 9 ) are l i s t e d on Table 1. F i g u r e s 16 and 17 show data p o i n t s and f i t t i n g l i n e s f o r 2 sample reaches. A F o r t r a n IVG program"L0GRE"was used to compute the Q - A, Q - v m , and Q - T r e g r e s s i o n s . Since v = 1/T and Q = Av m,and s i n c e these f a c t s were used to o b t a i n the r e g r e s s i o n data, any one of the r e g r e s s i o n s i s adequate, to e s t a b l i s h a l l 95 TABLE VI REGRESSION PARAMETERS OF STEADY FLOW Degrees "h C o r r e l a - Approx. S t . E r r o r of Reach of aA b A t l o n . Estimate {%) Freedom C o e f f i c i e n t Brockton 1-2 :8 8104 .3376 0 . 9 8 12 .5 Brockton 2-3 "'5 • 7183 . 2830. 0 . 9 9 4 . 0 P l a c i d 1-2 <2 1. 879 . 3 4 0 3 0 .96 3-2 P l a c i d 2-3 ••3 1. 930 .3065 0 . 9 5 6 . 2 P l a c i d 3--4 ::'4 3 . 943 .4656 0 . 9 9 5-3 Blaney 1-3 -.6 3. 375 . 4784 0 . 9 9 7 . 8 Blaney 3-5 • 16 3. 460 .5339 0.99 9-0 Blaney 5-4 7 3. 110 .4389 0 . 9 8 8 .9 P h y l l i s 1-2 J 3. 262 • 5413 0 . 9 8 9 . 0 P h y l l i s 2-3 .9 3. 202 .4577 0.99 3-6 P h y l l i s 3-4 .7 3 . 021 .4787 0 . 9 9 3 .9 P h y l l i s 4-6 "6 3. 199 . 4 0 2 3 0 . 9 7 7-8 P h y l l i s Lower 2 3. 207 .3557 0 . 9 9 7 . 0 96 TABLE V I I REGRESSION PARAMETERS OP STEADY FLOW (from Day, 1 9 6 9 ) Reach Degrees G o r r e l a t i o n o f : a A  13 C o e f f i c i e n t Freedom F u r r y 5 5 . 7 8 8 . 5 0 0 9 . 9 5 S l e s s e U 3 2 . 9 2 2 . 5 9 9 3 . 9 9 S l e s se M 4 4 . 1 7 9 . 5 2 1 3 . 9 7 S l e s s e L 4 2 . 9 2 6 . 5 7 6 5 1 . 0 J u n i p e r 3 3 - 3 7 7 . 4 0 7 5 . 9 9 Ewart U 3 4 . 0 9 1 . 4 1 8 4 . 9 9 Ewart L 4 3 . 2 5 3 . . 4 3 6 5 . 9 7 A s h n o l a U 3 3 . 092 " . 5 8 0 6 1 . 0 A s h n o l a . • M 3 4 . 6 3 8 . 4 6 4 7 1 . 0 A s h n o l a L 3 • 3 . 9 6 7 . 4 2 0 8 1 . 0 9 7 three and any s i g n i f i c a n t d e v i a t i o n of the r e g r e s s i o n c o e f f i c i e n t s from t h e i r t h e o r e t i c a l r e l a t i o n s i n d i c a t e s com- p u t a t i o n a l e r r o r s . With ' T = a TQ ' ... 5.12 and b v m= a vQ v ... 5.13 one obtains b = - b„ ... 5.14A v T a v ~&0~a~T . . . 5.14B b = 1 - b. ... 5.14C v A a _1 v a A ... 5.14D "L0GRE"also computes two sets of Q - T (T = a r r i v a l s s time of t r a c e r ) and Q - Tp (Tp = peak time) r e g r e s s i o n s , one u s i n g a l l data, and one u s i n g only those t e s t s f o r which the dye has been i n j e c t e d at the upstream end-point of the t e s t reach. The above r e g r e s s i o n s c o n s t i t u t e one of the main r e s u l t s of the f i e l d work, t h e r e f o r e a l l the"L0GRE"printouts and p l o t s with l i s t s of the data are i n c l u d e d i n the Appendix. I n s p e c t i o n of. the p l o t s ( F i g u r e s 1 6 and 1 7 , and Appen- dix) shows t h a t , over the range of discharges covered here, there appear to be no s i g n i f i c a n t d e v i a t i o n s 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 I I I 1 I I I • • • • .0001 .001 .01 Discharge in m 3 s-l + mean velocity t V m 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 m 3 s-1 10.0 4 mean velocity , V m © 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. I n p a r t i c u l a r , the Q - p l o t s of B l a n e y Creek,which has the b e s t coverage o f the d i s c h a r g e r a n g e , show no tendency towards l i n e a r b a s i n response (v '.independent o f Q) a t h i g h f l o w s as observed by P i l g r i m ( 1 9 6 6 ) . W i t h the almost t o t a l absence o f f l o o d p l a i n s a l o n g the t e s t reaches t h i s i s not s u r p r i s i n g . 5 . 3 D e t e r m i n i n g the Parameters of the Steady Flow E q u a t i o n The b a s i c f l o w e q u a t i o n ( 5.10) can be r e - w r i t t e n i n v a r i o u s n o n - d i m e n s i o n a l forms; one p o s s i b i l i t y , u s i n g terms s i m i l a r t o E q u a t i o n 5 . 2 i s " •»* ^ V r ' ... 5.10b o r , i f the concept o f a f o r m a t i v e d i s c h a r g e i s r e t a i n e d b A f - ( §—:) ... 5.10C D V' E i t h e r v e r s i o n i s s u i t a b l e f o r comparison w i t h the b a s i c s i m i l i t u d e c r i t e r i o n , E q u a t i o n 5 - 2 . The parameters o f the s t e a d y f l o w e q u a t i o n (a^' , A^, b^) can be c o n s i d e r e d " r e s u l t a n t measures" of the c h a n n e l f o r m i n g p r o c e s s , s i m i l a r t o c h a n n e l width,W^, depth,and roughness, D. Two p o s s i b l e forms o f the s i m i l i t u d e c r i t e r i o n 5 . 2 are t h e r e f o r e QD - S ' K ,f ( ' 5 / 3 , S , -|- , b ) = 0 ... 5 . 2 a 1 0 1 1/3 f ( > s > ' gf , a A' ) = 0 . . . 5 • 2b which shows that a^ and b^ should be determined by Q D and S, as a l l other v a r i a b l e s can be assumed constant. E x p l o r i n g t h i s p o s s i b i l i t y i n d e t a i l i s the main o b j e c t i v e of Day ( 1 9 6 9 ) . Some of h i s f i n d i n g s w i l l be summarized b r i e f l y here. Q D i s o b v i o u s l y not a " r e a d i l y a v a i l a b l e " parameter, but i n a r e g i o n with reasonably homogeneous c l i m a t e the con- t r i b u t i n g drainage area of a channel segment, DA, can be used i n s t e a d . As there i s v i r t u a l l y n othing known about the ph y s i c s of the process expressed by Equ a t i o n 5 . 2 , m u l t i p l e r e g r e s s i o n s of a A and b A on the independent v a r i a b l e s drainage area ( i n km ), and slope were t r i e d , i n c l u d i n g v a r i o u s combin- a t i o n s of transformed data. For the t e s t reaches i n the mountainous areas surround- i n g the lower F r a s e r V a l l e y the f o l l o w i n g two equations give best f i t a A = 1 . 7 3 8 , D A 0 : 2 9 2 2 " ... 5 . 1 5 b A = 0 . 2 8 : 8 8 S - ° - 1 0 a 6 D A 0 * 0 7 5 6 ' ... 5 . 1 6 They e x p l a i n 9 6 . 1 % and 6 9 . 7 % of the v a r i a n c e i n the data. With 11 degrees of freedom, both equations are s i g n i f i c a n t w e l l beyond the 1% l e v e l . The' 6 t e s t reaches i n the dry i n t e r i o r of'B. C. (J u n i p e r ,Creek, Ewart Creek and the Ashnola R i v e r , Tables 3 and 7) do not cover a wide enough range of the independent v a r i a b l e s to j u s t i f y a meaningful 102 r e l a t i o n s h i p f o r t h a t r e g i o n . I n g e n e r a l l y a p p l i c a b l e r e l a t i o n s f o r a^ and b^ d r a i n a g e a r e a can o b v i o u s l y not ta k e the p l a c e o f Q^'. I f i t i s t o be used i n the a n a l y s i s , some c o r r e c t i o n f a c t o r f o r r e g i o n a l v a r i a t i o n s i n the d r a i n a g e a r e a - r u n o f f r e l a t i o n has to be added. An a l t e r n a t i v e t o such a f a c t o r i s the use of a c o n s i s t e n t d i s c h a r g e v a l u e , such as the e s t i m a t e d mean an n u a l f l o o d . A f u r t h e r p o s s i b i l i t y w hich may prove i n t e r e s t i n g i n areas w i t h s p a r s e h y d r o - m e t e o r o l o g i c a l r e c o r d s , i s t o use width,W D,as an independent v a r i a b l e . The d a t a o f t h i s s t u d y g i v e good f i t t o r e g i o n a l l y c o n s t a n t r e l a t i o n s between w i d t h and d r a i n a g e area, and t h e s e r e l a t i o n s are e a s i l y e s t a b l i s h e d by measuring a few c h a n n e l w i d t h s on streams o f v a r i o u s s i z e s i n the r e g i o n o f i n t e r e s t . S l o p e appears t o have no e f f e c t on w i d t h . With Ŵ  and S as independent v a r i a b l e s and i n c l u d i n g the d a t a from a l l areas one o b t a i n s the f o l l o w i n g r e l a t i o n s f o r a^ and b A 0 ii 7 p a A = 0 .943 8 WD ^' : ... 5 . 1 7 0 . 1 3 6 8 _ Or: 0,82 2 b A = 0,2519 WD • S ' ... 5 . 1 8 They e x p l a i n &6:% and 6;3% o f the d a t a v a r i a n c e and are s t a t i s t i c a l l y s i g n i f i c a n t a t the 1% l e v e l , h a v i n g 18 degrees of freedom. I t i s not s u r p r i s i n g t h a t w i d t h and s l o p e d e t e r - mine b.A t o a l e s s e r degree t h a n a A > The f a c t o r a A , whi c h i s 3 - 1 i d e n t i c a l t o the f l o w a r e a a t Q = 1 m s can. v a r y over -a •. 1 0 3 l a r g e range and needs t o be p r e d i c t e d c l o s e l y . The range of b^ i s l i m i t e d t o a p p r o x i m a t e l y 0 . 2 5 < b^ < 0 . 6 5 , so t h a t a c c u r a t e p r e d i c t i o n o f b^ i s not e s s e n t i a l as l o n g as the f l o w s o f i n t e r e s t are o f the o r d e r o f 1 m^ s - 1 . More g e n e r a l p r e - d i c t i v e e q u a t i o n s s h o u l d be p o s s i b l e on the b a s i s o f E q u a t i o n 2 5 . 1 0 c by s u b s t i t u t i n g measured v a l u e s o f DA or f o r the u n c e r t a i n Q D (the r e l a t i o n between DA and WQ found by Day are 2 c o n s i s t e n t l y c l o s e t o DA <*" ) . 5 . 4 The F r i c t i o n Concept A p p l i e d t o Tumbling Flow A l t h o u g h the p r e s e n t s t u d y does not r e p l y on the f r i c t i o n - concept and the d a t a are l e s s t h a n i d e a l l y s u i t e d f o r a p p l i - c a t i o n o f g e n e r a l l y a c c e p t e d open c h a n n e l f r i c t i o n f o r m u l a s , a> b r i e f comparison between f r i c t i o n f o r m u l a s and the g e n e r a l f l o w e q u a t i o n ( 5 - 1 0 ) may be i n t e r e s t i n g and may f a c i l i t a t e c omparison w i t h o t h e r s t u d i e s . To•the w r i t e r ' s knowledge, 2 a l l p r e v i o u s work on very rough channels o r on t u m b l i n g f l o w i s based on the f r i c t i o n c o n c e p t , w h i c h , by r e q u i r i n g v i r t u a l l y u n o b t a i n a b l e roughness d a t a , tends t o y i e l d r e s u l t s t h a t cannot be a p p l i e d t o h y d r o l o g i c a l problems. ^Utah S t a t e U n i v e r s i t y appears t o have a c o n t i n u i n g r e s e a r c h program on t u m b l i n g f l o w . Some o f the r e s u l t s are p u b l i s h e d i n P e t e r s o n and Mohanty, I 9 6 0 and i n an e x t e n s i v e - number o f M.Sc. and Ph.D. t h e s e s , such as A l K a f a j i , 1 9 6 l ; J u d d , 1 9 6 3 ; A b d e l s a l a m , 1 9 5 6 . Other s t u d i e s on r o u g h , n a t - u r a l channels a r e : L e o p o l d , Bagnold e_t a l / , I 9 6 0 ; M i r a j g a o k e r and C h a r l u , 1 9 6 3 ; Johnson, 1 9 6 4 ; H e r b i c h , 1 9 6 4 ; A r g y r o p o u l o s , 1 9 6 5 ; K e l l e r h a l s , 1 9 6 7 ; Hartung and S c h e u e r l e i n , 1 9 6 7 ; S c h e u e r l e i n , 1 9 6 8 . 104 5 . 4 . 1 Open Channel Flow Formulas The problem t o be s o l v e d by an open c h a n n e l f l o w f o r m u l a i s o f the form v m = f (D, d s , S, g, ? w , /i , \ ) 5 . 1 9 i n which d % = A/W r e p l a c e s the more commonly used h y d r a u l i c r a d i u s (they are almost i n d i s t i n g u i s h a b l e i n most stream c h a n n e l s ) , ^ i s v i s c o s i t y and the X^ are n o n - d i m e n s i o n a l c o r - r e c t i o n f a c t o r s which v a n i s h i n the case of a b r o a d r e c t a n g u l a r c h a n n e l s e c t i o n and a p a r t i c u l a r shape of the roughness e l e - ments of d i a m e t e r D. Assuming t h i s t o be the case,one o b t a i n s the commonly used n o n - d i m e n s i o n a l form of E q u a t i o n 5 . 1 9 2 v v d« ~ "PUTS- - f ( S> — > d - ) 5 . 2 0 I t i s w e l l e s t a b l i s h e d t h a t S on the r i g h t s i d e o f ( 5 . 2 0 ) can be n e g l e c t e d as l o n g as steady f l o w does not l e a d t o the f o r m a t i o n o f s u r f a c e waves and e i t h e r one o f the two r e m a i n i n g parameters on the r i g h t i s o f t e n n e g l i g i b l e a l s o , depending on t h e i r r e l a t i v e s i z e . A f o r m u l a t i o n o f E q u a t i o n 5 . 2 0 ^ f o r the case of s t e e p , rough channels i s ( K e u l e g a n , 1 9 3 8 ) | A W = 6 . 2 5 + 5 . 7 5 l o g 1 0 ( ^ ) ... 5 . 2 1 which can be f i t t e d a p p r o x i m a t e l y w i t h e x p o n e n t i a l f u n c t i o n s o f the form 105 . 5 . 2 2 .3 .6 .8 1.0 2 4 6 8 10 20 FIGURE 1 8 . VALUES OF c 2 FOR BEST FIT TO EQUATION 5 . 2 1 C A I, - The commonly used Manning E q u a t i o n assumes a c„ o f 1 / 6 , which a , p r o v i d e s good f i t t o E q u a t i o n 5 - 2 1 over the range 7< — < 1 3 0 . I t i s i m p o r t a n t t o note t h a t E q u a t i o n 5 - 2 2 n e g l e c t s the terms v m d % / u > and S o f E q u a t i o n 5 . 2 0 . W h i l e the t h e o r y o f t u r b u l e n t boundary l a y e r s j u s t i f i e s the f o r m e r , t h e r e i s no a p r i o r i j u s t i f i c a t i o n f o r the l a t t e r i n cases where the 106 roughness elements a f f e c t the f r e e s u r f a c e , as i n tumbling flow. 5.4 . 2 Comparison with the Data The steady-flow data of t h i s study c o n s i s t of exponen- t i a l r e l a t i o n s between discharge and v e l o c i t y f o r c o n d i t i o n s of constant, but unknown roughness, constant known s l o p e , and constant, but unknown c r o s s - s e c t i o n a l shape. To t r a n s f o r m Equation 5 - 2 2 i n t o comparable form r e q u i r e s some assumptions r e g a r d i n g c r o s s - s e c t i o n a l shape. Two assumptions w i l l be used which should bracket the true s i t u a t i o n ( S e c t i o n 5 . 2 . 1 ) . ( i ) W = c 3  v s -3 v ( i i ) Wa Q° ' 2 s Equation 5 . 2 2 can be w r i t t e n as 1 -, c„ + 0 . 5 d„ <x v 2 * m and with Assumption (1) t h i s leads to e 2 + 0 . 5 c ? + 1 . 5 v m <* Q ... 5 . 2 3 and with A s s u m p t i o n ( i i ) i t leads to 0 . 8 ( c 2 + 0 . 5 ) cp+1.5 v «r Q ... 5.24 m • The Q-exponents of the l a s t two equations correspond to b of E q u a t i o n 5.13, and b i s r e l a t e d to b, of Tables 6 and v ^ ' v A 107 7 by b. = 1 - b ( E q u a t i o n 5-14C). The observed b cov e r V Jr\. \f the range 0 . 4 < b v < 0 . 7 2 . On F i g u r e 1 9 , the exponents o f E q u a t i o n s 5 . 2 3 and 5.24 have been p l o t t e d a g a i n s t c0. FIGURE 1 9 . EXPONENTS OF EQUATIONS 5 - 2 3 AND 5-24 v s . c 2 U s i n g F i g u r e 19 t o c o n v e r t the obser v e d b y t o c 2 , and F i g u r e 18 t o c o n v e r t c 2 t o apparent roughness r a t i o s (assuming E q u a t i o n 5 . 2 i s v a l i d ) one can see t h a t the d a t a c o v e r the approximate range of roughness r a t i o s from 0.4 t o 8 , w h i c h i s co m p a t i b l e w i t h the appearance o f the c h a n n e l s . From E q u a t i o n 5 . 1 8 one can o b t a i n an e x p l i c i t e q u a t i o n f o r b^ b y = 1 - 0 . 2 5 W D 1 4 S " - 0 8 . . . 5 . 2 5 w h i c h , i f used i n c o n j u n c t i o n w i t h F i g u r e s 18 and 19 i n d i c a t e s t h a t a t a f i x e d s l o p e , l a r g e channels are r e l a t i v e l y smoother t h a n s m a l l channels and, f o r a g i v e n c h a n n e l s i z e , s t e e p channels are rougher t h a n f l a t c h a n n e l s . T h i s a l s o agrees w i t h f i e l d o b s e r v a t i o n s . 108 I n c o n c l u s i o n , the d a t a o f t h i s s t u d y appear t o be c o m p a t i b l e w i t h the commonly a c c e p t e d l o g a r i t h m i c law f o r rough c h a n n e l s , E q u a t i o n 5.21, but w i t h o u t i n f o r m a t i o n on roughness s i z e and on shape of the f l o w s e c t i o n s , I t i s not p o s s i b l e t o d e c i d e whether t h i s e q u a t i o n g i v e s a m e a n i n g f u l r e p r e s e n t a t i o n o f f l o w i n e x t r e m e l y rough c h a n n e l s . 109 6 . _ UNSTEADY FLOW IN STEEP CHANNELS 6 . 1 K i n e m a t i c Waves and the Surge T e s t R e s u l t s 6 . 1 . 1 Some F e a t u r e s o f K i n e m a t i c Waves L i g h t h i l l and Whitham ( 1 9 5 5 ) i n t r o d u c e d the term " k i n e m a t i c wave" f o r a c l a s s o f waves which a r i s e I n one- d i m e n s i o n a l f l o w systems i f t h e r e i s a unique f u n c t i o n a l r e l a t i o n between: ( i ) the f l o w Q, ( i i ) the p o s i t i o n x, and ( i i i ) the q u a n t i t y p e r u n i t d i s t a n c e (A i n the case of a s t r e a m ) . The wave motions are t h e n governed by the e q u a t i o n o f con- t i n u i t y a l o n e . I t has l o n g been r e c o g n i z e d t h a t the movement of a f l o o d down a l o n g r i v e r can be approximated by t h i s type of wave (Seddon, 1 9 0 0 ; Masse', 1 9 3 5 ) . The e q u a t i o n of c o n t i n u i t y f o r unsteady f l o w i n a l o n g c h a n n e l i s AR + AA = o 6 i which can a l s o be w r i t t e n as o t d A d x b Q/d A has dimensions o f a v e l o c i t y and can only' depend on Q and on the p o s i t i o n x i f the assumptions f o r ' k i n e m a t i c waves are s a t i s f i e d . An o b s e r v e r moving a l o n g x a t speed 6Q/6 A 110 w i l l then observe no change In area or discharge (DA/Dt = 0 ) , which shows t h a t Equation 6 . 2 d e f i n e s a wave motion with d Q = c(x,Q) ... 6 . 3 6 A being the c e l e r i t y of these kinematic waves. In Chapters 8 and 9 of h i s book on open channel flow, Henderson ( 1 9 6 6 ) examines the c o n d i t i o n s under which k i n e - matic waves can approximate the movement of f l o o d waves. The equation of motion f o r a p r i s m a t i c channel can be w r i t t e n as (Henderson's Eq u a t i o n 8 . 5 ) '• v - dv <Sv , , Q _ -Q dd m .-.--m 1 m c u •• 1 2 3. ' h 5 2 2 4 / 3 i n which S„ i s the f r i c t i o n slope (e.g. n v /d« i f the f ^ & m m * Manning Equation i s a p p l i c a b l e ) . Terms 1 and 2 d e f i n e steady, uniform flow,' terms 1 to 4 d e f i n e steady, non-uniform flow, and the complete equation a p p l i e s to un-steady, non-uniform c o n d i t i o n s . I f S i s much l a r g e r than the 3 other terms on the r i g h t s i d e of E q u a t i o n 6 . 4 , the wave motion i s approximately k i n e m a t i c . Henderson shows t h a t t h i s c o n d i t i o n i s s a t i s f i e d i n a r e l a t i v e l y s t e e p , a l l u v i a l r i v e r , even d u r i n g a very r a p i d f l o o d r i s e . In tumbling flow channels the 5 t h term i s always one or more orders of magnitude s m a l l e r than the averaged S but may be comparable with the l o c a l S i n a few p l a c e s ( b i g p o o l s ) . Terms 3 and 4 however, may be of order S or more and, l i k e S, I l l they are h i g h l y and u n p r e d i c t a b l y v a r i a b l e with x. Equ a t i o n 6 . 4 does n o t , t h e r e f o r e , p e r m i t any d e f i n i t e c o n c l u s i o n s r e g a r d - i n g the a p p l i c a b i l i t y of kinematic wave theory to tumbling flow. Only experimental evidence can do t h i s . The equation f o r the kinematic wave c e l e r i t y at a f i x e d l o c a t i o n , c = dQ/dA, can a l s o be s t a t e d as d(vA) . .dv C = ~dA = v dA which shows t h a t c Increases with d i s c h a r g e i n n a t u r a l r i v e r channels. In a t r u l y kinematic channel, c(x) , i s t h e r e f o r e a unique and i n c r e a s i n g f u n c t i o n of A or Q. As a consequence, a kinematic wave cannot d i s p e r s e but the h i g h e r p a r t s of the wave w i l l tend to overrun the lower p a r t s , r e s u l t i n g i n a g r a d u a l l y steepening wave f r o n t of p o s i t i v e waves. In S e c t i o n 5 . 