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WRC RESEARCH REPORT NO . 149

SURCHARGE OF SEWER SYSTEMS

Ben Chie Yen

Ni c h o l a s Pa n s i c

Depar tment o f C i v i l Enginee r ing

Un i v e r s i t y o f I l l i n o i s a t Urb ana- Ch amp aig n

F I N A L R E P O R T

P r o e c t No. A-086-ILL

The work upon which t h i s pu b l i c a t io n i s based w a s sup por ted by funds

p r o v i d e d by t h e U. S . D ep ar tm en t o f t h e I n t e r i o r a s a u t h o r i z e d un d er

t h e Water Resou rces Research Act of 1964, P. L. 88-379

Agreem ent No. 14-31-0001-8015

UNIVERSITY OF ILLINOIS

WATER RESOURCES CENTER

2535 Hydrosystems Labora toryU rb an a, I l l i n o i s 6 18 01

March 1980

Contents of th is publicat ion do not necessari ly

ref lec t the views and pol icle s of the Off ice of

Water Research and Technology, U.S. Department o f

the Interior, nor does mention of trade names or

ccmmercial p roducts con st it ut e th ei r endorsement

.. or r e c mn da t i on f or use by t he U.S . Government.



ABSTRACT


Surcharge of a sewer is the situation in which the sewer entrance and exit are

submerged and the pipe is flowing full and under pressure. In this reportthe hydraulics of the surcharged flow as well as the open-channel flow

leading to and after surcharge is discussed in detail and formulated mathe-matically. The transition between open-channel and surcharge flows is also

discussed. This information is especially useful for those who intend tomake accurate advanced simulation of sewer flows. In this study an approximate

kinematic wave - surcharge model called SURKNET is formulated to simulate open-channel and surcharge flow of storm runoff in a sewer system. An example

application of the model on a hypothetical sewer system is presented.

Yen, Ben Chie, and Pansic, Nicholas


Research Report No. 149, Water Resources Center, University of Illinois,

Urbana, Illinois, March 1980, v+61 pp.

KEYWORDS--computer models/conduit flow/*drainage systems/flood routing/

hydraulics/hydrographs/mathematical models/ open-channel flow/*sewers/

sewer systems/*storm drains/storm runoff/*unsteady flow/urban drainage/urban runoff



CONTENTS

Page

LISTOFFIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .IST OF TABLES

CHAPTER . . . . . . . . . . . . . . . . . . . . . . . . .INTRODUCTION . . . . . . . . . . . . . . . . . . . . .1 RELATED PREVIOUS WORK . . . . . . . ..1 . Re la ted Work on Pr es su ri ze d Network Flow. . . . . . . . . . . . ..2 . Re la te d Work on Sewer Sur cha rge . . . . . . . . . . . . .. 3. O t he r Re l a t e d P r e v i o us S t ud i e s

I11. THEORY OF SEWER NETWORK HYDRAULICS . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . ..1 . F low i n a Sewer Pi pe

3 .1 .1 . C l a s s i f i c a t i o n o f Flow i n S i n g l e P i p e . . . . . .3 .1 .2 . H ydr a u l i c Be ha v i o r o f Flow i n S i n g l e P i p e . . . .3.1 .3 . Mathemat ica l Rep rese nta t ion of F low i n a P i p e . .3 .2 . F l o w i n a S e w e r N e t w o r k . . . . . . . . . . . . . . . . .3.2.1 . Sewer Jun c t i on s . . . . . . . . . . . . . . . . .3.2 .2. Sewer Networks . . . . . . . . . . . . . . . . .

I V . KINEMATIC WAVE . URCHARGE MODEL . . . . . . . . . . . . . . .4.1. Kin ema tic Wave Approximation . . . . . . . . . . . . . .4.2 . Fo rm ul at io no f SURKNETModel . . . . . . . . . . . . . .

4.2 .1. Open-Channel Flow . . . . . . . . . . . . . . . .4.2.2. Surcha rged Flow . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 2 . 3 . F l o w T r a n s i t i o n . . . . . . . . . . . . . . ..2 .4. Boundary Co nd i t i on s

. . . . . . . . . . . . . . . ..3 . SURKNET Compu ter Pr ogr am4. 4. Example A p p li ca ti o n of SURKNET . . . . . . . . . . . . .

4. 4. 1. Example Sewer Network . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 4 . 2 . Re s u l t s

V . CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . .REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii



Page

i v

16

28

Line

6 and 7

bottom

20

U I WRC Report 149

E r r a t a

in te rchange cap t ions o f F igs . 3 .1 and 3 .2

in te rchange cap t ions o f F igs . 3 .1 and 3 .2

change "when z i s t h e e l e v a t i o n o f th e junc t ion bo ttom." t o

"where z i s t h e e l ev a t i o n of t h e j u n c t i o n b o tt om,"

d e l e t e "a", s ho ul d r e ad " i n f i n i t e d i f f e r e n c e " -change "sec t ion" t o "sec t iona l "

chang e " r a t i o n " t o " r a t i o "

change "of a" t o "of an"

change "connected t o i t s upstream. For" t o

"connected a t th e upstream end. For the"

change "A2H2 - A l H l - A s which i s t h e change of wate r

volume on t h e ground." t o "A2H2 - A l H l = A s which i s t h e

change i n volume of wate r on t he ground."

change "p lo t s ' ' t o "p lo t"

change "purposely" t o "purposedly"

change "p. 60," t o "60 p."



LIST OF FIGURES

Page

F i g u r e . . . . . . . . . . . . . . . . . . . . . . . .. 1 P r e i s s m a n n S l o t 8

2 . 2 U .S. Ge o l o g i c a l Su rv ey C u l v e r t F lo w C l a s s i f i c a t i o n . . . . . . 12. . . . . . . . . . . . . . . . . . . . ..1 Sewer E xi t Flow Cases 16

3 .2 S e w e r E n t r a n c e F l o w C a s e s . . . . . . . . . . . . . . . . . . . 16. . . . . . . . . . . . . . .. 3 C l a s s i f i c a t i o n o f F lo w i n a Sewer 1 7. . . . . . . . . . . . . . . . . . . . .o l l aves i n a Sewer 19

Discha rge Ra t ing Curve f o r S teady Flow i n a

. . . . . . . . . . . . . . . . . . . . . . . . .i r c u l a r p i p e 2 1. . . . . . . . . . . .i r E n tr a in m en t I n s t a b i l i t y i n a Sew er 2 1. . . . . . . . . . . . . . . . .pen-Ch anne lFl owin a Sewer 23. . . . . . . . . . . . . . . . . .urcharged Flow i n a Sewer 26. . . . . . . . . . . . . .chemati c Drawing of Sewer Ju nc ti on 29

Backwater Ef f ec t on Disch arge of an Oakdale Avenue Sewer . . . 35

. . . . . . . . . . .xample o f Propa ga t i on o f F low Tra ns i t io n 42

. . . . . . . . . . . .low Ch ar t f o r Computer Pr og ra m SURKNET 48. . . . . . . . . . . . . . . . . . . . .xample Sewer Network 5 1. . . . . . . . .i scha rge and St orag e Graphs f o r E lement 1 -3 52

. . . . . . . . .i scha rge and S tora ge Graphs f o r E lement 2-3 53

. . . . . . . . .i sch arg e and St ora ge Graphs f o r Element 3-5 54. . . . . . . . .i scharg e and Sto rag e Graphs f o r Element 4-5 55

. . . . . . . . .i scharg e and Sto rag e Graphs f o r Element 5-6 56



LIST OF TABLES

Page

TABLE

3.1. P i p e E n t r a n c e C o n d i t i o n s . . . . . . . . . . . . . . . . . . . 14

3 .2 . P ip e Ex i t C ond i t ions . . . . . . . . . . . . . . . . . . . . . 14

3 . 3 . P i p e F l o w C o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . 1 5

3.4. In de nt i f ic a t io n of Bodha ine ls Types of P i pe Flow . . . . . . . 18

4.1. Sewer P ro p e rt i es of Example Network . . . . . . . . . . . . . 49

4.2. Manhole P ro p e rt i es of Example Network . . . . . . . . . . . . 49

4.3 . Example Manhole In fl ow Hydrograph . . . . . . . . . . . . . . 51



I. INTRODUCTION

In the terminology of sewerage engineering, surcharge is defined as

the condition that the sewer is flowing full and gravity-flow no longer

prevails. In hydromechanics, this condition is coqmonly referred to as

pressurized-conduit flow. Although sewers are traditionally designed assuming

open-channel flow, i.e., gravity flow in sewerage terminology (ASCE, 1969),

surcharge of sewers may well occur in both overloaded existing systems and

in new system designs. Some of the reasons for sewer surcharge are as

follows

(a) Underdesign resulting from inaccuracies in the design

equations, coupled with uncertainty in design parameters

(e.g., pipe roughness), can adversely affect system design.

(b) Hydrologic risk may cause surcharge because there is always

a probability, no matter how small, that the design dis-

charge may be exceeded one or more times during the service

life of the sewer.

(c) Construction errors and material deviations (e.g., tolerance

in the pipe dimensions), resulting in the sewer system in-

place not conforming to the design.

(d) In-line pumping stations that may be required due to system

constraints.

(e) In-line detention or retention storage resulting in

submergence of connecting pipes.



(f) Changing sewer system conditions after completion of

construction, such as blocking of manholes, deposition and

deformation of sewer pipes.

(g) Change of drainage basin characteristics after the design

and construction are completed.

A sewer system is characterized by a network of manholes and junctions

(nodes) connected by sewer pipes (links), usually of the dendritic type although

the loop-type networks are not uncommon. Storm sewer flow is time-varying

(i.e., transient or unsteady) in nature because all rainstorms have finite

durations and consequently the flood flow in the sewers changes with time.

If the flood is small, none of the sewer pipes are completely filled and the

flow remains as open-channel flow. However, for large floods, some or all

of the sewer pipes may change from open-channel flow to pressurized-conduit

flow during and near the time of the flood peak. Moreover, the flow in a

sewer is affected by the hydraulic conditions at both its upstream and

downstream ends. The rare exception is the case of supercritical gravity

flow for which only the upstream effect is important (Yen, 1977; Sevuk and

Yen, 1973). Hydrodynamically, the transition between open-channel flow and

surcharged flow in a sewer system is one of the most complicated unsolved

problems (Yen, 1978a).

In the management of sewer flow for pollution control and flood

mitigation, reliable prediction of the sewer flow is important. Obviously,

without an accurate evaluation of manhole surcharge, it is not possible to

predict reliably the level of flooding due to storm runoff.



I n r e c e n t ye a r s th e abatem e nt o f s to r m r unoff po l lu t io n i s an impor tant

c onc e rn . I n o r de r t o m ee t the r e qu i r e m e n t s o f t he Wa te r P o l lu t ion C on t r o l Act

Amendments of 1972 , P.L. 92-500, t h e co ns tr uc ti on of many new wastewater t re a t -

ment p l a n t s a nd the m od i f i c a t ion o f e x i s t i ng p l a n t s ha ve be en p r opose d o r a r e

underway i n t he Un i t ed S t a t e s . I t i s commonly acknowledged t h a t la rg e tr e a t -

m en t p l a n t s of th e s i z e r e qu i r e d t o ha nd le th e pe ak s to rm r unof f f rom u r ba n

a r e a s a r e ec on om i ca ll y u n j u s t i f i a b l e . Flow r a t e e q u i l i z a t i o n t hr ou g h t h e

use o f ups tr e am s to r a g e i s d e s i r a b l e . F u rt h er m o re , t r e at m e n t p l a n t s o p e r a t e

m os t e f f i c i e n t l y when the f low i s c o n s ta n t a t t h e d e si g n fl ow r a t e . Use of

o n - s i t e a n d / o r i n - l i n e d e t e n t i o n s t o r a g e h a s b ee n c o n s i de r e d a s an e f f e c t i v e

means of f low eq ui l i za t i on . For urban sewer sys tems th e sewers jo i n in g t he

i n - li n e d e t en t i o n f a c i l i t i e s a r e u su a l l y un de r s ur c ha r ge , p a r t i c u l a r l y d u ri ng

and immedia te ly a f t e r a heavy ra ins to rm. I f t h e su r c ha r ge flows ca nno t be

r e l i a b l y p r e di c t ed , i t i s m os t u n l i k e l y t h a t t h e p ur po s e o f f l o w e q u i l i z a t i o n

f o r u rb an r u n o ff p o l l u t i o n a ba te me nt c a n b e s a t i s f a c t o r i l y a c h ie v ed .

