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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.Sc, University of 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 i n THE FACULTY OF GRADUATE STUDIES The Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE © UNIVERSITY OF BRITISH COLUMBIA August, 1977 Patrick Alexander Neudorf, 1977 In p resent ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree t h a t permiss ion for e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my wr i t ten pe rm i ss ion . Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date A u g u s t 31. 1977 ABSTRACT The hydraulic design c r i t e r i a for c u l v e r t size s e l e c t i o n currently employed by most highways departments, including B r i t i s h Columbia's, can lead to economically non-optimal c u l v e r t s i z e choices. This thesis describes a method of economic analysis to determine the optimum sized c u l v e r t for any c u l v e r t s i t e , taking into d i r e c t account the uncertainty of the data. The method i s applied to a hypothetical c u l v e r t s i t e , assuming d i f f e r e n t hydro-l o g i c and economic s i t u a t i o n s . The uncertainty i n evaluating flood flows i s taken into account, and methods of c a l c u l a t i n g the value of better information are presented. The hydrologic, hydraulic, and economic aspects of c u l v e r t s e l e c t i o n and the problems and uncertainties i n c o l l e c t i n g data and making assumptions i n each of these areas are discussed before the r e s u l t s are presented. TABLE OF CONTENTS Chapter Page 1 INTRODUCTION . . . . 1 2 METHOD OF SOLUTION . . . 5 2.1 The Decision Tree 5 2.2 P r o b a b i l i t y Matrices 8 2.3 Calculations 11 3 EVALUATION OF FLOOD FLOWS . 14 3.1 Methods and Problems 14 3.2 Accounting for Uncertainty i n Flood Flows . . 17 4 CULVERT HYDRAULICS 26 4.1 Types of Culvert Flow 26 4.2 Entrance and E x i t Improvement 31 4.3 Mechanics of a Washout 33 4.4 Environmental Considerations 34 5 ECONOMICS • ..' 35 5.1 Ca p i t a l Cost 35 5.2 Flood Damage . 38 5.3 Annual Cost Comparison . . . . . 41 6 RESULTS 47 6.1 Annual Cost Curves for One Flood Frequency D i s t r i b u t i o n . . . . . 47 6.2 The E f f e c t of Uncertainty and the Value of Better Information . . . . . 47 6.3 S e n s i t i v i t y of the Optimal Decision to Changes i n the Discount Rate and the Service L i f e 57 6.4 The E f f e c t on the Optimal Decision of Changing the Damage Costs 60 7 CONCLUSION 68 LIST OF REFERENCES . 70 APPENDIX . . . 72 i i i LIST OF TABLES Table Page 3.1 Comparison of E f f e c t i v e Floods of Various' Return Periods for D i f f e r e n t D i s t r i b u t i o n s . . . . 25 5.1 C a p i t a l Costs of I n s t a l l e d Culverts . . . . . . . 36 6.1 Annual Costs f o r Flood Frequency D i s t r i b u t i o n Defined by Q 1 Q = 150 cfs and Q 1 0 Q = 220 cf s . . . 49 6.2 P r o b a b i l i t i e s of Incurring Some Headwater Damage and P r o b a b i l i t i e s of a Washout 50 6.3 The E f f e c t of Uncertainty i n Changing the Optimal Decision and the Value of Better Information 55 6.4 Optimum Culvert Diameters and Return Periods of S i g n i f i c a n t Headwater Levels for D i f f e r e n t Damage Costs 64 6.5 Comparison of Economic Analysis with the B r i t i s h Columbia Department of Highways' Design C r i t e r i a 65 i v ^ LIST OF FIGURES Figure Page 2.1 Decision Tree Used i n the Analysis 6 2.2 More Complicated Decision Tree 7 2.3 Hypothetical Function 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 of Annual Floods (1.0 Line: Q 1 Q = 150 c f s , Q1QQ = 220 cfs) . . . . 19 3.2 Flood Frequency Di s t r i b u t i o n s (1.0 Curve: Q 1 Q = 1 5 0 c f s ' Q i o o = 2 2 0 c f s * • • • • 2 0 3.3 Frequency Curves of Annual Floods (1.0 Line: Q1Q = 120 c f s , Q1QQ = 216 cfs) . . . . 24 4.1 Types of Culvert Flow 27 4.2 Headwater-Discharge Curves 29 5.1 Cap i t a l Costs of I n s t a l l e d Culverts 37 5.2 Headwater Damage Function . . . 39 5.3 Converting C a p i t a l Cost to Annual Cost 43 6.1 Annual Cost Curves for Flood Frequency D i s t r i b u t i o n Defined by Q 1 Q = 150 cfs and Q 1 Q 0 = 220 c f s . . . 48 6.2 Tot a l Annual Cost Curves for D i f f e r e n t Flood Frequency D i s t r i b u t i o n s (1.0 Curve: Q, n = 150 c f s and Q 1 0 Q = 220 cfs) 52 6.3 Annual Cost Curves (1.0 Curve: Q = 120 c f s and Q 1 0 Q = 216 cfs) 53 6.4 S e n s i t i v i t y of the Optimal Decision to Changes i n the Discount Rate and the Service L i f e . . . . 59 6.5 The E f f e c t on the Optimal Decision of Changing the Damage Costs 61 6.6 The E f f e c t on the Optimal Decision of Changing the Damage Costs; New Flood Frequency D i s t r i -bution 62 6.7 The E f f e c t of Uncertainty i n Changing the Optimal Decision at D i f f e r e n t Damage Costs 67 v ACKNOWLEDGEMENT T h e a u t h o r w i s h e s t o t h a n k D r . S . 0 . R u s s e l l f o r h i s g u i d a n c e i n t h e r e s e a r c h a n d p r e p a r a t i o n o f t h i s t h e s i s . T h a n k s a r e a l s o e x t e n d e d t o M r . R i c h a r d B r u n , who p r e p a r e d t h e f i g u r e s , a n d t o M r s . J a n e t B e r g e r o n , who t y p e d t h e t h e s i s . T h e a u t h o r w o u l d a l s o l i k e t o t h a n k t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a f o r t h e i r f i n a n c i a l s u p p o r t d u r i n g t h e p a s t two y e a r s . v i Chapter 1 INTRODUCTION The B r i t i s h Columbia Department of Highways presently selects culve r t sizes on the basis of two c r i t e r i a (1): (A) Culverts s h a l l carry the 10-year flood with head-water depths equal to the diameter of the c u l v e r t . (B) The culver t s h a l l carry a 100-year flood (1.8 x 10-year) by surcharge without headwater damage and without loss 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 to be rather a r b i t r a r y while the second c r i t e r i o n makes an attempt to weigh the cost of i n s t a l l i n g a larger pipe size against the savings from less frequent flood damage. The question i s , "Why was the 100-year flood chosen?" These c r i t e r i a can hardly be expected to re 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 for a l l cu l v e r t s i t e s i n a l l circumstances. For instance, for culverts under low f i l l s on low volume r u r a l high-ways, designing for the 25-year flood may be appropriate. In contrast, the 500-year flood could be appropriate for a long c u l v e r t under a major highway where substantial damages to upstream or downstream property could r e s u l t from flooding. Another problem i s , "What i s the 10-year flood or 100-year flood?" There i s often a great deal of uncertainty involved i n evaluating flood flows for small watersheds. In addition to hydrologic uncertainty, c u l v e r t design i s plagued by uncertainty i n areas such as the hydraulic performance of culverts, debris clogging, what flow w i l l cause washout, and estimating damage costs. The United States Bureau of Public Roads (USBPR) has stated that 44% of the highway drainage d o l l a r , or 15% of the highway 1 2 construction d o l l a r , i s spent for culverts (2). An analysis of sixteen 1961 p r o j e c t s i n B r i t i s h Columbia showed that 8.6% of the t o t a l cost was spent on culverts (1). C l e a r l y , these questions warrant att e n t i o n . This t h e s i s describes a method of economic analysis which can be used to determine the optimum cu l v e r t size for any c u l v e r t s i t e , taking into d i r e c t account the uncertainty of the data. The method i s applied to a hypothetical c u l v e r t s i t e where a 100 f t c u l v e r t i s to be placed on a 7% slope under a major r u r a l two-lane highway. The roadway width, including shoulders, i s 45 f t , and the highway embankments are sloped at 2:1. The roadway i s 10 f t above the cul v e r t i n v e r t at the entrance and 17 f t above the c u l v e r t i n v e r t at the e x i t . Reasonable flood frequency data, c u l v e r t costs, and flood damage costs were chosen. Only uncertainty i n the flood frequency data was considered i n the analysis, although uncertainty i n other areas i s discussed. The idea of applying economic analysis to determine the o p t i -mum s i z e c u l v e r t f o r a given s i t e i s not new. P r i t c h e t t (3) wrote a thesis e n t i t l e d Application of the P r i n c i p l e s of Engineering  Economy to the S e l e c t i o n of Highway Culverts (1964), and t h i s thesis i s often mentioned i n the l i t e r a t u r e . He concluded that substantial savings (15-20% i n the four examples presented) would be r e a l i z e d by applying economic analysis. The purpose of the present thesis i s to extend the analysis so that uncertainty i n the data can be accounted f o r . The e f f e c t on the optimal decision of uncertainty i n the flood frequency data i s studied. 3 A very important question when faced with uncertainty i s , "What i s the value of better information?" Or i n other words, "How much money, i f any, should be spent on a data gathering program to reduce uncertainty?" This question i s explored and possible solutions to the problem are presented. In addition, the s e n s i t i v i t y of the optimal decision to changes i n the discount rate and the service l i f e i s studied as i s the e f f e c t on the optimal decision of changing the damage costs. The only type of c u l v e r t i n s t a l l a t i o n considered i n the analysis i s a single round corrugated metal pipe (CMP) with a v e r t i c a l headwall and endwall. D i f f e r e n t materials and shapes may be advantageous i n some s i t u a t i o n s , but they are not considered 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 hydraulic e f f i c i e n c y , i s discussed but not incorpor-ated into the anal y s i s . The s t r u c t u r a l engineering aspect of c u l -v ert design i s not discussed. U t i l i t y , rather than monetary value, could have been used as the basis f o r culver t s e l e c t i o n . But since highway culverts are the r e s p o n s i b i l i t y of p r o v i n c i a l governments, monetary value was chosen. U t i l i t y would be more appropriate for culverts on private land c o n t r o l l e d by a firm or an i n d i v i d u a l with l i m i t e d f i n a n c i a l resources. In t h i s case the i n d i v i d u a l or firm may be more averse to severe fl o o d damage than the monetary value of the flood damage indicates. The following paragraph outlines the contents of the remaining chapters. Chapter 2 i l l u s t r a t e s the problem with a decision tree and outlines the formation and use of p r o b a b i l i t y matrices and vectors 4 which are used i n the c a l c u l a t i o n s . The next three chapters discuss various components of the decision tree. Chapter 3 discusses methods of evaluating flood flows and t h e i r inherent problems and presents the flood frequency d i s t r i b u t i o n s used i n the analysis. Types of culver t flow are discussed i n Chapter 4; Chapter 4 also includes short discussions of c u l v e r t entrance and e x i t improvement, the mechanics of a washout, and environmental considerations. Chapter 5 discusses the economic elements of the problem: the c a p i t a l costs of cu l v e r t s , flood damage costs, and how the c a p i t a l cost i s converted to an annual cost with emphasis on the question, "What i s the correct discount rate?" The results are presented and discussed i n Chapter 6, and conclusions are drawn i n Chapter 7. C h a p t e r 2 METHOD OF SOLUTION 2.1 T h e D e c i s i o n T r e e T h e c u l v e r t s e l e c t i o n p r o b l e m c a n be c o n v e n i e n t l y r e p r e s e n t e d w i t h a d e c i s i o n t r e e , a s shown i n F i g u r e 2.1. P o s s i b l e d e c i s i o n s ( f o r e x a m p l e , c u l v e r t s i z e ) a r e shown a s b r a n c h e s e m a n a t i n g f r o m a d e c i s i o n p o i n t , r e p r e s e n t e d b y a s q u a r e . E v e n t s w h i c h d e p e n d o n c h a n c e o r n a t u r a l o c c u r r e n c e ( f o r e x