0071449590_ar014

14.1. INTRODUCTION

Generally speaking urban drainage systems consist of three parts: the overland surfaceflow system, the sewer network, and the underground porous media drainage system.Some elements of these components are shown schematically in Fig. 14.1. Traditionallyno design is considered for the urban porous media drainage part. Recently porous mediadrainage facilities such as infiltration trenches have been designed for flood reduction orpollution control in cities with high land costs. For example, preliminary work on thisaspect of urban porous media drainage design can be found in, Fujita (1987), Morita et al.(1996), Takaaki and Fujita (1984) and Yen and Akan (1983). Much has yet to be devel-oped to refine and standardize on such designs; no further discussion on this undergroundsubject will be given in this chapter.

From a hydraulic engineering viewpoint, urban drainage problems can be classifiedinto two types: (1) design and (2) prediction for forecasting or operation. The requiredhydraulic level of the latter is often higher than the former. In design, a drainage facilityis to be built to serve all future events not exceeding a specified design hydrologic level.Implicitly the size of the apparatus is so determined that all rainstorms equal to andsmaller than the design storm are presumably considered and accounted for. Sewers,ditches, and channels in a drainage network each has its own time of concentration andhence its own design storm. In the design of a network all these different rainstormsshould be considered. On the other hand, in runoff prediction the drainage apparatus hasalready been built or predetermined, its dimensions known, and simulation of flow froma particular single rainstorm event is made for the purpose of real-time forecasting to beused for operation and runoff control, or sometimes for the determination of the flow ofa past event for legal purposes. The hydrologic requirements for these two types of prob-lems are different. In the case of prediction, a given rainstorm with its specific temporaland spatial distributions is considered. For design purposes, hypothetical rainstorms withassigned design return period or acceptable risk level and assumed temporal and spatial

CHAPTER 14HYDRAULIC DESIGN OF

URBAN DRAINAGE SYSTEMS

Ben Chie YenDepartment of Civil & Environmental Engineering

University of Illinois at Urbana-ChampaignUrbana, Illinois

A.Osman AkanDepartment of Civil and Enviromental Engineering

Old Dominion UniversityNorfolk, Virginia

14.1

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Source: HYDRAULIC DESIGN HANDBOOK

distributions of the rainfall are used. Table 14.1 lists some of these two types of designand prediction problems.

In the case of sanitary sewers, for design purposes the problem becomes the estima-tion of the critical runoffs in both quantity and quality, from domestic, commercial, and

14.2 Chapter Fourteen

FIGURE 14.1 Schematic of components of urban catch-ment. (From Metcalf & Eddy, Inc. et al., 1971).

TABLE 14.1 Types of Urban Drainage Problems (a) Design Problems

Type Design Purpose Hydro Information Required Hydraulic Sought Level

Sewers Pipe size (and slope) Peak discharge, Qp for Lowdetermination design return period

Drainage channels Channel dimensions Peak discharge, Qp for Low to moderatedesign return period

Detention/retention Geometric dimensions Design hydrograph, Q(t) Low to moderatestorage ponds (and outlet design)Manholes and Geometric dimensions Design hydrograph, Q(t) Low to moderatejunctions

Roadside gutters Geometric dimensions Design peak discharge, Low to moderateQp

Inlet catch basins Geometric dimensions Design peak discharge, LowQp

Pumps Capacity Design hydrograph Moderate to highControl gates Capacity Design hydrograph Moderate to highor valves



HYDRAULIC DESIGN OF URBAN DRAINAGE SYSTEMS

industrial sources over the service period in the future. For real-time control problems itinvolves simulation and prediction of the sanitary runoff in conjunction with the controlmeasures.

The basic hydraulic principles useful for urban drainage have been presented inChapter 3 for free surface flows, Chapters 2 and 12 for pipe flows, and Chapter 10 forpump systems. In the following, more specific applications of the hydraulics to urbandrainage components will be described. However, the hydraulic design for drainage ofhighway and street surfaces, roadside gutters, and inlets has been described in Chapter 13,design of stable erodible open channels in Chapter 16, and certain flow measurementstructures adaptable to urban drainage in Chapter 21; therefore they are not included inthis chapter.

14.2 HYDRAULICS OF DRAINAGE CHANNELS

Flows in urban drainage channels usually are open-channel flows with a free water surface.However, sewer pipes, culverts, and similar conduits under high flow conditions couldbecome surcharged, and pressurized conduit flows do occur. Strictly speaking, the flow is

Hydraulic Design of Urban Drainage Systems 14.3

TABLE 14.1 (continued) Types of Urban Drainage Problems (b) Prediction Problems

Type Purpose Hydro Input Hydro Information Required Hydraulic Sought Level

Real-time Real-time Predicted and/or just Hydrographs, Q(t, xi) Highoperation regulation measured rainfall,

of flow network data

Performance Simulation for Specific storm Hydrographs, Q(t, xi) Highevaluation evaluation of event, network data

a system

Storm event Determination Given past storm Hydrographs, Q(t, xi) Moderate-highsimulation of runoff at event or specified

specific locations input hyetographs,for particular past network dataor specified events

Flood level Determination Specific storm Hydrographs and Highdetermination of the extent hyetographs, stages

of flooding netwark data

Storm runoff Reduce and Event or continuous Hydrographs Q(t, xi) Moderate to highquality control of water rain and pollutant Pollutographs, c(t, xi)control pollution due to data, network data

runoff from rainstorms

Storm runoff Long-term, usually Long–term data Runoff volume Lowmaster large spatial scale Pollutant volumeplanning planning for

stormwater management




always unsteady, that is, changing with time. Nevertheless, in a number of situations, suchas in most cases of flow in sanitary sewers and for some rainstorm runoffs, change of flowwith time is slow enough that the flow can be regarded as approximately steady.

14.2.1 Open-Channel Flow

Open-channel flow occurs on overland, ditches, channels, and sewers in urban areas.Unsteady flow in open channels can be described by a momentum equation given belowin both discharge (conservative) and velocity (nonconservative) forms together with itsvarious simplified approximate models:

�g1A� �

∂∂Qt� � �

g1A� �∂

∂x�

�βAQ2

�

� �g1A� �o

σUxq1dσ � �

∂∂Yx� � S0 � Sf � 0. (14.1)

dynamic wavequasi-steady dynamic wave

noninertiakinematic wave

�1g� �

∂∂Vt�

� (2β � 1) �Vg� �

∂∂Vx�

� (β � 1) �gVA2� �

∂∂Ax�

�� ∂∂Yx�

� So � Sf � 0. (14.2)

where x � flow longitudinal direction measured horizontally (Fig. 14.2); A � flow cross-sectional area normal to x; y � vertical direction; Y � depth of flow of the cross section,measured vertically; Q � discharge through A; V � Q/A, cross-sectional average veloci-ty along x direction; So � channel slope, equal to tan θ, θ � angle between channel bedand horizontal plane; Sf � friction slope; σ � perimeter bounding the cross section A; q1

� lateral flow rate (e.g., rain or infiltration) per unit length of channel and unit length ofperimeter σ, being positive for inflow; Ux � x-component velocity of lateral flow when


FIGURE 14.2 Schematic of open, channel flow.




joining the main flow; g � gravitational acceleration; t � time; M � (gA)�1 �o

σ(Ux �

V)g1 dσ and β � Boussinesq momentum flux correction coefficient for velocity distri-bution:

β � �QA

2��o

σu�2dA (14.3)

u� � x-component of local (point) velocity averaged over turbulence.

The continuity equation is

�∂∂At� � �

∂∂Qx� � �o

σq1dσ (14.4)

If the channel is prismatic or very wide, such as the case of overland flow, Eq. (14.4) canbe written as

�∂∂Yt�

� �∂∂x�

(VY) � �1b� �o

σq1dσ (14.5)

where b is the water surface width of the cross section.In practice, it is more convenient to set the x and y coordinates along the horizontal lon-

gitudinal direction and gravitational vertical direction, respectively, when applied to flowon overland surface and natural channels for which So � tan θ. For human-made straightprismatic channels, sewers, pipes, and culverts, it is more convenient to set the x-y direc-tions along and perpendicular to the longitudinal channel bottom. In this case, the flowdepth h is measured along the y direction normal to the bed and it is related to Y by Y �h cos θ, whereas the channel slope So � sin θ.

The friction slope Sf is usually estimated by using a semiempirical formula such asManning’s formula

Sf � �n2

KV

2n

�V�� R �4/3 � �

nK

2

n

Q2

�AQ

2

�� R �4/3 (14.6)

or the Darcy-Weisbach formula

Sf � �8gfR�

V�V� � �8gfR� �

QA�Q

2

�� (14.7)

where n � Manning’s roughness factor, Kn � 1.486 for English units and 1.0 for SI units;f � the Weisbach resistance coefficient; and R � the hydraulic radius, which is equal toA divided by the wetted perimeter. The absolute sign is used to account for the occurrenceof flow reversal.

Theoretically, the values of n and f for unsteady nonuniform open-channel and pres-surized conduit flows have not been established. They depend on the pipe surface rough-ness and bed form if sediment is transported, Reynolds number, Froude number, andunsteadiness and nonuniformity of the flow (Yen, 1991). One should be careful that forunsteady nonuniform flow, the friction slope is different from either the pipe slope, the dis-sipated energy gradient, the total-head gradient, or the hydraulic gradient. Only for steadyuniform flow without lateral flow are these different gradients equal to one another.

At present, we can only use the steady uniform flow values of n and f given in the lit-erature as approximations. The advantage of f is its theoretical basis from fluid mechan-ics and its being nondimensional. Its values for steady uniform flow can be found fromthe Moody diagram or the Colebrook-White formula given in Chap. 2, as well as in stan-





dard hydraulics and fluid mechanics references. Its major disadvantage is that for a givenpipe and surface roughness, the value of f varies not merely with the Reynolds number butalso with the flow depth. In other words, as the flow depth in the sewer changes during astorm runoff, f must be recomputed repeatedly.

Manning’s n was originally derived empirically. Its major disadvantage is its trouble-some dimension of length to one-sixth power that is often misunderstood. Its main advan-tage is that for flows with sufficiently high Reynolds number over a rigid boundary witha given surface roughness in a prismatic channel, the value of n is nearly constant over awide range of depth (Yen, 1991). Values of n can be found in Chow (1959) or Chap. 3.

Other resistance coefficients and formulas, such as Chezy’s or Hazen-Williams’s, havealso been used. They possess neither the direct fluid mechanics justification as f nor inde-pendence of depth as n. Therefore, they are not recommended here. In fact, Hazen-Williams’s may be considered as a special situation of Darcy-Weisbach’s formula. A dis-cussion of the preference of the resistance coefficients can be found in Yen (1991).

Equations (14.6) and (14.7) are applicable to both surcharged and open-channelflows. For the open-channel case, the pipe is flowing partially filled and the geometricparameters of the flow cross section are computed from the geometry equations givenin Fig. 14.3.

The pair of momentum and continuity equations [Eqs. (14.1) and (14.4) or Eqs. (14.2)and (14.5)] with β � 1 and no lateral flow is often referred to as the Saint-Venant equa-tions or full dynamic wave equations. Actually, they are not an exact representation of theunsteady flow because they involve at least the following assumptions: hydrostaticpressure distribution over A, uniform velocity distribution over A (hence β � 1), and neg-ligible spatial gradient of the force due to internal stresses.

Those interested in the more exact form of the unsteady flow equations should refer toYen (1973b, 1975, 1996). Conversely, simplified forms of the momentum equation, name-ly, the noninertia (misnomer diffusion wave) and kinematic wave approximations of thefull momentum equation [Eq. (14.1)] are often used for the analysis of urban drainageflow problems.

Among the approximations shown in Eqs. (14.1) or (14.2), the quasi-steady dynamicwave equation is usually less accurate and more costly in computation than the noniner-tia equation, and hence, is not recommended for sewer flows. Akan and Yen (1981),among others, compared the application of the dynamic wave, noninertia, and kinematicwave equations for flow routing in networks and found the noninertia approximation gen-erally agrees well with the dynamic wave solutions, whereas the solution of the kinemat-ic wave approximate is clearly different from the dynamic wave solution, especially whenthe downstream backwater effect is important. Table 25.2 of Yen (1996) gives the properform of the equations to be used for different flow conditions.


FIGURE 14.3 Sewer pipe flow geometry. (From Yen, 1986a)




Analytical solutions do not exist for Eqs. (14.1) and (14.2) or their simplified formsexcept for very simple cases of the kinematic wave and noninertia approximations.Solutions are usually sought numerically as described in Chap. 12. In solving the differ-ential equations, in addition to the initial condition, boundary conditions should also beproperly specified. Table 14.2 shows the boundary conditions required for the differentlevels of approximations of the momentum equation. It also shows the abilities of theapproximations in accounting for downstream backwater effects, flood peak attenuation,and flow acceleration.

For flows that can be considered as invariant with time the steady flow momentumequations which are simplified from Eq. (14.2) for different conditions are given in Table14.3. The lateral flow contribution, mq, can be from rainfall (positive) or infiltration (neg-ative) or both. Instead of these equations, the following Bernoulli total head equation isoften used for flow profile computations:


TABLE 14.2 Theoretical Comparison of Approximations to Dynamic Wave Equation

Kinematic Noninertia Quasi-steady Dynamic wave dynamic wave wave

Boundary conditions required 1 2 2 2

Account for downstream No Yes Yes Yesbackwater effect and flow reversal

Damping of flood peak No Yes Yes Yes

Account for flow acceleration No No Only Yesconvective acceleration

TABLE 14.3 Cross-Section-Averaged One-Dimensional Momentum Equations for Steady Flow ofIncompressible Homogeneous Fluid

Prismatic channel �K � (K � K′) �DY

b

� � F2��ddyx� � So � Sf � �

Vg

2� �

d

d

βx� � Y �

ddKx� � mq

Constant piezometric (1 � F2) �ddYx� � So � (1 � F2) �

ddyx

s� � F2So

pressure distribution K � K′ � 1 � �Sf � F2 �

DB

b� �∂∂Bx� � �

Vg

2� �

d

d

βx� � mq

β � constant �ddYx� � or �

ddYx

s� �

K � K′ � 1Prismatic or wide channel

Definitions: F � mq � �g1A� �0

σ(U�x � 2βV)q�dσ

Db �� water surface width; K and K′ � piezometric pressure distribation correction factors for main an lateral flows;

V��g�D�b/�β�

(F2So � Sf � mq)��(1 � F2)

(So � Sf � mq)��(1 � F2)




�α

22Vg

22

� � Y2 �yb2 � �α21

gV2

1� � Y1 � yb1 � he � hq , (14.8)

where the subscripts 1 and 2 � the cross sections at the two ends of the computational reach,∆x, of the channel, Y � yb � the stage of the water surface where the channel bed elevationat section 1 is yb1 and that at section 2 is yb2 � yb1 � So ∆x; hq is the energy head from thelateral flow, if any; the energy head loss he � Se ∆x where Se is the slope of the energy line;and α � the Coriolis convective kinetic energy flux correction coefficient due to nonuniformvelocity distribution over the cross section (Chow, 1959; Yen, 1973). If there are other ener-gy losses, they should be added to the right-hand side of the equation. Methods of backwa-ter surface profile computation using these equations are discussed in Chap. 3.

If the flow is steady and uniform, Eqs. (14.1) and (14.4) or Eqs. (14.2) and (14.5)reduce to So � Sf and Q � AV. Hence, for steady uniform flow using Manning’s formula,

Q � 0.0496 �Kn

n� So

1/2 D8/3 �(φ �

φs2

i/3

nφ)5/3

�, (14.9)

where φ is in radians (Fig. 14.3). Correspondingly, the Darcy-Weisbach formula yields

Q � �18� ��

2�gf�S�o�� D5/2 �

(φ �

φs2

i/3

nφ)5/3

�. (14.10)

Figure 14.4 is a plot of these two equations that can be used to find φ.

14.2.2 Surcharge Flow

Sewers, culverts, and other drainage pipes sometimes flow full with water under pressure,often known as surcharge flow (Fig. 14.5). Such pressurized conduit flow occurs underextreme heavy rainstorms or under designed pipes. There are two ways to simulateunsteady surcharge flow in urban drainage: (1) The standard transient pipe flow approachand (2) the hypothetical piezometric open slot approach.

14.2.2.1 Standard transient pipe flow approach. In this approach, the flow is con-sidered as it is physically, that is, pressurized transient pipe flow. For a uniform sizepipe, the flow cross-sectional area is constant, being equal to the full pipe area Af;hence ∂A/∂x � 0. The continuity and momentum equations [Eqs. (14.4) and (14.2)with q1 � 0] can be rewritten as

Q � AfV (14.11)

�1g� �

∂∂Vt�

� �∂∂x�

�βgV2

� � �Pγ

a�

� �Sf , (14.12)

where Pa � the piezometric pressure of the flow and �–the specfic weight of the fluid. Ifthe pipe has a constant cross section and is flowing full with an incompressible fluidthroughout its length, then ∂V/∂x � 0. By further neglecting the spatial variation of β, inte-gration of Eq. (14.12) over the entire length, L, of the sewer pipe yields

��Pγ

a� exit

entrance

� Hu � Ku �2Vg

2� � Hd � Kd �

2Vg

2� � L

Sf � �

1g

� �∂∂Vt�

(14.13)





or

�gLA� �

∂∂Qt�

� Hu � Hd � (Ku � Kd) �2gQA

2

f2� � SfL , (14.14)

where Hu � the total head at the entrance of the pipe, Hd � the water surface outside thepipe exit, and Ku and Kd � the entrance and exit loss coefficients, respectively (Fig. 14.5).Equations (14.11) and (14.12) can also be derived as a special case of the commonly usedgeneral, basic, closed conduit transient flow continuity equation for waterhammer andpressure surge analysis, see, e.g., Chaudhry, 1979; Stephenson, 1984; Wood, 1980; Wylieand Streeter, 1983.


FIGURE 14.4 Central angle of water surfacein circular pipe (from Yen, 1986a).

FIGURE 14.5 Surcharge flow in a sewer. (AfterPansic, 1980).




�A1

� �ddAt� � �ρ

1� �

ddρt� �

∂∂Vx� � 0 (14.15)

or

�V1

� �∂∂Ht� � �

∂∂Hx� � �g

cV

2

� �∂∂Vx� � sin θ � 0 (14.16)

and the momentum equation

�1g� �

∂∂Vt�

� �Vg� �

∂∂Vx�

� �∂∂Hx�

� Sf � 0 (14.17)

where ρ � the bulk density of the fluid, H � Pa/γ � the piezometric head above the ref-erence datum, and c � the celerity of the pressure surge. The fact that Eqs. (14.11) and(14.12) can be derived from Eqs. (14.2) and (14.5) is the theoretical basis of thePreissmann hypothetical slot concept, which will be discussed below.

14.2.2.2 Hypothetical slot approach. This approach introduces hypothetically a contin-uous, narrow, piezometric slot attached to the pipe crown and over the entire length of thepipe as shown in Fig. 14.6. The idea is to transform the pressurized conduit flow situationinto a conceptual open-channel flow situation by introducing a virtual free surface to theflow. The idea was suggested by Preissmann (Cunge and Wegner, 1964). The hypotheti-cal open-top slot should be narrow so that it would not introduce appreciable error in thevolume of water. Conversely, the slot cannot be too narrow, with the aim of avoiding thenumerical problem associated with a rapidly moving pressure surge.

A theoretical basis for the determination of the width of the slot is to size the widthsuch that the wave celerity in the slotted sewer is the same as the surge celerity of the com-pressible water in the actual elastic pipe. The celerity c1 of the slot pipe is

c1 � �g�A�/b� (14.18)

where b � the slot width and A � the flow cross-sectional area. Neglecting the area con-tribution of the slot and hence A � Af � πD2/4 for a circular pipe, and equating c1 to thepressure wave speed c in the elastic pipe without the hypothetical slot, the theoretical slotwidth is


FIGURE 14.6 Preissmann hypo-thetical piezometric open slot.




b � πgD2/4c2 (14.19)

The surge speed in a pipe usually ranged from a few hundred feet per second to a fewthousand feet per second. For an elastic pipe with a wall thickness e and Young’s modulus ofelasticity Ep, assuming no pressure force from the soil acting on the pipe, the surge speed c is

c2 ��Eρf

f� /�1 � �

ηEE

p

f

eD

�� (14.20)

where Ef is the bulk modulus of elasticity and ρf is the bulk density, respectively, of theflowing water (Wylie and Streeter, 1983). Special conditions of pipe anchoring againstlongitudinal expansion or contraction and elasticity relevant to the surge speed c are givenin Table 14.4 where ω � Poisson’s ratio for the pipe wall material, that is, –ω is the ratioof the lateral unit strain to axial unit strain, and α is a constant to account for the rigiditywith respect to axial expansion of the pipe.

For small pipes, Eq. (14.19) may give too small a slot width, which would causenumerical problems. Cunge et al. (1980) recommend a width of 1 cm or larger.

The transition between part-full pipe flow and slot flow is by no means computation-ally smooth and easy, and assumptions are necessary (Cunge and Mazadou, 1984). Oneapproach is to assume a gradual width transition from the pipe to the slot. Sjöberg (1982)suggested two alternatives for the slot width based on two different values of the wavespeed c in Eq. (14.19). For the alternative applicable to h/D 0.9999, his suggested slotwidth b can be expressed as

�Db

� � 10�6 � 0.05423 exp[�(h/D)24] (14.21)

He further proposed to compute the flow area A and hydraulic radius R when the depthh is greater than the pipe diameter D as


TABLE 14.4 Special Conditions of Surge Speed in Full Pipe, Eq. (14.20)

Factor Condition

Pipe Anchor η � �2De� (1 � ω) � α �D

D� e�

Freedom of pipe Entirely free Only one Entire lengthlongitudinal (expansion joints at both ends) end anchored anchoredexpansion

Value of axial 1 1 � 0.5 ω 1 - ω2

expansion factor α

Elasticity E

Rigid pipe Air entrainment No air

Ep � ∞ ρf � ρwV–w + ρaV–a ρf � ρw

c2 � Ef /ρf Ef ��1 � V–a[(EE

w

w

/Ea) � 1� Ef � Ew

Subscript w denotes water (liquid); Subscript a � air; subscript f � fluid mixture; V– � volume.




A � (πD2/4) � (h � D)b (14.22)

R � D/4 (14.23)

A slight improvement to Sjöberg’s suggestion to provide a smoother computationaltransition is to use

A � A9999 � b(h � 0.9999D) (14.24)

for h/D � 0.9999 and assume that the transition starts at h/D � 0.91. Between h/D � 0.91and 0.9999, real pipe area A and surface width B are used. However, for h/D � 0.91, R iscomputed from Manning’s formula using pipe slope and a discharge equal to the steadyuniform flow at h � 0.91D, Q91; thus, for h/D � 0.91

R � (A91/A)R91 (14.25)

Because of the lack of reliable data, neither the standard surcharge sewer solutionmethod nor the Preissmann hypothetical open-slot approach has been verified for a singlepipe or a network of pipes. Past experiences with waterhammer and pressure surge prob-lems in closed conduits may provide some indirect verification of the applicability of thebasic flow equations to unsteady sewer flows. Nevertheless, direct verification is highlydesirable.

Jun and Yen (1985) performed a numerical testing and found there is no clear superi-ority of one approach over the other. Nevertheless, specific comparison between them isgiven in Table 14.5. They suggested that if the sewers in a network are each divided intomany computational reaches and a significant part of the flow duration is under surcharge,the standard approach saves computer time. Conversely, if transition between open-chan-nel and pressurized conduit flows occurs frequently and the transitional stability problemis important, the slot model would be preferred.


TABLE 14.5 Comparison Between Standard Surcharge Approach and Slot Approach

Item Standard Surcharge Approach Hypothetical Slot Approach

Concept Direct physical Conceptual

Flow equations Two different sets, one equation Same set of two equationsfor surcharge flow, two equations (continuity and momentum)for open-channel flow for surcharge and open-

channel flowsDiscretization for solution Whole pipe length for Divide into ∆x’s

surcharge flowWater volume within pipe Constant Varies slightly with slot

size, inaccurate if slot is toowide, stability problems ifslot is too narrow

Discharge in pipe at Same Varies slightly with ∆x, thusgiven time allows transition to progress

within pipeTransition between Specific criteria Slot width transition to avoid open channel flow and numerical instabilitysurcharge flow




14.3 FLOW IN A SEWER

14.3.1 Flow in a Single Sewer

Open-channel flow in sewers and other drainage conduits are usually unsteady, nonuni-form, and turbulent. Subcritical flows occur more often than supercritical. For slowly timevarying flow such as the case of the flow traveling time through the entire length of thesewer much smaller than the rising time of the flow hydrograph, the flow can often betreated approximately as stepwise steady without significant error.

The flow in a sewer can be divided into three regions: the entrance, the pipe flow, andthe exit. Figure 14.7 shows a classification of 10 different cases of nonuniform pipe flow


TABLE 14.5 (Continued)

Item Standard Surcharge Approach Hypothetical Slot Approach

Part full over pipe length Assume entire pipe length full Assume full or free ∆x by ∆xor free

Time accounting for transition Yes, specific inventory of No, implicitsurcharged pipes at different times

Programming efforts More complicated because of two Relatively simple because of sets of equations and time accounting one equation set and no and computer storage for transition specific accounting and

storage for transitionbetween open-channel andfull-pipe flows

Computational effort Depending mainly on accounting Depending mainly on space for transition times discretization ∆x

FIGURE 14.7 Classification of flow in a sewer pipe. (After Yen, 1986a).





