UBC Theses and Dissertations

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UBC Theses and Dissertations

Optimal culvert size selection Neudorf, Patrick Alexander 1977

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OPTIMAL CULVERT SIZE SELECTION by PATRICK ALEXANDER NEUDORF B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1975  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES The Department o f C i v i l  We a c c e p t t h i s  Engineering  t h e s i s as conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1977  ©  P a t r i c k Alexander Neudorf, 1977  In p r e s e n t i n g t h i s t h e s i s  in p a r t i a l  an advanced degree at the U n i v e r s i t y the L i b r a r y  s h a l l make i t  f u l f i l m e n t o f the requirements of B r i t i s h C o l u m b i a , I agree  freely available for  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e  r e f e r e n c e and copying o f t h i s  It  i s understood that copying or  thesis  Department of  The U n i v e r s i t y  Civil  Engineering  o f B r i t i s h Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  August  31.  1977  or  publication  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my wr i t ten pe rm i ss ion .  that  study.  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s r e p r e s e n t a t i v e s .  for  ABSTRACT The currently  hydraulic  lead to e c o n o m i c a l l y non-optimal c u l v e r t  optimum s i z e d c u l v e r t f o r any  British  size  culvert site,  i n t o d i r e c t account the u n c e r t a i n t y of the d a t a . to a h y p o t h e t i c a l  l o g i c and  of b e t t e r  economic s i t u a t i o n s .  The  method i s  uncertainty i n evaluating  The  hydrologic,  economic a s p e c t s of c u l v e r t s e l e c t i o n and i n c o l l e c t i n g d a t a and  these areas are  taking  methods of c a l c u l a t i n g the  i n f o r m a t i o n are p r e s e n t e d .  uncertainties  The  to  c u l v e r t s i t e , assuming d i f f e r e n t hydro-  flows i s taken i n t o account, and  and  selection  T h i s t h e s i s d e s c r i b e s a method of economic a n a l y s i s  determine the  applied  size  employed by most highways departments, i n c l u d i n g  Columbia's, can choices.  design c r i t e r i a f o r c u l v e r t  d i s c u s s e d b e f o r e the  flood value  hydraulic,  the problems  and  making assumptions i n each of r e s u l t s are  presented.  TABLE OF CONTENTS Chapter  Page  1  INTRODUCTION  2  METHOD OF SOLUTION . . .  5  2.1 2.2  The D e c i s i o n Tree P r o b a b i l i t y Matrices  5 8  2.3  Calculations  3  4  5  6  1  11  EVALUATION OF FLOOD FLOWS  .  14  3.1  Methods and Problems  14  3.2  Accounting  17  f o r U n c e r t a i n t y i n F l o o d Flows . .  CULVERT HYDRAULICS  26  4.1 Types o f C u l v e r t Flow 4.2 Entrance and E x i t Improvement 4.3 Mechanics o f a Washout 4.4 Environmental C o n s i d e r a t i o n s ECONOMICS 5.1 C a p i t a l Cost 5.2 F l o o d Damage 5.3 Annual Cost Comparison  26 31 33 34 35 35 38 41  • ..' . . . . . .  RESULTS 6.1 6.2 6.3 6.4  7  . . . .  47  Annual Cost Curves f o r One F l o o d Frequency Distribution . . . . . The E f f e c t o f U n c e r t a i n t y and the Value o f Better Information . . . . . S e n s i t i v i t y o f the Optimal D e c i s i o n t o Changes i n the D i s c o u n t Rate and the Service L i f e The E f f e c t on the Optimal D e c i s i o n o f Changing the Damage Costs  CONCLUSION  47 47 57 60 68  LIST OF REFERENCES .  70  APPENDIX . . .  72  iii  LIST OF TABLES Table 3.1  Page Comparison o f E f f e c t i v e Floods o f V a r i o u s ' Return P e r i o d s f o r D i f f e r e n t  5.1  Capital  6.1  Annual Costs  Costs o f I n s t a l l e d C u l v e r t s f o r F l o o d Frequency  6.3  6.4  6.5  . . . .  . . . . . . .  25 36  Distribution  D e f i n e d by Q = 150 c f s and Q = 220 c f s P r o b a b i l i t i e s o f I n c u r r i n g Some Headwater Damage and P r o b a b i l i t i e s o f a Washout The E f f e c t o f U n c e r t a i n t y i n Changing the Optimal D e c i s i o n and the Value o f B e t t e r Information 1 Q  6.2  Distributions  1 0 Q  . . .  49 50 55  Optimum C u l v e r t Diameters and Return P e r i o d s of S i g n i f i c a n t Headwater L e v e l s f o r D i f f e r e n t Damage Costs  64  Comparison o f Economic A n a l y s i s w i t h the B r i t i s h Columbia Department o f Highways' Design C r i t e r i a  65  iv  ^  LIST OF FIGURES Figure  Page  2.1  D e c i s i o n Tree Used i n the A n a l y s i s  6  2.2  More Complicated D e c i s i o n Tree  7  2.3  H y p o t h e t i c a l F u n c t i o n Y = f (X)  9  2.4  Truncated Skew Normal D i s t r i b u t i o n  9  3.1  Frequency Curves o f Annual F l o o d s (1.0 L i n e : Q = 150 c f s , Q = 220 c f s ) . . . .  3.2  F l o o d Frequency (1.0 Curve: Q 1 Q  3.3  19  1QQ  1 Q  =  Distributions ' ioo 1  5  0  c  f  s  Q  =  2  2  0  c  f  s  *  • • • •  2  0  Frequency Curves o f Annual F l o o d s (1.0 L i n e :  Q  1Q  = 120 c f s , Q  = 216 c f s ) . . . .  1QQ  24  4.1  Types o f C u l v e r t Flow  27  4.2  Headwater-Discharge  29  5.1  C a p i t a l Costs o f I n s t a l l e d C u l v e r t s  37  5.2  Headwater Damage F u n c t i o n  39  5.3 6.1  C o n v e r t i n g C a p i t a l C o s t t o Annual Cost Annual C o s t Curves f o r F l o o d Frequency D i s t r i b u t i o n D e f i n e d by Q = 150 c f s and Q = 220 c f s . . .  Curves  . . .  1 Q  6.2  1 Q 0  43 48  T o t a l Annual C o s t Curves f o r D i f f e r e n t F l o o d Frequency D i s t r i b u t i o n s (1.0 Curve: Q, = 150 c f s and Q = 220 c f s )  52  Annual C o s t Curves (1.0 Curve: and Q = 216 c f s )  53  n  1 0 Q  6.3  Q  = 120 c f s  1 0 Q  6.4 6.5 6.6  6.7  S e n s i t i v i t y o f the Optimal D e c i s i o n to Changes i n the D i s c o u n t Rate and the S e r v i c e L i f e . . . .  59  The E f f e c t on the Optimal D e c i s i o n o f Changing the Damage Costs  61  The E f f e c t on the O p t i m a l D e c i s i o n o f Changing the Damage C o s t s ; New F l o o d Frequency D i s t r i bution  62  The E f f e c t o f U n c e r t a i n t y i n Changing the Optimal D e c i s i o n a t D i f f e r e n t Damage Costs  67  v  ACKNOWLEDGEMENT  The  author  guidance  in  are  also  extended  and  to  would for  Mrs. also  their  the  wishes research  Janet like  to  Mr.  to  thank  financial  thank  S.  0.  and p r e p a r a t i o n Richard  Bergeron,  to  Dr.  the  support  who  Brun, typed  National during  vi  the  Russell  of  who the  this  two  Thanks  the The  Research Council past  his  thesis.  prepared thesis.  for  years.  figures, author of  Canada  Chapter 1 INTRODUCTION The  B r i t i s h Columbia Department o f Highways p r e s e n t l y s e l e c t s  c u l v e r t s i z e s on the b a s i s o f two c r i t e r i a (1): (A) C u l v e r t s s h a l l c a r r y the 10-year f l o o d w i t h headwater depths e q u a l t o the diameter o f the c u l v e r t . (B) The c u l v e r t s h a l l c a r r y a 100-year f l o o d (1.8 x 10-year) by surcharge without headwater damage and without l o s s through scour. E i t h e r c r i t e r i o n may govern. The  f i r s t c r i t e r i o n appears t o be r a t h e r a r b i t r a r y w h i l e the  second c r i t e r i o n makes an attempt t o weigh the c o s t o f i n s t a l l i n g a l a r g e r p i p e s i z e a g a i n s t the savings damage.  from l e s s f r e q u e n t  The q u e s t i o n i s , "Why was the 100-year f l o o d  flood  chosen?"  These c r i t e r i a can h a r d l y be expected t o r e s u l t i n s e l e c t i n g the optimal  c u l v e r t s i z e f o r a l l c u l v e r t s i t e s i n a l l circumstances.  For i n s t a n c e , f o r c u l v e r t s under low f i l l s ways, d e s i g n i n g  on low volume r u r a l  f o r the 25-year f l o o d may be a p p r o p r i a t e .  high-  In  c o n t r a s t , the 500-year f l o o d c o u l d be a p p r o p r i a t e f o r a long c u l v e r t under a major highway where s u b s t a n t i a l damages t o upstream o r downstream p r o p e r t y  c o u l d r e s u l t from f l o o d i n g .  Another problem i s , "What i s the 10-year f l o o d o r 100-year flood?"  There i s o f t e n a g r e a t d e a l o f u n c e r t a i n t y i n v o l v e d i n  e v a l u a t i n g f l o o d flows  f o r s m a l l watersheds.  In a d d i t i o n t o  h y d r o l o g i c u n c e r t a i n t y , c u l v e r t d e s i g n i s plagued by u n c e r t a i n t y i n areas  such as the h y d r a u l i c performance o f c u l v e r t s , d e b r i s c l o g g i n g ,  what flow w i l l cause washout, and e s t i m a t i n g damage c o s t s . The  U n i t e d S t a t e s Bureau o f P u b l i c Roads (USBPR) has s t a t e d  t h a t 44% o f the highway drainage  d o l l a r , or 15% o f the highway 1  2  construction  d o l l a r , i s spent f o r c u l v e r t s  s i x t e e n 1961  projects  t o t a l c o s t was  (2).  An  analysis  i n B r i t i s h Columbia showed t h a t 8.6%  s p e n t on c u l v e r t s  (1).  of  of  the  C l e a r l y , these q u e s t i o n s  warrant a t t e n t i o n . This  thesis describes  a method o f economic a n a l y s i s which  be used t o determine the optimum c u l v e r t s i z e f o r any taking  i n t o d i r e c t account the u n c e r t a i n t y  i s applied  to a h y p o t h e t i c a l  to be p l a c e d The  on  a 7%  slope  c u l v e r t s i t e where a 100  sloped  a t the e x i t .  a t 2:1.  The  17  f t culvert i s  the  highway  f t above  the  f t above the c u l v e r t i n v e r t  Only u n c e r t a i n t y  c o n s i d e r e d i n the  i n o t h e r areas i s  discussed.  idea of applying  economic a n a l y s i s  a thesis e n t i t l e d Application  new.  (15-20% i n the  literature.  i s to extend the accounted f o r .  e f f e c t on  i n the f l o o d f r e q u e n c y data i s  uncertainty  P r i t c h e t t (3)  wrote  He  (1964), and  this thesis  concluded t h a t  substantial  The  realized  purpose of the p r e s e n t  a n a l y s i s so t h a t u n c e r t a i n t y The  flood  to determine the o p t i -  f o u r examples presented) would be  economic a n a l y s i s .  and  of the P r i n c i p l e s o f E n g i n e e r i n g  Economy t o the S e l e c t i o n of Highway C u l v e r t s i s o f t e n mentioned i n the  i n the  a n a l y s i s , although  s i z e c u l v e r t f o r a g i v e n s i t e i s not  by a p p l y i n g  f t , and  roadway i s 10  frequency d a t a was  savings  method  Reasonable f l o o d frequency d a t a , c u l v e r t c o s t s ,  f l o o d damage c o s t s were chosen.  The  The  site,  under a major r u r a l two-lane highway.  c u l v e r t i n v e r t a t the entrance and  mum  of the d a t a .  roadway w i d t h , i n c l u d i n g s h o u l d e r s , i s 45  embankments a r e  culvert  can  i n the data can  the o p t i m a l d e c i s i o n of studied.  thesis be  uncertainty  3  A v e r y important q u e s t i o n when f a c e d with u n c e r t a i n t y i s , "What i s the v a l u e o f b e t t e r i n f o r m a t i o n ? "  Or i n o t h e r words,  "How much money, i f any, s h o u l d be spent on a data g a t h e r i n g program to reduce u n c e r t a i n t y ? "  T h i s q u e s t i o n i s explored and p o s s i b l e  s o l u t i o n s t o the problem a r e p r e s e n t e d . of  In a d d i t i o n , the s e n s i t i v i t y  the o p t i m a l d e c i s i o n t o changes i n the d i s c o u n t r a t e and the  service l i f e changing  i s s t u d i e d as i s the e f f e c t on the o p t i m a l d e c i s i o n o f  the damage c o s t s .  The o n l y type o f c u l v e r t i n s t a l l a t i o n considered i n the a n a l y s i s i s a s i n g l e round c o r r u g a t e d metal pipe v e r t i c a l headwall  and e n d w a l l .  (CMP) w i t h a  D i f f e r e n t m a t e r i a l s and shapes may  be advantageous i n some s i t u a t i o n s , b u t they are n o t c o n s i d e r e d here.  Entrance  improvement, which can r e s u l t i n a s i g n i f i c a n t  improvement i n h y d r a u l i c e f f i c i e n c y , i s d i s c u s s e d b u t n o t i n c o r p o r ated i n t o the a n a l y s i s .  The s t r u c t u r a l e n g i n e e r i n g a s p e c t o f c u l -  v e r t design i s not discussed. U t i l i t y , r a t h e r than monetary v a l u e , could have been used as the b a s i s f o r c u l v e r t s e l e c t i o n .  But s i n c e highway c u l v e r t s a r e  the r e s p o n s i b i l i t y o f p r o v i n c i a l governments, monetary v a l u e was chosen.  U t i l i t y would be more a p p r o p r i a t e f o r c u l v e r t s on p r i v a t e  l a n d c o n t r o l l e d by a f i r m o r an i n d i v i d u a l with l i m i t e d resources. to  financial  In t h i s case the i n d i v i d u a l o r f i r m may be more averse  severe f l o o d damage than the monetary value o f the f l o o d damage  indicates. remaining  The f o l l o w i n g paragraph  o u t l i n e s the contents o f the  chapters.  Chapter  2 i l l u s t r a t e s the problem w i t h a d e c i s i o n t r e e and  o u t l i n e s the f o r m a t i o n and use o f p r o b a b i l i t y m a t r i c e s and v e c t o r s  4  which a r e used i n the c a l c u l a t i o n s .  The next t h r e e c h a p t e r s  v a r i o u s components o f the d e c i s i o n t r e e .  discuss  Chapter 3 d i s c u s s e s  methods o f e v a l u a t i n g f l o o d flows and t h e i r i n h e r e n t problems and presents  the f l o o d frequency  d i s t r i b u t i o n s used i n the a n a l y s i s .  Types o f c u l v e r t flow a r e d i s c u s s e d i n Chapter 4; Chapter 4 a l s o i n c l u d e s s h o r t d i s c u s s i o n s o f c u l v e r t entrance the mechanics o f a washout, and environmental  and e x i t improvement, considerations.  Chapter 5 d i s c u s s e s the economic elements o f the problem: the c a p i t a l c o s t s o f c u l v e r t s , f l o o d damage c o s t s , and how the c a p i t a l c o s t i s converted  t o an annual c o s t w i t h emphasis on the q u e s t i o n ,  "What i s the c o r r e c t d i s c o u n t r a t e ? "  The r e s u l t s a r e presented  and d i s c u s s e d i n Chapter 6, and c o n c l u s i o n s are drawn i n Chapter 7.  