a m p l e , f l o o d s i z e ) a r e shown a s b r a n c h e s l e a d i n g f r o m a c h a n c e p o i n t , r e p r e s e n t e d b y a c i r c l e . P r o b a b i l i t i e s o f c h a n c e e v e n t s a r e a l s o g i v e n o n t h e b r a n c h e s . F i g u r e 2.1 c o n t a i n s o n l y o n e t r u e c h a n c e p o i n t s i n c e a u n i q u e h e a d -w a t e r l e v e l i s a s s i g n e d t o e a c h f l o o d s i z e f o r a g i v e n c u l v e r t d i a m e t e r . A p r o b a b l e damage c o s t , c a l c u l a t e d f o r e a c h h e a d w a t e r l e v e l , i s t h e f i n a l o u t c o m e a t t h e e n d o f e a c h f i n a l b r a n c h o f t h e d e c i s i o n t r e e . T h e s i m p l e d e c i s i o n t r e e shown i n F i g u r e 2.1 i s u s e d i n t h e a n a l y s i s p r e s e n t e d i n C h a p t e r 6. T h e d e c i s i o n t r e e c o u l d be c o m p l i -c a t e d t o i n c l u d e more d e c i s i o n s a n d m o r e c h a n c e e v e n t s . F i g u r e 2.2 i s a d e c i s i o n t r e e t o w h i c h t h e t y p e o f e n t r a n c e i m p r o v e m e n t h a s b e e n a d d e d a s a d e c i s i o n v a r i a b l e . U n c e r t a i n t y i n d e b r i s c l o g g i n g , c u l v e r t h y d r a u l i c s , a n d t h e h e a d w a t e r l e v e l w h i c h c a u s e s w a s h o u t h a v e a l s o b e e n a d d e d . T h e r e a r e o n l y two d e b r i s c l o g g i n g p o s s i b i l i -t i e s d e p i c t e d a l o n g w i t h t h e i r a s s o c i a t e d p r o b a b i l i t i e s : e i t h e r no d e b r i s c l o g g i n g o r c o m p l e t e d e b r i s c l o g g i n g w i t h no f l o w t h r o u g h t h e c u l v e r t . I n t e r m e d i a t e d e g r e e s o f d e b r i s c l o g g i n g c o u l d b e i n c l u d e d . T h e same s i t u a t i o n a l s o a p p l i e s t o w a s h o u t a t a p a r t i c u l a r 5 Data Deci s i on Culvert S i z e F l o o d Headwate r M e a n D a m a g e Cos t FIG.2.1 D E C I S I O N T R E E U S E D IN T H E A N A L Y S I S . F IG . 2 . 2 M O R E C O M P L I C A T E D D E C I S I O N T R E E . 8 headwater l e v e l ; intermediate degrees could also be included. These events are more f u l l y discussed i n Chapter 4. 2.2 P r o b a b i l i t y Matrices Matrices and vectors (one-dimensional matrices) are very use-f u l f or handling decision tree information and c a l c u l a t i o n s . The idea of representing a function bounded by upper and lower l i m i t s as a p r o b a b i l i t y matrix was developed by Russell and Hershman (4) and subsequently used by Nyumbu (5) and Brox (6). I t i s a useful concept for dealing with uncertainty. The formation of a p r o b a b i l i t y matrix i s i l l u s t r a t e d for the hypothetical function Y = f ( X ) , shown i n Figure 2.3. For any. value of X, the dependent variable Y i s not known with ce r t a i n t y but l i e s somewhere between the upper and lower limits.. The uncertainty about the true value of Y for a given value of X can be described by a pr o b a b i l i t y density function. In p r a c t i c e , the three curves of Figure 2.3 are u n l i k e l y to be known accurately, e s p e c i a l l y i n cases where there i s l i t t l e data av a i l a b l e . Determining the upper and lower bounds may be p a r t i -c u l a r l y d i f f i c u l t . However t h i s does not necessarily decrease the usefulness of the method since the decision maker can increase the separation between the upper and lower l i m i t s as his uncertainty increases. + Likewise, the shape of the p r o b a b i l i t y density function between the upper and lower bounds i s unl i k e l y to be known unless there i s s u f f i c i e n t data to analyze. A truncated skew normal d i s t r i b u t i o n , shown i n Figure 2.4, was deemed appropriate. This ' v a r i a t i o n of the normal d i s t r i b u t i o n was developed by Ward for f (X) V a l u e s of Y at X i assumed to follow a skew n o r m a l d i s t r i bu t i on S h a d e d area is p r o p o r t i o n a l to the probability that the value of Y at Xj is in the • in terva l DY . i DY T" Upper Bound Mos t P r o b a b l e L o w e r Bound F I G . 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 ) L M = 2 a , MU = 2 cr. Shaded a rea is proportional to proba bi lity that the value of X is in the in terva l D X M ( MODE) — I D X - — u FIG.2.4 T R U N C A T E D S K E W N O R M A L D I S T R I B U T I O N 10 Hershmari's the s i s . The d i s t r i b u t i o n i s a composite made up from two truncated normal d i s t r i b u t i o n s . The bounds are two standard deviations from the mode. The density function i s m u l t i p l i e d by 1 / (1 - .0456) to correct for the areas truncated at the ends of the d i s t r i b u t i o n . In the case where the upper and lower bounds are equidistant from the mode, the density function reduces to a trun-cated normal d i s t r i b u t i o n . Because t h i s d i s t r i b u t i o n i s easy to work with and can handle cases i n which the upper and lower bounds are not equidistant from the mode, i t was considered a reasonable choice. For any value of X the p r o b a b i l i t y that Y i s i n the i n t e r v a l D Y = Y2 ~ Y i c a n be found by integrating the pr o b a b i l i t y density function at X between Y 1 and Y 2 (see Figure 2.3). This i s the basis for forming a p r o b a b i l i t y matrix. The rows of the matrix represent discrete values of X and the columns represent Y i n t e r v a l s . An element of the matrix i s the p r o b a b i l i t y that the value of Y l i e s i n c e r t a i n i n t e r v a l DY for a s p e c i f i c value of X. The sum of the elements across any row necessarily equals 1.0. To simplify sub-sequent calculations the mid-points of a l l Y int e r v a l s are usually chosen to represent the columns. The discrete values of X are also commonly the mid-points of X i n t e r v a l s . In this way the information contained i n the three continuous curves of Figure 2.3 i s converted into d i s c r e t e pieces. Considering t h i s , some judgement must be used i n se l e c t i n g the size of the matrix. If the i n t e r v a l s are too large accuracy w i l l be l o s t . For example, given that DY i s an i n t e r v a l above the mode at a s p e c i f i c X, the p r o b a b i l i t y that the value of Y i s i n 11 the lower half of the i n t e r v a l i s greater than the p r o b a b i l i t y that the value of Y i s i n the upper half of the i n t e r v a l . The value of the mean u = /Yp(Y)dY, which i s somewhat below the mid-V point i n t h i s case, i s the correct choice f o r a representative value of Y for the i n t e r v a l . Thus using the i n t e r v a l mid-points r e s u l t s i n some inaccuracy. As the i n t e r v a l sizes (both X and Y) decrease, accuracy increases, but the number of computations involved i n forming and u t i l i z i n g the matrix increases. A computer program has been developed (Higgins 1975) for forming p r o b a b i l i t y matrices, but s t i l l the matrices' sizes should not be excessively large as the uncertainty involved i n p l o t t i n g the three curves of Figure 2.3 usually does not j u s t i f y large sized matrices. The flood p r o b a b i l i t y vector of Figure 2.1 was derived from a p r o b a b i l i t y matrix. The derivation i s discussed i n Chapter 3. The uncertainty i n culver t hydraulics could be described by a p r o b a b i l i t y matrix with flow pl o t t e d on the X axis of Figure 2.3 and headwater plotted on the Y axis. A flood damage p r o b a b i l i t y matrix could also be constructed from a graph s i m i l a r to Figure 2.3 with headwater plo t t e d on the X axis and damage cost plotted on the Y axis. Such a matrix i s only necessary 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 cost i s to be calculated. Only a single mean damage cost curve i s required to ca l c u l a t e the expected damage cost for each a. c u l v e r t diameter. 2.3 Calculations The decision tree c a l c u l a t i o n s for the e x i s t i n g data branch of Figure 2.1 are outlined below. The decision to gather more data 12 w i l l r e s u l t i n a new flood p r o b a b i l i t y vector. This topic i s more f u l l y discussed i n Chapters 3 and 6. Steps: 1. Calculate flood p r o b a b i l i t y vector for given flood frequency p l o t and flood i n t e r v a l vector (Chapter 3). 2. Choose culver t diameter (an annual investment charge f o r each cu l v e r t i s calculated, see Chapter 5). 3. Calculate a headwater l e v e l for each flood i n t e r v a l and l i s t the headwaters i n a vector (Chapter 4). 4. Calculate annual damage cost for each headwater l e v e l from 3 and l i s t the r e s u l t s .in a vector (damage cost = headwater damage cost + washout cost i f HW exceeds HWmax, see Chapter 5). 5. 'Calculate expected (or average) annual damage cost by multiply-ing each element i n the flood p r o b a b i l i t y vector by i t s corresponding element i n the damage cost vector and summing the products; i . e . , calculate the dot product of the two vectors. 6. Determine expected t o t a l annual cost by adding the annual investment charge and the expected annual damage cost. 7. Repeat steps 2 to 6 for a l l c u l v e r t diameters. 8. P l o t r e s u l t s and choose c u l v e r t with minimum expected t o t a l annual cost (Chapter 6). Costs are added on the basis of a s t a t i s t i c a l theorem which states that the expected value of the sum of two or more random variables i s equal to the sum of the expected values of the i n d i v i -dual random va r i a b l e s . A l l the calculations are handled by three computer programs: the f i r s t calculates the flood p r o b a b i l i t y vector, the second 13 calculates the headwater vector 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 calculations u t i l i z i n g the re s u l t s of the f i r s t two programs. Chapter 3 EVALUATION OF FLOOD FLOWS 3.1 Methods and Problems There i s a great deal of uncertainty associated with the evaluation of flood flows from small watersheds i n B r i t i s h Columbia. One of the main problems i s the lack of d i r e c t streamflow measure-ments for creeks on which culverts are to be located. Thus flood flows are normally evaluated i n d i r e c t l y . P r e c i p i t a t i o n - r u n o f f relationships are commonly used. Hetherington 1s pu b l i c a t i o n e n t i t l e d The 25-Year Storm and Culvert Size - A C r i t i c a l Appraisal (7) has a good discussion of the methods and problems of peak flow evaluation. Much of the discussion of t h i s section i s summarized from his paper. In order to evaluate peak flows i t i s necessary to understand the meteorological and physical processes which produce them. There are many d i f f e r e n t ways i n which a 25-year, 100-year, or any year peak flow could be generated. In coastal regions of B r i t i s h Columbia, major r a i n storms with durations of 12 to 36 hours or greater are the major cause of high peak runoff events. Rapid springtime melting of an above average winter snowpack i s a probable cause of high peak flows i n the I n t e r i o r . In very small watersheds of a few hundred acres or l e s s , high peak flows can also be generated by high i n t e n s i t y convective r a i n f a l l s (thundershowers) of duration less than 2 to 3 hours. Rain f a l l i n g on snow can cause high runoff events for both coastal and i n t e r i o r watersheds. Also a flood with a r e l a t i v e l y high return period can be generated when flow of lower return period i s temporarily blocked by a debris jam. 14 15 Storm runoff water backed up behind the debris jam i s released as a powerful surge when the dam collapses. The uncertainty about the conditions l i k e l y to cause high peak flows adds uncertainty to the i n d i r e c t evaluation of peak flows. 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 sort of p r e c i p i t a t i o n data i s usually available to apply to the watershed i n question. However, meteorological stations are widely scattered throughout the province and mostly located at low elevations Most stations c o l l e c t r a i n f a l l i n standard, non-recording gauges; hence, data on short duration r a i n f a l l i n t e n s i t i e s i s very l i m i t e d . 