C onsider a sewer de s ign pe r m i t t i ng a l i m i t e d de g r e e of su r c ha r ge f o r

a few sewers , w i th th e wate r con£ ined wi t h in manholes a i d ju nc t ion s and not

f loo din g th e ground and pavement. Under ce r t a i n c i rcums tances , ad di t i on a l

hydr a u l i c he a d w i l l b e a v a i l a b l e fr om t h e d i f f e r e n c e o f t h e w a te r s u r f a c e s

i n th e ups t ream and downstream manholes o r ju nc t i ons . The r e s u l t i n g h i g h e r

f lo w v e l o c i t y w i l l a l lo w the use of a smal le r sewer than would be th e case

f o r g r a v i t y f lo w . I f t h i s s u rc h a r ge c o n d i t i o n o c c u rs i n t h e m os t e x pe n si v e

sewer p ipe s i n th e sys te m , the s a v ings i n us ing sm a l l e r s e we r s c an be

con sid erab le , achievi ng a more economica l des ign f o r t he sewer sys tem. Con-

v e r s e l y , i f t h e s u rc h ar g e i s not prop er ly s imula t ed , th e sewer may be

o v e r s i z e d o r u n d e rs i z e d. I n t h e fo rm er c a s e i t w i l l be a waste of money



f o r t h e p r o t e c t i o n r e q u i r e d , w he re as f o r t h e l a t t e r c a s e t h e u n de r si ze d

sewers w i l l b e u n ab l e t o h an d le t h e d e s i g n s to rm ru n o f f , c au s i n g f r eq u en t

f lood ing .

The o b j e c t i v e of t h i s r e s e a r c h p r o j e c t i s t o develop an improved

s ew er s u rch a rge s i mu l a t i o n model w hich can b e us ed t o i n v es t i g a t e t h e e f f e c t s

o f su rcharge on bo th sewer des ign and d ra inage ope ra t ion . I n t h i s r e po r t ,

a f t e r a br ie f rev iew of re l a te d p rev ious work and hyd rau l i c theory , a

non l ine ar k in ema t ic wave sewer surch arge s im ula t ion scheme i s descr ibed and

an example i s presen ted .



11. RELATED PREVIOUS WORK

Many sewer f low s im ula t io n me thods have been p roposed i n t he pa s t ,

r ang ing f rom th e s impl e ra t i on a l me thod (ASCE, 1 9 69 ; Ye n , 1 97 8 b ) t o t h e h i g h l y

s o p h i s t i c a t e d com pute r-ba sed Sto rm Water Management Model (SWMM, Me tc al f G

Eddy e t a l . , 1 97 1) a nd I l l i n o i s S to rm Se we r Sy st em S i m u l a t i o n ( I SS) M odel

(Sevuk e t a l . , , 1973) . Mos t of t h e e x i s t i n g p i p e ne t wo r k f l o w m od el s a r e

e i t h e r p u r e l y o pe n- ch an ne l f l o w o r c o m p l e t e l y p r e s su r i z e d - c o n d u i t f l o w m o de l s.

R ea de rs a r e s ug g es t ed t o r e f e r t o p u bl i s he d r e f e r e n c e s ( e . g . , J am es F .

MacLaren, 1975; Chow and Yen, 1976, Br an d st e t te r , 1976; and Colyer and

P e t h i c k , 1 9 76 ) f o r a r e v i e w o f t h e e x i s t i n g op e n- c ha n ne l f l o w se wer s i m u l a t i o n

models .

2 .1 . Rel a te d Work on Pr es su r i ze d Network Flow

The e x i s t i n g p r e s s u r i z e d p i p e n e tw o rk f l o w s i m u l a t i o n mo de ls a r e p r i m a r i l y

s t e a d y f l o w m od el s d ev e lo p ed f o r wa t er - su p pl y n e tw o rk s a nd n o t s p e c i f i c a l l y f o r

se we r s . T h e se m od el s h a n d l e l o op - ty p e n e t wo r k s an d u s u a l l y so l v e t h e f l o w

e q u a t i o n s u s i n g o n e o f t h r e e a p p r o a c h e s : t h o se f o l l o wi n g C r o s s ' ( 19 36 ) co n-

c e p t o f s u c c e s s i v e r e l a x a t i o n a p p l i e d t o e a c h l o o p ( Adams, 1 9 61 ; D i l l i n g h a m ,

1967; Gra ves and Branscome, 1958; Hoag and Weinberg, 1957; Jac oby and Twigg,

1968) ; t ho se employ ing th e Newton-Raphson method f o r s ucc ess ive re l a xa t i on of

a l l l o o p s s i m u l t a n e o u s l y ( Epp an d Fo wl er , 1 9 70 ; Ha r t i n a n d Pe t e r s , 1 9 63 ;

L ek an e, 1 9 79 , L em ie ux , 1 9 7 2 ) ; an d t h o s e u s i n g l i n e a r i z a t i o n ( Ha rl ow e t a l . ,

1966; Wood and Ch ar le s, 1972 ). Althoug h i t has been shown th a t th e Darcy-

Weisbach or Colebrook-White re s i s t a nc e fo rmulas can be p rogrammed f o r so lu t i on

( F i e t z , 1 97 3; Le ka ne , 1 9 7 9 ) , m os t of t h e se m o de l s u se t h e l e s s d e s i r a b l e Hazen-

W i l l i a m ' s f o r m ul a . T h i s i s p a r t l y be ca us e t h e l a t t e r i s e a s i e r t o s o l v e , b u t



probably due more to the fact that Cross used the formula in his original dev-

elopment at the University of Illinois in 1936. Considering the wide range of

the Reynolds number of the flow in storm sewers and the later development in

fluid mechanics, the Hazen-William formula is not a preferred resistance formula

to be used. A review of the steady flow network models can be found in Jeppson

(1975) and Shamir (1973).

Of the existing pressurized pipe network transient flow models, the

majority are "water hammer1' odels emphasizing pressure surges (Wylie and

Streeter, 1978). The remaining few that handle unsteady flow in pressured

pipe networks cannot be applied directly to surcharged sewer systems. How-

ever, they are clearly useful in formulating a surcharged unsteady flow sewer

model. Stoner (1968), Wylie et al. (1974), and Vardy (1976) applied the

method of characteristics to model the unsteady flow of gas in pipe networks.

2.2. Related Work on Sewer Surcharge

Recently, approximate techniques for surcharge flow routing have been

incorporated into a number of sewer flow simulation models. Models using

only Manning's or Darcy-Weisbach's formulas for open-channel flow routing in

sewers can approximate surcharged flow. This is done by using the full

pipe diameter and hydraulic radius in the computations, but it must include a

means to estimate the available head for the flow. Examples of such models

are TRRL (1976) and ILLUDAS (Terstriep and Stall, 1974) for which the flow

simulation proceeds pipe by pipe from the upstream to downstream end of the

network in a cascading manner. Assumptions are made to estimate the

piezometric heads at the upstream and downstream ends of a pipe with no

direct interaction of the pressure and discharge between upstream and down-

stream pipes considered. In the Storm Water Management Model (SWMM, Metcalf

& Eddy et al., 1971), whenever surcharge occurs, the excess water is



assum ed t o s t o r e i n t h e u p s t r ea m m an ho le w i t h o u t a f f e c t i n g t h e d yn am ic s of

t h e f l o w .

A method proposed by Shu bin ski and Roesner (1973) adopte d Hardy Cros s '

( 1936 ) m et hod t o e s t i m a t e t h e f l ow ( w hi ch was i m p l i c i t l y a s sum ed s t e a d y w i t h i n

t h e c o m p u t a t i o n a l t i m e i n t e r v a l ) i n s u rc ha rg ed p i p e s. T he w a t e r d e p t h s i n t h e

m an ho le s a r e t h e n r e a d j u s t e d b as ed o n t h e c o n t i n u i t y p r i n c i p l e b e f o r e p ro -

c e e d i ng t o t h e ne x t t i m e i n t e r v a l c om pu ta t i on. They f ound t h i s m et hod un-

s t a b l e an d l a t e r pr op os ed a d i f f e r e n t v e r s i o n which i s i n c o r p o r a t e d i n t o

SWMM a s a p a r t of t h e WRE Tra nsp or t Block (EXTRAN). I n t h i s ve rs io n (1977

SWMM), when su r c h a r g e o c c u r s , t h e ma nh ol es c o n ne c te d t o t h e s u r c h ar g e d p i p e s

a r e assumed t o ha ve a r t i f i c i a l c r o s s s e c t i o n a l a r e a s , d e c r ea s i ng i n a r e a from

t h e p i p e c rown l i n e a r l y t o % of t h e p i p e c r ow n de p t h be low t h e g r ound. The

r o u t i n g t h e n p r o c e e d s as i n t h e n o n- su rc h ar ge d c a s e . E x ce ss w a t e r s p i l t o u t

f rom manhole on t o t h e ground i s a ss um ed l o s t a nd no t r e c ov e r a b l e . I t h a s

b e en f ou nd t h a t t h i s a pp r o ac h i s a l s o u n s a t i s f a c t o r y and l a c k s t h e o r e t i c a l

j u s t i f i c a t i o n .

I n t h e Fr en ch mode l CAREDAS de ve lo pe d by SOGREAH, s u r c h a r g e i s handled by

us i ng t h e " Pr ei ss m ann s l o t " t e c hn i que ( P re i ss m ann and Cunge , 1961) . I n t h i s

m e thod, t h e s u r c ha r ge d p i p e f l ow i s a r t i f i c i a l l y c on ve rt ed i n t o o pen -cha nn el

f l o w by a ss um in g t h e e x i s t e n c e o f a s l o t o n t o p and a l o n g t h e f u l l l e n g t h of

t h e p i p e ( F ig . 2 . 1 ) . The s l o t w i dt h i s s o n ar ro w t h a t i t s volume i s

ne g l i g i b l e . Cons e quen t l y , t h e open - channel f l ow dynam ic e qua t i on c a n be

a p p l i e d t o t h e s l o t -m o d i f ie d s u r c h a r g e f lo w . However, i f a t a ny g i ve n t i m e

many p i p e s a r e s u r c h a r g e d , t h e s o l u t i o n b eco mes v e r y e x p e n s i v e b ec a us e t h e

f l o w e q u a t i o n s f o r a l l s ur c ha r ge d p i p e s ( o f t e n f o r n on -s ur ch ar ge d p i p e s as

w e l l ) must be s o l ve d s i m u l t a ne ous l y . A ppr oxi ma te t e c hn i qu e s t o r e duc e t h e



s i m u l t a ne ous c om pu ta t i on , s uc h as t h e over- lapping-segment method (Sevuk e t a l . ,

1973) , are n o t v e r y r e l i a b l e when a p p l i e d t o t h e s l o t t e c hn i qu e b ec a us e p r e s s u r e

wav es i n s u r ch a r g e d p i p e s t r a n s m i t f a r t h e r up s tr e am t h a n i s t h e cas e of open-

channe l f low.I t

h a s a l s o b ee n f ou nd t h a t c o m p u t a t io n a l s t a b i l i t y p ro bl em s o c c ur

i f t h e assu med s l o t wi d t h i s t o o n ar ro w, a l t h ou g h n o t n e c e s s a r i l y i n f i n i t e s i m a l .

Song ( 1 97 6, 1 97 8) a p p l i e d t h e metho d o f c h a r a c t e r i s t i c s t o s o l v e b o t h

t h e open-channe l and surcha rged phases of a s imple sewer sys tem. H e assumed

t r a n s i t i o n f ro m op en -c ha nn el f l o w t o s u r c h a r g e d f l o w when t h e d e p t h i n t h e

c o nd u it e x ce ed ed a r e f e r e n c e s d e p t h s l i g h t l y smal ler t h a n t h e d i a m et e r o f

t h e p i pe . The j unc t i on o f t he p i pe s w e re as sume d as a p o i n t w i t h a common w a te r

s u r f ac e f o r a l l t h e j o i ni n g p i p es . J u n c t i o n s t o r a g e an d l o s s e s we re n o t d i r e c t l y

a c c oun t e d f o r .

Bettess e t a l . (1978) proposed an improved method t o hand le sur cha rge d

f l ow i n se we r sy s te m s. T he d i s c ha r ge o f a p i pe i n t h e se w er sy s t e m a t any

t i m e i s compared t o t h e p i p e - f u l l d i s c h a r g e . I f t h e f or me r e x ce e ds t h e l a t t e r ,

t h e p i pe i s assu med s u rc h a r g ed . I n t h i s m an ne r, a l l su r c h a rg e d p i p e s a t a

g i ve n t i m e i n t h e sy stem a r e i d e n t i f i e d . The subsys tem of surcha rged p ip es

a r e t he n s o l v e d s i m u l t a ne ous l y u s i n g Da rc y- We is ba ch 's f o r m u l a t o ge t he r w i t h

t h e uns t e a dy f l ow m anho le c on t i nu i t y e qu a t i on . T he m et hod i s r e a s o n a b l y

r e a l i s t i c , n o t e x c es s i v e l y s o p h i s t i c a t e d , a nd p r a c t i c a l . The m aj or u n ce r-

t a i n t i e s a re t h e m anh ole l o s s c o e f f i c i e n t s a nd t h e t r a n s i e n t c o n d i t i o n b etw een

open-channe l and surcha rged f l ow s .