FIGURE 14.8 Types of sewer entrance flow. (After Yen, 1986a).

based on whether the flow at a given instant is subcritical, supercritical, or surcharge.There are four cases of pipe entrance condition, as shown in Fig. 14.8 and below:

Case I is associated with downstream control of the pipe flow. Case II is associatedwith upstream control. In Case III, the pipe flow under the air pocket may be subcritical,supercritical, or transitional. In Case IV, the sewer flow is often controlled by both theupstream and downstream conditions.

Pipe exit conditions also can be grouped into four cases as shown in Fig. 14.9 andbelow:

Case Pipe exit hydraulic condition

A Nonsubmerged, free fall

B Nonsubmerged, continuous

C Nonsubmerged, hydraulic jump

D Submerged

Case Pipe entrance hydraulic condition

I Nonsubmerged entrance, subcritical flow

II Nonsubmerged entrance, supercritical flow

III Submerged entrance, air pocket

IV Submerged entrance, water pocket





TABLE 14.6 Pipe Flow Conditions

Possible Possible ExitCase Pipe Flow Entrance conditions Conditions

1 Subcritical I, III A, B

2 Supercritical II, III B, C

3 Subcritical → hydraulic drop → supercritical I, III B, C

4 Supercritical → hydraulic jump → subcritical II, III A, B

5 Supercritical → hydraulic jump → surcharge II, III D

6 Supercritical → surcharge II, III D

7 Subcritical → surcharge I, III D

8 Surcharge → supercritical IV B, C

9 Surcharge → subcritical IV A, B

10 Surcharge IV D

Source: From Yen (1986a).

FIGURE 14.9 Types of sewer exit flow. (AfterYen, 1986a).

In Case A, the pipe flow is under exit control. In Case B, the flow is under upstreamcontrol if it is supercritical and downstream control if subcritical. In Case C, the pipe flowis under upstream control while the junction water surface is under downstream control.In Case D, the pipe flow is often under downstream control, but it can also be under bothupstream and downstream control.

The possible combinations of the 10 cases of pipe flow with the entrance and exit con-ditions are shown in Table 14.6 for unsteady nonuniform flow. Some of these 27 possiblecombinations are rather rare for unsteady flow and nonexistent for steady flow, for exam-ple, Case 6. For steady flow in a single sewer, by considering the different mild-slope Mand steep-slope S backwater curves (Chow, 1959) as different cases, there are 27 possiblecases in addition to the uniform flow, of which six types were reported by Bodhaine (1968).




The nonuniform pipe flows shown in Fig. 14.7 are classified without considering thedifferent modes of air entrainment. The types of the water surface profile, equivalent tothe M, S, and A (adverse slope) types of backwater curves for steady flow, are also nottaken into account. Additional subcases of the 10 pipe flow cases can also be classifiedaccording to rising, falling, or stationary water surface profiles. For the cases with ahydraulic jump or drop, subcases can be grouped according to the movement of the jumpor drop, be it moving upstream or downstream or stationary. Furthermore, flow withadverse sewer slope also exists because of flow reversal.

During runoff, the change in magnitude of the flow in a sewer can range from only a fewtimes dry weather low flow in a sanitary sewer to as much as manyfold for a heavy rain-storm runoff in a storm sewer. The time variation of storm sewer flow is usually much morerapid than that of sanitary sewers. Therefore, the approximation of assuming steady flow ismore acceptable for sanitary sewers than for storm and combined sewers.

In the case of a heavy storm runoff entering an initially dry sewer, as the flow enters thesewer, both the depth and discharge start to increase as illustrated in Fig. 14.10 at times t1,t2, and t3 for the open-channel phase. As the flow continues to rise, the sewer pipe becomescompletely filled and surcharges as shown at t4 and t5 in Fig. 14.10. Surcharge flow occurswhen the sewer is underdesigned, when the flood exceeds that of the design return period,when the sewer is not properly maintained, or when storage and pumping occur.

Under surcharge conditions, the flow-cross-sectional area and depth can no longerincrease because of the sewer pipe boundary. However, as the flood inflow continuesto increase, the discharge in the sewer also increases due to the increasing differencein head between the upstream and downstream ends of the sewer, as sketched in thedischarge hydrograph in Fig. 14.10. Even under surcharge conditions while the sewer


FIGURE 14.10 Time variation of flow in a sewer. (AfterYen, 1986a).





diameter remains constant, the flow is usually nonuniform. This is due to the effects ofthe entrance and exit on the flow inside the sewer, and hence, the streamlines are not parallel.

As the flood starts to recede, the aforementioned flow process is reversed. The sewerwill return from surcharged pipe flow to open-channel flow, shown at t6 and t7 in Figure14.1. Since the recession is usually—but not always—more gradual than the rising of theflood, the water surface profile in the sewer is usually more gradual during flow recessionthan during rising.

The differences in the gradient of the water surface profiles during the rising and reces-sion of the flood bear importance in the self-cleaning and pollutant-transport abilities ofthe sewer. During the rising period, with relatively steep gradient, the flow can carry notonly the sediment it brings into the sewer but also erodes the deposit at the sewer bottomfrom previous storms. For a given discharge and gradient, the amount of erosion increas-es with the antecedent duration of wetting and softening of the deposit. During the reces-sion, with a flatter water surface gradient and deceleration of the flow, the sediment beingcarried into the sewer by the flow tends to settle onto the sewer bottom.

If the storm is not heavy and the flood is not severe, the rising flow will not reach sur-charge state. The flood may rise, for example, to the stage at t3 shown in Fig. 14.10 andthen starts to recede. The sewer remains under open-channel flow throughout the stormrunoff. For such frequent small storms, the flow in the sewer is so small that it is unableto transport out the sediment it carries into the sewer, resulting in deposition to be cleanedup by later heavy storms or through artificial means.

For a single-peak flood entering a long circular sewer having a diameter D and pipe sur-face roughness k, Yen (1973a) reported that for open-channel flow, the attenuation of theflood peak, Qpx, at a distance x downstream from the pipe entrance (x � 0) and the corre-sponding occurrence time of this peak, tpx, can be described dimensionlessly as

�QQ

p

p

0

x� � exp

�0.0771

�Dx

�

�Dk

�

0.17

�RD

b�

�0.42 �D2

Q

.5 �p0

g��

�0.16

�tp0

1.64 (tg � tp0) �Dg1

1

.

.

3

3

2

2�

�4

(14.26)

(tpx � tp0) �VD

w� �

6.03 log10

�QQ

p

p

0

x�

�0.18

�520

�Dk

�

�0.11

�RD

b�

0.66

� �Qb

Q4g

p

0

04.

.

4

2D�

0.1

tp00.68 (tg � tp0) �D

g0

0

.

.

8

8

2

2�

0.5, (14.27)

where Qp0 and tp0 � the peak discharge and its time of occurrence at x � 0, respectively;Qb is the steady base flow rate and Rb � hydraulic radius of the base flow; tg � the timeto the centroid of the inflow hydrograph at x � 0 above the base flow; g � the gravita-tional acceleration; and Vw � (Qb/Ab) � (gAb/Bb)1/2 � the wave celerity of the base flow,where Ab � the base flow cross � sectional area and Bb � the corresponding water-sur-face width. In both equations, the second nondimensional parameter in the right-hand sidek/D is a pipe property parameter; the third parameter Rb/D is a base flow parameter; thefourth nondimensional parameter represents the influence of the flood discharge; where-as the fifth and last nondimensional parameter reflects the shape of the inflow hydrograph.

The single – peak hydrograph shown in Fig. 14.10 is an ideal case for the purpose ofillustration. In reality, because the phase shift of the peak flows in upstream sewers andthe time–varying nature of rainfall and inflow, usually the real hydrographs are multipeak.




Because the flow is nonuniform and unsteady, the depth-discharge relationship, alsoknown as the rating curve in hydrology, is nonunique. Even if we are willing to considerthe flow to be steady uniform as an approximation, the depth-discharge relation is nonlin-ear, and within a certain range, nonunique, as shown nondimensionally and ideally in Fig.14.11 for a circular pipe. The nonunique depth-discharge relationship for nonuniform flow,aided by the poor quality of the water and restricted access to the sewer, makes it difficultto measure reliably the time-varying flow in sewers. Among the many simple and sophis-ticated mechanical or electronic measurement devices that have been attempted on sewersand reported in the literature, the simple, mechanical Venturi-type meter, which has sideconstriction instead of bottom constriction to minimize the effect of sediment clogging,still appears to be the most practical measurement means, that is, if it is properly designed,constructed, and calibrated and if it is located at a sufficient distance from the entrance andexit of the sewer. On the other hand, the hydraulic performance graph described in Sec.14.6.1 can be used to establish the rating curve for a steady nonuniform flow.


FIGURE 14.11 Rating curve for steady uniform flow incircular pipe.




Flow in sewers is perhaps one of the most complicated hydraulic phenomena. Even fora single sewer, there are a number of transitional flow instability problems. One of themis the surge instability of the flow in pipes of a network. The other four types of instabil-ities that could occur in a single sewer pipe are the following: The instability at the transition between open-channel flow and full conduit flow, the transitional instabilitybetween supercritical flow and subcritical flow in the open-channel phase, the water-sur-face roll-wave instability of supercritical open-channel flow, and a near dry-bed flowinstability. Further discussion on these instabilities can be found in Yen (1978b, 1986a). Itis important to realize the existence of these instabilities in flow modeling.

14.3.2. Discretization of Space-Time Domain of a Sewer for Simulation

No analytical solutions are known for the Saint—Venant equations or the surchargedsewer flow equation. Therefore, these equations for sewer flows are solved numericallywith appropriate initial and boundary conditions. The differential terms in the partial dif-ferential equations are approximated by finite differences of selected grid points on aspace and time domain, a process often known as discretization. Substitution of the finitedifferences into a partial differential equation transforms it into an algebraic equation.Thus, the original set of differential equations can be transformed into a set of finite dif-ference algebraic equations for numerical solution.

Theoretically, the computational grid of space and time need not be rectangular.Neither need the space and time differences ∆x and ∆t be kept constant. Nonetheless, it isusually easier for computer coding to keep ∆x and ∆t constant throughout a computation.For surcharge flow, Eq. (14.14) dictates the application of the equation to the entire lengthof the sewer, and the discretization applies only to the time domain. In an open-channelflow, it is normally advisable to subdivide the length of a sewer into two or three compu-tational reaches of ∆x, unless the sewer is unusually long or short. One computationalreach tends to carry significant inaccuracy due to the entrance and exit of the sewer and isusually incapable of sufficiently reflecting the flow inside the sewer. Conversely, too manycomputational reaches would increase the computational complexity and costs withoutsignificant improvement in accuracy.

The selection of the time difference ∆t is often an unhappy compromise of three crite-ria. The first criterion is the physically significant time required for the flow to passthrough the computational reach. Consider a typical range of sewer length between 100and 1000 ft and divide it into two or three ∆x, and a high flow velocity of 5–10 ft/s, a suit-able computational time interval would be approximately 0.2–2 min. For a slowly varyingunsteady flow, this criterion is not important and larger computational ∆t will suffice. Fora rapidly varying unsteady flow, this criterion should be taken into account to ensure thecomputation is physically meaningful.

The second criterion is a sufficiently small ∆t to ensure numerical stability. An often-usedguide is the Courant criterion

∆x/∆t V � �g�A�/B� (14.28)

In sewers, which usually have small ∆x compared to rivers and estuaries, this criterionoften requires a ∆t less than half a minute and sometimes 1 or 2 s.

The third criterion is the time interval of the available input data. It is rare to have rain-fall or corresponding inflow hydrograph data with a time resolution as short as 2, 5, oreven 10 min. Values for ∆t smaller than the data time resolution can only be interpolated.This criterion becomes important if the in-between values cannot be reliably interpolated.





In a realistic application, all three criteria should be considered. Unfortunately, in manycomputations only the second numerical stability is considered.

There are many, many numerical schemes that can be adopted for the solution of theSaint-Venant equations or their approximate forms [Eqs. (14.1)–(14.5)]. They can be clas-sified as explicit schemes, implicit schemes, and the method of characteristics. Many ofthese methods are described in Chap. 12, as well as in Abbott and Basco (1990), Cungeet al. (1980), Lai (1986), and Yen (1986a).

14.3.3 Initial and Boundary Conditions

As discussed previously and indicated in Table 14.1, boundary conditions, in addition toinitial conditions, must be specified to obtain a unique solution of the Saint-Venant equa-tions or their approximate simplified equations.

The initial condition is, of course, the flow condition in the sewer pipe when compu-tations start, t � 0, that is, either the discharge Q(x, 0), or the velocity V(x, 0), paired withthe depth h(x, 0). For a combined sewer, this is usually the dry-weather flow or base flow.For a storm sewer, theoretically, this initial condition is a dry bed with zero depth, zerovelocity, and zero discharge. However, this zero initial condition imposes a singularity inthe numerical computation. To avoid this singularity problem, either a small depth or asmall discharge is assumed so that the computation can start. This assumption is justifi-able because physically there is dry-bed film flow instability, and the flow, in fact, doesnot start gradually and smoothly from dry bed. Based on dry-bed stability consideration,an initial depth on the order of 0.25 in., or less than 5 mm, appears reasonable.

However, in sewers, this small initial depth usually is unsatisfactory because negativedepth is obtained at the end of the initial time step of the computation. The reason is thatthe continuity equation of the reach often requires a water volume much bigger than theamount of water in the sewer reach with a small depth. Hence, an initial discharge, or baseflow, that permits the computation to start is assumed. For a storm sewer, the magnitudeof the base flow depends on the characteristics of the inflow hydrograph, the sewer pipe,the numerical scheme, and the size of ∆t and ∆x used. For small ∆x and ∆t, a relativelylarge base flow is required, but may cause a significant error in the solution. In either case,it is not uncommon that in the first few time steps of the computation, the calculated depthand discharge decrease as the flood propagates, a result that contradicts the actual physi-cal process of rising depth and discharge. Nonetheless, if the base flow is reasonablyselected and the numerical scheme is stable, this anomaly would soon disappear as thecomputation progresses. An alternative to this assumed base flow approach to avoid thenumerical problem is to use an inverted Priessmann hypothetical slot throughout the pipebottom and assigning a small initial depth, discharge or velocity to start the computation.Currey (1998) reported satisfactory use of slot width between 0.001 and 0.01 ft.

As to boundary conditions, when the Saint-Venant equations are applied to an interiorreach of a sewer not connected to its entrance or exit, the upstream condition is simply thedepth and discharge (or velocity) at the downstream end of the preceding reach, which areidentical with the depth and discharge at the upstream of the present reach. Likewise, thedownstream condition of the reach is the shared values of depth and discharge (or veloc-ity) with the following reach. Therefore, the boundary conditions for an interior reachneed not be explicitly specified because they are implicitly accounted for in the flow equa-tions of the adjacent reaches.

For the exterior reaches containing either the sewer entrance or the exit, the upstreamboundary conditions required depend on whether the flow is subcritical or supercritical asindicated in Table 14.7.





For a sewer that is divided into M computational reaches and M � 1 stations, there isa continuity equation and a momentum equation written in finite difference algebraic formfor each reach. There are 2(M � 1) unknowns, namely, the depth and discharge (or veloc-ity) at each station. The 2(M � 1) equations required to solve for the unknowns come fromM continuity equations and M momentum equations for the M reaches, plus the twoboundary conditions. If the flow is subcritical, one boundary condition is at the sewerentrance (x � 0) and the other is at the sewer exit (x � L). If the flow is supercritical, bothboundary conditions are at the upstream end, the entrance, one of them often is a criticaldepth criterion. If at one instant a hydraulic jump occurs in an interior reach inside thesewer, two upstream boundary conditions at the sewer entrance and one downstreamboundary condition at the sewer exit should be specified. If a hydraulic drop occurs insidethe sewer, one boundary condition each at the entrance and exit of the sewer is needed;the drop is described with a critical depth relation as an interior boundary condition.Handling the moving surface discontinuity, shown schematically in Fig. 14.12, is not asimple matter. The moving front may travel from reach to reach slowly in different ∆t, orit may move through the entire sewer in one ∆t. If, for any reason, it is desired to computethe velocity of the moving front Vw between two computational stations i and i � 1 in asewer, the following equation can be used as an approximation;


FIGURE 14.12 Moving water surface discontinuity in asewer. (After Yen, 1986a).

TABLE 14.7 Some Types of Specified Boundary Conditions for Simulation of ExteriorReaches of Sewers

Location Upstream End of Sewer Downstream End of Sewer Entrance Reach (x � 0) Exit Reach (x � L)

Subcritical flow One of One ofQ(0, t) h(L, t); e.g. ocean tides, lakesh(0, t) Q(L, t); release hydrographV(0, t) Q(h); rating curve

V(h); storage-velocity relation

for all t to be simulated for all t to be simulated

Supercritical flow Two of the above None




Vw ��AiVi

A�

i �

A

1

i

�� 1

AV

i

i � 1� (14.29)

14.3.4 Storm Sewer Design with Rational Method

The most important components of an urban storm drainage system are storm sewers. A number of methods exist for designing the size of such sewers. Some are highly sophis-ticated, using the Saint-Venant equations, whereas others are relatively simple. In contrastto storm runoff prediction/simulation models, sophisticated storm sewer design methodsdo not necessarily provide a better design than the simpler methods, mainly because of thediscrete sizes of commercially available sewer pipes.

If the peak design discharge Qp for a sewer is known, the required sewer dimensionscan be computed by using Manning’s formula such that

AR2/3 � �Kn

n� �

�Q

S�p

o�� (14.30)

which can be obtained from Eq. (14.6) by assuming the friction slope Sf is equal to thesewer slope So. All other symbols in the equation have been defined previously. For a cir-cular sewer pipe, the minimum required diameter dr is

dr �3.208 �K

n

n� �

QSo

p�

3/8

(14.31a)

where kn � 1 for SI units and 1.486 for English units. If the Darcy-Wesibach formula (Eq.14.7) is used,

dr �0.811 �gS

f

o� Qp

2

1/5

(14.31b)

These two equations are plotted in Fig. 14.13 for design applications. The assumption So

� Sf essentially implies that around the time of peak discharge, the flow can well beregarded approximately as steady uniform flow for the design, despite the fact that theactual spatial and temporal variations of the flow are far more complicated as describedin Sec. 14.3.1.

In sewer designs, there are a number of constraints and assumptions that are commonlyused in engineering practice. Those pertinent to sewer hydraulic design are as follows:

1. Free surface flow exists for the design discharge, that is, the sewer is under “gravityflow” or open-channel flow. The design discharge used is the peak discharge of thetotal inflow hydrograph of the sewer.

2. The sewers are commercially available circular sizes no smaller than, say, 8 in. or 200mm in diameter. In the United States, the commercial sizes in inches are usually 8, 10,12, and from 15 to 30 inches with a 3-in. increment, and from 36 to 120 in. with anincrement of 6 in. In SI units, commercial sizes, depending on location, include mostif not all of the following: 150, 175, 200, 250, 300, 400, 500, 600, 750, 1000, 1250,1500, 1750, 2000, 2500, and 3000 mm.

3. The design diameter is the smallest commercially available pipe that has a flow capac-ity equal to or greater than the design discharge and satisfies all the appropriate con-straints.

4. To prevent or reduce permanent deposition in the sewers, a nominal minimum per-missible flow velocity at design discharge or at nearly full-pipe gravity flow is speci-






fied. A minimum full-pipe flow velocity of 2 ft/s or 0.5 m/s at the design discharge isusually recommended or required.

5. To prevent the occurrence of scour and other undesirable effects of high velocity �flow, a maximum permissible flow velocity is also specified. The most commonly usedvalue is 10 ft/s or 3 m/s. However, recent studies indicate that due to the improvedquality of modern concrete and other sewer pipe materials, the acceptable velocity canbe considerably higher.

6. Storm sewers must be placed at a depth that will allow sufficient cushioning to preventbreakage due to ground surface loading and will not be susceptible to frost. Therefore,minimum cover depths must be specified.

7. The sewer system is a tree-type network, converging toward downstream.

8. The sewers are joined at junctions or manholes with specified alignment, for example,the crowns aligned, the inverts aligned, or the centerlines aligned.

9. At any junction or manhole, the downstream sewer cannot be smaller than any of theupstream sewers at that junction, unless the junction has significantly large detention

FIGURE 14.13 Required sewer diameter. (m or ft)





storage capacity or pumping. There also is evidence that this constraint is unnecessaryfor very large sewers.

Various hydrologic and hydraulic methods exist for the determination of the designdischarge Qp. Among them the rational method is perhaps the most widely and simplestused method for storm sewer design. With this method, each sewer is designed individu-ally and independently, except that the upstream sewer flow time may be used to estimatethe time of concentration. The design peak discharge for a sewer is computed by using therational formula

Qp � i � Cjaj, (14.32)

where i � the intensity of the design rainfall; C � the runoff coefficient (see Chap. 5 forits values); and a is surface area. The subscript j represents the jth subarea upstream to bedrained. Note that �aj includes all the subareas upstream of the sewer being designed.Each sewer has its own design i because each sewer has its own flow time of concentra-tion and design storm. The only information needed from upstream sewers for the designof a current sewer is the upstream flow time for the determination of the time of concen-tration.

The rational formula is dimensionally homogenous and is applicable to any consistentmeasurement units. The runoff coefficient C is dimensionless. It is a peak discharge coef-ficient but not a runoff volume fraction coefficient. However, in English units usually theformula is used with the area aj in acres and rain intensity i in inches per hour. The con-version factor 1.0083 is approximated as unity.

The procedure of the rational method is illustrated in the following in English units forthe design of the sewers of the simple example drainage basin A shown schematically inFig. 14.14. The catchment properties are given in Table 14.8. For each catchment, thelength Lo and slope So of the longest flow path—or better, the largest Lo/�S�o� —shouldfirst be identified. As discussed Sec. 14.7, a number of formulas are available to estimatethe inlet time or time of concentration of the catchment to the inlet. In this example, Eq.(14.86) is used with K � 0.7 for English units and heavy rain, that is, to �0.7(nLo/�S�o�)0.6. The catchment overland surface texture factor N is determined fromTable 14.16

The design rainfall intensity is computed from the intensity-duration-frequency rela-tion for this location,

i(in./h) � �1td

0�

0T2r0

5

.2

� (14.33)

TABLE 14.8 Characteristics of Catchments of Example Drainage Basin A

Catchment Area Longest Overland Path Inlet Time Runoff(Acres) Length Lo Slope Surface Texture to Coefficient

(ft) N (min) C

I 2 250 0.010 0.015 6.2 0.8

II 3 420 0.0081 0.016 9.3 0.7

III 3 400 0.012 0.030 11.7 0.4

IV 5 640 0.010 0.020 12.9 0.6

V 5 660 0.010 0.021 13.1 0.6





where td � the rain duration (min) which is assumed equal to the time of concentration,tc, of the area described, and Tr � the design return period in years. For this example, Tr

� 10 years. Determination of i for the sewers is shown in Table 14.9a. The entries in thistable are explained as follows:

Column 1. Sewer number identified by the inlet numbers at its two ends.

Column 2. The sewer number immediately upstream, or the number of the catchmentthat drains directly through manhole or junction into the sewer being con-sidered.

Column 3. The size of the directly drained catchment.

Column 4. Value of the runoff coefficient for each catchment.

Column 5. Product of C and the corresponding catchment area.

Column 6. Summation of Cjaj for all the areas drained by the sewer; it is equal to thesum of contributing values in Column 5.

FIGURE 14.14 Sewer design example drainage basin A. (a) Layout (b) Profiles.





Column 7. Values of inlet time to the sewer for the catchments drained, that is, theoverland flow inlet time for directly drained catchments, or the time ofconcentration for the immediate upstream connecting sewers.

Column 8. The sewer flow time of the immediate upstream connecting sewer as givenin Column 9 in Table 14.9b.

Column 9. The time of concentration tc for each of the possible critical flow paths,tc � inlet time (Column 7) � sewer flow time (Column 10) for eachflow path.

Column 10. The design rainfall duration td is assumed equal to the longest of the dif-ferent times of concentration of different flow paths to arrive at theentrance of the sewer being considered, for example, for Sewer 31, td isequal to 13.9 min from Sewer 21, which is longer than that from directlycontributing Catchment V (13.1 min).

Column 11. The rainfall intensity i for the duration given in Column 10 is obtainedfrom the intensity-duration relation for the given location, in this case, Eq.(14.33) for the 10-year design return period.

Table 14.9b shows the design of the sewers for which the Manning n � 0.015, mini-mum soil cover is 4.0 ft, and minimum nominal design velocity is 2.5 ft/s. The exit sewerof the system (Sewer 31) flows into a creek for which the bottom elevation is 11.90 ft, theground elevation of its bank is 21.00 ft, and its 10-year flood water level is 20.00 ft.

Column 1. Sewer number identified by its upstream inlet (manhole) number.

Column 2. Ground elevation at the upstream manhole of the sewer.

Column 3. Length of the sewer.

Column 4. Slope of the sewer, usually follows approximately the average groundslope along the sewer.

Column 5. Design discharge Qp computed according to Eq. (14.32); thus, the productof Columns 6 and 11 in Table 14.9a.

Column 6. Required sewer diameter, as computed by using Eq. (14.31) or Fig. 14.13;for Manning’s formula with n � 0.015 and dr in ft, Eq. (14.31a) yields

dr �0.0324 ��

QS�p

o��

3/8

in which Qp, in ft3/s, is given in Column 5 and So is in Column 4.