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 recording gauges, have a very short period of record which r e s t r i c t s the r e l i a b i l i t y of return period c a l c u l a t i o n s . Extrapolating 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 well as v e r t i c a l l y , from observations taken at a single point 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 patterns are complex. The orographic 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 during major storms i n areas where mountain slopes are exposed d i r e c t l y to rain-bearing winds, such as on the western slopes of Vancouver Island. The network of snow survey s i t e s i s also sparse, and the extrapolation of snow survey data i s even more tenuous than for r a i n f a l l data. The simplest r a i n f a l l - r u n o f f models are empirical formulae r e l a t i n g peak flow to r a i n f a l l i n t e n s i t y and physiographic parameters of the watershed, such as drainage area or basin slope. The most popular formula i s the so-called " r a t i o n a l formula" (Q = CIA) which i s widely used by many agencies including 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 that they do not recognize the complexity of the runoff process. Each formula contains an empirical constant, C, usually c a l l e d the runoff c o e f f i c i e n t , which i s d i f f i c u l t to estimate for any watershed. C i s a constant i n the formula, but experience shows that i t s value varies widely from storm to storm ( 8 ) . The already questionable r e l i a b i l i t y of these formulae decreases as the watershed area increases. Models, such as the University of B r i t i s h Columbia Watershed Budget Model ( 9 ) , are much more accurate i n simulating the runoff process than simple formulae. These models also handle snowmelt and rain-on-snow conditions. C r i t i c a l sequences of d a i l y temperature as well as snowpack data are required to evaluate snowmelt runoff. A key aspect of the U.B.C. Watershed Model i s the d i v i s i o n of the watershed into area-elevation bands to account for the elevation dependence of p r e c i p i t a t i o n and temperature. In addition, other watershed c h a r a c t e r i s t i c s such permeability and groundwater storage are frequently elevation dependent. Some period of stream-flow record i s h e l p f u l i n evaluating the c a l i b r a t i o n parameters for 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 data, and not the 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 l i m i -t a t i o n on the r e l i a b i l i t y of the computed peak flow values i f the c a l i b r a t i o n parameters can be determined reasonably accurately. Besides using p r e c i p i t a t i o n - r u n o f f models, peak flow data for large streams could be transposed to smaller streams on a simple discharge per uni t area basis to estimate peak flows. The watersheds must have s i m i l a r physiographic and c l i m a t i c c h a r a c t e r i s t i c s . 17 Even so, t h i s approach i s l i k e l y to underevaluate small stream peak flows because of differences i n timing of runoff between large and small watersheds. A survey of 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 provide i n f o r -mation on peak flows that i s useful i n predicting flows for other watersheds. Crest-stage gauges i n s t a l l e d at c u l v e r t entrances and approach sections are very useful i n t h i s regard. The computed peak flow values along with the recorded p r e c i p i t a t i o n data can be used to assess p r e c i p i t a t i o n - r u n o f f formulae and watershed models. If the record i s long enough the return periods can also be estimated. Flows computed from discernable high-water marks are d i f f i c u l t to r e l a t e to a return period but s t i l l have some value i n assessing e x i s t i n g i n s t a l l a t i o n s . 3.2 Accounting for Uncertainty i n Flood Flows The uncertainty i n evaluating flood flows i s accounted for by placing upper and lower confidence l i m i t s , along with a most probable curve, on a flood frequency p l o t . The flood frequency d i s t r i b u t i o n chosen for specifying the three curves was the Gumbel d i s t r i b u t i o n . Other d i s t r i b u t i o n s , such as the log Pearson Type I I I , may be more appropriate and could be used equally w e l l . Both the- Gumbel and log Pearson Type III d i s t r i b u t i o n s consider only the annual f loods ,~ i . e . , the maximum flood peak i n each year. A p a r t i a l duration s e r i e s , which includes a l l independent flood events, d i f f e r s s u b s t a n t i a l l y from an annual series at low return periods (less than about 5 years). Thus the p a r t i a l duration series i s the more appropriate choice i f a culv e r t sustains damage at floods of a r e l a t i v e l y low return period. 18 The most probable curve i n the i n i t i a l analysis was s p e c i f i e d by s e t t i n g QlfJ = 150 cfs ( i . e . , the 10-year flood) and Q 1 Q 0 = 220 cfs, This l i n e i s l a b e l l e d 1.0 i n Figure 3.1. The lower and upper bounds were then simply s p e c i f i e d as multiples of the most probable curve, such as a lower bound of 0.5 and an upper bound of 1.5 times the most probable curve. Thus the difference between the bounds increases as the return period increases. Actually the bounds need not be s t r a i g h t l i n e s but could be any curves. For instance, i f the hydrologist has very l i t t l e confidence i n predicting high return period floods, the bounds w i l l diverge even more ra p i d l y with increasing return period than the s t r a i g h t l i n e bounds shown i n Figure 3.1. The flood p r o b a b i l i t y vector, which can be plotted as a p r o b a b i l i t y density function, i s e a s i l y computed f o r a singl e l i n e Gumbel p l o t by d i v i d i n g the v e r t i c a l axis into flood i n t e r v a l s and c a l c u l a t i n g the difference i n the p r o b a b i l i t i e s of the floods at the ends of each i n t e r v a l . The p r o b a b i l i t y density functions for the most probable curve, 1.0, and two multiple curves alone, 1.2 and 1.5, are shown i n Figure 3.2. The information conveyed by specifying a most probable curve with upper and lower bounds can also be converted in t o a single flood p r o b a b i l i t y vector and plotted as a p r o b a b i l i t y density functioi or an equivalent single curve Gumbel p l o t . F i r s t , a p r o b a b i l i t y matrix i s formed from the Gumbel p l o t with i t s upper and lower bounds exactly the same as for any bounded function as outlined i n Chapter 2. The horizontal scale of the Gumbel p l o t , which i s l i n e a r with _ e-b respect to the reduced variate b (the Gumbel equation i s P = e V a l u e s of b ( Gumbel d i s tr ibut ion ) - 2 . 0 -1.0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 1.01 I.I 1.52.0 5.0 10 20 50 100 200 500 1000 2000 5 0 0 0 1 0 0 0 0 Return Per iod ( yea r s ) FIG.3.1 F R E Q U E N C Y CU R V E S OF AN N U A L F L O O D S ( 1.0 LI NE « Qio= I50cfs , Qioo= 220cfs.) » .0130] . 0 1 2 0 . 0 1 1 0 . 0 I 0 0 | . 0 0 9 0 . 0 0 8 0 1 -. 0 0 7 0 . 0 0 6 0 . 0 0 5 0 -. 0 0 4 0 . 0 0 3 0 . 0 0 2 0 .0010 0 1 0.5-1.5 20 40 60 80 100 120 140 160 180 2 0 0 220 240 260 280 3 0 0 320 3 4 0 3 6 0 3 8 0 4 0 0 F l o o d M a g n i t u d e (c f s ) F IG. 3.2 FLOOD F R E Q U E N C Y DISTRI B U T I O N S ( 1.0 C U R V E =Q io= l50CFS ,Qioo= 2 2 0 C F S ) . 21 where P i s the p r o b a b i l i t y of equalling or exceeding a flood of a given s i z e ) , i s divided up into equally sized b i n t e r v a l s over a suitably large range of b. The b i n t e r v a l s are i n f a c t return period or p r o b a b i l i t y i n t e r v a l s , for example, one representing the 37 to 45 year return periods, and these p r o b a b i l i t i e s are calculated and temporarily stored i n a vector (sum = 1.0). The rows of the p r o b a b i l i t y matrix represent return period i n t e r v a l s , and each return period i n t e r v a l i s i n turn represented by the return period at the p r o b a b i l i t y mid-point of the i n t e r v a l since only one point i n each X i n t e r v a l i s used i n forming the matrix. The v e r t i -c a l scale of the Gumbel p l o t i s divided into flood i n t e r v a l s , for example, 250-255 c f s , and the columns of the matrix represent these flood i n t e r v a l s . An element of the matrix then represents the p r o b a b i l i t y that a flood of a given return period, say 40.6 years which i s at the p r o b a b i l i t y mid-point of the 37 to 45 year return period interval,, l i e s within a c e r t a i n range,, say 250-255 c f s . The sum of the elements across any row, as usual, equals 1.0. But i f the elements of each row are m u l t i p l i e d by the proba-b i l i t y of being i n the corresponding flood i n t e r v a l , for example, the elements of the 37 to 45 year return period i n t e r v a l row are m u l t i p l i e d by (.1/37) - (1/45) , then the sum of a l l elements i n the matrix w i l l equal 1.0. An i n d i v i d u a l element of the matrix then represents the o v e r a l l p r o b a b i l i t y that a flood both l i e s within a c e r t a i n range and belongs to a c e r t a i n return period i n t e r v a l . The p r o b a b i l i t y that a flood l i e s within a ce r t a i n range, regardless of what return period i n t e r v a l i t belongs to, i s obtained by summing the elements of the respective flood i n t e r v a l column of the new matrix. Thus the information conveyed by a bounded Gumbel p l o t i s converted into a single p r o b a b i l i t y vector which can i n turn be plotted as a p r o b a b i l i t y density function or a single 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 derived from bounded Gumbel p l o t s , along with three d i s t r i b u t i o n s derived from single l i n e s were used i n the i n i t i a l analysis with the most prob-able curve, 1.0, s p e c i f i e d by Q 1 Q = 1 5 0 cfs and Q 1 0 0 =220 c f s ( Q ^ Q Q / Q ^ Q = 1.47). Later, a d i f f e r e n t most probable curve with a Q J L Q Q to Q ^ Q r a t i o equal to 1.8 was considered to see what e f f e c t steepening the Gumbel curve would have on the decision tree r e s u l t s The 1.8 r a t i o i s used by the B r i t i s h Columbia Department of 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 vary consider-ably from watershed to watershed. For West Vancouver the Q 1 0 0 to Q ^ Q r a t i o i s about 1.6 (10). The new most probable curve was sp e c i f i e d by set t i n g Q 1 Q = 120 c f s and Q 1 0 Q = 216 c f s . In thi s cas one bounded d i s t r i b u t i o n , along with the most probable curve d i s t r i b u t i o n alone, was used i n the analysis. Figure 3.1 shows the equivalent Gumbel plots of the four bounded d i s t r i b u t i o n s , as well as some single l i n e Gumbel p l o t s , a l l based on a 1.0 l i n e with Q 1 Q = 150 cfs and Q 1 0 0 = 220 c f s . The curves derived from bounded d i s t r i b u t i o n s are l a b e l l e d by the multiple factors of the lower and upper bounds, such as 0.5-1.5, while single l i n e s are l a b e l l e d with a single multiple factor, such as 1.5. This l a b e l l i n g system i s used throughout the thes i s . Figure 3.2 shows some of the p r o b a b i l i t y density functions. Figure 2 3 3 . 3 shows Gumbel plots based on the new 1 . 0 l i n e defined by Q ^ Q = 1 2 0 cfs and Q ^ Q Q = 2 1 6 c f s . Table 3 . 1 summarizes some of the i n f o r -mation contained i n Figures 3 . 1 and 3 . 3 by l i s t i n g the e f f e c t i v e floods of eleven return periods for the d i f f e r e n t d i s t r i b u t i o n s . The term e f f e c t i v e flood i s used to denote the flood derived by converting a bounded Gumbel pl o t into a single equivalent curve. Looking at the r e s u l t s based on the 1 . 