2 .3 . O t he r Re l a t e d P r e v i ou s S t u d i e s

L i t t l e ha s be en done on a n e qua l l y i m por t a n t p r obl em o f s ew e r s u r c ha r ge ,

namely , t h e t r a n s i t i o n be tween surcharg ed and open-channe l f low. Hydrodynam-

i c a l l y , t h e t r a n s i t i o n phenomenon i s o ne of t h e m os t d i f f i c u l t p r ob le ms . I t



i nvo lv e s no t on ly uns te ady nonuni fo rm f low bu t a l so t he c om pl ic a t ions of a i r

ent ra inm ent , j un c t i on lo ss es , and moving bore s and surg es (moving hy dra ul i c

jumps). Ha ind l ( 1957) in ve s t ig a te d th e t r a ns i t io n o f s t e a dy f low f rom ope n-

c h an n el t o f u l l p i p e t h r o u gh a h y d r a u l i c jump. H e fou nd t h a t t h e t r a n s i t i o n

depends on th e pre -jump Froude number and a i r supp ly i n th e p ipe , and t h a t

t h e e ne rg y l o s s o f t h e r e s t r a i n e d h y d r a u l i c jump i s less t h a n o r e q u al t o

th e f r e e hy dr au li c jump havin g t h e same pre-jump Froude number. Mayer-Peter

and Fa vr e ( 1932) f i r s t d i s cu s s ed t h e t r a n s i e n t p ro blem i n t h e t a i l r a c e t u n n e l

of t h e Wett igen Hydropower Plan t . A br ie f r e v iew and d i sc us s ion of th e

t r a n s i e n t su r ge s i n a s im ple c ondu i t c an be found i n Wigger t ( 1972) . Zovne

(1970) s tud ie d th e p r opa ga t ion of bo r e s a nd hyd r a u l i c jum ps f o r uns te a dy

ope n- channe l f low us ing t he m ethod of c h a r a c te r i s t i c s . He concluded th a t th e

S a in t Venan t e qua t ions c a n be use d p r ov ide d c e r t a in p r e c a u t io ns a r e t a ken .

I n f o rm a t i on on l o s s e s of T ee j u n c t i o n s f o r p i p e s c a n b e f ou nd i n t h e

l i t e r a t u r e ( e . g . , M i l l e r , 1 97 1) . However, d a t a on j u n c t i o n l o s s e s i n m an ho les

a r e r a t h e r l i m i t e d f o r s t e a d y f l ow c a s e s and n o n e x is t e n t f o r u n st ea dy f lo w s.

S a n g s te r e t a l . (19 58) i n v e s t i g a t e d e x p e r i m e n t a l l y t h e m an ho le l o s s e s o f

surcharged p ip e f lows. Townsend and Pr in s (1978) presented some ex pe r i me nt a l - .

r e s u l t s on m an ho le l o s s c o e f f i c i e n t s u n de r s t e a d y f r e e - s u r f a c e f l ow s .

V o l k a rt (1 978 ) s t u d i e d e x p e r i m e n t a l l y t h e a i r e n t r ai n m e n t of s t e a d y fl o w

i n p a r t i a l l y f i l l e d p i pe s of s t e ep s lo p es . H i s r e s u l t s i n d ic a t e t h a t t h e a i r

ent ra inment depends on th e Froude number of th e f low, t h e depth t o p ipe

d ia m e te r r a t i o , a nd th e s lope and r oughness of th e p ipe . Kil len and Anderson

(1968) i n v e s t i g a t e d t h e i n t e r f a c e c h a r a c t e r i s t i c s a nd a i r e n tr a in m e nt of

f r e e s u r f a c e fl ow . D uk le r ( 1972hamong o t h e r s , d i s c u s s e d t h e s t a b i l i t y of t h e

i n t e r f a c e b et we en a i r and f l o w in g l i q u i d .



P e r h a ps t h e s t u d i e s mo st f a m i l l a r t o h y d r a u l i c c tn gi ne er s w hi ch a r e

r e l e v a n t t o s ew er f lo w are t h e c l a s s i f i c a t i o n s o f t y p e s o f f lo ws i n c u l v e r t s

r e po r t ed i n Chow (1959) , Po r t la nd Cement Ass oc ia t io n (1964 ) , and U.S. Geo log ica l

Survey (Bodha ine, 1969) . The l a s t c l a s s i f i c a t i o n i s r e p r oduc e d i n F i g . 2 .2 .

A l l t h e s e c l a s s i f . i c a t i o n s of d i f f e r e n t t y p e s of f lo ws a r e f o r s t e a d y f lo w i n

a s i n g l e p i p e . The a c t u a l f l o w c a s e s f o r se we r n e t wo r ks a r e , e s p e c i a l l y f o r

u n s t e a d y f l o w s , c o n s i d e r a b l y m or e c o m p l i c a t e d an d w i l l b e d i s c u ss e d i n t h e

f o l l o w i n g c h a p t e r .



C R I T I C A L D E P T HS U B M E R G E D

C R I T I C A L D E P T H a R A P I D F L O W

A T O U T LE T --

-: . '. ..3 6 ----,- Q = C A Q ~ ~(hl - ha- t ~ , ~ . ~ )

l L--T R A N Q U IL F L O W F U L L F L O W

-\

I' L

THROUGHOUT -,-r F R E E O U T F A L L --. ,..I

. . ' . 0 ' ' . ., - I

D ( 1 5hl-z 7 1.5

D 2.t ---TOT-----4 /D 2 1.0

h4/D; I0 s,'h4 hc, 1.0 . . . , : a '. ,, . / . '.

Fig . 2 . 2 . U.S. Geologica l Survey Culve r t F low Cl as s i f ic a t io n (Bodha ine , 1968)



111. THEORY OF SEWER NETWORK HYDRAULICS

B ec au se o f t h e t e m po r a l a nd s p a t i a l v a r i a t i o n s o f r a i n f a l l e v e n t s , s t or m

s ew er f l o w s a r e g e n e r a l l y u n s te a d y, i . e . , t i m e - va ry i ng . T he pa t t e r n o f s ew er

r uno f f due t o a s u r c ha r ge - ca us i ng hea vy r a i n s t o r m i s s uc h t ha t ope n - c ha nne l

f l ow o c c u r s i n t h e s ew er b e f o r e an d a f t e r t h e s u r c h a rg e . T h e re f o re , t h e e n t i r e

r a n g e of f l ow c o n d i t i o n s s h o u ld b e co n s i d e re d i f a c o m pl et e i n v e s t i g a t i o n o f

sewer surcharge i s d e s i r e d . A s a m a t t er of c on ve ni en ce , i n t h i s r e p o r t t h e

e n t i r e r an g e o f f l o w i s d i v i d e d i n t o t h r e e r e gi m es . They a r e open-channel o r

f r e e - s u r f a c e o r g r a v i t y f lo w , p r es s u r iz e d - c o n d ui t o r s u r ch a r ge d f l o w, a nd t h e

t r a n s i t i on bet we en t h e ope n -c hannel a nd s u r c ha r ge d f l ow s .

A sewer sys tem i s a network of manholes o r junc t io ns (nodes ) connec ted

by s ew e r p i p e s ( l i nk s ) . U s ua l l y s t o r m s ew e r ne t wor ks a r e c ons i de r e d a s t he

d e n d r i t i c t y p e n e t w o r k , a l t h o u g h l o ~ p - t y p e n e t w o r k s do e x i s t . T he re a r e , of

c o u r s e , o t h e r r e g u l a t o r y and c o n t r o l f a c i l i t i e s i n s ew er s y st e m s. However, i n

t h i s r e p o r t t h e e mp ha sis i s on the behav ior of th e manholes and p ip es , and th e

a u x i l i a r y f a c i l i t i e s a r e n o t c on si de re d.

3.1. Flow i n a Sewer Pip e

3.1.1. C l a s s i f i c a t i o n o f Flow i n S i n g l e F i p e

The f i r s t s t e p t ow ar d u n d e r s t an d i n g t h e f l o w i n a s ew er s y st em i s t o

u n d er s ta n d t h e f l o w i n a s i n g l e s e we r p i p e . The f lo w i n a s ew er p i p e , s i m i l a r

t o t h a t i n a c u l v e r t , h a s t h r e e r e g i o n s ; nam ely , t h e e n t r a n c e , t h e p i p e

f lo w, and t h e e x i t . T he re a r e f o u r c a se s o f e n t r a n c e c o n d i t i o n s a s l i s t e d

i n T a b le 3 . 1 and i l l u s t r a t e d i n F i g . 3 .1 . C ase I i s assoc ia ted wi th downs t ream

c o n t r o l of t h e p i p e f lo w . Case I1 i s a s s o c i a t e d w i t h up st re a m c o n t r o l of t h e

p i pe f l ow . I n Ca s e I11 t h e p i p e f l o w u n de r t h e a i r p oc k et may b e s u b c r i t i c a l ,

s u p e r c r i t i c a l , o r t r a n s i t i o n a l . I n C ase I V t h e p i p e f lo w i s o f t e n c o n t r o l l e d

by t h e downstream co nd i t io n bu t somet imes by bo th en tr an ce and downstream

c o n d i t i o n s .

1 3



TABLE 3.1. PIPE ENTRANCE CONDITIONS

Case Hydraul ic Condi t ion

I Nonsubmerged en tra nce , s u b c r i t ic a l f low

I I Nonsubmerged en t ra nce , su pe rc r i t i ca l f low

I11 Submerged en tra nce , a i r pocket

I V Submerged entrance, "water pocket"

TABLE 3.2. PIPE EXIT CONDITIONS

Case Hydraul ic Condi t ion

A Nonsubmerged, f r e e f a l l

B Nonsubmerged, continuous

C Nonsubmerged, h yd ra ul ic jump

D Submerged

The e x i t c o n d i t i o n s ca n a l s o b e g ro up ed i n t o f o u r c a s e s a s l i s t e d i n

T ab le 3 .2 and i l l u s t r a t e d i n F i g . 3 .2 . In Case A t h e p i p e f l ow i s u n d e r ex i t

co n t r o l . I n Case B t h e f l o w i s under ups t ream con t r o l i f it i s s u p e r c r i t i c a l

and dow nstream co n t r o l i f s u b c r i t i c a l . In Case C t h e p i p e f l o w i s under upstream

con t ro l wi th th e manhole water su r fa ce under downst ream con t ro l . In Case D

t h e p i p e f l ow i s of te n under downstream co nt ro l but can a l s o be under both

upstream and downstream control .

A s t o t h e f lo w w i t h i n t h e p i p e , i t can be s u b c r i t i c a l o r s u p e r c r i t i c a l

open-channel f low, uniform or nonuniform, w i th o r wi thout a hy dra ul ic jump o r

d ro p , g r a v i t y f l ow o r s u rch a rg ed , an d u s u a l l y t u rb u l en t . Wi thou t t ak ing in to

co n s i d e ra t i o n t h e d i f f e r e n t modes of a i r en t r a in men t, t h e p i p e f lo w can b e

c l a s s i f i e d i n t o t e n g r ou ps a s l i s t e d i n Ta ble 3 . 3 and i l l u s t r a t e d i n F i g. 3 .3 .

The p o s s i b l e en t r a n ce an d e x i t co n d i t i o n s f o r e ach o f t h e t en p i p e f l o w cas e sa



a r e a l s o g iv e n i n T ab l e 3. 3. T h e re f o r e, c o n s i d e r in g t h e p i p e f l o w t o g e th e r

w i t h i t s p o s s i b l e e n t r a n c e a nd e x i t c o n d i t i o n s , t h e r e a r e 29 p o s s i b l e c a s e s

a l t o g e t h e r j u s t f o r on e p i p e. F ur th er mo re , a d d i t i o n a l su b- ca se s e x i s t s i n c e

s to rm sewer f lows are uns teady . For example , f o r open-channel f low th e sub-

c a s e s c a n be w i t h a r i s i n g , f a l l i n g , o r s t a t i o n a r y f r e e s u rf a c e . F or t h e c a se s

w i t h a hy dr au li c jump o r dro p, t h e jump o r dro p may be moving upst rea m, down-

s t r e a m ,- o r s t a t i o n a r y . .

TABLE 3.3. PIPE FLOW CONDITIONS

P o s s i b l e P o s s i b l e

Case Pi pe F low Ent ra nce Case Ex i t Case

1 S u b c r i t i c a l I , I11 A Y B

2 S u b c r i t i c a l -t h y d r a u l i c d r o p -t s u p e r c r i t i c a l I , I11 B Y C

3 S u p e r c r i t i c a l 11, I11 B Y C

4 S u p e r c r i t i c a l -t hydrau l ic jump -t s u b c r i t i c a l 11, I11 A Y B

5 S u p e r c r i t i c a l -t hydrau l ic jump -t s u r c h a r g e 11, I11 D

6 Su rch a rg e -t s u p e r c r i t i c a l I V B Y C

7 Su rch a rg e -t s u b c r i t i c a l I V A y B

8 S u b c r i t i c a l -t s u r c h a r g e I , I11 D

9 S u p e r c r i t i c a l -t s u r c h a r g e

1 0 Su rch a rg e

I t s h o ul d b e m en ti on ed h e r e t h a t t h e f l o w c o n d i t i o n s g i v e n i n T ab l e 3 . 3

a p p l y t o s t e a d y f l o w as w e l l . However, Case 6 i s r a r e f o r u n st e ad y f l o w a nd

d o es n o t o ccu r a t a l l f o r s t e a d y f l ow . T h e r e f o r e , 27 p o s s i b l e s t e a d y f l o w

c a s e s e x i s t , o f w hi ch s ome are r a t h e r r a r e , e .g . , p i pe f low Cases 2 , 7 , and 9

s el do m o ccu r i n s t ead y f l o w. T he ca s e s d e s c r i b ed i n Chow (1 9 5 9) , Po r t l an d

Cement Asso c ia t ion Handbook (1964) , Bodhaine (1968) , and o t he r l i t e r a t u r e a r e

t h e m aj or c a s e s t h a t are o f t en o b s e rv ed . A s an example, Bodhaine 's ty pe s of

f l ow shown i n F i g . 2 .2 a r e i d e n t i f i e d i n Ta b le 3 .4 a c co r di ng t o t h e c l a s s i -

f i c a t i o n s i n T ab le 3.3.



