Flow Estimation (1)

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Flow Estimation and Routing

14 . 1 INTRODUCTION

This chapter presents methods and procedures for the estimation, routing, and attenuation of peakflows and flow volumes for sub-catchments as a prerequisite to the design of stormwaterconveyance systems, and detention and retention facilities. These methods and procedures may beused in conjunction with Chapter 16 to design stormwater system networks.

14 . 2 Design ARI

Ideally, the design ARI should be selected on the basis of economic efficiency. In practice,however, economic efficiency is typically replaced by the concept of level of protection. Theselection of this level of protection (or ARI), that actually refers to the exceedance probability of thedesign storm rather than the probability of failure of the drainage system, is largely based on localexperience. ARIs to be adopted for the design of minor and major stormwater systems are providedin Chapter 4.

A fundamental assumption which is made in flow calculations is that the design flow with a givenARI is produced by a design storm rainfall of the same ARI. Strictly speaking, the ARI of the flow

is influenced by other variable factors such as catchment antecedent wetness. However, the methodspresented in this Manual have been designed such that the above assumption is reasonable.

Regardless of the design basis, it is recommended that performance of the drainage system beexamined for a range of ARIs. A comparison of results for the various ARIs may indicate the needfor a different design basis.

14 . 3 STEPS IN FLOW ESTIMATION

The process of flow estimation generally involves the following main steps. These apply both forestimating design flows, and in estimating flows for historical events.

determination of time of concentration

rainfall estimation (see Chapter 13)

calculation of rainfall excess

conversion of rainfall excess to runoff

hydrologic/ hydraulic routing

Modern computer models generally use all of the above steps. Some earlier manual procedures,

such as the Rational Method, simplify or combine one or more steps to reduce computations. Thissimplification is acceptable on small, simple catchments but is a significant source of error in morecomplex situations.


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The time of concentration (tc) is often considered to be the sum of the time of travel to an inlet and

the time of travel in the stormwater conveyance system. In the design of stormwater drainagesystems, this can be the sum of the overland flow time and the times of travel in street gutters,roadside swales, stormwater drains, drainage channels, small streams, and other waterways. Anumber of methods, mostly using empirical equations, are provided below for estimating the time ofconcentration for urban catchments.

14.4.1 Components of Flow Time

Depending on the particular location, the calculation oftc

will include one or a number of

components as shown in Table 14.1.

Table 14. 1 Flow Time Components

Flow Type Components

Overland or 'sheet'flow

natural surfaces landscaped surfaces impervious surfaces

Roof to main pipesystem

residential roofs commercial/industrial

roofs

Open channel open drains kerbs and gutters roadside table drains monsoon drains engineered waterways natural channels

Underground pipe downpipe to streetgutter

pipe flow within lotsincluding roof drainage,car parks, etc

street drainage pipeflow

14.4.2 Calculation of Flow Time

This section gives procedures for the calculation for each of the components in Table 14.1. Some ofthese procedures are subject to uncertainty; the designer should therefore check that the results arephysically meaningful.


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Overland flow over unpaved surfaces initially occurs as sheet flow for a short time and distance afterwhich it begins to form a runnel or rill and travels thereafter in a natural channel form.

In urban areas, the length of overland flow will typically be less than 50 metres after which the flowwill become concentrated against fences, paths or structures or intercepted by open drains.

The formula shown below, known as Friends formula, should be used to estimate overland sheetflow times. The formulae was derived from previous work (Friend, 1954) in the form of anomograph (Figure 14.1) for shallow sheet flow over a plane surface.

( 14 . 1 )

where,

to = overland sheet flow travel time (minutes)

L = overland sheet flow path length (m)

n = Horton's roughness value for the surface

S = slope of surface (%)

Note : Values for Horton's 'n ' are similar to those for Manning's 'n ' for similar surfaces. Values aregiven in Table 14.2.

Some texts recommend an alternative equation, the Kinematic Wave Equation. However thistheoretical equation is only valid for uniform planar homogeneous flow. It is not recommended forpractical application.

Table 14. 2 Values of Mannings 'n' for Overland Flow

Surface Type Manningsn

Range

Concrete/Asphalt** 0.011 0.01-0.013

Bare Sand** 0.01 0.01-0.06

Bare Clay-Loam(eroded)**

0.02 0.012-0.033

Gravelled Surface** 0.02 0.012-0.03

Packed Clay** 0.03

Short Grass** 0.15 0.10-0.20

Light Turf* 0.20Lawns* 0.25 0.20-0.30

Dense Turf* 0.35


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Figure 14.1 Nomograph for Estimating Overland Sheet Flow Times (Source: ARR, 1977)

( ii ) Overland Flow Time over Multiple Segments

Where the characteristics of segments of a sub-catchment are different in terms of land cover or

surface slope, the sub-catchment should be divided into these segments, and the calculated traveltimes for each combined.

However, it is incorrect to simply add the values oft0for each segment as Equation 14.1 is based on

the assumption that segments are independent of each other, i.e. flow does not enter a segment fromupstream.

Utilising Equation 14.1, the following method for estimating the total overland flow travel time for

segments in series is recommended. For two segments, termed a and b (Figure 14.2):

( 14 . 2 a)

where,

La = length of flow for Segment a

Lb = length of flow for Segment b

ta,(La) = time of flow calculated for Segment a over

lengthLa

tb ,(...) = time for Segment b over the lengths indicated


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Figure 14.2 Overland Flow over Multiple Segments

For each additional segment, the following value should be added:

( 14 . 2 b)

where,

! = segment name

L total = total length of flow, including the current segment !

L!

= length of flow for current segment !

t!

