Hydraulic design criteria for rockfill closure of tidal gaps - vertical closure method

Si--v.2t W e e - i n ' . • V r . t s r b o u w k u n d e Po.3tbs.js 5 0 ^ 4 , 2 5 0 0 G A D E L F T

2 3 S E P . 1991

Hydraulic design criteria for rockfill closure

of tidal gaps

vertical closure method

I . . 1' L - i^ t t B ^ W t e .

I i i

••,->'n-Evaluation report

M 1 7 4 1 a r t IV B I B L I O T H E E K ' D i e n s t W e g - e n W a l a . ' - b o u v / k u n d e P o s t b u s 5 0 4 4 , 2 S Ö 0 G A D E L F T J u l y 1 9 8 5

CONTENTS SUMMARY page 1 . I n t r o d u c t i o n and c o n c l u s i o n s . 1 1.1 Scope o f t h e s t u d y 1 1.2 The v e r t i c a l c l o s u r e method 2 1.3 C o n c l u s i o n s 3 2. L i t e r a t u r e r e v i e w and a v a i l a b l e d a t a . 8 2.1 Low dam f l o w s i t u a t i o n 8 2.2 I n t e r m e d i a t e f l o w s i t u a t i o n 10 2 .3 H i g h dam f l o w s i t u a t i o n 12 2.4 Through f l o w s i t u a t i o n 15 3 . A n a l y s i s 17 3.1 S t a b i l i t y a p p r o a c h 17 3.2 C h a r a c t e r i s t i c f l o w s i t u a t i o n s 18 3.3 Low dam f l o w s i t u a t i o n 18 3.4 I n t e r m e d i a t e f l o w s i t u a t i o n 23 3.5 H i g h dam f l o w s i t u a t i o n 27 3.6 Through f l o w s i t u a t i o n 30 3.7 M u l t i c r e s t e d dam ( l o w e r c r e s t s t a b i l i t y ) 3 1

3 .8 F a i l u r e mechanism and damage m a r g i n 33

3.9 A d d i t i o n a l wave a t t a c k 35 4. A p p l i c a t i o n 37 4.1 I n d i c a t i v e and d e t a i l e d d e s i g n a p p r o a c h 37 4.2 Case s t u d i e s o f r e c e n t f a i l u r e s 38 4.2.1 I n t r o d u c t i o n . . . 3 8 4.2.2 S t a b i l i t y a n a l y s i s o f t h e Hoge Bekken o v e r f l o w w e i r f a i l u r e 39 4.2.3 S t a b i l i t y a n a l y s i s o f t h e M a r k i e z a a t s k a d e f a i l u r e 41 5. R e l a t e d i t e m s . , 44 5.1 D i s c h a r g e c h a r a c t e r i s t i c s 44 5.2 B o t t o m p r o t e c t i o n s t o n e s i z e r e q u i r e m e n t s li 48 5.3 I n f l u e n c e o f t h e a d j a c e n t ends o f r o c k f i l l banks 52

CONTENTS ( c o n t i n u e d ) i I 6. Recommendations 56 ? REFERENCES TABLES 1 Review o f i n d i c a t i v e d e s i g n c r i t e r i a f o r t h r e s h o l d c o n d i t i o n 2 Comparison o f computed and measured t h r o u g h f l o w d i s c h a r g e 3 B o t t o m p r o t e c t i o n s t a b i l i t y d a t a [ 1 ] and [ 3 5 ]

FIGURES

FIGURES 1 G r a d u a l c l o s u r e methods 2 T y p i c a l dam t y p e s 3 V e l o c i t y p r o f i l e examples, l o w dam f l o w 4 T y p i c a l p a r a m e t e r d i s c r e p a n c i e s 5 O v e r t o p p i n g h e i g h t c r i t e r i o n 6 O v e r f l o w b r e a k w a t e r s t a b i l i t y , c o l l a p s e damage S t a b i l i t j _ j ) l o t s _ 7 . _ . _ . 1 7 7 O v e r a l l r e s u l t s r o c k f i l l c l o s u r e dams 8 Broad c r e s t e d submerged embankment 9 Sharp c r e s t e d submerged embankment 10 B r o a d / n a r r o w c r e s t e d c l o s u r e dam 11 Round c r e s t e d c l o s u r e dam 12 M u l t i c r e s t e d c l o s u r e dam 13 M u l t i c r e s t e d c l o s u r e dam 14 O v e r f l o w embankment 15 O v e r t o p p i n g h e i g h t , t h r o u g h f l o w dam 16 T o t a l d i s c h a r g e , t h r o u g h f l o w dam 17 C o n c r e t e b l o c k s c l o s u r e dam £!}Z5ical_verif i c a t i o n _ j ) l o t s _ 1 8 18 Comparison o f c r i t i c a l v e l o c i t i e s a t downstream c r e s t l i n e U q 19 D e t e r m i n a t i o n o f k* and 20 S t a b i l i t y p l o t - t o t a l d r o p - , o v e r a l l r e s u l t s 21 Enlargement f a c t o r c r e s t c u r r e n t v e l o c i t y y 22 S t a b i l i t y p l o t , comparison o f s t a b i l i t y r e l a t i o n s 23 U q r e f e r r e d t o h^/AD, l o g a r i t h m i c f i t 24 U q r e f e r r e d t o h^/AD, e x p o n e n t i a l f i t

25 H i g h dam f l o w , c o m p a r i s o n o f Knauss w i t h data from [ 1 ]

26 M a t h e m a t i c a l l y m o d e l l e d f l o w t h r o u g h porous dambody, t y p i c a l f l o w p a t t e r n 27 C r i t i c a l d i s c h a r g e h i g h dam f l o w i n v e s t i g a t i o n s

28 E l a b o r a t e d s t a b i l i t y r e s u l t s t h r o u g h f l o w dam

29 M u l t i c r e s t e d dam-lower c r e s t s t a b i l i t y - , check o f proposed a p p r o a c h 30 F a i l u r e mechanism and causes

FIGURES ( c o n t i n u e d ) 3 1 O v e r a l l d a t a damage m a r g i n 3 2 Damage m a r g i n , t y p i c a l r e s u l t s from [ 1 ] 3 3 E q u i v a l e n t o v e r t o p p i n g h e i g h t ( H ' ) v e r i f i c a t i o n f o r combined c u r r e n t a n d wave a t t a c k 3 4 I n f l u e n c e o f c r e s t w i d t h f o r a t r a p e z o i d a l dam 3 5 I n f l u e n c e o f p o r o s i t y D/d f o r a round c r e s t e d dam 3 6 Comparison o f a r o u g h and a smooth c r e s t e d b r o a d dam

3 7 Hoge Bekken o v e r f l o w w e i r 3 8 M a r k i e z a a t s k a d e c l o s u r e dam 3 9 S t a b i l i t y a n a l y s i s Hoge Bekken o v e r f l o w w e i r 4 0 S t a b i l i t y a n a l y s i s M a r k i e z a a t s k a d e c l o s u r e dam 4 1 D i s c h a r g e c o e f f i c i e n t s o f t h r e s h o l d c o n d i t i o n 4 2 C h a r a c t e r i s t i c f r e e f l o w d i s c h a r g e c o e f f i c i e n t , t r a p e z o i d a l broad c r e s t e d w e i r 4 3 D i s c h a r g e - h e a d d i f f e r e n c e r e l a t i o n t h r o u g h f l o w dam 4 4 Bottom p r o t e c t i o n s t a b i l i t y b e h i n d a dam o r a s i l l 4 5 Model d a t a on d i v i n g j e t t r a n s i t i o n ( n o r e f e r e n c e i s made t o s t a b i l i t y ) 4 6 Bottom p r o t e c t i o n s t a b i l i t y r e l a t i v e t o dam s t a b i l i t y 4 7 S t a b i l i t y g r a p h a d j a c e n t r o c k f i l l bank f a c e

SYMBOLS a A B B' c C d D e g h

K

H H' i k k' m ^50 q Q u Uo u gap c l o s u r e stage = d/h (-) t o t a l c r o s s - s e c t i o n a l a r e a i n t h e gap based on t o t a i l w a t e r d e p t h (m^) c r e s t w i d t h (m) c r e s t w i d t h a t u p s t r e a m w a t e r l e v e l i n t e r s e c t i o n ( H < 0) (m) b o t t o m p r o t e c t i o n s t a b i l i t y c o e f f i c i e n t (-) Chezy roughness c o e f f i c i e n t = 18 l o g ( 6 h/D) ( m j / s ) dam h e i g h t (m) n o m i n a l s t o n e d i a m e t e r = (M^„/p (m) 50 s v o i d r a t i o (-) g r a v i t a t i o n a l c o n s t a n t = 9.81 (m/s^) t a i l w a t e r d e p t h (m) t a i l w a t e r d e p t h r e f e r r e d t o dam c r e s t ( = h^j ) t a i l w a t e r d e p t h r e f e r r e d t o l o w e r dam c r e s t ( m u l t i c r e s t e d p r o f i l e ) downstream c r e s t l i n e d e p t h r e f e r r e d t o dam c r e s t e x i t w a t e r d e p t h ( F i g . 4 3 ) upstream w a t e r d e p t h r e f e r r e d t o dam c r e s t ( " o v e r t o p p i n g h e i g h t " ) u p s t r e a m w a t e r d e p t h = H+d w a t e r s u r f a c e g r a d i e n t ( u n i f o r m f l o w ) f l o w a d j u s t m e n t f a c t o r ( n o n - l o g a r i t h m i c f l o w ) p e r m e a b i l i t y c o e f f i c i e n t « 0 . 2 / g F f o r r o c k f i l l t r a n s f e r e n c e f a c t o r f r o m t h e l o c a l u t o u gap f r e e f l o w d i s c h a r g e c o e f f i c i e n t = q/(1.7 H'-*^) mean s t o n e w e i g h t exceeded by 50% by w e i g h t t o t a l u n i t d i s c h a r g e t o t a l d i s c h a r g e t h r o u g h gap v e r t i c a l l y averaged c u r r e n t v e l o c i t y v e r t i c a l l y averaged c u r r e n t v e l o c i t y a t downstream c r e s t l i n e mean c l o s u r e v e l o c i t y = Q/A

t o t a l head over t h r o u g h f l o w dam (H < 0) = d+H-h

(m) (m) (m) (m) (m) (m) (-) (-) (m/s) (-) (-) ( k g ) (m2/s) (m3/s) (m/s) (m/s) (m/s) (m)

SYMBOLS ( c o n t i n u e d ) a dovmstream s l o p e a n g l e (°) a u upstream s l o p e a n g l e _(°) s i d e s l o p e a n g l e _(°) Y f l o w e n l a r g e m e n t f a c t o r a t t h e downstream c r e s t l i n e _(-) A r e l a t i v e s t o n e d e n s i t y i n w a t e r = ( p ^ - p ) / p _(-) A' r e l a t i v e s t o n e d e n s i t y i n a i r ^ e n t r a i n e d w a t e r = ^/<^^ <-) e n a t u r a l s l o p e a n g l e (') d i s c h a r g e c o e f f i c i e n t = q/(h^/2g(H-h^)') (-) ^ 2 = u^//2g(H"h, ) 0 D (-) P d e n s i t y o f t h e w a t e r (kg/m3) P s s t o n e d e n s i t y (kg/m3) a c a i : i ^ e n t r a i n m e n t f a c t o r _(-) p a c k i n g f a c t o r f o r s t o n e arrangement (-) S h i e l d s s t a b i l i t y f a c t o r _(-)

Note: u n l e s s s t a t e d d i f f e r e n t l y , boundary c o n d i t i o n symbols r e f e r t o t h r e s h o l d damage c o n d i t i o n .