TABLE 3.4. IDENTIFICATION OF BODMINE'S TYPES OF PIPE FLOW

~ o d h a i n e ' sType of C l a s s i f i c a t i o n Acco rd in g t o

Flow (Fi g. 2.2) Tab le 3.3.

1 11-3-B o r C

3.1 .2. Hydrau l ic Behavior of Flow i n Si ngl e Pi pe

There a r e a number of unsolved hydrodynamic problems encoun tered d urin q

t h e p ro ces s o f t h e f l o w i n a s t orm sewer s t a r t i n g from d r y o r n e a r l y d r y bed

t o s u rch a rg e an d t h en b ack t o t h e n ea r l y d ry bed. The p roblems o f a i r en t ra in -

ment, en t ranc e and e x i t los se s of th e p ipe a t connec t ing manholes , and a

moving sur fa ce di sc on t i nu i t y (moving hyd ra ul i c jump o r drop) have been

ment ioned i n Se c t ion 2 .3 .

Yen ( 1 9 7 8a ) d e sc r i b e d f i v e t y p e s o f h y d r a u l i c i n s t a b i l i t i e s i n sew er

system s, one of which i s t h e su rg e i n s t a b i l i t y of a n et wo rk . The o t h e r fo u r

t y p e s o f i n s t a b i l i t i e s o cc ur i n s i n g l e se we r p i p e s an d a r e d i s c us s e d b r i e f l y

a s f o ll o ws .

(A) A n ea r d ry -b ed f l ow i n s t a b i l i t y wh ich i s dominated by t he su r fa ce t e ns ion

e f f e c t . T hi s i n s t a b i l i t y i s not impor tan t f o r su rcharged f low.

(B) The t r a n s i t i o n i n s t a b i l i t y b etween s u p e r c r i t i c a l and s u b c r i t i c a l f lo w.

A sm en ti on ed p re v i ou s l y, t h i s i n s t a b i l i t y

i sp a r t i c u la r l y d i f f i c u l t t o

h a nd l e i f t h e f l o w i s u n s t ead y and t h e s u r f ac e d i s c o n t i n u i t y i s moving.

(C) W ate r- su rf ace ro ll -wav e i n s t a b i l i t y w hi ch i s d omin at ed by g r av i t y e f f ec t s

and usu al ly a ss oc ia te d wi th open-channel f low having a Froude number

g r e a t e r t h a n 2. F igure 3 .4 i s a s k e t c h of t h e r o l l wa ves. I f t h e h e ig h t of



Fig. 3.4. Roll Waves in a Sewer

the roll wave is large in comparison with the size of the sewer pipe, full-

pipe flow may occur intermittently because of the roll waves, especially

if tllc air entrainment problem occurs simultaneously.

(D) The instability at the transition between open-channel flow and full

conduit flow. This instability is most relevant to sewer surcharge.

There are several factors causing this instability, including (a) non-

unique discharge-depth relationship when the pipe is nearly full,

(b) insufficient air supply to maintain an air pocket at the pipe

entrance, (c) surface tension effect of the pipe crown when the pipe is

nearly full, and (d) surface waves, especially roll waves. These

factors may act individually or in combination to cause the instability

problen.

To illustrate the first factor of non-unique discharge-depth relationship,

consider the relatively simple case of steady flow in a circular pipe as an

example. The nondimensional discharge-depth relationship for steady, uniform,

open-channel flow and the discharge-piezometric pressure gradient relationshop

for steady uniform flow in a closed conduit is shown schematically in Fig. 3.5.

In the open-channel flow regime, the maximum discharge does not occur at the



d e p th , h , e q u a l t o t h e p ip e d ia m e t er , D. I t o ccu r s a t ap p ro xi m a t el y h = 0 . 9 4 D ,

v a ry i n g s l i g h t l y depend in g on t h e R ey no ld s number of t h e f low. T h i s d e c r e a s e

i n d i s ch a rg e when t h e p i p e i s n e a rl y f i l l e d i s d ue t o t h e r a p i d i n c r e a s e i n

we t t ed p e r i m e t e r as h approaches D , a nd t h e c o ns eq ue nt i n c r e a s e i n t h e p i p e

b ou nd ar y r e s i s t a n c e t o t h e f l ow . A s shown i n F i g . 3 . 5, t h e r e l a t i o n s h i p

b et ween t h e d i s ch a rg e and d ep t h o r p i ez o m e t r i c g rad i en t i s unique above point E

o r b e lo w p o i n t J . Between points J and E a g i v e n d i s c h a r g e c a n h a v e d i f f e r e n t

d e p t h s o r p i e z o m e t r i c g r a d i e n t . .TO i l l u s t r a t e t h e second f a c t o r of i n s u f f i c i e n t a i r s up pl y i n t h e p i pe

t o m a i n t ai n a s t a b l e a i r p o c k e t , c o n s i d e r t h e s i m p le c a s e of a submerged pipe

e n t r a n c e as shown i n Fig . 3 .6. Assume t h a t i n i t i a l l y t h e d i s c ha r g e e n t e r i n g

t h e p i p e i s Qe c o r r es p o nd i n g t o t h e m an hol e d e pt h a nd w a t e r s u r f a c e p r o f i l e

a w i t h t h e a i r pocket shown i n Fig . 3.6. T h is p r o f i l e i s c l a s s i f i e d as

Type 111-5-D wi th s ma ll e x i t submergence i n Table 3 .3 . S i n ce t h e s ewer

i s n o t ventilated and i t s downstream part i s s ea l ed by t h e h y d rau l i c jump i n

t h e p i p e , e n t r ai n m en t of t h e t r a pp e d a i r i n t o t h e f l o wi n g w a t e r c r e a t e s a low

p r e ss u r e i n t h e a i r p oc ke t ( a s i t u a t i o n s imi la r t o u nd e r - ve n t il a t e d w e i r o r

s l u i c e g a t e) . S u bs e qu e nt l y, t h e d i s c h a r g e i n t o t h e s e w er i n c r e a s e s w h i l e t h e

dep th i n t h e ups tream manhole d rops . O n e p o s s i b i l i t y i s t h a t t h e hi g h er

d i s ch a rg e (> Q ) p u sh e s t h e h y d r a u l i c jump o u t s i d e t h e p i p e , a l l o w i n g a i r t oe

e n t e r , r e s u l t i n g i n a tm os ph er ic p r e s s u r e f o r t h e a i r i n t h e p ip e. T h i s i s

shown as s u r fa c e p r o f i l e X i n F i g. 3.6 and i s Case 111-3-C i n Ta bl e 3.3.

C o ns equ ent l y, t h e d i s ch a rg e d ro ps (< Qe) , t h e h y d rau l i c jump o ccu r s i n s i d e

t h e p i p e a g a i n , a nd t h e c y c l e r e p e a ts .



Rat io o f Discharge to Pipe-Full Discharge, Q/Qf

Fig. 3.5. Discharge Rating Curve for Steady Flow

in a Circular Pipe

Fig. 3.6. Air Entrainment Instability in a Sewer



3.1.3. Mathematical Representation of Flow in a Pipe

The open-channel phase of the sewer flow can be represented mathematically

by a pair of partial differential equations of hyperbolic type (Yen, 1973,

1975)

-- l a fi 2 a 1 aA+-- - Q ) + cos0 - Kh) + (K - K') h cos0 - -

~ at g~ ax A ax A ax

in which Q is discharge; t is time; x is the distance along the pipe longi-

tudinal direction; A is flow cross sectional area perpendicular to x; h is

flow depth measured normal to x; 0 is the angle between the sewer axis and a

<

horizontal plane (Fig. 3.7); S = sin 8 is the sewer slope; S is the friction0 f

slope; f3 is a momentum flux correction factor; K and K' are correction

factors for nonhydrostatic pressure distribution; T represents the force due

to internal stresses acting normally to A; y is the specific weight of the

liquid, assumed incompressible and homogeneous; and g is gravitational

acceleration. Equation 3.1 is the continuity equation and Eq. 3.2 is the

momentum equation. They are derived from the principle of conservation of

mass and Newton's second law, respectively.

Because of the difficulties in solving Eqs. 3.1 and 3.2, in practice

they are simplified by assuming f3 = 1 (uniform velocity distribution over A),

hydrostatic pressure distribution (K = K' = l), and neglecting the last term

in Eq. 3.2 containing T. The result is the well known complete (but not

exact) dynamic wave or Saint Venant equations,



Alternatively, Eqs. 3.1 and 3 . 3 can be expressed in terms of the average

velocity, V, over the cross sectional area A, i.e.,

in which B is the water surface width.

In the surcharged phase, the flow cross-sectional area is csnstant equal

to the full pipe area Af. The continuity and momentum equations can be

written as

in which P is the piezometric pressure of the flow and other symbols area

as previously defined. For a pipe having constant cross-section and flowing

full throughout its length, av/ax = 0. By neglecting the spatial variation of

B and T, integration of Eq. 3 . 7 over the entire length, L, of the sewer

pipe yields

exitv2

= H - 1 av- K - = L (S + - - )u Hexit u 2g f g at



i n w hi ch H and Hexit a r e t h e t o t a l h ead a t t h e e n t r a n ce an d e x i t o f t h eU

p i p e , r e s p e c t i v e l y ( F i g . 3 . 8 ) ; a nd K i s t h e e n tr a nc e l o s s c o e f f i c i e n t .u

E qua t i ons 3 . 6 a nd 3 . 7 c an be de r i ve d a s a s p e c i a l c a s e o f E qs. 3 . 1 a nd 3 . 2.

T h i s , i n d e e d , i s t h e t h e o r e t i c a l b a s i s of t h e P re is sm an n s l o t t ec h n i q u e

(Preiss mann and Cunge, 19 61) .

The f r i c t i o n s l o p e , S f , i s us ua l l y e s t i m a t e d by u s i ng M a nn ing 's f o r m u l a

o r t h e Darcy-Weisbach formula ,

i n w h ic h n i s Manning 's roughness f a c t o r ; f i s t h e W ei sb ac h r e s i s t a n c e

c o e f f i c i e n t ; a n d R i s t h e h y d r a u l i c r a d i u s w hi ch i s e q ua l t c A d i v i de d by

t h e w e t t e d p e r i m e t e r . T he se two e q u a t i o n s a r e a p p l i c a b l e t o b o t h s u rc h ar g ed

a nd ope n- c ha nnel f l ow s . F o r t h e open - channel c a s e t h e p i pe i s f l ow i ng

p a r t i a l l y f i l l e d and t h e g e om e tr ic pa ra me te rs o f t h e f lo w c r o s s s e c t i o n a r e

computed a s fo l low s

D~A = - @ s i n @ )8

s i n @R = D- ( 1 - ---4 @

1

@R = D s i n -



i n w hi ch D i s t h e d i a me t e r of t h e p i p e a nd $ i s t h e c e n t r a l a n g le i n r a d ia n s

d es c r i b ed b y t h e wa t e r s u r f a ce h avi n g a w i d t h B ( F i g . 3 . 7 ) . I f t h e f l ow i s

assumed s te ady and uniform, Eqs. 3 .3 o r 3 .5 reduc e t o So = Sf and Q = AV.

Hence, from Eq. 3 .9 f o r s te ady uniform flow usi ng Manning's formula

i n which t h e c o n s t a n t C = 0 .0 73 7 f o r E n g li s h u n i t s a nd 0 .0 49 6 f o r S I u n i t s .

Corr espo nding ly , t h e Darcy-Weisbach formula (Eq. 3 . 1 0 ) y i e l d s

Advanced techn iques f o r so l v i ng s to rm sewer f low problems us ua l l y adop t

Eqs . 3 .1 and 3 .3 o r Eqs . 3 .4 and 3 .5 t o s imul a te t he open-channel f low phase

o f t h e s ewer f l o w and E qs . 3 .6 and 3 .8 t o s i m u l a t e t h e s u rcha rg ed f l o w.

