14 FLOW ESTIMATION AND ROUTING
14.1 INTRODUCTION..........................................................................................................141
14.2 DESIGN ARI................................................................................................................141
14.3 STEPS IN FLOW ESTIMATION......................................................................................141
14.4 TIME OF CONCENTRATION..........................................................................................141
14.4.1 Components of Flow Time...............................................................................141
14.4.2 Calculation of Flow Time.................................................................................142
14.4.3 Time of Concentration for Natural Catchment...................................................144
14.4.4 Time of Concentration for Small Catchments.....................................................144
14.5 RATIONAL METHOD....................................................................................................144
14.5.1 Rational Formula............................................................................................144
14.5.2 Analysis Procedure.........................................................................................145
14.5.3 Assumptions..................................................................................................145
14.5.4 Rainfall Intensity............................................................................................145
14.5.5 Runoff Coefficient...........................................................................................145
14.5.6 Variation of Subcatchment Conditions..............................................................145
14.5.7 The Partial Area Effect....................................................................................146
14.5.8 Limitations.....................................................................................................147
14.6 HYDROGRAPH METHODS.............................................................................................147
14.6.1 Basic Concepts...............................................................................................147
14.6.2 TimeArea Method..........................................................................................149
14.6.3 Kinematic Wave Method..................................................................................1411
14.6.4 Nonlinear Reservoir Method...........................................................................1411
14.6.5 Rational Method Hydrograph Method...............................................................1412
14.6.6 Generalised Analysis Procedure.......................................................................1413
14.7 FLOW ROUTING AND ATTENUATION PRINCIPLES.........................................................1413
14.8 FLOWTHROUGH POND AND RESERVOIR......................................................................1415
14.8.1 Hydrologic Routing.........................................................................................1415
14.8.2 TwoDimensional Hydraulic Routing.................................................................1416
14.9 FLOWTHROUGH POROUS MEDIA.................................................................................1418
14.9.1 Routing through Infiltration Facility..................................................................1418
14.9.2 Flow through Layered Media...........................................................................1419
14.9.3 Hydraulic of Aquifer Response to Infiltration.....................................................1419
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14.10 FLOW THROUGH CONVEYANCE....................................................................................1420
14.10.1 Hydrologic Routing.........................................................................................1420
14.10.2 Onedimensional Hydraulic Routing..................................................................1421
APPENDIX 14.A DESIGN CHARTS.............................................................................................1423
APPENDIX 14.B WORKED EXAMPLE..........................................................................................1427
14.B.1 Rational Method Calculation.............................................................................1427
14.B.2 TimeArea Calculation.....................................................................................1429
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14.1 INTRODUCTION
This chapter presents methods and procedures for the estimation, routing, and attenuation of peak flows and flow volumes for subcatchments as a prerequisite to the design of stormwater conveyance systems and quantity control facilities. These methods and procedures may be used 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. The selection of this level of protection (or ARI), that actually refers to the exceedance probability of the design storm rather than the probability of failure of the drainage system, is largely based on local experience. ARIs to be used for the design of minor and major stormwater systems are provided in Chapter 4.
A fundamental assumption, which is made in flow calculations, is that the design flow with a given ARI is produced by a design storm rainfall of the same ARI. Strictly speaking, the ARI of the flow is also influenced by other variable factors such as catchment antecedent wetness. However, the methods presented 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 be examined for a range of ARIs to ensure that the system will perform satisfactorily.
14.3 STEPS IN FLOW ESTIMATION
The process of flow estimation generally involves the following main steps. These apply both for estimating 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
• runoff 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. This simplification is acceptable on small, simple catchments but is a significant source of error in more complex situations.
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14.4 TIME OF CONCENTRATION
The time of concentration is the flow travel time from the most hydraulically remote point in the contributing catchment area to the point under study. The concept of time of concentration is important in all methods of flow estimation as it can be assumed that the rainfall occurring during the time of concentration is directly related to flow rate.
The time of concentration (£) is often considered to be the sum of the time of travel to an inlet plus the time of travel in the stormwater conveyance system. In the design of stormwater drainage systems, 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. A number of methods, mostly using empirical equations, are provided below for estimating the time of concentration for urban catchments.
14.4.1 Components of Flow Time
Depending on the particular location, the calculation of t_{c }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 drain system 
• 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 street gutter • pipe flow within lots including roof drainage, car parks, etc • street drainage pipe flow 
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14.4.2 Calculation of Flow Time
This section gives procedures for the calculation for each of the flow components in Table 14.1. Some of these procedures are subject to uncertainty; the designer must therefore check that the results are physically meaningful.
For very small, simple catchments, it is acceptable to adopt the simplified assumptions instead of performing detailed calculations.
(a) Overland Flow Time
Overland flow can occur on either grassed or paved surfaces. The major factors affecting time of concentration for overland flow are the maximum flow distance, surface slope, surface roughness, rainfall intensity, and infiltration rate.
Overland flow over unpaved surfaces initially occurs as sheet flow for a short time and distance after which 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 flow will become concentrated against fence, paths or structures or intercepted by open drains.
The formula shown below, known as Friend's formula, should be used to estimate overland sheet flow times. The formula was derived from previous work (Friend, 1954) in the form of a nomograph (Design Chart 14.1) for shallow sheet flow over a plane surface.
107. n.L^{1}'^{3}
where,
t_{0} = overland sheet flowravel time (minutes)
L = overland sheet flovpath length (m)
n = Manning's roughness value for the surface
5 = slope of overland surface (%)
Note: Values for Manning's 'n' are given in Table 14.2.
Some texts recommend an alternative equation, the Kinematic Wave Equation. However this theoretical equation is only valid for uniform planar homogeneous flow. It is not recommended for practical application.
(b) Overland Flow Time over Multiple Segments
Where the characteristics of segments of a subcatchment are different in terms of land cover or surface slope, the subcatchment should be divided into these segments, and the calculated travel times for each combined.
Table 14.2 Values of Manning's 'n' for Overland Flow
Surface Type 
Manning n 