Chapter  METHOD OF  The  2.1  Decision  The with (for  example,  decision chance  or  branches  water  diameter.  of  from  the  simple  presented  cated  include  to  a decision added  culvert  as  also  ties  depicted  the  only  to  Figure  shown a  as  are  true  flood  damage c o s t , at  the  which  depend  flood  size)  are  on  point  size  of  by  a  the  each  on  shown  as  circle.  a unique  a given for  a  branches.  since  for  calculated end  from  Events  given  chance  decisions  emanating  represented  also  represented  Possible  branches  example,  point,  each  conveniently  2.1.  square.  (for  one  outcome  debris  been  tree  included.  to  along  same  The  6.  head-  culvert  each  final  type  the  variable. the  of  headwater  branch  are  their  complete  only  situation  two  debris  also  5  of  of  could  in  debris  be  the  debris causes  to  no  clogging washout  compli-  has  washout possibilieither  flow could at  2.2  clogging,  clogging  with  the  Figure  probabilities:  debris  in  improvement  which  clogging  applies  used  events.  entrance  associated  degrees  tree  chance  level  is  2.1  Uncertainty  headwater  There  Figure  decision  which  Intermediate The  in  and more  with or  shown  decisions  and  added.  tree  Chapter  a decision  clogging  culvert.  in  more  hydraulics,  have  no  by  events  decision  analysis  been  are  a chance  assigned  final  in  c a n be  tree.  The  is  size)  chance  A probable  is  decision  is  shown  occurrence  contains  level  as  problem  represented  leading  2.1  level,  culvert  natural  Probabilities Figure  selection  tree,  point,  SOLUTION  Tree  culvert  a decision  2  a  through be  particular  Data Deci sion  Culvert Size  Flood  Headwater  Mean Damage Cost  FIG.2.1 D E C I S I O N  TREE  USED  IN  THE  ANALYSIS.  FIG.2.2  MORE  COMPLICATED  DECISION  TREE.  8  headwater l e v e l ; These events 2.2  i n t e r m e d i a t e degrees c o u l d a l s o be i n c l u d e d .  a r e more f u l l y  Probability  d i s c u s s e d i n Chapter 4.  Matrices  M a t r i c e s and v e c t o r s  (one-dimensional  matrices)  are v e r y use-  f u l f o r h a n d l i n g d e c i s i o n t r e e i n f o r m a t i o n and c a l c u l a t i o n s . i d e a o f r e p r e s e n t i n g a f u n c t i o n bounded by upper and lower as a p r o b a b i l i t y matrix was developed and  subsequently  concept  limits  by R u s s e l l and Hershman (4)  used by Nyumbu (5) and Brox (6).  I t i s a useful  f o r dealing with uncertainty.  The  formation of a p r o b a b i l i t y matrix  is illustrated  h y p o t h e t i c a l f u n c t i o n Y = f ( X ) , shown i n F i g u r e 2.3. of  The  f o r the  F o r any. v a l u e  X, the dependent v a r i a b l e Y i s not known with c e r t a i n t y but l i e s  somewhere between the upper and lower l i m i t s . .  The u n c e r t a i n t y about  the t r u e v a l u e o f Y f o r a g i v e n v a l u e o f X can be d e s c r i b e d by a p r o b a b i l i t y density function. In p r a c t i c e , the three curves o f F i g u r e 2.3 are u n l i k e l y to be known a c c u r a t e l y , e s p e c i a l l y i n cases where there i s l i t t l e available.  Determining  cularly d i f f i c u l t .  data  the upper and lower bounds may be p a r t i -  However t h i s does not n e c e s s a r i l y decrease the  u s e f u l n e s s of the method s i n c e the d e c i s i o n maker can i n c r e a s e the s e p a r a t i o n between the upper and lower l i m i t s as h i s u n c e r t a i n t y increases. Likewise,  + the shape o f the p r o b a b i l i t y d e n s i t y f u n c t i o n  between the upper and lower bounds i s u n l i k e l y to be known u n l e s s there i s s u f f i c i e n t data t o a n a l y z e .  A t r u n c a t e d skew normal  d i s t r i b u t i o n , shown i n F i g u r e 2.4, was deemed a p p r o p r i a t e . ' v a r i a t i o n o f the normal d i s t r i b u t i o n was developed  This  by Ward f o r  V a l u e s o f Y at X i a s s u m e d to f o l l o w a s k e w n o r m a l distribution Upper S h a d e d a r e a is p r o p o r t i o n a l to the probability that the v a l u e of Y at Xj is in the •  .i  interval  Most  DY  Bound  Probable  DY  T"  Lower  FIG.2.3  HYPOTHETICAL  FUNCTION  Bound  Y=f(X)  f(X)  LM  = 2 a,  MU  = 2 cr.  S h a d e d a r e a is proportional to p r o b a bi lity that the v a l u e of X is in the interval D X  M  ( MODE)  FIG.2.4 T R U N C A T E D  SKEW  —IDX-—  NORMAL  u  DISTRIBUTION  10  Hershmari's t h e s i s . two  The d i s t r i b u t i o n i s a composite made up from  t r u n c a t e d normal d i s t r i b u t i o n s .  deviations  from the mode.  1 / (1 - .0456) t o c o r r e c t the  distribution.  equidistant  The bounds a r e two standard  The d e n s i t y  function  i s m u l t i p l i e d by  f o r the areas t r u n c a t e d a t the ends o f  In the case where the upper and lower bounds are  from the mode, the d e n s i t y  c a t e d normal d i s t r i b u t i o n .  function  reduces t o a t r u n -  Because t h i s d i s t r i b u t i o n i s easy t o  work w i t h and can handle cases i n which the upper and lower bounds are n o t e q u i d i s t a n t  from the mode, i t was c o n s i d e r e d a reasonable  choice. For D Y  =  Y  2 ~  function for  Y  any v a l u e o f X the p r o b a b i l i t y t h a t Y i s i n the i n t e r v a l i  c  a  n  be found by i n t e g r a t i n g  a t X between Y  1  and Y  2  the p r o b a b i l i t y  (see F i g u r e 2.3).  forming a p r o b a b i l i t y m a t r i x .  density  T h i s i s the b a s i s  The rows o f the m a t r i x r e p r e s e n t  d i s c r e t e v a l u e s o f X and the columns r e p r e s e n t Y i n t e r v a l s . element o f the m a t r i x i s the p r o b a b i l i t y  t h a t the v a l u e o f Y l i e s  i n c e r t a i n i n t e r v a l DY f o r a s p e c i f i c v a l u e of X. elements a c r o s s any row n e c e s s a r i l y sequent c a l c u l a t i o n s  An  equals 1.0.  The sum o f the  To s i m p l i f y  sub-  the mid-points o f a l l Y i n t e r v a l s a r e u s u a l l y  chosen t o r e p r e s e n t the columns.  The d i s c r e t e v a l u e s o f X a r e a l s o  commonly the mid-points o f X i n t e r v a l s .  In t h i s way the i n f o r m a t i o n  c o n t a i n e d i n the three continuous curves o f F i g u r e 2.3 i s converted into discrete  pieces.  C o n s i d e r i n g t h i s , some judgement must be used i n s e l e c t i n g the  s i z e o f the m a t r i x .  w i l l be l o s t .  I f the i n t e r v a l s are too l a r g e  accuracy  F o r example, g i v e n t h a t DY i s an i n t e r v a l above the  mode a t a s p e c i f i c X, the p r o b a b i l i t y  t h a t the v a l u e o f Y i s i n  11  the  lower h a l f o f the i n t e r v a l i s g r e a t e r  than the p r o b a b i l i t y  t h a t the v a l u e o f Y i s i n the upper h a l f o f the i n t e r v a l .  The  v a l u e o f the mean u = /Yp(Y)dY, which i s somewhat below the midV p o i n t i n t h i s case, i s the c o r r e c t c h o i c e f o r a v a l u e o f Y f o r the i n t e r v a l . r e s u l t s i n some i n a c c u r a c y .  matrices, but s t i l l  b u t the number of computations  (Higgins  involved  A computer  1975) f o r forming p r o b a b i l i t y  the m a t r i c e s ' s i z e s should n o t be  l a r g e as the u n c e r t a i n t y  excessively  i n p l o t t i n g the t h r e e curves o f  2.3 u s u a l l y does n o t j u s t i f y l a r g e s i z e d m a t r i c e s . The  flood p r o b a b i l i t y vector  a p r o b a b i l i t y matrix. The  (both X and Y)  i n forming and u t i l i z i n g the m a t r i x i n c r e a s e s .  program has been developed  Figure  the i n t e r v a l mid-points  As the i n t e r v a l s i z e s  decrease, accuracy i n c r e a s e s , involved  Thus u s i n g  representative  uncertainty  of Figure  2.1 was d e r i v e d  The d e r i v a t i o n i s d i s c u s s e d  i n culvert hydraulics  i n Chapter 3.  c o u l d be d e s c r i b e d  by a  p r o b a b i l i t y m a t r i x w i t h flow p l o t t e d on the X a x i s o f F i g u r e headwater p l o t t e d on the Y a x i s . could  a l s o be c o n s t r u c t e d  from  2.3 and  A f l o o d damage p r o b a b i l i t y m a t r i x  from a graph s i m i l a r t o F i g u r e  2.3 w i t h  headwater p l o t t e d on the X a x i s and damage c o s t p l o t t e d on the Y axis.  Such a m a t r i x i s o n l y n e c e s s a r y i f a p r o b a b i l i t y d i s t r i b u t i o n  of damage c o s t i s t o be c a l c u l a t e d . curve i s r e q u i r e d  Only a s i n g l e mean damage c o s t  t o c a l c u l a t e the expected damage c o s t f o r each a.  c u l v e r t diameter. 2.3  Calculations The  of Figure  d e c i s i o n t r e e c a l c u l a t i o n s f o r the e x i s t i n g d a t a branch 2.1 are o u t l i n e d below.  The d e c i s i o n t o gather more d a t a  12  w i l l r e s u l t i n a new f l o o d p r o b a b i l i t y v e c t o r .  T h i s t o p i c i s more  f u l l y d i s c u s s e d i n Chapters 3 and 6. Steps: 1.  C a l c u l a t e f l o o d p r o b a b i l i t y v e c t o r f o r g i v e n f l o o d frequency p l o t and f l o o d i n t e r v a l v e c t o r (Chapter 3 ) .  2.  Choose c u l v e r t diameter  (an annual investment charge f o r each  c u l v e r t i s c a l c u l a t e d , see Chapter 5 ) . 3.  C a l c u l a t e a headwater l e v e l f o r each f l o o d i n t e r v a l and l i s t the  4.  headwaters  i n a v e c t o r (Chapter 4 ) .  C a l c u l a t e annual damage c o s t f o r each headwater l e v e l 3 and l i s t  the r e s u l t s .in a v e c t o r (damage c o s t = headwater  damage c o s t + washout c o s t i f HW exceeds HW 5. ' C a l c u l a t e expected ing  from  (or average)  max  , see Chapter 5 ) .  annual damage c o s t by m u l t i p l y -  each element i n the f l o o d p r o b a b i l i t y v e c t o r by i t s  c o r r e s p o n d i n g element i n the damage c o s t v e c t o r and summing the 6.  p r o d u c t s ; i . e . , c a l c u l a t e the d o t p r o d u c t o f the two v e c t o r s .  Determine  expected t o t a l annual c o s t by adding the annual  investment charge and the expected annual damage c o s t . 7.  Repeat  steps 2 t o 6 f o r a l l c u l v e r t d i a m e t e r s .  8.  P l o t r e s u l t s and choose c u l v e r t w i t h minimum expected annual c o s t  total  (Chapter 6 ) .  Costs a r e added on the b a s i s o f a s t a t i s t i c a l theorem  which  s t a t e s t h a t the expected v a l u e o f the sum o f two o r more random v a r i a b l e s i s e q u a l t o the sum o f the expected v a l u e s o f the i n d i v i d u a l random v a r i a b l e s . All the  the c a l c u l a t i o n s are handled by t h r e e computer programs:  f i r s t c a l c u l a t e s the f l o o d p r o b a b i l i t y v e c t o r , the second  13  calculates  the headwater v e c t o r f o r each c u l v e r t s i z e , and the  t h i r d program performs the remaining c a l c u l a t i o n s u t i l i z i n g the r e s u l t s o f the f i r s t two programs.  Chapter  3  EVALUATION OF FLOOD FLOWS 3.1  Methods and Problems There i s a g r e a t d e a l of u n c e r t a i n t y a s s o c i a t e d w i t h  the  e v a l u a t i o n of f l o o d flows from s m a l l watersheds i n B r i t i s h One  o f the main problems i s the l a c k of d i r e c t streamflow  ments f o r creeks on which c u l v e r t s are to be l o c a t e d . flows are normally e v a l u a t e d i n d i r e c t l y . r e l a t i o n s h i p s are commonly used. The  Columbia. measure-  Thus f l o o d  Precipitation-runoff  Hetherington s 1  publication  entitled  25-Year Storm and C u l v e r t S i z e - A C r i t i c a l A p p r a i s a l (7)  has  a good d i s c u s s i o n of the methods and problems of peak flow e v a l u a t i o n . Much o f the d i s c u s s i o n of t h i s s e c t i o n i s summarized from h i s paper. In o r d e r to e v a l u a t e peak flows i t i s necessary  to  understand  the m e t e o r o l o g i c a l and p h y s i c a l processes which produce them. are many d i f f e r e n t ways i n which a 25-year, 100-year, or any peak flow c o u l d be generated.  There year  In c o a s t a l r e g i o n s o f B r i t i s h  Columbia, major r a i n storms w i t h d u r a t i o n s o f 12 to 36 hours or g r e a t e r are the major cause of h i g h peak r u n o f f events.  Rapid  s p r i n g t i m e m e l t i n g o f an above average w i n t e r snowpack i s a probable cause of h i g h peak flows i n the I n t e r i o r .  In very s m a l l watersheds  of a few hundred acres or l e s s , h i g h peak flows can a l s o be generated  by h i g h i n t e n s i t y c o n v e c t i v e r a i n f a l l s  o f d u r a t i o n l e s s than 2 to 3 hours. h i g h r u n o f f events  (thundershowers)  Rain f a l l i n g on snow can cause  f o r both c o a s t a l and  i n t e r i o r watersheds.  a f l o o d w i t h a r e l a t i v e l y h i g h r e t u r n p e r i o d can be generated flow of lower r e t u r n p e r i o d i s t e m p o r a r i l y blocked by a d e b r i s 14  Also when jam.  15  Storm r u n o f f water backed up behind a powerful surge when the dam conditions l i k e l y  the d e b r i s jam i s r e l e a s e d as  collapses.  The  u n c e r t a i n t y about the  to cause h i g h peak flows adds u n c e r t a i n t y to the  i n d i r e c t e v a l u a t i o n o f peak f l o w s . P r e c i p i t a t i o n - r u n o f f models are commonly used because some s o r t o f p r e c i p i t a t i o n data i s u s u a l l y a v a i l a b l e t o apply to the watershed i n q u e s t i o n . s c a t t e r e d throughout  However, m e t e o r o l o g i c a l s t a t i o n s are w i d e l y  the p r o v i n c e and mostly  Most s t a t i o n s c o l l e c t r a i n f a l l  l o c a t e d a t low e l e v a t i o n s  i n standard, non-recording  hence, data on s h o r t d u r a t i o n r a i n f a l l  gauges;  i n t e n s i t i e s i s very  limited.  