(...) = time for the current segment a over the lengths indicated

( iii ) Roof Drainage Flow Time

While considerable uncertainty exists in relation to flow travel time on roofs, the time of flow in a lotdrainage system to the street drain, or rear of lot drainage system is generally very small forresidential lots and may be adopted as the minimum time of 5 minutes. However, for largerresidential, commercial, and industrial developments the travel time may be longer than 5 minutes in

which case it should be estimated using the procedures for pipe and/or channel flow as appropriate.

( iv ) Kerb Gutter Flow Time

The velocity of water flowing in kerb gutters is affected by:

the roughness of the kerb gutter and road surface

the cross-fall of the road pavement the longitudinal grade of the kerb gutter


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An approximate kerb gutter flow time can be estimated from Figure 14.3 or by the followingempirical equation:

( 14 . 3 )

where,

tg = kerb gutter flow time (minutes)

L = length of kerb gutter flow (m)

S = longitudinal grade of the kerb gutter (%)

Kerb gutter flow time is generally only a small portion of the time of concentration for a catchment.The errors introduced by these approximate methods of calculation of the kerb gutter flow time resultin only small errors in the time of concentration for a catchment, and hence only small errors in thecalculated peak flow.

( v ) Channel Flow Time

The time stormwater takes to flow along a open channel may be determined by dividing the length ofthe channel by the average velocity of the flow. The average velocity of the flow is calculated usingthe hydraulic characteristics of the open channel.

The Manning's Equation is suitable for this purpose:

( 14 . 4 a)

From which,

( 14 . 4 b)

Where,

V = average velocity (m/s)

n = Manning's roughness coefficient

R = hydraulic radius (m)

S = friction slope (m/m)

L = length of reach (m)

t = travel time (minutes)

Wh h l h i h d th it idth it b t


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Chapter 25 (derived from Mannings equation) where the flow, pipe diameter and pipe slope areknown. The time of flow, tp , is then given by:

( 14 . 5 )

where,

L = pipe length (m)

Where the pipe diameter is not known, the diameter can be first estimated given the flow at theupstream end of the pipe reach and the average grade of the land surface between its ends.

As is the case with kerb gutter flow time, pipe flow time is generally only a small portion of the timeof concentration for a sub-catchment. The error in the estimated pipe flow time introduced by theadoption of the wrong diameter or slope, or by the assumption that the pipe is flowing full when infact it is only flowing part full, will not introduce major errors into the calculated peak flow.

In many situations an experienced user will be able to estimate the velocity of flow in a pipe within areasonable accuracy. Therefore, the pipe flow time can be estimated directly from Equation 14.5.

14.4.3 Time of Concentration for Rural Catchment

For larger systems times of concentration should preferably be estimated on the basis of locallyobserved data such as the time of occurrence of flood peaks at or near the catchment outlet comparedwith the time of commencement of associated storms. In the absence of such information recoursemay be made to empirical formulae as for instance that of Bransby-Williams.


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Figure 14. 3 Kerb Gutter Flow Time

For rural catchments and mixed flow paths the time of concentration can be found by use of theBransby-Williams' Equation 14.6. In these cases the times for overland flow and channel or streamflow are included in the time calculated.

Here the overland flow time including the travel time in natural channels is expressed:

( 14 . 6 )

where,

tc

= the time of concentration (minute)

F = a coefficient, 58.5 when areaA in km2

= 92.5 when area in ha

L = Length (km) of flow path from catchment divided to outlet

A = Catchment Area (km2 or ha, based on value ofF)

S = Equal area slope of stream flow path (m/km).

14.4.4 Standard Time of Concentration for Small Catchments

Although travel time from individual elements of a system may be very short, the total nominal flowtravel time to be adopted for all individual elements within any catchment to its point of entry into the

stormwater drainage network shall not be less than 5 minutes.

For small catchments up to 0.4 hectare in area, it is acceptable to use the standard minimum times ofconcentration given in Table 14.3 instead of detailed calculation.

Table 14. 3 Standard Minimum Times of Concentration

Location Standard tc (minutes)

Roof and propertydrainage

5

Road inlet pits 5

Small areas < 0.4 hectare 10

14 . 5 RATIONAL METHOD

There are two basic approaches to computing stormwater flows from rainfall. The first approach is the Rational


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The Rational Formula is one of the most frequently used urban hydrology methods in Malaysia.The formula is:

( 14 . 7 )

where,

Qy = y year ARI peak flow (m3/s)

C = dimensionless runoff coefficient

y

It = y year ARI average rainfall intensity over time of concentration, tc , (mm/hr)

A = drainage area (ha)

Traditionally, design discharges for street inlets and stormwater drains have been computed usingthe Rational Method, although hydrograph methods also can be used for these purposes. Theprimary attraction of the Rational Method has been its simplicity. However, now that computerisedprocedures for hydrograph generation are readily available, computational simplicity no longer need

be the primary consideration.

Experience has shown that the Rational Method can provide satisfactory estimates of peak dischargeon most small catchments. For larger catchments, storage and timing effects can become significant,and therefore a hydrograph method is needed. Various methods have been devised to formpseudo-hydrographs based on the rational formula, but their reliability is uncertain and they shouldonly be used for the design of on-site stormwater detention and retention facilities.

14.5.2 Analysis Procedure

A procedure for estimating a peak flow from a single sub-catchment for a particular ARI using theRational Method is outlined in Figure 14.4. Peak flow estimates should be obtained for both theminor and major drainage systems. An example of peak flow estimation by rational method formulti-subcatchments is given in Table 16.A9 in Chapter 16.

14.5.3 Assumptions

Assumptions inherent in the Rational Method are as follows:

1. The peak flow occurs when the entire catchment is contributing to the flow.

2. The rainfall intensity is the same over the entire catchment area.

3. The rainfall intensity is uniform over a time duration equal to the time of concentration, tc..


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Figure 14. 4 General Procedure for Estimating Peak Flow for a Single Sub-catchment Using the Rational Method

4 The ARI of the computed peak flow is the same as that of the rainfall intensity i e a


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ponding of stormwater in the catchment might affect peak discharge.

the design and operation of large (and hence more costly) drainage facilities is to be undertaken,particularly if they involve storage.