2 i t was shown that channel reaches i n the tumbling flow regime obey equations of the form b A A = a^Q ( 5 . 1 0 ) . Kinematic waves i n such channels should t h e r e f o r e have the c e l e r i t y dq i ' , . y b A r - c = i i S . = ± A-- • " 6 5 c dA 1/b A . R ...DO a A A b A or, i n terms of discharge n 1-bA ° ~  aA°A . . . 6 . 6 S u b s t i t u t i n g b = 1 - b and a = 1/a. i n t o E quation 6 . 6 g i v e s v m b A . . . 6 . 7 112 which shows that the kinematic wave c e l e r i t y i s p r o p o r t i o n a l to v • m 6 . 1 . 2 I n d i c a t i o n s from the Surge Test R e s u l t s The three f e a t u r e s of kinematic waves which are s u i t - able f o r immediate comparison with the f i e l d data are: ( i ) the steepening of p o s i t i v e wave f r o n t s , ( i i ) the n o n - d i s p e r s i v e nature of kinematic waves, and ( i i i ) the wave c e l e r i t y c = v/b . A l l surge t e s t s show a c o n s i s t e n t downstream f l a t t e n i n g of p o s i t i v e and negative wave f r o n t s and a tendency towards i n c r e a s i n g l y smooth Q(t)--curves i n the downstream d i r e c t i o n . Both f a c t s are c l e a r evidence f o r d i s p e r s i v e e f f e c t s . A c c o r d i n g to Equation 6 . 7 , the kinematic surge c e l e r i t y p l o t s as a s t r a i g h t l i n e on l o g a r i t h m i c paper, p a r a l l e l to the Q - v l i n e . F i g u r e s 16 and 17 show these l i n e s f o r two m to t e s t reaches, together w i t h the observed surge c e l e r i t i e s based on the mid-points (over Q) of the observed r i s e or f a l l . The agreement between the t h e o r e t i c a l l i n e and the observed surge c e l e r i t i e s i s c o n s i s t e n t l y s i m i l a r to the s i t u a t i o n shown In F i g u r e s 16 and 1 7 . At in t e r m e d i a t e to high flows the agreement between observed and kinematic c e l e r i t i e s i s always c l o s e but at low flows, the observed c e l e r i t i e s tend to be s i g n i f i c a n t l y h i g h e r than kinematic. Symmetrical t e s t s , c o n s i s t i n g of an up-surge f o l l o w e d 113 by a s i m i l a r down-surge (or v i c e versa) are p a r t i c u l a r l y i n s t r u c t i v e on the mechanism of wave pr o p a g a t i o n ( F i g u r e 1 5 ) . The time l a g between the mid-points of the up and down- surges remains very c l o s e l y constant and i d e n t i c a l t o the o r i g i n a l l a g at the lake o u t l e t , with the c e l e r i t y of these mid-points being c l o s e to kinematic at a l l but the lowest flows and with the sharp changes i n discharge, becoming g r a d u a l l y smoother as noted above. From these comparisons one can conclude that the mechanism-' of wave pro p a g a t i o n at in t e r m e d i a t e to high flows through channels i n the tumbling flow regime i s e s s e n t i a l l y kinematic with a c e r t a i n d i s p e r s i v e e f f e c t added. The d i g r e s s i o n from kinematic c o n d i t i o n s at low flows can be e x p l a i n e d as f o l l o w s : at low stage l a r g e p a r t s of any tumbling flow channel are occupied b y ; . r e l a t i v e l y deep and slow-moving p o o l s , which are not kinematic a c c o r d i n g to the assumptions s t a t e d In S e c t i o n 6 . 1 . 1 . Changes i n discharge propagate through pools at the dynamic wave c e l e r i t y \/g d , which w i l l g e n e r a l l y be much l a r g e r than the c o r r e s - ponding kinematic wave c e l e r i t y . For example, at a flow of 3 — 1 0 . 0 1 m s Brockton Creek (Figure 1 6 ) contained a few pools with depth of more than 0 . 3 m and a l a r g e number of pools with depths between 0 . 1 and 0 . 3 m. The kinematic wave c e l e r i t y at t h i s flow i s 0 . 1 8 ms and the dynamic c e l e r i t i e s f o r 0 . 1 m - 1 - 1 and 0 . 3 m depth are 1 ms and 1 . 7 ms . The e f f e c t s of pools w i l l be examined i n g r e a t e r d e t a i l i n S e c t i o n 6 . 3 . 114 6.2 Kinematic Waves with Storage D i s p e r s i o n 6 . 2 . 1 D i s p e r s i o n through Secondary Dynamic E f f e c t s In the case of long r i v e r s with r e l a t i v e l y p r i s m a t i c channels and s u b - c r i t i c a l flow throughout, i t has long been re c o g n i z e d that the propagat i o n of f l o o d waves i s mainly kinematic with some d i s p e r s i v e e f f e c t s added. Hayami (1951) i n t r o d u c e d the equation d A ^ j n b_k _ <^2A r „ 4 t 2 dx c £ x 2 ... o.r f o r t h i s type of kinematic wave. This corresponds to Equation 6 .2 , with 3v m / 2 being the kinematic wave c e l e r i t y a c c o r d i n g to the Che'zy f r i c t i o n formula and D being an c undetermined d i s p e r s i o n c o e f f i c i e n t . Hayami a r r i v e d at Equation 6.7 by adding the e f f e c t of the changed water s u r f a c e s l o p e , which occurs d u r i n g the passage of a f l o o d wave, to the b a s i c f l o o d wave equation ( 6 . 2 ) . Without d e t a i l e d argument he claims f u r t h e r t h at the d i s p e r s i o n c o e f f i c i e n t c o n s i s t s of a sum of two terms, one accounting f o r the slope e f f e c t and the other accounting f o r wave d i s p e r s i o n i n storage elements, such as pools or permeable stream banks. He gives an e x p l i c i t s o l u t i o n f o r Equation 6 . 7 , based on the l i n e a r i z i n g assump- t i o n s of constant v and constant D , and found good agreement m c' . between computed and observed p r o p a g a t i o n of an a r t i f i c i a l l y produced symmetrical f l o o d wave. The main d i f f i c u l t i e s with 115 Hayami's s o l u t i o n s are the n o r m a l l y u n p r e d i c t a b l e s i z e o f D and v . c m L i g h t h i l l and Witham d i s c u s s s e v e r a l d i f f e r e n t forms of the d i s p e r s i o n term i n the k i n e m a t i c wave e q u a t i o n . They c l a i m t h a t , s i n c e the d i s p e r s i v e term i s p r o b a b l y s m a l l , when compared w i t h the terms on the l e f t o f E q u a t i o n 6 . 7 b A ^ _b_k "ST c b x and the d i s p e r s i o n term can t h e r e f o r e be s t a t e d i n any one o f the t h r e e forms: b2k/bx2, 1 / c 2 b 2 A / J t 2 , or 1/c b A/b x b t . The d i s p e r s i o n c o e f f i c i e n t , however, may be more e a s i l y e s t a b l i s h e d f o r one form t h a n f o r the o t h e r s , depending on c i r c u m s t a n c e s . Two methods f o r d e t e r m i n i n g the d i s p e r s i o n c o e f f i c i e n t are g i v e n , one based on observed f l o o d p r o f i l e s at f i x e d times (which i s r a r e l y p o s s i b l e ) , and the o t h e r based on the well-known (but d i f f i c u l t t o measure) h y s t e r e s i s e f f e c t , which o c c u r s i n s t a g e - d i s c h a r g e r a t i n g c urves d u r i n g the passage of a f l o o d wave. I n t u m b l i n g f l o w channels the d i s p e r s i o n term may not be s m a l l because of the e x t e n s i v e p o o l s t o r a g e , so t h a t the above t r a n s f o r m a t i o n o f the term c o n t a i n i n g second d e r i v a t i o n i s not j u s t i f i e d . Henderson ( 1 9 6 6 ) a l s o d i s c u s s e s the d i s p e r s i o n of k i n e m a t i c waves and shows t h a t h i s form of the e q u a t i o n of m o t i o n ( 6 . 4 ) can be t r a n s f o r m e d i n t o the k i n e m a t i c wave e q u a t i o n o f Hayami, E q u a t i o n 6 . 7 , i f terms 4 and 5 are n e g l e c t e d . 116 In c o n c l u s i o n i t appears that there i s a c o n s i d e r a b l e body of knowledge on d i s p e r s i v e kinematic waves, but i t i s based on, and a p p l i c a b l e to long, r e l a t i v e l y p r i s m a t i c channels, i n which the d i s p e r s i o n i s mainly the r e s u l t of d i f f e r e n c e s between steady . and unsteady slopes and the r e s u l t i n g h y s t e r e s i s i n the stage - d i s c h a r g e r e l a t i o n s . Tumbling flow i s c h a r a c t e r i z e d by frequent t r a n s i t i o n s from s u b - c r i t i c a l to s u p e r c r i t i c a l depths, which means t h a t , at l e a s t at the c r i t i c a l s e c t i o n , the sta g e - d i s c h a r g e r e l a t i o n i s unique. The d i s p e r s i o n i s t h e r e f o r e a t t r i b u t a b l e s o l e l y to storage. The consequences of t h i s do not appear t o have been i n v e s t i g a t e d b e f o r e . 6 . 2 . 2 The D i f f e r e n t i a l E quation of Kinematic Waves with Storage D i s p e r s i o n A channel, In which the r e l a t i o n between Q and A ; i s only unique at r e g u l a r l y spaced d i s c r e t e l o c a t i o n s , has the f o l l o w i n g Q - A r e l a t i o n at i n t e r m e d i a t e p o i n t s x. , <x<x. (see F i g u r e 2 0 ) l - l I & Q = f (A) I + ,<?WD -|-| ... 6 . 8 1 x. I s i n c e , at r i s i n g stage, some of the discharge at x w i l l go i n t o storage between x and x.. 117 FIGURE 2 0 . DEFINITION SKETCH FOR EQUATION 6 . 8 The c o e f f i c i e n t o f b A/b t has di m e n s i o n L, and w i t h Wp b e i n g the o n l y l e n g t h r e a d i l y a v a i l a b l e i n the p r e s e n t problem ( S e c t i o n 5 . 3 ) , i t i s c o n v e n i e n t t o use i t i n E q u a t i o n 6 . 8 , t o g e t h e r w i t h a n o n - d i m e n s i o n a l c o e f f i c i e n t fi , whose v a l u e w i l l have t o be det e r m i n e d l a t e r on. P h y s i c a l l y , the f a c t o r ft WD i s a l e n g t h measure i n d i r e c t i o n x, r e l a t e d t o the average s i z e o f p o o l s . I n some a l l u v i a l r i v e r s , p o o l - r i f f l e sequences s c a l e a p p r o x i m a t e l y w i t h w i d t h ( L e o p o l d e t a l . , 1 9 6 3 ) . By u s i n g WD i n ( 6 . 8 ) one assumes t h a t a s i m i l a r r e l a - t i o n h o l d s i n t u m b l i n g f l o w c h a n n e l s . Assuming a l o n g c h a n n e l w i t h d e n s e l y spaced c o n t r o l s e c t i o n s x^ , one may s u b s t i t u t e E q u a t i o n 6 . 8 i n t o the e q u a t i o n o f c o n t i n u i t y ( 6 . 1 ) , at l e a s t as an a p p r o x i m a t i o n , g i v i n g AA + AR AA + Q b2A _ n i t 6 A bx * A t <*xdt " u 118 i n which stands f o r & A/ 6t. Noting that d Q / d A = c and, according to Equation 6 . 8 , b Q / d A^ = Ŵ , one obtains o d t d x " D o xdt This i s the b a s i c equation f o r kinematic waves with storage d i s p e r s i o n . L i g h t h i l l and Whitham ( 1 9 5 5 ) a r r i v e at the same equation by con s i d e r i n g the e f f e c t of h y s t e r e s i s i n a stage- discharge r a t i n g curve, a p p l i c a b l e to the whole reach. Equation 6 . 8 a l s o defines a h y s t e r e s i s e f f e c t i n the Q - A r e l a t i o n , but i n a n a t u r a l tumbling flow channel t h i s i s probably h i g h l y v a r i a b l e and could c e r t a i n l y not lead to a p r a c t i c a l method f o r . e s t i m a t i n g /? , the one remaining f r e e parameter. An a l t e r n a t i v e i s to obta i n an e x p l i c i t s o l u t i o n f o r Equation 6 . 9 , with the r e l a t i v e l y simple I n i t i a l con- d i t i o n s of the surge t e s t s , and then to o b t a i n / ^ by f i t t i n g the s o l u t i o n to the observed surges. This w i l l be done i n the f o l l o w i n g two s e c t i o n s . 6 . 2 . 3 A S o l u t i o n f o r S t e p - l i k e Input An e x p l i c i t s o l u t i o n of Equation 6 . 9 i s only obtainable w i t h the l i n e a r i z i n g assumptions of constant c and constant A , which i s j u s t i f i a b l e f o r the surge t e s t s , since A Q < < Q . With these assumptions- Equation 6 . 9 becomes a l i n e a r , homo- geneous, p a r t i a l d i f f e r e n t i a l equation of second order. The s u b s t i t u t i o n 119 /S w F = A e D c t /3 W D 6. 10 t r a n s f o r m s the e q u a t i o n i n t o the compact, f i r s t c a n o n i c a l form a 2F dxd.t cF ^ 2 w r 2 = 0 6.11 D which has the c h a r a c t e r i s t i c s x = c o n s t , and t = c o n s t . The i n i t i a l c o n d i t i o n s and boundary c o n d i t i o n s of a surge t e s t can be approximated as: A(x k 0,0) = A . 6.12 A(0,0) = A o A(0, t > 0) = (1 + <* ) A A f t e r the above t r a n s f o r m a t i o n they become x <* « 1.0 F(x^ 0 , 0 ) F(0,0) = A e o = A c t F ( 0,t > 0) = (1 + °<)Aoe D 6.13 6.12a 6.13a oc « 1-0 S i n c e the i n i t i a l c o n d i t i o n s d e f i n e F on two o f i t s c h a r a c t e r i s t i c s , the problem t o be s o l v e d i s a s o - c a l l e d Goursat problem. Under the above c o n d i t i o n s i t has a unique s o l u t i o n ( M i k h l i n , 1966), w h i c h can be found by Riemann's method. 12.0 The Riemann F u n c t i o n B i s B = I [-v/4V xt ] i n which I (u) i s the " M o d i f i e d B e s s e l F u n c t i o n of the F i r s t o Kind of Order Zero". The s o l u t i o n of ( 6 . 9 ) i s of the form x c F ( x , t ) = F(0,0)B(0,0) + j B d F ^ | > 0 ) dj + J ' d F ( 0 , r ) d r d r 6.14 i n which J and T are dummy v a r i a b l e s i n l e n g t h and time coordi n a t e s r e s p e c t i v e l y . The l a s t term of"(6.14) cannot be evaluated i n the above form s i n c e the d e r i v a t i v e dF(0 5T)/dT" i s undefined at Y'= 0. P a r t i a l i n t e g r a t i o n of t h i s term gi v e s x r=t. F ( x , t ) = F ( O , O ) B ( O , O ) + CB d F ^ ? 0 ) d/ + B ( o , r ) P ( o , r ) j Q' r=o t - / f| P(.o,r) d r . . . 6 . 1 5 Since d I Q ( u ) / d u = I # 1 (u) and (-u) i s the " M o d i f i e d B e s s e l F u n c t i o n of the F i r s t Kind of Order One", s u b s t i t u t i n g f o r B. and the s t a r t i n g c o n d i t i o n s of F i n 6 . 1 5 gives 1 2 1 0 D t c r 2W& J 1 MM V , 1cl(t-rt7- 1 / 2 6 . 1 6 With A ( l , t ) = F exp(-l//?W D - ct^Wp) t h i s i s the e x p l i c i t s o l u t i o n of Equation 6 . 9 f o r a reach of l e n g t h 1. Under normal circumstances i t can be s i m p l i f i e d c o n s i d e r a b l y . For l a r g e -arguments u the B e s s e l f u n c t i o n s l 0 ( ! u ) and I^( u ) tend a s y m p t o t i c a l l y towards the f u n c t i o n \ / l / 2 T u exp(u) According to Jahnke and Emde ( 1 9 4 5 ) - the agreement i s w i t h i n 5% at u = 9 . With the 1 -values and time lags of the present surge data, the arguments w i l l always be much l a r g e r than 1 0 , so that the B e s s e l f u n c t i o n s can be r e p l a c e d by t h e i r more e a s i l y computed asymptote. The middle term on the l e f t of ( 6 . 1 6 ) i s n e g l i g i b l e , when compared with the two i n t e g r a l terms. Time l a g i s g e n e r a l l y g i v e n i n minutes, while a l l other data are i n meters and seconds. With these assumptions, Equation 6 . 1 6 becomes A 122 1 ^ ^ i r h / — — - ^ = r^ E X P ( P V ^ sf? + * f 0 120(1 +<*) A CI ^ f— : • ) dx + 5 p — / -i / = exp (p v / l V t - /9^IAL/ J / „ _ /TT;—ZTT" 6 0 c t ^ 0 J V 2 T P V ^ T T 6 0 c T 6 0 c t 1_ x , ^ ,„ + /?W /?W j a r •' • D / 2 2*" i n which p = ̂ /2k0o./fi . E q u a t i o n 6.17 poses.no computa- t i o n a l problems. The F o r t r a n IVG program "PD" computes A ( l , t ) f o r g i v e n v a l u e s o f Q Q,o<, fi, Ŵ , 1, a^, and b^ u s i n g E q u a t i o n 5.10a t o c o n v e r t d i s c h a r g e t o a r e a and v i c e v e r s a and E q u a t i o n 6.6 t o compute c. On o u t p u t , the program l i s t s s e v e r a l p a r t s o f (6.17). t o p e r m i t an assessment o f the c o n t r i b u t i o n o f the two terms on the r i g h t . The program, w i t h o p e r a t i n g i n s t r u c - t i o n s / and sample o u t p u t , i s l i s t e d i n the Appendix. D u r i n g the p e r i o d o f r a p i d l y changing A ( l , t ) , b o t h terms o f (6.17) are of s i m i l a r magnitude. The f i r s t term dominates b e f o r e that,when A ( 1 , t ) ~ A ,and the second term dominates the p e r i o d when A ( l , t ) ~ (1+ <xr ) A . Away from t = 1/c, the dominant terms become time independent. C l o s e t o t = 0, the s o l u t i o n f a i l s as a r e s u l t o f s u b s t i t u t i n g an a s y m p t o t i c f u n c t i o n f o r the two B e s s e l f u n c t i o n s . 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 w i t h F i e l d Data To e v a l u a t e the p r o b a b l e range of ft , which i s the only- f r e e parameter i n E q u a t i o n 6.17, a few surge t e s t s were com- pa r e d w i t h computed Q ( l , t ) curves c o v e r i n g a wide range of . F i g u r e 21 shows a t y p i c a l comparison. O b v i o u s l y the computed c of t h a t t e s t i s somewhat too s m a l l as . p o i n t e d out i n S e c t i o n 6.1.2. The b e s t f i t t i n g v a l u e s of f a l l c o n s i s t e n t l y between 0.5 and 1, w i t h the g r e a t e r v a l u e s o c c u r r i n g a t the l a r g e r d i s c h a r g e s . F i t was d e t e r m i n e d by i n s p e c t i o n . A l e a s t squares f i t of E q u a t i o n 6.17 t o e v a l u a t e o p t i m a l v a l u e s o f /3 , or /? and c i s f e a s i b l e , u s i n g the programs "NLIN2" ( S e c t i o n 3.3.1) and "PD", but i t would i n v o l v e e x c e s s i v e com- p u t e r t i m e . As noted i n S e c t i o n 5.2.1 the a c t u a l w a t e r s u r f a c e w i d t h o f n a t u r a l c h a n n e l s , W , i s g e n e r a l l y r e l a t e d t o Wn by f u n c t i o n s of t h e form w i t h the f a c t o r z p r o b a b l y f a l l i n g i n t o the range 0.1''" t o 0.2 i n the case of t u m b l i n g f l o w channels ( M i l l e r , 1 9 5 8 ) . T h i s suggests t h a t W might be a good e s t i m a t e o f ^ W n or A - ... 6.19 F i g u r e s 22a and 22b show a few t y p i c a l r e s u l t s o b t a i n e d on the b a s i s o f E q u a t i o n s 6.17 and 6.19, w i t h z t a k e n as 0.2 and Q D t a k e n as the e s t i m a t e d mean annual peak f l o w ( T a b l e 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 1 0 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 i m m e d i a t e l y below l a k e s can be used f o r com- p a r i s o n , because the surges on lower reaches do not f i t the i n i t i a l c o n d i t i o n s , as s t a t e d i n ( 6 . 1 2 ) and ( 6 . 1 3 ) . The w r i t e r knows of no o t h e r r o u t i n g method which c o u l d g i v e com- p a r a b l e f i t w i t h o u t h a v i n g t o e v a l u a t e some f r e e parameters from other- unsteady f l o w d a t a b e f o r e h a n d . E q u a t i o n 6 . 