Drainage

11% of exam

Highway hydrology (rational method, time of concentration, runoff) and hydraulics: gutter and inlet capacity, storm-drain pipe flow, open-channel flow, culvert headwater, energy dissipation, and detention.

6 concepts

Exam areas

A. Hydrology including runoff and water-quality mitigation

The Rational Method & Time of Concentration

Size storm drainage from peak runoff with Q = CiA, build a composite runoff coefficient, and find the intensity from an IDF curve at the time of concentration.

Almost every transportation drainage design starts with one number: the peak rate of runoff the system must carry. The rational method,

Q = CiA

, is the workhorse the PE Civil exam expects you to reach for on small urban catchments — roadway sub-areas, parking lots, interchange quadrants — typically under about 200 acres. It looks trivial, but the exam quietly tests whether you can build a defensible runoff coefficient, find the rainfall intensity at the ·right· duration, and keep the units honest. The handbook presents it in

\S6.5.2

Runoff Analysis, and the single most important footnote there is the unit identity that makes the formula work in customary units.

The rational formula and its hidden unit identity

The peak discharge equals the runoff coefficient times the rainfall intensity times the drainage area. The magic is that in US customary units the conversion factor from acre-inches per hour to cubic feet per second is

1.008

, so close to unity that it is dropped —

1\ \text{in/hr}

falling on

1\ \text{acre}

produces essentially

1\ \text{cfs}

. That coincidence is why you can multiply a dimensionless

C

, an intensity in

\text{in/hr}

, and an area in acres and read the answer directly in cfs. In SI the identity does not hold and a

0.00278

factor is required.

Q = CiA \qquad (1\ \text{acre}\cdot\text{in/hr} \approx 1\ \text{cfs})

The runoff coefficient and the composite C

The runoff coefficient

C

is the fraction of rainfall that becomes surface runoff; it rises with imperviousness, slope, and return period and falls with infiltration capacity. Pavement and roofs run

0.70

–

0.95

, lawns on sandy soil as low as

0.05

. A real catchment is a patchwork, so you weight

C

by area to get a single composite value. Always form the composite ·before· you multiply by intensity — averaging

C

unweighted, or applying the highest

C

to the whole area, is a classic over-design error.

C_w = \dfrac{\sum C_i A_i}{\sum A_i}

Time of concentration sets the intensity

The method assumes the storm lasts at least as long as the time of concentration

t_c

— the travel time of runoff from the hydraulically most distant point to the outlet — so the whole catchment contributes simultaneously and flow peaks. You therefore read the design intensity off the IDF curve at a duration equal to

t_c

. Because IDF curves fall steeply with duration, a too-long

t_c

understates

i

and undersizes the pipe; a too-short

t_c

does the opposite.

t_c

is the sum of the travel times through each flow regime along the longest path.

t_c = t_{\text{sheet}} + t_{\text{shallow}} + t_{\text{channel}}

Building t_c segment by segment

Overland sheet flow over the first

\sim\!100

–

300\ \text{ft}

is the slow part; the widely used NRCS TR-55 sheet-flow relation gives travel time (in hours) as a function of Manning roughness, length, the 2-year 24-hour rainfall

P_2

(in), and slope. (The handbook's

\S6.5.4

instead gives an intensity-based FHWA kinematic-wave form

t_i = (K_u/i^{0.4})(nL/\sqrt{S})^{0.6}

with

K_u = 0.933

USCS and the result in minutes — either is acceptable, but cite the one you use.) Flow then concentrates into shallow rills (velocity

V = k\sqrt{S}

, with

k\approx 16.1\ \text{ft/s}

unpaved,

20.3

paved) and finally into gutters or pipes where you use the Manning velocity. Add the segment times; each segment is length over velocity.

t_{\text{sheet}} = \dfrac{0.007\,(nL)^{0.8}}{P_2^{\,0.5}\,S^{0.4}}\ \text{(hr)}, \qquad t_{\text{seg}} = \dfrac{L}{V}