0 curve with Q ^ Q = 1 5 0 cfs and Q 1 Q 0 = 2 2 0 c f s , for the symmetrically bounded Gumbel plots the bounded d i s t r i b u t i o n s are more unfavourable than the l.Q d i s t r i b u t i o n above a return period of -about 2 . 3 years. The f a c t that they are more favourable below th i s return period has l i t t l e s i g n i f i c a n c e since i t i s un l i k e l y the design selected w i l l sustain damage at floods below the 2 . 3 - y e a r return period. The 0 . 8 - 1 . 2 d i s t r i b u t i o n d i f f e r s s u r p r i s i n g l y l i t t l e from the 1 . 0 d i s t r i b u t i o n . Increasing the steepness of the 1 . 0 l i n e results i n less difference 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 ; t h i s can be seen by comparing the 0 . 5 - 1 . 5 and 1 . 0 curves i n Figures 3 . 1 and 3 . 3 . V a l u e s of b (Gumbel d i s t r i bu t i on ) -2.0 -1.0 0 1.0 2.0 3.0 4 . 0 5.0 6.0 7.0 8.0 9.0 10.0 1.01 I.I 1.52.0 5.0 10 2 0 50 10 0 2 0 0 5 0 0 1000 2000 5 0 0 0 10 0 0 0 Return Per iod ( y e a r s ) FIG. 3.3 F R E Q U E N C Y C U R V E S OF A N N U A L FLOODS ( 1.0 LI NE = Qio= 120cf s , Qioo= 2 l 6 c f s ). TABLE 3.1 COMPARISON OF EFFECTIVE FLOODS OF VARIOUS RETURN PERIODS FOR DIFFERENT DISTRIBUTIONS 1.0 Flood Frequency Line Specified by Q 1 Q = 150 c f s and Q 1 Q 0 = 220 c f s D i s t r i b u t i o n Return Period (yr) 1.1 2.0 5.0 10 20 50 100 200 500 1000 10000 1.0 57 1 94 128 150 171 199 220 241 268 289 357 0.8-1.2 56 93 128 151 174 202 224 246 275 298 370 0.5-1.5 50 92 132 159 185 218 244 270 304 330 420 0.3-1.7 44 90 136 166 195 233 262 291 329 359 459 0.8-1.52 60 103 143 170 197 231 257 283 317 343 433 1.2 68 113 153 180 206 239 264 2 89 322 346 429 1.5 85 141 191 225 257 299 330 361 402 433 536 II . 1.0 Flood Frequency Line Specified by Q 1 0 = 120 cfs and ' Q100 = 216 cfs D i s t r i b u t i o n Return Period (yr) 1.1 2.0 5.0 10 20 50 100 200 500 1000 10000 1.0 0 43 89 120 149 187 216 244 282 310 404 0.5-1.5 0 41 89 122 155 198 232 265 311 345 463 ^ a l l e f f e c t i v e flood values are i n cfs 2 mean value of th i s truncated 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 of Culvert Flow The r e l a t i o n s h i p between the headwater depth and the discharge i s greatly influenced by the type of flow through the c u l v e r t . The type of c u l v e r t flow occurring at a given discharge may be determined by many variables including the i n l e t geometry; the slope, s i z e , and roughness of the c u l v e r t b a r r e l ; and the approach and tailwater conditions. For p r a c t i c a l purposes culve r t flow i s commonly c l a s s i -f i e d into s i x types. But by placing the c u l v e r t on a 7% slope and assuming the tailwater neither submerges the o u t l e t nor reaches a s u b c r i t i c a l depth causing backwater e f f e c t s at any discharge, the number of possible flow types was reduced to three, shown i n Figure 4J.1. Both the 7% slope and the tailwater assumptions are reasonable i n the mountainous and h i l l y t e r r a i n covering most of B r i t i s h Columbia. The hydraulic computations were based on equations and tables compiled by R. W. Carter i n 1957 (11). The equations for the three types of flow considered are given i n the appendix. A l l computations were done by computer since some cal c u l a t i o n s required tedious 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 requires a c o e f f i c i e n t of discharge, but the c o e f f i c i e n t of discharge i s a function of the headwater l e v e l f o r flow types 1 and 2. The cross-sectional area of the headwater pool i s assumed reasonably large so that the v e l o c i t y head i s n e g l i g i b l e . In addition the volume of water stored i n the headwater pool at any T y p e I ! C r i t i c a l Depth at I n l e t . T y p e 2- Rap id Flow at In le t . T y p e 3 s F u l l F l o w Free O u t f a l l . HW N O T A T I O N « D - Cu l ve r t -d i a ( m i n . d i a for C M P ) d c = c r i t i c a l depth h = p iezometr ic head above culvert invert at downstream end HW = depth of water in headwater pool H * = c r i t i c a l va lue for headwater depth ( H - I.5D used here) s c s c r i t i c a l s l ope for c u l v e r t b a r r e l s 0 = bed s lope of c u l v e r t b a r r e l F IG .4 .1 T Y P E S OF C U L V E R T F L O W 28 headwater l e v e l i s assumed small; so, e f f e c t i v e l y , at any time the discharge into the headwater pool equals the discharge through the culve r t . The headwater-discharge curves for several c u l v e r t diameters are shown i n Figure 4.2. The entrance of an ordinary c u l v e r t w i l l not be submerged i f the headwater i s less than a c e r t a i n c r i t i c a l value, designated by H*, while the o u t l e t i s not submerged. The value of H* varies from 1.2 to 1.5 times the c u l v e r t diameter, D, depending on the 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 conditions (12). Carter assumes H* = 1.5D, so t h i s value was used i n the c a l c u l a t i o n s . Chow (12) states, "For a preliminary analysis, the upper l i m i t H* = 1.5D may be used . . . because computations have shown that, where submergence was uncertain, greater accuracy could be obtained by assuming that the entrance was not submerged." Type 1 flow r e s u l t s when the headwater i s less than H*, the tailwater i s lower than the c r i t i c a l depth, and the c u l v e r t slope i s s u p e r c r i t i c a l . C r i t i c a l flow occurs at or near the c u l v e r t entrance, and the headwater depth depends only on the discharge, c u l v e r t s i z e , and entrance geometry. Thus, this i s an example of i n l e t c o n t r o l . Type 2 flow i s also an example of i n l e t c o n t r o l , but i n t h i s case with the entrance submerged. The i n l e t functions as an o r i f i c e with the flow entering the c u l v e r t contracting to a depth less than the diameter of 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 contraction of flow i n the form of a j e t under a s l u i c e gate. In the case of a square-ended culve r t set f l u s h with a v e r t i c a l head-wall and, indeed, with most cu l v e r t i n l e t s , type 2 flow follows 30 type 1 flow as the headwater depth increases with increasing discharge. However at high submergences of the o r i f i c e the culv e r t may suddenly f i l l and type 3 flow occurs. B l a i s d e l l (2) has found that the headpool l e v e l at which th i s occurs may be d i f f e r e n t each time the c u l v e r t f i l l s , making an exact determination d i f f i c u l t . At t h i s point there w i l l be a sudden increase i n flow through the c u l v e r t and a r e s u l t i n g decrease i n the headpool l e v e l as the control changes from the o r i f i c e to the pipe. A culv e r t i s considered h y d r a u l i c a l l y short 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. Carter has prepared charts to roughly d i s t i n g u i s h between these two flow types. The determination depends on many c h a r a c t e r i s t i c s such as culv e r t diameter, length, and slope; entrance geometry; headwater l e v e l ; entrance and o u t l e t conditions; etc. In practice i t turned out that, for a l l c u l v e r t diameters considered (3.5 to 7.0 ft) and over the headwater range of i n t e r e s t (up to 10 f t ) , i n a l l submerged i n l e t cases the flow was type 2. Also, the 7% slope was a steep slope i n a l l these cases although flow types 2 and 3 can occur on mild or steep slopes. In type 3 flow the culv e r t b a r r e l i s under suction with the piezometric head at the o u t l e t varying from a point below the centre to the top of the c u l v e r t . However, N e i l l (13) reports that the turbulent, aerated flow caused by the pipe corrugations may prevent the existence of sub-atmospheric pressures i n the c u l v e r t and cause the culv e r t to flow p a r t l y f u l l . This i s a v a r i a t i o n of type 3 flow and not type 2 flow. 31 4.2 Entrance.and E x i t Improvement Entrance improvement should always be considered since i t can increase the hydraulic e f f i c i e n c y of culverts and thus reduce the culvert size required. (An increase i n hydraulic e f f i c i e n c y means that at a given flow the headwater surface can be lowered; or stated conversely, at a given headwater depth the flow accommodated can be increased.) E x i t improvement may be required to prevent erosion problems. The primary purposes of a headwall are to r e t a i n the f i l l and protect the embankment from erosion. Wingwalls can be used i n addition to r e t a i n the f i l l and support the headwall. By re t a i n i n g the f i l l behind the headwall, endwall, and wingwalls, savings can be r e a l i z e d by a reduction i n the c u l v e r t length required. 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 design can be improved by making the entrance into a sloping apron (14). The c r i t i c a l depth occurs on the apron, and the flow i s accelerated along the apron and into the culvert . The sloping i n l e t has an appreciable e f f e c t as long as the culver t b a r r e l does not flow f u l l . Rounding or tapering the i n l e t increases the hydraulic e f f i c i e n c y by increasing the c o e f f i c i e n t s of discharge for a l l flow types. A more spectacular increase i n hydraulic e f f i c i e n c y can be obtained i n some circumstances by employing special i n l e t s , such as bell-mouth or hood i n l e t s . This advantage applies only when the cul v e r t entrance i s submerged and mainly to culverts on steep slopes. The s p e c i a l i n l e t prevents i n l e t o r i f i c e control (type 2 flow) and causes the pipe to flow f u l l (type 3 flow). B l a i s d e l l (2) has found i n experiments using a hood i n l e t that an intermediate flow type, 32 slug and mixture flow, consisting of alternating slugs of f u l l flow and a i r pockets, occurs before type 3 flow i s established. As the i n l e t j u s t becomes submerged, the ad d i t i o n a l head created by the short length of f u l l conduit draws the headpool down admitting a i r to the culvert . The a i r flow decreases as discharge increases u n t i l the c u l v e r t flows completely f u l l of water. There i s very l i t t l e increase i n the headpool depth u n t i l the discharge i s great enough to cause f u l l flow. Vortices at culver t i n l e t s can adversely a f f e c t c u l v e r t per-formance, p a r t i c u l a r l y during pipe control with low i n l e t submer-gences, and thus they can decrease the advantage of using s p e c i a l i n l e t s . Vortices form over the i n l e t and admit a i r to the cu l v e r t through the vortex core. The a i r replaces water i n the cul v e r t and reduces the discharge. Vortices can reduce the cu l v e r t capacity to anywhere between that obtained with pipe control and that obtained with o r i f i c e control. On the other hand, surface v o r t i c e s that do not have an a i r core may have l i t t l e e f f e c t on the cu l v e r t capacity. Vortices can be i n h i b i t e d by i n s t a l l i n g anti-vortex devices. Plugging of culverts i s considered by many to be one of the major problems associated with culverts (7). I t can lead to major flood damage, even i n cases of minor floods. Culverts should be designed to pass expected debris, keeping i n mind that any debris jams that occur*must be e a s i l y accessible by maintenance crews. Upstream debris racks are required i n some locations. Plugging by ice forming inside the c u l v e r t can be a problem i n B r i t i s h Columbia's I n t e r i o r . The o u t l e t end of a cu l v e r t should be designed to avoid 33 (1) blockage by debris, (2) damage by flow undermining the culver t and embankment, and (3) erosion of the downstream channel. The greater roughness of corrugated metal pipe as compared to concrete pipe i s an advantage i n reducing outlet v e l o c i t y . A s t i l l i n g basin or energy d i s s i p a t o r of some so r t may be required to reduce downstream erosion. 