Recommended 
Range 

Concrete/Asphalt** 
0.011 
0.010.013 
Bare Sand** 
0.01 
0.010.06 
Bare ClayLoam** (eroded) 
0.02 
0.0120.033 
Gravelled Surface** 
0.02 
0.0120.03 
Packed Clay** 
0.03 
0.020.04 
Short Grass** 
0.15 
0.100.20 
Light Turf* 
0.20 
0.150.25 
Lawns* 
0.25 
0.200.30 
Dense Turf* 
0.35 
0.300.40 
Pasture* 
0.35 
0.300.40 
Dense Shrubbery and Forest Litter* 
0.40 
0.350.50 
* From Crawford and Linsley (1966)  obtained by
calibration of Stanford Watershed Model.
** From Engman (1986) by Kinematic wave and storage analysis of measured rainfall runoff data.
However, it is incorrect to simply add the values of t_{0} for 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 from upstream.
Utilising Equation 14.1, the following method (Australian Rainfall & Runoff, 1998) for estimating the total overland flow travel time for segments in series is recommended. For two segments, termed A and B (Figure 14.1):
trotai = ta(l_{a}) + ^B(i_{A}+i_{B}) ~ t'b(La) (14.2a)
where,
L_{A} = length of flow for Segment A L_{B} = length of flow for Segment B
^a(la) = time of flow calculated for Segment A over
length Z_{A }t_{B(}j =time for Segment B over the lengths indicated
For each additional segment, the following time value should be added:
tadd = '/(W ~ tfU_{To}*/ ~ Li) d4.2b)
where,
t_{add} = time increment for additional segment
lTotai = total length of flow, including the current segment /
Lj = length of flow for segment /
£,(...) = time for the segment/over the lengths indicated
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Segment A
Segment B
Length ^
Travel Time
<r
<r
^{L}A(LA)
M
^{L}B(LA+LB)
^
<
^{L}B(LA)
M
Figure 14.1 Overland Flow over Multiple Segments
This procedure must be applied iteratively because the travel time is itself a function of rainfall intensity.
(c) Roof Drainage Flow Time
While considerable uncertainty exists in relation to flow travel time on roofs, the time of flow in a lot drainage system to the street drain, or rear of lot drainage system is generally very small for residential lots and may be adopted as the minimum time of 5 minutes (Chapter 23). However, for larger residential, 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.
(d) Kerbed Gutter Flow Time
The velocity of water flowing in kerbed gutters is affected by:
• the roughness of the kerb, gutter and paved surface
• the crossfall of the pavement
• the longitudinal grade of the kerbed gutter
• the flow carried in the kerbed gutter
The flow normally varies along the length of a kerbed gutter due to lateral surface inflows. Therefore, the flow velocity will also vary along the length of a gutter. As the amount of gutter flow is not known for the initial analysis of a subcatchment, the flow velocity and hence the flow time cannot be calculated directly. An initial assessment of the kerbed gutter flow time must be made.
An approximate kerbed gutter flow time can be estimated from Design Chart 14.2 or by the following empirical equation:
(14.3)
where,
t_{g} = kerbed gutter flow time (minutes)
L = length of kerbed gutter flow (m)
5 = longitudinal grade of the kerbed gutter (%)
Equation 14.3 should only be used for L < 100 metres. Kerbed 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 flow time result in only small errors in the time of concentration for a catchment, and hence high accuracy is not required.
(e) Channel Flow Time
The time stormwater takes to flow along a open channel may be determined by dividing the length of the channel by the average velocity of the flow. The average velocity of the flow is calculated using the hydraulic characteristics of the open channel.
The Manning's Equation is recommended for this purpose:
V= — R n
From which.
2/3 _{c}l/2
^{l}ch
^{n}'^{L} R^{2}'^{3} S^{1}'^{2 }60
(14.4a)
(14.4b)
40 Vs
where,
V = average velocity (m/s)
n = Manning's roughness coefficient
R = hydraulic radius (m)
5 = friction slope (m/m)
L = length of reach (m)
t_{ch} = travel time in the channel (minutes)
Where an open channel has varying roughness or depth across its width it may be necessary to sectorise the flow and determine the average velocity of the flow, to determine the flow time.
(f) Pipe Flow Time
The velocity 1/in a pipe running just full can be estimated from pipe flow charts such as those in Chapter 25, Appendix 25.B where the flow, pipe diameter, roughness and pipe slope are known. The time of flow through pipe, t_{p}, is then given by:
t_{p}=y (14.5)
where,
L = pipe length (m)
V = average pipe velocity (m/s)
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Where the pipe diameter is not known, the diameter can be first estimated given the flow at the upstream 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 time of concentration for a subcatchment. The error in the estimated pipe flow time introduced by the adoption of the wrong diameter or slope, or by the assumption that the pipe is flowing full when in fact 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 a reasonable accuracy. Therefore, the pipe flow time can be estimated directly from Equation 14.5.
14.4.3 Time of Concentration for Natural Catchment
For larger systems times of concentration should preferably be estimated on the basis of locally observed data such as the time of occurrence of flood peaks at or near the catchment outlet compared with the time of commencement of associated storms.
For natural/landscaped catchments and mixed flow paths the time of concentration can be found by use of the BransbyWilliams' Equation 14.6 (AR&R, 1987). In these cases the times for overland flow and channel or stream flow are included in the time calculated.
Here the overland flow time including the travel time in natural channels is expressed as:
where,
t_{c} = the time of concentration (minute)
F_{c} = a conversion factor, 58.5 when area A is in km^{2}, or 92.5 when area is in ha
L = length of flow path from catchment divide to outlet (km)
A = catchment area (km^{2} or ha)
5 = slope of stream flovpath (m/km)
14.4.4 Time of Concentration for Small Catchments
Although travel time from individual elements of a system may be very short, the total nominal flow travel 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 minimum times of concentration given in Table 14.3 instead of performing detailed calculation.
Table 14.3 Minimum Times of Concentration
Drainage Element 
Minimum £_{c} (minutes) 
Roof and property drainage 
5 
Road inlet 
5 
Small areas < 0.4 hectare 
10 
Note: the recommended minimum times are based on the minimum duration for which meaningful rain intensity data are available.
14.5 RATIONAL METHOD
There are two basic approaches to computing stormwater flows from rainfall. The first approach is the Rational Method, which relates peak runoff to rainfall intensity through a proportionality factor. The second approach starts with a rainfall hyetograph, accounts for rainfall losses and temporary storage effects in transit, and yields a discharge hydrograph. The hydrograph approach is discussed later in this Chapter.
14.5.1 Rational Formula
The Rational Formula is one of the most frequently used urban hydrology methods in Malaysia. It gives satisfactory results for small catchments only.
The formula is:
r ^{y}T A
_{=} L. l_{t}.A
^{y} 360
where,
Q_{y} = /year ARI peak flow (m^{3}/s)
C = dimensionless runoff coefficient
^{y}h = y year ARI average rainfall intensity over time of concentration, t_{Cl} (mm/hr)
A = drainage area (ha)
Traditionally, design discharges for street inlets and stormwater drains have been computed using the Rational Method, although hydrograph methods also can be used for these purposes. The primary attraction of the Rational Method has been its simplicity. However, now that computerised procedures for hydrograph generation are readily available, computational simplicity is no longer a primary consideration.
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Experience has shown that the Rational Method can provide satisfactory estimates of peak discharge on small catchments of up to 80 hectares. For larger catchments, storage and timing effects become significant, and a hydrograph method is needed. Various methods have been devised to form pseudohydrographs based on the Rational Method, but their reliability is uncertain and they should only be used for the design of onsite stormwater detention and retention facilities.
14.5.2 Analysis Procedure
A procedure for estimating a peak flow from a single subcatchment for a particular ARI using the Rational Method is outlined in Figure 14.2. Peak flow estimates should be obtained for both the minor and major drainage systems. An example of peak flow estimation by the Rational Method for multisubcatchments is given in Table 16.A9 in Chapter 16.
14.5.3 Assumptions
Assumptions used 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, t_{0}.
4. The ARI of the computed peak flow is the same as that of the rainfall intensity, i.e., a 5 year ARI rainfall intensity will produce a 5 year ARI peak flow.
Experience has shown that when applied properly, the Rational Method can provide satisfactory estimates for peak discharges on small catchments where storage effects are insignificant. The Rational Method is not recommended for any catchment where:
• the catchment area is greater than 80 hectares
• 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, 7, in the Rational Formula represents the average rainfall intensity over a duration 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 other hydrologic abstractions. The runoff coefficient accounts for the
integrated effects of rainfall interception, infiltration, depression storage, and temporary storage in transit of the peak rate of runoff. When estimating a value for the runoff coefficient, the roles played by these hydrologic processes should be considered. The runoff coefficient depends on rainfall intensity and duration as well as on the catchment characteristics. During a rainstorm the actual runoff coefficient increases as the soil become saturated. The greater the rainfall intensity, the lesser the relative effect of rainfall losses on the peak discharge, and therefore the greater the runoff coefficient.
Recommended runoff coefficient (C) values for rainfall intensities (I) of up to 200 mm/hr may be obtained from Design Chart 14.3 (urban areas) or 14.4 (rural areas), respectively. These design charts are based on Australian Rainfall & Runoff (1977). For 7> 400 mm/hr, a value of C = 0.9 should be used for all types of ground cover. For 7 values between 200 and 400 mm/hr, use linear interpolation between the applicable C values for 7=200 mm/hr and 7=400 mm/hr.
Design flow rates for stormwater inlets are calculated for local contributing subcatchments, while those for pipes and open drains are calculated for the accumulated areas draining through each pipe or open channel section or reach. Except for small lot drainage systems, it is inappropriate to simply add the separate flows from each subcatchment. This overestimates flow rates. When timestopeak differ, the total flow from a number of subcatchments will be less than the sum of the separate flows from each subcatchment.
The recommended procedure is to calculate the flow at each point along the drainage line from the Rational Formula, using average rainfall intensity and runoff coefficient values corresponding to the times of concentration at that point.
14.5.6 Variation of Subcatchment Conditions
Segments of different landuse or surface slope within a subcatchment can be combined to produce an average runoff coefficient. For example, if a subcatchment consists of segments with different landuse or surface
slope denoted by / = 1, 2,......, m; the average runoff
coefficient is:
m
C_{am}=^ (14.8)
avg _{m} \ i
;=1
where,
C_{avg}= average runoff coefficient
Q = runoff coefficient of segment /
A, = area of segment/(ha)
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Select design ARI
j
Discretise subcatchment
Estimate time of concentration, t_{c}
J 1
Determine average rainfall intensity, y/_{t}
J
Estimate runoff coefficients
Calculate average runoff coefficient
Calculate peak flow rate Q for the subcatchment
J
select design ARI for both minor and major drainage systems
divide subcatchment into segments of homogeneous land use or surface slope