The IDF curve

Intensity–duration–frequency curves encode the local rainfall statistics: for a chosen return period, intensity drops as duration grows. They are commonly fitted to a generic regional form

i = a/(t_c + b)^{\,c}

, or read from a chart; the handbook's

\S6.5.3

point-precipitation model uses a return-period form

i = cT^{m}/(t_d+f)^{e}

. Enter with duration

= t_c

and the design return period (10-yr for most storm drains, 25- to 50-yr for cross-culverts), and read the intensity. Never use a storm duration shorter than

t_c

— the catchment would not be fully contributing.

i = \dfrac{a}{(t_c + b)^{\,c}}

Assumptions and limits

The rational method gives a peak rate only — no volume, no hydrograph — so it cannot size detention storage (you need a hydrograph method for that). It assumes uniform rainfall over the whole area, a constant

C

during the storm, and a linear rainfall–runoff response, all of which degrade as area grows. Use it for drainage areas up to roughly

200\ \text{acres}

; beyond that switch to the NRCS curve-number/unit-hydrograph approach in

\S6.5.2.2

Q_{\text{peak}} = C\,i\,A \quad\text{(rate only — no storage volume)}

Exam strategy

Work it in order: composite

C

first, then

t_c

along the longest flow path, then read

i

from the IDF at duration

= t_c

, then multiply. If the problem gives sub-areas, weight

C

by area — do not average. Watch the units: acres and in/hr give cfs directly; if a problem hands you square feet or acres of a sub-area, keep everything consistent. If the question asks for storage or a full hydrograph, the rational method is the wrong tool — flag it and switch to NRCS. And when two sub-catchments combine, the controlling

t_c

for the downstream point is usually the longer path, but check both.

Key equations

Rational formula

Q = CiA

Peak discharge $Q$ (cfs) from runoff coefficient $C$ (dimensionless), intensity $i$ (in/hr), area $A$ (acres). In SI, $Q\,(\text{m}^3/\text{s}) = 0.00278\,CiA$ with $i$ in mm/hr, $A$ in hectares.

Composite runoff coefficient

C_w = \dfrac{\sum_{i} C_i A_i}{\sum_{i} A_i}

Area-weighted $C$ for a mixed-cover catchment; form it before applying $Q=CiA$ .

Time of concentration

t_c = \sum t_{\text{segment}} = t_{\text{sheet}} + t_{\text{shallow}} + t_{\text{channel/pipe}}

Travel time from the hydraulically most distant point to the outlet; sets the IDF duration.

Sheet-flow travel time (NRCS TR-55)

t_{\text{sheet}} = \dfrac{0.007\,(nL)^{0.8}}{P_2^{\,0.5}\,S^{0.4}}

TR-55 form, result in hours; $n$ overland roughness, $L$ length (ft, $\le\!300$ ft), $P_2$ 2-yr 24-hr rainfall (in), $S$ slope (ft/ft). The handbook $\S6.5.4$ uses an intensity-based FHWA form ( $K_u=0.933$ USCS, minutes) instead.

Shallow concentrated velocity

V = k\sqrt{S}

$k = 16.1\ \text{ft/s}$ unpaved, $20.3\ \text{ft/s}$ paved; then $t = L/V$ .

IDF intensity (fitted form)

i = \dfrac{a}{(t_c + b)^{\,c}}

Regional constants $a,b,c$ for a given return period; enter with duration $=t_c$ (min).

Customary unit identity

1\ \text{acre}\cdot\text{in/hr} = 1.008\ \text{cfs} \approx 1\ \text{cfs}

Why $Q=CiA$ returns cfs directly with $i$ in in/hr and $A$ in acres.

Worked examples

Composite C and peak discharge

Problem. A 23-acre highway catchment is 5 acres of impervious roof/structure ( $C=0.90$ ), 10 acres of landscaped lawn on heavy soil ( $C=0.22$ ), and 8 acres of asphalt pavement ( $C=0.85$ ). For the design storm the intensity at the time of concentration is $3.6\ \text{in/hr}$ . Find the composite $C$ and the peak discharge.