Many of the s t a t i o n s , p a r t i c u l a r l y those with r e c o r d i n g gauges, have a very s h o r t p e r i o d of r e c o r d which r e s t r i c t s of  the r e l i a b i l i t y  return period calculations. E x t r a p o l a t i n g p r e c i p i t a t i o n data, h o r i z o n t a l l y as w e l l as  vertically,  from o b s e r v a t i o n s taken a t a s i n g l e p o i n t i s a d i f f i c u l t  problem, p a r t i c u l a r l y i n mountainous t e r r a i n where p r e c i p i t a t i o n p a t t e r n s are complex.  The  o r o g r a p h i c e f f e c t s on p r e c i p i t a t i o n  can  be very pronounced e s p e c i a l l y d u r i n g major storms i n areas where mountain s l o p e s are exposed d i r e c t l y to r a i n - b e a r i n g winds, as on the western  s l o p e s of Vancouver I s l a n d .  The  such  network o f snow  survey s i t e s i s a l s o s p a r s e , and the e x t r a p o l a t i o n of snow survey data i s even more tenuous than f o r r a i n f a l l The  data.  s i m p l e s t r a i n f a l l - r u n o f f models are e m p i r i c a l formulae  r e l a t i n g peak f l o w to r a i n f a l l  i n t e n s i t y and p h y s i o g r a p h i c  parameters o f the watershed, such as drainage area or b a s i n s l o p e . The most p o p u l a r formula i s the s o - c a l l e d " r a t i o n a l  formula"  (Q = CIA) which i s w i d e l y used by many agencies i n c l u d i n g  the  16  B r i t i s h Columbia Department of Highways.  A l l these formulae  are  d e f i c i e n t i n t h a t they do not r e c o g n i z e the complexity o f the r u n o f f process.  Each formula c o n t a i n s an e m p i r i c a l c o n s t a n t , C, u s u a l l y  c a l l e d the r u n o f f c o e f f i c i e n t , which i s d i f f i c u l t to e s t i m a t e f o r any watershed.  C i s a c o n s t a n t i n the formula, but e x p e r i e n c e  t h a t i t s v a l u e v a r i e s w i d e l y from storm  to storm  q u e s t i o n a b l e r e l i a b i l i t y o f these formulae  (8).  decreases  The  shows  already  as the watershed  area i n c r e a s e s . Models, such as the U n i v e r s i t y of B r i t i s h Columbia Watershed Budget Model ( 9 ) , process  are much more a c c u r a t e i n s i m u l a t i n g the r u n o f f  than simple  formulae.  and rain-on-snow c o n d i t i o n s .  These models a l s o handle Critical  snowmelt  sequences o f d a i l y  temperature  as w e l l as snowpack data are r e q u i r e d to e v a l u a t e snowmelt r u n o f f . A key aspect o f the U.B.C. Watershed Model i s the of  the watershed i n t o a r e a - e l e v a t i o n bands to account  e l e v a t i o n dependence of p r e c i p i t a t i o n and  temperature.  other watershed c h a r a c t e r i s t i c s such p e r m e a b i l i t y and storage are f r e q u e n t l y e l e v a t i o n dependent.  division  f o r the In a d d i t i o n , groundwater  Some p e r i o d of stream-  flow r e c o r d i s h e l p f u l i n e v a l u a t i n g the c a l i b r a t i o n parameters f o r the model.  The  r e l i a b i l i t y of the p r e c i p i t a t i o n d a t a , and not  l i m i t a t i o n s of model i t s e l f , i s l i k e l y  to impose the major  the  limi-  t a t i o n on the r e l i a b i l i t y of the computed peak flow v a l u e s i f the c a l i b r a t i o n parameters can be determined  reasonably a c c u r a t e l y .  Besides u s i n g p r e c i p i t a t i o n - r u n o f f models, peak flow data for  l a r g e streams c o u l d be transposed  to s m a l l e r streams on a  d i s c h a r g e per u n i t area b a s i s to e s t i m a t e peak f l o w s . must have s i m i l a r p h y s i o g r a p h i c and c l i m a t i c  The  simple  watersheds  characteristics.  17  Even so, t h i s approach i s l i k e l y to underevaluate  s m a l l stream peak  flows because of d i f f e r e n c e s i n t i m i n g o f r u n o f f between l a r g e and s m a l l watersheds. A survey o f e x i s t i n g c u l v e r t i n s t a l l a t i o n s can p r o v i d e  infor-  mation on peak flows t h a t i s u s e f u l i n p r e d i c t i n g flows f o r other watersheds.  C r e s t - s t a g e gauges i n s t a l l e d a t c u l v e r t  entrances  and approach s e c t i o n s are v e r y u s e f u l i n t h i s r e g a r d .  The  computed  peak flow v a l u e s along w i t h the recorded p r e c i p i t a t i o n data can  be  used t o assess p r e c i p i t a t i o n - r u n o f f formulae and watershed models. I f the r e c o r d i s long enough the r e t u r n p e r i o d s can a l s o be  estimated.  Flows computed from d i s c e r n a b l e high-water marks are d i f f i c u l t  to  r e l a t e to a r e t u r n p e r i o d but s t i l l have some v a l u e i n a s s e s s i n g existing 3.2  installations.  Accounting The  f o r U n c e r t a i n t y i n F l o o d Flows  u n c e r t a i n t y i n e v a l u a t i n g f l o o d flows i s accounted f o r  by p l a c i n g upper and probable  lower c o n f i d e n c e  curve, on a f l o o d frequency  l i m i t s , along w i t h a most plot.  The  flood  d i s t r i b u t i o n chosen f o r s p e c i f y i n g the three curves was distribution. I I I , may  frequency the Gumbel  Other d i s t r i b u t i o n s , such as the l o g Pearson Type  be more a p p r o p r i a t e and c o u l d be used e q u a l l y w e l l .  the- Gumbel and  Both  l o g Pearson Type I I I d i s t r i b u t i o n s c o n s i d e r o n l y  the annual f loods ,~ i . e . , the maximum f l o o d peak i n each year.  A  p a r t i a l d u r a t i o n s e r i e s , which i n c l u d e s a l l independent f l o o d events, d i f f e r s s u b s t a n t i a l l y from an annual s e r i e s a t low r e t u r n periods  ( l e s s than about 5 y e a r s ) .  Thus the p a r t i a l d u r a t i o n  i s the more a p p r o p r i a t e c h o i c e i f a c u l v e r t s u s t a i n s damage a t f l o o d s of a r e l a t i v e l y low r e t u r n p e r i o d .  series  18  The most probable curve i n the i n i t i a l a n a l y s i s by  setting Q  lfJ  was  = 150 c f s ( i . e . , the 10-year flood) and Q  T h i s l i n e i s l a b e l l e d 1.0 i n F i g u r e 3.1.  specified  1 Q 0  = 220 c f s ,  The lower and upper  bounds were then simply s p e c i f i e d as m u l t i p l e s  o f the most probable  curve, such as a lower bound o f 0.5 and an upper bound of 1.5 times the most probable curve. i n c r e a s e s as the r e t u r n not  Thus the d i f f e r e n c e period  increases.  between the bounds  Actually  be s t r a i g h t l i n e s but c o u l d be any c u r v e s .  the h y d r o l o g i s t period  floods,  increasing Figure  has very l i t t l e  the bounds need  For i n s t a n c e , i f  confidence i n p r e d i c t i n g  return  the bounds w i l l d i v e r g e even more r a p i d l y w i t h  return  period  than the s t r a i g h t l i n e bounds shown i n  3.1. The f l o o d p r o b a b i l i t y v e c t o r , which can be p l o t t e d  p r o b a b i l i t y density  function,  c a l c u l a t i n g the d i f f e r e n c e  a t the ends o f each i n t e r v a l .  as a  i s e a s i l y computed f o r a s i n g l e  Gumbel p l o t by d i v i d i n g the v e r t i c a l a x i s and  high  into flood  line  intervals  i n the p r o b a b i l i t i e s o f the f l o o d s The p r o b a b i l i t y d e n s i t y  f o r the most probable c u r v e , 1.0, and two m u l t i p l e 1.2 and 1.5, a r e shown i n F i g u r e  functions  curves a l o n e ,  3.2.  The i n f o r m a t i o n conveyed by s p e c i f y i n g  a most p r o b a b l e curve  w i t h upper and lower bounds can a l s o be converted i n t o a s i n g l e f l o o d p r o b a b i l i t y v e c t o r and p l o t t e d  as a p r o b a b i l i t y d e n s i t y  or an e q u i v a l e n t s i n g l e curve Gumbel p l o t .  functioi  First, a probability  m a t r i x i s formed from the Gumbel p l o t w i t h i t s upper and lower bounds exactly 2.  the same as f o r any bounded f u n c t i o n  as o u t l i n e d  i n Chapter  The h o r i z o n t a l  s c a l e o f the Gumbel p l o t , which i s l i n e a r w i t h _ -b r e s p e c t t o the reduced v a r i a t e b (the Gumbel equation i s P = e e  Values -2.0  -1.0  1.01  I.I  0  1.0  1.52.0  of  2.0  5.0  10  b ( Gumbel d i s t r i b u t i o n ) 3.0  4.0  20  50  Return FIG.3.1  FREQUENCY  CU R V E S  Period  5.0  100  200  6.0  7.0  8.0  9.0  10.0  5 0 0 1000 2 0 0 0 5 0 0 0 1 0 0 0 0  (years)  O F AN N U A L F L O O D S ( 1.0 LI N E « Qio= I50cfs , Qioo= 2 2 0 c f s . )  »  .0130] .0120 .0110 .0I00| .0090 .00801.0070 .0060 .0050  -  .0040  0.5-1.5  .0030 .0020 .0010 0  1  20  40  60  80  100 120 140 160 180 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 3 2 0 3 4 0 3 6 0 3 8 0 4 0 0 Flood  FIG. 3.2 F L O O D  FREQUENCY  Magni tude (cf s )  D I S T R I B U T I O N S ( 1.0 C U R V E = Q i o = l 5 0 C F S , Q i o o =  220CFS).  21  where P i s the p r o b a b i l i t y o f e q u a l l i n g or exceeding a f l o o d o f a g i v e n s i z e ) , i s d i v i d e d up suitably  l a r g e range of b.  i n t o e q u a l l y s i z e d b i n t e r v a l s over a The  b i n t e r v a l s are i n f a c t  p e r i o d or p r o b a b i l i t y i n t e r v a l s , f o r example, one the  37 to 45 y e a r r e t u r n p e r i o d s ,  c a l c u l a t e d and  temporarily  and  representing  these p r o b a b i l i t i e s  stored i n a vector  rows of the p r o b a b i l i t y m a t r i x r e p r e s e n t and  return  (sum  are  = 1.0).  return period  The  intervals,  each r e t u r n p e r i o d i n t e r v a l i s i n t u r n represented  by  the  return  p e r i o d a t the p r o b a b i l i t y m i d - p o i n t o f the i n t e r v a l s i n c e o n l y p o i n t i n each X i n t e r v a l i s used i n forming the m a t r i x . cal  verti-  s c a l e of the Gumbel p l o t i s d i v i d e d i n t o f l o o d i n t e r v a l s , f o r  example, 250-255 c f s , and flood i n t e r v a l s . probability  An  the columns of the m a t r i x r e p r e s e n t  element o f the m a t r i x then r e p r e s e n t s  t h a t a f l o o d of a g i v e n r e t u r n p e r i o d , say  which i s a t the p r o b a b i l i t y m i d - p o i n t o f the  o f the elements across But  any  row,  i f the elements of each row  b i l i t y of b e i n g  by  years  250-255 c f s .  as u s u a l , equals  1.0.  are m u l t i p l i e d by  the  (.1/37) - (1/45) , then the sum An  The  proba-  are  of a l l elements i n the  i n d i v i d u a l element of the m a t r i x then  the o v e r a l l p r o b a b i l i t y  c e r t a i n range and probability  40.6  37 to 45 year r e t u r n p e r i o d i n t e r v a l row  m a t r i x w i l l e q u a l 1.0. represents  the  i n the c o r r e s p o n d i n g f l o o d i n t e r v a l , f o r example,  the elements o f the multiplied  these  37 to 45 year r e t u r n  p e r i o d i n t e r v a l , , l i e s w i t h i n a c e r t a i n range,, say sum  The  one  t h a t a f l o o d both l i e s w i t h i n  belongs to a c e r t a i n r e t u r n p e r i o d i n t e r v a l .  a The  t h a t a f l o o d l i e s w i t h i n a c e r t a i n range, r e g a r d l e s s  what r e t u r n p e r i o d  i n t e r v a l i t belongs t o , i s obtained  by  summing  the elements o f the r e s p e c t i v e f l o o d i n t e r v a l column of the  new  of  matrix.  Thus the i n f o r m a t i o n conveyed by a bounded Gumbel p l o t i s  converted i n t o a s i n g l e p r o b a b i l i t y plotted  as a p r o b a b i l i t y  v e c t o r which can i n t u r n be  density function  or a s i n g l e  equivalent  Gumbel curve. Four bounded d i s t r i b u t i o n s ,  i . e . , d i s t r i b u t i o n s d e r i v e d from  bounded Gumbel p l o t s , along w i t h three d i s t r i b u t i o n s d e r i v e d from single  l i n e s were used i n the i n i t i a l  a b l e curve, 1.0, s p e c i f i e d by Q (Q^QQ/Q^Q QJLQQ  to  = 1.47).  Q^Q  1  Q  analysis  w i t h t h e most prob-  = 1 5 0 c f s and Q  1 0 0  =220 c f s  L a t e r , a d i f f e r e n t most probable curve w i t h a  r a t i o equal t o 1.8 was c o n s i d e r e d t o see what e f f e c t  steepening the Gumbel curve would have on the d e c i s i o n The  results  1.8 r a t i o i s used by the B r i t i s h Columbia Department o f Highway  i n t h e i r design c r i t e r i a ,  although t h i s r a t i o can v a r y c o n s i d e r -  a b l y from watershed to watershed. Q ^ Q r a t i o i s about 1.6 (10). s p e c i f i e d by s e t t i n g Q one  tree  1 Q  F o r West Vancouver the Q  to  1 0 0  The new most probable curve was  = 120 c f s and Q  1 0 Q  = 216 c f s .  In t h i s cas  bounded d i s t r i b u t i o n , along w i t h the most probable curve  d i s t r i b u t i o n a l o n e , was used i n the a n a l y s i s . F i g u r e 3.1 shows the e q u i v a l e n t Gumbel p l o t s o f the f o u r bounded d i s t r i b u t i o n s , as w e l l a l l based on a 1.0 l i n e w i t h Q  as some s i n g l e l i n e Gumbel p l o t s , 1 Q  = 150 c f s and Q  curves d e r i v e d from bounded d i s t r i b u t i o n s multiple factors while single as  1.5.  1 0 0  = 220 c f s .  The  are l a b e l l e d by the  o f the lower and upper bounds, such as 0.5-1.5,  l i n e s are l a b e l l e d with a single m u l t i p l e f a c t o r ,  such  T h i s l a b e l l i n g system i s used throughout the t h e s i s .  F i g u r e 3.2 shows some of the p r o b a b i l i t y  density functions.  Figure  23  3.3  shows Gumbel p l o t s  based on the new 1 . 0 l i n e d e f i n e d by Q ^ Q =  120  c f s and Q ^ Q Q = 2 1 6 c f s .  Table 3 . 1 summarizes some o f the i n f o r -  mation c o n t a i n e d i n F i g u r e s 3 . 1 and 3 . 3 by l i s t i n g the e f f e c t i v e f l o o d s o f e l e v e n r e t u r n p e r i o d s f o r the d i f f e r e n t The  term e f f e c t i v e  flood  distributions.  