-2-1.2 The vertical closure method

Tidal basin closures have been carried out around the world for many centur-ies, mostly at a limited scale by trial and error execution and mainly based on experience. The increasing land reclamation, flood protection and fresh-water reservoir requirements in the beginning of this century stimulated, how-ever, enlargement in scale of the closure works. This has been possible be-cause of the improved insight into tidal hydrodynamics as well as by the de-. velopnent of large self-propelled equipnentde-.

During closure, the dumping of large stones, concrete cubes or other flow re-sistant elements into the gap, reduces its cross-section. At first, the total flow of water is basely reduced; consequently the flow velocity increases more or less proportionally with the decrease in cross-section, necessitating

larger units of material in the later stages of closure. Furthermore the sta-bility of the adjacent seabed, when it consists of erodab1e material, is en-dangered and there is a need for bottom protection to ensure a stable founda-tion to the closure dam.

After the dam crest emerges above the water surface, the core can be filled up with finer materials, e.g. sand or gravel, to reduce the permeability. Final-ly, covering layers are applied to seal the slopes of the dam and to provide protection against wave attack.

Some typical closure methods are representated schematically in Fig. 1. The vertical closure method, treated in this report, has some advantages compared with the horizontal method:

(i) Gap current ve10citles increase up to a maximtml value prior to the final closure in the free flow situation, and then reduce.

In contrast, when applying the horizontal closure method the velocities increase up to the final closure stage.

(ii) By controlling the cross-sectional area (see closure example given in Fig. 1), the scouring action can be minimized and controlled, whereas with the horizontal closure method a very extensive bottom protection is needed in the ultimate closure stage.

(iii) For large closure operations, a vertical closure using a cab1eway, bridge, helicopters or floating equipment, reduces the closing time considerably compared with horizontal, or combined horizontal/vertical

3

-methods. T h i s i s m a i n l y due t o t h e need f o r l e s s e x t e n s i v e b o t t o m pro-t e c pro-t i o n works and pro-t h e r e l a pro-t i v e l y f a s pro-t f i n a l c l o s u r e o p e r a pro-t i o n s ( h i g h c a p a c i t y , no bad w e a t h e r i n t e r r u p t i o n s ) .

The c o n c l u s i o n i s t h a t , f o r l a r g e c l o s u r e s , t h e v e r t i c a l method w i l l g e n e r a l l y be more f e a s i b l e t h a n t h e p u r e l y h o r i z o n t a l method. T y p i c a l dam p r o f i l e s , con-s i d e r e d i n t h e p r e con-s e n t i n v e con-s t i g a t i o n , a r e con-shown i n F i g . 2.

I n a d d i t i o n t o e x p e r i e n c e w i t h sudden c l o s u r e s , f o r example, c l o s u r e s w i t h s l u i c e c a i s s o n s w h i c h c a n be c l o s e d s i m u l t a n e o u s l y a t s l a c k t i d e , a l o t o f ex-p e r i e n c e has been g a i n e d i n a ex-p ex-p l y i n g t h e v e r t i c a l method i n v a r i o u s c l o s u r e schemes o f t h e D e l t a P r o j e c t i n t h e l a s t t w e n t y y e a r s . For t h e s e schemes many model i n v e s t i g a t i o n s and f i e l d e x p e r i m e n t s have been c a r r i e d o u t on new c l o

-s u r e p r o c e d u r e -s . The mo-st r e c e n t i n v e -s t i g a t i o n -s , connected w i t h p r e -s e n t and f o r t h c o m i n g c l o s u r e o p e r a t i o n s , a r e a l s o i n c o r p o r a t e d i n t o t h i s r e p o r t .

1.3 C o n c l u s i o n s

a. P a r a m e t e r d e f i n i t i o n s have been e l a b o r a t e d f o r t h e o u t l i n e d e s i g n o f r o c k -f i l l c l o s u r e dams. These p r o v i d e p r a c t i c a l s t a b i l i t y c r i t e r i a -f o r a l a r g e v a r i e t y o f dam t y p e s . F i g . 7.

The i n d e p e n d e n t p a r a m e t e r i s t h e t a i l w a t e r parameter, h.^/AD, i n w h i c h h. i s t h e t a i l w a t e r e l e v a t i o n r e l a t i v e t o t h e dam c r e s t ( i n s t e a d o f t h e

D

t a i l w a t e r d e p t h , com.m.only used up t o now) and AD i s t h e s t o n e s i z e para-m e t e r .

B a s i c a l l y , two s t a b i l i t y p a r a m e t e r s have been i n t r o d u c e d : t h e o v e r t o p p i n g h e i g h t p a r a m e t e r , H/AD

. t h e d i s c h a r g e p a r a m e t e r , q / ( g ° ( A D ) ^ • ^ ) , i n w h i c h q = t o t a l d i s c h a r g e .

b. Four t y p i c a l f l o w r e g i m e s have been d i s c u s s e d , dependent on ( s e e T a b l e 1 ) :

low dam f l o w (h_/AD > 4 ) : drowned f l o w , no i n f l u e n c e o f p o r o s i t y b

i n t e r m e d i a t e f l o w (-1 < h,/AD < 4 ) : f r e e f l o w , f l o w p e n e t r a t i o n i n t o b

t h e porous c r e s t

. h i g h dam f l o w (h,/AD < - 1 and H > 0 ) : submergence o f downstream c r e s t b

l i n e , r o u g h s h u t e f l o w on t h e i n n e r s l o p e

. t h r o u g h f l o w ( H < 0 ) : t h e f u l l d i s c h a r g e passes t h r o u g h t h e dam b o d y , o u t f l o w on t h e i n n e r s l o p e

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-c. I n t h e l o w dam f l o w s i t u a t i o n , a u n i f o r m f l o w a p p r o a c h , e.g. S h i e l d s ( 3 ) w i t h lj) = 0 . 0 4 f i t s t h e mean d a t a w e l l , so b o t h S h i e l d s and t h e mean d a t a c u r v e ( F i g . 7) can be t a k e n f o r o u t l i n e d e s i g n . Because o f t h e i n t e r r e l a -t i o n be-tween H and h^ i n drowned f l o w c o n d i -t i o n s -t h e use o f H/AD i s n o -t recommended. I n s t e a d , use s h o u l d be made o f ( H - h ^ ) / A D , w h i c h appears t o be more o r l e s s c o n s t a n t a t v a r y i n g h^/AD, F i g . 2 0 and T a b l e 1.

A f t e r t h e f r e e f l o w s i t u a t i o n i s r e a c h e d , r a i s i n g t h e dam w i l l l e a d t o an i n c r e a s i n g f l o w a t t a c k on t h e downstream p a r t o f t h e c r e s t and t h e i n n e r s l o p e , even though t h e d i s c h a r g e does n o t i n c r e a s e s i g n i f i c a n t l y . T h i s i s caused by f l o w p e n e t r a t i o n i n t o t h e porous c r e s t , c a u s i n g an i n c r e a s e o f t h e l o c a l c u r r e n t v e l o c i t y up t o v a l u e s f a r beyond t h e c r i t i c a l v e l o c i t y a t t h e i n s t a n t o f f r e e f l o w . T h i s phenomenon o f f l o w i n c r e a s e has been

s u b s t a n t i a t e d by measurements. S e c t i o n 3 . 4 .

Both t h e H and t h e q p a r a m e t e r s can be used f o r s t a b i l i t y p r e d i c t i o n s , F i g . 7; f r o m t h i s f i g u r e t h e d e c r e a s i n g s t a b i l i t y i n t h i s f l o w range ( 1 < h /AD < 4 ) as t h e dam i s r a i s e d ( e q u i v a l e n t t o l o w e r i n g t h e t a i l

-b

w a t e r e l e v a t i o n ) i s o b v i o u s .

The I z b a s h c r i t e r i o n ( 1 ) , f o r well-embedded stones ( f o r b r o a d c r e s t e d dams o n l y ) , i s h i g h l y d a n g e r o u s a t l o w t a i l w a t e r d e p t h s when t h e t h e o r e t i c a l 2 c r i t i c a l v e l o c i t y (/ j gH) i s s u b s t i t u t e d . I n c o n t r a s t , assuming t h a t t h e v e l o c i t y i s e q u a l t o t h e t h e o r e t i c a l d i s -charge (1.7 H^«5) d i v i d e d b y t h e a c t u a l t a i l w a t e r d e p t h ( h ^ ) , compensates t o some e x t e n t f o r t h e u n d e r e s t i m a t i o n o f t h e a c t u a l c u r r e n t v e l o c i t y and shows a r e a s o n a b l e f i t , p r o v i d e d t h a t a w a t e r d e p t h c o r r e c t i o n e q u a l t o D i s added t o a c c o u n t f o r t h e f l o w p e n e t r a t i o n . F i g . 2 2 , e q u a t i o n ( 1 8 ) . The same h o l d s f o r t r a n s f e r e n c e o f t h e I z b a s h c r i t e r i o n i n t o a d i s c h a r g e c r i -t e r i o n , when u . h^ i s i n s e r -t e d f o r q and -t h e same c o r r e c -t i o n w a -t e r d e p -t h i s a p p l i e d . F i g . 2 2 , e q u a t i o n ( 2 4 ) .