14.5.4 Rainfall Intensity

The rainfall intensity, I, in the rational formula represents the average rainfall intensity over aduration equal to the time of concentration for the catchment.

Refer to Chapter 13 for details on IDF relationships for estimating design rainfall intensity.

14.5.5 Runoff Coefficient

The runoff coefficient, C, in Equation 14.7 is a function of the ground cover and a host of otherhydrologic abstractions. The runoff coefficient accounts for the integrated effects of rainfallinterception, infiltration, depression storage, and temporary storage in transit of the peak rate ofrunoff. When estimating a value for the runoff coefficient, the roles played by these hydrologicprocesses should be considered. The runoff coefficient depends on rainfall intensity and duration aswell as the catchment characteristics. During a rainstorm the actual runoff coefficient increases asthe soil become saturated. The greater the rainfall intensity, the lesser the relative effect of rainfallabstractions on the peak discharge, and therefore the greater the runoff coefficient. Recommended

runoff coefficient values may be obtained for urban and rural catchments from Figure 14.5 andFigure 14.6, respectively.

14.5.6 Equivalent Impervious Area

Design flow rates for stormwater inlets are calculated for local contributing sub-catchments, whilethose for pipes and open drains are calculated for the accumulated areas draining through each pipeor open channel section or reach. Except for small lot drainage systems, it is inappropriate to simplyadd the separate flows from each sub-catchment. This over-estimates flow rates. Whentimes-to-peak differ, the total flow from a number of sub-catchments will have a maximum valueless than the sum of the separate flows from each sub-catchment.

A more accurate procedure is to sum the equivalent impervious areas, i.e. the product ofCand Avalues for each sub-catchment. Design flow rates can be calculated by multiplying total equivalentimpervious areas by average rainfall intensity values corresponding to the times of concentration atvarious points along the drainage line. Use of equivalent impervious areas also allows segments of

different land use or surface slope within a sub-catchment to be combined. For example, if asub-catchment consists of three segments with different land use or surface slope denoted by a, b,and c, the combined equivalent impervious area is:

( 14 . 8 )

where,

C = runoff coefficient

A = segment area (ha)


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intensityyIt (resulting from a lower tc), produces a greater peak discharge than that if the whole

upstream catchment is considered. This is known as the partial area effect.

Usually the above effect results from the existence of a sub-catchment of relatively small C.A,but aconsiderably longer than average tc . This can result from differences in the shapes and/or surface

slopes of sub-catchments within a catchment. Typical catchments that can produce partial areaeffects are shown in Figure 14.7.

It is important to note that particular sub-catchments may not produce partial area effects whenconsidered individually, but when combined at some downstream point with other sub-catchments,the peak discharge may result when only parts of these sub-catchments are contributing.

The onus is on the designer to be aware of the possibility of the partial area effect and to check as

necessary to ensure that the correct peak discharge is obtained.

14.5.8 Limitations

A principal limitation of the Rational Method is that only a peak discharge is produced. Therefore,the simple form of the Rational Method cannot be used to calculate the volume or shape of the runoffhydrograph, which is required for the design of facilities that use storage such as detention andretention basins.


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Figure 14. 7 Urban Catchments Likely to Exhibit Partial Area Effects


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14 . 6 HYDROGRAPH METHODS

14.6.1 Basic Concepts

This section discusses methods that should be used to develop a design hydrograph. Hydrographmethods must be used whenever rainfall spatial and temporal variations or flow routing/storageeffects need to be considered. Flow routing is important in the design of stormwater detention,water quality facilities, and pump stations, and also in the design of large stormwater drainagesystems to more precisely reflect flow peaking conditions in each segment of complex systems.

Hydrograph methods can be computationally involved and computer programs (refer Chapter 17)are usually used to generate runoff hydrographs.

( a ) Storm Intensity, Duration and Frequency

Design storm duration is an important parameter that defines the rainfall depth or intensity for agiven ARI, and therefore affects the resulting runoff peak. The design storm duration that producesthe maximum runoff peak traditionally defined as the time of concentration. Design storminformation is provided in Chapter 13.

The rainfall intensity, yIt, used in hydrograph methods represents the average rainfall intensity over aparticular duration t. This combination of average intensity and duration must have an ARI y equalto the desired ARI of the peak discharge. Rainfall intensity therefore depends on:

the desired ARI of the peak discharge

the duration under consideration

the local IDF relationship

Refer to Chapter 13 for details on IDF relationships for estimating design rainfall intensity.

Current practice is to select the design storm duration as one equal to or longer than the time ofconcentration for the catchment (or some minimum value when the time of concentration is short).Intense rainfalls of short durations usually occur within longer-duration storms rather than asisolated events. It is common practice (Packman and Kidd, 1980) to compute discharge for severaldesign storms with different durations, and then base the design on the "critical" storm whichproduces the maximum discharge. However the "critical" storm duration determined in this waymay not be the most critical for storage design.

Recommended practice is to compute the design flood hydrograph for several storms with differentdurations equal to or longer than the time of concentration for the catchment, and to use the onewhich produces the most severe effect on the pond size and discharge for design.

( b ) Spatial Distribution

Storm spatial characteristics are important for larger catchments. In general, the larger the catchmentd h h h i f ll d i h l if l h i f ll i di ib d h h

Spatial distributions can be represented in hydrograph methods However for areas of less than 10


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Spatial distributions can be represented in hydrograph methods. However for areas of less than 10

km2 in urban drainage systems the areal reduction factor can be neglected. It is also commonpractice to neglect any effects due to storm movement direction.

