261
KINEMATIC HYDROL06Y AND MODELLING
DEVELOPMENTS IN WATER SCIENCE. 26
OTHER TITLES IN THIS SERIES
7 COMPUTER SYSTEMS AND WATER RESOURCES
2 H.L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY
3
G. BUGLIARELLO AND F. GUNTER
V.V. HAIMES. W.A. HALL AND H.T. FREEDMAN MULTIOBJECTIVE OPTIMIZATION IN WATERRESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF-METHOD
4 J.J. FRIED GROUNDWATER POLLUTION
5 N. RAJARATNAM TURBULENT JETS
6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS
7 v. HALEK AND J. SVEC GROUNDWATER HYDRAULICS
8 J.BALEK HYDROLOGY AND WATER RESOURCES IN TROPICAL AFRICA
9 T.A. McMAHON AND R.G. MElN RESERVOIR CAPACITY AND YIELD
10 G. KOVACS SEEPAGE H Y DRAU LlCS
11 HYDRODYNAMICS OF LAKES: PROCEEDINGS OF A SYMPOSIUM 12-13 OCTOBER 1978, LAUSANNE, SWITZERLAND
12 CONTEMPORARY HYDROGEOLOGY: THE GEORGE BURKE MAXEY MEMORIAL VOLUME
SEEPAGE AND GROUNDWATER
14 D. STEPHENSON STORMWATER HYDROLOGY AND DRAINAGE
15 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in the series)
SYMPOSIUM ON GEOCHEMISTRY OF GROUNDWATER
W.H. GRAF AND C.H. MORTIMER (EDITORS)
W. BACK AND D.A. STEPHENSON (EDITORS)
13 M.A. MARIK~O AND J.N. LUTHIN
16 w. BACK AND R. L ~ T O L L E (EDITORS)
17 TIME SERIES METHODS I N HYDROSCIENCES
A.H. ELSHAARAWI (EDITOR) I N COLLABORATION WITH S.R. ESTERBV
18 J.BALEK HYDROLOGY AND WATER RESOURCES I N TROPICAL REGIONS
19 D. STEPHENSON PIPEFLOW ANALYSIS
20 I. ZAVOIANU MORPHOMETRY OF DRAINAGE BASINS
21 M.M.A. SHAHIN HYDROLOGY OF THE NILE BASIN
22 H.C. RIGGS STREAMFLOW CHARACTERISTICS
23 M. NEGULESCU MUNICIPAL WASTEWATER TREATMENT 24 L.G. EVERETT GROUNDWATER MONITORING HANDBOOK FOR COAL AND OIL SHALE DEVELOPMENT
25 W. KINZELBACH GROUNDWATER MODELLING: AN INTRODUCTION WITH SAMPLE PROGRAMS I N BASIC
KINEMATIC HYDROLOGY AND MODELLING
DAVID STEPHENSON
Department of Civil Engineering, University of the Witwatersrand, I Jan Smuts Avenue, 2001 Johannesburg, South Africa
and
MICHAEL E. MEADOWS Department of Civil Engineering, University of South Carolina, Columbia, SC 29208, U.S.A.
ELSEVIE R
Amsterdam - Oxford - New York - Tokyo 1986
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1,1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, N Y 10017, U.S.A.
Library of Congress Cataloging-in-Publication Data
Stephenson, Divid, 1943- Kinematic hydrology and modelling.
(Developments in water science ; 26) Bibliography: p. Includes indexes. 1. Runoff--Mathematical models. 2. Groundwater
flow--Mathematical models. I. Meadows. Michael E. 11. Title. 111. Series. GBg8O.S74 1986 551.48'8'0724 86-2175 ISBN 0-444-42616-7
ISBN 0444-42616-7 (Vol. 26) ISBN 044441669-2 (Series)
0 Elsevier Science Publishers B.V., 1986
All rights reserved. No part of th is publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other- wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./Science & Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands.
Special regulations for readers in the USA - This publication has been registed with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts o f this publication may be made in the USA. A l l other copyright questions, including photocopying outside of the USA, should be referred to the publisher, Elsevier Science Publishers B.V., unless otherwise specified.
Printed in The Netherlands
V
PREFACE
Many s t o r m w a t e r d e s i g n e n g i n e e r s and indeed h y d r o l o g i s t s w i I I b e
f r u s t r a t e d b y the l a c k o f h y d r a u l i c p r i n c i p l e s in some o f t h e c o n v e n t i o n a l
methods o f f l o o d c a l c u l a t i o n . The R a t i o n a l me thod and uni t h y d r o g r a p h
methods a r e easy t o a p p l y b u t l i m i t e d in a c c u r a c y and v e r s a t i l i t y .
K i n e m a t i c h y d r o l o g y i s t h e n e x t l o g i c a l s tep in s o p h i s t i c a t i o n b e f o r e the
f u l l h y d r o d y n a m i c e q u a t i o n s a r e r e s o r t e d to. The k i n e m a t i c e q u a t i o n s in
f a c t compr i se the c o n t i n u i t y e q u a t i o n and a h y d r a u l i c r e s i s t a n c e
e q u a t i o n . I n many cases s o l u t i o n of these e q u a t i o n s f o r f l o w r a t e s and
w a t e r dep ths i s s i m p l e a n d e x p l i c i t . I n more c o m p l i c a t e d p r o b l e m s the
e q u a t i o n s may b e u s e d to s i m u l a t e the r u n o f f p rocess .
U n f o r t u n a t e l y much o f t he l i t e r a t u r e o n the k i n e m a t i c method h a s been
h i g h l y ma themat i ca l a n d o f t e n o f an e x p e r i m e n t a l n a t u r e . The e q u a t i o n s ,
g r a p h s and mode ls p u b l i s h e d a r e t h e r e f o r e o f l i t t l e use to t h e p r a c t i c a l
e n g i n e e r , a n d may d i s c o u r a g e h i m f rom u s i n g t h i s method. In f a c t once
con f idence i s g a i n e d , t he method c a n b e a p p l i e d in s i m p l e fo rm t o a
v a r i e t y o f ca tchments . The te rm k i n e m a t i c r e f e r s to movement where
a c c e l e r a t i o n s a r e n e g l i g i b l e - w h i c h i s g e n e r a l l y a p p l i c a b l e t o o v e r l a n d
a n d s h a l l o w s t r e a m f l ow .
The book i s a i m e d a t b o t h t h e t h e o r e t i c i a n and the p r a c t i t i o n e r . Thus
the ma themat i ca l sec t i ons a r e u s e f u l i f m o d e l l i n g i s r e q u i r e d , b u t t h e
c h a p t e r s o n d e s i g n c h a r t s c o u l d b e r e a d w i t h v e r y l i t t l e ma themat i ca l
u n d e r s t a n d i n g o t h e r t h a n a b a s i c a p p r e c i a t i o n o f t he k i n e m a t i c method.
L i t t l e ma themat i ca l b a c k g r o u n d i s r e q u i r e d , a n d no compu te r k n o w l e d g e i s
necessa ry f o r those sec t ions . I t i s hoped t h a t t he p e a k f l o w c h a r t s w i l l
p r o v i d e a n a l t e r n a t i v e to the R a t i o n a l me thod a n d the SCS method f o r
e s t i m a t i n g r u n o f f . Simi l a r l y the d imens ion less h y d r o g r a p h s a r e
c o m p e t i t i v e w i t h u n i t h y d r o g r a p h methods. The u s e r w i l l g r a d u a l l y
become a w a r e o f t he f a c t t h a t t he k i n e m a t i c method i s f a i r l y easy to a p p l y
i f s i m p l e s o l u t i o n s a r e r e q u i r e d . I t a l s o p e r m i t s c o n s i d e r a t i o n of m a n y
more f a c t o r s t h a n some o t h e r methods o f f l o o d c a l c u l a t i o n , w h i c h in t u r n
c a n o n l y i m p r o v e a c c u r a c y a n d p r o v i d e f o r g r e a t e r u n d e r s t a n d i i g o f t he
r u n o f f process.
Of cou rse the k i n e m a t i c me thod i s n o t t he f i n a l a n s w e r in h y d r o l o g y .
There a r e many ques t i ons s t i l l to b e answered , a n d some degree o f
simp1 i f i c a t i o n i s s t i l l r e q u i r e d . A l t h o u g h the me thod p r o v i d e s a l o g i c a l
way of v i s u a l i z i n g r u n o f f , a c t u a l r u n o f f f rom m a n y ca tchmen ts compr i ses
v i
pa r t overland, subsurface and interface flow. The combined effect cannot
easi ly be modelled. Also water does not run off r u r a l catchments in a
sheet - i t frequently forms r i vu le t s and i s diverted by obstacles which can
be loosely termed roughness. Some of these factors can be accounted for by
adjust ing the hyd rau l i c factors used i n the equations, o r ca l i b ra t i ng
models.
Results of research and development a re now advanced and
experience in appl icat ion is required before general acceptance of the
kinematic method can be hoped for. I n pa r t i cu la r the a b i l i t y to select soi l
losses, roughnesses and catchment geometry to adequately describe the
hydrau l i cs of the system, can only be gained w i th experience.
The scope of the kinematic method is therefore unl imited from the
point of view of the researcher w i th an enqu i r ing mind. Some of the
theoretical considerations are taken fu r ther i n chapter 2 on kinematic
equations, 4 on assumptions and 5 on numerical theory for modelling.
On the other hand the pract i t ioner is probably more interested i n the
best answer ava i lab le . He may manage qu i te suf f ic ient ly reading only
chapter 3 on peak flows, chapter 6 w i th dimensionless hydrographs and
possibly chapter 7 on marginal effects and 9 w i th some examples of the
value of the techniques. Hopefully he w i l l be inspired to go into
modelling, which may b r i n g i n chapter 8 on flow i n conduits, and 10, 1 1
and 12 wi th examples of computer models of var ious catchments.
Much of the material i n th i s book i s der ived from notes for a course
presented by the authors. There is copious reference to previous
research in kinematic h'ydrology, as well as new material a r i s ing from
research by both authors. I n pa r t i cu la r the senior author was the
recipient of a research contract i n urban hydrology from the Water
Research Commission.
The manuscript was typed into i t s f i na l form by Janet Robertson, for
which the authors are most g ra te fu l .
v i i
CONTENTS
CHAPTER 1 . INTRODUCTION
H i s t o r i c a l r e v i e w
C l a s s i c a l h y d r o l o g y . H y d r o d y n a m i c e q u a t i o n s I nf i I t r a t i o n
So i l p h y s i c s mode ls Green and Ampt model H y d r o l o g i c i n f i l t r a t i o n
Def i n i t i ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mode I s . . . . . . . . . .
CHAPTER 2 . ANALYSIS OF RUNOFF
I n t r o d u c t i o n . . . . . . D y n a m i c e q u a t i o n s . . . . . .
C o n s e r v a t i o n o f mass . . . . . C o n s e r v a t i o n of momentum . . . .
S i m p l i f i e d e q u a t i o n s . . . . . . The k i n e m a t i c e q u a t i o n s . . . . . K i n e m a t i c f l o w o v e r impermeab le p l a n e s
R i s i n g h y d r o g r a p h . g e n e r a l s o l u t i o n T ime o f c o n c e n t r a t i o n . . . . . E q u i l i b r i u m d e p t h p r o f i l e . . . . The r e c e d i n g h y d r o g r a p h . . . .
F r i c t i o n e q u a t i o n . . . . . .
. . . . . . . .
. . . . . . . .
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CHAPTER 3 . HYDROGRAPH SHAPE AND PEAK FLOWS
Des ign p a r a m e t e r s . . . . So lu t i on o f k i n e m a t i c e q u a t i o n s f o r H y d r o g r a p h s f o r p l a n e s . . . D e r i v a t i o n o f p e a k f l o w c h a r t s .
L o n g ca tchmen ts . . . . M o d i f i c a t i o n f o r p r a c t i c a l u n i t s
E f fec t o f c a n a l i z a t i o n . . . E s t i m a t i o n o f a b s t r a c t i o n s . .
. . . . . . . . . . f l o w o f f a p l a n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
CHAPTER 4 . K I NEMAT I C ASSUMPT I ON5
N a t u r e o f k i n e m a t i c e q u a t i o n s . . . K i n e m a t i c a p p r o x i m a t i on to o v e r l a n d f l o w
G o v e r n i n g e q u a t i o n s . . . . . C o n d i t i o n s f o r t he k i n e m a t i c a p p r o x i m a t K i n e m a t i c f l o w n u m b e r . . . .
K i n e m a t i c a n d non-k inemat i c waves . Wave speed . k i n e m a t i c waves . . Cres t subs idence . . . . . . H y d r a u l i c geomet ry and r a t i n g c u r v e s
Non-k inemat i c waves . . . . . . Wave speed . . . . . . Cres t s u b s i d e n c e . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . . Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3 4 9
13 15 17 19
23 23 24 25 27 28 30 30 32 33 33 37
43 43 45 40 49 51 56 57
59 59 60 60 65 66 67 67 68 69 71 71
v i i i
Looped r a t i n g c u r v e s . . . . . . . . . . . . . 74 M u s k i n g u m r i v e r r o u t i n g . . . . . . . . . . . . . 76
K i n e m a t i c and d i f f u s i o n mode ls . . . . . . . . . . 78 E s t i m a t i o n o f mode l p a r a m e t e r s . . . . . . . . . . 77
CHAPTER 5. NUMERICAL SOLUTl ONS
Methods o f s o l u t i o n o f e q u a t i o n s of mo t ion . . . Method o f c h a r a c t e r i s t i c s . . . . . . . .
Numer i ca l i n t e g r a t i o n of c h a r a c t e r i s t i c e q u a t i o n s . F i n i t e d i f f e r e n c e methods . . . . . . . .
D i f f e r e n c e q u o t i e n t s . . . . . . . . . Numer i ca l s o l u t i o n . . . . . . . . . .
E x p l i c i t scheme . . . . . . . . . . I m p l i c i t scheme . . . . . . . . . .
Accuracy and s t a b i l i t y of n u m e r i c a l schemes . . . Ef fec t o f f r i c t i o n . . . . . . . . . . Choos ing an e x p l i c i t f i n i t e d i f f e r e n c e scheme f o r t h e
o f t he one-d imens iona l k i n e m a t i c e q u a t i o n s . . sol I
. . . 81
. . . 81
. . . 83
. . . 86
. . . 87
. . . 88
. . . 91
. . . 93
. . . 95
. . . 102
. . . 103 J t i o n
CHAPTER 6. D l MENSIONLESS HYDROGRAPHS
U n i t h y d r o g r a p h s . . . . L i s t of symbo ls . . . . K i n e m a t i c e q u a t i o n s . . .
Excess r a i n f a l I . . . . Dimens ion less e q u a t i o n s . . .
S l o p i n g p l a n e ca tchmen t . . C o n v e r g i n g s u r f a c e ca tchmen t V-shaped ca tchmen t w i t h s t ream
Use o f d imens ion less h y d r o g r a p h s P rob lem . . . . S o l u t i o n . . . .
Development and use o f g r a p h s . . . . . . . . . . . 105 . . . . . . . . . . 106 . . . . . . . . . . 107 . . . . . . . . . . 108 . . . . . . . . . . 108 . . . . . . . . . . 110 . . . . . . . . . . 110 . . . . . . . . . . 114 . . . . . . . . . . 115 . . . . . . . . . . 125 . . . . . . . . . . 125 . . . . . . . . . . 125
CHAPTER 7. STORM DYNAM I CS AND D I STR I BUT I ON
Des ign p r a c t i c e . . . . . . . . Storm p a t t e r n s . . . . . . . .
V a r i a t i o n in r a i n f a l l i n t e n s i t y d u r i n g a s to rm S p a t i a l d i s t r i b u t i o n . . . . . . . Storm movement . . . . . . . .
Numer i ca l mode ls . . . . . . . . K i n e m a t i c e q u a t i o n s . . . . . . . Numer i ca l scheme . . . . . . . .
So lu t i ons for d y n a m i c s to rms . . . . . T ime v a r y i n g s to rms . . . . . . . S p a t i a l v a r i a t i o n s . . . . . . . . M o v i n g s to rms . . . . . . . .
. . . . . . 130
. . . . . . 131 . . . . . 131
. . . . . . 132
. . . . . . 133
. . . . . . 134
. . . . . . 134
. . . . . . 135
. . . . . . 137
. . . . . . 137
. . . . . . 140
. . . . . . 142
i x
CHAPTER 8. CONDUIT FLOW
Kinemat i c e q u a t i o n s f o r n o n - r e c t a n g u l a r sec t i ons . . P a r t - f u l I c i r c u l a r p i p e s . . . . . . . . . . Computer p r o g r a m f o r d e s i g n o f s to rm d r a i n n e t w o r k .
P r o g r a m d e s c r i p t i o n . . . . . . . . . . Computer p r o g r a m f o r s to rm n e t w o r k p i p e s i z i n g . . Sample i n p u t . . . . . . . . . . .
Trapezo i da I c h a n n e I s . . . . . . . . . . Compar i son of k i n e m a t i c and t ime-sh i f t r o u t i n g i n
c o n d u i t s . . . . . . . . . . . Sect ion geomet ry and e q u a t i o n s f o r c o n d u t t s . . . Computer s i m u l a t i o n . . . . . . . . . . C r i t e r i a f o r choos ing be tween t ime s h i f t and k i n e m a t i c r o u t i n g . . . . . . . . . . . L a g t ime f o r r o u t i n g h y d r o g r a p h s u s i n g t ime s h l f t methods . . . . . . . . . . . Compar i son o f methods f o r e v a l u a t i n g l a g t ime . . Time l a g f o r t r a p e z o i d s . . . . . . . . .
CHAPTER 9. URBAN CATCHMENT MANAGEMENT
E f f e c t s o f u r b a n i z a t i o n . . . . . . . . E f fec t o n r e c u r r e n c e i n t e r v a l . . . . .
Examp le : c a l c u l a t i o n o f p e a k r u n o f f f o r v a r i o u s c o n d i t i o n s . . . . . . . . . Virgin ca tchmen t . . . . . . . . . Reduc t ion in in f i I t r a t i o n . . . . . . . Ef fec t o f r e d u c e d r o u g h n e s s d u e to p a v i n g . . Ef fec t o f c a n a l i z a t i o n . . . . . . . . Combined r e d u c e d r o u g h n e s s a n d r e d u c e d losses
De ten t ion s t o r a g e . . . . . . . . .
. . . 145
. . . 145
. . . 147
. . . 148
. . . 151
. . . 152
. . . 152
. . . 153
. . . 154
. . . 161
. . . 162
. . . 165
. . . 165
. . . 166
. . . . . 172
. . . . . 172
. . . . . 173
. . . . . 174
. . . . . 175
. . . . . 176
. . . . . 176
. . . . . 177
. . . . . 179 Channe l s t o r a g e . . . . . . . , . . . . . . 180 K i n e m a t i c e q u a t i o n s f o r c losed c o n d u i t sys tems . . . . . . 184 Computer p r o g r a m t o s i m u l a t e r e s e r v o i r l eve l v a r i a t i o n s
in a p i p e n e t w o r k . . . . . . . . . . . . . . 186 D a t a input . . . . . . . . . . . . . . 188 L i s t of s y m b o l s in p r o g r a m . . . . . , . . . . . 189 P r o g r a m l i s t i n g . . . . . . . . . . . . . . 191
CHAPTER 10. K I NEMAT I C MODELL I NG
I n t roduc t i on Storrnwater mode l l i n g . . . . . Mathemat i ca l mode ls . . . . . System d e f i n i t i o n . . . . . . Term ino logy and d e f i n i t i o n s . . . . M o d e l l i n g a p p r o a c h e s . . . . . Examp les o f p a r a m e t r i c and d e t e r m i n i s t i c Two-d imens iona l o v e r l a n d f l o w mode l I ing
Two-d imens iona l k i n e m a t i c e q u a t i o n s B o u n d a r y c o n d i t i o n s . . . . . I n i t i a l c o n d i t i o n s . . . . .
. . . . . . . . . . . f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
mode ls . . . . . . . . . . . . . . . . , . . . . . . . , . . . . . . . . . . . . .
194 194 195 197 198 200 20 1 204 204 206 206
X
CHAPTER 1 1 . APPLICATIONS OF K I NEMAT I C MODELL I NG
Approaches . . . . . . . A model f o r urban wa te rsheds . . . .
P a r a m e t r i c r a i n f a l I excess components . So i l m o i s t u r e a c c o u n t i n g . . . . . I n f i l t r a t i o n component . . . . . . I m p e r v i o u s a r e a r a i n f a l I excess component O p t i m i z a t i o n component . . . . . D e t e r m i n i s t i c r u n o f f r o u t i n g components . Channe l and o v e r l a n d f l o w segments . . R e s e r v o i r segments . . . . . . . E x a m p l e a p p l i c a t i o n . . . . . .
A model f o r r u r a l wa te rsheds . . . . P r e c i p i t a t i o n excess . . . . . . F low r o u t i n g . . . . . . . Model appl i c a t i o n . . . . . . .
O v e r l a n d f l o w and s t reamf low p r o g r a m . D a t a input . . . . . . . O v e r l a n d and s t reamf low p r o g r a m I i s t i n g I n f i l t r a t i o n and seepage . . . .
Rea l - t ime model I i n g . . . . . . .
. . . . . . . 209
. . . . . . . 209
. . . . . . . 209
. . . . . . . 210
. . . . . . . 210
. . . . . . . 214
. . . . . . . 214
. . . . . . . 215
. . . . . . . 215
. . . . . . . 217
. . . . . . . 219
. . . . . . . 222
. . . . . . . 222
. . . . . . . 223
. . . . . . . 224
. . . . . . . 229 . . . . . . . 231
. . . . . . . 232
. . . . . . . 234
. . . . . . . 235
CHAPTER 12 . GROUNDWATER FLOW
Genera l comments . . . . . . . . . . . . . . 237 F low in po rous m e d i a . . . . . . . . . . . . . 237 D i f f e r e n t i a l e q u a t i o n s in p o r o u s m e d i a . . . . . . . . 239 A n a l y s i s o f s u b s u r f a c e f l o w . . . . . . . . . . . 240 F low in u n s a t u r a t e d zone . . . . . . . . . . . . 241 F low in non-homogeneous s a t u r a t e d zone . . . . . . . . 242
AUTHOR INDEX . . . . . . . . . . . . . . 246
INDEX . . . . . . . . . . . . . . 248
1
CHAPTER 1
I NTRODUCT I ON
HISTORICAL REV1 EW
Kinemat i c hydrology provides a method for estimat ing stormwater
runoff rates and volumes. I t is pa r t i cu la r l y useful for flood calculat ion.
I t i s a re la t i ve ly new term embracing techniques which have been appl ied
for many decades. Kinematic hydrology is decidedly more hyd rau l i ca l l y
correct than some of the more common methods of f lood estimation such as
the rat ional method, t ime-area methods, the Soil Conservation Service (SCS)
method and u n i t hydrograph methods. The kinematic method i s based on
the cont inui ty equation and a flow resistance equation, both basic
hydraul ic equations.
I t was the American hydrologist , Horton, (general ly associated with
i n f i l t r a t i on ) who in 1934 car r ied out the ear l iest recorded scient i f ic studies
of over land f low. Later Keulegan (1945) appl ied the cont inui ty and
momentum equations conjunctively for over land flow analysis. He investi-
gated the magnitude of the various terms in the dynamic equation of St.
Venant and indicated that a s impl i f ied form of the equation, now, termed
the kinematic equation, would be adequate for over land flow.
An in-depth analysis of the d i f fe ren t ia l cont inui ty and resistance
equations was undertaken by L i g h t h i l l and Whitham (1955) to whom the
designation kinematic waves 'can be a t t r ibu ted . They also f i r s t studied the
phenomenon of kinematic shock which can be appl ied to discont inui t ies in
flow and water depth. Although they suggested the kinematic approach for
overland flow modelling, i t was Henderson and Wooding (1964) who
obtained ana ly t i ca l solutions to the kinematic wave equations for simple
plane and channel shapes. A general izat ion of the catchment stream model
was also described by Eagleson (1967).
The ful I dynamic equations for one-dimensional incompressible flow i n
open channels were set down by St. Venant i n 1871. These equations were
for g radua l ly var ied unsteady flow such as f lood waves. The idea of
graphical integrat ion using character ist ic I ines was f i r s t suggested by
Massau in 1889. On the other hand Greco and Panattoni (1977) indicate
that imp l ic i t solution by f i n i t e differences is the most ef f ic ient method by
computer, avoiding instabi I i ty and g i v ing r a p i d convergence. Various
numerical methods of solut ion of the kinematic equations were investigated
by Kibler and Woolhiser (1970). The step length in f i n i t e dif ference
schemes p lays an important role i n the s tab i l i t y of the solut ion (Singh,
2
1977). Non-convergence was investigated for plane cascades by Croley and
Hunt (1981 ) . Brakensiek (1966) used numerical solutions to the kinematic
wave equations for the ana lys is of surface runoff from r u r a l watersheds.
He probably d id not real ise the extent to which numerical modell ing would
advance in later years using the kinematic equation and square x-t gr ids.
The lat ter approach does not warrant appendage of the term 'wave' to
kinematic since discont inui t ies are lost i n the simp1 i f i ed numerical method.
Wooding (1965 and 1966) presented a comprehensive review of the
theory of kinematic waves and used numerical solutions to der ive equations
for the r i s i n g and f a l l i n g limbs of hydrographs for simple planes and
channel configurations. Dur ing the 1970's the equations were appl ied to
more complicated catchment shapes (Schaake, 1975), in pa r t i cu la r the
catchment-stream model, the converging catchment and cascades of planes.
Although ana ly t i ca l solutions are ava i lab le for some cases the major i ty of
solutions are numerical, and dimensionless hydrographs fac i l i t a te the use
o f the resul ts of the studies (Constantinides and Stephenson, 1982). Since
the studies by Henderson and Wooding (1964) and lwagaki (1955) the shock
wave phenomenon has not r e a l l y received much at tent ion and for th is
reason the use of the name kinematic theory i s now considered adequate as
i t implies a more general app l i cab i l i t y than to waves. I n fact Borah and
Prasad (1982) indicate shock waves may i n fact not exist i n some cases
where predicted using the kinematic equations. This is because the
kinematic equations may not apply where the spat ia l var ia t ion i n depth i s
large. Even the St. Venant equations may not suff ice to describe r a p i d
var ied f low, as vert ical 'accelerat ions are not considered.
Woolhiser and Liggett (1967) investigated the app l i cab i l i t y of the
kinematic equations and proposed a dimensionless parameter ind ica t ing
whether the equations are adequate for any pa r t i cu la r case w i th simple
geometry. More recent research (Morr is and Woolhiser, 1980) has
investigated in greater detai l the app l i cab i l i t y of the kinematic equations
to dif ferent conditions.
The appl icat ion of kinematic theory has more recently been extended
to problems such as dynamic storms (Stephenson, 1984a), detention storage
(Stephenson, 1984b), urban drainage networks (Green, 1984) and to the
effects of urbanizat ion and storm runoff (Stephenson, 1983).
There i s as yet l i t t l e general da ta ava i l ab le on surface water losses
( i n f i l t r a t i on , and retent ion) to be used w i th kinematic equations. Skaggs
(1982) reviewed in f i l t r a t i on mechanics including the popular Horton model
and more advanced Green-AmDt model.
3
The m a j o r i t y o f p a p e r s d i f f e r e n t i a t e be tween s u r f a c e and s u b s u r f a c e
f l ow , i.e, o v e r l a n d f l o w i s t r e a t e d i n d e p e n d e n t l y . Rovey e t at. (1977)
deve loped an i n t e r a c t i v e i n f i l t r a t i o n model to accoun t f o r n o n - u n i f o r m
so i l losses. A f u r t h e r deve lopment b y Freeze (1972) a l l o w s f o r c o n t r i -
b u t i o n s f rom r e - a p p e a r i n g s h a l l o w g r o u n d w a t e r f l o w in a s a t u r a t e d
a q u i f e r .
retent ion u
F i g . 1 . 1 S i m p l i f i e d ca tchmen t l o n g i t u d i n a l sec t i on
D e f i n i t i o n s
Some te rms used in t h i s t e x t a r e used in d i f f e r e n t c o n t e x t e l sewhere
so to a v o i d c o n f u s i o n p a r t i c u l a r l y w i t h respec t t o t imes, some d e f i n i t i o n s
a r e g i v e n be low.
T ime to e q u i l i b r i u m (t,) i s t h e t ime t a k e n f r o m t h e commencement o f
p r e c i p i t a t i o n u n t i l t he w a t e r p r o f i l e down t h e ca tchmen t i s in e q u i l i b r i u m
a n d i n f l o w e q u a l s o u t f l o w e v e r y w h e r e , i . e . r u n o f f r a t e i s e q u a l to excess
r a i n f a l I r a t e , a s s u m i n g s teady p r e c i p i t a t i o n a n d losses.
T ime o f c o n c e n t r a t i o n ( t c ) i s the t ime f rom the commencement of p r e c i p -
i t a t i o n u n t i l t he e f fec t o f excess p r e c i p i t a t i o n e v e r y w h e r e in t h e c a t c h -
ment h a s a p p e a r e d a t t he o u t l e t . I t i s e q u a l to t h e t ime to e q u i l i b r i u m
f o r s teady excess r a i n u s i n g k i n e m a t i c theo ry whereas i t i s e q u a l to
t r a v e l t ime w i t h t ime a rea - theo ry . I t i s demons t ra ted l a t e r t h a t f o r a
s i m p l e p l a n e , k i n e m a t i c theo ry y i e l d s
t = (L ie ’ -m/aj l ’m (1.1) m-1 c
where L i s the l e n g t h o f f l o w p a t h , a n d f l o w v e l o c i t y V = a y where
y i s w a t e r d e p t h a n d m and a a r e c o e f f i c i e n t s d e f i n e d b y the e q u a t i o n
q = a y where q i s the f l o w r a t e p e r u n i t w i d t h . m
4
0 t td
Fig . 1 .2 Catchment water balance
Travel time (t,) i s the time for a pa r t i c l e of water to proceed from the
most remote pa r t of the catchment to the discharge point. For a plane i t
i s not equal to time of concentration according to kinematic theory since
water moves slower than a hydrau l i c response which travels a t wave
speed. I t i s shown later that for a plane
tc = tt/m ( 1 . 2 )
tL = mtc/( l+m) (1.3)
Lag time ( t ) i s the time between 50% of p rec ip i ta t ion and 50% of runoff .
I t w i l l be shown that for a plane L
Storm durat ion td i s the time from commencement of precipi tat ion u n t i l i t
ceases. Frequently when storm records are analyzed for intensi ty-durat ion
relat ionships storm durat ion i s defined as the time dur ing which average
storm intensity i s a specified f igure , so that storms w i th in storms can
occur.
T i m e of excess runoff ( t ) i s the time measured from the commencement of
runoff. I t is therefore less than the t i m e t from the commencement of
precipi tat ion by tu = u / i where u i s i n i t i a l abstraction and i i s the
precipi tat ion ra te (see Fig. 3.3 on page 4 9 ) .
Units of time are general ly seconds i f the System Internat ional ( S . I . ) un i t s
of metres, seconds and ki lograms, or the old Engl ish system of foot,
seconds and pounds are adopted. Later herein modifications for more
___
pract ical un i ts e.g. r a i n f a l l in mm/h or inches per hour, a re introduced.
CLASSICAL HYDROLOGY
For various reasons f lood hydrology has been a f a i r l y s ta t i c subject
for many decades. The ra t iona l method which was invented over 100 years
ago, and hydrograph theory, developed over 50 years ago, are s t i l l used
extensively. I f we reconsider the assumptions and l imi tat ions behind these
methods we may be prepared to consider developing new techniques more
5
appropriate to our technology and more accurate.
The simple l inear hydrology methods were probably developed for ease
of manual calculat ion, and as many hydrologists do not have a strong
mathematical background. I t i s t rue that some of the standard methods
have been programmed for computers. This fac i l i ta tes the subdiv is ion of
catchments but does not el iminate the l imi tat ions of many of the
assumptions behind the methods.
The current ava i l ab i l i t i es of computers to a l l should considerably ease
the next step - breaking away from simple input-output methods and
introducing more sophisticated hyd rau l i c equations i n the i r stead. I t i s
possible to simulate water flow and water surface prof i les w i th
considerable accuracy wi th the a i d of computers, even micro computers.
There are various levels of sophist icat ion which can be adopted to su i t the
problem and the machine avai lable.
These methods are based on solut ion of f i n i t e difference versions of
the d i f fe ren t ia l equations of flow. Computations proceed in increments of
time at selected in te rva ls i n space. There hsve been numerous advances i n
numerical methods i n mathematics in para l le l w i th the developments i n
computers. On the other hand the approximation of d i f fe ren t ia ls by f i n i t e
increments can lead to in'accuracins unless cer ta in rules a re complied
with. Some of the common problems are i ns tab i l i t y , numerical d i f fusion or
accumulating errors. The correct f i n i t e increments can be selected to
approximate the d i f fe ren t ia ls to a f i r s t order, second order or greater
order i f necessary. There a re also methods fo r solv ing imp l ic i t equations
such as by gradient convergence o r successive approximation. Where a-
number of simultaneous equations have to be solved over a g r i d there are
matr ix methods and re laxa t ion methods avai lable.
One of the greatest a ids to the engineer nowadays may be the desk
top micro computer. Whereas pract i t ioners tend to shy away from main
frame computers ( i f they can access one a t a l l ) the problems of job control
language, queing batch jobs, formal programming and debugging and r i s k
of runaway costs are no longer of concern. The kinematic method i s
intermediate level technology app l icab le to micro computer solutions,
whether ana ly t i ca l solutions o r numerical model I ing is contemplated.
The basis fo r much of our hydrology probably or ig inated with an
I r i sh engineer, Mulvaney, in 1851. He proposed an equation for runoff , €I
= KA. K al lows for a r a i n f a l l intensi ty but t h i s was not a s igni f icant
var iab le in B r i t a in . The method was taken a step fu r ther by introducing
an equation for excess r a i n f a l I intensi ty, e.g. the Birmingham formula,
6
( 1 . 4 )
where i i s in inches p e r h o u r and t i s t h e s to rm d u r a t i o n in m inu tes .
No a l l o w a n c e i s made f o r ex t reme s to rms and t h i s e q u a t i o n i s f o r a 1
to 2 y e a r f r e q u e n c y s to rm. The 20 was accep ted b y some as r e p r e s e n t i n g
a t ime o f e n t r y i n m i n u t e s ( e q u i v a l e n t t o t h e d e f i n e d c o n c e n t r a t i o n t ime
o f o v e r l a n d f l o w ) .
I t was assumed t h a t 100% r u n o f f o c c u r r e d f r o m impermeab le a r e a s
and none f rom p e r v i o u s a r e a s . T h i s assumpt ion was no t a c c e p t a b l e in
a r e a s o f high r a i n f a l l i n t e n s i t y and in the U n i t e d S ta tes where K u i c h l i n g
in 1889 m o d i f i e d the r u n o f f e q u a t i o n t o Q = C iA where the c o e f f i c i e n t
C i s a f u n c t i o n o f t h e ca tchment .
The coe f f i c i en t C i s most s t r o n g l y assoc ia ted w i t h the a v e r a g e
p e r m e a b i l i t y o f t h e ca tchment - t h u s 100% r u n o f f w o u l d o c c u r i f C i s u n i t y
a n d no r u n o f f f o r a comp le te l y p e r m e a b l e ca tchmen t . M o d i f i c a t i o n s to C a r e
made to accoun t f o r ca tchment s lope, v e g e t a t i o n cove r a n d so on b y
v a r i o u s peop le . I t h a s a l s o been r e a l i z e d t h a t dn tecedent m o i s t u r e
c o n d i t i o n s and s e v e r i t y of t he s to rm ( r e p r e s e n t e d b y the r e c u r r e n c e
i n t e r v a l ) c a n a f fec t C. F o r i n s t a n c e Rossmi l l e r (1980) p roposed t h e
f o l l o w i n g e m p i r i c a l e q u a t i o n f o r C :
( 1 3) 1.48(.15-1) ~ + l e 7 . 2
C = 7 . 7 ~ 1 0 - ~ C ~ ' R ~ ~ ( . 0 1 C ~ ) - ~ ~ ( .001 CN) (T)
where R i s the r e c u r r e n c e i n t e r v a l , S i s b e d s lope in pe rcen t , I i s
r a i n f a l l i n t e n s i t y i n inches p e r h o u r , M i s the f r a c t i o n o f wa te rshed w h i c h
i s i m p e r v i o u s and CN the So i l C o n s e r v a t i o n S e r v i c e (SCS) c u r v e number .
The assumpt ion o f a u n i q u e ' C ' f o r a n y ca tchmen t c a n l e a d to
s i g n i f i c a n t e r r o r s and u n d e r e s t i m a t i o n o f f l o o d r u n o f f . T h i s i s demons t ra ted
b y F i g u r e 1.3. The r u n o f f r a t e p e r u n i t a r e a f o r case ' a ' i s C i l . I f t h e
same C i s used f o r case b, where a h i g h e r r a i n f a l l i n t e n s i t y occu rs , t h e
loss w i l l b e g r e a t e r and the r u n o f f p r o p o r t i o n a l . A loss w h i c h i s
i ndependen t of r a i n f a l l i n t e n s i t y however w o u l d p r o d u c e a r u n o f f a s f o r
case c , w h i c h i s p r o p o r t i o n a l l y g r e a t e r t h a n f o r case b . The assumpt ion
f o r case b t h u s r e s u l t s in a n u n d e r e s t i m a t e o f f l o o d r u n o f f .
I n g e n e r a l t hen , i t i s i m p l i e d i n the R a t i o n a l method t h a t r u n o f f
i n t e n s i t y i s I i n e a r l y p r o p o r t i o n a l to r a i n f a l l i n t e n s i t y . T h i s a l s o assumes
t h a t t he ca tchmen t h a s r e a c h e d a n e q u i l i b r i u m , so i t became necessa ry to
es t ima te t h e ' c o n c e n t r a t i o n t i m e ' of ca tchments . L loyd -Dav ies deve loped
t h i s i d e a in 1905 and p roposed t h a t t he max imum p e a k r u n o f f f rom a
ca tchmen t o c c u r r e d f o r a s to rm w i t h a d u r a t i o n e q u a l to the c o n c e n t r a t i o n
t ime o f t he ca tchmen t . A common e q u a t i o n used f o r c o n c e n t r a t i o n t ime i s
7
0.385 (1.6) t c = (0.87L3/H)
where t is in hours, L i s the length of catchment i n km and H the drop
in metres, or
t = (11 .6L3/H)0.385 ( 1 - 7 )
where L is i n miles and H i n f t .
The ra t iona l method does not produce a complete hydrograph capable
of rout ing and so un i t hydrograph theory was developed. The theory was
based on the assumption that two un i ts of excess r a i n produce a
hydrograph w i th ordinates twice those of a hydrograph produced by one
un i t of excess r a i n in the same time. The term l inear hydrology i s often
appl ied to th is theory. The time scale i s also incremented l inear ly . Two
successive un i t s of r a i n are assumed to produce two u n i t hydrographs i n
succession which can be added together a t a l l points in time. We thus
have the S-curve hydrograph which i s caused b y an i n f i n i t e l y long storm.
Unit hydrographs do not account for the non-l inear response of a
catchment to excess ra in . Neither i s the concentration time of any
catchment area a unique time, i t depends on the flow rate, as seen fo r
instance, i n the Manning equation ( 2 . 4 7 ) . I n any case the travel time i s
not the same as the reaction time which i s also a function of flow ra te .
Non l inear hydrograph theory on the other hand has met wi th l imi ted
response.
To some extent the error in assuming the travel time is the
concentration time is nu l l i f i ed by assuming a f u l l conduit for computation
of t ravel time. The upstream conduits flow a t a lower r a t e than those
downstream. When the design storm is occurr ing for a downstream conduit,
upstream conduits w i l l be f lowing at less than design capaci ty as the
storm durat ion w i l l be greater than the design storm fo r the upper
conduits. Thus the assumption of a higher flow and velocity than w i l l
occur makes the resu l t ing ra te of concentration more near ly that of the
true hydrodynamic system.
Another misconception i s that the fu l I catchment must contr ibute for
the maximum runoff rate. Besides odd shaped catchments which can by
analysed using the tangent method (Watkins, 1962) a t rue ana lys is would
show many catchments do not contr ibute from the farthest extremity a t
peak flow. This i s not shown up by the ra t iona l method which i nva r iab l y
assumes the en t i re catchment contr ibutes. I t can be demonstrated only i f
soi I-dependent losses are assumed, not rain-dependent losses (e.g. 'C'). I t
is shown in chapter 3 that i f loss i s independent of r a i n f a l l then a
shorter durat ion storm in many cases produce a greater runoff ra te than
one which is of durat ion equal to the time to equ i l ib r ium.
a
rainfall and runof f r a t e s per u n i t a r e a of c a t c h m e n t
rainfal l ra te i l
runoff C i ,
loss = ( I - C ) i l = f 1 I
t ime t
(a) Medium storm
rainfal l runof rate i ra in fa l l r a t e i,
runof f = C i 2
t
( b ) Intense storm assuming same C as in (a) above
rainfall runoff r a t e
I
I 1 loss. f
t
( c ) Intense storm with same loss as (a)
Fig . 1 . 3 Effect of c o n s t a n t C on r u n o f f
9
HYDRODYNAMIC EQUATIONS
The Navier -Stokes e q u a t i o n s f o r i ncompress ib le f l u i d f l o w in t h r e e
d imensions a r e
o(-+U-+V-+W-) au a u au au = x - 32 + I-’(-+-+-) 3 ’ ~ a2u azu
a t ax a y a z ax ax2 ay2 az2
a v av av a~ a z v a z v a z v
ax2 ay2 az2 a t ax a y a~
a 2 w a 2 w a 2 w
p (4u -+v -+w- ) = Y - 3 + p(-+-+-)
o(a”+ua”+va”+wa”) = z - - ap + I-’(--- ~ --)
aY
a t ax ay az az axz ay2 a z 2
(1.9)
(1.10)
where p i s the mass d e n s i t y o f the f l u i d , u , v , w , a r e the v e l o c i t y
components in the x , y , z d i r e c t i o n s r e s p e c t i v e l y , X , Y , Z , a r e the b o d y
forces p e r u n i t vo lume, p i s the p r e s s u r e a n d p i s v i s c o s i t y . I n a d d i t i o n
to these th ree d y n a m i c e q u a t i o n s we h a v e the c o n t i n u i t y e q u a t i o n
= o au + + LW
ax a Y az __ ( 1 . 1 1 )
A l t hough these f o u r e q u a t i o n s t h e o r e t i c a l l y desc r ibe f l ow in a n y s i t u a t i o n ,
f rom the p o i n t o f v i e w o f c i v i l a n d h y d r a u l i c engineers they s u f f e r a
number of d r a w b a c k s . Fo r i n s t a n c e v i scous forces s h o u l d b e r e p l a c e d b y
t u r b u l e n t momentum t r a n s f e r o r e v e n by a sern i -empir ica l f r i c t i o n d r a g
equa t ion , e .g. b y M a n n i n g o r D a r c y .
I t i s g e n e r a l l y p o s s i b l e to work in one d imens ion in c i v i l e n g i n -
e e r i n g h y d r a u l i c s . Then the Navier -Stokes e q u a t i o n s c a n b e r e p l a c e d b y
the St. Venant e q u a t i o n s , w h i c h a l s o compr i se a d y n a m i c e q u a t i o n a n d a
c o n t i n u i t y e q u a t i o n , namely
f - s o = o (1.12) _ - l a v + v a v + a v + s g at g ax a x
(1.13)
where S i s the b e d s lope ( p o s i t i v e down in the x d i r e c t i o n ) , Sf i s t he
energy g r a d i e n t , Q i s the f l ow r a t e , B the s u r f a c e w i d t h , A the cross
sect ional a r e a a n d P the wet ted Der imeter . I t w i l l be seen on c lose
i nspec t i on t h a t t he St. Venant e q u a t i o n s a r e s i m i l a r in m a n y te rms to the
Navier-Stokes eaua t ions.
10
The so lu t i on of the St. Venant equa t ion i s , however, a d i f f i c u l t
enough task f o r the hyd ro log i s t o r c i v i l engineer . The c lass i ca l so lu t i on
i s b y the method of c h a r a c t e r i s t i c s which can e a s i l y be p o r t r a y e d
g r a p h i c a l l y . Computer so lut ion of the equa t ion in v a r i o u s forms is now
more common. Rap id so lu t i on of a f i n i t e d i f f e rence form o f the St. Venant
equat ions i n a s i m p l i f i e d form can e a s i l y be under taken on, f o r instance,
micro computers.
For the m a j o r i t y of o v e r l a n d f low cases and i n many channel a n d
condui t f low s i t u a t i o n s the St. Venant equat ions can be rep laced b y the
fol lowing two equat ions (see chap te r 2 ) .
Con t inu i t y a a + , a v = i ax at e
( 1 . 1 4 )
Dynamics S = Sf ( 1 . 1 5 )
where i i s the i n p u t p e r u n i t a rea of su r face (e.g. excess r a i n f a l l
i n t e n s i t y ) .
These equat ions a r e termed the k inemat i c equat ions. Equat ion (1.15)
merely s ta tes tha t the bed slope can be subs t i t u ted f o r the energy
g rad ien t in a f r i c t i o n equat ion.
For ove r land sheet f low q pe r u n i t w i d t h these equat ions become
( 1 . 1 6 )
( 1 . 1 7 ) m
4 = aY
where i is the excess r a i n f a l l r a t e .
I t i s f u r t h e r a s imple ma t te r to t ransform the k inemat i c equat ions (1.14)
a n d (1.15) i n t o equat ions a p p l i c a b l e to s torage rese rvo i r s w i t h i n t e r l i n k i n g
condui ts :
( 1 . 1 8 ) ah A Q + A- = q at
and A H / L = KQm ( 1 . 1 9 )
Here A i s the r e s e r v o i r su r face area, Q i s the net i n f l ow from
connect ing p ipes a n d q i s the d rawo f f from a r e s e r v o i r w i t h water leve l h .
The second equa t ion i s a p p l i c a b l e to closed condu i t s and i n fact i s s imp le r
t han the open channel k inemat i c equa t ion s ince the v a r i a b l e f low dep th i s
el iminated.
When the common node between condu i t s i s an open r e s e r v o i r the
c o n t i n u i t y equa t ion w i l l p r e d i c t the r a t e o f change i n water l eve l . I f the
condu i t s o r p i p e s connect a t a closed node i t i s necessary to solve
s imul taneously f o r head a t the node a n d f low i n the connect ing p ipes.
1 1
Many methods are ava i l ab le for th is, but the l inear method (Stephenson,
1984b) is pa r t i cu la r l y suitable. That procedure requires minimal data
preparation and solut ion i s faster than the manual node i te ra t i ve
correction procedure of Hardy Cross because i t i s impl ic i t , that is heads
of a l l nodes are solved for simultaneously. The kinematic method of
continuous simulat ion is a versat i le technique for ana lys is of u rban storm
drainage and water supply pipe networks pa r t i cu la r l y w h e n operation of
storage reservoirs i s involved.
The I imi t ing assumptions behind the kinematic method should however
be recal led. Although the assumption that the x-di f ferent ia l terms i n the
dynamic equation i s zero i s cer ta in ly va l i d , the time d i f fe ren t ia l terms
may i n some cases not be zero. This effect i s magnif ied by introducing
closed conduits wi th unvary ing cross-sectional area. Pressure r ises due to
change in flow ra te can be large, g i v i n g r i s e to water hammer.
I n such situations, i.e. when rap id f luctuat ions in flow are
possible, an a l te rna t ive method of analysis, namely elast ic analysis, must
be employed. To analyse a network using the water hammer equations
involves simultaneously solv ing the character ist ics and cont inui ty equation
at each node. Aspects of f r i c t i on damping requ i re pa r t i cu la r attention wi th
th is method. I n pa r t i cu la r the ra t i o of f r i c t ion head loss to water hammer
head can have an important effect on the speed of solution. When the
analyst i s only concerned w i th steady state heads and flows he can
a r t i f i c i a l l y speed convergence by suppressing the wave speed i.e. reducing
the numerical value used i n the computations.
The analyst i s thus a l te r ing the f i t of the mathematical model to the
real system. There are approximations and consequently scope for
adjustment at a number of stages i n the modelling. The fol lowing stages
are related by the ana lys t :
Real system (conduits and reservoirs)
Imagined system (what can be visual ized)
Mathemat ical model (d i f fe ren t ia l equations)
Numerical model ( f i n i t e differences)
Computer model (successive equations)
By adjust ing the imagined system one i s able to speed convergence of
the solution. The f i n i t e differences have to be l imi ted according to the
Courant c r i te r ion (1956) and pa r t i cu la r l y when f r i c t i on i s involved,
another c r i te r ion proposed by Wiley (1970)
At < : A x / c ) ( l - S g A t / 2 v ) ” ‘ ( 1 . 2 0 )
12
Equation (1.20) indicates that f r i c t ion affects the stabi I i t y of
numerical solutions. This is however due to the numerical approximation i n
solv ing the equations exp l i c i t l y ra ther than an i ns tab i l i t y caused by
f r i c t ion . Fr ic t ion has general ly an important role in kinematic theory. I t
relates water depth to flow ra te i.e. i t provides the l i nk between the
cont inui ty equation and the hydrograph. Although f r i c t ion energy loss
relat ionships are well known for stream flow which i s f u l l y turbulent and
sub-surface flow which is laminar, the process of over land flow is not
f u l l y appreciated. Flow depths are small and the dimensions of
roughness are comparable w i th the flow depth. There are complicating
influences such as tortuous flow paths around and over boulders,
vegetation, structures and other surface disturbances. Rain drops are
reported to cause turbulence at lower Reynolds numbers than for conduit
flow. Overton and Meadows (1976) indicate turbulent flow persists for sheet
flow i f the Reynolds number i n terms of p rec ip i ta t ion rate, i L / " = 20 to
2000 where i is the precipi tat ion ra te (m/s), L i s the over land flow path
length and L) i s the kinematic viscosity of the l i q u i d (water ) . This would
indicate that the energy gradient i s proport ional to flow ra te to the power
of m = 5/3 i f the Manning equation (2.47) i s assumed together wi th the
1/6 power law for velocity d is t r ibu t ion . Horton (1938) on the other hand
found m was approximately 2 on na tura l surfaces implying near ly laminar
conditions for uniform flow (constant depth i n the direct ion of f low) .
Actual ly m = 3 for pure laminar flow.
TENSION CONTROLS
\ GRAVITY
_. TIME
Fig. 1.4 Typical f i e ld i n f i l t r a t i on curve
13
I NF I LTRAT I ON
A major component of a stormwater model i s the rout ine to determine
the r a i n f a l l excess. Abstractions or losses are subtracted from input
r a i n f a l l resu l t ing i n the r a i n f a l l excess which must be routed to the basin
out let .
The losses which must be abstracted from r a i n f a l l are:
1
2.
3.
4.
Intercept ion-rainfal I caught by vegetation p r io r to reaching the
ground. The amount caught i s a function of ( a ) the species, age,
and density of vegetation, ( b ) character of the storm, and (c ) the
season of the year. I t has been estimated that i n a r u r a l watershed
as much as 10 to 20 percent of the r a i n f a l l du r ing the growing
season i s intercepted and returned to the atmosphere by evaporation.
Depression storage-water caught in smal I surface pockets and voids
held there u n t i l i t in f i l t ra tes or evaporates.
Evaporation-water returned to the atmosphere through vapor izat ion.
Evaporation is most important when i t i s not r a i n i n g ; i t i s neg l ig ib le
dur ing r a i n f a l l events when a representative ra te i s 0.05 mrn/hr
(0.002 i n /h r ) (Overton and Meadows, 1976).
Inf i l t rat ion-water lost to the soi l . Typ ica l l y , i n f i l t r a t i on i s the
major abstract ion du r ing a r a i n f a l I event. Three d is t inc t processes
are involved: ( a ) the movement of water into the soi l across the
a i r -so i l interface ( i n f i l t r a t i o n ) ; ( b ) the movement of water through
the soi l under the inf luence of g rav i t y and soi l suction (percol-
a t i on ) ; and ( c ) the depletion of the ava i l ab le volume wi th in the
soi l (storage deplet ion).
There are two basic approaches to modell ing r a i n f a l l excess. Each
loss can be modelled separately and the models l inked together, or a
single model can be developed that lumps the important losses together,
usual ly into i n f i l t r a t i on . This la t te r approach i s often followed i n event
Simulation models. Kinematic stormwater models a re mostly event models;
therefore, we a re mostly concerned w i th i n f i l t r a t i on models fo r the ra in -
fall abstract ion model.
A typical f i e ld i n f i l t r a t i o n curve i s shown i n F igure 1.4. I n f i l t r a t i on
begins at an i n i t i a l h igh ra te and decreases w i th time to a steady f i na l
rate. The forces inf luencing the movement of water into and through the
14
soi l are suction and g rav i t y . Dur ing the ear ly stages, the upper soi l
layer i s " th i rs ty " and in f i l t r a t i on is dominated by suction. With time, the
upper centimetre, more o r less, of the soi l surface becomes saturated and
the i n f i l t r a t i on ra te reduces to that ra te a t which water moves through
the saturated soi l . At t h i s point , g rav i t y dominates. As long as the
r a i n f a l l rate exceeds the instantaneous in f i l t r a t i on rate, or water i s
ponded on the surface, i n f i l t r a t i on w i l l continue a t the maximum possible
rate, defined by Horton (1933) as the capacity i n f i l t r a t i on rate. The effect
of r a i n f a l l r a t e on the i n f i l t r a t i on curve i s next examined. Three general
cases for i n f i l t r a t i on dur ing a steady r a i n f a l l were proposed by Mein and
Larson (1973):
Case A : i < k . (The ra in fa l I rate, i, i s less than the saturated soi l
hyd rau l i c conduct iv i ty, k . ) Under th i s condit ion, runoff w i l l not
occur, regardless of r a i n f a l l durat ion, because a l l r a i n f a l l w i l l
i n f i I trate.
Case B: ks < i < f . (The r a i n f a l l r a te i s less than the capaci ty
i n f i l t r a t ion rate, fp, but i s greater than the saturated hyd rau l i c
conduct iv i ty.) For a short durat ion r a i n f a l l , where i remains less
than f a l l the r a i n in f i l t ra tes . But for a r a i n f a l l of long
durat ion, the i n f i l t r a t i on capacity w i l l decrease u n t i l i t equals i ,
and surface ponding occurs.
P
P'
Case C : k < f <i: (The r a i n f a l l r a te i s greater than the i n f i l t r a t i on -__ s p capaci ty.) Under th i s condit ion, runof f occurs.
Cases 0 and C can be considered as two d is t inc t cases; however,
i n f i l t r a t i on often occurs as a two-phase process combining the two cases.
Bodman and Colman (1943) evaluated soi I water d is t r ibu t ion du r ing
i n f i I t ra t ion into a uniform, re la t i ve ly d ry soi I under surface ponding
conditions and establ ished that the typical p ro f i l e can be d iv ided into
four zones as shown in F igure 1.5. The uppermost zone i s the saturat ion
zone and var ies l i t t l e i n thickness, regardless of the total depth of
i n f i l t r a t i on . Immediately below th i s zone, there i s a zone of r a p i d
decrease in the water content, which Bodman and Colman ca l led the
t rans i t ion zone; and below i t , there occurs a zone of near ly constant
moisture ca l led the t ransmit t ing zone. This zone increases i n length in
direct proport ion to the volume of i n f i l t r a ted water. Next, there is the
wett ing zone which moves downward w i th a constant shape as i n f i l t r a t i on
continues. The wett ing zone ends at the wett ing f ront, which i s the
15
boundary between water penetrat ion and soi l at the i n i t i a l moisture
content.
Soil Physics Models
There are two approaches to modell ing the i n f i l t r a t i on process, soi l
physics models and hydrologic models. Soil physics models a re deter-
minist ic models based on the physics of soi l moisture movement, whi le
hydrologic models a re conceptual and are based on a die-away ra te u n t i l
the f i na l steady r a t e i s reached. The advantage of soi l physics models
i s that the parameters are understood and are measurable; the dis-
advantage i s that soil physics models typ ica l l y requ i re a large amount
of data, inc lud ing s i te measures of soi l porosi ty, hyd rau l i c conduct iv i ty,
soi I layer ing, etc. I n comparison, hydrologic models general ly have fewer
parameters, requ i re less da ta and are easier to solve; however, the
parameters are not a l way phys ica l l y interpretable and cannot be
measured, hence they must be establ ished by ca l ib ra t ion . A fu r ther
cr i t ic ism of hydrologic models i s that they oversimpl i fy the i n f i l t r a t i on
process, pa r t i cu la r l y du r ing periods of unsteady r a i n and r a i n f a l I less
than the soi I saturated hydrau l ic conduct iv i ty.
The governing equations for i n f i l t r a t i on are the conservation of mass
and an equation of motion.
r a n
I- W
MOISTURE CONTENT
I I I TRANSMITTING I ZONE I I 3 I WElTINGZONE
I
BODMAN AND COLMAN
I I-
W a n
MOISTURE CONTENT
8; 8, - I I I I I I SATURATED I ZONE I I I I
I GREEN AND AMPT
Fig. 1.5 Comparison of Green and Ampt soi l moisture p ro f i l e w i th Bodman-Colman p ro f i l e
16
The conservation of mass equation i s
a v + a o = o a z at
(1 .21 )
where v i s the specif ic discharge (ve loc i ty ) ver t i ca l l y , 0 i s the vol-
umetric moisture content.
The equation of motion i s based on Darcy 's law for a saturated,
homogeneous soi I ,
dh dz
v = - k - (1 .22 )
where v i s velocity as defined previously, k i s hydrau l i c conduct iv i ty,
h i s the hyd rau l i c head, dh i s the change i n head i n the direct ion of
flow over the length dz. The negative s ign indicates flow i s i n the
direct ion of decreasing head.
Darcy 's law can be generalized to unsaturated flow by expressing
the hyd rau l i c head as a function of soi l tension o r suction, and g rav i t y .
Dur ing the i n i t i a l stages of i n f i l t r a t i on when the water content i s low,
the tension force i s much la rger than the g rav i t y force and the flow
process i s control led by tension. As the pores f i l l , tension i s reduced
and g rav i t y becomes important. The hyd rau l i c head i s then equal to
tension $ p lus g r a v i t y , z .
h = $ + z (1 .23 )
and Darcy's law as appl ied to unsaturated flow i s
( 1 . 24 )
By combining Eqs. 1.23 and 1.24 , we get the governing equation
for one-dimensional, ver t i ca l , unsaturated f low, known as R ichard 's
equation.
( 1 . 2 5 )
where k and $ are both functions of 0 . Due to the nonlinear relat ionship
between hyd rau l i c conduct iv i ty, suction and soi l moisture, there i s no
general ana ly t i ca l solut ion to Eq. 1.25.
The problem is fu r ther complicated by hysteresis in that the
relat ionship between suction and moisture content i s not unique and
s ing le valued. The relat ionship depends on whether the soi l i s wett ing
( i n f i l t r a t i o n i s occurr ing) or d ry ing (d ra inage i s occur r ing) . These
relat ionships are shown i n F igure 1.6. General ly, for a given water
content, suction i s lower dur ing wett ing than dur ing drainage and
minor hysteret ic loops can occur between the main hysteretic loops. The
hysteretic effect i s a t t r ibu ted to ( 1 ) geometric nonuniformity of ind iv idua l
pores, ( 2 ) var ia t ions in contact angle i n wett ing and drainage,
17
( 3 ) en t rapped a i r , a n d ( 4 ) s w e l l i n g ( H i l l e l , 1971). Conduc t i v i t y l i kew ise
e x h i b i t s a h y s t e r e t i c effect.
MOISTURE CONTENT
Fig. 1.6 Typ ica l so i l suct ion - moisture r e l a t i o n
Green and Ampt Model
A conceptual model u t i l i z i n g D a r c y ' s law was proposed b y Green
and Ampt (1911). Many s tud ies, i n c l u d i n g those b y Mein a n d La rson (1973),
have demonstrated the usefulness of the Green a n d Ampt model f o r
model l i n g i n f i l t r a t i o n . As methods fo r measur ing the model parameters
a r e made eas ier , i t can be expected the model w i l l be more w ide ly
app l i ed .
D a r c y ' s law can be w r i t t e n as
v = f = k ( h + L + $ f ) / L f (1.26)
where f i s the i n f i l t r a t i o n r a t e a n d v i s equal to the v e r t i c a l v e l o c i t y ,
h i s the su r face pond ing depth, Lf i s the depth to the we t t i ng f r o n t ,
and qf i s suc t i on at the we t t i ng f r o n t .
f - n
Several assumptions were necessary to w r i t e D a r c y ' s law in the form
of E q . 1.26, namely:
1 . There e x i s t s a d i s t i n c t a n d p rec i se l y d e f i n a b l e we t t i ng f r o n t .
2. Suction at the wet t ing f ron t , ii, f , remains essen t ia l l y constant ,
rega rd less of t ime a n d depth.
18
3. Above ( b e h i n d ) the w e t t i n g f r o n t , t he s o i l i s u n i f o r m l y wet and of
cons tan t h y d r a u l i c c o n d u c t i v i t y k .
4 . Below ( i n f r o n t o f ) the w e t t i n g f r o n t , t he s o i l m o i s t u r e con ten t i s
r e l a t i v e l y u n c h a n g e d f r o m i t s i n i t i a l m o i s t u r e con ten t , 0 _.
These assumpt ions , when checked a g a i n s t t h e a c t u a l soi I m o i s t u r e p r o f i l e
of Bodman and Colman i l l u s t r a t e t h e a p p r o x i m a t e n a t u r e of t h e Green
a n d Ampt model. T h i s i s shown in F i g u r e 1.5.
The a c c u m u l a t e d i n f i l t r a t i o n d e p t h , F, c a n b e o b t a i n e d b y i n t e g r a t -
i n g E q . 1.26.
f = dF /d t = k ( h + Lf + $ f ) / L f
o r more d i r e c t l y f r o m
F = (0 - O i ) L f = A O L f (1.28)
where 9 i s t he s a t u r a t e d m o i s t u r e con ten t and 0 i s t he i n i t i a l m o i s t u r e
con ten t . The measure o f m o i s t u r e con ten t , 0 , i s a v o l u m e t r i c measure ,
t h e r e f o r e A0 i s c a l c u l a t e d w i t h t h e r e l a t i o n s h i p
A0 = (0 - 0.) = r$ ( l - Si ) (1.29)
where 4 i s t he soil p o r o s i t y and S. i s t he i n i t i a l deg ree o f s a t u r a t i o n .
A p p l y i n g the r e l a t i o n s h i p s i n Eqs . 1.28 and 1.29 to Eq. 1.27 and
(1.27)
I
i n t e g r a t i n g t o o b t a i n F g i v e s
= F - ( $ A O ) I n [ I + F / ( I ~ ~ A O + h A 0 ) l k t f
(1.30)
w h i c h i s a n o n l i n e a r e q u a t i o n i m p l i c i t in F and t. An e x p l i c i t f o rmu-
l a t i o n to s o l v e f o r , t he i nc remen ta l i n f i l t r a t i o n vo lume d u r i n g an
inc remen ta l t ime i n t e r v a l , A t , i s o b t a i n e d b y r e w r i t i n g Eq. 1.30.
T h i s g i v e s
k A t - 2Ft 1 2 F - k A t (1.31) t
t / ( ) + 2 k A t ( h t A 0 + J, A 0 + F ) 2 f t AF =
where A F i s the i nc remen t i n t o t a l i n f i l t r a t i o n f r o m t ime t t o t ime
t + A t , a n d F and ht a r e the i n f i l t r a t i o n and p o n d e d dep th , r e s p e c t i v e l y ,
a t t ime t. The re fo re , t he t o t a l i n f i l t r a t i o n a f t e r t h e t ime inc remen t i s
Ft+At t o r
F t+ At
where i i s t he r a i n f a l l i n t e n s i t y . I f A F < i A t f o r a t ime s tep t h e n excess
i n t e n s i t y , ie, occu rs .
t
= F + AF; i f AF < i A t + h t (1 .32a)
= F t + iA t + h t ; i f AF > iA t + ht (1 .32b)
The inc remen ta l c u m u l a t i v e i n f i l t r a t i o n e q u a t i o n , E q . 1.32, was deve loped
assuming u n i f o r m s o i l p r o p e r t i e s . However , i t c a n b e a p p l i e d to l a y e r e d
19
so i ls , assuming each l a y e r has un i fo rm p roper t i es . The r e q u i r e d so i l
p rope r t i es , i .e . K,, if, 4, a n d S i , a n d the th ickness, d, must be known
for each laye r . A f te r computing the i n f i l t r a t i o n d u r i n g each t ime
i n t e r v a l , the cumu la t i ve i n f i l t r a t i o n volume, F, i s compared w i t h the
s torage capac i t y of uppermost l a y e r not yet sa tu ra ted . Once a l a y e r
becomes sa tu ra ted , the i n f i l t r a t i o n r a t e i s cont ro l led b y the condi t ions
in tha t l a y e r o r the nex t lower l a y e r , whichever g i ves the sma l le r ra te .