i s used t o denote the f l o o d d e r i v e d by  c o n v e r t i n g a bounded Gumbel p l o t i n t o a s i n g l e  equivalent curve.  Looking a t the r e s u l t s based on the 1 . 0 curve w i t h Q ^ Q = 150  c f s and Q  plots  1  Q  0  = 2 2 0 c f s , f o r the symmetrically bounded Gumbel  the bounded d i s t r i b u t i o n s  a r e more unfavourable than the l.Q  d i s t r i b u t i o n above a r e t u r n p e r i o d o f -about 2 . 3 y e a r s .  The f a c t  t h a t they are more f a v o u r a b l e below t h i s r e t u r n p e r i o d has l i t t l e significance  since i t i s unlikely  the d e s i g n s e l e c t e d  damage a t f l o o d s below the 2 . 3 - y e a r r e t u r n p e r i o d . distribution differs surprisingly  little  will  The  i n less  between a bounded d i s t r i b u t i o n and the 1 . 0 d i s t r i b u t i o n ; 0 . 5 - 1 . 5  and  1.0  0 . 8 - 1 . 2  from the 1 . 0 d i s t r i b u t i o n .  I n c r e a s i n g the steepness o f the 1 . 0 l i n e r e s u l t s  seen by comparing the  sustain  curves i n F i g u r e s  difference t h i s can be 3 . 1  and  3 . 3 .  Values  -2.0  -1.0  1.01  I.I  0  1.52.0  1.0  of  2.0  5.0  10  b (Gumbel 3.0  4.0  20  50  Return FIG. 3.3  FREQUENCY  CURVES  Period  OF A N N U A L  distribution) 5.0  10 0  200  6.0  7.0  8.0  9.0  10.0  5 0 0 1000 2 0 0 0 5 0 0 0 10 0 0 0  (years)  F L O O D S ( 1.0 LI NE = Qio= 1 2 0 c f s , Qioo= 2 l 6 c f s ).  TABLE 3.1 COMPARISON OF EFFECTIVE FLOODS OF VARIOUS RETURN PERIODS FOR DIFFERENT DISTRIBUTIONS 1.0 F l o o d Frequency L i n e S p e c i f i e d by Q Distribution  II.  2  = 220 c f s  57 56 50 44 60 68 85  1  2.0  5.0  10  20  50  100  200  500  1000  94 93 92 90 103 113 141  128 128 132 136 143 153 191  150 151 159 166 170 180 225  171 174 185 195 197 206 257  199 202 218 233 231 239 299  220 224 244 262 257 264 330  241 246 270 291 283 2 89 361  268 275 304 329 317 322 402  289 298 330 359 343 346 433  1.0 F l o o d Frequency L i n e S p e c i f i e d by  Distribution  1.0 0.5-1.5  1 Q 0  Return P e r i o d (yr) 1.1  1.0 0.8-1.2 0.5-1.5 0.3-1.7 0.8-1.5 1.2 1.5  = 150 c f s and Q  1 Q  Q  1 0  10000 357 370 420 459 433 429 536  = 120 c f s and' 1 0 0 = 216 c f s Q  Return P e r i o d (yr) 1.1  2.0  0 0  43 41  5.0 89 89  10  20  50  100  200  500  1000  120 122  149 155  187 198  216 232  244 265  282 311  310 345  10000 404 463  ^ a l l e f f e c t i v e flood values are i n c f s 2 mean value o f t h i s t r u n c a t e d skew normal d i s t r i b u t i o n = 1.108 x most probable  value  Chapter  4  CULVERT HYDRAULICS 4.1  Types o f C u l v e r t Flow The  r e l a t i o n s h i p between the headwater depth and  i s g r e a t l y i n f l u e n c e d by the type of flow through  the d i s c h a r g e  the c u l v e r t .  type of c u l v e r t f l o w o c c u r r i n g a t a g i v e n d i s c h a r g e may  be  The  determined  by many v a r i a b l e s i n c l u d i n g the i n l e t geometry; the s l o p e , s i z e , and roughness o f the c u l v e r t b a r r e l ; and the approach and t a i l w a t e r conditions.  For p r a c t i c a l purposes c u l v e r t flow i s commonly  f i e d into s i x types.  classi-  But by p l a c i n g the c u l v e r t on a 7% s l o p e  and  assuming the t a i l w a t e r n e i t h e r submerges the o u t l e t nor reaches  a  s u b c r i t i c a l depth c a u s i n g backwater e f f e c t s a t any d i s c h a r g e , the number o f p o s s i b l e flow types was 4J.1.  Both the 7% s l o p e and  i n the mountainous and h i l l y  reduced  to t h r e e , shown i n F i g u r e  the t a i l w a t e r assumptions are t e r r a i n c o v e r i n g most of  reasonable  British  Columbia. The h y d r a u l i c computations were based on equations and t a b l e s compiled  by R. W.  C a r t e r i n 1957  (11).  The equations  types o f flow c o n s i d e r e d are g i v e n i n the appendix.  f o r the three  A l l computations  were done by computer s i n c e some c a l c u l a t i o n s r e q u i r e d t e d i o u s i t e r a t i o n procedures.  For example, c a l c u l a t i o n of the headwater  depth r e q u i r e s a c o e f f i c i e n t of d i s c h a r g e , but the c o e f f i c i e n t of d i s c h a r g e i s a f u n c t i o n o f the headwater l e v e l f o r flow types 1 and 2.  The  c r o s s - s e c t i o n a l area o f the headwater p o o l i s assumed  reasonably  l a r g e so t h a t the v e l o c i t y head i s n e g l i g i b l e .  In  a d d i t i o n the volume of water s t o r e d i n the headwater p o o l a t  any  Type I  Type  !  Rapid  2-  Type 3  Critical  s  Full  Depth  Flow  Flow  at  at  Inlet.  Inlet.  Free O u t f a l l .  HW  NOTATION « D - Culver t-dia (min.dia  for  CMP)  d = c r i t i c a l depth h = p i e z o m e t r i c h e a d above culvert invert at downstream HW = d e p t h of water in h e a d w a t e r p o o l H * = c r i t i c a l v a l u e for h e a d w a t e r depth ( H - I.5D u s e d c  s  s c  s = 0  critical bed  FIG.4.1  slope  slope  TYPES  for c u l v e r t  of c u l v e r t  OF  barrel  barrel  CULVERT  FLOW  end here)  28 headwater l e v e l i s assumed s m a l l ; so, e f f e c t i v e l y , a t any  time  the d i s c h a r g e i n t o the headwater p o o l equals the d i s c h a r g e the c u l v e r t . diameters  The headwater-discharge  are shown i n F i g u r e  through  curves f o r s e v e r a l c u l v e r t  4.2.  The entrance o f an o r d i n a r y c u l v e r t w i l l not be submerged i f the headwater i s l e s s than a c e r t a i n c r i t i c a l v a l u e , d e s i g n a t e d by H*,  w h i l e the o u t l e t i s not submerged.  1.2  to 1.5  The value o f H* v a r i e s  times the c u l v e r t diameter, D, depending on the  from  entrance  geometry, b a r r e l c h a r a c t e r i s t i c s , and approach c o n d i t i o n s (12). C a r t e r assumes H* = 1.5D,  so t h i s v a l u e was  used i n the  Chow (12) s t a t e s , "For a p r e l i m i n a r y a n a l y s i s , the upper H* = 1.5D  may  be used  where submergence was  . . . because computations  calculations. limit  have shown t h a t ,  u n c e r t a i n , g r e a t e r accuracy c o u l d be o b t a i n e d  by assuming t h a t the entrance was  not submerged."  Type 1 flow r e s u l t s when the headwater i s l e s s than H*,  the  t a i l w a t e r i s lower than the c r i t i c a l depth, and the c u l v e r t s l o p e is supercritical.  Critical  flow occurs a t or near the c u l v e r t  e n t r a n c e , and the headwater depth depends only on the d i s c h a r g e , c u l v e r t s i z e , and entrance geometry. inlet  Thus, t h i s i s an example o f  control. Type 2 flow i s a l s o an example of i n l e t c o n t r o l , but i n t h i s  case w i t h the entrance submerged.  The  i n l e t f u n c t i o n s as an  orifice  w i t h the flow e n t e r i n g the c u l v e r t c o n t r a c t i n g to a depth l e s s the diameter  than  o f the c u l v e r t b a r r e l i n a manner s i m i l a r to the  c o n t r a c t i o n of flow i n the form of a j e t under a s l u i c e gate.  In  the case o f a square-ended c u l v e r t s e t f l u s h w i t h a v e r t i c a l headw a l l and,  indeed, w i t h most c u l v e r t i n l e t s , type 2 flow f o l l o w s  30  type 1 flow as the headwater depth i n c r e a s e s with i n c r e a s i n g discharge. may  However a t h i g h submergences o f the o r i f i c e the  suddenly  fill  and type 3 flow o c c u r s .  Blaisdell  t h a t the headpool l e v e l a t which t h i s occurs may time the c u l v e r t f i l l s ,  culvert  (2) has  found  be d i f f e r e n t each  making an exact d e t e r m i n a t i o n  difficult.  A t t h i s p o i n t there w i l l be a sudden i n c r e a s e i n flow through c u l v e r t and a r e s u l t i n g decrease  the  i n the headpool l e v e l as the c o n t r o l  changes from the o r i f i c e to the p i p e . A c u l v e r t i s c o n s i d e r e d h y d r a u l i c a l l y s h o r t i f the flow i s type 2 and h y d r a u l i c a l l y long i f the flow i s type 3. prepared  C a r t e r has  c h a r t s to r o u g h l y d i s t i n g u i s h between these two  The d e t e r m i n a t i o n depends on many c h a r a c t e r i s t i c s such as diameter, entrance  types.  culvert  l e n g t h , and s l o p e ; entrance geometry; headwater l e v e l ; and o u t l e t c o n d i t i o n s ; e t c .  f o r a l l c u l v e r t diameters  considered  headwater range o f i n t e r e s t cases the flow was all  flow  type 2.  these cases although  In p r a c t i c e i t turned out t h a t , (3.5 to 7.0  f t ) and over  (up to 10 f t ) , i n a l l submerged A l s o , the 7% s l o p e was  flow types 2 and  the  inlet  a steep s l o p e i n  3 can occur on m i l d or  steep s l o p e s . In type 3 flow the c u l v e r t b a r r e l i s under s u c t i o n w i t h  the  p i e z o m e t r i c head a t the o u t l e t v a r y i n g from a p o i n t below the c e n t r e to the top o f the c u l v e r t .  However, N e i l l  t h a t the t u r b u l e n t , a e r a t e d flow caused may  (13) r e p o r t s  by the pipe c o r r u g a t i o n s  p r e v e n t the e x i s t e n c e o f sub-atmospheric p r e s s u r e s i n the  c u l v e r t and cause the c u l v e r t to flow p a r t l y f u l l . v a r i a t i o n of type 3 flow and not type 2 flow.  This i s a  31  4.2  Entrance.and Entrance  E x i t Improvement  improvement should always be c o n s i d e r e d s i n c e i t can  i n c r e a s e the h y d r a u l i c e f f i c i e n c y o f c u l v e r t s and thus reduce the c u l v e r t s i z e r e q u i r e d . (An i n c r e a s e i n h y d r a u l i c e f f i c i e n c y means t h a t a t a g i v e n flow the headwater s u r f a c e can be lowered;  or stated  c o n v e r s e l y , a t a g i v e n headwater depth the flow accommodated can be increased.)  E x i t improvement may be r e q u i r e d t o p r e v e n t e r o s i o n  problems. The primary purposes o f a headwall and p r o t e c t the embankment from e r o s i o n . a d d i t i o n t o r e t a i n the f i l l the f i l l  behind  a r e t o r e t a i n the f i l l Wingwalls can be used i n  and support the headwall.  the headwall,  endwall,  and wingwalls,  By r e t a i n i n g savings can be  r e a l i z e d by a r e d u c t i o n i n the c u l v e r t l e n g t h r e q u i r e d .  Where  s u f f i c i e n t f a l l i s a v a i l a b l e , c u l v e r t d e s i g n can be improved by making the entrance  i n t o a s l o p i n g apron  occurs on the apron, i n t o the c u l v e r t .  (14).  The c r i t i c a l  depth  and the flow i s a c c e l e r a t e d a l o n g the apron and  The s l o p i n g i n l e t has an a p p r e c i a b l e e f f e c t as  long as the c u l v e r t b a r r e l does n o t flow  full.  Rounding o r t a p e r i n g the i n l e t i n c r e a s e s the h y d r a u l i c e f f i c i e n c y by i n c r e a s i n g the c o e f f i c i e n t s o f d i s c h a r g e f o r a l l flow types.  A more s p e c t a c u l a r i n c r e a s e i n h y d r a u l i c e f f i c i e n c y can be  o b t a i n e d i n some circumstances bell-mouth  o r hood i n l e t s .  c u l v e r t entrance The  by employing s p e c i a l i n l e t s ,  T h i s advantage a p p l i e s o n l y when the  i s submerged and mainly  s p e c i a l i n l e t prevents  such as  to c u l v e r t s on steep s l o p e s .  i n l e t o r i f i c e control  causes the p i p e t o flow f u l l  (type 3 f l o w ) .  (type 2 flow) and  Blaisdell  (2) has found  i n experiments u s i n g a hood i n l e t t h a t an i n t e r m e d i a t e flow  type,  32  s l u g and mixture  flow, c o n s i s t i n g o f a l t e r n a t i n g s l u g s of f u l l  and a i r p o c k e t s , occurs b e f o r e type 3 flow i s e s t a b l i s h e d .  flow  As the  i n l e t j u s t becomes submerged, the a d d i t i o n a l head c r e a t e d by the short length of f u l l to  the c u l v e r t .  c o n d u i t draws the headpool  The a i r flow decreases as d i s c h a r g e i n c r e a s e s u n t i l  the c u l v e r t flows completely f u l l o f water. i n c r e a s e i n the headpool to  cause f u l l  down a d m i t t i n g a i r  There i s v e r y  little  depth u n t i l the d i s c h a r g e i s g r e a t enough  flow.  V o r t i c e s a t c u l v e r t i n l e t s can a d v e r s e l y a f f e c t c u l v e r t p e r formance, p a r t i c u l a r l y d u r i n g p i p e c o n t r o l w i t h low i n l e t submergences, and thus they can decrease inlets.  V o r t i c e s form over the i n l e t and admit a i r to the c u l v e r t  through  the v o r t e x c o r e .  reduces  the d i s c h a r g e .  to  the advantage o f u s i n g s p e c i a l  The a i r r e p l a c e s water i n the c u l v e r t and V o r t i c e s can reduce  the c u l v e r t c a p a c i t y  anywhere between t h a t o b t a i n e d w i t h p i p e c o n t r o l and t h a t obtained  with o r i f i c e control.  On the o t h e r hand, s u r f a c e v o r t i c e s t h a t do  not have an a i r core may have l i t t l e e f f e c t on the c u l v e r t c a p a c i t y . V o r t i c e s can be i n h i b i t e d by i n s t a l l i n g a n t i - v o r t e x d e v i c e s . P l u g g i n g o f c u l v e r t s i s c o n s i d e r e d by many t o be one o f the major problems a s s o c i a t e d w i t h c u l v e r t s  (7).  f l o o d damage, even i n cases o f minor f l o o d s . designed  t o pass expected d e b r i s , keeping  I t can l e a d t o major C u l v e r t s should be  i n mind t h a t any d e b r i s  jams t h a t occur*must be e a s i l y a c c e s s i b l e by maintenance crews. Upstream d e b r i s racks a r e r e q u i r e d i n some l o c a t i o n s . ice  P l u g g i n g by  forming i n s i d e the c u l v e r t can be a problem i n B r i t i s h  Columbia's  Interior.  The o u t l e t end o f a c u l v e r t should be designed  to avoid  33  (1) blockage by d e b r i s , (2) damage by flow undermining  the c u l v e r t  and embankment, and (3) e r o s i o n of the downstream c h a n n e l .  