1 7 i s not a p r a c t i c a l r o u t i n g e q u a t i o n f o r r o u t i n e h y d r o l o g i c a l work. I t i s a means of o b t a i n i n g the d i s p e r s i o n c o e f f i c i e n t o f the b a s i c wave e q u a t i o n ( 6 . 9 ) i f c i r c u m s t a n c e s p e r m i t the c r e a t i o n o f a s m a l l , s t e p - l i k e surge To o b t a i n an o p e r a t i o n a l f l o w f o r e c a s t i n g system, E q u a t i o n 6 . would have t o be c o n s i d e r e d n o n - l i n e a r and programmed f o r n u m e r i c a l s o l u t i o n , p o s s i b l y u s i n g the methods d i s c u s s e d by L i g h t h i l l and Whitham ( 1 9 5 5 ) or Henderson ( 1 9 6 6 ) . 6 . 3 A P r a c t i c a l Approach t o Unsteady, Tumbling Flow As an a l t e r n a t i v e t o the r o u t i n g method of the l a s t s e c t i o n , w hich c o n s i d e r s the t u m b l i n g f l o w c h a n n e l as a l a r g e sequence of s t o r a g e elements w i t h unique Q-A r e l a t i o n s at t h e i r o u t l e t s - , i t appears worth i n v e s t i g a t i n g whether a sequence of a few r e s e r v o i r s and channels c o u l d r e p r e s e n t t u m b l i n g f l o w . The b a s i c steady f l o w E q u a t i o n 5 . 1 0 can be s a t i s f i e d p h y s i c a l l y e i t h e r by an i n c l i n e d rough c h a n n e l w i t h the a p p r o p r i a t e roughness elements or by a smooth, almost h o r i z o n t a l r e s e r v o i r - l i k e c h a n n e l w i t h a w e i r - l i k e o u t l e t . The f o l l o w i n g 3 s e c t i o n s w i l l e x p l o r e the consequences of 128 assuming that a channel reach i n the tumbling regime can be represented by a r e l a t i v e l y s m a l l number of a l t e r n a t i n g r e s e r v o i r s and channels, both meeting the steady flow equation b A A = a AQ ...5.10 6.3.1'' Unsteady Flow through a N o n - l i n e a r R e s e r v o i r Unsteady flow through a p r i s m a t i c r e s e r v o i r of len g t h A and area A has to s a t i s f y the c o n t i n u i t y ' r e l a t i o n Q u ( t ) - Q(t) = - | | ... 6.20 i n which Q u ( t ) i s the i n f l o w , and Q(t)the outflow. E v a l u a t i n g the d e r i v a t i v e dA/dt with the d i m e n s i o n a l l y homogeneous form of the steady flow equation ' 5•10b and r e p r e s e n t i n g a l l d i s - charges as f r a c t i o n s of Q D (Q = q Q^), leads to da = _ ! 5 ( l-b 2-b } dt H c b A \ q - Q } ... 6.21 which i s a separable but n o n - l i n e a r d i f f e r e n t i a l equation. The i n t r o d u c t i o n of i n t o (6.21) i s p u r e l y f o r ease i n c o n v e r t i n g the formulas to other systems of u n i t s ; i t does not l i m i t Q to values l e s s than To o b t a i n an e x p l i c i t s o l u t i o n f o r a constant q , one can w r i t e (6.21) as f o l l o w s v 1 D f ,. q- J dt = — + constant A A D b A q u - q 129 S u b s t i t u t i n g (q - q)/q = k g i v e s t V 1 r D A / dk , . = -q j —^— + constant J k i A V A U J k ( l + k ) - B A The i n t e g r a l on the r i g h t has e x p l i c i t s o l u t i o n s f o r r a t i o n a l values of b A (b^ = i / j ) , but the form of the s o l u t i o n can vary widely, depending on i and j (Edwards, 1 9 2 1 ) . T h i s approach i s t h e r e f o r e not s u i t a b l e f o r a p p l i c a t i o n s i n which b A may take a f a i r l y wide range of v a l u e s . In g e n e r a l i t i s most e f f i c i e n t to s o l v e E q u a t i o n 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 I f a tumbling flow channel i s r e p r e s e n t e d as a cascade of kinematic channels and n o n - l i n e a r r e s e r v o i r s , as shown i n F i g u r e 23,. Equation 6.21 becomes dq.(t) nQ 2-b - a t - = * i A ° b A <*i-i ( t-V ^ ~ ̂  A> ••• 6- 2 1 a The time l a g between p o o l s , T^, i s e v a l u a t e d by neg- l e c t i n g the time l a g between a p o o l and the f o l l o w i n g channel, by assuming t h a t the channels are kinematic flow systems,and by assuming f u r t h e r t h at the p o o l water-surface i s h o r i z o n t a l at a l l t i m e s . 1 With t h i s , T^ becomes "'"This i s i d e n t i c a l with the assumption of an i n f i n i t e dynamic wave c e l e r i t y i n the p o o l s . The wave c e l e r i t i e s quoted in•Section.6.1.1 would appear to j u s t i f y t h i s . 130 m = ( l - oO -i L n c . ( t ) l and w i t h a s u b s t i t u t i o n f o r c , a c c o r d i n g t o E q u a t i o n 6.6 ° 0 Channel i Pool i FIGURE 23. DEFINITION SKETCH FOR THE CASCADE OF CHANNELS AND RESERVOIRS = ( 1 -dQ 1 A p b A 'D . . 6.22 The two e q u a t i o n s (6.21a) and (6.22) d e f i n e a c h a n n e l r o u t i n g system s u i t a b l e f o r n u m e r i c a l e v a l u a t i o n . A F o r t r a n IV G program "SNLR'-' - w r i t t e n f o r t h i s purpose i s l i s t e d i n the Appendix. - - - I n '-"SNLR"; the parameter i s c a l l e d o< . 1 3 1 S u i t a b l e assumptions f o r the f r e e parameters n and d w i l l be d i s c u s s e d i n the f o l l o w i n g s e c t i o n . However, con- s i d e r a t i o n s o f computing economics and s t a b i l i t y o f the computations impose f a i r l y narrow l i m i t s on b o t h p a r a m e t e r s . S t a b i l i t y i s a s s u r e d as l o n g as the f o l l o w i n g two con- d i t i o n s are met.. R ± >> Qj_ A t . . . 6 . 2 3 T L ( J + 1) < T L ( J ) + A t ••• 6 ' 2 4 i n which A t I s the f i n i t e time s t e p i n the n u m e r i c a l i n t e - g r a t i o n o f ( 6 . 2 1 a ) , and the s u b s c r i p t j r e f e r s t o these time s t e p s . The e x a c t f o r m u l a t i o n of I n e q u a l i t y 6 . 2 3 depends on the i n t e g r a t i n g method, but b y - r e q u i r i n g e i t h e r l a r g e r e s e r v o i r s o r s m a l l time i n t e r v a l s , i t c e r t a i n l y narrows the range o f p o s s i b l e cf and n v a l u e s . V i o l a t i o n o f I n e q u a l i t y 6 . 2 4 would i n d i c a t e a tendency towards the f o r m a t i o n o f a b o r e . T h i s may be a r a t h e r remote p o s s i b i l i t y , s i n c e even the l a r g e s t surges o f . t h i s study do not come c l o s e t o v i o l a t i n g the i n e q u a l i t y . 6 . 3 . 3 E v a l u a t i o n o f the Free Parameters from F i e l d Data F i g u r e 23 shows t h a t the parameter n i s a s c a l e p a r a - meter, which s h o u l d e x p r e s s whether a r e a c h i s r e l a t i v e l y " l o n g " o r " s h o r t " . W ith the r a t i o 1/W^ b e i n g the most r e a s o n - a b l e and p r a c t i c a l measure of r e l a t i v e l e n g t h , n w i l l be 132 assumed a p r i o r i t o be a f u n c t i o n of 1/W^ a l o n e . I t i s i m p o r t a n t t h a t n be independent o f Q because i t would be d i f - f i c u l t t o change n i n the course o f a co m p u t a t i o n . The parameter i n d i c a t e s how much of a g i v e n c h a n n e l r e a c h i s a c t i n g l i k e a r e s e r v o i r . S i n c e p o o l s are prominent at low f l o w s and ten d t o d i s a p p e a r d u r i n g f l o o d s , i t seems r e a s o n a b l e t o expect d t o be an d e c r e a s i n g f u n c t i o n o f Q/Q^. Prom a p r a c t i c a l p o i n t o f view t h i s i s a f e a s i b l e a s s umption. The boundary between a c h a n n e l segment and the a d j o i n i n g r e s e r v o i r can be s h i f t e d d u r i n g a surge c o m p u t a t i o n as they have i d e n t i c a l A-Q r e l a t i o n s . The combined e f f e c t o f I n e q u a l i t y 6 . 2 3 and the above assumption on the r e l a t i o n between 6 and Q i s somewhat u n f o r t u n a t e , s i n c e (6.23) i n d i c a t e s a need f o r l a r g e r r e s e r - v o i r s w i t h i n c r e a s i n g f l o w w h i l e the proposed d e c r e a s e of d w i t h Q has the o p p o s i t e e f f e c t . The two c o n d i t i o n s can be met s i m u l t a n e o u s l y by d e c r e a s i n g the time s t e p o f the n u m e r i c a l i n t e g r a t i o n as -Q i n c r e a s e s . To g a i n a c l e a r e r p i c t u r e o f the e f f e c t s o f changes i n n and i n d on the computed downstream f l o w , the surge t e s t shown i n F i g u r e 15 was r o u t e d t h r o u g h the t h r e e Blaney Creek reaches w i t h v a r i o u s assumed c o m b i n a t i o n s o f n and d . The r e s u l t s are shown on F i g u r e s 24 a, b, and c. They i n d i c a t e t h a t good f i t can be a c h i e v e d w i t h a wide range of n- d c o m b i n a t i o n s . I f the r e a c h i s d i v i d e d i n t o many 1 2 0 0 13°° Gauge I (ft) 1.25- 1.2 15.3 Gauge 3 (ft) 15.25 15.2 • r observed N R = L / (40 WD) o o o o N R = 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) O G G O 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 m 3 s - 1 VARIABLE NUMBER OF RESERVOIRS AT (p = 0.7 Fig. 24c 136 r e s e r v o i r s (n l a r g e ) ' they have t o cover a l a r g e p a r t o f the l e n g t h (cf l a r g e ) and v i c e v e r s a . S i n c e computing time i n c r e a s e s more than, l i n e a r l y w i t h n due t o (6.23) and the i n c r e a s e i n computing time w i t h d e c r e a s i n g d i s r e l a t i v e l y s m a l l , i t i s o b v i o u s l y advantageous t o keep n as s m a l l as p o s s i b l e . The surge t e s t d a t a g i v e no c l e a r i n d i c a t i o n o f a lower l i m i t . The p r a c t i c a l s o l u t i o n appears t o be t o s e l e c t a "1/Wp" c r i t e r i o n f o r n t h a t r e s u l t s i n a s i n g l e r e s e r v o i r f o r the s h o r t e s t reaches o f i n t e r e s t , which w i l l depend on the s c a l e at whi c h one i s w o r k i n g . As can be seen from T a b l e 2 n = 1/40WD ... 6.25 i s a s u i t a b l e a ssumption f o r the p r e s e n t s e t o f d a t a . The second parameter,6 , was e s t i m a t e d by t r i a l and e r r o r , u s i n g a l l the surge t e s t s o f B l a n e y Creek. The r e s u l t i n g d - q r e l a t i o n i s n a t u r a l l y o n l y v a l i d i n combin- a t i o n w i t h '(6.25). As i n the case o f the k i n e m a t i c waves w i t h s t o r a g e d i s p e r s i o n , the f r e e parameter d c o u l d a g a i n be o b t a i n e d w i t h a n o n - l i n e a r l e a s t squares f i t ( S e c t i o n 6.2.4),. but the amount o f computing time r e q u i r e d would be even g r e a t e r h e r e . A s i m u l t a n e o u s f i t o f n and d would p r o b a b l y not l e a d t o s i g n i f i c a n t r e s u l t s , s i n c e the minimum, o f the squared r e s i d u a l s appears t o be an e l o n g a t e d v a l l e y , l y i n g more or l e s s i n a d i a g o n a l d i r e c t i o n a c r o s s the d -n p l a n e . 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 s e v e r a l t y p i c a l comparisons between computed and observed surges f o r Blaney Creek and f o r other streams, which were not used f o r d e r i v i n g the 6 -q r e l a t i o n of F i g u r e 2 5 . d i r e c t comparison of observed and computed H - T curves. Since s e v e r a l of the gauge r a t i n g curves are not too w e l l d e f i n e d , some of the d i f f e r e n c e s between observed and com- puted surges are the r e s u l t of t h i s , r a t h e r than of any d e f i - ciency i n the r o u t i n g method. The closeness of f i t i s com- parable to the r e s u l t obtained with E q u a t i o n 6 . 1 7 . At very low stage the surge c e l e r i t y i s again underestimated. With an i n c r e a s e i n 6 i t i s p o s s i b l e to achieve a c o r r e c t surge a r r i v a l t i m e 3 b u t t h i s leads to exc e s s i v e damping. As p o i n t e d out i n S e c t i o n 6 . 1 . 2 , t h i s e f f e c t i s probably the r e s u l t of n e g l e c t i n g dynamic waves. There i s , however, some s l i g h t evidence, p a r t i c u l a r l y i n the case of Brockton Creek ( F i g u r e 16) The program "SNLR" converts discharge to stage f o r 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 £ 0 0 10 Gauge 3 (ft) August 15, 1967 Q^O.OI m3 s"1, A Q-0.001 Sept. 14, 1968 0.05, A Q — 0 . 0 1 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 IO 0 0 II00 j 2oo I3 0 0 -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 K 3 2 - | 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- l 4oo 1.90 ' ' 1.854 1 1 1 1 1 1 1 1 1 1 1 1 r- I 5 0 0 , 6 0 0 1.80-1 n 1 • ' ' ' 1 50 | |00 10 20 30 40 50 | 2 0 0 Gauge 6 (ft) July 28, 1967 Q~0.33m3s-', A Q - O . O I Nov. 29, 1968 Q ~ 2.2 , A Q -0.3 COMPUTED AND OBSERVED SURGES, PHYLLIS CREEK Fig 26 c 141 that the true Q - A r e l a t i o n i s somewhat curved on double l o g paper, probably due to a r e s i d u a l area at zero flow,and t h i s would r e s u l t i n higher values f o r the kinematic c e l e r i t y c at s m a l l d i s c h a r g e s . The major d i f f e r e n c e between, the r o u t i n g method based, cn channels and r e s e r v o i r s and r o u t i n g based on E q u a t i o n 6.17 i s that the former makes no assumption about the shape of the surge input to the reach and proceeds with f i n i t e time steps,, which,In the case of a r u n o f f model,permits the a d d i t i o n of l o c a l i n f l o w . 142 7 CONCLUSIONS 7 . 1 The H y d r a u l i c s of Tumbling Flow The o b j e c t i v e s of t h i s i n v e s t i g a t i o n are s t a t e d i n the I n t r o d u c t i o n as: ( i ) f i n d i n g the laws that govern steady and unsteady tumbling flow, and ( i i ) e x p r e s s i n g these laws i n terms of parameters which can be r e l a t e d to r e a d i l y a v a i l a b l e b a s i n data. W i t h i n c e r t a i n l i m i t a t i o n s , to be d i s c u s s e d below, t h i s has been accomplished. The aspects of steady channel flow,which are s i g n i - f i c a n t i n a r u n o f f model,are completely d e s c r i b e d by the f u n c t i o n s A = f (Q) f o r a l l segments.of the channel network. The f i e l d data i n d i c a t e t hat these f u n c t i o n s can be a p p r o x i - b A mated by simple e x p o n e n t i a l s of the form A = a A Q . On the b a s i s of s i m i l a r i t y c o n s i d e r a t i o n s I t i s shown that the two parameters a A and b A depend on b a s i n parameters which can be considered " r e a d i l y a v a i l a b l e " . T h i s i s confirmed by the h i g h l y s i g n i f i c a n t c o r r e l a t i o n s appearing i n Day's ( 1 9 6 9 ) r e g r e s s i o n models. The p h y s i c s of the channel forming p r o c e s s , which i s i m p l i c i t i n Day's r e g r e s s i o n equations, remains unknown. The two proposed methods f o r extending the steady flow equations to unsteady flow r o u t i n g are shown to be capable of reproducing the downstream propag a t i o n of s t e p - l i k e surges. Since more complicated channel i n f l o w s can be approximated by a s e r i e s of s t e p s , both methods are capable of r o u t i n g a l l hydrograph shapes except i n those cases where the n o n - l i n e a r i t y 143 of the channel response leads to the formation of a bore. The data impose a number of l i m i t a t i o n s on the. c o n c l u s i o n s . None of the t e s t reaches i s a f i r s t order channel and only three reaches r e p r e s e n t s m a l l streams; Brockton 1-2 and 2-3, and P l a c i d 1-2. T h e i r steady flow behaviour d i f f e r s markedly because Brockton Creek l i e s at t r e e - l i n e and i s e s s e n t i a l l y d e b r i s f r e e while P l a c i d Creek i s s e v e r e l y choked by l o g g i n g s l a s h . The r e s u l t s of the study are t h e r e f o r e not s u i t a b l e f o r a p p l i c a t i o n to almost complete channel networks as they appear on l a r g e s c a l e maps such as the 1:2400 map used i n Fi g u r e 2. The channel networks appearing on 1:50,000 NTS maps rep r e s e n t the approximate lower l i m i t to which the r e s u l t s of t h i s study may be a p p l i e d . The f i r s t order and most second order channels have to be i n c l u d e d i n the land phase. The c o n c l u s i o n s are f u r t h e r l i m i t e d by the l a c k of unsteady flow t e s t s on the l a r g e r streams. The flow regime i n l a r g e streams i s r a r e l y "tumbling" over long reaches and i t i s q u e s t i o n a b l e whether the unsteady flow models, p a r t i - c u l a r l y the r u l e s f o r determining the parameters and & , apply to o r d i n a r y , rough t u r b u l e n t channel flow. There i s a twofold r e g i o n a l l i m i t a t i o n on the data: ( i ) The r e l a t i o n s between channel parameters and d r a i n - age area depend on c l i m a t i c f a c t o r s . T h i s can be overcome by e s t a b l i s h i n g r e l a t i o n s between DA and W-̂, and then p r e d i c t i n g the channel performance from and S. This r e q u i r e s some f i e l d work and may t h e r e f o r e not always be f e a s i b l e . 144 ( i i ) Climate and e l e v a t i o n are the main f a c t o r s determin- i n g v e g e t a t i o n and can t h e r e f o r e have a c o n s i d e r a b l e e f f e c t on the performance of the s m a l l e r streams. . I t f o l l o w s that l o g g i n g and c l e a r i n g operations are of s i m i l a r importance. 7 • 2 B a s i n L i n e a r i t y Both r o u t i n g models developed i n S e c t i o n 6 show that the n o n - l i n e a r response of a channel segment i s l a r g e l y a consequence of the dependency between c and Q, which, f o r the kinematic approximation,is p o s i t i v e e x p o n e n t i a l s i n c e b c = dQ/dA «x v and v •= a Q . The surge t e s t s do however m m v & i n d i c a t e a c o n s i s t e n t tendency towards c values l a r g e r than dQ/dA (Figures 26 a, b and c) at low flows, which may be i n t e r p r e t e d as a trend towards l i n e a r response of the channel system at low flows. That the high flows show no tendency towards l i n e a r i t y i s not s u r p r i s i n g s i n c e the only reason which i s g e n e r a l l y advanced f o r such a t r e n d , the r a p i d i n c r e a s e i n flow area as the stream channels overflow onto the f l o o d p l a i n , i s r a r e l y a p p l i c a b l e i n steep mountainous b a s i n s . One may even argue t h a t , s i n c e the bed m a t e r i a l of degrading tumbling flow channels moves o.nly under extreme f l o o d c o n d i t i o n s and s i n c e a mobile bed would probably o f f e r lower r e s i s t a n c e to the f l o w i n g water ( K e l l e r h a l s , 1 9 6 7 ) , there c o u l d be s t r o n g e r n o n - l i n e a r trends d u r i n g extreme r u n o f f events. 145 7 . 3 Towards an Operative Channel Runoff Model Ne i t h e r of the two r o u t i n g methods of S e c t i o n 6 r e p r e s e n t s an o p e r a t i v e channel r u n o f f model, but both can be considered as b u i l d i n g b l o c k s out of which a channel r u n o f f model can now be assembled. Since the w r i t e r plans to pursue t h i s a f t e r completion of the present study, a b r i e f note on what remains to be done may be i n order. ( i ) I f the r o u t i n g method based on Equation 6 . 