Bouwer (1966) de f i ned the Green a n d Ampt parameter k to be " the
ac tua l h y d r a u l i c c o n d u c t i v i t y in the wetted zone," wh ich i s less than
the sa tu ra ted h y d r a u l i c c o n d u c t i v i t y , ks . He concluded, based on
p rev ious work, t ha t k may be taken as about 0.5kS. The sa tu ra ted
h y d r a u l ic c o n d u c t i v i t y can be determined b y severa l s t a n d a r d l abo ra to ry
tests.
E f fec t i ve s a t u r a t i o n i s de f i ned as 0 - 0
r (1.34)
where Er i s the r e s i d u a l mois ture content, Brooks a n d Corey (1966)
observed a s t r a i g h t I ine r e l a t i o n s h i p
se = (vJ / J I ) ' " ; f o r J, > jib (1.35)
where J, i s c a p i l l a r y pressure head (suc t i on ) a t a g i v e n so i l mois ture
content, 0 ; $ i s termed b u b b l i n g pressure a n d i s de f i ned a t the i n t e r -
cept of a s t r a i g h t l i n e p l o t of e f fec t i ve s a t u r a t i o n a n d c a p i l l a r y pressure
head; a n d B i s an index of the po re size d i s t r i b u t i o n . Porous media
composed of s i n g l e g r a i n , ma te r ia l have p r i m a r y po ros i t y (po ros i t y
cons is t ing o n l y of spaces between the g r a i n s ) a n d tend to h a v e smal l
va lues of 8. Media h a v i n g secondary po ros i t y (po re spaces a l so a v a i l a b l e
fo r f low w i t h aggregates) have l a r g e va lues (>1.0) .
b c
b
The we t t i ng f ron t suct ion i s est imated u s i n g the fo l l ow ing r e l a t i o n s h i p
11 'b f = __ - r l - 1 2
(1.36)
where rl = 2+3/B
Hydro log ic I n f i l t r a t i o n Models
Horton (1939) proposed an i n f i l t r a t i o n equa t ion to represent the
t yp i ca l i n f i l t r a t i o n cu rves observed i n doub le - r i ng i n f i l t rometer tests.
I n these experiments, the water i s cont inuously ponded above the so i l ;
therefore, the supp ly i s not l i m i t i n g a n d i n f i l t r a t i o n proceeds a t the
maximum po ten t i a l ra te. He observed tha t the i n f i l t r a t i o n r a t e was
i n i t i a l l y h i g h a n d decreased in t ime to a steady f i n a l ra te . The die-away
fol lowed a nega t i ve exponent ia l v e r y c losely . H is equa t ion i s
20
(1.37) -k t f = f + ( f o - f c ) e
where f i s the c a p a c i t y i n f i l t r a t i o n r a t e a t t ime t , fo a n d f a r e the
i n i t i a l a n d f i n a l i n f i l t r a t i o n r a t e s , a n d k i s the i n f i l t r a t i o n cons tan t
w h i c h i s a l l e g e d l y a f u n c t i o n of s o i l and vege ta t i on . I n theo ry t h i s
e q u a t i o n assumes the a i r - s o i l i n t e r f a c e i s s a t u r a t e d a t a l l t imes. I n
p r a c t i c a l terms t h i s means t h a t i t i s assumed the r a i n f a l l r a t e i s a l w a y s
g r e a t e r t h a n i n f i l t r a t i o n c a p a c i t y r a t e s , a n d hence some p o n d i n g w i l l
a l w a y s r e s u l t . T h i s i s a m a j o r d i s a d v a n t a g e i n the use of H o r t o n ' s model
s i n c e n a t u r a l r a i n f a l I r a t e s a r e h i g h l y v a r i a b l e a n d the re fo re f r e q u e n t l y
f a l l below the c a p a c i t y r a t e s . T h i s may n o t be a p rob lem w i t h h i g h
i n t e n s i t y d e s i g n r a i n f a l l s o r r a i n f a l l s d i s t r i b u t e d in t ime to a l w a y s
exceed the c a p a c i t y i n f i l t r a t i o n r a t e s .
C
Hol ton (1961) proposed a conceptual model of i n f i l t r a t i o n b a c k e d b y
s u b s t a n t i a l f i e l d exper imen ta t i on . He recogn ized f r o m soi I p h y s i c s a s
the po res f i l l , the i n f i l t r a t i o n r a t e d i e s a w a y a n d approaches a s t e a d y
f i n a l r a t e . The f i n a l r a t e of i n f i l t r a t i o n f c was assoc ia ted w i t h the
g r a v i t y f o rce a t f i e l d c a p a c i t y ( a n d i s assumed to e q u a l the s o i l
s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y , k s ) . He then f o r m u l a t e d a model to
r e l a t e c a p a c i t y i n f i l t r a t i o n r a t e to the a v a i l a b l e s o i l mo is tu re s t o r a g e
volume r e m a i n i n g a t a n y t ime, F a s P '
f = aF" (1.38) P + f c
The pa ramete rs a a n d n were de te rm ined e x p e r i m e n t a l l y f rom i n f i l t r omete r
p l o t d a t a . The exponent was f o u n d to be abou t 1.4 f o r a l l p l o t s s t u d i e d
a n d the c o e f f i c i e n t s v a r i e d f rom 0.2 to 0.8 f o r the so i l - cove r complexes
s t u d i ed.
REFERENCES
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Bodman, G.B. a n d Colman, E.A. 1943. M o i s t u r e a n d energy c o n d i t i o n s d u r i n g d o w n w a r d e n t r y of wa te r i n t o so i l s . Proc. Soi l Science SOC. of Amer ica, Vol. 7, pp 116-122.
Bo rah , D.K. a n d P r a s a d , S.N., 1982. Shock s t r u c t u r e in k i n e m a t i c wave r o u t i n g . I n Ra in fa l l -Runo f f Re la t i onsh ips , Ed t . S ingh, V.P., Water Resources Pub1 i ca t ions, Co lo rado , 582 pp.
Bouwer, H. 1966. R a p i d f i e l d measurement of a i r e n t r y v a l u e a n d h y d r a u l i c c o n d u c t i v i t y of soi I as s i g n i f i c a n t pa ramete rs in f l ow system a n a l y s i s . Water Resources Research, Vol. 2, No. 4 , pp 729-738.
B r a k e n s i e k , D.L. , 1966. H y d r o d y n a m i c s of o v e r l a n d f l ow a n d n o n - p r i s m a t i c channe ls . T r a n s . ASAE 9 ( 1 1 , pp 119-122.
Brooks, R.H. a n d Coley, A.T. 1966. P r o p e r t i e s of po rous med ia a f f e c t i n g f l u i d f l o w . Jou rna l of the I r r i g a t i o n a n d D r a i n a g e D i v i s i o n , ASCE, V o l . 92, No. I R 2 , pp 61-88.
21
Constan t i n i d e s , C.A. and Stephenson, D., 1982. D imens ion less h y d r o g r a p h s u s i n g k i n e m a t i c t h e o r y , Repor t 5/1982. Water Systems Research Pro- gramme, U n i v e r s i t y o f t h e W i t w a t e r s r a n d , Johannesburg .
Couran t , R., F r i e d r i c h s , K. and Lewy , H., 1956. On the p a r t i a l d i f f e r - ence e q u a t i o n s o f ma themat i ca l p h y s i c s . N.Y. U n i v . I n s t . Ma ths .
C ro ley , T.E. and Hun t , B . , 1981. M u l t i p l e v a l u e d and non-convergen t s o l u t i o n s in k i n e m a t i c cascade mode ls , J. H y d r o l . , 49, pp 121-138.
Dunne, T . , 1978. F i e l d s t u d i e s of h i l l s l o p e f l o w processes. Ch. 7, H i l l - s lope H y d r o l o g y , Ed. K i r k b y , M.J., John Wi ley , N.Y.
Eag leson, P., 1967. A d i s t r i b u t e d l i n e a r model for p e a k ca tchmen t d i s - charge. I n t l . H y d r o l . Symp., Co lo rado S t a t e U n i v . , F o r t C o l l i n s , pp 1-18.
Freeze, R.A., 1972. Ro le o f s u b s u r f a c e f l o w in g e n e r a t i n g s u r f a c e r u n o f f . 2, Ups t ream source a reas . Water Resources Research , 8 ( 5 ) , pp 1272- 1283.
G a l l a t i , M. and Maione, U., 1977. P e r s p e c t i v e o n m a t h e m a t i c a l mode ls o f f l ood r o u t i n g , in M a t h e m a t i c a l Mode ls f o r Su r face Water H y d r o l o g y , E d t . C i r ian i , T.A., Ma ione, U. and W a l l i s , J.R., W i ley In te rsc ience , 423 pp.
Greco, F. and P a n a t t a n i , L., 1977. Numer i ca l s o l u t i o n methods o f t h e St. Venant e q u a t i o n s . I n Mathemat i c a l Mode ls f o r S u r f a c e Water H y d r o l o g y , Ed t . C i r i an i , T.A., Ma ione, U. and W a l l i s , J.R., W i ley In te rsc ience , 423
PP . Green, I .R.A., 1984. WITWAT s t o r m w a t e r d r a i n a g e p r o g r a m . Repor t 1/1984,
Water Systems Research Programme, U n i v e r s i t y o f t h e W i t w a t e r s r a n d , Johannesburg .
Green, W.H. and Ampt, G.A. 1911. S t u d i e s o f s o i l p h y s i c s , 1 . The f l o w o f a i r and w a t e r t h r o u g h so i l s . J. o f A g r i c u l t u r e Science, V o l . 4, No. 1 , pp 1-24
Henderson, F.M. and Wooding, R.A., 1964. O v e r l a n d f l o w and g r o u n d - w a t e r f l o w f r o m s t e a d y r a i n f a l l of f i n i t e d u r a t i o n . J. Geophys. Res. 69 ( 8 ) pp 1531-1539.
H i l l e l , D. 1971. Soi I and w a t e r - p h y s i c a l p r i n c i p l e s and processes, Academic Press
Holton, H.N. 1961. A concept of i n f i l t r a t i o n es t ima tes in wa te rshed e n g i n e e r i n g , U.S. Dept. o f A g r i c u l t u r e , A g r i c . Research Serv i ce , No. 41-51, Wash ing ton , D.C.
Hor ton , R.E. 1933. The r o l e o f i n f i l t r a t i o n in t h e h y d r o l o g i c cyc le . T r a n s . o f the Amer i can Geophys ica l Un ion , H y d r o l o g y Papers , pp 446-460
Hor ton , R.E., 1938. The i n t e r p r e t a t i o n and a p p l i c a t i o n o f r u n o f f p l o t exper imen ts w i t h re fe rence to s o i l e ros ion p rob lems . Proc . So i l Sci . SOC. Am. 3, pp 340-349.
Hor ton , R.E. 1939. A p p r o a c h t o w a r d a p h y s i c a l i n t e r p r e t a t i o n o f i n f i l t a t i o n c a p a c i t y . Proc . So i l Science SOC. o f Amer ica , Vol . 5, pp 399-417.
Hor ton , R.E., Leach , H.R., and Van V I i e t , R . , 1934, L a m i n a r sheet f l o w . Amer. Geophys. Un ion , T rans . , P a r t I I , pp 393-404.
I w a g a k i , Y . , 1955. F u n d a m e n t a l s t u d i e s o n t h e r u n o f f a n a l y s i s by c h a r a c t e r i s t i c s . 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 , B u l l e t i n 10, Kyo to U n i v . 25 pp.
Keu legan, G.H., 1945. S p a t i a l l y v a r i e d d i s c h a r g e o v e r a s l o p i n g p l a n e . Amer. Geophys. Un ion T r a n s . P a r t 6, pp 956-959.
K i b l e r , D.F. a n d Woolh iser , D.A., 1970. The k i n e m a t i c cascade a s a h y d r o l o g i c a l model. Co lo rado S ta te U n i v . p a p e r 39, F o r t C o l l i n s , 25 pp.
Kouwen, N., L i , R.M. and Simons, D.B., 1980. F low r e s i s t a n c e i n vege- ta ted w a t e r w a y s . M a n u s c r i p t , Co lo rado Sta te U n i v e r s i t y , F o r t C o l l i n s .
L i g h t h i l l , F.R.S. and Whi tham, G . B . , 1955. On k i n e m a t i c waves , I , F l o o d measurements i n l o n g r i v e r s . Proc . Roya l SOC. o f London , A, 229, pp 281-31 6.
22
L loyd -Dav ies , D.E., 1905. The e l i m i n a t i o n o f s to rm w a t e r f rom sewerage
Massau, J., 1889. L ' i n t C g r a t i o n g r a p h i q u e . Assoc. l n g e n i e u r s S o r t i s des Ecoles SpCciales des G a r d , Anna les . 435 pp.
Me in , R.G. and L a r s o n , C . L . 1973. M o d e l i n g i n f i l t r a t i o n d u r i n g a s t e a d y r a i n . Water Resources Research , Vol . 9, No. 2, pp 384-394.
M o r r i s , E.M. and Woolh iser , D.A., 1980. Uns teady one-d imens iona l f l o w o v e r a p l a n e : P a r t i a l e q u i l i b r i u m and recess ion h y d r o g r a p h s . Water Resources Research , 16 ( 2 ) , pp 355-360.
Over ton , D.E. and Meadows, M.E., 1976. S to rmwate r m o d e l l i n g , Academic Press, 358 pp.
Rossmi l l e r , R.L. , 1980. The R a t i o n a l f o r m u l a r e v i s i t e d . Proc . I n t l . Symp. Storm Runo f f , U n i v . of K e n t u c k y , L e x i n g t o n .
Rovey, E.W., Woo lh iser , D.A. and Smi th , R.E., 1977. A d i s t r i b u t e d k i n e m a t i c model o f u p l a n d watersheds . H y d r o l o g y P a p e r 93, Co lo rado S ta te U n i v . , F o r t Col t i ns , 52 pp.
Schaake, J.C., 1975. Su r face wa te rs . Rev iew o f g e o p h y s i c s and space p h y s i c s 13 (13 ) pp 445-451.
S i n g h , V.P., 1977. C r i t e r i o n t o choose s tep l e n g t h f o r some n u m e r i c a l methods used in h y d r o l o g y . J. H y d r o l . , 33, p p 287-299.
Skaggs, R.W., 1982. I n f i l t r a t i o n . Ch. 4, H y d r o l o g i c a l M o d e l l i n g o f Sma l l Watersheds, Ed t . Haan , C.T., Johnson, H.P. and B r a k e n s i e k , D.L., ASAE Monog. 5.
scs ( s o i l Conserva t i on S e r v i c e ) 1972. N a t i o n a l E n g i n e e r i n g Handbook , Secn. 4. H y d r o l o g y , Wash ing ton , D.C.
Stephenson, D. , 1983. The e f f e c t s of u r b a n i z a t i o n . Course on Modern S to rmwate r D r a i n a g e P r a c t i c e , SAICE, Cape Town.
Stephenson, D. , 1984a. K i n e m a t i c s t u d y o f e f f e c t s of s to rm d y n a m i c s of r u n o f f h y d r o g r a p h s . Water S.A. Vol . 10, No.4, Oct. 1984. p p 189-196.
Stephenson, D. , 1984b. K inemat i c a n a l y s i s f o r d e t e n t i o n s to rage . EPA/Users g r o u p mee t ing . D e t r o i t .
Wa tk ins , L.H., 1962. The Des ign of U r b a n Sewer Systems, Road Research Techn. p a p e r 55, HMSO, London .
Wooding, R.A., 1965a. A h y d r a u l i c model f o r t h e ca tchment -s t ream p r o b l e m 1 , K i n e m a t i c wave theo ry . J. H y d r o l o g y , 3. pp 254-267.
Wooding, R.A., 1965.b. A h y d r a u l i c model f o r t he ca tchmen t -s t ream p rob lem, I I , Numer i ca l so lu t i ons , J. H y d r o l . 3. pp 268-282.
Wooding, R.A., 1966. A h y d r a u l i c model f o r t h e ca tchment -s t ream p r o b l e m I I I , Compar i son w i t h r u n o f f o b s e r v a t i o n s , J. H y d r o l o g y , 4, p p 21-37.
Woolhiser, D.A. and L i g g e t , J.A. 1967. Uns teady one-d imens iona l f l o w o v e r a p l a n e - The r i s i n g h y d r o g r a p h . Water Resources Research , 3 ( 3 ) , pp 753-771.
Wy l i e , E.B., 1980. Uns teady f r e e s u r f a c e f l o w compu ta t i ons . Proc . ASCE, 96 (HY 1 1 ) pp 2241-2251.
systems. M i n . Proc. I n s t n . C i v i l Engnrs . , 164 (2 ) pp 41-67.
23
CHAPTER 2
ANALYSIS OF RUNOFF
INTRODUCTION
In th is chapter a s impl i f ied descript ion of the r a i n f a l l - runof f
mechanism is presented, i.e. one which can be described i n equation form.
The concept of mass balance whereby input equals outflow p lus change i n
storage, is appl ied to simple catchments. The bui ld-up of water depth
over the catchment when a storm occurs i s described as well as the
mechanism whereby runoff occurs. The relat ionship between water depth
and flow ra te forms an important p a r t in the predict ion of flow so the
equation of motion ( i n fact only a simple flow resistance equation i n the
case of kinematic f low) i s introduced.
This simple ana lys is i s confined to a rectangular plane catchment
s loping uni formly down i n the direct ion of flow, and flow is assumed
overland. The equations of con t inu i ty and flow are thus p a r t i c u l a r l y
simple. Nevertheless the o r ig in of and the assumptions behind the
simpli f icat ions are presented. A simple demonstration of the app l i cab i l i t y
of the kinematic equations is also given. Later other components of flow
e.g. sub-surface flow (Beven, 1982) and a more prac t ica l assessment of the
contr ibut ion to streamflow are introduced w i th model I ing. The di f ferent-
iat ion of surface and subsurface flow i s often more complicated than
assumed here (Dunne, 1978).
DYNAM I C EQUAT I ON5
The equations governing unsteady, one-dimensional over land and open
channel flow are der ived by app ly ing the pr inc ip les of conservation of
mass and momentum to elemental f l u i d control volumes. One-dimensional
equations ac tua l l y describe the change i n streamflow i n two dimensions:
vert ical and longi tudinal . They are c lassi f ied as one-dimensional since
only one spat ia l va r iab le occurs as an independent var iable.
The important assumptions are:
1 . The water surface p ro f i l e var ies g radua l l y , which i s equivalent to
s ta t ing the pressure d is t r ibu t ion is hydrostat ic, i .e. , ver t i ca l
accelerations are smal I ;
2. Resistance to flow can be approximated by steady flow formulae;
3. The veloci ty d is t r ibu t ion across the wetted area can be represented
with the cross-sectional average veloci ty;
4 . Momentum car r ied to the strearnflow from la te ra l inf low is neg l ig ib le ;
and
5. The s1oDe of the channel i s small.
I n addi t ion, for t h i s der ivat ion, the channel i s assumed rectangular.
This s impl i f ies the mathematics involved and has l i t t l e effect on the f i na l
form of the governing equations.
Conservation of Mass
The cont inui ty p r i nc ip le states that the net mass inf low to a control
volume must equal the ra te of change of mass stored w i th in the control
volume. Consider the elemental f l u i d volume shown in Figure 2.1, where Q
i s the volumetric f lowrate in m3/s or cfs, q. is the lateral inf low r a t e i n
rn3/s per rn or cfs per foot length of channel, y and A are depth and cross
sectional area of flow in metres and square metres (feet and square
feet), respectively, 0 i s the slope of the channel w i th respect to the
Fig. 2.1 Der ivat ion of cont inui ty equation
horizontal measured as an angle, and x and t a re the space and time
coordinates in metres ( feet) and seconds. The total inf low to the section is
Inf low = Q + qiAX (2.1 1 and the total outflow is
(2.2) Outflow = Q + 22 A X ax
25
The change i n volume stored in the section i s equal to the change i n
cross-sectional area of flow mul t ip l ied by the length of the section.
( 2 . 3 ) aA =Ax
Change i n volume stored =
Combining these quant i t ies according to the above stated pr inc ip le ,
d i v id ing by Ax, and rear rang ing , y ie lds the cont inui ty equation
(2.4) ax a t
Conservation of Momentum
This second equation i s given by Newton's second law of motion
which states that the ra te of change of momentum is equal to the app l ied
forces. The app l ied forces, as seen i n Figure 2.2, are ( 1 ) pressure, ( 2 )
g rav i t y , and (3) resist ive f r i c t iona l forces.
w t Fig. 2.2 Der ivat ion of momentum equation
Consider forces i n the downstream direct ion as posit ive. The pressure
downslope acts opposite to the pressure upslope and upon summing, the net
pressure force becomes
-pgA(aY/ax) A X
where p i s the mass density of water and g i s the grav i ta t iona l acceler-
a t ion.
S imi la r ly , i t can be shown that the g r a v i t y or weight force ac t ing on
the volume of water in the section i s g iven by pgAAx tan0
26
where, for gradual ly var ied f low, tan !! closely corresponds to the channel
slope, S o , and may be expressed as such. This i s ca l led the small slope
approxi mat ion.
F ina l l y , the f r i c t i on force act ing to re ta rd the flow i s expressed i n
terms of an average shear stress
- TPAX
where T i s shear stress and P i s wetted perimeter. From the relat ionship
formed by equat ing head (energy) loss to the work done by the shear
force we know that T = YRSf, where Sf i s the f r i c t ion slope and Y i s the
un i t weight of l i qu id . Subst i tut ing for T , and reca l l ing that R=A/P, the
fol lowing expression for the f r i c t i on force i s obtained.
- VRS PAX = - YS AAx (2 .5) f f The resul tant force on the f l u i d volume i n the direct ion of flow is the
summat ion of the three appl ied external forces.
PgAAx [ - (ay /ax) + so - s f l The change in momentum consists of two par ts , the local or temporal
momentum change and the spat ia l or convective momentum change. The
local momentum of the f l u i d i s pAAx v , and the local change i s jus t the
time der iva t ive
The spat ia l change i n momentum i s the ra te of momentum change across the
control surface. The momentum f l u x through the control surface i s p v Z A ,
and the spat ia l change i s the x-der ivat ive
- (Pv’A) L x = ( 2 Av .E + v z 2 ) A x p (2 .7) a ax ax ax
The total momentum change i s the sum of the temporal and spat ia l
momentum changes.
A X P ( A .LY + v z),+ V A X P ( V - aA + ZA 2) at at a x ax
Subst i tut ing the fol lowing equivalence from cont inui ty * a v + v - aA 5 q , - - aA
ax ax I a t ( 2 . 8 )
allows the r a t e of momentum change to be wri t ten as
Equating th i s expression with the summation of external forces gives
the fol lowing famil i a r form for the conservation of momentum equation.
- ax g ( S o - S ) - Vqi at ax f A
(2 .9) a v + v - a v + g & =
where 5 i s bed slope, S i s f r i c t i on slope and R i s hyd rau l i c rad ius and
is equal to A/P. f
27
Eqs. 2.4 a n d 2.9 c a n b e made a p p l i c a b l e t o a n y c ross sec t i on f o r
b o t h o v e r l a n d and open c h a n n e l f l o w , t h o u g h s t r i c t l y they a p p l y to
r e c t a n g u l a r c h a n n e l s o n l y in t h e p r e s e n t fo rm.
These e q u a t i o n s a r e n o n l i n e a r , h y p e r b o l i c , p a r t i a l d i f f e r e n t i a l
e q u a t i o n s a n d r e p r e s e n t a n o n l i n e a r , d e t e r m i n i s t i c , d i s t r i b u t e d , t ime
v a r i a n t system. They a r e sometimes r e f e r r e d to as the St. Venan t
eaua t ions.
S l M P L l F I ED EQUATIONS
E q u a t i o n s 2 . 4 and 2 . 9 a r e accep ted as f u l l y d e s c r i p t i v e o f one d imen-
s i o n a l o v e r l a n d a n d open c h a n n e l f l ow r o u t i n g . These e q u a t i o n s d e s c r i b e
b o t h t h e f o r w a r d or downs t ream w a v e p r o p a g a t i o n c h a r a c t e r i s t i c s as we1 I
as the b a c k w a r d o r u p s t r e a m c h a r a c t e r i s t i c s . I t i s assumed t h a t f l o o d
waves in s t reams move downs t ream and s i n c e h i l l s l o p e r u n o f f i s a l w a y s
downh i I I , t he b a c k w a r d c h a r a c t e r i s t i c s a r e s i m p l y b a c k w a t e r e f fec ts , and
in some f l o w r o u t i n g i ns tances , t hey c a n h a v e s u b s t a n t i a l impac t a n d
c o n t r o l o n t h e f l o w . As such, these e q u a t i o n s a r e k n o w n g e n e r a l l y a s t h e
dynamic w a v e e q u a t i o n s . As a r u n o f f h y d r o g r a p h passes t h r o u g h a c h a n n e l
r e a c h , t he comb ined e f fec ts o f c h a n n e l i r r e g u l a r i t y , poo l a n d r i f f l e
p a t t e r n s , n a t u r a l and manmade r o u g h n e s s a n d g r a v i t y f o rces a c t to r e d u c e
the h y d r o g r a p h p e a k a s s u m i n g l a t e r a l i n f l o w i s i n s i g n i f i c a n t w h i l e
l e n g t h e n i n g the t ime base. T h a t i s , t h e peak o f t he h y d r o g r a p h i s
a t t e n u a t e d w h i l e the shape i s d i s p e r s e d in t ime ( a l s o in s p a c e ) . T h e
d y n a m i c w a v e e q u a t i o n s accoun t we1 I f o r h y d r o g r a p h a t t e n u a t i o n . However ,
two d r a w b a c k s to the who lesa le g e n e r a l use o f these e q u a t i o n s a r e the
l a r g e data r e q u i r e m e n t s and the necess i t y f o r n u m e r i c a l i n t e g r a t i o n . Ve ry
o f ten , b a s e d on channe l geomet ry a n d a l i g n m e n t and f l o o d wave
c h a r a c t e r i s t i c s , i t i s p o s s i b l e to make v a l i d s i m p l i f y i n g assumpt ions t h a t
a l l o w one to u t i l i z e app r -ox ima t ions to the d y n a m i c w a v e e q u a t i o n s . When
t h i s i s p o s s i b l e , a d v a n t a g e s in te rms o f ease o f s o l u t i o n and d a t a
r e q u i r e m e n t s arc? o f t e n r e a l i z e d .
Two a p p r o x i m a t i o n s t h a t h a v e f o u n d w i d e a p p l i c a t i o n i n e n g i n e e r i n g
p r a c t i c e a r e t h e diffusion and kinematic wave mode ls . The d i f f u s i o n w a v e
model assumes t h a t t he i n e r t i a te rms in t h e e q u a t i o n o f mo t ion , Eq . 2 . 9 ,
a r e n e g l i g i b l e compared w i t h t h e p r e s s u r e , f r i c t i o n , a n d g r a v i t y terms.
Thus , t h e d i f f u s i o n model e q u a t i o n s a r e c o n t i n u i t y , Eq. 2 . 4 , and the
f o l l o w i n g s i m p l i f i e d f o r m o f t he c o n s e r v a t i o n of momentum e q u a t i o n .
( 2 . 1 0 )
28
For pr ismat ic channels, Eqs. 2.4 and 2.10 are often combined into the
s ingle equation
aQ a 2 Q - a Q + c - = at ax DaxZ ( 2 . 1 1 )
where c i s the wave celer i ty i n m/s ( fps ) and D i s a hydrograph
dispersion coefficient i n m’/sec ( f t * /sec) . Because Eq. 2.11 i s of the form
of the classical advection-dif fusion equation, i t i s commonly ca l led the
di f fusion wave model.
The kinematic model fu r ther assumes the pressure term i s neg l ig ib le ,
reducing Eq. 2.10 to
so = Sf (2 .12)
which means the equation of motion can be approximated by a uniform flow
formula of the general form
Q = ay (2 .13)
where a,b a re constants.
b
Although approximat ions, both the di f fusion and kinemat ic models have
been shown to be f a i r l y good descript ions of the physical phenomemona i n
a var ie ty of open channel and over land flow rou t ing cases. The kinematic
model has been successfully appl ied to over land f low, to small streams
dra in ing up land watersheds, and to slow-rising f lood waves. This la t te r
case occurs both in major streams such as the Mississippi River when long
durat ion f lood hydrographs resu l t ing from, as an example, spr ing snowmelt
in the U.S. Midwest and Canada, and i n small streams where the
streamflow hydrograph nesul ts p r i nc ipa l l y from lateral stormwater inf low.
THE KINEMATIC EQUATIONS
For over land flow and in many channel flow situations, some of the
terms in the dynamic equation (2 .9) are ins ign i f i can t . Neglecting the qi
component one can wr i te the equation as
(2 .14)
The order of magnitude of each of the f i ve terms i s evaluated below
for a shallow stream. I f the bed slope ( 2 ) is 0.01, the longi tudinal ra te
of change of water depth ( 3 ) i s un l i ke l y to exceed O.lm/lOOm = 0.001. The
longi tudinal velocity gradient term ( 4 ) w i l l also be less than
(lm/s/lOm/s‘)(lm/s/lOOm) = 0.001, and the time ra te of change i n
velocity term ( 5 ) w i l l i n a l l p robab i l i t y be less than ( l / l O ~ ( l / l O O s ~ =
0.001.
29
Terms ( 3 ) , ( 4 ) and ( 5 ) a re therefore at least an order of magnitude
less than (2 ) for depths up to lm, and fo r flow depths less than O.lm
they w i l l be two orders of magnitude less. Those terms can therefore be
neglected for the major i ty of over land flow problems. The inaccuracy i n
solutions orni t t ing these terms for runoff hydrographs was evaluated by
various researchers:
Woolhiser and Liggett (1967) investigated the accuracy of the
kinematic approximation and found i t to be very good i f the dimensionless
parameter for planes SoL/yLFL’ i s greater than 20 and reasonable i f
greater than 10. yL i s the depth at the lower end of the plane of length
L and slope So and FL is the Froude number VL/ (gyL)? . i.e. gSoL/VL2
> 10. Morr is and Woolhiser (1980) and Woolhiser (1981) later found the
addi t ional c r i te r ion S L/y > 5 i s also required.
1
O L The resu l t ing s impl i f ied dynamic equation omit t ing terms ( 3 ) , (4) and
(5 ) simply states that the f r i c t ion gradient i s equal to the bed gradient.
The f r i c t ion gradient can be evaluated using any sui table f r i c t ion
equation, e.g. that of Manning. The two equations referred to as the
kinematic equations are thus the cont inui ty equation which per un i t width
of over land flow becomes
(2.15)
and a f r i c t i on equation of the form q = a ym
where m is a coefficient and a is a function of the water propert ies,
surface roughness, bed slope and g rav i t y . Equations (2.15) and (2.16)
apply to a wide f l a t bottom channel o r over land flow. The flow q is per
un i t width and flow depth i s y .
(2.16)
The quasi-steady flow approximation was o r ig ina l l y termed the
kinematic wave approximation since waves can only t rave l downstream and
are represented en t i re ly by the cont inui ty equation. Since the dynamic
forces are omitted, the Froude number F = v / J ( g y ) is i r re levant , and i n
fact that the wave speed c i s not given by C = but may be derived
by f ind ing dx/dt for which dy/dt = ie
rn- 1 From the f r i c t i on equation (2.16) aq = dy = mcly a Y ax -TT z a2
Substi tut ing into the cont inui ty equation y ie lds
(2.17)
(2.18)
(2.19)
but Since ie = dy/dt , the left hand side of t h i s equation must also equal
dy/dt.
30
(2.20) m- 1 There fo re
w h i c h i s t h e speed a t w h i c h a w a v e o f u n v a r y i n g a m p l i t u d e ( i f i = 0 )
t r a v e l s down the p l a n e .
S ince v = a y , i t m a y b e deduced t h a t t h e w a v e speed i s r e l a t e d t o
dx = c = m a y dt
m- 1
w a t e r v e l o c i t y v b y t h e e q u a t i o n ; c = mv. (2.21)
K I NEMATl C FLOW OVER IMPERMEABLE PLANES
The k i n e m a t i c w a v e e q u a t i o n s h a v e an i m p o r t a n t a d v a n t a g e o v e r t h e
d y n a m i c and d i f f u s i o n wave e q u a t i o n s ; a n a l y t i c s o l u t i o n s a r e p o s s i b l e f o r
s i m p l e w a t e r s h e d geomet r ies . I n t h i s sec t i on , t h e k i n e m a t i c s o l u t i o n s a r e
deve loped f o r r u n o f f f r o m an impermeab le r e c t a n g u l a r p l a n e . U n d e r these
c o n d i t i o n s , we a r e n o t conce rned w i t h e s t i m a t i n g r a i n f a l l loss d u e to
i n f i l t r a t i o n , n o r w i t h r o u t i n g f l o w s f i r s t o v e r l a n d and then t h r o u g h a
complex s t o r m w a t e r d r a i n a g e system. Numer i ca l mode ls g e n e r a l l y a r e
r e q u i r e d when i n f i l t r a t i o n i s i m p o r t a n t or mu1 t i p l e r o u t i n g s a r e i n v o l v e d .
R i s i n g H y d r o g r a p h - G e n e r a l Solution
For the c a s e of a l o n g impermeab le p l a n e , A = b y , Q = bq and R =
y , where q i s the f l o w p e r u n i t w i d t h , hence Eqs. 2.4 and 2.13 c a n b e
w r i t t e n
(2.15)
and
m q = aY
where ie i s the r a i n f a l I excess i n t e n s i t y . S u b s t i t u t i n g E q
2.16 and p e r f o r m i n g the d i f f e r e n t i a t i o n y i e l d s
Eq. 2.22 s t a t e s t h a t t o an o b s e r v e r m o v i n g a t t h e speed
dx m-1 d t am’ - - the d e p t h o f f l o w c h a n g e s w i t h the r a i n f a l I r a t e
a = . d t ‘ e
(2.16)
2.15 i n t o Eq.
(2.22)
(2.23
(2.24
Eqs. 2.23 and 2.24 p r o v i d e the b a s i s f o r a me thod o f c h a r a c t e r i s t i c s
s o l u t i o n to s u r f a c e r u n o f f . F o r s t e a d y r a i n f a l l excess i n t e n s i t y , Eq. 2.24
c a n b e i n t e g r a t e d t o o b t a i n
y = yo + i e t (2.25)
w h e r e Y o i s t h e i n i t i a l w a t e r d e p t h when r a i n f a l l b e g i n s . Eq . 2.25 i s t h e
31
equation for depth along each character ist ic as that character ist ic moves
from some i n i t i a l posit ion toward the downstream end of the plane. The
posit ion on the character ist ic at any instant i n time i s determined w i th Eq
2.23. For an i n i t i a l l y d ry surface y = 0, hence y = iet. Subst i tut ing th i s
relat ionship into E q . 2.23 gives
- - dx - am( ie t lm- ’ dt
which integrates to
x = x + n i t m-1 m
(2.26)
(2.27)
or more simply m- 1
x = x + a y t
which specifies the downslope posi t ion of the depth y a f te r time t. x i s
the point from which the forward character ist ics emanate, i.e., the o r ig in
of the character ist ics at t = 0, and i s measured from the upslope end of
the plane.
(2.28)
The discharge at any point a long a character ist ic i s given by the
re la t ionsh i p m
q = a ( i t ) (2.29)
Two character ist ic paths are shown in F igure 2.3. The f i r s t emanates
at a point in te r io r to the plane and t rave ls the distance L-x dur ing the
time to. The depth and discharge at each point ( x , t ) along th i s
character ist ic i s determined from Eqs. 2.25, 2.28 and 2.29. The second
character ist ic begins at the upslope end of the plane and t rave ls the
length of the plane durin.g the time tC. I n th i s case, the depth a t the
upstream end i s zero, yo = 0, for a l l t. Therefore, as long as the
ra in fa l I intensi ty remains constant, once this i n i t i a l character ist ic has
reached the downstream end of the plane, the depth p ro f i l e along the
plane w i l l remain constant regardless of how long the r a i n f a l l persists,
i . e . , an equ i l ib r ium depth p ro f i l e w i l l be establ ished. The time required
for th is to happen is the concentration time tC. At equ i l ib r ium no
addi t ional r a i n f a l l i s being added to surface detention storage, and the
rate of outflow equals the r a i n f a l l rate.
Recognizing that general ly what is required i s the runoff hydrograph
at the end of the plane catchment, the concept of an equ i l ib r ium time and
f lowrate suggests a way to s imp l i f y the use of Eqs. 2.25, 2.28 and 2.29.
In the fo l lowing sections, solutions and examples are given for the time to
equi l ibr ium, equi l ib r ium depth p ro f i l e and r i s i n g outf low hydrograph.
32
0 i
Fig. 2.3 Kinematic solut ion domain for p lane catchment
Time of Concentration
One can solve for the time of concentration from Eq. 2.27 using the
condit ions that at t =. tc, x - x = L. Subst i tut ing and rear rang ing to
solve for concentration time tc which is equal to time to equ i l ib r ium t m'
tC = ( L/cc iem-l ) 1 /m
For Manning-kinematic f l o w , time of concentration in minutes i s
t = (6.9/ ie0'4)(nL/So0*5) O e 6
for i i n mm/hr and L i n metres and
t = (0 .928 / i~O.~ ) ( ~ L / S ~ O * ' ) O e 6
for i i n in /h r and L i n feet
EXAMPLE 1 . Estimate the time of concentration for a r a i n f a l l r a te of
25 mm/hr on an asphalt pa rk ing lot 50 metres long and sloped a t 1%.
Assume n = 0.023.
Using Eq. 2.31, we f i n d
(2.30)
(2.31 )
(2.32)
Oe60 = 8.2 minutes
33
Hence, a r a i n i n t e n s i t y o f 25 mm/hr w i l l b r ing t h e p a r k i n g l o t t o
e q u i l i b r i u m in 8.2 m inu tes .
Equi l ib r ium D e p t h P r o f i l e
An e x p r e s s i o n f o r t h e e q u i l i b r i u m d e p t h p r o f i l e i s f o u n d b y s o l v i n g
Eqs. 2.25 a n d 2.28 s i m u l t a n e o u s l y , and r e c a l l i n g t h a t a t x o = 0 ,
yo = 0.
The r e s u l t i n g e x p r e s s i o n i s
y ( x ) = ( i e x / a ) 1 /m
w h i c h f o r M a n n i n g - k i n e m a t i c f l o w in S I u n i t s becomes
(2.33)
EXAMPLE 2. E s t i m a t e t h e e q u i l i b r i u m d e p t h a t t h e e n d o f t h e a s p h a l t
p a r k i n g l o t i n E x a m p l e 1 .
We need t o b e c a r e f u l w i t h u n i t s . The r a i n f a l l r a t e i s i n mm/hr ; b u t
the u n i t s i m p l i c i t in the M a n n i n g e q u a t i o n a r e met res and seconds.
There fore , we need to d i v i d e t h e r a i n f a l l r a t e b y 3 . 6 ~ 1 0 .
i
From Eq. 2.34
6
= 25/(3.6xlO 6 ) = 6 . 9 ~ 1 0 - ~ m / s e c
0.6 0.023( 6 . 9 ~ 1 0-6 ) ( 5 0 )
] = 0.0034 met res (0.01
[ Y ( L ) =
o r y ( L ) = 3.4 mm
The Reced ing H y d r o g r a p h
Henderson and Wooding (1964) d e r i v e d t h e k i n e m a t i c e q u a t i o n s f o r t h e
f a l l i n g h y d r o g r a p h . There a r e two cases i n v o l v e d : I . when t h e r i s i n g
h y d r o g r a p h i s at e q u i l i b r i u m , and I I , when t h e r i s i n g h y d r o g r a p h i s a t a
f low less t h a n e q u i l i b r i u m , i.e., p a r t i a l e q u i l i b r i u m .
Case I . D u r a t i o n o f r a i n f a l I, t > = t . A f t e r t h e r a i n f a l l s tops , f r o m d
2.24, i t c a n b e seen that on a c h a r a c t e r i s t i c
d y / d t = 0 (2.35)
w h i c h i n t e g r a t e s to y = c, where c i s some c o n s t a n t . S u b s t i t u t i n g t h i s
r e l a t i o n s h i p i n t o Eq. 2.23 r e v e a l s that t h e c o r r e s p o n d i n g c h a r a c t e r i s t i c
t r a j e c t o r i e s a r e l i n e s p a r a l l e l to the p l a n e and that t h e d e p t h , d i s c h a r g e
a n d wave speed d x / d t , r e m a i n cons tan t a l o n g a c h a r a c t e r i s t i c . T h i s
means t h a t b e g i n n i n g w i t h a p o i n t o n t h e e q u i l i b r i u m p r o f i l e and r e a l i z i n g
t h a t t he f u t u r e c o o r d i n a t e s o f t h a t d e p t h w i l l l i e o n a s i n g l e
c h a r a c t e r i s t i c , Eq. 2.23 c a n b e u s e d to l oca te t h e p o i n t in space a t a n y
f u t u r e t ime. T h i s p r i n c i p l e i s i l l u s t r a t e d in F i g u r e 2.4.
34
The equ i l ib r ium depth p ro f i l e a t the cessation of r a i n f a l l is indicated as
the l ine A-B -C3. After some time A t the depth p ro f i l e i s A-B2-C2. T h e
depth at point B l , y l , has moved along a constant character ist ic pa th to
the point B
1
2'
A L
Fig. 2 . 4 Water depth pro f i le
The distance moved is given by
Ax = a m y At
The new x co-ordinate i s
x = x + A x 1
1
m-1
m-1 = x + a m y l ( t - t d )
( 2 . 3 6 )
( 2 . 3 7 )
( 2 . 3 8 )
where x 1 was the posit ion for point B1. Note that i f the storm durat ion
t l > tc the time to equ i l ib r ium, then x = x That is, once the equi-
l i b r i um depth pro f i le i s establ ished i t w i l l remaln constant as long as
the r a i n f a l l continues a t a steady rate. From Eq. 2.33 , the equ i l ib r ium
depth can be expressed as
Y , = i i e X 1 / " 1 1 / m
Subst i tut ing th is relat ionship into Eq. 2.38 gives
l e-
( 2 . 3 9 )
(m-1 ) / m ( t - t ) ( 2 . 4 0 ) 1 + amIi,xl/a] d
x = x
A t the downstream end of the piane x = L and q = ay
A f t e r subst i tut ing these ident i t ies into Eq. 2 . 4 0 , w e obtain the fo l lowing
relat ionship between discharge and time for the recession hydrograph
= ieL. 1
( t - t ) = 0 (2 .41 ) d
35
Case I I . D u r a t i o n o f r a i n f a l l , t d < tc.
r e a c h i n g e q u i l i b r i u m , t h e n the d e p t h p r o f i l e a t
one s i m i l a r to A-B1-Cl in F i g u r e 2.4. T h a t
p r o f i l e w i l l b e deve loped f r o m t h e u p s l o p e e n d
p o i n t x 1 g i v e n b y
f t h e r a i n s tops p r i o r to
t = t d w o u l d c o r r e s p o n d t o
i s , an e q u i l i b r i u m d e p t h
f t h e p l a n e a t x = O to some
(2.42)
The d e p t h a t p o i n t B1
of the p l a n e a t t ime t,. T h i s t ime i s e v a l u a t e d a s
w i l l move a t a c o n s t a n t r a t e and w i l l r e a c h t h e e n d
L - x,
dx/dt t:; = td t
I n c o r p o r a t i n g Eqs. 2.26, 2.27, and 2.30, Eq. 2.43 becomes
o1 im- l m aim-l m e t c - e t d
t, = td t .m-1 m e td
a m i
(2.43)
(2.44)
w h i c h c a n b e s i m p l i f i e d to
(2.45) 1 m
t, = t d 1 + - [ ( t c / t d I m - 1 }
The d i s c h a r g e at t h e e n d o f t h e p l a n e will r e m a i n c o n s t a n t be tween
t d 5 t S t, and w i l l b e
q = a ( i t
A f t e r t,, t he recess ion proceeds a c c o r d i n g to Case I and Eq. 2.41 a p p l i e s
(2.46) mi
e d
EXAMPLE 3. Determine t h e r u n o f f h y d r o g r a p h f r o m the p a r k i n g l o t i n
Examp le 1 f o r t h e same r a i n f a l l r a t e b u t o f 10 m i n u t e s d u r a t i o n . Use t h e
M a n n i n g k i n e m a t i c s o l u t i o n .
The s o l u t i o n r e q u i r e s t h a t we f i r s t de te rm ine the t ime to e q u i l i b r i u m
w h i c h was done i n E x a m p l e 1 . The n e x t s t e p i s to g e n e r a t e t h e r i s i n g
h y d r o g r a p h . I f t d t c t he r i s i n g h y d r o g r a p h w i l l b e an e q u i l i b r i u m
h y d r o g r a p h . F i n a l l y , we mus t d e t e r m i n e w h i c h case f o r recess ion a p p l i e s
a n d then de te rm ine the recess ion graph a c c o r d i n g l y .
F rom E x a m p l e 1 we know t = 8.2 m inu tes , t h e r e f o r e t h i s even t
s a t i s f i e s t h e c o n d i t i o n s f o r an e q u i l i b r i u m r i s i n g h y d r o g r a p h a n d Case 1
recession. Because t d >= t an e q u i l i b r i u m p r o f i l e w i l l e x i s t o n t h e p l a n e
during the t i m e i n t e r v a l f rom t=8.2 m i n u t e s u n t i l t=10 m inu tes . D u r i n g t h a t
t ime r u n o f f f rom the p l a n e w i l l b e cons tan t and e q u a l to the p e a k r a t e .
The r i s i n g graph i s g i v e n b y Eq. 2.29 and t h e recess ion graph b y Eq.
2.41.
C ’
36
F i r s t de te rm ine the e q u a t i o n f o r t he r i s i n g graph. The c o e f f i c i e n t
in Eq. 2.29 i s
c1
The d e p t h i n met res i s de te rm ined b y
= S o e 5 / n = (0.01)0.5/0.023 = 4.35
y = i e t / (6x10 4 )
3 where t i s in m i n u t e s ; and t h e d i s c h a r g e i n m /sec /m-wid th o f p l a n e i s
q = 4.35 [( iet ) / ( 6 x 1 0 ) ] Nex t , d e t e r m i n e t h e e q u a t i o n f o r t h e recess ion h y d r o g r a p h . A f t e r
t he a p p r o p r i a t e s u b s t i t u t i o n s and u n i t s c o n v e r s i o n , Eq. 2.41 becomes
q --
6 5/3
(5 /3 ) ( 4 . 3 5 ) 0 * 6 ( q ) 0 ‘ 4 ( 6 ~ ) ( t - l o ) = o 25(50) 25
3 . 6 ~ 1 0
TABLE 2.1 Runo f f H y d r o g r a p h O r d i n a t e s
D i s c h a r g e , m 3 /sec Time, M i n u t e s Depth , mm
5 0.0 x 10 0.0 0.0
1 .o 0.42 1 .o 2.0 0.83 3.2 3.0 1.25 6.3 4.0 1.67 10.2 5.0 2.08 14.8 6.0 2.50 20.0 7.0 2.92 25.9 8 .0 3.33 32.4 9.0 3.42 33.7
10.0 3.42 33.7
F i g . 2.5 K i n e m a t i c h y d r o g r a p h shape f o r s i m p l e p l a n e w i t h td = tC
37
FR I C T I ON EQUAT I ON
One of the kinematic equations i s a f r i c t ion energy loss equation.
There are many f r i c t i on equations in use i n hyd rau l i c engineering and a
generalized comparison i s made below. The most popular equation re la t i ng
flow ra te to f r i c t i on energy gradient i s perhaps that of Manning, which
may be wri t ten as
(2.47)
where Kl i s 1 in 5. I. un i t s (metre-kilogram-seconds) and 1.486 i n
English un i ts (foot-pounds-seconds), n i s the Manning roughness, A i s the
cross sectional area, R the hyd rau l i c rad ius A/P, P the wetted perimeter
and S the energy slope. The 5. I. system of un i ts i s adopted below but i t
should be noted that the equation i s not dimensionless, and the roughness
factor n is a function of g rav i t y . Written i n terms of flow per un i t w id th
of a wide rectangular channel, (as for an over land flow plane)
q = JS 5/3 n Y
(2.48)
1
since hyd rau l i c rad ius R yb/b = y and area yb = y. Hence ~1 = S'/n
and m = 5/3. The Manning roughness coefficient n is reputedly a constant
for any surface roughness. Th is holds for la rge Reynolds numbers and
f u l l y developed turbulent flow, but comparison w i th the Darcy Weisbach
equation indicates that n ac tua l l y increases for low Reynolds numbers ( y v /
v<lOOO where v
Manning equation may be compared w i th the Darcy equation employing
St r i ck le r ' s equation for roughness, n = 0.13K k'/6/gi where k i s a l inear
measure of roughness analogous t o the Nikuradse roughness for pipes ( i n
metres i f K, = 1 ) . Subst i tut ing into Manning's equation yields
i s the kinematic viscosity of water, about 10-6mz/s). The
1
Q = 7.7(R/k)'/6A(SRg
I f t h i s equation
1 1
Q = (8/ f ) ' A(SRg)'
i t w i l l be seen Str i
1 2 -
i s compared with Darcy 's equation i n
< l e r i n effect used a Darcy f r i c t ion
0.135(k/R)'/3. (Note B r i t i sh pract ice is to use h i n place of
fo r a di f ferent factor.) According to Colebrook and White,
(2.49)
the form
(2.50)
actor equal to
as they use f
2.5
14.8R Re f 1 = -2 log ( - + 7 - 1 __ Ji
(2.51 )
38
where Re i s the Reynolds number, fo r pipes VD/v, or 4VR/V fo r non
c i r cu la r cross sections. Whereas the Colebrook-Whi te equation predicts
higher values of the Darcy f r i c t ion coefficient f fo r low Reynolds number
and any re la t i ve roughness k/R, the St r i ck le r equation assumes f depends
only on the re la t i ve roughness k/R. The St r i ck le r and Manning equations
can therefore be expected to underpredict roughness for low Reynolds
numbers. Higher values of n should therefore be used fo r over land flow
than for channel flows.
I n general, the value of n and hence flow depth has to be deter-
mined by t r i a l (assuming the Colebrook-White equation to app ly and not
S t r i ck le r ' s ) . I t i s therefore probably easier to use the Darcy equation for
th is purpose but since an exp l i c i t equation is required for ana ly t i ca l
solutions to the kinematic equations and the var ia t ion i n n i s less than
the var ia t ion in f w i th y , the Manning equation is preferred.
Table 2.2 indicates values of n and f wi th va ry ing water depths i n a
wide channel w i th a slope of 0.0025 and absolute roughness k = 0.0125 m.
The values of f a re calculated from the Colebrook-White equation using a
f i r s t estimate of Re from Manning's equation, and then n is re-calculated
from n = (f/8g)'R'/6 i.e. as for S t r i ck le r ' s equation.
TABLE 2.2 - Var iat ion of f and n wi th depth
Water Depth, m Reynolds No. Darcy f Manning n
1 .o 0.1
0.01
2 x lo6 50 000
1 000
0.03
0.09
0.60
0.02
0.023
0.04
The Chezy equation is often used i n preference to the Manning equation
i n American practice. This equation is
v = K,CJ(RS) (2.52)
where C i s known as the Chezy coefficient and K, i s 1 i n f t - second
un i ts and 0.552 i n S . I . uni ts. I n fact the Chezy equation i s very s imi la r
to the Darcy equation in the form
V = J (8g / f ) J ( R S ) (2.53)
and i t w i l l be seen that C = J m / K 2 ,
also for turbulent flow from (2.51) l/df'2 log (14.8R/k) (2.54)
39
Hence C = ( 2 / K 2 ) flg log (14.8/k)
or v J3gRs' log (14.8/k)
(2.55)
(2.56)
This equation stems from the log velocity d is t r ibu t ion across a section
whereas the St r i ck le r equation follows from a 1/6 power law f i t to the
ve loc i ty d i st ri but ion.
Resistance to r a i n f a l l induced over land flow over na tura l and man-
made surfaces is influenced by several factors inc lud ing surface rough-
ness, raindrop impact, vegetation, wind and i n f i I t ra t ion . Although there
have been many laboratory and f i e ld investigations to determine the
re la t i ve importance of these factors, the appropr iate resistance formula,
and methods fo r parameter estimation, i n pract ice the convention has been
to use either the Darcy-Weisbach equation modif ied for ra indrop impact, o r
the t rad i t iona l forms of the Manning or Chezy equations.
I n laminar flow studies of over land flow the approach has been to
assume the Darcy-Weisbach resistance law is appropr iate, i .e.
v = J(+ Sy)
and to estimate the f r i c t i on factor, f , from the theoretical re lat ionship
between f and Reynolds number Re,
f = K/Re (2.57)
where K is a parameter related to the surface roughness character ist ics
and ra indrop impact. The parameter K i s approximated by
K = K + A i
where KO i s the parameter for surface roughness and A and b a re
empir ical parameters. When. i i s i n inches per hour, the coefficient A i s of
the order of 10 and the exponent b i s approximately un i t y . Typical values
for K O , Manning's n, and Chezy's C are given i n Table 2.3. These
values are ranges found i n the l i te ra tu re and were obtained u t i l i z i n g data
from control led experiments or from smal I experimental watersheds.
(2.58) b
TABLE 2.3 Overland Flow Resistance Parameters
Surface Laminar Flow
KO
Concrete o r Asphalt 24 - 108 Bare Sand 30 - 120 Gravel led Surface 90 - 400 Bare Clay to 100 - 500
Sparse Vegetation 1000 - 4000 Short Grass 3000 - 10000
Loam Soil
Bluegrass Sod 7000 - 40000
Turbulent Flow Manning n Chezy C
0.01 - 0.013 73-38 0.01 - 0.016 65-33 0.012 - 0.03 38-1 8 0.012 - 0.03 36-1 6
0.053 - 0.13 11-5 0.10 - 0.20 6.5-3.6 0.17 - 0.48 4.2-1.8
40
I n the case of turbulent f low, ei ther the Manning or Chezy equation
is used. The Manning equation is probably the more popu lar equation and
i s used more often i n watershed simulat ion studies. The reasons for t h i s
are obviously i t s wide-spread acceptance in open channel hydrau l i cs and
the a v a i l a b i l i t y of extensive tables of n-values for most channel types and
conditions.
Reported research indicates that low flows are laminar and that h igh
flows are turbulent; bu t the location of the t rans i t iona l Reynolds number
i s indeterminate which makes i t d i f f i c u l t to app ly the Darcy-Weisbach
resistance formulation throughout the ent i re hydrograph. Transi t ion from
laminar to turbulent flow has been reported a t Reynolds numbers rang ing
from 20 to 2,000; w i th the range 300<Rer500 being the most f requent ly
reported . Overton (1972) analyzed 214 equ i l ib r ium hydrographs from an ea r l i e r
study of a i r f i e l d drainage conducted by the U.S. Army Corps of Engineers
(1954). He noted that these hydrographs supported the argument for low
flows being laminar and h igh flows being turbulent. I n almost every case,
the r i s i ng hydrograph i n i t i a l l y rose very slowly ind ica t ing viscous laminar
f low, and then became turbulent as the flow increased. Overton analyzed
the r i s i n g port ion of a l l 214 hydrographs i n dimensionless form. He
normalized the discharge by the r a i n rate, and time by a lag time
parameter, which he defined as the time from the occurrence of 50% of
the r a i n f a l l to 50% of the runoff volume. The normalized (dimensionless)
hydrographs were plotted on transparent paper and superimposed. I t was
apparent, that w i th in a small e r ro r , a s ing le dimensionless r i s i n g
hydrograph could represent a l I 214 hydrographs. The average dimensionless
r i s i n g hydrograph was then plotted against the laminar, Manning and
Chezy dimensionless r i s i n g hydrographs as shown i n F igure 2.6. Flows
appear to be laminar du r ing the f i r s t ha l f of the per iod of r i se and
turbulent du r ing the second ha l f . A n e r ro r ana lys is indicated a 15%
standard er ro r i n f i t t i n g the en t i re r i s i n g hydrograph for the Manning
kinematic solut ion, and 19% for both the laminar and Chezy solutions.
tL,
To i l l us t ra te the effect of using only the Manning equation to
represent the flow equation hence the flow resistance, throughout the
ent i re hydrograph, consider Izzard 's (1946) laboratory experimental r u n
Nos. 136 and 138. Run No. 136 consisted of two burs ts wi th r a i n f a l l
intensity of 3.56 in/hr interrupted by a two minute l u l l . The f i r s t burst
lasted for 10 minutes and the second, 1 1 minutes. The r a i n f a l l event
produced a maximum Froude number of 0.55 and a minimum kinematic flow
number of 156. Run No. 138 consisted of two bursts: the f i r s t was 1.83
in/hr for 8 minutes and the second was 3.55 in/hr for 8 minutes. This
41
1. I
KINEMATIC WAVE MODELS
~ i ~ . 2.6 Compar i son o f T u r b u l e n t and L a m i n a r K i n e m a t i c Wave I
Solu
even t p r o d u c e d
m in imum k i n e m a
i o n s w i t h Observed R i s i n g H y d r o g r a p h s
a p p r o x i m a t e l y t h e same max imum F r o u d e n u m b e r a n d
i c f l o w n u m b e r a s R u n No. 136. The r u n o f f s u r f a c e was
a n a s p h a l t p l a n e w i t h t h e f o l l o w i n g p h y s i c a l c h a r a c t e r i s t i c s :
L = 72 f t ; M a n n i n g n - v a l u e = 0.024; and So = 0.01.
B l a n d f o r d and Meadows (1983) a n a l y z e d these e v e n t s w i t h a f i n i t e
element f o r m u l a t i o n o f t h e k i n e m a t i c o v e r l a n d f l o w model a n d o b t a i n e d
the r e s u l t s shown in F i g u r e 2.7. F o r Run No. 136, t h e p r e d i c t e d r i s i n g
and f a l l i n g l i m b s o f t h e h y d r o g r a p h l a g t h e o b s e r v e d l i g h t l y , w h i l e t h e r e
i s n e a r p e r f e c t agreement a t t h e h i g h e r f l ows . F rom Run No. 138, o n l y
the s i m u l a t e d f a l l i n g l i m b l a g s t h e obse rved ; t he r e s t o f t h e h y d r o g r a p h
matches the o b s e r v e d v e r y we l l .
REFERENCES
Beven, K., Dec. 1982. On s u b s u r f a c e s to rmf low , P r e d i c t i o n s w i t h s i m p l e k i n e m a t i c t h e o r y f o r s a t u r a t e d and u n s a t u r a t e d f l ows . Water Resources Res. 18 ( 6 ) p p 1627-33.
B l a n d f o r d , G.E. and Meadows, M.E. 1983, F i n i t e E lement S i m u l a t i o n o f K inemat i c S u r f a c e Runo f f , P roceed ings , F i f t h I n t e r n a t i o n a l Sympos ium on F i n i t e E lements in Water Resources, U n i v e r s i t y o f Vermont, B e n n i n g - ton , Vermont .
Dunne, T. 1978. F i e l d s t u d i e s o f h i l l s l o p e f l o w processes. Ch. 7, H i l l s l o p e H y d r o l o g y , Ed. K i r k b y , M.J. John Wi ley , N.Y.
Henderson, F.M. and Wooding, R.A. 1964 , O v e r l a n d F l o w and Ground- w a t e r f r o m a Steady R a i n f a l l o f F i n i t e D u r a t i o n , J o u r n a l o f Geo- p h y s i c a l Research , 69 ( a ) , pp. 1531-1540.
42
I z z a r d , C.F., 1946. H y d r a u l i c s o f Runof f f rom Developed Sur faces, H igh - way Research Board , P roceed ings o f t he 26th A n n u a l Mee t ing pp. 1 29-1 50.
M o r r i s , E.M., a n d Woolhiser, D.A., A p r i l 1980. Uns teady one-d imensional f l o w o v e r a p l a n e : p a r t i a l e q u i l i b r i u m a n d recess ion h y d r o g r a p h s . Water Resources Research, 16(2) pp 355-366.
Over ton , D.E., 1972. A V a r i a b l e Response O v e r l a n d F low Model, Ph.D. Disse r ta t i on . Dept. o f C i v i l E n g i n e e r i n g , U n i v . o f M a r y l a n d .
Over ton, D.E. a n d Meadows, M.E., 1976 Stormwater M o d e l l i n g , Academic Press, New York .
St. Venant , A.J.C. B a r r e de, 1848. Etudes Theor iques e t P r a t i q u e s s u r l e Mouvement de E a u x Courantes. ( T h e o r e t i c a l and P r a c t i c a l S tud ies o f Stream F l o w ) , P a r i s .
U.S. Army Corps of Eng ineers 1954. D a t a Repor t , A i r f i e l d D r a i n a g e I n v e s t - i g a t i o n s , Los Angeles D i s t r i c t , O f f i ce o f t he Ch ie f of Eng ineers , A i r f i e l d s B r a n c h E n g i n e e r i n g D i v i s i o n , M i l i t a r y Cons t ruc t i on .
Woolhiser, D.A. 1981. P h y s i c a l I l y based models of wa te rshed r u n o f f , p p . 189-202 in S ingh , V.P., (Ed . ) R a i n f a l l Runof f Re la t i onsh ips . Water Resources P u b l i c a t i o n s , Colorado, 582 pp.
Woolhiser, D.A. a n d L i g g e t t , J.A., 1967. Uns teady one-d imensional f l o w o v e r a p l a n e . The r i s i n g h y d r o g r a p h . Water Resources Research, 3 ( 3 ) pp 753-771.
-OBSERVED 4r L c \ e
w (3
0 4 8 12 16 20 24 28 32
TIME( m i n 1
F i g . 2.7 Ou t f l ow H y d r o g r a p h f o r I z z a r d ' s R u n No. 136
43
CHAPTER 3
HYDROGRAPH SHAPE AND PEAK FLOWS
DESIGN PARAMETERS
Knowledge of the runof f process enables flow rates and volumes
to be predicted. Hydrograph character ist ics are of interest to researchers,
planners, designers and managers of drainage systems. The drainage
engineer w i l l be most concerned with peak flows for design purposes.
I t i s also frequently useful to have the hydrograph shape especial ly
i f detention storage or rou t ing can reduce the peak flow. Expressions
for hydrograph shape and peak flows as a function of excess r a i n f a l l
intensi ty can be der ived as follows for over land flow of f simple planes.
The kinematic equations summarized below are used for th is purpose.
Continuity
( 3 . 2 ) m
Energy 9 = CLY
where x i s the direct ion of f low, t i s time, i i s the excess r a i n f a l l
r a te i - f , f i s the loss r a t e and q i s the discharge r a t e per u n i t
catchment width. For the present a l l un i t s must be assumed consistent.
Later un i ts w i l l be introduced i n order to render the numerical values
more meaningful. I t w i l l be assumed in the fo l lowing analysis that i
d‘ and f are uniform in time’ and space for the durat ion of the storm t
q i s the flow ra te per u n i t width of plane, y i s flow depth, CL i s a
coefficient and m i s an exponent.
SOLUTION OF KINEMATIC EQUATIONS FOR FLOW OFF A PLANE
The kinematic equations can be solved ana ly t i ca l l y for some simple
cases. . I n pa r t i cu la r the runoff from a rectangular plane catchment
subject to uni form excess r a i n can be studied i n de ta i l and expressions
for the time to equ i l ib r ium and hydrograph shape can be derived.
The fol lowing ana lys is demonstrates the simp1 i c i t y of a r r i v i n g at
an equation for runof f for the catchment from a simple p lane catchment
sloping in the direct ion of flow. The analysis i s handled more r igorously
in chapter 2.
One s ta r t s wi th the general ized (one-dimensional) kinematic
equations for over land flow namely 3.1 and 3.2.