Column 7. The nearest commercial nominal pipe size that is not smaller than therequired size is adopted.

Column 8. Flow velocity computed as V � Q/Af; that is, it is calculated as Column 5multiplied by 4/π and divided by the square of Column 7. As discussed inYen (1978b), there are several ways to estimate the average velocity of theflow through the length of the sewer. Since the flow is actually unsteadyand nonuniform, usually the one used here, using full pipe cross section,is a good approximation.

Column 9. Sewer flow time is computed as equal to L/V, that is, Column 3 divided byColumn 8 and converted into minutes.





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0.01

0033

.02.

432.

506.

71.

04.

0027

.85

25.3

523

.85

21.3

5

3128

.70

500

0.01

4444

.42.

532.

757.

2023

.85

21.1

016

.65

13.9

0

(31

28.7

050

00.

0156

44.4

2.50

2.50

7.80

23.8

521

.35

16.0

513

.55)





Column 10. Product of Columns 3 and 4; this is the elevation difference between thetwo ends of the sewer.

Column 11. The upstream pipe crown elevation of Sewer 11 is computed from theground elevation minus the minimum soil cover, 4.0 ft, to save soil exca-vation cost. In this example, sewers are assumed invert aligned except thelast one (Sewer 31), which is crown aligned at its upstream (23.85 ft forupstream of Sewer 31 and downstream of Sewer 21) to reduce backwaterinfluence from the water level at sewer exit.

Column 12. Pipe invert elevation at the upstream end of the sewer, equal to Column 11minus Column 7.

Column 13. Pipe crown elevation at the downstream end of the sewer, equal to Column11 minus Column 10.

Column 14. Pipe invert elevation at the downstream end of the sewer, equal to Column13 minus Column 7. For the last sewer, the downstream invert elevationshould be higher than the creek bottom elevation, 11.90 ft.

The above example demonstrates that, in the rational method, each sewer is designedindividually and independently, except the computation of sewer flow time for the purposeof rainfall duration determination for the next sewer, that is, the values of tf in Column 8of Table 14.9a are taken from those in Column 9 of Table 14.9b.

The profile of the example designed sewers are shown as the solid lines in Fig. 14.14b.If the water level of the creek downstream of Sewer 31 is ignored, theoretically a cheap-er design could be achieved by putting the exit Sewer 31 on a slightly steeper slope, from0.0144 to 0.0156 to reduce the pipe diameter from 2.75 to 2.50 ft. The new slope can beestimated from

So ��4π10

2

/3� �K

n2

n2� Qp

2

/d16/3. (14.34)

This alternative is shown with the parentheses in Table 14.9b and as dashed lines in Fig.14.14b. However, one should be aware that the water level of a 10-year flood in the creekis 20.00 ft and hence, the last sewer is actually surcharged and its exit is submerged. Thesewer will not achieve the design discharge unless its upstream manhole is surcharged byalmost 4 ft (20.00–16.05). Therefore, the original design of 2.75 ft diameter is a safer andpreferred option in view of the backwater effect from the tailwater level in the creek. Infact, Sewer 21-31 may also be surcharged due to the downstream backwater effect.

Sometimes, a backwater profile analysis is performed on the sewer network to assessthe degree of surcharge in the sewers and manholes. In such an analysis, energy losses inthe pipes and manholes should be realistically accounted for. However, the intensity-dura-tion-frequency-based design rainfall used in the rational method design is an idealistic,conceptual, simplistic rain and the probability of its future occurrence is nil. The actualperformance of the sewer system varies with different actual rainstorms, each having dif-ferent temporal and spatial rain distributions. But it is impossible to know the distribu-tions of these future rainstorms, whereas the ideal rainstorms adopted in the design of therational method are used as a consistent measure of protection level. Although designingsewers using the rational method is a relatively simple and straightforward matter, check-ing the performance of the sewer system is a far more complex task requiring thoroughunderstanding of the hydrology and hydraulics of watershed runoff. For instance, check-ing the network performance by using an unsteady flow simulation model would requiresimulation of the unsteady flow in various locations in the network accounting for losses




in sewer pipes as well as in manholes and junctions (the latter will be discussed in the nextsection).

Moreover, for a given sewer network layout, by using different sewer slopes, alterna-tive designs of the network sewers can be obtained. A cost analysis should be conductedto select the most economic feasible design. This can be done with a system optimizationmodel such as Illinois Least-Cost Sewer System Design Model (ILSD) (Yen et al., 1984).

14.4 HYDRAULICS OF SEWER JUNCTIONS

There are various auxiliary hydraulic structures such as junctions, manholes, weirs,siphons, pumps, valves, gates, transition structures, outlet controls, and drop shafts in asewer network. Information relevant to design of most of these apparatuses are welldescribed in standard fluid mechanics textbooks and references, particularly in theGerman text by Hager (1994) and Federal Highway Administration (FHWA, 1996). In thissection, the most important auxiliary component in modeling the sewer junctions are dis-cussed. For sewers of common size and length, the headloss for the flow through a seweris usually two to five times the velocity head. Thus, the head loss through a junction iscomparable to the sewer pipe loss, and is not a minor loss.

14.4.1 Junction Classifications

A sewer junction usually has three or four sewer pipes joined to it. Under normal flowconditions, one downstream pipe receives the outflow from the junction and other pipesflow into the junction. However, junctions with only two or more than four joining pipesare not uncommon. The most upstream junctions of a sewer network are usually one-wayjunctions having only one sewer connected to a junction. The horizontal cross section ofthe junction can be circular or square or may be another shape. The diameter or horizon-tal dimension of a junction normally is not smaller than the largest diameter of the join-ing sewers. To allow the workers room to operate, usually junctions are not smaller than3 ft (1 m) in diameter. For large sewers, the access to the junction can be smaller than thediameter of the largest joining sewer.

Sewers may join a junction with different vertical and horizontal alignments, and theymay have different sizes and slopes. Vertically, the pipes may join at the junction withtheir centerlines or inverts or crowns aligned, or with any line of alignment in between.There is no clearly preferred alignment that could simultaneously satisfy the requirementsof good hydraulics at low and high flows without complicating either construction cost ordesign. The bottom of the junction is usually at or slightly lower than the lowest invert ofthe joining sewers.

In the horizontal alignment, often the outflow sewer is aligned with one (usually themajor) inflow sewer in a straight line with other sewers joining at an angle. For cities withsquare blocks, right-angle junctions are most common. Typical sewer benching and flowguides in junctions are shown in Fig. 14.15.

With the alignment of the joining pipes and the shape and dimensions of junctions notstandardized, the precise, quantitative hydraulic characteristics of the junctions vary con-siderably. As a result, there are many individual studies of specified junctions, but a gen-eral comprehensive quantitative description of junctions is yet to be produced.

For the purpose of hydraulic analysis, junctions can be classified according to the fol-lowing scheme (Yen, 1986a):





1. According to the geometry: (a) one-way junction, (b) two-way junction, (c) three-way junction—merging (two pipes flow into one pipe) or dividing (one pipe flows intotwo pipes), and (d) four- or more-way junction—merging, dividing, or merging anddividing.

2. According to the flow in the joining pipes: (a) open-channel junction (with open-channel flow in all joining pipes), (b) surcharge junction (with all joining pipes sur-charged), and (c) partially surcharged junction (with some, but not all, joining pipessurcharged).

3. According to the significance of the junction storage on the flow: storage junction orpoint junction.

Hydraulically, the most important feature of a junction is that it imposes backwatereffects to the sewers connected to it. A junction provides, in addition to a volume—how-ever small—of temporal storage, redistribution and dissipation of energy, and mixing andtransfer of momentum of the flow and of the sediments and pollutants it carries. The pre-cise, detailed hydraulic description of the flow in a sewer junction is rather complicatedbecause of the high degree of mixing, separation, turbulence, and energy losses. However,correct representation of the junction hydraulics is important in realistic simulation andreliable computation of the flow in a sewer system (Sevuk and Yen, 1973).


FIGURE 14.15 Junction benching of sewers and flow guides.




14.4.2 Junction Hydraulic Equations

The continuity equation of the water in a junction is

∑0

0Qi � Qj � �

ddst� (14.35)

where Qi � the flow into or out from the junction by the i-th joining sewer, being positivefor inflow and negative for outflow; Qj � the direct, temporally variable water inflow into(positive) or the pumpage or overflow or leakage out from (negative) the junction, if any;s � the storage in the junction; and t � time. For a two-way junction, the index i � 1, 2;for a three-way junction, i � 1, 2, 3, and so on.

The energy equation in a one-dimensional analysis form is

∑0

0Qi

�2Vgi2

� � �Pγ

i� � Zi

� QjHj � s �ddYt� � ∑

0

0QiKi �2

Vgi2

� , (14.36)

where Zi, Pi, Vi � the pipe invert elevation above the reference datum, piezometric pres-sure above the pipe invert, and velocity of the flow at the end of the section of the ith sewerwhere it meets the junction, respectively; Hj � the net energy input per unit volume of thedirect inflow expressed in water head; Ki � the entrance or exit loss coefficient for the ithsewer; Y � the depth of water in the junction; and g � the gravitational acceleration. Thefirst summation term in Eq. (14.36) is the sum of the energy input and output by the join-ing pipes. The second term at the left-hand side of the equation is the net energy broughtin by the direct inflow. The first term to the right of the equal sign is the energy stored inthe junction as its water depth rises. The last term is the energy loss.

The momentum equations for the two horizontal orthogonal directions x and z are

∑0

0(QiVix) � �0

0

g�Pγ

x� dA (14.37)

and

∑0

0(QiViz) � �0

0

g�Pγ

z� dA , (14.38)

where px and pz � the x and z components of the pressure acting on the junction bound-ary, respectively, and A � the solid and water boundary surface of the junction. The directflow Qj is assumed entering the junction without horizontal velocity component. Theright-hand side term of Eqs. (14.37 and 14.38) is the x or z component force, where theintegration is over the entire junction boundary surface A. The left-hand side term is thesum of momentum of the inflow and outflow of the joining pipes. Note that for a three-way merging junction, two of the Qi’s are positive and one Qi is negative, whereas for athree-way dividing junction, two of the Qi’s are negative.

Joliffe (1982), Kanda and Kitada (1977), Taylor (1944), and others suggested the useof momentum approach to deal with high velocity situations. To illustrate this approach,consider the three-way junction shown in Fig. 14.16. The control volume of water at thejunction enclosed by the dashed line is regarded as a point, and there is no volume changeassociated with a change of depth within it. One of the two merging sewers is along thedirection of the downstream sewer, whereas the branch sewer makes an angle ϕ with it.When one assumes that the pressure distribution is hydrostatic and the flow is steady, theforce-momentum relation can be written as





γh�2A2 � γh�3A3 cos ϕ � γh�bAb sin ϕ � γh�1A1 � F (14.39)

� ρQ1V1 � ρQ2V2 � ρQ3V3 cos ϕ ,

where A � the flow cross-sectional area in a sewer; h� � depth of the centroid of A; γ� the specific weight; ρ � the density of water; Q � the discharge; V � Q/A � thecross-sectional mean velocity; and F � the sum of other forces that are normallyneglected. Some of these neglected forces are the component of the water weight in thecontrol volume along the small bottom slope, the shear stresses on the walls and bot-tom, and the force due to geometry of the junction if the sewers are not invert alignedor the longitudinal sewers are of different dimensions. The subscripts 1, 2, and 3 iden-tify the sewers shown in Fig. 14.16, and b represents the exposed wall surface of thebranch in the control volume shown as ab in the figure. For the special case of invertaligned sewers with the branch (pipe 3) joining at right angle, ϕ � 90º, Eq. (14.39) canbe simplified as

A2(gh�2 � V22) � A1(gh�1 � V1

2) (14.40)

or

�QQ

2

1� �

�AA

2

1�

�((ggh�h�

1

2

//VV

1

2

2

2

))

�

�

11�

1/2

(14.41)

Based on experimental results of invert-aligned equal-size pipes merging with ϕ � 90º,Joliffe (1982) observed that the upstream depth h1 � h2 and proposed that

�hh

c

3

1� � �h

h

c

2

1� � ξF3

�b (14.42)

where hc1 � the critical depth in the downstream sewer, F3 � the Froude number of theflow in the branch sewer, and

ξ � 0.999 � 0.482

�QQ

2

1�

� 0.381

�QQ

2

1�

2. (14.43)

b � 0.514 � 0.067

�QQ

2

1�

� 0.197

�QQ

2

1�

2

� 0.122

�QQ

2

1�

3

(14.44)

The equation describing the load of sediment or pollutants, expressed in terms of con-centration c, can be derived from the principle of conservation as

�ddt� �0

s

cds � ∑ Qici � Qjcj � G , (14.45)


FIGURE 14.16 Control volume of junctionfor momentum analysis.




where G � a source (positive) or sink (negative) of the sediment or pollutants in thejunction.

Equations (14.35–14.44) are the theoretical basic equations for sewer junctions. Theyare applicable to junctions under surcharge as well as open-channel flows in the joiningpipes. However, more specific equations can be written for the point-type and storage-typejunctions.

14.4.3 Experiments on Three-Way Sewer Junctions and Loss Coefficients

Proper handling of flow in sewer networks required information on the loss coeffi-cients at the junctions. Unfortunately, there exists practically no useful quantitativeinformation on energy and momentum losses of unsteady flow passing through a junc-tion. Therefore, steady flow information on sewer junction losses are commonly usedas an approximation.

Table 14.10 summarizes the experimental conditions of three-way merging, surcharg-ing, top-open sewer junctions conducted by Johnston and Volker (1990), Lindvall (1984),and Sangster et al. (1958, 1961). Also listed in the table are the experiments by Blaisdelland Mason (1967), Serre et al. (1994), and Ramamurthy and Zhu (1997); these experi-ments were not conducted on open-top sewer junctions but on three-way merging closedpipes. They are listed in the table as an example because these tests were conducted withdifferent branch and main diameter ratios and with different pipe alignments. Hence, theymay provide helpful information for sewer junctions. There exists considerably more information on merging or dividing branched closed conduits than on sewer junc-tions. The reader may look elsewhere (e.g., Fried and Idelchik, 1989, Miller, 1990) forinformation about centerline-aligned three-way joining pipes as an approximation tosewer junctions.

The loss coefficients K2 1 and K3 1 for the merging flow are defined as

kij � . (14.46)

Figure 14.17 shows the experimental results of (1987) and Sangster et al. (1958) andLindvall for the case of identical pipe size of the main and 90º merging lateral. The cor-responding curves suggested by Miller (1990) and Fried and Idelchik (1989) for three-way identical closed pipe junctions are also shown as a reference. The values of the losscoefficients in a sewer junction that is open to air on its top are expected to be slightlyhigher than the enclosed pipe junction cases given by Miller because of the water volumeat the junction above the pipes.

The effect of the relative size of the joining branch pipe is shown in Fig. 14.18. The exper-imental data of Sangster et al. (1961) have identical upstream pipe sizes, D2 � D3 for four dif-ferent values of lateral branch to downstream main pipe area ratio, A3/A1. The data of Johnstonand Volker (1990) on surcharged circular open-top sewer junction are not plotted in Fig. 14.18because the mainline pipe area ratio A2/A1 � 0.41 instead of unity in the figure. Conversely,as a comparison, the smoothed curve of K21 for A3/A1 � 0.5 of the three-way pipe junction ofSerre et al. (1994) with A1 � A2 is plotted in Fig. 14.18a, and their experimental curves of K3

1 for A3/A1 � 0.21 and 0.118 are plotted in Fig. 14.18b. Also shown in the figure, as reference,are the three-way pipe junction curves for different values of A3/A1 suggested by Fried andIdelchik (1989) and Miller (1990) for identical size of main pipes, A2 � A1. The experimentsof Sangster et al. (1961) indicated that for a given A3/A1, the effect of A2/A1 on the loss coeffi-cients is minor. Therefore, their curves should be comparable with those of Fried and Idelchik,

�V2g

i2

� � hi � Zi

�

�2Vgj2

� � hj � Zj

��V2g

12

�





Hydraulic Design of Urban Drainage Systems 14.35T

AB

LE

14.

10E

xper

imen

tal S

tudi

es o

n T

hree

-way

Jun

ctio

n of

Mer

ging

Sur

char

ged

Cha

nnel

s

Ref

eren

ceTy

pe o

f Ju

ncti

onSh

ape

of C

hann

els

Cha

nnel

P

ipe

Ali

gnm

ent

at J

unct

ion

Type

of

Rem

arks

Slop

eV

ertic

alL

ongi

tudi

nal

Flo

w

Sang

ster

et a

l. Sq

uare

,C

ircu

lar,

Hor

izon

tal

Flus

hed

botto

mSt

raig

ht th

roug

h an

d St

eady

Als

o te

sts

of o

ppos

ed

(195

8,19

61)

rect

angu

lar,

or

D�

3.0

in.3

.75

in.

one

90º

mer

ging

cha

nnel

late

ral p

ipes

;ro

und

box

4.75

in. o

r 5.

72 in

.te

sts

with

gra

tein

flow

into

junc

tion

Lin

dval

lR

ound

box

Cir

cula

r,H

oriz

onta

lC

ente

r al

igne

dSt

raig

ht th

roug

h an

d on

eSt

eady

Los

s co

effi

cien

t (1

984,

1987

)D

mai

n�

144

mm

,90

º m

ergi

ng c

hann

elde

pend

ent o

n D

br/D

mai

n�

1.0,

junc

tion

diam

eter

,0.

686,

or 0

.389

late

ral p

ipe

diam

eter

,an

d fl

ow r

atio

John

ston

and

Sq

uare

box

Cir

cula

r,H

oriz

onta

lFl

ushe

d bo

ttom

Cen

terl

ine

alig

ned

with

Stea

dyV

olke

r (1

990)

Dm

ain

d�

70 m

m,

slig

ht d

efle

ctor

for

Dm

ain

up/D

mai

nd

�0.

64,

late

ral i

n ju

nctio

nD

br/D

mai

nd

�0.

91

Bla

isde

ll an

dE

nclo

sed

pipe

Cir

cula

r,C

ente

r or

top

Stra

ight

thro

ugh

and

one

Stea

dyR

eyno

lds

num

ber

Mas

on (

1967

)ju

nctio

nD

br/D

mai

n�

0.25

�1.

0al

igne

dm

ergi

n ch

anne

l at 1

5º–1

65º

effe

ct in

sign

ific

ant

by 1

5º in

crem

ents

Serr

e et

al.

Enc

lose

d pi

peC

ircu

lar,

(Hor

izon

tal)

Cen

ter

alig

ned

Stra

ight

thro

ugh

and

one

Stea

dy(1

994)

junc

tion

Dm

ain

�44

4 m

m,

90º

mer

ging

cha

nnel

Dbr

/Dm

ain

�0.

14,0

.23,

0.34

,or

0.46

Ram

amur

thy

Enc

lose

d

Rec

tang

ular

,(H

oriz

onta

l)Sa

me

heig

htSt

raig

ht th

roug

h an

d on

eSt

eady

and

Zhu

re

ctan

gula

r4.

14 m

m h

igh,

90º

emer

ging

bra

nch

(199

7)co

ndui

t jun

ctio

nm

ain

wid

th 9

1.5

mm

,br

anch

wid

th

20.4

mm

,70.

5 m

m,

or 9

1.5

mm





FIGURE 14.18 Headloss coefficients for surcharged 3-way junction with 90o merging lateral of differ-ent sizes. (a) Mainline loss coefficient K21. (b) Branch loss coefficient K31.

FIGURE 14.17 Experimental headloss coefficients for surcharged 3-way sewer junction with identicalpipe sizes and 90o merging lateral. (a) Mainline loss coefficient K21; (b) Branch loss coefficient K31.

Miller, and Serre et al. However, Fig. 14.18 depicts considerable disagreement among the dif-ferent sources, indicating the need for more reliable investigations.

The joining angle of the lateral branch is a significant factor affecting the loss coeffi-cients, particular on K3 1. The values of the loss coefficients decrease if the joining anglemore or less aligns with the flow direction of the main, and increase if the lateral flow isdirected against the main. The references of Fried and Idelchik (1989) and Miller (1990)provide some idea on the adjustment needed for the K values due to the joining angle.





Listed in Table 14.11 is a summary of experiments on steady flow in three-way merg-ing open-channel junctions. Most of the studies were done with point-type junctions. Theexperimental subcritical flow results of storage-type junctions by Marsalek (1985) andTownsend and Prins (1978) are plotted in Fig. 14.19 for lateral joining 90º to the same sizemainline pipes. Yevjevich and Barnes (1970) gave the combined main and lateral losscoefficient but not the separate coefficients, making the result difficult to be used in rout-ing simulation. The points in the figure scatter considerably, but they are generally in thesame range of the loss coefficient values for surcharged three-way 90º merging junctionexcept K3 1 for Townsend and Prins’ data. It is interesting to note that the most frequentlyencountered sewer junctions are three- and four–way box junctions with unsteady sub-critical flow in the joining circular sewers. None of the open-channel experiments wasconducted under these conditions. All were tested with steady flow. It is obvious thatexisting experimental evidence and theory do not yield reliable quantitative informationon the loss coefficients of three-way sewer junctions. Before more reliable information isobtained, provincially for design and simulation of three joining identical size sewers, forK2 1 a curve drawn between that of Lindvall and that of Sangster et al. can be used as anapproximation. For K3 1, the curve of Lindvall can be used. For joining pipes of unequalsizes, the curves of Sangster et al. appear to be tentatively acceptable.

14.4.4 Loss Coefficient for Two-Way Sewer Junctions

Two-way junctions are used for change of pipe slope, pipe alignment, or pipe size.Experimental studies on two-way, surcharged, top-open sewer junctions are listed in Table

FIGURE 14.19 Headloss coefficients for 3-way open-channel sewer junction with identical pipesizes and 90E merging lateral. (a) Mainline loss coefficient K-21 (B) Branch loss coefficient K-31

(After Yen, 1987).




14.38 Chapter FourteenT

AB

LE

14.

11E

xper

imen

tal S

tudi

es o

n T

hree

-Way

Jun

ctio

n of

Mer

ging

Ope

n C

hann

els

Pip

e A

lign

men

t at

Jun

ctio

nTy

pe o

f F

low

Ref

eren

ces

Type

of

Shap

e of

Cha

nnel

U

pstr

eam

Dow

nstr

eam

Rem

arks

Junc

tion

Cha

nnel

sSl

ope

Vert

ical

Lon

gitu

dina

lP

ipes

Pip

e

Tayl

or (

1944

)Po

int

Rec

tang

ular

,H

oriz

onta

lFl

ushe

d St

raig

ht th

roug

h Su

bcri

tical

Subc

ritic

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lso

theo

retic

al a

naly

sis

iden

tical

wid

th,

botto

man

d on

e m

ergi

ng

base

d on

mom

entu

m,

B�

4 in

.ch

anne

l at 4

5º

good

agr

eem

ent w

ith

or 1

35º

45º

mer

ging

but

not

w

ith 1

35º

mer

ging

Bow

ers

(195

0)Po

int

Tra

pezo

idal

,0.

0062

,0.0

12Fl

ushe

dSt

raig

ht th

roug

h Su

perc

ritic

alSu

perc

ritic

alSt

ruct

ure

P7,h

ydra

ulic

iden

tical

wid

th,

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d on

e m

ergi

ng

jum

ps f

orm

ed u

pstr

eam

B

� 7

.2 in

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anne

l at 5

1ºof

junc

tion,

othe

r st

ruct

ures

with

late

ral

botto

m u

p to

3 f

t ab

ove

mai

n

Beh

lke

and

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ecta

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ar o

rE

ach

chan

nel

Flus

hed

Stra

ight

thro

ugh

Supe

rcri

tical

Supe

rcri

tical

Use

of

tape

red

wal

l in

Pritc

hett

trap

ezoi

dal

slop

e va

ried

botto

man

d on

e m

ergi

ng

the

junc

tion

to d

imin

ish

(196

6)(s

ide

slop

e 1:

1)in

depe

nden

tlych

anne

l at 1

5º,

diag

onal

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e an

d

30º,

or 4

5ºpi

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p pr

oble

ms

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ber

and

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tR

ecta

ngul

ar,

Hor

izon

tal

Flus

hed

Stra

ight

thro

ugh

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ritic

alSu

bcri

tical

Gre

ater

loss

es a

ssoc

iate

dG

reat

ed (

1966

)B

� 5

in.

botto

man

d on

e m

ergi

ng

with

incr

easi

ng m

ergi

ngch

anne

l at 3

0º,

angl

es o

f br

anch

cha

nnel

60º,

or 9

0º

Yev

jevi

ch a

ndSq

uare

C

ircu

lar,

0.00

008

Flus

hed

Stra

ight

thro

ugh

Subc

ritic

alSu

bcri

tical

Gre

ater

loss

for

the

case

B

arne

s (1

970)

box

Dm

ain

� 6

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14.12. All the experiments were conducted with the same size upstream and downstreampipes joining the junction. Only Sangster et al. (1958, 1961) tested also the effect of dif-ferent joining pipe sizes. These experimental results show that for a straight-through, two-way junction, the value of the loss coefficient is usually no higher than 0.2. Alignment ofthe joining pipes and benching in the junction are also important factors to determine thevalue of the loss coefficient.