3.2. Flow i n a Sewer Network

3 . 2 . 1 Sewer J u n c t i o n s

The s ewers i n a n e t work a r e j o i n ed by m an ho le s an d j u n c t i o n s . T h e re a r e

one-way m a n ho l e s. o r j u n c t i o n s , f o r w hi ch t h e r e i s on ly one p ip e connec ted t o

th e manho le . Th is i s t h e c a s e f o r t h e m os t u p st r eam manh ol e whi ch r ece i v es

s u r f ac e ru n o f f d i r ec t l y t h ro u g h t h e i n l e t s . Two-way j u n c t i o n s h ave two p i p es

co n nec t ed t o a j u n c t i o n . They a r e u s u a l l y pro v i ded f o r a chan ge i n a l i g n m en t ,

p i p e s i z e , o r s l o p e . A t hr ee -w ay j u n c t i o n ( o r Y - ju nc t io n) h a s t h r e e p i p e s

co n nec t ed t o i t . A fou r-way (o r fo r k ) j u n c t i o n h as fo u r p i p es co n n ec ted t o

t h e j u n c t i o n . J u n c t i o ns j o i n i n g more th a n f o u r p i p e s e x i s t i n s p e c i a l s i t u a t i o n s .

H y d r a u l i c a ll y , a j u nc t i o n im pos es t h r e e m aj or e f f e c t s . F i r s t , i t p ro v i d es

a s p a c e f o r s t o r a g e . S ec on d, i t d i s s i p a t e s t h e k i n e t i c e ne rg y of t h e f l ow

27



f rom jo in ing sewers . T h i r d , i t i mp os es b a ck wa te r e f f e c t s t o t h e s e we rs

connected a t t h e j un c t i on . The p r e c i s e h y d r a u l i c d e s c r i p t i o n of t h e fl ow a t

s ew er j u n c t io n s i s r a t h e r c o m p l ic a te d and d i f f i c u l t b e c au s e o f t h e h i g h d e g r e e

o f f l o w mixin g, s ep a r a t i o n , t u r b u l en ce , an d en e r gy lo s ses . Y et co r r e c t

r e p r e s e n t a t i o n of t h e j u n c t i o n h y d r a u l i c s i s i mp or ta nt i n r e a l i s t i c a nd

r e l i a b l e c o m pu ta ti on o f f l o w i n sewer sys tems .

M a t he m a ti c a ll y , t h e j u n c t i o n h y d r a u l i c c o n d i t i o n i s u s u a l l y d e s c r i b e d

by a co nt in u i ty equa t ion and somet imes a ide d by an energy equat i on . The

momentum equation i s r a r e l y u s e d b ec a u se i t i s a v e c t o r r e l a t i o n s h i p a nd t h e

chan ges of momentum and f o r ce s a r e d i f f i c u l t t o ev a lu a t e i n a junct ion . The

p r i n c i p l e o f c o n s e r v a t i o n o f mass g i v e s t h e f o l l o w in g c o n t i n u i t y e q u a t i o n

i n w hi ch Q i s t h e f l o w i n t o o r o u t f rom t h e j u n c t i o n b y t h e i - t h j o i n i n gi

s ew e r, b e i n g p o s i t i v e f o r i n f l o w a nd n e g a t i v e f o r o u t f l o w ; r e p r e s e n t s t h e

d i r e c t , t e m po r al l y v a r i a b l e w a t e r i n f l ow i n t o ( p o s i t i v e ) o r t h e pumpage o r

l e a k a ge o ut from ( n e g a t i v e ) t h e j u n c t i o n , i f a n y ; s i s t h e s t o r a g e i n t h e

j u n c t i o n ; a n d t i s t ime. For a man ho le o f co n s t a n t c r o s s - s ec t i o n a l a r e a ,

A s = A.Y w h er e Y i s t h e d e pt h i n t h e j u n c t i o n ( F ig . 3 . 9 ) . N ot in g t h a t Yj ' J

i s r e l a t e d t o t h e man hole wa t er s u r f a c e e l e v a t i o n H by H = Y + z ( F ig . 3 . 8 ) ,

when z i s t h e e l e v a t i o n o f t h e j u n c t i o n b o tt om .

ds- - dl?

d t - Aj ;it-

I n r e g a r d t o t h e e n er g y r e l a t i o n s h i p , a n e x a c t e n er g y b u dg e t a cc o un t o f

th e f l o w th r o u g h ju n c t io n i s i m p r a c t i c a l ; i f n o t i m p o ss i bl e . I n s t e a d ,



i

v0



approximate energy expressions are assumed. For a submerged entrance from the

junction into a surcharged downstream pipe (Case IVY Fig. 3.1) the flow

2behaves like an orifice and the head loss through the entrance is K V /2g

U

where K is the entrance loss coefficient and V is the velocity in theu

downstream pipe. Accordingly, the instantaneous discharge from the junction

into the downstream pipe can be estimated from Q = AV where the velocity V is

given by Eq. 3.8. For a sharp-edged abrupt entrance, the value of K isu

approximately equal to 0.5 (Rouse, 1950).

For Case I11 in Fig. 3.1, where the entrance of the out-flowing downstream

pipe is submerged but the pipe is not filled, the flow near the entrance

behaves somewhat like a sluice gate. The outflow rate from the junction is

Q = AV where A and V are the flow cross sectional area and velocity at the

vena contracta, respectively. The veloctiy can be estimated by using

in which AH is the piezometric head difference between the water surface in the

junction and the vena contracta. The corresponding entrance head loss is

For the case of flow from the junction into a downstream sewer with a non-

submerged entrance (Cases I and I1 in Fig. 3.1) the water depth in the junction

is assumed equal to the entrance loss plus the specific energy of the flow at

the pipe entrance. Thus,



i n wh ich z i s t h e h e ig h t o f t h e p ip e i n v e r t ab ov e th e r e f e r en ce d atum and y

i s t h e d e p t h o f f lo w m ea su re d v e r t i c a l l y a t t h e e n t r a n c e o f t h e p i p e . N ote

t h a t f o r Case I1 i n F i g. 3 . 1 y = y a t t h e s ew er e n t ra n c e .C

. .

Now c o n s i d e r t h e i n f l o w s i n t o a j u n c t i o n . F o r a n u p s tr e am p i p e d i s c h a r g i n g

i n t o t h e j u n ct i on , i f t h e e x i t of t h e p i p e i s submerged (Case D , F i g . 3 . 2 . ) ,

t h e w a t e r s u r fa c e i n t h e j u n ct i o n i s assum ed e q u a l t o t h e t o t a l h ea d o f t h e

f lo w a t t h e p i p e e x i t minus t h e e x i t l o s s . Thus,

v22

v2

H = -- v -- K -1 e x i t+ I b ; - K d t - H e x i t d 2 g

i n w hi ch th e p i ezo met r i c h ead P /y i s measured f rom the r ef er en ce da tum, V i sa

t h e v e lo c i t y a t t h e p i pe e x i t , and K i s t h e e x i t l o s s c o e f f i c ie n t . I f t h ed

k i n e t i c e n er gy o f t h e j e n c t i o n i n f l o w i s assumed t o b e co mp le te ly l o s t ,

K = 1. T h i s i s a g r o s s a s su mp tio n b ecau se , u n l e s s t h e j u n c t io n i s v e r y l a r g e ,d

p a r t o f t h e k i n e t i c e n er g y may b e r e c ov e r ed i n t h e o u t f lo w fro m t h e j u n c t i o n

i n t o th e downstream sewer . This energy reco very depends on, among ot h er

f a c t o r s , t h e a li g n ment o f u p s t ream an d d ow ns tr eam p ip es and th e s i z e o f t h e

j u n c t i o n .

I f t h e u p st r eam in f lo w in g p ip e i s no t submerged a t i t s e x i t i n t o t h e

j u n c t i o n a nd t h e f l ow i n t h e p i p e i s s u b c r i t i c a l ( C a s e A and B i n F i g . 3 . 2 ),

t h en

i f y + z < H

o t h e r w i s e



i n which z i s t h e h ei g ht of t h e p i pe i n v e r t a t i t s e x i t measured above th e

ref ere nce datum, y i s t h e p i p e f l o w de p th m eas ur ed v e r t i c a l l y a t i t s e x i t , a n d

Yci s t h e c r i t i c a l d e p th c o rr es po nd in g t o t h e i n st a nt a ne o u s f lo w r a t e Q a t

t h e p ip e e x i t ( F ig . 3 . 9) .

For t h e abo ve t h r ee ca s e s o f f l o w i n t o a j u n c t i o n (submerged e x i t o r

s u b c r i t i c a l f l o w ) , t h e f l o w i n t h e u p st ream p i p e i s d i r e c t l y a f f e c t e d by t h e

water dep th i n th e junc t ion , excep t fo r the cond i t ion of Eq. 3 .22b (Case A ,

Fig . 3 .2 ) . Th i s i s commonly known a s th e downstream backwater ef f e c t . I f th e

f lo w a t t h e e x i t o f t h e u p st re am p i p e i s s u p e r c r i t i c a l ( C a s e s B an d C i n

Fig . 3 .2 ) , t h e f l o w i n t h e u ps tr eam p i p e i s n o t a f f ec t ed by t h e w a t e r d ep th i n

t h e j u n c t i o n . The w a t e r d ep th i n t h e j u n c t i o n i s determined by t he j unc t io n

con t inu i ty equa t ion (Eq. 3 .17) and th e energy equa t ion , i . e . ,

i f n o h y d rau l i c jump o ccu rs i n t h e j u n c t io n . I f a h y d rau l i c jump o ccu r s i n t h e

junc t ion which i s a h i g h l y u n l i k e l y ca s e ,

i n w hi ch I! r ep re s en t s t h e h ead l o s s of t h e h y d ra u l i c jump.f-jump

3.2. 2. Sewer Networks

W ith t h e h y d r au l i c s o f i n d i v i d u a l s ew er s d e s c r i b ed ma t hema t ica l ly a s

d i s cu s sed i n Sec t i o n 3 .1 and t h e h y d rau l i c s o f i n d i v i d u a l j u n c t i o n s ma th ma ti ca ll y



r e p r e s e n t e d a s d i s c u s s e d i n S e c t i o n 3 . 2 .1 , th e problem of s torm runof f i n

a se we r n et wo rk ca n t h e o r e t i c a l l y b e s o lv e d u s i ng p h y s i c a l p r i n c i p l e s . I n

t r u t h , no t a s i ng le se we r ne twork c ons i s t ing o f more tha n a f ew ( sa y 10 ) p ipe s

h a s s o f a r b ee n s o l ve d s a t i s f a c t o r i l y u s i n g t h e b a s i c h ydrodyn amic p r i n c i p l e s

wi thou t making use o f any as sum ptions o r a r t i f i c a l l y imposed c on t r o l s . There

a r e many r ea s o ns f o r t h i s d i f f i c u l t y , s uc h a s

( a ) th e S a in t Venan t e qua t ions a r e no t e xa c t (Yen, 1973, 1975) ;

( b) t h e fl ow r e s i s t a n c e c o e f f i c i e n t , w he th er i t i s i n th e Manning,

Chezy, o r Darcy-Weisbach form , i s unknown fo r unst eady nonuniform flow;

( c ) t h e e n er gy l o s s c o e f f i c i e n t s a t t h e j u n ct i o n s (Ku f o r p i pe e n t r a n ce

l o s s and Kd f o r p ip e e x i t lo ss ) a re geometry dependent and unknown

f o r uns tea dy f low;

( d ) t h e m a th e ma t ic a l d i f f i c u l t i e s o f s o l v i n g t h e S a i n t V en an t e q u a t i o n s

o r s i m i l a r h yp e rb o li c t y pe p a r t i a l d i f f e r e n t i a l e qu a ti o ns ;

( e ) t h e h y d r a u l i c i n s t a b i l i t y p ro bl em s i n c l u d i n g p r e s s u r e s u r g e and t h e

change between open-channel and pr es su r i ze d con duit f lows; and

( f ) t h e ba ckwate r e f f e c t , i . e . , t h e m utual depende nc e o f the f low i n

t h e c o n n e c t i n g p i p e s .

Some of t he se reaso ns do no t impose se r io us problems. For ins ta nce , th e

S a i n t Ve na nt e qua t i ons , a l though no t e xa c t , ha v e be e n shown t o be good

approximat ions even f o r su p er c r i t ic a l f low (Zovne, 1970) . The s teady- f low

r e s i s t a n c e and e ne rgy c o e f f i c i e n t s c a n be use d a s a ppr ox im a t ions . M a the ma t ic al

d i f f i c u l t i e s have been reduced wi th t h e improvement and development of computer

c a pa b i l i ty a nd num e r ic a l t e c hn ique s .

The hydraul ic s t a b i l i t y p r o b l e m s f o r a sewer p ip e have been d isc ussed

i n S e c t ion 3 .1 .2 . F o r a sewer ne twork wi t h the p ipe s su r c ha r ge d , t he r e i s a

p r e s s u re s u r ge i n s t a b i l i t y d ue t o t h e i n t e r a c t i o n betw een p r e s s u r i z e d c o n du i ts .



Su r g es i n t h e s ew ers a r e p r e s su r e w av es s im i l a r t o wa terhammer i n p ip e n e t -

w ork s f o r w hich p r ev io u s s t u d i e s have been r ev iew ed b r i e f l y i n Ch ap ter 2.

The surg es may be due to th e meeting of f l oo d waves f rom di f f e r e n t sewer

b r an ch es a t a j u n c t io n , d ue t o sud den su r ch a r g es o f man ho les o r p ip es , o r d ue

t o any o th e r ab r u p t ch an ge o f t h e f l o w . I t has even been observed i n many

lo c a t i o n s t h a t su r g es o f w a te r s p i l l ed o u t f ro m man ho les o n to t h e gr ou nd su r f a ce .