44
I f the Manning equation i s assumed to hold then Q = K S $/n and m
= 5/3 where So i s the slope of the plane i n the direct ion of f low, and n
is the Manning roughness. K i s 1 .O i n 5.1. (metre) un i ts and 1.486 i n f t -
sec units.
l o
After ra in fa l I commences, the water depth near the downstream end of
the catchment w i l l increase at a r a t e i the excess r a i n f a l l rate. The
water surface pro f i le then w i l l be para l le l to the plane a t the
downstream end before equ i l ib r ium i s reached, which i s assumed to occur
before the r a i n stops, i.e. t < tc 5 td, where t i s the time to equ i l ib -
r ium, usual ly referred to as the concentration time of the catchment.
e’
Start ing at the top or upstream end of the catchment where water
depth and discharge ra te w i l l be zero, a negative surge due to a non-
zero dy/dx w i l l t rave l down the catchment over land increasing i n depth as
r a i n continues to fa l l . Then a t any point in time downstream of the surge
the water surface i s increasing in depth at a r a t e i but upstream the
water depth i s at equ i l ib r ium since aq/ax = ie(see F ig . 2.4 l ine ABIC1).
Eventual ly the whole catchment w i l l reach an equ i l ib r ium wi th input
i L p e r u n i t width equal to discharge qL. At the instant the catchment
reaches equ i l ib r ium
(3.3)
t i s the concentrat ion time of the catchment, which i s a function of the
catchment length L, slope So, roughness n and excess r a i n f a l l r a te i The
la t te r effect (i,) ra re l y appears in time of concentration formulae
associated w i th the ra t iona l method.
e‘
Dur ing the time of flow bu i ld -up the water depth a t the ex i t i s i t,
and the corresponding discharge r a t e q L = a ( i e t ) m (3.4)
The speed a t which the reaction from upstream travels down the
catchment before equi l ibr ium, i s obtained from the cont inui ty equation. At
the wave front the ra te of increase i n flow depth is
(3.5)
where dx/dt i s the ra te of t rave l of the wave front.
dy/dt = ie a t the wave front point (and downstream of i t ) . One also
has from the cont inu i ty equation by expanding the aq/ax term
- 3 Y + 3 dy = i at ay dx e
dx By comparing w i th av + - * - i
one must have - dx = at the wave front
at dt ax e
dt a y
45
(3.7) m- 1
and from 3.2, dx/dt = may
Since y = iet,
dt
which is the speed of the wave front at any time t 5 tC. Also du r ing
equ i l ib r ium the discharge r a t e at any point x from the upstream water-
shed i s q = i x
(3.8) m-1 - dx = mcr(iet)
( l i n e ABlC3 i n Fig. 2.4). Hence from 3.2 e
x = aym/i (3.9)
An expression for the discharge ra te a f te r the storm stops, which
i s assumed to be a f te r the time to equ i l ib r ium ( t 2 t z t ) , i s obtained
by considering the water depth p ro f i l e along the catchment again. After
the r a i n stops the effect of a l l upstream depths travels down to the ex i t
at a speed dx/dt g iven by 3.7. To predict when the depth a t the ex i t
i s ' y ' , imagine a series of waves t rave l l i ng from the water p ro f i l e curve
in a downstream direct ion a t a constant speed dx/dt = aq/ay = mayrn-'
Integrat ing, x = x + maym-'(t- td) (3.10)
(3.11) = q/ ie i mq
since y = (q/a)l/m. (3.12)
I n pa r t i cu la r at the ex i t ,
L = q/ ie + mq 1-1 /mat /m b t d )
which i s an imp1 i c i t expression for the f a l l i n g l imb of the hydrograph.
The f u l l hydrograph shape i s thus as i n Fig. 3.1.
d - c
l-I/m l / m a ( t - t 1 d
(3.13)
HYDROGRAPHS FOR PLANES
Expressions for the r i s i n g and f a l l i n g l imbs of the hydrograph
off a simple rectangular catchment were der ived previously. The
discharge a t the mouth before time t o r td i s reached, .is g iven by
q = a ( ie t ) " ' (3.14)
I f r a i n continues af ter t = tc i .e. td > tc, then the hydrograph top i s
horizontal as indicated i n Fig. 3.1 case I l l .
I f on the other hand r a i n stops a t t = t then the hydrograph
fa l l s immediately af ter t (case 1 1 ) . I n e i ther case i t may be shown
that the f a l l i n g limb of the hydrograph i s obtained from the imp l ic i t
(3.13)
The total depth of excess r a i n has been kept constant i n each case
i n Fig. 3.1 so that i = p/ td where P i s the depth of precipi tat ion.
46
c-
Flow I n u d q
+ Time 1
F i g . 3.1 Outf low h y d r o g r a p h shape f o r d i f fe ren t storm dura t ions b u t s i m i l a r to ta l excess r a i n .
A l so i l l u s t r a t e d in F ig . 3.1 i s the case of the h y d r o g r a p h f o r a
shor t storm ( t < t ) (case I ) . Af te r t ime td the downstream depth ( a n d
hence f lowrate) remains constant u n t i l the in f luence of the upstream end
reaches the e x i t .
The upstream l i m i t of y = i t i s a t
d c
e d m-1 m
t e d
x = q / ie = aym/ ie = a i
I f t h i s po in t t r a v e l s a d is tance L - x a t a speed
dx /d t = maym-'
(3.15)
(3.16)
rn-l i t w i l l reach the e x i t in time
At = Ax/ma(ietd) m-1- - (L-a iern- l tdm)/maie m-1 td m-1 L / a ( ie td) - td
- (3.17) m
C' Assume i i s the excess r a i n f a l l r a t e f o r a storm d u r a t i o n equal to t
Since ie td = iectC f o r equal volume of r a i n , a n d ec
At = (tc-td)/m (3.18)
This i s the d u r a t i o n of the f l a t top of the h y d r o g r a p h I in F ig . 3.1.
I t should be noted tha t the f a l l i n g l imbs of the h y d r o g r a p h in F ig . 3.1
omit losses a f t e r r a i n stops. I f i n f i l t r a t i o n ( f ) continues the h y d r o g r a p h
w i l l look l i k e those in F ig . 3.2. I t i s genera l l y necessary to model such
system numer ica l l y to get the h y d r o g r a p h shape (see Wooding, 1965).
47
I D
9 i L -
0 . 5
0 I I J 1 5
t / f ,
F i g . 3.2 Effects of i n f i l t r a t i o n on catchment d ischarge
48
DERIVATION OF PEAK FLOW CHARTS
I f i t can be assumed that the r a i n f a l l intensi ty-durat ion relat ion-
ship for a specif ied frequency of exceedance can be approximated by a
formula of the form
a I =
(C+td)P (3.19)
where i is the r a i n f a l l ra te i n mm/h o r inches per hour and td i s the
storm durat ion i n hours, then a simple estimate of peak flow can be
d derived. I t I S assumed that i i s constant du r ing the storm durat ion t
and uniform over the catchment. The storm i s also assumed stat ionary.
c i s a time constant unique for a pa r t i cu la r r a i n f a l l region and p is
an exponent, also a unique function of the region. Thus for temperate
regions i n South Afr ica i t was found that c = 0.24h and p = 0.89 whi le
for coastal regions c = 0.20h and p = 0.75 (Op ten Noort and Stephenson,
1982).
a i s a function of r a i n f a l l region, mean annual precipi tat ion, MAP,
i n mm and recurrence in te rva l T i n years e.g. a = (b+e.MAP)Toe3 where
b and e are regional constants. a i s not dimensionless and i f c and
td are i n hours, then i i s i n mm/h or inches per hour. An areal
reduction factor i s also necessary for la rge catchments (e.g. Stephenson,
1981).
Losses are subdiv ided into two components, an i n i t i a l loss u i n
mm o r inches and a uniform in f i l t r a t i on loss ra te f i n mm/h o r inches
per hour. A typical r a i n f a l I I D F ( intensi ty-durat ion-frequency)
relat ionship and the corresponding hyetograph with losses and excess
runoff indicated i s presented i n Fig. 3.3.
The ra te of excess r a i n f a l l i s :
i = i - f (3.20)
where f i s the i n f i l t r a t i on ra te and the durat ion of excess r a i n i s :
te = td - tu = td - u / i
where u i s the i n i t i a l abstract ion (measured i n terms of a depth of
r a i n ) .
(3.21)
For small catchments the maximum peak runoff ra te occurs when
the durat ion of excess r a i n equals the concentration time, t . For p l a i n
rectangular catchments the concentration time is a function of excess
r a i n f a l l r a te
where t = te (both i n hours here) (3.23)
a = r S / n (S.1 un i ts ) o r 1.486 &/n ( fps un i t s ) (3.24)
49
hyetogroph
{ I \>( I
ntial abstrac-
tion
t 4 ~-t+----4
Fig. 3.3 Excess flow hyetograph derived from IDF curve
i i s excess r a i n f a l l intensi ty, L i s catchment length, 5 i s the
downstream slope, m i s an exponent, 5/3 i n Manning equation, n i s the
Manning roughness, q i s the discharge ra te per un i t width.
The fol lowing expression may then be der ived for ie/a from
equations (3.19) to (3.22):
(3.25)
Thus the maximum runoff ra te per un i t width of catchment
q = i L (3.26)
may be obtained i n terms of a. Equation (3.25) may be solved i te ra t i ve ly
e.g. using the Newton Raphson procedure, for ie/a, or solut ion may be
obtained w i th the a i d of graphs of ie/a plotted against ( L / a am-').
The storm durat ion td corresponding to the peak runoff may be obtained
from (3.21) and (3.22).
Long Catchments
For very long catchments the theoretical concentration time tc i s
high. I n such cases the corresponding excess r a i n f a l l r a te for a durat ion
equal to tc i s low and i n fact could conceivably be less than the
i n f i l t r a t i on ra te f. I t is thus apparent that i n such cases the maximum
runoff ra te may coincide w i th a storm of shorter durat ion than the
concentration time of the catchment. The en t i re catchment w i l I thus not
50
be contr ibut ing at the time of the peak i n the ( r i s i n g ) hydrograph.
I f a local, intense storm turns out to be the design storm, the areal
reduction factor appl ied to point r a i n f a l l intensi ty relat ionships may
be less s igni f icant. The factor is general ly closer to un i ty the smaller
the lateral extent of the storm, but on the other hand shorter durat ion
storms have a more s ign i f i can t reduction factor ( less than long storms).
These facts w i l l not b e revealed using the Rational method with r a i n f a l l
propor t iona I losses.
Before equ i l ib r ium is reached the runoff p e r un i t width a t the
mouth of the catchment at any time t af ter the commencement of excess
r a i n or runoff i s m
q = a ( ie te )
where t - - td - u/ i m-1 l / m
<tc = (L /a ie )
( 3 . 2 7 )
( 3 . 2 8 )
( 3 . 2 9 )
( 3 . 3 0 )
( 3 . 3 1 )
( 3 . 3 2 )
q / a a m i s plotted against td i n Figs. 3 .4 to 3.6 ( the f u l l l ines) for
d i f ferent values of the dimensionless parameters U = u/a and F = f /a.
For a l l cases of F > 0 the l ines exh ib i t a peak runoff and the correspond-
ing storm durat ion td for an i n f i n i t e l y long catchment. For most catch-
ments i t i s necessary to establ ish whether t i s less than t i.e. whether
the peak occurs before the catchment has reached equi l ibr ium.
I n fact , t = td - t ( 3 . 3 3 )
Therefore for t = c te
( 3 . 3 5 )
U
L/aam-l may therefore be plot ted against td for selected values of u/a
and f /a as on the r i g h t hand side of Figs. 3 . 4 to 3 . 6 . Each chart i s
for a di f ferent i n i t i a l abstraction factor U, and i t may be necessary
to interpolate between graphs for intermediate values of U. Lines for
var ious F are plot ted on each graph.
Now the peak runoff w i l l be the maximum of either
( a )
(b ) that for t < tc for long catchments.
that corresponding to t = tc for short catchments o r
51
I n order to ident i fy which condit ion applies, enter the chart for
the correct U = u/a wi th L /aa on the r i g h t hand side and using the
dotted l ine corresponding to the correct F read off the corresponding
+ t on the abscissa. I t may occur that the equ i l ib r ium tc i s t d = tc off the chart to the r i g h t i n which case i t i s probably of no interest
since the fol lowing case applies. Select the f u l l l i ne wi th the F = f /a
and decide whether i t s maximum l ies a t o r to the left of the value t d
previously established. I f the peak l ies to the lef t , read the revised
design storm durat ion td corresponding to the peak, and the correspond-
i ng peak flow parameter q/aa on the left hand ordinate.
m-1
m
Modif icat ion for Prac t ica l Uni ts
The preceding equations assume dimensional homogeneity. Unfortun-
ately both the Manning resistance equations and the I-D-F relat ionships
are empir ical and the coefficients depend on the un i ts employed. I n the
Manning form of equation (3.241, q i s i n m 2 / s i f iete i s i n metres.
a i s 6 / n i n S . I . un i ts where S i s the dimensionless slope and n is the
Manning roughness.
I t i s most convenient to work w i th td i n hours and i and a i n
mm/h. T h e numbers are then more rea l i s t i c . I n equation (3.32) i f q i s
i n m3/s/m, a i n m-s uni ts, a in mm/h and t i n hours then the lef f
hand side should be replaced by d
q l ~ ~ ~ m - 1 0 ~ ~ 5/3 aam o a
- -
= I O ~ Q 5/3 Baa
(3.36a)
w h e r e Q i s total runoff ra te off a catchment of width B metres. Note that
the r i g h t hand side of (3.32) i s i n hm i f td i s i n h , so no correction
is made to the above factor to convert a to secs, only to convert mm
to m. This i s what the left hand ax i s of Figs. 3.4 to 3.6 represent i f
a i s i n mm/h. I t is referred to as the runoff- factor, QF. S imi la r ly the
left hand side of equation (3.35) i s L /aam- l i n homogeneous uni ts, o r
i f a i s i n mrn/h, L i n metres and a i n m-s uni ts, then i t should be
replaced b y :
a looom - L - - L
a 3600000 m 2 / 3 36aa (3.37a) -
This is termed the length factor LF.
I f q i s i n f t ’ /s / f t , a i n ft-sec un i ts , a i n inches per hour and td i n
hours then the expression for Q should be replaced by
63q ( 3.36b ) aa ’ I 3
52
... LENGTH W l O A T t 9.8
u= 0.004 c= 0.zL.l0 P= 0.890
0. I
a. I 0 I 2 3
5JDRl bCRAlIOU IN HA5
Y. Y
q.0
3 . 6
3 .2
L -1
0 I- v R
I t-
W -I
2.8
2.q LL
2.0 9
1.6
I . I
I1.B
0.9
6.0
F i g . 3.4 Peak runoff f ac to rs fo r U = 0.00
53
-- AUNDff Ffl(1UR . . . LENGTH fRClOR
E I 2 51ORR DURATION IN HR5
9.6
q . O
3.6
3.2
L -.l
0 I- U [r
I I- D
2.8 fy:
2.q L
7.0 z
4
1.6
1.2
0. H
0.q
0.0
Fig . 3.5 Peak runof f factors fo r U = 0.20
54
: . 2
I .a
1 . 9
0.9
LL
13
K 0
0.7
I- LJ
21.6 LL
L LL
7 ct:
g 0.s
O.Y
0 . 3
0.2
0. i
a.n
-- RUNUFf FR(1UR . . . LENETH FACTOR Y.B
4.0
3.6
3 .2
L -I
2.9 ~y:
n I- V a: 2.Y L
I I-
irr 2.0 z w -I
I .6
I .2
0.9
0. q
0.0
Fig . 3.6 Peak runoff factors f o r U = 0.40
55
a n d the length fac to r expression (3.37b) i s rep laced b y
687a a‘’’
Figs. 3.4 to 3.6 a r e p lo t ted in terms of the dimensioned expressions f o r
LF a n d QF, i.e. use td i n hours a n d the other terms in the chosen metr ic
o r Eng l ish u n i t s above. F u r t h e r c h a r t s were pub l ished b y Stephenson
(1982).
EXAMPLE
Consider a p l a n e r e c t a n g u l a r catchment w i t h the fo l low ing charac ter is t i cs :
o v e r l a n d f low length L = 800m
w i d t h 0 = 450 m
slope S = 0.01
Manning roughness n = 0.1
i n l a n d reg ion , MAP = 620mm/annum
20 year recurrence i n t e r v a l storm
r a i n f a l l factor a = (7.5 + 0.034 x 620)20°‘3 = 70mm/h
c = 0.24h, p = 0.89 in i mm/h = a/ (c+t )p
i n f i l t r a t i o n r a t e f = 14mm/h
i n i t i a l abs t rac t ion u = 14mm
a= JT/n = 1.0
F = f / a = 0.2
u = u /a = 0.2
d
2/3 = 1.30 800
L F = * = 36 x 1 x 7Q
From F ig . 3.5 concentrat ion t ime t = 3.0h
I t w i l l be noted tha t there may be two so lut ions fo r storm d u r a t i o n
td. The longer one corresponds to a very low p r e c i p i t a t i o n r a t e a n d i s
of l i t t l e in terest . Even the shor ter t ime to e q u i l i b r i u m is longer than
the storm r e s u l t i n g in the peak runof f . I n t h i s case i t appears tha t the
peak runof f corresponds to a 1.3 hour storm (shor te r than the time to
e q u i l i b r i u m ) a n d the corresponding
QF = 0.30 = 1 0 5 61/~aa5/3
therefore Q = 0.30 x 450 x 1 x = 1 . 6 0 m 3 / s
The corresponding p r e c i p i t a t i o n r a t e i s :
i = a / (c+ t )’ = 70/(0.24 + 1.3)o’89 = 48mm/h
The equ iva len t r a t i o n a l coef f ic ient C i s
1.60/(450 x 800 x 48/3600000) = 0.33
I t may also be conf i rmed tha t the storm d u r a t i o n corresponding to time
to e q u i l i b r i u m of the catchment i s 2.8 hours :
d
56
I f i
t = (L /a ie )
= {800/1 x (2.90 x I O - ~ ) ~ / ~ ~ ~ / ~ = 9050s = 2.5h
t = u / i = 14/48 = 0.3h
= 2.8h Storm durat ion t
The corresponding runoff would be only:
i BL = 2.9 x 10 x 800 x 450 = l.Om’/s
i.e. the peak runoff corresponds to a storm of shorter durat ion than
to time to equ i l ib r ium of the catchment.
= 70/(0.24+3.0)0.89-14 = 10.5mm/h = 2.9x1OP6m/s m-1 1-m
U - d
-6
EFFECT OF CANALIZATION
The charts presented are for the case of over land f low. I t fre-
quent ly occurs that runoff reaches channels, and thus flows to the mouth
of the catchment faster than i f overland. The c r i t i c a l storm durat ion
may thus be shorter and the peak flow higher than with no canal izat ion.
A n estimate for the concentration time of a catchment w i th a wide
rectangular channel down the middle may be made using th is chapter
i f over land flow time can be neglected. The effect ive catchment width
i s taken as b , the stream width, and both r a i n f a l l r a te i and losses
f and u should be increased b y the factor B/b where B is the t rue
catchment width. The charts herein can then be app l ied as i n the
example.
I n many si tuat ions both over land flow and stream flow are s ign i f i -
cant and the problem cannot be solved as simply as herein. The hydro-
logist must then resort to t r i a l and er ro r methods using dimensionless
hydrographs for catchment - stream systems as presented later.
‘channel
Fig . 3.7 Rectangular catchment wi th central col lect ion channel.
57
EST I MAT I ON OF ABSTRACT IONS
The losses to be deducted from precipi tat ion include interception
on vegetation and roofs, evapotranspirat ion, depression storage and
in f i l t r a t i on . The remaining losses may be d iv ided into i n i t i a l retention
and a time-dependent i n f i l t r a t i on . The losses are rea l l y functions of many
var iables, inc lud ing antecedent moisture condit ions and ground cover.
I n f i l t r a t i on i s time-dependent and an exponential decay curve is often
used. The in f i l t r a t i on typ ica l l y reduces from an i n i t i a l ra te of about
50 mm/h down to 10 mm/h over a per iod of about an hour. T h e rates,
especial ly the terminal loss rate, w i l l b e higher for coarse sands than
for clays.
The time-decaying loss ra te could be approximated b y an i n i t i a l
loss p lus a uniform loss over the durat ion of the storm. Values of i n i t i a l
and uniform losses are tentat ively suggested below. The mean uniform
loss rates are average f o r storms of 30 minutes durat ion and the i n i t i a l
losses include the i n i t i a l 10 minutes r a p i d i n f i l t r a t i on o r saturat ion
amount, I n the case of ploughed lands, and other especial ly absorpt ive
surfaces an addi t ional loss of up to l O m m or more may be included.
Allowance must also be made for reduced losses from covered areas
(paved or roofed).
TABLE 3.1 Surface Loss Parameters
I n i t i a I abstract ion In f i l t r a t i on ra te
Max Soil
Surface retention moisture def ic i t
mm inches mm/h inches/h -~ mm inches - - Paved up to 5 0.2 0 0 0 0 Clay up to 5 0.2 15 0.8 2 - 5 0.1 - 0.2 Loam up to 7 0.3 20 1.2 5 - 15 0.2 - 0.6 Sandy up to 10 0.4 30 1.5 15 - 25 0.6 - 1 Dense up to 15 0.6 5 0 . 2 5 - 15 0.2 - 0.6
vegetation
58
REFERENCES
O p ten Noort , T.H. and Stephenson, D., 1982. F l o o d Peak C a l c u l a t i o n i n South A f r i c a . Water Systems Research Programme, U n i v e r s i t y of t h e W i t w a t e r s r a n d , Repor t No. 2/1982.
Stephenson, D., 1981. Stormwater H y d r o l o g y a n d D r a i n a g e , E l s e v i e r , Amsterdam. pp 276.
Stephenson, D., 1982. "Peak F lows f rom Smal l Catchments U s i n g K inemat i c H y d r o l o g y , ' ' Water Systems Research Programme, Repor t 4/1982. U n i v e r s i t y of the W i t w a t e r s r a n d , Johannesburg.
Wooding, R.A., 1965. A H y d r a u l i c Model f o r t h e Catchment-s t ream Problem, I I , Numer ica l So lu t i ons , Jou rna l of H y d r o l o g y , 3, p 268-282.
59
CHAPTER 4
K I NEMAT I C ASSUMPT IONS
NATURE OF K I NEMATIC EQUATIONS
The k i n e m a t i c f l o w a p p r o x i m a t i o n h a s p r o v e d t o b e v e r y u s e f u l
in s t o r m w a t e r m o d e l l i n g and in the deve lopment o f a b e t t e r u n d e r s t a n d i n g
o f the r u n o f f p rocess . K i n e m a t i c mode ls a r e d e t e r m i n i s t i c mode ls and
rep resen t a d i s t r i b u t e d , t i m e - v a r i a n t system. They c a n , t he re fo re , b e
c o u p l e d w i t h o t h e r p rocess mode ls t o i n v e s t i g a t e the e f f e c t s o f l a n d use
change , tempora l and s p a t i a l v a r i a t i o n s in r a i n f a l I and wa te rshed
c o n d i t i o n s , a n d p o l l u t a n t washo f f .
S t a r t i n g w i t h the f o r m u l a t i o n o f t he k i n e m a t i c w a v e t h e o r y b y
L i g h t h i l l a n d Wh i tham (19551, k i n e m a t i c o v e r l a n d f l o w mode ls h a v e been
u t i l i z e d i n c r e a s i n g l y in h y d r o l o g i c i n v e s t i g a t i o n s . The f i r s t a p p l i c a t i o n
to wa te rshed m o d e l l i n g was b y Henderson and Wooding (1964) . The
c o n d i t i o n s u n d e r w h i c h the k i n e m a t i c f l o w a p p r o x i m a t i o n h o l d s f o r s u r f a c e
r u n o f f were f i r s t i n v e s t i g a t e d b y Woolh iser and L i g g e t t (1967) ; t h e y
f o u n d i t i s an a c c u r a t e a p p r o x i m a t i o n to the f u l I e q u a t i o n s f o r most
o v e r l a n d f l o w cases . S ince then , a n a l y t i c a l s o l u t i o n s h a v e been o b t a i n e d
f o r r u n o f f h y d r o g r a p h s during s t e a d y r a i n f a l I on s i m p l e geomet r i c s h a p e d
wa te rsheds ; and n u m e r i c a l mode ls h a v e been deve loped f o r a p p l i c a t i o n
to more complex wa te rsheds and u n s t e a d y r a i n f a l l . W i th the easy
a v a i l a b i l i t y o f m ic ro-computers , t he n u m e r i c a l mode ls a r e r e a d i l y access-
i b l e . Success fu l use o f these mode ls r e q u i r e s a f a m i l i a r i t y w i t h compu te rs
a n d an u n d e r s t a n d i n g o f k i n e m a t i c o v e r l a n d f l o w .
K I NEMAT I C APPROXIMATION TO OVERLAND FLOW
K i n e m a t i c o v e r l a n d f l o w o c c u r s when the d y n a m i c te rms in the
momentum e q u a t i o n a r e neg l i g i b l e . T h e r e i s no a p p r e c i a b l e b a c k w a t e r
e f fec t a n d d i s c h a r g e c a n b e exp ressed as a u n i q u e f u n c t i o n o f t h e d e p t h
o f f l o w a t a l l d i s t a n c e x and t ime t. T h a t i s ,
Q = b a y " (4.1
where Q i s t he d i s c h a r g e , y i s t h e d e p t h of f l o w , b the w i d t h a n d ci , m
a r e cons tan ts .
The l a t t e r c o n c l u s i o n c a n b e e s t a b l i s h e d b y n o r m a l i z i n g t h e
momentum e q u a t i o n b y t h e s t e a d y u n i f o r m d i s c h a r g e Qn. The momentum
e q u a t i o n t h e n becomes
60
where So i s the bed slope, v i s flow velocity, g i s g rav i t y and A i s
cross sectional area. I f the sum of the terms to the r i g h t of the minus
sign is much less than one, then
Q Qn ( 4 . 3 )
which means that g radua l ly -var ied flow may be approximated by a
uniform flow formula such as Manning’s equation. I f one writes
Manning‘s equation for a wide rectangular cross-section such as an
over land flow plane, since the hydrau l i c rad ius can be approximated
by the depth of flow, one obtains the fo l lowing expression (SI un i ts )
Q = - b y y 1 2/3s f
or Q/b = A S y5 I3
( 4 . 4 )
( 4 . 5 )
where Q/b i s discharge per u n i t width. For runof f from a plane surface
with uniform roughness and slope, n and So are constant. Eq. 4.5 can
therefore be wr i t ten in the same form as Eq. 4.1 with cx = S “n and
rn = 5/3; and for the c i ted conditions discharge can indeed be expressed
as a unique funct ion of the depth of flow.
1
Governing Equations
The governing equations for the kinematic over land flow approx-
imation are Eq. 4.1 and the equation for con t inu i ty
where q. is the inf low per u n i t length x .
Conditions for the Kinematic Approximation
The conditions under which the kinematic approximation holds for
over land f i o w can best be i l l us t ra ted b y app ly ing the fuf i equations
to runoff from a long, uni formly sloped plane of u n i t width as shown
in F igure 4.1. The p lane i s of length L and slope So. Rain fa l l occurs
over the plane at the r a t e i ( x , t ) , and i n f i l t r a t i o n i s at the ra te f ( x , t ) .
By wr i t i ng the r a i n f a l I and i n f i l t r a t i on rates in terms of x and t , we
include the effects of spat ia l and temporal var ia t ions i n r a i n f a l l and
soi l . The cont inu i ty and momentum equations are wr i t ten as
( 4 . 7 )
and
61
v i
Y ( 4 . 8 ) - av a v ay -
a t + v z + g= - g(S0 - 5 ) - e f
where i e ( x , t ) i s the r a i n f a l l excess r a t e a t distance x and time t and
the other terms have been defined previously.
UNIFORM RAINFALL
F L -x x Fig. 4.1 Uniform r a i n over a long impermeable plane
For the purpose of t h i s discussion, Sf i s conveniently defined by
the Chezy equation
( 4 . 9 )
C being the Chezy coefficient which equals Jf/sg where f i s the Darcy
f r i c t i on factor. By wr i t i ng Eqs. 4.7 and 4.8 i n dimensionless form, the
number of parameters are reduced from f i ve to two with obvious advant-
ages. Woolhiser and Liggett (1967) f i r s t presented the fol lowing dimen-
sionless equations
au a T ax ax - ' H + " E + H - = 1 (4 .10 )
and
(4 .11 )
where
H = y/yo, U = v/vo, X = x/L, T = tvo/L ( 4 . 1 2 )
and y and v a re the normal depth and velocity, respectively, at the
end of the p lane for a given steady r a i n f a l l excess ra te , i The normal-
i z i n g parameters a re related by :
i L = v y (4 .13 )
e'
0 0
62
and 2 2
vo/coYo = so ( 4 . 1 4 )
The two independent parameters in Eqs. 4.10 and 4.11 are the normal
flow Froude number, Fro, vo/ J ( g y o ) , and the kinematic flow number,
k (Woolhiser and Liggett , 1967) .
k = - 2
yoFro
( 4 . 1 5 )
Woolhiser and Liggett (19671, Brutsaert (19681, Morr is (1979) and
Vieira (1983) solved Eqs. 4.10 and 4.11 for the r i s i n g hydrograph for
a range of F r and k values under normal depth, c r i t i ca l depth and
zero depth gradient downstream boundary conditions. The solutions were
started using an ana ly t i c solut ion for simple cases and numerical
solutions i n the other three character ist ic solut ion zones. The resu l ts
of al I studies were qu i te s imi lar . Sample resul ts a re shown in Figures
4.2 and 4.3 .
0 .I 2 3 .4 .5 .6 .7 .8 9 1.01.1 1.21.31.41.51.6 T
Fig. 4.2 Effect of va ry ing k on dimensionless r i s i n g hydrograph
3 , 764, 1967, American Geophysical Union). (Woolhiser and Liggett , Water' Resources Research,
63
As seen i n F igure 4 .2 , as the kinematic flow number increases,
the solut ion converges very r a p i d l y toward the solut ion for k equal to
i n f i n i t y . Woolhiser and L igget t noted that for Fro = 1 the maximum er ror
between the r i s i n g hydrographs for k = 10 and k equal i n f i n i t y i s about
10 percent. The effect of va ry ing Fro whi le holding k constant i s shown
i n Figure 4.3. Simi lar to the resul ts obtained for increasing k , as F r
increases, the solut ion converges to the solut ion for k equal to i n f i n i t y .
What i s the s igni f icance of k equal to i n f i n i t y ? I f one div ides
Eq. 4.11 by k, the momentum equation reduces to the fo l lowing expression.
1 - U2/H = 0
Hence
U2 = H ( 4 . 1 7 )
Subst i tut ing Eq. 4.17 into the dimensionless cont inu i ty equation, Eq.
4.10, the k inemat ic wave equation i s obtained.
( 4 . 1 6 )
= 1 - aH a H 2 / 3 aT + x ( 4 . 1 8 )
Solving Eq. 4.18 for an i n i t i a l l y d r y surface, we get
H = T ( 4 . 1 9 )
and from Eq. 4.17
1 / z U = T
Thus, the r i s i n g hydrograph is given by
I .5
Q* :$ AC k =I0
PA RAM E TER : Fro /
. I 1 '0 .I .2 .3 .4 .5 .6 .7 .8 .9 1.01.1 1.2 1.31.41.5 1.6
T
( 4 . 2 0 )
( 4 . 2 1 )
Fig. 4.3 Effect of va ry ing F r on dimensionless r i s i n g hydrograph (Woolhiser and Liggc?tt, Water Resources Research, 3 ,
764, 1967, American Geophysical Union).
64
m'
10-
where Q c i s discharge normalized by the excess r a i n f a l l intensi ty. The
resul t i n Eq. 4.21 suggests that a l l r i s i n g hydrographs for steady excess
r a i n on uniform planes can be represented by a s ing le dimensionless
hydrograph. This resul t also suggests there i s a unique relat ionship
between depth and discharge, and the depth i s the normal depth for
uniform flow at that discharge. When k i s la rge the solut ion to the f u l l
equation can be closely approximated by the kinematic solution. This
i s the kinematic approximation which has been described in de ta i l by
several invest igators ( L i g h t h i l l and Whitham, 1955; Wooding, 1965;
Woolhiser and Ligget, 1967; Morr is and Woolhiser, 1980; Vieira, 1983).
Woolhiser and L igget t (1967) stated that the kinematic wave
approximation may be used instead of the fu l I equations i f k > 2 0 and
Fro> 0.5. Overton and Meadows (1976) noted that Eq. 4.21 i s appl icable
to character ist ic zone A (Fig. 5.1) and of the solut ion shown i n F igure 5.1,
zone A constitutes substant ia l ly a l l of the solut ion for kinematic flow
numbers of 10 or greater. Therefore, they recommend the kinematic wave
approximation be used only when k>10 , regardless of the Froude number
value. Morr is and Woolhiser (1980) re-evaluated Eqs. 4.10 and 4.11 and
suggested that F r k ' 5 i f the kinematic approximation i s used. I t i s
interesting to note th i s i s equivalent to the o r ig ina l c r i t e r i a suggested
by Woolhiser and Liggett , except that i t al lows the kinematic approxi-
mation to be used for F r < 0.5, provided k >20. With these condit ions i n
mind, and using the resul ts obtained i n h i s own study, V ie i ra (1983)
developed the plot i n F igure 4.4 as a guide to determine when the
kinematic and d i f fus ion wave approximations may be used.
2
0
FULL SAINT VENANT
k
KINEMATIC APPROX.
Fig . 4.4 App l icab i l i t y of kinematic, d i f fusion and dynamic wave models (After Vieira, 1983)
65
Kinematic Flow Number
The k inemat ic f low number can be p laced in terms of the p h y s i c a l
a n d h y d r a u l i c charac ter is t i cs of a p l a n e b y e l i m i n a t i n g y a n d Fro from
Eq. 4.15 us ing Eqs. 4.5 a n d 4.13. The r e s u l t i n g r e l a t i o n s h i p i s
1 .2s0.4L0.2 gn 0 k =
. 0.8 (4.22)
e For r a i n f a l l in tens i ty in mm/hr a n d length in meters, Eq. 4.22 becomes
1 .2s0.4L0.2
. 0 . 8 k = 1 . 7 ~ lo6 "_
e
a n d fo r r a i n f a l l in tens i ty in in/hr a n d length in feet
1.2 0.4 0.2
- 5 " L . 0.8 e
k = 10
(4.23)
(4.24)
In general, h i g h k va lues a r e produced on rough, steep, long p lanes
w i t h low r a i n ra tes.
S i m i l a r l y , the q u a n t i t y k F r 2 can be expressed in terms of the
phys ica l a n d h y d r a u l i c charac ter is t i cs of a p lane. From Eq. 4.15
k F r 2 = ~ (4.25)
I f we w r i t e Eq. 4.21 in dimensional form u s i n g Eq. 4.12, M a n n i n g ' s
res is tance law instead o f ' Chezy's, a n d the fo l low ing d e f i n i t i o n fo r Q+
0 Y O
Q Q, = - e
we obta in the equat ion
t v
e L 0 5 / 3
Q = i A (-)
(4.26)
(4.27)
where A i s the c o n t r i b u t i n g watershed area. For a steady r a i n f a l l excess
ra te , the f low i s a maximum a n d equal to i when the terms ins ide the
parentheses a r e equal to one, tha t is , when time i s equal to the time
of concentrat ion, tC, o r p r e f e r a b l y , the time to e q u i l i b r i u m . The q u a n t i t y
L /vo i s one d e f i n i t i o n fo r the time of concentrat ion used in peak r u n o f f
estimates. According to Eq. 4.27, f o r a steady excess ra te , a t the time
of concentrat ion, the r u n o f f r a t e i s a maximum a n d equal to i . In other
words, one d e f i n i t i o n fo r the time of concentrat ion i s tha t i t i s the time
requ i red fo r a watershed to reach e q u i l i b r i u m fo r a steady r a i n f a l l
excess. Th is occurs when
tC = L /vo (4.28)
66
Substi tut ing Eq. 4.28 into Eq. 4.13 we get
y o = i t
which, when substi tuted into Eq. 4.25 yields
e c
2 oL kFo = - i t
e c
(4.29)
(4.30)
Using the def in i t ions of Eqs. 4.28 and 4.29, and Manning's equation,
one obtains the desired expression, for r a i n f a l I i n mm/hr,
and
460Sb*3 L O e 4
kF: = 0.6i0.2 e
(4.30a)
(4.30b
for r a i n f a l l i n in /h r .
2 I n general, kFo values are h igh for smooth, steep, long planes w i th
low r a i n f a l l rates. This resul t i s s im i la r to the expression for k, except
that the effect of roughness on the Froude number suggests the kinematic
model may be more appl icable to u rban watersheds with smooth impervious
surfaces.
To i l l us t ra te the hydrological app l i cab i l i t y of these results,
consider an asphal t pa rk ing lo t w i th the fo l lowing character ist ics:
L = 50 meters; So = 0.005; n = 0.022. For an average excess intensi ty
of 50 m m / h r , k = 200 and kFr2 = 31.
K I NEMAT I C AND NON-K I NEMAT I C WAVES
I t was noted in Chapter 2 that the di f fusion and kinematic wave
models may be used instead of the f u l l dynamic wave equations i f cer ta in
assumptions can be made. I n th is section, condit ions under which the
two models can be appl ied to flood rou t ing i n streams are examined.
The material presented here should g ive the reader a better under-
standing of the physical nature of kinematic and non-kinematic waves.
The physical s igni f icance of kinematic and non-kinematic waves and
the major differences between the respective models are better understood
i f the wave speed and crest subsidence (hydrograph dispersion) charac-
ter ist ics a re known.
67
Wave Speed - Kinematic Waves
The kinematic wave speed i s determined by comparing the cont inui ty
equation wi th no lateral inf low
with the de f in i t ion of the total der iva t ive of Q
dQ - aQ dx aQ dt ax dt
+ -
By rewr i t ing E q . 4.31 as
a Q aA dQ dt = ax at dA dx - + - - -
to an observer moving with wave speed, c,
the flow ra te would appear to be constant, i.e.,
( 4 . 3 1 )
( 4 . 3 2 )
( 4 . 3 3 )
( 4 . 3 4 )
( 4 . 3 5 )
This resul t follows from the de f in i t ion of the total der iva t ive , Eq. 4.32,
and the equation of cont inui ty, Eq. 4.31.
For most channels where the flow i s in-bank
( 4 . 3 6 )
where B i s the channel top width i n meters ( fee t ) ; and since Q i s a
unique function of y
Q = c l ym ( 4 . 3 7 )
the kinematic wave speed i s given as
( 4 . 3 8 )
This relat ionship i s analogous to that of Seddon (1900) who observed
that the main body of f lood waves on the Mississippi River moved a t
a r a t e given by Eq. 4.38.
Eq. 4.38 implies that equal depths on both the leading and
recession limbs of a hydrograph t rave l a t the same speed. Since greater
depths move at faster rates, i t follows that the leading l imb of the
hydrograph w i l l steepen and the recession limb w i l l develop an elongated
ta i l . Eq. 4.38 also shows that kinematic waves are propagated down-
stream only, i.e. Eq. 4.38 i s a forward character ist ic. Kinematic flow
does not exist where there are backwater effects.
Crest Subsidence
Combining Eqs. 4.32 and 4.35, and subs t i tu t ing fo r Q using Eq
68
4.37 i t can be shown that to an observer moving w i th wave speed c
Manipulat ing th is equation y ie lds
- d Y = V + Q dt = o dx ax at dx
( 4 . 3 9 )
( 4 . 4 0 )
which establishes that theoret ical ly, the kinematic wave crest does not
subside as the wave moves downstream.
These resul ts show that a kinematic wave can a l te r i n shape bu t
does so without crest subsidence. Further, the maximum discharge ra te
occurs wi th the maximum depth of f low. (Th is i s the assumption imp l ic i t
i n the slope-area method for est imating flood discharges from h igh water
marks).
H y d r a u l i c Geometry and Rating Curves
One important aspect of the kinematic wave model i s the replacement
of the momentum equation wi th a uniform flow formula, which i s nothing
more than a s ing le valued r a t i n g between discharge and depth (or area)
at a point in the stream. As discussed previously, the fact that na tura l
channels are not pr ismat ic leads to subsidence and dispersion of a hydro-
graph, suggesting that the discharge ra t i ng relat ionship i s not unique
but var ies over the hydrograph. I f the dispersive character ist ics are
small such that a va r iab le r a t i n g relat ionship does not d i f f e r s ign i f i -
can t ly from the s ing le valued ra t i ng , the conclusion can be drawn that
the main body of a
kinematic model (o r
simul at ion purposes.
tat ional requirements
I t i s evident
and discharge f i r s t
hydrograph moves kinematical ly. I n which case, the
the di f fusion model) should be suff ic ient for most
This represents an economy of da ta and compu-
over the dynamic wave model.
from the relat ionships between hyd rau l i c geometry
set for th by Leopold and Maddock ( 1 9 5 3 ) that the
flow in many streams i s essential ly kinematic. T h e fact that the channel
character ist ics of na tura l streams seemed to consti tute an interdependent
system which could be described by a series of graphs hav ing a simple
geometric form suggested the term "hydraul i c geometry". Subsequent
studies have ver i f ied and expanded on th i s i n i t i a l work w i th the resu l t
that hyd rau l i c geometry equations may be used to estimate general
channel character ist ics at any locat ion w i th in the drainage system.
As a resu l t of the i r ana lys is of the va r ia t i on of hyd rau l i c
character ist ics at a pa r t i cu la r cross-section in a r i v e r , Leopold and
Maddock proposed that discharge be related to other hyd rau l i c factors
69
i n the fol lowing manner.
(4.41a) b w = aQ
(4.41b) f d = CQ
v = kQm ( 4 . 4 1 ~ )
where w i s width, d i s depth, v i s cross-sectional mean veloci ty, Q i s
discharge, and a, b, c, f , k , and m are best f i t constants. I t follows
that since width, depth, and mean velocity a re each functions of
discharge, then b + f + m = 1.0; and ack = 1.0. Betson (1979) noted
that a fourth relat ionship also can be presented (4.41d)
A = nQp
where A i s the cross-sectional area of flow. Betson also noted that
f = p - b and m = 1 - p. T h e relat ionship in Eq. 4.41 are for ind i -
v idua l stat ions i n that they re la te channel measures to concurrent
discharge.
The resu l ts from several studies are shown i n Table 4.1. I t i s
notable that the values do not va ry widely, p a r t i c u l a r l y for the depth
discharge relat ionship. These resu l ts reinforce the use of s ing le valued
ra t i ng curves and simp1 i f ied rou t ing models.
NON-K I NEMAT I C WAVES
The resu l t in Eq. 4.40 frequent ly does not agree with nature.
Rather, due to previously mentioned factors, flow peaks are seen to
subside which suggests the appl icat ion of the kinematic model i s l imited,
and that ei ther the d i f fus ion or dynamic wave model i s preferred. I t
i s important then to examine the non-kinematic wave models and to
establ ish how they d i f f e r from the kinematic model.
Differences between the two non-kinematic models can be invest-
igated by examining the signi f icance of each of the dynamic terms i n
the momentum equation. The discharge at a point in a stream is
Q = vA (4.42
The momentum equation can be rewri t ten as follows:
- Q A2 ax A3 ax A at A2 at ax o f A
a Q Q 2 - aA + - 1 - aQ - - Q - aA + ay = g (s -5 ) - - q i (4.43
The p a r t i a l der iva t ive of A w i th respect to time is removed i n terms of
the spat ia l der iva t ive of Q using the cont inui ty expression. After th is
subst i tut ion and rear rang ing , Eq. 4.43 becomes
s -sf 28 aQ Q2 aA + - a B + a v 1 =
gA2 ax
gA3 ax gA at ax (4.44)
70
TABLE 4.1 Typ ica l Stat ion Exponent Terms f o r Geomorphic Equat ions
Exponents
LOCAT I ON OF BASIN
w i d t h depth ve loc i ty a rea Reference b f m P
Midwest 0.26 0.40 0.34 0.66 Leopold, et al. (1954)
Brandywine, P.A 0.04 0.41 0.55 0.45 d i t t o
158 Stations in U.S. 0.12 0.45 0.43 0.57 d i t t o
B i g Sandy River , K Y 0.23 0.41 0.36 0.64 Sta l l a n d Yang (1976)
Cumberland Plateau, KY 0.245 0.487 0.268 0,732 Betson (1979)
Johnson C i ty , TN 0.08 0.43 0.49 0.51 Weeter a n d
T heore t i ca I 0.23 0.42 0.35 0.67 Leopold a n d Meadows (1 979)
Langbe in (1962)
At any cross-section Eq. 4.36 holds; and fo r most n a t u r a l channels , the
wave speed ( c e l e r i t y ) i s approx imated b y the k inemat ic wave speed.
I f Chezy's res is tance equa l ion i s assumed
38 2A
c = - (4.45)
Drawing on these two re la t ionsh ips and the d e f i n i t i o n fo r Froude number
2 g 2 8
gy gA3 (4.46) F r 2 = __
the var ious terms in Eq. 4.44 can be r e w r i t t e n as
n
a n d
(4.47c)
Trac ing back , the cont r ibu t ion of each term in the momentum equat ion
i s found (Meadows, 1981).
a n d
71
1 av 2 a Y
g at ax = (-0.75 F r - _ - (4.48b)
which allows the momentum equation to be wri t ten as
( 1 - 0.25 Fr2) a = ax ' 0 - 'f
(4.49)
An equivalent expression was found by Dooge (1973).
Examination of Equations 4.48 and 4.49 reveals that the convective
and temporal accelerat ion terms essential ly are of equal magnitude but
opposite sign, and hence, act to near ly cancel each other. These two
terms are s ign i f i can t for Froude numbers greater than 0.60, where s igni-
ficance i s taken as 10 percent of the coefficient value i n Equation 4.49.
Evidence of Froude numbers less than 0.60 for unsteady events i n small
streams i s documented in the l i terature, e.g. (Gburek and Overton,
1973). Further, using the theoretical values for hyd rau l i c elements of
Leopold and Langbein (1962), i t was shown by Meadows (1981) that
F r oi
demonstrat ing that Froude number i s la rge ly insensit ive to increasing
discharge i n most na tura l streams for flow i n bank. These resul ts suggest
the di f fusion wave model can be confidently appl ied to most f lood rou t ing
events.
Wave Speed
Based on the method of characterist ics, i t was shown that dynamic
waves propagate both downstream ( fo rward character ist ic) and upstream
(backward charac ter is t i c ) . The di f fusion wave speed i s given by the
kinematic wave speed. As such, the di f fusion wave model has only a
forward character ist ic meaning that wave forms are propagated only
downstream and that backwater effects a re negl ig ib le. I t i s left to the
reader to confi rm this.
C r e s t Subsidence
Both the dynamic and d i f fus ive wave models simulate a dispersing
hydrograph, hence, a subsiding wave crest. To i I lustrate, consider the
modified d i f fus ive wave equation, Eq. 4.49. For the fo l lowing develop-
ment, a rectangular cross section i s assumed. As w i th the der ivat ion
of most over land and open channel flow equations, t h i s assumption
great ly simp1 i f ies the mathematics, yet does not a l te r appreciably the
f i na l form of the equations being developed.
72
Approx imat ing the f r i c t i o n slope w i t h Chezy's equat ion, Eq. 4.49
becomes
Q2 (1 - 0.25 F r 2 ) a = S - -
ax o c2A2R
T a k i n g the p a r t i a l d e r i v a t i v e w i t h respect to t ime
2 a ay Q2 - 2 a Q 2 aA 1 aR ( 1 - 0.25 F ~ ) - (--) = - __ 1 at ax c2A2R Q a t A a t R a t J
From cont inu i ty
o r
Genera l ly , over a reach, aq/ax = 0. Thus,
For a p r ismat ic section
dA _ = B d y
such tha t
- a A * B d y g B a a x dA a x which, when subs t i tu ted in to Eq. 4.54 y ie lds
(4.50)
(4.51)
(4.52)
(4.53)
(4.54)
(4.55)
(4.56)
In o b t a i n i n g Eq. 4.56, the assumption was made tha t aB/at = 0;
which i s sa t is fac to ry i f the channel i s r e c t a n g u l a r o r the f lood wave
r i ses s lowly . The momentum equat ion can now be w r i t t e n
(4.57)
For a wide r e c t a n g u l a r channel ( w > l O y ) , the h y d r a u l i c r a d i u s , R,
i s approx imate ly equal to the depth of f low, y . Us ing t h i s approx imat ion
a n d c o n t i n u i t y f o r a r e c t a n g u l a r geometry
(4.58a)
(4.58b)
the r i g h t hand s ide of Eq. 4.57 i s r e w r i t t e n as
73
Comb in ing s i m i l a r terms a n d r e c o g n i z i n g t h a t the c o e f f i c i e n t terms a r e
mere l y S 0’
s [ - - - - 2 aQ 3 qi j o a a t A a x A
The whole e q u a t i o n t h u s becomes
M u l t i p l y i n g b y Q / 2
F o r Chezy ’s e q u a t i o n
3Q 2A
c = -
M a k i n g t h i s s u b s t i t u t i o n i n t o Eq. 4.60
(4.59)
(4 .60)
(4.35)
(4.61)
w h i c h i s a c o n v e c t i v e - d i f f u s i v e e q u a t i o n f o r u n s t e a d y s t reamf low . T h i s
e q u a t i o n i l l u s t r a t e s the o r i g i n of t he d i f f u s i v e wave l a b e l . The presence
of t he d i s p e r s i o n te rm (second p a r t i a l d e r i v a t i v e ) c o n f i r m s t h a t the
d i f f u s i v e wave model s imu la tes a s u b s i d i n g peak .
One v e r y i n t e r e s t i n g p r o p e r t y of t he c res t r e g i o n of a d i f f u s i v e
wave c a n b e d e r i v e d b y r e w r i t i n g Eq . 4.50 in terms of Q a s f o l l o w s
Q = C y B ,/y[So-(l - 0.25 F r 2;2]
where h y d r a u l i c r a d i u s h a s been a p p r o x i m a t e d b y y . T a k i n g the d e r i v a -
t i v e w i t h respect to x a n d e q u a t i n g to zero y i e l d s
(4.62) a x
(4.63)
In the r e g i o n o f t he c res t , the shape of the h y d r o g r a p h i s concave
downward , a n d a 2 y / a x 2 0, a n d the re fo re , b y Eq. 4.63, a y / a x < 0,
a l so . Tha t i s , t he peak f l o w r a t e does not o c c u r where d e p t h i s a
maximum, b u t a t a p o i n t in a d v a n c e o f the maximum dep th .
74
Looped Rating Curves
Eq. 4.62 c lear ly demonstrates that a s ing le valued r a t i n g between
discharge and depth (a rea) does not hold for non-kinematic waves. An
approximate expression for the va r iab le (looped) r a t i n g curve i s given
by
- _ Q - 4- ( 1 - 0.25 Fr')
'n ax
( 4 . 6 4 )
where Q i s the uniform flow at a given depth. This expression i s
rendered more useful i f the spat ia l der iva t ive i s replaced by some
al ternate quant i t y , deductible from in-si tu conditions.
Using the kinemat ic relat ionship
ay - 1 - ay ax c a t
Eq. 4.64 can be wr i t ten as
( 4 . 6 5 )
2 ( 1 - 0.25 F r )
a t ( 4 . 6 6 )
I t must be noted that Eq. 4.66 i s not s t r i c t l y correct since the kinematic
relat ionship was included.
A typical looped r a t i n g curve i s shown in the F igure 4.5. Com-
par ison with the associated discharge hydrograph i l lus t ra tes that as
a f lood hydrograph passes a point , the maximum discharge i s f i r s t
observed, then the maximum depth, and f i n a l l y a point where the flow
is uniform. The uniform 'flow occurs when the f lood wave i s essent ia l ly
horizontal and therefore has a slope, dy/dx, that i s very small re la t i ve
to the bed slope. Th is obviously w i l l occur close to the region of max-
imum depth. The occurrence of uniform flow i s i l l us t ra ted graph ica l l y
as the point of intersection of the looped r a t i n g curve w i th the s ing le
valued uniform flow r a t i n g curve.
I t should be noted that the scale i s exaggerated for c l a r i t y . The
three points in question are more l i ke l y to occur much closer together
than indicated by the f igure.
The usefulness of the looped r a t i n g curve compared with a s ingle
valued r a t i n g curve is determined b y how wide the loop is re la t i ve to
the s ingle valued curve. I t should be noted however, that most publ ished
streamflow data and associated r a t i n g curves determined from f i e ld
discharge measurements general ly a re better approximated by a s ingle
valued relat ionship. Looped curves can be approximated using Eq. 4.64
or 4.66 and time series records of r i v e r stage a t a stat ion.
UNIFORM FLOW RATING W R V E d Q
TIME
Fig. 4.5 Loop stage-discharge r a t i n g curve and associated discharge hydrograph for at tenuat ing wave.
76
MUSK I NGUM R I VER ROUT I NG
Flood rou t ing refers to a set of models used to predict the temporal
and spat ia l var ia t ions of a f lood wave ( runof f hydrograph) as i t t rave ls
through a channel reach. Routing techniques are classed into two
categories: hydrau l ic and hydrologic. The kinemat ic, d i f fusion and
dynamic wave models a re hyd rau l i c rou t ing models. The hydrologic models
are based on cont inu i ty and an empi r i ca l l y der ived relat ionship between
channel storage and discharge; therefore, they are not as r igorous as
the hydrau l i c models and represent a fu r ther s impl i f icat ion to the f u l l
equations for open channel flow.
Perhaps the best known and most widely used of the hydrologic
models i s the Muskingum rou t ing model. This model was developed
o r ig ina l l y for f lood rou t ing on the Muskingum River in Central Ohio,
hence the o r ig in of the name. T h e model u t i l i zes cont inu i ty
dS l + Q L - 0 = - dt
where I i s inf low to a r i v e r reach, QL is lateral inf low ( = q A x ) , 0 i s
outflow and S is the storage w i th in the reach; and the storage relat ion-
(4 .67 )
ship
S = K [Z l + ( l - Z ) O ] (4 .68 )
where K i s a character ist ic storage time approximated as the t rave l time
through a reach, and 2 i s a weight ing coefficient.
For attenuating waves, z<o-5.
Equations 4.67 and' 4.68 are solved using a f i n i t e di f ferencing
technique. Defining I 1 = l ( t ) and I = I ( t + A t ) , and s im i la r l y , 01 , 02, and
S1 and S2, the fo l lowing approximation to Eq. 4.67 i s wr i t ten 2
01+02 s -s (4 .69 )
11+12+&--=- 2 1 2 2 At
where QL i s the average lateral inf low dur ing the time in te rva l At. The
inf low hydrograph provides I, and 1 2 , and O2 i s the desired quant i t y .
O1 i s known from ei ther i n i t i a l condit ions or a previous calculat ion.
S and S are expressed in terms of I and 0 as follows
S - S = K[Z(12-11) + ( 1 - Z) (02-01)] ( 4 . 7 0 )
1 2
2 1
Substi tut ing Eq. 4.70 into Eq. 4.69 and s imp l i f y ing gives
o2 = C 0 l 2 + C , I , + c20, + c3QL
where
( 4 . 7 1 )
( 4 . 7 2 a ) -KZ + 0.5At '0 - K-KZ + 0.5At
KZ + 0.5At c 1 = K-KZ t 0.5At
K-KZ - 0.5At '2 = K-KZ + 0 . 5 ~ 1
a n d
'3 - K-KZ + 0 . 5 ~ t A t
77
(4.72b)
( 4 . 7 2 ~ )
(4 .72d)
K a n d t must h a v e the same t ime u n i t , a n d the f i r s t t h r e e c o e f f i c i e n t s
sum to 1.0.
Estimation of Model Parameters
The M u s k i n g u m model i s q u i t e s e n s i t i v e to the se lec t i on of model
pa ramete rs . H i s t o r i c a l l y , K a n d Z h a v e been es t ima ted b y m a t c h i n g model
o u t p u t w i t h a c t u a l i n f l ow-ou t f l ow reco rds . The o b v i o u s sho r t coming i s
t ha t the model i s l i m i t e d to g a u g e d st reams. Of tent imes, we need to r o u t e
f l o o d h y d r o g r a p h s a l o n g u n g a u g e d st reams. To do so r e q u i r e s t h a t we
h a v e a means o f e s t i m a t i n g model p a r a m e t e r s f rom a v a i l a b l e c h a n n e l a n d
h y d r o g r a p h c h a r a c t e r i s t i c s .
F o l l o w i n g the techn ique o f Cunge (1969) a n d u s i n g a T a y l o r se r ies
e x p a n s i o n to each of the terms in Eq . 4.67, i t i s t rans fo rmed to a n
e q u i v a l e n t e q u a t i o n o f the c o n v e c t i v e - d i f f u s i v e fo rm
- - 'a + A? ?-!? = b x ( l - Z ) c ( Q ) - - I - hXL ]- a % a t K a x
QL ' 2 K 2 + T
a x (4.73)
where A x i s the r e a c h leng th . Compar ison o f t h i s e q u a t i o n w i t h Eq. 4.68
shows t h a t
= $Q) (4.74a)
a n d
2 = - Q(1-0.25Fr ) ]
BSoA x c (Q )
(4.74b)
Cunge (1969) a n d l a t e r r e s e a r c h e r s developed s i m i l a r exp ress ions to Eqs.
4.74. Ponce a n d Y e v j e v i c h (1978) cons ide red the v a r i a t i o n o f K a n d
Z w i t h Q; w h i l e Dooge (1973) i n c l u d e d the c o r r e c t i o n f o r d y n a m i c ef fects ,
( I -0 .25Fr 1 , in the e q u a t i o n for 2 , b u t cons ide red c ( Q ) to b e cons tan t
a n d no t a f u n c t i o n of 8. Therefore, Eqs. 4.74 a r e the most g e n e r a l
exp ress ions f o r K a n d Z (Meadows, 1981).
2
Another v e r y i m p o r t a n t f e a t u r e o f Eq. 4.73 i s t h a t i t demonstrates
the Musk ingum r o u t i n g model i s d i f f u s i v e for Z<O.5, a n d o f f e r s the same
a d v a n t a g e s o f the d i f f u s i o n wave model. I f , however , Z=O.5, the
78
Muskingum model predicts pure t ranslat ion, and i s equivalent to the
kinematic wave model. Typical values for Z for na tura l streams are 0.3
to 0.4, and for pr ismat ic channels, 0.4 to 0.5.
K I NEMATIC AND D I FFUS ION MODELS
We have discussed the kinematic and di f fusion wave models as
approximations to the dynamic wave model and have shown they are
appl icable for certain f lood wave and channel condit ions.. As users, we
need c r i t e r i a or guidel ines for selecting which model to use. Two notable
works toward establ ishing such guidel ines are Henderson (1963) and
Ponce, et a l . ( 1 9 7 8 ) .
Henderson conducted a theoretical examination of the governing
equations s imi la r to that presented in the previous sections. He compared
theoretical resu l ts wi th a l imited number of f lood hydrographs, and noted
that subsidence i s most pronounced in the v i c in i t y of the wave crest.
Generally, he may be credited wi th efforts to c lassi fy f lood waves
according to the magnitude of So into waves broadly character ist ic of
steep, mi Id and intermediate slopes. However, he cautioned that t h i s
c lassi f icat ion i s not exhaustive, but should suf f ice for most floods in
na tura l waterways. He d i d not offer specif ic guidel ines to define mi ld,
intermediate and steep, although he concluded the kinematic model i s
appl icable i n steep channels; the d i f fus ion i n m i ld and steep channels;
and the dynamic to a l l three. The ra t iona le for t h i s conclusion i s that
he considered Fr2 > > 1 in steeply sloped channels; hence, according to
Eq. 4.49 , the momentum equation w i l l become S = Sf . For m i ld slopes,
F r2 < < I , and the momentum equation becomes :he s t r i c t d i f fusion wave
model. For intermediate slopes a l l terms in the momentum equations are
si gnif i cant.
Ponce, et a l . , appl ied l inear s tab i l i t y ana lys is in an ef for t to
propose a theory that accounts for wave celer i ty as well as attenuation
characterist ics. To do so required they use a l inearized, therefore
somewhat simp1 i f ied, version of the governing equations. Assuming a
sinusoidal wave, they compared the propagat ion character ist ics of the
kinematic, d i f fusion and dynamic wave models. As expected, the dynamic
wave model i s app l i cab le to the en t i re spectrum of waves that can be
routed with a one-dimensional model. For Fr<2, the celer i ty of a dynamic
wave i s greater than the kinematic wave celer i ty. For Fr=2, r o l l waves
w i l l form. Thus, for p r imary waves (main body of a f lood wave), Fr=2
i s the threshold d i v id ing attenuation and amp1 i f icat ion. For secondary
waves, Fr= l i s the threshold d i v i d i n g the propagation upstream or
79
downstream; fo r F r = l they remain s ta t ionary o r p ropagate downstream
o n l y ; and fo r F r < 1 , they propagate on ly downstream. A phys ica l
observat ion b y Stoker (1957) e x p l a i n s t h i s conclusion r e g a r d i n g secondary
waves:
"What seems to happen i s the fo l low ing : smal l forerunners of a
d is turbance (wave) t r a v e l w i t h the speed fi r e l a t i v e to the f l o w i n g
stream, b u t the r e s i s t i v e forces act i n such a way as to decrease the
speed of the main p o r t i o n of the d is tu rbance f a r below the va lues g i v e n
b y a.. . I '
Ponce, et a l . , d i d o f fe r f i r s t generat ion c r i t e r i a fo r app l i ca t ion
of the k inemat ic a n d d i f f u s i o n wave models:
Kinematic:
D i f fus ion :
TBSo > 171
T B S o ( e ) '30
where TB i s the d u r a t i o n of the f lood wave, So i s the channel slope,
v a n d y a r e the i n i t i a l ve loc i ty a n d depth of f low, respect ive ly , and
g i s g r a v i t y . Based on these c r i t e r i a , the k inemat ic model app l ies to
shal low f low o n steep slopes (hence the steep channel of Henderson a n d
sur face r u n o f f from h i l l s lopes) a n d to long d u r a t i o n f lood waves (slow
r i s i n g floods on major r i v e r s as observed b y Seddon). The d i f f u s i o n model
i s a p p l i c a b l e to these as we l l as a w ider range. When these two models
b reak down, the dynamic model appl ies.
P
These c r i t e r i a a r e si,gnif icant in that they r e l a t e model a p p l i c a t i o n
to channel slope a n d h y d r o g r a p h charac ter is t i cs . The reader i s caut ioned
that these a r e o n l y f i r s t generat ion formulae.
REFERENCES
Betson, R.P., 1979. A geornorphic model fo r use in streamflow rou t ing , Water Resources Research, Vo l . 15, No. 1 , pp. 95-101.
Brutsaer t , W. 1968. The i n i t i a l phase of the r i s i n g h y d r o g r a p h of tu rbu len t free sur face flow w i t h unsteady l a t e r a l in f low. Water Resources Research, V o l . 4, p p 1189-1 192.
Cunge, J.A., 1969. O n the subject of a f lood propagat ion computation method (Muskingum Method). J. Hydr . Res., V o l . 7, No. 2, pp . 205-230.
Dooge, J .C . I . 1973. L i n e a r theory of hydro log ic systems. U.S. Dept. of Agr icu l tu re , Agr i . Res. Ser. Tech. B u l l . No. 1968.
Gburek, W.J. a n d Overton, D.E., 1973. Subcr i t i ca l k inemat ic f low in a s tab le stream. J. Hydr . Div. ASCE. V o l . 99, No. HY9, pp. 1433-1447.
Henderson, F.M., 1963. Flood waves in pr ismat ic channels. J. Hydr . Div . ASCE, V o l . 89, No. H Y 4 , pp. 39-67.
Henderson, F.M. a n d Wooding, R.A., 1964. Over land f low a n d groundwater from a steady r a i n f a l l o f f i n i t e durat ion. Journal of Geophysical Research, V o l . 69, No. 8, p p . 1531-1540.
80
Leopold, L.B., e t a l . , 1954. F l u v i a l Processes in Geomorphology. N.H. Freeman, San F ranc i sco , Cal .
Leopold, L.B. a n d L a n g b e i n , W.B., 1962. The concept o f e n t r o p y in landscape e v o l u t i o n . U.S. Geologica l Su rvey P ro f . P a p e r 500-A.