9 i s to be used, a c o n s i d e r a b l e programming problem remains to be solved. One can e i t h e r adopt the present s o l u t i o n f o r step - l i k e input ( 6 . 1 7 ) to u n s p e c i f i e d input shapes or one might choose a p u r e l y numerical s o l u t i o n based on s u c c e s s i v e a p p r o x i - mations to Q (x,t) i n a f i n i t e g r i d on the x - t planes of a l l channel segments. ( i i ) The r o u t i n g method based on n o n - l i n e a r r e s e r v o i r s i s c l o s e r to being o p e r a t i o n a l . The grad u a l change i n d with Q remains to be i n c o r p o r a t e d i n the program "SNLR". ( i i i ) A mathematical f o r m u l a t i o n f o r the drainage network w i l l have to be devised to permit proper r e p r e s e n t a t i o n of the channel parameters i n a computer. The model r e c e n t l y pro- posed by Surkan ( 1 9 6 9 ) appears to be adaptable to t h i s purpose. ( i v ) A s u i t a b l e r e p r e s e n t a t i o n of the land-phase input to the channel system w i l l have to be found. Hydrographs of a few small source areas may be a c c e p t a b l e , p o s s i b l y i n com- b i n a t i o n with m e t e o r o l o g i c a l r e c o r d s . 146 (v) With i n c r e a s i n g b a s i n s i z e , the channel phase becomes dominant over the land phase. I t i s important, t h e r e f o r e , to extend the r o u t i n g methods to l a r g e r streams than those con- s i d e r e d i n the present study. I t w i l l probably become necessary to o b t a i n the b a s i c A(Q)-equation from e x t e n s i v e r i v e r surveys r a t h e r than with t r a c e r methods. D i s p e r s i o n c o e f f i c i e n t s and wave c e l e r i t i e s w i l l have to be e s t a b l i s h e d through c o n t r o l l e d r e l e a s e s from s e v e r a l major dams. Only a f t e r completion of a l l t h i s w i l l i t be p o s s i b l e to pass a f i n a l judgement on the u s e f u l n e s s of the two-phase approach to r u n o f f . However, the r e s u l t s presented here leave l i t t l e room f o r doubt that i t w i l l be p o s i t i v e . 147 8 . BIBLIOGRAPHY Aastadj J . and Sdgnen, R. 1 9 5 4 . "Discharge Measurements by- Means of a S a l t Solution., 'the R e l a t i v e S a l t D i l u t i o n Method'," I n t e r n a t i o n a l A s s o c i a t i o n of S c i e n t i f i c Hydrology. P u b l i c a t i o n No. 3 8 . Assemble I n t e r n a t i o n a l e d'Hydrologle de Rome. Tome I I I . Abdelsalam, W. W. , 1 9 6 5 . Flume Study of the E f f e c t of Con- c e n t r a t i o n and S i z e of Roughness Elements on Flow i n High Gradient N a t u r a l Channels. Ph.D. T h e s i s , Utah State U n i v e r s i t y . Ackers, P eter, 1 9 5 8 . "Resistance of F l u i d s Flowing i n Channels and Pi p e s , " Department of S c i e n t i f i c and In- d u s t r i a l Research, Hydr. Research S t a t i o n , Hydr. Research Paper No. 1 , London. A l K h a f a j i , Abbas Nasser, 1 9 6 1 . The Dynamics of Twc-Dlmensional Flow i n Steep N a t u r a l Streams. Ph.D. T h e s i s , Utah State U n i v e r s i t y . Amorocho, J . , and Hart, W. E. 1 9 6 4 . "A C r i t i q u e of Current Methods i n H y d r o l o g i c Systems I n v e s t i g a t i o n s , " American Ge o p h y s i c a l Union, T r a n s a c t i o n s ? V o l . 4 5 , No. 2 , pp. 3 0 7 - 3 2 1 . Argyrcpoulos, P r a x i t e l i s A., 1 9 6 5 - "High V e l o c i t y Flow i n I r r e g u l a r N a t u r a l Streams," I n t e r . Assoc. f o r H y d r a u l i c . Research, 1 1 t h Meeting, Leningrad, Paper 1 - 2 6 . B a r r , David I. H. - 1 9 6 8 . " D i s c r i m i n a t i n g F o r m ulation of n-term Non-Dimensional F u n c t i o n a l Equations from (n+ 1 ) - term Dimensionally Homogeneous Equations with P a r t i c u l a r Reference t o Incompressible Viscous Flow," I n s t i t u t i o n of C i v i l E n gineers,Proc., V o l . 3 9 , pp. 3 0 5 - 3 1 2 . B a r r , David 1. H., and Herbertson, John G. 1 9 6 8 . " S i m i l i t u d e Theory A p p l i e d to C o r r e l a t i o n of Flume Sediment Transport Data," Water Resources Research, V o l . 4 , No.2 ,p.3 0 7 . Church,Michael. 1 9 6 7. "Observations of Turbulent D i f f u s i o n i n a N a t u r a l Channel," Canadian J o u r n a l of E a r t h S c i e n c e s , V o l . 14. Church,: M. . and K e l l e r h a l s , R. 1 9 6 9 . "Stream Gauging i n I s o l a t e d Areas Using P o r t a b l e Equipment," T e c h n i c a l B u l l e t i n No. , Inla n d Waters Branch, Canada Dept. of Energy, Mines, and Resources, ( i n p r e s s ) . 148 Day, T., 1 9 6 9 - The Channel Geometry of Mountain Streams. -•.M. A. T h e s i s , University,'Of B r i t i s h Columbia ( i n p r e s s ) . D i a c h i s h i n , A. N., 1 9 6 3 . "Dye D i s p e r s i o n S t u d i e s , " J o u r n a l of the S a n i t a r y E n g i n e e r i n g D i v i s i o n , American S o c i e t y of C i v i l E n gineers,Proceedings. V o l . 8 9 , No. SA1, pp. 2 9 - 4 9 . .Diskin, M. H., 196-7. "A. D i s p e r s i o n Analog Model f o r Watershed Systems," Proceedings of the I n t e r n a t i o n a l Hydrology - ' Symposium. F o r t C o l l i n s . V o l . 1 . Edwards, Joseph, 1 9 2 1 . A T r e a t i s e on the I n t e g r a l C a l c u l u s . Macmillan and Co., London. V o l . 1 . E l d e r , J . W. , 195-9. "The D i s p e r s i o n of Marked F l u i d i n Turbulent Shear Flow," J o u r n a l of F l u i d Mechanics, V o l . 5 . F i s c h e r , H. B., 1 9 6 6 . " L o n g i t u d i n a l D i s p e r s i o n i n Laboratory and N a t u r a l Streams," Report No..KH-12-12, W. M. Kech H y d r o l o g l c Laboratory, C a l i f o r n i a I n s t i t u t e of Technology, Pasadena, C a l . Hartung, F r i t z , and S c h e u e r l e i n , Helmut, 1 9 6 7 . "Macrotur- bu l e n t Flow i n Steep Open.v'Channels with High N a t u r a l Roughness," Proceedings of the I n t e r n a t i o n a l Hydrology Symposium, F o r t C o l l i n s . V o l . 1. Hayami, S h o i t i r o , 1 9 5 1 . "On the Propagation of F l o o d Waves," D i s a s t e r P r e v e n t i o n Research I n s t i t u t e , Kyoto U n i v e r s i t y , B u l l e t d n s , No . 1 . Hays, J . R., 1 9 6 6 . Mass Transport Mechanism i n Open Channel Flow. Ph.D. T h e s i s , V a n d e r b i l t U n i v e r s i t y . Henderson, F. M., 1 9 6 6 . Open Channel Flow. Macmillan Co., New York H e r b i c h , John B.., and S h u l l t s , Sam, Nov. 1 9 6 4 . "Large-Scale Roughness i n Open;Channel Flow," J o u r n a l of the H y d r a u l i c D i v i s i o n , ASCE, V o l . 9 0 , N 0 . H Y 6 , pp. 203-236~7 ; ~~' H e r r i o , John G..., 1 9 6 3 . . Methods of Mathematical A n a l y s i s and Computation.. John Wiley & Sons, Inc. New York. Jahnke, Eugene, and Emde, F r i t z , 19 4 5 . Tables of F u n c t i o n s . Dover P u b l i c a t i o n s , New York. Johnson, M a r t i n L., 1 9 6 4 . "Channel Roughness i n Steep Moun- t a i n Streams," American Geophysical Union, Ann. Meeting. Judd, H a r l E., 1 9 6 3 - A Study of Bed- C h a r a c t e r i s t i c s i n R e l a t i o n to Flow i n Rough, H i g h - g r a d i e n t , N a t u r a l Channels. Ph.D. T h e s i s , Utah State U n i v e r s i t y . 149 K e l l e r h a l s , R o l f , 1 9 6 7 . " S t a b l e Channels With Gravel-Paved Beds," J o u r n a l of Waterways and Harbors D i v i s i o n , ASCE, V o l . 9 3 , No. WW1, Proc. Paper 5 0 9 1 . pp. 6 3 - 8 4 . Larson, C. L., 1 9 6 5 . "A Two-Phase Approach to the P r e d i c t i o n of Peak Rates and Frequencies of Runoff f o r Small Ungauged Watersheds," T e c h n i c a l Report No. 5 3 , Dept. of C i v i l E n g i n e e r s , S t a n f o r d . Leopold, L. B., Wolman, M. Gordon, and M i l l e r , John P., 1 9 6 3 . ' F l u v i a l Processes i n Geomorphology. W. H. Freeman and Co., San F r a n c i s c o and London. Leopold, L. B., Bag^old, R. A., and Wolman, M. Gordon, and Bush,, L. M. , i 9 6 0 . "Flow R e s i s t a n c e i n Sinuous or I r r e g u l a r ; Channels," U n i t e d States G e o l o g i c a l Survey, P r o f e s s i o n a l Paper, -No . 2 82-D . L i g h t h i l l , M. J . and Whitham, G. B., 1 9 5 5 - On Kinematic Waves: I. F l o o d Movement i n Long R i v e r s , Proc. Royal Soc. of London (A) , V o l . 2 2 9 , pp.. ' 2 8 1 - 3 1 . 6 . Marquardt, D. W., 1 9 6 4 . "Least Squares E s t i m a t i o n of Non- L i n e a r Parameters," A Computer Program i n F o r t r a n IV Language,•IBM Share L i b r a r y , D i s t r i b u t i o n , N o . 3 0 9 4 . Masse', P i e r r e , 1 9 3 5 . • Hydrodynamlque f l u v l a l e , regimes v a r i a b l e s . ' • Hermann et C i e , P a r i s . M i k h l i n , S. G., 1 9 6 7 . ' L i n e a r Equations of Mathematical P h y s i c s . H o l t , Rhinehart, and Winston, Inc. M i l l e r , J . P., 1 9 5 8 . "High Mountain .Streams ^ E f f e c t s of Geo- logy on Channel C h a r a c t e r i s t i c s and Bed M a t e r i a l , " New Mexico, State Bureau of Mines and M i n e r a l Resources, Memoir No. 4 . Mirajgaoker, A. G., and C h a r l u , K.L.N., Sept. 1 9 6 3 . " N a t u r a l Roughness E f f e c t s i n R i g i d Open Channels," J o u r n a l of the H y d r a u l i c s D i v i s i o n , ASCE, V o l . 8 9 , No. H Y 5 , Proc. Paper 3630~ pp. 2 9 - 4 4 . Morisawa, M. 1 9 5 7 - "Accuracy of Determination of Stream Lengths from Topographic Maps," American Ge o p h y s i c a l Union T r a n s a c t i o n s , V o l . 3 8 , No. 1 . 0 s t r e m , G., 1 9 6 4 . "A Method of Measuring Water Discharge i n Turbulent Streams," Geographical B u l l e t i n No. 2 1 , Geographical Branch, Dept. of Mines and T e c h n i c a l Surveys, Ottawa. 150 Overton, D. E., 1 9 6 7 . " A n a l y t i c a l S i m u l a t i o n of Watershed Hydrographs from R a i n f a l l , " Proceedings of the I n t e r - n a t i o n a l Hydrology Symposium, F o r t C o l l i n s , V o l . 1. Peterson, D. F. and Mohanty, P. K., i 9 6 0 . "Flume Studies i n Steep, Rough Channels," J o u r n a l of the H y d r a u l i c s D i v i s i o n , Proceedings of the American S o c i e t y of C i v i l E n g ineers, V o l . 8 6 , No. HY9, pp. 5 5 - 7 6 . P i l g r i m , D. H., 1 9 6 6 . " R a d i o a c t i v e T r a c i n g of Storm Runoff on a Small Catchment," J o u r n a l of Hydrology. V o l . 4 , pp. 2 8 9 - 3 2 6 . Replogle, J . A. et a l . , 1 9 6 6 . "Flow Measurements with F l u o r e s c e n t T r a c e r s , " J o u r n a l of the H y d r a u l i c s D i v i s i o n , American S o c i e t y of C i v i l E n gineers, V o l . 9 2 , No. HY5, Proc. Paper 4 8 9 5 , pp. 1 - 1 5 . Scheidegger, A. E., 1 9 6 6 . " E f f e c t of Map Scale on Stream Orders," B u l l e t i n of the I n t e r n a t i o n a l A s s o c i a t i o n of S c i e n t i f i c Hydrology, V o l . 1 1 , No. 3 . S c h e u e r l e i n , Helmut, 1 9 6 8 . "Der Rauhgerinneabfluss," B e r i c h t No. 14 der V e r s u c h s a n s t a l t f u r Wasserbau der Technischen Hochschule Munchen, Oskar V. M i l l e r I n s t i t u t . Seddon, J . A., 1 9 0 0 . " R i v e r H y d r a u l i c s , " T r a n s a c t i o n s , American S o c i e t y of C i v i l E n gineers, V o l . 4 3 , pp. 1 7 9 - Stewart, John H., and LaMarche, Valmore, C., 1 9 6 7 . " E r o s i o n and D e p o s i t i o n Produced by the F l o o d of December 1 9 6 4 on Coffee Creek, T r i n i t y County, C a l i f o r n i a , " United States G e o l o g i c a l Survey, P r o f e s s i o n a l Paper, No. 422-K. Sugaware, M., 1 9 6 7 . "Runoff A n a l y s i s and Water Balance A n a l y s i s By a S e r i e s Storage Type Model," Proceedings of the I n t e r n a t i o n a l Hydrology Symposium, F o r t C o l l i n s , V o l . 1 . Surkan, A. J . 1 9 6 9 . " S y n t h e t i c Hydrographs: E f f e c t s of Network Geometry." Water Resources Research, V o l . 5 , No. 1 , pp. 1 1 2 - 1 2 8 . . T a y l o r , G. I . , 1 9 5 4 . "The D i s p e r s i o n of Matter i n Turbulent Flow Through a Pipe," Proceedings of the Royal S o c i e t y of London, V o l . 2 2 3 - Thackston, E. L. et_ a l . , 1 9 6 7 . "Least Squares E s t i m a t i o n of Mixing C o e f f i c i e n t s , " J o u r n a l of the S a n i t a r y E n g i n e e r i n g D i v i s i o n , American S o c i e t y of C i v i l E n gineers, Proceedings, V o l . 9 3 , No. SA 3 , PP- 4 7 - 5 8 . 1 5 1 Thackston, E. L. and Krenk e l , P. A., 1 9 6 7 . " L o n g i t u d i n a l Mixing i n N a t u r a l Streams," J o u r n a l of the S a n i t a r y E n g i n e e r i n g D i v i s i o n , American S o c i e t y of C i v i l E n gineers, V o l . 9 3 , No. SA 5 , pp. 6 7 - 9 0 . Wilson, J . P., 1 9 6 8 . " F l u o r o m e t r i c Procedures f o r Dye T r a c i n g , i n Techniques of Water Resources I n v e s t i g a t i o n s of the U. S. G e o l o g i c a l Survey, Chapter A 1 2 , Book 3 . Y a l i n , M. S., Jan. 1 9 6 6 . "A T h e o r e t i c a l Study of S t a b l e A l l u v i a l Systems," Golden J u b i l e e Symposium, C e n t r a l Water and Power Research S t a t i o n , Poona. PHOTOGRAPHS 152 PHOTOGRAPH 2. PLACID CREEK, ALONG REACH PI 3 - 4, LOOKING DOWNSTREAM. TYPICAL LOG JAM IN FOREGROUND. PHOTOGRAPH 4. BLANEY CREEK, AT B l 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 1 1 . RECORDING CONDUCTIVITY BRIDGE, WITH ELECTRONIC INTERVAL TIMER. PHOTOGRAPH 12. CONTROL STRUCTURE AT THE OUTLET OF BLANEY LAKE. THREE FLASHBOARDS IN PLACE. PHOTOGRAPH 1 3 . 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 l i s t i n g Sample p l o t o f r a t i n g curve Sample p r i n t o u t DQV Source l i s t i n g Sample p r i n t o u t TAILEX Source l i s t i n g Sample p r i n t o u t Sample p l o t s w i t h example f o r d e t e r m i n a t i o n o f A, B, and D QVEL . Source l i s t i n g , i n c l u d i n g s u b r o u t i n e s f o r n u m e r i c a l i n t e g r a t i o n Three sample o u t p u t s PLOTGA Source l i s t i n g Sample p l o t s , w i t h and w i t h o u t - e x t e n s i o n LOGRE Source l i s t i n g , i n c l u d i n g t h r e e s u b r o u t i n e s P r i n t o u t and p l o t s f o r a l l 13 t e s t r e a c h e s PD Source l i s t i n g , i n c l u d i n g two s u b r o u t i n e s Sample p r i n t o u t SNLR Source l i s t i n g , i n c l u d i n g one s u b r o u t i n e Sample p r i n t o u t 161 168 170 176 185 189 249 2 5 3 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, T S T , (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 , D l L 1 0 , ( 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 P R I N T O U T OF I N P U T DATA c P R I N T O U T . O F C O N V E R T E D D A T A , WITH C O N C E N T R A T I O N . I N ..PPM c OF THE P R I M A R Y S O L U T I O N . c O P T I O N A L , c PUNCHED DATA C A R O S C O N T A I N I N G C O N C E N T R A T I O N Y C , AND c T I M E X T , AS R E Q U I R E D FOR PROGRAM DQV, ( 2 F 9 . 3 ) , c PLOT OF R A T I N G C U R V E . C A L L P L O T S D I M E N S I O N XT ( 2 0 0 ) , YC ( 2 0 0 ) , . T I T L E ( 7 ) , C C ( 2 0 ) ,R FAD (20) 1 FORMAT ( 1 3 , F 7 . 2 , 7 A 4 , 2 X , 3 E 1 0 . 5 ) 2 ' FORMAT ( 2 F 9 . 3 ) ' ; > "-3 FORMAT ( 1 3 , 4 F 1 0 . 0 , 2 1 3 ) 5 FORMAT!. 8H0RATING,4X, 4HSTEP, 9X ,2HCC, 8X,. THREADING ,/ K 1 3 X , 12 , 7X, F6.0 , 4 X , F9.4 ) ) 7 FORMAT (12) 16 FORMAT (18H1C0NTR0L CARD 1 = / I X , 13, 2X, F7.2, 7A4, 1 2X, 3E12.6 / 18H0C0NTR0L CARD 2 = / . I X , 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 C C d ) = ( C C d ) /(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 = C C d ) * REAO(I) + SYX 12 C C SXX =READ(I)**2 + SXX BB = SYX / SXX . ' TION ADJUST Y C d ) TO A ZERO BACKGROUND AND CONVERT TO CONCEMTRA- DBACK =(BACKST - BACKND )/ (XT(K) -TST) OD 11 1 = 1 , K 11 IF ( X T U ) .LE. TST ) Y C d ) =0.0 * DBACK IF (XT(I) .GT. TST ) Y C d ) = Y C d ) - BACKST +((XTd)-TST) Y C d ) = BB* Y C d ) ,. 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 C A L L S C A L E ( R E A D , N , 7 . 0 , R X M I N , R D X , 1) C A L L S C A L E { CC , N , 9 . 0 , C Y M I N , C D Y , 1) CCMAX = ( C C M A X - C Y M I N ) / CDY C A L L A X I S ( 0 . 0 , 0 . 0 , 7 H R E A D I N G , - 7 , 7 . 0 , 0 . 0 , R X M I N , R D X ) CALL A X I S I 0 . 0 , 0 . 0 , 2 0 H C O N C E N T R A T I O N IN P P M , + 2 0 , 9 . 0 , 9 0 . 0 , 1 C Y M I N , CDY) CALL .SYMBOL..:.(. 1 . 0 , 9 . 5 , 0 . 2 1 , T I T L E . , 0 . 0 , . . 2 8 ) CALL SYMBOL ( 1 . 0 , 9 . 0 , 0 . 2 1 , 7 H T E M P . = , 0 . 0 , 7 ) CALL NUMBER ( 2 . 4 , 9 . 0 , 0 . 2 1 , TEMP , 0 . 0 , 1 ) CALL SYMBOL ( 3 . 5 , 9 . 0 , 0 . 2 1 , 4 H B 8 = , 0 . 0 , 3 ) . C A L L NUMBER ( 4 . 6 , 9 . 0 , 0 . 2 1 , B B , 0 . 0 , 4 ) DO 13 I = 1 t " N " • CALL...SYMBOL... i READ( I ) , CCC I ) , 0 . 14 . . , 4 , 0 . 0 v -1 ) CALL P LOT ( 0 . 0 « 0 . 0 , + 3 ) CALL PLOT { R EAD {N ) , CCMAX , +2 ) CALL PLOT t 9.6 , 0 . 0 , - 3 ) CONTINUE PUNCH'. DATA .CARDS .. ... _ IF .< NPUNCH <, N E . 0 ) GO T O 18 WRITE «7»1 J K » 1 S T , T I T L E , A , , B , D WRITE < '7,2) ( Y C U ) , X T ( I ) , 1= 1,K ) C O N T I N U E CALLS _Tn_ FITTING. AND P L O T T I N G . ROUT INES . ... _ THE CALL CARDS FOR N U M E R I C A L I N T E G R A T I O N OF T H E C - T C U R V E AND FOfl PITTING A GAMMA-DISTRIBUTION E X T E N S I O N GO IN H E R E , f.A\L.JJSJ. LEX IK; TST, X T , YC , T I T L E ) 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 A P I 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 ( 5 4 4 4 . 0 0 1 . 6 3 9 0 55 4 5 . 0 0 1 . 6 3 8 0 166 56 4 6 . 0 0 1 . 6 3 7 0 5 7 4 7 . 0 0 1 . 6 3 6 0 58 4 8 . 0 0 1 . 6 3 5 0 L 5 9 4 9 . 0 0 1 . 6 3 4 0 ' 6 0 5 1 . 0 0 1 . 6 3 2 0 61 5 3 . 0 0 1 . 6 3 0 0 62 . 5 5 . 0 0 1. 6 3 0 0 , 63 6 0 . 15 1 . 6 2 8 0 R A T I N G S T E P C C R E A D I N G 1 0 . 1 . 5 9 7 0 2 1 0 . 1 . 8 7 9 0 3 . 2 0 . 2 . 1 4 6 0 4 3 0 . 2 . 4 1 7 0 5 4 0 . . 2 . 7 0 0 0 6 5 0 . 2 . 9 7 8 0 C O N V E R T E D F I N A L D A T A DATA NO XT ;" R E A D I N G . . . . 1 1 4 . 5 0 0 . 0 2 1 5 . 5 0 0 . 0 4 0 2 3 1 6 . 0 0 0 . 2 5 4 5 4 1 6 . 5 0 0 . 7 7 3 7 . 5 1 7 . 0 . 0 . . . 1 . 8 5 8 9 6 1 7 . 5 0 3 . 9 4 5 6 7 1 8 . 17 7 . 4 6 8 1 8 1 8 . 5 0 1 0 . 6 8 8 1 9 1 9 . 0 0 1 5 . 7 3 5 7 10 1 9 . 5 0 1 9 . 2 5 9 4 11 2 0 . 0 0 2 3 . 6 1 0 3 12 2 0 . 5 8 2 7 . 5 2 5 2 13 2 1 . 0 0 2 9 . 6 5 6 0 14 2 1 . 5 0 3 0 . 9 1 5 4 15 2 2 . 0 0 3 1 . 1 2 9 8 16 2 2 . 5 0 3 0 . 5 6 0 3 17 _ 2 3 . 0 0 2 9 . 2 0 7 2 . 18 2 3 . 5 0 2 7 . 6 7 9 7 19 2 4 . 0 0 2 5 . 3 6 8 6 20 2 4 . 5 0 2 2 . 8 3 9 7 21 2 5 . 0 0 2 0 . 6 1 5 7 22 2 5 . 5 0 1 8 . 2 6 1 0 23 . 2 6 . 0 8 1 6 . 4 2 8 2 24 2 6 . 5 0 1 4 . 9 8 8 5 25 2 7 . 0 0 1 3 . 2 8 6 9 26 2 7 . 5 0 1 1 . 4 9 8 3 2 7 2 8 . 5 0 8 . 7 4 8 3 28 2 9 . 0 0 7 . 3 9 5 2 29 2 9 . 5 0 6 . 6 5 1 5 3 0 3 0 . 0 0 5 . 8 2 0 8 31 3 0 . 5 0 5 . 2 9 5 0 32 3 1 . 0 0 4 . 6 3 8 4 33 3 1 . 5 0 4 . 1 1 2 5 34 3 2 . 0 0 3 . 7 1 7 3 . 