4.3 Mechanics of a Washout An assumption i s made i n the analysis that the roadway w i l l wash out as soon as the road i s overtopped. It i s further assumed that the washout r e s u l t s i n the same damage to the roadway, no matter what flow caused the washout, and the culvert i t s e l f i s not damaged i n the process. These assumptions are not completely v a l i d but were made to simplify the analysis. The roadway i s l i k e l y to withstand some overtopping, with minimal damage, before washing out. The washout mechanism may s t a r t with gravel being eroded at both the upstream and downstream embankments, eventually leading to the undermining and collapse of the road surface. Once the road surface collapses the flow rate over the road surface w i l l increase dramatically, and the washout w i l l proceed quickly. Given the uncertainties 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 to estimate at what point a road w i l l wash out. A culvert i s l i k e l y to sustain some damage during a washout, although a headwall and endwall may prevent i t from being washed away. Scour under the c u l v e r t w i l l mean that the c u l v e r t has to be l i f t e d out and r e - i n s t a l l e d . Highway embankments are not designed as dams. If ponding i s allowed for i n the design of a cu l v e r t , provision must be made so that seepage through the embankment w i l l not lead to f a i l u r e by piping or other means. Also the slopes of the embankments must not be so great that they collapse when saturated. 4.4 Environmental Considerations Environmental considerations might be c a l l e d intangibles i n an economist's terms. I t i s d i f f i c u l t to place a monetary value on f i s h i n a stream because they may be worth much more than t h e i r commercial value. I f f i s h and other aquatic organisms are to be preserved i n streams passing through c u l v e r t s , economic analysis for c u l v e r t design may have to be supplemented by analysis of the ef f e c t s of the proposed design on the organisms involved. High flow v e l o c i t i e s i n culverts are common and may prevent f i s h from moving upstream. Reinforced concrete pipe, with i t s low roughness c o e f f i c i e n t , i s more of a problem than corrugated metal pipe. B a f f l e s might be needed to reduce the v e l o c i t y . E x i t f a c i l i t i e s , for example, 5 foot drops, often i n h i b i t f i s h access to the c u l v e r t . One approach to the enti r e problem i s to preserve the natural 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 large arch structure, although i t i s bound to be much more expensive than a pipe culve r t . • •\ Chapter 5 ECONOMICS 5.1 Ca p i t a l Cost The approximate c a p i t a l costs of i n s t a l l e d culverts are shown i n Table 5.1 and Figure 5.1. These costs are for 100 f t lengths of asbestos bonded, asphalt coated corrugated metal pipe (CMP) cul v e r t s , with v e r t i c a l concrete headwalls and endwalls, as used i n the analysis. The i n s t a l l e d CMP costs are from the D i s t r i c t of  West Vancouver Drainage Survey by Dayton and Knight Ltd., Consulting Engineers (10). The i n s t a l l a t i o n cost i s based on "average" conditions i n West Vancouver and represents the cost of i n s t a l l i n g a culve r t under an e x i s t i n g highway. Consequently the i n s t a l l a t i o n cost w i l l be somewhat less for a new highway construction project, p a r t i c u l a r l y under f i l l s , as l i t t l e or no excavation w i l l be required The costs can only be taken as approximate because they depend to a large extent on the conditions at each culvert s i t e . The trans-portation cost to the s i t e i s also a variable factor that must not be overlooked. The cost l e v e l s used i n the Dayton and Knight report are equivalent to an Engineering News-Record (ENR) Construction Cost Index of 2500 for 1975. The costs i n Table 5.1 and Figure 5.1 have been adjusted to an ENR index of 3000 for 1977. The headwall-endwall set costs were calculated from C a l i f o r n i a D i v i s i o n of Highways values presented i n P r i t c h e t t ' s thesis (3) by multiplying by the r a t i o of the ENR index i n 1977 to that i n 1964 (3000/900). This method of updating costs i s only approximate as the ENR index 35 TABLE 5.1 CAPITAL COSTS OF INSTALLED CULVERTS Culvert Pipe Headwall To t a l Diameter Cost* & Endwall Cost (feet) ($) Cost ($) ($) 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 *for 100 f t length 2 4 0 0 0 2 2 0 0 0 1 -2 0 0 0 0 1 -I 8 0 0 0 H -1 6 0 0 0 1 — 1 4 0 0 0 1 -I 2 0 0 0 h -1 0 0 0 0 1 — 8 0 0 0 H 6 0 0 0 h -4 0 0 0 1 — 2 0 0 0 1 -4.0 4.5 5.0 5.5 6.0 C u l v e r t D i ameter ( feet ) 7.0 FIG.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 38 represents the cost of a group of items consisting of fixed quantities of labour, cement, s t e e l , and lumber, and not the cost of purchasing and i n s t a l l i n g c u l v e r t s . There w i l l also 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 costs. 5.2 Flood Damage The flood damage cost at a p a r t i c u l a r headwater l e v e l i s the sum of two items: the headwater damage cost and the washout cost i f the road washes out. Headwater damage i s the r e s u l t of water backing up and f l o o d -ing p u b l ic or private property upstream of the c u l v e r t . Damage to the highway embankment, such as erosion of gravel caused by high headwater, i s included under headwater damage. Upstream flooding i s l i k e l y to be a problem only i n populated areas where development encroaches on the stream, or i n flood plains where substantial ponding can take place and inundate large areas of 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 analysis i s shown i n Figure 5.2. The shape of the curve was chosen a r b i t r a r i l y with marginal flood damage f i r s t increasing then decreasing. A t y p i c a l flood damage vs. depth curve for urban property i s shown by James and Lee (15) as a combination of three s t r a i g h t l i n e s with the f i r s t segment having the : greatest slope and the f i n a l segment a slope of zero. The damage i s assumed to be a function of headwater l e v e l only and not of culve r t s i z e . This may not be true i n the case of damage to the highway embankment as v e l o c i t y and turbulence around the c u l v e r t i n l e t at a given headwater w i l l vary for d i f f e r e n t c u l v e r t diameters. 0 5 6 7 8 9 H e a d w a t e r Depth ( f e e t ) F I G . 5.2 H E A D W A T E R D A M A G E F U N C T I O N 40 The roadway i s assumed to wash out i f the headwater overtops the highway ( i . e . , exceeds 10 f t i n t h i s case). The washout i s assumed to r e s u l t i n extensive damage to the roadway but no damage to the c u l v e r t and i t s headwall and endwall. The v a l i d i t y of these assumptions was discussed i n Chapter 4. The washout cost i s the sum of (1) the cost of r e p a i r i n g the highway, (2) expenses for flagmen, barricades, f l a r e s , and signing for t r a f f i c detours, and (3) the cost of interrupting t r a f f i c , which i s borne by the road-users themselves. The r e p a i r cost w i l l depend on the a v a i l a b i l i t y of labour, materials, and machinery, as well as the extent of damages. The cost of i n t e r r u p t i n g t r a f f i c i s more d i f f i c u l t to deter-mine. I t includes the increased motor vehicle operating cost for detour mileage, slowdowns, stops, and vehicle washing; the cost of increased t r a v e l time; and the cost of increased accident proba-b i l i t y . These costs w i l l vary from vehicle to v e h i c l e , p a r t i c u l a r l y between trucks and cars; therefore a weighted average must be used. The value of time l o s t for occupants of vehicles not on business i s often evaluated at one-third the average wage. The volume of t r a f f i c , time required to repair the road, and type of detour route a v a i l a b l e a l l influence the magnitude of the cost of interrupting t r a f f i c . I f no detour i s a v a i l a b l e on a major . highway, the cost w i l l be very high. Conversely, the cost w i l l be. low for minor highways. A washout cost of $15,000 i s used i n the i n i t i a l a n alysis. The cost borne by the highways department for repairing the road and providing flagmen, barricades, etc. i s assessed at $5000, and 41 the cost borne by the road-users at $10,000. The road-user cost i s roughly calculated as the product of the average d a i l y t r a f f i c (ADT), the time required to repair the road i n days, and the average cost of delay per v e h i c l e . The average d a i l y t r a f f i c i s the average 24-hour volume for a given year, counting both di r e c t i o n s of t r a v e l . A t y p i c a l ADT of 2500 for a major r u r a l two-lane highway i s assumed, and the time required to repair the highway i s estimated at 2 days. The average cost of delay per v e h i c l e , including both increases i n operating cost and t r a v e l time, i s set at $2.00 per v e h i c l e . This low cost per vehicle implies a r e l a t i v e l y minor detour. I t might be argued that road-user costs should not be included i n the economic analysis since the highways department does not compensate motorists for the delay. However, looking at the problem from a broad s o c i a l point of view, which a government should always do, these costs are r e a l and must be included since highways are public e n t i t i e s and not p r i v a t e l y owned. Some mention of maintenance cost should be made, although i t was not included i n the analysis. P r i t c h e t t , i n his t h e s i s , assumes an equal average maintenance cost for pipe culverts from 18 to 96 i n . on the basis that the larger culverts have a larger area of brush to c l e a r at the entrance and e x i t of the pipe, but less sand and debris to clean out as compared to the smaller 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 decision w i l l not be affected by the maintenance cost. 5.3 Annual Cost Comparison Before an economic analysis for choosing c u l v e r t s i z e can be completed, the c a p i t a l cost and expected annual damage cost, 42 computed as outlined i n Chapter 2, must be placed on a comparable basis so they can be added. The equivalent uniform annual cost method, i n which the investment cost i s converted to an annual cost, i s used i n t h i s case. The present value method, which involves combining the investment cost and expected annual damage cost into a single present worth sum, could equally well be used and would y i e l d the same r e s u l t as the equivalent uniform annual cost method. The factor to convert an investment cost into an equivalent annual cost i s designated as the capital-recovery factor and may be computed from the expression r ( l + r ) n / ( ( l + r ) n - 1), where r i s the discount rate per annum and n i s the estimated service l i f e of the c u l v e r t or highway, whichever i s shorter. The equation i s for a series of n year-end payments, as shown i n Figure 5.3, although the capital-recovery factor 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 series of n mid-year payments, as long as n i s not too small. The question of what i s the correct discount rate to use i n computing the capital-recovery factor i s a matter of considerable debate. I t i s a very important question as a change of 1% i n the discount rate ( i . e . , from 4% to 5% or from 7% to 6%) w i l l often change the project selected. A low discount rate with a long service l i f e w i l l favour designs with a high c a p i t a l cost since the annual investment charge w i l l be lower than i n the case where the discount X, rate i s high or the service l i f e i s low. The term discount rate 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 rate. Discount rate, r, as used here, i s the r e a l rate of i n t e r e s t as opposed to the money rate of i n t e r e s t , x. The discount rate can be computed as r = (x - i ) / ( l + i) or approximately r = x - i , where Method l ! U n i f o r m S e r i e s A A A A A 43 I n • 21 n - I I n A C - C R F C R F = r ( l + r ) n where r = x - i I +i ( l + r ) n - l C = c a p i ta I cos t A = equivalent annual cost in base year dollars (i.e.dollars at C R F = cap i t a l - recovery factor beginning of year I) r = d i scount rate x - money rate of in teres t i = i n f l a t i o n rate Method 2 : E x p o n e n t i a l S e r i e s A n - 3 A n-2 'n- l A 0 " i L L i l i . . , n-2 A 0= C- E C R F E C R F (T^ )[(7fr)-'] A x = annual cost in dol lars of year x •, A x = A 0 ( I + i ) x E C R F = exponentia l series cap i ta l - recovery factor N . B . A Q i s not included in the summation for ca lcu lat ing the E C R F , in conformi ty with the p e r i o d - e n d step convent ion. It c a n be eas i l y proven that C R F = E C R F if r as de f ined above is used in c a l cu l a t i n g the C R F . Therefore the two methods are e q u i v a l e n t . F I G . 