( c ) Temporal Distribution

Commonly used approaches to distribute rainfall within a design storm were discussed inChapter 13. The temporal distribution adopted can have a significant effect on the shape of therunoff hydrograph, and on the peak discharge.

The recommended design temporal patterns are presented in Chapter 13.

( d ) Rainfall Excess

Not all of the rainfall that falls on a catchment, produces runoff. Some rainfall losses occur, such asevaporation, infiltration and depression storage. The remaining rainfall after subtracting rainfalllosses is called the rainfall excess.

In the Rational Method, described in the previous section, the concept of rainfall excess is not useddirectly. Instead the runoff coefficient allows for rainfall losses. Rainfall excess concepts are usedin most hydrograph methods and this discussion focuses mainly on those methods.

The physical processes of interception of rainfall by vegetation, infiltration of water into the soilsurface, and storage of water in surface depressions are commonly termed rainfall abstractions.Although these three processes are physically complex, simplified modeling procedures have beenfound give acceptable results for urban stormwater drainage.

Values can be derived by analysing observed rainfall and runoff data. Since individual values aredependent on the particular rainfall and catchment wetness characteristics of the event, individual

values have little meaning except as indicators of those particular events. For design, an averagevalue is usually needed, and since there is no reason for expecting loss rate values for a catchment toconform to any particular distribution, the median of the derived values is probably the mostappropriate for design.

In discussing losses it is important to distinguish between directly connected impervious areas(DCIA) and pervious areas. The main rainfall losses only apply to pervious areas. Imperviousareas that are not directly connected to the pipe system, such as tennis courts and concrete paths that

are surrounded by pervious (grassed) surfaces, are also subject to losses because water must passover these and possibly infiltrate before reaching a point of entry to a pipe or open drain.

(i) Evaporation Losses

Evaporation is generally insignificant and is neglected during the short-duration storms of concern instormwater drainage design.

(ii) Interception/ Depression Storage Losses

The predominant form of loss on pervious surfaces is by infiltration. Some of the most frequently


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The predominant form of loss on pervious surfaces is by infiltration. Some of the most frequentlyused types of loss models are illustrated in Figure 14.8.

Figure 14. 8 Typical Loss Models for Estimating Rainfall Excess

These five types of loss models are described below.

(a) loss (and hence runoff) is a constant fraction of rainfall in each time period : this is similar tothe Rational Method runoff coefficient concept.

(b) constant loss rate : where the rainfall excess is the residual left after a selected constant rate ofinfiltration capacity is satisfied.

(c) initial loss and continuing loss: which is similar to (2) except that no runoff is assumed tooccur until a given initial loss capacity has been satisfied, regardless of the rainfall rate. Thecontinuing loss is at a constant rate. A variation of this model is to have an initial loss followed by a

It should be noted that loss values derived according to one of the models are not directly


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g ytransferable to other models. The choice of loss model therefore depends in part on the choice offlow estimation method. In most urban stormwater drainage applications this is not a seriousproblem as the losses are generally only small in comparison to rainfall, and therefore a high degreeof accuracy in estimation is not necessary.

Loss values are derived by analysing observed rainfall and runoff data. Since individual values are

dependent on the particular rainfall and catchment wetness characteristics of the event, individualvalues have little meaning except as indicators of those particular events. For design, an averagevalue is usually needed, and since there is no reason for expecting loss rate values for a catchment toconform to any particular distribution, the median of the derived values is probably the mostappropriate for design.

(iv) Choice of Infiltration Loss Model

Choice and validity of the 5 models depend on the data available and the likely runoff processes.

Models (a) and (b) are not often used in current practice. Model (e), the U.S. Soil ConservationService approach is not very suitable for urban drainage. It has given only fair results when tested inthe United States. Models (c) or (d) are recommended for use in Malaysian urban drainagesituations. Table 14.4 contains recommended values for use by drainage designers.

Table 14. 4 Recommended Loss Models and Values for Pervious Areas (Note 1)

Condition Loss Model (Note2)

Recommended Values

Urban areasgenerally

Initial loss-Lossrate

Initial loss: 10 mm Loss rate: 5mm/hr

Urban areas

>80 hectares

As above, or Initial

loss proportionalloss

Initial loss: 10 mm Proportional

Loss: 20% ofrainfall

Continuoussimulation

Horton model withregeneration ofinfiltration

Initial Infiltration Capacity f0

A. DRY soils (little or no vegetation)

Sandy soils: 125 mm/hrLoam soils: 75 mm/hrClay soils: 25 mm/hr

For dense vegetation, multiply valuesgiven in A by 2

B. MOIST soils

Soils which have drained but not driedout: divide values from A by 3

Soils close to saturation: value close to

saturated hydraulic conductivity

Soils partially dried out: divide valuesfrom A by 1.5-2.5

Hydrologic SoilGroup (Note 3),UltimateInfiltration Ratefc (mm/hr)

A 10 7.5B 7.5 3.8C 3.8 1.3D 1.3 - 0

If a subcatchment contains areas of different surface condition or land use weighted average values


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If a subcatchment contains areas of different surface condition or land use, weighted average valuesof losses for different conditions or land uses, such as proportions of pervious and imperviousareas, can be derived using methods similar to those in Section 14.5.6.

For continuous modeling, the choice of loss model becomes much more significant. The modelmust provide for infiltration capacity to be 'restored' during dry periods in between rainfall events.

The Horton model with regeneration is recommended, as used in SWMM.