estimate overland flow time estimate flow times for all other flow components within the subcatchment such as kerb gutters, pipe, and channels, etc.
calculate ^for design ARI of /years and duration t equal to the time of concentration, from IDF data for area of interest
estimate C values for each segment if there are different land covers
use Equation 14.8
calculate peak flow rate from Equation 14.7
Figure 14.2 General Procedure for Estimating Peak Flow for a Single Subcatchment Using the Rational Method
14.5.7 The Partial Area Effect
In general, the appropriate t_{c} for calculation of the peak flow at any point in a catchment is the longest time of flow to that point. However, in some situations, the peak flow may occur when only part of the upstream catchment is contributing, i.e. the product of a lesser C.A and a higher
intensity ^{Y}I_{t} (resulting from a lower t_{c}), 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 subcatchment of relatively small C.A, but a considerably longer than average t_{c}. This can result from differences in
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the shapes and/or surface slopes of subcatchments within a catchment. Typical catchments that can produce partial area effects are shown in Figure 14.3.
It is important to note that particular subcatchments may not produce partial area effects when considered individually, but when combined at some downstream point with other subcatchments, the peak discharge may result when only parts of these subcatchments 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 Rational Method cannot be used to calculate the volume or shape of the runoff hydrograph, which is required for the design of facilities that use storage such as detention and retention basins.
14.6 HYDROGRAPH METHODS
14.6.1 Basic Concepts
This section discusses methods that should be used to develop a design hydrograph. Hydrograph methods must be used whenever rainfall spatial and temporal variations or flow routing/storage effects 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 drainage systems 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 a given ARI, and therefore affects the resulting runoff peak. The design storm duration that produces the maximum runoff peak traditionally is defined as the time of concentration. Design storm information is provided in Chapter 13.
The rainfall intensity, ^{v}I_{t}, used in hydrograph methods represents the average rainfall intensity over a particular duration fwith an average recurrence interval (ARI) of /.
Intense rainfalls of short durations usually occur within longerduration storms rather than as isolated events. These are reflected in the temporal patterns used. It is common practice to compute discharge for several design storms with different durations, and then base the design on the "critical" storm, which produces the maximum discharge. However the "critical" storm duration determined in this way may not be the most critical for storage design.
Recommended practice is to compute the design flood hydrograph for several storms with different durations equal to or longer than the time of concentration for the catchment, and to use the one, which 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 catchment and the shorter the rainfall duration, the less uniformly the rainfall is distributed over the catchment. For any specified ARI and duration, the average rainfall depth over an area is less than the point rainfall depth.
The ratio of the areal average rainfall with a specified duration and ARI to the point rainfall with the same duration and ARI is termed the areal reduction factor, and is discussed in Section 13.2.4 of Chapter 13.
(a) Nonurban area upstream
Partial area
(b) Narrow nonurban area upstream
Figure 14.3 Urban Catchments Likely to Exhibit Partial Area Effects
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Spatial distributions can be represented in hydrograph methods. However for areas of less than 10 km^{2} in urban drainage systems the areal reduction factor can be neglected. It is also common practice to neglect any effects due to storm movement direction.
(c) Temporal Distribution
Commonly used approaches to distribute rainfall within a design storm were discussed in Chapter 13. The temporal distribution adopted can have a significant effect on the shape of the runoff 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 as evaporation, infiltration and depression storage. The remaining rainfall after subtracting rainfall losses is called the rainfall excess.
In the Rational Method, described in the previous section, the concept of rainfall excess is not used directly. Instead the runoff coefficient allows for rainfall losses. Rainfall excess concepts are used in 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 soil surface, and storage of water in surface depressions are commonly termed rainfall abstractions. Although these three processes are physically complex, simplified modelling procedures have been found give acceptable results for urban stormwater drainage.
Values can be derived by analysing observed rainfall and runoff data. Since individual values are dependent 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 average value is usually needed, and since there is no reason for expecting loss rate values for a catchment to conform to any particular distribution, the median of the derived values is probably the most appropriate 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. Impervious areas 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 pass over 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 shortduration storms of concern in stormwater drainage design.
(ii) Impervious Area Losses
Impervious surfaces cause small rainfall losses due mainly to depression storage. Recommended values are given in Table 14.4.
(iii) Pervious Area Infiltration Loss Models
The predominant form of loss on pervious surfaces is by infiltration. Some of the most frequently used types of loss models are illustrated in Figure 14.4. These five types of loss models are described as follows:
1. loss (and hence runoff) is a constant fraction of rainfall in each time period : this is similar to the Rational Method runoff coefficient concept.
2. constant loss rate : where the rainfall excess is the residual left after a selected constant rate of infiltration capacity is satisfied.
3. initial loss and continuing loss, which is similar to constant loss rate except that no runoff is assumed to occur until a given initial loss capacity has been satisfied, regardless of the rainfall rate. The continuing loss is at a constant rate. A variation of this model is to have an initial loss followed by a loss consisting of a constant fraction of the rainfall in the remaining time periods (the initial lossproportional loss model).
4. infiltration curve or equation : representing capacity rates of loss decreasing (usually exponentially) with time. The Horton Loss Model is of this type.
5. standard rainfallrunoff relation : such as the U.S. Soil Conservation Service relation.
It should be noted that loss values derived according to one of the models are not directly transferable to other models. The choice of loss model therefore depends in part on the choice of flow estimation method. In most urban stormwater drainage applications this is not a serious problem as the losses are generally only small in comparison to rainfall, and therefore a high degree of 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, individual values have little meaning except as indicators of those particular events. For design, an average value is usually needed, and since there is no reason for expecting loss rate values for a catchment to conform to any particular distribution, the median of the derived values is probably the most appropriate for design.
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□ Rainfall □ Loss
Time Time
(a) Constant Fraction (b) Constant Loss Rate
Time lime
(c) Initial LossLoss Rate (d) Infiltration curve
(e) U.S. SCS relations
Figure 14.4 Typical Loss Models for Estimating Rainfall Excess
(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 Conservation Service approach requires further research for Malaysian urban drainage application. Models (c) or (d) are recommended for use in Malaysian urban drainage situations. Table 14.4 contains recommended loss values for use by drainage designers.
If a subcatchment contains areas of different surface condition or landuse, weighted average values of losses for different conditions or landuses, such as proportions of pervious and impervious areas, can be derived using methods similar to those in Section 14.5.6.
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Horton's equation is widely used for describing infiltration capacity in a soil. It describes the decrease in capacity as more water is absorbed by the soil, and has the form:
f = f_{c}+{f_{Q}f_{c}).e^{u} (14.9)
where,
f = the infiltration capacity (mm/hr) at time t
f_{0}= the initial rate of infiltration (mm/hr)
f_{c} = the final constant rate of infiltration (mm/hr)
k= a shape factor (per hr)
t= the time from the start of rainfall (hr).
Recommended parameter values for the Horton's equation are given in Table 14.4. The GreenAmpt equation can also be used. (Chow, 1988)
14.6.2 TimeArea Method
Timearea methods utilise a convolution of the rainfall excess hyetograph with a timearea diagram representing the progressive area contributions within a catchment in set time increments. Separate hydrographs are generated for the impervious and pervious surfaces within the catchment. These are combined to estimate the total flow inputs to individual subcatchment entries to the urban drain network.
The timearea method dates from the research of Ross in 1922. Networked urban drainage adoptions of the procedure however only date back to 1963. This computerised program known as the TRRL Method was developed by the UK Transport and Road Research Laboratory (TRRL), described by Watkins (1963). In the U.S., Terstriep and Stall (1974) further developed the method to include pervious runoff. In South Africa Watson (1981) made a number of additional changes particularly to the way infiltration was estimated. Between 1982 and 1986 Watson's model was used through extensive changes, to formulate a computerised package known as ILSAX (O'Loughlin, 1988). The subcatchment runoff estimating procedure still utilises the basic timearea method to estimate both pervious and impervious portion of runoff.
This method assumes that the outflow hydrograph for any storm is characterised by separable subcatchment translation and storage effects. Pure translation of the direct runoff to the outlet via the drainage network is described using the channel travel time, resulting in an outflow hydrograph that ignores catchment storage effects.
To apply the method, the catchment is first divided into a number of time zones separated by isochrones or lines of equal travel time to the outlet (Figure 14.5b). The areas
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between isochrones are then determined and plotted against the travel time as shown in Figure 14.5c.
The translated inflow hydrograph ordinates q, for any selected design hyetograph (Figure 14.5d) can now be determined. Each block of storm in Figure 14.5a should be applied (after deducting losses) to the entire catchment; the runoff from each subarea reaches the outflow at lagged intervals defined by the timearea histogram. The simultaneous arrival of the runoff from areas A_{t}, A_{2},...for storms It, I_{2},...should be determined by properly lagging and adding contributions, or generally:
Qi ^{=I}i^{A}i
I_{i}__{1}.A_{2}
I,.A,
(14.10)
where,
g, = the flow hydrograph ordinates (m^{3}/s)
It = excess rainfall hyetograph ordinates (mm/hr)
A, = timearea histogram ordinates (ha)
/ = number of isochrone area contributing to the outlet
For example, the runoff from storms Ij on A_{3l} I_{2} on A_{2} and Is on At arrive at the outlet simultaneously, and q_{3} is the total flow. The total inflow hydrograph (Figure 14.5d) at the outlet can be obtained from Equation 14.10.
Table 14.4 Recommended Loss Models and Values for Hydrograph
Condition 
Loss Model 
Recommended Values 