The  g r e a t e r roughness of c o r r u g a t e d metal p i p e as compared t o c o n c r e t e p i p e i s an advantage i n r e d u c i n g o u t l e t v e l o c i t y . or  energy  A s t i l l i n g basin  d i s s i p a t o r o f some s o r t may be r e q u i r e d t o reduce  downstream  erosion. 4.3  Mechanics o f a Washout An assumption  i s made i n the a n a l y s i s t h a t the roadway  wash out as soon as the road i s overtopped.  will  I t i s f u r t h e r assumed  t h a t the washout r e s u l t s i n the same damage to the roadway, no matter what flow caused damaged  the washout, and the c u l v e r t i t s e l f i s not  i n the p r o c e s s .  These assumptions are not completely  but were made t o s i m p l i f y  the a n a l y s i s .  The roadway i s l i k e l y minimal  to withstand  damage, b e f o r e washing o u t .  s t a r t w i t h g r a v e l being eroded embankments,  some o v e r t o p p i n g , w i t h  The washout mechanism  may  a t both the upstream and downstream  e v e n t u a l l y l e a d i n g to the undermining  the road s u r f a c e .  valid  and c o l l a p s e o f  Once the road s u r f a c e c o l l a p s e s the flow r a t e  over the road s u r f a c e w i l l w i l l proceed q u i c k l y .  i n c r e a s e d r a m a t i c a l l y , and the washout  Given the u n c e r t a i n t i e s of the s i t u a t i o n ,  i t may be very d i f f i c u l t t o e s t i m a t e a t what p o i n t a road w i l l wash out. A culvert i s l i k e l y although a headwall away.  t o s u s t a i n some damage d u r i n g a washout,  and endwall may prevent i t from being washed  Scour under the c u l v e r t w i l l mean t h a t the c u l v e r t has t o be  l i f t e d out and r e - i n s t a l l e d .  Highway embankments a r e not designed allowed  as dams.  I f ponding i s  f o r i n the d e s i g n o f a c u l v e r t , p r o v i s i o n must be made so  t h a t seepage through  the embankment w i l l not l e a d t o f a i l u r e by  p i p i n g o r o t h e r means.  A l s o the s l o p e s o f the embankments must not  be so g r e a t t h a t they c o l l a p s e when s a t u r a t e d . 4.4  Environmental Environmental  Considerations c o n s i d e r a t i o n s might be c a l l e d i n t a n g i b l e s i n  an economist's terms. f i s h i n a stream  I t i s d i f f i c u l t t o p l a c e a monetary v a l u e on  because they may be worth much more than  commercial v a l u e .  their  I f f i s h and o t h e r a q u a t i c organisms are to be  p r e s e r v e d i n streams p a s s i n g through  c u l v e r t s , economic a n a l y s i s  f o r c u l v e r t d e s i g n may have t o be supplemented by a n a l y s i s o f the e f f e c t s o f the proposed d e s i g n on the organisms i n v o l v e d . High flow v e l o c i t i e s i n c u l v e r t s are common and may f i s h from moving upstream.  prevent  R e i n f o r c e d c o n c r e t e p i p e , w i t h i t s low  roughness c o e f f i c i e n t , i s more o f a problem than c o r r u g a t e d pipe.  B a f f l e s might be needed t o reduce the v e l o c i t y .  facilities, the c u l v e r t .  metal  Exit  f o r example, 5 f o o t drops, o f t e n i n h i b i t f i s h access t o One approach t o the e n t i r e problem i s t o p r e s e r v e  the n a t u r a l streambed by i n s t a l l i n g a s u f f i c i e n t l y l a r g e arch s t r u c t u r e , although pipe c u l v e r t . •  •\  i t i s bound t o be much more expensive  than a  Chapter  5  ECONOMICS 5.1  C a p i t a l Cost The approximate  i n Table 5.1  c a p i t a l c o s t s of i n s t a l l e d c u l v e r t s are shown  and F i g u r e 5.1.  These c o s t s are f o r 100  f t lengths  of asbestos bonded, a s p h a l t coated c o r r u g a t e d metal p i p e c u l v e r t s , w i t h v e r t i c a l c o n c r e t e headwalls i n the a n a l y s i s .  The  i n s t a l l e d CMP  West Vancouver Drainage Engineers  (10).  The  (CMP)  and endwalls, as used  c o s t s are from the D i s t r i c t o f  Survey by Dayton and Knight L t d . , C o n s u l t i n g  i n s t a l l a t i o n c o s t i s based on  "average"  c o n d i t i o n s i n West Vancouver and r e p r e s e n t s the c o s t of i n s t a l l i n g a c u l v e r t under an e x i s t i n g highway.  Consequently  c o s t w i l l be somewhat l e s s f o r a new p a r t i c u l a r l y under f i l l s ,  the  installation  highway c o n s t r u c t i o n p r o j e c t ,  as l i t t l e o r no e x c a v a t i o n w i l l be r e q u i r e d  The c o s t s can o n l y be taken as approximate  because they depend to  a l a r g e e x t e n t on the c o n d i t i o n s a t each c u l v e r t s i t e .  The  trans-  p o r t a t i o n c o s t t o the s i t e i s a l s o a v a r i a b l e f a c t o r t h a t must not be  overlooked. The c o s t l e v e l s used i n the Dayton and Knight r e p o r t are  e q u i v a l e n t to an E n g i n e e r i n g News-Record Index o f 2500 f o r 1975.  (ENR)  The c o s t s i n T a b l e 5.1  have been a d j u s t e d t o an ENR  C o n s t r u c t i o n Cost and F i g u r e  index of 3000 f o r 1977.  The  5.1 headwall-  endwall s e t c o s t s were c a l c u l a t e d from C a l i f o r n i a D i v i s i o n of Highways v a l u e s p r e s e n t e d i n P r i t c h e t t ' s thesis (3) by m u l t i p l y i n g by the r a t i o o f the ENR  index i n 1977  to t h a t i n 1964  T h i s method o f updating c o s t s i s o n l y approximate 35  (3000/900).  as the ENR  index  TABLE 5.1 CAPITAL COSTS OF INSTALLED CULVERTS  Culvert Diameter (feet)  Pipe Cost* ($)  Headwall & Endwall Cost ($)  Total Cost ($)  3.0  5280  1270  6550  3.5  6300  1570  7870  4.0  7560  1870  9430  4.5  9120 •  2170  11290  5.0  10680  2470  13150  5.5  12480  2770  15250  6.0  14400  3080  17480  6.5  16560  3400  19960  7.0  19200  3700  22900  * f o r 100 f t l e n g t h  24000  220001-  200001-  I8000H-  160001—  140001-  I2000h-  100001—  8 0 0 0 H  6000h-  40001—  20001-  4.0  4.5  Culvert  FIG.5.1  CAPITAL  COSTS  5.0 Diameter  OF  5.5  6.0  (feet)  INSTALLED  CULVERTS  7.0  38  r e p r e s e n t s the c o s t of a group of items c o n s i s t i n g o f f i x e d q u a n t i t i e s of l a b o u r , cement, s t e e l , and lumber, and not the c o s t of  p u r c h a s i n g and i n s t a l l i n g c u l v e r t s .  There w i l l a l s o be  d i s p a r i t i e s between C a l i f o r n i a and B r i t i s h Columbia 5.2  F l o o d Damage The  sum if  costs.  f l o o d damage c o s t a t a p a r t i c u l a r headwater l e v e l i s the  o f two  items: the headwater damage c o s t and the washout c o s t  the road washes out. Headwater damage i s the r e s u l t of water backing up and  ing  p u b l i c or p r i v a t e p r o p e r t y upstream o f the c u l v e r t .  flood-  Damage to  the highway embankment, such as e r o s i o n o f g r a v e l caused by h i g h headwater, i s i n c l u d e d under headwater damage. is likely  to be a problem  encroaches ponding  Upstream f l o o d i n g  o n l y i n p o p u l a t e d areas where development  on the stream, or i n f l o o d p l a i n s where s u b s t a n t i a l  can take p l a c e and inundate l a r g e areas o f r e s i d e n t i a l or  a g r i c u l t u r a l land. The headwater damage curve used i n the a n a l y s i s i s shown i n F i g u r e 5.2.  The  shape of the curve was  m a r g i n a l f l o o d damage f i r s t  chosen  a r b i t r a r i l y with  i n c r e a s i n g then d e c r e a s i n g .  A  typical  f l o o d damage v s . depth curve f o r urban p r o p e r t y i s shown by James and Lee  (15) as a combination of t h r e e s t r a i g h t l i n e s w i t h the  first  segment having the g r e a t e s t s l o p e and the f i n a l segment a s l o p e o f :  zero.  The damage i s assumed to be a f u n c t i o n o f headwater  o n l y and not o f c u l v e r t s i z e .  T h i s may  level  not be t r u e i n the case of  damage t o the highway embankment as v e l o c i t y and t u r b u l e n c e around the c u l v e r t i n l e t a t a g i v e n headwater w i l l vary f o r d i f f e r e n t culvert  diameters.  0  5  6  7  Headwater  F I G . 5.2  HEADWATER  Depth  DAMAGE  8  9  (feet)  FUNCTION  40  The  roadway i s assumed t o wash out i f the headwater overtops  the highway  (i.e.,  exceeds 10 f t i n t h i s c a s e ) .  assumed to r e s u l t i n e x t e n s i v e to the c u l v e r t and assumptions was sum  of  washout i s  damage to the roadway but no damage  i t s headwall and  discussed  The  endwall.  i n Chapter 4.  The  The  v a l i d i t y of  these  washout c o s t i s the  (1) the c o s t of r e p a i r i n g the highway, (2) expenses f o r  flagmen, b a r r i c a d e s ,  f l a r e s , and  s i g n i n g f o r t r a f f i c detours,  (3) the c o s t o f i n t e r r u p t i n g t r a f f i c , which i s borne by users themselves.  The  repair cost w i l l  depend on the  the  and  road-  availability  of l a b o u r , m a t e r i a l s , and machinery, as w e l l as the extent  of  damages. The mine.  cost of i n t e r r u p t i n g t r a f f i c  I t i n c l u d e s the i n c r e a s e d motor v e h i c l e o p e r a t i n g  detour mileage, slowdowns, s t o p s , increased bility.  t r a v e l time; and  value  cost for  accident  proba-  from v e h i c l e to v e h i c l e , p a r t i c u l a r l y  c a r s ; t h e r e f o r e a weighted average must be  at one-third  volume o f t r a f f i c ,  the average wage. time r e q u i r e d to r e p a i r the road,  type o f detour r o u t e a v a i l a b l e a l l i n f l u e n c e the magnitude of cost of i n t e r r u p t i n g t r a f f i c .  and the  I f no detour i s a v a i l a b l e on a major .  highway, the c o s t w i l l be very h i g h . low  used.  o f time l o s t f o r occupants of v e h i c l e s not on b u s i n e s s i s  often evaluated The  deter-  and v e h i c l e washing; the c o s t of  the c o s t o f i n c r e a s e d  These c o s t s w i l l vary  between t r u c k s and The  i s more d i f f i c u l t t o  C o n v e r s e l y , the c o s t w i l l  be.  f o r minor highways. A washout c o s t of $15,000 i s used i n the i n i t i a l  The  c o s t borne by  and  providing  analysis.  the highways department f o r r e p a i r i n g the  road  flagmen, b a r r i c a d e s , e t c . i s assessed a t $5000,  and  41  the c o s t borne by the road-users a t $10,000. i s roughly  The  road-user c o s t  c a l c u l a t e d as the product of the average d a i l y  (ADT), the time r e q u i r e d  to r e p a i r the road i n days, and  c o s t of delay per v e h i c l e .  The  A t y p i c a l ADT  the  average  i s the  average  both d i r e c t i o n s o f  travel.  average d a i l y t r a f f i c  24-hour volume f o r a g i v e n y e a r , c o u n t i n g  traffic  o f 2500 f o r a major r u r a l two-lane highway i s assumed,  and  the time r e q u i r e d to r e p a i r the highway i s e s t i m a t e d a t 2 days.  The  average c o s t of d e l a y per v e h i c l e , i n c l u d i n g both i n c r e a s e s  operating low  c o s t and  t r a v e l time, i s s e t a t $2.00 per v e h i c l e .  in  This  c o s t per v e h i c l e i m p l i e s a r e l a t i v e l y minor d e t o u r . I t might be argued t h a t road-user c o s t s should  not be  included  i n the economic a n a l y s i s s i n c e the highways department does not compensate m o t o r i s t s  f o r the d e l a y .  However, l o o k i n g a t the problem  from a broad s o c i a l p o i n t o f view, which a government should do,  these c o s t s are r e a l and must be  p u b l i c e n t i t i e s and  i n c l u d e d s i n c e highways are  not p r i v a t e l y owned.  Some mention o f maintenance c o s t should was  not i n c l u d e d  always  i n the a n a l y s i s .  be made, although i t  P r i t c h e t t , i n h i s t h e s i s , assumes  an e q u a l average maintenance c o s t f o r p i p e c u l v e r t s from 18 on the b a s i s t h a t the  to 96 i n .  l a r g e r c u l v e r t s have a l a r g e r area o f brush  to c l e a r a t the entrance and  e x i t of the p i p e , but  l e s s sand  and  d e b r i s to c l e a n out as compared t o the s m a l l e r diameter c u l v e r t s . Using t h i s assumption, the c u l v e r t s i z e d e c i s i o n w i l l by 5.3  not be a f f e c t e d  the maintenance c o s t . Annual Cost Comparison Before an economic a n a l y s i s f o r choosing c u l v e r t s i z e can  completed, the c a p i t a l c o s t and  expected annual damage c o s t ,  be  42  computed as o u t l i n e d i n Chapter 2, must be placed on a comparable b a s i s so they can be added.  The  e q u i v a l e n t uniform annual c o s t  method, i n which the investment c o s t i s converted to an annual c o s t , is  used i n t h i s case.  The  present  combining the investment c o s t and a s i n g l e present  worth sum,  value method, which  involves  expected annual damage c o s t i n t o  c o u l d e q u a l l y w e l l be used and  would  y i e l d the same r e s u l t as the e q u i v a l e n t uniform annual c o s t method. The  f a c t o r to c o n v e r t  annual c o s t i s designated  an investment c o s t i n t o an  as the c a p i t a l - r e c o v e r y f a c t o r and  be computed from the e x p r e s s i o n is  the d i s c o u n t  r(l + r) /((l n  r a t e per annum and  + r)  may  - 1 ) , where r  n  n i s the estimated s e r v i c e  o f the c u l v e r t or highway, whichever i s s h o r t e r . for  equivalent  The  life  equation i s  a s e r i e s o f n year-end payments, as shown i n F i g u r e  5.3,  although  the c a p i t a l - r e c o v e r y f a c t o r w i l l not be s i g n i f i c a n t l y d i f f e r e n t f o r a s e r i e s of n mid-year payments, as long as n i s not too The  question  of what i s the c o r r e c t d i s c o u n t  r a t e to use  computing the c a p i t a l - r e c o v e r y f a c t o r i s a matter of debate.  I t i s a very  discount  rate  (i.e.,  important q u e s t i o n from 4%  change the p r o j e c t s e l e c t e d . life will  favour  small.  