35 . 3 2 . 5 0 . . . 3 . 2 3 5 0 .. 3 6 3 3 . 0 0 2 . 9 2 6 8 3 7 3 3 . 5 8 2 . 5 7 4 5 t 38 3 4 . 0 0 2 . 3 9 7 5 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 1 4 . OOBR R2 G 1 U P - i - 2 X t AUG 1 5 , 6 7 A» B, AND D = . 1 6 3 2 3 E 0 1 . 1 0 0 0 0 E 0 1 . 8 6 0 0 0 E - 0 1 DATA NO XT YC 1 1 2 . 0 0 0 . 0 \ 2 1 5 . 0 0 0 . 1 0 0 I 3 1 8 . 0 0 0 . 6 0 0 4 2 1 . 0 0 1 2 . 1 0 0 5 .... 2 4 . 0 0 . 3 8 . 0 0 0 _L 6 2 7 . 0 0 5 0 . 5 0 0 7 2 9 . 0 0 4 7 . 0 0 0 8 3 1 . 0 0 3 8 . 0 0 0 n Q V 9 3 3 . 0 0 2 7 . 8 0 0 10 3 5 . 0 0 2 0 . 2 0 0 . 11 3 7 . 0 0 1 4 . 1 0 0 . S A M P . L E . . P R I N T O U T 12 4 1 . 0 0 7 . 3 0 0 13 4 5 . 0 0 4 . 5 0 0 14 4 9 . 0 0 3 . 1 0 0 15 5 4 . 0 0 2 . 4 0 0 16 5 9 . 0 0 1. 8 0 0 17 . 6 9 . 0 0 . . _ - 1 . 0 0 0 170 T A I L E X c c c S U B R O U T I N E T A I L E X ( K , T S T , X T I N . Y C , T I T L E ) T H I S F O R T R A N / 3 6 0 S U B R O U T I N E I S C A L L E D BY DQV OR BY N A C L , I F A GAMMA E X T E N S I O N I S TO BE F I T T E D . IT P L O T S T H E C C C C - T DARA IN THE FORM ( L O G C - B * L O G T ) V S . ( T ) , FOR B - V A L U E S OF - 1 , 0 , 1, 2 , 3 , A N D 4 . D I M E N S I O N X T 1 2 0 0 ) , Y C ( 2 0 0 ) , 1 X T I N ( 2 0 0 ) , X T P L 0 1 2 0 0 ) DO 1 J = 1 , K B X L ( 2 0 0 ) , Y L ( 2 0 0 ) , T I T L E l 7 f , X T ( J ) = X T I N ( J ) - T S T X T P L O I J ) = XT ( J ) I F ( X T P L O < J ) . L T . 0 . 0 ) X T P L O ( J ) * 0 . 0 I F ( Y C ( J > . L E . 0 . 0 ) Y L ( J ) = - 1 0 . 0 I F ( Y C ( J > . G T . 0 . 0 ) Y L ( J ) = A L O G ( Y C ( J ) ) C O N T I N U E UUIMI inut C A L L S C A L E ( X T P L O , K , 1 4 . 0 , X T M I N , DXT , 1 ) <XTIV C A L L A X I S ( 0 . 0 , 7 . 0 , 1 5 H X T - T S T , ( M I N ) , - 1 3 , 1 4 . 0 , 0 ^ 1 DXT ) .XTMIN, N = 0 . 0 DO 2 J = 1, K Y D I F F = Y C ( J + l ) - Y C ( J ) The parameters, A  B, and D, of 'TAILEX' correspond 2 .... 4 I F ( Y D I F F . L T . 0 . 0 ) N = N+ l I F (N . E Q . 3 ) GO TO 3 C O N T I N U E to YI.YZ.and Y3 of text. WRITE ( 6 , 4 ) T I T L E , T S T , J , X T ( J ) FORMAT ( 1 8 H 1 S U B R O U T I N E T A I L E X , / 9 H RUN N O . , 7 A 4 , 2 X , 1 18H S T A R T I N G T I M E , F 7 . 2 , / "2 2 8 H T A I L E X S T A R T S A T P O I N T NO ,I 2 , 1 0 X , 7 H X T I M E = , F 8 . 2 ) L O O P FOR D I F F E R E N T B V A L U E S DO 6 J 5 = 0 , 5 _ _ J 2 = J 5 - ] B = J 2 DO 7 J 3 = J K A X = X T ( J 3 ) I F ( AX . G T . 0 . 0 1 ) B X L ( J 3 ) = B * A L O G ( A X ) I F ( AX . L E . 0 . 0 1 ) BXL ( J 3 ) = - 1 0 . 0 Y ( J 3 ) = - B X L ( J 3 ) I F ( IWR ITE ( 6 , 8 ) T S T + Y L ( J 3 ) J 2 . E Q . 1 . O R . J 2 . E Q . 4 ) NXT(M) , B , ( M , Y C ( M ) , Y U M ) , B X L (M) , Y(MT> 2 , M = J , K ) FORMAT ( 1 6 H 0 I N I T I A L T I M E = , F 7 . 2 , / 1 5 4 H NO C O N C . L O G C B * L O G T 5H B = , F 6 . 2 , / Y XT - T I N , / 2 ( I X , 12 , F 8 . 2 , F 9 . 4 , F 1 0 . 4 , F 1 0 . 4 , F 9 . 2 ) ) JNC.) E L I M I N A T I O N OF D A T A P O I N T S TO SECOND P O I N T B E Y O N D PEAK(NOT"> I F ( J 2 . N E . ( - l ) ) G 0 TO 1 0 • _ _ YMAX = 4 . * ALOG ( A X ) I F ( Y L ( K ) . L E . 0 . 0 . A N D . Y L ( K ) . G T . ( - 9 . ) ) Y M A X = YMAX + Y L ( K ) DY = 0 . 5 I F (YMAX . G T . 3 . 5 ) DY = 1 . 0 I F (YMAX . G T . 7 . 0 ) DY = 2 . 0 I F (YMAX . G T . 1 4 . ) DY = 5 . 0 I F (YMAX . G T . 3 5 . 0 ) DY = 1 0 . 0 YLMAX = Y ( J ) I F ( Y ( K ) . G T . Y ( J ) ) YLMAX = Y ( K ) 171 DX = 0 . 5 I F ( Y LMAX - G T . 1 . 5 ) DX = 1 . 0 I F ( YLMAX . G T . 3 . 0 ) DX = 2 . 0 I F ( YLMAX . G T . 6 . 0 ) DX = 5 . 0 I F ( DX . G T . DY ) DY = DX YMIN = - 7 . 0 * DY YMAX = * 3 . 0 * DY . , ; ; C A L L A X I S ( 0 . 0 , 0 . 0 , 1 5 H L 0 G C - B . L O G T , +15 , 1 0 . 0 , 9 0 . 0 , 1 Y M I N , OY ) C A L L SYMBOL ( 1 . 0 , 1 . 0 , 0 . 2 1 , T I T L E , 0 . 0 , 2 8 ) C A L L SYMBOL ( 1 . 0 , 0 . 5 , 0 . 2 1 , 1 5 H S T A R T I N G T I M E = , 0 . 0 , 15 ) C A L L NUMBER ( 4 . 0 , 0 . 5 , 0 . 2 1 , T S T , 0 . 0 , 2 ) 10 0 0 12 J 4 = J , K 12 ~Y(J4) = Y ( J 4 ) / DY + 7 . 0 C A L L S Y M B O L ( X T P L 0 ( J ) » Y ( J ) , 0 . 1 4 , 3 , 0 . 0 , -1 ) KS= J +1 DO 11 M = KS , K I F ( Y L ( M ) . L T . ( - 9 . 0 V ) G O T O 11 I F ( Y ( M ) . G T . 1 0 . 0 . O R . Y ( M ) . L T . 0 . 0 )G0 TO 13 C A L L S Y M B O L ( X T P L 0 ( M ) , Y ( M ) , 0 . 1 4 , 3 , 0 . 0 , - 2 ) GO TO 11 (_2) 13 I F ( Y ( M ) . G T . I O . O ) C A L L SYMBOL ( X T P L O ( M ) , 1 0 . , 0 . 14 , 7 , 0 . 0 ,) • I F ( Y ( M ) . L T . 0 . 0 ) C A L L S Y M B O L ( X T P L O ( M ) , 0 . 0 , 0 . 1 4 , 5 . 0 . 0 ,) 11 C O N T I N U E ^2) 6^ _ C O N T I N U E . . C A L L P L O T ( 1 6 . 0 , 0 . 0 , - 3 ) R E T U R N 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._ _ 2 7 - 8 0 _ _ 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 1 3 - 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 ".. . --• - S A M PI F P R I N T - O U T  174 ' T A I L E X ' o in S A M P L E P L O T I sL B L R 2 9 . 3 - 5 - 4 X . J U N E 1 3 . 6 8 S T A R T I N G T I M E = 1 1 5 . 0 0 CD I o a m CM- SAMPLE PLOT BR R2 G1UP-1-2X- AUG 15.67 STARTING TIME = 14-00 176 QVE l S U B R O U T I N E Q V E L ( K K , T S T , X , Y , T I T L E , A , B, 0 , N R W T , N I G A ) C C S U B R O U T I N E FOR N U M E R I C A L I N T E G R A T I O N AND N U M E R I C A L E V A L U A T I O N C OF F I R S T MOMENTS OF T I M E - C O N C E N T R A T I O N C U R V E S « E X T E N S I ON TO C I N F I N I T E T I M E , B A S E D ON A D E C L I N E O F C S I M I L A R TO A GAMMA C D I S T R I B U T I O N , I S O P T I O N A L . C .. T H I S PROGRAM R E Q U I R E S .4.. S U B R O U T I N E S , G A U S S 1 , G A U S S 2 , A U X 1 , C AND AUX2 C £ INPUT [ C KK IS T H E NUMBER OF D A T A P O I N T S , C A L L E D K IN N A C L AND D Q V , C T S T IS S T A R T I N G T I M E , AS B E F O R E , C X...AND „Y. A R E . THE T - C D A T A , . . C A L L E D .XT A N D . . Y C . IN NACL AND DQV , C T I T L E IS AS B E F O R E , C A , B , AND D ARE T H E P A R A M E T E R S OF T H E GAMMA E X T E N S I O N , C NRWT " C 0 I F RHWT T E S T C 1 IF NA CL T E S T , 5 0 L I T E R TANK C 2 IF NA C L . J P S T , . 16 L L T E R TANK J . _ C N I G A , C 0 I F A , B, D ARE NOT G I V E N C 1 I F A , B, D ARE A V A I L A B L E FOR E X T E N S I O N TO I N F . C C O U T P U T C. . „ I N T E G R A L S . . A N D . F I R S T MOM. OVER T H E D A T A . PO I N T S , U S I N G C F I R S T AND SECOND ORDER M E T H O D S . MEAN T I M E I S ( F I R S T MOMENT C S S T A R T I N G T I M E , T S T ) . C O P T I O N A L , C WITH N IGA = 1, I N T E G R A L S AND MOMENTS W ITH E X T E N S I O N TO C I N F I N I T E T I M E . * TNT TO X T ( K ) « I S T H E N E G L E C T E D PART OF C . _ THE.. I N T E G R A L OVER T H E GAMMA D I S T R I B U T I O N , UP TO _.. C T I M E X T ( K ) . ' F A C T O R F A M • IS T H E A D J U S T M E N T TO A , TO C A C H I E V E C L O S E S T F I T TO T H E L A S T 3 DATA P O I M T S . C * F A C T O R A ' I S T H E C O R R E C T E D V A L U E OF A . A * F A M . C WITH NRWT = 1 OR 2 , T H E PROGRAM C O M P U T E S T H E D I S C H A R G E . C D I M E N S I 0 N _ X T ( 2 6 0 I . , _ „ Y C . ( 2 0 0 ) , „ X ( 2 a ) , . Y ( 2 0 ) . . , _ I I . T L E ( 7 . ) , Y C 0 ( 3 ) , 1 F A C T 0 R ( 3 ) D O U B L E P R E C I S I O N DGAMMA , G X , G Y R E A L M T , MEAN T I , MEAN T 2 , M E A N T 3 ; ._ C C E L I M I N A T I O N OF S U P E R F L U O S D A T A P O I N T S DO .9 . . . J _=_1_^__K_K I F < X ( J ) . G T . T S T ) GO TO 10 9 C O N T I N U E 10 K = K K - J + 2 DO 11 J l = 1 , K I = J1 + J -1 XT ( J 1-H) = X ( I ) .-. .TST. 11 YC ( J l + 1) = Y "(I) XT ( 1 ) = 0 . 0 YC ( 1 ) = 0 . 0 C C. I N T E G R A L OF D A T A P O I N T S , F I R S T AND S E C O N D ORDER METHODS 1 7 7 C F I R S T ORDER 1 C T MT 1 = 0 . 0 C T INT1 = 0 . 0 DO 4 J = 2 T K T R A P E Z = U Y C ( J - 1 ) + Y C ( J ) ) * ( X T ( J ) - X T U - U J I / 2 . 0 C T INT1 = C T I N T 1 + T R A P E Z - CT~MT_l....=._.C.T-MT-_.l„+__T.RA.P..E2._*_(-_XT.(.J^.l.) +„.„(.(.2 . * Y.C ( J ) + Y C ( J - l ) ) / 1 ( 3 . 0 * ( Y C ( J - 1 ) + Y C ( J ) ) ) ) * ( X T ( J ) - X T ( J - i ) ) ) 4 C O N T I N U E F I R S T M = C T MT 1 / C T I NT 1 MEAN T l = F I R S T M + T S T C C S E C O N D - O R D E R 1 . . _ _ ^ ... K T E S T = (K / 2 ) * 2 KL IM = K - 1 C T INT2 a 0 . 0 ; CT MT 2 = 0 . 0 IF ( K T E S T . N E . K ) K L I M = K „ DO 5 J _ = _ 3 _ , _ . K L I M » 2 .. D T I = XT ( J - l ) - X T ( J - 2 ) DT2 = XT ( J ) - X T U - 2 ) FO = ( ( ( ( D T 2 * * ? ) * 0 T l ) / 2 . ) - ( P T 2 * * 3 ) / 6 . 0 ) / ( D T 1 * D T 2 ) F l = 1 ( D T 2 * * 3 ) / ( ~ 6 . 0 ) ) / ( ( D T 1 * * 2 ) - ( D T 1 * D T 2 ) ) t - p T I F2 = ( U D T 2 * * 3 ) / 3 . 0 ) - ( ( ( D T 2 * * 2 ) * D T I ) / 2 . 0 ) ) / ( ( D T 2 * * 2 V ... * . D T 2 ) )... D INT = FO * YC ( J - 2 ) + F I * YC ( J - l ) + F 2 * Y C ( J ) C T I N T 2 = C T I N T 2 + D INT U =. Y C ( J - 2 ) / ( D T I * D T 2 ) V = Y C ( J - l ) / ( D T 1 * * 2 - D T 1 * D T 2 ) W = Y C ( J ) / ( D T 2 * * 2 - D T 1 * 0 T 2 ) C T . .MT_2—=_CT_MT_ .2- .+ . .D I N T . * (±DT_2)) 1 ( XT ( J - 2 ) +( ( ( U * V + W ) * ( D T 2 * * 4 ) ) / 4 . 0 - ( ( U * ( D T l ) ? + ( V * D T 2 ) +(W * D T D ) * ( O T 2 * * 3 ) / 3 . 0 + ( U * D T I * 3 ( D T 2 « * 3) ) / 2 . 0 ) / D I N T ) , 5 C O N T I N U E I F ( K T E S T . N E . K ) GO TO 6 T R A P E Z = _ ( . { y . C - < K - l . J * „ y C . ( K ) . . l * U X . T . . - . ( . K l = = _ X T - ( K - = l ) ) . ) . . . / - _ 2 . 0 CT INT 2 = T R A P E Z + C T INT 2 CT MT 2 = C T MT 2 + T R A P E Z * ( X T ( K - l ) + ( 2 . 0 * Y C ( K ) + Y C ( K - 1 ) ) / 1 ( 3 . 0 * ( Y C ( K ) + Y C ( K - H ) ) * ( X T ( K ) - X T ( K - I ) ) ) 6 F I R S T N = C T MT2 / C T INT 2 MEAN T2 = T S T + F I R S T N C WR ITE R E S U L T S (MEANT2 WRITE ( 6 , 7 ) C T INT 1 • F I R S T M » MEAN T l , C T I N T 2 , F I R S T N , ) V 7 FORMAT ( 3 2 H 1 1 N T E G R A T I O N OF MEASURED P O I N T S . / 1 1 9 H 0 I N T E G R A L C T 1 t F 1 5 . 5 t 1 0 H P P B * MIN t / 2 19H F I R S T MOMENT ( 1 ) = t F 1 5 . 5 * 4 H M I N T / . ..... 3 .. ... 19 H _M EAN_t.IJM.E_.' (1 ) =— -•?- - F 1 5 . 5 4 H M I N „ ... f / 4 19H I N T E G R A L C T 2 t F 1 5 . 5 » 1 0 H P P B * MIN f / 5 19H F I R S T MOMENT ( 2 ) = t F 1 5 . 5 • 4 H M I N t / 6 19H MF AN T I M F ( ? ) t . F 1 5 . 5 t 4HMTN 1 (INF. C I N T E G R A T I O N OF D A T A P O I N T S C O M B I N E D WITH F I T T E D E X T E N S I O N TO) IF ( N I G A . E Q . 0 ) GO TO 3 1 7 8 C C O R R E C T I O N F A C T O R K3 = K - 3 DO 8 J = 1 , 3 I = K3 + J YCO ( J ) = A * ( X t ( I ) * * B ) * EXP< - D * X T ( I ) ) 8 F A C T O R ( J ) = Y C I I I / Y C O ( J ) - F A M_=JJj? A C.TO.R_. . ).„_+.„ 2...0„JL., J£A C T OR (2 ) _ t _ 3 . . 0 _ * F A C T OR (3 I . ) / 6 . 0 C C I N T E G R A T I O N OF GAMMA D I S T R I B U T I O N R = B + 1 . 0 GX = R GY = DGAMMA(GX ) .. G = GY _I __ GAMMA = A * G / ( D * * R ) X M O l = R / D C T I N T F = (D * * " . * ) / G F I R S T 3 = 0 . 0 X I N T = XT IK ) / 2 0 . 0 C T I N T 3 = 0 . 0 . 1 . _ ; DO 12 J = 1 , 2 0 COUNT = J XUL = X T ( K ) - X I N T * ( C O U N T - 1 . 0 ) , X L L = XUL - X I N T C A L L G A U S S K X L L , XUL , D B , A , B , D ) C A L L. .GAUSS 2 (._ X L . L _ i _ X U L _ _ t _ D A.t..A B , D.) C T I N T 3 = C T I N T 3 + DB F I R S T 3 = F I R S T 3 + DA 12 C O N T I N U E ; D E B C T = ( C T I N T 3 * A ) / GAMMA D I N T 3 =(GAMMA - C T I N T 3 * A ) * FAM DFM03 = ( ( X M O l * GAMMA) - ( F I R S T 3 * A ) ) * FAM C T INT 4 = C T I N T 1 + D I N T 3 F I R m = ( C T M T 1 + D F M 0 3 ) / C T INT .4 C T I NT 5 = C T I N T 2 + D I N T 3 F I R M5 = ( C T M T 2 + D F M 0 3 ) / C T INT 5 MEAN T 3 = F I R M5 + T S T C WR ITE R E S U L T S " W R I T E ( 6 , 1 3 ) D I N T 3 , DFMO 3 , C T I N T 4 , F I R M 4 , C T I N T 5 , F I R M 5 1 , MEAN T 3 , D E B C T , FAM 13 FORMAT ( 4 5 H 0 I N T E G R A T I O N OF D A T A P O I N T W ITH F I T T E D E X T . , / I 1 8 H 0 A R E A C O R R . , F 1 5 . 5 , 7 H P P B * M I N , / 2 18H_.FIR.SX_.MaM.-.C0RR.=_ , . F 1 5 . 5 , . 15H M I N * * . 2 ^ * „ P P B , / / 3 18H AREA BY T R A P E Z = , F 1 5 . 5 , 7 H P P B * M I N , / 4 18H F I R S T MOM. ( T R ) = , F 1 5 . 5 , 7HMIN , / 5 18H AREA BY P A R A B . = , F 1 5 . 5 , 7 H P P B * M I N , / 6 18H F I R S T M O . BY P A . = , F 1 5 . 5 , 7 H M I N / 7 18H MEAN T I M E 3 , F 1 5 . 5 , 7HMIN // 8 ... 18H INT. TO X T ( K ) =.. , F 1 5 . 5 __../_ _.. 9 18H F A C T O R FAM , F 1 5 . 5 ) A = FAM * A WR ITE ( 6 , 1 6 ) A 16 FORMAT { 18H F A C T O R A = , F 1 5 . 5 ) 3 C O N T I N U E 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 ~ . I F _ 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 S E C . ) RFTURN ; ; ; END .' .. ' 1 8 0 S U B R O U T I N E G A U S S 1 ( A , B, A R E A , X A , X B , XD) C C S U B R O U T I N E IN F O R T R A N / 3 6 0 C A L L E D BY S U B R O U T I N E Q V E L . C D I M E N S I O N A X I 4 ) , H ( 4 ) D O U B L E P R E C I S I O N A X , H - AXi.l)_.^Jl..a6Q2B9_8.56.49753^ A X { 2 ) = 0 . 7 9 6 6 6 6 4 7 7 4 1 3 6 2 7 A X ( 3 ) = 0 . 5 2 5 5 3 2 4 0 9 9 1 6 3 2 9 A X t 4 ) •= 0 . 1 8 3 4 3 4 6 4 2 4 9 5 6 5 0 H Q ) = 0 . 1 0 1 2 2 8 5 3 6 2 9 0 3 7 6 H ( 2 ) = 0 . 2 2 2 3 8 1 0 3 4 4 5 3 3 7 4 . H ( 3 ) = „ 0 . 3 1 3 7 0 6 6 4 5 8 7 7 8 8 7 H ( 4 ) = 0 . 3 6 2 6 8 3 7 8 3 3 7 8 3 6 2 P = ( B + A ) * 0 . 5 Q = ( B - A ) * 0 . 5 SUM = 0 . 0 DO 3 0 J = 1 , 4 R = A X ( J ) * Q _ _ _ _ _ X = P+R C A L L A U X l ( X , Y , X A , X B , XD) Z ' = Y X = P-R C A L L A U X l ( X , Y , X A , X B , XD) 30 _SUM_j=. „ .SUM_+_H ( J ) * ( Z+Y ) J J :_. A R E A = 0 *SUM R E T U R N END " S U B R O U T I N E A U X l ( X , Y , A , B, D) C C C A L L E D BY G A U S S 1. C : Y = ( X * * B l ' * E X P ( (- D ) * X ) R E T U R N 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 I N T E G R A T I O N OF M E A S U R E D P O I N T S I N T E G R A L C T 1 F I R S T MOMENT ( 1 1 = MEAN T I M E ( 1 ) = I N T E G R A L C T 2 6 7 8 . 4 9 8 0 5 P P B * M IN 1 7 . 4 1 5 3 7 M I N 3 1 . 4 1 5 3 7 M I N 6 7 4 . 1 2 2 0 7 P P B * M IN F I R S T MOMENT ( 2 ) = MEAN T I M E ( 2 ) 1 7 . 3 8 9 0 2 M I N 3 1 . 3 8 9 0 2 M I N I N T E G R A T I O N OF D A T A P O I N T WITH F I T T E D E X T . AREA C O R R . 1 3 . 5 4 7 0 8 P P B * M I N F I R S T MOM. C O R R . = 9 3 0 . 0 9 4 2 4 M I N # * 2 . * P P B AREA BY.__TRAP.EZ i F I R S T MOM. ( T R ) AREA BY P A R A B . F I R S T M O . BY PA.- .6.9 2 . 0 4 4 9 2 P_PB*_M I N. 1 8 . 4 1 8 4 4 M I N 6 8 7 . 6 6 8 9 5 P P B * M I N 1 8 . 3 9 8 9 9 M I N MEAN T I M E 3 3 2 . 3 9 8 9 9 M I N I NT JO J K T t K ) . F A C T O R FAM F A C T O R A . 0 . 9 4 9 4 2 . 1 . 2 1 3 6 5 1 . 9 8 1 0 6 1 Q V E L 1 SAMPLE PRINTOUT FOR 'B R R 2 , GI UP I - 2X' I N T E G R A T I O N OF M E A S U R E D P O I N T S I N T E G R A L C T 1 F I R S T MOMENT ( 1 ) = MEAN T I M E ( 1 ) I N T E G R A L C T 2 6 1 5 3 . 1 9 9 2 2 P P B * MIN 9 6 . 2 5 4 7 0 M I N 2 1 1 . 2 5 4 7 0 M I N 6 1 4 9 . 3 8 6 7 2 P P B * M IN J 8 3 F I R S T MOMENT ( 2 ) = MEAN T I M E ( 2 ) 9 6 . 2 1 2 6 9 M I N 2 1 1 . 2 1 2 6 9 M I N I N T E G R A T I O N OF D A T A P O I N T WITH F I T T E D E X T . AREA C O R R . 4 5 9 . 5 9 2 5 3 P P B * M I N F I R S T MOM. C O R R . = 1 7 6 8 5 6 . 8 7 5 0 0 M I N * * 2 * PPB A R E A . BY. T R A P E Z F I R S T MOM. ( T R ) AREA BY P A R A B . F I R S T M O . BY P A . . 6 6 1 2 . 7 8 9 0 6 P P B * M I N . 1 1 6 . 3 0 9 6 5 M I N 6 6 0 8 . 9 7 6 5 6 P P B * M I N 1 1 6 . 2 8 2 1 4 M I N MEAN T I M E 3 2 3 1 . 2 8 2 1 4 M I N INT TO X T ( K ) F A C T O R FAM F A C T O R A . 0 . 8 5 2 0 2 . 0 . 9 9 5 3 1 0 . 4 4 7 2 2 D I S C H A R G E 0 . 1 2 6 0 9 C U B I C M PER S E C , Q V E L' SAMPLE PRINTOUT FOR J BL. R 2 9, G 3 - 5 - 4X' I N T E G R A T I O N OF M E A S U R E D P O I N T S 184 I N T E G R A L C T 1 F I R S T MOMENT < 1 ) : MEAN T I M E ( 1 ) I N T E G R A L G T 2 2 6 8 . 6 1 5 4 8 P P B * MIN 9 . 8 9 9 0 5 M I N 2 4 . 8 9 9 0 3 M I N 2 6 8 . 4 3 8 2 3 P P B * M IN F I R S T MOMENT ( 2 ) ; MEAN T I M E ( 2 ) 9 . 9 0 4 5 2 M I N 2 4 . 9 0 4 5 1 M I N D I S C H A R G E 3 . 1 0 4 2 5 C U B I C M PER S E C 'Q V E L* _.. SAMPI F PRINTOUT FOR 'PH R2 8, G3- 4 - 6X' 1 8 5 PI 0 T fi A S U B R O U T I N E P L O T G A (Kt T S T , X T I N , Y C I N , A , B, D, T I T L E ) C C T H I S S U B R O U T I N E IN F O R T R A N / 3 6 0 I S C A L L E D BY T H E MAIN C PROGRAMS NACL AND DQV TO P L O T T H E C - T C U R V E S . O P T I O N A L L Y C IT WILL A L S O P L O T T H E GAMMA E X T E N S I O N S . I N T H I S C A S E IT SHOULD C BE C A L L E D A F T E R T H E S U B R O U T I N E Q V E L HAS B E E N C A L L E D , AS Q V E L C IMPROVE . S _ _THE . . E S _T IMAXE_OF_A . C D I M E N S I O N X T ( 2 0 0 ) , Y C ( 2 0 0 ) , T I T L E ( 7 ) , X T I N I 2 0 0 ) , Y C I N ( 2 0 0 ) DO 11 I =1 , 2 0 0 ' X T ( I ) = X T I N ( I ) 11 Y C ( I ) = Y C I N U ) C A L L . S C A L E ( X T , K, 1 0 . 0 , . . X T M I N , D X T , 1 ) C A L L S C A L E ( Y C , K, 9 . 0 , Y C M I N , D Y C , 1 ) ( D X T ) C A L L A X I S ( 0 . 0 , 0 . 0 , 1 5 H T I ME IN M I N U T E S , - 1 5 , 1 3 . 0 , 0 . 0 , XTMIN, ) C A L L A X I S ( 0 . 0 , 0 . 0 , 2 0 H C 0 N C E N T R A T I O N IN P P B , » 2 0 , 9 . 0 , ) 1 Y C M I N , DYC ) (90 .0 , DO 1 J = 1, K 1 C A L L . SYMBOL . ( X T . ( J ) , _YC ( J L . „ , . „ 0 . . 1 A . , 2. ,. „ 0 . 0 , - 1 .) C A L L SYMBOL ( 6 . 0 , 9 . 0 , 0 . 2 8 , T I T L E , 0 . 0 , 30 ) C A L L SYMBOL ( 6 . 0 , 8 . 5 , 0 . 1 4 , 5 H T S T = , 0 . 0 , 5 ) C A L L NUMBER ( 7 . 5 , 8 . 5 , 0 . 1 4 , T S T , 0 . 0 , 2 ) I F (A . E Q . 0 . 0 ) GO TO 6 C A L L P L O T ( 0 . 0 , 0 . 0 , +3 ) . . .YMAX__=.LY.CM. IN_£OYXJ_•__.9. . .0 . : : 0 0 2 I = 1, 131 F = I- 1 X . = F / 1 0 . 0 T = X T M I N + X * DXT - T S T IF { T . L T . 0 . 0 ) T = 0 . 0 Y _ _ = ( {... A* ( _ t _ * * . . B l ) * . E X P ( - D * T ) - Y C M I N ) / DYC IF ( Y . L E . YMAX ) GO TO 5 C A L L SYMBOL ( X , 9 . 0 , 0 . 0 7 , 13 , 0 . 0 , - 2 ) GO TO 2 5 C A L L P L O T ( X , Y , + 2 ) 2 C O N T I N U E C A L L SYM80L._ (...6 , 0 . . , _ 8 . 0 . . . , _ _ 0 . 14. .. 24HGAMMA . . PARAM. A , B , D = 1 0 . 0 , 24 ) C A L L NUMBER ( 9 . 0 , 8 . 0 , 0 . 1 4 , A , 0 . 0 , 6 ) C A L L NUMBER ( 1 0 . 3 , 8 . 0 , 0 . 1 4 , B , 0 . 0 , 2 ) C A L L NUMBER ( 1 1 . 6 , 8 . 0 , 0 . 1 4 , 0 , 0 . 0 , 6 ) 6 C A L L PLOT ( 1 5 . 0 , 0 . 0 , - 3 ) R E T U R N • END PH R 2 8 . 3 - 4 - 6 X . SEPT . 1 7 . TST = 15.00 A A A A A A A A A A A ' P L O T G A* S A M P L E P L O T 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 2 9 . 3-5- 4 X . JUNE 13. 68 TST = 115.00 GAMMA PARAM. A. B. D 0.447722 1.00 0.012499 'PL 0 T G A' S A M P L E P L O T A 80.0 120.0 160.0 280.0 TIME 320.0 360.0 IN MINUTES BR R 2 G 1 U P - 1 - 2 X . RUG 1 5 . 