Figure 14.20a shows the headloss coefficiet of a surcharged two-way open-top junc-tion connecting two pipes of identical diameters aligned centrally given by the experi-ments of Archer et al. (1978), Howarth and Saul (1984), Johnston and Volker (1990) andLindvall (1984). Noticeable is the swirl and instability phenomena when the junction sub-mergence (junction depth to pipe diameter ratio) is close to two and the correspondinghigh head loss coefficient. The ranges of loss coefficient given by Ackers (1959),Marsalek (1984), and Sangster et al. (1958) are also indicated in Fig. 14.20a, but the dataon the variation with the pipe-to-junction–size ratio was not given by these investigators.Sangster et al. (1958) also tested the effect of different sizes of joining pipes for sur-charged two-way junction. Some of their results are plotted in Fig. 14.20b. They did notindicate a clear influence of the effect of the size of the junction box. However, BoPedersen and Mark (1990) demonstrated that the loss coefficient of a two–way junctioncan be estimated as a combination of the exit headloss due to a submerged discharging jetand the entrance loss of flow contraction. They suggested that the loss coefficient Kdepends mainly on the size ratio between the junction and the joining pipes of identicalsize. For an infinitely large storage junction, the theoretical limit of K is 1.5. For the junc-tion–diameter to pipe-diameter ratio, DM/D less than 4, they proposed to estimate the Kvalues according to benching as shown in Fig. 14.21.

14.4.5 Storage Junctions

For a storage- (or reservoir-) type junction, the storage capacity of the junction is rela-tively large in comparison to the flow volume and hydraulically it behaves like a reservoir.A water surface, and hence, the depth in the junction can be defined without great diffi-culty. A significant portion of the energy carried in by the flows from upstream sewers isdissipated in the junction. If the horizontal cross-sectional area of the junction Aj remainsconstant, independent of the junction depth Y, the storage is s � AjY. Hence, ds/dt �Aj(dY/dt) � Aj(dH/dt), where H � Y � Z � the water surface elevation above the refer-ence datum, and Z � the elevation of the junction bottom. Therefore, from Eq. (14.35),

∑Qi � Qj � Aj �ddHt�

(14.47)

Either the energy equation (Eq. 14.36) or the momentum equations [Eqs. (14.37) and(14.38)] can be used as the dynamic equation of the junction. If the energy loss coefficientKi in Eq. (14.36) can be determined, use of the energy equation is appropriate. On theother hand, if the pressure on the junction boundary can be determined, the momentumequation is also applicable. If the junction were truly a large reservoir, both the loss coef-ficients and the pressure could reasonably be estimated on the basis of information onsteady flow entering or leaving a reservoir.

Customarily for the convenience of computation, instead of Eq. (14.36), the junctionenergy relationship is divided for each joining sewer by relating the total head of the sewerflow to the total head in the junction. Assuming that the energy contribution from thedirect lateral inflow Qj is negligible, the component of Eq. (14.36) for each joining seweri can be written as





FIGURE 14.20 Headloss coefficient for surcharged 2-way open-top straight-through sewer junction. (a) Same size sewers upstream and downstream. (b)Different joining pipe sizes. (After Yen, 1987).





H � (1 � Ki)(Vi2/2g) � (Pi/γ) � Zi (14.48)

For open-channel flow in the joining pipes, the piezometric term Pi/γ is

Pi/γ � hi (14.49)

where hi is the open-channel flow depth of the ith pipe at the junction. It should be cau-tioned that Eq. (14.48) is applicable only when there is no free surface discontinuitybetween the junction and the sewer. In other words, they are applicable to Cases B and Din Fig. 14.9 and all four cases in Fig. 14.8. The flow equations for these pipe exit andentrance cases are given in Table 14.13.

14.4.6 Point Junctions

A point-type junction is the one whose storage capacity is negligible, and the junction istreated as a single confluence point. Hence, Eq. (14.35) is reduced to

∑Qi � Qj � 0 (14.50)

For subcritical flow in the sewers emptying into the point junction, the flow can dis-charge freely into and without the influence of the junction only when a free fall existsover a nonsubmerged drop at the end of the pipe (Case A in Fig. 14.9). Otherwise, the sub-critical flow in the sewer is subject to backwater effect from the junction. Since the junc-tion is treated as a point, the dynamic condition of the junction is usually represented bya kinematic compatibility condition of common water surface at the junctions for all thejoining pipes without a free fall (Harris, 1968; Larson et al., 1971; Roesner et al., 1984;Sevuk and Yen, 1973; and Yen, 1986a). Thus, by neglecting the junction storage and forsubcritical sewer flow into the junction,

hi � hic if Zi � hic � Zo � ho (Case A in Fig. 14.9)(14.51)

FIGURE 14.21 Effect of benching on loss coefficient of surcharged two-way sewer junctionaccording to Bo Pedersen and Mark (1990).





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hi � Zi � ho � Zo otherwise (Case B in Fig. 14.9)

where Zo and ho � the invert elevation and depth of the flow at the entrance of the down-stream sewer taking the outflow from the junction, respectively.

For a supercritical flow in a sewer flowing into a point junction, Case C in Fig. 14.9would not occur. Only a subcase of Case B in Fig. 14.9 with hi hic exists where Eq.(14.51) applies. For Case D of Fig. 14.9 with submerged exit,

Zi � (Pi/γ) � (Po/γ) � Zo � Zi � Di (14.52)

The flow in the downstream sewer, which takes water out from the junction, may besubcritical, supercritical, or submerged, depending on the geometry and flow condi-tions. The flow equations are the same as the storage junction outflow Cases I–IV givenin Table 14.13.

14.5. HYDRAULICS OF A SEWER NETWORK

Hydraulically, sewers in a network interact, and the mutual flow interaction must beaccounted for to achieve realistic results. In designing the sewers in a network, the con-straints and assumptions on sizing sewers as discussed in Sec. 14.3.4 should be noted.

The rational method is the most commonly and traditionally used method for thedesign of sewer sizes. As described in Sec. 14.3.4, each sewer is designed independentlywithout direct, explicit consideration of the flow in other sewers. This can be done becauseto design a sewer, only the peak discharge, not the entire hydrograph, of the design-stormrunoff is required. As previously explained in Sec. 14.3.4, each sewer has its own designstorm. The information needed from upstream sewers is only for the alignment and burydepth of the sewer and the flow time to estimate the time of concentration for the deter-mination of the rainfall intensity i for the sewer to be designed.

Contrarily, in simulation of flow in an existing or predetermined sewer network forurban stormwater control and management, often the hydrograph, not merely the peak dis-charge, is needed, and a higher level of hydraulic analysis that considers the interaction ofthe sewers in the network is required. This network system analysis involves combiningthe hydraulics of individual sewers as described in Sec. 14.3, together with the hydraulicsof junctions described in Sec. 14.4.

A sewer network can be considered as a number of nodes joined together by a numberof links. The nodes are the manholes, junctions, and network outlets. The links are thesewer pipes. Depending on their locations in the network, the nodes and links can be clas-sified as exterior or interior. The exterior links are the most upstream sewers or the lastsewer having the network exit at its downstream end. An exterior sewer has only one endconnected to other sewers. Interior links are the sewers inside the network that have bothends connected to other sewers. Exterior nodes are the junctions or manholes connectedto the upstream end of the most upstream sewers, or the exit node of the network. An exte-rior node has only one link connected to it. Interior nodes (junctions) inside the networkhave more than one link connected to each node.

A systematic numeric representation of the nodes or links is important for computersimulation of a network. One approach is to number the links (sewers) according to thebranches and the order of the pipes in the branch, similar to Horton’s numbering of riversystems. Another approach is to identify the links by the node numbers at the two ends ofthe link. Using the node number identification system, a numbering order technique sim-ilar to topographic contour lines called the isonodal line method, can be used. This method






was proved effective for computer manipulation of sewer network simulation (Mays,1978; Yen et al., 1976).

There are special hydraulic features of the networks that are often ignored to simulateflow in sewers. Usually, much attention is given to single sewers in the network but littleattention to the junctions, and the mutual effects among the sewers are often neglected. Ifthe pipes in a sewer network are all surcharged, it is obvious that the network should besolved as a whole similar to water supply networks. Conversely, if all the pipes carrysupercritical flow, one can simply solve the flow starting from the most upstream sewers,complete the solution for the upstream sewers before proceeding to the solution of the downstream sewers in sequence. Likewise, if each one of the sewers in the networkhas a drop of sufficient height at its exit to ensure a free fall at its downstream end, thedownstream boundary condition is specified when the flow is subcritical. Hence, the solu-tion can still progress sewer by sewer starting from the most upstream exterior sewers,having the solution for the upstream sewers completed before proceeding to the nextdownstream sewer.

However, more often than not, flows in sewer networks do not fall into one of thesethree types just mentioned. For example, for a subcritical sewer flow that can be mathe-matically represented by the Saint-Venant equation or its simplified approximations, thedownstream boundary condition at the sewer exit depends on the hydraulic condition ofthe downstream junction. Except for the network exit, this downstream junction conditionby itself is unknown, and its solution is affected by the sewers joining to it. The junctioncontinuity equation is obtained from Eq. (14.47) written in a finite difference form usingthe average values between time levels n and n � 1:

∑(Qi,n � Qi,n � 1) � Qj,n � Qj,n � 1 � �2∆At

j� (Hn � 1 � Hn) � 0 (14.53)

where the summation of i is over the number of the sewers connected to the junction. Fora point-type junction, the last term vanishes. It should be cautioned that if surcharge flowis considered, point-type junctions should not be used. When the junction is completelyfilled and ground flooding occurs, Aj is so large that Hn � 1 practically equals Hn. The junc-tion dynamic equation depends on the condition between the junction and the connectingpipes, as shown in Table 14.13 and discussed in Secs. 14.4.5 and 14.4.6.

Thus, for a simple network consisting of a single four-way junction with threeupstream sewers and one downstream sewer and having each sewer divided into two com-putational reaches, by using an implicit numerical scheme the matrix of the equations tobe solved for the 24 unknowns is shown in Fig. 14.22. There is one continuity equationand one momentum equation in its complete or simplified form for each reach. There aretwo boundary conditions for each sewer, which are described by the junction equations foran interior junction, and by the network outlet or inflow conditions for an exterior junc-tion. Therefore, in general, if there are N sewers in the network and each sewer is dividedinto M reaches for open-channel flow computation, there are N(2M � 2) algebraic equa-tions to solve for the N(2M � 2) unknowns.

If all the sewers of the simple four-way junction network are surcharged and thePreissmann hypothetical slot is not used, there are nine algebraic equations to be solvedfor the nine unknowns, that is, one surcharge equation and one upstream junction conti-nuity equation for each sewer, plus the network outlet boundary condition. The matrix forimplicit solution of the simple network is shown in Fig. 14.23.

Likewise, if the simple four-way junction network has one sewer surcharged and theother three sewers under open-channel flow for which each sewer is divided into two com-putational reaches, there are 20 equations for the 20 unknowns. The corresponding matrixis shown in Fig. 14.24. In general, if a network has No sewers under open-channel flow,each divided into M computational reaches, and Ns sewers under surcharge flow, the total





FIGURE 14.22 Matrix configuration for 4-way junction with all four sewers having open-channel flow.(After Yen, 1986a).

FIGURE 14.23 Matrix configuration for 4-way junction with all four sew-ers having surcharged flow. (After Yen, 1986a).

FIGURE 14.24 Matrix configuration for four-way junction with three sewers having open-channel flow and one sewer surcharged. (After Yen, 1986a).





number of equations is No(2M � 2) � 2Ns if the last (exit) pipe of the network is not sur-charged, or No(2M� 2) � 2Ns � 1 if the exit pipe of the network is surcharged.

Thus, it is obvious that the number of unknowns and equations to be handled can eas-ily become large for a sewer network consisting of many sewers. Consequently, themethod of solution becomes important to achieve the efficient solution of the entire net-work. The network solution techniques can be classified into four groups as follows:

1. Cascade method. In this method, the solution is sought sewer by sewer, starting fromthe most upstream sewers. Each sewer is solved for the entire duration of runoffbefore moving downstream to solve for the immediately following sewer: that is, theentire inflow hydrograph is routed through the sewer before the immediate down-stream sewer is considered. For the noninertia, quasi-steady dynamic wave anddynamic wave equations, this cascading routing is possible only for the following twoconditions: (1) the flow in all the sewers is supercritical throughout, or (2) the exitflow condition of each sewer is specified, independent of the downstream junctioncondition—for example, a free fall over a drop of sufficient height at the downstreamend of each sewer. However, in some sewer models, the downstream condition of thesewer is arbitrarily assumed—for example, using a forward or backward differenceand approaching to normal flow, so that the cascading routing computation can pro-ceed, and the solution, of course, bears no relation to reality. Conversely, in the kine-matic wave approximation, only one boundary condition is needed and it is usuallythe inflow discharge or depth hydrograph of the sewer. At the junction, only thedownstream sewer dynamic equation and junction continuity equations are used. Thejunction dynamic equations for the upstream sewers are ignored. Thus, the computa-tion can proceed downstream sewer by sewer in a cascading manner, completelyignoring the downstream backwater effects. This method of solving each sewer indi-vidually for the entire hydrograph before proceeding to solve for the next downstreamsewer is relatively simple and inexpensive. But, it is inaccurate if the downstreambackwater effect is significant.

2. One-sweep explicit solution method. In this method, the flow equations of the sewersand junctions are formulated by using an explicit finite difference scheme such that theflow depth, discharge, or velocity at a given computation station x � i∆x and the cur-rent time level t � n ∆t can be solved explicitly from the known information at the pre-vious locations x i ∆x at the same time level, as well as known information at theprevious time level t � (n � 1) ∆t. Thus, the solution is sought reach by reach, sewerby sewer, and junction by junction, individually from upstream to downstream, over agiven time level for the entire network before progressing to the next time level foranother sweep of individual solutions of the sewers and junctions for the entire net-work. In this method, only one or a few equations are solved for each station, avoid-ing the large matrix manipulation as in the implicit simultaneous solution of the entirenetwork, and computer programming is relatively direct and simple compared to thesimultaneous solution and overlapping segment methods to be discussed later.Nevertheless, this one-sweep method bears the drawback of computational stabilityand accuracy problems of explicit schemes that requires the use of small ∆t. An exam-ple of this approach for sewer flow is the Extended Transport (EXTRAN) Block ofStormwater Management Model (SWMM) (Roesner et al., 1984) in which each seweris treated as a single computational reach. Variations of this one-sweep approach doexist. For example, each reach of a sewer can be solve explicitly, or the sewer is solvedimplicitly, and then the junction flow condition is solved explicitly.

3. Simultaneous solution method. When the implicit difference formulations of thedynamic wave, quasi-steady dynamic wave, or noninertia equations and the corre-





sponding junction equations are applied to the entire sewer network and solved for theunknowns of flow variables together, simultaneous solution is ensured. The simulta-neous solution usually involves solving a matrix similar to those shown in Figs.14.22–14.24, but much bigger for most networks. There are numerically stable andefficient solution techniques for sparse matrices of implicit schemes. Nonetheless,since the matrix is not banded, solution of the sparse matrix may still require high com-putational cost and large computer capacity if the network is large.

4. Overlapping segment method. To avoid the costly implicit simultaneous solution forlarge networks and still preserve the advantages of stability and numerical efficiencyof the implicit schemes, Sevuk (1973) and Yen (1973a) proposed the use of a techniquecalled the overlapping segment method. Similar to the well–known Hardy Crossmethod for solving flow in distribution networks, this method is a successive iterationtechnique. Unlike the Hardy Cross method which applies only to looped networks, theoverlapping segment method can be applied to dendritic as well as loop-type networks.It decomposes the network into a number of small, overlapped, subnetworks or seg-ments, and solutions are sought for the segments in succession. Thus, it is suitable tobe used in programming for models to be applied for simulation of large sewer net-works in different locations.

A simple, single-step overlapping segment example of a network consisting of threesegments is shown in Fig. 14.25. Each segment is formed by a junction together withall the sewers joining to it. Thus, except for the most upstream and downstream (exte-rior) sewers, each interior sewer belongs to two segments—as a downstream sewer forone segment and then as an upstream sewer for the sequent segment, that is, “over-lapped.” Each segment is solved as a unit.

The flow equations are first applied to each of the branches of the most upstreamsegment for which the upstream boundary condition is known and solved numericallyand simultaneously with appropriate junction equations for all the sewers in the seg-ment. If the flow is subcritical and the boundary condition at the lower exit of thedownstream sewer of the segment is unknown, the forward or backward differences,depending on the numerical scheme, are used as an approximate substitution.Simultaneous numerical solution is obtained for all the sewers and junctions of the seg-ment for each time step, repeating until the entire flow duration is completed.For example, for the network shown in Fig. 14.25a, solutions are first sought for eachof the two segments shown in Fig. 14.25b. Since the downstream boundary conditionof the segment is assumed, the solution for the downstream sewer is doubtful and dis-carded, whereas the solutions for the upstream sewers are retained. The computationnow proceeds to the next immediate downstream segment (e.g., the segment shown inFig. 14.25c). The upstream sewers of this new segment were the downstream sewersof each of the preceding segments for which solutions have already been obtained. The

FIGURE 14.25 Overlapping segment scheme for network solution. (After Yen, 1986a).





inflows into this new segment are given as the outflow from the junctions of the pre-ceding segments. With the inflows known, the solution for this new segment can beobtained. This procedure is repeated successively, segment by segment in the down-stream direction, until the entire network is solved. For the last (most downstream)segment of the network, the prescribed boundary condition at the network exit is used.The method of overlapping segments reduces the requirements on computer size andtime when solving for large networks. It accounts for downstream backwater effect andsimulates flow reversal, if it occurs. Its accuracy and practical usefulness have beendemonstrated by Sevuk (1973) and Yen and Akan (1976). Solution by the single-step overlapping segment method accounts for the downstreambackwater effect of subcritical flow only for the adjacent upstream sewers of the junc-tion, but it cannot reflect the backwater effect from the junction to sewers fartherupstream if such case occurs. However, by considering the length-to-depth ratio of actu-al sewers, in most cases the effect of backwater beyond the immediate upstream branch-es is small, and hence, this imposes no significant error in routing of sewer flows. Forthe rare case of two junctions located closely, the overlapping segment method can bemodified to perform double iteration by recomputing the upstream segment after theapproximate junction condition is computed from the first iteration of the downstreamsegment. Alternatively and perhaps better, a two-step overlapping procedure can beemployed by making a segment containing two neighboring junctions, forming the seg-ment with three levels of sewers instead of two, and retaining only the solutions of thetop level sewers for each iteration. Thus, interior sewers are iterated twice. The over-lapping segment method can be modified to account for divided sewers or loop net-works, in addition to tree-type networks.The entrance and exit loss coefficients of the sewers at a junction vary with the sub-

mergence Y/D at the exit or entrance of the sewers. The values of these loss coefficientsmay be approximated from the information given in Sec. 14.4.2. In order to minimize thechance of computation instability due to discrete values of the loss coefficient K, Pansic(1980) assumed an S-curve-type smooth transition starting from a minimum value K1 atincipient submergence Y/D � 1. The loss coefficient will attain a maximum value Km atsome maximum submergence Ym/D, for example, at the level of ground flooding. The vari-ation in between is

K � K1 � α

�DY

� � 1

2

for 1 �DY

� � �Ym

2�

DD

� (14.54)

and

K � Km � α

�DY

� � �12�

�YD

m� � 1

2

for �Ym

2�

DD

� � �DY

� � �YD

m� , (14.55)

where

α � 2 . (14.56)

14.6 CAPACITY AND BOTTLENECK DETERMINATION

One of the most useful pieces of information to solve urban drainage problems is know-ing the flow carrying capacity of the channel or sewer. There are actually various kinds ofcapacities. There is the capacity for a single sewer. There is the capacity of the sewer net-

Km �K1��

�YD

m� � 1

2





work as a system which usually is different from the capacity of individual sewers. For anopen channel, often the maximum steady uniform flow that the channel can carry withoutspilling over bank is quoted as its capacity. For a sewer, the just about full gravity (open-channel) steady uniform flow is usually quoted.

In fact, for a subcritical open-channel flow, the discharge that the channel can carrydepends on the downstream water level. For a channel with a range of possible exit waterlevels, this would require repeated backwater profile computations. For a channel or sewernetwork that has a number of connected channels, the number of backwater computationscan easily become very large making it nearly impossible, if not impractical, for the net-work capacity determination. Yen and González (1994) developed a method to summarizethe backwater information of a channel into a hydraulic performance graph from whichthe network capacity can be determined. Knowing the channel or sewer capacities allowsa new approach to solve flood drainage problems by separating them into two parts: Thedemand part of how much water needs to be drained, which is essentially a hydrologyproblem. The supply part of how much can the channel or sewer handle, that is, the capac-ity, which is a hydraulic problem. The flood drainage problem can be analyzed by com-paring the two parts and searching for a solution.

14.6.1 Hydraulic Performance Graph

A hydraulic performance graph (HPG) is a plot of a set of curves showing the rela-tionship between water depths, yu and yd, at the upstream and downstream ends of thechannel reach for different specified discharges, that is, yu � F(yd, Q). Depicted inFigs. 14.26 and 14.27 are the HPGs for a mild-slope channel and a steep-slope chan-nel, respectively.

For a channel with mild slope (that is, yn � yc where yn is the normal flow depth and yc

is the critical depth for the given Q), the HPG has the following main characteristics:

1. The hydraulic performance curves, each for a given discharge, never intercept eachother. The curves with higher discharges are located above those with lower discharges.

2. The left bound of the curves indicated as “C-curve” in Fig. 14.26 represents the locusof critical flow condition at the downstream end of the channel reach. The downstream

FIGURE 14.26 Hydraulic performance graph formild–slope channel. (From Yen and Gonzalez, 1994).





depth yd is computed by using the following equation with yd � yc(Q),:

�ATc

c3

� � �Qg

2

� (14.57)

where Ac and Tc � the flow cross-sectional area and water surface width correspond-ing to the critical depth yc, respectively, and g � the gravitational acceleration.

3. The hydraulic performance curves are bounded at the right by the 45º straight line yu

� yd � SoL, designated as Z-line in Fig. 14.26, which is the line representing a hori-zontal water surface and no flow. The performance curves approach asymptotically tothe Z-line for very large values of yd or yu where the flow has very small convectiveacceleration.

4. The locus of normal flow depths for the possible discharges is name the N-line. It canbe expressed as yu � yd and is located at a distance SoL left to the Z-line.

5. The N-line divides the hydraulic performance curves into two regions. The regionbetween the C-curve and the N-line contains all the possible pairs of upstream anddownstream end water depths for which the backwater profiles are M2-type (see Table3.5 or Chow, 1959), whereas the water depth pairs within the region between the N-line and the Z-line correspond to all the possible M1-type profiles.

The HPGs for horizontal-slope and adverse-slope channels are similar to those in Fig.14.26 without the N-line because for these two cases, the normal depth is infinity andimaginary, respectively The HPG of S1 profiles in a steep-slope channel (Fig. 14.27) hasthe following major characteristics:

1. The hydraulic performance curves of S1 profiles, each for a given discharge, neverintercept each other. The curves with higher discharges are located below those withlower discharges.

2. The right bound of the curves indicated as Cu-curve in Fig. 14.27 represents the locusof critical flow condition at the upstream end of the channel reach. The upstream depthyu is computed by using Eq. (14.57) with yu � yc(Q).

FIGURE 14.27 Hydraulic performance graph forsteep–slope channel. (From Yen and Gonzalez, 1994).





3. The region of S1-profile hydraulic performance curves is bounded on the left by thehydraulic performance curve that corresponds to the discharge Qs which starts at theCu curve and is asymptotic to the Z-line for large yu and yd. The threshold discharge Qs

is the discharge for which the channel slope is critical. For smaller discharges, thechannel slope becomes mild instead of steep. The value of Qs is determined by settingthe critical depth equal to the normal depth, or equivalently the critical discharge equalto the normal flow discharge with yn � yc, which yields

�Rs

4

A

/3

s

Ts� � �

Kg

n

n2S

2

o� (14.58)

For a given channel reach, the procedure to establish the hydraulic performance graphHPG is as follows:

1. Determine the ranges of depths or water surface elevations to be considered at the twoends of the channel reach.

2. Determine and plot the Z-line for which the water surface elevations are equal at theupstream and downstream ends, or yd � yu � SoL, where So is the channel slope and Lis the reach length.

3. Determine and plot the N-line which is the 45º line at a distance equal to SoL to the leftof the Z-line, and on which yd � yu.

4. For a mild-slope channel with M1- or M2-type backwater profiles, choose a discharge Q,

a. Compute the normal flow depth yn � yu � yd and mark this point on the N-line forthis Q.b. Compute the critical depth, yc, at the downstream end of the channel reach byusing Eq. (14.57).c. Perform the backwater computation for the given Q and yc at the downstream endto determine the corresponding upstream depth yu, the backwater computation can bedone by using the standard step method, direct step method, or any other methodsdescribed in Sec. 3.5 or in Chow (1959).d. Plot the result of this set of (yu, yd � yc) for the specified Q as one point of the C-curve on the HPG; it is also the beginning point of the hydraulic performance curve ofthe chosen discharge.

For a steep-slope channel with S1-type backwater profiles,

a. Determine the value of Qs which corresponds to the condition at which the criticaldepth is equal to the normal depth, that is, yn � yc, or equivalently the critical dischargeis equal to the normal flow discharge, that is, Qn � Qc. The value of yn � yc can beobtained by using Eq. (14.58). This depth is the water depth that corresponds to theminimum discharge Qs, for which the channel slope remains steep, and with this depth,Qs can be computed by using Eq. (14.57).

b. For a chosen Q Qs, use Eq. (14.57) to compute the critical depth yc at theupstream end of the channel reach.

c. Perform the backwater computation for the specified Q starting with the corre-sponding yc at the upstream end of the reach to determine the downstream depth yd.

d. Plot the result of this set of (yd, yu � yc) for the specified Q as one point of the Cu-curve on the HPG, and it serves as the beginning point of the hydraulic performancecurve of the chosen discharge.