considerable  as a change of 1%  to 5% or from 7% to 6%)  will  i n the often  A low d i s c o u n t r a t e w i t h a long  designs w i t h a h i g h c a p i t a l c o s t s i n c e the  investment charge w i l l be  in  lower than i n the case where the  service  annual discount  X,  r a t e i s h i g h or the s e r v i c e l i f e The rate.  term d i s c o u n t  is  low.  r a t e i s used to d i s t i n g u i s h i t from i n t e r e s t  D i s c o u n t r a t e , r , as used here, i s the r e a l r a t e o f i n t e r e s t  as opposed to the money r a t e of i n t e r e s t , x. be computed as r =  The  discount  rate  can  (x - i ) / ( l + i ) or approximately r = x - i , where  Method  l  !  Uniform A  43  Series  A  A  A  A n • 21 n - I I n  I  C-CRF  A  CRF =  r(l + r )  n  where  (l + r ) - l  r =  n  x-i I +i  C A CRF r  = c a pita I cost = e q u i v a l e n t annual cost in base year dollars (i.e.dollars at = capital-recovery factor beginning of year I) = discount rate  x i  - m o n e y rate of i n t e r e s t = i n f l a t i o n rate  Method  2  Exponential  :  Series A -3  A -2 'n- l n  n  A  0  " i  LLili..,  n-2  A = C- E C R F 0  ECRF  A  (T^)[(7fr)-']  = annual cost  x  in  dollars  of y e a r  x •, A  E C R F = exponential series capital-recovery N . B . A i s not i n c l u d e d  in the summation  Q  in c o n f o r m i t y  x  =A  0  the  CONVERTING  CAPITAL  COST  TO  ECRF,  convention.  It c a n be e a s i l y p r o v e n that C R F = E C R F if r as d e f i n e d is u s e d in c a l c u l a t i n g the C R F . T h e r e f o r e the two m e t h o d s equivalent.  FIG.5.3  x  factor  for c a l c u l a t i n g  with the p e r i o d - e n d step  ( I + i )  ANNUAL  above are  COST.  44  i  i s the r a t e o f i n f l a t i o n .  T h i s e q u a t i o n c o r r e c t s the money r a t e  of i n t e r e s t f o r the e f f e c t o f i n f l a t i o n . A discount  r a t e o f 4% was chosen f o r the i n i t i a l  analysis.  T h i s f i g u r e was based on an i n t e r e s t r a t e f o r r i s k f r e e investment, such as government bonds, e q u a l t o about 10% and a r a t e o f i n f l a t i o n equal t o about 6%.  In f a c t , both the money i n t e r e s t r a t e and the  i n f l a t i o n rate are l i k e l y life  to fluctuate considerably  of the c u l v e r t o r highway.  r a t e a r e u s u a l l y much s m a l l e r , i n t e r e s t adjusts  over the s e r v i c e  But f l u c t u a t i o n s i n the r e a l i n t e r e s t as i n the long r u n the money r a t e o f  t o account f o r the i n f l a t i o n r a t e .  i n t e r e s t r a t e s on government savings bonds i n c r e a s e d in  the e a r l y 1960s t o 8 t o 10% i n the 1970s.  As an example, from about 5%  But the c a l c u l a t e d  r e a l i n t e r e s t r a t e h e l d steady f o r 1965 to 1972 a t a moderate of 3% b e f o r e i t f e l l An e q u i v a l e n t  level  i n 1973 (16). method o f h a n d l i n g  the problem o f i n f l a t i n g  c o s t s i s i l l u s t r a t e d by the e x p o n e n t i a l  s e r i e s i n F i g u r e 5.3.  Here the c a p i t a l c o s t i s converted t o an e x p o n e n t i a l  s e r i e s o f annual  c o s t s i n c r e a s i n g a t the r a t e o f i p e r c e n t per annum, as opposed to a s e r i e s o f u n i f o r m annual c o s t s . c o s t i s a l s o assumed t o i n c r e a s e per  c e n t p e r year; t h e r e f o r e ,  The expected annual damage  exponentially  a t the r a t e o f i  the two s e r i e s o f annual c o s t s can  be added t o determine the s e r i e s o f t o t a l annual c o s t s  f o r a given  c u l v e r t diameter. A c t u a l l y , only  the annual c o s t s a t the b e g i n n i n g o f the base  year need be computed s i n c e a l l annual c o s t s i n c r e a s e  a t the same  . r a t e , i . Hence the c u l v e r t s i z e d e c i s i o n can be made by comparing the t o t a l annual c o s t s i n the base y e a r .  The money r a t e o f i n t e r e s t ,  45 x, i s used t o compute the annual investment charge a t the b e g i n n i n g o f the base year s i n c e the e f f e c t o f i n f l a t i o n i s taken i n t o account directly.  In f a c t , the annual investment charge computed a t the  b e g i n n i n g o f the base year w i l l be same f o r the e x p o n e n t i a l method and the e q u i v a l e n t u n i f o r m annual c o s t method (1 + i ) ) ; t h e r e f o r e The  the two methods a r e e x a c t l y  d i s c u s s i o n o f the e x p o n e n t i a l  series  (r = (x - i ) /  equivalent.  s e r i e s i s meant t o p o i n t o u t  the importance o f t a k i n g the r a t e o f i n f l a t i o n i n t o account.  It  would be a s e r i o u s e r r o r t o c a l c u l a t e the c a p i t a l - r e c o v e r y f a c t o r f o r the e q u i v a l e n t u n i f o r m s e r i e s method on the b a s i s o f the money rate of i n t e r e s t with i t s b u i l t - i n i n f l a t i o n f a c t o r .  T h i s would  amount t o adding a uniform s e r i e s , the annual c a p i t a l c o s t , t o an exponentially  i n c r e a s i n g s e r i e s , the expected annual damage c o s t .  I f the e q u i v a l e n t  uniform s e r i e s method i s a p p l i e d , the money r a t e  of i n t e r e s t must be c o r r e c t e d  f o r the e f f e c t o f i n f l a t i o n so t h a t  there w i l l be two uniform s e r i e s , both i n base year d o l l a r s . The  foregoing  d i s c u s s i o n assumes t h a t the expected annual  damage c o s t i n c r e a s e s  a t the same r a t e as i n f l a t i o n , o r i n o t h e r  words, remains the same i n r e a l terms.  Factors  such as upstream  land development and highway t r a f f i c growth w i l l r e s u l t i n a r e a l increase  i n the expected annual damage c o s t .  a l t e r n a t e routes  Construction of  o r switches to other modes o f t r a n s p o r t a t i o n (due  to r a p i d l y i n c r e a s i n g g a s o l i n e p r i c e s , e t c . ) w i l l r e s u l t i n a r e a l decrease i n the expected annual damage c o s t .  I t i s often  difficult  to f o r e c a s t these changes, p a r t i c u l a r l y over a long p e r i o d o f time such as 20 o r 30 y e a r s , I t should  b u t some attempt should  be made.  be mentioned t h a t annual c o s t c a l c u l a t i o n s a r e  46  v a l i d r e g a r d l e s s of the f i n a n c i n g scheme employed to pay the c a p i t a l c o s t , as long as the d i s c o u n t r a t e i s a p p r o p r i a t e stances  f o r the circum-  (17) .  A c u l v e r t s e r v i c e l i f e o f 30 years was used i n the i n i t i a l analysis. asbestos  A c t u a l l y t h i s value i s c o n s e r v a t i v e as a p r o p e r l y bonded, a s p h a l t coated  CMP  can be expected t o l a s t much  l o n g e r ; p a r t i c u l a r l y i f i n a d d i t i o n the i n v e r t i s paved w i t h o r concrete  to guard a g a i n s t sediment a b r a s i o n .  asphalt  F a c t o r s such as  the c o r r o s i o n p o t e n t i a l a t the proposed c u l v e r t s i t e , pated highway s e r v i c e l i f e ,  installed,  the a n t i c i -  and c o s t w i l l i n f l u e n c e the c u l v e r t  m a t e r i a l , m a t e r i a l t h i c k n e s s , and type o f p r o t e c t i v e treatment selected.  For example, f o r temporary roadways such as l o g g i n g  o n l y simple  g a l v a n i z e d CMP  c u l v e r t s would be j u s t i f i e d .  This  d e c i s i o n c o u l d a l s o be i n c l u d e d i n the d e c i s i o n t r e e o f F i g u r e w i t h d i f f e r e n t m a t e r i a l s or p r o t e c t i v e c o a t i n g s having service l i v e s .  A f u r t h e r complication i s introduced  roads,  2.2  different  i f culvert  damage i s a n t i c i p a t e d when the roadway washes out s i n c e the s e r v i c e l i f e o f the c u l v e r t may be shortened  or terminated  by damage.  There may be a g r e a t d e a l o f u n c e r t a i n t y i n e s t i m a t i n g the s e r v i c e l i f e o f a highway o r c u l v e r t . noted t h a t i f n i s i n i t i a l l y  In t h i s r e g a r d i t should  be  l a r g e , say 30 y e a r s , a l a r g e i n c r e a s e  i n n, say to 100 y e a r s , w i l l o n l y moderately change the c a p i t a l recovery  factor.  The d i f f e r e n c e i n the c a p i t a l - r e c o v e r y f a c t o r w i t h  i n c r e a s i n g n w i l l decrease as the d i s c o u n t r a t e , r , i n c r e a s e s .  Chapter  6  RESULTS 6.1  Annual Cost Curves f o r One  F l o o d Frequency  Distribution  The annual c o s t curves f o r the s i n g l e l i n e Gumbel p l o t d e f i n e d by s e t t i n g Q ^ g = 1 5 0 c f s and Q - ^ Q Q = 2 2 0 c f s are shown i n F i g u r e 6 . 1 . The expected t o t a l c o s t curve f o r convenience)  (the word "expected" i s o f t e n o m i t t e d  shows t h a t the optimum c u l v e r t diameter i s 5 . 0 f t  w i t h s m a l l e r diameter c u l v e r t s becoming l e s s c o m p e t i t i v e more r a p i d l y than l a r g e r diameter c u l v e r t s . was  The c o s t data from which F i g u r e 6 . 1  p l o t t e d , as w e l l as some a d d i t i o n a l i n f o r m a t i o n , i s g i v e n i n  Table 6 . 1 . The  s o - c a l l e d m a r g i n a l investment c o s t s (MIC)  listed in  Table 6 . 1 are the d i f f e r e n c e s i n annual investment c o s t between g i v e n s i z e d c u l v e r t s and c u l v e r t s of the next s m a l l e r s i z e .  Similar-  l y m a r g i n a l s a v i n g s (MS)  annual  i s the d i f f e r e n c e i n expected t o t a l  damage c o s t between a g i v e n s i z e d c u l v e r t and the c u l v e r t of the next s m a l l e r s i z e .  The use o f these m a r g i n a l c o s t s and s a v i n g s i s  f u l l y e x p l a i n e d i n the t h i r d s e c t i o n of t h i s c h a p t e r . Table 6 . 2 g i v e s the annual p r o b a b i l i t y of i n c u r r i n g some .headwater damage and the annual p r o b a b i l i t y o f a washout f o r each  culvert  diameter, f i r s t u s i n g the Gumbel p l o t d e f i n e d above and then u s i n g the Gumbel p l o t d e f i n e d by Q ^ Q = 1 2 0 c f s and Q - ^ Q Q 6.2  =  2 1 6cfs.  The E f f e c t of U n c e r t a i n t y and the Value of B e t t e r I n f o r m a t i o n The e f f e c t , o f u n c e r t a i n t y i n the f l o o d frequency p l o t w i t h  the most probable curve s p e c i f i e d by Q  47  n n  = 1 5 0 c f s and Q  i n r i  = 2 2 0 cfs  2000  18 0 0  4.0  4.5  5.0 Culvert  FIG.6.1  ANNUAL  COST  DISTRIBUTION Qioo = 2 2 0 c f s .  5.5 Diameter  6.0  6.5  7.0  (feet)  CURVES  FOR F L O O D  DEFINED  BY  FREQUENCY  Qio=l50cfs  AND  TABLE 6.1 ANNUAL COSTS FOR FLOOD FREQUENCY DISTRIBUTION DEFINED BY Q  Culvert Diameter (ft) ,  Investment Cost • ($) 1  M a r g i n a l Headwater Investment Damage Cost Cost ($/size) ($)  1 Q  = 150 CFS AND Q  Margina^ Washout T o t a l Cost Damage Savings Cost ($/size) ($) ($)  Total Cost ($)  1 0 Q  = 220 CFS  Increase % Increase in Total in Total Cost from Cost from Optimum (.$) Optimum  4.0  545  90  348  786  1134  1998  1679  810  93. 2  4.5  653  108  146  209  356  778  1009  140  16.1  5.0  760  107  61  47  108  - 248  869  —  —  5.5  882  122  29  12  41  67  923  54  6.2  6.0  1011  129  15  3  18  23  1029  160  18.4  6.5  1154  143  8  0  9  9  1163  294  33.8  7.0  1324  170  5 •  0  5  4  1329  460  52.9  ^based  on r = 4% and n = 30 y r  MIC = (annual investment culvert size)  c o s t o f g i v e n c u l v e r t s i z e ) - (annual investment  c o s t o f next s m a l l e r  3 MS = (annual t o t a l damage c o s t o f next s m a l l e r c u l v e r t s i z e ) - (annual t o t a l damage c o s t o f given c u l v e r t size) vo  50  TABLE 6.2 PROBABILITIES OF INCURRING SOME HEADWATER DAMAGE AND PROBABILITIES OF A WASHOUT  F l o o d Frequency D i s t r i b u t i o n S p e c i f i e d by: Q Q  Culvert Diameter (ft)  1 0 1 0 0  = 150 c f s , = 220 c f s  Probability HW > 5 f t  Probability HW > 10 f t  Q Q  1 Q 1 0 0  = 120 c f s , ='216 c f s  Probability HW > 5 f t  Probability HW > 10 f t  4.0  ,48705  .05242  ,17658  .03051  4.5  ,33201  .01396  ,12592  .01157  5.0  ,21641  .00310  08901  .00386  5.5  ,13704  .00081  06253  .00145  6.0  10000  .00018  04930  .00048  6.5  06170  .00002  03441  .00011  7.0  04450  .00000  02705  .00003  JO,  51  is  shown i n F i g u r e 6.2.  as i n Chapter  3.  The curve l a b e l l i n g system  i s the same  The e f f e c t o f u n c e r t a i n t y i n changing the o p t i m a l  d e c i s i o n from t h a t of the most probable curve alone appears r a t h e r minimal.  F o r a symmetric d i s t r i b u t i o n  to be  (upper and lower  bounds e q u i d i s t a n t from the most probable c u r v e ) , the bounds must be somewhat f u r t h e r a p a r t than 0.5-1.5 b e f o r e a s w i t c h t o a 5.5 f t c u l vert i s indicated.  The expected  t o t a l c o s t curve f o r the asymmetri-  c a l l y bounded d i s t r i b u t i o n , 0.8-1.5, i s v e r y s i m i l a r to t h a t of the 0.3-1.7 d i s t r i b u t i o n . F i g u r e 6.3 shows the r e s u l t s f o r the two d i s t r i b u t i o n s w i t h the new most probable curve s p e c i f i e d by Q ^ Q = 120 c f s and Q - ^ Q Q = 216 c f s .  As a r e s u l t o f the s t e e p e r most probable c u r v e , the t o t a l  c o s t s o f c u l v e r t diameters  l e s s than the optimum diameter i n c r e a s e  l e s s r a p i d l y than i n the cases shown i n F i g u r e 6.2, although a g a i n c u l v e r t s s m a l l e r than the optimum become l e s s c o m p e t i t i v e more r a p i d l y than c u l v e r t s l a r g e r than the optimum. u n c e r t a i n t y i n changing  The e f f e c t o f  the o p t i m a l d e c i s i o n i s l e s s w i t h the new  most probable curve, as can be seen by comparing the 0.5-1.5 and 1.0 curves o f F i g u r e s 6.2 and 6.3. Two methods o f c a l c u l a t i n g the v a l u e o f b e t t e r i n f o r m a t i o n are d i s c u s s e d i n the f o l l o w i n g paragraphs.  The second method i s  the b e t t e r o f the two, and although t h i s method was not a c t u a l l y used i n the a n a l y s i s , i t warrants The  a full  discussion.  