6 7 J TST = 14.00 GAMMA PARAM. A. B. D = 1.981554 1.00 0.086499 A A ' P L 0 T G A* S A M P L E P L O T A 189 L 0 G R E ' r c C T H I S F O R T R A N / 3 6 0 PROGRAM COMPUTES THE L I N E A R R E G R E S S I O N S C ON LOG Q ( D I S C H A R G E ) OF T H E F O L L O W I N G V A R I A B L E S= C LOG TM (MEAN T R A C E R T R A V E L T I M E ) C LOG A ( C R O S S E C T I O N A L A R E A ) C LOG V ( V E L O C I T Y ) C LOG TS _ ( S T A R T I N.G.TIME) . C LOG TP ( P E A K T I M E ) C LOG T S S ( S T A R T I N G T I M E , BUT O M I T T I N G RUNS WITH T R A C E R C I N J E C T I O N ABOVE T H E R E A C H ) C LOG T P P ( P E A K T I M E , O M I T T I N G RUNS AS FOR T S S ) C T H E A C T U A L R E G R E S S I O N A N A L Y S I S I S DONE BY A S U B R . •REGR* . C TWO P L O T T I N G . S U B R O U T I N E S ..CAN. ALSO. BE. C A L L E D _ F R O M T H I S PROGRAM. C C I N P U T C F I R S T CONTROL CARD, ONE PER J O B S U B M I S S I O N , = C NO. OF S E T S , ( 1 2 ) . C SECOND CONTROL C A R D , ONE P E R DATA S E T , = C T I T L E , ( 6 X , ? A 4 ) C DATA C A R D S , ONE P E R T E S T RUN, = C RUN NO. ( 1 2 ) COL 1 £ 2 C I D E N T I F I C A T I O N ( I D COL 5 J C T H I S I S I FOR RUNS WITH I N J E C T I O N OF T R A C E R AT U P S T R E A M C END OF T E S T R E A C H , 0 FOR OTHER RUNS. C DATA .._ . ( 6 F 6 . 0 ) COL 7 . £ ON, ( Q , TS , TP , TM , A , V ) C Q I N L / S , T I M E S I N M I N . , A I N SO M, C V I N M/S, 0 I S C O N V E R T E D TO CU M/S. C C OUTPUT C P R I N T O U T OF DATA, C L I N E A R R E G R E S S I O N E Q U A T I O N S , STANDARD ERROR OF E S T I M A T E , C C O R R E L A T I O N C O E F F . , D E G R E E S OF F REEDOM, F - R A T I O . C C A L L P L O T S D I M E N S I O N N O I 3 0 ) , I D ( 3 0 ) , Q ( 3 0 ) , T S ( 3 0 ) , TP ( 3 0 ) , T M ( 3 0 ) , 1 A ( 3 0 ) , V ( 3 0 ) , T S S I 3 0 ) , T P P ( 3 0 ) , T I T ( 7 ) ,QQ(30) 2,Q1 ( 3 0 ) .. C C. LOOP FOR NUMBER OF SETS , READ ( 5 , 1 ) KTOT 1 FORMAT ( 1 2 ) DO 2 K S = 1 , KTOT C R E A D I N G AND PR T N T I N G OF ~DATA~ READ ( 5 , 3 ) T I T 3 FORMAT ( 6 X , 7 A 4 ) DO 4 K - 1, 3 0 ~~ (V(k) READ ( 5 , 5 ) N O ( K ) , I D ( K ) , Q I K ) , T S ( K ) , T P ( K ) , TM ( K ) , A (K) ,) 5 FORMAT ( 1 2 , 2 X T . I I , I X , 12F. 6.0 ) i I F ( N O . ( K ) . L E . 0 ) GO TO 6 4 C O N T I N U E 6 K = K - 1 • WRITE ( 6 , 7 ) T I T 7 FORMAT ( 1 H 1 , 7 A 4 , / 1 58H0NO..I0. Q.J L / S ) TS TP_ TM. A v/) 190 WR ITE ( 6 , 8) < N 0 ( I 1 ) , 1 0 ( 1 1 ) , Q U I ) , T S U I ) , T P U 1 ) , T M Q 1) » 1 A C I D , V ( I 1 ) , I I = 1, K ) 8 FORMAT ( I H , 1 1 2 , 1 3 , F 1 0 . 2 , 3 F 9 . 2 , 2 F 9 . 4 ) r K I D = 0 c T R A N S F O R M A T I O N T O . LOGS...... . . E V A L U A T I O N . . O F . _ .N0 . . _D£ S I M P L E . RUNS DO 9 12 = 1, K Q( 12 ) = A L O G 1 0 ( Q U 2 J / 1 0 0 0 . ) Q 1 U 2 ) = Q ( I 2 ) T S ( I 2 ) = A L 0 G 1 0 ( T S ( 1 2 ) ) T P ( 1 2 ) = A L 0 G 1 0 ( T P ( I 2 ) ) TM ( 1 2 ) = A L O G 1 0 (TM (1 2 ) ) .. A ( 1 2 ) = A L O G 1 0 ( A ( I 2 )> V U 2 ) = A L 0 G 1 0 ( V ( I 2 ) ) 9 K I D = K I D + ID ( 1 2 ) C C R E G R E S S I O N S QN O R I G I N A L D A T A 10 . .FORMAT „ ( 1H0 /._1_8H L O G T M . V S . L O G Q ) . WR ITE ( 6 , 1 0 ) C A L L REGR ( Q , TM , K , A T M , BTM ) 11 FORMAT ( 1 H 0 / 1 8 H L O G A V S . L O G Q ) WR ITE ( 6 , 1 1 ) C A L L REGR ( Q , A , K , A A , BA ) 12 FORMAT J 1 H 0 / 18H L O G V V S . L O G Q ) WR ITE ( 6 , 1 2 ) C A L L REGR ( Q , V , K , A V , BV ) I F ( K I D . E Q . K ) GO TO 18 13 FORMAT ( 1 H 0 / 18H L O G T S V S . L O G Q ) WR ITE ( 6 , 1 3 ) C A L L REGR ( Q , T S , K , A T S , BTS ) 14 FORMAT ( 1 H 0 / 1 8 H L O G T P V S . L O G Q ) WR ITE ( 6 , 1 4 ) C A L L REGR ( Q ", T P , K , A T P , BTP ) r I F ( K I D . E Q . 0 ) GO TO 20 L C T P AND TS R E G R E S S I O N S _ _ 18 N ID = 0 DO 15 1 5 = 1 , K I F ( I D ( 1 5 ) . L E . 0 ) GO TO 15 N ID = N ID + 1 T S S ( N I D ) = TS 115 ) T P P ( N I D ) = TP ( 1 5 ) . „ _ . Q Q ( N I O ) = Q ( 1 5 ) 15 C O N T I N U E I F ( N I D . N E . K I D ) WR ITE ( 6 , 1 6 ) KS 16 FORMAT ( 1 H 1 , 17H N ID ERROR IN S E T , 12 ) 17 FORMAT ( 1 H 0 / 18H L O G T S S V S . LOG Q ) WR ITE ( 6 , 17 ) '_ : C A L L REGR ( Q Q , T S S , K I D , A T S S , 8 T S S : ' k 19 FORMAT ( 1 H 0 / 18H L O G T P P V S . LOG Q ) WR ITE ( 6 , 1 9 ) C A L L REGR ( Q Q , T P P , K I D , A T P P , B T P P ) GO TO 21 191 C C A L L S TO P L O T T I N G S U B R O U T I N E S C 20 C O N T I N U E ( B T M , C A L L T P L O I Q , TS , TP, TH , K , K , A T S , A T P , A T M , B T S , BTPT) 1 T IT,Q) GO TO 22 21 C O N T I N U E . _ ... . _ ... . ( B T S S , C A L L T P L O ( Q , T S S , T P P , TM , K , K I D , A T S S , A T P P , ATM ,) 1 B T P P , BTM , T I T , QQ ) 22 C O N T I N U E ' C A L L H Y P L O IQ1, A , V , K , A A , AV , BA , BV , T I T ) C 2 C O N T I N U E . • _ C A L L P LOTNO S T O P 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 S U B R O U T I N E T P L O ( Q , T S S , T P P , TM , K , K I D , A T S S , A T P P ,ATM, 1 B T S S , B T P P , BTM , T I T , QQ ) (LOGRE T H I S F O R T R A N / 3 6 0 S U B R O U T I N IS C A L L E D FROM T H E MAIN PROGRAM) I T P L O T S T H E R E G R E S S I O N S OF T S S , T P P , TM ON Q , INCLUDING D A T A P O I N T S . DI MENS I ON ol 3 0 ) , T S S ( 30 F, T P P ( 3 0 ) " , TM ( 3 0 ) , T ( 9 0 ) , TI T (7) I , Q Q ( 3 0 ) S C A L E O A T A DO 1 I - 1, K T ( I ) . _ = TM .( I ) DO ION = 1, K I D K5 = N + K T ( K5 ) = T P P ( N ) N l = K + K I D K6 = N + N l T ( K 6 _ ) = T S S ( N ) _ _ __. N2 =N1 + K I D C A L L S C A L E ( t , N2 , 5 . 0 , T M I N , DT , 1 ) K7 = K + 2 * K I D DO 100 J 5 = 1 , K7 T ( J 5 ) = T ( J 5 ) + 3 . 0 DO 2 J _ = 1, K T M ( J ) = T ( J ) DO 11 J l = 1, K I D K7 = J l + N l T S S ( J l ) = T ( K7 ) K8 = J l + K T P P ( J l ) = T ( . K8 ) C A L L S C A L E ( Q , K , 5 . 0 , QMIN , OQ , 1 ) DO 2 0 0 J J 7 = 1 , K I D Q Q ( J J 7 ) ( Q Q ( J J 7 ) - QMIN ) / DQ DRAW A X I S (DO) C A L L A X I S ( 0 . 0 , 3 . 0 , 1 6 H Q I N CU M / S E C , - 1 6 , 5 . 0 , 0 . , Q M l l ^ ) C A L L A X I S ( 0 . 0 , 3 . 0 , 1 2 H T I M E IN M I N . , +12 , 5 . 0 , 9 0 ^ P L O T P O I N T S (TMIN , DT) DO 3 1 1 = 1 , K . C A L L SYMBOL ( Q ( I I ) , T M ( I 1 ) , 0 . 0 7 , 2 , 0 . 0 , -I ) DO 4 12 = 1 , K I D C A L L SYMBOL ( Q Q ( I 2 ) , T S S ( I 2 ) , 0 . 0 7 _ , 3 , 0 . 0 , -1 ) C A L L SYMBOL I Q Q ( 1 2 ) , T P P ( I 2 ) , 0 . 0 7 , 4 , 0 . 0 , -1 ) P L O T R E G R E S S I O N L I N E S TM VS Q L I N E YB = ( ( ( A T M + B T M * Q M I N ) . - . TMIN) . . / D T ) + J 3 . 0 XB = 0 . 0 I F < YB . L E . 8 . 0 ) GO TO 101 YB = 8 . 0 : . - . XB = ( 5 . 0 * DT + TMIN -• ATM - Q M I N * BTM ) / ( DQ * BTM) YE = ( ( ( A T M + B T M * (QMIN + 5 . 0 * D Q ) ) - T M I N ) / DT ) + 3 . 0 XE = 5 . 0 1 9 4 I F ( Y E . GE . 3 . 0 ) GO TO 102 Y E = 3 . 0 XE = ( T M I N - ATM - QMIN * BTM J / ( DQ * BTM ) 102 C A L L P L O T < XB , YB , +3 ) C A L L P L O T ( X E , YE , +2 ) C c T S VS Q L I N E . YB = ( ( ( A T S S + B T S S * QMIN) - T M I N ) / D T ) + "3.0" XB = 0 . 0 I F I YB . L E . 8 . 0 ) GO TO 1 0 3 YB = 8 . 0 XB = ( 5 . 0 * DT + TMIN -r A T S S - Q M I N * B T S S ) / ( DQ * BTSS ) 1 0 3 Y E = ( ( ( A T S S + B T S S * . (QMIN ..+ 5 . 0 * . DQ) ) - T M I N ) / DT. » + 3 . 0 XE = 5 . 0 IF ( Y E . GE . 3 . 0 ) GO TO 1 0 4 YE = 3 . 0 XE = ( T M I N - A T S S - QMIN * B T S S ) / { DQ * B T S S ) C A L L P L O T ( X B , YB , +3 ) C A L L P LOT ( X E ... , YE , + 2 ) . . . C TP VS Q L I N E YB = ( U A T P P + B T P P * QMIN) - T M I N ) / DT ) + 3 . 0 XB = 0 . 0 I F ( YB . L E . 8 . 0 ) GO TO 105 YB _„=.._8.0. _ _ .. .. .. XB = ( 5 . 0 * DT + T M I N - A T P P - Q M I N * BTPP ) / ( DQ * BTPP ) 1 0 5 YE = I U A T P P + B T P P * (QMIN + 5 . 0 * D Q ) ) - T M I N ) / DT ) + 3 . 0 XE = 5 . 0 IF { Y E . GE . 3 . 0 ) GO TO 1 0 6 YE = 3 . 0 XE = ( T M I N . - . . A T P P - QMIN .*... BTPP. . ) . . . / ( DQ * B T P P ) 1 0 6 C A L L P L O T ( X B , YB , +3 ) C A L L P L O T ( X E , Y E , +2 ) C C WR ITE T I T L E AND L E G E N D C A L L SYMBOL ( 0 . 5 , 9 . 0 , 0 . 2 1, T I T , 0 . 0 , 28 ) C A L L SYMBOL ( .1 . 0 ., 1 .5 , . 0 . 0 7 , 2 . . , . 0 . 0 . , - 1 ) . C A L L SYMBOL ( 1 . 0 , 1 . 0 , 0 . 0 7 , 4 , 0 . 0 , -I ) C A L L SYMBOL ( 1 . 0 , 0 . 5 , 0 . 0 7 , 3 , 0 . 0 , -1 ) C A L L SYMBOL ( 1 . 5 , 1 . 5 , 0 . 1 4 , 9HMEAN T I M E , 0 . 0 , 9 ) C A L L SYMBOL ( 1 . 5 , 1 . 0 , 0 . 1 4 , 9 H P E A K T I M E , 0 . 0 , 9 ) C A L L SYMBOL ( 1 . 5 , 0 . 5 , 0 . 1 4 , 1 3 H S T A R T I N G T I M E , 0.0 , 13) C : C C O M P L E T E O U T L I N E , MOVE ON C A L L PLOT ( 0 . 0 , 8 . 0 , +3 ) C A L L P L O T ( 5 . 0 , 8 . 0 , +2 ) C A L L PLOT ( 5 . 0 , 3 . 0 , +1 ) C A L L P LOT ( 1 1 . 0 , 0 . 0 , - 3 ) R E T U R N . „ . ; „ END 195 S U B R O U T I N E H Y P L O (Q , A , V , K , AA , A V , BA , BV , T I T ) C (LOGRE C T H I S F O R T R A N / 3 6 0 S U B R O U T I N E I S C A L L E D FROM THE MAIN PROGRAM ) C IT P L O T S T H E R E G R E S S I O N S OF A , AND V t ON Q . C D I M E N S I O N Q ( 3 0 ) , A ( 3 0 ) , V ( 3 0 ) , T I T ( 7 ) C _ . ._ C S C A L E DATA C A L L S C A L E I Q , K , 5 . 0 , QMIN , OQ ,1 ) C A L L S C A L E ( V , K , 5 . 0 , VM IN , DV ,1 ) C A L L S C A L E ( A , K , 5 . 0 , AM IN , DA ,1 ) C C .. D R A W . A X I S . _' (QMIN, C A L L A X I S ( 0 . 0 , 3 . 0 , 16HQ IN C U M / S E C , - 1 6 , 5 . 0 , 0.0, I 1 DQ ) (AMIN, . C A L L A X I S ( 0 . 0 , 3 . 0 , 1 4 H A R E A IN SQ M . , H 4 ,5.0,90.0,; 1 DA ) (VMIN. C A L L A X I S ( 5 . 0 , 3 . 0 , 1 7 H V E L 0 C I T Y IN M/SEC , — 1 7 , 5 . 0 , 9 U. U,) 1 0V..1 „ _ C C P L O T P O I N T S DO 1 I = 1 , K A ( I ) = A ( I ) + 3 . 0 V( I ) - V ( I ) + 3 . 0 C A L L SYMBOL ("_Q( I )..,.. A ( I ) _ , 0 . 0 7 , 2 0 . 0 _ , _ - 1 ) 1 C A L L SYMBOL ( Q ( I ) , V ( I ) , 0 . 0 7 , 3 , 0 . 0 , -1 ) C C P L O T R E G R E S S I O N L I N E C C Q V S . A L I N E YB = ( ( ( A A . + B A * „ Q M I N ) - A M I N ) / O A ) + 3 . 0 _ . . XB = 0 . 0 IF ( YB . G E . 3 . 0 ) GO TO 101 YB = 3 . 0 XB = (AM IN - AA - QMIN * BA ) / (DQ * B A ) 101 XE = 5 . 0 YE = ( ( ( AA + B A * ( QMIN + 5 . 0 * D Q ) ) - A M I N ) A OA ) + 3 . 0 IF ( YE . L E . 8 . 0 ) GO TO 102 Y E = 8 . 0 XE = ( 5 . 0 * DA + AMIN - AA - Q M I N * BA ) / (OQ * BA ) 102 C A L L P L O T ( XB , YB , + 3 ) C A L L P LOT ( XE , YE , + 2 ) c ..;„ C Q VS V L I N E XB = 0 . 0 YB = (<( AV + B V * Q M I N ) - V M I N ) / D V ) + 3 . 0 I F ( YB . G E . 3 . 0 ) GO TO 103 YB = 3 . 0 XB = ( VMIN - AV - QMIN * BV . ) / (DQ * BV) 103 XE = 5 . 0 YE = ( ( ( AV + BV * ( Q M I N + 5 . 0 * D Q ) ) - V M I N )/ DV ) + 3 . 0 IF ( YE . L E . 8 . 0 ) GO TO 1 0 4 YE = 8 . 0 XE = ( 5 . 0 * DV + VMIN - AV - Q M I N * BV ) / ( DQ * BV ) 104 C A L L P L O T ( XB:.,__YB . , + 3 . J > 3r9-6 r C A L L P L O T ( XE , YE , +2 ) c T I T L E AND L E G E N D C A L L SYMBOL ( 0 . 5 , 9 . 0 , 0 . 2 1 , T I T , 0 . 0 , 28 ) C A L L SYMBOL ( 1 . 0 , 1 . 6 , 0 . 1 4 , 2 , 0 . 0 , -1 ) C A L L SYMBOL ( 1 . 0 , 1 . 1 , 0 . 1 4 , 3 , 0 . 0 , -1 ) ... C A L L SYMBOL . I 1 . 5 , 1 . 5 , . 0 . 1 4 , . . .9HFL0W AREA . ., . .. 0 . 0 . , .9 .) r C A L L SYMBOL ( 1 . 5 , 1 . 0 , 0 . 1 4 , 9 H V E L 0 C I T Y , , 9 ) C C O M P L E T E O U T L I N E , MOVE ON C A L L P L O T i 0 . 0 , 8 . 0 , +3 ) C A L L P LOT I 5 . 0 , 8 . 0 , +2 ) . C A L L PLOT.. ( 8 . 0 , 0 . 0 , - 3 . ) ... R E T U R N END B R O C K T O N C K , R E A C H 1-2 197 > NO ID 0 ( L / S ) TS T P TM A V 2 1 9 . 2 0 1 7 . 0 0 2 5 . 5 0 3 0 . 1 0 0 . 1 4 0 0 0 . 0 6 6 0 4 1 5 . 6 0 2 4 . 0 0 3 7 . 3 0 4 4 . 5 0 0 . 1 2 6 0 0 . 0 4 5 0 7 0 0 . 6 7 1 1 0 . 0 0 1 8 0 . 0 0 1 9 3 . 0 0 0 . 0 6 5 0 0 . 0 1 0 3 9 0 0 . 1 6 3 5 0 . 0 0 5 4 0 . 0 0 5 6 0 . 0 0 0 . 0 4 5 0 0 . 0 0 3 5 ... . 10 . _ 1 __1...2.3 . 7 . 0 . 0 0 l.l.7.._00_..'. 1159... 5.0. - .0 . .Q99Q ... _ 0 . 0 1 2 5 16 1 5 8 . 0 0 4 . 9 0 8 . 1 0 9 . 0 8 0 . 2 6 6 0 0 . 2 1 8 0 18 1 5 0 . 0 0 7 . 2 0 1 1 . 6 0 1 2 . 5 8 0 . 3 1 7 0 0 . 1 5 8 0 19 1 1 5 8 . 0 0 2 . 5 0 5 . 0 0 5 . 4 0 0 . 4 3 0 0 0 . 3 6 8 0 21 1 9 2 . 0 0 4 . 2 0 7 . 3 0 8 . 3 3 0 . 3 8 6 0 0 . 2 3 80 23 1 1 3 7 . 0 0 3 . 2 0 5 . 5 0 6 . 9 7 0 . 4 8 1 0 0 . 2 8 5 0 LOG TM V S . LOG Q SSX 1 0 . 1 9 0 3 S SY 4 . 5 0 0 8 SP ^ 6 . 7 5 3 6 R E G R E S S I O N E Q U A T I O N Y= 0 . 2 0 5 6 + - 0 . 6 6 2 8 X STANDARD ERROR OF E S T I M A T E 0 . 0 5 5 8 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 2 D E G R E E OF FREEDOM 9 .£,_.= . 1 4 3 9 . 3 . 0 9 8 — _ . :L.0 G R E COMPLETE PRINTOUT L O G A V S . L O G Q SSX 1 0 * 15LQ.3 SSY . 1 . 1 8 5 6 SP ..... 3 . 4 3 9 9 . R E G R E S S I O N E Q U A T I O N Y= - 0 . 0 9 1 3 •+ 0 . 3 3 7 6 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 5 5 3 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 8 9 6 DEGRE E_ O F „ F R E E DOM . .9 F = 3 7 9 . 7 5 1 0 LOG V V S . LOG Q SSX 1 0 . 1 9 0 3 SSY 4 . 5 0 9 5 SP 6 . 7 5 9 9 R E G R E S S I O N E Q U A T I O N Y= 0 . 0 9 3 4 + 0 . 6 6 3 4 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 5 6 1 C O R R E L A T I O N C O E F F . I C I E N T . 0 . 9 9 7 2 _ „ D E G R E E OF FREEDOM 9 F = 1 4 2 3 . 0 2 9 8 LOG TS V S . L O G Q SSX 1 0 . 1 9 0 3 SSY 4 . 7 9 8 7 SP - 6 . 9 7 7 8 * , 198 r R E G R E S S I O N E Q U A T I O N Y= - 0 . 1 1 7 3 * S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 5 0 9 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 8 D E G R E E O F F R E E D O M 9 ^ F = 1 8 4 1 . 9 0 1 1 - 0 . 6 8 4 7 X ? L O G T P V S . L O G Q . S S X 1 0 . 1 9 0 3 S S Y 4 . 5 4 1 9 S P - 6 . 7 8 4 0 R E G R E S S I O N E Q U A T I O N Y = 0 . 1 3 9 0 + S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 5 6 6 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 2 D E G R E E O F F R E E D O M 9 - 0 . 6 6 5 7 .X F = 1 4 1 2 . 1 2 7 9 L O 6 " Y S S ^ S 7 " L O G ~ Q " S S X 4 . 0 3 4 9 S S Y 1 . 7 4 9 8 S P - 2 . 6 4 7 2 R E G R E S S I g^YoljATl O N V - T " " -076787 + S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 4 6 6 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 6 3 - 0 . 6 5 6 1 X D E G R E E O F F R E E D O M 7 F = 7 9 8 . 5 0 8 1 L O G T P P V S . L O G Q S S X 4 . 0 3 4 9 S S Y 1 . 6 0 3 9 S P - 2 . 5 3 2 2 R E G R E S S I O N E Q U A T I O N Y = 0 . 1 8 9 6 + - 0 . 6 2 7 6 X S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 4 9 6 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 5 4 D E G R E E O F F R E E D O M 7 F = _ _ . „ 6 4 6 . 0 5 7 1 . . . 7 1 9 9 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 B R O C K T O N C K , R E A C H 2-3 NO 10 Q ( L / S ) T S T P 1 1 6 . 7 0 1 3 . 6 0 3 3 . 0 0 4 0 6 . 8 0 2 7 . 0 0 3 4 . 5 0 8 0 0 . 7 3 7 6 . 0 0 1 6 0 . 0 0 11 1 1 . 2 0 5 3 . 0 0 9 7 . 0 0 1.7. _ l . 3 4 . 0 0 „,L 3 . 0 0 7 . 2 0 20 1 8 9 . 0 0 2 . 2 0 4 . 7 0 22 1 1 1 0 . 0 0 1 . 8 0 4 . 0 0 TM 3 5 . 0 0 3 4 . 17 0 . 1 7 4 0 0 . 1 7 2 0 0 . 0 3 9 0 0 . 0 4 0 0 1 7 2 . 0 0 1 1 9 . 8 0 8 .5 .0 5 . 3 9 4 . 5 3 0 . 0 9 3 5 0 . 1 0 6 5 0 . 3 4 0 . 0 . 0 . 3 5 6 0 0 . 3 6 9 0 0 . 0 0 7 8 0 . 0 1 1 3 0 . 1 5 9 0 0 . 2 5 0 0 0 . 2 9 8 0 L O G TM V S . L O G Q SSX 4 . 7 0 5 4 S SY 2 . 4 1 7 4 SP - 3 . 3 7 1 6 R E G R E S S I O N E Q U A T I O N Y= - 0 . 0 1 2 9 • ST ANDARD ERROR O F . E ST I M A T E 0 . 0 1 7 9 . C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 7 D E G R E E O F FREEDOM 6 F = 7 5 0 6 . 9 6 0 9 - 0 . 7 1 6 5 X LOG...A V S . L.QJS_Q_ SSX 4 . 7 0 5 4 SSY 0 . 3 7 8 4 SP 1 . 3 3 1 6 R E G R E S S I O N . EQUAT ION . . . Y= - 0 . 1 4 3 7 + STANDARD ERROR OF E S T I M A T E 0 . 0 1 7 9 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 7 9 D E G R E E OF FREEDOM 6 . 0 . 2 83.0 X F = 1 1 8 2 . 4 1 4 3 L O G V V S . L O G Q SSX 4 . 7 0 5 4 S SY 2 . 4 1 6 7 SP 3 . 3 7 1 0 R E G R E S S I O N E Q U A T I O N Y= 0 . 1 4 4 2 + S T A N D A R D ERROR O F E S T I M A T E 0 . 0 1 8 5 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 9 6 0 . 7 1 6 4 X D E G R E E OF FREEDOM F = 7 0 7 2 . 7 9 6 9 LOG TS V S . L O G Q SSX 4 . 7 0 5 4 SSY 2 . 7 5 6 2 SP - 3 . 5 5 7 7 R E G R E S S I O N E Q U A T I O N Y= - 0 . 4 4 2 2 + - 0 . 7 5 6 1 X S T A N D A R D ERROR O F E S T I M A T E 0 . 1 1 5 1 202 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 8 7 9 D E G R E E OF FREEDOM 6 F = 2 0 3 . 1 1 1 7 L O G TP V S . L O G Q SSX 4 . 7 0 5 4 SSY 2 . 4 5 9 6 SP - 3 . 3 9 8 0 R E G R E S S I O N E Q U A T I O N Y= - 0 . 0 7 1 0 + - 0 . 7 2 2 1 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 3 4 2 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 8 8 D E G R E E OF FREEDOM 6 F = 2 1 0 2 . 5 3 1 2 LOG T S S V S . L O G Q SSX 2 . 8 7 0 0 SSY 1 . 5 7 5 0 SP - 2 . 1 2 5 3 R E G R E S S I O N E Q U A T I O N Y= - 0 . 4 5 3 3 + S T A N D A R D ERROR O F E S T I M A T E 0 . 0 1 9 4 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 6 - 0 . 7 4 0 5 X D E G R E E OF FREEDOM 4 F = 4 1 9 8 . 3 0 4 7 LOG T P P V S . L O G Q SSX 2 . 8 7 0 0 SSY 1 . 4 4 7 7 SP - 2 . 0 3 6 4 R E G R E S S I O N E Q U A T I O N Y= - 0 . 0 6 0 6 + S T A N D A R D ERROR OF E S T I M A T E 0 . 0 3 0 5 . C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 0 D E G R E E OF FREEDOM 4 F = 1 5 5 7 . 8 1 5 9 - 0 . 7 0 9 5 X - - - • — - - BROCKTON CK.'.REACH 2-3 Q IN CU M / SEC A MEAN TIME . PEAK TIME + STARTING TIME 204 P L A C I D C R E E K , R E A C H 1-2 20 5 NO ID Q ( L / S ) T S T P TM A V 2 1 5 0 . 3 0 1 2 2 . 0 0 ^ 13 1 3 5 . 4 0 1 6 5 . 0 0 1 6 4 . 0 0 2 1 0 . 0 0 0 . 6 6 0 0 0 . 2 0 8 . 0 0 2 8 0 . 6 0 0 . 6 1 8 0 0 . 0 7 6 3 0 5 7 3 ? 25 1 6 4 . 6 0 1 1 5 . 0 0 28 1 9 5 . 9 0 9 4 . 0 0 1 5 9 . 0 0 1 8 0 . 6 0 0 . 7 2 8 0 0 . 1 3 0 . 0 0 1 4 4 . 2 0 0 . 8 6 3 0 0 . 0 8 9 0 1110 I C G TM V S . L O G Q SSX 0 . 0 9 9 7 SSY 0 . 0 4 4 1 SP - 0 . 0 6 6 0 R E G R E S S I O N E Q U A T I O N Y= S T A N D A R D ERROR OF E S T I M A T E 1 . 4 7 5 8 + 0 . 0 1 4 7 - 0 . 6 6 2 2 X C O R R E L A T I O N C O E F F I C I E N T D E G R E E OF FREEDOM F = - 0 . 9 9 5 1 3 2 0 3 . 3 2 2 6 L O G A V S . LOG Q SSX 0 . 0 9 9 7 SSY 0 . 0 1 1 9 SP 0 . 0 3 3 9 R E G R E S S I O N E Q U A T I O N Y= 0 . 2 7 3 9 + 0 . 3 4 0 3 X STANDARD ERROR OF E S T I M A T E C O R R E L A T I O N C O E F F I C I E N T D E G R E E OF FREEDOM 0 . 0 1 4 1 0 . 9 8 3 3 3 F = 5 8 . 3 3 3 8 L O G V V S . L O G Q SSX 0 . 0 9 9 7 SSY 0 . 0 4 3 7 SP 0 . 0 6 5 6 R E G R E S S I O N E Q U A T I O N Y= STANDARD ERROR OF E S T I M A T E C O R R E L A T I O N C O E F F I C I E N T - 0 . 2 7 4 8 + 0 . 0 1 4 7 0 . 9 9 5 0 0 . 6 5 8 7 X D E G R E E OF FREEDOM F = 3 1 9 9 . 5 1 7 2 LOG T S S V S . L O G Q SSX 0 . 0 9 9 7 SSY 0 . 0 3 0 7 SP - 0 . 0 5 4 1 R E G R E S S I O N E Q U A T I O N •Y= S T A N D A R D ERROR OF E S T I M A T E 1 . 4 1 2 1 + 0 . 0 2 5 7 - 0 . 5 4 2 5 X C O R R E L A T I O N C O E F F I C I E N T D E G R E E OF FREEDOM I F = - 0 . 9 7 8 2 3 4 4 . 3 0 1 1 2 0 6 LOG TPP VS. LOG Q SSX 0.0997 SSY 0.0210 SP -0.0448 > REGRESSION EQUATION Y- 1.6546 + -0.4498 X S T A N D A R D 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 P L A C I D C R E E K , R E A C H 2-3 NO ID Q ( L / S ) TS 209 5 10 1 5 . 4 0 7 2 . 6 0 1 6 0 . 0 0 6 2 . 0 0 TP 2 3 6 . 0 0 9 4 . 0 0 TM 3 5 1 . 3 0 1 2 1 . 4 0 0 ^ 5 3 4 0 0 . 8 6 9 0 V 0 . 0 2 8 9 0 . 0 8 3 4 r 18 1 1 1 4 . 5 0 4 8 . 0 0 7 1 . 0 0 2 4 1 1 2 1 . 2 0 4 7 . 5 0 7 4 . 0 0 2 7 1 1 8 1 . 