A f t e r t h e emergence o f t h e downstream c r e s t h e i g h t , t h e p o r o s i t y o f t h e r o c k f i l l dam i s s t i l l such t h a t t h e r e i s a p o s i t i v e o v e r t o p p i n g h e i g h t : h i g h dam f l o w s i t u a t i o n . I n t h i s s i t u a t i o n a c h a r a c t e r i s t i c c u r r e n t v e l o c -i t y a t t h e -i n n e r s l o p e c a n n o t be c l e a r l y d e f -i n e d , because o f t h e e x t r e m e l y r o u g h , a e r a t e d t y p e o f f l o w , comparable t o rough shute f l o w on r o c k f i l l s p i l l w a y s and upper r i v e r r e a c h e s .

5

-I The s t a b i l i t y o f t h e i n n e r s l o p e , w i t h a p o t e n t i a l damage r e g i o n near t h e J i n t e r s e c t i o n w i t h t h e t a i l w a t e r l e v e l , proved t o be d e s c r i b e d f a i r l y w e l l b y t h e Knauss r e l a t i o n s h i p ( 2 7 ) f o r s t e e p s h u t e f l o w , p r o v i d e d t h a t t h e t o t a l d i s c h a r g e ( o v e r Snd t h r o u g h t h e dam) i s t a k e n . F i g . 25. The c o r r e s -ponding s l o p e a n g l e i s i n t h e range 1:2 t o 1:3. At s t e e p e r s l o p e a n g l e s t h e Knauss r e l a t i o n s h i p seems t o be t o o c o n s e r v a t i v e , w h i l e f o r g e n t l e r s l o p e s a d e v i a t i n g i n d i c a t i v e d e s i g n c u r v e i s proposed as shown i n F i g . . ' 2 7 .

The q - c r i t e r i a m e n t i o n e d above can be t r a n s f e r r e d i n t o H - c r i t e r i a u s i n g q = m 1.7 H^*^ ( T a b l e 1 ) .

The assessment o f t h e d i s c h a r g e c h a r a c t e r i s t i c s i s i m p o r t a n t i n t h i s f l o w r e g i o n , because o f t h e d o m i n a n t p o r o s i t y i n f l u e n c e ( D / d ) ; a s i m p l e compu-t a compu-t i o n a l p r o c e d u r e , as o u compu-t l i n e d i n S e c compu-t i o n 5.1, d e v i a compu-t e s compu-t o o much f o r ' p r a c t i c a l use. I t i s e n v i s a g e d t h a t d i s c h a r g e measurements i n a s c a l e

model w i l l be needed f o r a t y p i c a l dam t y p e under d e s i g n . F i g . 41 can o n l y be used f o r c e r t a i n s e p c i f i c g e o m e t r i e s .

The t h r o u g h f l o w s i t u a t i o n w i l l n o r m a l l y be s t a b l e , i f t h e i n n e r s l o p e i s n o t t o o s t e e p , because o f t h e h i g h l y reduced d i s c h a r g e ( n o o v e r t o p p i n g ) . F o r a dam w i t h a v e r y s t e e p s l o p e , e.g. a t an a n g l e o f repose « 1:1.25, a s t a b i l i t y c r i t e r i o n has been o b t a i n e d f r o m t h e e x p e r i m e n t a l r e s u l t s o f P r a j a p a t i ( 2 9 ) , F i g . 2 8 . Note t h a t i n t h i s case t h e a c t u a l t a i l w a t e r d e p t h ( h a p p e a r s and n o t h, . C o n v e r s i o n o f t h i s i n t o a H - c r i t e r i o n l e a d s t o t h e

b

e x p r e s s i o n shown i n F i g . 15, see a l s o T a b l e 1. These c r i t e r i a a r e v a l i d f o r D/d = 0.02 t o 0.05, t h u s f o r dams o f r e l a t i v e l y f i n e m a t e r i a l s .

d. For d e t a i l e d d e s i g n two approaches a r e recommended: i" . F u r t h e r a n a l y s i s o f r e l e v a n t d a t a .

I . A d d i t i o n a l model t e s t s ( s t a b i l i t y and d i s c h a r g e c h a r a c t e r i s t i c s ) f o c u s ¬ ; sed on t h e most c r i t i c a l b u i l d i n g s t a g e s and f l o w s i t u a t i o n s .

A n a l y s i s o f r e l e v a n t d a t a may p r o v i d e u s e f u l a d d i t i o n a l i n f o r m a t i o n see f o r i n s t a n c e F i g s . 3 4 , 35 and 36 d e a l i n g w i t h t h e ( s e c o n d a r y ) i n f l u e n c e o f [ c r e s t w i d t h , p o r o s i t y and c r e s t roughness, r e s p e c t i v e l y . I I n cases o f d e v i a t i n g g e o m e t r i e s , o t h e r f l o w c i r c u m s t a n c e s , e t c . , model [ t e s t i n g w i l l be e s s e n t i a l f o r o p t i m i z a t i o n o f t h e d e s i g n . Because t h e ' i n d i c a t i v e d e s i g n a p p r o a c h . w i l l t r a c e t h e c r i t i c a l s i t u a t i o n s d u r i n g •

-6-c l o s u r e , su-6-ch i n v e s t i g a t i o n may be l i m i t e d t o t h e s e -6-c r i t i -6-c a l s i t u a t i o n s o n l y by w h i c h i t w i l l r e m a i n r e l a t i v e l y cheap.

e. No g e n e r a l boundary can be i n d i c a t e d between t h e above t h r e s h o l d c o n d i -t i o n s and e x -t e n s i v e o r f a i l u r e damage s i n c e -t h i s i s h i g h l y dependen-t o n t h e a c t u a l dam geometry ( c o n t r a r y t o t h e t h r e s h o l d c o n d i t i o n ) , see f o r i n s t a n c e F i g . 3 1 . Some f a i l u r e model t e s t s , i n c o r p o r a t e d i n t h e o p t i m i z a -t i o n and c h e c k i n g -t e s -t s , w i l l be i n d i s p e n s a b l e f o r -t h e f i n a l d e s i g n .

f . I n t h e case o f a m u l t i c r e s t e d dam l a y o u t t h e l o w e r c r e s t s t a b i l i t y can be r o u g h l y a p p r a i s e d i n a way analogous t o t h e s t a b i l i t y o f t h e h i g h e s t

c r e s t , p r o v i d e d t h a t t h e t a i l w a t e r e l e v a t i o n i s r e f e r r e d t o t h e l o w e r c r e s t h e i g h t under c o n s i d e r a t i o n and t h e o v e r t o p p i n g h e i g h t i s r e f e r r e d t o t h e h i g h e s t c r e s t . T h i s approach i s e l a b o r a t e d i n S e c t i o n 3.7 and F i g . 29.

g. A d d i t i o n a l wave a t t a c k ( o r d e r o f magnitude o f wave a m p l i t u d e s m a l l e r t h a n o v e r t o p p i n g h e i g h t ) can be t a k e n i n t o a c c o u n t by a d d i n g 1/3 o f t h e s i g n i -f i c a n t wave h e i g h t t o t h e o v e r t o p p i n g h e i g h t ; -f o r t h e s t a b i l i t y a n a l y s i s t h e r e s u l t i n g e q u i v a l e n t o v e r t o p p i n g h e i g h t (H') c a n be c o n s i d e r e d as t h e a c t u a l o v e r t o p p i n g h e i g h t , [ 2 0 ] and F i g s . 6 and 33. For c o n c r e t e b l o c k s 1/4 can be t a k e n i n s t e a d o f 1/3 [ 2 0 ] .

h . T h r e e - d i m e n s i o n a l e f f e c t s on t h e s t o n e s t a b i l i t y due t o t h e presence o f abutments, a r e expected t o be s m a l l when t h e c l o s u r e dam i s i n t h e i n t e r -m e d i a t e o r h i g h da-m f l o w s t a t e , S e c t i o n 5.3; a t l o w e r h e i g h t s o f t h e da-m these e f f e c t s w i l l i n c r e a s e b u t no q u a n t i t a t i v e i n f o r m a t i o n i s a v a i l a b l e . The s t a b i l i t y o f an a d j a c e n t r o c k f i l l bank f a c e i t s e l f , can be assessed u s i n g F i g . 4 7 , based on t h e e x p e r i m e n t a l d a t a o f N a y l o r [ 3 0 ] .

i . The s u r f a c e l a y e r s t a b i l i t y o f t h e b o t t o m p r o t e c t i o n i s r e v i e w e d i n Sect i o n 5.2 i n w h i c h a s i m p l e a p p r o a c h i s p r o p o s e d , based on an e x p e r i m e n -t a l l y d e -t e r m i n e d d i s -t u r b a n c e parame-ter R,, F i g . 44. I n -t h i s a p p r o a c h , -t h e i n f l u e n c e o f dam geometry i s p r a c t i c a l l y r u l e d o u t . I n a d d i t i o n , a compar-i s o n compar-i s made o f t h e s t a b compar-i l compar-i t y o f t h e b o t t o m p r o t e c t compar-i o n r e l a t compar-i v e t o t h e s t a b i l i t y o f t h e dam ( c r e s t ) , p r o v i d e d t h a t t h e same s t o n e s i z e i s used ( F i g . 4 6 ) .

A case s t u d y , i l l u s t r a t i n g t h e a p p l i c a b i l i t y o f t h e p r e s e n t d e s i g n c r i t e -r i a , i s p -r e s e n t e d i n S e c t i o n 4.2, w h i c h -r e f e -r s t o two p -r o t o t y p e f a i l u -r e s . F i g s . 37 t o 40.

Recommendation o f s u b j e c t s f o r f u r t h e r s t u d y a r e reviewed i n Chapter 6, two a r e e s p e c i a l l y emphasized h e r e :

E x t e n s i o n t o c l o s u r e s w i t h t r a n s p o r t a b l e m a t e r i a l ("dynamic s t a b i l i t y c l o s u r e s " ) .

S y s t e m a t i c i n v e s t i g a t i o n i n t o two- and t h r e e - d i m e n s i o n a l d i s c h a r g e c h a r a c t e r i s t i c s o f c l o s u r e dams.