Hortons equation is widely used for describing infiltration capacity in a soil. It describes thedecrease in capacity as more water is absorbed by the soil, and has the form:

( 14 . 9 )

where,

f= the infiltration capacity (mm/hr) at time t,

f0 andfc = the initial and final constant rates of infiltration (mm/hr)

k= a shape factor (recommended value of 4 /hr)

t= the time from the start of rainfall (hours)

14 6 2 Ti A M h d


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14.6.2 Time-Area Method

Time-area methods utilise a convolution of the rainfall excess hyetograph with a time-area diagramrepresenting the progressive area contributions within a catchment in set time increments. Separatehydrographs are generated for the impervious and pervious surfaces within the catchment. These arecombined to estimate the total flow inputs to individual sub-catchment entries to the underground

urban drain network.

The time-area method dates from the research of Ross in 1922. Networked urban drainage adoptionsof the procedure however only date back to 1963. This computerised program known as the TRRLMethod was developed by the UK Transport and Road Research Laboratory (TRRL), described byWatkins (1963). In the US Terstriep and Stall (1974) further developed the method to includepervious runoff. In South Africa Watson (1981) made a number of additional changes particularlyto the way infiltration was estimated. Between 1982 and 1986 OLoughlin (1988) using Watsonsmodel as a basis carried out extensive changes once again to formulate a computerised package known

as ILSAX. The sub-catchment runoff estimating procedure still utilises the basic time-area method toestimate both pervious and impervious portion runoff.

The peak discharge, Qp , is the sum of flow contributions from subdivisions of the catchment defined

by time contours (called isochrones), which are lines of equal flow time to the point where Qp is

required. The method is illustrated in Figure 14.9. The flow from each contributing area bounded bytwo isochrones (T- !T, T) is obtained from the product of the mean intensity of effective rainfall ( i )from time T- !T to the area (!A). Thus, QA,, the flow atXat time 4 hours is given by:

( 14 . 10 a)

i.e.,

( 14 . 10 b)

As with the Rational Method, the whole catchment is taken to be contributing to the flow after Tequals Tc..

Using the above nomenclature, the peak flow at X when the whole catchment is contributing to theflow, a period Tc after the commencement of rain, is:

( 14 . 11 )

where,

n = the number of incremental areas between successive isochrones, given by (Tc/ !T)

k = a counter

The unrealistic assumption made in the Rational Method of uniform rainfall intensity over the wholecatchment during the whole ofTc is avoided in the timearea method, where the catchment

contributions are subdivided in time. The varying intensities within a storm are averaged over discrete

constructed. From the beginning of the flow in drain 1 at T = 0 there is a steady increase in area


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ycontributing until T= T1 which is the value ofTc for area A 1. Drain 2 begins to contribute to the

outfall flow at T= T3 before T= T1. After a further period, T2 , area 2 reaches its own Tc at time T=

T2 + T3. Between times T3 and T1 , both drains have been flowing and the joint contributing area (at

C) at T= T1 is given by:

(14. 12 )

Figure 14. 9 TimeArea Method

From T= (T2 + T3), both areas are contributing fully. The timearea curve for the combined drains is

the composite line OBCD.

The principle of the TRRL Hydrograph Method is outlined in Figure 14.10. In Figure 14.10 (a), acatchment area, divided into four sub-areas, is drained by a single channel to the outfall where thehydrograph is required. Sub-area 1 begins contributing to the flow first, to be followed sequentiallyby the other three sub-areas. The individual timearea curves are shown in Figure 14.10 (b) and

i th b t ib ti t l ti i t l d th it f th h l

rainfall will contribute to the direct runoff from the catchment area. Excess rainfall rates are computedi d i h S i 14 6 1(d) Fi 14 10 ( ) h h ff i f i f ll f


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in accordance with Section 14.6.1(d). Figure 14.10 (c) shows the effective rates of rainfall excess foreach time unit i1, i2, i3, etc, for the storm duration (9 time units).

The discharge rates after each time interval are given by:

Figure 14.10 (d) shows the sequence of discharges forming the runoff hydrograph at the outfall.

Figure 14.10 TRRL Hydrograph Method

Two further considerations are necessary. A time of entry from the onset of the storm rainfall to the

time of flow into the pipe is usually taken to be 2 minutes and must be allowed for in thecomputations. Secondly, experience has shown that there is a certain amount of retention of water inthe channel, and amendments to the hydrograph must be made to account for channel storage(Watkins, 1962).

14 .6.3 Kinematic Wave Method

The kinematic-wave method is a hydraulic method for routing runoff across planar surfaces and

through small channels and pipes. The kinematic-wave formulation couples the continuity equationwith a simplified form of the momentum equation that includes only the bottom-slope andfriction-slope terms. Kinematic-wave theory is only valid for one-dimensional overland flow on aplanar surface. It therefore has little practical use in urban drainage applications.

Figure 14.11 shows the catchment conceptualised as a reservoir with rainfall as inflow, and infiltrationand surface discharge as outflows The depth y represents the average depth of surface runoff and the


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and surface discharge as outflows. The depth y represents the average depth of surface runoff, and thedepth ydrepresents the average depression storage in the catchment. The continuity relationship for

this system is:

( 14 . 13 )

where,

A = catchment area

I = rainfall intensity

f = the infiltration rate

Q = the discharge at the catchment outlet

The model assumes uniform overland flow at the catchment outlet at a depth equal to the differencebetween y andyd. Based on the Manning friction relationship, the catchment discharge, Q is given

by:

( 14 . 14 )

where,

W = a representative width for the catchment

n = the average value of the Manning roughness coefficient for the catchment

S = average surface slope

Substituting Equation 14.14 into Equation 14.13 yields a non-linear differential equation for y. A simple finite

difference form of the equation is used to solve for the depth y at the end of each time step. Thisequation is:

( 14 . 15 )

where,

!t = time step increment

y1 = depth at the beginning of the time step

y2 = depth at the end of the time step

I = average rainfall rate over the time step


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Figure 14. 11 Definition Sketch for Non-linear Reservoir Model

For each time step, three separate calculations are performed. First, an infiltration equation is used tocompute the average potential infiltration rate over the time step (in the SWMM program, the userselects either the Green-Ampt or the Horton infiltration equation), then Equation 14.15 is solvediteratively for y

2, and, finally, Equation 14.14 yields the corresponding discharge.

