1.

Course Number: CE 365K

Course Title: Hydraulic Engineering Design
Course Instructor: R.J. Charbeneau

Subject: Urban Drainage Systems

Topics Covered:
1. Hydraulics of sheet flow (overland flow)
2. Hydraulics of gutters and drainage inlets
3. Design storm: estimating peak discharge
4. Design of storm sewer systems

1.2

Urban Storm Drainage System

ASCE, 1992

1.3

Topic 1 Hydraulics of Sheet

(Overland) Flow
Overland flow as sheet flow
Continuity equation for overland flow
Determine Time of Concentration
[equilibrium time]

1.4

Sheet Flow on a Rough Surface:

Turbulent Boundary Layer

Rh y

1.5

Manning Equation for Sheet Flow

q = unit discharge (L2/T)

Manning equation developed for open channel flow how well does
it work for sheet flow?
Does rainfall impact the calculation of sheet flow discharge?

1.6

CRWR Research for TxDOT

(Highway Drainage through Superelevation Transitions)

Rainfall Simulator Setup

1.7

CRWR Research for TxDOT

1.6

Typical Data:

No significant effect from simulated rainfall

1.9

Usual Approach for Sheet Flow

Calculations in Engineering Design
Use Manning equation
So = slope of
surface
(pavement)

Sheet flow equation has the form

= So1/2 / n
m = 5/3

Next step: add continuity

1.10

Continuity
Equation

rainfall

I
yt+t

Flow in Flow out = Storage Increase

yt

qx

(qx + I x) t (qx+x) t =
(yt+t yt) x

qx+x

x+x

For small x and t:

Per unit width,

q=vy

Continuity Equation for

Overland Flow:

Both y and q
are variables

1.11

Kinematic Wave Theory

Express flow (q) as a function of the storage (y)
Manning equations can be written:

The continuity equation becomes

Now only y
is a variable

Wave
celerity:

Conclusion: the
drainage wave moves
faster than the water!!

1.12

Method of Characteristics
Continuity equation

MOC formulation

First-order (hyperbolic) PDE

Equivalent system of ODEs;

one set of ODEs applies
along each characteristic

Calculate solution
along drainage
path

y(x,t) along
characteristic

x
t

other base
characterisic

base
characteristic

1.13

Application to Overland Flow

Constant rainfall excess Io starting at time 0

Drainage response time

(Time of Concentration, Tc)
at x = L:

1.14

Example #1 - Runoff from Pavement

Surface
Paved asphalt surface (n = 0.015)
Slope S = 0.01
Length L = 200 ft
Constant rainfall intensity I = 10 inch/hr (0.000231 ft/s).
Paved Surface

Flow is the same down

each strip of the pavement;
depth y(x,t) and discharge
q(x,t) depend only on
distance and time

Flow

200

= S1/2/n
= 1.49 (0.01)1/2/0.015
= 10

1.15

Questions:
1. What is the maximum discharge at edge of pavement (per unit width)?
q = I L = 0.000231 ft/s * 200 ft = 0.0462 ft2/s
2. How long does it take to reach this discharge (Tc)?
Tc = (L / ( I2/3) )3/5 = (200 / (10 * 0.0002312/3) )3/5 = 172 sec.
3. What is the maximum depth at edge of pavement?

Kinematic wave

Manning equation

y = I Tc = 0.000231 ft/s * 172 sec = 0.0397 ft = 0.48 inch

y = (q / )3/5 = (0.0462 / 10)0.6 = 0.0397 ft

to

1.16

Example #1: Drainage Wave

Profiles
t = 50 sec

Base Characteristic Plane

y = Io t

y = Io t
50
t = 150 sec

150
Characteristic
from (0,0)

172

Increasing y(x)
y(0) = 0

t = 172 sec

Drainage Depth
Profiles at different
times (50, 150, 172
seconds)

1.17

Example #1: Depth and Flow at

Pavement Edge
Steady conditions reached
at t = 172 seconds

y = Io t

q = y5/3

1.18

Conclusions
Steady-state watershed response occurs at the
Time of Concentration, Tc, which should be used
as the design rainfall duration (Td = Tc)
Kinematic wave model for time of concentration
(Manning Eq.):

Overland
Flow

1.19

Topic 2 Hydraulics of Gutters and

Drainage Inlets
Simple and composite gutters
Curb inlets on grade
Curb inlets in sag

1.20

Simple and Composite Gutter

Curb

T
Roadway

T = spread
Y = depth at curb
T
Qw

Qs

1.21

Simple Gutter Discharge (Izzards Eq.)