5 . 3 C O N V E R T I N G C A P I T A L C O S T TO A N N U A L C O S T . 44 i i s the rate of i n f l a t i o n . This equation corrects the money rate of i n t e r e s t f o r the e f f e c t of i n f l a t i o n . A discount rate of 4% was chosen for the i n i t i a l a n alysis. This figure was based on an i n t e r e s t rate for r i s k free investment, such as government bonds, equal to about 10% and a rate of i n f l a t i o n equal to about 6%. In f a c t , both the money i n t e r e s t rate and the i n f l a t i o n rate are l i k e l y to fluctuate considerably over the service l i f e of the c u l v e r t or highway. But fluctuations i n the r e a l i n t e r e s t rate are usually much smaller, as i n the long run the money rate of in t e r e s t adjusts to account for the i n f l a t i o n rate. As an example, i n t e r e s t rates on government savings bonds increased from about 5% i n the early 1960s to 8 to 10% i n the 1970s. But the calculated r e a l i n t e r e s t rate held steady for 1965 to 1972 at a moderate l e v e l of 3% before i t f e l l i n 1973 (16). An equivalent method of handling the problem of i n f l a t i n g costs i s i l l u s t r a t e d by the exponential series i n Figure 5.3. Here the c a p i t a l cost i s converted to an exponential series of annual costs increasing at the rate of i per cent per annum, as opposed to a series of uniform annual costs. The expected annual damage cost i s also assumed to increase exponentially at the rate of i per cent per year; therefore, the two series of annual costs can be added to determine the series of t o t a l annual costs for a given cul v e r t diameter. Ac t u a l l y , only the annual costs at the beginning of the base year need be computed since a l l annual costs increase at the same .rate, i . Hence the culver t size decision can be made by comparing the t o t a l annual costs i n the base year. The money rate of i n t e r e s t , 45 x, i s used to compute the annual investment charge at the beginning of the base year since the e f f e c t of i n f l a t i o n i s taken into account d i r e c t l y . In f a c t , the annual investment charge computed at the beginning of the base year w i l l be same for the exponential series method and the equivalent uniform annual cost method (r = (x - i ) / (1 + i ) ) ; therefore the two methods are exactly equivalent. The discussion of the exponential series i s meant to point out the importance of taking the rate of i n f l a t i o n i n t o account. I t would be a serious error to calc u l a t e the capital-recovery factor for the equivalent uniform ser i e s method on the basis of 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 . This would amount to adding a uniform s e r i e s , the annual c a p i t a l cost, to an exponentially increasing s e r i e s , the expected annual damage cost. If the equivalent uniform series method i s applied, the money rate of i n t e r e s t must be corrected for the e f f e c t of i n f l a t i o n so that 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 discussion assumes that the expected annual damage cost increases at the same rate as i n f l a t i o n , or i n other 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 cost. Construction of alternate routes or switches to other modes of transportation (due to r a p i d l y increasing gasoline p r i c e s , etc.) w i l l r e s u l t i n a r e a l decrease i n the expected annual damage cost. I t i s often d i f f i c u l t to forecast these changes, p a r t i c u l a r l y over a long period of time such as 20 or 30 years, but some attempt should be made. I t should be mentioned that annual cost c a l c u l a t i o n s are 46 v a l i d regardless of the financing scheme employed to pay the c a p i t a l cost, as long as the discount rate i s appropriate for the circum-stances (17) . A c u l v e r t service l i f e of 30 years was used i n the i n i t i a l analysis. Actually t h i s value i s conservative as a properly i n s t a l l e d , asbestos bonded, asphalt coated CMP can be expected to l a s t much longer; p a r t i c u l a r l y i f i n addition the inv e r t i s paved with asphalt or concrete to guard against sediment abrasion. Factors such as the corrosion p o t e n t i a l at the proposed culvert s i t e , the a n t i c i -pated highway service l i f e , and cost w i l l influence the c u l v e r t material, material thickness, and type of protective treatment selected. For example, for temporary roadways such as logging roads, only simple galvanized CMP culverts would be j u s t i f i e d . This decision could also be included i n the decision tree of Figure 2.2 with d i f f e r e n t materials or protective coatings having d i f f e r e n t service l i v e s . A further complication i s introduced i f c u l v e r t damage i s anticipated when the roadway washes out since the service l i f e of the c u l v e r t may be shortened or terminated by damage. There may be a great deal of uncertainty i n estimating the service l i f e of a highway or c u l v e r t . In this regard i t should be noted that i f n i s i n i t i a l l y large, say 30 years, a large increase i n n, say to 100 years, w i l l only moderately change the c a p i t a l -recovery factor. The difference i n the capital-recovery factor with increasing n w i l l decrease as the discount rate, r , increases. Chapter 6 RESULTS 6 . 1 Annual Cost Curves f o r One Flood Frequency D i s t r i b u t i o n The annual cost curves for the single l i n e Gumbel p l o t defined by s e t t i n g Q ^ g = 1 5 0 cfs and Q - ^ Q Q = 2 2 0 cfs are shown i n Figure 6 . 1 . The expected t o t a l cost curve (the word "expected" i s often omitted for convenience) shows that the optimum culver t diameter i s 5 . 0 f t with smaller diameter culverts becoming less competitive more r a p i d l y than larger diameter c u l v e r t s . The cost data from which Figure 6 . 1 was plotted, as w e l l as some a d d i t i o n a l information, i s given i n Table 6 . 1 . The so-called marginal investment costs (MIC) l i s t e d i n Table 6 . 1 are the differences i n annual investment cost between given sized culverts and culverts of the next smaller s i z e . Similar-l y marginal savings (MS) i s the difference i n expected t o t a l annual damage cost between a given sized c u l v e r t and the c u l v e r t of the next smaller s i z e . The use of these marginal costs and savings i s f u l l y explained i n the t h i r d section of t h i s chapter. Table 6 . 2 gives the annual p r o b a b i l i t y of i n c u r r i n g some .head-water damage and the annual p r o b a b i l i t y of a washout for each c u l v e r t diameter, f i r s t using the Gumbel p l o t defined above and then using the Gumbel p l o t defined by Q ^ Q = 1 2 0 cfs and Q - ^ Q Q = 2 1 6 c f s . 6 . 2 The E f f e c t of Uncertainty and the Value of Better Information The effect, of uncertainty i n the flood frequency p l o t with the most probable curve s p e c i f i e d by Qn n = 1 5 0 c f s and Q i n r i = 2 2 0 c f s 47 2 0 0 0 18 0 0 4.0 4.5 5.0 5.5 6.0 6.5 7.0 C u l v e r t D i a m e t e r ( f e e t ) FIG.6.1 A N N U A L C O S T C U R V E S FOR F L O O D F R E Q U E N C Y D I STR IBUT ION D E F I N E D BY Q i o = l 5 0 c f s A N D Qioo = 2 2 0 c f s . TABLE 6.1 ANNUAL COSTS FOR FLOOD FREQUENCY DISTRIBUTION DEFINED BY Q 1 Q = 150 CFS AND Q 1 0 Q = 220 CFS Culvert Investment Marginal Headwater Washout Total Margina^ Total Increase % Increase Diameter Cost 1 • Investment Damage Cost Damage Savings Cost i n Total i n Total (ft) , ($) Cost Cost ($) Cost ($/size) ($) Cost from Cost from ($/size) ($) ($) 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 yr MIC = (annual investment cost of given culvert size) - (annual investment cost of next smaller culvert size) 3 MS = (annual t o t a l damage cost of next smaller culvert size) - (annual t o t a l damage cost of given culvert size) vo 50 TABLE 6.2 PROBABILITIES OF INCURRING SOME HEADWATER DAMAGE AND PROBABILITIES OF A WASHOUT Flood Frequency D i s t r i b u t i o n Specified by: Q 1 0 = 150 c f s , Q 1 0 0 = 220 cfs Q 1 Q = 120 c f s , Q 1 0 0 ='216 cf s Culvert P r o b a b i l i t y P r o b a b i l i t y P r o b a b i l i t y P r o b a b i l i t y Diameter HW > 5 f t HW > 10 f t HW > 5 f t HW > 10 f t (ft) 4.0 4.5 5.0 5.5 6.0 6.5 7.0 ,48705 ,33201 ,21641 ,13704 10000 06170 04450 .05242 .01396 .00310 .00081 .00018 .00002 .00000 ,17658 ,12592 08901 06253 04930 03441 02705 .03051 .01157 .00386 .00145 .00048 .00011 .00003 JO, 51 i s shown i n Figure 6.2. The curve l a b e l l i n g system i s the same as i n Chapter 3. The e f f e c t of uncertainty i n changing the optimal decision from that of the most probable curve alone appears to be rather minimal. For a symmetric d i s t r i b u t i o n (upper and lower bounds equidistant from the most probable curve), the bounds must be somewhat further apart than 0.5-1.5 before a switch to a 5.5 f t c u l -vert i s indicated. The expected t o t a l cost curve for 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 very s i m i l a r to that of the 0.3-1.7 d i s t r i b u t i o n . Figure 6.3 shows the r e s u l t s for the two d i s t r i b u t i o n s with the new most probable curve s p e c i f i e d by Q ^ Q = 120 cfs and Q - ^ Q Q = 216 c f s . As a r e s u l t of the steeper most probable curve, the t o t a l costs of culver t diameters less than the optimum diameter increase less r a p i d l y than i n the cases shown i n Figure 6.2, although again culverts smaller than the optimum become less competitive more rap i d l y than culverts larger than the optimum. The e f f e c t of uncertainty i n changing the optimal decision i s less with the new most probable curve, as can be seen by comparing the 0.5-1.5 and 1.0 curves of Figures 6.2 and 6.3. Two methods of c a l c u l a t i n g the value of better information are discussed i n the following paragraphs. The second method i s the better of the two, and although t h i s method was not a c t u a l l y used i n the analysis, i t warrants a f u l l discussion. The f i r s t method assumes that the most probable flood frequency curve i s i n f a c t the true curve. Then the true t o t a l cost curve i s the 1.0 curve of Figure 6.2 or 6.3. The value of better information i s simply the difference between the t o t a l costs on the 52 2 6 0 0 2 4 0 0 2 2 0 0 (A i _ O O TJ o o o C ^ 1 0 0 0 5.5 6.0 D i a m e t e r ( f e e t ) 7.0 F IG.6.2 T O T A L A N N U A L C O S T C U R V E S FOR D I F F E R E N T F L O O D F R E Q U E N C Y Dl STR I BUTI ON S ( 1. 0 CUR V E = Q io= I 5 0 c f s AND Qioo= 2 2 0 c f s . ) 2 0 0 0 3.5 4.0 4.5 5.0 5.5 C u l v e r t D i a m e t e r (feet) 6.0 F IG.6.3 A N N U A L C O S T C U R V E S ( 1.0 C U R V E = Qio= 120 c f s AND Qioo = 2 l 6 c f s ). 