5. For the Q chosen in Step 4, select a feasible downstream depth yd and perform a back-water computation to determine the upstream depth yu. This pair of (yd, yu) constitutesa point of the hydraulic performance curve for this Q.

6. Repeat Step 5 for a few selected yd’s to provide sufficient pairs of (yd, yu) values toplot the hydraulic performance curve for the chosen Q. The curve starts at the C-curve and approaches asymptotically to the Z-line for large yu or yd. For a mild-slope channel, the constant Q curve crosses the N-line between the downstream Cd-curve and the Z-line. For a steep-slope channel, the constant Q curve starts at theupstream Cu-curve.

7. Select different feasible discharges and repeat Steps 4–6 to establish the hydraulic per-formance curves for different Q’s.

8. Connect the critical-depth C points computed for different discharges as the C-curveon the HPG.


FIGURE 14.28. Example HPG for Reach 1 of Boneyard Creek.





A typical M-type HPG is shown in Fig. 14.28 as an example.

14.6.2 Flow Capacities of a Channel Reach

The following representative flow capacities can be defined for an individual channelreach (Yen and Gonzalez, 1994):

1. Absolute maximum carrying capacity of a channel reach (Qamax)—the largest dischargethe reach is able to convey when the water depth at its exit cross section is criticalwhile there is no bank overflow.

2. Maximum uniform flow capacity (Qnmax)—the maximum steady, uniform, flow dis-charge that the reach can convey either as the flow just about to spill overbank, or asthe flow reaching a surcharged condition, with the free surface parallel to the channelbottom.

3. Maximum flow capacity for a given exit water level (Qexmax)—for a given tailwaterstage, the maximum steady gradually varied open-channel flow discharge that thereach can carry without spilling overbank. Obviously, this capacity varies with the tail-water lever, having Qamax as its upper limit.

4. Maximum surcharged-flow capacity (Qsmax)—for a channel reach with a top cover suchthat under high flow the open-channel flow changes to pressurized conduit flow, thiscapacity is the discharge this closed-top reach can convey when the upstream watersurface is at the bank�full stage and the downstream water elevation is at the crownlevel of the opening of the bridge, culvert, or sewer.

The water surface profiles corresponding to these four different capacities are shownin Fig. 14.29 for the closed-top Reach 2 in Fig. 14.30 as an example. The value of Qsmax

is determined from the closed-conduit flow rating curve shown in Fig. 14.31, whereas theopen–channel flow capacities are determined from Fig. 14.32. The values of Qamax andQnmax for the other three open-channel reaches individually are also listed in Table 14.14,which are read from their individual HPG shown in Figs. 14.28, 14.33, and 14.34.

The HPGs of channels of similar geometries can be nondimensionalized for more general uses. Shown in Fig. 14.35 is such a graph for open-channel flows in circular sew-ers with SoL/D � 0.05 and Qf /�g�D�5� � 0.224 where So, L, and D � the slope, length, anddiameter of the sewer pipe, respectively; Qf � the just-full steady uniform flow sewer

FIGURE 14.29 Water surface profiles for different hydraulic capacities of closed-topbridge Reach 2 of Boneyard Creek.





FIGURE 14.30 Example flow capacities and water surface profiles of Boneyard Creek fortwo exit water levels.

FIGURE 14.31 Rating curve for Reach 2 of Boneyard Creek.





FIGURE 14.32. Example HPG for Reach 2 of Boneyard Creek (7 + 375 to 7 + 435).

capacity; y � the depth of flow, subscripts 1 and 2 � upstream (entrance) and downstream(exit) cross section of the sewer, respectively; and the subscript n � normal (steady uni-form) flow.

14.6.3 Bottleneck and Channel System Capacity Determination

The bottleneck of network of drainage channels or sewers is the critical location withinthe network where the water is about to spill overbank or violate specified restriction.Therefore, for a given exit water level, the bottleneck determines the capacity of the chan-nel network as a system without flooding. Different exit water levels may have differentbottleneck locations and different system capacities. Because the flows in the channels of

TABLE 14.14 Maximum Capacities of Individual Reaches of Boneyard Creek

Channel Reach Qamax Qnmax Qsmax

(ft3/s) (ft3/s) (ft3/s)

Reach 1: Lincoln Ave.—Gregory St. 1130 930

Reach 2: Gregory St. Bridge 1025 807 1371

Reach 3: Gregory St.—Footbridge 1100 800

Reach 4: Footbridge—Loomis Lab (7 � 945) 1370 1070





the network mutually interact, this interaction must be accounted for in the system capac-ity determination. The set of the HPGs and rating curves for the individual reaches of thesystem can be used together in sequence to determine the bottleneck and capacity of thechannels as a system. To apply the HPG method of Yen and Gonzalez (1994) to determinethe bottleneck and flow capacity of a drainage channel system, the channel system is firstsubdivided into reaches such that within each reach the geometry, alignment, and rough-ness are approximately the same and there is no significant lateral inflow within the reach.For subcritical flow, the system capacity is a function of the tailwater level at the exit ofthe most downstream reach. The maximum system capacity occurs when depth at the sys-tem exit is critical. While the flow capacity of each reach can be determined individuallyfrom its HPG, the flow capacity of the channel flowing as a part of the system should be

FIGURE 14.36. Schematic of lateral runoff contributionalong a channel. (After Yen and Gonzalez, 1994).

FIGURE 14.35. Nondimensional hydraulic performance graph for a sewer pipe. (Yen, 1987).




determined by accounting for the backwater effect between the reaches, the losses at thejunctions, if any, and the significant lateral flow joining the channel.

Concerning the lateral flow entering the channel system at the junctions betweenreaches, a simple approximate method of Yen and Gonzalez (1994) can be used if nobetter method is available. As shown schematically in Fig. 14.36; the area drained bychannel reach j�1 is Au and the corresponding discharge is Qu � Cui Au, where Cu is therunoff coefficient for the area drained. The lateral flow joining the channel at the down-stream end of reach j�1 is QL(j�1). The sewer delivering the lateral flow peak dischargeQL(j�1) drains an incremental local area AL(j-1) having a CL(j1) runoff coefficient. Therefore,under the rainstorm with intensity i, QL(j � 1) � CL(j � 1)iAL(j � 1). At the junction betweenreaches j�1 and j, Qj � Qu � QL(j�1). Hence, the ratio between the lateral flow and theflow in reach j�1 is

� (14.59)

For a given tailwater level at the system exit, for each reach the upstream water levelis determined from the HPG knowing the downstream water level, progressing reach byreach toward upstream, with junction head loss included if it exists. If the reach is sur-charged, the rating curve is used instead of HPG. It may require a trial of several dis-charges to locate the bottleneck and identify the system capacity iteratively. Details andexamples of this procedure can be found in Yen and Gonzalez (1994). By applying thisprocedure to the example four-reach system shown in Fig. 14.30, with the rating curve ofFig. 14.31 and HPGs of Figs. 14.28, 14.33, and 14.34, the network capacity for exit tail-water level equal to 711.0 ft is determined as 515 ft3/s. This capacity is controlled by thebottleneck of spilling water overbank near the upstream end of reach 1 as shown in Fig.14.30. The water surface profile is also shown in this figure.

The channel system hydraulic capacity and the location of the bottleneck vary withthe water surface elevation at the exit. When the exit water level is low, the bottleneckstend to locate in the upstream parts of the system. As the exit tailwater level rises, thebottlenecks tend to move downstream as the exit tailwater level rises, and the systemcapacity decreases as demonstrated by the example shown in Fig. 14.37. Identificationof the most likely range of exit tailwater levels and removal of the bottlenecks in thisrange may be a relatively simple and effective way for system capacity improvement.For the example system shown in Figs. 14.30 and 14.37, removal (raising) of theGregory Street Bridge (the first two obstacles) improves the system capacity as indicat-ed by the dashed line in Fig. 14.37.

If the flood frequency (discharge-return period) relationship is known, the systemcapacity curve and the locations of bottlenecks shown in Fig. 14.37 can be converted intoa system capacity curve in terms of return period versus exit water level as demonstratedin Fig. 14.38 for the system shown in Fig. 14.30. It can be seen that with the improvementof the Gregory Street Bridge removal, the system absolute maximum capacity is increasedfrom 860 to 970 ft3/s, or a return period improvement from 25 to 40 years. At a likely exitwater level of 710.75 ft, the system capacity is increased from 655 to 730 ft3/s, or a returnperiod improvement from 11 to 15 years.

Thus, it is obvious that when the connecting reaches are considered interacting as asystem, the overall channel capacity is different from any of the capacity values of theindividual reaches. For an open-channel system, the backwater effects of connectingreaches usually prevent the exit depth of interior reaches to become critical. Therefore, theabsolute maximum capacity, Qamax, of a reach serves as the upper bound provided open-

CL(j � 1)AL(j � 1)

��CuAu

QL(j � 1)�

Qu






FIGURE 14.38 Example flood stage frequency of Boneyard Creek.

FIGURE 14.37 Example hydraulic capacity curve of Boneyard Creek.





FIGURE 14.39 Depth-velocity relation for steady open-channel plane flow. (From Yen, 1986b).

channel flow prevails in the reach and also in adjacent reaches upstream and downstream.For a closed-top reach, the upper bound is the larger of Qamax and Qsmax. For an open-chan-nel reach connected to a closed-top reach at either its upstream or downstream, or both,the upper bound is the larger between Qamax and the largest discharge allowed under sub-merged exit or entrance condition of the open-channel reach.

The capacity of the system of reaches as a whole cannot exceed the smallest of theupper bound of the individual reaches just mentioned, adjusted for lateral flow enteringthe interior reaches in the channel system. However, the location of the bottleneck, whichdetermines the capacity of the channel as a system, may not and often is not in this reachof smallest Qamax upper bound, and in such a case, the system capacity is smaller than thissmallest upper bound.

14.7 HYDRAULICS OF OVERLAND FLOW

14.7.1 Overland Flow and Resistance Equations

Runoff on overland surface is usually nonuniform, unsteady, open-channel flow. For thedepth and velocity ranges encountered in urban surface runoff, the flow can be laminar orturbulent, subcritical or supercritical, stable or unstable, as depicted in Fig. 14.39 for





steady flow as an approximation. Mathematically, the flow can be described by the one-dimensional continuity relationship [Eq. (14.4) or Eq. (14.5)] and momentum relationship[Eq. (14.1) or Eq. (14.2)] or their approximations, if applicable] or the corresponding two-dimensional flow equations (see Yen, 1996, Tables 25.6 and 25.7). These equations areequally applicable to impervious and pervious surfaces, gutters, and pavement. For a high-ly pervious surface where the infiltrating subsurface flow interacts with the free surfaceflow, a conjunctive surface-subsurface flow simulation such as those developed by Akanand Yen (1981a) and Morita et al. (1996) should be used.

Conversely, for drainage design of streets, roads, roadside gutters and inlets, the flowtime of concentration is so short that often the flow can be regarded as steady and thedesign can proceed as such. Traditionally used design procedures with this assumptionhave been described in Chap. 13.

The resistance coefficient in Weisbach form can be determined from Fig. 14.40 (Yen,1991) which is a modified form of the Moody diagram for two-dimensional steady uni-form flow over rigid impervious boundary. The curves can be expressed in equation formfor the Reynolds number R � VR/v � 500 as

f � 24/R (14.60)

for 500 R � 30,000 as

f � 0.224/R0.25 (14.61)

and for R � 30,000 with ks/R 0.05 as

f � �14�

�log

�1

k2

s

R��

1R.9

0.

59�

�2 (14.62)

Correspondingly, expressed in terms of Manning’s n, for R � 500

n � R1/6 ��R3

�� (14.63)Kn��g�

FIGURE 14.40 Weisbach resistance coefficient for steady uniform flow in open channels. (FromYen,1991).




for 500 < R �30,000 as

n � R1/6 �0�.0�2�8� R�1/8 (14.64)

and for R > 30,000 with ks/R < 0.05 as

n ��log

�1

k2

s

R��

1R.9

0.

59�

�1

(14.65)

For the third region of fully developed turbulent flow (Eq. 14.65), it is well known that ncan be regarded approximately as a constant (Yen, 1991) and its value can be estimatedfrom standard tables such as in Chow (1959) or Table 3.3.

For shallow overland flow under rainfall, raindrops bring in mass, momentum, andenergy input into the flow, and hence, the resistance coefficient is modified. Based on theresult of a regression analysis by Shen and Li (1973), the values of n and f for R 900can be estimated from the following nondimensional relationship

f � 8��Kn

g��

R

n1/6�

2

� 24 � 660

0.4

(14.66)

For a higher Reynolds number, Eq. (14.62) or constant n applies.One of the frequent purposes for overland flow simulation is to determine the peak dis-

charge and its time of occurrence. For a continuous rainfall, the time to reach the peak dis-charge when all the areas within the watershed contribute is one definition of the time ofconcentration. A popular method of solving such overland flow problems is the kinematicwave approximation of the Saint-Venant equations because it is relatively simple, easy tosolve, and requires only one boundary condition for solution, whereas the other two high-er level approximations require two boundary conditions (Sec. 14.3.4). Its biggest draw-back is its incapability to account for the backwater effect from downstream. Such back-water effect does exist in urban subcritical overland flow, for example, when the street sur-face flow joins the gutter flow, and the backwater effect from inlet catch basin.

Despite the heterogeneous nature of urban catchments, a fundamental understandingof the overland surface runoff process can be gained through the consideration of therunoff of rainfall excess on a sloped, homogeneous, relatively smooth, plane surface. Afterthe initial losses are satisfied, rain water starts to accumulate on the surface. Initially whenthe amount of water is small and the surface tension effect is predominant, the water maybe held as isolated pots without occurrence of flow, as one would observe on a glass sur-face with a small amount of water. As rain water supply continues, the surface tension canno longer overcome the gravity force and the momentum input of the raindrops along theslope of the surface. The individual water pools merge and flow starts downslope.

One should be wise and extremely careful to select the appropriate simplified equa-tions to solve overland flow problems. For instance, if the geometry of a short street gut-ter is well defined, the hydraulic characteristics of the inlet catch basin downstream of thegutter are known, and a relatively reasonable accurate result is desired, the kinematic waveapproximation will not be acceptable and at least the noninertia approximation should beused. Conversely, when simulating a whole block conceptually as a flow plane, there is noreason to use the Saint–Venant or noninertia equations because the grossly aggregatedinformation of the block is incompatible with the sophisticated equations. Likewise, when

i��

3

g�v�

Kn��g�

R1/6

�4 �2�

Kn��g�






representing a long flow surface as an impervious or pervious surface which is describedmerely by its length, width, slope, and overall average surface roughness type, the kine-matic wave approximation usually suffices, whereas the noninertia or Saint-Venant equa-tions overkill because the downstream backwater effect is insignificant except for a smallstretch at the very downstream.

14.7.2 Kinematic Wave Modeling of Overland Flow

Despite its heterogeneous nature, urban overland surface is often hypothetically conceivedas a collection of wide planes in modeling. For most overland flows, the depth is relativelysmall compared to flow length and the downstream backwater effect is insignificant;hence, the kinematic water approximation is applicable. For a wide open channel wherethe hydraulic radius R is equal to the flow depth Y, the kinematic wave momentum equa-tion, So � Sf, can be simplified as

V � aYm �1 (14.67)

or in terms of discharge per unit width of the channel q1 as

q1 � aYm (14.68)

where m � 5/3 and a � �K�nS�o� /n for the Manning formula, m � 3/2 and a � (8gSo/f)1/2

for the Darcy-Weisbach formula, and m � 3/2 and a � �C�S�o� for the Chezy formula.If either Eq. (14.67) or Eq. (14.68) is solved together with the continuity relationship

[Eqs. (14.4) or (14.5)] in a nonlinear form, the simplification is a nonlinear kinematicwave approximation, often simply referred to as kinematic wave. If Eq. (14.67) or Eq.(14.68) is solved together with a simplified, linear form of Eqs. (14.4) or (14.5), the sim-plification is a linear kinematic wave approximation.

Combining Eq. (14.67) with Eq. (14.5) and assuming a to be independent of x, we obtain

�∂∂Yt�

� �∂∂x� (aYm) � �

∂∂Y

t� � maYm �1 �

∂∂Yx�

� ie (14.69)

where the rainfall excess ie is

ie � i � f'/ (14.70)

where i � the rain intensity on the water surface and f'/ � infiltration rate at channel bot-tom, that is, land surface.

Since from Eq. (14.68) with a independent of x, ∂q1/∂t � maYm�1(∂Y/∂t). Substitutionof this relation into the continuity equation yields another popular form of the kinematicwave approximation used in modeling:

�∂∂qt1

� � �1c� �

∂∂qx

1� � �

ice� (14.71)

where

c � maYm�1 (14.72)

Assuming that a and m are both constants, Eqs. (14.69) and (14.71) yield

�ddxt� � c � maYm�1 � mV (14.73)





�ddqx

1� � ie (14.74)

�d

d

q

t1

� � iemV (14.75)

�d

d

Y

t� � ie (14.76)

and

�ddYx� � �

mie

V� (14.77)

For an initially dry surface (Y � 0 for 0 � x � L at t � 0) under constant rainfallexcess, ie and zero depth at the upstream end, integration of Eq. (14.76) yields

Y � iet (14.78)

Substituting this equation into Eq. (14.73) and integrating, one has

x � x0 � aiem�1 tm (14.79)

Let xO � 0, the equilibrium peak discharge time, te, can be obtained with x � L where Lis the total length of the overland surface:

te ��ai

L

em�1�

1/m

(14.80)

At this time, the discharge per unit width from the overland surface is

q1L � ieL (14.81)

and the discharge for 0 t te is

q1L � a(iet)m (14.82)

FIGURE 14.41 Sketch of kinematic-wave water surface profileduring buildup times.





The water surface profile during the buildup period 0 t te based on the above kine-matic wave analysis is shown in Fig. 14.41. Generalized charts are available in the litera-ture to determine the peak kinematic overland flow rate from infiltrating surfaces forwhich ie is not constant (Akan, 1985a, 1985b, 1988).

14.7.3 Time of Concentration

The equilibrium time given in Eq. (14.80) for Manning’s formula is often referred to asthe kinematic wave time of concentration:

tc � 0.6

i�0.4 (14.83)

For infiltrating overland flow planes, Akan (1989) obtained numerical solution to the kine-matic overland flow and the Green and Ampt infiltration equations and fitted the follow-ing equation to the numerical results by regression:

tc � 0.6(i � K)�0.4 � 3.10K1.33Pf φ(1 � Si)i �2.33 (14.84)

where K is the soil hydraulic conductivity, φ is porosity, Pf is characteristic suction head,and Si is the initial degree of saturation of the soil. Note that this equation reduced to Eq.(14.83) for impervious surfaces with K � 0.

Morgali and Linsley (1965) numerically solved the Saint-Venant equations instead ofthe kinematic wave approximation for a number of runoffs from idealized catchment sur-face, and the results were regressed to give the following equation for the time of con-centration in minutes,

tc � K �nS

0

o

.

0

6

.

0

3

5

8

Lie

0

0

.

.

3

5

8

9

8

3

� (14.85)

where K � 0.99 for English units with L in feet, and i in in./h. Izzard (1946) provided thefollowing equation based on his experiments of artificial rain on sloped surfaces:

tc � 410.0007i1/3 � �i2

k/3�

�C

L2So�

1/3

(14.86)

for English units with iL 500, L in ft and i in in./h, and C is the rational formula runoffcoefficient. In practical applications, often the rain intensity i is unknown a priori. Hence,tc of Eqs. (14.83)–(14.86) is computed iteratively with the aid of a rainfall intensity rela-tionship.

For overland surfaces of regular geometry beyond the two-dimensional surface justdiscussed, formulas for peak discharge and time of equilibrium estimation can be foundin Akan (1985c) and Singh (1996).

In addition to the hydraulic-based equations for time of concentration, a number ofhydrologic-based empirical time of concentration formulas also exist (Kibler, 1982).Equations (14.83) and (14.85), when applied to actual catchments, usually yield a tc valuesmaller than found empirically. There are a number of possible reasons to cause this dis-crepancy (Yen, 1987), including the following:

1. The catchment surface is usually not homogeneous as is assumed in the derivation ofEqs. (14.83) and (14.85), the surface undulation is far more than the sand-equivalentroughness assumed in the derivation.

nL�Kn �S�o�

nL�Kn �S�o�





TABLE 14.15 Values of K for Yen and Chow Formula

Light rain Moderate rain Heavy Rain

Rain intensity (in./h) 0.8 0.8–1.2 �1.2

(mm/h) 20 20–30 �30

For Lo in feet with to in hours 0.025 0.018 0.012

to in min 1.5 1.1 0.7

For Lo in metres with to in hours 0.050 0.036 0.024

to in min 3.0 2.2 1.4

Source: From Yen and Chow (1983).

2. For shallow depth Manning’s n is not a constant (Yen, 1991) and raindrop impactincreases n.

3. The sensitivity to rain input i0.4 is far more than reality. In the derivation, the input i isassumed as evenly distributed over the surface and without momentum, a pattern dif-ferent from real rainfall.

4. Equations (14.83) and (14.85) are based on the time reaching the peak flow consider-ing the influence of the flood wave propagation, different from the water particle travel

time along the longest (or largest L/�S�o�) flow path.

5. The hydraulic time of peak flow measured from the commencement of rainfall excess,whereas the hydrologic time of concentration measured from the commencement of rainfall.

Considering the aforementioned factors and to eliminate the rain intensity iterationprocess, Yen and Chow (1983) proposed the following formula for the overland flow timeof concentration:

tc � K

0.6

(14.87)

where K is a constant and N is an overland texture factor, similar to Manning’s n butmodified for heterogeneous nature of overland surfaces. The values of K and N, modi-fied slightly from their originally proposed values, are given in Tables 14.15 and 14.16,respectively.

14.8 MODELING OF CATCHMENT RUNOFF

14.8.1 Scientific Fineness versus Practical Simplicity

Urban catchment runoff comes mostly from rainfall excess, that is, rainfall minus abstrac-tions. The contribution of prompt subsurface return flow is usually negligible. (This is notnecessarily the case for sewers, where leakage through joints and cracked pipes could be

NL��S�o�





TABLE 14.16 Overland Texture Factor N for Eq. (14.86)

Overland Surface Low Medium High

Smooth asphalt pavement 0.010 0.012 0.015

Smooth impervious surface 0.011 0.013 0.015

Tar and sand pavement 0.012 0.014 0.016

Concrete pavement 0.012 0.015 0.017

Tar and gravel pavement 0.014 0.017 0.020

Rough impervious surface 0.015 0.019 0.023

Smooth bare packed soil 0.017 0.021 0.025

Moderate bare packed soil 0.025 0.030 0.035

Rough bare packed soil 0.032 0.038 0.045

Gravel soil 0.025 0.032 0.045

Mowed poor grass 0.030 0.038 0.045

Average grass, closely clipped sod 0.040 0.050 0.060

Pasture 0.040 0.055 0.070

Timberland 0.060 0.090 0.120

Dense grass 0.060 0.090 0.120

Shrubs and bushes 0.080 0.120 0.180

Land use

Business 0.014 0.022 0.035

Semibusiness 0.022 0.035 0.050

Industrial 0.020 0.035 0.050

Dense residential 0.025 0.040 0.060

Suburban residential 0.030 0.055 0.080

Parks and lawns 0.040 0.075 0.120

Source: From Yen and Chow (1983).

considerable.) Among the abstractions, infiltration is by far the most significant. On arainstorm-event basis, evapotranspiration is relatively negligible. Interception varies withland use and seasons. Depression storage is a matter of definition and subsequent methodof estimation. Quantitative information on the initial losses—interception and depressionstorage—can be found in, for example, Chow (1964) or Maidment (1993). At any rate, fora heavy rainstorm, the amount of initial losses is relatively small. However, it should benoted that in terms of pollution, or runoff on an annual basis, the contributions of lightrainstorms are also significant.

The hydrologic characteristics of urban catchments vary with land uses and seasons.The surface may range from the relatively impervious surfaces such as streets, sidewalks,driveways, parking lots, and roofs to pervious surfaces such as lawns, gardens, bare soil,





and parks. Rainfall excess on these surfaces are drained directly or indirectly through adja-cent different types of surfaces and gutters into inlet catch basins, and then into sewers orchannels (Fig. 14.42)]. The geometric composition of these different types of surfaces informing a city block or catchment is usually random. This random heterogeneous natureof urban catchment surface imposes great difficulty in precise simulation of rainstormrunoff. Essentially, each surface requires a special, individual, “custom made” treatmentwhich is costly and impractical in terms of both data and computation requirements. Fromthe scientific viewpoint, existing knowledge appears to allow detailed scientific and quan-titative simulation of each of the rainfall abstraction and surface flow processes than thecurrent practice in urban drainage. Such simulation has not been incorporated in engineer-ing practice mainly due to the conflicts between the detailed truthfulness in the scientificapproach and the need for efficiency, simplicity, and tolerable accuracy in the practical pro-cedures. In practice, various assumptions are explicitly or implicitly made so that somedegree of simplification can be achieved for the sake of practical application and analysis.