Curb

Roadway

Gutter longitudinal
slope = So
Pavement cross-slope
Sx = 1/z = Y/T

Area A = (Y T)/2 = SxT2/2

Wetted Perimeter P = T (1 + Sx2)0.5 + T Sx
Typically Sx ~ 0.02 and P = 1.020 T P = T

Izzards equation for gutter flow:

Izzard, 1946

1.22

Travel Time (Tc) in Gutters

Lateral inflow (qn) from pavement surface causes the
gutter discharge to increase along the drainage length
Pavement Crown

x
qn
QB
Inlet

Sheet
Flow
Gutter Flow

Curb

x=L
qn

W
Spread

Q = QB + qn x

Inlet

Gutter Flow Travel Time:

1.23

Drainage Inlets

Types of inlets
Performance of curb inlets
Performance of depressed curb inlets
Curb inlets in sag

1.24

Drainage Inlets
Curb inlets are used most
often on urban streets and
highways. These are usually
depressed, and may also be
recessed back from the
curb.
Gutter (grate) inlets can be
used on bridges, but have
issues with safety (bicycles)
and clogging.
Slotted drains can be used in
parking lots to collect
overland sheet flow.
ASCE, 1992

1.25

Curb Inlets
Design questions: how does the capture and bypass of gutter
discharge depend on inlet length L and inlet depression?

1.26

Curb Inlets

1.27

Analysis of Flow to Curb Inlets

Inlet Opening

VL

Spread, Tmax

Plan View

Elevation View
Curb
Inlet Opening

Ymax

h(x)
x

L
LT

1.28

Analysis of Curb Inlets (Izzard, 1950)

The specific energy E is equal to the approach velocity head plus
the curb depth Y: E = Y + VL2/2g. However, the approach velocity
VL does not contribute to flow entering the inlet.
Only Y is significant in causing flow into the inlet. The edge of the
inlet acts as a control point, and the flow at the inlet lip is critical
(based on Y).
Critical flow: yc = (2/3) Y

and

Vc2/2 g = (1/3) Y

Unit discharge q based on curb depth Y:

1.29

Variable Depth Along Inlet

Because of flow into the inlet, the depth of water will vary along the
inlet opening. Izzard (1950) assumed that the hydraulic head varies
linearly along the length of the inlet opening according to
Y = Ymax [1 x/LT]
LT = theoretical length of the inlet for complete capture of gutter flow

Elevation View
Curb
Inlet Opening

Ymax

Y(x)
x

L
LT

1.30

Results
These assumptions with variable inlet depth give

The integral gives

If L = LT, then all of the approach discharge QA is captured by the curb

inlet

10

1.31

Inlet Efficiency, E
The curb inlet efficiency, E, is the ratio of the curb discharge that is
captured by an inlet of length L, to the total approach discharge:

FHWA studies (HEC-12) suggest that this theoretical relationship

should be modified as

FHWA

The bypass discharge, QB, is calculated from

QB = QA QCI = QT (1 E)

1.32

Inlet Capture Length

The length of curb inlet required to capture all of the approach gutter
discharge is calculated from

Combine this with Izzards equation for Ymax:

In U.S. Customary units, this is often written as

LT - (ft); QA - (ft3/s)

FHwA

1.33

Example #2
A gutter (n = 0.014) carries a discharge of 5 cfs down a street with longitudinal
slope of 2% and cross-slope 1:33 (Sx = 0.03). What fraction of the gutter
discharge will be captured by a curb inlet that is 10 ft long?
Approach: 1) find Ymax; 2) find LT; 3) find E; 4) and find QCI = E QA
1) With Izzards equation for gutter flow we have
T = Ymax / Sx
= 0.26/0.03
= 8.5 ft
2) The inlet capture length is calculated using