0 0 3 6 . 1 0 5 4 . 8 0 8 3 . 3 0 0 . 9 2 . 1 0 I. 6 2 . 1 0 1 . 9 3 9 0 0 . 1 0 0 0 0 . 1 1 0 0 0 . 1 2 2 0 1 1 0 0 1 6 3 0 L O G TM V S . L O G 0 SSX 0 . 6 9 5 5 SSY 0 . 3 3 6 3 SP - 0 . 4 8 21 R E G R E S S I O N E Q U A T I O N Y= 1 . 2 9 1 9 + - 0 . 6 9 3 2 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 2 6 5 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 6 9 D E G R E E OF FREEDOM 4 F = 4 7 7 . 4 1 9 4 L O G A V S . L O G Q SSX 0 . 6 9 5 5 SSY 0 . 0 6 7 4 SP 0 . 2 1 3 1 R E G R E S S I O N E Q U A T I O N Y= 0 . 2 8 5 5 S T A N D A R D ERROR O F E S T I M A T E 0 . 0 2 6 6 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 8 4 2 + 0 . 3 0 6 5 X D E G R E E OF FREEDOM 4 F = 9 2 . 4 1 0 2 LOG V V S . L O G Q SSX 0 . 6 9 5 5 SSY 0 . 3 3 5 6 SP 0 . 4 8 1 5 R E G R E S S I O N E Q U A T I O N Y= - 0 ; 2 8 6 8 STANDARD ERROR OF E S T I M A T E 0 . 0 2 7 0 + 0 . 6 9 2 4 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 6 7 D E G R E E OF FREEDOM 4 F = 4 5 6 . 2 5 9 5 L O G T S S V S . L O G Q SSX 0 . 6 9 5 5 SSY 0 . 2 4 9 9 SP - 0 . 4 1 6 8 R E G R E S S I O N E Q U A T I O N Y= 1 . 1 1 7 0 S T A N D A R D ERROR O F E S T I M A T E 0 . 0 0 7 4 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 7 , D E G R E E OF FREEDOM 4 + - Q . 5 9 9 3 X y 210 F = 4 5 8 8 . 8 8 2 8 L O G TPP V S . L O G Q SSX 0 . 6 9 5 5 S SY 0 . 2 3 9 7 S P ~ - 0 . 4 0 7 7 R E G R E S S I O N E Q U A T I O N Y= 1 . 3 1 0 1 + - 0 . 5 8 6 2 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 1 4 6 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 8 7 D E G R E E OF FREEDOM 4 F = 1 1 1 8 . 6 2 1 3 211 PLACID CREEK. REACH .2-3 \ MEAN TIME K PEAK TIME STARTING TIME PLACID CREEK. REACH 2-3 A FLOW AREA + VELOCITY P L A C I D C R E E K , R E A C H 3-4 NO ID Q ( L / S ) T S T P TM A V < 17 1 1 4 . 5 0 6 7 4 . 0 0 18 1 2 4 5 . 0 0 1 8 8 . 0 0 1 0 0 0 . 0 0 1 2 0 0 . 0 0 2 2 3 . 0 0 2 5 5 . 3 0 0 . 5 6 5 0 0 . 2 . 0 3 0 0 0 . 0 2 5 6 1 2 1 0 ? 19 1 2 6 8 . 0 0 1 6 2 . 0 0 21 1 7 0 . 0 0 3 0 2 . 0 0 _ 2 4 1. 2 . 6 7 . 0 0 1 5 7 . 5 0 1 9 9 . 0 0 2 3 8 . 9 0 3 8 6 . 0 0 4 8 1 . 6 0 2 0 0 . 0 0 2 5 4 . 0 0 2 . 0 0 0 0 0 . 1 3 4 0 1 . 0 9 5 0 0 . 0 6 3 8 2 . 2 0 0 0 0 . 1 2 1 0 27 1 4 0 4 . 0 0 1 2 1 . 0 0 1 6 9 . 0 0 2 0 9 . 0 0 2 . 7 4 0 0 0 . 1 4 8 0 L O G TM V S . L O G Q SSX 1 . 4 8 9 4 SSY 0 . 4 2 1 3 SP - 0 . 7 9 0 8 R E G R E S S I O N E Q U A T I O N Y= 2 . 0 9 0 2 + S T A N D A R D ERROR O F E S T I M A T E 0 . 0 1 8 7 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 8 3 - 0 . 5 3 1 0 X D E G R E E OF FREEDOM F = 5 1 2 0 3 . 0 8 0 6 L O G A V S . L O G Q SSX 1 . 4 8 9 4 SSY 0 . 3 2 5 0 SP 0 . 6 9 3 4 R E G R E S S I O N E Q U A T I O N Y= 0 . 5 9 5 8 + S T A N D A R D ERROR OF E S T I M A T E 0 . 0 2 3 2 0 . 4 6 5 6 X C O R R E L A T I O N C O E F F I C I E N T D E G R E E OF FREEDOM F = 0 . 9 9 6 7 5 : -'• 5 9 9 . 9 7 4 9 L O G V V S . L O G Q SSX 1 . 4 8 9 4 SSY 0 . 4 2 9 6 SP 0 . 7 9 8 0 R E G R E S S I O N E Q U A T I O N Y= - 0 . 5 9 4 7 + 0 . 5 3 5 8 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 2 3 0 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 7 5 D E G R E E OF FREEDOM 5 F = 8 1 0 . 3 3 8 9 LOG T S S V S . L O G Q SSX 1 . 4 8 9 4 SSY 0 . 3 6 6 1 SP - 0 . 7 3 4 5 \— R E G R E S S I O N E Q U A T I O N Y= 1 . 9 2 2 7 + S T A N D A R D ERROR OF E S T I M A T E 0 . 0 3 1 2 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 4 7 - 0 . 4 9 3 1 X DEGREE OF FREEDOM F = 5 3 7 2 . 5 4 0 0 2T? LOG T P P V S . L O G Q SSX i . 4 8 9 4 S S Y 0 . 4 2 5 2 SP - 0 . 7 9 3 9 R E G R E S S I O N E Q U A T I O N Y= 2 . 0 0 3 5 + - 0 . 5 3 3 0 X STANDARD ERROR OF E S T I M A T E 0 . 0 2 2 6 . C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 6 D E G R E E OF FREEDOM 5 F = _ 8 2 6 . 4 0 0 ? _ _____ PLOT T A P E S U C C E S S F U L L Y W R I T T E N DONE STOP 0 E X E C U T I O N T E R M I N A T E D $ S I G 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 B L A N E Y C R E E K , R E A C H 1 - 3 217 N O I D Q ( L / S ) T S T P T M A v 1 1 1 5 0 0 . 0 0 3 0 . 0 0 '> 4 4 . 5 0 V 1 2 1 1 7 5 0 . 0 0 1 7 . 0 0 2 3 . 5 0 5 4 . 7 0 2 7 . 4 8 2 . 3 9 0 0 4 . 2 2 0 0 0 . 2 0 9 0 0 . 4 1 6 0 R 1 5 1 1 1 5 0 0 . 0 0 7 . 5 0 1 0 . 6 0 1 6 1 1 6 3 0 . 0 0 1 9 . 5 0 2 6 . 4 0 17 1 1600.00 1 8 . 3 0 2 5 - 5 0 1 2 . 1 0 3 0 . 1 6 2 9 . 6 3 1 2 . 1 5 0 0 4 . 3 2 0 0 4 . 2 5 0 0 0 . 9 4 7 0 0 . 3 8 0 0 0 . 3 8 7 0 1 9 1 1 9 5 0 . 0 0 1 5 . 0 0 2 2 . 5 0 2 4 1 1 2 0 . 0 0 7 1 . 0 0 1 0 1 . 0 0 3 1 1 1 4 6 . 0 0 5 7 . 0 0 9 0 . 0 0 2 5 . 3 7 1 2 3 . 0 0 1 1 7 . 0 0 4 . 3 2 0 0 1 . 2 9 0 0 1 . 4 9 0 0 0 . 4 5 1 0 0 . 0 9 6 0 0 . 0 9 8 0 3 5 1 2 8 5 . 0 0 4 0 . 5 0 6 1 . 0 0 36 1 7 4 1 . 0 0 2 4 . 0 0 3 4 . 0 0 7 1 . 2 0 4 0 . 4 0 1 . 7 7 0 0 2 . 6 2 0 0 0 . 1 5 5 0 0 . 2 8 3 0 L O G T M V S . L O G Q S S X 3 . 2 5 3 7 S S Y 0 . 8 9 9 4 S P - 1 . 7 0 1 8 R E G R E S S I O N E Q U A T I O N Y = 1 . 5 8 5 4 S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 3 4 0 + - 0 . 5 2 3 0 X C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 4 8 D E G R E E O F F R E E D O M 9 F = 7 7 0 . 1 7 7 5 L O G A V S . L O G Q S S X 3 . 2 5 3 7 S S Y . . . 0 . 7 5 3 7 S P 1 . 5 5 6 6 R E G R E S S I O N E Q U A T I O N Y = 0 . 5 2 8 3 + 0 . 4 7 8 4 X S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 3 3 5 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 4 0 D F G R F F O F F R E E D O M 9 F = 6 6 1 . 6 2 2 1 L O G V V S . L O G Q S S X 3 . 2 5 3 7 S S Y 0 . 8 9 4 5 S P 1 . 6 9 8 1 R E G R E S S I O N E Q U A T I O N Y = - 0 . 5 2 7 0 S T A N D A R D E R R O R O F E S T I M A T E 0 . 0 3 2 1 C O R R E L A T I O N . C O E F F I C I E N T 0 . 9 5 5 4 • 0 . 5 2 1 9 X D E G R E E O F F R E E D O M 9 F = 8 5 8 . 6 0 8 4 L O G T S S V S . L O G Q S S X 3 . 2 5 3 7 S S Y 0 . 7 6 4 2 S P . - 1 . 5 6 7 8 218 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 J L 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 C O O . 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 J L 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 S S Y 1 .7186 S P •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 = 6 8 7 . 9 8 0 2 L O G T S V S . LOG Q SSX 7 . 5 6 1 9 SSY 1 . 4 0 0 3 SP - 3 . 0 3 7 5 R E G R E S S I O N E Q U A T I O N Y= 1 . 0 0 4 7 + STANDARD ERROR OF E S T I M A T E 0 . 1 0 6 1 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 3 3 4 - 0 . 4 0 1 7 X D E G R E E OF FREEDOM F = 17 1 0 8 . 3 3 7 8 LOG TP V S . L O G Q SSX 7 . 5 6 1 9 SSY 1 . 5 9 8 0 SP - 3 . 4 3 3 7 R E G R E S S I O N E Q U A T I O N Y= 1 . 2 0 0 3 + STANDARD ERROR O F E S T I M A T E 0 . 0 4 9 3 - 0 . 4 5 4 1 X C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 8 7 8 D E G R E E OF FREEDOM 17 F = . _ 6 4 . 1 . 9 3 1 6 . L O G T S S V S . L O G Q SSX . . 5 . 5 5 . 1 1 S S Y . . 1 . 1 2 7 4 _S.P_ - 2 . 4 8 2 7 R E G R E S S I O N E Q U A T I O N Y= 0 . 9 4 0 7 + - 0 . 4 4 7 2 X STANDARD ERROR OF E S T I M A T E 0 . 0 3 9 4 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 2 4 0EGR.EE CF . . FREEDOM _ 1 . 2 1_ F = 7 1 6 . 6 1 7 7 LOG T P P V S . L O G Q SSX 5 . 5 5 1 1 SSY 1 . 2 1 7 7 SP - 2 . 5 7 4 8 R E G R E S S I O N E Q U A T I O N Y= 1 . 1 8 4 6 + - 0 . 4 6 3 8 X STANDARD ERROR OF E S T I M A T E 0 . 0 4 6 1 C 0 RR.E LA T I O N . C O E F F I C I E N J ±0 . . 9.904 , . D E G R E E OF FREEDOM 12 F = 5 6 1 . 8 8 0 1 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 . B L A N E Y C R E E K , R E A C H 5-4 2 2 5 NO ID Q ( L / S ) TS TP TM 10 12 0 0 5 7 0 . 0 0 1 9 6 0 . 0 0 4 6 . 0 0 2 5 . 0 0 5 7 . 5 0 2 9 . 0 0 6 3 . 10 3 0 . 8 0 2 . 3 2 0 0 3 . 9 0 0 0 0 . 2 4 6 0 0 . 5 0 3 0 13 0 1 1 0 0 0 . 0 0 19 0 2 2 C 0 . 0 0 .21 ... 1 .JULQ.JOXL 23 1 28 1 29 0 1 2 5 . 0 0 2 8 0 . 0 0 1 4 0 . 0 0 1 3 . 5 0 2 4 . 0 0 3 8 . 5 0 9 8 . 0 0 5 6 . O C 9 3 . 80 1 3 . 9 0 2 6 . 0 0 5 1 . 4 0 1 3 7 . 0 0 8 1 . 0 0 1 3 3 . 5 0 1 4 . 4 0 3 0 . 3 0 ...J5iu.6iL 1 6 5 . 0 0 9 3 . 9 0 1 5 8 . 0 0 1 0 . 2 0 0 0 4 . 3 0 0 0 _2....850.Q„ 1 . 3 3 0 0 1 . 7 0 0 0 1 . 4 3 0 0 1 . 0 8 0 0 0 . 5 1 2 0 . 0 . 2 7 4 0 0 . 0 9 4 0 0 . 1 6 5 0 0 . 0 9 8 0 3 7 0 1 3 5 0 . 0 0 2 8 . 5 0 3 2 . 7 0 3 5 . 9 0 3 . 1 3 0 0 0 . 4 3 2 0 L O G TM V S . LOG Q SSX 3 . 1 4 4 4 SSY 0 . 9 9 8 7 SP - 1 . 7 6 2 7 R E G R E S S I O N E Q U A T I O N Y= 1 . 6 8 2 8 + - 0 . 5 6 0 6 X STANOARD ERROR OF E S T I M A T E 0 . 0 3 8 9 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 4 7 D E G R E E OF FREEDOM 8 F = 6 5 4 . 3 9 7 9 L O G A V S . L O G Q SSX 3 . 1 4 4 4 SSY 0 . 6 1 6 2 SP 1 . 3 8 0 2 R E G R E S S I O N E Q U A T I O N Y= 0 . 4 9 2 7 + 0 . 4 3 8 9 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 3 8 6 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 1 5 D E G R E E OF FREEDOM 8 _ F _ = : 4 0 6 . 3 1 . 1 8 . . . LOG V V S . LOG Q . SSX 3_, 1 4 4 4 SSY 1 ...OjOi)7_ . S_P_ 1 . 7 6 4 6 . R E G R E S S I O N E Q U A T I O N Y= - 0 . 4 9 2 2 + 0 . 5 6 1 2 X STANDARD ERROR OF E S T I M A T E 0 . 0 3 8 7 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 4 8 D E G R E E OF FREEDOM JL_ F = 6 6 2 . 0 3 5 4 L O G TS V S . L O G Q SSX 3 . 1 4 4 4 SSY 676478" SP -1.4164 v. ( 869 .0930 2 2 6 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 F R E E D O M ' . 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 P H Y L L I S C R E E K , R E A C H 1-2 NO ID Q < L / S ) T S T P TM 3 6 9 . 0 0 3 8 . 5 0 5 6 . 0 0 6 9 . 9 0 3 1 2 . 0 0 4 3 . 5 0 6 1 . 0 0 7 6 . 4 0 1 . 9 9 0 0 1 . 8 3 0 0 V 0 . 1 8 5 0 0 . 1 7 1 0 1 4 7 0 . 0 0 2 4 0 0 . 0 0 J . 3 9 8 . 0 0 . 9 4 5 . 0 0 9 4 5 . 0 0 8 2 6 . 0 0 2 1 . 2 5 1 6 . 5 0 OSL-.0JCL 2 4 . 0 0 2 2 . 5 0 2 3 . 0 0 2 8 . 5 0 2 2 . 5 0 2 8 . 5 0 3 3 . 0 0 3 3 . 0 0 3 5 . 4 0 3 6 . 1 9 2 7 . 7 9 3 6 . 6 6 3 9 . 8 0 3 8 . 7 5 4 6 . 3 9 4 . 1 0 0 0 5 . 1 5 0 0 .3 . . .9600 2 . 9 0 0 0 2 . 8 3 0 0 2 . 9 6 0 0 0 . 3 5 8 0 0 . 4 6 7 0 . 0 . 3 5 4 0 0 . 3 2 6 0 0 . 3 3 4 0 0 . 2 7 9 0 3 4 8 0 . 0 0 1 4 . 5 0 2 0 . 0 0 2 5 . 8 1 6 . 9 4 0 0 0 . 5 0 2 0 LOG TM V S . L O G Q SSX 0 . 9 3 5 5 S S Y 0 . 2 0 4 4 SP - 0 . 4 3 0 6 R E G R E S S I O N E Q U A T I O N Y= 1 . 6 2 6 2 + S T A N D A R D ERROR O F E S T I M A T E 0 . 0 2 9 6 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 8 4 9 - 0 . 4 6 0 3 X D E G R E E OF FREEDOM F = 8 2 2 5 . 9 6 2 3 LOG A V S . L O G Q SSX 0 . 9 3 5 5 S SY 0 . 2 8 0 2 SP 0 . 5 0 6 4 R E G R E S S I O N E Q U A T I O N Y= 0 . 5 1 3 5 + S T A N D A R D ERROR OF E S T I M A T E 0 . 0 2 9 4 0 . 5 4 1 3 X C O R R E L A T I O N C O E F F I C I E N T 0 . 9 8 9 1 D E G R E E OF FREEDOM 8 F = _ 3 1 7 . 0 2 2 0 . L O G V V S . L O G Q SSX 0...93.55 SSY., Q.. 20.32. S . P . 0 . 4 2 9 5 R E G R E S S I O N E Q U A T I O N Y= - 0 . 5 1 3 3 + 0 . 4 5 9 1 X S T A N D A R D ERROR O F E S T I M A T E 0 . 0 2 9 4 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 8 5 0 D E G R E E OF FREEDOM _ . _ 8 F = 2 2 8 . 4 5 9 7 L O G T S S V S . L O G Q S S X 0 . 9 3 5 5 S SY 0 . 1 9 5 8 SP - 0 . 4 1 9 1 R E G R E S S I O N E Q U A T I O N Y= 1 . 3 7 5 1 + - 0 . 4 4 8 0 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 3 3 9 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 7 9 2 D E G R E E OF FREEDOM 8 F = 1 6 2 . 9 6 2 2 f L O G T P P V S . L O G Q • ) SSX 0 . 9 3 5 5 S SY 0 . 2 0 7 9 SP - 0 . 4 3 8 2 R E G R E S S I O N E Q U A T I O N Y= 1 . 5 2 8 8 + S T A N D A R D ERROR OF E S T I.MAT E 0 . 0 1 9 5 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 3 6 D E G R E E OF F R E E D O M 8 F = 5 3 8 . 0 9 8 9 - 0 . 4 6 8 4 X — — ~ — •. — — - - — - - — — — • — _ „ — PHYLLIS CREEK. REACH 1-2 Q IN CU M / SEC A MEAN T I M E - PERK TIME STARTING TIME 232 PHYLLIS CREEK. REACH 1-2 Q IN CU M / SEC A + FLOW flREft VELOCITY P H Y L L I S CREEK t R E A C H 2-3 NO ID Q ( L / S ) T S T P TM 7 4 8 . 0 0 2 5 . 0 0 3 8 . 0 0 4 3 . 5 2 3 5 2 . 0 0 3 7 . 5 0 5 7 . 0 0 6 4 . 7 6 233 2 . 7 2 0 0 1 . 9 1 0 0 0 . 2 7 8 0 0 . 1 8 4 0 2 2 8 . 0 0 1 5 9 0 . 0 0 .255.Q..JD.GL 2 4 1 5 . 0 0 9 8 5 . 0 0 1 8 8 0 . 0 0 4 3 . 0 0 1 8 . 7 5 1 5 . 0 0 1 4 . 2 5 2 2 . 4 0 1 7 . 0 0 7 0 . 0 0 2 9 . 5 0 . 2 1 . 0 0 2 1 . 2 0 3 3 . 5 0 2 4 . 0 0 8 7 . 6 2 2 8 . 0 5 2 3 , 4 3 2 4 . 1 5 4 0 . 0 0 2 6 . 7 8 1 . 6 7 2 0 3 . 9 3 0 0 5 . . O 0 0 0 . 4 . 8 7 0 0 3 . 3 0 0 0 4 . 2 2 0 0 0 . 1 3 6 2 0 . 4 0 50 . 0 . 5 1 0 0 0 . 4 9 6 0 0 . 2 9 8 0 0 . 4 4 5 0 8 4 0 . 0 0 1 1 9 4 . 0 0 3 6 1 0 . 0 0 2 3 . 5 0 2 0 . 0 0 1 2 . 5 0 3 5 . 5 0 3 1 . 2 0 .17 . . 2 0 . 4 0 . 8 4 3 5 . 4 8 1 8 . 6 5 2 . 9 6 0 0 3 . 5 5 0 0 5 . 6 3 0 0 0 . 2 8 4 0 0 . 3 3 6 0 0 . 6 4 1 0 L O G TM V S . L O G Q SSX . . 1 . 3 5 1 7 JSSAL . 0 . 4 0 0 5 SP — 0 . 7 3 4 0 R E G R E S S I O N E Q U A T I O N Y= 1 . 5 7 9 8 + - 0 . 5 4 3 0 X STANDARD ERROR OF E S T I M A T E 0 . 0 1 4 5 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 6 D E.GR EE_.OF__ERJE£.O.OM 10 F = 1 9 0 2 . 6 9 1 2 LOG A V S . LOG Q SSX 1 . 3 5 1 7 S SY 0 . 2 8 4 4 SP 0 . 6 1 8 7 R E G R E S S I O N E Q U A T I O N Y= 0 . 5 0 5 7 + STANDARD ERROR OF E S T I M A T E 0 . 0 1 1 5 CO R RE L AT I O N . C Q E F F IC I E NT 0 . 9 9 7 9 . . . D E G R E E OF FREEDOM 10 F = 2 1 4 4 . 6 5 5 3 0 . 4 5 7 7 X L O G V V S . L O G Q SSX 1 . 3 5 1 7 S SY 0 . 3 9 8 6 SP 0 . 7 3 2 7 R E G R E S S I O N E Q U A T I O N Y= - 0 . 5 0 5 5 + 0 . 5 4 2 1 X S T A N D A R D .ERRO R O F . E S T IM A T E 0 . 0 1 2 2 C O R R E L A T I O N C O E F F I C I E N T 6 . 9 9 8 3 D E G R E E OF FREEDOM 10 F = 2 6 4 8 . 8 0 0 0 L O G T S V S . LOG Q f" f R E G R E S S I O N E Q U A T I O N Y- 1 . 3 4 8 1 • S T A N D A R D ERROR O F E S T I M A T E 0 . 0 1 3 5 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 1 L D E G R E E OF FREEDOM 10 - 0 . 4 5 4 6 X 2 3 4 1- ^ F = 1 5 2 2 . 2 8 0 0 r L O G TP V S . L O G Q SSX 1 . 3 5 1 7 SSY 0 . 3 4 1 1 SP - 0 . 6 7 6 4 f" R E G R E S S I O N E Q U A T I O N Y= 1 . 5 2 5 7 • S T A N D A R D ERROR O F E S T I M A T E 0 . 0 1 7 0 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 6 2 - 0 . 5 0 0 4 X ;•" D E G R E E OF FREEDOM 10 F = 1 1 6 6 . 0 4 8 6 f- L O G T S S V S . L O G Q ( SSX 1 . 0 2 9 4 SSY 0 . 2 1 8 5 SP • 0 . 4 7 2 9 ( R E G R E S S I O N E Q U A T I O N Y= 1 . 3 4 5 7 + S T A N D A R D ERROR OF E S T I M A T E 0 . 0 1 4 8 - 0 . 4 5 9 3 X r C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 0 D E G R E E OF FREEDOM 7 F = 9 8 5 . 6 1 9 6 r LOG T P P V S . L O G Q (".. SSX 1 . 0 2 9 4 _ _ S S Y 0 . 2 6 1 5 . _ SP - 0 , 5 1 8 6 R E G R E S S I O N E Q U A T I O N Y= 1 . 5 2 1 9 + - 0 . 5 0 3 7 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 0 7 1 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 4 D E G R E E OF FREEDOM 7 F = 5 1 3 9 . 0 6 6 4 . 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 P H Y L L I S C R E E K , R E A C H 3-4 237 NO ID Q ( L / S ) TS T P TM V 3 5 0 1 7 5 0 . 0 0 3 3 9 . 0 0 1 8 . 0 0 3 2 . 0 0 3 7 . 0 0 4 5 . 0 0 3 9 . 5 6 5 2 . 4 6 2 . 8 8 0 0 1 . 7 4 0 0 0 . 2 6 0 0 0 . 1 9 5 0 6 10 .13 . 19 27 28 0 0 1 1 3 3 8 . 0 0 2 2 8 . 0 0 24.9il.00.. 1 0 7 0 . 0 0 1 2 0 0 . 0 0 3 1 0 0 . 0 0 3 9 . 5 0 5 3 . 0 0 1 1 . 5 0 1 9 . 0 0 1 7 . 0 0 1 0 . 3 0 4 7 . 0 0 6 4 . 0 0 -1J&-.JB.0. 2 6 . 5 0 2 4 . 5 0 1 4 . 5 0 5 3 . 3 4 6 4 . 7 0 . 1 8 . 8 3 2 9 . 8 8 2 8 . 5 2 1 7 . 3 3 1 . 7 5 0 0 1 . 4 8 7 0 J t . 5 6 0 0 . 3 . 1 2 0 0 3 . 3 3 0 0 5 . 2 2 0 0 0 . 1 9 3 0 0 . 1 6 0 8 0 . 5 4 6 Q . 6 . 3 4 4 0 0 . 3 6 1 0 0 . 5 9 5 0 29 1 3 6 9 0 . 0 0 8 . 9 0 1 3 . 5 0 1 5 . 4 1 5 . 5 3 0 0 0 . 6 5 5 0 L O G TM V S . LOG Q SSX 1 . 5 9 5 2 SSY 0 . 4 2 5 0 SP - 0 . 8 2 1 1 R E G R E S S I O N E Q U A T I O N Y= 1 . 4 9 0 3 + STANDARD ERROR OF E S T I M A T E 0 . 0 1 8 4 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 7 2 - 0 . 5 1 4 7 X D E G R E E O F FREEDOM F = 8 1 2 4 8 . 8 0 6 2 LOG A V S . LOG Q SSX 1 . 5 9 5 2 SSY 0 . 3 6 7 6 SP 0 . 7 6 3 6 R E G R E S S I O N E Q U A T I O N Y= 0 . 4 8 0 1 + S T A N D A R D ERROR OF E S T I M A T E 0 . 0 1 7 3 0 . 4 7 8 7 X C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 7 1 D E G R E E OF FREEDOM 8 F..= 1 2 1 9 . 0 8 4 5 L O G V V S . LOG Q SSX . . 1 , 5 9 5 2 „__S5Y_ . 0 , 4 1 8 3 . SP 0 . 8 1 4 6 R E G R E S S I O N E Q U A T I O N Y= - 0 . 4 7 8 6 + 0 . 5 1 0 6 X S T A N D A R D ERROR OF E S T I M A T E 0 . 0 1 8 2 C O R R E L A T I O N C O E F F I C I E N T 0 . 9 9 7 2 D E G R EE OF F R E E DOM 81 F = 1 2 6 0 . 8 4 5 2 L O G TS V S . LOG Q SSX 1 . 5 9 5 2 SSY 0 . 5 7 9 2 SP - 0 . 9 4 5 2 238 r R E G R E S S I O N E Q U A T I O N Y= 1.2809 + -0.5925 X S T A N D A R D ERROR OF E S T I M A T E 0.0523 C O R R E L A T I O N C O E F F I C I E N T -0.9833 D E G R E E OF FREEDOM 8 F = 204.5374 > : ; ; : LOG TP V S . LOG Q SSX 1.5952 SSY 0.4718 SP -0. 8616 R E G R E S S I O N E Q U A T I O N Y= STANDARD ERROR OF E S T I M A T E 1.4390 + 0.0303 -0.5401 X C O R R E L A T I O N C O E F F I C I E N T D E G R E E OF FREEDOM F = -0.9932 8 506.9019 LOG TSS V S . L O G Q SSX 0.7485 SSY 0.2131 SP -0. 3982 R E G R E S S I O N E Q U A T I O N Y= S T A N D A R D ERROR OF E S T I M A T E C O R R E L A T I O N C O E F F I C I E N T D E G R E E OF FREEDOM 0 1.2698 + 0.0173 .9972 5 .-0..532O. _x_ : ~ - - F = 706 .3044 L O G T P P V S . L O G Q SSX 0.7485 SSY 0.1959 SP -o. 3825 R E G R E S S I O N E Q U A T I O N Y= STANDARD ERROR OF E S T I M A T E C O R R E L A T I O N C O E F F I C I E N T 0 "1.4235 + 0.0113 .9987 -0.5109 ~x D E G R E E CF FREEDOM F = 1521 5 .2705 P L G T _ J AP_E_SUCCESS .F UL L Y WR I T T E N _ DONE S T O P 0 E X E C U T I O N T E R M I N A T E D PHYLLIS CREEK. REACH 3-4 A + MEAN TIME PERK TIME STARTING TIME PHYLLIS CREEK. REACH 3-4 PHYLLIS CREEK, REACH 4-6 2 i | 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 . i s . ...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 S T A N D A R D ERROR O F E S T I M A T E 0 . 0 8 3 1 C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 6 6 6 D E G R E E OF FREEDOM 7 F = 8 5 . 3 6 3 4 242 LOG TP V S . L O G Q SSX 1 . 4 6 3 3 SSY 0 . 5 4 4 7 SP - 0 . 8 8 9 8 R E G R E S S I O N E Q U A T I O N Y= 1 . 1 6 7 2 + S T A N D A R D ERROR O F E S T I M A T E 0 . 0 2 4 6 C O R R E L A T I O N _ C O E F F I C I E N T - 0 . 9 9 6 7 _ _ D E G R E E OF FREEDOM 7 F = 8 9 3 . 3 0 1 3 - 0 . 6 0 8 1 X LOG T S S V S . L O G Q SSX 0 . 3 3 2 6 SSY 0 . 1 1 0 8 SP - 0 . 1 9 1 9 R E G R E S S I O N E Q U A T I O N Y= 0 . 9 2 5 0 + S T A N D A R D E R R O R _ O F E S T I M A T E ^ . _ 0 . 0 0 2 8 _ C O R R E L A T I O N C O E F F I C I E N T - 1 . 0 0 0 0 D E G R E E O F FREEDOM 2 F = 1 4 6 1 3 . 2 9 6 9 - 0 . 5 7 7 1 X L O G . T P P V S . L O G . S SX 0 . 3 3 2 6 SSY 0 . 1 2 3 0 SP - 0 . 2 0 2 0 R E G R E S S 10N_ E.QUATION Y=. S T A N D A R D ERROR O F E S T I M A T E C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 1 D E G R E E OF FREEOOM 2 1 . 1 4 4 7 + 0 . 0 1 5 2 - 0 . 6 0 7 6 X F = 5 2 9 . 8 9 0 9 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 P H Y L I S LOWER 245 NO ID Q CL/S) TS TP TM 10 20 3 6 5 . 0 0 2 6 9 . 