Impervious 
Initial lossLoss rate 
Initial loss: 1.5 mm 
Loss rate: 0 mm/hr 
Areas 

Pervious 
Initial loss  
Initial loss: 10 mm 
Proportional Loss: 
Areas 
proportional loss, or 
20% of rainfall 

Initial lossLoss rate, 
Initial loss: 10 mm for all soils 
Loss rate: 

or 
(i) Sandy open structured soil 
10  25 mm/hr 

(ii) Loam soil 
3  10 mm/hr 

(iii) Clays, dense structured soil 
0.5  3 mm/hr 

(iv) Clays subject to high shrinkage and in a 
4  6 mm/hr 

cracked state at start of rain 

Horton model 
Initial Infiltration Capacity f_{n} 
Ultimate Infiltration 

A. DRY soils (little or no vegetation) 
Rate f_{c} (mm/hr), for Hydrologic Soil 

Sandy soils: 125 mm/hr 
Group (see Note) 

Loam soils: 75 mm/hr Clay soils: 25 mm/hr 
A 107.5 B 7.53.8 

For dense vegetation, multiply values given in A by 2 
C 3.81.3 

B. MOIST soils 
D 1.30 

Soils which have drained but not dried out: divide 

values from A by 3 

Soils close to saturation: value close to saturated 

hydraulic conductivity 

Soils partially dried out: divide values from A by 1.52.5 

Recommended value of Aris 4/hr 
Note: Hydrological Soil Group corresponds to the classification given by the U.S. Soil Conservation Service. Well drained sandy soils are "A"; poorly drained clayey soils are "D". The texture of the layer of least hydraulic conductivity in the soil profile should be considered. Caution should be used in applying values from the above table to sandy soils (Group A). Source: XPSWMM Manual (WPSoftware, 1995).
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At <> 