3) The capture efficiency is calculated from

E = 1 (1 L/LT)1.8 = 1 (1 10/38.5)1.8 = 0.42
4) The captured curb inlet discharge is QCI = 0.42 x 5 = 2.1 cfs

11

1.34

Depressed Curb Inlet

Top of curb

Sw = Sx + a/W
Sx
a

Eo = fraction of total Q in the

depressed gutter section

Sw
W

Equivalent cross slope, Se = Sx + (a/W) Eo

Total Interception Length:

Continue with inlet capture calculations using new LT value.

1.35

Example #3
Repeat Example #2 for depressed inlet with 4-inch depressing (a = 0.33 ft) over
depressing width 2 ft (W = 2 ft).
Approach: 1) find equivalent cross slope Se; 2) find LT; 3) find E; and 4) find QCI = E QA

1) Sw = Sx + a/W = 0.195 Eo = 0.74 Se = 0.03 + (0.33/2) * 0.74 = 0.15

2) The inlet capture length is calculated using

3) The capture efficiency is calculated from

E = 1 (1 L/LT)1.8 = 1 (1 10/14.7)1.8 = 0.87
4) The captured curb inlet discharge is QCI = 0.87 x 5 = 4.4 cfs

1.36

Curb Inlet in Sag

water enters inlet from both sides

h = height of curb inlet opening
Y = depth at curb inlet opening
do = depth to inlet centroid

If Y < h inlet acts as a weir

with critical flow at inlet:
Q = (2/3)1.5 g0.5 Y1.5 L
If Y > 1.4 h inlet acts as an
orifice (Cd ~ 2/3)
Q = Cd (h L) (2 g do)0.5

do

Sx

For h < Y < 1.4 h interpolate

12

1.37

Example #4: Curb Inlet in Sag

h

do

Sx

Determine the length of a curb

inlet in sag for a design
discharge Qdesign = 18 cfs. The
following parameters hold.
T = 12 ft

At the curb,
Y = T Sx + a = 12 (0.025) + (4/12)
= 0.633 ft = 7.6 inch

Sx = 0.025
h = 6 inch
a = 4 inch
W = 2 ft

Y/h = 7.6/6 = 1.27 neither criteria works! Need to try both.

1.38

Example (Cont.)
Weir-type flow:

Orifice-type flow:
do = Y h/2 = 7.6 4/2 = 5.6 inch = 0.467 ft

To be conservative, assume weir-type flow and

L = Qdesign / (Q/L) = 18 / 1.56 = 11.5 ft (use 12 ft)

1.39

Topic 3: Estimating Peak Discharge

Design of many stormwater hydraulic structures depend
on the discharge they are required to control
Examples include:
Inlets
Drains
Culverts

Peak discharge is estimated using the Rational Method,

limited by a minimum Time of Concentration (generally 6
to 10 minutes)

13

1.40

Rational Method Calculation

Rational Method: Q = C I A
Q = Peak discharge (cfs)
C = Runoff coefficient (dimensionless)
I = Rainfall intensity (in/hr) from IDF curve
A = Drainage area (acres)

Units: 1 acre-inch/hr = 1.008 cfs

Useful conversions:

1 acre = 43,560 ft2

1 sq. mile = 640 acres

1.41

Rational Method - Assumptions

The Rational Method is based on the following
assumptions:
a) The peak discharge at any location is directly proportional to the
average rainfall intensity during the time of concentration (for that
location)
b) The time of concentration is the travel time from the most remote (in
travel time, not necessarily distance) point in the contributing area to
the location under consideration
c) The contributing area can be the entire drainage area upstream of the
location or some subset of this area, such as only the directly
connected impervious portion of the drainage area

1.42

Runoff Coefficient, C
Fraction of the rainfall intensity (I) that contributes to
the peak discharge (depends on rainfall intensity and
duration)
Typical values (ASCE):
Pavement
Urban business
Neighborhood
Residential
Single family
Suburban
Industrial
Light
Heavy

0.70 0.95
0.70 0.95
0.50 0.70
0.30 0.50
0.25 0.40
0.50 0.80
0.60 0.90

14

1.43

Rainfall Intensity: IDF Curves

Variability in rainfall intensity with duration can be
described by a model called the Intensity-DurationFrequency (IDF) curve.
Frequency refers to the return period of the event. The
rainfall intensity with a 10 minute duration for an event
with a 10 year return period will be greater than the
corresponding intensity of an event with a 2 year return
period.