In the design of the size of most drainage facilities, usually knowing the design peakdischarge, Qp, suffices. Conversely, for operation, planning, stormwater quality controland design involving runoff volume (such as detention storage), the discharge or stagehydrograph of the design rainstorm is needed. For the former, Qp, traditionally simplehydrologic methods such as the rational method can be used. For the latter, the runoffhydrograph can be determined using a hydrologic or hydraulic simulation model.Hydraulic-based simulation models employ a momentum or energy equation [either Eqs.(14.1) or (14.2), or any of the simplified approximations], together with the continuityequation; whereas hydrologic models do not consider momentum or energy equations.

14.8.2 Modeling Procedure

Physically based simulation of the catchment rainfall-runoff process considers the watertransport processes in the elements or components (Fig. 14.42) and their relative distribu-tion within the catchment. Because of the number of elements in a catchment and the

FIGURE 14.42. Elements of urban catchment. (After Yen, 1987).





amount of computations involved, such simulations are usually done using a computer-based model. In formulating a physically based model, the following process phases areconsidered after the rainfall input has been determined:

1. Decomposition of the catchment. In this phase, one determines the level of subdivi-sion of the catchment and components used to represent a subcatchment. Should thecatchment be divided merely into subcatchments? Should the pervious and impervioussurfaces be considered separately? Should the street gutters and inlet catch basins beconsidered specifically? How should the roof contribution in the model be treated?How are the different types of surfaces in a subcatchment related to one another?Should the detention ponds be treated separately and individually? The more the dif-ferent surfaces and components are aggregated as a unit, the simpler the model, but thegreater the loss of physical reality.

2. Methods for abstractions. In this phase, the methods to calculate the losses due tointerception, depression storage, and infiltration are selected. One should consider ifthe abstraction values are assigned catchment wide or if they should be allowed to varyfor different subcatchments and different types of surfaces. The latter is more physi-cally satisfactory, but it also requires more input information. It should be decided ifthe water detained on the overland surface contributes to infiltration when rain supplyis insufficient. If multiple-event continuous modeling is being considered, methods tocalculate evapotranspiration and infiltrability recovery should also be included.

3. Runoff from subcatchments. In this phase, the method of transforming rain excesswater to runoff and the routing of runoff on the surfaces and subcatchments to the inletcatch basins is selected in accordance with the level of subdivision of the catchment.In hydraulically based models, for routing runoff in a catchment, in addition to the con-tinuity equation, a flow momentum equation of some form is used. The continuityequation can be based on the cross-sectional averaged form [Eqs. (14.4) or (14.5)]. Themomentum equation can be the full dynamic wave equation or any of its simplifica-tions shown in Eqs. (14.1) or (14.2). These equations are applied to the elements of anurban catchment, step by step in sequence as shown in Fig. 14.42 for each time step toyield the runoff hydrograph of the catchment.

It is not necessary to use the same routing method for the different elements andtypes of surfaces in a catchment. For example for an aggregated pervious surface, thetime-area method, at most a kinematic wave routing usually would suffice because ofthe gross representation of its hydraulic characteristics. Conversely, for a street pave-ment, gutter, and inlet catch basin system, a kinematic wave routing may not be suffi-cient to provide realistic results because of its relatively well�defined geometric proper-ties and the mutual backwater effects, and hence a noninertia routing may be desirable.

It is obviously impractical to apply the highly sophisticated routing methods to each ofthe overland surfaces and gutters. The construction cost per unit length of such surfacesand gutters is small. And the cost of collection of data needed for such sophisticated com-putation methods is high. Moreover, the hydraulic characteristics of the overland surfacechange with time, depending on cleaning and season. On the other hand, the total lengthof streets and gutters in a catchment and in a city may be considerable. It is also desirableto have reasonably accurate inlet hydrographs as the input to the sewer system. Besides,for pollution control, the estimate of pollutant transport depends on the runoff estimation.Therefore, selection of the most appropriate overland routing method for a model and acatchment is a difficult task requiring delicate balance.

Hydrologic simulation models for catchment runoff are not discussed here. They canbe found in Chow (1964), Maidment (1993), and many other hydrology books. They




range from distributed system model similar to the hydraulic-based model describedabove in Steps 1–3, but the flow velocity in Step (3) is estimated by using some empiri-cal techniques; or in the lumped hydrologic system models such as the conceptual reser-voir-channel models of the Nash/Dooge type or the unit hydrograph methods the runoff isestimated from an assumed relationship without considering the physical process (Yen,1986b). In most cases, synchronized runoff and rainfall data are needed for derivation ofthe catchment unit hydrograph or calibration of the lumped system model; single�eventdata sets are difficult to obtain and if available often are not sufficiently reliable.

For a city or a region, the urban surfaces tend to have some degree of similarity, espe-cially in the United States where many cities have standardized square or rectangularblocks. It is, therefore, possible to group the urban surfaces into typical blocks. Reliablesimulations can be made to establish unit hydrographs for the typical blocks or subcatch-ments. A few of such typical unit hydrographs should be sufficient for a catchment. Inlater applications, the use of the unit hydrographs provides relatively accurate resultsavoiding the repetitive costly sophisticated routing computation for individual rainstormsand blocks. Yen et al. (1977) first proposed this approach for a catchment in San Franciscousing a nonlinear kinematic wave routing for the surface and gutter flow. Akan and Yen(1980) developed nondimensional unit hydrographs for street-gutter-inlet systems usingdynamic wave routing and considering specifically the inlet capacity allowing by passflow. Harms (1982) took a similar concept using a semiempirical approach to establish1–min unit hydrographs.

However, the unit hydrograph theory suffers from the linearity assumption betweenrainfall excess and surface runoff, making it inaccurate when the depth of the simulatedrainstorm is significantly different from that of the rainstorm the unit hydrograph isderived. Lee and Yen (1997) introduced a hydraulic element of kinematic-wave based flowtime determination on geomorphologically represented catchment subdivision for deriva-tion of the catchment instantaneous unit hydrograph, making it a hydraulic distributedmodel and allowing derivation of unit hydrographs for ungaged catchments.

14.8.3 Selected Catchment Hydraulic Simulation Models

There exist many urban rainfall-runoff models. A summary of the important features ofselected hydraulic-based urban catchment models, mostly nonproprietary, is given inTable 14.17. Some models cover both catchment surface and sewer network parts; onlythe catchment part is summarized in this table. The sewer part is given in Tables 14.21 and14.22. One should refer to the original references for details and objectives of these mod-els. Similar hydrology-based watershed models that have also been applied to urbancatchments, such as SCS-TR55 and TR-20, Hydrologic Engineering Center (HEC-1) (orits replacement HEC-HMS), and RORB are not presented here.

14.8.4 Verification and Calibration of Models

Models should never be used without being tested and verified. It has happened again andagain that in the enthusiasm in model development, models are used without verification.All models have their own assumptions and simplifications. Moreover, most urban runoffmodels contain coefficients, exponents, or adjustment factors that require calibration withdata to determine their values.

Besides verification and application for predictions, there are other operational modesof models such as those shown in Table 14.18. In calibration, we try to determine the most






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S





suitable values of the coefficients of the parameters (variables) knowing the input and out-put from observed data. In verification, we have the parameters and their coefficient val-ues all determined for the model, and we have the data on both input and output. The inputis run through the model to produce output, which is compared to the known output in thedata set to verify the agreement between the computed and observed outputs. On the otherhand, verification is different from validation. Validation is to ascertain if the correct equa-tion or model is used to solve the problem. Verification is to find out if the equation ormodel is solved correctly.

No model can do everything. For example, a good flow simulation model may not pro-duce a good design of the drainage system. Conversely, a good design model may not—andoften need not—be an accurate flow simulation model. Therefore, models should be veri-fied and validated according to their objectives and their applications. In verifying a model,the verification criteria should be setup to confirm with the model objectives. For example,the verification can be made according to the peak discharge, time to peak, or to the fittingof the hydrograph as desired by the objective. Various verification fitting error measureshave been suggested in the literature (ASCE Task Committee, 1993; Yen, 1982). Some mea-sures are listed in Table 14.19. In the table, the magnitude parameter Q can be discharge,depth, velocity, or concentration as appropriate to the problem investigated. The subscript pdenotes the peak magnitude of the time graph. The subscript m represents the measured ortrue values used as the gauge for the curve fitting and verification of the model simulation.

The selection of the error measures to evaluate the merit of simulation models dependson the objective of the simulation. For example, if the accuracy of the peak rate and peak-ing time are of paramount importance, εQp

and εtpwould be the most appropriate error

measures. If the overall fitting of the curves is the main objective, εRMS would be the mostimportant measure, while ετ1

, εva, εtp

, and εQpcould be used as auxiliary measures.

In calibration, since the reliability of a single set of data is uncertain, the more sets of

TABLE 14.18 Modes of Operation of Models

Mode Input TransformationParameters Coefficient Values Output

Prediction Known Known Known ?

Calibration Known Known ? Known

Verification Known Known Known ?(/Known?)

Validation Known Known Known Applicable ?

Detection ? Known Known Known

Parameter Known ? Known (?) Knownidentification

Sensitivity Known Known Known ?(Per unit changeof parameter or

coefficient)

Reliability Known Known Known ?(Over likely ranges of

parameters)

Source: From Yen (1986b).





AB

LE

14.

19Si

mul

atio

n or

Mea

sure

men

t Err

or M

easu

res

Err

or m

easu

reD

efin

itio

nR

emar

ks*

Mag

nitu

de e

rror

sPe

ak r

ate

erro

r� Q

p�

(Qp

�Q

pm)

/ Q

pm

Mea

n ra

te e

rror

� d�

(Q��

Q�m)

/ Q�

m

Cum

ulat

ive

volu

me

erro

r� V

�(V

�V

m)

/ Vm

V��t 0

Qdt

��0 i

Qi�

t

Vm

��t 0

Qmdt

��0 i

Qm

i�t

Abs

olut

e vo

lum

e er

ror

� va

� � V1 m�

�t 0�Q

�Q

m�dt

�� V1 m�

�0 i�QQ

i�

Qm

i��t

Rat

e m

omen

t err

or� Q

r�

(QQr�

Qrm

)/

Qrm

Qr�

� V1 ��t 0

�1 2�Q

2 dt

� � 21 V�

�0 i

Qi2 �

t

Qrm

�� V1 m��t 0

�1 2�Q

m2dt

� � 2V1

m��0 i

Q2 m

i�

t

Roo

t-m

ean-

squa

re� R

MS

�� VT m�

� T1 ��T 0

(Q�

Qm)2

dt 1/

2

� �� V

mT� � �t i

(Qi�

Qm

i)2�

t 1\2

Tim

e er

rors

Peak

-rat

e tim

e er

ror

� tp

�(t

p�

t pm)/

t pm

Peak

tim

e fi

rst-

mom

ent e

rror

� τ1

�(τ

1�

τ 1m)/

τ 1mτ i

�� V1 ��t 0

tQdt

� � V1 �

�t i

Qi

t i�

�� 2t � �

t

Gra

ph d

ispe

rsio

n er

ror

� g�

(G�

Gm)/

Gm

Seco

nd m

omen

ts w

ith r

espe

ct to

t pm:

G�

� V1 ��T 0

Q(t

�t p

m)2

dt

Gm

�� V1 m�

�T 0

Qm(t

m�

t pm)2

dt

Sour

ce:

From

Yen

(19

82).

*(

Vor

Vm

beco

mes

tota

l vol

ume

if t

�fl

ow d

urat

ion

cons

ider

ed,T

.Sub

scri

pt m

5 m

easu

red

or r

efer

ence

bas

e va

lues

;

subs

crip

ti�

sum

mat

ion

inde

x; s

ubsc

rit

p�

pea

k m

agni

tude

of

the

time

grap

h.





data used, the better. Different sets of data would produce different sets of coefficient val-ues. Normally, a weighted average (e.g., through optimization) of the values is adopted foreach of the coefficients.

It is not infrequent to see a model misused or abused. Sometimes this is due to the lackof understanding about how the model works. Sometimes it is due to the lack of appreci-ation of the operational modes. For example, data used for calibration should not be usedagain for verification. Yet, this situation happens again and again. In such a case of usingthe same data for calibration and verification, the difference between the model output andthe recorded data is simply a reflection of the numerical errors and the deviation of theparticular data set from the weighted average situation.

Not all models require calibration. Presumably, some strictly physically based modelshave their coefficient values assigned based on available information and no calibration isneeded. However, in rainfall-runoff modeling, some degree of spatial and temporal aggre-gation of the physical process is unavoidable. Therefore, calibration is desirable, if notnecessary.

14.9 DETENTION AND RETENTION STORAGE

Detention and retention basins are widely used to control the increased runoff due tourbanization of undeveloped areas. These basins can also offer excellent water quality ben-efits since pollutants are removed from the stormwater runoff through sedimentation,degradation, and other mechanisms, as the runoff is temporarily stored in a basin.Detention basins are sometimes called dry ponds, because they store runoff only duringwet weather. The outlet structures are designed to completely empty the basin after a stormevent. Retention basins are sometimes called wet ponds since they retain a permanent pool.

FIGURE 14.43 Routing of runoff through detention basin.

Post development hydrograph (Pond inflow)

Required pond volume

Pre–developmentpeak flow rate

Routed post–developmenthydrograph (pond outflow)

Time

Flow

rat

e





FIG

UR

E 1

4.44

Bas

ic e

lem

ents

of

dete

ntio

n ba

sin.

Em

erge

ncy

spill

way

Cla

y co

re

Wat

er s

urfa

ce

Stor

mw

ater

man

agem

ent s

trac

ture

with

hin

ged

grat

e

Flow

dir

ectio

n

Prot

ectio

n

Mai

nten

ance

Sho

ulde

r

Ant

i–se

ep c

olla

r

Pipe

bed

ding

Ove

r–si

zeba

rrel

Rip

rap

ener

gydi

ssip

ator

Des

ign

Tailw

ater

End

wal

lGeo

text

ile T

reat

men

t





FIGURE 14.45 Detention–outlet structures.





14.9.1 Detention Basins

The primary function of a detention basin is to control the quantity of stormwater runoff.Most stormwater management policies require that the postdevelopment peak flow rates bereduced to predevelopment peak flow rates for one or more specified design return periodssuch as 2, 10, and 25 years. Peak flow reduction is achieved by routing the postdevelopmentrunoff through a detention basin, that is by detaining the runoff temporarily in a basin.Figures 14.43 illustrates the effect of a detention basin on storm water runoff.

The schematic diagram given in Fig. 14.44 shows the basic elements of a detentionbasin. In addition, sediment forebays are often used for partial removal of sediments fromthe stormwater runoff before it enters the detention basin. Energy dissipating structuressuch as baffle chutes are used at inlets. Most detention basins also have a trickle flow ditchor gutter in the bottom sloped towards the outlet to provide drainage of the pond bottom.

14.9.1.1 Detention basin design guidelines. Specific design criteria for detention basinsvary in different local ordinances. Some general guidelines are summarized herein.Similar guidelines can be found elsewhere in the literature [ASCE, 1996; FederalHighway Administration (FHWA), 1996; Loganathan et al. 1993; Stahre and Urbonas,1990; Urbonas and Stahre, 1993; Yu and Kaighn, 1992].

The main objective of a detention basin is to control the peak runoff rates. The outfallstructures should be designed to limit the peak outflow rates to allowable rates. A detentionbasin should also provide sufficient volume for temporary storage of runoff. The inlet, outlet,and side slopes should be stabilized where needed to prevent erosion. The side slopes shouldbe 3H/1V or flatter. The channel bottom should be sloped no less than 2 percent toward thetrickle ditch. Detention basin length to width ratio should be no less than 3.0. Outlets shouldhave trash racks. Coarse gravel packing should be provided if a perforated riser outlet is used.An emergency spillway should be built to provide controlled overflow relief for large storms.A 100-year storm event can be used for the emergency spillway design.

14.9.1.2 Outlet structures. Detention basin outlet structures can be of orifice-type, weir-type, or combinations of the two. Schematics of basic outlet structures are shown in Fig. 14.45.

Discharge through an orifice outlet is calculated as

Q � koao�2�g�H�o�, (14.88)

where ao � the orifice area, ko � the orifice discharge coefficient, and Ho � the effectivehead. If the orifice is submerged by the tailwater, Ho is the difference between the head-water and tailwater elevations. If the orifice is not submerged by the tailwater, it isassumed that Q � O if the headwater is below the centroid of the orifice. Otherwise, Ho

is set equal to the difference between the headwater elevation and the orifice centroid. Thisapproximation is acceptable for small orifices. To account for partial flow in lanrge ori-fices, the inlet control culvert flow formulation discussed in Chap. can be used to deter-mine the orifice flow rates. Short outflow pipes smaller than 0.3 m (1.0 ft) in diameter canalso be treated as an orifice provided that Ho is greater than 1.5 times the diameter. Typicalvalues of ko are 0.6 for square-edge uniform entrance conditions, and 0.4 for ragged edgeorifices (FHWA, 1996).

Weir-type structures include sharp-crested weirs, broad�crested weirs, spillways, andv-notch weirs. Flow over spillways, broad–crested weirs, and sharp crested weirs with noend contractions is expressed as

Q � kwLc�2�g�Ho1.5 (14.89)





where kw � the weir discharge coefficient, Lc � the effective crest length, and Ho � thehead over the weir crest. The weir discharge coefficient depends on the type of the weirand the head. Discharge coefficients for various types and structures are tabulated inChapter. The head over the weir is the difference between the water surface elevation in the detention basin and the weir crest. For sharp crested weirs with end contractions

Q � kw (Lc � 2Ho)�2�g�Ho1.5 (14.90)

and for V-notch weirs

Q � kv (�185�)�2�g�(tan �

φ2

�) Ho2.5, (14.91)

where kv � the V-notch discharge coefficient, φ � the notch angle, and Ho � the head overthe notch bottom.

Riser pipes act like a weir at low heads and like an orifice at higher heads. It is alsopossible that the flow will be controlled by the outflow barrel at even higher heads. Inmany applications, the outflow barrel is oversized to avoid the flow control by the barrel.In that case the outflow through the structure is calculated for a given head both using theweir and orifice flow equations, and the smaller of the two is used. If a trash rack isinstalled, the clear water area should be used in the calculations. It should be noted thatmany engineers design riser pipes so that orifice-type flow will not occur, because it isoften observed that vortices form in the structure under orifice flow conditions.Sometimes antivortex structures are installed to avoid this problem.

Multiple outlets are used if the design criteria require that more than one design stormbe considered. Figure 14.46 displays schematics of several multiple�outlet structures.

14.9.1.3 Stage–storage relationships. The stage-storage relationship is an importantdetention basin characteristic. For regular�shaped basins, this relationship is obtainedfrom the geometry of the basin. For instance, for trapezoidal detention basins that has arectangular base of W � L and a side slope of z, the relationship between the volume, S,and the flow depth d is

S � L W d � (L � W) zd2 � �43

� z2d3 (14.92)

FIGURE 14.46 Example multiple outlet structures.





For irregular-shaped detention basins, first the surface area, As, versus elevation, h,relationship is obtained from the contour maps of the detention basin site. Then

S2 � S1 � (h2 � h1) �As1 �

2As2

� (14.93)

where S1 and As1 correspond to elevation h1, and S2 and As2 correspond to h2. Equation(14.93) is applied to sequent elevations. A more accurate relationship is

S2 � S1 � �h2 �

3h1

� AS1 � As2 � �A�s1� A�s2� (14.94)

The stage-storage relationship for most human-made and natural basins can also beapproximated by

S � bhc (14.95)

where b and c � fitting parameters. Figure 14.47 displays approximate relationshipsbetween the parameter b and c, the base area, length to width ratio, and the side slope fortrapezoidal basins.

14.9.1.4 Detention pond design aids. The conventional procedure for the hydraulic designof a detention basin is a trial-and-error procedure, and it consists of the following steps:

1. Calculate the detention basin inflow hydrograph(s) for the design return period(s)being considered. A rainfall-runoff model, such as HEC1, TR-20, or SCSHYDRO, canbe used for this purpose. For urbanizing areas, the inflow hydrograph(s) are normallythose calculated for postdevelopment conditions.

FIGURE 14.47 Detention basin stage-storage parameters. (After Currey and Akan, 1998).

1 2 3 480,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

80,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

1 2 3 4

Side slope (H/V)Side slope (H/V)

bottom area(m2 or ft2)

bottom area(m2 or ft2)





Tabl

e 14

.20

Des

ign

Aid

Equ

atio

ns f

or D

efin

ition

and

Ret

entio

n St

orag

e

Equ

atio

nN

umbe

r O

utle

t R

emar

ksR

efer

ence

of O

utle

tsTy

pes(

s)

Not

N

ot

Bas

ed o

n nu

mer

ical

sim

ulat

ions

Wyc

off

and

Sing

h sp

ecif

ied

spec

ifie

dT

b�

time

base

of

infl

ow h

ydro

grap

h(1

976)

Not

N

otT

rian

gula

r in

flow

hyd

rogr

aph,

Abt

and

Gri

gg

spec

ifie

dsp

ecif

ied

trap

ezoi

d ou

tflo

w h

ydro

grap

h(1

978)

Not

N

otT

rian

gula

r in

flow

and

out

flow

Bak

er (

1979

)sp

ecif

ied

spec

ifie

dhy

drog

raph

Not

N

otFo

r SC

S 24

-h T

ypes

I a

nd I

ASo

il C

onse

rvat

ion

spec

ifie

dsp

ecif

ied

rain

fall

Serv

ice

(198

6)

Not

N

ot

For

SCS

24-h

Typ

es I

I an

d II

ISo

il C

onse

rvat

ion

spec

ifie

dsp

ecif

ied

rain

fall

Serv

ice

(198

6)

t d�

stor

m d

urat

ion,

Tc

�tim

e of

con

cent

ratio

nN

otN

ot

Aro

n an

d K

ible

rsp

ecif

ied

spec

ifie

dT

rape

zoid

al in

flow

hyd

rogr

aph,

risi

ng(1

990)

limb

of o

utfl

ow h

ydro

grap

h is

line

ar,

Wei

rC

onst

ant r

eser

voir

sur

face

are

a,va

lid f

orK

essl

er a

nd D

iski

nSi

ngle

type

0.2

�Q I pp �

0.

9(1

991)

Ori

fice

Con

stan

t res

ervo

ir s

urfa

ce a

rea,

valid

for

Kes

sler

and

Dis

kin

Sing

lety

pe0.

2

�Q I pp �

0.9

(199

1)

�S Sm Rax ��

0.87

2�

0.86

1 �Q I pp �

�S Sm Rax ��

0.93

2�

0.79

2�Q I pp �

S max

�I p

t d�

Qp(�t d

� 2T

c�

)

�S Sm Rax ��

0.68

2�

1.43

�Q I pp �

�1.

64 �Q I pp �

2�

0.80

4 �Q I pp � 3

�S Sm Rax ��

0.66

0�

1.76

�Q I pp �

�1.

96 �Q I pp �

2�

0.73

0 �Q I pp � 3

�S Sm Rax ��

1�

�Q I pp �

�S Sm Rax ��

1�

�Q I pp � 2

�S Sm Rax ��

1.29

1(1

�Q

p/I

p)0.

753

��

�(T

b/t p

)0.4

11




14.84 Chapter FourteenTa

ble

14.2

0(C

ontin

ued)

Equ

atio

nN

umbe

r O

utle

t R

emar

ksR

efer

ence

of O

utle

tsTy

pes(

s)

Sing

leW

eir

McE

nroe

type

Gam

ma

func

tion

infl

ow h

ydro

grap

h(1

992)

Sing

leO

rifi

ceM

cEnr

oety

peG

amm

a fu

nctio

n in

flow

hyd

rogr

aph

(199

2)

Sing

leW

eir

Gam

ma

func

tion

infl

ow h

ydro

grap

h,C

urre

y an

d A

kan

h max

�

1/c

type

stag

e-st

orag

e re

latio

nshi

p:S

� b

hc,

(199

8)h

� s

tage

,Lc

� w

eir

cres

t len

gth,

k w�

wei

r di

scha

rge

coef

fici

ent,

g�

gra

vitio

nal a

ccel

erat

ion

Lc

�

1.5/

c

h max

�

1/c

Sing

leO

rifi

ceG

amm

a fu

nctio

n in

flow

hydr

ogra

ph,

Cur

rey

and

Aka

n�S Sm Rax �

�0.

847

�0.

841

�Q I pp �

type

stag

e-st

orag

e re

latio

nshi

p:S

� b

hc,

(199

8)

h�

sta

ge,a

o�

ori

fice

are

a,k o

= o

rfic

e di

scha

rge

coef

fici

ent,

g�

gra

vita

tiona

l a o

�

0.5/

c

acce

lera

tion

Qp

� k O�

2�g�b

��

�

0.84

7S R

�o.

841

�Q I pp �S R

o.84

7S R

�8.

841

�Q I pp �S R

��

�b

Qp

� k w�

2�g�b

��

�

0.92

2S R

�0.

787

�Q I pp �S R

0.92

2S R

�0.

787

�Q I pp �S R

��

�b

�S Sm Rax ��

0.92

2�

0.78

7 �Q I pp �

�S Sm Rax ��

0.97

�1.

42�Q I pp �

�0.

82 �Q I pp �

2�

0.46

�Q I pp � 3

�S Sm Rax ��

0.98

�1.

17�Q I pp �

�0.