8

-2. L i t e r a t u r e r e v i e w and a v a i l a b l e d a t a

An e x t e n s i v e l i t e r a t u r e c o m p i l a t i o n on v e r t i c a l c l o s u r e stone s t a b i l i t y has been r e p o r t e d i n a r e c e n t DHL i n v e s t i g a t i o n , M 1741 P a r t I [ 3 ] , i n w h i c h much a t t e n t i o n has been paid on t h e " i n t e r m e d i a t e f l o w " and " h i g h dam f l o w " s i t u a -t i o n . An a n a l y s i s o f o n g o i n g e x p e r i m e n -t s was n o -t i n c l u d e d i n -t h i s r e p o r -t and, c o n s e q u e n t l y , a comprehensive p i c t u r e was n o t o b t a i n e d .

An o u t l i n e o f r e l e v a n t l i t e r a t u r e g i v e n i n [ 3 ] , and newly t r a c e d l i t e r a t u r e i s r e v i e w e d below. The l i t e r a t u r e i s d i v i d e d i n t o t h e r e l e v a n t f l o w s i t u a t i o n s ( s e e a l s o l i s t o f " D e f i n i t i o n s " ) d i s c u s s e d i n Chapter 3.

2.1 Low dam f l o w s i t u a t i o n (h,/AD > 4 ) _^

I z b a s h [ 4 ] gave a v e r y s i m p l e e m p i r i c a l r e l a t i o n s h i p f o r t h e c r i t i c a l c u r r e n t v e l o c i t y f o r t h r e s h o l d damage: u//gAD = 1 . 7 ( 1 ) f o r a w e l l embedded stone and u//gAF = 1.2 ( 2 ) f o r an i s o l a t e d stone on t o p o f a dam.

A l t h o u g h t h e roughness i n f l u e n c e f o r d i f f e r e n t w a t e r depths i s i g n o r e d , these e x p r e s s i o n s have been used w i d e l y . No r e f e r e n c e i s made t o t h e a c t u a l f l o w s i -t u a -t i o n (drowned o r f r e e f l o w ) .

For u n i f o r m f l o w c o n d i t i o n s DHL has e s t a b l i s h e d s e v e r a l v i s u a l l y d e t e r m i n e d i n s t a b i l i t y l e v e l s i n I n v e s t i g a t i o n M 648/863 [ 5 ] . Based on t h e w e l l - k n o w n S h i e l d s e x p r e s s i o n seven c r i t e r i a were f o u n d , i n f a c t , f o r i j ; .

9

-i n w h -i c h

C = 18 l o g (6h/D) (^•niite-Colebrook f o r m u l a ) ^ = S h i e l d s parameter

I n case o f a dam, u and h r e f e r t o t h e downstream c r e s t l i n e .

I t was observed t h a t even a t v a l u e s o f ^ c o n s i d e r a b l y l o w e r t h a n t h e a c t u a l S h i e l d s v a l u e o f 0.057 some t r a n s p o r t o f m a t e r i a l t o o k p l a c e . These observa-t i o n s were c o n f i r m e d q u a n observa-t i observa-t a observa-t i v e l y by e x observa-t e n s i v e observa-t r a n s p o r observa-t measuremenobserva-ts by • P a i n t a l [ 6 ] .

A c c o r d i n g t o P a i n t a l , f o r a z e r o t r a n s p o r t , t h e ^ v a l u e has t o go down t o

about 0.02! A r e l a t i v e l y h i g h e r t\) can be s e l e c t e d f o r dam s t a b i l i t y , f o r i n -s t a n c e i n t h e range o f 0.03-0.04, becau-se o f r e - -s t a b i l i z i n g t e n d e n c i e -s a f t e r a c l o s u r e dam has been damaged by t h e r e m o v a l o f some s t o n e s . To a c c o u n t f o r a bed s l o p e o f a i n t h e f l o w d i r e c t i o n , f r o m s i m p l e s t a b i l i t y a n a l y s i s i t can be f o u n d t h a t t h e c r i t i c a l v e l o c i t y has t o be r e d u c e d by a f a c t o r /sin(<{i-a)/sin(j) ( 4 ) and, a c c o r d i n g l y , f o r a s i d e s l o p e o f 3, by a f a c t o r B o t h e x p r e s s i o n s r e f e r t o u n i f o r m f l o w w i t h 0 = n a t u r a l a n g l e o f repose o f t h e m a t e r i a l (" 40° f o r r o c k f i l l s t o n e s ) . At t h e c r e s t o f o v e r f l o w dams t h e c u r r e n t p a t t e r n i s no l o n g e r u n i f o r m l y d i s -t r i b u -t e d , as i n d i c a -t e d i n F i g . 3 [ 7 ] , and -t h e r e f o r e -t h e above u n i f o r m f l o w exp r e s s i o n s a r e n o t a exp exp l i c a b l e u n c o n d i t i o n a l l y . DHL, t h e r e f o r e , experformed i n v e s -t i g a -t i o n s i n -t o -t h e s -t a b i l i -t y o f a w i n -t e r s i l l ( c l o s u r e dam under submerged f l o w c o n d i t i o n s ) . From M 711-11 [ 8 ] , f o r a b r o a d - c r e s t e d dam w i t h a r e l a t i v e c r e s t w i d t h o f B/d > 5, t h e f o l l o w i n g f o r m u l a was f o u n d by c u r v e f i t t i n g f o r t h e submerged f l o w c o n d i t i o n . F i g . 4: ( 5 ) u^//gAD'= 1.4 l o g ( 3 . 5 h^/D) ( 6 )

-10-and f r o m M 7 1 1 - I I I [ 9 ] f o r a s h a r p - c r e s t e d dam:

u /•gAD'= 1.4 l o g ( 1 . 5 h /D) ( 7 )

o- ° o

Here and h ^ r e f e r t o t h e dovmstream c r e s t l i n e . The c o e f f i c i e n t s 3.5 and 1.5 i n d i c a t e a more o r l e s s undeveloped boundary l a y e r .

I n ( 6 ) and ( 7 ) , h/D i s an a d d i t i o n a l p a r a m e t e r , as i n ( 3 ) , compared t o t h e . I z b a s h f o r m u l a e ( 1 ) and ( 2 ) . For h/D a p p r o x i m a t e l y = 5 , ( 6 ) and ( 7 ) a r e e q u i -v a l e n t t o ( 1 ) and ( 2 ) r e s p e c t i -v e l y .

E x p r e s s i o n ( 6 ) a l s o a g r e e s w i t h ( 3 ) when i|) = 0.04 t o 0.05 o v e r t h e a r e a under i n v e s t i g a t i o n (h/D = 5 t o 2 2 ) , whereas ( 7 ) c o r r e s p o n d s t o a i|; v a l u e o f 0.02 t o 0.03 (h/D = 9 t o 2 9 ) . I n p r a c t i c e , t h e r e f o r e , a b r o a d c r e s t e d submerged o v e r -f l o w dam can s a -f e l y be d e s i g n e d w i t h t h e u n i -f o r m -f l o w s t a b i l i t y a p p r o a c h when f o r i|j 0.04 i s t a k e n a g a i n s t 0.02 f o r t h e s h a r p - c r e s t e d dam a t s u b c r i t i c a l f l o w . A p r a c t i c a l p r o b l e m , however, i s t h e d e t e r m i n a t i o n o f t h e a c t u a l w a t e r d e p t h a t t h e downstream c r e s t l i n e h^. More c o n v e n i e n t i s t o t a k e t h e t a i l -w a t e r d e p t h r e l a t i v e t o t h e dam c r e s t h^.

I n t h e l o w dam f l o w s i t u a t i o n t h e d i f f e r e n c e between h^ and h ^ i s s m a l l and h ^ may be t a k e n as w e l l f o r i n d i c a t i v e c o m p u t a t i o n s . I n a d d i t i o n , a b e t t e r

a p p r o a c h i s t o d e t e r m i n e t h e c r i t i c a l d i s c h a r g e q, because t h e n d i f f e r e n c e s between h^ and h^ a r e e l i m i n a t e d t o some e x t e n t ( S e c t i o n 3.3).

When a c l o s u r e dam i s r a i s e d f u r t h e r , t h e f l o w r e g i m e w i l l become s u p e r c r i t i -c a l ( f r e e f l o w -c o n d i t i o n ) . T h i s means t h a t any subsequent l o w e r i n g o f t h e t a i l w a t e r l e v e l , r e l a t i v e t o t h e dam c r e s t , ( i n f a c t t h e c r e s t i s r a i s e d ) , may be c o n s i d e r e d as n o t a f f e c t i n g t h e d i s c h a r g e over t h e c r e s t . L o c a l l y , however, a t t h e downstream edge o f t h e c r e s t , t h e a c t u a l v e l o c i t y i s s t i l l i n c r e a s i n g w i t h t h e l o w e r i n g o f t h e t a i l w a t e r l e v e l because o f s t r e a m l i n e c u r v a t u r e and

f l o w p e n e t r a t i o n i n t o t h e permeable c r e s t , see F i g . 2 1 . I n a d d i t i o n t h e d i s -c h a r g e t h r o u g h t h e permeable dam body i s a l s o i n -c r e a s i n g .

A l t h o u g h t h e c u r r e n t v e l o c i t y e x e r t s t h e a c t u a l d e s t a b i l i z i n g f o r c e , i t i s d i f f i c u l t t o c h a r a c t e r i z e t h i s v e l o c i t y and t h e c o r r e s p o n d i n g l o c a l w a t e r d e p t h i n t h e f r e e o v e r f l o w s i t u a t i o n .

1 1

T h i s e x p l a i n s why w a t e r l e v e l parameters based on upstream and t a i l w a t e r e l e -v a t i o n s , a l t h o u g h more i n d i r e c t , a r e more f e a s i b l e .

The DHL i n v e s t i g a t i o n w i t h a n a r r o w c r e s t e d dam, M 731 P a r t I I [ 1 0 ] , conse-q u e n t l y r e l a t e s t o t h e d e t e r m i n a t i o n o f t h e c r i t i c a l w a t e r e l e v a t i o n s w i t h r e s p e c t t o t h e dam c r e s t .