Leopold, L.B. a n d Maddock, T. , Jr . 1953. The h y d r a u l i c geometry o f s t r e a m channe ls a n d some p h y s i o g r a p h i c i m p l i c a t i o n s . U.S. Geologica l Su rvey P ro f . P a p e r 252.
L i g h t h i l l , M.J. and Whitham, G.B., 1955. On k i n e m a t i c waves: I. F l o o d movement in l o n g r i v e r s . Proc. Roya l Society , London, Vol. 229, No. 1178, pp. 281 -
Meadows, M.E., 1981. M o d e l l i n g the impac t o f s to rmwate r r u n o f f , in Proceedings, I n t e r n a t i o n a l Symposium on U r b a n H y d r o l o g y , H y d r a u l i c s a n d Sediment Con t ro l , U n i v e r s i t y o f K e n t u c k y , L e x i n g t o n , Ken tucky , pp. 31 3-31 9.
M o r r i s , E.M., 1979. The e f fec t o f t he s m a l l s l ope a p p r o x i m a t i o n a n d lower b o u n d a r y c o n d i t i o n s on s o l u t i o n s o f t he S a i n t Venant Equa t ion . Jou rna l of H y d r o l o g y , Vol . 4 0 , pp. 31-47.
M o r r i s , E.M. a n d Woolhiser, D.A., 1980. Unsteady one-d imensional f l ow o v e r a p l a n e : p a r t i a l e q u i l i b r i u m a n d recess ion h y d r o g r a p h s . Water Resources Research, 16 ( 2 ) , pp 355-366.
Over ton, D.E. a n d Meadows, M.E., 1976. Stormwater M o d e l l i n g . Academic Press, N.Y.
Ponce, V.M., L i , R.M. a n d Simons, D.B., 1978. A p p l i c a b i l i t y o f k i n e m a t i c a n d d i f f u s i o n wave models. J. H y d r . D i v . ASCE, Vol . 104, No. HY3, pp. 353-360.
Ponce, V.M. a n d Yev jev i ch , V. 1978. Muskingum-Cunge Method w i t h v a r i a b l e pa ramete rs . J. H y d r . D i v . ASCE, Vol. 104, No. HY12, pp. 1663- 1667.
Seddon, J.A., 1900. R i v e r h y d r a u l i c s . T r a n s : ASCE, Vol . 43, p. 179. Sta l I , J.B. a n d Yang , C.T., 1970. H y d r a u l i c geometry o f I I I i n o i s s t reams.
Research Repor t No. 15, Water Resources Research Center , U n i v e r s i t y o f I I I i no i s , U r b a n a , I I I i no i s .
V i e i r a , J.H.D., 1983. Cond i t i ons g o v e r n i n g the u s e of a p p r o x i m a t i o n s f o r the S a i n t Venant e q u a t i o n s f o r s h a l l o w s u r f a c e w a t e r f low. J o u r n a l o f H y d r o l o g y , Vol. 60, pp. 43-58.
Weeter, D.W. a n d Meadow's, M.E., 1978. Water Q u a l i t y Mode l i ng f o r R u r a l Streams. F i r s t Tennessee - V i r g i n i a Development D i s t r i c t , Johnson C i t y , 104 p.
Wooding, R.A., 1965. A h y d r a u l i c model f o r the catchment steam problem, I . Kinemat i c wave theo ry . Jou rna l of H y d r o l o g y , Vol . 3, pp 254-267.
Woolhiser, D.A. a n d L i g g e t t , J.A. 1967. Unsteady one d imens iona l f l o w o v e r a p l a n e - the r i s i n g h y d r o g r a p h . Water Resources Research, Vol. 3, NO. 3, pp. 753-771.
81
CHAPTER 5
NUMER I CAL SOLUT IONS
METHODS O F SOLUTION OF EQUATIONS OF MOTION
T h e r e a re no known general ana ly t i ca l solut ions to the hyd rau l i c
equations
(5.1
They must therefore be solved using the method of character ist ics or
numerical integrat ion techniques. Ava i lab le numerical methods include
f i n i t e di f ferencing and f i n i t e elements.
F in i t e di f ferencing techniques are founded on the classical def in i t ion
for a continuous der iva t ive term. Use of these methods transforms the
set of p a r t i a l d i f fe ren t ia l equations into an equal number of approximate
algebraic equations which then a re solved according to the ru les of
I inear algebra.
The f i n i t e element method i s a re la t i ve l y recent approach to solv ing
pa r t i a l d i f fe ren t ia l equations that govern hyd rau l i c processes. The basis
of f i n i t e element integrat ion i s approximating polynomials. I n essence,
the polynomial coeff icients a re adjusted to minimize an er ro r term whi le
sat isfy ing known boundary. conditions. The resu l t ing polynomials express
the unknown var iab les i n terms of the known (independent) var iables.
The detai Is of t h i s method are beyond the scope of these notes; however,
appl icat ion of t h i s method to kinematic over land flow i s i l l us t ra ted i n
a later chapter.
Given the many and var ied ways of in tegra t ing the f lood rou t ing
equations, one can log ica l l y ask which technique to choose. Some csn
be discarded as being inaccurate o r unstable o r too time consuming;
others seem to reproduce solutions re la t i ve l y well. However, there i s no
s ing le answer to which method i s "best". Indeed, the answer to that
question depends on the pa r t i cu la r appl icat ion, and perhaps on the
ava i lab le comput ing equipment.
METHOD OF CHARACTER I ST I C S
The method of character ist ics may be described as a technique
whereby the problem of solv ing two simul taneous p a r t i a l d i f fe ren t ia l
82
equations (cont inui ty and momentum) can be replaced by the problem
of solv ing four o rd inary d i f fe ren t ia l equations. This method has been
known for many years; i t was devised long before the computer as a
means for g raph ica l l y in tegra t ing the unsteady streamflow equations.
T h e character ist ic equations are no longer solved graph ica l l y , but are
solved numerical ly using the computer.
By making the subst i tut ion
( 5 . 3 ) 2
c = SY
into Eqs. 5.1 and 5.2 and then by wr i t i ng f i r s t the sum, and then t
difference, of the two new equations, we obtain the two equations
q i (,-,)a0 ax + a0 at = g (so - S f ) - ( v+c) - A
(5.4a
(5.4b
e
which are two equations i n the form of di rect ional der ivat ives of
v + 2c. Recall ing the def in i t ion of a total der iva t ive , i t can be shown
that for
-
(5.5) - _ z ; - v + c
then
dx a(vk2.c) a(V'2.c) d (v*2c) dt a x at dt
which gives the desired set of o rd inary d i f fe ren t ia l equations to replace
the p a r t i a l d i f fe ren t ia l equations. The character ist ic roots (direct ions)
are given b y Eqs. 5.5, and along each direct ion the respective. total
der ivat ives i n Eqs. 5.6 hold. The resu l t ing equations can be rewr i t ten :
c + . _ dx - - v + c (5.5a) ' dt
(5.6a) d (v+2c)
(5.5b)
(5.6) - - + - = ~
q i - - - g(S0 - S f ) - ( v - c ) - dt A
- dx c . _ = v - c ' dt
(5.6b) d(v-2c) 'i
where c and c- symbolical ly designate forward and backward character-
g(S0 - S f ) - ( v + c ) - _ _ _ - A d t +
i s t i c respective1 y.
Physical ly, the character ist ic roots represent the pa th i n time
and space followed by a disturbance, e.g. flood wave. The speed of
propagation i s given by the slope dx/dt ; and the state of the system
(values of the dependent var iab les) i s given by the total der ivat ives
that hold along the character ist ic paths.
Mathematical ly, the character ist ics a re loci of possible discont inui t ies
In the temporal and spat ia l der ivat ives of the dependent var iables. Thus,
one may think of a character ist ic curve as a l ine of separation between
83
two regions of somewhat di f ferent physical conditions. Th is i s important
when modelling unsteady flows where the boundary conditions va ry w i th
time, since the solut ion at inter ior points of the ( x , t ) domain are
dependent on the boundary information. The solut ion at those points
above a character ist ic curve requires more boundary information than
the solution below the character ist ic. I n fact , the character ist ics i n
Eqs. 5.5 define four unique solution zones as shown i n F igure 5.1.
A
UPSTREAM I t
BOUNDARY
x= 0 X=
I
UPSTREAM DOWNSTREAM BOUNDARY BOUNDARY
x= 0
DOWNSTREAM BOUNDARY
- x :L
F ig . 5.1 Zones of solution domain defined by character ist ics.
(Woolhiser and Liggett , Water Resources Research, 3, 755, 1967, American Geophysical Union).
These zones are formed by the intersection of the forward and backward
character ist ics ernanat ing from the upstream (x=O) and downstream (x=L)
end of the channel reach at the i n i t i a l time. The solut ion in Zone A
requires only the i n i t i a l values ( the beginning state of the system a t
a l l x ) ; whi le the solut ion i n Zones B and C requires both i n i t i a l values
and a boundary condit ion. This i s because these zones l i e above the
backward and fo rward character ist ics, respectively. Zone B requires the
downstream boundary condit ion, and Zone C , the upstream. F ina l l y , Zone
D, which l ies above both character ist ics requires the i n i t i a l values and
both boundary conditions.
Numerical Integrat ion of Character ist ic Equations
The objective of the method of character ist ics is to f i l l the ( x , t )
plane with character ist ics as shown i n Figure 5.2. The unknowns are
determined at the intersections where the four equations 5.5a, 5.5b, 5.6a
84
and 5.6b are satisf ied. The extent to which solutions can be obtained
over the ( x , t ) p lane i s dependent on the amount of i n i t i a l value, (x,O),
and boundary condit ion, (0 , t ) and ( L , t ) , information that i s specif ied
beforehand. I n i t i a l conditions are the velocity and depth of flow at a l l
x at the beginning of the simulat ion, usua l ly designated time zero. The
upstream boundary condit ion typical ly i s the known inf low hydrograph
that i s to be routed downstream. Values for v and y are obtained with
known ra t i ng relat ionships. Usual l y the outflow hydrograph at the down-
stream end (boundary) i s the desired resu l t ; hence, Q, v and y a t the
downstream boundary a re unknown. However, i f r a t i n g relat ionships are
known, they can be used as the downstream boundary condit ion.
With the boundary information specified, the solut ion for v and y
at a suf f ic ient number of intermediate points and at the downstream
boundary i s obtained at the intersection of forward and backward
character ist ics and at the intersection of forward character ist ics and
the downstream boundary, respectively. The number of solution points
must be suff ic ient to adequately describe the movement of a f lood wave
downstream and i s determined by the number of character ist ics inscribed
on the ( x , t ) plane. Usual ly, the a v a i l a b i l i t y of data l im i ts the number
of character ist ics; however, i t should be noted that a better solut ion
general ly i s obtained when more character ist ic curves are involved.
Mathematical ly, a complete solut ion i s obtained i f a l l the boundary
informat ion i s ut i I ized.
The usual procedure for solv ing Eqs. 5.5 and 5.6 simultaneously
i s shown in F igure 5.2. Consider the points numbered 2, 5, 6, 10, 1 1
and 12. There i s both a forward and a backward character ist ic emanating
from points 2, 5 and 10. The intersection of the forward character ist ic
out of point 10 w i th the backward character ist ic out of point 5 specifies
the conditions (values of v and y ) at point 1 1 . S imi la r ly , point 6
conditions are determined at the intersection of the respective character-
is t ics out of po in ts 5 and 2. The fo rward character ist ic out of point 10
i s continued downstream i n time and space u n t i l i t intersects w i th the
backward character ist ic out of point 2, thereby determining v and y
at point 12. The procedure continues for a1 I forward and backward
character ist ics unt i I they intersect ei ther the downstream or upstream
boundary.
At each intersection there are four unknowns x, t, v, and y. These
are uniquely determined by the simultaneous solut ion of the four
equations given by Eqs. 5.5 and 5.6. At the downstream boundary x
i s given and i s no longer an unknown. The other three unknowns are
determined by the simultaneous solution of Eqs. 5.5a and 5.6a and r a t i n g
85
Fig. 5.2 Characterist ics i n ( x , t ) Plane
curves that re la te v and y.
A method for solv ing the character ist ic equations according to th i s
procedure i s out l ined as follows. With reference to Figure 5.3, i t i s
assumed the values for v and y a re known at L and R and are desired
at M. Eqs. 5.5 a re approximated as
L
and XM = XR + ( t M - tR) ( V - c ) ~
These two equations can be easi ly solved for the two unknowns XM and
tM. Once these a re known Eqs. 5.6 a re solved by the same approach.
where A = g(So-Sf)L - (v-c), - q i (5.11)
A L
86
Fig. 5.3 Characterist ic Solution for Point M
The method out l ined by E q s . 5.7 through 5.13 is l inear and can
be solved for v M, cM hence yM. The boundary condit ion, inf low, i n i t i a l
values and downstream r a t i n g curve must be known. A more stable and
accurate solut ion can be obtained with a nonl inear formulation (Overton
and Meadows, 1976; and Mahmood and Yevjevich, 1975).
When solv ing problems using the method of character ist ics, check
whether the, flow i s subcr i t i ca l o r supercr i t ical . When the flow is sub-
c r i t i c a l , v < c and the forward character ist ic has a posi t ive slope dx/dt
i n the ( x , t ) p lane while the backward character ist ic has a negat ive
slope, as shown i n F igure 5.4a. When, however, the flow is supercr i t i ca l ,
v > c and both character ist ics have a posi t ive slope i n the ( x , t ) plane,
F igure 5.4b.
+ +
F ig . 5.4 Characterist ic Lines for Subcr i t ical and Supercr i t ical Flows
L FIN I TE DIFFERENCE METHODS
F in i te di f ferencing involves replacing the continuous der iva t ive
terms with approximate f i n i t e dif ference quotients, thereby transforming
the set of d i f fe ren t ia l equations into a set of e i ther l inear o r non-
l inear algebraic equations which can b e solved more read i l y . These
87
algebraic equations re la te unknown dependent va r iab le values a t nodal
points on a f i n i t e g r i d over lay ing the continuous solution domain to
known in i t i a l values and boundary conditions. Solution of these equations
i s e i ther direct o r through a root determining scheme such as the Newton-
Raphson Method. In ei ther case, depending on the manner i n which the
replacements are made, mat r ix techniques may also be required.
Difference Quotients
F in i te dif ference quotients are obtained by d i v i d i n g the dif ference
between two values of a function by the corresponding two values of
the independent var iable. For the case of a function of a s ingle
var iable, e.g. f ( x ) , the dif ference quotient i s given b y
f (x+Ax) - f ( x )
The l im i t ing value, as Ax->O, is the def in i t ion of the der iva t ive
A X
d f ( x ) - l im f (x+Ax) - f ( x ) ~- dx Ax
a x->o ( 5 . 1 4 )
Thus the f i n i t e dif ference quotient i s an approximation to the continuous
der ivat ive as long as Ax i s kept small.
Several dif ference quotients can be defined to approximate p a r t i a l
der ivat ives. To i l l us t ra te some of them, consider a function of two
independent var iables, say U ( x , t ) . With reference to the f i n i t e dif ference
g r i d i n Figure 5 . 5 , the most commonly used dif ference quotients are
defined as follows. The forward difference approximation to the f i r s t
pa r t i a l der iva t ive for U wi th respect to x i s
- 2U = U(x+Ax,t) - U(x , t ) ax Ax
( 5 . 1 5 )
Physical ly, one can thin< of an observer standing at the point ( x , t ) ,
looking ahead ( fo rward) to the point (x+ Ax , t ) , and using the elevation
( funct ion value) dif ference between the two points d iv ided by the distance
to evaluate the slope (va lue of the de r i va t i ve ) . The backward dif ference
approximation i s
- - au - u ( x , t ) - u ( x - A x , ~ ) ax A X
The centered (o r cen t ra l ) dif ference approximation is
- au = U(x+Ax,t) - U(x-Ax,t) ax 2 ax
( 5 . 1 6 )
( 5 . 1 7 )
88
Fig. 5.5 F in i t e Difference Grid for x , t Solution Domain
NUMER I CAL SOLUTION
There are two basic f i n i t e dif ference schemes used i n solv ing the
streamflow rou t ing equations. They a re the exp l i c i t and imp1 i c i t schemes.
Exp l i c i t schemes u t i l i ze i n i t i a l value and left hand side (upstream)
boundary information and solve for the remaining g r i d points one a t a
time. They are subject to s tab i l i t y l imi tat ions on the al lowable g r i d
in te rva l size which means exp l i c i t schemes t yp i ca l l y have large da ta
requirements. However, exp l i c i t methods often resu l t i n l inear algebraic
equations from which the unknowns can be evaluated d i rec t l y without
i te ra t i ve computations. Imp l ic i t schemes u t i l i ze i n i t i a l value and both
left and r i gh t hand side boundary information, and solve for the
unknown g r i d po in ts at the next time level simultaneously. Therefore,
imp1 i c i t schemes often requ i re mat r ix techniques. Imp l ic i t methods
t yp i ca l l y involve nonl inear algebraic f i n i t e dif ference equations whereby
the solut ion i s at ta ined by i terat ion. Both schemes can be and have been
used in solv ing the governing equations for over land and open channel
flow.
89
Most e x i s t i n g methods f o r n u m e r i c a l s o l u t i o n o f e q u a t i o n s c a n b e
c l a s s i f i e d i n t o t h e f o l l o w i n g g r o u p s :
( a ) E x p l i c i t f i n i t e d i f f e r e n c e methods
( b ) I m p l i c i t f i n i t e d i f f e r e n c e methods
( c ) F i n i t e e lement methods
The use o f the f i r s t two methods was summar i sed b y L i g g e t t a n d
Woolh iser (1967). They r e v i e w e d d i f f e r e n t e x p l i c i t f i n i t e d i f f e r e n c e
schemes. The schemes were:
a ) me thod o f c h a r a c t e r i s t i c s
b ) u n s t a b l e me thod
c ) d i f f u s i o n me thod
d ) Lax -Wendro f f me thod
e ) l e a p - f r o g me thod
The method o f c h a r a c t e r i s t i c s uses an i r r e g u l a r g r i d f o l l o w i n g the
c h a r a c t e r i s t i c c u r v e s w h i l e the o t h e r s use a r e c t a n g u l a r g r i d f o r t he
s o l u t i o n o f t he e q u a t i o n s .
The me thod o f c h a r a c t e r i s t i c s emp loys the f a c t t h a t f l o w conforms
to c e r t a i n r e l a t i o n s h i p s a l o n g c h a r a c t e r i s t i c c u r v e s a n d t h e r e f o r e t h e
s o l u t i o n i s p e r f o r m e d a l o n g the c h a r a c t e r i s t i c c u r v e s . The m a i n
a d v a n t a g e s o f t he c h a r a c t e r i s t i c me thod i s t h a t i t i s a c c u r a t e a n d f a s t .
I t i s t he most a c u r a t e me thod f o r t he same i n i t i a l p o i n t s p a c i n g o f a l l
methods. I t s a c c u r a c y i s a consequence o f f o l l o w i n g the c h a r a c t e r i s t i c
c u r v e s w h i c h d e s c r i b e t h e p a t h o f t he d i s t u r b a n c e s in the f l o w . I t a l s o
c o v e r s the x - t p l a n e f a s t e r t h a n a n y o t h e r me thod w i t h t h e same i n i t i a l
p o i n t s p a c i n g . The m a i n d i s a d v a n t a g e o f t he me thod o f c h a r a c t e r i s t i c
i s t h a t d a t a a t i n t e r m e d i a t e p o i n t s in the x - t p l a n e i s d i f f i c u l t t o
o b t a i n i n an a c c e p t a b l e fo rm, r e q u i r i n g ted ious i n t e r p o l a t i o n techn iques .
I f t he me thod i s a p p l i e d to a two-d imens iona l p r o b l e m the use of t h e
c h a r a c t e r i s t i c me thod becomes e v e n more d i f f i c u l t . More r e c e n t l y more
e l a b o r a t e methods o f c h a r a c t e r i s t i c s were deve loped. A b b o t t a n d Verwey
(1970) used a f o u r - p o i n t me thod o f c h a r a c t e r i s t i c s , i .e . u t i l i s i n g t h r e e
d i f f e r e n t p o i n t s in f i x i n g the p r o p e r t i e s of a f o u r t h p o i n t . T h i s s o l u t i o n
c o u l d o n l y b e used w i t h the d y n a m i c e q u a t i o n s a s the k i n e m a t i c e q u a t i o n s
do no t h a v e n e g a t i v e c h a r a c t e r i s t i c s r e q u i r e d f o r t h i s method.
The i m p l i c i t me thod o f s o l u t i o n i n v o l v e s s imu l taneous s o l u t i o n o f
a l l the f l ow p r o p e r t i e s b y s o l v i n g a m a t r i x ; i t s m a i n a d v a n t a g e i s t h a t
the r a t i o o f space to t ime i n t e r v a l , Ax/A t , i s no t g o v e r n e d b y a n y
s t a b i l i t y c r i t e r i a a n d the me thod i s c o n s i d e r e d to b e s t a b l e f o r a n y
cho ice o f A x and At. Most p r e v i o u s i n v e s t i g a t o r s c o n s i d e r e d t h i s to b e
90
an advantage. Liggett and Woolhiser (1967) report , however, that they
were unable to make pract ical use of t h i s 'advantage ' . I f they increased
Ax/At ra t i o more than would be al lowed for i n an exp l i c i t f i n i t e d i f -
ference scheme, inaccuracy resul ted and sometimes stabi I i t y problems
occurred. They suggest that the imp l ic i t methods seemed to be more
advantageous when deal ing wi th r i v e r problems but pointed out that
attention should be pa id to the accuracy of the resu l ts obtained.
Only a few invest igators have used f i n i t e element methods i n solv ing
the St. Venant equations. The main reason for nor being used i s that
f i n i t e element programs are expensive to r u n and accuracy and s tab i l i t y
c r i t e r i a can become tedious to app ly .
Exp l i c i t f i n i t e dif ference schemes have been widely used i n the past
for the solution of the one-dimensional S t . Venant equations. They d i f f e r
from each other in the way they define the i r discharge and depth
gradients, but they a l l express the flow propert ies at a certain time
as a function of the flow propert ies at a previous time thus permit t ing
an exp l i c i t solution. They are simple to use as they use a f i xed regu la r
g r i d and i t i s easier to follow the var ia t ion of the flow propert ies along
the catchment as the solut ion i s performed exp l i c i t l y . They have been
found to be accurate and economical when proper ly used. The main
problems accompanying the choice and the use of an exp l i c i t f i n i t e
difference scheme are, however, those of accuracy and s tab i l i t y . Choosing
the most proper scheme and using i t accordingly is, therefore, important
in obtaining stable and accurate results.
The main exp l i c i t f i n i t e dif ference schemes which have been used
previously are summarised i n F igure 5.6 in terms of the points used at
a time in te rva l to propagate information at the next time in te rva l .
The propert ies of the di f ferent schemes are summarized by L igget t
and Woolhiser (1967). T h e unstable method was found to be unre l iab le
whi le the rest shcwed signs of i ns tab i l i t y when used i n certain cases.
The Lax-Wendroff scheme tended to dampen out ins tab i l i t ies and produce
better results.
Various other invest igators were faced w i th s imi la r problems when
using such di f ferent schemes for the solut ion of the kinematic equations.
Constant inides (1982), however, argued that as the na ture of informat ion
propagation for the kinematic equations d i f fe rs from that of the St.
Venant equations a l te rna t ive dif ference schemes had to be developed.
Furthermore, he argued that the scheme to be used should propagate
numerical ly, information i n a s imi la r manner as suggested by the kine-
mat i c character ist ic equations. Using th i s he developed a scheme shown
to be accurate, stable and fast (Table 5.1, p. 103).
91
X- AX a x+ & X X-AX X X + A X
( a ) Unstable method ( b ) Di f fusing method
( d ) Lax Wendroff method
Uses for d i f fus ing scheme for the f i r s t time in te rva l and the leap-frog scheme for subsequent t ime in te rva ls
X- AX X X+hX
( c ) Leap-frog method
P : Point where flow propert ies w i l l be calculated ( x , t ) + : Points used for de f in ing discharge gradients
4: Direction a t which information is propagated
-: Direct ion a t which information i s propagated
: Points used i n the depth gradient def in i t ion
fo r discharge
for depth
F ig . 5.6 Exp l i c i t f i n i t e difference scheme used i n the solut ion of the one-dimensional St. Venant equations.
Exp l i c i t Scheme
The appl icat ion of the exp l i c i t method to the unsteady flow equations
i s p r imar i l y the outcome of pioneering work by J.J. Stoker; a complete
descript ion is found in Isaacson, et at. (1956). The exp l i c i t scheme
92
shown h e r e i s f rom t h a t r e p o r t . A r e c t a n g u l a r c h a n n e l w i t h no l a t e r a l
i n f l o w i s assumed.
A ne twork o f node p o i n t s i s shown in F i g u r e 5.7 f o r s o l v i n g the
g o v e r n i n g e q u a t i o n s u s i n g the e x p l i c i t method. The v a r i a b l e s a r e k n o w n
a t p o i n t s L, M and R, and a r e to be de te rm ined f o r p o i n t P. Us ing a
centered d i f f e r e n c e quo t ien t to a p p r o x i m a t e the s p a t i a l d e r i v a t i v e s a n d
a f o r w a r d d i f f e r e n c e q u o t i e n t to a p p r o x i m a t e the tempora l d e r i v a t i v e s ,
the f o l l o w i n g a p p r o x i m a t i o n s a r e made a t p o i n t M :
(5 .18)
(5.19)
( 5 . 2 0 )
F i g . 5.7 Network o f P o i n t s f o r E x p l i c i t Method
Simi l a r a p p r o x i m a t ions a r e made to the o t h e r d e r i v a t i v e terms. When
these a p p r o x i m a t i o n s a r e i n s e r t e d i n t o Eqs . 5 .1 and 5 . 2 , v ( P ) and y ( P )
c a n be s o l v e d d i r e c t l y a s
(5.21 )
( 5 . 2 2 )
93
t
The s o l u t i o n p r o c e d u r e i s to use the i n f o r m a t i o n a t t ime leve l t a n d
so lve f o r the unknowns a t each o f the g r i d p o i n t s a t t ime leve l t t
A t . Once t h i s row o f v a l u e s h a s been determined, a d v a n c e the com-
p u t a t i o n s to t ime leve l t t 2At. The v a l u e s a t t ime leve l t + A t become
the i n i t i a l v a l u e s f o r d e t e r m i n i n g the unknowns a t t h i s a d v a n c e d t ime
leve l . The s o l u t i o n proceeds i n t h i s f a s h i o n u n t i l a l l the g r i d p o i n t s
in the s o l u t i o n doma in h a v e been determined.
To ensu re s t a b i l i t y , the g r i d s izes Ax a n d a t a r e chosen to s a t i s f y
the c o n s t r a i n t
pstream Downstream Boundary Boundary
Condi t ion : o n d i t . i o n 2 m /
A t 1
(5.23)
T h i s c r i t e r i o n f o r compu ta t i ona l step sizes, k n o w n as the Couran t
c o n d i t i o n , i n s u r e s t h a t the t ime increment i s se lected s u c h t h a t the node
p o i n t P l i e s w i t h i n the a r e a bounded b y the f o r w a r d a n d b a c k w a r d
c h a r a c t e r i s t i c s genera ted f rom node p o i n t s L a n d R . As d i scussed
p r e v i o u s l y , t h i s ensu res t h a t p o i n t P i s w i t h i n s o l u t i o n zone A a n d c a n
be f u l l y de te rm ined u s i n g o n l y the i n i t i a l v a l u e i n f o r m a t i o n c o n t a i n e d
a l o n g the l i n e f rom L t o R.
Imp I ici t Scheme
A ne twork of node p o i n t s i s shown in F i g u r e 5.8 f o r s o l v i n g the
uns teady f l ow e q u a t i o n s u s i n g a n i m p l i c i t method. t he cen te red f o u r p o i n t
d i f f e r e n c e scheme i s i I l u s t r a t e d (Amein a n d F a n g , 1969).
I
X
F i g . 5.8 Network o f P o i n t s f o r I m p l i c i t Method
94
The fol lowing approximat ions
Sf = 1, 1 [ S f ( l ) + S f ( 2 ) + Sf(3) + Sf(4)]
to the der iva t ive terms are made:
(5.24)
(5.25)
(5.26)
(5.27)
(5.28)
(5.29)
These approximations are used to replace the respective terms i n
Eqs. 5 . 1 and 5.2. Hydrau l i c var iab les at node points 1 , 2 and 3 a re
known from boundary condit ions and i n i t i a l values, hence the unknowns
are Q(4) , v ( 4 ) , y ( 4 ) , A(4) and Sf (4 ) . Since y (4 ) and A(4) are related
by the cross-sectional geometry and Q ( 4 ) , v ( 4 ) and A(4) are related by
cont inui ty, ' there are ac tua l l y three unknowns and two equations. Since
there i s the need for another equation, the dif ference scheme i s wr i t ten
for a l l of the distance steps at g iven time level u n t i l the downstream
boundary i s reached. I n F igure 5.8 there are 12 g r i d boxes, meaning
there w i l l be 24 equations to be wr i t ten but there w i l l be 27 unknowns.
The three add i t iona l equations are specif ied by the downstream boundary
condit ion which most often i s a r a t i n g curve between discharge and area
(depth 1. The resu l t ing set of a lgebraic f i n i t e dif ference equations i s non-
l inear and must be solved using an i te ra t i ve root- f inding scheme. Amein
and Fang (1969) found that the Newton scheme could b e used to l inear ize
the equations which they then solved using mat r ix techniques.
The solut ion procedure i s to solve for a l l the unknowns at one
advanced time level before proceeding to the next. A l l values are
determined simultaneously, and must sat isfy a l I boundary conditions.
Therefore, t h i s method avoids the s tab i l i t y requirements of the exp l i c i t
method meaning tha t la rger x and t g r i d in te rva l sizes can be used
which requires less input data.
95
ACCURACY AND STAB I L I TY OF NUMERICAL SCHEMES
There are two approximations in numerical modelling. One needs
to ask the questions: "How well i s the na tura l system modelled b y the
di f ferent ia l equations?", and, "How well i s the solut ion to the d i f fe ren t ia l
equations represented by the computational algori thm?". I n the analysis
here more at tent ion is pa id to the second question. The f i r s t question
can only be answered by studying the behaviour of the na tura l system
and comparing i t to the equations appl ied to i t . Therefore i t w i l l be
assumed here that the d i f fe ren t ia l equations approximate the system we1 I
despite the fact that i t has been noticed that th is i s not necessarily
the case. Abbott (1974) noticed that a dif ference scheme considerably
di f ferent from the d i f fe ren t ia l equations used to describe a system, can
y ie ld more accurate resul ts than a difference scheme s imi la r to the
d i f fe ren t ia l equations when compared w i th experimental resul ts.
There are three possible sources of e r ro r associated with f i n i t e
difference solutions to p a r t i a l d i f fe ren t ia l equations. I t i s important
that one understands these sources, their consequence i f not control led,
and means for con t ro l l ing them. These three sources of e r ro r are:
truncation, d iscret izat ion, and round-off. Truncation er ro r occurs when
a der iva t ive is replaced with a f i n i t e dif ference quotient; d iscret izat ion
er ro r i s due to the replacement of a continuous model ( funct ion) w i th
a discrete model; and round-off e r ro r i s essential ly machine er ro r i n
that the algebraic f i n i t e dif ference equations are not always solved
exact I y . For f i n i t e dif ference solutions to be accurate, they must be con-
sistent and stable. Consistency simply means that the t runcat ion errors
tend to zero as Ax and At - > 0, i.e., as Ax and At - > 0 the f i n i t e
dif ference equation becomes the o r ig ina l d i f fe ren t ia l equation. This i s
examined i n the fo l lcwing paragraphs. S tab i l i t y implies the control led
growth of round-off error. S tab i l i t y considerations apply p r i nc ipa l l y to
exp l i c i t schemes to be discussed later. Any numerical scheme that al lows
the growth of e r ro r , eventual ly "swamping" the t rue solut ion, is unstable.
96
Generally, to ensure s tab i l i t y requires that l im i ts b e placed on the
al lowable sizes for Ox and At, The c r i te r ion for establ ishing the al low-
ab le sizes i s that they b e chosen such that the forward and backward
character ist ics w i l l not t ravel the distance Ax in the time in te rva l At .
This insures that the solut ion at the advanced point i n time can be f u l l y
determined from ava i l ab le i n i t i a l value information; i.e. the g r i d point
being solved i s i n solut ion Zone A . General ly, i f a numerical scheme
i s both consistent and stable, i t s solut ion w i l l be convergent (accurate)
wi th the solut ion of the p a r t i a l d i f fe ren t ia l equation.
The truncat ion er ro r i s examined with a Tay lo r ' s series expansion
for U ( x , t ) at the point (x,O), i.e. time i s held constant. 2
U(x+Ax,t) = U ( x , t ) + Ax% + % Lu 2 + * ' * a x 2 ! a x
(5.31)
where the der iva t ives are evaluated at x , t . D iv id ing Eq. 5.31 by ,x,
and rearranging, gives the series equivalent to the forward dif ference
quotients, Eq. 5.8
(5.32)
which shows that replacing aU/ ax wi th the forward dif ference quotient
introduces an er ro r of approximation equal to those terms on the r i g h t
hand side of Eq. 5.32 a f te r a U/ ax. This e r ro r i s proport ional to th?
f i r s t power of Ax ; we cal I t h i s f i r s t order e r ro r (o r approximat ion).
Simi lar ly, i t can be shown that the backward dif ference quotient has
f i r s t order e r ro r , and the centered dif ference has second order e r ro r .
Consider the fo l low ing p a r t i a l d i f fe ren t ia l equation
aa aa at ax - + c - = o
One f i n i t e dif ference approximation to th i s equation i s
Q(x+Ax,t+At) + Q(x, t+At) - Q(x+Ax,t) - Q(x , t )
2 At
Q(x+A.x,t+At) - Q(x,t+At) = + C
A x
(5.33)
(5.34)
Examination of the Tay lo r ' s series residuals reveals the absolute value
of the truncation er ro r i s
(5.35) 2 2
2 Error = -. A t - a 2 Q +O(Cx ,At )
2 ax
where the last term indicates a second order of approximation. On
inspection i t appears that Eq. 5.34 is consistent wi th Eq. 5.33 as A t - 0. However, for t h i s pa r t i cu la r solut ion, s tab i l i t y considerations
requ i re that
Ax c 5 - A t
(5.36)
97
S u b s t i t u t i n g t h i s i n e q u a l i t y i n t o Eq. 5.34 t r a n s f o r m s t h e e r r o r t e rm i n t o
(5.37)
w h i c h i n d i c a t e s a s m a l l e r r o r term, b u t one t h a t c a n become s i g n i f i c a n t
i f A t - > 0 f a s t e r t h a n A x 2 - > 0. S ince A x a n d At a r e f i n i t e and a r e
no t a p p r o x i m a t e l y ze ro , Eq. 5.34 a p p r o x i m a t e s Eq. 5.33 w i t h second o r d e r
a c c u r a c y b u t w i t h a te rm i n t r o d u c i n g a r t i f i c i a l ( n u m e r i c a l ) d i s p e r s i o n .
T h i s e x a m p l e was chosen because i t i l l u s t r a t e s how k i n e m a t i c mode ls
c a n s i m u l a t e a d i s p e r s i n g h y d r o g r a p h . Eq. 5.33 i s m e r e l y the k i n e m a t i c
w a v e e q u a t i o n f o r no l a t e r a l i n f l o w w h i c h , t h e o r e t i c a l l y , c a n n o t p r e d i c t
h y d r o g r a p h d i s p e r s i o n . Eq. 5.34 i s one o f t he f i n i t e d i f f e r e n c e mode ls
used to s o l v e the k i n e m a t i c model. Because o f t he presence o f t he
t r u n c a t i o n e r r o r , i t s i m u l a t e s a d i s p e r s i n g h y d r o g r a p h , t h e r e b y demon-
s t r a t i n g t h a t a n u m e r i c a l k i n e m a t i c model c a n s i m u l a t e a d i s p e r s i n g
h y d r o g r a p h .
Numer i ca l d i s p e r s i o n o r d i f f u s i o n i s t h e p rocess i n w h i c h t h e E r r o r
i s formed. I t i s t h e deve lopment o f t he t r u n c a t i o n e r r o r . to the e r r o r
t h r o u g h the n u m e r i c a l t e c h n i q u e used.
L a x ' s (1954) theo ry , p r o v e d b y R i c h t m y e r and Mor ton (1967) s ta tes
t h a t f o r l i n e a r e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s o p e r a t i n g on u n i f o r m l y
c o n t i n u o u s i n i t i a l a n d b o u n d a r y d a t a the f o l l o w i n g theorem h o l d s . G i v e n
a p r o p e r l y posed i n i t i a l - v a l u e p r o b l e m and f i n i t e d i f f e r e n c e a p p r o x i m a t i o n
to i t t h a t s a t i s f i e s the cons is tency c o n d i t i o n s , s t a b i l i t y i s t he necessa ry
a n d s u f f i c i e n t c o n d i t i o n f o r convergence. T h i s i s however p r o v e d o n l y
f o r l i n e a r e q u a t i o n s a n d a c c o r d i n g to Abbo t t (1979) i t b r e a k s down when
t h e r e a r e d i s c o n t i n u i t i e s in f l ow .
S ince one i s d e a l i n g w i t h n o n - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s
( p . d . e ' s ) t h e r e i s no r i g o r o u s p r o o f s p e c i f y i n g s t a b i l i t y c r i t e r i a . F o r
l i n e a r p .d .e ' s , however , s t a b i l i t y a n a l y s e s e x i s t . Von Neuman (1949)
was f i r s t to d e v i s e a p o w e r f u l t e c h n i q u e f o r d e t e r m i n i n g s t a b i l i t y c r i t e r i a
f o r l i n e a r p . d . e ' s . He made use o f t he f a c t t h a t j u s t a b o u t a n y f u n c t i o n
c a n b e r e p r e s e n t e d b y a F o u r i e r se r ies . The l i n e a r s t a b i l i t y a n a l y s i s
method e s s e n t i a l l y de te rm ines how t h e F o u r i e r c o e f f i c i e n t s b e h a v e ( g r o w ,
decay , o r s t a y c o n s t a n t ) w i t h t ime f o r a n y te rm i n the F o u r i e r se r ies .
F o r s t a b i l i t y to o c c u r t h e r a t i o o f a F o u r i e r c o e f f i c i e n t o f a n y te rm a t
a n y t ime o v e r t h e F o u r i e r c o e f f i c i e n t o f t he same te rm a t a p r e v i o u s t ime
must b e less t h a n one.
The e f fec t o f A x and A t on s t a b i l i t y and a c c u r a c y a r e summar i zed
in F i g u r e 5.9. F rom F i g u r e 5.9 one c a n deduce t h a t t he m a i n c r i t e r i a
in the se lec t i on o f Ax and A t v a l u e s f o r a n e x p l i c i t f i n i t e d i f f e r e n c e
scheme a r e :
98
solution i s solution is stable
___)
Accuracy of solution decreases due to numerical diff sion
For fixed (6X/At), accuracy for smaller A X and At I ncreases
Fig. 5.9 Effect of value of A x and A t on s tab i l i t y and accuracy for and exp l i c i t f i n i t e dif ference scheme.
a ) that the scheme shal I proceed under stable condit ions
( 5 . 3 8 )
Ax Ax sha l l be close to (-) to minimise di f fusion er ro rs and obtain
b, at At c r
c ) the dif ference scheme’shal I be convergent. Th is could be ascertained
by runn ing the scheme wi th di f ferent A x ‘ s and At ’s and comparing
with ana ly t i ca l resul ts i n a simple case.
( A X / At,]cr
optimal accuracy.
has been shown to be the speed of wave disturbance or
information as i t i s propagated. Th is can be demonstrated by considering
the manner in which information i s propagated along the character ist ic
curves. For example, consider a central dif ference scheme, simi l a r to
the di f fusion method, for solv ing the St. Venant equations. Let i
represent a space in te rva l , and k represent a time in te rva l as shown
in Figure 5.10. The point i n question, i.e. where the flow propert ies
a re to be calculated, has the co-ordinates ( i , k ) . Information about the
flow propert ies i s sought from the previous time in te rva l . I n F igure 5.10
( a ) the true propagation speed i s smaller than the numerical propagation
speed whi le in F igure 5.10 ( b ) the converse i s true. Numerical propa-
gat ion l ines are l ines that have a slope Ax/A t i n the x - t p lane whi le
true propagation l ines have a slope dx/dt in the x - t plane. I n F igure
99
5.10 ( a ) information is obtained w i th in the i - I , i + I range by the
true propagation l ines. In F igure 5.10 information i s sought b y the t rue
propagation l ines outside the i - I, i + I range.
Since information outside th i s range i s not propagated by the
numerical scheme, i t cannot be found and thus i ns tab i l i t y w i l l resul t .
A more detai led explanat ion i s given by Stoker (1957).
For stabi I i t y of an exp l i c i t f i n i t e dif ference scheme the fol lowing
must therefore hold:
Lix dx At = d t - > - (5.39)
This i s referred to as the "CFL condit ion" a f te r Courant, Fr iedr ichs
and Lewy (1928), or simply the Courant c r i te r ion for s tab i l i t y .
t t
1-1 i i+1
( a ) ax > dx At dt
1-1 i i+i
Numerical propagation l ines; slope ( A x / A t )
- - - - True propagation l ines; slope (dx /d t )
Fig. 5.10 Comparison of numerical and theoretical propagation of information in a central dif ference scheme
I t has been noticed, however, that even i f one satisf ies the CFL
conditions i t i s not necessarily true that the solut ion of the dif ference
scheme i s inherent ly stable (e.g. by LAX, 1954; Richtmyer and Morton,
1967; Abbott, 1974). There are two poss ib i l i t i es which could g i ve r i se
to i ns tab i l i t y . There could be a physical d iscont inui ty in the flow, e.g.
100
a bore or a hyd rau l i c jump or pa ras i t i c waves could be generated w i th in
the difference scheme.
I n terms of character ist ics a physical d iscont inui ty imp1 ies the inter-
section of two or more characterist ics. Theoretical ly th is resul ts i n
dif ferent values of flow propert ies for a f i xed place and time. I n a
difference scheme w i th a f i xed g r i d th is theoretical mult ivaluedness
cannot b e accounted for and i n the solut ion i s present i n the form of
osci l lat ions. I f the difference scheme tends to amp1 i f y these osci l lat ions
i ns tab i l i t y w i l l occur. I f however, these osci l lat ions get damped s tab i l i t y
w i l l resul t and our scheme is referred to as a dissipat ive dif ference
scheme.
The dif ference scheme being used can also cause osci l lat ions ca l led
paras i t i c waves. I t has been noticed (e.g. by Abbott, 1974) that the
paras i t i c waves do not only occur when a physical d iscont inui ty occurs
but can ar ise out of the numerical procedure used. Therefore cer ta in
difference schemes have been found to produce pa ras i t i c waves whi le others
do not when considering the same physical problem.
There are two ways these problems can be overcome. I f a physical
discontinuity exists, i t can be located, the laws governing the discon-
t i nu i t y can be appl ied, and the laws governing continuous flow can be
appl ied to each side.
I t i s also possible to adjust any difference scheme to dampen instead
of amp1 i f y paras i t i c waves. The solutions obtained from these "dissip-
at ive dif ference schemes", a re ca l led "weak solutions", as i n th i s way
s tab i l i t y i s obtained a t the loss of accuracy (see Lax , 1954). Abbott
(1974) describes the dissipat ive schemes and the amount of accuracy lost
extensively.
I f one considers the method of sett ing up a d iss ipa t ive scheme, one
w i l l also i l l us t ra te the pr inc ip le of the weighted averages which is based
on averaging flow propert ies at a certain time in te rva l by l inear inter-
polat ion according to where the character ist ic curves intersect a t t =
constant l ine. Consider for example a backward dif ference scheme as shown
i n Fig. 5.11 and the way information about depth ( y ) is propagated. k;l k-1
Depth at time t = k - 1 i s taken to be as 4 ( 1 - r,)y I + v i - 1 (see
Figure 5.11) .
Suppose now one wants to propagate the depth at point Q. Then
interpolat ing l i nea r l y between points A and B one must use depth a t Q a t
time t = k-1 as (1 - r ) y I + r y i - 1 , where r i s the r a t i o of distance QB
over distance AB i n f i g u r e 5.11.
kT1 k-1
101
t
1 T at
1-1 I
-4 i+l
- t
Fig. 5.11 T h e p r i nc ip le of weighted averages for information prop-
agation i n a backward dif ference scheme.
I f one uses the fact that information is t ru l y propagated as a speed
of - dt then the slope of l i ne QP, shown i n Figure 5.11 should be the
value of - dx a t point Q (representing a point i n space a t a pa r t i cu la r
time) denoted as (dx /d t l g
S t r i c t l y speaking the value of r should therefore be
dx
dt
( 5 . 4 0 ) dx AX
r = ( 3 l Q / t
A dissipat ive dif ference scheme i s one as described above but wi th
r chosen i n such a way as to dampen osci l lat ions. The discrepancy
between r chosen and r i n equation (5 .40 ) w i l l resul t i n loss of accuracy
i n the solut ion of the dif ference scheme.
102
EFFECT OF FRICTION
Because the f r i c t i o n term in the f low equa t ion i s non l i n e a r i t makes
so lut ion of i m p l i c i t t ype equat ions more d i f f i c u l t t han wi thout the
f r i c t i o n term. A number of methods of accoun t ing f o r the f r i c t i o n term
was descr ibed b y Cunge et a l . (1980) : The f r i c t i o n g r a d i e n t i s assumed
to be of the form
sf = Q I Q I / K ~ (5.41)
where K = ARZ'3/n (Mann ing , S.I. u n i t s ) (5.42)
R = A/P (5.43)
I f a n e x p l i c i t scheme i s not acceptable, f o r instance i f Sf i s l a r g e
compared w i t h a y / a x , then some form of a v e r a g i n g of Sf in t ime i s
requ i red . Cunge et a l . suggest t a k i n g the ave rage Q ove r the d i s tance
i n t e r v a l a n d s q u a r i n g t h a t , r a t h e r t h a n the ave rage of the squares of
the Q l s over the i n t e r v a l , i.e.
(5.44)
An a l t e r n a t i v e wh ich produces a l i n e a r equa t ion a n d a l so y i e l d s
the correct s i g n of Q was suggested b y Stephenson (1984) f o r closed
condui ts : ,
[ ( Q " / K " ) ~ + (Q" /K" 1 ' 3 (5.45) + 1 - 0
j j j+l j + l 4
Stre lkof f (1970) i nd i ca tes tha t the d i r e c t e x p l i c i t scheme i s i n h e r e n t l y
unstable. He ind i ca tes the L a x t ype scheme shou ld s a t i s f y the Courant
c r i t e r i o n . For i m p l i c i t schemes he suggests t h a t to ensure s t a b i l i t y in
f r i c t i o n
At < A g / T K O
(5.46)
where KO = AoC 1% = Qo/ (5.47)
.'. At < gsf
(5.48)
Wyl ie (1970) suggested tha t f o r a s imple l i n e a r e x p l i c i t system f o r
open channels tha t for s t a b i l i t y
At 5 (Ax/c) (1 - gSfAt/2V)'" (5.49)
Even t h i s does not guarantee s t a b i l i t y acco rd ing to Wylie.
103
CHOOSING AN EXPLIC IT F I N I T E DIFFERENCE SCHEME FOR THE SOLUTION
OF THE ONE-D I MEN5 I ONAL K I NEMAT I C EQUAT I ONS
Constan t i n i d e s (1982) used v a r i o u s schemes f o r s o l v i n g the one-
d imens iona l k inemat i c e q u a t i o n s in an a t t e m p t to choose t h e most s u i t a b l e
scheme. The d i f f e r e n c e schemes men t ioned e a r l i e r a s w e l l a s new p r o p o s a l
schemes were used. The e q u a t i o n s were s o l v e d f o r d i f f e r e n t p r o b l e m s
w h i c h c a n a l s o b e s o l v e d w i t h a n a l y t i c a l methods. The a n a l y t i c a l sol-
u t i o n s were then compared w i t h r e s u l t s f r o m t h e n u m e r i c a l so lu t i ons . The
s u i t a b i l i t y o f t he v a r i o u s d i f f e r e n c e schemes was t h e n e v a l u a t e d on the
b a s i s o f a c c u r a c y and s t a b i l i t y . The cho ice o f a d i f f e r e n c e scheme was
done b y the p rocess o f e l i m i n a t i o n a s more c o m p l i c a t e d p r o b l e m s were
cons ide red . A new p r o p o s a l scheme, shown in T a b l e 5.1 was f o u n d to y i e l d
e x t r e m e l y a c c u r a t e r e s u l t s , t o b e s t a b l e as l o n g a s t h e Couran t c r i t e r i o n
i s s a t i s f i e d and to b e f a s t and economic to run. The scheme i s summar i sed
in T a b l e 5.1 b y d e f i n i n g the d i s c h a r g e r a t e a n d d e p t h a t a t ime i n t e r v a l .
TABLE 5.1 B a c k w a r d - c e n t r a l e x p l i c i t d i f f e r e n c e schemes
D i f f e r e n c e Scheme D i s c h a r g e R a t e D e p t h y a t
t = k - 1 - a t t = k - 1 ax
i - l i i + l
INDEX
x p o i n t s where f l o w p r o p e r t i e s a r e t o b e c a l c u l a t e d
+ p o i n t s used f o r c a l c u l a t i n g d i s c h a r g e a t t ime t = k - 1
0 p o i n t s used f o r c a l c u l a t i n g d e p t h a t t ime t = k - 1
The e x p l i c i t f i n i t e d i f f e r e n c e scheme shown i n T a b l e 5 . 1 a l t h o u g h
chosen b y t r i a l and e r r o r a s b e i n g t h e most e f f i c i e n t scheme, becomes
a p p a r e n t when one c o n s i d e r s t h e method o f c h a r a c t e r i s t i c s d e s c r i b e d
e a r l i e r . The schemes p r o p a g a t e i n f o r m a t i o n downs t ream o n l y a s i s
sugges ted b y t h e c h a r a c t e r i s t i c e q u a t i o n .
104
REFERENCES
Abbot t , M.B., 1974. Con t inuous f l ows , d i s c o n t i n u o u s f l o w s and n u m e r i c a l
Abbo t t , M.B., 1979. Computa t i ona l h y d r a u l i c s . P i t m a n P u b l . L t d . L o n d o n Abbo t t , M.B. and Verwey, A., 1970. F o u r - p o i n t method o f c h a r a c t e r i s t i c s .
J. Hyd. D iv . , ASCE, HY12, Dec. 1970. Amein , M. and F a n g , C.S. (19691, S t reamf low r o u t i n g - w i t h a p p l i c a t i o n s to
N o r t h Card l ina R i v e r s . Repor t No. 17, Wate r Resources Research I n s t i t u t e , U n i v e r s i t y o f N o r t h C a r o l i n a , Chape l H i l l , N o r t h C a r o l i n a .
Cons tan t i n ides , C.A., 1982. Two-d imens iona l k i n e m a t i c model l i n g o f t h e r a i n f a l I - r u n o f f process. Water Systems Research Programme, Repor t 1/1982. U n i v . o f t he W i t w a t e r s r a n d .
C o u r a n t , R., F r i e d r i c h s , K.O. and Lewy , H., 1928. Uber d i e p a r t i e l l e n D i f f e r e n t i a l g l e i c h u n g e n d e r Ma themat i schen P h y s i k , M a t h . A n n , 100.
Cunge, J.A., H o l l y , F.M. and Verwey, A., 1980. P r a c t i c a l Aspec ts o f Compu ta t i ona l R i v e r H y d r a u l i c s . P i t m a n s , Bos ton , 420 pp.
Isaacson, E., S tocker , J.J., and Troesch, B.A., 1956. Numer i ca l s o l u t i o n o f f l o o d p r e d i c t i o n and r i v e r r e g u l a t i o n p rob lems . I n s t . Ma th . Sci . Repor t No. IMM-235, New Y o r k U n i v e r s i t y , New Y o r k .
L a x , P.D., 1954. Weak s o l u t i o n s f o r non- l i n e a r h y p e r b o l i c e q u a t i o n s and t h e i r n u m e r i c a l a p p l i c a t i o n s . Comm. P u r e A p p l . M a t h . 7.
L i g g e t , J.A. and Woolh iser , D.A., 1967. D i f f e r e n c e s o l u t i o n s o f t h e s h a l l o w w a t e r e q u a t i o n . J. Eng. Mech. D i v . ASCE, A p r i l .
L i g h t h i l l , F.R.S. and Whi tham, C.B., M a y 1955. On k i n e m a t i c waves 1 . F l o o d movement in l o n g r i v e r s . P roc . Roy . SOC. London , A, 229.
Mahmood, K . and Y e v j e v i c h , Eds., 1975 , U n s t e a d y f l o w in open c h a n n e l s , Vols. I and I I , Water Resources P u b l i c a t i o n s , F o r t C o l l i n s , Co lo rado .
Over ton , D.E. and Meadows, M.E., 1976. S to rmwate r Model I i n g . Academic Press , New Y o r k .
R ich tmyer , R.D. and Mor ton , K.W.. 1967. D i f f e r e n c e methods o f i n i t i a l v a l u e prob lems. 2 n d Ed. In te rsc ience , New Y o r k .
Stephenson, D. 1984. P ipe f l ow A n a l y s i s . E l s e v i e r , Amsterdam, 274 p. Stoker , J.J. 1957. Water Waves. I n t e r s c i e n c e Press , New Y o r k . S t r e l k o f f , T., 1970. Numer i ca l s o l u t i o n o f Sa in t -Venan t e q u a t i o n s . Proc .
Von Neuman, J . , 1963. Recent t h e o r i e s o f t u r b u l e n c e . Co l l ec ted Works
Wyl ie, E.B., Nov. 1970. Uns teady f r e e - s u r f a c e f l o w compu ta t i ons . Proc .
a n a l y s i s . J. Hyd . Res., 12, No. 4.
ASCE. J. H y d r . D i v . 96(HY1) , 223-252.
(1949/1963) e d i t e d b y A.H. T a u b , 6 , Pergamon, O x f o r d .
ASCE, J. H y d r . D i v . , 9 6 ( H Y l l ) , 2241-2251.
105
CHAPTER 6
DIMENSIONLESS HYDROGRAPHS
UNIT HYDROGRAPHS
I n the same way tha t the peak f low g raphs in Chapter 3 can rep lace
the Rat ional equa t ion , so k inemat i c theory can be used to generate u n i t
hyd rog raphs fo r l a r g e r catchments. The s impl i f y i n g assumptions in the
Rat ional method a n d the peak f low c h a r t s a r e of ten i naccu ra te when i t
comes to l a r g e r catchments. A n extens ion of the Ra t iona l method became
necessary f o r l a r g e catchments a n d u n i t h y d r o g r a p h theory was developed.
The h y d r o g r a p h shape was needed for r o u t i n g too. A n analogous procedure
i s developed below f o r se lect ing h y d r o g r a p h s fo r v a r i o u s catchment
con f igu ra t i ons . A n advan tage ove r the u n i t h y d r o g r a p h methods i s t h a t the
hyd rog raphs here a r e dimensionless a n d a l low fo r v a r i o u s s impl i f i e d
catchment con f igu ra t i ons . T h i s i s o f fset b y a s l i g h t l y more compl icated set
of ca l cu la t i ons . As w i t h u n i t h y d r o g r a p h procedures however, the
catchment storm d u r a t i o n i s selected b y t r i a l .
The dimensionless hyd rog raphs presented below a r e synthes ized f o r
selected un i fo rm storm du ra t i ons . The catchments selected have v a r y i n g
shape a n d topography rep resen t ing the m a j o r i t y of smal l catchments. The
hyd rog raphs , be ing dimensionless, a r e presented as func t i ons o f r a i n f a l I
i n tens i t y a n d should therefore f i n d i n te rna t i ona l a p p l i c a b i l i t y . The user
must select r a i n f a l I r a t e s corresponding to des i red r e t u r n pe r iods as we l l
as i n i t i a l abs t rac t i on a n d i n f i l t r a t i o n ra tes a p p l i c a b l e to the catchment i n
quest ion.
The hyd rog raphs a r e in tended fo r use b y des ign engineers where not
on l y the h y d r o g r a p h peak f low r a t e b u t the shape of the h y d r o g r a p h i s
impor tant . The app l icat ion to d i f f e ren t catchments of v a r y i n g shape a n d
topography i n deve lop ing the hyd rog raphs makes t h e i r use more a d v a n t -
ageous over o ther techniques, as e x p l a i n e d below.
The l a g effect due to o v e r l a n d f low leng th , su r face roughness a n d
slope i s i n v a r i a b l y i nc luded i n the g r a p h s presented. The r e s u l t i s a more
r e a l i s t i c a n d e f fec t i ve h y d r o g r a p h f o r the des igner than i s poss ib le w i t h
p rev ious methods. The ef fect of f low concentrat ion i n streams a f t e r f l o w i n g
ove r land cannot be r e a d i l y assessed u s i n g isochronal methods ( o r any
other s t a n d a r d method). Nei ther can the effect of c h a n g i n g g round slope or
converg ing f low wh ich can a l l be accounted f o r w i t h the k inemat i c models
used here.
For peak discharge computation storms of durat ion smaller or equal
to the time of equ i l ib r ium of the catchment a re important, as a storm
could produce maximum peak discharge of f the catchment. Higher f lood
peaks may resul t from a shorter storm. T h e c r i t i ca l storm durat ion, i .e.
the storm durat ion that w i I I produce maximum peak discharge, w i l l depend
on two factors, these being the way the catchment responds to storms of
durat ion less than the catchment's time of equ i l ib r ium, the r a i n f a l l
characterist ics and the retent ive propert ies of the catchment's soi Is . Storms
of durations longer than the catchment's time of equ i l ib r ium are also
important, especial ly in cases where runoff volume i s of importance.
Neither a s ing le value of peak discharge ra te nor total runoff volume
are general ly suf f ic ient for a l l the purposes of the drainage engineer. The
time the catchment takes to reach i t s peak discharge as well as the
complete hydrograph shape are general ly of prime importance. I n cases
where runoff hydrographs have to be combined from di f ferent catchments o r
a re routed through hydraul i c conduits, the complete runoff hydrograph
shape i s essential for accurate design.
The hydrograph shape i s also important i n designing hyd rau l i c
structures to cope w i th floods of h igher re tu rn periods than those which
they were designed to ca r ry . The p a r t of the hydrograph not car r ied b y
the hydrau l i c conduit structure, i f known, can be diverted by sui table
means, while, i t s backwater effects upstream and the force on the
structure could also be evaluated.
The volume under the hydrograph i s of pa r t i cu la r importance when
detention or retention storage are contemplated. The rou t ing effect and
peak flow attenuation are pa r t i cu la r l y sensit ive to the hydrograph shape
as opposed to the peak.
I n general the dimensionless hydrographs should be of pa r t i cu la r
interest to the urban drainage engineer who w i l l wish to study stormwater
management and the effects of urbanisat ion - changing surface configur-
a t ion , roughness and permeabil i ty on flow rates.
DEVELOPMENT AND USE OF GRAPHS
I n developing runoff hydrographs for a catchment i t i s important to
understand how the catchment w i l l react to di f ferent storms. The volume of
surface runof f i s p r imar i l y a funct ion of r a i n f a l I and i n f i l t r a t i on
characterist ics, whi le the hydrograph shape is a function of catchment
shape, roughness and topographical characterist ics.
107
Computer models can account fo r any time and space va r ia t i on of
ra in fa l I and catchment character ist ics as described later. T h e i r use entai I s
substantial computer time and the model has to be used i n conjunction
wi th various storm inputs to ensure c r i t i ca l storm input. I n th i s section,
runoff hydrographs of f catchments of f i xed shapes and w i th spat ia l l y
var ied catchment character ist ics are presented. The resu l t ing hydrographs
are dimensionless, i.e. in terms of catchment size and r a i n f a l l rate,
al lowing the use of d i f ferent catchment dimensions and di f ferent roughness
and catchment slope parameters. The design engineer can use these
hydrographs for na tu ra l catchments which have simi l a r shapes to the model
catchments studied and where the roughness and slope character ist ics are
consistent. The design engineer s t i l l has to use h i s judgement i n
approximat ing catchment shapes and in averaging roughness and slope
pa ramet ers.
The kinematic equations have been used to prepare the hydrographs
presented by Constantinides and Stephenson (1982). Computer solut ion of
the f i n i t e dif ference form of the equation of motion and the flow
resistance equation was performed for numerous si tuat ions. With the use of
dimensionless parameters the number of var iab les i s reduced considerably
and a few graphs present a range of hydrographs covering the range of
parameters normal l y encountered.
Runoff hydrographs off three model catchments a re presented, these
being the fol lowing:
( a ) A sloping plane catchment
( b ) A converging surface catchment
( c ) A V-shaped catchment wi th stream
Design hydrographs may be obtained by comparing dimensional runof f
hydrographs fo r di f ferent storm durations, and selecting I the one resu l t ing
in maximum flow r a t e ( i f the unattenuated peak i s of concern) o r greatest
volume required to attenuate the flood i f storage is to be designed, or any
other relevant c r i t i ca l parameter.
L i s t of Symbols
x space ax i s along over land p lane (m or f t )
z space ax i s along channel (m or f t )
L length of over land p lane (m or f t )
Ls length of channel o r stream (m or f t )
So bed slope of over land p lane
108
n roughness coefficient of over land planes
n roughness coefficient of channel or stream
0 angle describing converging surface catchment ( rad ians )
r ra t i o describing converging surface catchment
w width of over land flow in converging surface catchment ( m or f t )
H depth of channel ( m o r f t )
b width of channel (m or f t )
yo depth of over land flow
qo discharge per u n i t width of over land flow (m’/s or f t ’ /s)
ys depth of channel flow (m or f t )
Q discharge of channel flow (m3/s or f t 3 / s )
Q discharge of converging surface (m3/s or f t 3 / s )
( m or f t )
Kinematic equations
The one-dimensional kinematic equations for flow have already been
presented and a re merely stated here. They consist of the cont inui ty
equation and an equation re la t i ng hydrau l ic resistance to flow.
aQ aA + - - - ax at - q~ (6.1 1 and q = uym ( 6 . 2 )
where Q i s the flow ra te (m’/s or f t ’ / s ) , A i s the cross sectional area
(m‘ or f t ’ ) , t i s time (secs), x i s the space a x i s (m or f t ) , q i s la te ra l
inf low per un i t length along the x - ax i s (m’/s or f t ‘ / s ) , q i s the
average discharge across a section per u n i t width (m’/s or f t ‘ /s) and y
i s the depth of water (m or f t ) . u , m are coefficients dependent on surface
roughness and bed slope.
L
EXCESS RA I NFALL
I n developing runof f hydrographs of f the simple catchments already
outl ined, an excess r a i n f a l l d is t r ibu t ion is required. I n th is case, excess
r a i n f a l l intensi ty i s assumed to be uniform in space, and constant du r ing
the storm and equal to a negative constant (being a constant i n f i l t r a t i on
ra te ) a f te r the storm. Fig. 6.2 depicts the assumed excess r a i n f a l l input
and F ig . 6 . 1 shows the assumed r a i n f a l l input and loss d is t r ibu t ion for
obtaining the excess r a i n f a l l d is t r ibu t ion shown i n Fig. 6 .2.
109
(mm/h
i l
U V f C
*
- * u - +4 t e d t (h)
* ). I i
l d
* t e
Fig. 6.1 Assumed r a i n f a l l input and d is t r ibu t ion losses
In Figs. 6 . 1 and 6 . 2 i i s r a i n f a l l intensi ty ra te (mm/h o r inches/h), i is
excess ra in fa l I intensi ty r a t e (mrn/h o r inches/h), td i s storm durat ion
( h ) , ted i s excess r a i n f a l l durat ion ( h ) , fc i s f i n a l i n f i l t r a t i on r a t e
(rnm/h or inches/h),f i s uniform in f i l t r a t i on ra te (mm/h o r inches/h) and u
is i n i t i a l abstraction (rnm or inches).
The f i na l i n f i l t r a t i on rate, fc i s a function of so i l type and
vegetation cover or land use. The excess r a i n f a l l intensi ty, ie, i s a
function of excess r a i n f a l l durat ion, t, which depends on local r a i n f a l l
characterist ics and on catchment soi I and vegetation cover propert ies.