0 0 4.58 4.75 1 0 . 3 3 1 2 . 0 0 13 . 7 9 1 8 . 1 8 2 . 3 3 0 0 2 . 3 1 9 0 0.1565 0 . 1 1 5 9 30 40 4 3 5 8 . 0 0 3 4 1 5 . 0 0 1.56 1.75 2.55 3.00 2.78 3.60 5 . 6 1 4 0 5 . 6 9 1 0 0.7760 0 . 6 0 0 0 LOG TM V S . LOG Q SSX 1.2032 SSY 0 . 5 0 2 7 SP - 0 . 7 7 6 5 REGRESSION EQUATION Y= 0.8764 + - 0 . 6 4 5 4 X STANDARD ERROR OF ESTIMATE 0.0280 CORRELATION C O E F F I C I E N T - 0 . 9 9 8 4 DEGREE OF FREEDOM 3 F = 6 3 9 . 7 9 6 6 LOG A VS. LOG SSX 1.203 2 SSY 0 . 1 4 8 9 S_P Q . 4 2 0 6 REGRESSION EQUATION Y= 0 . 5 4 4 9 + 0 . 3 4 9 6 X STANDARD ERROR OF ESTIMATE 0 . 0 3 1 0 CORRELATION C O E F F I C I E N T 0 . 9 9 3 5 DEGREE^OF FREEJDOM„ 3__I_ F = 1 5 2 . 6 9 7 8 LOG V VS. LOG Q 'SS X " " 1 . 2 0 3 2 SSY 0 . 5 1 1 4 S P 0 . 7 8 2 9 REGRESSION EQUATION Y= - 0 . 5 4 5 1 + 0 . 6 5 0 7 X STANDARD ERROR OF ESTIMATE 0.0309 CORR E L AT I ON „C0 EFF I C I ENT 0 . 9 9 8 1 __„._ . DEGREE OF FREEDOM 3 F = 5 3 2 . 2 0 3 9 LOG TSS VS. LOG Q SSX 1.2032 SSY 0.2045 SP - 0 . 4 9 5 1 REGRESSION EQUATION Y= 0 . 4 6 0 4 + - 0 . 4 1 1 5 X STANDARD ERROR 0F_ ESTIMATE 0.0197 _ CORRELATION "COEFFICIENT - 0 . 9 9 8 1 DEGREE OF FREEDOM 3 F = 5 2 5 . 7 3 4 6 -2-46- L G G T P P V S . L O G Q SSX 1 . 2 0 3 2 S S Y 0 . 3 7 0 4 SP - 0 . 6 6 7 5 > R E G R E S S I O N E Q U A T I O N Y= 0 . 7 6 7 1 + - 0 . 5 5 4 8 X STANDARD ERROR O F E S T I M A T E 0 . 0 0 7 3 __• C O R R E L A T I O N C O E F F I C I E N T - 0 . 9 9 9 9 D E G R E E OF FREEDOM 3 F = 6 9 2 7 . 8 6 7 2 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 S i : . ' •." C = ( QO ** (l.-BA)) / (AA*BA )••',• P ' .= SQRTU240.* C) / ((BE * ' W ) * * 2 M 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 IN IT IAL DATA WRITE ( 6 , 3) T IT 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 ( * IN IT IAL CONDIT IONS ' / l » IN IT IAL 0 (QOI , ALPHA ( A L ) , BETA (BE) = 3 F 1 0 . 3 / 2' START OF TIME COUNT ( I T O ) , T S , T E , DT = », 16 , 3 F 8 . 1 / 3». WIDTH ( W) .. , LENGTH (i. ) , AA, BA 4 F 1 0 . 3 / 4« IN IT IAL AREA ( A O ) , CELER ITY (C) 2 F 1 0 . 3 // 5« COEF1 , C O E F 2 , P 3 E 1 4 . 5 ) WRITE ( 6 , 5 ) 5 FORMAT ( 1 0 TIM E 1ST EXPO. 1ST INTEG . 1ST TERM 2ND E X P U c c 12ND I NT E• . 2ND TERM A(T ) Q (T ) '/) DO LOOP FOR N VALUES OF A IT ) DO 6 I = I, N T = TS+ ( 1 - 1 ) * DT C c F IRST 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 A l l =. _FGAU16(LS , L E , AUXl ) A I NT 1 = AINT1 + A l l CONTINUE FT1 = C0EF1 * A lNT 1 / (T * * 0 . 2 5 ) C C SECOND INTEGRAL OVER T A l NT 2. = 0 . 0 '.; TTE = 0 . 0 TO IFF = T / 2 0 . 0 DO 8 J = 1 , 2 0 C C TT S = TTE TTE = TTS + TDIFF K _ = 1 AT ? = FGAU16( T T S , AINT2 = AINT2 + AI2 CONTINUE TTE AUX2 ) FT2 = C0EF2 * AINT2 A = FT 1 + FT2 0 . = ( A. / A A ) * * U . O /BA) . ;._ WRITE RESULTS WRITE ( 6 , 9 ) T , E X 1 , A INT 1 , FT 1 , EX2 , A I N T 2 , FT2 , A 9 FORMAT ( IX , F 4 . 0 , 8 E 1 2 . 4 ) 6 CONTINUE STOP 251 FUNCTION • AUX l { X ) C C FUNCTION CALLED BY THE LIBRARY PROGRAM FGAU16 C REAL L COMMON L» P» BE , H,. C , T , E X 1 , E X 2 , K . EX1 = P. *SQRT_( ( L -X ) *T ) + ( X -L - ( 6 0 . * ..C. * T) ) / . ( B E * W ) AUX1= EXP (EX1) /(( L-X}*# 0.25) K = 0 RETURN •-' •- E N D FUNCTION AUX2 { Z) c C FUNCTION CALLED BY THE LIBRARY PROGRAM FGAU16 r _ _ - L COMMON L, P T BE, W, C , T, E X 1 , EX 2 ,K E X 2 = P * SORT ( L * ( T - I) ) + ( 6 0 . * C * Z - 60.*C*T - L ) I, AU X 2 = EXP ( E X ? ) / ( ( T - Z ) * * 0 . 7 5 ) ^ BE*W) K = 0 RE TURN END •p. D: SAMPLE OUTPUT P H Y L L I S C R E E K , J U N E 2 2 , 1 9 6 8 . D O W N S U R G E I N I T I A L C O N D I T I O N S I N I T I A L Q I Q O ) , A L P H A I A D , 8 E T A ( B E ) = S T A R T O F T I M E C O U N T U T O » , T S , T E , O T = W I D T H ( W ) , L E N G T H ( L I , A A . B A I N I T I A L * A R E A ( A O ) , C E L E R I T Y t C ) ; 0 . 8 1 5 - 0 . 0 6 5 0 . 5 4 0 1 3 0 0 1 2 . 0 6 0 . 0 2 . 0 1 1 . 5 0 0 7 7 7 . 0 0 0 3 . 2 6 0 2 . 9 1 8 0 . 5 1 6 0 . 5 4 1 C 0 E F 1 , C 0 E F 2 , P 0 . 1 4 0 0 5 E 0 0 0 . 3 8 4 8 4 E 0 1 0 . 1 7 9 1 9 F 0 1 T l M F 1 S T E X P O . 1 S T I N T E G . 1 S T T E R M 2 N D E X P O . 2 N D I N T E . 2 N D T E R M A m Q ( T ) 1 2 . - 0 . 3 4 6 1 E 0 2 C . 3 8 7 4 E 0 2 0 . 2 9 1 5 E 0 1 - 0 . 9 8 1 3 E 0 2 0 . 4 5 3 8 E - 0 6 0 . 1 7 4 6 E - 0 5 0 . 2 9 1 5 E 0 1 0 . 8 1 3 4 E 0 0 1 4 . - 0 . 4 2 2 9 E 0 2 0 . 4 0 2 7 F 0 2 0 . 2 9 1 5 E 0 1 - 0 . 9 6 1 1 E 0 2 0 . 2 5 7 2 E - 0 4 0 . 9 8 9 9 E - 0 4 0 . 2 9 1 6 E 0 1 0 . 8 1 3 6 E 0 0 1 6 . - 0 . 5 0 1 2 E 0 2 0 . 4 1 6 1 E 0 2 0 . 2 9 1 4 E 0 1 - 0 . 9 4 2 5 E 0 2 0 . 5 7 2 7 E - 0 3 0 . 2 2 0 4 E - 0 2 0 . 2 9 1 6 E 0 1 0 . 8 1 3 7 E 0 0 1 8 . _-_0...5m9.E_ _ Q 2 L _ 0 . 4 . 2 . 5 . 3 E _ Q 2 _ —.0*23320. _0_1_ - 0 . 9 2 5 1 E _ 0 2 _ C . 6 0 Q 9 E - 0 ? 0 _ . _ 2 3 . 1 2 E r 0 1 0 . 2 9 1 5 E 0 1 0 . 8 1 3 1 E 0 0 2 0 . - 0 . 6 6 1 6 E 0 2 0 . 4 1 9 2 E 0 2 0 . 2 7 7 6 E 0 1 - 0 . 9 0 8 9 E 0 2 0 . 3 4 C 8 E - 0 1 0 . I 3 1 2 E 0 0 0 . 2 9 0 8 E 0 1 0 . 8 0 9 5 E 0 0 2 2 . - 0 . 7 4 3 3 E 0 2 0 . 3 7 6 8 E 0 2 0 . 2 4 3 7 F 0 1 - 0 . 8 9 3 6 E 0 2 0 . 1 1 6 8 E 0 0 0 . 4 4 9 6 E 0 0 0 . 2 8 8 7 E 0 1 0 . 7 9 8 7 E 0 0 2 4 . . - 0 . 8 2 5 7 E 0 2 0 . 2 8 8 1 E . 0 2 0 . 1 8 2 3 E 0 1 - 0 . 8 7 9 1 F 0 2 C . 2 6 6 4 E 0 0 0 . 1 0 2 5 E 0 1 0 . 2 8 4 8 E 0 1 0 . 7 7 9 2 E 0 0 2 6 . - 0 . 9 0 8 9 E 0 2 0 . 1 7 7 5 F 0 2 0 . I 1 0 1 E 0 1 - 0 . 8 6 5 3 E 0 2 0 . 4 4 2 3 E 0 0 0 . 1 7 0 2 E 0 1 0 . 2 8 0 3 E 0 1 0 . 7 5 6 5 E 0 0 2 8 . - 0 . 9 9 2 7 E 0 2 0 . 8 5 7 0 F 0 1 0 . 5 2 1 8 E 0 0 - 0 . 8 5 2 1 E 0 2 0 . 5 8 3 3 E 0 0 0 . 2 2 4 5 E 0 1 0 . 2 7 6 7 E 0 1 0 . 7 3 8 4 E 0 0 3 0 . - 0 . 1 0 7 7 E 0 3 0 . 3 2 2 2 E 0 1 0 . 1 9 2 8 E 0 0 - 0 . 8 3 9 5 E 0 2 0 . 6 6 3 4 E 0 0 0 . 2 5 5 3 E 0 1 0 . 2 7 4 6 E 0 1 0 . 7 2 8 3 E 0 0 3 2 . - 0 . 1 1 6 2 E 0 3 0 . 9 4 7 4 E 0 0 0 . 5 5 7 9 E - 0 1 - 0 . 8 2 7 4 E 0 2 0 . 6 9 6 8 E 0 0 0 . 2 6 8 1 E 0 1 0 . 2 7 3 7 E 0 1 0 . 7 2 4 0 E 0 0 3 4 . - 0 . 1 2 4 7 E 0 3 0 . 2 2 0 2 E 0 0 0 . 1 2 7 7 E - 0 1 - 0 . 8 1 5 7 E 0 2 C . 7 0 7 3 E 0 0 0 . 2 7 2 2 E 0 1 0 . 2 7 3 5 E oi 0 . 7 2 2 8 E 0 0 3 6 . - 0 . 1 3 3 3 E 0 3 0 . 4 0 9 8 E - 0 1 0 . 2 3 4 3 E - 0 2 - 0 . 8 0 4 5 E 0 2 0 . 7 0 9 8 E 0 0 0 . 2 7 3 2 E 0 1 0 . 2 7 3 4 E 0 1 0 . 7 2 2 4 E 0 0 3 8 . - 0 . 1 4 1 9 E 0 3 0 . 6 1 8 7 E - 0 2 0 . 3 4 9 0 E - 0 3 - 0 . 7 9 3 6 E 0 2 0 . 7 1 0 3 E 0 0 0 . 2 7 3 3 E 0 1 0 . 2 7 3 4 E 0 1 0 . 7 2 2 4 E 0 0 4 0 . - 0 . 1 5 0 5 E 0 3 0 . 7 6 7 7 E - 0 3 0 . 4 2 7 5 E - 0 4 - 0 . 7 8 3 1 E 0 2 0 . 7 1 0 3 E 0 0 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 7 2 2 4 E 0 0 4 2 . - 0 . 1 5 9 2 E 0 3 0 . 7 9 2 6 E - 0 4 Q . 4 3 6 0 E - 0 3 - 0 . 7 7 . 2 9 E _ 0 . 2 _ _ _ Q . . J _ L O A E _ _ . 0 . Q _ 0_ .2_7_i _ t E _ 0 1 _ 0 _ ^ 2 J _ 3 4 E _ 0 1 0 . 7 2 2 3 E 0 0 4 4 . - 0 . 1 6 7 9 E 0 3 0 . 6 8 8 5 E - 0 5 0 . 3 7 4 4 E - 0 6 - 0 . 7 6 3 1 E 0 2 0 . 7 1 0 4 E 0 0 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 7 2 2 3 E 0 0 4 6 . - 0 . 1 7 6 7 E 0 3 0 . 5 0 8 6 E - 0 6 0 . 2 7 3 5 E - 0 7 - 0 . 7 5 3 4 E 0 2 0 . 7 1 0 4 E 0 0 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 7 2 2 3 E 0 0 4 8 . - 0 . 1 8 5 4 E 0 3 0 . 3 2 2 4 E - C 7 0 . 1 7 1 6 E - 0 8 - 0 . 7 4 4 1 E 0 2 0 . 7 1 0 4 E 0 0 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 7 2 2 4 E 0 0 5 0 . - 0 5 2 . - 0 5 4 . - 0 5 6 . - 0 5 8 . - 0 . 1 9 4 2 F 0 3 . 2 0 3 0 E 0 3 . . . 2 . L 1 2 . E _ 0 3 _ . 2 2 0 7 E 0 3 • 2 2 9 6 E 0 3 0 . 0 . J Q . 0 . 0 . 1 7 7 0 E - 0 8 8 4 8 6 F - 1 0 3 5 7 6 F - 1 1 1 3 3 4 E - 1 2 4 4 3 5 E - 1 4 0 . 9 3 2 5 E - 1 0 0 . 4 4 2 6 E - 1 1 J 3 _ . . 1 _ 8 A 8 . E _ _ L L 2 _ 0 . 6 8 3 2 E - 1 4 0 . 2 2 5 1 E - 1 5 - 0 . 7 3 5 1 E 0 2 - 0 . 7 2 6 2 S 0 2 _ i 0 _ . . J _ l _ 7 6 E _ Q 2 . - 0 . 7 0 9 2 E 0 2 - 0 . 7 0 1 0 E 0 2 0 . 7 1 0 4 E 0 . 7 1 0 4 E JD .7_L04E_ 0 . 7 1 0 4 E 0 . 7 1 0 4 E 0 0 0 . 0 0 0 . J D _ 0 Q _ _ . 0 0 0 . 0 0 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 0 . 2 7 3 4 E 0 1 2 7 3 4 E 0 1 2 7 3 4 _ E _ _ Q J L . 2 7 3 4 6 0 1 2 7 3 4 E 0 1 0 . 7 2 2 4 E 0 0 0 . 7 2 2 4 E 0 0 J L . _ I 2 2 _ t _ E . _ 0 . Q . . . 0 . 7 2 2 4 E 0 0 0 . 7 2 2 3 E 0 0 ro ro TSN L R * 2 5 3 C PROGRAM FOR FLOOD ROUTING THROUGH SEQUENCES OF NONLINEAR C RESERVOIRS AND KINEMATIC CHANNELS, WRITTEN I N FORTRAN / 3 6 0 . _C C 1 CONTROL CARD PER CHANNEL ( C O N S I S T I N G OF SEVERAL R E A C H E S ) : C NO OF REACHES, KR, ( 1 2 ) fi T I T L E OF CHANNEL, T I T R , ( 1 0 A 4 ) . C C 3 CONTROL CARDS PER REACH; C _C NO 1 CONTAINS: C NO OF RESERVOIRS, N, ( 1 2 ) ; fi FACTOR ALPHA , AL, ( F 6 . 0 ) ; C LOCAL INFLOW ALONG REACH, QINC, ( F 6 . 0 ) . C _ _ •. c NO 2 CONTAINS: Parameter "Sigma" of text C T I T L E FOR REACH, T I T , ( 10AA) . . u A 1 . II . ! _ . . „ D i f is Alpha in SNLR . __ NO 3 CONTAINS: C LENGTH OF REACH, L, ( F 6 . 0 ) ; STEADY FLOW PARAMETERS, AA AND C BA, ( F 6 . 0 ) ; FORMATIVE DISCHARGE, QD, ( F 6 . 0 ) „ _ C RATING CURVE PARAMETERS, AH1, BH1, HOI, AH2, C BH2, H02, ( F 6 . 0 ) ; STARTING T I M E , T S , ( F 6 . 0 ) ; TIME INT. _C OF H-DATA, P E L T , ( F 6 . 0 ) ; 2X ; NO OF I N I T I A L H-DATA, I N , C ( 1 2 ) ; NO OF INTERVALS TO BE COMPUTED, K, ( F 6 . 0 ) . C R C DATA CARDS: C C I N I T I A L H-DATA , ( 1 2 F 6 . 0 ) _C C EXPLANATIONS: C C ALPHA = L ( R E S E R V O I R ) / 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 I S THE STARTING TIME IN MINUTES OF THE INPUT DATA. C DATA : THE PROGRAM READS » I N ' H-DATA, AND ASSUMES THAT C ALL FURTHER ( K - I N ) DATA POINTS ARE_EQUAL_TO.THE LAST C INPUT VALUE C C OUTPUT: C C THE PROGRAM PRINTS THE CONTROL CARD DATA AND I N I T I A L C H-DATA; IT THEN ROUTES THE FLOW THROUGH SUCC E S S I V E REACHFS, C WITH THE OUTPUT OF REACH ( 1 - 1 ) + QINC BECOMING INPUT OF C REACH ( I ) . C . . •• • C THE OUTFLOW OF EACH REACH I S PRINTED, AND, I F THE RATING C CURVE PARAMETERS ARE G I V E N , THE GAUGE READINGS. DIMENSION Q ( 1 0 0 ) , Y ( 2 ) , F ( 2 ) , T E M P ( 2 ) , H ( I O O ) , T ( 1 0 0 ) , Q 1 ( 1 0 0 ) , 1 T I C 100) , T I T ( I O ) , TITR ( 1 0 ) COMMON FQ, QIM1, BA REAL L 2 5 4 C L O O P FOR R E A C H E S R E A D ( 5 , 4 0 ) K R , T I T R 4 0 FORMAT ( 1 2 , 1 0 A 4 ) DO 41 KK = I ,KR C C R E A D C O N S T A N T S READ ( 5 , 3 0 ) . ..N.. , AL ,_.Q.I.NC_ _ . _ 30 FORMAT ( I 2 , 2 F 6 . 0 ) R E A D ( 5 , 2 0 ) T I T 20 FORMAT ( 1 0 A 4 ) ; ; (H02, R E A D (5 , 1) L , AA , B A , QD , AH1 , B H 1 , H O I , A H 2 , BH? ,) 1 TS , D E L T , IN , K 1 F O R M A T ! 1 2 F 6 . 0 , 2 X , 2 12 .) ... . .._ :.. KL= K K= K+1 I F ( KK . G T . 1 ) GO TO 4 3 C C R E A D I N I T I A L H I G H T DATA READ ( 5 , 2 ) ( H ( I ) , 1= . . 1 , IN ) 2 F O R M A T ( 1 2 F 6 . 0 ) I F ( I N . E Q . K ) GO TO 4 C ; C C O M P L E T E I N I T I A L ARRAY DO 3 I = I N , K 3 H( I ) = H( IN)... ._ _ . C C C O N V E R T H I G H T TO D I S C H A R G E , C O M P U T E T I M E S 4 DO 5 I = 1 , K T( I ) = T S + ( I - l ) * D E L T Q ( I )•=( (H( I ) -H01 ) / A H l ) * * ( 1 . / B H D 5 C O N T I N U E ..... .. ._ _ i GO TO 4 5 C C A D J U S T FOR KK . G T . 1 AND K L A . N E . K 4 3 IN = K IF < K L A . E Q . K ) GO TO 45 DO 90 NI . = K L A , . K . 9 0 Q ( N 1 ) = Q ( K L A ) C C I N I T I A L C O N S T A N T S 4 5 AD = A A * QD * * B A F T T = ( L * A D * BA ) / ( N * Q D * 6 0 . 0 ) F T = F T T . * j .1 . 0 . - „ AL .) _ FQ = ( 1 . - AL )i / ( F T * AL ) T C = ( F T T * AL ) / ( Q ( l ) * * ( 1 . 0 - B A ) ) Q F A C = Q ( l ) / QD IDT = 2 . 0 * D E L T / TC I F ( IDT . L T . 1 ) IDT = 1 I F ( O F A C . G T . 0 . 1 5 ) . I O T = 2 * I D T _ C C WR ITE I N I T I A L C O N D I T I O N S WR ITE ( 6 , 6 ) T I T R , KK 6 FORMAT < * 1NON-L I NEAR R E S E R V O I R R O U T I N G ' / ' I N I T I A L C O N D I T I O N S ' 1 / 1 0 A 4 , ' R O U T I N G OVER ' , 1 2 , ' . R E A C H •) WR ITE ( 6 , 2 1 ) T I T _ 255 21 F O R M A T ( 1 H 0 , 1 0 A 4 ) WR ITE ( 6 , 31> N • AL , Q I N C 31 FORMAT C O N = NO OF R E S . = • , 1 5 , • A L ( P H A ) = L ( R E S ) / L = \ 1 F 1 0 . 5 / ' Q I N C R E M E N T « ' , F 1 0 . 5 ) I T S = TS WR ITE ( 6 , 7 ) K L , I N , A A , B A , D E L T , I D T , I T S * t » Q D , 1 A H 1 , B H 1 , H 0 1 , A H 2 , B H 2 , H 0 2 , A O , F Q , F T 7 FORMAT P O K , IN • , 2 1 7 , / • AA , BA • , 2 F 1 2 . 5 , / 1 * DT OF QO IN MIN , N O . S T E P S PER DT • F 1 0 . 5 , 18 / 2 ' S T A R T I N G T I M E , L E N G T H (M) , QO ' , 3 X , I 4 , 2 F 1 2 . 3 / 3 ' A H 1 , B H 1 , H 0 1 , « , 3 F 1 5 . 6 / 4 « A H 2 , B H 2 , H 0 2 , « , 3 F 1 5 . 6 / / 5 ' A D , F Q , F T , • , 5 X , 3 E 1 5 . 6 / / ) I F ( KK . G T . 1 ) GO TO 6 5 W R I T E ( 6 , 8 ) ( I , T ( I ) , H ( I ) , Q ( I ) , I = 1, IN ) 8 FORMAT ( ' N O T I M E L E V E L D I S C H A R G E ' / / 1 < I X , 1 2 , I X , F 7 . 1 , 2 X , F 7 . 3 , 2 X , F 1 0 . 5 >) C C C O N V E R T Q TO NONDI MENS IONAL Q 65 C O N T I N U E DO 4 6 1= 1 , K 46 Q l l ) = ( Q ( I ) + Q I N C ) / QD C C A D V A N C E S O L U T I O N BY 1 R E S E R V O I R DO 9 I = 1 , N Y ( l ) = T ( l ) + FT / ( Q ( 1 ) * * ( l . - B A J ) Y ( 2 ) = Q ( l ) Q l ( l ) = Y ( 2 ) T K l l = Y d ) A I N = IDT D = D E L T / A I N OO 10 J= 2 , K C C C O M P U T E Q ( I - l ) : DT = F T / ( ( ( Q ( J - l ) + Q l ( J - l ) ) / 2 . ) ^ * ( 1 . - B A ) ) T O = T K J - l ) + D E L T / 2 . - DT DO 82 M = 1 , K I F ( T ( M ) . G T . TO ) GO TO 85 82 C O N T I N U E 85 I F (M . E Q . l ) GO T 0 B 3 Q IM1 = Q ( M - i ) + ( ( Q ( M ) - Q ( M - 1 ) ) / D E L T ) # ( T 0 - T ( M - l ) ) GO TO 8 9 83 QIM1 = Q ( 1 ) • _ _ _ _ _ _ 89 C A L L RK ( Y , F , T E M P , D , 2 , I D T ) 12 FORMAT ( 6 E 1 4 . 4 , 2 1 5 ) Q K J ) = Y ( 2 ) T1 I J I = Y d ) 10 C O N T I N U E ! DO 13 I I = 1,K ." _ . Q ( I I ) = Q 1 ( 1 1 ) 13 T ( I I ) = T K I I ) 9 C O N T I N U E ' C C P R I N T OUT R E S U L T S A F T E R 1 R E A C H DO 50 I = 1 ,K . _ _ 256 Q( I ) = Q( I ) * QD I F { BH2 . L F . 0 . 0 ) GO TO 50 H U ) = AH2 * ( Q ( I ) * * B H 2 )+ H02 50 C O N T I N U E WR ITE ( 6 , 1 1 ) N 11 F O R M A T ( • O D I S C H A R G E A F T E R « , I 2 , » R E S E R V O I R S * //) I F (. BH2 . L E . _Q. .Q ) WR ITE . . ( . .6 ,61 ...I (_J_ ,. T ( I ) , . Q( I ) , 1 = l,K) 61 FORMAT ( » NO T I M E L E V E L D I S C H A R G E « / / 1 ( I X , I 2 , 1 X , F 7 . W 2 X , 7 X , 2 X , F 1 0 . 5 ) ) I F ( BH2 . G T . 0 . 0 . WR ITE ( 6 , 8 ) ( I, T ( I ) , H ( I ) , Q ( I ) , I = I,K) K L A = K C 41 C O N T I N U E WR ITE ( 6 , 7 0 ) 70 FORMAT ( 1H1 ) STOP C END S U B R O U T I N E AUXRK ( Y , F ) C D I M E N S I O N Y ( 2 ) , F ( 2 ) COMMON F O , Q 1 M 1 , BA . I F ( Y ( 2 ) . L E . 0 . 0 ) STOP 6 4 F ( 2 ) = F Q * ( Q I M 1 * ( Y ( 2 ) * * ( 1 . - B A ) ) - Y ( 2 ) * * ( 2 . - BA ) ) 11 R E T U R N A END NON-LINEAR RESERVOIR ROUT IMG IN IT IAL CONDITIONS 8LANFY CREEK, OCT . 1 3 , 1 9 6 8 , 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 . 0 2 0 0 0 0 . 7 0 0 0 0 K , IN 50 36 AA , BA 3 ,37500 0 . 4 7 8 0 0 DT .OF QO IN MIN , NO. STEPS PER DT 1 .00000 STARTING TIME , LENGTH (M5 , QD AH 1, B H U H O l , 1. 320000 AH2» 8H2, H02 , 1. „230000 AD 14 6 8 6 . 0 0 0 1 2 . 0 0 0 0 . 4 4 5 0 0 0 - 0 . 1 1 0 0 0 0 0 . 3 5 4 0 0 0 1 4 . 0 0 0 0 0 0 FQ , F T , 0 , 11069 3F 02 0 . 2 5 5 0 3 7 E 01 0 . 1 6 8 0 4 3 E 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 . 9 5 7 9 4 4 5 1 7 . 0 1 8 . 0 1. 180 1. 180 0 . 9 4 9 6 5 0 . 9 4 9 6 5 'SNLR i 6 19 .0 1. 180 0 . 94965 7 2 0 . 0 1.181 0 . 9 5 1 3 0 Printout for one of the 8 9 2 1 . 0 2 2 . 0 1.182 1.183 0 . 9 5 2 9 6 0 . 9 5 4 6 2 computations of Figure 24 10 2 3 . 0 1. 184 0 . 95628 11 2 4 . 0 1.185 0 . 9 5 7 9 4 12 2 5 . 0 1 .186 0 . 9 5 9 6 0 13 2 6 . 0 1. 186 0 .95960 14 2 7 . 0 1 .187 0 .96 127 15 2 8 . 0 1.188 0 .96 29 3 16 2 9 . 0 1. 188 0 . 9 6 2 9 3 17 3 0 . 0 1. 189 0 . 9 6 4 6 0 18 3 1 . 0 1 .189 0 . 9 6 4 6 0 19 3 2 . C 1. 190 0 . 9 6 6 2 7 20 3 3 . 0 1. 190 0 . 9 6 6 2 7 21 3 4 . 0 1. 191 0 . 9 6 7 9 4 22 3 5 . C 1.192 0 . 9 6 9 6 2 23 3 6 . 0 1. 193 0 . 9 7 1 2 9 24 3 7 . 0 1.193 0 .97 129 25 3 8 . 0 1. 194 0 . 9 7 2 9 7 26 3 9 . 0 1. 195 0 . 9 7 4 6 4 27 4 0 . 0 1. 195 0 . 9 7 4 6 4 28 4 1 . 0 1. 196 0 .97632 29 4 2 . 0 1.197 0 .97 800 ' 30 4 3 . 0 1 .197 0.97800- 31 4 4 . 0 1.198 0 . 9 7 9 6 9 32 4 5 . C 1. 199 0 .98 137 33 4 6 . 0 1.285 1. 13 223 34 47 .0 I. 290 1. 14137 35 4 8 . 0 1.293 1 .14687 36 4 9 . 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 IN IT IAL 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 . R E S . / L = 0 .700 00 QINCREMENT = 0 . 0 8 0 0 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 1 2 . 0 0 0 AH 1 , B H 1 , H 0 1 , 1. 230000 0 . 3 5 4 0 0 0 1.4.000000 AH 2 , BH2, H02 _ 1. 459999 0 . 331000 9 .50 0000 AD, F Q , F T , 0 . 1 2 9 0 2 7 E 02 0 . 2 7 0 9 2 5 E 01 0 . 1 5 8 1 8 8 F 00 DISCHARGE AFTER 6 RESERVOIRS NO TIME LEVEL 0 I SCHARGE 1 2 2 . 0 1 1.0 49 1. 1.9 60 7 2 2 3 . 0 1 1.049 1. 19606 3 2 4 . 0 1 1.049 1 . iQ605 4 2 5 . 0 1 1.049 1 . 19 604 5 2 6 . 0 1 1.C49 1. 19 603 6 2 7 . C 1 1.049 L . 19 602 7 2 8 . 0 1 1.049 1. 19 601 8 2 9 . 0 1 1.049 1. 1959 3 9 3 0 . 0 1 1 .049 1.19 59 2 10 3 1 . 0 1 1.049 1. 19576 11 3 2 . 0 1 1 .049 1 . 1953 9 12 3 3 . 0 1 1 .049 1. 1 9 46 6 13 3 4 . 0 1 1.048 1. 19332 14 3 5 . 0 1 1. 047 1.19110 15 3 6 . 0 1 1.046 1 .18775 16 37.0 . 1 1.044 1.18 306 17 3 8 . 0 1 1.041 1.17693 18 3 9 . 0 1 1.038 1.16 942 19 4 0 . 0 1 1.034 1.16 071 20 A l .0 1 1.03C 1.1510 9 2L 4 2 . 0 1 1.025 1.1409 I 22 4 3 . 0 1 1.021 1. 13055 23 44 .C I 1.016 1 . 12033 24 4 5.0 1 1.012 1.11071 25 4 6 . C 1 1.008 1 .10179 26 4 7 . 0 1 1.00 4 1 .09379 2 7 4 8 . C 1 1.00 I 1.08 63 3 28 4 9 . 0 1 0 . 9 9 8 1. 0 8 09 4 29 5 0 . 0 1 0 .996 1.07609 3C 5 1 . 0 1 0 . 9 9 4 1 .07224 31 5 2 . 0 1 0.99.3 1.069 2 9 32 5 3 . 0 1 0. 992 1.06713 3 3 5 4 . 0 1 0 .991 1 . 0656 5 34 5 5 . 0 1 0 .991 1 .06474 35 5 6 . 0 1 0 . 9 9 0 1 .06429 2 6 0 ( 3 6 57.G 1 0 . 9 9 0 • 1 .06 42 2 3 7 58 .0 1 0 . 9 9 0 1 . C 6 4 4 5 ' 3B 5 9 . 0 1 0 . 9 9 1 1.06 4 9 1 3 9 6 0 .0 1 0 . 9 9 1 1.06 5 5 7 4 0 6 1 .0 1 0 . 9 9 1 1 . 0 6 6 4 2 V 41 6 2 . 0 1 0 . 9 9 2 1 . 0 6 7 4 9 f 42 6 3 . 0 10.99 3 1. 0 6 88 9 4 3 6 4 . 0 10.99 3 1 . 0 7 0 8 1 44 6 5 . 0 1 0 . 9 9 5 1 . 0 7 3 5 3 45 6 6 . 0 1 0 . 9 9 6 1 . 0 7 7 4 0 46 6 7 . 0 1 0 . 9 9 9 1.08 28 2 4 7 6 8 . 0 1 1.00 2 1 . 0 9 0 0 9 48 6 9 . C 1 1 . 0 0 7 1.G9 9 4 ? 49 7 0 . 0 1 1 . 0 1 2 1 . 1 1 0 7 9 5 0 7 1 . 0 1 1 . 0 1 8 1.12 394 51 7 2 . 0 1 1 . 0 2 4 1 . ] 3 H 3 7 52 7 3.0 1 1 . 0 3 1 1 . 1 5 3 4 5 5 3 7 4 . 0 1 1.0 37 1.1 ', * '* 3 54 7 5 . 0 1 1 . 0 4 3 1 . 18 2 5 8 55 7 6 . 0 1 1 . 0 4 9 1 . 1 9 5 21 5 6 7 7 . 0 1 1 . 0 5 3 1.20 58 2 5 7 78 .0 1. 1. 0 5 7 1 . 2 1 4 2 0 5 8 7 9 . C 1 1 . 0 6 0 1.22"4 -\ 59 HO . 0 1 1.061 1 . 2 2 4 8 1 6 0 B 1 .0 1 1.0 6 3 1.22 7 74 6 1 8 2.0 1 1 . 0 6 3 1 . 2 2 ^ 6 2 ' NGN-LI NEAR RESERVOIR ROUTING INITIAL CONDITIONS BLANEY CREEK, OCT. 13,1968, 1-3-5-4 2 6 1 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 1 1 ? 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 4 2 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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