(V 

h 
h 

c 2 
h 
k 
Area A
0 At 2At 3At 4At Time t
4At 

Isochrones 
\ 

/ \ 
......._{?} .....,. 
A, 
^^_ o "H 
•*4f 

\* ^{v} J 
\ 

^* \ 
xs 

s ^{A}l 
^ 
(a) Rainfall Histogram
(b) Catchment Isochrones
u
At 2At 3At 4At
Time t
(c) TimeArea Curve
o c 
/ / 111 y ^ i 
q_{2} i 
\ 
<^{3 }\ <—> \ ^{5} 
Time t
(d) Runoff Hydrograph
Figure 14.5 TimeArea Method
14.6.3 Kinematic Wave Method
The kinematicwave method is a hydraulic routing method for routing runoff across planar surfaces and through small channels and pipes. The kinematicwave formulation couples the continuity equation with a simplified form of the momentum equation that includes only the bottomslope and frictionslope terms (Chapter 12). Kinematicwave theory is only valid for onedimensional overland flow on a planar surface. It has little or no practical use in urban drainage applications.
14.6.4 Nonlinear Reservoir Method
In the nonlinear reservoir method, the catchment is conceptualised as a very shallow reservoir. The discharge from this hypothetical reservoir is assumed to be a nonlinear function of the depth of water in the reservoir. Computer models such as RAFTS and SWMM can optionally use a nonlinear reservoir approach to compute surface runoff hydrographs. The SWMM procedure is described here (Huber and Dickinson, 1988).
Figure 14.6 shows the catchment conceptualised as a reservoir with rainfall as inflow, and infiltration and surface discharge as outflows. The depth / represents the average depth of surface runoff, and the depth y_{d} represents the average depression storage in the catchment. The continuity relationship for this system is:
dy_ dt
A (lf)Q
(14.11)
where,
A = catchment area
/ = 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 difference between / and y_{d} . Based on the Manning friction relationship, the catchment discharge, Q is given by:
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Q = ^{yy_{d})%s%
(14.12)
where,
1/1/ = a representative width for the catchment
n = Manning roughness coefficient for the catchment
y_{d} = average depth of depression storage
5 = average surface slope
Substituting Equation 14.12 into Equation 14.11 yields a nonlinear differential equation for /. A simple finite difference form of the equation is used to solve for the depth /at the end of each time step. This equation is:
/2/1
At
_ cws^{/2}
An
/l+/2
Yd
(14.13)
where, At = /i =
/2 =
/ =
f =
time step increment depth at the beginning of the time step depth at the end of the time step average rainfall rate over the time step average infiltration rate over the time step
I
Rainfall rate
I
I
T7777777777777T
Discharge Q
Yd
Figure 14.6 Definition Sketch for Nonlinear Reservoir Model
For each time step, three separate calculations are performed. First, an infiltration equation is used to compute the average potential infiltration rate over the time step, then Equation 14.13 is solved iteratively for y_{2}, and, finally, Equation 14.12 yields the corresponding discharge.
Unlike the timearea method, which uses excess rainfall as input, the nonlinearreservoir method couples the processes of infiltration and surface runoff. The nonlinearreservoir model assumes that infiltration occurs at the potential rate over the entire surface area whenever the ponded depth is nonzero. The excessrainfall models, on the other hand, entirely neglect infiltration of ponded water. This difference becomes important following cessation of rainfall, or whenever the rainfall intensity drops below the potential infiltration rate. In reality, infiltration does continue for some time after rainfall ceases, but the area over which infiltration continues to occurs after rainfall ceases would tend to be underestimated by the nonlinear reservoir methods and
overestimated by the excessrainfall methods, though the difference will depend on the degree of discretisation used, since a more detailed schematisation can partially account for the phenomenon of decreasing area of infiltration.
14.6.5 Rational Method Hydrograph Method
This procedure extends the Rational Method to the development of runoff hydrographs. This method is only recommended for the sizing of small onsite detention and retention storages, and must NOT be used under any circumstances for the sizing of community or regional detention and retention facilities.
The RMHM relies on the assumptions of the Rational Method. Therefore, the RMHM is subject to the same misunderstandings and misapplications as the Rational Method. An important additional assumption is that the rainfall intensity averaging time used in the RMHM equals the storm duration. As introduced in the preceding discussion of the Rational Method, this assumption means that the rainfall, and the runoff generated by the rainfall that occurs before or after the rainfall averaging period are not accounted for. Accordingly, the RMHM may underestimate required storage volume. However, for the design of small onsite facilities, this underestimation will not generally be significant enough to warrant using one of the more complex hydrograph methods.
The RMHM also is based on the assumption that the runoff hydrograph can be approximated as being either triangular or trapezoidal in shape. This is equivalent to assuming that the runoff from the onsite catchment area increases linearly with time or that there is a linear areatime relationship for the subcatchment.
RMHM hydrograph shape assumptions may be explained using the subcatchment illustrated in Figure 14.7. The subcatchment has an area of 0.1 hectares, a runoff coefficient C of 0.4 and a time of concentration t_{c} of 10 minutes. As illustrated in Figure 14.8, three types of hydrographs may be developed for the subcatchment using the RMHM procedure. Hydrograph type is a function of the length of the rainfall averaging time, d, with respect to the subcatchment time of concentration. The following three types are possible:
Type 1 : d is greater than t_{c}. The resulting trapezoidal hydrograph has a uniform maximum discharge of Ci_{T}A, as determined from the Rational Method. The linear rising and falling limbs each has a duration of t_{c}.
Type 2 : d is equal to t_{c} . The resulting triangular hydrograph has a peak discharge of Ci_{T}A. The linear rising and falling limbs each have a duration of t_{c}.
Type 3 : d is less than t_{c}. The resulting trapezoidal hydrograph has a uniform maximum discharge of Ci_{T}A times (d/t_{c}). The linear rising and falling portions of the hydrograph each have a duration of d
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C=0.4
t_{c} = 10 minutes
Potential Pond Site
Subbasin
Existing Development
Discharge Point
Figure 14.7 Sub catchment and Physical Parameters Needed to Apply the RMHM
^{Jl} d 
d>^{f} 

tc 1 
tc . 
^{L} C 
/ ^{Ci}T^{A} / 1 
Time (a) Type 1
 ^{d} I d=t_{c}
, tc , , tc , / C\_{T}A \
Time (b) Type 2
d<t_{r}
T
Ci_{T}A (d/t_{c})
Time (c) Type 3
In summary, hydrograph type in the RMHM is determined by the relationship between rainfall averaging time and the time of concentration of the subcatchment. Given the hydrograph type, the peak discharge is determined using the Rational Method formula.
14.6.6 Generalised Analysis Procedure
A procedure for estimating a runoff hydrograph from a single subcatchment for a particular ARI and duration is outlined in Figure 14.9. Hydrographs should be obtained for both the minor and major drainage systems.
14.7 FLOW ROUTING AND ATTENUATION PRINCIPLES
Flow routing is the process of converting a hydrograph that passes through some part of a flow system to allow for the changes that occur during its passage. There are three main types of flow routing:
• catchment routing, which converts a rainfall excess hyetograph into a hydrograph at the catchment outlet, allowing for the distribution of rainfalls over the catchment surface, and various lags or delays along flow paths;
• channel routing, which allows for the changes in hydrographs as they flow along river or channel reaches, caused by variations in the channel geometry which result in storage effects; and
• reservoir routing, which allows for storage effects in a concentrated, "level pool" reservoir.
Figure 14.8 Hydrograph Types for the RMHM
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Select hydrograph model or method