A t y p i c a l r e s u l t i s shown i n F i g . 4. The b r o k e n l i n e s , caused by t h e mixed-t y p e w a mixed-t e r l e v e l p a r a m e mixed-t e r , make g e n e r a l use d o u b mixed-t f u l .

P r i o r t o t h e p r e s e n t s t u d y , DHL p e r f o r m e d an i n v e s t i g a t i o n i n t o t h e s t a b i l i t y o f t h e M a r k i e z a a t s k a d e , a s e c o n d a r y c l o s u r e dam o f t h e D e l t a P r o j e c t :

M 1741, P a r t I I [ 1 ] . Two dam t y p e s , one w i t h a b r o a d c r e s t and one w i t h a nai^-row c r e s t , were i n v e s t i g a t e d b o t h i n t h e i n t e r m e d i a t e and h i g h dam f l o w r a n g e .

Much e f f o r t has been p u t i n s e l e c t i n g t h e most a p p r o p r i a t e s t a b i l i t y para-m e t e r s . Because t h e dapara-m t y p e i n f l u e n c e was weak, o n l y t h r e e parapara-meters were

found t o g o v e r n t h e s t a b i l i t y , see F i g . 5:

( i ) Upstream w a t e r l e v e l r e f e r r e d t o t h e s t o n e d i m e n s i o n s : H/AD ( i i ) Downstream w a t e r l e v e l r e f e r r e d t o t h e dam h e i g h t : h/d

( i i i ) Stone d i a m e t e r r e f e r r e d t o t h e dam h e i g h t ( p e r m e a b i l i t y p a r a m e t e r ) : D/d These f i n d i n g s have been compared w i t h t h e d a t a processed from t h e e a r l i e r DHL i n v e s t i g a t i o n , M 731 P a r t I I , w h i c h i n d i c a t e d t h e same t e n d e n c i e s , F i g . 5.

A v e r y u s e f u l , l a r g e s c a l e i n v e s t i g a t i o n was performed by Brogdon and Grace [11] i n t o w i d e c r e s t e d o v e r f l o w r o c k f i l l embankments i n r i v e r s , w i t h and w i t h -o u t an access r -o a d . U n f -o r t u n a t e l y t h e y d i d n -o t succeed i n -o b t a i n i n g a f e a s i b l e d i m e n s i o n l e s s r e p r e s e n t a t i o n . T h e i r d a t a , a f t e r a d a p t i o n t o t h e parameter

c h o i c e above, a l s o proved t o f i t t h e o v e r a l l p i c t u r e f a i r l y w e l l , F i g . 5.

DHL has a l s o i n v e s t i g a t e d t h e s t a b i l i t y o f c o n c r e t e b l o c k s , M 731 P a r t X [ 1 2 ] . The c o n c r e t e b l o c k s used f o r t h e s e t e s t s had an e l l i p t i c a l c y l i n d e r shape and a normal cube shape. By i n t r o d u c i n g a n o m i n a l d i a m e t e r D f o r t h e s t o n e s t a b i l -i t y f o r a l l c u r r e n t - r e s -i s t a n t e l e m e n t s ( = M ^q / p^ ) , a dominant shape i n f l u -ence on t h e s t a b i l i t y b e h a v i o u r was more o r l e s s e l i m i n a t e d . The t e s t r e s u l t s f o r t h e c o n c r e t e b l o c k s c l o s u r e dam a r e , t h e r e f o r e , i n some degree comparable t o t h e a c t u a l r o c k f i l l d a t a , see Chapter 3.

1 2

-2.3 H i g h dam f l o w s i t u a t i o n ( h ^ / A D < - 1 and H > 0 )

The i n v e s t i g a t i o n s m e n t i o n e d above, D H L i n v e s t i g a t i o n s M 7 3 1 P a r t I I , M 1 7 4 1 P a r t I I and Brogdon and Grace's r e s u l t s , a l s o a p p l y f o r t h e h i g h dam f l o w s i

-t u a -t i o n , w i -t h a downs-tream w a -t e r l e v e l c o n s i d e r a b l y l o w e r -t h a n -t h e dam c r e s -t .

A n o t h e r c l o s u r e dam r e s e a r c h p r o j e c t s h o u l d a l s o be m e n t i o n e d , c a r r i e d o u t by Meermans [ 1 3 ] . He i n v e s t i g a t e d t h e s t a b i l i t y b e h a v i o u r o f a s h a r p - c r e s t e d dam. The r e s u l t s , e x p r e s s e d i n t h e parameters H/AD, h/d and D/d, have a l s o been p l o t t e d i n F i g . 5 , and show even n e g a t i v e c r i t i c a l upstream w a t e r l e v e l s a t l o w downstream w a t e r d e p t h s , A l o t o f complementary i n f o r m a t i o n i s a v a i l a b l e f o r t h i s f l o w r e g i o n f r o m i n -v e s t i g a t i o n s i n t o t h e s t a b i l i t y o f r o c k f i l l on s p i l l w a y s , and u p p e r r i -v e r reaches ( s t e e p s h u t e f l o w ) . Other i n v e s t i g a t i o n s r e l a t e t o t h e (downstream) s l o p e p r o t e c t i o n f o r f r e e f l o w c o n d i t i o n s and a r e r e l a t e d t o t h e c r i t i c a l u n i t d i s c h a r g e . The a c t u a l f l o w i n t h i s s i t u a t i o n cannot be c h a r a c t e r i z e d by t h e u n i f o r m f l o w a p p r o a c h because o f the extreme i n f l u e n c e o f r o u g h n e s s , i n c l u d i n g a e r a t i o n e f f e c t s . For t h i s r e a son t h e d e f i n i t i o n o f a c r i t i c a l c u r r e n t v e l o c i t y i s n o t p r a c t i c a l i n t h i s s i -t u a -t i o n , see S e c -t i o n 3 . 5 .

L i n f o r d and Saunders [ 1 4 ] i n v e s t i g a t e d o v e r f l o w r o c k f i l l dams w i t h an imper-v i o u s s e a l i n g a t t h e u p s t r e a m s l o p e and a t t h e c r e s t . Much a t t e n t i o n was p a i d

t o v a r y i n g t h e stone a r r a n g e m e n t , c h a r a c t e r i z e d by t h e i n t r o d u c t i o n o f a " p a c k i n g f a c t o r " . A c l o s e p a c k i n g o f m a n u a l l y edgeplaced s t o n e s , f o r i n

-s t a n c e , proved t o be a b l e t o w i t h -s t a n d a d i -s c h a r g e o f more t h a n t h r e e t i m e -s the d i s c h a r g e o f a f l a t p l a c e d arrangement ( t h e l a t t e r b e i n g even l e s s r e s i s -t a n -t -t h a n a n a -t u r a l dumped l a y e r ) .

edge placed f l a t placed

I n a d d i t i o n t h e a u t h o r s f o u n d a 3 5 % r e d u c t i o n o f t h e c r i t i c a l d i s c h a r g e f o r rounded g r a v e l compared t o r o c k f i l l ( a f t e r a d a p t i o n t o t h e n o m i n a l d i a m e t e r D ) .

-13-The i n v e s t i g a t i o n o f H a r t u n g and S c h e u e r l e i n [ 1 5 ] r e f e r r i n g t o t h e h y d r a u l i c s o f s t e e p and r o u g h open c h a n n e l f l o w , s h o u l d a l s o be m e n t i o n e d .

The i n v e s t i g a t i o n s [ 1 4 ] and [ 1 5 ] have been a n a l y s e d by Knauss [ 1 6 ] . Knauss proposed a s i m p l i f i e d r e l a t i o n s h i p f o r t h e c r i t i c a l u n i t d i s c h a r g e , m a i n l y based on t h e more p r a c t i c a l r e s u l t s o f H a r t u n g and S c h e u e r l e i n :

q = 0.84 f~G^ (1.9 + 0.8<)) - 3 s i n a ) s i n w h i c h : q = c r i t i c a l s p e c i f i c d i s c h a r g e (m^/s) G = average stone w e i g h t (kN!) s (J> = p a c k i n g f a c t o r r a n g i n g f r o m =i 0.6 f o r " n a t u r a l p a c k i n g " t o - 1.1 f o r "manual p a c k i n g " a = a n g l e o f t h e downstream s l o p e A f t e r i n s e r t i n g D as t h e n o m i n a l d i a m e t e r and i n t r o d u c i n g A as e q u i v a l e n t t o D (A = 1.7 f o r the above Knauss e x p r e s s i o n ) :

q = 1.95 (AD)^'5 (1.9+0.8 <j) - 3 s i n a ) ( 8 )

I n [ 1 6 ] Knauss assumes t h a t n a t u r a l p a c k i n g can be adopted f o r a dumped pack-i n g . T h pack-i s a s s u m p t pack-i o n seems n o t t o be v e r y w e l l f o u n d e d , however, as pack-i n a n o t h e r p u b l i c a t i o n o f Knauss [ 1 7 ] a mean v a l u e o f (j) = 0.5 was t a k e n f o r n o r m a l l y p i t c h e d stone r e v e t m e n t s . T h i s must be k e p t i n m i n d , t h e r e f o r e , e s p e c i a l l y because o f t h e u n s a f e a p p r o a c h .

A b a s i c element i n t h i s r e l a t i o n s h i p i s t h e r e l a t i v e i n c r e a s e o f A by a e r a t e d f l o w , which o c c u r s on s l o p e s s t e e p e r t h a n about 1:10, by w h i c h the f l o w r e -s i -s t a n c e i -s i n c r e a -s e d . I t -s h o u l d be n o t e d t h a t t h e t a i l w a t e r d e p t h i -s l e f t o u t o f t h i s p i c t u r e , s i n c e o n l y e q u i l i b r i u m f l o w a t t h e downstream slope i s con-s i d e r e d .

Lysne and T v i n n e r e i m [ 1 8 ] i n v e s t i g a t e d f l o w o v e r an i m p e r v i o u s w e i r s i l l a t f u l l s c a l e w i t h a r e l a t i v e l y s h o r t downstream s l o p e . They observed t h a t t h e c u r r e n t a t t a c k was maximum b e l o w t h e p o i n t where f u l l y developed f l o w o c c u r r -ed, a t some d i s t a n c e b e l o w t h e c r e s t . P r o v i d e d t h a t t h e t a i l w a t e r d e p t h was below t h i s zone, no i n f l u e n c e on t h e s t o n e s t a b i l i t y was o b s e r v e d . The s t a b i l i t y was found t o f i t t h e u n i f o r m f l o w s t a b i l i t y f a i r l y w e l l on g e n t l e

-14-dovmstream s l o p e s , v a r y i n g from 1:6 t o 1:12. T h i s may m a i n l y be due t o t h e i n s i g n i f i c a n t a e r a t i o n because o f t h e r e l a t i v e l y g e n t l e s i l l s l o p e a n g l e s . T h i s i m p l i e s b e t t e r c u r r e n t v e l o c i t y and water d e p t h measurement

p o s s i b i l i t i e s , i n c o n t r a s t t o s t e e p s l o p e s .