110
Rainfall intensit: (mm/h)
t (hi 1 -
t (hi
1 - t t time runoff stops time runoff stops
Fig . 6 .2 Excess r a i n f a l l input
D I MENS IONLFSS EQUAT I ONS
I t i s evident from kinematic theory that if any catchment i s
subjected to a constant excess r a i n f a l l intensi ty i for a period equal to
or longer than i t s time of equ i l ib r ium i t w i l l produce a peak discharge
equal to i mu l t ip l ied by the area of the catchment. In deciding on
dimensionless parameters to be used for developing runoff hydrographs i t
therefore seems logical to plot the ra t i o o f discharge d iv ided by excess
r a i n f a l l intensity and area against a ra t i o of time div ided by the time of
equ i l ib r ium of a simple catchment, namely the s loping plane catchment.
Sloping Plane Catchment
For the s loping p lane catchment depicted i n Fig. 6 .3 the cont inu i ty
equation becomes:
(6.3a)
= - f c fo r t ? ted (6.3b)
The uniform flow equation can also be expressed as:
m qo = uoyo
1
where a = Soy/no
1 1 1
16.4)
and n o,So are the Manning coefficients and bed slope respectively.
Expressing y i n terms of qo from equation (6.4), d i f fe ren t ia t ing w i th
respect to t and subs t i tu t ing i n equation (6.3) y ie lds:
The fol lowing dimensionless var iab les are then defined:
x = x LO
P = qo
i eLo
tc 0
tCO
-
T = mte -
mt TD = ed
~ (6.9)
F = fc (6.10) - I e
where t is the time of concentration of a s loping plane i n kinematic co theory and is given by :
tco = (--- Lo l /m (6.11)
Subst i tut ing for qo, x , t , ted and f i n equation (6.5) and manipulat ing
y ie lds the fo l lowing equation:
m-1 'oie
1 ap aP
aT ax _ _ _
D
> TD
+ - = 1 f o r T < T 4
= -F for T (6 .12)
F ig . 6 . 3 Sloping plane catchment
P
&
F i g . 6 .& D i m e n s i o n l e s s r u n o f f hydrographs f o r the s l o p i n g p l a n e catchment
F = 0.0
F=O. 5
D Pmax versus' T
F i g . 6 .5 Dimensionless r u n o f f h y d r o g r a p h s f o r t he s l o p i n g p l a n e catchment
F = 0 . 5
114
where (m- l ) /m = 0.4 fo r m = 5/3
Equat ion (6.12) i s solved fo r f low P as a f u n c t i o n of t ime r a t i o T a t
D the ou t l e t end of the catchment p lane . Th is i s repeated fo r d i f f e ren t T
va lues. Di f ferent p lo t s a r e ob ta ined f o r d i f f e ren t F values in F igs . 6.4
a n d 6.5. The theory of Overton (1972) was a l so adap ted to cascades o f
p lanes b y K i b l e r and Woolhiser (1970).
Converging surface Catchment
For the converg ing su r face depic ted in F i g . 6.6 the c o n t i n u i t y
eq. (6.1) becomes (Woolhiser, 1969):
(6.13)
= - w f f o r t > ted (6. o c
where w = ( L o - x)B
(6. a n d Qo =
Express ing y i n terms of €Io f rom equat ion (6 .4 ) , d i f f e r e n t i a t i n g
respect to t a n d s u b s t i t u t i n g i n equa t ion (6.13) y i e l d s :
m WoaoYo
4 )
5 )
w i t h
l - l / rn
ed = i w f o r t j t
= - w f f o r t > t d
aQo wo 5 e o 7- 0 0 a t
- + ax
o c (6.16)
I n a d d i t i o n to dimensionless v a r i a b l e s de f i ned in equat ions (6.8) to
(6.10) the f o l l o w i n g dimensionless v a r i a b l e s a r e de f i ned (Singh, 1975):
F i g . 6 . 6 Converg ing su r face catchment
115
(6.17)
(6.18)
w h e r e ( 1 - r 2 ) / 2 i s the area of the catchment a n d r the r a t i o of bottom
segment to the to ta l catchment r a d i u s .
For the converg ing sur face tCO i s def ined as the time of e q u i l i b r i u m
fo r a s lop ing catchment of leng th L (1-r), i.e.
t i 0 - la Oiem-l] (6.19)
Subs t i tu t ing f o r x , Qo,wo,t, ted,fc a n d m in equat ion (6.16) a n d
man i p u I a t i n g y ie lds :
L o ( l - r ) 1 /m
-~
(6.20)
(6.21)
Equations (6.20) a n d (6.21) were so lved numer ica l l y to g i v e S as a
funct ion of T a t the ou t le t f o r d i f fe ren t T,, values. Plots a r e f o r v a r i o u s
r a n d F va lues as presented in F igs. 6.7 a n d 6.8.
V-Shaped Catchment w i t h Stream
I n the V-shaped catchment ( F i g . 6.9) the d ischarge from o v e r l a n d
f low i s used as input in the channel . Kinematic theory i s used to route
over land f low runof f through the channel. I t i s assumed tha t bo th
over land f low p lanes a r e s i m i l a r . From k inemat ic theory the c o n t i n u i t y
equat ion in the channel would be:
__ = 2q0L a Q ~ + b a y s
az at (6 .22)
A bas ic assumption i n equat ion (6.22) i s tha t the n a t u r a l depth of the
channel i s a lways grea ter than the water depth in the channel . Another
assumption i s tha t the channel area i s small compared to the p l a n e area.
The un i fo rm f low res is tance equat ion f o r the channel may be wr i t ten :
(6.23) m
Qs = baSYs
Express ing y in terms of Q from (6 .23) , d i f f e r e n t i a t i n g w i t h respect
to t a n d s u b s t i t u t i n g in to (6.22) y i e l d s :
S
Fig . 6 .7 D i m e n s i on I e s s r u n off h y d rog r alp h s f o r the con v e r g i n g s u r f ace c a t c h m e n t
R = 0.05 F = 0.00
.... m
0, - R=O .05 F=O. 50
5-
Smax versus T D
S
F i g . 6 .8 Dimensionless r u n o f f h y d r o g r a p h s fo r the converg ing s u r f a c e ca tchment
R = 0.05 F = 0.50
118
( 6 . 2 4 )
I n a d d i t i o n to the dimensionless v a r i a b l e s de f i ned i n equat ions ( 5 . 6 )
a n d (6.10) the fo l l ow ing dimensionless v a r i a b l e s a r e de f i ned :
Q = Q , / ~ L ~ L ~ ~ ~ ( 6 . 2 5 )
2 = z/L!j ( 6 . 2 6 )
where t i s the s a m e as fo r the s l o p i n g p lane , i .e. equa t ion ( 6 . 1 1 ) .
Subs t i t u t i ng fo r Q ,z,t ,qo, a n d rn in equat ion (6 .24 ) a n d r e - a r r a n g i n g
y i e l d s :
LO
2L5 0.6 ba 0.6 where G = (c )
Equa t ion ( 6 . 1 2 ) i s solved to y i e l d P as a
5 2L0
i n c t i o n c
the p lanes. P i s used as i n p u t i n equat ion ( 6 . 2 7 ) 10
( 6 . 2 7 )
( 6 . 2 8 )
T a t X .- 1 f o r
so lve for Q as a
func t i on of T a t the out le t f o r d i f f e r e n t va lues fo r F G G a n d the r e s u l t s
appended a t F igs . 6.10 a n d 6.13. The same problem was hand led i n a
d i f f e ren t ' way b y Wooding (1965).
H>ys a t a1
Ca t c hme n t
F-ig. 6 .9 V-shaped catchment w i t h stream
9.- d
9- d
Q -
5-
9- d
61- d
3- d
8- d
R- d
3- d
F i g . 6 .10 D i m e n s i o n l e s s runoff hydrographs for the V-shaped catchment with stream
G = 0.5 F = 0.0
Fig . 6 .ll D i m e n s i o n l e s s runoff h y d r o g r a p h f o r t h e V - s h a p e d c a t c h m e n t w i t h s t r e a m
G = 0.5 F = 0 . 5
>
N 0
I
F i g . 6 . 1 2 D i m e n s i o n l e s s r u n o f f h y d r o g r a p h s f o r t h e V - s h a p e d c a t c h m e n t b v i t h s t r e a m
G = 2 . 0 F = 0.0
G=2.0
Fig. 6 .13 Dimens ionless runoff hydrographs fo r the V-shaped catchment wi th s t r e a m
G = 2.0 F = 0.5
123
F i g . 6 . 1 4 E x a m p l e : Catchment w i t h s t ream
124
O v e r l a n d f l o w
Channel f l o w
X w
Cover M a n n i n g ' s n Slope
Medium g r o w t h 0.15 5%
Medium g r o w t h 0.15 1.2%
meadow
meadow
0 0 2 4 6 8 10
EXCESS STORM DURATION (HOURS]
F i g . 6.15 Examp le : Excess i n t e n s i t y - d u r a t i o n r e l a t i o n s h i p
TABLE 6 .1 Examp le : M a n n i n g ' s roughness c o e f f i c i e n t s
a n d b e d s lopes
125
USE OF D I MENS I ONLESS HYDROGRAPHS
The procedure for using the dimensionless hydrographs i s i l l us t ra ted
by means of an example.
Problem
Consider the na tura l catchment out l ined i n Fig. 6.14 and the 5
year recurrence in te rva l excess IDF relat ionship shown i n Fig. 6.15.
Obtain the runoff hydrograph producing the worst peak discharge off
the catchment. The excess IDF relat ionship given al lows for the storm
spat ia l d is t r ibu t ion (which has been reduced from the point excess
r a i n f a l l IDF re la t ionsh ip ) and has been developed using local r a i n f a l l
data and catchment characterist ics. The average f i na l i n f i l t r a t i on ra te
of the soi l ( f ) i s 1.5 mm/h.
So I u t i on
The natura l catchment shown i n F ig . 6.14 i s approximated by a
V-shaped catchment wi th stream. The main waterway i n the catchment
has a length of 1350 metres and subdivides the catchment approximately
i n the middle. The other waterways are minor and most of the catchment
flow is i n the form of over land flow f lowing perpendicular ly to the
waterway. The waterway . is assumed to be a rectangular channel 31-17 wide.
The assumed V-shaped catchment w i th stream is i l l us t ra ted i n F ig . 6.16.
Manning's roughness coefficients are shown i n Table 6 .1 whi le bed slopes
are averaged using the contour l ines from Fig. 6.14 and summarized i n
Table 6.1. Parameter G must be evaluated using (6.28):
G = Z(1350)- 2 (308.9)
Figs. 6.10 and 6.11 with G = 0.5 a re used for choosing the c r i t i ca l
runoff hydrographs. T h e i n f i l t r a t i on parameter F i s a function of the
excess r a i n f a l l rate.
Table 6.2 shows the calculat ions i n choosing a c r i t i ca l runoff hydrograph
and dimensioning i t . The table refers to Figure 6.10.
126
Outlef
Scale I :7500
Fig . 6 .76 E x a m p l e : Assumed c a t c h m e n t
TABLE 6 .2 Exarrple : Choosing and dimensioning runoff hydrograph w i t h m9x lmum peak c i scha rge
- -. r----------- -___
1 . 2
1 . 4
1 2 . 7 0 U . 1 1 8
1 1 . 6 3 0.12' ,
0 . 6 ~ 0 . 4 ( 3 . 6 ~ 1 O b )
3 6 0 0 0 1
t C O
a Qs F a c t o r s t.o d imeno j .on I r u n o f F hydrograph
Mu 1 t l p l y var j.a bX e
___~.
source
var j.a bX e
units hours m / h r
source guess excess IDF's
1 . o 1 3 . 9 9 0 . 1 0 7
0 . 5 1 7 . 5 5 0.086 --
a x is
m' I s hours hours I I
dirnensionl.ess hydrographs
0 . 9 9 5
0 .go9
1 . 6 7 5
1 . 1 0 1
I
1 .034 1.934 1 1 . 6 6 2 . 7 0 1 P . 6 2 0 I 2 . 9 4 2
1 1 . 0 7 1 2.178 1 1 . 2 9
_" ....,
Critical storm has an CXCPSF: duration of 1 . j : h o u r s prodiicIny a dlsclrarqc peak of 1 .70 c u m ~ c ' i .
~ ~ . . - - - , . - - . _ . . . . . . - . - - - I - - _ -
3.0
2 -5
n E
I .o
0.5
0.0 1.0 2.0 3.0 Time (hours)
!Fig. 6 . 1 7 Examp le : C r i t i c a l runo f f h y d r o g r a p h
4.0 5.0
129
As c a n b e seen f r o m T a b l e 6 . 2 the s to rm p r o d u c i n g the max imum p e a k
d i s c h a r g e o f f t he ca tchmen t h a s an excess s to rm d u r a t i o n o f 1.2 h o u r s a n d
p roduces a p e a k d i s c h a r g e o f 2.70 cumecs. The comp le te r u n o f f h y d r o g r a p h
i s o b t a i n e d f r o m F i g . 6.10 f o r a v a l u e of TD = 1.93. The h y d r o g r a p h i s
r e n d e r e d d i m e n s i o n a l b y m u l t i p l y i n g t h e two a x e s o f F i g . 6.10 b y t h e
v a l u e s g i v e n i n T a b l e 6.2 a n d i s shown in F i g . 6.17.
REFERENCES
Constan t i n ides , C.A. and Stephenson, D., 1982. D imens ion less h y d r o g r a p h s u s i n g k i n e m a t i c theo ry , Water Systems Research Programme, Repor t 5/1982, U n i v e r s i t y o f t he W i t w a t e r s r a n d .
K i b l e r , D.F. a n d Woolh iser , D.A., 1970. The k i n e m a t i c cascade a s a h y d r a u l i t model. H y d r o l . p a p e r 39, Co lo rado S ta te U n i v e r s i t y , F o r t Col I i ns .
Over ton , D.E., 1972. K i n e m a t i c f l o w o n l o n g impermeab le p l a n e s , Water Res. B u l l . 8 ( 6 ) .
S i n g h , V.P., 1975. H y d r i d f o r m u l a t i o n o f k i n e m a t i c w a v e model o f wa te rshed r u n o f f , J. H y d r o l . 27.
Wooding, R.A., 1965. A h y d r a u l i c model f o r t he ca tchmen t s t ream p r o b l e m ,
Woolh iser , D.A., 1969. O v e r l a n d f l o w o n a c o n v e r g i n g su r face . T r a n s . I I . Numer i ca l So lu t i ons . J . H y d r o l . 3.
Am. SOC. A g r . E n g r . 12 (4 ) , 460-462.
130
CHAPTER 7
STORM DYNAM I CS AND D I STR I BUT I ON
DES I GN PRACT I CE
I t i s common pract ice to design stormwater systems for uniform
intensi ty, uni formly distr ibuted, stat ionary storms. Lack of data often
makes any other basis for design d i f f i cu l t . There i s l i t t l e information
ava i lab le on instantaneous precipi tat ion rates, storm cel l size and cel l
movement. Time average precipi tat ion ra te or p rec ip i ta t ion depth can be
predicted from intensity-duration-frequency curves (e.g. Van Wyk and
Midgley, 1966) or equations such as that of Bel l (1969). The most common
method of abstract ing data from r a i n f a l l records is to select a durat ion
and calculate the maximum storm precipi tat ion i n that period. The so-
defined storm may include times of low r a i n f a l l intensity immediately
preceding and succeeding a more intense prec ip i ta t ion rate.
Such simp1 i f icat ions in data render runoff calculat ion s impl ist ic.
Even when employing numerical models i t i s simplest to use a uniform
intensity hyetograph for every point on the catchment. Although time
va ry ing storms a re sometimes used, the prec ip i ta t ion pattern i s seldom
related to the maximum possible runoff rate.
Warnings have been made against s impl i f icat ion in ra in fa l I
patterns. For example, James and Scheckenberger (1983) indicated that
storm movement can affect' the runoff hydrograph s ign i f i can t ly . Eagleson
(1978) has expounded on the spat ia l v a r i a b i l i t y of storms and Huff (1967)
studied the time v a r i a b i l i t y of storms.
Although much research has been done on storm v a r i a b i l i t y ,
re la t i ve ly l i t t l e has been publ ished on the resu l t ing effects on runoff
hydrographs (Stephenson, 1984). Research appears to have concentrated on
models of pa r t i cu la r (monitored) storms over pa r t i cu la r catchments. The
design engineer or hydrologist does not have suff ic ient guidance as to
what storm pattern to design for. Presumably cer ta in ra in fa l I sequences,
spat ia l var ia t ions and storm movement w i l l resu l t i n a higher ra te of
runoff than other r a i n f a l l patterns for a pa r t i cu la r catchment. Apart from
an indicat ion of what storm pattern produces the worst flood, one needs an
indicat ion of what storm pattern could be expected for the design
catchment. Such da ta should be ava i l ab le on a frequency basis i n order to
estimate the l ikel ihood of the worst hyetograph shape, spat ia l storm
d is t r ibu t ion and movements occurring. Although isolated catchments have
been studied a t many research centres considerably more information is
required for the country as a whole. Analysis and use of such data
131
i n di f ferent combinations would requ i re many t r i a l s before the worst storm
patterns would emerge. An a l te rna t ive approach i s a determinist ic one.
Before ca lcu la t ing runoff, the analyst determines the fol lowing in order
to select the correct design storm:
i ) The storm durat ion. For small catchments th is i s usua l ly equated
to the time of concentration of the catchment.
i i ) Var iat ion i n p rec ip i ta t ion ra te du r ing the storm
i i i ) Spat ia l d is t r ibu t ion of the storm; and the
i v ) Direct ion and speed of movement of the storm.
The above information could be employed i n numerical modell ing
of the design storm. Al ternat ively, fo r minor structures, s impl ist ic
methods such as the Rational method could be employed. Since data
shortage often l im i ts the accuracy of modelling, the la t te r , manual
approach, i s often su f f i c ien t ly accurate. The guides presented below
may assist both the modeiler b y p rov id ing information on which design
storm would produce the highest runoff ra te and the formula orientated
solut ion by prov id ing factors to account fo r storm v a r i a b i l i t y .
STORM PATTERNS
Variation in rainfal l intensity during a storm
I n order to understand the reasons for and extent of v a r i a b i l i t y
(spa t ia l and temporal) of r a i n f a l l , i t i s useful to describe the physical
process of cloud formation and precipi tat ion. Convective storm clouds
or ig ina te from r i s i n g a i r masses. The size and shape of the r i s i n g a i r
mass depends on the topography and the a i r masses w i l l usua l l y be of
smaller scale than the a i r mass which has been brought by advection
and which contains suf f ic ient moisture fo r ra indrops to precipi tate.
Mader (1979) concluded from radar observations of storms in South Afr ica
that storm areas, durat ion and movement were related to mean 500 mb
winds, thermal i ns tab i l i t y and wind shear.
Most recorded hyetographs indicate that r a i n f a l I intensi ty i s
highest somewhere i n the middle of the storm durat ion. Huff (1967)
presented extensive data on r a i n f a l l rates fo r storms of va ry ing intensi ty
ind ica t ing a time d is t r ibu t ion somewhat between convex upward and
t r i angu la r . In order to create a hyetograph which could be used for
simple design of interconnecting stormwater conduits, Keifer and Chu
(1957) proposed an exponential d is t r ibu t ion termed the Chicago storm.
132
The posit ion of the peak intensity could be var ied and was observed
to occur about 0.375 of the storm durat ion from the s ta r t .
i
mm/h
I
'1
Fig. 7.1 Hyetograph with peak near beginning
Spat i a I d is t r ibu t ion
The nature of storm cel ls w i th in a potent ia l r a i n area has been
documented by many researchers e.g. Waymire and Gupta (1981). The
persistence of storms observed i n the northern hemisphere has not been
found i n countr ies south of the equator however (Carte, 1979). The la rger
a i r mass w i th in which storm cel ls occur i s referred to as the synoptic
area (see F ig . 7.2). The synoptic area can last f o r 1 to 3 days and
the size i s general ly greater than 104km2. Within the synoptic area are
la rge mesoscale areas (LMSA) of lo3 to lo4 km2 which have a l i f e of
several hours. Sometimes small mesoscale areas (SMSA) of lo2 to lo3 k m 2 can
exist simultaneously. Within the mesoscale areas or sometimes on the i r
own, convective cel ls, which are regions of cumulus convective precipi-
tat ion, exist. These may have an area extent of 10 to 30 km2 and have
an average l i f e of several minutes to ha l f an hour. These cel ls a re of
concern to the hydrologist involved i n stormwater design. By comparing
the storm cel l size w i th the catchment size he can decide whether the
133
cell scale i s s ign i f i can t i n inf luencing spat ia l d is t r ibu t ion over the
catchment. There may be over lapping cel ls which could resul t i n greater
intensity of p rec ip i ta t ion than for the s ingle cel ls. Eagleson (1984)
investigated the stat ist ics of storm cel l occurrences i n a catchment and
found the poss ib i l i t y
lapping small storms.
Synoptic
of large storms can be computed assuming over-
a r e a 7
Fig. 7.2 Areal d is t r ibu t ion of a convective storm
The shape of the storm cel l has s igni f icance for catchments la rger
than the cel l . Scheckenberger (1984) indicates that the cel ls are e l l i p -
t ical which may be related to storm movement. The r a i n f a l l intensi ty
i s highest a t the centre and decreases outwards. The intensi ty has been
shown to decrease exponential ly, r a d i a l l y outwards from the focus, i n
various local i t ies as i n Fig. 7.3 (Wilson et al., 1979). Generally the
v a r i a b i l i t y i n intensi ty does not necessarily cause higher runoff intensit ies
but on small catchments near the centre of the cel l the average precipi-
tat ion can be higher than for a la rger catchment, and as a ru le , the
r a i n f a l l depth increases the smaller the storm area.
Storm movement
Clouds general ly t ravel w i th the wind at the i r elevation. As the
r a i n f a l l s i t goes through lower speed wind movements so that the most
s igni f icant speed is that of the clouds. The direct ion of lower winds
can also d i f f e r from the general d i rect ion of movement of the upper strata.
This may be the reason Changnon and Vogel (1981) observed s l i gh t l y
dif ferent direct ions for storm and cloud movements. Dixon (1977) analysed
storm data and indicated storm cel ls have a c i rcu la t ion i n addi t ion to
a general forward movement.
134
NUMERICAL MODELS
The effect of storm dynamics and d is t r ibu t ion can be studied
numerical ly and the resul ts for simple plane catchments are presented
below. The kinematic equations are employed i n the numerical scheme.
Although these solutions are no subst i tute for detai led catchment modell ing
when there are suf f ic ient data, they do indicate which var iables are
l i ke l y to be the most important i n storm dynamics. I t must be pointed
out that the fo l lowing studies are s impl i f ied to the extent of assuming
constant speed storms with unvary ing spat ia l d is t r ibu t ion . True storms
are considerably more complex as explained i n the above reference.
Fig. 7.3 I l l us t ra t i on of spat ia l d is t r ibu t ion of p rec ip i ta t ion intensi ty
Kinematic equations
The one-dimensional kinematic equations are for a simple plane
catchment (Brakensiek, 1967):
The cont inui ty equation; a y + a q = i and at ax
Flow resistance; q = a y m
y is water depth on the plane, q is discharge ra te per un i t width of
plane, ie i s excess r a i n f a l l rate, t i s time, x i s longi tudinal distance
down the plane, a i s assumed a constant and m i s a coefficient. Employ-
ing the Manning discharge equation i n S . I . un i ts a = J (So) /n where
So i s the slope of the plane, n i s the Manning roughness coeff icient,
and m is 5/3.
135
The number o f v a r i a b l e s c a n be reduced to f a c i l i t a t e s o l u t i o n
b y r e - w r i t i n g t h e e q u a t i o n s in terms of t h e f o l l o w i n g d imension less
v a r i a b l e s :
x = x / L
T = t / t c
I = i / i
Q = q/iaL e a
where L i s t h e l e n g t h o f o v e r l a n d f l ow , ia i s t h e t ime and space a v e r a g e d
excess r a i n f a l l r a t e and tc i s t h e t ime to e q u i l i b r i u m , o r t ime o f
c o n c e n t r a t i o n , f o r a n a v e r a g e excess r a i n f a l l i . S u b s c r i p t c r e f e r s to
t ime o f concen t ra t i on , d to s to rm d u r a t i o n , a to t ime and space a v e r a g e
a n d p to peak . Then t h e f o l l o w i n g exp ress ion f o r t c a n b e d e r i v e d :
a
I n g e n e r a l t he d imension less v a r i a b l e s a r e p r o p o r t i o n a l t o t h e
d imensioned v a r i a b l e s . Thus Q i s t h e p r o p o r t i o n o f max imum f l o w a t
e q u i l i b r i u m . S u b s t i t u t i n g y = ( q / a ) f r om t h e r e s i s t a n c e e q u a t i o n and
f o r X ,T , I and Q f rom t h e e q u a t i o n s f o r t he d imens ion less terms, the
f o l l o w i n g e q u a t i o n r e p l a c e s t h e c o n t i n u i t y equa t ion .
1 -m
T h i s s i n g l e e q u a t i o n c a n be s o l v e d f o r Q i n s teps of T and X f o r v a r i o u s
d i s t r i b u t i o n s o f I and m = 5 / 3 .
F i g . 7 . 4 P l a n e r e c t a n g u l a r catchment s t u d i e d w i t h s to rm
Numerical Scheme
A l t h o u g h i t a p p e a r s a s i m p l e m a t t e r to r e p l a c e d i f f e r e n t i a l s b y
f i n i t e d i f f e r e n c e , t h e r e c a n b e p rob lems of conve rgence and speed of
s o l u t i o n u n l e s s t h e c o r r e c t n u m e r i c a l scheme i s employed. The s imp les t
136
f i n i t e dif ference schemes are exp l i c i t , employing values of Q at a
previous T to estimate new values a t the next time T . This method i s
not recommended as i t i s often unstable when discont inui t ies i n r a i n f a l l
intensi ty occur. Upstream differences are usua l ly taken i n such schemes,
as downstream effects cannot be propagated upstream according to
Huggins and Burney (1982). I t i s also necessary to l im i t the value of
AT/AX to ensure s tab i l i t y .
Woolhiser (1977) documented var ious numerical schemes inc lud ing
very accurate methods such as Lax-Wendroff ' s . Brakensiek (1967)
suggested 3 schemes: four point , imp l ic i t and exp l i c i t . His second scheme
( imp l i c i t ) i s adopted here as i t i s accurate and r a p i d for the examples
chosen.
I M -'l M * X
Fig. 7.5 X-T g r i d employed i n numerical solut ion
Employing the notat ion i n the g r i d i n Fig. 7.5,
aQ - Q1-Q2 ax AX
Q1+Q -Q -Q aQ - 2 3 4 - - 2T 2AT
Since aQ/ aT i s not sensit ive to Q2/5, ( the power i s less than one),
Q2/5 i s approximated by ((Q3+Q4)/2)2/5, i.e. an exp l i c i t form i s employed
here or else the resu l t ing equations would be d i f f i cu l t to solve. The
f i n i t e difference approximation to the d i f fe ren t ia l equation is thus:
137
Q 2 + Q +Q -Q 3 4 2 Q +€I 0 . 4 5 (3) ( 1 + - )
2 AX 2A T s o l v i n g f o r Q 1 : Q , = -
3
1 5 / 3 Q3+Q4 0 . 4 (- ) _ _ - ~ .
2AT A X 2
S t a r t i n g a t t h e u p s t r e a m e n d o f t h e ca tchmen t w h e r e Q2 = 0 and r e p l a c i n g
Q2 a t the n e x t p o i n t b y Q1 a t t h e p r e v i o u s p o i n t , a l l t h e v a r i a b l e s o n
t he r i g h t hand s i d e a r e k n o w n a n d one c a n s o l v e f o r Q , . The d imens ion -
less t ime s tep used was 0.05. The d i f f e r e n c e f o r s m a l l e r t ime s teps was
f o u n d b y t r i a l to b e u n n o t i c e a b l e .
SOLUT!ONS FOR DYNAMIC STORMS
T ime varying s to rms
One o f t h e most f r e q u e n t l y used s i m p l i f y i n g assumpt ions , b u t a
d a n g e r o u s assumpt ion , i n m a n y r a i n f a l I - r u n o f f mode ls i s t h a t o f c o n s t a n t
p r e c i p i t a t i o n r a t e t h r o u g h o u t the s to rm d u r a t i o n . The tempora l v a r i a t i o n
of p r e c i p i t a t i o n i n t e n s i t y f o r s to rms o v e r I l l i n o i s was documented b y
H u f f (1967) whose f i n d i n g s were o f t e n e x t r a p o l a t e d to o t h e r r e g i o n s . He
sugges ted i d e n t i f y i n g t h e q u a r t i l e o f max imum p r e c i p i t a t i o n and f u r t h e r
e m p l o y i n g p r o b a b i l i t i e s o f t he r a i n s o c c u r r i n g sooner o r l a t e r t h a n the
med ian . H u f f p l o t t e d h i s r e s u l t s a s mass r a i n f a l l c u r v e s so i t i s no t
easy t o d i s c e r n t h e s h a p e o f t h e h y e t o g r a p h s u n l e s s h i s c u r v e s a r e
d i f f e r e n t i a t e d w i t h respec t to t ime. I n g e n e r a l t hey a r e f o u n d t o b e
convex u p w a r d s . A p a r t f rom K e i f e r a n d C h u ' s (1957) s y n t h e t i c hye to -
g r a p h , ev idence p o i n t s to convex up h y e t o g r a p h s . The assumpt ion o f a
t r i a n g u l a r h y e t o g r a p h i s t h u s ex t reme a s a r e a l s to rm w o u l d t e n d t o
b e less ' p e a k y ' t h a n a t r i a n g u l a r one. The g e n e r a l t r i a n g u l a r - s h a p e d
r a i n f a l l r a t e v e r s u s t ime r e l a t i o n s h i p d e p i c t e d in F i g . 7.6 i s t h e r e f o r e
s t u d i e d . The t ime o f t h e p e a k i s v a r i e d ' be tween t h e s t a r t o f t h e s to rm
(TP = 0 ) and t h e e n d ( T = 1 ) . P
' I
T 1 T P
F i g . 7.6 T e m p o r a l l y v a r y i n g s to rm
138
Simple models of hyetographs assume a single peak i n r a i n f a l l
intensi ty. Storms with mult ip le major peaks can be synthesized from over-
lapp ing compound storms. I t i s a s ingle peak-storm which is considered
here and the time of the peak intensi ty permitted to va ry .
Q .
Constant exce'
0 1 2 T
Fig . 7.7 Simulated dimensionless hydrographs caused by storms with
time va ry ing r a i n f a l l intensi t ies (F ig . 7.6) bu t the same total p rec ip i ta t ion
Design storms for flood estimation general ly peak i n intensi ty i n
the f i r s t ha l f of the storm. T h i s i s an a l l ev ia t i ng factor i n peak runof f ,
as indicated i n Fig. 7.7. That i s a plot of hydrographs from the simple
catchment depicted i n F ig . 7 . 4 wi th var ious hyetographs imposed, i . e .
a rectangular hyetograph and t r i angu la r hyetographs with various peak
times were employed. The ordinate i n F ig . 7.7 i s the discharge ra te
expressed as a f ract ion of the mean excess prec ip i ta t ion rate, and the
abscissa i s time as a f ract ion of the time of concentration for a uniform
storm with p rec ip i ta t ion ra te equal to the mean ra te over the storm for
each of the t r i angu la r hyetographs.
I t w i l l be observed from F ig . 7.7 that i f the storm intensi ty peaks
in the f i r s t p a r t of i t s durat ion ( T 50.5) the peak runoff i s less than
that for a uniform storm of the same average intensi ty. This holds for
peaks up to 80% of the durat ion af ter commencement of r a i n . Only for
the peak a t the end of the storm (e.g. T = 1.0) does the peak runoff
exceed that for a uniform intensi ty storm. Then the peak runoff i s
approximately 10% greater than for a uniform storm of the same durat ion.
P
P
139
Q . T d = 0.4 0.6 0,8 1 1.2
I . _ . . . _ . _ . a
0 1 2 1
Fig. 7.8 Simulated dimensionless hydrographs caused by late peaking storms of constant volume and va ry ing durat ion
I f the storm durat ion i s not equal to the time of concentration for
a uniform storm however, the peak can be higher. Fig. 7.8 i s for a
storm of constant volume peaking a t i t s termination ( T P = 1 ) and for
durat ions represented by Td = 0.4 to 1.2. These hydrographs are for
storms of equal volume so that the shorter durat ion storms are of a
higher intensi ty than longer durat ion storms. Depending on the IDF curve
then a short durat ion storm may or may not resul t i n a higher runoff
ra te than for one of durat ion equal to the concentration time of the
catchment.
I t should be recal led that a l l other hydrographs plotted are for
a specif ied excess ra te of precipi tat ion. That is, i f the hyetograph i s
uniform so are the abstractions. I n practice, losses w i l l be higher a t
the beginning of a storm, resu l t ing i n a late peak i n excess r a i n even
for a uniform prec ip i ta t ion rate. This has the same effect as a storm
peaking i n the la t te r pa r t as i t increases the peak runoff . The effect
i s compounded as a storm which peaks near the end w i l l occur on a
re la t i ve l y saturated catchment so a greater proport ion of the higher ra te
of r a i n w i l l appear as runoff n?ar the end. This tends to make the
excess r a i n versus time graph concave upwards i f the hyetograph was
a s t ra igh t - l ined t r iang le . This effect i s not modelled here bu t a l l the
effects resul t i n a higher peak than for a uniform input. Scheckenberger
i n fact indicates peaks up to 30% greater than for uniform storms due
to the sum of these effects.
140
Spatial var ia t ions
I t appears that areal d is t r ibu t ion of the storm i s less effect ive
than temporal d is t r ibu t ion i n inf luencing peak runoff rate. Fig. 7.10
represents the simulated runoff from a 2-dimensional p lane subjected to
various d is t r ibu t ions of a steady excess ra in . The storm durat ion was
made in f i n i t e i n case the time to equ i l ib r ium exceeded the storm durat ion.
The spat ia l (o r longi tudinal i n th is case) d is t r ibu t ion was assumed
t r iangu lar , the peak va ry ing from the top to the bottom of the catchment
as i n F ig . 7.9.
I
'P =2
I , =1
peak intensity
_ -
Fig. 7.9 Catchment wi th long i tud ina l l y va ry ing storm
The same example would app ly to a uniform intensi ty storm over
a wedge-shaped catchment, the catchment width increasing I inear ly to
X and then decreasing l inear ly towards the outlet where X = 1 . P
0 1 2 1
F ig . 7.10 Simulated dimensionless hydrographs caused by steady semi-
i n f i n i t e storms of va ry ing d is t r ibu t ion down catchment (F ig . 7.9).
141
F i g . 7.10 d e p i c t s the r e s u l t i n g s i m u l a t e d h y d r o g r a p h s w h i c h i n d i -
ca te t h a t the r u n o f f n e v e r exceeds t h a t f o r a r e c t a n g u l a r s p a t i a l d i s t r i -
b u t i o n of r a i n f a l l . The r e s u l t i n g d imens ion less t ime to e q u i l i b r i u m i s
n e a r l y u n i t y f o r a l l cases, i m p l y i n g the same t ime o f c o n c e n t r a t i o n h o l d s
f o r uneven d i s t r i b u t i o n a s f o r u n i f o r m d i s t r i b u t i o n o f r a i n . The re i s
t he re fo re no t a chance o f a s h o r t e r d u r a t i o n s torm w i t h a h i g h e r
i n t e n s i t y c o n t r i b u t i n g to a g r e a t e r p e a k t h a n the u n i f o r m storm ( u n l e s s
the i n t e n s i t y - d u r a t i o n c u r v e is a b n o r m a l l y s teep) s i n c e the t ime to e q u i l -
i b r i u m i s no t reduced r e l a t i v e to a u n i f o r m storm.
X=x/L cotchmcnt lcngth
F i g . 7.11 Catchment w i t h a s torm m o v i n g down i t
9
1
1 1
1 2 1
F i g . 7.12 S imu la ted d imension less h y d r o g r a p h s caused b y u n i t s teady u n i f o r m storms m o v i n g down catchment a t d i f f e r e n t speeds (see F i g . 7.11)
142
Moving storms
Fig. 7.12 represents simulated hydrographs from a storm wi th a
constant p rec ip i ta t ion ra te and spa t ia l l y uniform travel I i ng down the
catchment. The long i tud ina l extent of the storm cel l i s the same as the
length of the catchment since in general smaller area storms are reputed
to be more intense than la rger cel ls. C i s X/Tc o r the speed d iv ided
b y the ra te of concentration. For slow storms ( C S 1 ) the dimensionless
hydrograph peak i s un i t y whi le for faster storms the peak is less. The
faster storms do not f a l l on the catchment long enough to reach equi-
I i b r i um.
Q 1
I
c = S
0 1 ' 2 7
F ig . 7.13 Simulated dimensionless hydrographs caused by steady uniform semi-inf inite storms moving down catchment a t d i f ferent speeds
Fig. 7.13 indicates there is also no increased peak for storms of
semi-inf inite longi tudinal extent (never ending once they enter the catch-
ment). A l l peaks converge on un i t y and there is no peak greater than
un i t y . Thus movement does not appear to resul t i n a hydrograph peak
greater than for a stat ionary storm.
For storms of l imi ted extent t rave l l i ng up the catchment, the peak
flow was observed ' to be less than for a stat ionary storm and the faster
the speed of t ravel of the storm the smaller the peak runoff .
143
I t h a s been demons t ra ted u s i n g n u m e r i c a l s o l u t i o n s t o t h e k i n e m a t i c
e q u a t i o n s f o r s i m p l e ca tchmen ts t h a t n o n - u n i f o r m i t y in r a i n f a l l i n t e n s i t y
c a n a f f e c t p e a k r u n o f f r a t e s . Tempora l v a r i a t i o n in excess p r e c i p i t a t i o n
r a t e c a n i n c r e a s e r u n o f f r a t e a b o v e t h a t f o r a s t e a d y r a t e o f r a in . S ince
s to rms u s u a l l y p e a k sometime a f t e r commenc ing and t ime d i m i n i s h i n g
a b s t r a c t i o n s t e n d to cause a l a t e r p e a k in excess r a i n f a l l r a t e , t h e
assumpt ion o f s t e a d y r a i n f a l l c a n b e d a n g e r o u s a s p e a k r u n o f f i s
underes t ima ted.
Uneven s p a t i a l d i s t r i b u t i o n o f a s to rm does n o t d i r e c t l y c o n t r i b u t e
to a h i g h e r p e a k r u n o f f u n l e s s i t r e s u l t s in a s h o r t e r d u r a t i o n s to rm
b e i n g t h e d e s i g n s to rm. S torm movement reduces t h e p e a k f l o w u n l e s s
t h e movement i s down-ca tchment , when t h i s model shows n o c h a n g e in
p e a k r u n o f f r a t e . A s m a l l e r , more i n tense s to rm t h a n t h e one t o e q u i l i -
b r i u m f o r t h e ca tchmen t m a y however r e s u l t in a h i g h e r p e a k r u n o f f r a t e .
REFERENCES
B r a k e n s i e k , D.L., 1967. K i n e m a t i c f l o o d r o u t i n g . T r a n s Amer. SOC. A g r i c . Engs . lO (3 ) p 340-343.
Be1 I , F.C., 1969. Genera l i zed r a i n f a l I - d u r a t i on - f requency r e l a t i o n s h i p s . Proc . Amer. SOC. C i v i l E n g r s . 95 (HY1) 6537, p 311-327.
C a r t e , A.E. 1979. S u s t a i n e d s to rms o n the T r a n s v a a l H i g h v e l d . S.A. Geogr. J o u r n a l , 6 1 ( 1 ) p. 39-56.
Changnon, S.A. and Voge l , J.L. , 1981. H y d r o c l i m a t o l o g i c a l c h a r a c t e r - i s t i c s of i s o l a t e d seve re r a i n s t o r m s . Water Resources Research 17 (6 ) p 1694-1700.
D i x o n , M.J., 1977. P roposed Mathemat i ca l Model f o r t he E s t i m a t i o n o f A r e a l P r o p e r t i e s o f H i g h D e n s i t y Shor t D u r a t i o n Storms. Dept . Water A f f a i r s , Tech. Rept. TR78, P r e t o r i a .
Eag leson, P.S., 1978. C l ima te , s o i l and vege ta t i on . 2. The d i s t r i b u t i o n o f a n n u a l p r e c i p i t a t i o n d e r i v e d f r o m o b s e r v e d s to rm sequences. Water Resources Research 14 (5 ) p 713-721.
Eag leson, P.S., 1984. The d i s t r i b u t i o n o f ca tchmen t c o v e r a g e b y s t a t i o n a r y r a i n s t o r m s . Water Resources Research , 20(5) P 581-590.
H u f f , F.A., 1967. Time d i s t r i b u t i o n o f r a i n f a l l in h e a v y s to rms. Water Resources Research , 3( 14) p 1007-1019.
H u g g i n s , L.F. and B i r r n e y , J.R., 1982. S u r f a c e r u n o f f , s t o r a g e and r o u t i n g . I n H y d r o l o g i c M o d e l l i n g of Sma l l Watersheds . Ed. Haan , C.T., Johnson, H.P. and B r a k e n s i e k , D.L., Amer. SOC. A g r i c . Engrs . Mono- g r a p h No.5.
James, W. and Scheckenberge r , R., 1983. S torm d y n a m i c s model fo r urban r u n o f f . I n t l . Symp. U r b a n H y d r o l o g y , H y d r a u l i c s and Sediment c o n t r o l , L e x i n g t o n , K e n t u c k y . p 11-18.
K e i f e r , C.J. and Chu , H.H. 1957. S y n t h e t i c s to rm p a t t e r n s f o r d r a i n a g e des ign . Proc . Amer. SOC. C i v i l E n g r s . 83 ( H Y 4 ) p 1332-1352,
Mader , G.N., 1979. Numer i ca l s t u d y o f s to rms in t h e T r a n s v a a l . S.A. Geogr. J o u r n a l , 6 1 ( 2 ) p 85-98.
N a t u r a l E n v i r o n m e n t Research Counc i I, 1975. F l o o d S t u d i e s Repor t , Vol . 1 . H y d r o l o g i c a l S tud ies , London , 5 vo lumes.
Scheckenberger , R., 1984. D y n a m i c s p a t i a l l y v a r i a b l e r a i n f a l I mode ls fo r s t o r m w a t e r management . M. Eng. Repor t , McMaster U n i v e r s i t y , Hami I ton.
144
Stephenson, D., 1984. Kinematic study of effects of storm dynamics on runoff hydrographs. Water S.A. October, Vol. 10, No. 4, pp 189-196.
Van Wyk, W. and Midgley, D.C., 1966. Storm studies i n S.A. - Smal I area, h igh intensi ty r a i n f a l l . The C i v i l Eng. in S.A., June, Vol. 8 No.6, p 188-197.
Waymire, E. and Gupta, V.L. 1981. The mathematical structure of r a i n f a l l representations 3, Some appl icat ions of the point process theory to ra in fa l I processes. Water Resources Research, 1 7 ( 5 ) , p 1287-1294.
Wilson, C.B., Valdes, J.B. and Rodrigues, 1 . 1 . ) 1979. On the inf luence of the spat ia l d is t r ibu t ion of r a i n f a l l i n storm runoff . Water Resources Research, 1 5 ( 2 ) , p 321-328.
Woolhiser, D.A., 1977. Unsteady free surface flow problems. I n Math- ematical Models for Surface Water Hydrology. Ed. by C i r i an i , T . A . Maione, U. and Wal I is, J.R., John Wiley G Sons, 423 pp.
145
CHAPTER 8
CONDUIT FLOW
KINEMATIC EQUATIONS FOR NON-RECTANGULAR SECTIONS
The a n a l y s i s o f f l o w i n c o n d u i t s i s more c o m p l i c a t e d t h a n f o r
o v e r l a n d f l o w on accoun t o f s i d e f r i c t i o n . N o n - r e c t a n g u l a r c ross sec t i ons
e.g. t r a p e z o i d s a n d c i r c u l a r d r a i n s a r e more d i f f i c u l t t h a n r e c t a n g u l a r
sec t i ons to a n a l y z e . Su r face w i d t h and h y d r a u l i c r a d i u s become a
f u n c t i o n o f w a t e r d e p t h . The s i d e s o f t h e c h a n n e l (and top in the c a s e
o f c losed c o n d u i t s ) i n c r e a s e f r i c t i o n d r a g . As f a r a s t h e f o r m o f t h e
b a s i c k i n e m a t i c e q u a t i o n s i s conce rned t h e m a t h e m a t i c a l e x p r e s s i o n s
become more c o m p l i c a t e d , a n d n u m e r i c a l s o l u t i o n s a r e necessa ry in t h e
m a j o r i y y o f cases.
The c o n t i n u i t y e q u a t i o n r e m a i n s
o r e x p a n d i n g t h e second te rm,
where t h e f i r s t t e rm i s the r a t e o f r i s e , t h e second p r i s m s t o r a g e a n d
t h e t h i r d wedge s t o r a g e .
The d y n a m i c e q u a t i o n reduces t o
( 8 . 3 ) M Q = aAR
where Q i s t h e d i s c h a r g e r a t e , a I S a f u n c t i o n of c o n d u i t roughness ,
q i s i n f l o w p e r u n i t l e n g t h , B i s t h e s u r f a c e w i d t h , A i s t h e c r o s s
sec t i ona l a r e a o f f l o w and R i s t h e h y d r a u l i c r a d i u s A/P where P i s
t h e we t ted p e r i m e t e r . E m p l o y i n g M a n n i n g ' s f r i c t i o n e q u a t i o n ,
a = K S /n a n d M = 2/3 ( 8 . 4 )
where
LIZ
1 K 1 = l (S .1 . u n i t s ) and 1.486 ( i t - sec u n i t s )
n = M a n n i n g ' s r o u g h n e s s c o e f f i c i e n t
O w i n g t o t h e g r e a t e r d e p t h s i n c o n d u i t s i n c o m p a r i s o n w i t h o v e r l a n d
f l o w , l ower v a l u e s o f n a r e a p p l i c a b l e . The a b o v e e q u a t i o n s c a n b e
s o l v e d f o r s p e c i a l cases o f n o n r e c t a n g u l a r c o n d u i t s a s i n d i c a t e d be low.
PART-FULL C I RCULAR P I PES
The c ross s e c t i o n a l a r e a o f f l o w i n a c i r c u l a r c o n d u i t ( F i g . 8 . 1 )
r u n n i n g p a r t f u l l (S tephenson, 1981) i s
146
0 a . a 4 2 2 2
A = - D 2 ( - -cos-sin-)
a n d P = DO 2
Thus i f one takes 0 as the v a r i a b l e , the c o n t i n u i t y equat ion becomes
aA a o + aa aa at
- a x = q ; _ -
a n d
;’ ( 1 + s i n 2 ~ - cos2 0 a o i@ = 2 % ) a t + ax
I n f i n i t e d i f ference form, s o l v i n g f o r 0 a f t e r a t ime i n t e r v a l A t ,
a = o + ( q - - ) G Q 8 Gt A X
2 1 ~ 2 ( l + s i n 2 5 - cos’g) 2 2
a n d in terms of the new , s ince 61 = aARZ3
0 . a D2 0 2 2 4 2 2
. a cos-sin- 2 3
o’} Q = a- (3 - cos-sln-) { z ( l -
( 8 . 7 )
( 8 . 8 )
(8 .9)
I n o rde r to s imulate f low a n d dep th v a r i a t i o n s in pipes, the l a t t e r
two equat ions a r e a p p l i e d a t successive po in ts fo r successive t ime
i n t e r v a Is.
I n a d d i t i o n to a n a l y s i s of f lows i n p ipes , the methods can b e
a p p l i e d to des ign b y successive a n a l y s i s . When des ign ing storm d r a i n
co l lect ion systems there a r e many approaches (Yen and Sevuk, 1975) .
I t i s i n normal p r a c t i c e not necessary to consider su rcha rged cond i t i ons
in a des ign unless a d u a l system (ma jo r a n d m ino r condu i t s ) i s employed.
I f p ipes a r e designed to r u n j u s t f u l l a t t h e i r des ign c a p a c i t y , then
they w i l l r u n p a r t f u l l f o r any other des ign storm d u r a t i o n . The h i g h e r
up the leg a p i p e l eng th i s , the shor ter w i l l b e the concentrat ion t ime,
o r t ime to f low e q u i l i b r i u m . The des ign storm d u r a t i o n w i l l equa l the
concentrat ion t ime of the d r a i n s down to the p i p e i n question. Any
subsequent p ipes w i I I have l a r g e r concentrat ion times a n d consequently
a lower storm in tens i t y .
F i g . 8.1 Cross section th rough p a r t - f u l l p i p e
147
COMPUTER PROGRAM FOR DESIGN OF STORM DRAIN NETWORK
The p r e c e d i n g scheme was employed i n a p r o g r a m f o r a n a l y s i n g
the f l o w in e a c h p i p e in a d r a i n a g e n e t w o r k the p l a n o f w h i c h i s
s p e c i f i e d b y t h e d e s i g n e r . The e n g i n e e r must p re -se lec t t h e l a y o u t , sub-
d i v i s i o n o f ca tchmen t , p o s i t i o n o f i n l e t s a n d g r a d e s . The g r a d e s w i l l
in g e n e r a l con fo rm to t h e s lope o f t h e g r o u n d .
I t i s necessa ry to s i m u l a t e o v e r l a n d f l o w and e a c h u p p e r dra in
i n o r d e r to s i z e a n y l ower drain. Such a n a l y s i s c a n o n l y b e done
p r a c t i c a l l y b y d i g i t a l compu te r u s i n g n u m e r i c a l s o l u t i o n s o f t h e f l o w
e q u a t i o n s . M a n y c a l c u l a t i o n s a r e necessa ry f o r comp lex n e t w o r k s . A
l i m i t a t i o n o n t h e max imum t ime i n t e r v a l f o r n u m e r i c a l s t a b i l i t y i m p l i e s
m a n y i t e r a t i o n s u n t i l e q u i l i b r i u m f l o w c o n d i t i o n s a r e r e a c h e d f o r e a c h
p i p e d e s i g n . I n a d d i t i o n , a number of d i f f e r e n t s to rm d u r a t i o n s mus t
b e i n v e s t i g a t e d f o r e a c h p i p e . A s i m p l e and e f f i c i e n t i t e r a t i v e p r o c e d u r e
was t h e r e f o r e sough t in o r d e r to m i n i m i z e compu te r t ime. The k i n e m a t i c
fo rm of t he f l o w e q u a t i o n was emp loyed to e n s u r e t h i s . The emphas is
t h r o u g h o u t t h e p r o g r a m i s s i m p l i c i t y o f d a t a i n p u t a n d m i n i m i z a t i o n o f
c o m p u t a t i o n a l e f f o r t . Some a c c u r a c y i s s a c r i f i c e d b y t h e s i m p l i f i c a t i o n s
b u t t h e o v e r r i d i n g assumpt ion of p r e c i p i t a t i o n p a t t e r n i s p r o b a b l y more
i m p o r t a n t .
The d e s i g n method (S tephenson, 1980) p roceeds f o r success i ve p i p e s ,
t h e d iamete rs o f w h i c h a r e c a l c u l a t e d p r e v i o u s l y . I t i s assumed t h e
n e t w o r k l a y o u t i s spec i f i ed , . and t h e p i p e g r a d e s a r e d i c t a t e d b y t h e
g r o u n d s lope. S t a r t i n g a t t h e top e n d s o f a d r a i n a g e system, t h e p r o g r a m
s izes success i ve l y l ower p i p e s . Thereby e a c h p i p e u p s t r e a m o f t he one
to b e des igned i s p re -de f i ned . I t i s necessa ry t o i n v e s t i g a t e s to rms o f
d i f f e r e n t d u r a t i o n a n d c o r r e s p o n d i n g i n t e n s i t y of f l o w to de te rm ine t h e
d e s i g n s to rm r e s u l t i n g in max imum f l o w for t h e n e x t p i p e .
I t i s assumed t h a t t h e d e s i g n s t o r m r e c u r r e n c e i n t e r v a l i s p r e -
se lec ted . The i n t e n s i t y - d u r a t i o n r e l a t i o n s h i p i s t h e n assumed t o b e o f
t h e f o r m
i =a e b + t d (8.10)
B y s e l e c t i n g s to rms o f v a r y i n g d u r a t i o n td, and s i m u l a t i n g t h e
f l o w b u i l d u p down t h e d r a i n s , t h e p r o g r a m c a n se lec t a s to rm w h i c h
w i l l r e s u l t in t h e max imum p e a k f l o w f r o m t h e l ower e n d o f t h e system.
T h a t d i s c h a r g e i s t h e one t o use f o r s i z i n g t h e n e x t l ower p i p e . T h u s
t h e p r o g r a m p roceeds f r o m p i p e t o p i p e u n t i l t h e e n t i r e n e t w o r k i s
des igned .
148
The program is l imited i n appl icat ion to selection of d r a i n pipe
diameters for a simple g rav i t y col lect ing system, and uses kinematic
theory and the l imi tat ions of the theory should be recal led. I t should
be noted that for major pipes i t may become necessary to al low for
backwater and rou t ing effects (Barnes, 1967). The program does not
optimize the layout (Argamon et al. 1973; Mer r i t t and Bogan, 1973). Nor
i s surcharge (Mar t in and King, 1981) o r detention storage considered
here.
\ Lsubcotchment boundary
* drain 4 ‘drain number
Fig. 8.2 Layout p lan of drainage network sized i n example
Program description
Pipes are assumed to flow i n i t i a l l y a t a depth corresponding to
a subtended angle of 0.2 radians at the centre. The corresponding flow
is very low, bu t th is assumption avoids an anomaly for the case of
zero depth when the numerical solut ion of the exp l i c i t equation i s
impossi b le.
Inf low from subcatchments i s assumed to occur along the f u l l length
of the respective pipe, i.e. subcatchment breadth is assumed to be equal
to pipe length. This affects over land flow time to some extent. I f
necessary ( i f flow is sensit ive to storm dura t ion) the subcatchment
f r i c t ion factor could be adjusted to g ive the correct over land flow time.
The computer program, wr i t ten i n FORTRAN for use i n conversational
mode on a terminal connected to an IBM 370 machine, i s appended. The
149
input format i s described below. Data is read i n free format and can
be input on a terminal as the program stands.
F i rs t l ine of da ta :
M, A , B, E, I N , I R , 1 1 , G .
Second and subsequent l ines of data (one l ine for each length of p ipe ) :
x ( I ) , s ( I ) , z ( I ) , C ( I ) , S O ( I ) , EO(I), I B ( I ) .
The input symbols are explained below:
M - The number of pipes: the number of pipes should be minimized
fo r computational cost minimization. For computational accuracy
the pipes should be d iv ided into lengths of the same order
of magnitude. I t i s convenient to make the pipe lengths equal
to the distance between in lets. In lets between 10 and 200m
apar t are normally suf f ic ient for computational accuracy.
There should be a t least two pipes i n the system.
A , B - Precipi tat ion ra te i i s calculated from an equation of the
form i = A / ( B + td) where td i s the storm durat ion and B
is a regional constant (both i n seconds). A i s a function
of storm re tu rn per iod and catchment location and i t s un i ts
are i n m i f S I un i ts are used, and f t i f ft-lb-sec un i ts a re
used.
E - Pipe roughness. This i s analogous to the Nikuradse roughness
and E i s measured i n m o r f t . I t i s assumed i n the program
that a l l pipes have the same roughness. A conservative f igure
of a t least 0.001 m (0.003 f t ) i s suggested to account for
surface deteriorat ion w i th time due to erosion, corrosion or
deDos i ts .
IN, - For each pipe siz ing computation var ious storm durat ions
I R are investigated, rang ing from I U l to IU2 i n steps of I R ( a l l
i n seconds). The smallest storm durat ion IU1 i s set equal
to the over land flow time for an upper pipe of the previous
pipe design storm durat ion for subsequent pipes down a leg.
The number of storm durat ions investigated is specif ied by
I N and the increment in t r i a l storm durat ion is specified b y
I R . Thus IU2 = IU1 + IN";I I . The accuracy of the computations
i s affected b y the number of t r i a l storm durations. A value
150
of I N between 3 and 10 i s usua l ly sat isfactory. The upper
l im i t can be estimated beforehand from experience or by t r i a l
( i f a l l design storm durat ions tu rn out to be less than the
IU2 specif ied then the I N selected i s sat isfactory).
The computational time and cost i s affected by the time incre-
ment of computations I I (seconds). The maximum possible
value is dependent on the numerical s tab i l i t y of the compu-
tat ions. A value equal to the minimum value of
w i l l normal ly be sat isfactory (of the order of 60 to 300
seconds) .
Gravi tat ional acceleration (9.8 i n S I un i ts and 32.2 i n ft-sec
un i t s ) .
The pipe data are next read i n l i ne by l i ne for M pipes.
As the program stands, 98 ind iv idua l pipes are permitted,
and any number of legs subject to the maximum number of
pipes.
The pipe length i n m or f t , whichever un i ts are used. An
upper l im i t on ind iv idua l pipes of 200m is suggested for
computational accuracy and a lower l im i t of 10m for opt imizing
computer time.'
The slope of the p ipe i n m per m or f t per f t .
The surface area cont r ibu t ing runoff to the pipe i n m or f t
The proport ion of precipi tat ion which runs off (analogous
to the ' C ' i n the Rational formula).
The over land slope of the cont r ibu t ing area, towards the in le t
a t the head of the pipe.
EO( 1 ) The equivalent roughness of the over land area i n m or f t
depending on un i ts employed.
IB(I) The number of the pipe which is a branch into the head of
p ipe I .
For no branch, pu t I B ( I ) = 0
For a header pipe a t the top of a leg, pu t I B ( I ) = - 1 .
Only one branch pipe per in let i s permitted.
More must be accommodated by insert ing short dummy pipes
between.
The order i n which pipes are tabulated should be obtained
as fol lows:
151
Computer Program for Storm Network Pipe Sizing
L.OOO1 L.0002 L.0003 L.0004 5 L.0005 10 L.0006 L.0007 L.0008 L. 0009 L . O O 1 O L.OO1l L.0012 12 L.0013 L.0014 L.0015 13 ~ . 0 0 1 6 I 5 L.0017
L.0019 L .0520 1 .0021 L.0022 L.5023 L .5024 L.0025 20 ~ . 0 0 2 6 L.OU27
L.0029 L.0030 L.UJ31 2 3 L.0032 L.UJ33 L.0024 L.0533 L.3036 L.i)037 L.UO3J L.0039 L.0040 L.0041 L . 0 0 4 2 30 L.0063 L.JU44 L.0045 L . 0 0 4 0 L . J J 4 7 L.304d 32 L.0049 L.305S 3 5 L.J05l 40 L.>il52 45 L.dOb3 5u L . 3 0 5 4 L.3055 L.5056 100 L.3057 L.5058 1 1 0 L.0054 120 L.0060
L . O O I ~
~ . 5 0 2 a
L e d 0 6 1 L.SO62 L-SJt3 L.0064 201) L.0065 L.0066 L.0067 L.0068 L.0069 290 L.0070 300 L.0071 L.0072 350 L.0073 L.0074 60 L.0075 L.0076 400 L.5077 70 L.0078 L.0079 60 L.00eO S I O P L.0081 END
L.5001 STUdM SEWER C€SIGN L.0002 P I P E LENGTh C I A CRAOE DSFLC/S STORM S AREA ~ . 0 0 0 3 I 100. -576 .0020 - 2 4 4 1016. 20006. L . O O 0 ' 4 2 150. -514 -004C .155 Y11. 20000.
44(3./4. I
L.JJU> 3 200. - 6 4 3 -0040 .162 206d. 40000 . L.0306 4 100. - 4 1 5 . 0 0 2 0 - 1 0 2 772. 10000+ ~ . 0 0 0 7 5 100. .574 .0040 .342 20od. 40000. ~ . u 0 0 6 6 2 0 0 - .613 .0040 .u17 2 0 6 8 . 10000. L . 0 0 U ' i 7 2CO. -253 .0020 .o9b 2068. 4 0 0 0 0 . 1.. i) J 10 @ 100. -505 .0050 1.287 2068. 20000. L a O J 1 1 DATA @ .0751440..0010 2563 301) 60
152
After d rawing out a p lan of the catchment wi th each pipe, the
longest leg possible i s marked, s ta r t i ng from the ou t fa l l , then success-
i ve ly shorter legs on f i r s t the longest, then successively shorter pipes.
Now the pipes are numbered i n the reverse over, s ta r t i ng a t the top
of the shortest leg etc. Proceed down each leg w i th the numbering u n t i l
a junct ion i s reached. Never proceed past a branch which has not been
tabulated previously. I n th is way a l l pipes leading into a pipe w i l l
have had the i r diameters calculated before the next lower pipe i s
designed.
Sample Input
The data are i n metres and are taken from Fig. 8.2
8
100
150
200
100
100
200
200
100
.075
.002
.004
.004
.002
.004
.004
.002
.005
TRAPEZOIDAL CHANNELS
1440
20000
20000
40000
10000
40000
10000
,40000
20000
.001
.4
.4
.4
. 3
.4
.5
.4
.4
3
.005
.003
.003
.005
.003
.005
.002
.003
300
.01
. O l
. O l
.02
.Ol
.01
.01
.01
60 9.8
-1
-1
1
-1
-1
4
0
3
b
Fig. 8.3 Trapezoidal channel geometry
153
For trapezoidal channels the hydrau l i c equations become
A = Y ( b
P = b + y 1 J(1 + 1 / S 1 2 ) + i ( l + 1 /S2 ’ ) ) l
I n pa r t i cu la r for a vert ical sided rectangular channel of l imi ted width
b , employing the Manning equation.
A yb,
P = b + 2 y
+ y/S1 + Y/SI)
Q - ayb yb )23
(b+2y
= ~ i ( y b ) ~ ’ ~ / (b+2y IZ3
The ana lys is of flow i n channels must general ly be done numer-
i ca l l y . The channel i s d iv ided into reaches and a sui table time step
selected to simulate flow and depth var iat ions. The cont inui ty and
f r i c t ion equations are appl ied conjunctively to calculate increase i n water
depth and flow ra te respectively. The method can be employed for
catchment channel flow simulations. Many na tura l channels can be
approximated by a trapezoid, o r else a number of trapezoids. A channel
p lus flood plane can be represented by two trapezoids a t d i f ferent bed
levels, the flood plane being a t the top of the banks of the channel.
The roughness, and hydrau l i c rad ius , and consequently the velocity w i l l
d i f f e r from channel to overbank and this can be accounted for.
COMPAR I SON OF K I NEMAT I C AND T I ME-SH I FT ROUT I N G I N CONDU I TS
Whereas over land flow time lag may be predicted qu i te d i f fe ren t ly
using kinematic or time lag methods, i n the case of conduits, time lag
often provides a su f f i c ien t ly accurate assessment of flow. That is, owing
to the confined cross section of a conduit, flow i s more incl ined to
emerge at the same ra te that i t enters a conduit, and travel time
approximates react ion time su f f i c ien t ly well .
In stormwater drainage, runof f hydrographs from over land flow
consti tute the essential input to hyd rau l i c conduits; e.g. pipes, channels,
culverts etc. The over land flow hydrographs are attenuated fu r ther as
they t ravel through the conduits. I n a stormwater drainage network,
where conduits and manholes are in te r l inked to ca r ry water from di f ferent
subcatchments on to a major outlet, hydrograph attenuation through the
conduits i s very important. Hydrographs from conduits leading to the
same manhole have to be summated for designing hyd rau l i c structures
or conduits downstream or for studying the behaviour of an ex is t ing
network under certain conditions. T h e magni tude of the hydrograph peaks
154
as well as their re la t i ve time posit ions are important for the accurate
assessment of design flows.
Various methods exist for rou t ing runoff hydrographs through closed
conduits. The most commonly used are time sh i f t methods. A time sh i f t
method sh i f t s the en t i re hydrograph i n time without any storage consider-
at ions for attenuation. The time sh i f t or l ag time i s calculated by
d i v id ing the length of the conduit by the velocity of the water i n the
conduit. This velocity i s usual ly taken to be the velocity of water i n
the conduit when the conduit i s almost f u l l under steady condit ions.