I
J
Determine average rainfall intensity
select model or method appropriate for
the purpose of the design, e.g. subcatchment
flow, storage facility design, etc.
select design ARI for both minor and major drainage systems
estimate pervious and impervious areas for the subcatchment
initially select a duration that is less than the estimated time of concentration for the subcatchment
select temporal pattern applicable to the selected design ARI and duration of rainfall
calculate ^{f}7 for design ARI of /years and duration ffrom IDF data for site
Select rainfall loss model and determine design parameters
Obtain flow hydrograph using selected model or method
Yes
)
Select hydrograph that produces the maximum peak flow rate or storage volume
J
determine average rainfall intensity for the selected ARI and duration from the IDF data for the area of interest
determine rainfall excess by subtracting rainfall losses from design rainfall hyetograph
a range of hydrographs with different durations will need to be developed to determine the maximum peak flow rate, or in the case of a storage facility, the hydrograph that produces the maximum storage
Figure 14.9 General Procedure for Estimating a Hydrograph for a Single Subcatchment
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Broadly defined, flow routing is an analytical procedure intended to trace the flow of water through a hydrological system, pond, conveyance, or porous media, given some runoff event hydrograph as input. 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 a flood, then the procedure is specifically known as flood routing. Routing by lumped system methods is called hydrological routing. These methods calculate the flow as a function of time alone. Routing by distributed system methods is called hydraulic routing, and the flow is calculated as a function of both space and time throughout the system.
For hydrologic routing the inflow I(t), outflow O(t), and storage S(t) are related by the continuity equation:
dS_ dt
l(t)0(t)
(14.14)
the change in storage over the j time interval jAt,
Sj_{+1}  Sj, which can be rewritten as:
^{S}M ~ ^{s}j
<7 ^{+} ^{I}M
At
^{{}0_{J+}0_{J+}^
At (14.15)
The inflow values at the beginning and end of the y'^{th} time interval are Ij and I_{j+ll} respectively, and corresponding
outflow values are <3_{7}and O_{j+1}.
The values of 7_{7} and I_{j+1},
are known because they are prespecified (i.e. the inflow hydrograph ordinates). The values Oj and Sj are known at the y'^{th} time interval. Hence Equation 14.15 contains two unknowns, O_{j+1} and S_{j+ll} which are isolated by multiplying Equation 14.15 by 2/L\t and rearranging the result to produce:
25
7+1
At
O
7+1
(^7+^{/}7+l)
At
O,
(14.16)
If an inflow 1(f) is known Equation 14.14 cannot be solved directly to obtain the outflow O(t), because both O and 5 are unknown. A second relationship, the storage function, is required to relate /, 5, and O. Coupling the continuity equation with the storage function provides a solvable combination of two equations and two unknowns.
The specific form of the storage functions to be employed in hydrologic routing depends on the nature of the system being analysed. In reservoir routing by the levelpool method (see Section 14.8.1), storage is a nonlinear function of Oonly, i.e. 5 = f(0), and the function, f(0), is determined by relating reservoir storage and outflow to reservoir water level. In channel routing by the Muskingum method (see Section 14.10.1), storage is linearly related to / and O. Similarly, in porous media, storage is a function of outflow, which depends on storage in the media and the underlying soils.
14.8 FLOW THROUGH POND AND RESERVOIR
14.8.1 Hydrologic Routing
Levelpool routing is a procedure for calculating the outflow hydrograph from a pond reservoir, assuming a horizontal water surface, given its inflow hydrograph and storagedischarge characteristics. When a reservoir has a horizontal water surface, its storage is a function of its watersurface elevation, or depth in the pool. Likewise, the discharge is a function of the water surface elevation, or head on the outlet works. Combining these two functions yields the invariable singlevalued function.
Integration of the continuity equation (Equation 14.14) over the discrete time intervals provides an expression for
In order to calculate the outflow O_{j+1} from Equation 14.16, a storagedischarge function relating 2S/At +0 and O is needed. The method of developing this function using stagestorage and stagedischarge relationship is shown in Figure 14.10.
The relationship between watersurface elevation and reservoir storage can be obtained using topographic maps or from field surveys. The stagedischarge relationship is derived from hydraulic equations relating head and discharge for various types of spillway and outlet works. The value of At is the same as the time interval of the inflow hydrograph.
For a given watersurface elevation, the values of storage 5 and discharge O are determined. Then, the value of 2S/At+0 is calculated and plotted against O. In routing the flow through the j^{th} time interval, all terms in the righthandside of Equation 14.16 are known, and so the value of 2Sj+i/At+0 can be computed. The corresponding value of Oj_{+1} can be determined from the storagedischarge function 2S/At+0 versus O. To set up the data for the next time interval, the value 2Sj_{+1}/AtOj_{+1} is calculated by:
25
7+1
At
■°M^r^{+0}^
■20_{M}
(14.17)
The computation is repeated for subsequent routing periods. Input requirements for this routing method are:
the storagedischarge relationship
the storageindication relationship
the inflow hydrograph
initial values of the outflow rate (O_{x}) and storage (50
the routing interval (A;)
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An analysis procedure for hydrologic routing is shown in Figure 14.11.
14.8.2 TwoDimensional Hydraulic Routing
Twodimensional hydraulic routing is especially useful for pond and lake design that involves structures and water quality. The governing equations of a depthaveraged twodimensional model for a wellmixed shallow pond or lake are derived from the principles of mass and momentum conservation (Chapter 12). In the case of mass conservation, the net mass flux through a control volume is balanced by a change of fluid density. In
conservation of momentum, a momentum flux balance is performed on a control volume. Details of the derivation can be found in textbooks (Ippen, 1966).
With the assumption that vertical accelerations and velocities are negligible compared to horizontal ones, and lake water is completely homogeneous, incompressible, and viscous, the continuity and momentum equations for depthaveraged two dimensional flow are given as:
Continuity Equation
d^ duH dvH
dt dx
dy
(14.18)
Inflow (/)
Volume = {I+I )At/2
Outflow (0)
Volume = (0+0_{M})At/2
Storage AS
AS= S(t_{]+1})  S(L)
Inflow (I)
Outflow (OJ
SECTION AA
Figure 14.10 Development of the StorageDischarge Function for Hydrologic Pond Routing
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Select hydrograph model or method
Calculate next time step
I
J
Calculate next value of outflow hydrograph
Yes
Select hydrograph that produces the maximum peak flow rate or storage volume
select model or method appropriate for the purpose of the design e.g. storage routing, facility design, flood routing etc.
select design ARI for both minor
and major drainage systems from Table 4.1
At should be the same as the time step of the inflow hydrograph
Equation 14.18
a range of hydrographs with different durations will need to be developed to determine the hydrograph that produces the maximum storage
Figure 14.11 General Analysis Procedure for Hydrologic Routing
Momentum Equations x  direction:
du du du dC 1
—+U—+V—+g— +
dt dx dy dx pH_{x}
^{T}bx
1
7h_{x}
dhh_{vv} dhh.
dx
xy
:0
dy (14.19a)
y  direction:
dv dv — + u
dt dx
dv
where,
H = h+ <;
d^
dy dy _{P}H
'^{T}by~
PHy
dHT„, dHz„
xy
dy
dy (14.19b)
0
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h = still water depth
£ = free surface displacement
T_{bx} and r_{by} are bottom shear stresses in which it is
generally assumed that:
^{1} bx
^{T}by
pgn^{2} 
u\u^{2} 
+ v^{2} 
_{H}m " X 