A d d i t i o n a l i n f o r m a t i o n on r e l a t i v e l y g e n t l e s l o p e rough s h u t e h y d r a u l i c s , comes f r o m l o w head r i v e r c o n t r o l s t r u c t u r e s w h i c h c o n s i s t s o f e n e r g y d e s t r o y s i n g , l o c a l l y r o u g h , s l o p e s . These are w e l l - k n o w n i n Germany and a r e c a l l e d " B l o c k s t e i n r a m p e n " . There have been many model and f i e l d i n v e s t i g a t i o n s on

r o u g h s l o p e s o f 1:8 and l e s s over t h e past t w e n t y y e a r s . I t s h o u l d be n o t e d , however, t h a t t h e s e i n v e s t i g a t i o n s were g e n e r a l l y r e s t r i c t e d t o a f a l l o f a few m e t r e s and c o n s e q u e n t l y , f u l l y developed f l o w d i d not o c c u r , and t h e r e f o r e maximum c u r r e n t a t t a c k was n o t e n c o u n t e r e d . I n a l l cases manual p a c k i n g was

i n v o l v e d , r a n g i n g from <j) = 0.5 (easy s t o n e p i t c h i n g ) up t o (j) = 1.0 ( s p e c i a l dense p i t c h i n g ) [ 1 7 ] ,

P l a t z e r [ 1 9 ] d e a l s w i t h many a s p e c t s i n t h e d e s i g n o f these r e l a t i v e l y g e n t l e s l o p e s , e.g. s t o n e s t a b i l i t y , jtimp/backwater p e r f o r m a n c e , w a t e r l e v e l u n d u l a -t i o n s downs-tream o f -t h e jxomp, energy l o s s e f f i c i e n c y and s c o u r h o l e a c -t i o n downstream. The s t o n e s t a b i l i t y t e s t s , however, were v e r y r a r e compared t o t h e i n v e s t i g a t i o n s d e a l t w i t h by Knauss [16 and 1 7 ] ,

For t h i s t y p e o f s t a b i l i t y Knauss [ 1 7 ] proposes t h e f o l l o w i n g c r i t i c a l u n i t d i s c h a r g e ( f o r i n i t i a t i o n o f m o t i o n )

..^..'•'[ui.^.io.eis-^,,] ( 9 ,

i n w h i c h : a = a n g l e o f t h e s l o p e <j) = p a c k i n g f a c t o r , r a n g i n g from 0.5 ( e a s y s t one p i t c h i n g ) t o 1.0 ( s p e c i a l dense p i t c h i n g ) T h i s r e l a t i o n s h i p i s r e s t r i c t e d t o s l o p e s o f 1:8 t o 1:15. Because a i r e n t r a l n -ment can be n e g l e c t e d f o r s l o p e s g e n t l e r t h a n about 1:10, t h i s r e l a t i o n s h i p can be compared w i t h Lysne and T v i n n e r e i m ' s r e s u l t s , and a l s o w i t h a p a r t o f t h e i n v e s t i g a t i o n o f L i n f o r d and Saunders, see S e c t i o n 3.5.

A s p e c i a l t y p e o f h i g h dam f l o w was i n v e s t i g a t e d r e c e n t l y by DHL, M 1631 P a r t I [ 2 0 ] r e l a t i n g t o o v e r t o p p i n g f l o w and waves i n s t o r m s u r g e c o n d i t i o n s o f t h e

-15-c o n n e -15-c t i n g b r e a k w a t e r dams on b o t h s i d e s o f t h e O o s t e r s -15-c h e l d e Storm Surge B a r r i e r s e c t i o n s . The u s u a l b r e a k w a t e r a p p r o a c h d i d n o t a p p l y , however, because o f t h e h i g h head d r o p a c r o s s t h e v e r y permeable dam s e c t i o n s when t h e storm s u r g e b a r r i e r was i n t h e c l o s e d mode.

The model r e s u l t s i n d i c a t e d t h a t , i n t h e case o f dominant o v e r f l o w a c t i o n , r e l a t i v e t o t h e wave a c t i o n , t h e a d d i t i o n a l wave a c t i o n can be a c c o u n t e d f o r by a d d i n g an e x t r a u p s t r e a m head t o t h e upstream s t i l l w a t e r l e v e l o f some 1/3 ( r o c k f i l l ) o r 1/4 ( c o n c r e t e b l o c k s ) o f t h e s i g n i f i c a n t wave h e i g h t ( d e n o t e d . b y Hg = " e q u i v a l e n t o v e r t o p p i n g h e i g h t " , see F i g . 6 ) . The stone and c o n c r e t e cube d i m e n s i o n s were v a r i e d on a l a r g e r a n g e . The c o l l a p s e o v e r f l o w h e i g h t ( i n c l u d -i n g wave a c t -i o n ) p r o v e d t o be r e l a t e d t o t h e stone o r b l o c k d -i m e n s -i o n s (AD) t o t h e power 5/6, a l t h o u g h t h e s c a t t e r was f o u n d t o be s u b s t a n t i a l . F i g . 6.

F i n a l l y i t s h o u l d be n o t e d t h a t i n p r a c t i c e no o r p r a c t i c a l l y no o v e r t o p p i n g w i l l be a l l o w e d w i t h o v e r f l o w dams w h i c h a r e r e l a t i v e l y i m p e r v i o u s . A m i n o r o v e r t o p p i n g can l e a d , i f n o t e s p e c i a l l y a c c o u n t e d f o r , t o a t o t a l c o l l a p s e o f t h e dam. T h i s happened w i t h a c l o s u r e dam i n South A f r i c a as d e s c r i b e d by Odendaal and van Z l j l [ 2 1 ] . The p r e v e n t i o n o f o v e r t o p p i n g has a l s o been s t r e s s e d by S a r k a r i a and Dworsky [ 2 2 ] who i n v e s t i g a t e d t h e wire-mesh s c r e e n r e i n f o r c e m e n t o f t h e downstream s l o p e f o r t h e b a r r a g e t y p e dam as p r o t e c t i o n a g a i n s t o v e r t o p p i n g . C o n s e q u e n t l y t h e r e i s no o v e r t o p p i n g h e i g h t c r i t e r i o n f o r i m p e r v i o u s dams and a d i s c h a r g e c r i t e r i o n must be used i n d e p e n d e n t l y o f t h e t a i l w a t e r d e p t h ; t h e r e s u l t s o f L i n f o r d and Saunders, Lysne and T v i n n e r e i m , and Rnauss, t h e r e f o r e , have been a n a l y s e d s e p a r a t e l y i n S e c t i o n 3.5.

2.4 Through f l o w s i t u a t i o n (H<0)

A f t e r c o m p l e t i o n o f t h e c l o s u r e o f a r o c k f i l l dam, and p r i o r t o t h e d e f i n i t e s e a l i n g and f i l l i n g o f s l o p e s and c o r e , t h e r e i s a t h r o u g h f l o w s i t u a t i o n . T h i s s i t u a t i o n i s c h a r a c t e r i z e d by a n e g a t i v e o v e r t o p p i n g h e i g h t , i . e . t h e upstream w a t e r l e v e l i s b e l o w t h e dam c r e s t . Note t h a t d u r i n g t h e f i n a l h i g h dam f l o w s t a g e t h e downstream c r e s t l i n e i s a l r e a d y r u n n i n g d r y , a l t h o u g h t h e r e i s s t i l l a p o s i t i v e o v e r t o p p i n g h e i g h t . I n p r a c t i c e t h e t h r o u g h f l o w s t a g e i s n o t , n o r m a l l y , a c r i t i c a l s t a g e f o r stone s t a b i l i t y as g e n e r a l l y t h e maximum s t o n e d i m e n s i o n s a r e a p p l i e d i n t h i s u l t i m a t e s t a g e . No s p e c i a l i n v e s t i g a t i o n has t h e r e f o r e been i n i t i a t e d t o d a t e by DHL.

-16-More i n f o m a t i o n can be o b t a i n e d from t h r o u g h f l o w r o c k f i l l b a r r a g e s , i n w h i c h the d i s c h a r g e i s passed t h r o u g h t h e p e r v i o u s c o r e , t h u s a v o i d i n g t h e need f o r s p i l l w a y s . Time was n o t a v a i l a b l e f o r an e x t e n s i v e s t u d y on t h i s f l o w s i t u a -t i o n , so r e f e r e n c e s i n -t h i s e x -t e n s i v e f i e l d o f i n v e s -t i g a -t i o n , e.g. v a r i o u s i n v e s t i g a t i o n s by J.K. W i l k i n s and by A.K. P a r k i n s , have n o t been r e v i e w e d e x c e p t f o r P r a j a p a t i [ 2 3 ] , P r a j a p a t i s t u d i e d , by e x t e n s i v e e x p e r i m e n t s , a t h r o u g h f l o w r o c k f i l l dam and d e t e r m i n e d t h e t h r e s h o l d u n i t d i s c h a r g e ( c r i t i c a l d i s c h a r g e f o r t h e o n s e t o f i n s t a b i l i t y ) as a f u n c t i o n o f t a i l w a t e r d e p t h and s t o n e s i z e . The a p p l i c a t i o n i s r e s t r i c t e d , however, t o r o c k f i l l p l a c e d a t t h e a n g l e o f r e p o s e ( 1 : 1 , 2 5 ! ) . P r a j a p a t i ' s r e s u l t s i n d i c a t e a c r i t i c a l u n i t d i s c h a r g e p r o p o r t i o n a l t o t h e t a i l w a t e r d e p t h t o t h e power 1/3 and s t o n e s i z e t o t h e power 7 / 6 . The t a i l w a t e r d e p t h dependency i s p a r t i c u l a r l y u s e f u l compared t o t h e Knauss r e l a t i o n -s h i p w h i c h r e f e r -s o n l y t o i m p e r v i o u -s o v e r f l o w dam-s, -see S e c t i o n 3.6.