1.44

IDF Curves
Statistical (frequency) analysis of local rainfall
records to identify how intensity (I) varies
with rainfall duration (Td) for different
probability of occurrence or return period (TR)
IDF model equation:

1.45

TxDOT IDF Curves for Travis County

Return
Period

2 year

5 year

10 year

25 year

50 year

100 year

56

69

77

87

91

103

8.1

8.6

8.6

8.6

8.6

8.1

0.796

0.780

0.775

0.766

0.751

0.752

I (in/hr)
Td (min)

15

1.46

Travis County IDF Curves, including

data from two rainfall events
100-yr

1981 Memorial Day

Flood rainfall event

2-yr

Note: actual events do not follow IDF curves

1.47

Example #5
What is the rainfall intensity and depth for an event with 8 minute
duration and 5 year return period?
From the IDF curve

The corresponding depth is

P = 7.71 x (8/60) = 1.03 inches

1.48

P-Td-TR Curves
Precipitation depth (P), duration (Td), return period (TR)

16

1.49

Example #6 Tc Using Kinematic Wave

An inlet captures drainage from a 3 acre watershed (30% impervious).
What is the peak discharge for a 5 year return period event?
Data.

Pervious area: n = 0.4, L = 250 ft, S = 0.025, C = 0.25

Impervious (pavement): n = 0.014, L = 500 ft, S = 0.01, C = 0.95

Use of the kinematic wave equation to estimate the time of concentration

requires the rainfall intensity, which in turn depends on the time of
concentration. Approach is to choose a duration, calculate the rainfall intensity,
then calculate the time of concentration, and finally compare with the assumed
duration.
Assume Td = 8 minutes (which gives I = 7.71 in/hr = 0.00018 ft/s; see
Example #3). With the data above, the kinematic wave model gives
(impervious) tc = 315 sec = 5.3 min; and (pervious) tc = 1180 sec = 19.7 min.
Repeat (see helpful hint on next page).

1.50

Example #6 (Cont.)
To find the time of concentration, we are combining the kinematic wave
equation with the IDF curve equation. Taking into account unit conversions,
these may be combined as follows:
(Both Tc and
Td in minutes)

Trial and error give (impervious) Tc = 4.95 minutes and (pervious) Tc =

24.5 minutes. These durations correspond to rainfall intensities
(impervious) I = 9.04 in/hr and (pervious) I = 4.50 in/hr.
We next need to determine whether the runoff from only the impervious
area gives a greater peak discharge than runoff from the entire area.

1.51

Example #6 (Cont.)
Considering only the impervious area:
Q = 0.95 x 9.04 x 0.9 = 7.73 cfs
Considering the total drainage area, the rational method gives
Q = (0.95 x 0.9 + 0.25 x 2.1) x 4.50 = 6.21 cfs
The discharge from only the impervious area is larger and would be used to
design the inlet and storm sewer for a peak discharge of 7.73 cfs.