77 �Q I pp �

2�

0.46

�Q I pp � 3





2. Set the hydraulic design criteria. In most applications, the postdevelopment peak(s) arerequired to be reduced to the magnitude(s) of the predevelopment peak(s) for thedesign return period(s). If predevelopment peak(s) are not available, a rainfall–runoffmodel can be used to calculate them. The hydraulic design criteria may also restrict themaximum water surface elevation in the detention basin.

3. Trial design a detention basin. A trial design consists of the stage-storage relationship,and the types, sizes and elevations of the outlet structures.

4. Route the inflow hydrograph(s) through the trial-designed detention basin and checkif the design criteria set are met. If not go back to Step 3. Also, if the criteria are met,but the outflow peak(s) are much smaller than the allowable value(s), then the trialbasin is overdesigned. Again, go back to Step 3. The level-pool routing procedure isadequate for most detention basin design situations. This procedure is based on thesolution of the hydrologic storage routing equation

�ddSt� � I � Q, (14.96)

where I� inflowrate and t � time. Unless a computer program is used, Eq. (14.96) issolved by employing a semigraphical method like the storage indication method, whichcan be found in any standard hydrology textbook.

Obviously a good trial design is the key in this procedure. Designing a detention basincan become a tedious and lengthy task if the trial designed basins are not chosen carefully.Various charts and equations are available in the literature that can be used as trial designaids. Most of these aids are based on predetermined solutions to Eq. (14.96) in dimen-sionless form (Akan, 1989, Akan, 1990; Currey and Akan, 1998; Kessler and Diskin,1991; McEnroe, 1992). Others are based on assumed inflow and outflow shapes (Abt andGrigg, 1978; Aron and Kibler, 1990) or results of numerous routings for many detentionbasins (Soil Conservation Service, 1986; Wycoff and Singh, 1976). Table 14.20 presentsvarious design aid equations, where Ip � the peak inflow rate (peak discharge of postde-velopment hydrograph), Qp � the allowable peak outflow rate, Smax � the required stor-age volume, and SR � the volume of runoff.

The use of these design aids can be illustrated through a simple-example. Suppose therainfall excess resulting from a design rainfall is 3.5 in over a 2,178,000 ft2 urban water-shed, and the runoff hydrograph has a peak of Ip � 212ft3/s occurring at tp � 30 min �1800 s. A detention basin is to be designed to reduce the peak flow rate to Qp � 120 ft3/s.A weir-type outlet will be used that has kw � 0.40. It is also required that the depth ofwater above the weir crest not to exceed 6.50 ft. A trapezoidal detention basin is suggest-ed width a length-to-width ratio of 4 and sideslopes of 3H/1V. To size the required basin,let the surface area of the detention basin at the weir crest elevation be 40,000 ft2. Thenfrom Fig. 14.47, b � 42,500 and c � 1.055. By definition, SR � (3.5/12)(2,178,000) �635,250 ft3. Using the equations suggested by Currey and Akan (1998) from Table 14.20,we obtain Smax � 302,715 ft3, hmax � 6.43 ft, and Lc � 2.29 ft. Note that hmax 6.50 ft.,so the suggested basin with a base area of 40,000, ft2 should work.The Soil Conservation Service (SCS), (1986) equations given Table 14.20 are recom-mended if the standard SCS design rainfall hyetographs are to be used. Also, these equa-tions are not restricted to single outlet detention basins. Suppose a detention basin isrequired to control the stormwater runoff for 2-year and 25-year events. Given for the 2-year event are Ip � 91 ft3/s, Qp � 50 ft3/s, SR � 408,480 ft3, and for the 25 �year event Ip

� 360 ft3/s, Qp � 180 ft3/s, SR � 928,750 ft3. A two-stage weir outlet is suggested with kw

� 0.40, and the maximum water depth above the lower weir crest is not allowed to exceed5 ft. The design is to be based on SCS 24-h Type II rainfall. Suppose a trapezoidal deten-





tion basin with a length-to-width ratio of 4 and side slopes of 3H/1V is suggested. FromTable 14.22, Smax � 928,750 [0.682 � 1.43(180/360) � 1.64(180/360)2 � 0.804(180/360)3] � 256,750 ft3. Likewise, Smax � 105,400 ft3 for the 2–year event. To deter-mine the base area (or the surface area of the detention basin at the lower crest elevation),use Eq. (14.92) with S � 256,800 ft3, L � 4W, z � 3, and d � 5 ft. Solving the equationfor W, we obtain approximately W � 104 ft, and then L � 416 ft. To size the lower crestfor the 2-year event use Smax � 105,400 ft3. Now substituting S � 105,400 ft3, W � 104ft, L � 416 ft and z � 3 in Eq. (14.92) and solving for d, we obtain the maximum headover the lower crest for the 2-year event as being 2.25 ft. Next, using the weir equation(Eq. 14.89) with Q � 50 ft3/s, kw � 0.40, h � 2.25 ft, and g � 32.2 ft/s2, we obtain Lc �4.61 ft for the lower crest. Let the upper crest be placed 2.30 ft above the lower crest. Tosize the upper crest, the 25-year event is considered. The maximum head over the lowercrest will be 5 ft. At this head the lower crest will discharge 165 ft3/s [from Eq. (14.89)].Therefore, the upper crest should be sized to pass (180 � 165) � 15 ft3/s under a head of(5.00 � 2.30) � 2.70 ft. From the weir formula [Eq. (14.89)] we obtain Lc � 1.05 ft forthe upper crest.

14.9.2 Extended Detention Basins

Extended detention basins are effective means of removing particulate pollutants fromurban storm water runoff. As shown in Fig. 14.48, an extended detention pond has twostages. The bottom stage is expected to be inundated frequently. The top stage remains dryexcept during large storms.

FIGURE 14.48 Extended detention basin. (After Schueler, 1987).





14.9.2.1 Detention volume and time. An extended detention basin is designed todetain a certain quantity of runoff, sometimes referred to as the water quality volume,for a certain period of time to achieve the targeted level of pollutant removal. The vol-ume to be detained and the duration over which this volume to be released vary in dif-ferent stormwastet management policies. For example, Hampton Roads PlanningDistrict Commission (1992) requires that a quantity of runoff calculated as 0.5 inchtimes the impervious watershed area be released over 30 h in southeastern Virginia.Prince George County Department of Environmental Resources (1984) requires therunoff volume generated from the 1�year, 24-hour storm be released over a minimumof 24 h.

American Society of Civil Engineers (1998) outlines a procedure to size extended basinsserving up to 1.0 km2 (0.6 m3) watersheds. In this procedure, the volume of water to bedetained per unit watershed area, Po, is estimated as

Po � ar(0.858i3 � 0.78i2 � 0.774i � 0.04)P6, (14.97)

where ar � a regression coefficient, i � the watershed imperviousness expressed as a frac-tion, and P6 � the mean storm precipitation depth that can be obtained from Fig. 14.49.The value of the regression coefficient ar is 1.109, 1.299, and 1.545 for detention volumerelease times of 12, 24, and 48 h, respectively. Interpolation of these values is allowed fordurations between 24 and 48 h.

The use of this procedure can be illustrated by a simple example. Suppose anextended detention basin is to be designed for a 150-acre watershed in Norfolk,Virginia that is 40 percent impervious. Determine the required size if the detainedrunoff is to be released over 36 hours. From Fig. 14.49, P6 � 0.67 in for Norfolk,Virginia. Because the watershed is 40 percent impervious, i � 0.40. Also, interpolat-ing the ar values between 24 and 48 h, ar � 1.422 for 36 h. Then, from Eq. (14.97), weobtain Po � 0.27 in. Therefore, the volume of runoff to be detained is (150) (0.27/12)� 3.38 aq-ft � 147,233 ft3. It is advisable to increase this volume by about 20 percentfor sedimentation.

FIGURE 14.49 Mean storm precipitation depth in inches. (After ASCE, 1998).





14.9.2.2 Extended detention outlet structures. The outlets for extended detention basinsare designed to slowly release the captured runoff from the basin over the specified emp-tying time to allow settling of particulate pollutants. We sometimes refer to these outletsas water quality outlets. Low-flow orifices are often used as outlet structures. Figure 14.50displays various methods for extending detention times. As pointed out by Schueler(1987) and ASCE (1998), however, extended detention outlet structures are generallyprone to clogging. This makes the design of outlet structures difficult since the hydraulicperformance of a clogged outlet will be uncertain and different from what it is designedfor. Regular cleanouts must be performed.

A hydrograph routing approach is probably the best way to size an extended detentionbasin and the water quality outlet. However, this requires an inflow hydrograph. In prac-tice, as discussed in the preceding section, only the volume of captured runoff is consid-ered for pollutant removal. There are no broadly accepted procedures to convert this vol-ume to an inflow hydrograph. Therefore, the water quality outlets are often sized by usingapproximate hydraulics. This can be illustrated by a simple example.

FIGURE 14.50 Extended detention pond outlets. (After Schueler, 1987).





Suppose an extended detention basin has a bottom length of 80 ft, a width of 20 ft, andside slopes of 3:1(H:V). The outlet is to be sized so that it will release a water quality vol-ume of 10,200 ft3 over a period of 40 h. To determine the depth of water corresponding tothis volume, Eq. (14.92) is written as

10,200 � (80)(20)d � (80 � 20)3d2 � (4/3)32d3.

By trial and error, d � 3.6 ft. Let the outlet structure be comprised of 1/2-in circularragged edge orifice holes cut around a riser pipe. Let the average elevation of the holes be1 ft above the pond bottom. Therefore, the average head over the orifice holes is (3.6 �1.0)/2 � 1.3 ft. To empty 10,200 ft3 over 40 h � 144,000 s, the average release rate is10,200/144,000 � 0.0708 ft3/s. Noting that the orifice area of a 1/2�in hole is 0.00136 ft2,and ko � 0.40 for ragged edge orifices, we can write Eq. 14.88 as

0.0708 � N(0.40)(0.00136)�2�(3�2�.2�)��1�.3�,

where N is the number of orifice holes. Solving for N we obtain N � 14.22. Therefore, weuse 14 holes evenly distributed.

14.9.2.3 Extended detention basin design considerations. Additional design considera-tions for extended detention basins can be found in various publications (ASCE, 1998;FHWA, Schueler, 1987; Urbonas and Stahre, 1993). Briefly, the basin should graduallyexpand from the inlet, toward the outlet. A length-to-width ratio of 2 or higher is recom-mended. Side slopes should not be steeper than 3:1 (H:V) and flatter than 20:1 (H:V). Ariprap, concrete, or paved low-flow channel is required to convey trickle flows. A twostagedesign is recommended with a 1.5- to 3.0 ft-deep bottom stage and a 2.0- to 6.0-ft-deepupper stage. A wetland marsh created in the bottom stage will help remove soluble pollu-tants that cannot be removed by settling. The detention basin inlet should be protected toprevent erosion. If the outlet is not protected by a gravel pack, some form of trash rackshould be used. A sediment forebay is recommended to encourage sediment deposition tooccur near the point of inflow

14.9.3 Retention Basins

Retention basins or wet ponds retain a permanent pool during dry weather as shown inFig. 14.51. A high removal rate of sediment, biological oxygen demand (BOD), organicnutrients, and trace metals can be achieved if stormwater is retained in the wet pond longenough. During wet weather, the incoming runoff displaces the old stormwater from thepermanent pool from which significant amounts of pollutants have been removed. Thenew runoff is retained until it is displaced by subsequent storms. The permanent pooltherefore will capture and treat the small and frequently occurring stormwater runoffwhich generally contain high levels of pollutant loading. The storage volume providedabove the permanent pool is used to control the runoff peaks caused by the specifieddesign storm events.

14.9.3.1 Permanent pool volume. Among all the factors influencing the pollutantremoval efficiency of a retention basin, the size of the permanent pool is the most impor-tant. As pointed out by Schueler (1987), in general, “bigger is better.” However, after athreshold size is reached, further removal by sedimentation is negligible.





FIGURE 14.51 Retention basin. (After Yu and Kaighn, 1992).

The required size of the permanent pool in relation to the contributing watershed areavaries in different stormwater management policies. For example, FHWA (1996) and Yuand Kaign (1992) recommend a permanent pool size three times the water quality volumedefined for extended detention basins. Montgomery County Department of EnvironmentalProtection (1984), Maryland, requires a volume greater than 0.5 in times the total water-shed area. ASCE (1998) recommends that Eq. (14.97), be used with a drain time of 12hours to determine the permanent pool volume. It is also recommended that a surchargeextended detention volume, equal to the permanent pool volume, be provided above thepermanent pool. U.S. Environmental Protection Agency (1986) provides geographicallybased design curves to determine the permanent pool surface area as percent of the con-tributing watershed area (see Fig. 14.52). Hartigan (1989) and Walker (1987) treat a reten-tion basin as a small euthrophic lake and employ empirical models to size the retentionpond. This procedure is outlined by ASCE (1998).

U.S. Environmental Protection Agency (1986) presented a procedure to evaluate thelong-term pollutant removal efficiency of retention basins depending on the basin size andthe rainfall statistics of the project area. This procedure was developed by DiToro andSmall (1979), and is outlined in various publications (Akan, 1993; Stahre and Urbonas,1990; Urbonas and Stahre, 1993).

14.9.3.2 Retention basin design considerations. Wet ponds can be designed to controlthe peak runoff rates from rare and large storm events if additional storage volume is pro-vided above the permanent pool. The size of the additional volume can be determined byusing the procedures described for detention basins.

The outlet structures for retention basins include a low flow outlet to control the runofffrom frequent storm events and overflow devices to control the runoff from larger storms.





FIGURE 14.52 Design curves for solids settling. for low–density residential land use. (After USEPA, 1986).

FIGURE 14.53 Retention basin outlet structures. (After Schueler, 1987).





Typical outflow structures are shown in Fig. 14.53 (Schueler, 1987).Additional design considerations have been presented by Schueler (1987). In sum-

mary, the pond should be wedge-shaped, narrowest at the inlet and widest at the outlet.A minimum length to width ratio of 3:1 should be used. The pond depth should average3–6 ft, with a shallow underwater bench around the pond’s perimeter. Side-slopesshould be no steeper than 3:1 (H:V) and not flatter than 20:1 (H:V). If the soils at thepond site are highly permeable, the pond’s bottom should be lined by impervious geot-extile or a 6-in clay liner. The inlets and outfalls should be protected by riprap or othermeans to prevent erosion. Wet ponds should be surrounded by a 25-ft buffer strip plant-ed with water-tolerant grasses and shrubs. A sediment forebay should be constructednear the inlet of the pond with extra storage equal to the projected sediment trappingover a 20� to 40-year period.

14.9.4 Computer Models for Detention and Retention Basin Design

As discussed in the preceding sections, a trial-and-error procedure is used for hydraulicdesign of retention and detention basins. A basin is first trial-designed, and then the designhydrographs are routed through the basin to verify if the design criteria are met. Therefore,any reservoir routing computer program can be useful for designing detention and reten-tion basins. The widely known TR-20 (Soil Conservation Service, 1986) and HEC-1(Hydrologic Engineering Center, 1990), for instance, have reservoir routing schemes andcan be used for pond design. These models are in public domain. Commercially availablepond routing software are a lot more user-friendly, and they include Watershed ModelingStandalone (www.eaglepoint.com). The commercially available models allow a variety ofdifferent outlet structures and simulation of multiple storm events. Also available arePONDOPT (www.cahh.com) and BASINOPT (www.cahh.com) which include an analy-sis option for reservoir routing as in the other pond models. These two models also havea unique design option which performs all the iterations internally. The ponds are sizedand the outlet invert elevations and sizes are determined by the program formultiple–return periods.

14.10 SEWER HYDRAULIC SIMULATION MODELS

A model is defined here as a method or simulation algorithm that has been coded into a com-puter program for computations and applications. Numerous models have been developedfor sewer networks. These sewer models can be classified in different ways as follows:

1. According to the purpose of the model: (a) design models—hydraulic design, or Optimaldesign, risk-based design; (b) evaluation/predictions models, (c) planning models.

2. According to the objective of the project: (a) flood control or (b) pollution control.

3. According to the extent of space consideration: (a) overland surface only, (b) sewersystem only, or (c) sewer system and overland surface.

4. According to the nature of wastewater: (a) sanitary sewer models or (b) storm andcombined sewer models.

5. According to water-quality considerations: (a) quantity only, (b) quality only (rare),or (c) quantity and quality.

6. According to time considerations of rainfall input: (a) single-event models or(b) multiple-event continuous models.





7. According to probability considerations: (a) deterministic or (b) probabilistic—purestatistical or stochastic.

8. According to systems concept: (a) lumped system or (b) distributed system.

9. According to hydrologic principles considered: (a) hydrologic (principle of mass con-servation) or (b) hydraulic (principles of conservation of mass and momentum orenergy).

In the first classification, the design models are for the determination of the size ofthe sewers and perhaps also their slope and layout of either a new sewer systems or anextension or modification of an existing system. The evaluation/prediction models arethose used to simulate the flow in an existing or predetermined sewer system for whichthe size, slope, and layout are already specified. Their use is to compute the flow in thesewers to check the adequacy of sewer capacity, system performance, operation, man-agement of pollution abatement, flood mitigation, and so forth. Or, the model may beincorporated as part of a real-time operation system. The planning models are thosemodels used for strategy planning and decision making for urban or regional storm andwaste water management, usually applied to a larger time and spatial frame than thedesign or evaluation models.

The design models design the sewers in a network for a hypothetical future eventwhich is represented by the design storms of specified return period or risk level. The eval-uation/prediction models simulate the runoff produced by a rainstorm of the past, present,future, or the flow from other sources. The planning models usually consider a relativelylong continuous period of time covering many rainstorms and dry periods in between. Theplanning models utilize the least hydraulic consideration of flow on overland areas and insewers. Often, a simple water budget balance suffices. A typical example is the STORMmodel (Hydrologic Engineering Center, 1974). Supposedly, for the purpose of reliableflow simulation, the evaluation/prediction models require the highest level of hydraulicsophistication and accuracy. However, many lower level models do exist. Due primarilyto the discrete sizes of commercial pipes, usually a moderate level of hydraulics is ade-quate for the design models (Yen and Sevuk, 1975; Yen et al., 1976). Most of the existingsewer models are evaluation/prediction models. Aside from the design models derivedfrom the rational method, there are actually very few true sewer design models; amongthem only two models, ILSD (Yen et al., 1976, 1984) and WASSP (Price, 1982b), havepublished user’s guides and arrangements for release of programs. Some of the evalua-tion/prediction models have the ability to compute the diameter required for gravity flowof a specific discharge. However, they are not true design models because different sew-ers should be designed for different rainstorms of different durations corresponding to thedifferent time of concentration of the sewers. Hence, many computer runs are required tocomplete the design of a network using these models.

In the last classification, the hydraulic models can further be classified according to thelevel of hydraulics shown in Eq. (14.1) or (14.2) as follows: dynamic wave models, non-inertia models, nonlinear kinematic wave models, and linear kinematic wave models. It isimpossible to summarize and report the hydraulic properties of all the exiting sewer mod-els in this chapter. Therefore, only selected models are made in this presentation. Sincethis article deals with the hydraulics of sewers, in the following section, only thehydraulics of selected models are discussed. For models that allow more than onehydraulic level for flow routing, they are presented according to their respective highesthydraulic level. For information and comparison of the nonhydraulic aspects of the mod-els, the reader should refer to other references such as those by Brandstetter (1976), Chowand Yen (1976), and Colyer and Pethick (1976) in addition to the original model develop-ers’ reports or papers. Models without hydraulic consideration of the sewer flow, such as





the rational method models, are excluded. When water quality transport simulation issought, nearly all the models perform the flow routing first and allow another pollutanttransport model—usually in concentration form—coupled with the routing result for sim-ulation. Only the Storm Water Management Model (SWMM) has the quality portion inte-grated in the model as a modular block.

14.10.1 Hydraulic Properties of Selected Dynamic Wave Sewer Models

Well�known models, in which the highest hydraulic level, the dynamic wave simula-tion is employed, are listed in Table 14.21. In the table, the subscript o denotes the sewerreceiving outflow from the junction. All these models were developed for flow simula-tion rather than for design of sewers in a network. CAREDAS, UNSTDY,HYDROWORKS, and MOUSE are proprietary models. Among the nonproprietarymodels, only two [ISS and Stormwater Management Models Extended Transportation(SWMM-EXTRAN)] have user’s manuals published and available to the public. Fordynamic wave and noninertia models, the junction conditions and surcharge transitionconditions—if surcharge is allowed—are important for reliable and realistic simulationof the flow. However, for most of the models listed in Table 14.21, information aboutthe details and assumption on the surcharge transition and on junctions is inadequatelygiven. Also, except ISS which cannot handle flow having a Froude number greater than1.6, it is not known whether the other models can handle supercritical flow with rollwaves, and if so, what assumptions are involved. In the following, dynamic wave mod-els listed in Table 14.21 are briefly discussed in three groups, namely, the explicitscheme model (SWMM-EXTRAN), the models that handle only open-channel sewerflows, and the models that handle both open-channel and surcharge sewer flows. Theallowed network size given in Table 14.21 is that indicated in the quoted literature. Formost models, this number has been increased with later developments.

14.10.1.1 Explicit scheme model: SWMM-EXTRAN. The Storm Water ManagementModel (SWMM) developed under continuous support of the U.S. EnvironmentalProtection Agency is one of the best known among all the sewer models. The ExtendedTransport block (EXTRAN) (Roesner and Shubinski, 1982; Roesner et al., 1984) wasadded to the SWMM Version III to provide the model with dynamic wave simulationcapability. The entire sewer length is considered as a single computational reach, and thedynamic wave equation is written in backward time difference between the time levels n� 1 and n for the sewer, and expressed explicitly as

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ace

or(d

oubl

e sw

eep)

and

So

ftwar

e W

OR

KS

Man

ning

criti

cal d

epth

h o�

H�

(KV2 /2

g)(1

991,

1997

)

.





Hn � 1 � Hn � �∆A

t

j

� (∑Qi,n � Qj,n) (14.99)

where the subscript j indicates junotion. The junction dynamic relation is simplified as acommon water surface [Eq. (14.51)]. Equations (14.98) and (14.99) are solved explicitlyby using a modified Euler method and half-step and full-step calculations. Courant’s sta-bility criterion is adopted to select the computational ∆t.

In EXTRAN, when a junction is surcharged, instead of properly applying the conti-nuity equation (Eq. 14.53), it assumes the point-type junction continuity relationship(Eq. 14.50) applies. On the basis of this point-type junction continuity equation, anexpression of the junction water head is derived through an improper application of thechain rule of differentiation, for which a Taylor expansion would have been more appro-priate. The unsatisfactory result was apparently recognized, and remedies were attempt-ed through the introduction of an adjustment factor and the assumption on the numeri-cal iterations to either reach a maximum number set by the user or the algebraic sum ofthe inflows and outflows of a junction being less than a tolerance. In an earlier versionof EXTRAN that was applied to a project in San Francisco, California, an attempt wasmade to artificially modify the geometry of the junction so that numerical solutioncould be obtained.

The SWMM-EXTRAN, with its explicit difference formulation, solves the flow sewerby sewer by using the one-sweep explicit solution method with no need for simultaneoussolution of the sewers of the network. Therefore, it is relatively easy to program.Nonetheless, because of the assumptions on the surcharge condition, and also the stabili-ty and convergence (accuracy) problems of the explicit scheme for the open-channel con-dition, on a theoretical basis EXTRAN is inferior to other dynamic wave models listed inTable 14.21. The other models, of course, have their share of problems concerning theassumptions on the transitions between open–channel and surcharge flows, betweensupercritical and subcritical flows, and on roll waves.

14.10.1.2 Dynamic wave model handling only open-channel flow: ISS. The IllinoisStorm Sewer System Simulation (ISS) model (Sevuk et al. 19973) solves the dynamicwave equation using the first-order scheme of the method characteristics. The Saint-Venant equations [Eqs. (14.2) and (14.5)] or similar type partial differential equations aretransformed mathematically into two sets of characteristic equations, each set consistingof a pair of ordinary differential equations which are solved numerically using a semi-implicit scheme. The formulation can be found in Sevuk and Yen (1982).

The junction conditions used for a storage junction are Eq. (14.47), together with theequations in Table 14.15, for sewer exits, and for sewer entrances Eq. (14.48) with Ki �0, that is,

H � (V2/2g) � h � Z (14.100)

For a point junction, the equations are Eq. (14.50), together with Eqs. (14.51) or (14.52).The ISS model program considers direct backwater effects for up to three sewers in a junc-tion. For junction with more than three joining sewers, the excess sewers (preferably thosewith small backwater effects from the junction) are treated as direct inflow, that is, Qj inEqs. (14.47) or (14.50). The flow in the network is solved by using the overlapping seg-ment method. The outlet of the network can be any one of the following: (1) a free fall,(2) flow continuing to approach normal flow, (3) a stage hydrograph h � f(t), (4) a ratingcurve Q � f(h), (5) a velocity-depth relationship v � f(h), and (5) a discharge-time rela-tionship Q � f(t).





When used to compute the required pipe diameter of a sewer, ISS is the only modelamong those listed in Table 14.21 that uses a maximum depth criterion to ensure gravityflow in the sewer for the design situation. Other models compute the required pipe diam-eter on the basis of the peak discharge that does not guarantee gravity flow because, forunsteady flow, maximum depth usually does not occur at the same time as the maximumdischarge in a sewer. The ISS model can easily be modified to account also for surchargeflow by adding the Preissmann hypothetical slot.