-17-3. A n a l y s i s

3.1 S t a b i l i t y approach

As shovm b e l o w , two d i f f e r e n t ways can be f o l l o w e d i n o r d e r t o a r r i v e a t t h e r e q u i r e d s t o n e d i m e n s i o n s i n c l o s u r e dam d e s i g n .

Method A r e q u i r e s a t h o r o u g h knowledge o f t h e d e t a i l e d c u r r e n t p a t t e r n , espe-c i a l l y a t t h e downstream espe-c r e s t l i n e r e g i o n , i n espe-c l u d i n g : . d i s c h a r g e c h a r a c t e r i s t i c s and f l o w c o n t r a c t i o n phenomena w a t e r s u r f a c e p r o f i l e i n c r e a s e o f f l o w v e l o c i t y t h r o u g h f l o w p e n e t r a t i o n i n t h e porous c r e s t . v e r t i c a l f l o w d i s t r i b u t i o n . I f a l l t h e u n c e r t a i n t i e s a r e t a k e n i n t o a c c o u n t , t h i s can e a s i l y l e a d t o an o v e r e s t i m a t e o f t h e s t o n e d i m e n s i o n s . Method B l e a d s d i r e c t l y t o a p r e c i s e d e s i g n o f t h e r e q u i r e d s t o n e d i m e n s i o n s , p r o v i d e d t h a t s u f f i c i e n t e x p e r i m e n t s a r e c a r r i e d o u t .

For o u t l i n e d e s i g n , however. Method A can be a p p l i e d t o g e t h e r w i t h Method B, p r o v i d e d t h a t t h e r e i s s u f f i c i e n t e x p e r i m e n t a l d a t a and i n s i g h t i n t o t h e phys-i c a l phenomena phys-i n v o l v e d . The p r e s e n t r e p o r t phys-i s phys-i n t e n d e d as a c o n t r phys-i b u t phys-i o n t o t h i s a p p r o a c h .

For optimum d e s i g n , i n cases where t h e geometry and c o n d i t i o n s v a r y l a r g e l y , s p e c i f i c e x p e r i m e n t s w i l l anyhow be r e q u i r e d u n c o n d i t i o n a l l y f o r t h e t i m e be-i n g .

-18-3-2 C h a r a c t e r i s t i c f l o w s i t u a t i o n s

Two t y p e s o f f l o w s i t u a t i o n s a r e d i s c u s s e d i n S e c t i o n 2:

drowned o r s u b - c r i t i c a l f l o w - m o d u l a r o r c r i t i c a l f l o w

These f l o w c o n d i t i o n s a r e r e l a t e d t o t h e head and t a i l w a t e r e l e v a t i o n s . For. t h e t h r e s h o l d c o n d i t i o n o f s t o n e s t a b i l i t y t h e head and t a i l w a t e r e l e v a t i o n s a r e a l s o i n t e r r e l a t e d . C o n s e q u e n t l y , an a t t e m p t has been made t o r e l a t e t h e t r a n s i t i o n between drowned and m o d u l a r f l o w t o t h e t a i l w a t e r e l e v a t i o n ( r e l a -t i v e -t o -t h e dam c r e s -t ) o n l y . T h i s proved -t o be p o s s i b l e , w i -t h i n -t h e l i m i -t s o f t h e i n v e s t i g a t i o n ; t h e c h a r a c t e r i s t i c v a l u e o f h ^ , made d i m e n s i o n l e s s by d i v i -s i o n by t h e -s t o n e -s t r e n g t h p a r a m e t e r AD, proved t o l a y between 3 and 5, w i t h a an average v a l u e o f 4.

When t h e c r e s t l e v e l i s r a i s e d above t h e t a i l w a t e r l e v e l , a t a c e r t a i n i n s t a n t t h e downstream c r e s t l i n e w i l l emerge, o c c u r r i n g a t a p p r o x i m a t e l y h^/^D = - 1 . C o m p l e t i o n o f t h e c l o s u r e can be d e f i n e d as when H=0; t h e c o r r e s p o n d i n g h^/AD v a l u e w i l l l a r g e l y depend on t h e p e r m e a b i l i t y o f t h e dam body; f o r t h e p r e s e n t i n v e s t i g a t i o n [ 1 ] , an i n d i c a t i v e v a l u e o f -5 was f o u n d .

I n t h e a n a l y s i s t h e f o l l o w i n g f l o w s i t u a t i o n s have been d e s i g n a t e d : . Low dam f l o w (h^/AD > 4)

. I n t e r m e d i a t e f l o w (-1 < h,/AD < 4) b

. H i g h dam f l o w (h,/AD < - 1 ) b

. Through f l o w (H < 0)

3.3 Low dam f l o w s i t u a t i o n (h^/AD > 4) b

For u n i f o r m f l o w c o n d i t i o n s , a w i d e l y used e x p r e s s i o n f o r r e l a t i n g t h e c r i t i -c a l v e l o -c i t y u^ t o t h e s t o n e d i a m e t e r D, i s e q u a t i o n ( 3 ) g i v e n i n Chapter 2, w h i c h has been d e r i v e d f r o m t h e S h i e l d ' s diagram f o r t h e i n i t i a t i o n o f m o t i o n :

-19-i n w h -19-i c h : g = g r a v i t a t i o n a l c o n s t a n t = 9.81 m/s^ f = S h i e l d s parameter C = 18 l o g ( 1 2 h/k) m V s ( W h i t e - C o l e b r o o k ) k = 2D f o r n a t u r a l dumped r o c k f i l l

For a dam u and h r e f e r t o t h e downstream c r e s t l i n e

The v a l u e o f \() i s dependent on t h e s t a b i l i t y r e q u i r e m e n t s o f t h e r o c k f i l l s t r u c t u r e under c o n s i d e r a t i o n . For b o t t o m p r o t e c t i o n d e s i g n , f o r i n s t a n c e , i|; = 0.03 i s a d o p t e d by DHL as a p r a c t i c a l v a l u e f o r t h e i n i t i a t i o n o f s t o n e d i s p l a c e m e n t . At t h e c r e s t o f an o v e r f l o w dam, however, t h e c u r r e n t i s n o t u n i f o r m l y d i s t r i -b u t e d ( c u r v i - l i n e a r f l o w , a c c e l e r a t e d f l o w ) and a c u r r e n t v e l o c i t y a d a p t i o n f a c t o r k i s i n t r o d u c e d t o t a k e i n t o a c c o u n t t h i s i n f l u e n c e . T h i s f a c t o r has t o be d e t e r m i n e d e x p e r i m e n t a l l y . W i t h r o c k f i l l o v e r f l o w dams t h e downstream c r e s t e x p e r i e n c e s t h e h e a v i e s t c u r r e n t a t t a c k , and so t h e l o c a l v e r t i c a l l y -averaged c u r r e n t v e l o c i t y u^ i s t a k e n as t h e r e f e r e n c e v e l o c i t y . The c r i t i c a l l o c a l c u r r e n t v e l o c i t y u^ now r e a d s :

^^^^ry (10)

/AgD /g

A f t e r some damage has o c c u r r e d , t h e r e m a i n i n g dam body becomes more s t a b l e t h a n b e f o r e , because o f t h e g e o m e t r i c a l d e f o r m a t i o n . T h i s i m p l i e s t h a t f o r s t a b l e dam d e s i g n t h e v a l u e o f ij) may be somewhat h i g h e r t h a n i n d i c a t e d above. Assuming t and k^ t o be f a i r l y c o n s t a n t f o r one dam geometry, u^//AgD

remains a f u n c t i o n o f C, i . e . o f t h e l o c a l d e p t h parameter h^/D ( o r h^/AD) o n l y . E q u a t i o n s ( 6 ) and ( 7 ) do, i n f a c t , show a dependency on h^/D o n l y . I n a d d i t i o n , t h e c o e f f i c i e n t s , 3.5 and 1,5, i n d i c a t e an undeveloped boundary l a y e r a t t h e dam c r e s t .

P l o t t i n g e q u a t i o n ( 6 ) i n F i g , 18, and a l s o e q u a t i o n ( 1 0 ) w i t h k^ = 1 and ij^ = 0,04 t h e r e i s a r e m a r k a b l e agreement between t h e u n i f o r m f l o w approach ( n o t e t h a t k^ = 1) and t h e w i d e c r e s t e d dam r e s u l t s , w i t h i n t h e l i m i t s o f t h e i n v e s -t i g a -t i o n . F u r -t h e r a n a l y s i s o f -t h e a c -t u a l k^ v a l u e ( w i -t h ij^ = 0.04) i n d i c a -t e s an average v a l u e o f 1.1 and 0.9 f o r t h e b r o a d and s h a r p c r e s t e d d a t a , r e s p e c t i v e -l y . F i g . 19. For t h e s h a r p c r e s t e d dam, e q u a t i o n ( 1 0 ) can be f i t t e d w i t h equa-t i o n ( 7 ) when a v a l u e o f a p p r o x i m a equa-t e l y 0,7 i s f a k e n f o r k^ equa-t o equa-t a k e i n equa-t o acr-c o u n t t h e h i g h a acr-c acr-c e l e r a t i o n and acr-c u r v i - l i n e a r i t y o f t h e f l o w .

-20-A c c o r d i n g t o t h e I z b a s h ' e x p r e s s i o n s ( 1 ) and ( 2 ) [ 4 ] , f o r t h e s t a b i l i t y o f stones on t o p o f a dam: u o = 1.7 f o r a w e l l embedded s t o n e ( w i d e c r e s t ) /AgD and u o = 1 . 2 f o r an i s o l a t e d s t o n e on s t o p o f a dam ( s h a r p c r e s t ) , t h e w a t e r d e p t h dependency i s o b v i o u s l y be i g n o r e d .

When these e x p r e s s i o n s a r e p l o t t e d i n F i g . 18 i t appears t h a t t h e DHL equa-t i o n s ( 6 ) and ( 7 ) i n equa-t e r s e c equa-t e q u a equa-t i o n s ( 1 ) and ( 2 ) a equa-t abouequa-t h^/D = 5 ( o r h^/AD - 3 ) . Because I z b a s h d i d n o t v a r y t h e w a t e r d e p t h a p p r e c i a b l y , h i s

r e s u l t s u n d e r e s t i m a t e t h e c r i t i c a l c u r r e n t v e l o c i t y a t l a r g e r w a t e r d e p t h s .