54 1.0 curve of the c u l v e r t diameter chosen under uncertainty and the true optimum cu l v e r t diameter. The values of better information calculated i n t h i s manner for the most probable curve s p e c i f i e d by Q^g = 150 cfs and Q - ^ Q Q = 220 cfs are given i n Table 6.3. Using t h i s method, better information only has a value i f the optimum decision under uncertainty i s d i f f e r e n t from the optimum decision of the most probable curve alone. I f the percentage increase i n t o t a l cost of the next larger size above the optimum i s small, such as 0.52% for the 0.5-1.5 d i s t r i b u t i o n , the decision maker w i l l l i k e l y choose the larger s i z e , changing the value of better information. The method ju s t discussed i s fundamentally unsound because the true flood frequency curve i s never known. In f a c t the value of better information may be substantial even i f the optimum decision under uncertainty i s the same as that of the most probable curve alone. For instance, taking 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 that the true t o t a l cost curve i s the 1.2 curve of Figure 6.2. I n s t a l l i n g a 5.0 f t diameter culvert then r e s u l t s i n a t o t a l cost of $84/yr more than the optimum for a 5.5 f t c u l v e r t . S i m i l a r l y , i f the 0.8 curve i s the true curve the optimum culver t diameter w i l l l i k e l y be 4.5 f t , and thus i n s t a l l i n g a 5.0 f t c u l v e r t re s u l t s i n a greater t o t a l cost than the optimum. These examples suggest a way to calculate the value of perfect information. A number of t o t a l cost curves could be calculated for d i f f e r e n t multiples of the most probable curve between the upper and lower bounds. For example, i f the bounds are 0.8 and 1.2, t o t a l cost curves could be calculated for the curves: 0.80, 0.85, 0.90, ... 1.20. A curve i s then plotted of the t o t a l cost of the optimum cu l v e r t vs. 5 5 TABLE 6 . 3 THE EFFECT OF UNCERTAINTY IN CHANGING THE OPTIMAL DECISION AND THE VALUE OF BETTER INFORMATION Flood Frequency D i s t r i b u t i o n Optimum Culvert % Increase i n Total Cost Value of Better Information ($) Diameter (ft) of Next 2 Larger Size Annual Value Present Value 4 1 . 0 5 . 0 ' 6 . 2 1 0 . 8 - 1 . 2 5 . 0 5 . 3 2 0 0 0 . 5 - 1 . 5 5 . 0 0 . 5 2 0 0 0 . 3 - 1 . 7 5 . 5 5 . 8 3 5 4 9 3 0 0 . 8 - 1 . 5 5 . 5 6 . 2 9 5 4 9 3 0 1 . 2 5 . 5 4 . 6 7 5 4 9 3 0 1 . 5 6 . 0 0 . 3 1 1 6 0 2 7 7 0 1 . 0 curve: Q - ^ Q = 1 5 0 c f s , Q 1 0 0 = 2 2 0 cfs 2 using t o t a l annual cost curve ed (see Figure 6 . 2 ) for d i s t r i b u t i o n being consider— assuming better information r e s u l t s i n the true curve beinq i d e n t i f i e d as the 1 . 0 curve Present Value = Annual Value / CRF CRF = . 0 5 7 8 3 ; r = 4 % , n = 3 0 yr CRF = capital-recovery factor r = discount rate n = service l i f e 56 the multiple of the most probable curve. The p r o b a b i l i t i e s that the true curve l i e s within small i n t e r v a l s of multiples of the most probable curve (for example, 0.80-0.81, 0.81-0.82, 1.19-1.20) are then calculated from the truncated skew normal or normal d i s t r i b u t i o n . The t o t a l cost of the optimum c u l v e r t at the mid-point of each i n t e r v a l i s calculated from the previously constructed optimum cost curve and m u l t i p l i e d by the p r o b a b i l i t y that the true curve i s i n that i n t e r v a l . The sum of these products over a l l i n t e r v a l s y i e l d s the expected t o t a l cost with-perfect information. The value of perfect information i s the difference between the expected t o t a l cost of the'optimum culve r t chosen under uncertainty and the expected t o t a l cost with perfect information. I t i s i n t e r e s t i n g to note that the expected t o t a l costs with uncertainty of Figures 6.2 and 6.3 could also be calculated i n a manner si m i l a r to that for the expected t o t a l cost with perfect information, rather than by reducing a bounded flood frequency p l o t to a single curve as outlined i n Chapter 3. The only difference i s that the cost of the c u l v e r t size being considered i s used for a l l i n t e r v a l s instead of the cost of the optimum sized c u l v e r t . In p r a c t i c e , no data gathering program w i l l eliminate a l l uncertainty; so the value of perfect information fi x e s an uppermost l i m i t to the value_ of better information. The value of better information i n reducing the uncertainty l i m i t s from 0.5-1.5 to 0.8-1.2 might be estimated by subtracting the values of perfect information i n the two cases. This i s only an estimate because i t cannot be known beforehand how the better information w i l l change the uncertainty l i m i t s and the most probable curve. 57 After 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 value of better information for a p a r t i c u l a r c u l v e r t s i t e can be calculated by subtracting the t o t a l cost of the cu l v e r t size chosen a f t e r the data gathering from the t o t a l cost of the culv e r t size that would have been chosen before the data gathering, both these t o t a l costs being from the new curve. If th i s i s done for a large number of c u l v e r t s i t e s , such as along a proposed new highway route, then a f a i r l y accurate monetary value of a data gathering program may r e s u l t . The estimate of the value of the program made before i t was i n s t i t u t e d can then be compared to the calculated value of the program a f t e r i t i s completed to see how accurate the estimation procedure was. The rough figures of Table 6 . 3 show that the value of data gathering can be sub s t a n t i a l . Keeping i n mind that these are for a single c u l v e r t s i t e , i t may be very worthwhile to i n s t a l l a network of 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 recording gauges i n some streams, before s e l e c t i n g c u l v e r t sizes for a new highway. 6 . 3 S e n s i t i v i t y of the Optimal Decision to Changes i n the Discount  Rate and the Service L i f e The s e n s i t i v i t y of the optimal decision to changes i n the discount rate and the service l i f e was investigated by using marginal investment cost, MIC, and marginal savings, MS, curves. These curves are s i m i l a r to an economist's marginal cost and marginal revenue curves that are used i n analyzing a firm's revenue, cost, and p r o f i t p i c t ure. A firm seeking to maximize i t s p r o f i t produces to the point where marginal revenue ( i . e . , the revenue gained from the l a s t u n i t of output) equals marginal cost ( i . e . , the 58 cost of producing the l a s t u n i t of output). S i m i l a r l y , s t a r t i n g with a small c u l v e r t s i z e , larger c u l v e r t sizes are selected u n t i l the point where the marginal investment cost of moving to the next larger s i z e i s greater than the marginal savings gained by moving to the next la r g e r c u l v e r t s i z e . Figure 6.4 was constructed using the r e s u l t s for the case where the most probable curve i s s p e c i f i e d by Q ^ Q = 150 cfs and QlOO = 220 c f s with uncertainty bounds of 0.5-1.5. There are four marginal investment cost curves representing d i f f e r e n t i n t e r e s t rates and service l i f e s , along with one marginal savings curve, shown in Figure 6.4. The optimal size c u l v e r t for a p a r t i c u l a r marginal savings, marginal cost curve combination i s the f i r s t c u l v e r t size to the l e f t of the i n t e r s e c t i o n of the two curves. Because there are only a l i m i t e d number of c u l v e r t sizes available (4.0, 4.5, 5.0 f t , etc.) and because the marginal curves were constructed using incremental differences i n costs and savings between c u l v e r t s i z e s rather than by taking instantaneous slopes on continuous curves, the i n t e r s e c t i o n point does not indicate the optimum diameter. An i n t e r s e c t i o n point near one of the fixed diameters, such as that for the r = 4%, n = 30 yr MIC curve which in t e r s e c t s the MS curve near a cu l v e r t diameter of 5.5 f t , indicates instead that the 5.5 f t cu l v e r t has nearly the same t o t a l cost as the 5.0 f t c u l v e r t . The difference between the MIC and MS curves at a p a r t i c u l a r c u l v e r t diameter i s the difference i n t o t a l cost between that c u l v e r t diameter and the next smaller culv e r t diameter. Thus i t i s r e l a t i v e l y easy to see how competitive the optimum sized culvert i s with culverts of smaller and larger s i z e . 59 4 8 0 *~ 4 4 0 Q) N CO 2 4 0 0 o o o —- 3 6 0^ CO CP c '> o to 3 2 0 o c cn 2 8 0 o 2 . i _ o 2 4 0 to o o c . 2 0 0 cu e CO <u > 1 6 0 c r -H o c 120 o> o _ 8 0 o 3 C C < 4 0 0 [2 difference in total annual cost between 5.0ft .and 4.5ft. c u l v e r t s 0 1 4.0 f lood frequency d i s t r ibu t ion : 0 . 5 - 1.5 with 1.0 curve 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 0 r = discount rate n = serv ice life in years. , 0 3 0 4.5 5.0 5.5 6 .0 C u l v e r t D i a m e t e r ( fee t ) 6.5 7. 0 F IG. 6.4 SENS IT IV ITY OF THE OPT IMAL DEC IS ION TO C H A N G E S IN T H E D I S C O U N T R A T E AND T H E S E R V I C E L I F E . 6 0 Looking at Figure 6 . 4 , the optimal decision 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 discount rate or to changes i n the service l i f e with the discount rate f i x e d at 4 % . (Incidentally the MIC curve for n = o o and r = 4% l i e s about one quarter of the way between the r = 0 % , n = 3 0 yr curve and the r = 4 % , n = 3 0 yr curve, being closer to the lower curve.) I f the MS curve were f l a t t e r then the optimal decision would be more sensi-t i v e to changes i n the i n t e r e s t rate and the service l i f e . 6 . 4 The E f f e c t on the Optimal Decision of Changing the Damage Costs Marginal savings and marginal investment cost curves were again used to determine the e f f e c t on the optimal decision of varying the washout cost and the headwater damage curve. Figure 6 . 5 shows the r e s u l t s for the most probable flood frequency curve s p e c i f i e d by Q ^ Q = 1 5 0 cfs and Q - ^ Q Q = 2 2 0 cfs with uncertainty l i m i t s of 0 . 5 - 1 . 5 . The marginal cost curve represents the standard case with r = 4% and n = 3 0 yrs. Figure 6 . 6 shows the r e s u l t s for the most probable curve s p e c i f i e d by Q 1 Q = 1 2 0 cfs and Q 1 0 0 = 2 1 6 c f s with the same uncertainty bounds as before. A l l the marginal savings curves are for the standard headwater damage curve, except one i n each fi g u r e . The marginal savings curve for any washout cost and any multiple of the standard headwater damage curve could e a s i l y be plotted from the curves presented i n Figures 6 . 5 or 6 . 6 . The greater spread of the MS curves of Figure 6 . 6 compared to Figure 6 . 5 indicates that the optimal decision w i l l vary more with changing damage costs with the steeper most probable curve used i n Figure 6 . 6 . Table 6 . 4 summarizes some of the information of Figures 61 4 8 0 f lood f requency distr i bution : 0 .5 - 1.5 with 1.0 curve s p e c i f i e d by Qio = I50cf s and Qioo= 2 2 0 c f s 5.0 5.5 6.0 C u l v e r t Di a m e t e r ( feet ) 6.5 7.0 FIG. 6.5 T H E E F F E C T ON T H E O P T I M A L DEC IS ION OF C H A N G I N G T H E D A M A G E C O S T S . 4 8 0 i 4 4 0 4 0 0 f lood frequency distr ibut ion 0 . 5 - 1.5 with 1.0 curve s p e c i f i e d by Q io = 120 c f s and Qioo= 21 6 c fs 3 6 0| 3 2 0 , 2 8 0 2 4 0 200< TO 10 o s o c o o O O O O O I 6 0 I 2 0 8 0 4 0 4.0 4.5 5.0 5.5 6.0 C u l v e r t 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 OPT IMAL DECIS ION OF CHANG ING T H E D A M A G E C O S T S ; NEW F L O O D F R E Q U E N C Y D I S T R I B U T I O N . 