f i r s t method assumes t h a t the most probable  frequency curve i s i n f a c t the t r u e c u r v e . curve i s the 1.0 curve o f F i g u r e 6.2 o r 6.3.  flood  Then the t r u e t o t a l c o s t The v a l u e of b e t t e r  i n f o r m a t i o n i s simply the d i f f e r e n c e between the t o t a l c o s t s on the  52 2600  2400  2200  (A i _  O O TJ  o o o C  ^  1000  5.5  6.0  Diameter  FIG.6.2  TOTAL FLOOD  ANNUAL  COST  FREQUENCY  Qio= I 5 0 c f s  AND  7.0  (feet)  CURVES  FOR  DIFFERENT  Dl S T R I B U T I O N S ( 1. 0  Qioo=  220cfs.)  CUR V E =  2000  3.5  4.0  4.5  5.0  Culvert  FIG.6.3  ANNUAL AND  COST  5.5  6.0  D i a m e t e r (feet)  CURVES  Qioo = 2 l 6 c f s ).  ( 1.0 C U R V E = Qio= 120 c f s  54  1.0  curve of the c u l v e r t diameter chosen under u n c e r t a i n t y and  t r u e optimum c u l v e r t diameter.  The  v a l u e s of b e t t e r  c a l c u l a t e d i n t h i s manner f o r the most probable Q^g  c f s and Q - ^ Q Q  = 150  =  220  t h i s method, b e t t e r i n f o r m a t i o n o n l y has  information  curve  c f s are g i v e n i n Table  the  specified  6.3.  by  Using  a value i f the optimum  d e c i s i o n under u n c e r t a i n t y i s d i f f e r e n t from the optimum d e c i s i o n of the most probable  curve  alone.  I f the percentage i n c r e a s e i n  t o t a l c o s t of the next l a r g e r s i z e above the optimum i s s m a l l , such as 0.52%  f o r the 0.5-1.5 d i s t r i b u t i o n , the d e c i s i o n maker w i l l  choose the l a r g e r s i z e , changing the value of b e t t e r The  method j u s t d i s c u s s e d i s fundamentally  the t r u e f l o o d frequency of b e t t e r i n f o r m a t i o n may  curve  i s never known.  information.  unsound because In f a c t the  value  be s u b s t a n t i a l even i f the optimum d e c i s i o n  under u n c e r t a i n t y i s the same as t h a t of the most probable alone.  likely  curve  For i n s t a n c e , t a k i n g the 0.8-1.2 d i s t r i b u t i o n as an example,  there i s a chance t h a t the t r u e t o t a l c o s t curve F i g u r e 6.2.  I n s t a l l i n g a 5.0  i s the 1.2  curve  diameter w i l l l i k e l y be 4.5  i s the t r u e curve f t , and  thus i n s t a l l i n g a 5.0  m u l t i p l e s of the most probable  curves A curve  f t culvert  These examples  to c a l c u l a t e the value o f p e r f e c t i n f o r m a t i o n .  A number o f t o t a l c o s t curves  bounds.  f t culvert.  the optimum c u l v e r t  r e s u l t s i n a g r e a t e r t o t a l c o s t than the optimum. suggest a way  of  f t diameter c u l v e r t then r e s u l t s i n  a t o t a l c o s t o f $84/yr more than the optimum f o r a 5.5 S i m i l a r l y , i f the 0.8  curve  c o u l d be c a l c u l a t e d f o r d i f f e r e n t  curve between the upper and  For example, i f the bounds are 0.8 c o u l d be c a l c u l a t e d f o r the c u r v e s :  and 0.80,  1.2, 0.85,  total 0.90,  lower cost ...  1.20.  i s then p l o t t e d of the t o t a l c o s t of the optimum c u l v e r t v s .  55  TABLE 6 . 3 THE EFFECT OF UNCERTAINTY IN CHANGING THE OPTIMAL DECISION AND THE VALUE OF BETTER INFORMATION  Flood Frequency Distribution  1.0  Optimum Culvert Diameter (ft)  % Increase i n T o t a l Cost o f Next Larger Size  Value o f B e t t e r Information ($)  2  Annual Value  Present Value 4  1.0  5 . 0  0 . 8 - 1 . 2  5 . 0  5.32  0  0  0 . 5 - 1 . 5  5 . 0  0.52  0  0  0 . 3 - 1 . 7  5 . 5  5 . 8 3  54  930  0 . 8 - 1 . 5  5 . 5  6.29  54  930  1.2  5 . 5  4.67  54  930  1.5  6 . 0  0 . 3 1  curve: Q - ^ Q  =  '  6 . 2 1  cfs,  1 5 0  2  u s i n g t o t a l annual c o s t ed (see F i g u r e 6 . 2 )  Q  1  0  0  =  2 2 0  curve f o r  160  cfs  d i s t r i b u t i o n being c o n s i d e r —  assuming b e t t e r i n f o r m a t i o n r e s u l t s i n the t r u e curve i d e n t i f i e d as the 1 . 0 curve P r e s e n t Value = Annual Value / CRF CRF  =  . 0 5 7 8 3 ;  r  =  4%,  n  =  30  CRF = c a p i t a l - r e c o v e r y f a c t o r r = discount rate n = service l i f e  2770  yr  beinq  56  the m u l t i p l e of the most probable  curve.  The  probabilities  that  the t r u e curve  l i e s w i t h i n s m a l l i n t e r v a l s of m u l t i p l e s of the most  probable  ( f o r example, 0.80-0.81, 0.81-0.82,  curve  are then c a l c u l a t e d from the t r u n c a t e d distribution.  The  1.19-1.20)  skew normal or normal  t o t a l c o s t o f the optimum c u l v e r t a t the mid-  p o i n t of each i n t e r v a l i s c a l c u l a t e d from the p r e v i o u s l y c o n s t r u c t e d optimum c o s t curve curve  and m u l t i p l i e d by  i s i n that i n t e r v a l .  The  sum  the p r o b a b i l i t y t h a t the of these products  i n t e r v a l s y i e l d s the expected t o t a l c o s t w i t h - p e r f e c t The  true  over a l l information.  value of p e r f e c t i n f o r m a t i o n i s the d i f f e r e n c e between the  expected t o t a l c o s t of the'optimum c u l v e r t chosen under u n c e r t a i n t y and  the expected t o t a l c o s t w i t h p e r f e c t i n f o r m a t i o n . I t i s i n t e r e s t i n g to note t h a t the expected t o t a l c o s t s  u n c e r t a i n t y of F i g u r e s 6.2  and  6.3  with  c o u l d a l s o be c a l c u l a t e d i n a  manner s i m i l a r t o t h a t f o r the expected t o t a l c o s t w i t h p e r f e c t i n f o r m a t i o n , r a t h e r than by r e d u c i n g a bounded f l o o d frequency t o a s i n g l e curve  as o u t l i n e d i n Chapter 3.  The  plot  only d i f f e r e n c e i s  t h a t the c o s t of the c u l v e r t s i z e b e i n g c o n s i d e r e d  i s used f o r a l l  i n t e r v a l s i n s t e a d o f the c o s t of the optimum s i z e d c u l v e r t . In p r a c t i c e , no data g a t h e r i n g program w i l l e l i m i n a t e a l l u n c e r t a i n t y ; so the v a l u e of p e r f e c t i n f o r m a t i o n f i x e s an uppermost l i m i t to the value_ o f b e t t e r i n f o r m a t i o n . information i n reducing  v a l u e of b e t t e r  the u n c e r t a i n t y l i m i t s from 0.5-1.5 to  0.8-1.2 might be estimated i n f o r m a t i o n i n the two  The  by s u b t r a c t i n g the v a l u e s of p e r f e c t  cases.  T h i s i s o n l y an estimate  i t cannot be known beforehand how the u n c e r t a i n t y l i m i t s and  because  the b e t t e r i n f o r m a t i o n w i l l change  the most probable  curve.  57  A f t e r the new  data i s a c t u a l l y c o l l e c t e d and a new  t o t a l cost  curve i s drawn, the v a l u e of b e t t e r i n f o r m a t i o n f o r a p a r t i c u l a r c u l v e r t s i t e can be c a l c u l a t e d by s u b t r a c t i n g the t o t a l c o s t of the c u l v e r t s i z e chosen a f t e r the data g a t h e r i n g from the t o t a l c o s t o f the c u l v e r t s i z e t h a t would have been chosen before the data g a t h e r i n g , both these t o t a l c o s t s b e i n g from the new  curve.  If  t h i s i s done f o r a l a r g e number o f c u l v e r t s i t e s , such as along a proposed  new  highway r o u t e , then a f a i r l y accurate monetary v a l u e of  a data g a t h e r i n g program may  result.  the program made b e f o r e i t was  The estimate o f the v a l u e o f  i n s t i t u t e d can then be compared t o  the c a l c u l a t e d v a l u e o f the program a f t e r i t i s completed a c c u r a t e the e s t i m a t i o n procedure  s i n g l e c u l v e r t s i t e , i t may  how  was.  The rough f i g u r e s of Table 6 . 3 g a t h e r i n g can be s u b s t a n t i a l .  t o see  show t h a t the v a l u e of data  Keeping  i n mind t h a t these are f o r a  be v e r y worthwhile  to i n s t a l l a network  o f p r e c i p i t a t i o n gauges, or even i n s t a l l weirs and r e c o r d i n g gauges i n some streams, 6.3  b e f o r e s e l e c t i n g c u l v e r t s i z e s f o r a new  highway.  S e n s i t i v i t y o f the Optimal D e c i s i o n to Changes i n the  Discount  Rate and the S e r v i c e L i f e The  s e n s i t i v i t y of the o p t i m a l d e c i s i o n t o changes i n the  d i s c o u n t r a t e and the s e r v i c e l i f e was m a r g i n a l investment  c o s t , MIC,  i n v e s t i g a t e d by u s i n g  and m a r g i n a l s a v i n g s , MS,  These curves are s i m i l a r to an economist's  curves.  marginal c o s t and  m a r g i n a l revenue curves t h a t are used i n a n a l y z i n g a f i r m ' s revenue, c o s t , and p r o f i t p i c t u r e .  A f i r m seeking to maximize i t s p r o f i t  produces to the p o i n t where m a r g i n a l revenue gained from the l a s t u n i t of output)  ( i . e . , the revenue  equals m a r g i n a l c o s t ( i . e . ,  the  58  c o s t o f p r o d u c i n g the l a s t u n i t of o u t p u t ) .  Similarly, starting  w i t h a s m a l l c u l v e r t s i z e , l a r g e r c u l v e r t s i z e s are s e l e c t e d  until  the p o i n t where the m a r g i n a l investment c o s t o f moving to the next l a r g e r s i z e i s g r e a t e r than the m a r g i n a l savings gained by moving to the next l a r g e r c u l v e r t s i z e . F i g u r e 6.4 was c o n s t r u c t e d u s i n g the r e s u l t s f o r the case where the most p r o b a b l e curve i s s p e c i f i e d by Q ^ Q = 150 c f s and QlOO = 220 c f s w i t h u n c e r t a i n t y  bounds o f 0.5-1.5.  m a r g i n a l investment c o s t curves r e p r e s e n t i n g r a t e s and s e r v i c e i n F i g u r e 6.4.  lifes,  There a r e f o u r  different interest  a l o n g w i t h one m a r g i n a l s a v i n g s c u r v e , shown  The o p t i m a l s i z e c u l v e r t f o r a p a r t i c u l a r m a r g i n a l  s a v i n g s , m a r g i n a l c o s t curve combination i s the f i r s t c u l v e r t  size  to the l e f t o f the i n t e r s e c t i o n o f the two c u r v e s . Because t h e r e a r e o n l y a l i m i t e d number o f c u l v e r t available  (4.0, 4.5, 5.0 f t , e t c . ) and because the m a r g i n a l curves  were c o n s t r u c t e d u s i n g i n c r e m e n t a l d i f f e r e n c e s between c u l v e r t s i z e s r a t h e r on  sizes  than by t a k i n g  i n costs  and savings  instantaneous slopes  continuous c u r v e s , the i n t e r s e c t i o n p o i n t does not i n d i c a t e the  optimum diameter.  An i n t e r s e c t i o n p o i n t  diameters, such as t h a t  near one o f the f i x e d  f o r the r = 4%, n = 30 y r MIC curve which  i n t e r s e c t s the MS curve near a c u l v e r t diameter o f 5.5 f t , i n d i c a t e s instead the  t h a t the 5.5 f t c u l v e r t has n e a r l y  5.0 f t c u l v e r t .  The d i f f e r e n c e  the same t o t a l c o s t as  between the MIC and MS  a t a p a r t i c u l a r c u l v e r t diameter i s the d i f f e r e n c e  i n total  curves cost  between t h a t c u l v e r t diameter and the next s m a l l e r c u l v e r t diameter. Thus i t i s r e l a t i v e l y easy t o see how c o m p e t i t i v e the optimum c u l v e r t i s w i t h c u l v e r t s o f s m a l l e r and l a r g e r s i z e .  sized  59  480  *~ Q)  flood frequency d i s t r i b u t i o n : 0 . 5 - 1.5 with 1.0 c u r v e s p e c i f i e d by  440  N  CO  2 o  [2  Qio = I 5 0 c f s  400  and  Qioo = 2 2 0 c f s  o o —CO  CP c '> o to  0  3 6 0^ r = discount  rate  n = s e r v i c e life in years. 320  ,  d i f f e r e n c e in total a n n u a l cost between 5 . 0 f t . a n d 4.5ft. culverts  o c cn 2 8 0 o 2 .  0  i_  o  240  to o o c. 200 0 cu  e  CO  <u > c r-H  160  o c o> o  120  _  30  80  o 3  C C  <  40  0 4.0  1  4.5  5.0  5.5  Culvert  FIG. 6.4  Diameter  6.0  OF T H E O P T I M A L  CHANGES  THE  THE  SERVICE  DISCOUNT LIFE.  7. 0  (feet)  SENSITIVITY IN  6.5  DECISION  RATE  AND  TO  60  Looking a t F i g u r e 6 . 4 , the o p t i m a l d e c i s i o n does not appear to be p a r t i c u l a r l y s e n s i t i v e to changes i n the d i s c o u n t r a t e or t o changes i n the s e r v i c e l i f e ( I n c i d e n t a l l y the MIC q u a r t e r o f the way  w i t h the d i s c o u n t r a t e f i x e d a t 4 % .  curve f o r n = o o and r = 4 % l i e s about  between the r = 0 % , n = 3 0 y r curve and  r = 4 % , n = 3 0 y r curve, b e i n g c l o s e r t o the lower curve.) MS  curve were f l a t t e r  the I f the  then the o p t i m a l d e c i s i o n would be more s e n s i -  t i v e to changes i n the i n t e r e s t r a t e and the s e r v i c e 6.4  one  life.  The E f f e c t on the Optimal D e c i s i o n of Changing the Damage Costs M a r g i n a l savings and m a r g i n a l investment c o s t curves were  again used to determine the e f f e c t on the o p t i m a l d e c i s i o n of v a r y i n g the washout c o s t and the headwater damage c u r v e . shows the r e s u l t s f o r the most probable f l o o d frequency s p e c i f i e d by Q ^ Q = 1 5 0 c f s and Q - ^ Q Q of  0 . 5 - 1 . 5 .  =  Figure 6 . 5 curve  2 2 0 c f s with u n c e r t a i n t y l i m i t s  The m a r g i n a l c o s t curve r e p r e s e n t s the standard case  w i t h r = 4 % and n = 3 0 y r s .  F i g u r e 6 . 6 shows the r e s u l t s f o r the  most probable curve s p e c i f i e d by Q  1  Q  = 1 2 0 c f s and Q  w i t h the same u n c e r t a i n t y bounds as b e f o r e .  1  0  0  = 2 1 6 cfs  A l l the marginal s a v i n g s  curves are f o r the standard headwater damage curve, except one i n each f i g u r e .  The m a r g i n a l s a v i n g s curve f o r any washout c o s t and  any m u l t i p l e of the standard headwater damage curve c o u l d e a s i l y  be  p l o t t e d from the curves p r e s e n t e d i n F i g u r e s 6 . 5 or 6 . 6 . The g r e a t e r spread of the MS  curves o f F i g u r e 6 . 6 compared to  F i g u r e 6 . 5 i n d i c a t e s t h a t the o p t i m a l d e c i s i o n w i l l v a r y more w i t h changing damage c o s t s w i t h the s t e e p e r most probable curve used i n Figure 6 . 6 .  T a b l e 6 . 