Storage balance methods are also used for rout ing. They apply mass
balance equations across the conduit. Such equations are solved by ei ther
exp l i c i t or imp l ic i t schemes. Both time sh i f t and storage rou t ing methods
ignore non-uniform flow and dynamic effects i n the system. Other methods
for hydrograph rou t ing include rou t ing through conduits using the
kinemat i t equations or even the dynamic equations of flow.
The use of the kinemat ic equations for rou t ing requires comparat ively
large computational ef for t i n comparison w i t h time sh i f t as the equations
have to be solved at close g r i d po in ts along the conduit over short time
increments. Most exist ing drainage models use time shi f t methods and
since the solut ion of the kinematic equation i s tedious i t may i n some
cases be unwarranted.
Section Geometry and Equations for Conduits
Two section configurations are studied here, one a c i r cu la r section
and the other a trapezoid. Both sections are assumed to be p a r t l y f u l l
as dynamic effects of the system are not studied. For the pipe th i s
implies that the depth of flow i s always less than the pipe diameter
whi le for the trapezoid i t s sides are assumed to be h igh enough to al low
any depth of water.
For p a r t i a l l y f i l l e d closed conduits, i.e. where no lateral inf low
exists along the conduit, the kinematic cont inui ty equation is :
(8.11)
where q i s discharge ( m 3 / s ) , a i s cross sectional area of flow (m’ ) ,
x i s distance along the conduit from the in le t (m) and t i s time ( 5 ) .
I n kinematic theory discharge can be assumed to be a function of
flow depth as the f r i c t i on slope i s assumed to equal the bed slope. T h i s
enables the use of uniform flow equations expressed i n terms of bed slope
instead of f r i c t i on slope. Such equations are usua l ly described i n the
fol lowing form:
155
( a ) Pipe
I ( b ) Trapezoid
F ig . 8.4 Condui t Sections
(8.12) m-1
q = a a R
where c( and m a r e f r i c t i o n f low coef f ic ients depending, on the un i fo rm
f low equat ion used, R i s the h y d r a u l i c r a d i u s of the section, i.e. a /p
(m) a n d p i s the wetted per imeter of the section ( m ) .
0 = 1 5 112 a n d m = 5/3
where n = M a n n i n g ' s roughness coef f ic ient a n d S = bed slope.
(8.13) n
I n s e r t i n g the va lues of a a n d m f rom 8.13 in equat ion 8.12 y ie lds :
1 4 .5/3
p2-/'3 q = - s (8.14)
The geometry of the condu i t s i s desc r ibed b y equat ions 8.15 - 8.18
I56
(8.15)
A = b y + y2 tan (90 - 0
P = b + 2 y sec (90 - 0) Trapezoid
(8.16)
(8.17)
(8.18)
The equat ions 8.11 a n d 8.14 were reduced to a dimensionless form
b y Constant in ides (1983) w i t h the choice of s u i t a b l e va r iab les . The
dimensionless equat ions a r e then so lved f o r d i f f e r e n t condu i t sections
a n d i n p u t hyd rog raphs . The k inemat i c equat ions a r e solved in t h e i r
dimensionless form to f a c i l i t a t e genera l i za t i on of r e s u l t s in terms of
constant parameters tha t a r e func t i ons of the i n p u t parameters. The
use of the dimensionless equat ions reduces computat ional e f f o r t as the
number of cases to s tudy reduces g r e a t l y .
The v a r i a b l e s q, a, x and t a r e reduced to the dimensionless
v a r i a b l e s Q, A, X a n d T b y d i v i d i n g them b y a p p r o p r i a t e v a r i a b l e s w i t h
i den t i ca l u n i t s as fol lows:
For the p ipe,
Q = q/qm (8.19)
A = a/d2 (8.20)
P = p / d (8.21)
Y = y / d (8.22)
For the t rapezoid,
Q = q/qc (8.23)
A = a/b2 (8.24)
P = p/b (8.25)
Y = y /b (8.26)
a n d fo r both sections
x = x/L (8.27)
T = t / t (8.28) k
where q i s the maximum flow c a p a c i t y of the p i p e ( m 3 / s ) , 7 0 . 335285~h d 8 /3 m qc i s a d i scha rge v a r i a b l e , be ing a func t i on of f r i c t i o n coef f ic ients
0 , m a n d bottom w id th of t rapezoid, b (m’/s) i.e.
i s a t ime constant ( 5 ) a n d L i s the length of the condu i t (rn).
qc = 1 5 112 b 8 1 3
n ’ t k
To def ine the d i scha rge a n d time constants a p p r o p r i a t e l y the
dimensionless k inemat i c equat ions a r e ob ta ined b y s u b s t i t u t i n g the
dimensionless v a r i a b l e s in the c o n t i n u i t y equa t ion i.e. f o r the p ipe,
(8.29)
157
R e a r r a n g i n g y i e l d s :
(8.30)
Fu r the rmore b y d e f i n i n g the t ime cons tan t a s in e q u a t i o n 8.31
reduces e q u a t i o n 8.30 to the d imension less e q u a t i o n 8.33. S i m i l a r l y f o r
t he t r a p e z o i d the t ime cons tan t i s d e f i n e d in e q u a t i o n 8.32.
F o r the p i p e :
Ld2 t = -
qm
For the t rapezo id :
LbZ t = -
qc
(8.31)
(8.32)
where the d imens iona l c o n t i n u i t y equation i s :
aQ aA -
S i m i l a r l y , t he u n i f o r m f low e q u a t i o n 8.14 can be reduced to i t s d imension-
less form, i .e.
f o r the p i p e :
(8.33) - a x + - - aT
(8.34)
where the maximum c a r r y i n g c a p a c i t y of a p i p e c a n b e shown to b e
qm = 0.335282 Sf d8/3 (8.35)
S u b s t i t u t i n g in e q u a t i o n 8.34 a n d r e a r r a n g i n g y i e l d s :
Q = A5/3 - 1
0.335282 p2/3
F o r the t r a p e z o i d the u n i f o r m f l o w e q u a t i o n reduces
Q q c = - S z - 1 (Ab‘ )5 /3
(Pd )2 /3
D e f i n i n g q a s in e q u a t i o n 8.38 reduced e q u a t i o n C
(8.36)
to:
(8.37)
8.37 to the d imension-
less f l ow e q u a t i o n f o r the t r a p e z o i d g i v e n in e q u a t i o n 8.39:
‘ c - n - t ,t b8/3 (8.38)
Fo r the t r a p e z o i d : A 5/3
Q = - ,,2/3
(8.39)
E q u a t i o n s f o r t f o r b o t h sect ions c a n b e o b t a i n e d b y s u b s t i t u t i n g
e q u a t i o n 8.35 a n d 8.38 i n t o e q u a t i o n 8.31 a n d 8.32. S i m i l a r l y , f o r
o b t a i n i n g the d imension less a r e a a n d pe r ime te r v a r i a b l e s ( A a n d P )
e q u a t i o n s 8.20, 8.21 a n d 8.22 a r e s u b s t i t u t e d in e q u a t i o n s 8.15 to 8.18.
k
The r e s u l t i n g exp ress ions a r e summar ised below
158
Pipe 1 -1 1 4 2
A = - cos (1-2Y) - (- - Y ) . (‘f - Y 2 ) 1 / 2
- 1 P = cos (1-2Y)
L t =
( 8 . 4 0 )
( 8 . 4 1 )
( 8 . 4 2 )
Channel
A = Y + Y2 tan (90-0)
P = 1 + 2Y sec (90-0)
( 8 . 4 3 )
( 8 . 4 4 )
( 8 . 4 5 )
Two shapes of inf low hydrographs are routed through the conduits,
one a uniform and the other a t r i angu la r time d is t r ibu t ion . These two
time d is t r ibu t ions were chosen as they represent extreme cases, i.e. a
na tura l runoff hydrograph, from over land flow, would have a shape
between these two extremes depending on the r a i n f a l l and catchment
character i st i cs.
I n add i t ion to the shapes the hydrographs were assumed to have
a var iety of durat ions and intensit ies. Fig. 8.5 i l l us t ra tes the inf low
hydrographs i n the i r dimensionless form. I
Q I M ‘ I M
Q I M i s the maximum discharge factor or inf low factor and TD = td/tk,
where td i s durat ion.
Fig. 8.5 Different dimensionless inf low hydrographs
The dimensionless equation fo r speed of propagation i s
2. 4
2. 0
_ - dT C 1
PIPE
-
where C 1 = 0.335262 for the pipe
= 1.0 for the trapezoid C1
where for pipes:
3A TP = 2 ( Y - Y Z )
and for t rapezoi ds :
a A - ’ cos (90-0) + Y sin (90-0) 2P 2
159
( 8 . 4 6 )
( 8 . 4 7 )
( 8 . 4 8 )
Equation 8.46 i s a function of the depth coeff icient, Y. i t was solved
in terms of Y using a computer model.
DEPTH/DIhMETER
F ig . 8.6 Dimensionless propagation speed of a disturbance i n p a r t i a l l y f i l l e d pipes
160
0
C 0
(0
-t N
0
m
(0
9
N
d
... 4
4
4
4
cu 0
d
.LP/XP
F i g . 8.7 Dimensionless propagation speed of a d i s t u r b a n c e in trapezoids
T y p i c a l r e s u l t s a r e g i v e n i n F i g . 8.6 a n d 8.7 . I t c a n been seen f rom
F i g . 8.6 t he max imum d imens ion less p r o p a g a t i o n speed in a p i p e i s 1.63
a n d o c c u r s when the d e p t h o v e r d i a m e t e r r a t i o i s 0.62. F i g . 8.7 shows
t h a t f o r t he t r a p e z o i d t h e d imens ion less p r o p a g a t i o n speed inc reases w i t h
an i n c r e a s e in the d e p t h o v e r bo t tom w i d t h r a t i o . I t i s necessa ry to k n o w
f o r b o t h the p i p e and the t r a p e z o i d the max imum d e p t h o v e r d i a m e t e r and
d e p t h o v e r bo t tom w i d t h r a t i o s r e s p e c t i v e l y in o r d e r t o assess the max imum
p r o p a g a t i o n speed, ( d X / d T ) m , during a n y s i n g l e s i m u l a t i o n . I t s h o u l d b e
no ted t h a t f o r t h e p i p e a n y s i m u l a t i o n , where the d e p t h o v e r d i a m e t e r
r a t i o exceeds 0 .62 , w i l l h a v e a ( d X / d T ) of 1.63. m
The max imum d e p t h o f f l o w i n the c o n d u i t t o b e encoun te red d u r i n g
s i m u l a t i o n w i l l b e a f u n c t i o n o f t he max imum i n f l o w d i s c h a r g e a t t he
i n l e t , a s the h y d r o g r a p h w i l l a t t e n u a t e as i t t r a v e l s a w a y f r o m t h e
i n l e t . The max imum d imens ion less d e p t h o f f l o w in the c o n d u i t ( Y ) i s
r e l a t e d to the max imum d imens ion less i n f l o w d i s c h a r g e , o r i n f l o w f a c t o r
( Q ) , b y e q u a t i o n s 8.36 a n d 8.39 . I M
E q u a t i o n 8.46 y i e l d s max imum p r o p a g a t i o n speeds f o r d i f f e r e n t i n f l o w
f a c t o r s .
Computer S i mu I a t ion
A compu te r model was deve loped f o r s o l v i n g the d imens ion less k i n e -
m a t i c e q u a t i o n s f o r c l o s e d c o n d u i t s . The mode l r o u t e s d imens ion less i n f l o w
h y d r o g r a p h s t h r o u g h the c o n d u i t s t o p r o d u c e d imens ion less o u t f l o w h y d r o -
g r a p h s a t t he ou t let . The d imens ion less h y d r o g r a p h s were then s t u d i e d
to e v a l u a t e the e f f e c t s t h a t a sec t i on o f f i x e d geomet ry and l e n g t h h a s
in a t t e n t u a t ing i n f l o w h y d r o g r a p h s o f v a r y i n g d i s c h a r g e a n d d u r a t i o n .
F o r e v e r y i n f l o w f a c t o r and i n f l o w h y d r o g r a p h d i s t r i b u t i o n d i f f e r e n t
d imens ion less s to rm d u r a t i o n s were assumed. The d imens ion less s to rm
d u r a t i o n s w e r e assumed. The d imens ion less s to rm d u r a t i o n s a r e d e f i n e d
as the s to rm d u r a t i o n o v e r the t ime cons tan t r a t i o , i .e .
‘d t k
TD = - ( 8 . 4 9 )
V a l u e s o f TD v a r i e d f rom 0.2 t o 10 a c c o r d i n g to the i n f l o w t ime
d i s t r i b u t i o n and sec t i on t ype . The f o l l o w i n g o b s e r v a t i o n s a r e made :
a ) S i m u l a t i o n s i n d i c a t e d the l a g t ime o f t h e o u t f l o w r e l a t i v e t o t h e
i n f l o w h y d r o g r a p h decreases w i t h an i n c r e a s e o f i n f l o w h y d r o g r a p h
d u r a t i o n (for a c o n s t a n t i n f l o w f a c t o r ) . The reason f o r t h i s i s t h a t
l o n g e r d u r a t i o n i n f l o w s i m p l y h i g h e r i n f l o w vo lumes. H y d r o g r a p h s w i t h
l ower vo lumes t e n d t o s p r e a d more w i t h i n t h e c o n d u i t r e s u l t i n g in l ower
w a t e r dep ths w h i c h in t u r n r e s u l t i n l ower f l o w v e l o c i t i e s and p r o p a g a -
I62
t ion of disturbance speeds. This inev i tab ly increases the i r lag time.
The same argument explains the second observation, i.e.
b ) The ra t i o of peak at the out let over peak at the in let increases
with increasing storm durat ion ( fo r a constant inf low factor) o r i n other
words inf low hydrographs of smal ler storm durat ion undergo higher
discharge attenuation than hydrographs of longer duration. The reason
for th is i s the same as i n a ) , i.e. lower volumes spread more than
b igger volumes resu l t ing i n lower depths of flow and thus lower dis-
charges.
C ) The lag time for an inf low hydrograph of f i xed durat ion decreases
with higher inf low factors. The reason for t h i s i s ident ical to a ) as
higher inf low factors imply higher volumes of water.
d ) Peak flow attenuation i s h igher for small inf low factors ( fo r a
constant inf low dura t ion) than for h igh inf low factors, the reason being
the same as for observation b ) .
Further deductions from the resul ts can be made by representing
the pr intout resul ts i n the form of graphs. This i s done i n subsequent
sect ions.
C r i t e r i a for choosing between Time Shi f t and Kinematic Routing
One of the main objectives of t h i s study was to develop a method
for assessing whether time shi f t methods can be used without hav ing
to resort to rou t ing methods. T h e main assumption behind time sh i f t
methods i s the preservation of the hydrograph ordinates without any
attenuation. To accept time sh i f t methods, therefore, the hydrograph
attenuation that would happen in a real l i f e s i tua t ion must be small .
One must therefore decide what a re acceptable l im i ts of attenuation.
As hydrograph attenuation d i f fe rs throughout the hydrograph durat ion
one usual ly refers to peak attenuation. I n th i s study a peak attenua-
t ion of 10% i s taken to be the maximum peak attenuation that can be
ignored. This value, although a r b i t r a r i l y defined, i s based on the fact
that more accurate determination of runof f i s not j us t i f i ed due to the
corresponding inaccuracies i n input determination. Furthermore, in a
drainage system consist ing of var ious conduits in te r l inked i n a network,
to lerat ing a b igger peak attenuation can resu l t in a gross overestimation
of the outflow peak. This occurs since a small peak attenuation i s
propagated downstream through various conduits and doing that increases
i n magni tude.
163
Inflow hydrograph d u r i t i o n / t c
Fig . 8.8 Diagram ind ica t ing when time sh i f t rou t ing can be used w i th p a r t i a l l y f i l l e d pipes.
Fig. 8.9 Diagram ind ica t ing when time sh i f t rou t ing can b e used w i th trapezoids a t angle of side to horizontal of 30°
8
6 -
9
' U n i f o r m lnpui g/ /' K I nemal I c r u u t 1 n g R",t b e used T l r n e S t 1 1 f i ' , - 1 L N T E K M E U l A l t A R E A /routing ndy b e u s e d
/'
1 . 2
1 . 1
1 . o
0 . 9
I . ," 0 . 8 - E
0.7
0 . 6
0 . 5
\ \ P I P E
I 0
I 1 I 0 . 2 0,Q 0 . 5 0 . 8
inflow p e a k d > s c h a r g e / q m
0
164
Fig. 8.10 Diagram ind ica t ing when time sh i f t rou t ing can be used
w i th trapezoids at angle of side to horizontal of 90'
inflow p e a k d > s c h a r g e / q m
Fig. 8.11 Time lag for hydrographs routed through p a r t i a l l y f i l l e d pipes
165
Having decided on an acceptable peak attenuation to be neglected
i t i s assumed that kinematic rou t ing describes accurately rou t ing i n a
real l i f e s i tuat ion. The resul ts obtained by kinematic rou t ing are
employed to assess the condit ions under which time sh i f t methods are
acceptable, i.e. i n t h i s case the conditions under which the peak
attenuation i s lower than 10%. To do th i s the resul ts were used to
obtain a dimensionless inf low dura t ion for a 0.9 outflow to inf low peak
ra t i o for every type of section and inf low factor. The dimensionless
inf low dura t ion was obtained e i ther by I inear interpolat ion or whenever
thought necessary by p lo t t ing dimensionless durat ion against outflow to
inf low peak r a t i o and obtaining the dimensionless durat ion for a peak
ra t i o of 0.9, the peak ra t i o of 0.9 corresponding to a 10% peak
attenuation. The resul ts are summarised in Figs. 8.8 - 8.10.
Lag Time for Routing Hydrographs Using Time Shif t Methods
Using a s imi la r method to the above the dimensionless time lag of
hydrographs w i th a peak attenuation of 10% was obtained. The dimension-
less lag times are surnmarised i n Figs. 8.11 to 8.13 for pipes and
selected trapezoids.
A dotted l ine represents l ag times as obtained by time sh i f t methods
for comparison purposes.
Comparison of Methods for E v a l u a t i n g L a g Time
Two assumptions are cur ren t ly popular for ca lcu la t ing the time lag
of a hydrograph to be routed by time sh i f t methods. The time lag is
ei ther assumed to be the length of the conduit d iv ided by the velocity
of the water when the conduit i s d ischarging at f u l l capacity o r i t i s
assumed that the time lag i s the length of the conduit d iv ided by the
velocity of water i n the conduit corresponding to the maximum discharge
of the inf low hydrograph.
Method 1 (TLp = L / (qm/am) . ) The time constant ( t k ) for the pipe i s
given by equation 8.31. The dimensionless time lag i s thus
tLP - a rn - - dZ
rn
k t
where a /d2 i s the dimensionless flow area for a pipe discharging at
maximum capaci ty. Subst i tut ing for arn/dz
‘LP - = 0.7653
k t
(8.50)
166
This gives the dimensionless lag time for a pipe and plots i n Fig.
8.11 as a s t ra igh t l ine. As can be seen the l ag time calculated b y th i s
method over-estimates the true value for h igh inf low factor values (b igger
than 0.6) and grossly underestimates i t for low inf low factor values
( lower than 0 .25 ) . For intermediate inf low factor values th i s method
yields time lags which l i e between the range set up by the two di f ferent
input distr ibut ions.
Method 2 ( tLp = L/(qim/qm).) The fol lowing relat ionship holds for the
dimensionless time lag:
(8.51)
where A. i s the dimensionless flow area corresponding to the maximum
discharge of the inf low hydrograph ( 4 . 1 . I
i m Equation 8.51 was solved i n the fo l lowing way to express ( t / t )
i n terms of the inf low factor (qim/qm). The dimensionless water depth
( Y i ) corresponding to the flow depth A i i s solved knowing the inf low
factor and a Newton-Raphson i te ra t i ve scheme, using equation 8.36.
i s used to solve for A. and consequently for tLp/tk. The calcu-
lated values of tLp/tk are plotted i n Fig. 8.11 for comparison. I t can
be seen that t h i s present method yields time lags closely resembling the
resul ts obtained from kinemat ic theory using uniform input hydrographs.
Th is occurs as uniform input hydrographs (wh ich do not attenuate
s ign i f i can t ly - 10% on ly ) maintain an approximately constant depth
through the i r t ravel through the conduit, thus hav ing a speed of flow
simi lar to that calculated by the ex is t ing method. The fact that time
lag as developed by kinematic rou t ing i s s l i gh t l y less than that using
the present method i s because some attenuation (10%) occurs du r ing
rou t ing for producing the resul ts.
LP k
Y .
I
Time Lag for Trapezoids
Method ( 1 ) as out l ined above i s not app l i cab le for trapezoids i n
th i s study as they are assumed to be deep enough to accommodate
incoming hydrographs of any discharge. As the i r depth i s not restr icted
one cannot t a l k of maximum discharge through trapezoids. Method ( Z ) ,
however, can be. used to express the dimensionless lag time (tLp/tic) i n
terms of the inf low factor (4 . /qc ) to compare time lags w i th the present
method with the resul ts shown i n Figs. 8.12 and 8.13. im
The time constant t i s g iven by equation 8.32 for the trapezoid. k
Th is yields the dimensionsless time lag.
167
2.5
i . c
_' i .5
-
1 . 0
i
T i m e s h i f t m e t h o d
Unlforrn I n p u t
_I
0 7 4 6 8
Inflow p e a k d i r c h a r g e / q c
Fig. 8.12 Time lag for hydrographs routed through trapezoids wi th angle of side to horizontal of 30°
168
\
T R A P E Z O I D
A N G L E - 90'
\\ --------
Fig. 8.13
/ T r i a n g u l a r I n p u t
I I I 6
I n f l o w p e a k d > s c h a r g e / q ,
Time lag for hydrographs roubed through trapezoids w i th angle of side to horizontal of 90
LP Ai t ~- t k -0
169
(8.52)
This equation i s solved to y ie ld the r a t i o t / t for d i f ferent values
of the inf low factor. Note that the relat ionship w i l l d i f f e r for d i f ferent
angles for the trapezoid as the dimensionless flow area i s a funct ion
of the side angle. The resul ts a re plot ted in Figs. 8.12 and 8.13 together
wi th the kinematic rou t ing resul ts for comparison purposes. As can be
seen ( the dashed l ines) the time lags from the present method are s l i gh t l y
higher but closely resemble the ones from kinematic rou t ing using a
uniform input. Note that th is was also the case for the pipe. The reasons
for the i r resemblance are s imi la r to those for the pipe and are discussed
in the previous section.
Lp k
I t can be seen from Figs. 8.8 to 8.10 that the dimensionless inf low
durat ion i s much more c r i t i ca l than the dimensionless inf low peak
discharge for determining whether time sh i f t methods can be used. This
i s more apparent i n the case of trapezoids where the 10% peak attenua-
t ion curves appear almost vert ical for dimensionless inf low peak dis-
charge values greater than 2.0.
Furthermore, i t can be seen that the dimensionless infow durat ion
decreases with increasing inf low factor. This i s expected as inf low
hydrographs w i th a s imi la r inf low factor need b igger durat ions than
ones with a h igher inf low factor for both inf low hydrographs to have
s imi la r volumes. As was discussed ear l ie r , h igher inf low volumes w i l l
imply smal ler peak attenuation, other parameters being constant, one
exception to th i s observation being the pipe for inf low factors h igher
than 0.8. I t can be seen from Fig. 8.8 that as the inf low factor
approaches un i ty the dimensionless durat ion (causing a 10% peak
attenuation to the inf low hydrograph) increases.
This i s probably due to the fact that a pipe discharges more when
not f lowing fu l I as already discussed.
I t w i l l also be noted that for trapezoids and discharge inf low
factors of less than 2.0 the dimensionless inf low durat ion increases
sharply as the dimensionless discharge decreases. This i s probably due
to the fact that at low depths of flow side f r i c t i on effects cause a
stabi I i t y effect on the flow h igh l y at tenuat ing peak discharges. This
in tu rn implies higher inf low durat ions for maintaining a peak
attenuation of 10%.
For a constant inf low factor, the inf low dura t ion ( imp ly ing a 10%
peak attenuation of the routed hydrograph) i s b igger for the t r i angu la r
d is t r ibu t ion than for the uniform one. This i s to be expected as a
170
t r i angu la r d is t r ibu t ion has a lower inf low volume than a uniform one,
both d is t r ibu t ions hav ing the same dura t ion and inf low factors.
The t r i angu la r d is t r ibu t ion would therefore need a greater durat ion
( fo r a constant inflow factor) or a greater inf low factor ( fo r a constant
dura t ion) to y ie ld a s imi la r resu l t to the uniform d is t r ibu t ion . Note that
a constant volume w i l l not imply identical resu l ts between the two
distr ibut ions as the shape also p lays an important ro le i n the rou t ing ;
for examp I e:
From Fig. 8.8, for a pipe and an inf low factor of 0.6, the corres-
ponding dimensionless durations resu l t ing i n a 10% peak attenuation of
the inf low hydrograph, are found to be 0.18 for a uniform input d i s t r i -
but ion and 0.82 for a t r i angu la r d is t r ibu t ion . This implies that the
t r iangu lar d is t r ibu t ion has a b igger inf low volume than the uniform one
in the ra t i o o f :
This ra t i o ( t r i a n g u l a r to uniform volume) var ies depending on the
inf low factor and type of section bu t i s always found to be more than
un i t y . This implies fu r ther that a uniform time d is t r ibu t ion i s attenuated
less than a t r i angu la r one even i f both have the same volume when
routed through a closed conduit.
A fu r ther comparison of the effects the inf low d is t r ibu t ion has on
the resul ts i s shown i n Figs. 8.11 to 8.13. The uniform hydrograph takes
more time to t rave l along the conduit ( i t has a b igger lag time) than
the t r iangu lar one (both hydrographs attenuated at the i r peak by 10%).
The reason for t h i s i s that for a constant inf low factor and a constant
peak attenuation the t r iangu lar d is t r ibu t ion has a much b igger dura t ion
than the uniform one. Furthermore, i n the case of the t r i angu la r d i s t r i -
but ion, the peak discharge i n the outflow hydrograph corresponds to
the peak of the inf low hydrograph which l ies, i n time, in the middle
of i t s durat ion. I n the case of the uniform d is t r ibu t ion however, the
outflow hydrograph peak w i l l correspond to the inf low peak at a much
ea r l i e r stage of the d is t r ibu t ion , i.e. at the beginning of the inf low
hydrographs. Th is implies a la te r entry time ( i n the condui t ) for the
peak of the uniform d is t r ibu t ion resu l t ins i n a longer lag time.
The engineer faced with the problems of rou t ing a runof f hydrograph
through a pipe or a channel w i l l f i nd the resul ts presented here of
d i rect use. The runof f hydrograph could be the resul t of over land flow
or the outflow from another conduit. Figs. 8.8 to 8.10 can be used to
establ ish the necessity of rou t ing whi le Figs. 8.11 to 8.13 can be used
171
to c a l c u l a t e a l a g t ime f o r t he cases f o r w h i c h t ime s h i f t r o u t i n g i s
shown to b e adeaua te .
REFERENCES
Argaman , Y., Shami r , U. and S p i v a k , E. 1973. Des ign o f o p t i m a l sewerage systems, Proc . ASCE, ( 9 9 ) , EE5, Oct., p 703-716.
Ba rnes , A.H., 1967. Compar i son o f compu ted and o b s e r v e d f l o o d r o u t i n g in a c i r c u l a r c ross sec t ion . I n t l . H y d r o l . Sympos. Co lo rado S ta te U n i v . , F o r t C o l l i n s , p p 121-128.
Cons tan t i n ides , C.A., 1983. Compar i son o f k i n e m a t i c and t ime s h i f t r o u t i n g in c l o s e d c o n d u i t s . Repor t 3/1983. Water Systems Research Programme, U n i v e r s i t y o f t he W i t w a t e r s r a n d .
Green, I .R.A., 1984. WITWAT s t o r m w a t e r d r a i n a g e p r o g r a m . Wate r Systems Research Programme, Repor t 1/1984. U n i v e r s i t y o f t h e W i t w a t e r s r a n d . 67p
M a r t i n , C . and K i n g , D., 1981. A n a l y s i s o f s to rm sewers u n d e r s u r - c h a r g e . Proc . Conf. U r b a n S to rmwate r , I I I i no i s . pp 74-183.
M e r r i t t , L.B. and Bogan , R.H., 1973. Computer b a s e d o p t i m a l d e s i g n of sewer systems. Proc . ASCE, ( 9 9 ) , EE1, Feb. pp 35-53.
Stephenson, D., 1980. D i r e c t d e s i g n a l g o r i t h m f o r s to rm drain ne tworks . Proc . I n t . Conf. U r b a n Storm D r a i n a g e , U n i v . K e n t u c k y , L e x i n g t o n .
Stephenson, D., 1981. S to rmwate r H y d r o l o g y and D r a i n a g e , E l s e v i e r , 276 PP.
Yen, B.C. and Sevuk , A.S., 1975. Des ign o f s to rm sewer ne tworks . Proc . ASCE, 101, EE4, Aug. 535-553.
172
CHAPTER 9
URBAN CATCHMENT MANAGEMENT
EFFECTS OF URBAN I ZAT I ON
I n nature a semi-equil ibr ium exists between precipi tat ion, runoff
and in f i l t r a t i on into the ground. Over years the water table f luctuates
about a mean. I t recedes dur ing droughts when seepage into watercourse
exceeds replenishment rates, and r ises when i t ra ins . The depth of soi l
above the water tab le i s general ly not excessive or else vegetation dies,
the ground dr ies out and wind blows the soi l away. The amount of water
which r ises up i n the soi l under cap i l l a ry action or in vapour form i s
l imited by the depth of water table.
The construction of impermeable ba r r i e rs on the surface, such as
roads and bu i ld ings , reduces the ra te of ground water replenishment.
The water runs o f f easier and the l imited permeable area res t r i c ts i n f i l -
t rat ion. The groundwater level w i l l therefore drop and the zone above
the water tab le w i l l g radua l ly d r y out. Vegetation and the soi l charac-
ter ist ics w i l l change. I f we a re not to affect our environment adversely
we should attempt to re tu rn some of the stormwater we channel o f f our
urban area back to the ground. This can be accomplished by ensur ing
adequate permeable surfaces, and by direct ing stormwater into special ly
selected or constructed seepage areas. We w i I I then not only maintain
the regime but also minimize design flow rates downstream.
The deplet ion of groundwater w i l l also a l te r the relat ionship
between r a i n f a l l and runoff . After a d r y spel l more water w i l l be needed
to saturate the ground so that the i n i t i a l abstract ion may be greater
than before the development occurred. This i s offset to an extent by the
impermeable ground cover. The net effect i s to make a more extreme
hydrology i.e. a greater dif ference between floods and droughts than
before deve lopme n t .
Effect on Recurrence In te rva l
Urban Development affects the ra in fa l I pa t te rn and stat ist ics as
well as the runof f pattern. I t has been al leged that b lanket ing effects
due to solar shields affect evaporation and hence the resul tant p rec ip i -
tat ion. The blanket of smog, dust, fumes etc., may also affect the place
in which the clouds release the i r moisture, so the effect of urbanizat ion
on r a i n f a l l i s d i f f i c u l t to estimate and the stat ist ical propert ies of
173
r a i n f a l l records (e.g. the mean, coefficient of variance, frequency and
d is t r ibu t ion) w i l l be affected as well to some extent. Ra in fa l l i s reputed
to f a l l more on the leeward side of c i t ies due to the heat ing up of the
a i r over the c i t y and up to 15% more precipi tat ion has been a t t r ibu ted
to th is effect. (Huff and Changnon, 1972; Colyer, 1982). Apart from this,
the relat ionship between r a i n f a l l and runoff i s affected.
Some of the s impl ist ic methods of assessing runoff suppose that
the recurrence in te rva l of a calculated f lood i s the same as the recur-
rence in te rva l of the causi t ive r a i n f a l l for the design storm durat ion.
I t could be that t h i s assumption i s borne i n mind i n the choice of the
Rational coefficient. That i s the use of the ra t iona l method gives a
certain recurrence in te rva l of runoff (equal to that of the selected storm
in fac t ) but i t does not imply that the design storm i s the one which
w i l l produce that runoff . This i s a gross s impl i f icat ion and i t i s ra re l y
that the recurrence in te rva l of a storm and i t s resu l t ing f lood coincide.
This i s due to the predominating effect of abstract ion or losses. I t w i l l
be recal led that general ly the Rational coefficient C i s nearer 0 than
1 imply ing losses are greater than runoff . That i n tu rn means that
losses, which in tu rn a re mostly soi l moisture abstraction, affect runof f
more than r a i n f a l l . Hence the runof f and i t s re tu rn per iod should be
more related to soi l moisture conditions than to r a i n f a l l . A study by
Sutherland (1982) indicates l i t t l e correlat ion between ra in fa l I recurrence
in te rva l and the recurrence in te rva l of the f lood when assessed i n terms
of the peak flow rate. He proposed that antecedent moisture condit ions,
measured in terms of the total precipi tat ion i n preceding days, should
be a parameter i n runoff-duration-frequency relat ionships. His content ion
is that the p robab i l i t y of a certain runoff intensi ty i s more related to
the probab i l i t y of the soi l being a t a certain saturat ion than the r a i n f a l l
intensi ty.
How does urbanizat ion affect the argument? I n fact i t counters the
above ideas. The more the na tura l surface cover i s replaced by imper-
meable surfaces the more runof f becomes a direct response function to
r a i n f a l l . I n the l im i t for 100% runoff , soi l does not feature and the
recurrence in te rva l of runof f i s equal to that of the storm causing i t .
EXAMPLE :
CALCULATION OF PEAK RUNOFF FOR VARIOUS CONDITIONS
The effect of urbanizat ion on runoff can be i l l us t ra ted with the
fol lowing example. I n pa r t i cu la r i t w i l l be seen that the peak flows
174
increase ( a s w e l l as the volume of r u n o f f ) .
F i g . 9.1. The ef fect o f u r b a n i z a t i o n on r u n o f f
i ) V i r g i n C a t c h m e n t I
The s imp le r e c t a n g u l a r catchment dep ic ted in F i g . 9.2 w i l l be
s tud ied to i n d i c a t e the v a r i o u s e f fec ts of u r b a n development on the s torm
r u n o f f peak. The ef fects computed a r e reduced roughness, impermeable
cover a n d channe l i za t i on . A constant f requency, u n i f o r m r a i n f a l I i n t e n s i t y
d u r a r i o n r e l a t i o n s h i p as fo l lows i s used:
i (mm/h ) = a
(0.24+td)
where t d i s the storm d u r a t i o n in hours.
T h i s i s t y p i c a l o f a temperate a rea , a n d the v a l u e of ' a ' f o r t h i s r e g i o n
i s est imated to be 70 mm/h f o r storms w i t h a 20 y e a r recu r rence i n t e r v a l
o f exceedance.
The catchment i s assumed to h a v e a constant s lope o f 0.01 a n d
i n i t i a l l y the cove r i s grass. The rep resen ta t i ve M a n n i n g roughness f o r
o v e r l a n d f low i s est imated to be 0.1. The i n i t i a l a b s t r a c t i o n ( s u r f a c e
re ten t i on a n d mo is tu re d e f i c i t make up) i s 30 mm a n d subsequent mean
i n f i l t r a t i o n r a t e ov,er a storm, 10 mm/hr.
Thus c1 = J S /n = JO.01 /0.1 = 1.0
I n f i l t r a t i o n r a t i o F = f / a = 10/70 = 0.143
I n i t i a l loss r a t i o U = u /a = 30/70 = 0.429
Leng th fac to r in S I U n i t s LF = L/36aa2'3 = 2 0 0 0 / 3 6 x 1 ~ 7 0 ~ ~ = 3.27
175
C a s e 1 , 2 a n d 3
F ig . 9.2 Simple catchment a n a l y z e d
From F ig . 3.6 ( f o r U = 0.40) r e a d e q u i l i b r i u m te > 4h ( o f f the graph) b u t
the peak r u n o f f f ac to r f o r t h i s F i s QF = 0.23 wh ich corresponds to a
storm d u r a t i o n of t d = 2.2h. The peak r u n o f f r a t e i s
Qp = 0.23Baa5"/10
same storm d u r a t i o n i s
= 17.6m3/s A i =
so the r a t i o n a l coef f ic ient C = 2.74/17.6 = 0.16.
= 0.23x1000x1x70 5'3/105 = 2.74m3/s
The to ta l p r e c i p i t a t i o n r a t e o v e r the catchment of a rea A f o r the
70 x 1000 x 2000
(0.21++2.2)-*~ x 3600 x 1000
Note however tha t the f u l l catchment i s not c o n t r i b u t i n g a t the
time of peak r u n o f f f o r the des ign storm, so C does not o n l y represent
the reduc t i on in r u n o f f due to losses, i t also accounts f o r o n l y p a r t o f
the catchment c o n t r i b u t i n g . The r u n o f f f o r the f u l l catchment would be
less as the storm d u r a t i o n would be longer than 2.2 h so the i n t e n s i t y
would be less a n d the losses r e l a t i v e l y h ighe r .
i i ) Reduction in Infiltration
I f the i n f i l t r a t i o n a n d i n i t i a l abs t rac t i ons a r e reduced b y u r b a n i z -
a t ion, the peak r u n o f f increases. The cons t ruc t i on of b u i l d i n g s a n d roads
cou ld reduce i n f i l t r a t i o n r a t e to 7 mm/h a n d in i t ia l abs t rac t i on to 14
mm. For F = 7/70 = 0.1 a n d U = 14/70 = 0.20 (F ig . 3.5) then f o r LF
= 3.27 as f o r case ( i ) , the t ime to e q u i l i b r i u m i s o f f t he c h a r t b u t the
c r i t i c a l storm has a d u r a t i o n of 2.2 hou rs a n d the corresponding peak
f low i s
176
= 0.44 x 1000 x 1 .O x 70 / l o 5 = 5.24m3/s QP
The corresponding runof f coefficient C works out to be 0.30
C a s e 4
Fig. 9.3 Catchment wi th channel
i i i ) Effect of Reduced Roughness due to Paving
With the construction of roads, pavements and bu i l d ing the na tu ra l
retardat ion of the surface runof f i s el iminated and concentrat ion time
reduces. That is, the system response i s faster and as a resu l t shorter,
sharper showers a re the worst from the point of view of runoff peak.
For the sample catchment the effect ive Manning roughness could qu i te
easi ly be reduced to 0.03. Then a = 3.33 and LF = 0.98. The time to
equ i l ib r ium would therefore be 3h bu t the peak intensi ty storm has a
durat ion of 2.2h as before. I n th i s case extent of the storm over the
catchment i s greater however, and the peak runof f i s
Q = 0.23 x 1000 x 3.33 x 705’3/10 = 9.12m3/s
The corresponding increase i n C i s from 0.16 to 0.52 an appreciable
increase i f i t i s borne i n mind th i s i s only due to reduced roughness
and does not account for reduced in f i l t r a t i on . I t w i l l be noted that the
effect of reducing roughness i s even greater than decreasing i n f i I t rat ion
for t h i s case. The same effect i s magnif ied in the fo l lowing example.
P
i v ) Effect of Canalization
The effect of a stream down the centre of the catchment i s i l l us -
t rated in the fo l lowing example. The same surface roughness ( n = 0.1)
177
a n d p e r m e a b i l i t y ( f = 10 rnm/h, u = 30 mm) a s f o r case ( i ) a r e assumed.
The o v e r l a n d f l ow cross s lope i s t aken a s 0.04 a n d 0.01 f o r a 8 m w ide
channel down the catchment. The d imension less h y d r o g r a p h s in Chap te r
s i x a r e used a g a i n . 2L u.6 b a u.6
The stream catchment r a t i o G = (2 ) 2 ba s 2Lo
( 2 x 2000 o.6 8 x 2 o.6 = o.50 - 8 x 1 2 x 500
By t r i a l , guess storm d u r a t i o n r e s u l t i n g in peak r u n o f f of 1.5h, t hen
70 i = __ (o.24 1.5).m - 10 = 42.7-10 = 32.7 mm/h e
ted = t d - t
F = 10/32.7 = 0.31
= 1.5 - 30/42.7 = 0.80h
3/5
2860s = 0.80h L o ) l / m = 500
m-1 2 x ( 32.7/3600000 ) 2'3 tCO = i 011
TD = (5 /3) ted/ tco = (5/3)0.8/0.8 = 1.67
Therefore td = t
I n t e r p o l a t i n g F i g s . 6.10 a n d 6.11 the peak fac to r Q = 0.85
Peak f low Q
Ra t iona l coe f f i c i en t C = 15.4/(42.7x2/3.6) = 0.65
+ t = 0.8 + 30/42.7 = 1.50 h wh ich agrees w i t h guess ed u
= QAie = 0 . 8 5 ~ 2 ~ 1 0 6 x32.7/3.6x106 = 15.4m3/s
v ) Combined reduced roughness and reduced losses
I f roughness is reduced b y p a v i n g to 0.03 then a = 3.33 a n d L F
= 0.98 as f o r case ( i i i ) . The reduced loss fac to rs become F = 0.1 a n d
U = 0.2 as f o r case ( i i ) . From F i g . 3.5 t = 1.7 h a n d the co r respond ing
P F = 0.43.
Hence the peak f l ow Q = 0.43 x 1000 x 3.33 x 70 5 3 = 17.0m3/s. The
r a i n f a l l r a t e f o r a storm of t h i s d u r a t i o n i-s
70 x 1000 x 2000 (0.24 + 1 . 7 ) 0 8 9 x 36000 x 1000 = 21.6 m 3 / s so C = 0.79.
The r e l a t i v e ef fect of each v a r i a b l e on peak r u n o f f c a n be compared
w i t h the a i d of Tab le 9.1. The ef fect o f r e d u c i n g i n f i l t r a t i o n 30% a n d
i n i t i a l a b s t r a c t i o n 40% i s to doub le the peak r u n o f f . The c r i t i c a l s torm
d u r a t i o n was not a f fected b u t the e f fec t i ve a r e a c o n t r i b u t i n g increased
s l i g h t l y . The ef fect of r e d u c i n g su r face roughness i s even more
remarkab le however. Even m a i n t a i n i n g the same losses ( b o t h i n i t i a l a n d
a b s t r a c t i o n a n d i n f i l t r a t i o n ) a s f o r the n a t u r a l catchment the r u n o f f peak
178
increased by a factor of 4. The area cont r ibu t ing increased noteably
although the c r i t i c a l storm durat ion was not affected. Reducing roughness
even more would not necessarily increase runoff much as prac t ica l l y the
en t i re catchment contr ibutes for case ( i i i ) whereas the area cont r ibu t ing
i n case ( i ) was much less. Only for case ( v ) wi th reduced roughness
and losses i s the concentration time equal to the c r i t i c a l storm durat ion.
TABLE 9.1 Showing effect of d i f ferent surface conf igurat ions on peak runoff from a 2000m long by lOOOm wide catchment.
So = 0.01, i = 70 rnm/h/(0.24h + t ) 0.89
d
CASE n f mm/h u mm t h t d h i mm/h Q m'/s C __ - ~ - ~ _ _ ~ ~ ~ P
i ) V i rg in 0.1 10 30 5 2.2 36.7 2.74 0.16 catchment
i i ) Reduced 0.1 7 14 4 2.2 36.7 5.24 0.30 losses
i i i ) Reduced 0.03 10 30 3 2.2 36.7 9.12 0.52 roughness
i v ) Canaliz- 0.1 10 30 0.8 1.5 42.7 15.4 0.65 at ion (stream width 3m)
V ) Reduced 0.03 7 14 1.7 1.7 38.8 17.0 0.79 losses and roughness
The effect of canal izat ion is somewhat s imi la r to reducing roughness
- water velocit ies, and concentration rates, are faster. This i s due to
the greater depth i n channels ( Q = 6 Jsy"/n). Consequently a greater
area contr ibutes to the peak.
Not much sense can be made out of comparing the resu l t ing ra t iona l
coefficients ( r a t i o of peak runoff ra te to r a i n f a l l r a te times catchment
a rea ) . That i s because the time of concentration for each case is
di f ferent due to d i f f e r i ng roughness, r a i n f a l l r a te etc. I n any case i t
i s i r re levant when i t comes to c r i t i c a l storm durat ion which is shorter
than the time to equi l ibr ium.
179
DETENTION STORAGE
Although the kinemat ic equations as presented previously cannot
accommodate reservoir storage they may be rearranged to i I lustrate the
storage components in them. The St. Venant equations w h i c h include terms
for storage when water surface i s not pa ra l l e l to the bed, a re
aA - aa a t ax -- ( 9 . 1 1
( 9 . 2 )
The f i r s t equation i s the cont inu i ty equation and the second the
so-called dynamic equation. The f i r s t equation does not give the total
storage i n the reach, i t represents the ra te of change i n cross sectional
area of flow as a function of inf low and outflow. The second equation
contains more about the d is t r ibu t ion of storage. The last two terms
represent the wedge component of storage, which are absent in the
kinematic equations. The kinematic equations therefore treat storage as
a prism, wi th storage in blocks and no allowance for dif ference i n slope
between bed and water surface i s made. Since the second equation i s
replaced by a f r i c t i on equation and So = Sf i n the kinematic equations,
only the f i r s t equation i n the case of the kinematic equations can be
used to calculate storage changes.
The cont inui ty equation may be wr i t ten as
0-1 A z - A i - + - = o A x At (9 .3 )
where 0 i s outf low, I i s inf low over a reach of length A x , and A t and
A2 are the cross sectional areas before and a f te r A t respectively.
I f 0 = ( 0 + 0 2 ) / 2 and I = ( I , + 12) /2 and AAx is replaced by S, the
storage which i s a function of A l and Ao, which i n tu rn a re functions
of f lowrate, e.g. S = X I + ( l - X ) O , then equation (9 .3 ) becomes the one
frequently used for open channel rout ing,
1
0 2 = c , I I + c 2 1 2 + caO1 ( 9 . 4 )
where c I , c and cg are functions of A x and A t. The la t te r equation
i s referred to as Muskingum's equation used i n rou t ing floods along
channels. i f X = 0 the rou t ing equation corresponds to level pool o r
reservoir rout ing. The more general equation w i th X = 1/2 represents
a 4-point numerical solut ion of the cont inui ty equation as employed i n
kinemat i c models (Brakensiek, 1967).
180
CHANNEL STORAGE
Channel storage performs a s imi la r function to pond storage i n
re ta rd ing flow, and there are many analogies which can be drawn
between the two. Channel storage i s a function of f r i c t ion resistance
and channel shape and can be control led in var ious ways.
The form of f r i c t i on equation, as well as the f r i c t ion factor, affect
the reaction speed of a catchment and the volume stored on the catchment.
The excess r a i n stored on the catchment, whether in channels or on
planes, i s a form of detention storage, and as such, affects the con-
centrat ion time and consequently the peak ra te of runoff . Some f r i c t ion
formulae used i n stormwater drainage pract ice are l is ted below.
S . I . un i ts Engl ish un i ts
Darcy-Weisbach Q = (8/f)1bA(RSg)1fi Q = (8 / f ) lR A(RSg)'n
Chezy Q = 0.55CA(RS)'" Q = CA(RS)'/*
Manning Q = AR2/3S1/2/n Q = 1 .486AR 2'3S v2 / n
Str ickler Q = 7.7A(R/k) (RSg)V2Q = 7.7A(R/k)v6 (RSg)Ih
R i s the hyd rau l i c rad ius A/P where A i s the area of flow and P
9.5)
9.6)
9.7)
9.8)
the
wetted perimeter. R can be approximated by depth y for wide rectangular
channels. S i s the energy gradient, f i s the f r i c t ion factor and k i s
a l inear measure of roughness analogous to the Nikuradse roughness.
Both the roughness. coefficient CY and the exponent m o f R o r y i n
the general flow equation (9.11) affect the peak flow o f f a catchment.
This i s la rge ly due to the at tenuat ing effect of f r i c t ion resu l t ing i n a
larger time to equi l ibr ium. A r a i n f a l I excess intensi ty-durat ion relat ion-
ship i s required to evaluate the effect of each coefficient on peak runoff
ra te and maximum catchment storage. The fol lowing expression for excess
ra in fa l I intensi ty i s assumed:
(9.9)
I n th i s equation i t i s customary to express i and a in mm/h or inches
per hour and b and td in hours where td i s the storm durat ion assumed
equal to time of concentration tc for maximum peak runoff of a simple
ca tchrnent.
Start ing wi th the kinemat i c equation for cont inui ty
a v . 3 at ax = 'e (9.10)
181
and a general flow resistance equation
q = aym (9.11)
then i t may be shown that tc = (L /a iem- ' ) ' /m where q i s the runoff r a t e
per u n i t width of the catchment and y i s the
limb of the hydrograph i s given by the equation
q = CY ( i t ) m
1
1 2 ' 0 p . , , , , , , , . . , , , , ,
flow depth. The r i s i n g
(9.12)
m Fig. 9.4 Hydrograph shapes for di f ferent values of m in q = a y
and another expression may be der ived from the f a l l i n g limb (see Chapter
2 ) . I n Fig. 9.4 are plotted dimensionless hydrographs to i l l us t ra te the
effect of m on the shape, of the hydrograph. The graphs are rendered
dimensionless by p lo t t ing Q = q/ ieL against T = t / tc. m i s used as a
parameter. Thus m = 1/2 represents closed conduit or o r i f i ce f low, m
= 1 represents a deep vert ical sided channel, m = 3/2 represents a wide
rectangular channel according to Darcy or a rectangular weir, m = 5/3
represents a wide rectangular channel i f Manning's equation i s employed,
and rn = 5/2 represents a t r i a n g u l a r . w e i r . T h e graphs immediately
indicate the effect of m on catchment detention storage since the area
under the graph represents storage.
The smaller m, the greater storage. Thus provided storage i s
economical by th ro t t l i ng outflow one may increase storage and increase
concentration time thereby reducing discharge r a t e (which i s not immed-
iately apparent from these graphs as they are plot ted re la t i ve to excess
ra in fa l I in tens i ty ) . I n pract ice the concentration time increases the
greater the storage so that the lower intensi ty storms become the design
storms. Th is has a compound effect in reducing flow rates since total
volume of losses increases and i t i s possible that the en t i re catchment
w i l l not contr ibute at the peak flow time.
182
A general solut ion of peak flow and storage i n terms of intensi ty-
durat ion relat ionships i s der ived below. Solving (9.9) wi th td = tC for
maximum ra te of runoff per u n i t area and general iz ing by d i v i d i n g by
a.
1 -_ 1 L/a(a/3600000)m-1 I
I c + I P 3600 ( ie/a ) '- ' lm
(9.13)
m-1 The term L / u a is referred to as the length factor. The constants are
introduced for a i n mm/h, and time of concentration i n hour uni ts. The
maximum peak flow factor ie/a i s plot ted against length factor i n Fig.
9.5, since i t i s not easy to solve (9.13) d i rec t l y for i /a
i e /a and s/a
max
c=o. 9 b=0.25h 1 2
s * . . . I I I , * ] I n 1 I 1 I . . . jC rn- 1 L 1.i a 00
, / f )
I0
F ig . 9.5 Peak flow and storage versus length factor
An expression for the corresponding catchment storage is der ived
below. At equ i l ib r ium the flow per un i t width a t a distance x down the
catchment i s
183
q = i x e
= ay m
therefore y = ( i e x / a ) l I m
Integrat ing y w i th respect to x y ie lds the total volume on the catchment
o r in terms of the
m+l a
average depth of storage s = V / L
1 /m L 1/m , ( ) - - - a ( a/3600000) m-' 3600
(9.14)
where s i s i n mm, and i and a a re i n mm/h. s/a i s also plotted against
length factor i n Fig. 9.5. I t w i l l be obzerved that average storage depth
does not increase in proport ion to L/aam-'. I n fact the ra te of increase
reduces beyond L/aam-' = 50, and the ra te of reduction i n peak flow
ie/a also decreases beyond the f igure, ind ica t ing reducing advantage
in increasing channel length or roughness ( 01 = K 1 J ( S ) / n ) . Since total
channel cost i s a direct function of storage capaci ty i t would appear
to be an optimum at some intermediate value of L/aam-' i f there i s a
cost associated with peak discharge e.g. cu lver ts or f looding downstream
(see Fig.
minimum
c o s t $
Fig. 9.6
9.6).
Optimum catchment storage volume.
Note that i n f i l t r a t i o n a f te r the r a i n f a l l stops, i s neglected i n the
above analysis. Inclusion of that effect would lower the ie/a and s/a
l ines to the r i gh t , imply ing a la rger L/aam-' i s best. The model provides
an indicat ion of total storage i n the system. The location (and volume)
184
o f s t o r a g e c o u l d b e f u r t h e r o p t i m i z e d u s i n g d y n a m i c p r o g r a m m i n g methods
o r b y d e t a i l e d m o d e l l i n g . I t s h o u l d b e f o u n d g e n e r a l l y t h a t i t i s most
economical to p r o v i d e p o n d s t o r a g e ( m = 1 /2 ) a t t he o u t l e t , whereas
c h a n n e l o r ca tchmen t s t o r a g e ( m = 5/3) i s most economica l a t t h e h e a d
o f the system.
K I NEMATIC EQUAT I ONS FOR CLOSED CONDU I T SYSTEMS
I f the open c h a n n e l k i n e m a t i c e q u a t i o n s a r e a p p l i e d to c losed
c o n d u i t f l o w the p r o b l e m becomes a s t e a d y s t a t e f l o w one s i n c e f l o w
r a t e s become independen t o f c r o s s sec t i on . T h i s i s p r o v i d e d t h e c o n d u i t s
r e m a i n f u l I a n d t h e r e a r e no s t o r a g e ponds a t nodes j o i n i n g c o n d u i t s .
I f one p e r m i t s s t o r a g e v a r i a t i o n a t nodes o n e h a s the r e s e r v o i r - p i p e
s i t u a t i o n encoun te red in w a t e r s u p p l y w h i c h i s o f t e n a n a l y z e d e m p l o y i n g
pseudo-s teady f l o w equa t ions .
F i g . 9.7 I n p u t - o u t p u t node s t o r a g e
The c o n t i n u i t y e q u a t i o n becomes (see F i g . 9.7)
d h i (9.15) d t
(Q i+ , -Q i ) - qi + Ai- = O
where the r e s e r v o i r s u r f a c e a r e a A . r e p l a c e s B d x i n t h e open c h a n n e l
c o n t i n u i t y e q u a t i o n where B i s t h e ca tchmen t w i d t h . q i s the r e s e r v o i r
i n f l o w here . The d y n a m i c e q u a t i o n i s r e p l a c e d b y
Q. = aAm (9 .16a)
where A i s t he ( c o n s t a n t ) c o n d u i t c r o s s s e c t i o n a l a r e a . S ince the k i n e -
m a t i c e q u a t i o n s o m i t t he dependency of Q o n h e a d d i f f e r e n c e h, t h e l a t t e r
e q u a t i o n assumes the h e a d g r a d i e n t a l o n g t h e p i p e e q u a l s t h e p i p e
g r a d i e n t , i.e. f r e e - s u r f a c e j u s t f u l l f l o w . S ince A i s a c o n s t a n t i t i s
r e l a t i v e l y easy to r e p l a c e the l a s t e q u a t i o n b y one of the f o r m
185
Q . = 01 Ah.m (9.16b)
This equation is app l i cab le to free discharge from an or i f i ce o r over a
weir. One more app l icab le to condui t flow would be
Q = aA(hi- l -h i ) ( 9 . 1 6 ~ )
Any one of the above three equations could be appl icable in storm-
water drainage. For channel or over land flow (9.16a) appl ies, for
complete storage control (9.16b) appl ies and for closed conduit control
( 9 . 1 6 ~ ) i s appl icable. The la t te r form of equation has i n fact been
employed in water re t i cu la t ion pipe network analysis. I t can b e appl ied i n
storm drainage to closed systems (not of great interest i n stormwater
management pract ice) or to pipe-reservoir problems. Surface detention and
a r t i f i c i a l detention storage ponds can be handled i n an overa l l flow
balance employing the closed conduit kinematic method. I t should be
noted that the numerical instabi I i t y problems associated with solut ion of
the open channel kinematic equations are absent. Time steps can be much
la rger than for open channel kinemat ic model I ing. Storage f luctuat ions
may be computed i n steps and the effect of changes i n pond water levels
on flows i n conduits can be accounted for.
m
One possible appl icat ion of such a program i s to an inter-connected
pond system wi th reversible flows i n conduits. Overload from one pond
can be forced back to another pond. Such si tuat ions can read i l y a r ise
from spat ia l l y va r iab le storms and possibly for t rave l l i ng storms.
Off-channel storage can also be accounted for. Such ponds have
the advantage that water level var ia t ions are not as marked as the head
var iat ions in the d ra in pipes (which may in fact be surcharged). This
i s due to the revers ib le head loss between the main conduit and the pond.
3 3 1.4m /s
1. Om
Fig. 9.8 Conduit and storage storm d ra in network.
The s imp l i f ied layout in F ig . 9.8 was analyzed employing the
accompanying k inematic closed conduit continuous simulat ion program.
Input and output a re appended to i l l us t ra te the s imp l ic i t y i n th i s type of
anlysis. Flow reversal , pond level var ia t ions and the large attenuation i n
peak flow w i l l be observed due to the ponds (from 5.6m3/s down to
1.5m3/s). By ad jus t ing ind iv idua l pond areas and conduit sizes an
optimum design could be achieved for any design storm input. A
sensi t iv i ty ana lys is for a l te rna t ive storms such as di f ferent storm
durations or ones with spat ia l v a r i a b i l i t y would then be performed.
COMPUTER PROGRAM TO SIMULATE RESERVOIR LEVEL V A R I A T I O N S I N A PIPE
NETWORK
Closed conduit drainage networks can as explained above, be used to
ameliorate peak flows by d i rec t ing water into storage. Flow can be i n
ei ther direct ion and depends on the dif ference i n water levels a t the two
ends of the conduit, not on the conduit gradient as for open channels.
Apart from this, the pr inc ip les a re the same as for open channel kinematic
flow. That i s steady state condit ions (head loss/flow equations) are used
together wi th the cont inui ty equations. The accompanying computer program
wri t ten in HP 85 'BASIC' w i l l simulate the var ia t ions in water level i n
reservoirs in add i t ion to performing a network flow balance.
The program is based on the l i nea r node method (Stephenson, 1984)
network ana lys is wi th an add i t iona l var iab le , area of reservoir for each
' f i xed head' o r , i n th is case, ' reservoir type' node. I f the simulat ion
durat ion T 4 i n hours and time increment T5 are input , for example 24 and
1 , then the heads a t each node and water level i n each reservoir w i l l be
pr in ted out every hour. The actual network i terat ions each time in te rva l
af ter the f i r s t should be minimal since the network flows are balanced i n
the f i r s t i terat ion and only unbalanced due to reservoir level changes
which w i l l have to be corrected a t subsequent time intervals. Although
drawoffs a re time-fixed in the present program, they could be al tered a t
pauses i n the runn ing or inserted i n equation form.
The output, namely level var ia t ions , could be used to estimate
required reservoir depths (using t r i a l reservoir surface areas) and i n fact
to see at which reservoir locations the storage i s most required. Data
requirements a re s im i la r to the ana lys is program w i th the fo l lowing
addit ions.
187
I n the f i r s t da ta l ine a f te r the name, the simulat ion durat ion and
increment in hours i s added at the end of the l ine. I n the pipe data,
the f i r s t pipes should be from the var ious reservoirs wi th the surface
areas of the up-stream reservoirs i n square metres given at the end of
the pipe da ta lines. I n order to d isp lay the reservoir levels i n the
biggest reservoir i t i s necessary to have a supply pipe from a pseudo
f i xed head, very large, reservoir to represent a pumped supply feeding
into the actual biggest level reservoir i n the d is t r ibu t ion system.
The selection of 'upper ' and ' lower ' nodes for any pipe, numbered
1 and 2 i s somewhat a r b i t r a r y . I f the incorrect flow direct ion i s
assumed, a negative flow number w i l l appear i n the answers thus ind i -
ca t ing the flow direct ion i s from node 2 to node 1 as specified.
When da ta i s put in , the order of pipes i s to a l imi ted extent
a r b i t r a r y , but the 'node 1 ' of any pipe should have been defined as
a 'node 2 ' i n some previous pipe. Th is does not apply to the f i r s t p ipe
which w i l l o r ig ina te at a reservoir. The order of pipes enables da ta
on successive nodes, i .e. i n i t i a l estimates of heads and f lows, to proceed
down the system from previously defined nodes.
Node numbering i s also open to the user except the reservoir- type
nodes (w i th specif ied i n i t i a l water levels) should be numbered f i r s t ,
from 1 to J3.
There i s scopc for set t ing a l l Darcy f r i c t i on factors the same to
minimize data requirements, or to va ry each factor. Note i f o ld da ta
in f i les i s used those f r i c t ion factors, not the 'common' factor, i s used
even i f a common factor i s fed in. To p r i n t out ' o ld da ta ' i n f i l e , i t
i s necessary to go into revis ion mode ( 2 ) of p ipe data input. To get
out of revis ion mode, type 0 for p ipe number to be revised.
Par t of the da ta i s read interact ively on the keyboard. The f i r s t
l ines of data (name, durat ion, no. of nodes, reservoir data and pump
data) i s typed for each run. The pipe and node data can
or retr ieved from a f i l e or ammended i n a f i le .
The time increment between i terat ions for simulatiot
be small enough to avoid large var ia t ions i n water levels
between i terat ions. The reservoir surface area and flows
this.
Addit ional pipes can be added in ed i t ( 2 ) mode and
stored in the da ta f i le . Pipes can only be removed by
b e typed i n
mode must
n reservoirs
w i l l control
d i l l then be
l im i t i ng the
number of p ipes i n the i n i t i a l l ines to el iminate those not required a t
the end. The other way i s to put a very small diameter for a pipe to
be removed from the network. New nodes or reservoirs can be added by
re typ ing in data.
188
When reading i n i n i t i a l da ta however, no more than the number
of pipes i n the da ta f i l e should be specified, The number w i l l auto-
mat ical ly be increased when more data l ines are added.
The last specif icat ion of any drawoff i s retained i f a node happens
to be specif ied more than once in input. One should also make sure each
node i s specif ied (as a N2) at least once to define i t s drawoff.
Data Input
Each l ine may contain more than one u n i t of data separated b y
commas.
L ine 4
L ine 5
Lines 6 . . .
L ine 7
Lines 8.. .
L ine 9
L ine 10
L ine 1 1
Lines 12...
L ine 1 Name of network (and r u n no.)
L ine 2 Analysis ( 0 ) or simulat ion ( 1 ) - type 0 o r 1
L ine 3 Drawoff durat ion i n minutes, thus i f drawoff i s over
8 hours, type 480. Simulation durat ion mins. I f 24
hours, type 1440, Time increment DT, mins. Suggest
30 - 120.
Constant ( 0 ) or various ( 1 ) Darcy f ’ s - type 0 or 1
No. pipes,
No. nodes ( total inc lud ing reservoirs)
No. reservoir type nodes.
(one for each reservoir node i n successive order )
I n i t i a l water level, rn
Surface area of reservoir , mz
Old ( 0 ) or new ( 1 ) o r revised pipe data ( 2 ) ;
type 0 , l or 2.
(one for each pipe i n new pipe da ta)
Node 1 no.
Node 2 no.
Pipe length rn
Pipe inside d ia . , rn
Drawoff at node 2, m3/s
(Darcy f r i c t ion factor i f l i ne 4 i s 1 )
I f l ine 7 i s 2 , w i l l ask pipe no. for revision.
Pipe da ta for new pipes as for L ine 8 inc luding Darcy
f r i c t ion factor.
No. of pumps or pressure reducing valves (one per
p ipe ) .
Pipe no. i n which pump or PRV i s instal led, pumping
head or PRV head loss ( - ) in rn.
189
L i s t of Symbols in Program
1 = a n a l y s i s , 2 = s i m u l a t i o n
0 = c o n s t a n t f , 1 = v a r y i n g D a r c y f .