Unlike the time-area method, which uses excess rainfall as input, the non-linear-reservoir methodcouples the processes of infiltration and surface runoff. The non-linear-reservoir model assumes thatinfiltration occurs at the potential rate over the entire surface area whenever the ponded depth isnon-zero. The excess-rainfall models, on the other hand, entirely neglect infiltration of pondedwater. This difference becomes important following cessation of rainfall, or whenever the rainfallintensity drops below the potential infiltration rate. In reality, infiltration does continue for some timeafter rainfall ceases, but the area over which infiltration continues to occurs after rainfall ceases would

tend to be underestimated by the non-linear-reservoir methods and overestimated by theexcess-rainfall methods, though the difference will depend on the degree of discretisation used, since amore detailed schematisation can partially account for the phenomenon of decreasing area ofinfiltration.

14.6.5 Generalised Analysis Procedure

A procedure for estimating a runoff hydrograph from a single sub-catchment for a particular ARI and

duration is outlined in Figure 14.12. Hydrographs should be obtained for both minor and majordrainage systems.


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Flow routing is the process of converting a hydrograph that passes through some part of a flow systemto allow for the changes that occur during its passage. There are three main types of flow routing:


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g g p g yp g

catchment routing, which converts a rainfall excess hyetograph into a hydrograph at thecatchment outlet, allowing for the distribution of rainfalls over the catchment surface, and various lagsor delays along flow paths;

channel routing, which allows for the changes in hydrographs as they flow along river or channelreaches, caused by variations in the channel geometry which result in storage effects,

reservoir routing, which allows for storage effects in a concentrated, level pool reservoir.Broadly defined, flow routing is an analytical procedure intended to trace the flow of water through ahydrological system, pond, conveyance, or porous media, given some runoff event hydrograph asinput. The procedure determines the flow hydrograph at a point downstream, from known or

assumed flow hydrographs at one or more points upstream. If the flow is a runoff event such as aflood, then the procedure is specifically known as flood routing. Routing by lumped system methodsis called hydrological routing. These methods calculate the flow as a function of time alone. Routingby distributed system methods is called hydraulic routing, and the flow is calculated as a function ofboth space and time throughout the system.

For hydrologic routing the input I (t), output Q(t), and storage S(t) are related by the continuityequation:

( 14 . 16 )

If an inflow I(t) is known Equation 14.16 cannot be solved directly to obtain the outflow Q (t),because both Q and Sare unknown. A second relationship, the storage function, is required to relateI ,S, and Q. Coupling the continuity equation with the storage function provides a solvablecombination of two equations and two unknowns.

The specific form of the storage functions to be employed in hydrologic routing depends on thenature of the system being analysed. In reservoir routing by the level-pool method, storage is anon-linear function ofQ only, i.e. S= f(Q ), and the function, f(Q ), is determined by relating reservoirstorage and outflow to reservoir water level. In channel routing by theMuskingum method, storage islinearly related to Iand Q. Similarly, in porous media, storage is a function of outflow which dependson storage in the media and the underlying soils.

14 . 8 FLOW THROUGH POND AND RESERVOIR

14. 8 .1 Hydrologic Routing

Level-pool routing is a procedure for calculating the outflow hydrograph from a pond reservoir,assuming a horizontal water surface, given its inflow hydrograph and storage-dischargecharacteristics. When a reservoir has a horizontal water surface, its storage is a function of itswater-surface elevation, or depth in the pool. Likewise, the discharge is a function of the watersurface elevation, or head on the outlet works. Combining these two functions yields the invariablesingle-valued function.

Integration of the continuity equation (Equation 14.16) over the discrete time intervals provides an

are pre-specified (i.e. the inflow hydrograph ordinates). The values Qj and Sj are known at the jth

time interval Hence Equation 14 17 contains two unknowns Q j +1 and S j +1 which are isolated by


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time interval. Hence Equation 14.17 contains two unknowns, Qj +1 and Sj +1, which are isolated by

multiplying Equation 14.17 by 2/!tand rearranging the result to produce:

( 14 . 18 )

In order to calculate the outflow Qj +1 from Equation 14.18, a storage-discharge function relating

2S /!t+ Q and Q is needed. The method of developing this function using stage-storage andstage-discharge relationship is shown in Figure 14.13.

For a given water-surface elevation, the values of storage Sand discharge Q are determined. Then,

the value of2S /!t+Q is calculated and plotted against Q. In routing the flow through thejth time


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S Q p g Q g g jinterval, all terms in the right-hand-side of Equation 14.18 are known, and so the value of2Sj+1/! t+Q can be computed. The corresponding value ofQj+1 can be determined from the

storage-discharge function 2S /!t+Q versus Q. To set up the data for the next time interval, the value2Sj+1/! t -Qj+1 is calculated by:

( 14 . 19 )

The computation is repeated iteratively for subsequent routing periods. Input requirements for thisrouting method are:

the storage-discharge relationship the storage-indication relationship the inflow hydrograph initial values of the outflow rate (Q1) and storage (S1)

the routing interval (!t)An analysis procedure for hydrologic routing is shown in Figure 14.14.