14.10.1.3 Dynamic wave models handling both open-channel and surcharge flows.Among the seven models belonging to this group listed in Table 14.21, four of them—CAREDAS, UNSTDY, Joliffe, and HYDROWORKS—are numerically similar, using afour-point implicit scheme and adopting the Preissmann fictitious open slot to simulatesurcharge flow. Details of the four-point implicit scheme can be found in Liggett andCunge (1975) and Lai (1986). In fact, the same four-point numerical scheme is also usedin SURDYN (Pansic, 1980). SURDYN is the only model in this group of seven that sim-ulates the surcharge flow by using the standard pressurized conduit approach and solvingit simultaneously with the open–channel flow. The surcharge equation used in this modelis a quasi-steady dynamic equation obtained by dropping the local acceleration (∂V/∂t)term in Eq. (14.2). For the rising transition from open-channel flow, surcharge is assumedto occur when the discharge exceeds Qf or when the pipe exit is submerged. Falling tran-sition from surcharge to open–channel flow is assumed to occur when the pipe entranceis not submerged, when the discharge falls less than Qf or when the pipe exit is not sub-merged. Pansic (1980) reported that the model simulates the unsteady flow reasonablywell. But oscillations often occur at transitions between open-channel and surcharge con-ditions. This oscillation problem is partly numerical, partly hydraulic, and partly due toassumptions.

Among these models, HYDROWORKS, MOUSE, UNSTDY, and CAREDAS are pro-prietary. They are briefly introduced in the following:

1. HYDROWORKS. The dynamic wave sewer flow routing option of HYDROWORKS isbased on an earlier model SPIDA from the same company, Hydraulics Research, inEngland. HYDROWORKS also contains noninertial (WALLRUS) and nonlinear kine-matic wave (WASSP-SIM) sewer routing options. The model can handle a looped-typenetwork as well as a dendritic type. For dynamic wave routing, the inertia terms arelinearly phased out from a Froude number equal to 0.8-1.1. Essentially, for supercriti-cal flow, the noninertia approximation is used. For pressurized flow, the hypotheticalslot width is assumed one-twentieth of the maximum pipe diameter.

2. MOUSE. This model was upgraded from Danish Hydraulic Institute’s (DHI) System11-sewer (S11-S) model. It was first released in 1985 and subsequently updated withpersonal computer PC technology advancements. It uses the Abbott-Ionescu six-pointimplicit scheme (Abbott and Basco, 1990) which is relatively stable and consistent butcostly in computation. The model allows loop network. In addition to dynamic waverouting, it also has noninertia (identified in the model as diffusion wave) and kinemat-ic wave routing options for sewers.

3. UNSTDY. The UNSTDY model uses four-point noncentral implicit schemes to solvethe Saint-Venant equations for subcritical flow. Supercritical flow is simulated byusing the kinematic wave approximation. The model can solve a looped network in thesystem.





TA

BL

E 1

4.22

Sum

mar

y of

Hyd

raul

ic P

rope

rtie

s of

Non

iner

tia S

ewer

Net

wor

k M

odel

s

Mod

elO

pen-

Cha

nnel

Flo

w

Inte

rior

Jun

ctio

ns

Surc

harg

e R

efer

ence

s

Num

eric

al

Para

met

ers

S fSe

wer

Dow

nstr

eam

So

luti

on

Det

enti

on

Junc

tion

F

low

Sche

me

Con

diti

onSc

hem

eSt

orag

eE

quat

ion

HV

Mh,

QC

oleb

rook

-U

nspe

cifi

ed o

r Pi

pe b

y pi

peN

oΣQ

� 0

and

Stan

dard

Gei

ger

and

Whi

tera

ting

curv

eh i

� h

opr

essu

rize

dD

orsc

h (1

980)

;pi

pe f

low

Kly

m e

t al.

(197

2); V

ogel

and

Kly

m(1

973)

DA

GV

L-

Impl

icit,

six-

poin

th,

QM

anni

ngJu

nctio

n w

ater

Sim

ulta

neou

sY

esΣQ

� d

s/dt

and

Prei

ssm

ann

slot

Sjöb

erg

(198

2)D

IFF

cont

inui

ty

surf

ace

or(d

oubl

e sw

eep)

h o�

H�

(KV

2 /2g

) or

four

–poi

nt m

omen

tum

criti

cal d

epth

ΣQ�

0an

dh i

� h

0

w�

0.5

5

MO

USE

Six-

poin

t im

plic

ith,

QM

anni

ngJu

nctio

n w

ater

Sim

ulta

neou

sY

esΣQ

� 0

and

Prei

ssm

ann

slot

DH

I (1

994)

w�

0.5

surf

ace

(dou

ble

swee

p)h i

� h

0-

NIS

NFo

ur-p

oint

impl

icit

h,Q

Man

ning

Junc

tion

wat

erSi

mul

tane

ous,

Yes

ΣQ�

0 a

nd h

i�

h0

orPr

eiss

man

n sl

otPa

glia

ra a

nd

surf

ace

or

over

-lap

ping

ΣQ

� d

s/dt

and

Yen

(19

97)

criti

cal d

epth

segm

ent

h o�

H�

(KV

2 /2g

)





4. CAREDAS. This is one of the earliest full dynamic wave sewer flow routing modelsdeveloped by SOGREAH at Grenoble, France. This is the first model to incorporate thePreissmann slot to simulate surcharge flow. In applying CAREDAS, a sewer network isfirst checked for the sewers with sufficiently steep slope for which the kinematic waveequation can be applied as an adequate approximation. The dynamic wave model isapplied to each group of the connected, gently sloped sewers.

14.10.2 Hydraulic Properties of Noninertia Sewer Models

The noninertia approximation of the unsteady flow momentum equation [Eq. (14.1)] isprobably the most efficient option among the dynamic-wave momentum equation optionsto solve unsteady open-channel sewer flow problems. It accounts for downstream back-water effect, and it allows reversal flow. Computationally, it is much simpler than the fulldynamic wave option. It is only for rare highly unsteady cases that the noninertia optionis inadequate and the full dynamic wave or the exact momentum options are required.However, only a few noninertia sewer models have been developed; only four are report-ed in the literature and they are summarized in Table 14.22.

The proprietary HVM-QQS model was developed by Dorsch Consult (Klym et al.,1972; Vogel and Klym, 1973) at Munich, Germany. It has been misquoted as a dynam-ic wave model (Brandstetter, 1976). Examination of the equations [Eqs. (3) and (4) inVogel and Klym, 1973] reveals that, in fact, it is a noninertia model. It was stated thatto avoid simultaneous solution of all the sewers in the network, further assumptionswere made. One assumption is to let Sf � So(Q/Qo)2, where Qo is defined as a normalflow discharge corresponding to So, but it is not clear what depth is used in computingQo. Another issue that the sewer downstream boundary condition at the exit is eitherunspecified or a rating curve h � h(Q), or the exit depth hydrograph h(t) is known. Infact, with unspecified downstream boundary condition, this model does not reallyaccount for the backwater effect, and thus, it omits one of the important advantageousproperties of the noninertia model. No information is given on whether the flow equa-tions are solved implicitly or explicitly.

The DAGVL-DIFF model was developed at the Chalmers University of Technology(Sjöberg, 1982) at Göteborg, Sweden. The equations in the model are solved in a mannersimilar to the dynamic wave model DAGVL-A and were found generally satisfactory. Nofurther development or support of the DAGVL models has been provided since the devel-opment of S11S/MOUSE.

The proprietary DHI (1994) model MOUSE contains noninertia and kinematic wavesewer routing options in addition to dynamic wave routing. The noninertia option simu-lates the flow the same way as the dynamic wave option; thus, it does not take full advan-tage of the simplicity and computational efficiency of the noninertia modeling.

The NISN model (Pagliara and Yen, 1997) utilizes the overlapping segment method tosolve for the flow in a network. For each segment, the flow equations are solved simulta-neously using the Preissmann four-point implicit scheme. Junction storage and headlossare allowed. There is no network size limit for this model.

14.10.3 Nonlinear Kinematic Wave Models

Unlike the dynamic wave and noninertia sewer models, there exist many kinematic wavemodels. Only a few nonlinear kinematic wave models are listed in Table 14.23 for dis-





AB

LE

14.

23Se

wer

Hyd

raul

ic P

rope

rtie

s of

Sel

ecte

d N

onlin

ear

Kin

emat

ic W

ave

Mod

els

Mod

elO

pen-

chan

nel

flow

Inte

rior

Tran

siti

onSu

rcha

rge

Flo

w R

efer

ence

s

Sew

er

Para

met

ers

S fN

umer

ical

Solu

tion

Ju

ncti

on

Con

diti

onSu

rcha

rge

Inte

rior

Solu

tion

Hyd

raul

ics

Sche

me

Sche

me

Hyd

raul

ics

Junt

ion

Sche

me

USG

SN

onlin

ear

A,Q

Man

ning

Exp

licit

Cas

cade

ΣQ�

ds/d

tQ

� Q

fQ

�Q

fSt

ore

exce

ssPi

pe b

y pi

peD

awdy

et

al.

kine

mat

icw

ater

,rel

ease

(197

8)w

ave

late

r

ILSD

-B2

Non

linea

r h,

QM

anni

ngFo

ur-p

oint

Cas

cade

ΣQ�

0N

AN

AN

AN

AY

en a

nd

kine

mat

ic

impl

icit

Sevu

k (1

975)

,w

ave

Yen

et

al.

(197

6)

ILSD

-B3

Mus

king

um-

h,Q

Man

ning

Four

-poi

ntC

asca

deΣQ

�0

NA

NA

NA

NA

Yen

et

al.

Cun

geim

plic

it(1

976)

WA

SSP-

Mus

king

um-

h,Q

Dar

cy-

Qua

siO

ne s

wee

p,ΣQ

�ds

/dt

Q�

Qf

Uns

tead

yH

calc

ulat

ed,

Impl

icit

Pric

e SI

MC

unge

Wei

sbac

h ex

plic

itpi

pe b

y pi

peor

ass

umed

dyna

mic

head

loss

essi

mul

tane

ous

(198

2a,b

)an

d su

bmer

genc

eeq

uatio

nco

nsid

ered

re

laxa

tion

Col

ebro

ok-

cond

ition

sW

hite

SWM

M-

Impr

oved

A

,QM

anni

ngFo

ur-p

oint

One

sw

eep,

ΣQ�

0Q

� Q

fQ

� Q

fSt

ore

exce

ss

Pipe

by

pipe

Hub

er a

nd

TR

AN

S-no

nlin

ear

impl

icit,

pipe

by

unle

ss s

tora

gew

ater

,rel

ease

H

eane

y PO

RT

kine

mat

icw

�0.

55pi

pe

bloc

k is

use

dla

ter

(198

2),H

uber

wav

eet

al.

(198

4);

Met

calf

&E

ddy

Inc.

et

al (

1971

)





cussion. All of the models listed in these tables, except the USGS model, have provisionto compute the required diameter for a specified discharge using the Manning or Darcy-Weisbach formula. All the nonlinear kinematic wave models listed in Table 14.23 consid-er the backwater effect from upstream (entrance) of the sewer within the realm of a singlesewer and not beyond, and not the backwater effect from downstream (sewer exit).

The kinematic wave models, unable to compute reliably the sewer flow cross-sectionarea A, depth h, and velocity V, are of questionable usefulness in coupling with a water-quality equation for water-quality evaluation. Unless the downstream backwater effect isalways insignificant, otherwise a water-quantity model having a hydraulic level of nonin-ertia approximation or higher should be used.

Nonlinear kinematic wave models may be classified further according to the man-ner the flow equations are formulated for solution. The first group includes the modelssolving directly the nonlinear kinematic wave equations. The first two models in Table14.23, [U.S. Geological Survey’s Distributed Routing Rainfall-Runoff Model (USGS)(Dawdy et al., 1978)] and [Illinois Least-Cost Sewer System Design Model, option B2(ILSD-B2) (Yen et al., 1976)] belong to this group. The second group includes themodels that solve an explicit linear algebraic equation of the Muskingum equationform. The Illinois Least-Cost Sewer System Design Model, option B3 (ILSD-B3) (Yenet al., 1976) and the British Hydraulics Research’s Wallingford Storm Sewer Designand Analysis Package Simulation Method (WASSP-SIM) (Price, 1982a,b) belong tothis group. The third group consists of the models using other modified nonlinear kine-matic wave equations for solution such as the TRANSPORT Block in SWMM (Metcalfand Eddy. et al., 1971).

1. ILSD-B2 and USGS models. In the first group, the continuity equation is written as afinite difference algebraic equation of one variable (usually h or Q) or two variables(e.g., h, Q or A, Q) and solved iteratively with the aid of the simplified momentumequation, So � Sf, where Sf is approximated by Manning’s or similar formulas to relatethe depth or area to discharge. A formulation used in Yen and Sevuk (1975) and adopt-ed in ILSD-B3 is given in the following as an example. Noting that B(h) � ∂A/∂h andG(h) � ∂Q/∂h, Eq. (14.4) can be rewritten as

B(h) �∂∂ht�

� G(h) �∂∂hx�

� 0 (14.101)

For partially filled circular pipes (Fig. 14.3),

B(h) � D sin

�φ2�

(14.102)

and by using Manning’s formula

G(h) � �Kn

n� So

1/2R2/3 �B3�

5 � �sin 2

1(φ/2)�

�sin

φφ

� �1

(14.103)

� �0.13

n2Kn� So

1/2D5/3 1 � �sin

φφ

�2/3 5 sin �

φ2� � �sin (

1φ/2)� �

siφnφ� � 1

where the central angle φ in radians is (Fig. 14.3):

φ � 2 cos �1 [1 � (2h/D)] (14.104)

Consider the four computational grid points boxed by the time levels n and n � 1 andspace levels i and i � 1, Eq. (14.101) can be transformed into the following implicitfour-point forward–difference equation:





�21∆t� [(Bi,n � 1 � Bi � 1,n � 1)(hi,n � 1 � hi � 1,n � 1 � hi,n � hi � 1,n)]

� �∆1x� [(Gi,n � 1 � Gi � 1,n � 1)(hi � 1,n � 1 � hi,n � 1)] � 0 (14.105)

This equation is nonlinear only with respect to the unknown flow depth hi � 1, n � 1 sinceBi � 1, n � 1 and Gi � 1, n � 1 are both expressed in terms of the depth [Eqs. (14.102) and(14.103)], and hence it can readily be solved by using Newton’s iteration method. Thesolution proceeds sewer by sewer from upstream toward downstream. Within eachsewer, the flows for all the reaches are determined for a given time before proceedingto the next time step.In ILSD, there are actually several sewer flow routing schemes of different hydrauliclevels, including options B2 and B3 listed in Table 14.23 and the option of hydrographtime lag adopted in ILSD-1 and 2. The objective of ILSD is to develop an efficient andpractical optimization model for the least cost system design of sewer networks.Therefore, the sensitivity and significance of the sophistication of hydraulics on opti-mal design of sewer systems were investigated. It was found that for the purpose ofdesigning sewers, because of the discrete sizes of commercially available pipes, unso-phisticated hydraulic schemes often suffice, and hence the hydrograph time lagmethod, instead of options B2 and B3, is adopted in ILSD-1 and 2. Yen and Sevuk(1975) also arrived at a similar conclusion that for design, a low hydraulic level rout-ing method is often acceptable, whereas for evaluation and simulation of flow in sew-ers, a high hydraulic level routing is usually required.In the USGS model, the finite difference equation is formulated from Eq. (14.47) sim-ilar to ILSD-B2. However, the nonlinear relation between Q/Qf and A/Af is approxi-mated by a straight line, and the flow area A is expressed explicitly as

Ai � 1,n � 1 � f(Ai,n � 1, Ai � 1,n) (14.106)

Hence, solution for all the reaches within a sewer must be obtained at each time for thetime increments. However, for the sewers in a network, the solution technique can beeither the cascade method or the one-sweep method. No information on which one isused in the model is given in the literature.

2. SWMM-TRANSPORT. Only one model in the third group of modified nonlinear kine-matic wave models is listed in Table 14.23. The SWMM is a comprehensive urbanstorm water runoff quality and quantity simulation model for evaluation and manage-ment. A good summary of the model is given in Huber and Dickinson (1988), Huberand Heaney (1982), and Metcalf and Eddy et al. (1971). It has two sewer flow routingoptions, TRANSPORT and EXTRAN, not counting the crude gutter-type routing inthe RUNOFF block. EXTRAN was discussed above. TRANSPORT is the originalsewer-routing submodel built in the progam. In TRANSPORT, the continuity equationis first normalized using the just-full steady uniform flow discharge Qf and area Af, thenthe equation is written in finite differences and expressed as a linear function of thenormalized unknowns A/Af and Q/Qf at the grid point x � (i � 1)∆x andt � (n � 1)∆t:

(Q/Qf)i � 1,n � 1 � C1(A/Af)i � 1,n � 1 � C2 � 0 (14.107)

where C1 and C2 � functions of known quantities. From the simplified dynamic equa-tion Sf � So and Manning’s formula, we have




(Q/Qf) � AR2/3/AfRf2/3 � f(A/Af) (14.108)

Accordingly, curves of normalized discharge-area relationship Q/Qf versus A/Af forsteady uniform flow in pipes of different cross-sectional geometries are establishedand solved together with Eq. (14.107) for Q/Qf and A/Af. In the kinematic wave methodof solving Eqs. (14.107) and (14.108), in addition to the initial condition, only oneboundary condition is needed, which is usually the inflow hydrograph at the sewerentrance. No downstream boundary condition is required, and hence, no backwatereffect from the downstream can be accounted for if the flow is subcritical. However, inTRANSPORT through a formulation of friction slope calculation using the previoustime values at the spatially forward point, the downstream backwater effect is partial-ly accounted for at one time step behind. In routing the unsteady nonuniform flow byusing Eqs. (14.107) and (14.108), the value of Qf is not calculated as the steady uni-form full-pipe discharge. Instead, it is adjusted by assuming that

Sf � So � �∂∂hx�

� �Vg� �

∂∂Vx�

� So � �hi � 1

∆,n

x� hi,n� � �

Vi2

�

21

g,n

∆�

xVi

2,n

� (14.109)

To improve computational stability, it is further assumed in TRANSPORT that at anyiteration k, Qfk is taken as the average of previous and current values: that is,

Qfk � �12� Qf(k � 1) � AfRf

2/3

(14.110)

� �So∆x � hi,(k � 1) � hi � 1,(k � 1) � �Vi

2,2(k

g� 1)� � �

Vi2

�

21

g,(k � 1)��1/2

where all the values of h and V are those at the previous time n∆t that are known if theone–sweep or implicit solution method is used to solve for the flow in individual sew-ers at incremental times. Incorporating Eq. (14.109) for Sf in Manning’s formula yieldsa quasi-steady dynamic wave approximation instead of the kinematic wave. Thus, useof Eq. (14.110) to compute Qf indirectly gives a partial consideration of the down-stream backwater effect with a time lag. This improvement of the kinematic waveapproximation makes SWMM TRANSPORT hydraulically more attractive than thestandard nonlinear kinematic wave models. Presumably, the partial accounting of thedownstream backwater effect is effective as long as the flow does not change rapidlywith time, and no hydraulic jump or hydraulic drop is allowed. A hydraulic compari-son of EXTRAN to improvement and advantages over TRANSPORT has not beenreported and would be interesting.Nonetheless, since the downstream boundary condition is not truly accounted for, it isrecommended in SWMM�TRANSPORT that for a sewer with a large downstreamstorage element from which the backwater effect is severe, the water surface is assumedas horizontal from the storage element going backward until it intercepts the sewerinvert. Moreover, when the sewer slope is steep, presumably implying high-velocitysupercritical flow, the flood may simply be translated through the sewer without rout-ing, that is, shifting of the hydrograph without time lag. Also, if the backwater effect isexpected to be small and the sewer is circular in cross section, the gutter flow routingmethod in the RUNOFF Block may be applied to the sewer as an approximation.In SWMM, large junctions with significant storage capacity and storage facilities arecalled storage elements, equivalent to the case of storage junction (that is, ds/dt � 0),which was discussed above. Only the continuity equation, Eq. (14.35), is used in stor-age element routing. No dynamic equation is considered except for the cases with weiror orifice outlets. Small junctions are treated as point-type junctions with ds/dt � 0.

Kn�






3. ILSD-B3 and WASSP-SIM. In the second group of nonlinear kinematic wave models,both ILSD-B3 and WASSP-SIM adopt the Muskingum-Cunge method. The Muskingum routing formula can be written for discharge at x � (i � 1)∆x andt � (n � 1)∆t as

Qi � 1,n � 1 � C1Qi,n � C2Qi,n � 1 � C3Qi � 1,n (14.111)

in which

C1 ��K(1

K�X �

X)0�.5

0∆.t5�t

� (14.112a)

C2 ��K(10.

�5∆

Xt)�

�K

0X.5∆t� (14.112b)

and C3 ��KK

((11

��

KX)

)��

00..55∆∆

tt

� (14.112c)

where K is known as the storage constant having a dimension of time and X a factorexpressing the relative importance of inflow. Cunge (1969) showed that by taking K and∆t as constants, Eq. (14.111) is an approximate solution of the nonlinear kinematic waveequation [Eqs. (14.4) and (14.102) or Eq. (14.104)]. He further demonstrated that Eq.(14.111) can be considered as an approximate solution of Eq. (14.104) if

K � ∆x/c (14.113)and

X � �12

� � (ε/c∆x) (14.114)

where ε is the “diffusion” coefficient and c is the celerity of the flood peak that can beapproximated as the length of the reach divided by the flood peak travel time through thereach. Assuming K � ∆t and denoting α � 1 � 2X, Eq. (14.111) can be rewritten as

Qi � 1,n � 1 � �22

��

αα� Qi, n � �2 �

αα� Qi, n � 1 � �2 �

αα� Qi � 1, n (14.115)

In the traditional Muskingum method, X and, consequently, α are regarded as constant. Inthe Muskingum method as modified by Cunge, α is allowed to vary according to the chan-nel geometry and is computed as

α � KQ/So(∆x)2B (14.116)

in which B is the surface width of the flow and So the sewer slope. The values of α arerestricted to being between 0 and 1 so that C1, C2, and C3 in Eq. (14.112) will not be neg-ative. It is the variation of α, and hence C1, C2, and C3, that classifies the Muskingum-Cunge method as a nonlinear kinematic wave approximation.

The Muskingum-Cunge method offers two advantages over the standard nonlinearkinematic wave methods. First, the solution is obtained through a linear algebraic equa-tion [Eq. (14.111) or Eqs. (14.115) and 14.116)] instead of a partial differential equation,permitting the entire hydrograph to be obtained at successive cross sections instead ofsolving for the flow over the entire length of the sewer pipe for each time step as for thestandard nonlinear kinematic wave method. Second, because of the use of Eq. (14.116), alimited degree of wave attenuation is included, permitting a more flexible choice of thetime and space increments for the computations as compared to the standard nonlinearkinematic wave method.





In ILSD-B3, the coefficient α in Eq. (14.115) is computed at each grid point by usingEq. (14.116), while B and K both change with respect to time and space. The values of Kare computed by using Eq. (14.113) with the celerity c evaluated by

c � ∂Q/∂A (14.117)

or for a partially filled pipe using Manning’s formula

c � �0.13

n2Kn� So

1/2 D2/3 1 � �sin

φφ

�2/3�5 � sin �2 �φ2

��sinφ

φ� � 1� (14.117)

The initial flow condition is the specified base flow as in ILSD-B2. The upstreamboundary condition of the sewer is the given inflow hydrograph. The flow depth and othergeometric parameters at the sewer entrance can be computed from the geometric equa-tions given in Fig. 14.3. The junction condition used is the continuity relationship, Eq.(14.53). The solution is obtained over the entire time period at a flow cross section beforeproceeding to the next cross section. The solution then proceeds downstream section bysection and then sewer by sewer in a cascading sequence. More details on the computa-tional procedure of ILSD-B3 can be found in Yen et al. (1976).

The British model WASSP is a sewer design and analysis package consisting of foursubmodels (Price, 1982b): A modified rational method for design of sewers, a hydrographmethod for design of sewers using the Muskingum-Cunge routing, an optimal designmethod, and a simulation method using the fixed parameter Muskingum-Cunge techniquefor open-channel routing in sewers and the unsteady dynamic equation for surcharge flowcomputations. Open-channel flow is routed using Eq. (14.111) with the coefficients C1,C2, and C3 expressed as functions of c and µ � Q/2BSo. In computation, c is taken as thefull-pipe velocity and µ is evaluated at h/D � 0.6. Sewers under open-channel flow aresolved pipe by pipe, using a directionally explicit algorithm to calculate the discharge atthe sewer exit. The space increment ∆x along the sewer is selected automatically in termsof ∆t to enhance computational accuracy. Connected surcharged sewers are solved simul-taneously. For surcharge flow, a time increment as small as a few seconds may be neces-sary if surges occur. The transition between open-channel flow and surcharge flow isassumed to occur when the discharge exceeds Qf, when the sewer entrance and exit aresubmerged, or when the water depth in the junction is higher than the sewer flow depthplus the entrance or exit headloss (Bettess et al num., 1978). At a junction, only the con-tinuity equation is considered for open-channel flow. For surcharge flow, in addition to thecontinuity equation, junction headloss is considered and incorporated into the surchargeunsteady dynamic wave equation. The headloss coefficient is assumed to be 0.15 for ajunction with straight pipes, 0.50 for 30º bend pipes, and 0.90 for 60º bend pipes. Somedetails of WASSP-SIM are reported in Price (1982b).

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