4 summarizes some of the i n f o r m a t i o n of F i g u r e s  61  480 flood f r e q u e n c y d i s t r i bution : 0 . 5 - 1.5 with 1.0 curv e s p e c i f i e d by Qio = I 5 0 c f s and Qioo= 2 2 0 c f s  5.0  5.5  Culvert  Di a m e t e r ( f e e t )  FIG. 6.5 T H E E F F E C T CHANGING  6.0  ON T H E O P T I M A L THE  DAMAGE  6.5  DECISION  COSTS.  7.0  OF  4 80i f l o o d frequency d i s t r i b u t i o n 0 . 5 - 1.5 with 1.0 c u r v e s p e c i f i e d by  440  Q io = 120 c f s  and  Qioo= 21 6 c f s  400  3 6 0| 10  o  320,  s  o c  280  o o  240  TO  O O O O O  200<  I 60  I 20  80  40  4.0  4.5  5.0  5.5  6.0  Culvert  D i a m e t e r (feet)  6.5  FIG. 6.6 T H E E F F E C T ON T H E O P T I M A L D E C I S I O N CHANGING  THE DAMAGE  FREQUENCY  C O S T S ; NEW  DISTRIBUTION.  OF  FLOOD  63  and  6.5  different for  and  the  of if  return  culvert.  design  period  flood  British B,  diameter  5.0  ft  5.5  Table  given  curve. would  6.5  shows  design  used  thesis.  this  Substantial Highways'  return  a 5.0  costs  costs  with  is  c a n be  moderate  that  damage  the  ft  culvert  case  they  the  to  case  the  are  reflect  incurred traffic  than  in  by  cost  low  or  flood  to  meet  design  the  criterion  A and B.  Thus  a  derived  single  equivalent  5.5  ft  second d i s t r i b u t i o n ,  the  the  B.  rural  or  very  highways  there  cases  all  is  where  a the  method  cases.  Department  low  a  Highways  assumed i n  very  where  A and  economic a n a l y s i s  using  in  damage  select  the  is  cases.  only  first  the  would  using  volume  cases  volume  the  required  damage i s  incurred  washout  the  hydraulic  however  some  meets c r i t e r i a  consequences of rather  in  meet c r i t e r i a  of  it  cases;  hydraulic  the  optimum  h e a d w a t e r damage  all low  of  Highways use  is  washout)  diameter  Highways'  (i.e.,  is  Highways'  chosen as  the  very  in  No h e a d w a t e r  could  is  in  the  the  periods  head-  causing  to  the  return  which  none o f  there  period  required  criteria  if  at  depth equal  since  each of  the  headwater  actually  no'headwater  the  costs  criteria  Low w a s h o u t  delay  extra  and the  damage c o s t  In  be  along with  Columbia Department of  a washout),  Department's  washout  is  culvert  culvert,  culvert  in  table,  diameter for  headwater  headwater  Columbia Department of  frequency  ft  (the  starts  100-year  distribution,  British  flood  is  cases,  ft  Columbia Department of  and a  the  the  culvert  introduction)  there  can occur  frequency  British  (see  two  the  the  headwater  Assuming that  for  Looking at  below  the  applicable,  criteria  floods  in  optimal  a n d 10.0  5.0  meet the  100-year  the  damage c o s t s  damage,  culverts  at  listing  headwaters  water  of  by  6.6  of  high. while  high  substantial traffic  64  TABLE 6.4 OPTIMUM CULVERT DIAMETERS  AND RETURN PERIODS OF SIGNIFICANT  HEADWATER LEVELS FOR DIFFERENT DAMAGE COSTS I.  F l o o d Frequency D i s t r i b u t i o n : 0.5-1.5 w i t h 1.0 curve by Q = 150 c f s and Q = 220 c f s 1QQ  1 Q  Washout Cost ($)  5000n 5000 15000 25000 50000 100000  II.  Optimum Culvert Diameter (ft)  Return P e r i o d (yr) HW > 5.0 f t  4.5 5.0 5.0 5.5 5.5 6.0  3.0 4.2 4.2 6.2 6.2 8.0  '  HW>10.0 f t  45 135 135 390 390 1300  F l o o d Frequency D i s t r i b u t i o n : 0.5-1.5 w i t h 1.0 curve by Q = 120 c f s and Q, = 216 c f s 1 n  Washout Cost ($)  5000n 5000 15000 25000 50000 100000  2  D = culvert  specified  HW =2 D  2.1 4.2 4.2 10.4 10.4 31  specified  nfl  Optimum Culvert Diameter (ft) 4.0 4.5 5.0 5.0 5.5 6.0  3  Return P e r i o d (yr) HW > 5.0 f t 5.7 7.8 10.7 10.7 14.6 18.1  HW > 10.0 f t 28 64 162 162 370 900  HW>  D  1  3.4 5.7 10.7 10.7 22 52  diameter  n = no headwater damage; otherwise standard headwater damage curve ( F i g u r e 5.2) i s used 'The curves o f F i g u r e 6.6 i n d i c a t e t h a t the optimum c u l v e r t diameter i s 4.5 f t , winning by a s l i g h t margin over the 4.0 f t culvert. But the MIC curve was drawn as a smooth curve which does n o t e x a c t l y pass through a l l the data p o i n t s . Using the a c t u a l data p o i n t s , the 4.0 f t c u l v e r t wins by a s l i g h t margin.  65  TABLE 6.5 COMPARISON OF ECONOMIC ANALYSIS WITH THE BRITISH COLUMBIA DEPARTMENT OF HIGHWAYS' DESIGN CRITERIA I.  F l o o d Frequency D i s t r i b u t i o n : 0.5-1.5 w i t h 1.0 curve s p e c i f i e d by Q = 150 c f s and Q 0 =  1 Q  Washout Costl ($)  2  2  0  c  f  s  1Q  Optimum Expected Expected E x t r a Expected E x t r a C u l v e r t T o t a l Annual Annual C o s t i f Annual C o s t i f Diameter Cost C u l v e r t Diameter C u l v e r t Diameter (ft) ($) Selected i s Selected i s 5.0 f t ($) 5.5 f t ($) 2  5000 15000 25000 50000 100000  4.5 5.0 5.0 5.5 6.0  778 871 945 1009 1088  3  117 49 1  19 121 412  48  I I . F l o o d Frequency D i s t r i b u t i o n : 0.5-1.5 w i t h 1.0 curve s p e c i f i e d by Q = 120 c f s and Q = 216 c f s 1  i r m  Washout Cost ($) 1  5000 15000 25000 50000 100000  Optimum Culvert Diameter (ft)  Expected T o t a l Annual Cost ($)  4.0 5.0 5.0 5.5 6.0  Expected E x t r a Annual C o s t i f C u l v e r t Diameter Selected i s 5.0 f t ($)  726 853 914 1019 1122  3  65 _ 50 255  no headwater damage assumed 'meets B.C. Dept. o f Highways' c r i t e r i o n B o n l y meets B.C. Dept. o f Highways' c r i t e r i a A and B (see I n t r o duction f o r c r i t e r i a )  66  volume alone  i s very h i g h .  are j u s t not " r i g h t "  The Highways Department's  criteria  f o r a l l roads under a l l c o n d i t i o n s .  F i g u r e 6.7 i l l u s t r a t e s t h a t the e f f e c t o f u n c e r t a i n t y i n changing the o p t i m a l d e c i s i o n i s g r e a t e r when the damage c o s t i s g r e a t e r , as the s e p a r a t i o n between the 1.0 and 0.5-1.5 MS i n c r e a s e s w i t h i n c r e a s i n g damage c o s t . better information i s l i k e l y than f o r low damage c o s t s .  curves  Consequently the v a l u e o f  t o be g r e a t e r f o r h i g h damage c o s t s  67  480  4.0  F I G . 6.7  4.5  5.0  5.5  Culvert  D i a m e t e r (feet)  THE EFFECT OPTIMAL  6.0  OF U N C E R T A I N T Y  6.5  IN C H A N G I N G  D E C I S I O N AT D I F F E R E N T  DAMAGE  7.0  THE COSTS.  Chapter 7 CONCLUSION T h i s t h e s i s has  d e s c r i b e d a method of economic a n a l y s i s to  determine the optimum s i z e d c u l v e r t f o r any  culvert site.  method takes u n c e r t a i n t y i n t o account and  i s capable of  the value of b e t t e r i n f o r m a t i o n .  aspects  Various  The  estimating  of the c u l v e r t  s e l e c t i o n problem: h y d r o l o g i c , h y d r a u l i c , and economic were d i s cussed, and  the method was  a p p l i e d to a h y p o t h e t i c a l c u l v e r t s i t e ,  assuming d i f f e r e n t h y d r o l o g i c and economic s i t u a t i o n s . The  p o t e n t i a l advantages of employing economic a n a l y s i s i n  c u l v e r t s e l e c t i o n appear so g r e a t t h a t one wonders why to be used. difficulty  L i n s l e y and F r a n z i n i (17) i s t h a t of e s t i m a t i n g  i n excess of c u l v e r t c a p a c i t y . " can s o l v e the problem.  s t a t e , "The  i t has  practical  the probable damages from T h i s i s very  t r u e , but  I t would not be d i f f i c u l t to conduct  a d d i t i o n to experiment, o b s e r v a t i o n s  of c u l v e r t s i n the  f l o o d c o n d i t i o n s and  s i t e s a f t e r washouts w i l l mates.  flows  research  experiments to f i n d out what causes a c u l v e r t to wash out.  operating during  yet  In  field  c l o s e i n s p e c t i o n s of c u l v e r t  l e a d t o much improved damage c o s t  esti-  Even i f there i s much u n c e r t a i n t y i n v o l v e d i n e s t i m a t i n g  damage c o s t s , t h i s u n c e r t a i n t y c o u l d be accounted f o r i n the economic  a n a l y s i s , and  estimates  o f the value of b e t t e r i n f o r m a t i o n  in  t h i s area c o u l d be made. Another argument t h a t might be made i s t h a t the engineering  extra  c o s t i n v o l v e d i n a p p l y i n g economic a n a l y s i s to c u l v e r t  s e l e c t i o n w i l l outweigh the savings  68  from the program.  T h i s i s very  69  u n l i k e l y i f a l l c a l c u l a t i o n s are handled by computer. the i n i t i a l c o s t o f d e v e l o p i n g to handle any i n the  s i t u a t i o n may  long run.  a good general- program t h a t i s able  be h i g h , i t i s bound to pay  Initially  the c o s t o f  considered  only s i m p l i f i e d ,  be  obtaining up.  hypothetical  although s e v e r a l u s e f u l r e s u l t s were obtained.  If further  i s done, i t would be worthwhile to c o n s i d e r r e a l s i t u a t i o n s  and  to complicate  and  estimating  improvement and be g i v e n  will  be h i g h , but i t w i l l decrease as a data bank i s b u i l t  T h i s t h e s i s has  research  itself  f o r example, damage c o s t e s t i m a t e s ,  s i m i l a r f o r many c u l v e r t s i t e s .  cases,  for  More i n p u t data i s r e q u i r e d f o r an economic  a n a l y s i s , but t h i s data,  data may  Although  the problem.  The  problems of d e b r i s  damage c o s t s deserve more a t t e n t i o n . d i f f e r e n t c u l v e r t m a t e r i a l s and  consideration.  x.  clogging  Entrance  shapes should  also  L I S T OF REFERENCES  N e s b i t t , M. C. (1963) Handbook of C u l v e r t H y d r a u l i c s , Design, and I n s t a l l a t i o n , B.C. Department o f Highways, M a t e r i a l s T e s t i n g , Design, and P l a n n i n g Branch, V i c t o r i a , B.C. B l a i s d e l l , F. W. (1966) " H y d r a u l i c E f f i c i e n c y i n C u l v e r t Design J o u r n a l o f the Highway D i v i s i o n , A.S.C.E., 92 (HW1): 11-22, P r o c . Paper 4 7 09. P r i t c h e t t , H a r o l d D. (1964) A p p l i c a t i o n of the P r i n c i p l e s o f E n g i n e e r i n g Economy t o the S e l e c t i o n of Highway C u l v e r t s , Master's t h e s i s , S t a n f o r d U n i v e r s i t y , Department o f C i v i l Eng. Hershman, S t a n l e y (1974) An A p p l i c a t i o n of D e c i s i o n Theory to Water Q u a l i t y Management, Master's t h e s i s , U.B.C, Department o f C i v i l Eng. Nyumbu, Inyambo L. (1976) The E f f e c t o f U n c e r t a i n t y i n I r r i g a t i o n Development, Master's t h e s i s , U.B.C, Department of C i v i l Eng. Brox, Gunter H. (1976) Water Q u a l i t y i n the Lower F r a s e r R i v e r B a s i n : A Method t o E s t i m a t e the E f f e c t of P o l l u t i o n on the S i z e o f a Salmon Run, Master's t h e s i s , U.B.C, Department o f C i v i l Eng. H e t h e r i n g t o n , E. D. (1974) The 25-Year Storm and C u l v e r t S i z e , F e d e r a l Department o f the Environment, Canadian F o r e s t r y S e r v i c e P a c i f i c F o r e s t Research C e n t r e , V i c t o r i a , B.C., Report BC-X-102. L i n s l e y , R. K., M. A. K o h l e r , and J . L. H. Paulhus Hydrology f o r E n g i n e e r s , McGraw-Hill, New York.  (1958)  Quick, M. C and A. P i p e s (1975) A Combined Snowmelt and R a i n f a l l Runoff Model, unpublished l e a f l e t , U.B.C, Department of C i v i l Eng. D i s t r i c t o f West Vancouver Drainage Survey, Dayton & K n i g h t L t d . , C o n s u l t i n g E n g i n e e r s , 1973. C a r t e r , R. W. .(1957) Computation o f Peak Discharge a t C u l v e r t s , U n i t e d S t a t e s G e o l o g i c a l Survey C i r c u l a r 376. Chow, V. T. (1959) New York.  Open-Channel H y d r a u l i c s , McGraw-Hill,  N e i l l , C R. (1962) H y d r a u l i c T e s t s on Pipe C u l v e r t s , Research C o u n c i l o f A l b e r t a , A l b e r t a Highway Research Report 62-1. Oglesby, C. H. and L. I . Hewes (1963) John Wiley & Sons, New York. 70  Highway E n g i n e e r i n g ,  71  15.  James, L. D. and R. R. Lee (1971) P l a n n i n g , McGraw-Hill, New York.  Economics o f Water Resources  16.  Samuelson, P. A. and A. S c o t t Ryerson L i m i t e d , Toronto.  17.  L i n s l e y , R. K. and J . B. F r a n z i n i (1972) E n g i n e e r i n g , McGraw-Hill, New York.  (1975)  Economics, McGraw-Hill Water-Resources  APPENDIX  HEADWATER D E P T H  -  see  Type  Reference  1 Flow: HW =  11 f o r  Critical  additional  Depth  (Q/cA ) /(2g) 2  c  where  c  is  a  CALCULATIONS  at  + d  function  Inlet -  c  information  w^/{2q)  of  + 'h  (HW/D)  2 v,  /(2g)  a n d h^  were assumed 1.2 i s s o l v e d by f i r s t  1  negligible.  1  The  equation  •Type  5 Flow:  Rapid Flow  HW =  (Q/cA ) /(2g)  where  c  Type  o  6 Flow:  is  d  c  (Q/A  = v  c  )  Inlet  2  a  Full  at  calculating  function  Flow  Free  of  (HW/D)  Outfall  2 assuming  v.  /(2g)  and h , 1.2  1  h  x  =  (Q/cA ) V ( 2 g ) Q  where then However,  are  negligible  f  c  is  a constant  HW = h-j_  h^  + h  cannot  - S  Q  be  easily  +  3  for  h  f  2  >  3  a particular  inlet  configuration  L determined.  h^ was i n f a c t c a l c u l a t e d f r o m d i m e n s i o n l e s s r a t i o c h a r t s w h i c h a r e b a s e d on e x p e r i m e n t , r a t h e r t h a n from the above e q u a t i o n . In a d d i t i o n HW f o r t y p e  t o D, ,Q, 6 flow.  and c;  n,  L , and  s  are  required  to  calculate  Notation S u b s c r i p t s 1, 2, in Figure 4.1. A  o  = area of  3,  culvert  and  4 denote  barrel  72  location  of  section  as  shown  73  Notation A  c  (cont.)  area o f flow a t c r i t i c a l  section  c  c o e f f i c i e n t of discharge  D  c u l v e r t diameter (min. d i a . f o r CMP)  d  c  h  c r i t i c a l depth = p i e z o m e t r i c head above c u l v e r t i n v e r t a t downstream  h^ = head l o s s due to f r i c t i o n HW = depth o f water i n headwater p o o l L  = l e n g t h of c u l v e r t  n  = Manning's roughness c o e f f i c i e n t  Q  = discharge  s  o  v v  = bed s l o p e o f c u l v e r t = velocity  c  barrel  = critical  velocity  barrel  end  

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