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Figure 14. 14 General Analysis Procedure for Hydrologic Routing

14.8.2 Two-Dimensional Hydraulic Routing

Two-dimensional hydraulic routing is especially useful for pond and lake design that involvesstructures and water quality The governing equations of a depth averaged two dimensional model

Continuity Equation


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( 14 . 20 )

Momentum Equations

x - direction:

( 14 . 21 a)

y - direction:

( 14 . 21 b)

where,

H = h + !

h = still water depth

! = free surface displacement

and are bottom shear stresses in which it is generally assumed that:

( 14 . 21 c)

( 14 . 21 d)

where,

u and v are depth-averaged velocity components

n = the Manning roughness coefficient

For flows in small and medium size waterbodies such as detention ponds, the effects of wind and earthrotation are insignificant compared to the driving forces neglected.

The equations are normally solved by finite differences or finite elements methods. Theirnon-linearity is handled by the Newton Raphson technique. There are only a few general softwareprograms such as WASP4 (USEPA, 1987) commercially available. Others are available from R&Dinstitutes such as RESPOND from the University of Virginia USA (Wu 1992) The flow equations

The constant infiltration model provides a method of analysis and evaluating the hydraulics ofsoakage pits. This model can be used to calculate the maximum water level occurring in a giveninfiltration system during a design storm event, thus allowing the required depth of a system to be


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infiltration system during a design storm event, thus allowing the required depth of a system to becalculated. This section introduces equations for a maximum water level for both plane andthree-dimensional infiltration drainage systems.

Applying the flow balance approach (Equation 14.16) to infiltration drainage facilities where,

S = S(h ) , the volume of water stored in the infiltration system

Q in = I(t ) , the inflow of runoff

Qout = O (t ) , the outflow through infiltration

and,

h = water level above the depth of water in the infiltration system

t = time since the start of the rainfall event

Considering the Storage

The storage available depends on whether or not the facility is rubble-filled.

( 14 . 22 )

where,

V = V(h) , the volume of the water filled part of the facility and is a function of the geometry of thesystem

n = effective porosity of any fill material and is unity if the system is not rubble-filled

The hydrological balance equation may therefore be written as:

( 14 . 23 )

The water level in the system, h(t) , can be found at any time, t, by inserting appropriate functions ofQ in (t), Qout(t), and V(h) into Equation 14.23, rearranging for h , and integrating with respect to time.

Considering the Inflow

It is assumed that the facility receives a constant rate of runoff, estimated from the Rational Formula.

, for a duration D ( 14 . 24 )

where,

I = intensity of the rainfall

A D = impermeable area drained

AW= wetted infiltration surface area of the facility which depends on the geometry of the system and

is a function of the water depth, h


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qf = the infiltration coefficient which includes an appropriate factor of safety

With these assumptions, a flow balance equation may be written for h from Equation 14.23:

( 14 . 26 )

The design method requires integration of Equation 14.26 in order to derive formulae describing thewater level, h, at given times, t. These formulae depend on the geometrical functionsAW(h) and

V(h). For simple geometries, Equation 14.26 may be solved analytically. For complex geometries, it

can only be solved using an approximate method or numerical integration.

Once a solution to Equation 14.26 is found for a particular type of infiltration system, the highestwater level, hmax , can be found for a particular storm event. Using the constant inflow function

described above, the highest water level will occur at the end of the storm event when tequals thestorm duration, D.

14.9.2 Hydraulic Routing

Infiltration drainage systems are founded in the unsaturated zone above the water table. Adiagrammatic representation of the conceptual model of soakage pit hydraulics in which groundwaterflow in the unsaturated zone is taken into account is shown in Figure 12.13. It is envisaged that a'bulb' of saturation becomes established around the pit. As groundwater flows away from the pit areathrough which it passes increases due to the three-dimensional nature of the flow and the soil becomesunsaturated. Thus to provide a realistic description of the hydraulic behaviour of infiltration system,both saturated and unsaturated groundwater flows have been considered in this section.

( a ) Saturated Groundwater Hydraulics

Allowing for storage to occur in the soil for non-steady flow conditions, the general 3-dimensionalequations of motion for homogeneous-isotropic saturated phreatic groundwater flow is:

( 14 . 27 )

where,

N = stormwater recharge (+ve) or abstraction (-ve) rate

S = the specific yield/ effective porosity

t = time

h = phreatic water level

k = saturated hydraulic conductivity

Unsaturated groundwater flow differs from saturated groundwater flow in that the hydraulic propertiesof the soil, the hydraulic conductivity and the storage coefficient both depend on the degree ofsaturation of the soil. As there is both water and air present in the soil, capillary action exerts a force,


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or soil suction, and so the water in the soil is below atmospheric pressure. The degree of soil suction,or negative pressure, also depends on the degree of saturation and the soil properties.

From Darcys equation (Chapter 12), the one-dimensional infiltration equation of motion forunsaturated groundwater flow is:

( 14 . 28 )

where,

! = the volumetric moisture content

z = elevation head

k = unsaturated hydraulic conductivity

The equation is often referred to as the modified Richards equations (1931) and can be simplyextended to 3-dimensions for real world problems. The hydraulic conductivity - pressure headfunction may be defined as:

( 14 . 29 )

where,

k = the hydraulic conductivity

ks = the saturated hydraulic conductivity

kr = the relative hydraulic conductivity

The moisture content - pressure head function may be defined as:

( 14 . 30 )

where,

! = moisture content

n = porosity

q r = relative saturation (Sr < !r


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14 . 10 FLOW THROUGH CONVEYANCE

14.10.1 Hydrologic Routing

A widely used hydrologic method for routing flows in conveyance systems is the Muskingum method.It models the storage volume of flow in a channel reach by a combination of wedge andprism storage(Figure 14.15). When the flood wave is advancing, inflow exceeds outflow and a positive wedge ofstorage is produced. When the flood is receding, outflow exceeds inflow and a negative wedgeresults. In addition, a prism of storage is formed by a volume of (approximately) constantcross-section along the length of the channel reach.