The t h e o r y t o a n a l y z e t h e s u r g e i n s t a b i l i t y h a s be en d e ve lo pe d. I t needs on ly

t o b e r e f in ed and ap p l i ed t o s ew er n e tw o r k s. I t should be mentioned t ha t

s i n c e p r e s s u r e i s t r an s mi t t ed immed ia t e ly , t h e su r g es i n t h e s ew er s an d man ho les

a r e m u tu a ll y r e l a t e d and t h e r e i s a p o s s i b l i t y o f r e so n an c e.

The pr im a ry r e a s o n f o r t h e d i f f i c u l t i e s i n s o l v i n g s ew er n et wo rk f l o w

a c c u r a t e l y u s i ng t h e p h y s i c a l p r i n c i p l e s i s t h e m u t ua l b a ck wa te r e f f e c t s

b etween th e sew er s. I n f a c t , a ma jo r d i f f e r e n c e b etween th e f lo w th r ou g h a

culver t and a sewer network i s t h e n et wo rk e f f e c t f o r t h e l a t t e r . The e f f e c t

of backwater i n a sewer ne twork cannot be over-emphasized , pa r t ic u l a r l y fo r

th e case o f su r cha r g ed s ew er s , a s can b e d emo n s tr a t ed t hr o ug h t h e f o l l o w in g

r a t h e r s i m p l i f i e d e x a m p l e .

Figure 3 .10 i s a schem atic drawing of a sewer i n t he Oakdale Avenue

Drainage Basin i n Chicago. The 10- in . sewer i s 1 70 f t l o ng r u n n in g n o r t h

a lo n g L ec l a i r e A venue f ro m an a l l e y t o t h e i n t e r s e c t i o n o f O a kd al e a nd

Le cl ai re Avenues. The sewer has a s lo pe of 0 .71% and a Manning roughness

f a c t o r n = 0 .0 14 . I n a c o n v e n t i o n a l c a l c u l a t i o n , t h e s ew er f l ow i s computed

u s i n g t h e s ew er s l o p e a s t h e f lo w s l o p e , i . e . , c o rr e sp o nd i ng t o t h e w a te r

s u r f a c e s U and C i n th e upst r eam and downst ream manholes , r e sp ec t i ve ly .

Using Manning's formula and assuming s t ea dy uniform f low, th e computed dis -

charge i s 1 .7 2 c f s . I n r e a l i t y , even f o r a given upstream manhole having



Ups t reammanho le

S 1op e---

Downst eammanhole

Di s cl-Ia r a e

c f s

I)

1.22

1.72

Fi g. 3.10. Backwater Ef fe ct on Di sc har ge of an Oakdale Avenue Sewer

a w a t e r s u r f a c e a t U , t h e d ow ns tream manh ole l e v e l can b e l ow er o r h i g h e r

t h an l e v e l C. I f t h e l e v e l i s l o we r , t h e d i s c h a r g e w i l l b e g r e a t e r t h an

1 .7 2 c f s ; w he re as i f t h e l e v e l i s h i g h e r , t h e d i s c h a r g e w i l l b e sma l l e r .

O b v iou s ly , i f t h e do wn st ream l e v e l i s a t A , t h e s ame e l e v a t i o n as i n t h e up-

s t r eam manhole , th e re w i l l b e no f l o w i n t h e s ewer .

The ac tu a l hydr au l ic phenomena, o f c ours e , a r e cons ider ab ly more compl ica ted

t h a n t h i s s i m p l i f i e d i d e a l i z e d e xa mp le b e ca u se t h e f l o w i s u n s t ead y and th e r e i s

more than one sewer i n th e ne twork impos ing mutual backwater e f f e c ts . Moreover,

i f t h e m an ho le s a r e f u l l y s u r c h ar g e d , t h e o v e r l a nd s u r f a c e f l o w b et we en t h e

man h o lesw i l l i n t e r a c t w i th t h e s ew er f l ow . I n mos t en g in ee r in g co mpu tat i on

o f s ewer f l o w s , t h e e f f e c t s of f l o w u n s t ead i n es s , o f t h e ch an ges o f w a te r

s u r f a c e s i n t h e c o n ne c t i ng j u n c t i o n s , and o f t h e i n t e r a c t i o n b et we en s u r -

charged sewer and sur fa ce f lows a r e s imply ignored . The sewer ca pa ci t y i s

simply computed as f u l l - p i p e g r a v i t y f l ow , e q u a l t o t h e f l ow u nd er t h e he ad

d i f f e r e n c e be tw ee n w a t e r s u r f a c e s U and C shown i n Fig. 3 .10. The excess ive

f low above th e computed sewer ca pac i t y i s o f t e n a ssumed t o s t o r e i n t h e

i mm ed ia te u p st r ea m j u n c t i o n . L i ke w is e , t h e w a t e r l e v e l i n t h e j u n c t i o n i s n o t

co mpu ted w h i l e t h e ex ces s s t o r ed w a te r i s assumed t o impose no e f f e c t on the



f lo ws i n t h e o t h e r sewers c o nn e ct i ng t o t h e j u n c t i o n . H a n d li n g t h e s u r -

charge i n suc h a manner i s simply errone ous, no t an approximation a s many

engi neers have thought , an d i t c an r e s u l t i n d an ge ro us a nd c o s t l y c o n c lu s i o ns

f o r urban sewer fl ow management.

A s d i s c u s s e d i n S e c t i o n 3 .1 . 1 , f o r a s i n g l e sewer t h e r e a r e 29 p o s s i b l e

f low cas es (Table 3 .3) . For a two-way ju nc ti on th er e ar e 29' = 8 4 1 p o s s i b l e

3c a se s . F o r a thr ee -way junc t ion , t h e r e a r e 29 = 24,389 c a se s . I n ge ne r a l ,

i

i f t h e r e a r e N pos s ib le f low c a se s i n ea c h se we r, f o r a n m-way junc t ion t h e '

p o s s i b l e c a s e s a r e fl a ssum ing no r e v e r sa l f low oc c ur s i n a ny one of th e

m sewers . For a three -way jun c t i qn , i f two of th e sewers a re inf low sewers

f o r which t he o r de r o f oc c u r r e nc e of th e f lows i n th e two se we r s a r e im m a te r ia l

an d i nt e r c h a n g ab l e ( e . g . , i n t h e f l ow i d e n t i f i c a t i o n , t h e f l o w c a s e o f 11-3-B

i n Sewer 1 and IV-10-B i n Sewer 2 i s c o ns i de re d a s t h e d u p l i c a t e o f t h e

re ve rs e ca se of IV-10-B i n Sewer 1 and 11-3-B i n Sewer 2) , t h e number of

2p o s s i b l e c a s e s i n N (N+1)/2 = 12,615. Likewise , f o r a four -way jun c t i on not

c oun t ing the dup l i c a te c a se s , and wi thou t r e ve r s a l f low i n any one of th e

2

p ipe s , t h e number o f po ss i b le f low c a se si s N

(N+l)(N+2)/6=

130,355. When

th e network s i z e expands from one junc t ion t o many ju nc t ion s , th e number of

p o s s i b l e c a s es i n c r e a s e s a s t ro n o m ic a l ly . T h is d em o ns tr a te s t h e d i f f i c u l t y

i n s o l v i n g p r e c i s e l y t h e s t or m r u n of f i n se we r n et wo rk s and i l l u s t r a t e s a

m a jo r d i f f e r e nc e i n c ons ide r ing a s in g l e p ipe and a sewer syst e m. Obviously,

i t i s n o t p o s s i b l e t o ac co un t f o r a l l t h e c a s e s i n any a t t em p t of u s i n g

Eqs. 3. 1 and 3.3 (o r 3.4 and 3.5) , 3.6 and 3.7 o r 3.8, and 3.17 throu gh 3.24,

whenever a pp l i c a b l e , t o so lve f o r t he s to r m wate r flow i n a sewer networ k .

T h e r e f o re , a s su m pt i on s an d s i m p l i f i c a t i o n s a r e n e c e ss a r y t o e x cl u d e t h e l e s s

impor tant cas es so t h a t th e problem becomes managable and so lva ble .



I V . KINEMATIC WAVE - SURCHARGE MODEL

4.1. Kinem atic Wave Appro ximat ion

B ecau se o f t h e co m p lex i ty i n s o l v i n g t h e Sa i n t Venant eq u a t i o n s f o r

unstea dy open-channel f lo ws, a number of app roxima tions have been used i n

so l v in g eng ine er i ng p roblems (Yen, 1977) . A p o p u l a r s i m p l i f i c a t i o n i s

t he k ine mat ic wave approx imatio n which has a s i mp li f ie d momentum equ at io n

o b t a i n e d by dr o pp i ng a l l b u t t h e l a s t two s l o p e t er m s i n Eq. 3 . 3 o r 3 . 5 , i . e . ,

The f r i c t i o n s l o p e , S f , i s nor mal ly approxim ated by th e Darcy-Weisbach formula

(Eq. 3 .10) o r Manning's formula (Eq. 3 .9) . The kin ema tic wave appro ximati on

i n v o l v e s s o l v i n g Eq. 4 . 1 w i t h a p p r o p r i a t e i n i t i a l an d b ou nd ar y c o n d i t i o n s ,

t o g e t h e r w i th a c o n t i n u i t y e q u at i o n

i n w hi ch t h e t e rm s a r e a s d e f i n e d p r e v i o u s l y i n Ch a pt er 3 . T h i s e q u a t i o n

c an b e i n t e g r a t e d o v e r a r e a c h o f t h e s ew er p i p e h av i ng a l e n g t h AL t o y i el d

i n w hi ch I and Q a r e t h e i n f l o w i n t o an d o u t f lo w fr om t h e r e a c h , r e s p e c t i v e l y :

s i s t h e w a te r s t o r a g e i n t h e r e a c h , and d s / d t i s t h e t i me r a t e o f c h an ge

o f s t o r ag e . T h i s eq u a t i o n i s general ly known as t h e s t o r a g e e q u at i o n i n

hydrology.

The s i m p l i f i ed f l o w eq u a t i o n s f o r a s u rch a rg ed p i p e f l o w cor re s p on d i n g t o

t h e open-channel k in ema tic wave eq ua t i ons can be obta ine d from Eq. 3 .8 by



n e g l e c t i ng t h e av/at t e rm. Wi th th e downs tream jun ct i on wa ter s u r fa ce ,Hd'

r e l a t e d t o H by Eq. 3 .21 (Fig. 3 .81, Eq. 3 .8 y i e l d se x i t

t o g e t h e r w i t h t h e p i p e f l o w c o n t i n u i t y e q u a t i o n

A l l t h e terms have been def in ed p rev iou s ly . The j u n c t i o n c o n t i n u i t y e q u a t i o n

2i s gi ve n by Eq. 3.18. Combining Eqs. 4 .3 and 3.6 and no t i ng t h a t A f = nD /4

where D i s t h e p i p e d i a m e t e r , o n e o b t a i n s

i n w hi ch

Only f o r s p e c i a l s i m p le c a s es t h a t a n a n a l y t i c a l s o l u t i o n c a n b e o bt a i ne d

f o r the open-channel f low k ine mat i c wave equat i ons . (Eqs . 3 . 1 o r 3.4 and 4 .1 ) .

I n g e n e r a l t h e s e e q u a t i o n s a r e s o l v e d nu m e ri c a l ly . In t h i s c h a p t er a

s i mp li f i ed kin ema tic wave-surcharge model c a l l e d SURKNET i s f o r mu la t ed t o

s imu l a t e ap p r o x ima tely t h e o pen- ch an ne l an d s u r ch a r g e f l ow s i n a sewer network.

A more s o p h i s t i c a t e d an d ac cu r a t e mo de l bas ed o n t h e dy namic wave eq u a t i o n s w i l l

b e p r e s e n t e d i n a s e p a r a t e r e p o r t w h i c h i s t h e j u n i o r a u t h o r ' s M.S. t h e s i s .



4. 2. For m u la ti on of SURKNET Model

The SURKNET Model i s fo rmu la ted based on Eqs . 4 . 1 and 4 .2 f o r open

channe l f low and Eqs . 4 . 3 a nd 3 . 6 f o r s u rc h a r g ed p i p e f l o w , t o g e t h e r w i t h

M an ni ng 's f o r mu l a (Eq. 3 . 9 o r 3 .1 5) t o e s t i m a t e t h e f r i c t i o n s l o p e . D e t a i l s

a r e d e s cr i be d a s f o ll o w s.