0 = o l d d a t a , 1 = new d a t a , 2 = r e v i s e o l d d a t a
0 = n o d a t a l i s t i n g r e q u i r e d , 1 = r e q u i r e d
h e a d Ios s/Q I Q 1 ZH f o r each SOR
C A F
p i p e d iamete r ( m )
o ld v a l u e o f H ( I )
D a r c y f r i c t i o n f a c t o r e.g. 0.012 l a r g e dia. c l e a n p i p e
0.03 s m a l l t u b e r c u l a t e d p i p e
common D a r c y f a c t o r
h e a d a t node o r j u n c t i o n I
node c o u n t e r
n u m b e r of nodes
u p p e r node number o f p i p e
lower node n u m b e r on p i p e
n u m b e r of r e s e r v o i r t y p e nodes
i te ra t i on
p i p e c o u n t e r
node c o u n t e r
p i p e c o u n t e r
number o f c o n n e c t i n g p i p e s
M 2 ( L , M l ( L ) ) p i p e n u m b e r c o n n e c t i n g
N$ a lphanc rmer i c name of system, up to 12 c h a r a c t e r s
NO max imum n u m b e r m a i n i t e r a t i o n s p e r m i t t e d e.g. 4: t 5
N1 max imum number SOR (success i ve o v e r - r e l a x a t ion o f
s imu l taneous e q u a t i o n s ) i t e r a t i o n s e.g. 4T t 10
N2 c o u n t e r f o r m a i n i t e r a t i o n s
N3 c o u n t e r f o r SOR i t e r a t i o n s
P n u m b e r of p i p e s
P1 numer o f p i p e s and P R V ' s ( 1 p e r p i p e max imum)
Q ( K ) f l ow in p i p e
Q1 d r a w o f f m3 / s
Q2( I ) d r a w o f f m3 /s
R ( k ) pump h e a d i n m, ( o r p r e s s u r e r e d u c i n g v a l v e h e a d in m
i f n e g a t i v e )
S g n2 / 8
190
S ( 2 ) I C K i j
s 3 CHj
S4( I CK i jH j
55 o l d Q ( K ) f o r a v e r a g i n g
T3 d r a w o f f d u r a t i o n , m i n s e .g . 8 h x 60 = 480
T4 s i m u l a t i o n d u r a t i o n , m i n s e.g. 24 x 60 = 1440
T 5 t ime inc remen t i n s i m u l a t i o n , m i n s e.g. 60
TO t o l e r a n c e on h e a d i n m e . g . 0.0001
T1 t o l e r a n c e on SOR in m e.g. 0.01
W-SOR f a c t o r e.g. 1.3 (1-2)
X ( K ) p i p e l e n g t h m
REFERENCES
B r a k e n s i e k , D . L . , 1967. K i n e m a t i c f l o o d r o u t i n g . T r a n s Am. SOC. A g r i c . E n g r s . lO(3) p 340-343.
Co lye r , P.J., 1982. The v a r i a t i o n o f r a i n f a l l o v e r an urban ca tchmen t . Proc. 2nd I n t I. Cong. U r b a n Storm D r a i n a g e . U n i v e r s i t y o f I I l i n o i s .
H u f f , F.A. and Changnon , S.A., 1972. CI i m a t o l o g i c a l assessment o f urban e f f e c t s o n p r e c i p i t a t i o n a t St. L o u i s . J. A p p l . Me teo ro logy , 1 1 , p 823-842.
Stephenson, D . , 1984. K i n e m a t i c a n a l y s i s o f d e t e n t i o n s to rage . Proc . S torm Water Management and Qua1 i t y u s e r s Group Mee t ing , USEPA, D e t r o i t .
Stephensdn, D . , 1984. P i p e f l o w A n a l y s i s , E l s e v i e r , Amsterdam, 204 p p . S u t h e r l a n d , F.R., 1983. An i m p r o v e d r a i n f a l I i n t e n s i t y d i s t r i b u t i o n f o r
h y d r o g r a p h s y n t h e s i s . Water Systems Research Programme, Repor t 1/1983, U n i v e r s i t y o f t he W i t w a t e r s r a n d .
191
Program Listing
16 ! NETSIFI KINEMHTJC~CONTIN SI MULN OF NETWORKS WITH STORAG E
26 ASSIGN# 1.~0 "DF~TNET* ! CREA TE"DATNE1 J 166988
,S2(56),S4(56>,F(56>
), R(96) J Ill (58) J M2<58,5)
38 DIN C(5b,,Q<?6~,H<5b).Q2(56)
46 DIM J1<5b), JE(56>,D(58), X(56
58 DISP "NAPlE OF NETYORK'; 66 INPUT NS 76 DISP "ANALYSIS OR SIPlULATION
( 112) " i 88 INPUT ~i 56 IF' A1=2 THEN 146
l 6 b T3=1 116 T4=1 126 T5=1 136 GOTO 166 146 DISP 'DRAWOFF DURATIONnin,SI
16b Q2<1>=6
M DURNm in, DTm in" i 158 INPUT T3,T4,T5
176 T3=T3$6b 186 T4=T4X60 196 T5=T5*6b 268 DISP 'VARYING fs<B/l)"; 216 INPUT A2 226 IF R2=8 THEN 2C;n . - . . - - - 238 DISP-~NPIPES, NODES, NRESS"; 246 INPUT P,J,J3 258 GOTO 286
276 INPUTUP, J, J3,Fl 286-DISP INITL WATER LEVELm,SUR
256 FOR L=l TO J3 386 DISP L; 316 INPUT H(L),A(L) 326 NEXT L
FACE AREA m2";
336 346 356
366 376 388 356
468 416 426 436 446
458
6=5.8 S=3.14155*2tG/S DISP "OLD OR NEW OR REVISE P IPEOATA(6/1/2) " i - INPUT A4 IF A4=1 THEN 436 f NElJ DATA FOR K=l TO P ! OLD DATA
,D(K>,Q2(JZ<K)>,Ftk) NEXT K IF H4=2 THEN 586 GOTO 746 IF H2=6 THEN 466 DISP "NODE1, NODE2,Lm,Dm,DRAW OFFEm3/s,DARCYf " j
GOTO 476
READ# 1,K J Jl(K),JP(K),XCK>
466 DISP " N ~ D E 1 , N O l l E 2 ~ L m , D m ~ D R A W
47B FOR K=l TO P ! P I P E Dt3TH 0 F 2 m3 / s '' ;
546 556
566 578 586 598 668 616 614 616 628
636
646
122 \ J C ( E r' I =O 1 PRINT# 1,K i JI(K>,JE(K>,X(K
NEXT K GOTO 748 FOR Kl=l TO 168 DISP "REVISE PIPE NO."; INPUT K IF K=b THEN 666 IF K<=P THEN 626 P=K DISP "NODEl.HODE2,Lm,Dn,DRA~ DFF~NIJ/s,DARCY~"~ INPUT Jl(K),J2(K),X(K),D(K>, Q2( J2(K) ), F < K ) PRINT# 1,K ;'Jl<K),JE(K>,X<K
),D<K),Q2<J2~K?),F(K>
),D(K),QZ(J2(K)),F<K> 653 NEXT F l 666 DISP DATALIST REQD (b/ 67b INPUT A 5 686 IF R5=! THEN 748 696 PRINT NODE1 N2 Xra Dm
/s f 788 FOR ~ = i TO P 718 PRINT USING 736 i Jl(K>
),X<K),D<K),Q2<J2<Kj>,F 728 NEXT K 7 7 c * V Y n r r nnr.n nnnn nnnnn n nnn
758 Q(K)=3.14159*D(K>*2/4 766 R<K)=6 776 C<K)=S*D(K>*S/F(KZ/X(K) ! I/
786 IF J2<K)<=J3 THEN 866 K
798 H ( J 2 ( K > ) = H ( J l < K ) ) - l / C < K ) * Q ( K
)"i
Qm3
J2(K K)
) *2 866 NEXT K 816 DISP "NO.PUtlPS/PRVs'i 826 INPUT P1 83b FOR P;=l TO P1 846 DISP PIPEN,+HEADm Nl-N2"iP2
856 866 876 886 896 ~~
566 918 926 938
INPUT K,R(K) NEXT P2 FOR L=l TO J M1 (L)=6 FOR H=l TO P IF Jl<M)=L THEN 526 IF JE<M)<>L THEN 546 Hl(L)=Ml(L>+l HE(L,Hl<L)>=H
946 NEXT M 956 NEXT L '166 W = l . 3 ! SOR FHCTOR 376 T6=.6B61 ! TOLERANCE n Y8B Tl=.Bl ! S O R TUL rn 996 NB=SQH<J)+5 ! ITNS PIPES lB6O Nl=SQR(Jj+lB ! ITNS SOR 1618 N2=6 1626 N3=6 1638 PRINT "PIPEtiET" I NZ i64B FOR T6=T5 TO T4 STEP T5 1056 I F Td<=T3 THEN la%@ 1068 FOE L=i Tit J 1678 122cL:i=@ 1888 NEXT L le98 FOR 1=1 Ti? t.rS
192
1110 1128 1138 1 1 4 0 1150
1168
1170 1188 1190 1280 1210 1220 1238 1 2 4 8 1250 1260 1270 1288
1298 1388
1310
1338
1348 1350 1360 1370 1388 1398 1 4 0 0 1 4 1 0
1320
1 4 2 8 1430 1 4 4 0 1 4 5 0 1468 1 4 7 0
1 4 8 8 1 4 5 0 1588 1510 ~~~~
1520 1530 1540 1550 1560 1 5 7 0 1580 1558 1680 1 6 1 0
1 6 2 8
1630 1 6 4 8
1658 1 6 6 B
1678 1688
NEXT K
S2(Jl<K))=S2<Jl(K))-C(K)/hB S t Q C K ) ) S 2 ( J 2 ( K ) ) = S 2 < J 2 < K ) > - C < K ) / A B S ( Q ( K ) ) NEXT K FOR K = l TO N1 C2=8 S3=0 N3=N3+1 I F J3+1>J THEN 1380 FOR L=J3+1 TO J S 4 <L)=0 FOR fl3=l TO M l ( L ) Fl=H2<L,M3)
FOR K = l TO P
/hB
/AB
NEXT M 3 OE=H<L>
NEXT L I F C2/S3<=T1 THEN 1390 NEXT K FOR K = l TO P ! NEW FLOtJS
NEXT K I F C3/P<=T0 THEN 1510 NEXT I FOR L=l TO ~3 ! RES LEVEL H ( L ) = H ( L ) - Q 2 < L ) * T S / H ( L ~ ~- FOR M 3 = 1 TO M l < L > M = M 2 < L , M 3 ) IF J l (M)<>L THEN 1578 H(L)=H(L) -Q(M>*T5/A(L) I F J2<M)<>L THEN 1590 H(L)=H(L)+Q(M)*T5/A<L) NEXT M 3 NEXT L PRINT USING "K,DDDDOD,X,K DDDD . O D " i "Ts=" , T6, " H i = " ( 1 )
.s
P R I N T "NODE1 NZ Xm Om 12 m 3 / s H2m ''
PRINT CLSING 1668 j J l ( K ? , J 2 i K ) , X ( K ) , D ( K > , Q ( K ) , H ~ J ~ ~ ~ ) ) NEXT k
FOR K=I T O r
IMAGE ODD. ~ D D O ~ O D D D D , O D . D D D , O D D . DDD, ODDDO. n STOP NEXT TE.
ASSIGN# 1 TO X END
193
l O O m
1 -
1000mx0.15
0.09
F i g . 9.9 P i p e n e t w o r k a n a l y z e d
8,3,768, .2, . 0 3
3,2,668, .2,8 N O . PUMPS/PRVs? 1 PIPEN,+HEADm Nl-t42 1 ? 1 > 1
9 'i
F'l PENET T E S T S I m 1 Tc=i.4.413O h1= 1138.813 tr0OEl t12 XDI Dm Grm3,s H?m
1 5 1E:OB .3WB .143 7 4 . 8 5 t. 868 .i06 .137€. 5c.l 5 8 550 ,258 .6i9 ee.3 d 7 1088 .15@ -.El14 5 4 . 3 8 7 S 8 8 ,288 -654 5 4 . 3 5 4 788 4 3 988 8 3 708 3 2 688
Ts=28888 H1= t4.IODEl N2 Xm
.156 . 818
.158 .018
.268 -.835
.288 - . B 5 5
Dm G r m 3 / s 10B.88
72.5 78.7 78.7 7 8 . 8
H2m 1 5 1808 73.1 5 6 888 49.8
8 958 E 5 . 1 2 7 1888 . l5G -.614 53.2 8 7 8Bb .2@8 . 8 5 4 5 3 . 2 5 4 708 - 1 5 8 .816 71 .4 4 3 988 .1SO .81b 69.2 8 3 788 . Z B B - .634 65.2 3 2 688 . 2 B U -.853 76.1
Ts=43286 H1= 188.88
.368 .146 . 2 8 Q .876 .ZSB .678
NODE1 t42 Xn Drn Q m S / s H2m 1 5 1588 .JBB - .810 5 B . 2 5 6 888 .280 .689 97.9 5 8 958 .258 .837 95.9 i 7 1088 .158 96.' Y 7 808 . 2 B 8 -.888 9 6 . 3 '. -
5 4 76P 4 3 98B 8 3 768 3 2 688
T s = 5 7 6 8 1 3 H 1 = H-lnDEl N2 X m
1 5 1888 5 6 860 5 8 958 rj 7 I8Sb 8 7 888 5 4 7QB 4 3 588 8 3 ;fit3 3 T'
.is(? , 1 5 0 -288 . 288 180.
Om
.288
.2513
.I58
.280
.150
.150
.2QB
. 2 0 0 ,
, 0 1 7 53 .9 817 8:3. 3 Ei46 88.3 ,863 78.4
U N I ~ / C H2m - . 817 95 .4
, 885 5 9 . 6 .,836 97.2
97.5 - .Be8 97.5
,816 95.3 .81G 50.6 .845 r e . @
88
, 8 6 1 S8.G
194
CHAPTER 10
K I NEMAT I C MODELL I NG
INTRODUCTION
Kinematic flow holds for those cases when a unique relat ionship
exists between the depth of flow and the volumetric f lowrate. Model
equations are der ived through simp1 i f icat ions to the fu l I equations
governing gradual ly-var ied, unsteady over land and open channel f low.
When appl ied to one-dimensional over land flow, i f the r a i n f a l l r a te i s
steady and the watershed geometry i s a regu la r geometry such as a
plane, ana ly t i ca l solutions can be obtained for the equivalent character-
i s t i c form of the governing equations. Otherwise, one must use numerical
solut ion techniques. Current ly, kinematic models best apply to h igh l y
impervious (u rban) and/or smal I watersheds. However, research i s on-
going in several countr ies to extend the app l i cab i l i t y to la rge and
mu1 t ip le land use watersheds.
The study of kinematic hydrology and modell ing must begin w i th
the der ivat ion of the f u l l equations governing over land and open channel
f low, followed with an examination of model s impl i f icat ions and when they
can be invoked, development of the character ist ic roots, and then proceed
with ana ly t i ca l and numerical solutions and example applications. I n
th is chapter, a discussion is given of general modell ing concepts and
def in i t ions to provide insights and understanding of the role of kinematic
model l i n g as one approach to hydrologic model I ing.
STORMWATER MODELL I NG
Kinematic model I i ng fa1 Is under the umbrel la of stormwater model I ing.
Stormwater i s defined as the direct watershed response to r a i n f a l l
(Overton and Meadows, 1976). I t i s the runoff which enters a d i tch ,
stream or storm sewer which does not have a s igni f icant base flow
component. This def in i t ion does not assume that a1 I stormwater reaches
an open channel by the over land flow route, although i n u rban areas
the direct response i s mostly through over land flow due to the h igh
degree of imperviousness. I n contrast, i n r u r a l watersheds, an over land
flow component may be nonexistant and direct storm response may be
only near the stream and occur as shallow subsurface flow.
As defined, stormwater i s associated with small up land or headwater
watersheds where base flow i s not a s ign i f i can t port ion of the total
195
streamflow du r ing periods of r a i n f a l I. Therefore, the emphasis of
stormwater modell ing i s on the storm hydrograph and not the streamflow
hy drograp h .
MATHEMAT I CAL MODELS
A mathematical model i s simply a quant i ta t i ve expression of a
process or phenomenon one i s observing, analyzing, o r predict ing. Since
no process can be completely observed, any mathematical expression of
a process w i l l involve some element of stochasticism, i.e. uncertainty.
Hence, any mathematical model formulated to represent a process or
phenomenon w i l l be conceptual to some extent and the r e l i a b i l i t y of the
model w i l l be based upon the extent to which i t can be or has been
ver i f ied. Model ver i f i ca t ion i s a function of the da ta ava i lab le to test
the model sc ien t i f i ca l l y and the resources ava i l ab le (time, manpower,
and money) to perform the tests. Since time, manpower, and money always
have f i n i t e l imi ts, decisions must be made as to the degree of complexity
the model i s to have, and the extensiveness of the ver i f i ca t ion tests that
are to be performed.
The i n i t i a l task of the modeler then i s to make decisions as to which
model to use or to bu i l d , how to ver i f y i t , and how to determine i t s
stat ist ical r e l i a b i l i t y in appl icat ion, e.g., feas ib i l i t y , p lann ing , design,
or management. This decision-making process I S in i t ia ted by c lear ly
formulating the objective of the modell ing endeavour and p lac ing i t i n
the context of ava i lable resources.
. .
I f the i n i t i a l model form does not achieve the intended objective,
then i t simply becomes a matter of rev is ing the model and repeating the
experimental ver i f icat ions u n t i l the project objective i s met. Hence,
mathematical modell ing is by i t s nature heur ist ic and i terat ive. The
choice of model revisions as well as the i n i t i a l model structure w i l l also
be heavi ly affected by the range of choice of modelling,concepts ava i l ab le
to the modeler, and by the s k i l l which the modeler has or can develop
i n app ly ing them.
Figure 10.1 i s a schemat i c representat ion of the model I ing process.
The modelling process i s not new but i s nothing more than a modern
expression of the classical scient i f ic thought processes involved i n the
design of an experiment. What i s new i s that today a very large number
of concepts can be evaluated e f f i c ien t ly i n a very small amount of time
at a re la t i ve ly small expense using computers and the body of ana ly t i ca l
techniques termed systems analysis.
CONCEPT OBJECTIVE- OF - OEFl NED HYDROLOGIC
PROCESSES
F i g . 10.1 T h e rnodeiling process
MODEL EXPERIMENTAL 08 JECTIVE VERl FlCATl ON - FORMULLTION .
, t FEEDBACK
L FEEDBACK
197
SYSTEM DEF I N I T I ON
Dooge (1976) h a s deve loped a good w o r k i n g d e f i n i t i o n of a sys tem
as b e i n g a n y s t r u c t u r e , dev i ce , scheme, o r p r o c e d u r e t h a t i n t e r r e l a t e s
a n i n p u t t o an o u t p u t in a g i v e n t ime re fe rence.
The k e y concep ts o f a sys tem a r e :
1 . A sys tem c o n s i s t s o f p a r t s connected toge the r in acco rdance w i t h
some s o r t o f p l a n , i.e. i t i s an o r d e r e d a r r a n g e m e n t .
2. A sys tem h a s a t ime f r a m e
3. A sys tem h a s a cause-e f fec t r e l a t i o n .
4. A sys tem h a s t h e m a i n f u n c t i o n t o i n t e r r e l a t e an i n p u t and o u t p u t ,
e.g., s to rm r a i n f a l l and s torm r u n o f f .
In the s t r i c t e s t d e f i n i t i o n , the systems a p p r o a c h i s a n o v e r a l l one
and does n o t conce rn i t s e l f w i t h d e t a i l s w h i c h may o r m a y n o t b e
i m p o r t a n t and w h i c h , in a n y case, m a y no t b e known. T h i s seeming ly
l i m i t s the sys tems a p p r o a c h to an a t tempt to ge t a r o u n d the comp lex
geomet ry and p h y s i c s of t he h y d r o l o g i c sys tem. I f we were s o l e l y
concerned w i t h p rob lems of i d e n t i f i c a t i o n ( d e f i n e d b y Doodge as the
r e c o g n i t i o n o f t he o v e r a l l n a t u r e of a s y s t e m ' s o p e r a t i o n , b u t n o t a n y
d e t a i l s o f t he n a t u r e o f t he sys tem i t s e l f ) , t h i s a t t i t u d e o f i g n o r i n g t h e
d e t a i l s o f t he sys tem w o u l d b e a r e a s o n a b l e one. However , when we a r e
g o i n g t o s i m u l a t e a h y d r o l o g i c sys tem and i t s response, the e lemen ts
of physicaz h y d r o l o g y become i m p o r t a n t . F o r i n s t a n c e , i f we bu i ld o r
use a model t h a t i s in c o n f l i c t w i t h the p h y s i c a l r e a l i t i e s , t h e n we c a n
h a r d l y expec t to o b t a i n good r e s u l t s f rom s u c h a mode l , o r e v e n t o b e
a b l e to c a l i b r a t e the model t o a c h i e v e good r e s u l t s . Thus , t he sys tems
a p p r o a c h t o s t o r m w a t e r m o d e l i n g mus t c o n s i d e r t h e a s s i m i l a t i o n of p rocess
mode ls i n t o an o v e r a l l r e p r e s e n t a t i o n o f t h e h y d r o l o g i c c y c l e , o r p o r t i o n s
thereo f . How we1 I the model components mus t r e p r e s e n t the d i f f e r e n t
p rocesses depends on the p u r p o s e f o r the model and how m u c h data a r e
a v a i l a b l e w i t h w h i c h to v e r i f y t he mode l .
I n c o n c l u s i o n , t he essence of sys tems a n a l y s i s a s a p p l i e d t o s to rm-
w a t e r modelling i s to i n t e r r e l a t e ra in fa l l ( i n p u t ) to s t o r m w a t e r ( o u t p u t )
w i t h a r e l i a b l e model in a c o m p u t a t i o n a l l y e f f i c i e n t manner .
TERM I NOLOGY AND DEF I N I T IONS
There has been an evolut ion of systems jargon, and i t i s important
to review the main pa r t s to better understand hydrologic modell ing.
A variable has no f i xed value (e.g., discharge) whereas a parameter
i s a constant whose value var ies wi th the circumstances of i t s appl icat ion
(e.g., Manning n-value).
The dist inct ion between Linear and nonlinear systems i s of paramount
importance i n understanding the mechanism of hydrological model I ing .
A l inear system i s defined mathematical ly by a l inear d i f fe ren t ia l
equation, the p r inc ip le of superposit ion appl ies and system response i s
only a function of the system i tse l f . An example of a l inear system
representation i s the u n i t hydrograph model. A nonl inear system i s
represented by a nonl inear d i f fe ren t ia l equation and system response
depends upon the system i tsel f and the input intensi ty. An example of
a nonl inear system representation i s kinematic over land flow. I t i s well
known that real world systems are h igh l y nonl inear, but l inear
representations have often been made because the system i s not under-
stood we1 I or because of the pressures exerted by resource constraints.
The s t a t e of a system i s defined as the values of the var iab les of
the system'at an instant i n time. Hence, i f we know exact ly where a l l
of the stormwater i s and i t s f lowrate i n a basin, then we know the state
of the system. The state of a stormwater system i s determined ei ther from
histor ical da ta or by assumption.
System memory i s the length of time i n the past over which the input
affects the present state. i f stormwater from a basin today i s affected
by the stormwater flow yesterday, the system (watershed) i s said to have
a f i n i t e memory. I f i t i s not affected at a l l , the system has no memory;
and, i f i t i s affected by storm flows since the beginning of the world,
the system i s said to have i n f i n i t e memory. Memory of surface water flow
systems i s mostly a function of antecedent moisture condit ions.
A time-invariant system i s one i n which the input-output re la t ion
i s not dependent upon the time a t which the input i s appl ied to the
system. To i I lustrate, u n i t hydrograph models represent the catchment
as a t ime- invar iant system because the same u n i t hydrograph (response
funct ion) i s maintained throughout the storm regardless of var ia t ions
in watershed conditions. Usual l y , time-variance i s considered among
storm events, seasons of the year, etc., and not w i th in i nd i v idua l
storms. Time-invariance indicates constant land use, ground cover, and
drainage system conf igurat ion and capaci ty, and ignores soi l moisture
var ia t ions and the effects of erosion.
199
A lwnped va r iab le or parameter system i s one in which the
var iat ions i n space ei ther do not exist o r have been ignored. Conversely,
a d i s t r i b u t e d parameter system recognizes spat ia l var iat ions. The input
i s said to be lumped i f r a i n f a l l into a system i s considered to be
spatial ly uniform. Lumped systems are represented by ord inary dif feren-
t ia l equations and d is t r ibu ted systems are represented by p a r t i a l d i f fe r -
ent ia l equations.
A system i s said to be s t o c h a s t i c i f for a given input there i s an
element of chance or probabi I i t y associated w i th obtaining a certain
output. A d e t e r m i n i s t i c system has no element of chance in i t , hence
for a given input a completely predictable output resul ts for g iven
i n i t i a l and/or boundary values. A pure ly random process has no deter-
minist ic component and output i s completely given to chance. A para-
metric or conceptual model does have an element of chance b u i l t into
i t since there alway w i l l be er ro rs i n ver i f y ing i t on real data. I t does
therefore have a stochastic component. A alack boz model relates input
to output by an a r b i t r a r y function, and has no inherent physical
signif icance.
Model Optimization i s the objective determination of the "best" values
for the model parameters using hydrologic data for the type of watersheds
and range of hydrologic condit ions for which the model has been
designed. This funct ion i s I imited to parametric stormwater models, and
i s appl ied in the regional izat ion process. To reg iona l i ze a model means
to develop a scient i f ic basis for p red ic t ing the model parameters on
ungauged watersheds from hydrologic and physiographic character ist ics
of that watershed. Regionalization can be accomplished only i f there are
enough benchmark watersheds w i th adequate periods of record that a
stat ist ical inference can be drawn, i.e., s ta t i s t i ca l l y s ign i f i can t
parameter predict ion equations can be developed.
Model cat ibra t ion bas ica l l y i s the f ine-tuning of model parameter
values to achieve the best f i t between observed and predicted runoff
hydrographs. T o ver i fy a model i s to compare model predict ions w i th
observed runof f values without adjust ing parameter values to confirm
the model i s doing a reasonable job in s imulat ing the t rue watershed
response to known input.
Two concepts that a re frequently confused (misused) are a n a l y s i s
and s i m u h t i o n . The confusion with ana lys is stems from what i t i s being
used to describe. As i t re lates to stormwater models, ana lys is i s the
procedure used to ca l i b ra te a model to the data. I t i s an attempt to
improve the state-of-the-art and i s fundamental l y a research and develop-
200
ment tool. Simulation, by contrast, u t i l i zes the resul ts of previous
analyses (and regional izat ion methods) to synthesize (p red ic t ) stormwater
runoff from ei ther design or real time r a i n f a l l on ungauged watersheds.
Simulations also can be performed at gauged watersheds to generate
runoff data for design events or events not contained in the ava i l ab le
record. Analysis i s often applied, for example, i n the context of studying
the probable performance of a storm sewer system du r ing design storm
events. We are prone to say that we have analyzed the system and
found that i t should work! Actual ly, what w e a re doing i s using
simulat ion resul ts to predict the probable performance character ist ics
of the storm sewer system.
MODELL I NG APPROACHES
There are two conceptual approaches that have been used i n develop-
ing stormwater models. An approach often employed i n urban p lann ing
has been termed deterministic modeling or system simulat ion. These models
have a theoretical structure based upon physical laws and measures of
i n i t i a l and boundary conditions. When conditions are adequately
specified, the output from such a model should be known with a h igh
degree of cer ta in ty . I n r e a l i t y , however, because of the complexity of
the stormwater flow process, the number of physical measures required
would make a complete model intractable. Simp1 i f icat ions and approx-
imations must therefore be made. Since there are always a number of
unknown model coeff icients and parameters that cannot be d i rec t l y o r
easi ly measured, i t i s required that the model b e ver i f ied. T h i s means
that the resu l ts from usable determinist ic models must b e ver i f ied by
being checked against real watershed data wherever such a model i s
to be appl ied.
The second conceptual stormwater modeling approach has been termed
parametric modell inq. I n th i s case, the models are somewhat less
r igorously developed and general ly simpler in approach. Model parameters
are not necessarily defined as measurable physical ent i t ies although they
are general ly ra t iona l . Parameters for these models a re determined b y
f i t t i n g the model to hydrologic da ta wi th an optimization technique.
Application of parametr ic models to ungauged watersheds i s possible only
i f regional ized parameter predict ion equations a re ava i lab le and are
based on data from watersheds w i th in the same geographical area and
with simi la r geomorphic and land use character ist ics as the watershed
being considered. As w i th determinist ic models, user confidence stems
from ver i f i ca t ion studies using local data.
20 1
EXAMPLES OF PARAMETRIC AND DETERMINIST I C MODELS
An e x c e l l e n t e x a m p l e o f a p a r a m e t r i c s t o r m w a t e r model i s t h e TVA
S to rmwate r Model (Betson, e t a l . , 1980). The model i s an even t s i m u l a t i o n
model f o r m u l a t e d w i t h a v a r i a t i o n o f t he SCS c u r v e n u m b e r r u n o f f model
f o r d e t e r m i n i n g r a i n f a l l excess a n d a u n i t h y d r o g r a p h mode l . An even t
model s i m u l a t e s the r u n o f f f r o m a one- t ime r a i n f a l l e v e n t , whereas a
con t inuous mode l s i m u l a t e s a t ime s e r i e s o f d a i l y f l o w s a n d h y d r o g r a p h s .
The c u r v e n u m b e r model was m o d i f i e d somewhat to i n c l u d e a c o n s t a n t
a b s t r a c t i o n r a t e w h i c h a l l o w s f o r i n f i l t r a t i o n during l u l l s and a f t e r t h e
c e s s a t i o n o f r a i n f a l l b u t b e f o r e r u n o f f was ceased. T h i s i n t r o d u c e d a
new p a r a m e t e r , P H I , w h i c h i s a n a l o g o u s t o t h e s o i l s a t u r a t e d h y d r a u l i c
c o n d u c t i v i t y , b u t w h i c h i s d e t e r m i n e d s o l e l y t h r o u g h o p t i m i z a t i o n s t u d i e s .
The u n i t h y d r o g r a p h shape i s d e s c r i b e d w i t h two t r i a n g l e s , t h e so -ca l l ed
d o u b l e t r i a n g l e u n i t h y d r o g r a p h , and r e q u i r e s f o u r p a r a m e t e r s , t he p e a k
f l o w r a t e a n d t ime t o p e a k o f t he f i r s t t r i a n g l e , t he t ime b a s e f o r b o t h
h y d r o g r a p h s , a n d t h e t ime t o p e a k o f t he second t r i a n g l e . A f i f t h model
p a r a m e t e r , t he p e a k o r d i n a t e o f t he second t r i a n g l e , i s d e t e r m i n e d f r o m
the c o n s t r a i n t t h a t t he vo lume u n d e r the u n i t h y d r o g r a p h e q u a l one b a s i n
i n c h o r mm o f r u n o f f .
The TVA deve loped r e g i o n a l p r e d i c t i o n e q u a t i o n s f o r e a c h of t h e
model p a r a m e t e r s u s i n g d a t a f r o m o v e r 500 e v e n t s o n 38 r u r a l , u r b a n
and s u r f a c e m i n e d wa te rsheds in the Tennessee V a l l e y r e g i o n . U s i n g these
e q u a t i o n s , t h e model c a n b e a p p l i e d to o t h e r wa te rsheds w i t h i n t h e same
p h y s i o g r a p h i c r e g i o n s w i t h r e a s o n a b l e success, as demons t ra ted in
v e r i f i c a t i o n s t u d i e s b y Betson, e t a l . (1981) . However , t h i s model s h o u l d
no t b e used o u t s i d e the l i m i t s o f i t s r e g i o n a l i z a t i o n . T h i s was demon-
s t r a t e d b y Meadows, e t a l . (1983) in a s t u d y o f t h e a p p l i c a t i o n o f f o u r
u n i t h y d r o g r a p h mode ls to wa te rsheds in 14 p h y s i o g r a p h i c p r o v i n c e s
ac ross the U n i t e d States. The r e s u l t s w h i c h they o b t a i n e d f o r e a c h model
b a s i c a l l y were a c c e p t a b l e o n l y w i t h i n the r e g i o n s of t h e i r deve lopment .
Most d e t e r m i n i s t i c mode ls a r e f o r m u l a t e d w i t h t h e k i n e m a t i c r u n o f f
mode l , o f w h i c h t h e r e a r e s e v e r a l , i n c l u d i n g the EPA Storm Water
Management Model ( M e t c a l f and E d d y , e t a l . , 1971). t he USGS D i s t r i b u t e d
R a i n f a l l - R u n o f f R o u t i n g Mode l ( D a w d y , e t a l . , 1978), and WITWAT (Green,
1984), t o name a few. These mode ls d i f f e r , b u t e a c h i s f o r m u l a t e d w i t h
a 2 o r 3 p a r a m e t e r model f o r i n f i l t r a t i o n , and k i n e m a t i c o v e r l a n d and
c h a n n e l r o u t i n g . The i n f i l t r a t i o n mode l p a r a m e t e r s g e n e r a l l y c a n b e
es t ima ted f r o m s i t e measures o r a s t y p i c a l v a l u e s in tex tbooks and
pub1 i shed r e p o r t s . S i m i l a r l y , t he r o u t i n g p a r a m e t e r s , e.g. M a n n i n g ' s
n - v a l u e , c a n b e e s t i m a t e d f rom p u b l i s h e d sources . Thus , these mode ls
202
are appl icable to an ungauged si te because model parameters general ly
a re measurable or typical values a re known. Confidence i n model simu-
la t ion i s h igh , but should be confirmed through ver i f i ca t ion studies once
local data became avai lable.
The best of both worlds i s i l l us t ra ted by the USGS model. I t can
be appl ied d i rec t l y as a determinist ic model, o r i f local ca l ib ra t ion da ta
are ava i lab1 e, the soi I-moi sture accounting and i n f i I t rat ion parameters
can be optimized. The USGS terms th i s version of the model a parametric-
determinist ic runof f model (A1 ley, et a l . , 1980).
Engineers have designed drainage systems for decades using the
we1 I-known Rational Method, and have simulated watershed runoff w i th
u n i t hydrograph models, e.g. SCS cu rv i l i nea r u n i t hydrograph. Why i s
i t necessary, or even useful, to work wi th kinematic stormwater models
now? The answer to th i s question l ies in an examination of what kine-
matic models w i l l do for the engineer - and perhaps i t also l ies i n what
the other methods w i l l not do.
F i rs t , the ro le of models i n general should be acknowledged.
Engineering design of drainage systems and environmental impact assess-
ment of land use change requ i re informat ion about watershed response
to prescribed "design" events which most often are extreme events.
Since most small basins are not gauged for both r a i n f a l l and streamflow,
l i t t l e hydrologic data i s ava i l ab le to quant i f y the necessary response
characterist ics. Further, i f a watershed i s gauged, i t i s un l i ke l y that
a sui table "design" event, i s contained in the record unless the gauge
has been i n operation for many years. Even so, the data are for the
watershed response in i t s current land use condit ion and are not a t rue
measure of the watershed response fol lowing land use change. To proper ly
quant i fy the watershed response for "af ter" development conditions, the
record would have to be extended for several years to insure the probab-
i l i t y of an adequate number of acceptable events. But the land use
change must s t i l l be planned, the associated drainage system designed,
and impact statements prepared. W e do not have the l uxu ry of being
able to wai t for the data to be collected, so we must resort to predict ion
methods.
I t i s widely accepted that mathematical stormwater models are the
only ava i lab le means of making re l i ab le prGdictions of watershed response
to design events and of the effects of land use change on stormwater
runof f and qua l i t y . I t must be stated emphatical ly, however, that models
are not a subst i tute for f i e ld gathered da ta or knowledge of the
hydrologic/hydraul ic and water qua1 i t y processes on the p a r t of the user.
No model can p red i t how a na tura l system behaves as dependably as
direct measurements of the system i tsel f . The p r inc ipa l use of models
i s in situations where direct measurements a re ei ther impossible o r
impractical, such as the "af ter" development conditions. When a drainage
system is under design, for example, a model w i l l let the designer look
at many a l te rna t ive configurations. More important ly, the designer can
answer the "what i f " questions, and can do so w i th in a reasonable
framework of time and costs. Models also permit a more accurate ana lys is
of complex watershed and drainage systems. T h e advent of models has
changed the engineer from a cookbook a r t i s t who re l ied heav i l y on
judgement to a serious analyst and planner.
The selection of a model t yp i ca l l y i s a statement of user confidence,
which has been defined as "the bel ief in the r e l i a b i l i t y or c red ib i l i t y
of the resul ts and exists ei ther consciously or subconsciously i n the
minds of the model user or cl ientele" (ASCE, 1983). This bel ief i s der ived
from experiences i n the use, development, o r test ing of a model, from
user understanding of watershed hydrologic processes and model repre-
sentation of these processes, and from confidence in au thor i ty , e.g.,
textbooks, technical journals, and federal agency endorsement. Ul t imately,
confidence i s founded on ver i f i ca t ion studies at the watershed where the
simulations are required.
The keyword i s "rel iable". When using a model, one must remember
the model i s merely a mathematical expression of the t rue system and
cannot account for a l l the subtlet ies of the var ious phenomena (processes)
involved. Reliable resul ts are those on which the model user c a n foster
the bel ief that i f such an e'vent occurs, the probably runoff hydrograph
w i l l be very much l i ke the model predict ions.
So why use kinematic models? Perhaps the best answer i s that they
are determinist ic, d is t r ibu ted parameter models that can account for the
spat ia l watershed and ra in fa l I var ia t ions and the non l inear i ty of the
runoff process. I n other words, kinematic models a re a better model of
the true process. Because kinematic models a re based' on the physics
of the runoff process; the model structure i s ra t i ona l , the parameters
are measurable or a re ava i l ab le from publ ished studies and textbooks,
and the model can be appl ied w i th a minimum of ca l ib ra t ion data. (They
can be appl ied i n the absence of ca l ib ra t ion data; user confidence i s
supported by the extensive test ing and documentat ion of kinemat i t
models.) Though young i n evolut ion, there are now several models
ava i l ab le for computer use, even personal computer use. Thus, kinematic
models are as read i l y used as other models and have the advantages
offered by determinist ic models.
204
Fig . 10.2 Contour p lo t of topography
TWO - D I MENS I ONAL OVERLAND FLOW MODELL I NG
Topography and catchment surface character ist ics can not be proper ly
accounted for in one-dimensional models in a l l cases. The cone shaped
catchment i s a typical example. A l s o the effect of va ry ing surface
roughness, slope and losses i s often two-dimensional. Storm patterns
cannot be accounted for proper ly and the assumption of a rectangular
hyetograph over the en t i re catchment i s often dangerous. I t may be
necessary i n the case of complex catchments o r r a i n to resort to two-
dimensional model l ing.
Two-dirnensiona I k i nernat ic equations
One -dimensional equations can be extended into two dimensions as
fol lows:
205
F ig . 10.3 Topography in 3-Dimensions
The cont inui ty equation becomes
(10.1)
where q i s the flow i n the x direct ion ( m 2 / s ) and qz is the flow i n
the z direct ion ( m ’ / s ) .
A proof of t h i s equation can be found in Dronkers ( 1 9 6 4 ) . For two
dimensional flow two motion equations are required. I n kinematic theory
these are obtained by assuming
sox = sfx (10 .2)
soz = Sfz (10.3)
where S i s the bed slope in the x direction, So* is the bed slope i n
the z direction, S f x i s the f r i c t ion slope i n the x direction and 5 i s
the f r i c t ion slope in the z direction.
ox
fz
For the general form of headloss equation one can obtain
( 1 0 . 4 ) 1 m qx = 4 b,Y ) z
( 1 0 . 5 )
( 1 0 . 6 )
and a x = funct ion of 5
a z = funct ion of S ox
02
This idea for two dimensional flow was used b y Orlob (1972). I t
w i l l be noticed that q t i s a lways posi t ive whi le q and q can b e
Positive O r negative as ( a x ) 2 and are functions of s and Soz ox respective1 y.
Boundary conditions
There are two boundary condit ions that can be used on watersheds.
One can assume that the water depth at the boundary i s always zero
and that a l l the water enter ing the o r ig in leaves i t i n the form of a
discharge. This has been assumed in a l l ex is t ing theories. One must
then define
(10.7)
yk = 0.0 (10.8) I
so qk = i ~ x / 2 (10.9) I e
One could a l te rna t ive ly assume that the discharge at the o r ig in i s
control led by the depth of water at the o r ig in as assumed for the rest
of the points. For t h i s case we must then use the same equations as
wi th the other point's. The effect of using the two di f ferent boundary
conditions w i l l be shown later.
I ni t ia I conditions
After the f i r s t time step i t may be assumed that the water depth
at al I points, except at the o r ig in i n the case of the f i r s t boundary
condit ion, for the case of an i n i t i a l l y d ry catchment
(10.10)
The proposed equations may be solved at g r i d points over a p lane
provided runof f i s adequately described by the kinemat ic equations.
Where there are flow concentrations such as an in land depression, storage
w i l l not be accounted for except w i th a separate rout ine to account for
net volume stored. I f outflow eventual ly occurs when the depression i s
f i Iled, again a separate rout ine i s needed to detect th is.
The effect of channelization, for example r i l l s and furrows i n which
runoff collects, can be accounted f o r by reducing the effect ive dx o r
dy over which runoff occurs. Where channel side f r i c t ion i s app l i cab le
however condui t equations may be required.
The effect of spa t i a l l y va ry ing soi l types and cover can also affect
losses to a s ign i f i can t extent. l n f i I t rat ion, and possible re-emergence
of interface flow can be accounted for wi th a two-layer model wi th per-
meable interface. A sample of such a model i s discussed later.
F ig . 10.4 F l o w direct ions 0 from model
207
2000 00
.o +.P +.o .+.o. -0.0 +1eo.o+lso.o+~eo.o+rso.o+1so.o 1900 f
+reo.o+-a6.13+.0 +o.o +iae.e--1500
+ i i i . 7 +ieo.o +12s.s +ias.o+1213.9--1400
+71.8 -0.0 + l ~ E . E + l 7 l . E + ? * 8 . , 0 - - ~ 3 0 0
+70.6 4-7E.e -0.0 +155.4+1EO.E--1200
+?ie.8+107.6+7i.6 +o.o +?se.7--1 1 0 0
+i4i.e+iee.e+se.a +o.o +iss.4--1 OD0
a.e +47.7 +o.o +7e.e +7e.e +4.e +o.o +i4?.a+?4,e+s~.~ 70.0 f '
208
REFERENCES
A l l e y , W.M., D a w d y , D.R. and Schaake, J.C., J r . , 1980. P a r a m e t r i c - d e t e r m i n i s t i c urban wa te rshed mode l . J. H y d r . D i v . ASCE, Vol . 106, No. HY5, pp. 679-690.
ASCE, 1983. Q u a n t i f i c a t i o n o f land use c h a n g e e f f e c t s u p o n h y d r o l o g y , b y the Task Commit tee on Q u a n t i f y i n g Land Use Change E f f e c t s , R.P. Betson, Chmn., p r e s e n t e d a t t h e J u l y 20-22, 1983. ASCE I r r i g a t i o n and D r a i n a g e D i v i s i o n S p e c i a l t y Conference, h e l d a t Jackson, Wyoming.
Betson, R.P., Ba les , J . and P r a t t , H.E., 1980. Users Gu ide t o TVA- HYSIM, A h y d r o l o g i c p r o g r a m f o r q u a n t i f y i n g l a n d u s e c h a n g e e f fec ts . EPA-600/7-80-048, Tennessee Va l ley A u t h o r i t y , Wate r Systems Development B r a n c h , N o r r i s , Tennessee.
Betson, R.P., Ba les , J. and Deane, C.H., 1981. Me thodo log ies f o r assess- ing s u r f a c e m i n i n g impac ts . Repor t No. WR28-1-550-108, Tennessee V a l l e y A u t h o r i t y , Wate r Systems Development B r a n c h , N o r r i s , Tennessee.
Dawdy , D.R., Schaake , J.C., J r . and A l l e y , W.M., 1978. U s e r ' s g u i d e f o r d i s t r i b u t e d r o u t i n g r a i n f a l I - r u n o f f model. U.S. Geo log ica l S u r v e y Water Resources I n v e s t i g a t i o n s 78-90.
DJoge, J.C.I., 1973. L i n e a r t h e o r y o f h y d r o l o g i c systems. U.S. Dept. o f A g r i c u l t u r e , A g r i c u l t u r a l Research Serv i ce , Tech. Bu l I . No. 1468.
D r o n k e r s , J . J . , 1964. T i d a l compu ta t i ons in r i v e r s and c o a s t a l w a t e r s . N o r t h H o l l a n d P u b l i s h i n g Co., Amsterdram.
Green, I .R.A., 1984. WITWAT s t o r m w a t e r d r a i n a g e p r o g r a m - T h e o r y , A p p l i c a t i o n s and U s e r ' s M a n u a l . Repor t No. 1/1984, Water Systems Research Programme, Dept. o f C i v i I E n g i n e e r i n g , U n i v e r s i t y o f t h e W i t w a t e r s r a n d , Johannesburg , South A f r i c a .
Meadows, M.E., Howard , K.M. and Ches tnu t , A.L. , 1983. Development o f mode ls f o r s i m u l a t i n g s t o r m w a t e r r u n o f f f rom s u r f a c e c o a l m i n e d l a n d s : U n i t h y d r o g r a p h models. Repor t No. G5115213, Vol . 1 , U.S. Dept. o f t h e I n t e r i o r , O f f i c e o f S u r f a c e M i n i n g , D i v i s i o n of Research , Wash ing ton , D.C.
O r l o b , G.T., 1972. Ma themat i ca l m o d e l l i n g o f e s t u a r i a l systems. I n t e r n a - t i o n a l Sympos ium on ma themat i ca l m o d e l i n g techn iques in w a t e r resources systems, E d i t o r As i t K . B i swas . P roceed ings Volume 1 .
M e t c a l f and E d d y , Inc.,. U n i v e r s i t y o f F l o r i d a , and Water Resources Eng ineers , 1971. Storm w a t e r management model. U.S. E n v i r o n m e n t a l P r o t e c t i o n Agency , Wash ing ton , D.C.
Over ton , D.E. and Meadows, M.E., 1976. S to rmwate r M o d e l i n g , Academic P ress , New Y o r k , N.Y.
F i g . 10.5 Water d e p t h v a r i a t i o n a t t = 8 m i n o v e r t h e ca tchmen t
209
CHAPTER 1 1
APPL I CAT IONS OF K I NEMAT I C MODELL I NG
APPROACHES
T h i s c h a p t e r c o n t a i n s examp les o f k i n e m a t i c s t o r m w a t e r s i m u l a t i o n
mode ls and t h e i r a p p l i c a t i o n to r u r a l and urban wa te rsheds . These
mode ls were se lec ted f rom t h e r a n g e o f a v a i l a b l e mode ls because t h e y
a r e s i m p l e in concept a n d s t r u c t u r e , h a v e been tes ted e x t e n s i v e l y , and
a r e r e p r e s e n t a t i v e o f the a p p r o a c h e s t a k e n in d e v e l o p i n g k i n e m a t i c
wa te rshed models. F o r these reasons , t hey s h o u l d h e l p the r e a d e r t o more
f u l l y u n d e r s t a n d k i n e m a t i c mode ls and t h e i r a p p l i c a t i o n s .
The r e a d e r i s r e m i n d e d t h a t w i t h a n y w a t e r s h e d mode l , a p p r o x -
i m a t i o n s a n d s i m p l i f i c a t i o n s a r e made. P r e v i o u s l y , we h a v e seen t h a t
k i n e m a t i c mode ls a r e s i m p l i f i c a t i o n s t o the d y n a m i c w a v e mode ls ; and
t h a t t h e i r s o l u t i o n , whe the r a n a l y t i c a l o r n u m e r i c a l , r e q u i r e s a p p r o x -
i m a t i o n s to the w a t e r s h e d geomet ry , d r a i n a g e l a y o u t , r a i n f a l I p a t t e r n ,
e tc . The examp les in t h i s c h a p t e r i l l u s t r a t e d i f f e r e n t a p p r o a c h e s t o
m a k i n g these a p p r o x i m a t ions,
A MODEL FOR URBAN WATERSHEDS
A model t h a t h a s been s u c c e s s f u l l y a p p l i e d t o urban wa te rsheds i s
t he U.S. Geo log ica l S u r v e y mode l , DR3M ( D a w d y , e t a l . , 1978). T h i s mode l
combines the soi I m o i s t u r e a c c o u n t i n g and r a i n f a l I excess components
o f the model deve loped b y Dawdy and o t h e r s (1972) w i t h the k i n e m a t i c
w a v e r o u t i n g components o f t he mode l deve loped b y LeC le rc a n d Schaake
(1973) . I n p u t to the model i n c l u d e s d a i l y r a i n f a l l , s to rm r a i n f a l l , d a i l y
pan e v a p o r a t i o n and a p h y s i c a l d e f i n i t i o n o f t he d r a i n a g e b a s i n
d i s c r e t i z e d i n t o a s m a n y as 50 segments, i n c l u d i n g o v e r l a n d f l o w ,
c h a n n e l and r e s e r v o i r segments. D u r i n g s to rm d a y s , t h e model g e n e r a t e s
a s i m u l a t e d d i s c h a r g e h y d r o g r a p h b a s e d o n i n p u t d a t a f rom a s m a n y
a s t h r e e r a i n gauges . The model c o n s i s t s o f two m a i n se ts o f components :
p a r a m e t r i c r a i n f a l I excess and d e t e r m i n i s t i c r u n o f f r o u t i n g components.
Parametric Rainfal I Excess Components
The p a r a m e t r i c r a i n f a l I excess components a r e a so i l m o i s t u r e
a c c o u n t i n g component , an infi I t r a t i on component, an i m p e r v i o u s a r e a
r a i n f a l I excess component, and an o p t i m i z a t i o n component. A s u b s t a n t i a l
210
pa r t of the ra in fa
developed by Dawdy
ca l ib ra t ion to estab
I excess components was adopted from a model
et a l . (1972). This component i s used du r ing model
ish opt irnal parameter values for s i te i n f i I t rat ion
and soi I moisture storage functions.
Soil Moisture Accounting
The soi l moisture accounting component determines the effect of ante-
cedent condit ions on i n f i l t r a t i on . Soil moisture i s modelled as a two
layered system, one representing the antecedent base moisture storage
(EMS), and the other, the upper wetted p a r t caused by i n f i l t r a t i on into
a saturated moisture storage (SMS).
Dur ing r a i n f a l l days, moisture i s added to SMS based on the Ph i l i p
i n f i l t r a t i on equation (Ph i l ip , 1954). On other days, a specif ied proport ion
of da i l y r a i n f a l l ( R R ) i n f i l t r a tes into the soi l . I r r i ga t i on ( fo r example,
lawn water ing) can be included in the d a i l y water balance. This i s
achieved through user supplied i r r i ga t i on rates for each month. I f a
da i l y p rec ip i ta t ion i s less than the da i l y i r r i ga t i on rate, the d a i l y
precipi tat ion i s set equal to the i r r i ga t i on rate.
Evapotranspirat ion takes place from SMS, based on a v a i l a b i l i t y ,
otherwise from BMS, wi th the r a t e determined from pan evaporation mul t i -
p l ied by a pan coefficient ( E V C ) . Moisture i n SMS dra ins into BMS wi th
a control l ing parameter (DRN) determining the rate. Storage i n BMS has
a maximum value (BMSN) equivalent to the f i e ld capaci ty moisture storage
of an act ive zone. Zero storage in BMS i s assumed to correspond to
w i l t i ng point condit ions i n the act ive soi l zone. When storage i n BMS
exceeds BMSN, the excess i s sp i l led to deeper storage. These s p i l l s could
be the bas is for rou t ing interf low and baseflow components, i f desired.
However, t h i s opt ion i s not included in the present version of the model.
A schematic flow char t of the soi l moisture accounting component i s shown
i n F igure 1 1 . 1 .
Infiltration Component
I n f i l t r a t i on i s computed with the Ph i l i p equation (Ph i l i p , 1954),
which i s merely a var ia t ion to the Green and Ampt equation. One form
of the Green and Ampt equation i s
( 1 1 . 1 )
where F i s the accumulated i n f i l t r a t i on depth, K i s the effect ive
21 1
, DRAINAGE BMS
t
I EVAPO- I TRANSPI RATION
USE BMS WITH RGF
TO COMPUTE TO.-
PS 1
RAINFALL I N PUT
COMPUTE: IN FILTRATION
1
SPILL TO: DEEPER STORAGE
F i g . 1 1 . 1 Schemat ic of DR3M s o i l m o i s t u r e a c c o u n t i n g component
21 2
hydrau l i c conduct iv i ty, H i s the depth of water ponded on the soi l
surface, P i s the wett ing f ront section, and Z i s the depth to the wett ing
f ront. Using the relat ionship
F z = - 0 - 0.
S I
(11.2)
in which B S i s the volumetric soi l moisture content at saturat ion and
0 . i s the i n i t i a l (unsaturated) moisture content, Eq. 11.1 i s transformed
into the Phi I ip equation.
1 H + P(3s-0 i )
F dF - = K [ l + dt
(11.3)
Since the wett ing front suction i s general ly several orders of magnitude
greater than the depth of ponded water, the H term may be ignored.
The mnemonic ident i f ie rs used to designate the resu l t ing i n f i l t r a t i on are
FR = KSAT ( 1 + z) SMS (11.4)
in which FR=dF/dt, KSAT=K, PS=P(@ - @ . ) , and SMS=F. S I
The wett ing front suction i s not constant, but var ies wi th the soi l
moisture condit ion. The effect ive value of PS i s assumed to va ry l inear ly
between a w i l t i ng point and f i e ld capaci ty, and i s computed with the
relat ionship
PS = P S P ~ R G F - (RGF - 1 b$$ (11.5)
in which BMS i s the i n i t i a l moisture storage i n the soi l column; BMSN
i s the moisture storage i n the soi l column at f i e ld capaci ty; PSP i s the
effective value of PS at f i e ld capaci ty; and RGF i s the ra t i o of PS a t
w i l t i ng point to that at f i e ld capaci ty. This relat ionship i s shown i n
Figure 11.2.
Point potential i n f i l t r a t i on (FR) computed by the Ph i l i p equation
i s converted to effect ive i n f i l t r a t i on over the basin using the scheme
of Crawford and Linsley (1966). Let t ing SR represent the supply ra te
of ra in fa l I for i n f i l t r a t i on and OR represent the r a t e of generation of
ra in fa l I excess, the equations are
OR = -’ i f SR < FR (11.6a) SR . 2F R
( 1 1 .6b) FR . 2
OR = SR - -; If SR > FR
A schematic of these relat ionships is shown i n F igure 11.3. The r a i n f a l l
excess rate, OR, i s represented by the area between the dashed SR l i ne
and the l inear i n f i l t r a t i on capaci ty curve. The parameters for soi l
moisture accounting and in f i l t r a t i on are enumerated i n the fo l lowing:
21 3
F i g . 11.2
LL 0 w 3 B w > 5 w LL LL w
RGF X PS P
PS P WILTING FIELD POINT CAPACITY
(BMS= BMSN 1 (BMS = 0 1
SOIL- MOISTURE CONTENT
R e l a t i o n s h i p d e t e r m i n i n g e f f e c t i v e v a l u e o f so i l -mo is tu re p o t e n t i a l (PS)
w,
z - a a
t I I 1
/tF v, RAINFALL
INFILTRATION
0 25 50 75 100
'R
PERCENTAGE OF AREA WITH INFILTRATION CAPACITY EQUAL TO OR LESS THAN INDICATED VALUE
F i g . 11.3 R e l a t i o n s h i p d e t e r m i n i n g r a i n f a l l excess (OR) a s f u n c t i o n o f maximum i n f i l t r a t i o n c a p a c i t y (FR) a n d s u p p l y r a t e o f r a i n f a l l ( S R )
214
1 . Soil Moisture Accounting. The parameters consist of : ( a ) DRN -
A constant drainage ra te for red is t r ibu t ion of soi l moisture between SMS
and BMS, i n inches per day; ( b ) EVC - A pan coeff icient for convert ing
measured pan evaporation to potential evapotranspirat ion; ( c ) RR - The
average proport ion of da i l y r a i n f a l l that i n f i l t r a tes into the soi l for
the per iod of simulat ion excluding storm days; and ( d ) BMSN - Soil
moisture storage a t f i e ld capaci ty, in inches.
2. I n f i l t r a t i on . The parameters consist o f : ( a ) KSAT - The hyd rau l i c
conduct iv i ty at na tu ra l saturat ion, i n inches per hour; ( b ) RGF - Ratio
of suction at the wett ing front for soi l moisture at w i l t i ng point to that
at f i e ld capaci ty.
Impervious Area Ra in fa l I Excess Component
Two types of impervious surfaces are considered by the model. The
f i r s t type, effect ive impervious surfaces, a re those impervious areas that
are d i rec t l y connected to the channel drainage system. Roofs that d ra in
onto driveways, streets and paved pa rk ing lots that d ra in onto streets
are examples of effect ive impervious surfaces. The second type, non-
effect ive ihpervious surfaces, a re those impervious areas that d ra in to
pervious areas. An example of a noneffective impervious area i s a roof
that drains onto a lawn.
The only abstract ion from r a i n f a l l on effect ive impervious areas i s
impervious retention. Thi's retention, which i s user specified, must be
f i l l e d before runof f from effect ive impervious areas can occur. Evapo-
ra t i on occurs from impervious retention du r ing periods of no r a i n f a l I .
Rain f a l l i n g on noneffective impervious areas i s assumed to runoff
onto the surrounding pervious area. The model assumes th i s occurs
instantaneously and that the volume of runof f i s uni formly d is t r ibu ted
over the cont r ibu t ing pervious area. This volume i s added to the r a i n
fa1 l ing on the pervious areas p r i o r to computation of pervious area
ra in fa l I excess.
Optimization Component
An option i s jncluded i n the model to ca l i b ra te the soi l moisture
and i n f i I t ra t ion parameters for drainage basins hav ing measured r a i n f a l I
runoff data. The method of determining optimum parameter values i s
based on an opt imizat ion technique devised by Rosenbrock (1960).
21 5
Impervious area i s not included as a parameter to be optimized,
but i s a parameter to which simulated runof f volumes are very sensit ive.
Therefore, values of imperviousness should be determined accurately
before using the optimization option. I f i n i t i a l estimates of imperviousness
are grossly in e r ro r , resu l t ing volumes and peaks w i l l be grossly i n
error. I n that case, estimates of imperviousness must be adjusted by
the modeler. This adjustment may include rev is ing estimates of the
effective and noneffective impervious areas, perhaps by t r i a l and er ro r .
Determini st i c Runoff Routing Components
After determining "opt imum" parameter values and computing the
time series of r a i n f a l l excess, control i n the model i s t ransferred to the
runoff rou t ing component. The mathemat ical representation of an urban
basin requires discret izat ion of the total drainage area into a set of
segments. There a re three basic types of segments defined for the model:
channel segments, over land flow segments and reservoir segments. There
i s wide f l e x i b i l i t y to the approach one can take i n d i v id ing a basin
into segments for runof f computations. Guide1 ines for basin segmentation
are presented by Al ley and Veenhuis (1979) and Dawdy, et a l . , (1978).
Channel and Over land Flow Segments
A channel segment i s permitted to receive upstream inf low from as
many as three other segments, inc lud ing other channel segments and
reservoir segments. I t also may receive la te ra l inf low from over land flow
segments. The over land flow segments receive uni formly d is t r ibu ted la te ra l
inf low from r a i n f a l I excess. A schemat ic i l l us t ra t i ng the relat ionships
between channel and over land flow segments i s shown in Figure 11.4 - 5.
Kinematic wave theory i s appl ied i n the . r a i n f a l l runof f model to
both over land flow anci channel rout ing. The necessary equations to be
solved for each channel and over land flow segment a re
aa aA - ax + at = q ;
b Q = aA
(11.7)
(11.8)
in which the terms a re as previously defined.
F in i te dif ference approximations are used to solve Eqs. 11.7 and
11.8. To avoid the convergence and s tab i l i t y problems that can occur
wi th pa r t i cu la r numerical g r i d spacings ( i .e. the re la t i ve sizes of A t
and Ax ) , two f i n i t e dif ference methods of solut ion are used to solve for
216
JOVERIAND FLOW[ [oEFNDi I
PHYSICAL CHARACTERISTICS OF COMPONENTS IN THE SCHEMATIC REPRESENTATION
F ig . 11.5 Discret izat ion of u r b a n catchment in to segments
21 7
Q and A a t t he u n k n o w n gr id p o i n t s . The chosen s o l u t i o n p r o c e d u r e i s
made in t h e model p r o g r a m and depends u p o n the r a t i o , G , o f t h e
k i n e m a t i c w a v e speed to Ax/At.
(11 .9)
in w h i c h Q3 i s t he d i s c h a r g e a t node p o i n t 3 in the f i n i t e d i f f e r e n c e
g r i d as shown i n F i g u r e 11.6. I f 0 i s g r e a t e r t h a n o r e q u a l t o u n i t y ,
t he e q u a t i o n s u s e d a r e
T h i s i n v o l v e s o n l y mesh p o i n t s 1 , 2 and 4. I f 0 i s l ess t h a n u n i t y , t h e
e q u a t i o n s used a r e
At A 4 = A + q t + - 3 Ax ('1 - '3)
(11 .12)
( 1 1 .13) b 4
Q4 = aA
Ax a n d At v a l u e s a r e chosen to e n s u r e a b o u t 10 nonzero o r d i n a t e s
u n d e r the r i s i n g l i m b o f a n e q u i l i b r i u m h y d r o g r a p h and to keep
computa t i o n a l e r r o r s w i t h i n a c c e p t a b l e bounds . The U.S.G.S. recommends
t h a t A t b e se lec ted as
1 (11.14) At = 0.1 ( t
where t and tec a r e the k i n e m a t i c o v e r l a n d a n d c h a n n e l t imes of
e q u i l i b r i u m , r e s p e c t i v e l y . C o m p u t a t i o n a l e r r o r s h o u l d b e m i n i m i z e d i f
Ax and At a r e se lec ted so t h a t t h e c h a r a c t e r i s t i c p a s s i n g t h r o u g h p o i n t
1 a l s o passes t h r o u g h p o i n t 4. A c c o r d i n g l y , i t i s recommended t h a t u x
eo +
eo
b e se lec ted a s
At (11 .15a) a x = - LO
e o
( 1 1 .15b) L c a t Ax = - C
ec
fo r t he o v e r l a n d and c h a n n e l segments, r e s p e c t i v e l y . When Eqs. 11.14
and 11.15 r e s u l t in n o n - i n t e g e r v a l u e s , t he u s e r mus t r o u n d to t h e
neares t i n t e g e r .
Reservoir Segments
P r o v i s i o n i s made in the model f o r r e s e r v o i r r o u t i n g b a s e d o n t h e
c o n t i n u i t y e q u a t i o n . E i t h e r o f two r o u t i n g methods c a n b e used. One
method i s l i n e a r s t o r a g e r o u t i n g
21 8
Fig . 11.6 Four point f i n i t e dif ference g r i d
5 = co (11 .16)
i n which S i s the storage; Q i s the outf low; and C i s a constant.
Al ternat ively, the modif ied Puls rou t ing method can be used
-+ o2 = I ) + l 2 + - - At A t
( 1 1 .17)
in which I i s the inf low to the reservoir and the subscr ipts 1 and 2
refer to the beginning and end of the time in te rva l , A t , respectively.
The modif ied Pu ls method u t i l i zes a tab le of storage outflow values as
supplied by the user.
There are many ways of accounting for storage with the kinematic
method, a l l of which are, due to the l imi tat ion of the kinematic method,
approximations. That is, due to the fact that the dynamic equation omits
acceleration and deceleration of the water i n time and space, wedge
storage is omitted. Storage can only be included as channel type storage
(see Chapter 9) i f the discharge relat ionship can be described i n terms
of a kinematic type of discharge - depth equation.
T h e cont inui ty equation in the kinematic equations can however
be used to account for the l ag effect of storage i n one of two ways.
Inf low to a reach can be spread over the f u l l surface area of the reach
as demonstrated i n the example la te r i n th is chapter, o r the stream (o r
over land f low) surface area can be replaced b y a storage area at the
junctions of reaches with conduit reaches as indicated i n the program
i n Chapter 8.
Example Application
The model was appl ied to the Sand Creek T r ibu ta ry watershed near
Denver, Colorado. Th is drainage basin i s a 183 acre area of predom-
inant ly s ingle fami ly resident ia l land use w i th some mul t i fami ly land
use, a church, a recreational center, a f i r e stat ion, and two small parks.
The basin has some storm sewers in i t s upper end but re l ies mostly on
street gutters and concrete l ined open ditches for flow conveyance.