A d d i t i o n a l t o a c u r r e n t v e l o c i t y a p p r o a c h , t h e g e n e r a l r e l a t i o n s h i p ( 1 0 ) can be expressed i n terms o f a w a t e r l e v e l d i f f e r e n c e over t h e dam, Hh, by i n t r o -d u c i n g a c o e f f i c i e n t • u = / 2 g ( H - h J - ( 1 1 ) o 2 b Combining e q u a t i o n s ( 1 1 ) and ( 1 0 ) y i e l d s AD . ,2u2 ( 1 2 ) w h i c h i s now t h e g e n e r a l e x p r e s s i o n f o r t h e c r i t i c a l d r o p over t h e s t r u c t u r e .

Since and ^ a r e n e a r l y c o n s t a n t f o r one dam g e o m e t r y , t h e c r i t i c a l d r o p i s m a i n l y a f u n c t i o n o f C, a n d , t h e r e f o r e , o f h^/D ( o r h^/AD) and y^. A g e n e r a l tendency i s t h a t i n c r e a s e s w i t h i n c r e a s i n g v a l u e s o f h^/AD, as shown i n F i g . 19 f o r a b r o a d and s h a r p c r e s t e d dam. C w i l l a l s o i n c r e a s e w i t h h^/AD, so i n e q u a t i o n ( 1 2 ) t h e y b a l a n c e each o t h e r t o some e x t e n t . T h i s b a l a n c i n g can be seen i n F i g . 2 0 , i n w h i c h t h e t o t a l d r o p has been p l o t t e d a g a i n s t t h e t a i l -w a t e r d e p t h f o r t h e d a t a f r o m v a r i o u s i n v e s t i g a t i o n s . T h i s i s , i n f a c t , a b e t t e r p r e s e n t a t i o n o f t h e l o w dam f l o w s i t u a t i o n t h a n t h e p r e s e n t a t i o n o f o v e r f l o w h e i g h t a g a i n s t t a i l w a t e r d e p t h i n F i g . 7 e t c . The reason f o r t h i s i s t h a t i n t h e submerged f l o w s i t u a t i o n t h e u p s t r e a m and t a i l w a t e r e l e v a t i o n s a r e

2 1

-i n t e r r e l a t e d v -i a t h e d -i s c h a r g e c h a r a c t e r -i s t -i c s . A l t h o u g h t h e s c a t t e r -i n F -i g . 7 seems t o be s m a l l , t h e s c a t t e r i n t h e c r i t i c a l d r o p i s s u b s t a n t i a l .

A c l o s e r e x a m i n a t i o n o f F i g . 30 j . n d i c a t e s t h e f o l l o w i n g t e n d e n c i e s w h i c h can be used f o r o u t l i n e d e s i g n :

For b r o a d and n a r r o w c r e s t e d dams w i t h a compact p r o f i l e (M 1741-11), t h e r e i s a c o n s t a n t mean v a l u e f o r (H-h, )/AD o f 1 . 5 , ( f o r h,/AD > 4 ) , i n c r e a s i n g

b b t o about 2 t o 3 f o r v e r y b r o a d c r e s t e d dams (M 1711-11). However, t h e Brogdon and Grace r e s u l t s i n d i c a t e a v a l u e o f 1.5 t o 2 f o r v e r y b r o a d c r e s t e d dams w i t h h i g h p o r o s i t y .

For round c r e s t e d dams (M 731-11) (H-h.)/AD has a mean v a l u e o f about 2, b

v a l i d up t o v e r y h i g h tailx-zater d e p t h s (h,/AD = 2 0 ! ) . I t must be s t r e s s e d , b

however, t h a t t h e s c a t t e r i s r a t h e r l a r g e .

A somewhat d e v i a t i n g p i c t u r e i s o b t a i n e d f o r t h e s h a r p c r e s t e d dam; t h e mean v a l u e o f (H-h,)/AD i s about 2 f o r h./AD = 4 and i n c r e a s e s l i n e a r l y t o

b D a b o u t 3 f o r v e r y h i g h t a i l w a t e r depths (h^/AD « 2 0 ) .

b

I t must be remarked t h a t t h e above c r i t i c a l d r o p parameter (H-h^)/AD c a n be a p p l i e d i n t h e l o w dam s i t u a t i o n , i n a d d i t i o n t o t h e c r i t i c a l c u r r e n t o r d i s -charge c r i t e r i o n ( t h e l a t t e r w h i c h i s d e a l t w i t h b e l o w ) . The o v e r t o p p i n g h e i g h t p a r a m e t e r H/AD, w h i c h i s v e r y u s e f u l i n t h e i n t e r m e d i a t e and h i g h dam f l o w s i t u a t i o n ( f r e e f l o w c o n d i t i o n s ) , must be dissuaded i n t h e low dam f l o w r e g i o n , because t h e good agreement o f t h e d a t a p o i n t s i n F i g . 7 i s , i n f a c t , m a i n l y a p p a r e n t .

For completeness' sake t h e I z b a s h and t h e S h i e l d s e q u a t i o n s have been compared i n F i g . 22. For t h e I z b a s h e q u a t i o n f o r u , /2g(Hh^)' has been s u b s t i t u t e d ( a s -suming y = 1) . The S h i e l d s e q u a t i o n ( 1 2 ) has been p l o t t e d , w i t h y = 1 , k^ = 1 and il = 0.04.

From F i g . 22 i t can be seen t h a t w i t h y = 1, t h e S h i e l d s approach i s s l i g h t l y too o p t i m i s t i c a t h i g h h^/AD. C o n t r a r y , t h e I z b a s h f o r m u l a , w i t h i n s e r t i o n o f y = 1 remains t o o c o n s e r v a t i v e ; a q u a n t a t i v e assessment o f t h i s d e v i a t i o n f r o m t h e o v e r a l l d a t a i s h a r d t o g i v e however, because o f t h e u n f e a s i b l e p r e s e n t a -t i o n (H/AD i n s -t e a d o f ( H - h , ) / A D ) .

b

The advantage o f a c r i t e r i o n based on w a t e r e l e v a t i o n on b o t h s i d e s o f t h e dam o n l y i s t h a t i t p r o v i d e s a v e r y p r a c t i c a l c r i t e r i o n f o r m o n i t o r i n g t h e dam s t a b i l i t y d u r i n g t h e c l o s u r e o p e r a t i o n s .

-22-I n a d d i t i o n t o t h e c r i t i c a l v e l o c i t y method and t h e c r i t i c a l head d r o p method a t h i r d p o s s i b i l i t y can be c o n s i d e r e d , namely a p p l y i n g t h e t o t a l s p e c i f i c d i s

-c h a r g e q. Analogous t o t h e f i n d i n g s i n F i g . 18 t h a t t h e -c r i t i -c a l v e l o -c i t y parameter i s a f u n c t i o n o f h^/AD, i t i s assumed now t h a t t h e c r i t i c a l d i s -c h a r g e i s a l s o a f u n -c t i o n o f h^/AD f o r ea-ch dam geometry.

For t h e l o w dam f l o w s i t u a t i o n t h e t o t a l d i s c h a r g e can be assumed t o pass o v e r t h e downstream c r e s t l i n e , because t h e f l o w t h r o u g h t h e porous dam body i s r e -l a t i v e -l y n e g -l i g i b -l e . The d i s c h a r g e r e -l a t i o n s h i p r e a d s : q = h / 2g(H-h^) , ( 1 3 ) and f r o m e q u a t i o n ( 1 1 ) , b e i n g r e l a t e d a c c o r d i n g t o - h,/h^ ( 1 4 ) I n f a c t , i n F i g . 19, i s d e r i v e d from y w i t h t h e a i d o f e q u a t i o n ( 1 4 ) . The v a l u e o f y^ proved t o be r a t h e r i n s e n s i t i v e t o t h e c r e s t w i d t h f o r l o w dam f l o w . F i g . 42. y^ i s l a r g e r f o r t h e wide c r e s t e d dam t h a n f o r t h e s h a r p c r e s t e d dam, because o f t h e s m a l l e r h fh, v a l u e ( 0 . 9 . a g a i n s t 1,0). T h i s ' O b

d i f f e r e n c e i n h / h ^ compensates, t o some e x t e n t , f o r t h e d i f f e r e n c e between

o b

t h e c r i t i c a l d i s c h a r g e s o f t h e two dam t y p e s i n comparison w i t h t h e c r i t i c a l v e l o c i t i e s w h i c h d e v i a t e much more.

The c r i t i c a l d i s c h a r g e parameter i s o b t a i n e d by making q d i m e n s i o n l e s s b y d i -v i d i n g i t by (AD)1«^; t h e j u s t i f i c a t i o n o f t h e exponent 1.5 i s g i -v e n i n t h e

i n v e s t i g a t i o n o f Knauss [ ^ , d e a l i n g w i t h s t e e p shute f l o w h y d r a u l i c s .

The m i n i m i z a t i o n o f t h e i n f l u e n c e o f dam geometry i s i l l u s t r a t e d i n F i g . 7, i n w h i c h many t y p e s o f dams have been t a k e n i n t o c o n s i d e r a t i o n . As a r e s u l t , t h e d i s c h a r g e p l o t i n F i g . 7 p r e s e n t s , as does F i g . 20, a u s e f u l t o o l f o r t h e o u t -l i n e d e s i g n o f t h e r e q u i r e d stone d i m e n s i o n s f o r a wide range o f dam t y p e s f o r drowned f l o w c o n d i t i o n s .

The r e a s o n a b l e agreement o f a l l t h e a v a i l a b l e d a t a p o i n t s p l o t t e d i n F i g . 7, compared w i t h t h e r a t h e r random s c a t t e r o f t h e d a t a , r u l e s o u t a s i g n i f i c a n t i n f l u e n c e o f o t h e r parameters w h i c h may r e s u l t f r o m d i m e n s i o n a n a l y s i s w i t h i n the l i m i t s o f t h e i n v e s t i g a t i o n s , such as h/d ( w i t h d = dam h e i g h t ) and D/d ( p e r m e a b i l i t y p a r a m e t e r ) . A dam geometry i n f l u e n c e , w i t h emphasis on t h e