4.2 .1. Open-Channel Flow

I n s t e a d o f s o l v i n g t h e k in e m at i c wave e q u a t io n s d i r e c t l y a s i n a t r u e

ki ne ma tic wave model (Yen, 197 7), Eq. 4.2 i s r e w r i t t e n i n a f i n i t e d i f f er e nc e

f orm f o r a r e a c h ,

i n wh ich t h e s u b s c r i p t s 1 a nd 2 r e p r e s e n t t h e t i m es t and t = t + A t . I n1 2 1t h i s e q ua ti on , a l l t h e q u a n t i t i e s a t t im e t a r e known fr om t h e r e s u l t s o f t h e1p r e v i o u s t i me - st e p c om p ut at io n o r i n i t i a l c o n d i t i o n . The i n f l o w I 2 i s know from

th e ou t f low of th e p reced ing rea ch o r manhole . The two unknowns a r e

Q 2 and s To e s t i m a t e t h e s t o r a g e s , an approx ima t ion p roposed by Thol in2 'a nd Ke i f e r ( 19 60 ) b ase d on s p a t i a l i n t e g r a t i o n o f M an ni ng 's f o r m u l a i s

a d o pt e d a nd m o d i f ie d f o r p a r t - f u l l p i p e f l o w ,

E q u at i on s 4 . 6 a nd 4 .7 a r e s o l v e d n u m e r i c a l l y a s f o l l o w s .

(A) S e l e c t a t r i a l v a l u e o f Q and compute s from Eq. 4.7.2 2

(B) S u b s t i t u t e t h e t r i a l Q a n d c o r r e sp o n d i n g s i n t o Eq. 4 . 6 .2 2

(C) A d ju s t t r i a l v a lu e of Q2 '

(D) Repea t s t ep s A , By and C u n t i l c o n v e r g e n c e .

A t t h e e n t r a n c e o f a s e wer p i p e f ro m t h e m a nh o le , t h e i n f l o w , I , i n t o

t h e f i r s t r e ac h of t h e pi p e i s computed by usin g Manning 's form ula fo r

39



p a r t f u l l f l ow , Eq. 3 . 15 , w he re t h e c e n t r a l a n g l e 4 i s a f u n c t i o n o f t h e f low,

d e p t h a nd p i p e d i a m e t e r a s e x p r e s se d i n Eq. 3 .1 4.

The change o f wa t e r dep th i n a manhole i s computed f rom th e co n t in u i ty

equa t ion (Eq . 3.17) w r i t t e n i n a f i n i t e d i f f e r e n c e fo rm ,

i n w hi ch C i n d i c a t e s s umm atio n of a l l t h e i n f l o w s i n t o t h e m an ho le an d o t h e r

t e r m s a r e as d e f i n e d p r e v i o u s l y .

4 .2 .2 . Surcharged Flow

Under su rcha rge d con di t io n , t h e manhole co n t i nu i ty e qua t ion (Eq. 3 .17)

i n f i n i t e d i f f er e n c e form i s

i n w hi ch t h e m an ho le c r o s s s e c t i o n a r e a a t t i m e s t an d t may b e d i f f e r e n t2 1

when i t i s c o m p l e t e l y f i l l e d . C o n s i d e r i n g a sewer p ipe t o g e t h e r w i t h i t s

u p s t r ea m m an ho le a s a n e l e m en t , e l i m i n a t i n g HU2 by combining Eqs. 4.4 a nd

4 . 9 y i e l d s

i n w hi ch H i s t h e w a t e r s u r f a c e e l e v a t i o n i n t h e u p st re am man ho le a t t h eu l

t ime tl , Hd2 i s t h e w a t er s u r f a c e e l e v a t i o n i n t h e d ow ns tr ea m m an ho le a t t h e

t ime t and A and A a r e t h e u ps tr ea m ma nh ole c r o s s s e c t i o n a l a r e a s a t t2 ' 1 2 1

and t r e s p e c t i v e l y .'

Through t h e H t e r m , t h i s e q u a ti o n c l e a r l y i n d i c a te s t h a t t h e f lo ws i nd2

t h e s u r c h a r g e d p i p e s a r e i n t e r r e l a t e d a nd s h o u ld b e c on s i d e re d a s a s y st em .



However, f o r th e p res en t model, i n o rde r t o make t he p ip e computa t ion sequence

co mp a tib l e w i th t h e k in emat i c mod el , n amely so lv in g t h e p ip e s i n a cascad in g

manner, p ip e by pi pe , from upstrea m towards downstream, i t was decided th a t

an approx imat ion i s assumed on H SO t h a t each p ip e can b e so lv ed in d ep en d -

d2

e n t l y . A f u r t h e r r e a s o n i s t h a t SURKNET i s only an approximate model and a

more sop hi st ic at ed dynamic wave i s a l s o b e in g d ev e lo p ed an d w i l l b e r ep o r t ed

soon. The assum ption adopte d i s t h a t H i n Eq. 4.10 i s approximated by t hed2

d e pt h a t t h e p r e v i ou s t i m e , Hdl. The consequence of t h i s assumpt ion may be

sev e r e u n de r u n fav o r ab l e co n d i t i o n s .

When th e wate r s ur fa ce i n a manhole r eache s th e g round , su r fa ce ponding

i s assumed with out volume l i m it at io n. The impounded wat er on th e sur fa ce i s

assumed t o r e t u r n t o t h e s ame man ho le a t a l a t e r t ime w i th o u t any lo s s es . No

in t e r- man h ole su r f ac e fl o w i s d i r e c t l y a l l o w e d .

With t h e a s su mp t ion s j u s t d esc r ib ed , Eq. 4 . 10 can ea s i l y b e so lv ed a s a

q u a d r a t i c f u n c t i o n o f Q But v io l a t i o n s of t h e a s su mp tio n s i n t r o d u ce2

co mp l i ca t i o n s i n to t h e metho d. The f o l lo w in g d e sc r ib e s t h e p o s s ib l e con seq u en ces

of t he above fo rmula t ion .

1. The a ss u mp t io n s a r e s a t i s f i e d an d t h e a l g o r i t h m c on ve rg e s t o a c o r r e c t

v a lu e w i th in a f i n i t e number o f i t e r a t i o n s . No p ro blems a r i s e , an d th e r o u t in g

i s c om p le te f o r t h a t e l em en t a t t h e c u r r e n t t im e .

2. The a s su m p ti o ns a r e n o t s a t i s f i e d an d t h e r e q u i r e d d i s c h a r g e Q i s s o2

l a r g e t h a t i t d r a i n s more w a te r t h an t h a t s t o r ed i n t h e u ps t ream manh ole.

T h i s wou ld i n d i ca t e a d r y b ed s i t u a t i o n i n t h a t e l emen t w hich i s n o t

a l lowed . H en ce , t h e d i s ch a r g e , Q 2 , i s s e t t o t h e b a s e fl ow v a lu e , and t h e

r o u t i n g i s complete.

3 . The a s su m pt i on s a r e n o t s a t i s f i e d , a nd t h e r e q u i r e d d i s c h a r g e Q i s2

n o t l a r g e en ou gh to d r a in away a s u f f i c i en t amount o f s t o r ag e w i t h in t h e



s p e c i f i e d t im e i n t e r v a l . I n t h i s c a s e , t h e computed Q2 i s a c c e p t e d as t h e

s u r c ha r ge d e l e m en t d i s c h a r g e and t he ups t re a m m anho le de p t h a nd s t o r a ge va l ue s

a r e s e t t o t h e va l ue s c o n s is t e nt w i th Q2 '

The r e a d e r s hou l d be aw ar e t h a t t h e i m pe t us f o r m ak ing t h e a bove s i m -

p l i f y i n g a ss um p ti on i s t h e na t u r e of t h e k i ne m a t i c wave r o u t i ng sc he me . S i nc e

t h e downst ream boundary c on di t io n i s n o t a cc ou n te d f o r i n t h e a n a l y s i s , t h e

downstream manhole wat er dep th a t t h e new t im e i s n o t known. Th ere fo re , some

r e a s ona b l e a s s um pt i on m ust be m ade t o c a r r y o u t t h e s u r c ha r ge d f l ow com-

pu t a t i on s . H i ghe r o r de r sc he me s s uc h a s t h e d i f f u s i on wave o r dynam ic wave

do n o t e x h i b i t t h i s p ro bl em .

4 .2 .3 . F low Tr an s i t i on

F l o w t r a n s i t i o n , as d ef in ed i n t h i s s t u d y, i s the dynamic process

w he re by t h e f l ow c on d i t i on i n a g i ve n p i p e ' r e ve r t s be tw e en ope n -c ha nne l a nd

s u r c ha r ge d f l ow . For a s u f f i c i e n t l y heavy r a i n f a l l e ve nt , t h e f l ow w i l l

i n i t i a l l y b e o pe n- ch an ne l, t h e n make t h e t r a n s i t i o n t o s ur c ha r ge d f l ow . A s

t h e s t o r m p a s se s , t h e p r e s s u r i z e d c o n d i t i o n i s r e l i e v e d and t h e t r a n s i t i o n

back t o open-channe l f low i s made. The t r a n s i t io n problem i s a f u n c t i o n o f

c h a n n el g e om e tr y , s l o p e , r o u gh n e ss , l e n g t h , s u r f a c e t e n s i o n , r o l l w av es , a nd

t h e d e g r e e o f a i r e n tr a in m en t . I n t h e t r a n s i t i o n co m pu ta ti on , o f t e n b o t h

t h e hydrodynamic and numer i ca l i n s t a b i l i t i e s a r e involved . I n th e SLTRKNET

f o rm u la t io n , t h e i n s t a b i l i t y e f f e c t s o f r o l l w a v e s , s u r f a c e t e n s i o n , a nd a i r

e n t r ai n m e n t a r e n e g l e c t e d .

I n a s ew e r c o n s i s t i n g of a number o f r e a c h e s , t r a n s i t i o n i s assumed t o

p r opa g a t e t ow a r ds e i t h e r ups t r ea m o r downst re am r e a c h by r e a c h i n s uc c e s s i o n

( e . g . , F i g . 4 . 1 ) . However, i f a l ong c om pu t a t i ona l t i m e i n t e r v a l i s us e d ,

t h e t r a n s i t i o n may o c c u r s i m u l t a ne o u s ly i n m ore t h a n o ne r e a c h . D ur i ng t he



transition propagation, surcharged reaches are assumed to carry the just-full

pipe discharge until transition is completed for the entire sewer.

Transition between open-channel flow and surcharge flow is assumed to

follow the trajectory JFE of the discharge rating curve shown in Fig. 3.5,

adjusted for flow unsteadiness, instead of JGFE. However, to avoid repeated

computations of the rating curves for different degrees of unsteadiness, it is

arbitrarily assumed that the transition point J occurs at the flow depth to

pipe diameter ration, h / ~ , qual to 0.91. The same assumption was made in the

French model CAREDAS mentioned in Chapter 11. A better assumption would be to

assume the transition from open-channel flow to surcharged flow to follow

.TGE and from surcharged flow to open-channel flow to follow EFJ. However,

because of the loop nature of JGEFJ for the rising and falling discharges, a

computational stability problem may occur if the discharge changes slowly near

this transition region. Therefore, the simple version is assumed in SURKNET.

For the case of downstream propagation of the transition, when the out-

flow of a reach approaches the just-full pipe discharge, transition is assumed

to occur in that particular reach.

For the case of upstream propagation of transition to surcharge, when

surcharge occurs at the upstream end of a reach, the immediate upstream reach

is check for transition. If the upstream reach is not already surcharged or

undergoing transition for surcharging, this upstream reach is assumed to be

surcharged during one time increment of computation. If the flow in the up-

stream reach is originally supercritical, this moving transition is essentially

a simplified view of a upstream-moving hydraulic jump.

For the upstream propagation of transition from surcharged flow to open-

channel flow, instead of considering the transition reach by reach, an

assumption is made that when the water surface at the exit of the sewer drops



be low t h e p ip e c rown, t h e e x i t i s no l ong e r s ubm er ge d and t h e e n t i r e s ew er

w i l l become open-channel f low i n one t ime increme nt . T h i s a s s um pt i on i s

nece ssa ry because t h e k inemat ic wave approximat ion i s u n a b le t o a cc o un t f o r

t h e dow ns tr ea m ba c kw at e r e f f e c t .

4 .2.4. Boundary Con di t i ons

I n SURKNET t h e sewer network i s d i v i de d i n t o e l e m e n t s . Each e l em e n t

c o n s i s t s o f a s ew er t o g e t h e r w i t h t h e m an ho le c o n n e ct e d t o i t s ups t ream. For

s u r c ha r ge d c ond i t i o n , t h e e l em e nt i s a na l yz e d as a u n i t by s o l v i n g E q . 4.10.

F o r o pe n- ch an ne l a nd t r a n s i t i o n c o n d i t i o n s , t h e f l o w i s s o l ve d s e p a r a t e l y f o r

t h e m an ho le a n d r e a c h e s of s e we r s u s i n g t h e a p p r o p r i a t e e q u a t i o n s g i v e n i n

S e c t i on 4 . 2. 1 . The s o l u t i on depe nds on t h e boundar y c on d i t i o ns , i . e . , t h e f l ow

c ond i t i o ns i mpos ed a t t he e nds of t h e m anhol es o r sewers. T he i nhe r e n t

a s su m pt i on s o f t h e k i n em a t i c a p pr o xi m at i on p r e c l u d e t h e c o n s i d e r a t i o n of t h e

downst ream boundary e f f ec t s . However, i n SURKNET a n a t t e m p t i s made t o account

f o r d ow ns tr ea m b a ck w at e r e f f e c t when t r a n s i t i o n t o s u r c h a r g e i s pr opa ga t i ng

towards ups t ream as d i s c u s s e d i n S e c t i o n 4 . 2. 3.