Solution. Composite coefficient (area-weighted):

C_w = \dfrac{(5)(0.90)+(10)(0.22)+(8)(0.85)}{23} = \dfrac{4.50+2.20+6.80}{23} = \dfrac{13.50}{23} = 0.587

. Peak discharge with

i = 3.6\ \text{in/hr}

A = 23\ \text{acres}

Q = C_w\,i\,A = (0.587)(3.6)(23) = 48.6\ \text{cfs}

. Sanity check: an all-pavement 23-acre site would give

\approx 0.85\times3.6\times23 = 70

cfs; the lawn pulls the composite down, so

48.6

cfs is reasonable. Units:

(\text{in/hr})(\text{acre}) \to \text{cfs}

via the

1.0

identity. Answer:

C_w = 0.587

Q = 48.6\ \text{cfs}

Q = (0.587)(3.6)(23) = 48.6\ \text{cfs}

Time of concentration then intensity

Problem. Runoff reaches a storm inlet along three segments: (1) sheet flow $150\ \text{ft}$ over dense grass ( $n=0.15$ ) at $S=0.02$ , with $P_2 = 3.2\ \text{in}$ ; (2) shallow concentrated flow $900\ \text{ft}$ , unpaved, at $S=0.015$ ; (3) gutter/pipe flow $1200\ \text{ft}$ at $V=4.5\ \text{ft/s}$ . The IDF curve is $i = 100/(t_c+10)^{0.85}$ ( $t_c$ in min). For a 12-acre area with $C=0.55$ , find $t_c$ , $i$ , and $Q$ .

Solution. Segment 1 (sheet):

t_1 = \dfrac{0.007\,(0.15\times150)^{0.8}}{3.2^{0.5}\,0.02^{0.4}} = 0.226\ \text{hr} = 13.6\ \text{min}

. Segment 2 (shallow, unpaved):

V = 16.1\sqrt{0.015} = 1.98\ \text{ft/s}

, so

t_2 = 900/1.98 = 455\ \text{s} = 7.6\ \text{min}

. Segment 3 (pipe):

t_3 = 1200/4.5 = 267\ \text{s} = 4.4\ \text{min}

t_c = 13.6 + 7.6 + 4.4 = 25.6\ \text{min}

. Intensity:

i = 100/(25.6+10)^{0.85} = 100/20.8 = 4.80\ \text{in/hr}

. Peak:

Q = CiA = (0.55)(4.80)(12) = 31.7\ \text{cfs}

. Sanity check: sheet flow dominates

t_c

as expected for dense grass;

i

\sim\!26

min is a plausible mid-range value. Answer:

t_c = 25.6\ \text{min}

i = 4.80\ \text{in/hr}

Q = 31.7\ \text{cfs}

Common pitfalls

•Reading the IDF intensity at the storm's total duration instead of at $t_c$ . The design duration for a rational-method peak IS the time of concentration — a longer duration gives a smaller $i$ and undersizes the system.
•Averaging the runoff coefficients of sub-areas unweighted, or applying the largest $C$ to the entire catchment. Always area-weight: $C_w = \sum C_iA_i / \sum A_i$ .
•Forgetting the $1.0$ unit identity only holds in customary units. In SI you must include the $0.00278$ factor ( $i$ in mm/hr, $A$ in hectares) — dropping it is a 360-fold error.
•Using the rational method to size a detention basin. $Q=CiA$ gives a peak rate, not a volume or hydrograph; storage design needs a hydrograph (NRCS) method.
•Stretching the method past its valid range. Beyond roughly $200\ \text{acres}$ the uniform-rainfall and constant- $C$ assumptions fail; switch to the NRCS curve-number approach.
•Mismatched $t_c$ units in the IDF formula — the fitted form usually takes minutes, while the sheet-flow equation returns hours. Convert before substituting.
•Taking the shorter flow path as the time of concentration. $t_c$ is the longest travel time to the outlet; check every contributing path and use the maximum.

References

NCEES PE Civil Reference Handbook — §6.5.2 Runoff Analysis (Rational Formula Method)
NCEES PE Civil Reference Handbook — §6.5.4 Time of Concentration (FHWA intensity-based sheet-flow form) and §6.5.3 Point Precipitation (IDF)
NRCS TR-55, Urban Hydrology for Small Watersheds — sheet-flow, shallow-concentrated, and channel travel-time relations
FHWA HEC-22, Urban Drainage Design Manual (3rd ed.) — runoff coefficient and time-of-concentration practice

Detention & Water-Quality Mitigation

Attenuate the peak with detention storage routing, size the water-quality (first-flush) volume from the runoff coefficient, and select BMPs for stormwater treatment.