Assuming that the cross-sectional area of the flow is directly proportional to the discharge at the

section, the volume of prism storage is KQ , where K is a coefficient of proportionality. The volumeof wedge storage is assumed to be equal to KX(I-Q), whereX is a weighting factor having the range0!X!0.5. The total storage is defined as the sum of the two storage components S=KQ+KX(I-Q)which can be rearranged to give the linear storage function for the Muskingum method:

( 14 . 31 )

Figure 14.15 Prism and Wedge Storage in a Channel Reach

The value ofXdepends on the shape of the modeled wedge storage, X= 0 corresponds to reservoir

(level-pool) storage and Equation 14.31 reduces to S= KQ.X= 0.5 for a full wedge. In most naturalstream channels, X is between 0 and 0.3, with a mean value near 0.2. Great accuracy in determiningXis usually not necessary because the Muskingum method is not sensitive to this parameter.

The parameter K is the time of travel of the flood wave through the reach. Although techniques existthat allow for the values ofKandX to vary according to flow rate and channel characteristics (e.g.,the Muskingum-Cunge method), for hydrologic routing the values ofKandXare assumed to bespecified and constant throughout the range of flow. The change in storage over the time interval !t(fromj toj+1) is:

( 14 .32 )

( 14 . 34 a )


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( 14 . 34 b )

( 14 . 34 c )

Note that C1 +C2 +C3 = 1.

The values ofKandX are determined usingobserved inflow and outflow hydrographs in thechannel reach. By assuming various values ofXand known values of inflow, successive values ofKcan be computed using:

( 14 . 35 )

Equation 14.35 is derived from Equations 14.17and 14.32. The computed values of thenumerator and denominator of Equation 14.35are plotted for each time interval !t, with thenumerator on the vertical axis and thedenominator on the horizontal axis. This usuallyproduces a graph in the form of a loop (see

Figure 14.16). The value ofX that produces aloop closest to a single line is taken to be thecorrect value for the reach. The parameter K,from Equation 14.35 is the slope of the line forthe value ofX.

Figure 14.16 Procedure for DeterminingXand K

Values

14.10.2 One-dimensional Hydraulic Routing

Procedures for distributed-flow hydraulic routing are popular because they compute flow rate andwater level as functions of both space and time. The methodologies are based upon the Saint-Venantequations of one-dimensional flow. In contrast, the lumped hydrologic-routing procedures discussedin the previous sections compute flow rate as a function of time alone

( 14 . 36 a)


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Momentum:

( 14 . 36 b)

where,

x = longitudinal distance along the conveyance

t = time

A = cross-sectional area of flow

A0 = cross-sectional area of dead storage (off-channel)

q = lateral inflow per unit length along the conveyance

h = water-surface elevation

vx = velocity of lateral flow in the direction of flow

Sf = friction slope

Se = eddy loss slope

B = width of the conveyance at the water surface

Wf = wind shear force

! = momentum correction factor

g = acceleration due to gravity

The Saint-Venant equations operate under the following assumptions:1. The flow is one-dimensional with depth and velocity varying only in the longitudinal direction of

the conveyance. This implies that the velocity is constant and the water surface is horizontal acrossany section perpendicular to the longitudinal axis.

2. There is gradually varied flow along the channel so that hydrostatic pressure prevails and verticalaccelerations can be neglected.

3. The longitudinal axis of the channel is approximated as a straight line.4.

The bottom slope of the channel is small and the bed is fixed, resulting in negligible effects ofscour and deposition.

5. Resistance coefficients for steady uniform turbulent flow are applicable, allowing for a use ofManning's equation to described resistance effects

Hydrologic (lumped) routing methods may not perform well in simulating the flow conditions whenbackwater effects are significant and the drain or channel slope is mild, because these methods have nohydraulic mechanisms to describe upstream propagation of changes in the flow momentum.


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APPENDIX 14 .A WORKED EXAMPLES

Time-Area Calculation

A worked example is shown in Table 14.A.1. There are four increments of area (ha) resulting in a time of

concentration for the catchment equivalent to four time units. The storm duration extends over 10 time units. A

runoff coefficient of 0.64 has been assumed and thus the total areal rainfalls in column 2 have been multiplied by 0.64

to give the corresponding effective rainfalls (i mm/hr). The values ofq for each increment (a ) and effective rainfall

rate (i ) are calculated from:

where q is in m3/s, i is in mm/hr, a is in hectares, and 360 is the conversion factor for the units used.

Table 14.A1 The TRRL Hydrograph Method (Runoff Coefficient = 0.64)

Time Unit

Areal rate of rain Increment of area (ha)Discharge

q (l/s)Total Effective i

(mm/hr)

a1

0.25

a2

0.82

a3

0.92

a4

0.34

1 13.7 8.8 6.1 6.1

2 90.0 57.6 40.0 20.0 60.0

3 59.4 38.0 26.4 131.2 22.5 180.14 18.3 11.7 8.1 86.6 147.2 8.3 250.2

5 16.3 10.8 7.5 26.7 97.1 54.4 185.7

6 13.7 8.8 6.1 24.6 29.9 35.9 96.5

7 5.3 3.4 2.4 20.0 27.6 11.1 61.1

8 5.1 3.3 2.3 7.7 22.5 10.2 42.7

9 6.1 3.9 2.7 7.5 8.7 8.3 27.2

10 4.6 2.9 2.0 8.9 8.4 3.2 22.5

11 6.6 10.0 3.1 19.7

12 7.4 3.7 11.1

13 2.7 2.7

The summation of the rows across a1 to a4 gives the discharge values after each time increment, and thus the requiredhydrograph. It will be noted that the peak flow occurs after the fourth time interval, the time of concentration of the

catchment. This does not always happen, e.g. with late peaking rainfalls.

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