pgn^{2} 
v~\u^{2} 
+ v^{2} 
H
1/3
(14.19c)
(14.19d)
where,
u and v are depthaveraged velocity components and n is the Manning roughness coefficient
For flows in small and medium size waterbodies such as detention ponds, the effects of wind and earth rotation are insignificant compared to the driving forces. The equations are normally solved by finite differences or finite elements methods. Their nonlinearity is handled by the Newton Raphson technique. The flow equations provide convective velocity effects to solving the transport equation for pond water quality analysis and design (see Section 15.6.5).
14.9 FLOW THROUGH POROUS MEDIA
14.9.1 Routing through Infiltration Facility
The constant infiltration model provides a method of analysis and evaluating the hydraulics of soakage pits. This model can be used to calculate the maximum water level occurring in a given infiltration system during a design storm event, thus allowing the required depth of a system to be calculated. This section introduces equations for a maximum water level for both plane and threedimensional infiltration drainage systems.
Applying the flow balance approach (Equation 14.15) to infiltration drainage facilities where,
S = S(h), the volume of water stored in the infiltration system
Q_{jn} = I (t), the inflow of runoff
Qout= O (t), the outflow through infiltration
and,
h = water depth above the bed level of 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.
where,
V = V(h) , the volume of the water filled part of the facility and is a function of the geometry of the system
n = effective porosity of any fill material and is unity if the system is not rubblefilled
The hydrological balance equation may therefore be written as:
dV dh _ _
(14.21)
The water level in the system, h(t), can be found at any time, t, by inserting appropriate functions of Q/„ft), Qout(t), and V(h) into Equation 14.21, 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.
Q_{in} = C.I.A_{D} , for a duration D
(14.22)
where,
/ = intensity of the rainfall A_{D} = impermeable area drained
Considering the Outflow
It is assumed here that the infiltration rate (flow per unit surface area) is constant, thus:
Qout = Qf A
(14.23)
where.
A_{w} = wetted infiltration surface area of the facility which depends on the geometry of the system and is a function of the water depth, h
q_{f} = the infiltration coefficient which includes an appropriate factor of safety
With these assumptions, a flow balance equation may be written for h from Equation 14.21:
_{h{t)}__:IA_{D}o,A{h) ^{v} ^{;} J riVihW
dV(h) dh
(14.24)
The design method requires integration of Equation 14.24 in order to derive formulae describing the water level, h, at given times, t These formulae depend on the geometrical functions A^h) and V(h). For simple geometries, Equation 14.24 may be solved analytically. For complex
S = nV
(14.20)
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geometries, it can only be solved using an approximate method or numerical integration.
14.9.3 Hydraulic of Aquifer Response to Infiltration
Once a solution to Equation 14.24 is found for a particular type of infiltration system, the highest water level, h_{maxi }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 t equals the storm duration, D.
14.9.2 Flow through Layered Media
For the two layered materials shown in Figure 14.12, which are typically found in practice, vertical flow (infiltration or filtration ) through the layers are:
<7i_{2} = AK_{X}
g_{2}__{3} = AK_{2}
A
7
Pi
AK,
At\
AzT
Pi
Pi
z,  z.
For homogeneous media and continuity of flow:
AH
q = AK_{z}
AZ
where.
K_{z}
AZ_{1} + AZ_{2 }AZ, AZ_{2}
/f,
k,
(14.25a)
AK_{2}^ (14.25b)
Az,
(14.26)
AH = Aft_{1}+Aft_{2} ,
Infiltration drainage systems are found in the unsaturated zone above the water table. A diagrammatic representation of the conceptual model of soakage pit hydraulics in which groundwater flow in the unsaturated zone is taken into account is shown in Figure 14.12. It is envisaged that a 'bulb' of saturation becomes established around the pit. As groundwater flows away from the pit area through which it passes increases due to the threedimensional nature of the flow and the soil becomes unsaturated. 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 nonsteady flow conditions, the general 3dimensional equations of motion for homogeneousisotropic saturated phreatic groundwater flow is:
^(h—\ —\h—) —(ft — dx y dx) dy \ dy) dz \ dz
N__S_dh_ k~ k dt
(14.27)
where,
N = stormwater recharge (+ve) or abstraction (ve) rate
5 = the specific yield/ effective porosity
t = time
ft = phreatic water level
k = saturated hydraulic conductivity
and AZ = AZ_{1} +AZ_{2}
Pi/Y + Zi
Pi/y + Ti P3/Y + Z3
Datum
Figure 14.12 Flow through a Saturated Layered System
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The equation can be solved numerically for saturated groundwater flow problems if the initial and boundary conditions are defined and the hydraulic conductivity and specific storage are known. Derived velocity values will serve as convective terms in the solution of solute transport in Chapter 15. The equation for saturated confined flow is similar and applicable when stormwater is recharged into a deeper aquifer system.
(b) Unsaturated Groundwater Hydraulics
Unsaturated groundwater flow differs from saturated groundwater flow in that the hydraulic properties of the soil, the hydraulic conductivity and the storage coefficient both depend on the degree of saturation of the soil. As there is both water and air present in the soil, capillary action exerts a force, 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 Darcy's equation (Chapter 12), the onedimensional infiltration equation of motion for unsaturated groundwater flow is:
Aw^ _{+} ^kl _{=} if.£vi _{(1428)}
dZ ^{v} ' dZ dZ dy/ dt
where,
0 = 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 simply extended to threedimensions for real world problems. The hydraulic conductivity  pressure head function may be defined as:
k(_{w}) = k_{5}k_{r}(_{w}) (14.29)
where,
k = the hydraulic conductivity
k_{s} = the saturated hydraulic conductivity
k_{r} = the relative hydraulic conductivity
The moisture content  pressure head function may be defined as:
0{_{w}) = ne_{r}{_{w}) (14.30)
where,
0 = moisture content
n = porosity
0_{r} = relative saturation (5_{r} < 0_{r} < 1)
S_{r} = specific retention, i.e. the moisture content of the soil which will not drain under the influence of gravity
Various analytical models have been proposed to describe the relationship between the relative hydraulic conductivity and relative moisture content with pressure. These functions involve the use of one, two, three or four curve fitting coefficients for a particular soil.
In practice k(\//) and 0(\//) are hysteresis functions depending on whether the soil is wetting or drying at the time. Since the infiltration problem is concerned mainly with wetting then the hysteresis of soil properties is ignored.
(c) Coupled SaturatedUnsaturated Hydraulics
The integrated solution of Equations 14.27 and 14.28 are useful for the planning and design of larger stormwater recharge schemes under more complex environments. They are also important for use in municipal subsurface drainage design especially in vulnerable hillslope vadose areas.
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 and prism storage (Figure 14.13). When the flood wave is advancing, inflow exceeds outflow and a positive wedge of storage is produced. When the flood is receding, outflow exceeds inflow and a negative wedge results. In addition, a prism of storage is formed by a volume of (approximately) constant crosssection along the length of the channel reach.
Assuming that the crosssectional area of the flow is directly proportional to the discharge at the section, the volume of prism storage is KO, where K\s a coefficient of proportionality. The volume of wedge storage is assumed to be equal to KX(IO), where X is a weighting factor having the range 0< X<Q.5. The total storage is defined as the sum of the two storage components S=KO+KX(I0) which can be rearranged to give the linear storage function for the Muskingum method:
S = K[X I + (lX)0] (14.31)
The value of X depends on the shape of the modelled wedge storage, X= 0 corresponds to reservoir (levelpool) storage and Equation 14.31 reduces to S = KOX= 0.5 for a full wedge. In most natural stream channels, X is between 0 and 0.3, with a mean value near 0.2. Great
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accuracy in determining X\s usually not necessary because the Muskingum method is not sensitive to this parameter.
K
0.5Af[(/_{>+1}+/,)(0_{>+1+}0j X[l_{j+i}Ij)_{+}(lXlO_{j+i}Oj)
(14.35)
10
Wedge storage = KX{I0)
Prism storage = KO
Equation 14.35 is derived from Equations 14.16 and 14.32. The computed values of the numerator and denominator of Equation 14.35 are plotted for each time interval At, with the numerator on the vertical axis and the denominator on the horizontal axis. This usually produces a graph in the form of a loop (see Figure 14.14). The value of X that produces a loop closest to a single line is taken to be the correct value for the reach. The parameter K, from Equation 14.35 is the slope of the line for the value of X.
Figure 14.13 Prism and Wedge Storage in a Channel Reach
The parameter K is the time of travel of the flood wave through the reach. Although techniques exist that allow for the values of /fand X to vary according to flow rate and channel characteristics (e.g., the MuskingumCunge method), for hydrologic routing the values of /fand X are assumed to be specified and constant throughout the range of flow. The change in storage over the time interval A£(fromy toy' + l) is:
S_{J+1} Sj=K [XI_{J+1} +(lX) 0_{J+1} ]  [XTj +(lX) Oj ]
(14.32)
The change in storage can also be expressed by Equation 14.16 and combining Equations 14.32 and 14.16 yields the Muskingum routing equation:
O
7+1
C\Ij_{+}\ +C_{2}Ij +C_{3}.Oj
where.
At2KX
^{1} ~ 2K(lX) + At
At + 2KX
^{2} ~ 2K(lX) + At
_{c} _ 2K{lX)At
^{3} 2K(lX) + At
Note that Q +C_{2} +C_{3}= 1.
(14.33)
(14.34a) (14.34b) (14.34c)
The values of /fand Xtov a single reach are determined using observed inflow and outflow hydrographs in the channel reach. By assuming various values of X and known values of inflow, successive values of K can be computed using:
o x
+
i—i
X
(a)
O
x
+
i—i
X
(b)
O
+
i—i
X
^ ^
(c)
Figure 14.14 Procedure for Determining Xand K Values
14.10.2 Onedimensional Hydraulic Routing
Procedures for distributedflow hydraulic routing are popular because they compute flow rate and water level as functions of both space and time. The methodologies are based upon the SaintVenant equations of onedimensional flow. In contrast, the lumped hydrologicrouting procedures discussed in the previous sections compute flow rate as a function of time alone.
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Unsteady flow is described by the conservation form of the SaintVenant equations. This form provides the flexibility to simulate a wide range of flows from gradual flood waves in rivers to pipe flows, and can simulate lateral inflows or outflows such as weirs and pumping. The equations are given below (Chow et al, 1988).
Continuity: 