63 6.5 a n d 6.6 b y l i s t i n g t h e o p t i m a l c u l v e r t d i a m e t e r f o r e a c h o f t h e d i f f e r e n t damage c o s t s i n t h e two c a s e s , a l o n g w i t h t h e r e t u r n p e r i o d s f o r h e a d w a t e r s o f 5.0 a n d 10.0 f t ( t h e h e a d w a t e r a t w h i c h h e a d -w a t e r d a m a g e , i f a p p l i c a b l e , s t a r t s a n d t h e h e a d w a t e r c a u s i n g w a s h o u t ) a n d t h e r e t u r n p e r i o d f o r t h e h e a d w a t e r d e p t h e q u a l t o t h e d i a m e t e r o f c u l v e r t . L o o k i n g a t t h e t a b l e , a c t u a l l y none o f t h e o p t i m u m c u l v e r t s m e e t t h e B r i t i s h C o l u m b i a D e p a r t m e n t o f H i g h w a y s ' h y d r a u l i c d e s i g n c r i t e r i a ( s e e i n t r o d u c t i o n ) s i n c e t h e r e i s h e a d w a t e r damage a t f l o o d s b e l o w t h e 100 - y e a r r e t u r n p e r i o d i n a l l c a s e s ; h o w e v e r t h e 100 - y e a r f l o o d h e a d w a t e r damage c o s t i s v e r y low i n some c a s e s . A s s u m i n g t h e r e i s n o ' h e a d w a t e r damage ( i . e . , t h e o n l y damage t h a t c a n o c c u r i s a w a s h o u t ) , i n t h e c a s e o f t h e f i r s t f l o o d f r e q u e n c y d i s t r i b u t i o n , a 5.0 f t c u l v e r t i s r e q u i r e d t o m e e t t h e B r i t i s h C o l u m b i a D e p a r t m e n t o f H i g h w a y s ' h y d r a u l i c d e s i g n c r i t e r i o n B , a n d a 5.5 f t c u l v e r t i s r e q u i r e d t o m e e t c r i t e r i a A a n d B . T h u s t h e B r i t i s h C o l u m b i a D e p a r t m e n t o f H i g h w a y s w o u l d s e l e c t a 5.5 f t d i a m e t e r c u l v e r t , g i v e n t h a t t h e y u s e t h e d e r i v e d s i n g l e e q u i v a l e n t f l o o d f r e q u e n c y c u r v e . I n t h e c a s e o f t h e s e c o n d d i s t r i b u t i o n , a 5.0 f t c u l v e r t w o u l d be c h o s e n a s i t m e e t s c r i t e r i a A a n d B. T a b l e 6.5 shows t h e c o n s e q u e n c e s o f u s i n g t h e H i g h w a y s D e p a r t m e n t ' s d e s i g n c r i t e r i a r a t h e r t h a n t h e e c o n o m i c a n a l y s i s m e t h o d u s e d i n t h i s t h e s i s . No h e a d w a t e r damage i s a s s u m e d i n a l l c a s e s . S u b s t a n t i a l e x t r a c o s t s a r e i n c u r r e d by u s i n g t h e D e p a r t m e n t o f H i g h w a y s ' c r i t e r i a i f t h e w a s h o u t c o s t i s v e r y l o w o r v e r y h i g h . Low w a s h o u t c o s t s c o u l d r e f l e c t l o w v o l u m e r u r a l h i g h w a y s w h i l e h i g h w a s h o u t c o s t s c a n be i n c u r r e d i n c a s e s w h e r e t h e r e i s a s u b s t a n t i a l d e l a y w i t h m o d e r a t e t r a f f i c v o l u m e o r i n c a s e s w h e r e t h e t r a f f i c TABLE 6.4 64 OPTIMUM CULVERT DIAMETERS AND RETURN PERIODS OF SIGNIFICANT HEADWATER LEVELS FOR DIFFERENT DAMAGE COSTS I. Flood Frequency D i s t r i b u t i o n : 0.5-1.5 with 1.0 curve s p e c i f i e d by Q 1 Q = 150 cf s and Q1QQ = 220 cfs Washout Optimum Return Period (yr) Cost Culvert ($) Diameter (ft) HW > 5.0 f t HW>10.0 f t HW =2 D 5000n 4.5 3.0 45 2.1 5000 5.0 4.2 135 4.2 15000 5.0 4.2 135 4.2 25000 5.5 6.2 390 10.4 50000 5.5 ' 6.2 390 10.4 100000 6.0 8.0 1300 31 II . Flood Frequency D i s t r i b u t i o n : 0.5-1.5 with 1.0 curve s p e c i f i e d by Q 1 n = 120 cfs and Q,nfl = 216 cf s Washout Cost ($) Optimum Culvert Diameter (ft) Return Period (yr) HW > 5.0 f t HW > 10.0 f t HW> D 1 5000n 2 4.0 3 5.7 28 3.4 5000 4.5 7.8 64 5.7 15000 5.0 10.7 162 10.7 25000 5.0 10.7 162 10.7 50000 5.5 14.6 370 22 100000 6.0 18.1 900 52 D = culvert diameter n = no headwater damage; otherwise standard headwater damage curve (Figure 5.2) i s used 'The curves of Figure 6.6 indicate that the optimum cu 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 c u l v e r t . But the MIC curve was drawn as a smooth curve which does not exactly pass through a l l the data points. Using the actual data points, the 4.0 f t cu 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. Flood Frequency D i s t r i b u t i o n : 0.5-1.5 with 1.0 curve s p e c i f i e d by Q 1 Q = 150 cfs and Q 1 Q0 = 2 2 0 c f s Washout Optimum Expected Expected Extra Expected Extra C o s t l Culvert Total Annual Annual Cost i f Annual Cost i f ($) Diameter Cost Culvert Diameter Culvert Diameter (ft) ($) Selected i s Selected i s 5.0 f t 2 ($) 5.5 f t 3 ($) 5000 15000 25000 50000 100000 4.5 5.0 5.0 5.5 6.0 778 871 945 1009 1088 19 121 412 117 49 1 48 II . Flood Frequency D i s t r i b u t i o n : 0.5-1.5 with 1.0 curve s p e c i f i e d by Q1 = 120 cfs and Q i r m = 216 cfs Washout Optimum Expected Expected Extra Cost 1 Culvert Total Annual Annual Cost i f ($) Diameter Cost Culvert Diameter (ft) ($) Selected i s 5.0 f t 3 ($) 5000 4.0 726 65 15000 5.0 853 25000 5.0 914 _ 50000 5.5 1019 50 100000 6.0 1122 255 no headwater damage assumed 'meets B.C. Dept. of Highways' c r i t e r i o n B only meets B.C. Dept. of Highways' c r i t e r i a A and B (see Intro-duction for c r i t e r i a ) 66 volume alone i s very high. The Highways Department's c r i t e r i a are just not " r i g h t " for a l l roads under a l l conditions. Figure 6.7 i l l u s t r a t e s that the e f f e c t of uncertainty i n changing the optimal decision i s greater when the damage cost i s greater, as the separation between the 1.0 and 0.5-1.5 MS curves increases with increasing damage cost. Consequently the value of better information i s l i k e l y to be greater for high damage costs than for low damage costs. 67 4 8 0 4.0 4.5 5.0 5.5 6.0 C u l v e r t D i ameter (feet) 6.5 7.0 FIG. 6.7 T H E E F F E C T OF U N C E R T A I N T Y IN CHANGING T H E O P T I M A L D E C I S I O N AT D I F F E R E N T DAMAGE C O S T S . Chapter 7 CONCLUSION This thesis has described a method of economic analysis to determine the optimum sized c u l v e r t f o r any c u l v e r t s i t e . The method takes uncertainty into account and i s capable of estimating the value of better information. Various aspects of the c u l v e r t s e l e c t i o n problem: hydrologic, hydraulic, and economic were d i s -cussed, and the method was applied to a hypothetical c u l v e r t s i t e , assuming d i f f e r e n t hydrologic 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 analysis i n culvert s e l e c t i o n appear so great that one wonders why i t has yet to be used. Lin s l e y and F r a n z i n i (17) state, "The p r a c t i c a l d i f f i c u l t y i s that of estimating the probable damages from flows i n excess of c u l v e r t capacity." This i s very true, but research can solve the problem. I t would not be d i f f i c u l t to conduct experiments to f i n d out what causes a c u l v e r t to wash out. In addition to experiment, observations of culverts i n the f i e l d operating during flood conditions and close inspections of c u l v e r t s i t e s a f t e r washouts w i l l lead to much improved damage cost e s t i -mates. Even i f there i s much uncertainty involved i n estimating damage costs, t h i s uncertainty could be accounted for i n the econo-mic analysis, and estimates of the value of better information i n th i s area could be made. Another argument that might be made i s that the extra engineering cost involved i n applying economic analysis to c u l v e r t s e l e c t i o n w i l l outweigh the savings from the program. This i s very 68 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. Although the i n i t i a l cost of developing a good general- program that i s able to handle any s i t u a t i o n may be high, i t i s bound to pay for i t s e l f i n the long run. More input data i s required for an economic analysis, but t h i s data, for example, damage cost estimates, w i l l be s i m i l a r for many c u l v e r t s i t e s . I n i t i a l l y the cost of obtaining data may be high, but i t w i l l decrease as a data bank i s b u i l t up. This thesis has considered only s i m p l i f i e d , hypothetical cases, although several useful r e s u l t s were obtained. I f further research i s done, i t would be worthwhile to consider r e a l s ituations and to complicate the problem. The problems of debris clogging and estimating damage costs deserve more attention. Entrance improvement and d i f f e r e n t c u l v e r t materials and shapes should also be given consideration. x. LIST OF REFERENCES Nesbitt, M. C. (1963) Handbook of Culvert Hydraulics, Design, and I n s t a l l a t i o n , B.C. Department of Highways, Materials Testing, Design, and Planning Branch, V i c t o r i a , B.C. B l a i s d e l l , F. W. (1966) "Hydraulic E f f i c i e n c y i n Culvert Design Journal of the Highway D i v i s i o n , A.S.C.E., 92 (HW1): 11-22, Proc. Paper 4 7 09. P r i t c h e t t , Harold D. (1964) Application of the P r i n c i p l e s of Engineering Economy to the Selection of Highway Culverts, Master's thesis, Stanford University, Department of C i v i l Eng. Hershman, Stanley (1974) An Application of Decision Theory to Water Quality Management, Master's thesis, U.B.C, Department of C i v i l Eng. Nyumbu, Inyambo L. (1976) The E f f e c t of Uncertainty i n I r r i g a t i o n Development, Master's thesis, U.B.C, Department of C i v i l Eng. Brox, Gunter H. (1976) Water Quality i n the Lower Fraser River Basin: A Method to Estimate the E f f e c t of P o l l u t i o n on the Size of a Salmon Run, Master's thesis, U.B.C, Department of C i v i l Eng. Hetherington, E. D. (1974) The 25-Year Storm and Culvert Size, Federal Department of the Environment, Canadian Forestry Service P a c i f i c Forest Research Centre, V i c t o r i a , B.C., Report BC-X-102. Lins l e y , R. K., M. A. Kohler, and J . L. H. Paulhus (1958) Hydrology for Engineers, McGraw-Hill, New York. Quick, M. C and A. Pipes (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 of West Vancouver Drainage Survey, Dayton & Knight Ltd., Consulting Engineers, 1973. Carter, R. W. .(1957) Computation of Peak Discharge at Culverts, United States Geological Survey C i r c u l a r 376. Chow, V. T. (1959) Open-Channel Hydraulics, McGraw-Hill, New York. N e i l l , C R. (1962) Hydraulic Tests on Pipe Culverts, Research Council of Alberta, Alberta Highway Research Report 62-1. Oglesby, C. H. and L. I. Hewes (1963) Highway Engineering, John Wiley & Sons, New York. 70 71 15. James, L. D. and R. R. Lee (1971) Economics of Water Resources Planning, McGraw-Hill, New York. 16. Samuelson, P. A. and A. Scott (1975) Economics, McGraw-Hill Ryerson Limited, Toronto. 17. L i n s l e y , R. K. and J . B. Fra n z i n i (1972) Water-Resources Engineering, McGraw-Hill, New York. APPENDIX HEADWATER DEPTH CALCULATIONS - s e e R e f e r e n c e 11 f o r a d d i t i o n a l i n f o r m a t i o n T y p e 1 F l o w : C r i t i c a l D e p t h a t I n l e t HW = ( Q / c A c ) 2 / ( 2 g ) + d c - w^/{2q) + ' h w h e r e c i s a f u n c t i o n o f (HW/D) 2 v , / ( 2 g ) a n d h^ w e r e a s s u m e d n e g l i g i b l e . 1 1 1 . 2 T h e e q u a t i o n i s s o l v e d b y f i r s t c a l c u l a t i n g d c ( Q / A c = v ) •Type 5 F l o w : R a p i d F l o w a t I n l e t HW = ( Q / c A o ) 2 / ( 2 g ) w h e r e c i s a f u n c t i o n o f (HW/D) T y p e 6 F l o w : F u l l F l o w F r e e O u t f a l l 2 a s s u m i n g v . / ( 2 g ) a n d h - a r e n e g l i g i b l e 1 , f 1 . 2 h x = (Q/cAQ) V ( 2 g ) + h 3 + h f 2 > 3 w h e r e c i s a c o n s t a n t f o r a p a r t i c u l a r i n l e t c o n f i g u r a t i o n t h e n HW = h-j_ - S Q L H o w e v e r , h^ c a n n o t be e a s i l y d e t e r m i n e d . 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 o n e x p e r i m e n t , r a t h e r t h a n f r o m t h e a b o v e e q u a t i o n . I n a d d i t i o n t o D , ,Q, a n d c ; n , L , a n d s a r e r e q u i r e d t o c a l c u l a t e HW f o r t y p e 6 f l o w . N o t a t i o n S u b s c r i p t s 1 , 2 , 3 , a n d 4 d e n o t e l o c a t i o n o f s e c t i o n a s shown i n F i g u r e 4 . 1 . A = a r e a o f c u l v e r t b a r r e l o 72 73 Notation (cont.) A c area of flow at c r i t i c a l section c c o e f f i c i e n t of discharge D cu l v e r t diameter (min. dia. for CMP) d c c r i t i c a l depth h = piezometric head above culver t i n v e r t at downstream end h^ = head loss due to f r i c t i o n HW = depth of water i n headwater pool L = length of c u l v e r t b a r r e l n = Manning's roughness c o e f f i c i e n t Q = discharge s = bed slope of cu l v e r t b a r r e l o v = v e l o c i t y v = c r i t i c a l v e l o c i t y c 

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