Detailed records of ra in fa l I and streamflow a re collected at 5 minute
intervals. A stage discharge re la t ion was developed using flow p r o f i l e
analysis and discharge measurements made du r ing storm runoff .
Two sample runs are discussed. The f i r s t r u n was an optimization
r u n to ca l i b ra te the model on an antecedent per iod of record. In the
second run, the soi I moisture accounting and i n f i l t ra t ion parameters were
set to the i r f i n a l values from the f i r s t r u n and ten storm events were
simulated.
Before any simulat ions were performed, the watershed was delineated
into sub-basins (over land flow segments) and a drainage network
(channel segments). A schematic showing how the watershed was
approximated with the over land and channel segments i s given i n
Figure 11.7.
The ra t iona le behind the basin segmentation i s as follows: s ta r t i ng
at the basin outfal I, i t was f i r s t noted that the major drainage system
of the basin consisted of concrete l ined ditches which were located i n
the positions marked by channel segments CH20, CH21 and CH22. In
analyzing the reach of concrete l ined d i tch comprised of CH20 and CH21,
i t was noted that a street, which drained 14 acres of land Q(F03, inter-
sected th i s reach. Therefore, t h i s reach of concrete l ined d i tch was sub-
d iv ided into channel segments CH20 and CH21, and the intersecting street
was designated as channel segment CH23. Overland flow segments QIFOl,
220
,c)Fo2. ---- CH23 CHANNEL SEGMENT JTOl JUNCTION AND NUMBER - GENERAL DIRECTION OF OVERLAND FLOW
OVERLAND FLOW SEGMENT AND NUMBER
F i g . 11.7 Schematic representat ion of Sand Creek t r i b u t a r y watershed
22 1
OFO2, OF03, OF04 and OF09 were then delineated based on th i s channel
segmentation. I t should be noted that over land flow segment F04 does
not have balanced lengths of over land flow to CH21. To fu r ther subdiv ide
th i s over land flow segment would also requ i re that segments CH21 and
OF09 be fu r ther subdivided.
The unallocated concrete l ined d i tch was then assigned as channel
segment CH22. Overland flow segments OF06 and OF05 were then delineated.
To avoid the need to subdiv ide channel segment CH20 which would requ i re
subd iv id ing over land flow segments OF01 and OF02, channel segment CH25
was used to bypass channel segment CH20. A junct ion segment, JTOl,
was required to sum the flow from the two channel segments at the outlet
of the basin. F ina l l y , the remaining pa r t of the basin was drained by
a street which was assigned as channel segment CH24.
Once the basin was segmented, the sub-basin boundaries w e r e f i e ld
checked and representative channel cross sections determined. Channel
slopes were determined from the drainage maps, and over land flow slopes
were estimated from the U.S. Geological Survey topographic map for the
area and the street corner elevations shown on the City of Denver
drainage maps. Sub-basin areas w e r e planimetered and lengths of over-
land flow were computed by d i v id ing the area of each sub-basin, i n
square feet, by the length, in feet of the channel segment into which
i t contr ibutes lateral inf low.
TABLE 11.1 Model Simulation Results for Sand Creek Tr ibu tary Watershed
Runoff Runoff Volume Peak Flow Event Number
i n c f s . .
Date in inches Measured Simulated Measured Simulated
1
2 3 4 5 6 7 8 9
10
7-7 2-73 7-1 9-73 7-22-73 7-24-73 7-30-73 8-07-73 9-1 1-73 9-1 1-73 7-22-74 7-30-74
0.08
0.16 0.055 0.33 0.063 0.70 0.073 0.23 0.20 0.53
0.08 0.19 0.052 0.28 0.082 0.76 0.14 0.16 0.32 0.47
32 68 22
104 32
2 36 48
143 98
251
23 74 14 97 28
2 80 58 68
117 2 16
222
The per iod of record to be simulated was July 12, 1973 to July, 30,
1974. To establ ish i n i t i a l moisture condit ions for the beginning data,
an optimization r u n was conducted for the per iod May 1 , 1973 to July
12, 1973. Using input values for r a i n f a l l d a i l y pan evaporation and
recorded runoff the model was "cal ibrated" b y determining optimal values
for the i n f i l t r a t i on and soil moisture accounting parameters. A second
r u n was then conducted to simulate watershed runoff (hydrographs) du r ing
the specified simulat ion period. Results are shown i n Table 1 1 . 1 .
A MODEL FOR RURAL WATERSHEDS
The model described for u rban stormwater simulat ion could be appl ied
easi ly to r u r a l watersheds. I n th is section, however, let us examine
another model which has been appl ied only to r u r a l watersheds, and
which uses f i n i t e elements to solve the kinematic equation instead of
f i n i t e di f ferencing. The fol lowing model was developed for r u r a l water-
sheds i n ag r i cu l tu ra l land use, but i t has been suggested i t also can
be appl ied to surface mining disturbed watersheds.
A f i n i t e element storm hydrograph model (FESHM) has been developed
at V i rg i7 ia Polytechnic Ins t i tu te and State Universi ty as p a r t of a
program to develop a d is t r ibu ted parameter model to simulate flow on
ungauged watersheds (Ross, et a l . , 1978 and 1982). Spatial v a r i a b i l i t y
was a requirement because a long range goal i s to be able to simulate
not only runoff from mixed land use watersheds, but also waterborne
pol lutants. Thus, FESHM was developed to integrate spat ia l and temporal
var iat ions in c l imatic and watershed character ist ics.
The model consists of two major components: a p rec ip i ta t ion excess
generator and a f lood rou t ing algor i thm which routes the excess along
over land flow elements and down the stream channel elements.
Precipitation Excess
The calculat ion of r a i n f a l I excess depends on the spat ia l d is t r ibu t ion
of two watershed character ist ics, land use and soi l mapping uni ts. A
map of land use patterns i s superimposed on the watershed s i te map,
def in ing soi ls to create a hydrologic response u n i t (HRU) map. Each area
with a unique land use and soi l mapping combination i s referred to as
an HRU. A given ra in fa l I on the watershed w i l l resul t i n a di f ferent
amount of p rec ip i ta t ion excess from each HRU.
223
1 I2
The amount of p r e c i p i t a t i o n excess i s d e t e r m i n e d u s i n g Hol t a n ’ s
i n f i l t r a t i o n e q u a t i o n ( H o l t a n , 1961). T h i s i n f i l t r a t i o n model h a s been
a p p l i e d to a w i d e r a n g e of d a t a b y S h a n h o l t z and L i l l a r d (1970) and
H o l t a n , e t a l . , (1975) w i t h r e a s o n a b l e success. I t was i n c l u d e d in FESHM
p r i m a r i l y because the d a t a necessa ry to d e f i n e model p a r a m e t e r s c l o s e l y
p a r a l l e l t h e concep ts f o r d i v i d i n g a d r a i n a g e a r e a i n t o HRUs.
, = o
Flow Routing
The second p a r t o f FESHM r o u t e s p r e c i p i t a t i o n excess t o t h e o u t l e t
o f t he wa te rshed . To a c c o m p l i s h t h i s , t h e w a t e r s h e d i s d i v i d e d i n t o
o v e r l a n d and s t reamf low e lemen ts a s shown in F i g u r e 11.8. The n u m b e r
o f e lements to b e used depends o n the h y d r a u l i c and h y d r o l o g i c he tero-
g e n e i t y o f t he wa te rshed . The HRUs t h a t o c c u r in each o v e r l a n d f l o w
element a r e c a t a l o g u e d , and the r a i n f a l l excess f rom e a c h HRU i s
w e i g h t e d b y i t s f r a c t i o n a l a r e a in t h e e lement . T h i s r a i n f a l l excess
i s t h e n r o u t e d t h r o u g h o v e r l a n d f l o w e lements u s i n g a f i n i t e e lement
a p p r o x i m a t i on o f t h e k i n e m a t i c w a v e mode l .
U s i n g t h e G a l e r k i n t e c h n i q u e ( L a p i d u s and P i n d e r , 1982) and l i n e a r
v a r i a t i o n of p a r a m e t e r s w i t h i n an e lement , t h e e lement e q u a t i o n becomes
(11.18)
where s c r i p t “ell i s t he e lement l e n g t h , e q u i v a l e n t to t h e t l o w l e n g t h
ac ross e a c h e lement . The t ime d i f f e r e n t i a l o f a r e a i s r e p l a c e d b y a
s i m p l e f i n i t e d i f f e r e n c e a p p r o x i m a t i o n . Thus
2A - A ( t + A t ) - A ( t ) a: At
- _
The f i n a l e lement e q u a t i o n i s
(11.19)
(11.20)
224
Model Application
FESHM has been tested o n watersheds in seven s tates c o v e r i n g a
wide r a n g e of l a n d use, topography a n d c l i m a t i c cond i t i ons . D r a i n a g e
areas r a n g e d from approx ima te l y 2 acres to 193 s q u a r e mi les. The s i ze
F i g . 1 1 .8 Watershed map showing o v e r l a n d a n d channel f i n i t e elements
225
o f the w a t e r s h e d did n o t l i m i t i t s a p p l i c a t i o n ; t h e a c c u r a c y o f s i m u l a t i o n s ,
however , most l i k e l y was a f u n c t i o n o f t h e q u a l i t y and r e s o l u t i o n o f t h e
a v a i l a b l e d a t a .
The model was a p p l i e d to s i x wa te rsheds in V i r g i n i a , ranging in s i z e
f r o m 183 to 1,058 a c r e s ( 1 a c r e = 0.4047 h e c t a r e s ) . S i x t e e n s t o r m e v e n t s o n
these wa te rsheds w e r e s imu l a t e d and t h e s i m u l a t e d h y d r o g r a p h s compared
w i t h the obse rved . These r u n s were made w i t h o u t a n y e f f o r t s to c a l i b r a t e
the mode l , and t h e r e f o r e a r e t y p i c a l o f t h e m o d e l ' s p e r f o r m a n c e o n
u n g a u g e d wa te rsheds . The r e s u l t s a r e summar i zed in T a b l e 11.2. T h e mean
e r r o r in the p r e d i c t e d s to rm vo lume was 4.4 p e r c e n t , w i t h a s t a n d a r d
d e v i a t i o n o f 49.9 percen t . The mean e r r o r in the p r e d i c t e d p e a k d i s c h a r g e
was 22.6 p e r c e n t , w i t h a s t a n d a r d d e v i a t i o n of 50.1 percen t . The l a r g e
s t a n d a r d d e v i a t i o n i s d u e l a r g e l y to the i n c l u s i o n o f some v e r y s m a l l
( vo lume) s to rms. The model t ends to s i m u l a t e bes t those s to rms w i t h a
r e t u r n p e r i o d o f 20 y e a r s o r g r e a t e r .
An e x a m p l e o f t he s i m u l a t i o n r e s u l t s o f a s to rm even t on P o w e l l ' s
Creek i s shown i n F i g u r e 11.9. The HRU map i s g i v e n i n F i g u r e 11.10, and
the s u b d i v i s i o n of the w a t e r s h e d i n t o e i g h t f i n i t e e lements i s shown in
F i g u r e 1 1 . 1 1 . For t h i s s to rm, the e r r o r in p e a k d i s c h a r g e was 3.4 p e r c e n t ,
and the e r r o r in s to rm vo lume was 2.1 percen t . A s e n s i t i v i t y a n a l y s i s o f
e lement s i z e i n d i c a t e d l i t t l e improvement to u s i n g s m a l l e r e lements .
TABLE 11.2 Compar i son o f . R u n o f f Volume and Peak F low f o r
S i m u l a t e d and Recorded F l o w s in V i r g i n i a Watersheds
(Ross, 1978)
S t o r m Runof f Vo lume ( i n ) Peak F low ( c f s) Watershed Even t Recorded S i m u l a t e d Recorded S i m u l a t e d
Powe l l s Creek 10/10/59 0.73 0.46 109.88 114.34 5/31/62 0.92 1.34 241.10 359.71 7/11/65 2.06 2.38 419.82 775.13
6/12/58 0.44 0.46 83.29 78.32 6/24/58 0.43 0.44 91.26 94.44 9/19/60 0.73 0.47 45.74 59.95
Pony M o u n t a i n B r a n c h
Rocky Run B r a n c h
C r a b Creek
B r u s h Creek
Ches tnu t B r a n c h
7/23/70 1.44 1 .13 327.80 544.32 O/ 5/72 5.79 3.26 609.77 551 .OO
8 /21/66 0.19 0.26 179.20 231.16 0/24/71 0.57 0.30 180.54 137.59 6/16/76 0.49 0.31 205.58 148.34
7 /22 /59 0.44 1 .08 791.47 1,741.56 9/30/59 1.14 0.86 924.70 434.69 5/28/73 0.27 0.43 296.38 623.58
8/23/67 0.67 0.63 416.08 595.84 8 / 4 /74 0.43 0.49 339.65 448.43
226
4 50
400
3 50
300
250 v) LL 0
200 <3
I
a a 0 150 v, 0
100
50
0
POWELLS CREEK WATERSHED 71 I 1/65 STORM EVENT
-SIMULATED HYDROGRAPH X- RECORDED HYDROGRAPH
3.4% PEAK ERROR 2. I '7" VOL. ERROR
I I
1.0 2.0 3.0 4.0
TIME, HRS
F i g . 11.9 Compar i son of s i m u l a t e d and r e c o r d e d h y d r o g r a p h s Powe l l s Creek wa te rshed , V i r g i n i a
53
F i g . 11.10 HRU m a p of Powe l l s Creek wa te rshed
227
228
POWELCS CREEK HALIFAX COUNTY,VA
WATERSHED BOUNDARY
CHANNEL ELEMENT
OVERLAND FLOW ELEMENT NP
SCALE IN FEET
Fig . 1 1 . 1 1 F i n i t e element map f o r Powells Creek watershed
229
OVERLAND FLOW AND STREAMFLOW PROGRAM
The simpl ic i ty wi th which a computer program can be assembled from
the basic kinematic equations i s demonstrated here. The program wr i t ten
in BASIC for an HP85 micro computer i s appended. I t I S a s impl ist ic
program incorporat ing over land flow and channel flow in series. Con-
secut ive channels can feed into designated downstream channels, enabl ing
branches to be included.
The channels a re assumed for s impl ic i ty to be rectangular i n cross
section wi th f lood planes on ei ther side (F ig . 11 .12 ) , which i n fact can
form the overland flow planes. Flow planes are assumed to be rectangular
and of uniform slope and roughness.
An assembly of planes and channels of the type envisaged i s
i l l us t ra ted i n Fig. 11.13.
The model was developed to study the effect of bank storage on f lood
rout ing. The f lood planes act as dead storage - flow on them i s assumed
lateral - from or to the channel where the longi tudinal flow occurs.
Channel depressions can a l x , be included - by widening the stream bed.
Flow i s obviously assumed to be kinematic - that i s backwater effects
and unsteady flow are neglected.
Overland flow of sub-catchments i s calculated using the kinematic
equations for rectangular planes. Inf low i s the net r a i n f a l l and outflow
i s assumed over the f u l l width B of the plane. Flow r a t e i s calculated
from an equation of the form: ,
Q = 0 “ y ( 1 1.21 ) m
where c1 = s Ifl/n
S = slope in direct ion of flow
n = Manning roughness coefficient
m = 5/3
Inf low to channel reaches can be from both sub-catchments and
upstream channels, but not r a i n , as the width i s assumed neg l ig ib le ,
as well as losses along the channel.
A l l storage i n channels and over land i s assumed to be pr ism storage
i.e. no wedge storage or dif ference between bed slope and water surface
slope i s permitted. This i s i n accordance with the kinematic s impl i f icat ion
but provided distance in te rva ls are l imited, i t i s not inaccurate. Flow
of f the sub-catchments i s also assumed to be a function of the average
depth at the outlet end, so i f the var ia t ion i n depth i s l i ke l y to be
signi f icant a cascade of planes or planes leading into wide channels
representing planes may be preferable.
230
The kinematic equations a re ab le to accommodate storage by re-
wr i t ing as follows the cont inui ty equation
LBdy = d t (A ie + Qi - Q o ) (11 .23)
where A i s the area of the sub-catchment w i th excess r a i n f a l l r a t e i e
and Qi - Q i s the upstream inf low minus downstream outflow which i n
tu rn are functions of the water depth y and LB i s the surface length
times width of the channel p lus f lood planes (based on the water depth
at the previous time in te rva l as a simple exp l i c i t solut ion i s used). I n
fact the term A i e f a l l s away for the channel storage computations and
LB i s equal to A for over land flow computations.
Since the simulat ion i s not s t r i c t l y a f i n i t e dif ference solut ion to
the d i f fe ren t ia l cont inui ty equation bu t merely a flow balance at successive
nodes or junctions the speed of propagation of disturbances i s not s t r i c t l y
corrrect. Thus some numerical d i f fusion i s bound to occur unless a l l
reach lengths are proport ional to dx/dt , the wave speed. The advantages
of a va r iab le reach length, however, appear to outweigh the disadvant-
ages. That i s da ta can be fed in in na tura l (unequal) channel lengths,
and time in te rva ls can be extended above what i s normal ly required for
kinematic sirnul at ion. I
Fig. 11.12 Channel section
23 1
Subcatchment w i t h ove r land f l ow
lLC1 Flood p lane
7 Storage b a s i n \\\
Channel \\\y Q4
Fig . 11.13 Possible layout of sub-catchments and channels for kinematic flood Dlane model
Data input
The basic program requests interact ively the fo l lowing data. The
da ta for each l ine i s typed i n consecutively w i th commas separating the
numbers.
F i r s t Input L ine
Simul at ion durat ion, minutes
In te rva l between success;ve flow calculat ions, minutes
In te rva l between tabulat ion of flows and depths for each channel, minutes.
Second Input L ine
Preceding week’s r a i n f a l l i n mm
Channel no. at which a hydrograph i s required
Estimate of maximum flow i n channel (m’/s) for hydrograph plot a x i s
scal ing.
Th i rd Series of Input (each Time in te rva l )
Rainfal I intensi ty i n mm/h (assumed uni formly d is t r ibu ted)
Data Lines at end of program (stored as a f i l e to save input each r u n ) .
Downstream channel no. into which the sub-catchment discharges
Sub-catchment surface area, mZ
232
O v e r l a n d and streamf low p r o g r a m I i s t i n g
5 P R I N T 16 P R I N T "OVERLAND b CHANNEL K I
NEMfT IC FLOW SIRULATION CHT S I R
F(38) ,U<38 , ,N(38 ) ,0 (30 ) ,X (36 ),2(38>,B138,,V(JB,,R(30)
36 D I M G < J B ) , E ( 3 9 ) , H ( J B ) , U ( 3 8 ) ,
46 D I S P S IMULATIONt , INTERUAL.T
5 6 INPUT ,, T 1, T2 , T 4 52 D I S P PPECEEDING WK KHINmra,H
28 DIN J ( 3 6 , , H ( 3 8 > , L ( 3 8 ) , S < 3 8 ) ,
Q<38,:Y<38,,P<36,6, ,C(38,6)
H B U L H T J O N t ( a l 1 m i n s > " ;
YDROGPH CHANL no, PMAXrn3/'s"; 53 I N P U T R5,K5,(;11 55 P R I N T 68 P R I N T 'CHANL Qm3s DEPTHm" 78 Il=B . - -~ 88 A l = 6 85 R=5 /3 98 FOR 1=1 TO 56
1 8 6 READ J ( I ) , A ( I > , L ( I ) , S [ I , , R 3 ,
161 I F I > 1 THEN 185 105 I F J ( I > = 8 THEN 2 8 6 116 S ( I ) = S < I > " . 5 / R 3 126 Y ( I ) = 8 1 3 6 Q ( I i = 8 146 H l = A l + A < I > 158 I l = I 1 + 1 168 NEXT I 20B K 1 = @
F t I j
2 4 5 Kl=Kl+l 2 5 8 I F O<K>=B THEN 280 2 7 8 NEXT K 286 FOR K = l TO K 1 282 I F N ( K > O K 5 THEN 298 284 K5=k 298 K2=B - - . . . - - 380 FOR 1=1 TO I 1 318 I F J < I > < > N ( K > THEN 356 326 K 2 = K 2 + 1 338 P ? K , K Z ? = I JSO NEXT I 368 P(K ,K2+1>=0 365 K4=6 376 FOR K3=1 TO K 1 388 I F O ( K 3 ) < i N ( k J THEN 4 1 8 396 K4=K4+1 4 8 8 C(K,K4)=K3 4 1 8 NEXT K 3
426 C ( K > K 4 + 1 > = 6 436 NEXT K 45@ GCLEAR 468 SCALE - T 2 , T l * - ( Q 1 / 2 0 ) , Q l 4 x 1 XAXIS e , m , e , ~ i 486 Y w I s 8, 160 ,8 ,~1 496 MOVE T2,-((;11/?6> 588 LABEL ' T h o u r s
515 Q l = I N T < Q l ) 516 MOVE - T 2 > Q l t . 9
528 LABEL 'Q!3/s- " h V A L S ( Q l > & ' I
546 FOR T 6 = 6 TO T1-T4 STEP T 4 545 FOR T5=T2 TO T 4 STEP T 2 5 4 8 T=T6+T5 558 D I S P 'RHINms/h" 555 INPUT R ( 1 ) 578 FOR 1=1 TO I 1 5 7 2 R ( I ) = R ( l ) 575 U(I>=F<I)*(l+<il686/(R5+16~~
~ . 5 - 1 ) * 2 . 7 1 8 ~ < - ( . 8 8 1 * T S 6 8 1 ) ) 586 Y(I>=(R(I)-U(I)>/3686806%T2%
6 8 + Y ( I ) - P C I ) / A ( I ) * T 2 * 6 6 596 I F Y<I)>B THEN 628 686 Q<I)=6
N CHFINL L V A L S ( K 5 )
616 GOT0 6 3 6 626 Q(I)=S(I)*YCl)*~tH(I~~L~I~ 636 NEXT I 6 4 6 FOR K = l TO K 1 656 Ql=-U(KJ 666 FOR K2=1 TO 6
680 Q l = Q l + Q ( P ( K , K 2 ) , 698 NEXT KZ
716 I F i < K , K 4 > = 8 THEN 7 4 8 726 81 =Ql+Ld< C'K, k 4 > > 738 NEXT K4 748 B l = B ( K > 742 I F H ( K ? < = E ( K / THEN 7 5 6 7 4 4 B l = B l + ( H t K ) - E < K ) ) * G ( K ) 756 H ( K , = H ( K ) + Q l S T 2 * 6 @ / X ( K ' , B 1 752 I F H ( K ) ) 8 THEN 766 7 5 4 H < K > = 81 766 NEXT K 778 FOR K = l TO K 1 788 U(K>=Z(K>*H<K>*#*BtK) 782 I F H < I O < = E ( K ) THEM 888
578 I F P < K t K 2 ) = 6 THEN 768
788 FOR ~ 4 = i T O tj
785 U ( K ) = U ( h , + V ( h ) * t t K ) * C H t k , - E c K ) ) * C M + 1 )/2AFl
795 IMAGE JD,DDDD m,nm DUD B 0 B NEXT K 862 Q2=6 883 FOR 1=1 TO I 1 884 Q Z = Q Z + H ( I > * ( P ( I ) - F ~ I , , , 3 6 g 8 8
88 805 NEXT I Se7 MOVE T,uz 868 PLOT T,Q2 818 MOVE T,W(K5)
233
828 PLOT T , W < K 5 ) 825 NEXT T 5 838 PRINT "TIRE m i n s " , ~ 831 FOR K = l TO K 1 832 PRINT USING 795 i N(k>,U(K>,
H < K > 833 NEXT K 834 NEXT T 6 - 835 COPY 846 END l e l b . DATA 1.6b8bbbbb,166bb,.62,.
1,5
1,s
1,5
1,s
1826 DATA 2 ~ 6 8 0 0 6 6 0 6 ~ 1 6 8 8 8 , . 6 2 ~ .
l b 3 0 DATA 3 , 6 6 8 6 8 b b 6 ~ 1 8 8 b b , . 6 1 ~ .
1848 DATA 4,68Bbbbbb,lbBb8,.61,.
1658 DATA 5,6bb6b6b6~1b88b,.b1,.
, l 6 b
, l b b 1118 DATA 2,4,1b@bb, .bZ, .b5,1b,2
1120 DATA 3,5,1bbb6, .b2, .b5,1b,2
1136 DATA 4,5,2b86b, .62, .85,16,2 I 160
I 168 1148 UHTA 5 ~ 8 ~ 2 b 0 6 8 , .61, .84,15,1
> 28b
S I MULA T I ON ? , I NTERV AL ., T AEUL A T I i t N t (all m i n s , ? 360 I 3u I68 PRECEEDING WK RAINmm,HYDROGPH CH ANL no,QMAXm3/c? 1b0,5, l @ 8 8 K!AINmm,'n .? 38 i'i I Nmm /h
34 PHI Nmm/ h
38 R A 1 Nmm h
38 PAIkmm, t i
36 PA I Nmm,'h
I
I
,
OUEHLANEl &. CHANNEL KINEMATIC F L OW SII'IULHTION CATSIPI
CHANL Qn3s DEPTHm T I M E m i n s 66
1 3.53 .28T 2 7 . 8 b ,462 3 5 . 8 1 .354 4 2 .95 .25? 5 2 .58 ,201
1 2 5 . 1 2 931 2 58.92 1.427 3 32.95 1.P% 4 33 .54 1.188 5 21.21 .716
1 76.63 1.819 2 164.12 2 .186 3 74.53 1 . 7 m 4 173.12 2 .965 5 143.32 2.235
T I H E mins 128
TIPlE m i n s 186
T I N E m i n s 1 76 .79 2 72.81 3 56.89 4 2 1 i . 3 3 5 346. li
1 6 1 . 8 3 2 54.62 3 44 38 2 178.29
2 9 5 . 2 2
I 46.93
T I N E m inz
T IME m i n s
1 . 8 2 1 1.764 1.5b8 3 . 3 9 9 3.794
1 . ddb = '- .- 1 .484 1 .318 3.818 3 .477
34 RA 1 Nmm 1 h
234
Overland flow distance, m.
Average slope over the sub-catchment in the direct ion of flow towards
the channel, m/m
Manning roughnesses of the sub-catchment (note roughness for f i r s t
catchment i s also used for a1 I f lood planes)
Steady uI timate i n f i l t r a t ion loss in mm/h
The end of the sub-catchment da ta i s iden t i f ied by typ ing i n a l ine of
s ix zeros separated by commas.
Data Lines; Channel reaches;
Channel number
Downstream channel number
Length of channel, m
Slope m/m
Man n i n g roughness
Bed width
Channel depth, m
Flood plane width per m depth
The last channel must be the lowest (downstream) channel which
is ident i f ied by i t s downstream channel no. ( the second item of data
in the l ine) being zero.
The program commences p r i n t i n g a table of channel nos, flow rates
in m 3 / s and water depths i n m, at each successive time in te rva l . A plot
of the hydrograph at the designated node appears on the screen
simul taneously and th i s hydrograph i s subsequently plotted on paper.
I n f i l t r a t i on and Seepage
I n the previous example, excess r a i n f a l l was routed overland. That
i s the user has to insert an i n f i l t r a t i on r a t e for each sub-catchment
in the data. I n fact losses include an i n i t i a l abstract ion and then a
time decreasing i n f i I t rat ion. The Horton (1933) equation suggests an
exponential l y decreasing loss, whereas the Green-Ampt (191 1 ) equation
indicates a less r a p i d decrease i n absorption. The la t te r equation i s
based on a s impl ist ic model of soil pores and the decrease in i n f i l t r a t i o n
i s based on saturat ion of the soi l pores. Ei ther of these equations could
be programmed read i l y and the indices made functions of preceding
r a i n f a l l or moisture conditions.
A port ion of the i n f i l t r a t i on reaches the water table. The water
below th i s level flows la te ra l l y under a hyd rau l i c gradient. The deeper
the groundwater the higher the flow rate, which can however be exceed-
i ng l y slow and may not contr ibute to the hydrograph due to a storm
except as a f a i r l y steady basic f low. Where the water table i s h igh
however, or i f there i s a perched water tab le or h igh rock level, inter-
face flow may occur dur ing or soon af ter a storm. That i s seepage
emerges on the surface or into streams short ly a f te r a storm commences
and th i s must therefore be included in the model. Ground water flow
can be model led using the kinematic equations too.
REAL-T I M E MODELL I NG
A model such as that described above was employed on a real-t ime
bas is to predict inf lows into a reservoir feeding a hydro-electr ic stat ion
(Stephenson, 1986). Rainfal I s igna ls from t ipping-bucket gauges d i s t r i -
buted over the 5000 square ki lometre catchment were telernetered to a
central processing u n i t l i nked to two micro computers. One computer
processed the data, f i l ed i t on f loppy discs and at the same time gave
a visual d isp lay and pr in tou t of r a i n f a l l over the last hour, and
summaries of totals for preceding week etc. The other computer had the
catchment model which predicted flow ra te into a reservoir. Both storm
runof f on a short term bas is and low flows over a longer time span were
predicted enabl ing the reservoir to be operated to optimize hydro-power
generat ion.
The hardware and da ta gather ing system were thus low cost bu t
compatible wi th the accuracy which could be expected. The computer
program, based on the kinernat i c equations and 15 sub-catchments was
also at a level matching the da ta and accuracy which could be expected.
The program i s also ab le to p red ic t ahead the flows based on al ter-
na t ive assumed r a i n f a l l rates. The system lag was 12 to 24 hours, which
was general ly suf f ic ient to operate gates, but not for conservation of
water over the d r y season.
236
REFERENCES
Al ley, W.M. a n d Veenhuis, J.E., 1979. Determination of b a s i n character- i s t i cs for an urban d i s t r i b u t e d rou t ing , r a i n f a l I - runof f model, in Proceedings, Stormwater Management Model (SWMM) Users Group Meeting,
Crawford, N.H. a n d L ins ley , R.K., 1966. D i g i t a l s imu la t ion in hydro logy : Stanford Watershed Model I V . Technical Report 39, C i v i l Eng ineer ing Department, Stanford Un ivers i ty , Cal i fo rn ia .
Dawdy, D.R., L i c h t y , R.W. a n d Bergman, J.M., 1972. A r a i n f a l l runof f s imulat ion model fo r est imat ion of f lood peaks fo r smal l d r a i n a g e bas ins. U.S. Geological Survey Professional Paper 506-8.
Dawdy, D.R., Schaake, J.C., Jr . a n d A l ley , W.M., 1978. User 's gu ide f o r d i s t r i b u t e d r o u t i n g r a i n f a l I runof f model. U.S. Geological Survey-Water Resources Inves t iga t ions 78-90.
Green, W.H. a n d Ampt, G.A., 1911. Studies of so i l phys ics, I , Flow of a i r a n d water through soi ls. J. Agr ic . Science, 4 ( 1 ) , p 1-24.
Hol ton, H.N., 1961. A concept fo r i n f i l t r a t i o n estimates in watershed engineer ing. U.S. Dept. of Agr icu l tu re , A g r i c u l t u r e Research Service, ARS-4 1 -45, Wash i n g ton, D . C . U.S. Dept. of Agr icu l tu re , A g r i c u l t u r a l Research Service, Technical B u l l e t i n No. 1518, Washington, D.C.
Horton, R.E., 1933. The r o l e of i n f i l t r a t i o n in the hydro log ica l cycle. Trans. Am. Geophys. Union., Hydro logy papers, p 446-460.
Lap idus , L. a n d Pinder , G.F., 1982. Numerical so lu t ion of p a r t i a l d i f f e r e n t i a l equat ions i n science a n d engineer ing. John Wiley a n d Sons, New York, N.Y.
LeClerc, G. a n d Schaake, J.C., J r . , 1973. Methodology f o r assessing the po ten t ia l impact of u rban development on u r b a n runof f a n d the r e l a t i v e ef f ic iency of r u n o f f cont ro l a l te rna t ives . Ralph M. Parsons Labora tory Report No. 167, Massachusetts I n s t i t u t e of Technology, Cambridge, Mass.
P h i l i p , J.R., 1954. An i n f i l t r a t i o n equat ion w i t h phys ica l s ign i f icance. Proceedings of the Soil Science Society of America, Vol. 77, pp . 153-157.
Rosenbrock, H.H., 1960. An automatic method of f i n d i n g the greatest o r least va lue of a funct ion. Computer Journal, Vol. 3, p p . 175-184.
Ross, B.B., et a l . 1978. A model f o r e v a l u a t i n g the effect of l a n d uses on f lood flows. B u l l e t i n 85, V i r g i n i a Water Resources Research Center, V i r g i n i a Poly technic I n s t i t u t e a n d State Un ivers i ty , B lacksburg , V i r g i n i a .
Ross, B.B., et al. 1982. Model f o r s imu la t ing r u n o f f a n d erosion in ungauged watersheds. Bul le t in 130, V i r g i n i a Water Resources Research Center, V i r g i n i a Poly technic I n s t i t u t e a n d State U n i v e r s i t y , B lacksburg , V i r g i ni a.
Shanholtz, V.O. a n d L i l l a r d , J.H., 1970. A s o i l water model f o r two contrast ing t i l l a g e systems. B u l l e t i n 38, V i r g i n i a Water Resources Research Center, V i r g i n i a Poly technic I n s t i t u t e a n d State U n i v e r s i t y , B lacksburg, V i r g i n i a .
Stephenson, D., 1986. Real-time k inemat ic catchment model fo r h y d r o operat ion. Proc. ASCE Energy Journa I .
pp . 1-27.
Holton, H.N. et a l . 1975. USDAHL-74 Revised model of watershed hydro logy .
237
CHAPTER 12
GROUNDWATER FLOW
GENERAL COMMENTS
Although the major i ty of the book analyses surface runoff the same
theory i s app l i cab le to sub-surface flow. That i s kinematic ana lys is can
be used to study a major i ty of groundwater flow problems that re la te
to catchment y ie ld and response to storms.
The largest problem i s re la t ing the contr ibut ion to groundwater flow
to in f i l t ra t ion . Although prac t ica l l y a l I lateral flow underground occurs
in the zone beneath the water table, a l l water permeating i n from the
surface does not reach the water table. Some i s held by cap i l l a ry forces
onto soi l par t i c les in the semi saturated zone.
There are also i n some si tuat ions bar r ie rs to flow under the surface.
Ei ther obstacles i n the path of the flow stop lateral flow, (Har r , 1977)
or horizontal impermeable layers create perched water tables which
resul ts in more than one lateral flow path underground (Weyman, 1970).
The water table can i n some instances emerge on the surface, ei ther for
the rest of the flow path down to a stream, o r to disappear again when
a more porous aqu i fe r i s reached. The ana lys is below i s therefore
somewhat simp1 is t i c but demonstrates the pr inc ip les of kinemat ic hydrology
can be used to estimate groundwater flows. The accuracy of the ana lys is
i s more l i ke ly to be l imi ted by lack of da ta on the aqui fer than the
ana ly t i ca l method. The mechanism of groundwater contr ibut ion on slopes
was explained by Dunne (1978). Further ana lys is of the ro le of sub-
surface flow i s given by Freeze (1972).
FLOW I N POROUS M E D I A
Whereas fo r over land flow the Manning equation was found most
appl icable for est imating flow-depth relat ionships, flow through ground
i s general ly laminar.
I n the flow equation
(12.1) m
q = “ Y
for laminar over land flow
CY = gs/3v (12.2)
and m = 3 where V i s the kinematic viscosity of the f l u id , g i s g rav i t y
and S i s the energy gradient, and for tu rbu len t and over land flow
238
a = S "/N for the Mann ing equa t ion
a n d 01 = CS " for the Chezy equa t ion
(12.2b)
(12.2c)
For f low th rough porous media the head loss i s genera l l y l a m i n a r b u t
the genera l equa t ion i s
S = av
where b i s 1 for l a m i n a r f low, i nc reas ing th rough 1.85 fo r coarse
p a r t i c l e s (Ahmed a n d Sunada, 1969) to 2 fo r t u rbu len t f low th rough l a r g e
rocks (Stephenson, 1979). A genera l express ion i s
S = KvZ /gdn2 (12.4)
where K = Klev/vd + Kz (12.5)
here e i s the po ros i t y , v the apparen t ve loc i t y q/h a n d d a rep resen ta t i ve
g r a i n size. For most a q u i f e r s Kz i s n e g l i g i b l e so
S = K1vv/ged2 (12.6)
general l y an equa t ion of the fo l l ow ing form i s used
S = v / k (12.7)
where k i s the permeabi I i ty ,
(12.3) b
then q = kSh (12.8)
so = kS a n d m = 1 (12.9)
I t i s assumed the scale of the system i s such tha t su r face tension
forces can be neglected. A l though these may be s i g n i f i c a n t above the
water t a b l e the l a t e r a l f low in t h i s r e g i o n i s u s u a l l y n e g l i g i b l e .
F i g . 12.1 De f in i t i on sketch fo r f low over a s lop ing p lane .
239
Fig. 12.2 Steady groundwater flow over a s loping plane into a stream
DIFFERENTIAL EQUATIONS I N POROUS MEDIA
Since flow velocit ies in porous media a re as a r u l e very small , the
depth of saturated aqui fer can be large to discharge the flow. The
slope of the water table may therefore d i f f e r from the slope of the
impermeable bed and the kinematic dynamic equation may requ i re
modif icat ion. The di f fusion equation i s thus often used. I n Fig. 12.1
the aqui fer i s assumed to over lay an impermeable plane. The only outside
contr ibut ion i s assumed to be the i n f i l t r a t i on from above, i (m/s). The
r a t e w i l l be assumed constant i n the ana lys is below. Varying aqui fer
t ransmissiv i ty and allowance for p a r t i a l saturat ion i s made later. Owing
to the slow velocities, omission of the dynamic terms i s even more
jus t i f ied than for over land flow.
The cont inui ty equation becomes
- a g + e g = i ax
The dynamic equation becomes ah
q = k ( S h - h -1 ax
This i s termed the Dupuit-Forchheimer equation. The f i r s t term on
the r i gh t hand side gives the Darcy equation and the second term i s
the correction for water surface gradient, as i n the di f fusion equation.
( 1 2.10)
El iminat ion of q from the cont inui ty and di f fusion equation y ie lds
( 1 2 . 1 1 ) k a t ax
240
Henderson and Wooding (1964) s o l v e d t h i s e q u a t i o n f o r c e r t a i n cases
to show the d e p t h o f emergence i s i n f l u e n c e d b y the downs t ream con-
d i t i o n s (see F i g . 12.2).
ANALYSIS OF SUBSURFACE FLOW
Freeze (1972) i n d i c a t e d t h a t s u b s u r f a c e f l o w c o u l d o n l y g e n e r a t e
storrnf low i .e. a r e l a t i v e l y s h o r t t ime to p e a k h y d r o g r a p h in cases w h e r e
the s o i l i s v e r y permeab le . Where the s lope i s v e r y steep however , Beven
(1981) i n d i c a t e s i n t e r f l o w ( f l o w u n d e r a n d o v e r s u r f a c e s u c c e s s i v e l y ) c o u l d
o c c u r w h i c h w o u l d a c c e l e r a t e the c o n c e n t r a t i on process .
Beven (1981) e x t e n d e d Henderson and Wood ing ' s (1964) a n a l y s i s f o r
k i n e m a t i c s u b s u r f a c e f l o w u s i n g a d imens ion less fo rm o f t h e e x t e n d e d
Dupu i t -Forchhe imer e q u a t i o n s :
where
x = X / L
H = 2h /L t a n 0
T = k s i n 0 t /2eL
A = 4 i cos 0 /k s i n 2 0 - 4 i / kS
O m i t t i n g the d i f f u s i o n te rm
1.0 -
H
9.5 -
I 0 0.5 1 . 0
X
(12.12)
(12.13)
(12.14)
(1 2.15)
(12.16)
(12.17)
Fig. 12.3 A c o m p a r i s o n o f s t e a d y s t a t e w a t e r t a b l e p r o f i l e s p r e d i c t e d b y t h e ex tended Dupu i t -Fo rchhe imer ( b r o k e n l i n e s ) and k i n e m a t i c wave ( s o l i d l i n e s ) e q u a t i o n s f o r d i f f e r e n t v a l u e s of X . ( K . Beven, Water Resources Research , 17, 1422, 1981, C o p y r i g h t Amer i can Geophys ica l U n i o n ) .
24 1
resul ts in the kinematic equation which may be integrated to g ive the
r i s i n g hydrograph at x = L
H = A T ( T < Tc = 0.5) (12.18)
A comparison of th is solut ion wi th numerical solutions of the
extended Dupuit-Forchheimer equation was made by Beven (1981) who
used an imp l ic i t f i n i t e difference method and i te ra t i ve relaxat ion solution.
The resul ts are given i n Fig. 12.3 and 12.4. They indicate the kinematic
equation holds reasonably for A < 1 .
Fig. 12.4 Rising hydrographs predicted by the extended Dupuit-Forchheimer (broken l ines) and kinematic wave (so l i d l ine) equations for di f ferent values. of X . ( K . Beven, Water Resources Research, 17, 1423, 1981, Copyright American Geophysical Union).
FLOW I N UNSATURATED ZONE
Beven (1982) extended h i s ana lys is of the saturated zone to al low
for vert ical flow i n the unsaturated zone above the water table. He also
accounted for var ia t ions i n porosity and hyd rau l i c conduct iv i ty w i th
depth.
Start ing from Campbell's (1974) model the hyd rau l i c conduct iv i ty
i s based on the equations
k = K ( O ) = S 2B+3
K S
(12.19)
( 12.20)
w h e r e k i s the re la t i ve hyd rau l i c conduct iv i ty, K ( 0 ) i s hyd rau l i c
conduct iv i ty at moisture content 0 , subscript s refers to saturat ion
conditions, S i s re la t i ve saturat ion, $ i s c a p i l l a r y tension, qb i s tension
242
a t a i r e n t r y a n d B is a pore size parameter, which var ies from about
4 for sands to 1 1 for c lays. I t i s assumed tha t
K5 = K, ( D - Z)" = K,hn
= :k ( D - 2 ) = _hm
(12.21
( 12.22)
where Z i s depth below s o i l sur face perpend icu la r to the slope, D i s
the constant depth of s o i l , K,, 0 :!, n a n d m a r e constants ( n " 2 m ) .
S tar t ing a t t = o w i t h constant i n i t i a l c a p i l l a r y tension
*o a n d h = 0,
Darcy 's law fo r unsatura ted f low i s
( 12.23)
- - K ( O ) ?!!! + K ( O ) cos $ (12.24) qz - az
where q i s the volume f low perpend icu la r to the slope a n d $ i s the
slope ang le to the hor izon ta l .
Assuming water due to i n f i l t r a t i o n moves down w i t h a n in te r face
p a r a l l e l to the slope, a t a r a t e
dz = I - dt O w ( Z ) -dz , t=O
where Ow(z) i s the water content a t which K(C,z )cos$ = i
Then "(2) = , (K,kcos 1 $ 1/(2B+3) (D-Z)m-n/(2B+3)
b = a (D-Z)b = a h
0 ( i ) 1 / ( 2 ~ + 3 ) = n 7 ': Kd.cosa . 2B +3 where a =
( 12.25)
(12.26
(12.27
(12.28
The above equat ions a p p l y u n t i l the wetted f ron t reaches a po in t a t
which i = KS(z)cos$
Below th is depth hw = (i/K,cos$) 1 /n (12.29)
dz = I ( 12.30) - dt O S ( Z ) - o(z , t=o)
I n t e g r a t i n g the two equations fo r z g ives the time tUZ a t which the
wet t ing f ron t reaches the bottom of the pro f i le .
tUZ = 7 {t+b ( D - hw l+m w 1 +m l+b) + 0. (h l+m 0 " ) l /BDl+m ) } (12.31) 1 a l'+b
FLOW IN NON-HOMOGENEOUS SATURATED ZONE
I f i t can be assumed the water tab le i s p a r a l l e l to the impermeable
bed, the h y d r a u l i c g rad ien t i s s i n @ a n d n+l
h K*s in@ h 9, = 1 K s ( h ) s i n $ d z =
0 n+l ( 12.32)
243
The k inemat ic wave equat ion thus becomes
" & + i (12.33) e ( h ) = -K,: s i n $I h
where t > t . I f i n p u t continues u n t i l t > t the steady s tate
ah ax
uz U Z
unsaturated p r o f i l e has poros i ty e ( h ) = 0 (h) - O w ( h ) ( S
= O ; h I h w (
= O,hm-ahb ; h > hw (
Subs t i tu t ing i n t o the prev ious equat ion g ives
ah - ,K': s i n $ hW ah I
b b ) - + a t O+hm-ah ' m
a t 0, h -ah
2.34)
2.35)
2.36)
2.37)
I n t e g r a t i n g the charac ter is t i c equat ion y ie lds
t = t
When the charac ter is t i c from the top of the catchment reaches the out le t ,
a steady s tate i s estab l ished a n d
(12.38) + % (hl+m - hw 1 +m ) - Itb a (h l + b - hw'+b) 1 uz I l+m
( 12.39)
a n d the time of concentrat ion i s
(12.40)
A so lut ion f o r the f a l l i n g l imb of the h y d r o g r a p h i s a lso poss ib le
i f i t can be assumed a d r y i n g f r o n t descends u n i f o r m l y once r a i n f a l l
stops.
Af ter sur face i n f i l t r a t i o n ceases the d r y i n g f r o n t continues to f a l l
a t a r a t e
(12.41)
The wetted p r o f i l e i s g iven b y
(12.42) 1 ) 1/(2B+3) (D-Z) m-n/(2B+3) ' W = 'a ( K.:cos $,
a n d when d r y i n g
0 ( z ) = O * h m (-!A ) 1 /B
I n t e g r a t i n g the two equat ions y i e l d s the time a f t e r cessation of r a i n u n t i l
the d r y i n g f r o n t reaches the water tab le
( 12.43) d qd
where z = o a t t = t
F ig . 12.5 depic ts the e q u i l i b r i u m hydrographs fo r d i f fe ren t r a i n f a l l ra tes.
244
Time , hours
F i g . 12.5 E q u i l i b r i u m h y d r o g r a p h s r e s u l t i n g f rom r a i n f a l l s o f d i f f e r e n t i n t e n s i t i e s a n d d u r a t i o n . Model pa ramete rs a r e as in T a b l e 12.1 w i t h L = 10 m. A i s 0.002 m/h r f o r 30 h o u r s ; B 0.001 m/h r f o r 50e hours : C 0.0006 m/h r f o r 70 h o u r s a n d D 0.0002 m/h r f o r 120 hours. (K . Beven, Water Resources Research, 18, 1631, 1982, C o p y r i g h t Amer ican Geophysica l U n i o n ) .
TABLE 12.1 Model pa ramete rs w i t h v a l u e s used in examp le c a l c u l a t i o n s
~~
Parameter Symbol Va I u e
S o i l dep th
E f fec t i ve s lope l e n g t h
H y d r a u l i c c o n d u c t i v i t y pa ramete r
H y d r a u l i c c o n d u c t i v i t y pa ramete r
Po ros i t y parameter
Po ros i t y pa ramete r
Soi l mo is tu re c h a r a c t e r i s t i c
Soi l mo is tu re c h a r a c t e r i s t i c pa ramete r
pa ramete r
I n i t i a I moi s t u r e tens i on
D r y i n g mo is tu re tens ion
Slope a n g l e
D
L
K
e
0
e.
m
'b
0
$ 0
'd @
0.6m
10.0, 14.0111
2.0057 m/h r
2.73
0.8035
1.135
10.0 cm
5.0 cm
500.0 cm
300.0 cm
15O
245
REFERENCES
Ahmed, N. and Sunada , D.K., 1969. N o n l i n e a r f l o w in p o r o u s med ia . Proc . ASCE, J. H y d r . D i v . HY6, Nov. p p 1847-1857.
Beven, K. 1981. K i n e m a t i c s u b s u r f a c e s to rmf low . Water Resources Research 1 7 ( 5 ) , p p 1419-1424.
Beven, K. 1982. O n s u b s u r f a c e s to rmf low : P r e d i c t i o n s w i t h s i m p l e k i n e m a t i c t h e o r y f o r s a t u r a t e d and u n s a t u r a t e d f l ows . Water Resources Research , 1 8 ( 6 ) , pp 1627-1633.
Campbe l l , G.S. 1974. A s i m p l e me thod f o r d e t e r m i n i n g u n s a t u r a t e d c o n d u c t i v i t y f rom m o i s t u r e r e t e n t i o n d a t a . So i l Sc i . , 117, pp 311-314.
Dunne, T. 1978. F i e l d s t u d i e s o f h i l l s l o p e f l o w processes. I n H i l l s l o p e H y d r o l o g y , Ed. M.J. K i r b y , John Wi ley , N.Y.
Freeze, R.A. 1972. Ro le o f s u b s u r f a c e f l o w in g e n e r a t i n g s u r f a c e r u n o f f , 2, Ups t ream source a r e a . Water Resour . Res. 8, p p 1272-1283.
H a r r , R.D. 1977. Water f l u x in s o i l a n d s u b s o i l on a steep f o r r e s t e d s lope, J. H y d r o l . 33, p p 37-58.
Henderson, F.M. a n d Wooding, R.A. 1964. O v e r l a n d f l o w and g r o u n d w a t e r f l o w f rom a s t e a d y r a i n f a l l o f f i n i t e d u r a t i o n . J. Geophys. Research , 69 ( 8 ) , pp 1531-1540.
Stephenson, D. 1979. Rock f i I1 i n H y d r a u l i c E n g i n e e r i n g , E l s e v i e r , 215 p. Weyman, D.R. 1970. T h r o u g h f l o w on h i l l s l o p e s and i t s r e l a t i o n to t h e
s t ream h y d r o g r a p h . B u l l . I n t . Assoc. Sci. H y d r o l . 15 ( 2 ) , pp 25-33.
246
AUTHOR INDEX
Abbot t , M.B. 94, 97 Ahmed, N. 238 A l l e y , W.M. 201, 215 Arnein, M. 92 Arnpt, G.A. 235 Argaman , Y. 148
Ba les , J. 200 Barnes , A.H. 148 B e l l , F.C. 130 Bergman, J.M. 209 Betson, R.P. 70, 200 Beven, K. 23, 237, 240, 241 B l a n d f o r d , G.E. 41 Bogan, R.H. 148 Borah , D.K. 2 B r a k e n s i e k , D.L . 2, 134, 152 B r u t s a e r t , W. 62 B u r n e y , J.R. 136
Campbe l l , G.S. 241 C a r t e , A.E. 132 Changnon, S.A. 133, 146 Chestnu t , A.R. 200 Chu, H.H. 137 C o l y e r , P.J. 146 C o n s t a n t i n i d e s , C.A. 2, 103,
C o u r a n t , R. 10, 100 C r a n f o r d , N.H. 214 C r o l e y , T.E. 2 Cunge, J.A. 77, 102
107, 154
D a w d y , D.R. 201, 209 Dean, C.H. 200 D ixon , M.J. 133 Dooge, J.C. 1 , 77, 197 Dronkers , J.J. 204 Dunne, T. 5 , 23, 237
Eag leson, P.S. 1 , 131
F a n g , C.S. 92 Freeze, R.A. 237 F r i e d e r i c h s , K.O. 10, 100
G a l l a t i , M. 2 G b u r e k , W.J. 72 Greco, F. 1 Green, I .R.A. 153, 200
Green, W.H. 235 Gup ta , V .L . 132
Haan , C . T . 22 H a r r , R . D . 237 Henderson, F .M. 2, 33, 78, 240 Ho l ton , H.N. 224 Hor ton , R.E. 1 , 1 1 , 235 Howard , K.M. 200 Huf f , F.A. 130, 137, 146 H u g g i n s , L.F. 136 H u n t , B. 10
I saacson , E. 90 I w a g a k i , Y. 1 I z z a r d , C.F. 40
James, W. 130
K e i f e r , C.J. 131 Keu legan , G.H. 1 K i b l e r , D.F. 1 , 114 K i n g , D. 148 K i r k b y , M.J. 21 Kouwen, N. 11
L a n g b e i n , W.B. 79 L a p i d u s , L. 224 L a x , P.D. 101 Leach , H.R. 1 L e C le rc , G. 209 Leopo ld , M.B. 70 L e w y , H. 10, 100 L i , R.M. 1 1 , 78 L i c h t y , R.W. 209 L i g g e t , J.A. 2, 29, 195 L i g h t h i l l , F.R.S. 1 L i g h t h i l l , M.J. 59, 165 L i l lard, J.H. 224 L i n s l e y , R.K. 214 L loyd -Dav ies , D . E . 6
Mader , G.N. 131 Maddock , 1. 68 Mahmood, K . 90 Maione, U. 2 M a r t i n , C. 148 Massau , J. 1 Meadows, M.E. 41, 70, 90, 200 M e r r i t , L .B . 148
247
M i d g l e y , D.C. 130 M o r r i s , E.M. 2, 29, 62 Morton, K.W. 97
Op ten Noort , T.H. 43 Or lob , G.T. 204 Over ton, D.W. 1 1 , 40, 72, 90,
114, 194
P a n a t t a n i , L. 1 P h i l i p , J.R. 210 P i n d e r , G.F . 224 Ponce, V.M. 78 P r a s a p , S . N . 2 P r a t t , H.E. 200
Richt rnyer , R.D. 97 Rodr igues, 1 . 1 . 133 Rosenbrock, H.H. 214 Ross, B.B. 223 Rossrni l ler , R .L . 6 Rovey, E.W. 2
S a i n t Venant 27 Schaake, J.C. 2, 201, 209 Scheckenberger , R. 130 Seddon, J.A. 67 Sevuk, A.S . 146 Shanhol tz , V.O. 224 Sirnons, D.B. 11, 78 S i n g h , V.P. 1 , 114 Skaggs , R.W. 2 S t a l l , J.B. 70 Stephenson, D. 2, 43, 107, 129
145, 159, 238 Smi th, R.E. 2 Stoker, J.J. 90, 100 S t r e l k o f f , T. 104 S u t h e r l a n d , F.R. 146 Sunada, D.K. 238
Troesch, B.A. 90
Watk ins , L.H. 7 Wayrnire, E. 132 Weeter, D.W. 70 Weyrnan, D.R. 237 Whitharn, G.B. 1 , 59, 65 Wi lson, C.B. 133 Wooding, R.A. 2, 33, 59, 64,
Woolhiser, D.A. 2, 29, 59, 95,
Wy l i e , E.B. 10, 102
118
114, 136
Yang, C .T . 70 Yen, B.C. 146 Yev jev i ch , V. 77, 190
Valdes, J.B. 133 Van V l i e t , R . 1 Van Wyk, W. 130 Veenhuis , J.E. 215 Verwey, A . 94 V i e i r a , J . H . D . 64 Von Neuman, J. 98
248
INDEX
A b s t r a c t i o n , 15, 49, 57, 139,
A c c u r a c y , 29, 85, 96 Advec t i on , 28 Amer ican, 1 An tecedent m o i s t u r e c o n d i t i o n , 6 ,
15, 57 A p p l i c a t i o n s , 209 A p p r o x i m a t i o n , 12, 29, 59 A q u i f e r , 238 A r e a l d i s t r i b u t i o n , 140 A r e a l r e d u c t i o n , 49 Assumpt ions , 23, 58 A t t e n u a t i o n , 27
175, 237
B a c k w a t e r , 27 Ba lance , 4 B i r m i n g h a m f o r m u l a , 5 B l a c k b o x , 199 B l a n k e t i n g , 172 B o u n d a r y c o n d i t i o n s , 88, 206
C a l c u l a t o r , 5 C a l i b r a t i o n , 199 C a n a l i z a t i o n , 56, 176 C a p i l l a r y , 237 Cascades o f p l a n e s , 2 Catchment , 2, 49, 176, 205 Catchment -s t ream, 115 Channe l , 24, 56, 68, 115, 145, 215 Channe l s t o r a g e , 180 C h a r a c t e r i s t i c s , 31, 81, 89 Chezy , 38, 65, 72, 180 C l a s s i c a l methods, 4 C losed c o n d u i t s , 184 Co lebrook Whi te , 38 Computer , 1 , 5, 82, 108 Computer p r o g r a m , 147, 186, 229 Concen t ra t i on , 3, 6, 7, 49, 65,
C o n d i t i o n , 60 Conduct i v i t y , C o n t i n u i t y , 24 C o n t r i b u t i o n , 23 Con t ro l vo lume, 24 Conserva t i on , 24, 25 C o n v e r g i n g ca tchmen t , 114 Co-ord ina te , 24 Cor rec t i on , 1 1 C o u r a n t c r i t e r i o n , 1 1 , 92, 100 Cover, 15, 109
135, 147, 178
241
Cres t subs idence , 67 C r i t i c a l , 98 C r i t i c a l d u r a t i o n , 178 Cross sec t i on , 24, 179
D a r c y , 37, 180, 187 D e f i n i t i o n , 3, 198 Des ign , 105, Des ign s to rm, 130 De ten t ion s t o r a g e , 2, 179 D e t e r m i n i s t i c , 201, 215 D iamete r , 146 D i f f e rence q u o t i e n t s , 84 D i f f e r e n t i a l , 1 , 5, 85, 97 D i f f u s i o n , 28, 69, 78, 240 D imens ion less h y d r o g r a p h s , 105 D imens ion less v a r i a b l e s , 105 D imens ion less e q u a t i o n s , 105 D i r e c t i o n , 25, 204 D i s c h a r g e , 69 D i s c r e t i z a t i o n , 85 D i s p e r s i o n , 27, 68 D i s s i p a t i v e , 101 D i s t r i b u t i o n , 130, 133, 140 D r a i n , 146, 221 D u a l system, 146 D u r a t i o n , 6, 35, 46, 105, 139, 173 Dynamic , 1 , 27 D y n a m i c s to rm, 2, 130
E n e r g y , 37 E n g i n e e r , 105 E q u a t i o n s o f mot ion , 23 E q u i l i b r i u m , 32, 50 E r r o r , 40, 97 E v a p o r a t i o n , 1 , 209 Excess ra in , 108, 209 E x p l i c i t s o l u t i o n , 92
F a l l i n g l i m b , 45, 243 F i n i t e d i f f e r e n c e , 1 , 81, 94 F i n i t e e lement , 222 F lood , 27, 77 F o u r i e r , 98 FORTRAN, 148 F r e q u e n c y , 6, 173 F r i c t i o n , 26, 27, 37, 180 F roude , 62
249
Geometry, 68, 154 Govern ing equat ions, 61 Gradient , 6 Graph ica l , 1, 106 Groundwater, 13, 172, 237
H is to ry , 1 Hyd rau l i cs , 5, 25, 68, 145 H y d r a u l i c r a d i u s , 37, 73 Hydrodynamic, 9 H y d r o g r a p h , 40, 43, 45, 63, 71,
Hydro log i s t , 6, 132 Hydro logy , 1 , 4 Hyetograph, 49, 132
76, 105, 129, 138, 181, 231
I I I ino is , 137 Impermeable, 28, 172 Imperv ious, 2, 214 I m p l i c i t , 90, 136 I n f i l t r a t i o n , 3, 13, 19, 47, 48,
57, 109, 177, 210, 234, 242 I n i t i a l abs t rac t i on , 4 I n i t i a l cond i t i on , 206 I n t e n s i t y , 6, 8, 31, 48, 174, 231 I n t e n s i t y - d u r a t i o n , 4, 48, 147,
I n te rcep t ion , 57 Isochronal , 105 I t e r a t i v e , 90
180
Job con t ro l l anguage , 5
Kinematic, 1 , 27, 28, 59, 60, 65,
K inemat ic f l ow number, 62, 69 Kinemat ic waves, 66
78, 87, 108, 145, 179, 194, 209
L a g t ime, 4 L a m i n a r , 37, 237 L a n d use, 198 L a t e r a l i n f l ow , 24, 221 Lax-Wendroff , 94 Leap- f rog, 94 L i n e a r , 5, 10, 186, 198 Long catchment, 49 Looped r a t i n g c u r v e , 74 Losses, 13, 57, 139, 177, 240
Mann ing , 38, 44, 65, 111, 134,
Matri;, 87 145 180
Memory, 198 Mesoscale, 132 Method of so lu t i on , 81 M iss i ss ipp i R i v e r , 22, 67 Model, 77, 194, 224 Momentum, 9, 25, 59 Mot ion, 25 Mov ing storm, 142 Muskingum, 76, 179
N a v i e r Stokes, 9 Networks, 147, 186 Newton Raphson, 49, 84, 93 N iku radse , 37, 149, 180 Nodes, 186 Non-convergence, 2, 81 Non r e c t a n g u l a r sect ions, 145 Non u n i f o r m f l ow , 4 Numer ica l d i f f u s i o n , 96, 97 Numer ica l scheme, 87 Numer ica l so lu t ions, 2, 81
Obstruct ions, 12 One-dimensional , 1 , 23, 95, 103,
Optimum, 183, 214 Order of magn i tude , 28 Osci I l a t i o n , 102 O v e r l a n d f l ow , 1 , 43, 115, 229
108, 194
Paramet r i c , 191, 209 P a r a s i t i c wave, 101 P a r t f u l l p ipes , 105 Pav ing , 176 Peak f low, 43, 173, 176 Permeab i l i t y , 12, 172, 238 Perv ious , 2, 9, 237 P i p e s , . 105, 146, 186 P lane , 1 , 32, 43, 114, 135 Poros i t y , 1 , 237 Posi t ion, 31, 34 Powel I s Creek, 225 P r a c t i t i o n e r , 12 P r e c i p i t a t i o n , 3, 1 1 , 15, 131,
172, 222 Pressure, 25 Profi le., 35, 239
Management, 172 Radar , 131
250
R a i n f a l l , 33, 65, 108, 131, 172,
R a i n f a l l i n t e n s i t y , 48 R a i n f a l l impact , 39 Ra t iona l coef f ic ients , 175 Ra t iona l method, 1, 4, 50, 105,
R a t i n g , 68, 74, 93, Real-time, 235 Receding h y d r o g r a p h , 33 Recurrence i n t e r v a l , 6, 172, Relaxat ion, 93, 241 Reservoirs, 10, 179, 185 Retent ion, 57, 214 Re ta rda t i on , 172 Reynolds number, 37 Rout ing, 180, 223 R i s i n g h y d r o g r a p h s , 30, 45 R ive r , 224 Roughness, 37, 176 Runoff , 1 , 23, 43, 49, 56, 59,
R u r a l watershed, 222
209
131, 172
107, 141,
Sa in t Venant, 9, 95, 179 Sand Creek, 220 Sa tu ra t i on , 57, 242 Section, 26 Seepage, 13, 237 Shock waves, 2 Simulat ion, 40, 161, 200, 231 Slope, 24, 68 S lop ing p lane , 110 S o i l , 15 Soi l Conservat ion Service, 1 , 6 So i l mois ture, 210 Soi l phys i cs , 15 Solut ion, 102, 136 S p a t i a l d i s t r i b u t i o n , 134 S t a b i l i t y , 81, 96 S t a n d a r d methods, 5 Stochast ic, 199 Storage, 179 Storm, 131, 147 Storm ce l l , 130 Stormwater, 106, 147, 194, 209 Streamflow, 23, 221, 229 S t r i c k l e r , 37, 180 Sub-surface, 3, 240 Surface, 2 Sur face water , 3 Symbols, 107 Synopt ic, 132 System, 197
T a y l o r ser ies, 77 Tennessee V a l l ey , 201 Terminology, 198 Time a rea , 1 Time dependent, 57, 131 Time of concentrat ion, 3 Time to e q u i l i b r i u m , 3 Time s h i f t , 153, 165, 176 T rapezo ida l channe l , 152, 166 T r a v e l l i n g storm, 142 T r a v e l t ime, 4 T r i a n g u l a r storm, 137 T runca t ion , 97 T u r b u l e n t , 37, 237 Two-dimensional , 191
Un i t s , 51 Uni form f l ow , 1 U n i t h y d r o g r a p h , 7, 105 Un i ted States, 6 Unsaturated, 240 U r b a n d r a i n a g e , 2 U r b a n watershed, 209 Urban iza t i on , 209
Vegetat ion, 12, 57, 172 V e r i f i c a t i o n , 199 V i r g i n catchment, 174 Viscos i ty , 237 Volume, 106 V-shaped catchment , 115
Water hammer, 11 Watershed, 65, 200, 209, 222 Water sur face, 23, 68 Wave, 2, 4, 68, 78 Waves, k i n e m a t i c 66 Wave speed, 67 Weak so lu t i on , 101 Weighted ave rage , 101 Wetted per imeter , 26, 37 Wet t ing f r o n t , 17 Wind, 133