dQ _{} d{A + A_{Q}) dx dt 

Momentum: 

d 
(fiQ^{2}) 
13<? 1 
A \ J 
A dt A 
dx 
q = 0
(14.36a)
dh_
dx
■s_{f}+s_{e}
pqv_{x} +W_{f} B = 0 (14.36b)
where,
x = longitudinal distance along the conveyance
t = time
A = crosssectional area of flow
A_{0} = crosssectional area of dead storage (offchannel)
q = lateral inflow per unit length along the conveyance
h = watersurface elevation
v_{x} = velocity of lateral flow in the direction of flow
S_{f} = friction slope
S_{e} = eddy loss slope
B = width of the conveyance at the water surface
W_{f} = wind shear force
P = momentum correction factor
g = acceleration due to gravity
The SaintVenant equations operate under the following assumptions:
1. The flow is onedimensional 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 across any section perpendicular to the longitudinal axis.
2. There is gradually varied flow along the channel so that hydrostatic pressure prevails and vertical accelerations 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 of scour and deposition.
5. Resistance coefficients for steady uniform turbulent flow are applicable, allowing for a use of Manning's equation to described resistance effects.
6. The fluid is incompressible and of constant density throughout the flow.
The momentum equation consists of terms for the physical processes that govern flow momentum. When the water level or flow rate is changed at a certain point in a channel with a subcritical flow, the effects of these changes propagate back upstream. These backwater effects can be incorporated into distributed routing methods through the local acceleration, convective acceleration, and pressure terms.
Hydrologic (lumped) routing methods may not perform well in simulating the flow conditions when backwater effects are significant and the drain or channel slope is mild, because these methods have no hydraulic mechanisms to describe upstream propagation of changes in the flow momentum.
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APPENDIX 14.A DESIGN CHARTS
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60 50 40 30 20 10 5 4 3 2 15 10 20 50 100 200 500 1000
Time of Travel Over Surface (min.) Length of Overland Flow (m)
Design Chart 14.1 Nomograph for Estimating Overland Sheet Flow Times (Source: AR&R, 1977) (Overland Sheet Flow Times  Shallow Sheet Flow Only)
1000
en
c
100
10
1.0 r 4 ft 
(\\0 
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10
Slope (%)
Design Chart 14.2 Kerb Gutter Flow Time
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Flow Estimation and Routing
l.o
0.9
0.8
0.7
0.6
32 'u
it
o u
it o
c
0.5
0.4
0.3
0.2
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( 

\2J City Areas Full and Solidly Built Up (^\ Surface Clay, Poor Paving, Sandstone Rock \___/ Commercial & City Areas Closely Built Up { 3 J Semi Detached Houses on Bare Earth f~2\ Bare Earth, Earth with Sandstone Outcrops V___) Urban Residential Fully Built Up with Limited Gardens ( 5 J Bare Loam, Suburban Residential with Gardens /'T~N Widely Detached Houses on Ordinary Loam V__) Suburban Fully Built Upon Sand Strata ( 7 J Park Lawns and Meadows 

\ I 11 Id 

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III III/ 
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III III/ 
v_ 

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
Rainfall Intensity, / (mm/hr)
Design Chart 14.3 Runoff Coefficients for Urban Catchments Source: AR&R, 1977
Note: For I> 200 mm/hr, interpolate linearly to C= 0.9 at /= 400 mm/hr
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l.o
0.9
0.8
0.7
0.6
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C
0.4
0.3
0.2
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( E ) 

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\___/ Clay Soil  Open Crop, Close Crop or Forest f BJ Medium Soil  Open Crop ( C J Medium Soil  Close Crop /T~\ Medium Soil  Forest \__J Sandy Soil  Open Crop ( E ) Sandy Soil  Close Crop 

 

l l l bandy boil  Forest 

X 
r^ "^ 
/ 

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
Rainfall Intensity, / (mm/hr)
Design Chart 14.4 Runoff Coefficients for Rural Catchments Source: AR&R, 1977
Note: For I> 200 mm/hr, interpolate linearly to C= 0.9 at /= 400 mm/hr
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Flow Estimation and Routing
APPENDIX 14.B WORKED EXAMPLE
14.B.1 Rational Method Calculation
To determine the design peak for flow generated from a minor drainage of medium density residential area of 10 hectares in Kuala Lumpur. Assume 80 m of overland flow followed by 400 m of flow in an open drain. Catchment area average slope = 0.5%. The catchment is shown in Figure 14.Bl.
Figure 14. Bl Catchment Area
From Table 4.1, minor system design ARI = 5 years, major system design ARI = 100 years. Determine fr
From Design Chart 14.1 for paved surface, t_{0}= 8 minutes.
Average velocity in the open drain should be assessed using Manning's equation. Assume V= 1.0 m/s. Then t_{d} = L/V= 400 sec. = 6.7 minutes, say = 7 minutes. Therefore total t_{c} = 8 + 7 = 15 minutes. Determine /and C For ARI of 5 years, from Table 13.Al using Equation 13.1 for £=30 minutes;
ln(I) = 5.1086 + 0.5037ln(30)  0.2155[ln(30)]^{2} + 0.0112[ln(30)]^{3} = 4.7698
^{5}/_{30} = 117.9 mm/hr
Similarly ^{5}/_{50} = 75.7 mm/hr
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Solution: Step (1) :
Step (2):
Flow Estimation and Routing
Convert to rainfall depths. ^{5}P_{30}= 115.3/2 = 587.9 mm ^{5}P_{60}= 75.7/1 = 75.7 mm
From Equation 13.3,^{5}P_{15} = 57.65  0.8*(72.957.65) ^{5}Pi5= 45.5 mm, therefore ^{5}/_{15} = 182 mm/hr From Design Chart 14.3, C = 0.87 (category (3)) Step (3): Determine Peak Flow ((7)
From Equation 14.7; 

Cx^{5}/,_{s}xA ^{Qs} ~ 360 
0.87 x 182 x 10 360 
= 4.4 m^{3}/s 
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14.B.2 TimeArea Calculation
Develop the runoff hydrograph from a 10 hectare medium density residential area located in Kuala Lumpur. The hydrograph is required to design the minor stormwater systems of the area. Given that the time of concentration, £ at the outlet is 15 minutes. The study area is shown in Figure 14.B2(a).
Solution: From Table 4.1, minor system design ARI = 5 years.
Step (1): Delineate subcatchments based on the isochrones
The isochrones were approximated and drawn to subdivide the whole catchment into subareas, as shown in Figure 14.B2(a). Total area for each subcatchment within the two adjacent isochrones was estimated. The cumulative timearea curve is given in Column E of Table 14.Bl.
Step (2): Calculate design rainfall I for the whole catchment
The design rainfall intensity, /was taken as 182 mm/hr. The rainfall distribution (Column B of Table 14.B1) and rainfall losses (Column C, taken from Table 14.4) for the design storm duration (15 minute), were assumed and shown in Figure 14.B2(b). Rainfall losses were deducted from the corresponding rainfall and the resulting effective rainfall is given in Column D of Table 14. Bl.
Step (3): Calculate ordinates of the hydrograph
Runoff generated from individual subcatchment (from Column F to J) due to each incremental effective rainfall amount (Column D) is calculated using Equation 14.10. These ordinates are added to get the ordinates of the total hydrograph at the outfall (Column K). The peak runoff discharge is calculated as 4.7 m^{3}/s. The total hydrograph is shown in Figure 14.B3(c).
Table 14.Bl Calculation of the TimeArea Method
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
K 
Time (min) 
Rainfall (mm) 
Losses (mm) 
ER (mm) 
TimeArea Curve (m^{2}) 
Runoff Generated by the Effective Rainfall (in mm) 
Hydrograph (m^{3}/s) 

9.9 
15.9 
9.1 
6.8 
2.3 

0 
0.0 
0.0 
0.0 
0 
0.00 
0.00 

3 
11.4 
1.5 
9.9 
27000 
1.48 
0.00 
1.48 

6 
15.9 
0.0 
15.9 
50000 
1.26 
2.39 
0.00 
3.65 

9 
9.1 
0.0 
9.1 
69000 
1.04 
2.03 
1.37 
0.00 
4.44 

12 
6.8 
0.0 
6.8 
85000 
0.88 
1.68 
1.16 
1.02 
0.00 
4.75 
15 
2.3 
0.0 
2.3 
100000 
0.82 
1.42 
0.96 
0.87 
0.34 
4.41 
18 
0.00 
1.33 
0.81 
0.72 
0.29 
3.15 

21 
0.00 
0.76 
0.61 
0.24 
1.61 

24 
0.00 
0.57 
0.20 
0.77 

27 
0.00 
0.19 
0.19 

30 
0.00 
0.00 
Note: ER  Effective Rainfall
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(a) Study Area with Subcatchment and Isochrones
Figure 14.B2 Components of TimeArea Method
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