17

1.52

Time of Concentration, Tc
Description: Tc = drainage time from most
hydraulically distant location within watershed to
location of interest. Generally includes the following:

Tc = Toverland

flow

+ Tconcentrated flow + Tchannel flow

Kerby-Kirpick Method: Use Kerby equation for

overland/concentrated flow and Kirpick equation for
channel flow
Approximation for small to moderate size watersheds: use
Kirpick equation to estimate Tc for channel flow and add 30
minutes to account for overland and shallow concentrated
flow

1.53

Kerby (1959) Method (Overland Flow)

Area < 10 acre; slope < 0.01

L = overland flow path length to defined channel (ft)

S = average watershed slope (ft/ft)
N = flow retardance factor:
N = 0.02

smooth impervious surface

N = 0.10

smooth, bare packed soil

N = 0.20

poor grass; moderately rough bare surface

N = 0.40

average grass

N = 0.60

deciduous forest

N = 0.80

dense grass; coniferous forest; deep ground liter

1.54

Kirpich (1940) Channel Flow

Multiply Tc by 0.4 if overland flow path is concrete or asphalt

Multiply Tc by 0.2 if channel is concrete-lined

18

1.55

Topic 4 Design of Storm Sewer Systems

Inlet spacing
Design criteria
Watershed delineation
Design procedure

1.56

Inlet Spacing
Inlet spacing is determined by limitations on spread of
stormwater gutter flow across the roadway surface.
Some inlet bypass (carry-over) of gutter flow is OK,
except at roadway intersections where cross flow
should be avoided.
Read Section 8.VII (Street and Intersection Design;
pg 250-260) from ASCE (1992) for classification of
streets and limitations on pavement encroachment
and cross flow.

1.57

Gutter Flow (Steady, Nonuniform)

Pavement Crown

x
Critical Location
for Time of
Concentration, Tc

qn
QB

Inlet

Sheet
Flow
Gutter Flow

Curb

x=L
qn

W
Spread

Q = QB + qn x

Maximum spread determines inlet spacing:

Inlet
qn = C I W

Izzards Equation

19

1.58

Example #7 Inlet Spacing

Highway crown

If the spread is
limited to the gutter
section (Tmax = 8 ft),
q
W = 20 ft
W
what is the maximum
inlet spacing? Design
frequency = 25 yrs;
Inlet, L =
Gutter Section 8 ft wide
Inlet
10 ft, no
use minimum Tc = 6
depression
min; and assume inlet
bypass based on
Approach: 1) calculate QA based on maximum spread; 2)
same conditions
calculate inlet efficiency and bypass; 3) assume rainfall
intensity based on limiting Tc and calculate inlet spacing; 4) upstream.
check Tc based on L; 5) use new I and repeat 3) etc.
L=?

Pavement: n = 0.016;
Sx = 0.02; So = 0.01;
C = 0.95

1) Using Izzards eq., QA = (3/8) [1.5 (0.01)1/2/0.016] x (0.02)5/3 x 88/3 = 1.3 cfs
2) The total inlet capture length is

1.59

Example #7 (Cont.)
2) (Cont.) The inlet efficiency is E = 1 (1 10/21)1.8 = 0.69. With this inlet
efficiency, the bypass discharge is QB = (1 E) QA = 0.4 cfs.
3) For a 25-yr event with Td = Tc (assumed Tc = 6 minutes for now), the
rainfall intensity is I = 87/(6 + 8.6)0.766 = 11.2 in/hr = 0.000258 ft/s. The
pavement drainage path length is approximately W = 20 [1 +(0.01/0.02)2]0.5
= 22.4 ft. The normal lateral flow rate into the gutter section qn = 0.95 x
0.000258 x 20 = 0.0049 ft2/s. With this lateral flow rate, the inlet spacing is
calculated from

1.60

Example #7 (Cont.)
4) The pavement drainage slope is S = ( Sx2 + So2)0.5 = 0.0224. With Io =
C I = 0.95 x 11.2 = 10.6 in/hr = 0.000248 ft/s, the time of concentration for
lateral flow from the pavement is calculated from

For the flow through the gutter, g = 0.945 (1.5) (0.02)0.33 (0.01)0.5/0.016 =
2.44 and qn = 0.0050 ft2/s. The time of concentration for gutter flow is

The total time of concentration is Tc = 37 + 97 = 134 sec = 2.2 min

5) This Tc value is less than Td = 6 minutes, so there is no need to repeat the
calculations. Use L = 180 feet.