The ups t ream boundary con di t io n f o r any e lement i s th e manhole in f lows

d i r e c t l y i n t o t he ma nhol e fr om s u r f a c e a nd fr om upst r ea m s e w e rs . T he upst r ea m

boundary co nd i t i on of t h e sewer i s t h e o u t f l o w f ro m t h e m an ho le . I f t h e

sewer i s c om pl e t e l y s u r c ha r ge d a nd i t s upstream end submerged, t h e e lement i s

so lv ed as a un i t (Eq . 4 .10) and th e boundary con di t ion be tween th e manhole

a nd t he f o l l o w i ng s e w e r i s not needed . F p r t r a n s i t i o n a nd open -c ha nne l f l ow

i n t h e p i p e, t h e bo un da ry c o n d i t i o n s a r e e l a b o r at e d a s f o l l ow s .

( a ) The upstream manhole i s f i l l e d b elow t h e crown a t t h e e n t r a n c e o f

t h e f o ll o wi n g p i p e . - I n t h i s c a s e , t h e s ewer ( a t l e a s t f o r i t s

e n t r a n c e r e a ch ) i s open-channel flo w. The upstrea m manhole wa te r

s u r f a c e e l e v a t i o n i s computed by usi ng Eq. 4.8 . The i n i t i a l i ~ f l o w s ,



o u t f l o w , a nd w a t e r s u r f a c e e l e v a t i o n of t h e m an ho le a t t h e t i m e

t = t a r e a l l known. T he i n f l o ws i n t o t h e m an ho le a t t h e t i m e1t = t + A t a r e a l s o known a s t h e b o un d ar y c o n d i t i o n . T he m an ho le

2 1o u t f l o w , Q 2 , a t t h e t i me t i s approxim ated by u sin g Manning 's

2

f o rm u la f o r p a r t - f u l l p i p e f l o w , Eq. 3 . 15 , i n w hi ch $ i s a known

f u n c t i o n of t h e w a te r s u r f a c e e l e v a t i o n H a t t h e t i m e t2 2 '

Ac c o r d i n g l y , H 2 an d Q 2 c an b e s o l v e d a nd i n t u r n u se d as t he up-

s t r e a m b ou nd ar y c o n d i t i o n f o r t h e e n t r a n c e r e a c h o f t h e o p en -c ha nn el

se wer f lo w . T h e f l o w i n t h e s ewe r r e a c h i s d e t e r m i n e d by so l v i n g

Eqs . 4 .6 and 4 .7.

The manhole i s f i l l e d ab ov e t h e c ro wn o f t h e f o l l o w i n g p i p e b u t

b el ow t h e g ro u nd s u r f a c e ( C ase M i n F ig . 3 . 9 ). - I n t h i s c a s e, t h e

sewer f low can be open-channel (Case I11 i n F i g . 3 . 1) o r s u r ch a r ge d

(Case I V i n F i g. 3 . 1 ) . F or t h e l a t t e r , t h e e le me nt i s su r c h a r g e d ,

Eq. 4.10 i s a p p l i c a b l e a nd no b ou nd ar y c o n d i t i o n a t t h e e n t r a n c e o f

t h e s e we r i s r e q u i r e d . F o r t h e f o r m er , t h e m an ho le w a t e r s u r f a c e

e l e v a t i o n H a t t im e ti s

given by Eq. 4 . 8 i f Q i s p r o v i d e d .2 2 2

S t r i c t l y s p ea k in g , t h e d is c h a r g e i n t o t h e p i p e i s a f u n c t i o n of t h e

w a t e r s u r f a c e e l e v a t i o n i n t h e u p s tr e am m an ho le a s w e l l a s t h e d own-

s t re a m c o n d i t io n . T h e o r e t i c a l l y , Q can be computed by using a2

s l u i c e g a t e f or mu la , e q u al t o t h e f l ow c r o s s s e c t i o n a l a r e a A a t

t h e v en a c o n t r a c t a m u l t i p l i e d b y t h e v e l o c i t y g i v e n by Eq. 3 .1 9.

However, s i n c e t h e t i m e o f o c c u r r e n c e of t h i s p h ase i s u s u a l l y s h o r t

i n co m pa ri so n t o t h e t im e i n cr e m en t o f t h e r o u t i n g , f o r t h e s a k e of

s i m p l i c i t y , t h e m an ho le o u t f lo w i s assumed e q u a l t o t h e j u s t - f u l l

p ip e d i sc ha rg e g iven by Manning ' s fo rmula . Th i s manhole ou t f low i s

t h e n u s ed a s t h e u p s tr e a m b ou nd ar y c o n d i t i o n f o r t h e f o l l o w i n g

sewer reach .



( c ) The manhole i s f i l l e d and p on di ng o ccu r s on g ro un d s u r fa ce (Case N

i n F ig . 3 .9 ) . - The manho le water budget i s de sc ri be d by Eq. 4 . 9 i n

which H i s e s s e n t i a l l y e q u al t o t h e g ro und e l e v a t i o n , b u t

A2H2-

AIHl-

AS whichi s

t h e change of w at er volume on t he ground.

This volume, A s , can be computed i f Q i s known. Again , a s de sc ri be d2

i n t h e p r e ce d in g p a r a g ra p h, Q s h o u ld b e e v a l u a t e d by u s i n g t h e s l u i c e2

g a t e e q u a ti o n w i t h t h e h ea d e q u a l t o t h e d i f f e r e n c e be tw een t h e

g ro un d e l e v a t i o n and w a t e r s u r f a c e e l e v a t i o n a t t h e ve n a c o n t r a c t a .

Once more fo r th e sake of s i mp l i c i ty , th e sewer ups t ream boundary

c o n d i t i o n i s approximate i n SURKNET by th e j u s t - f u l l p i pe d isc har ge

giv en by Manning's for mul a.

4. 3. SURKNET Computer Pr og ram

The SLTRKNET co mputer pr og ra m was w r i t t e n i n ASA FORTRAN f o r running on

th e Uni ver s i ty o f I l l i n o i s CDClCYBER 175 d i g i t a l computer sys tem. I t c o n s i s t s

o f n ea r l y 1 00 0 s t a t em en t s . A f l o w ch a r t o f t h e prog ram i s shown i n F ig . 4 .2 .

S i n ce no u s e r ' s gu ide was p repare d , th e program i s n ot l i s t e d i n t h i s r ep o rt .

However, a copy o f t he l i s t i n g i s av a i l a b l e a t t h e Hy dros ys tems L ab o ra t o ry o f

t h e U n iv e r s it y o f I l l i n o i s a t Urbana f o r i n s p ec t i o n.

4. 4. Example A p p l ic a ti o n of SURKNET

The SLTRKNET model was a p p li e d t o a h yp o th e t ic a l 5-p ipe sewe r netw ork a s

an example. The n e tw or k p r o p e r t i e s and t h e s i m u l a t i o n r e s u l t s a r e p r e s e nt e d

i n t h i s s e c t i on .

4. 4. 1. Example Sewer Network

The examp le s ewer n e t wo rk co n t a i n s f i v e s ewers of d i f f e r en t l e n g t h s ,

d ia me te rs , and s l op es a s shown i n Fig . 4 .3 and Tab les 4 .1 . The Manning rough-

n e s s f a c t o r i s 0 .0 12 f o r a l l t h e s ew e r s. The m anh ole p r o p e r t i e s a r e g i v en i n



+

r s e t UP network 1,

k As sol ate element I

F i g . 4 . 2 . Flow Ch ar t f o r Computer Program SURKNET

48



T a b le 4. 2. The s ew e rs a r e i n v e r t - a l i g n e d a nd t h e i n v e r t s a r e 6 i n c h e s a bo ve

t h e b ot to m o f t h e c o nn e c t in g m an ho le . The p i p e i n v e r t e l e v a t i o n a t t h e o u t l e t

( ro o t n o d e ) i s l o ca t ed s u ch t h a t t h e minimum s o i l co v e r r eq u ir em ent abo ve

t h e s ew er pi p e c an be s a t i s i f i e d .

I n t h e e x a m p l e , i d e n ti c a l i n f l o w hy dr og ra ph s a r e a p p l i e d t o e a c h of t h e

f i v e m an ho le s. No d i r ec t i n f l o w h y dro graph en t e r s t h e o u t l e t , (Ro ot Node

n u mb e r6 ) . The v a l u e s of t h e hy dr og ra ph a r e l i s t e d i n T a b le 4. 3.

I n t h e n um e r i ca l c o m pu t at i on , t h e t i m e i n t e r v a l u s e d i s A t = 3 0 s ec

a nd t h e s p a ce i n t e r v a l i s AL = 1 00 f t f o r a l l s ew er s e x ce p t Sewe r 2-3 f o r

wh ic h AL = 50 f t .

TABLE 4 .1. SEWER PROP ERTI ES OF EXAMPLE NETWORK

Sewer Length Slo pe Diamet er

( f t ) ( f t >

TABLE 4.2 . MANHOLE PROPERTIES OF EXAMPLE NETWORK

Manhole Ground Manhole Manhole

E leva t ion Dep th Diameter

( f t ) ( f t ) ( f t )

1 51.10 14.0 4

4 48.00 12. 0 3

5 46.70 11.0 6

6 -- Out le t (Roo t node)

p i p e i n v e r t e l e v a t i o n = 3 5. 45 f t .



4 .4 .2 . R e su l t s

The computer s im ul at io n re s u l t s of t h e example us in g t h e SURKNET model

a r e p l o t t e d i n F ig s. 4 . 4 t h ro u gh 4 .8 , r e s p e c t i v e l y , f o r t h e f i v e s e we r s. I n

e a ch of the se f i gu r e s , t h e d i sc ha r ge p l o t shows th e in f low hydr ogr aph in t o

th e ups t ream manhole and th e out f low hydrograph a t th e ex i t of t he sewer. The

s to r a g e p l o t s shows th e combined i n - l ine s to r a g e of wa te r i n the sewer a nd

th e manhole connec ted t o i t s upst r ea m e nd, a s we l l a s t he s u r f a c e ponding

above ground when t h e manhole i s c o m p l e t e l y f i l l e d .

The manhole inf l ow hydrographs were purpose ly se le c t ed w i th a h igh peak

d i s ch a r ge i n o r d er t o g e n e r at e s e ve r e s ur c ha r g e i n a l l t h e s e we rs . C on si de ri ng

t h a t t h e SURKNET model was meant o nly a s an approx imate si mu la ti on of th e flo w,

th e r e su l t s a r e inde e d r e a sona b le and a c c e p ta b le . The sha r p pe aks of t he ou t -

flow hydrogra phs fo r Sewers 3-5, 4-5, and 5-6 d u r in g t h e p e r i o d t = 10 min t o

15 m in a r e th e c ombined r e s u l t o f num e r ic al e r r o r s a nd hydr a u l i c a s sum pt ions .

An anomaly i s t h e e a r l y p ea k of t h e i n - l i n e s t o r a g e i n F ig . 4 .8 f o r

Element 5-6. However, becau se of t h e computer money and time co n s t r a i n t s ,

f u r t h e r i n v e s t i g a t i o n was n o t m ade.



Sp-

1'



0

.

\

\-

\f

\t

\Iw

L\

\w

\

\II \V

\



O



.O



9

O

\

\

.-\-,

\\\

Y.aU

\

I

\

\

\

\

s a-

\

\

;

0

0

0

1ncID

0--0

W-r

W --

o

(

0z



V. CONCLUDING REMARKS

Hydrodynamica l ly , s torm sewer f low i s one of t he most comp l ica ted and

d i f f i c u l t p ro bl em s. The b a s i c p h y s i c a l p r i n c i p l e s o f s u r c h a r g ed a nd o pe n-

c h a nn e l f l ow s a r e d e s c r i b e d i n C h a p te r 3 . The ph ys ic al phenomena of t h e

t r a n s i t i o n b et we en t h e s e two f l ow p h a s e s a r e a l s o d i s c u s s e d . I n v i ew o f t h e

s t a t e - o f - a r t i n s i m u l a t i ng s u r c ha r ge d s e we r f l ow , a n a t t e m p t ha s be e n made t o

de ve l op i mprove d m ode ls f o r s i m u l a t i on o f s uc h f l ow . I t i s shown i n t h i s

re po rt t h a t a ki ne ma t ic wave model , SURKNET, can be form ula ted . Ne ve rt he les s ,

b e c au s e of t h e i n h e r e n t l i m i t a t i o n of t h e k i n e m a t i c wave a pp r o x im a t io n ,

SURKNET can onl y be c ons ide red a s a sm al l s t e p forw ard i n t h e advancement of -

t h e te c h n i q u es i n r e l i a b l e s im u l a t io n . I n a c om panion r e p o r t u nd er p r e p a r a t i o n ,

a dynamic wave - s u r c h a r ge m ode l w i l l b e p r op o se d . N e v e r t h e l e s s , f u r t h e r

r e s e a rc h , p a r t i c u l a r l y v e r i f i c a t i o n o f t h e m od els u si n g r e l i a b l e f i e l d d a t a

i s m o s t d e s i r a b l e .



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