20

1.61

Storm Sewer Design

The basic approach for storm sewer design is similar to
that used for design of inlet spacing for highway runoff.
1) A watershed area is delineated
2) time of concentration estimated
3) and the design rainfall is estimated using IDF curves
and a duration equal to the time of concentration
4) Peak discharge is calculated using the rational method
5) Storm sewer pipe size is determined based on the
peak discharge using Mannings equation, assuming that
the sewer pipe flows full
6) Storm sewer network layout follows topography to
the extent that is practical.

1.62

General Criteria: Storm Sewer

Design
Storm sewer size is determined by application of
Mannings equation for the design peak discharge that
the sewer pipe will carry; this discharge includes both
inlet flow plus upstream sewer discharge
Slope must be sufficient to maintain a velocity greater
than 2-3 ft to prevent significant sedimentation in sewer
pipe
Manholes should be utilized at sewer junctions and at
locations with significant changes in direction
Slopes should be uniform between manholes; size
increases generally occur at manhole (pipe size should
never decrease in the downstream direction)
There should be at least 3 ft of cover over the crown of
the pipe to support earth and external loads

1.63

Circular Section, Diameter D

Central Angle, :

T
y

21

1.64

Circular Pipe (conduit)

Pipe-full discharge:

D
y

Q > Qf for y > 0.8 D

Q = 1.076 Qf for y/D = 0.938

1.65

Size of Storm Sewer Pipe

Mannings equation for (circular) pipe:

This assumes that the pipe is flowing full under design

conditions
The discharge must include the peak flow from both
upstream sewer pipe plus inlet flow
US Customary units

1.66

Watershed Delineation
Watershed delineation includes estimation of
contributing area size, slope, and land use classification.
Drainage areas are delineated using topographic data,
that is generally available from a number of sources.
USGS topo maps may be accessed from a number of
sources. A useful online source is
http://www.topozone.com
Modern practice uses digital elevation model (DEM)
data and tools that help automate watershed
delineation. An alternative is to manually delineate the
watershed this is the approach assumed herein.

22

1.67

Drainage Area Delineation

Roadway
Inlet

530
540
550
560

1.68

Area Estimation
Herons formula

U = 0.5 (a + b + c)

Roadway
Inlet

b
c

530
540
550

560

1.69

General Polygon
3

Apply trapezoidal rule

clockwise around the
figure

1
5
E

23

1.70

Storm Sewer Design Procedure

Delineate the watershed contributing flow to each inlet. Calculate
(CA) for each inlet.
Moving from upstream to downstream through the storm sewer
drainage network, for each inlet calculate Tc including overland flow
plus concentrated flow. Compare with Tc from upstream drainage
areas including overland flow, concentrated flow, and channel
(conduit) flow travel time. Select the largest Tc value for the inlet.
Set Td = Tc and calculate the rainfall intensity using the IDF curve.
Calculate the design discharge from Q = (CA) I.
Estimate the required conduit size based on the design discharge,
using the next larger commonly available size.
Adjust and verify all calculations considering selected conduit size,
design discharge, and calculated travel times through conduits.
Check for desired hydraulic grade line elevations for final design.

1.71

Example Sewer Inlet and Pipe Layout

Example
from
Gribbin,
2002

1.72

First Delineate Drainage Areas for Each

Inlet

24

1.73

Inlet Drainage Areas - 1

1.74

Inlet Drainage Areas - 2

1.75

Analysis of Travel Time to Pipe

Segment
Three possible
drainage paths
leading to pipe
segment 3-5. To
find Tc, select the
one with the
longest travel
time.

25

1.76

Suggested References
Drainage of Highway Pavement, Hydraulic Engineering
Circular No. 12, FHWA, March 1984.
Urban Drainage Design Manual, Hydraulic Engineering
Circular No. 22, FHWA, November 1966 (2nd Ed., August
2001).
Design and Construction of Urban Stormwater
Management Systems, ASCE Manuals and Reports of
Engineering Practice No. 77, ASCE, 1992.
Stormwater Conveyance Modeling and Design, Haestad
Methods, Haestad Press, 2003.
Izzard, C.F., Hydraulics of Runoff from Developed
Surfaces, Proc. 26th Annual Meeting, Highway Research
Board, 1947.

26