Hydrology

A. Storm characteristics

Storm Characteristics and Probability of Exceedance

Design-storm depth-duration-frequency, the return period, annual exceedance probability, the risk of one or more exceedances over a design life, and reading an IDF curve.

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B. Runoff analysis

NRCS (SCS) Curve-Number Runoff

Curve number from land use and hydrologic soil group, initial abstraction, the NRCS runoff-depth equation, antecedent moisture adjustment, and area-weighted CN.

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C. Hydrograph development and applications

Unit and Synthetic Hydrographs

The unit-hydrograph concept, convolution of a storm hydrograph, the S-curve for changing rainfall duration, and synthetic methods (SCS dimensionless and Snyder).

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E. Time of concentration

The Rational Method, Time of Concentration and IDF

Q = CiA for peak discharge on small catchments: composite runoff coefficient, time of concentration by flow segment, and selecting intensity from an IDF curve.

The rational method is the most-used peak-flow tool in civil practice — storm sewers, inlets, small culverts, and parking-lot drainage all start with

Q = CiA

. Its deceptive simplicity hides three sub-problems that decide whether the answer is right: choosing the runoff coefficient

C

, computing the time of concentration

t_c

, and reading the correct intensity

i

from an IDF curve at the design return period. Points evaporate in the unit bookkeeping (

A

in acres,

i

in in/hr, answer in cfs) and in the circular dependence between

t_c

and

i

. The method is set out in PE Civil Reference Handbook §6.5.2.1 (Rational Formula Method) with

t_c

in §6.5.4.

The rational formula

The peak discharge from a small catchment is the product of a dimensionless runoff coefficient

C

, the design rainfall intensity

i

, and the drainage area

A

. The conceptual heart is that the design intensity must have a duration equal to the time of concentration: only when the storm lasts at least

t_c

does the whole watershed contribute to the outlet simultaneously, giving the true peak. In USCS units the conversion is a near-identity —

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

— so with

A

in acres and

i

in in/hr the answer drops out in cfs.

Q = C\,i\,A

The runoff coefficient and composite C

The coefficient

C

is the fraction of rainfall that becomes peak runoff, lumping infiltration, depression storage, and surface type into one number — near

0.95

for pavement and roofs,

0.10

–

0.25

for parks and lawns. For a mixed catchment, area-weight the sub-area coefficients into a composite

C

. Steeper slopes and rarer return periods push

C

upward because losses become a smaller share of a bigger storm. Always weight by area, never simple-average.

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

Time of concentration: the flow path in segments

The time of concentration is the travel time from the hydraulically most distant point to the outlet, and it is built up segment by segment along the flow path. A typical path is sheet (overland) flow, then shallow concentrated flow, then open-channel or pipe flow — each segment has its own travel-time formula, and

t_c

is their sum. Underestimating

t_c

selects too high an intensity and oversizes everything; overestimating does the reverse.

t_c = \sum t_i = T_{ti} + t_{sc} + t_{ch}

Sheet, shallow, and channel travel times

Sheet flow over the first

\le 300\,\text{ft}

uses the kinematic-wave form

T_{ti} = \dfrac{K_u}{i^{0.4}}\left(\dfrac{nL}{\sqrt{S}}\right)^{0.6}

— that is

K_u (nL)^{0.6}/(i^{0.4}S^{0.3})

— with

K_u = 0.933

(USCS, minutes;

6.92

SI), exactly as the handbook prints it in §6.5.4.2, where

n

is the overland roughness and

S

the slope. Note that

i

appears here — the very intensity you are trying to find — so sheet-flow

t_c

is iterative. Shallow concentrated flow uses a velocity

V = k\sqrt{S}

(with

k\approx20.3

paved,

16.1

unpaved) so

t_{sc}=L/(60V)

; channel/pipe flow uses Manning's velocity and

t_{ch}=L/(60V)

, in minutes.

T_{ti} = \dfrac{K_u}{i^{0.4}}\left(\dfrac{nL}{\sqrt{S}}\right)^{0.6}, \qquad t_{seg} = \dfrac{L}{60\,V}

Selecting intensity from the IDF curve

With

t_c

in hand, enter the IDF curve for the design return period at duration

= t_c

and read the intensity. Because the sheet-flow term contains

i

, the practical procedure is a short iteration: assume an

i

, compute

t_c

, read a new

i

from the IDF, and repeat once or twice until they agree. Many jurisdictions supply a fitted form

i = c\,T^m/(t_c+f)^e

(

t_c

in minutes) so the read becomes a calculation.

i = \dfrac{c\,T^{m}}{(t_c + f)^{e}}

Assumptions and limits

The rational method assumes a uniform storm over a small, mostly impervious area with a constant

C

, and it yields only the peak — no volume, no hydrograph shape. It is generally restricted to drainage areas under about 200 acres (some agencies say 1 mi²); beyond that, spatial rainfall variation and storage make the linearity break down, and you switch to a unit-hydrograph or NRCS approach. Knowing this boundary is itself an exam point: a 500-acre basin is a signal to abandon

Q=CiA

.

Exam strategy

Work the chain: composite

C

→ segment travel times →

t_c

→ IDF read at the return period →

Q=CiA

. If sheet flow is present, iterate

i

and

t_c

to consistency (usually two passes). Check units relentlessly —

A

in acres,

i

in in/hr,

t_c

feeding the IDF in minutes,

Q

in cfs. If the area exceeds ~200 acres or the problem asks for a volume or hydrograph, the rational method is the wrong tool. A quick bound:

Q

can never exceed

C_{max}\,i\,A

with

C\to1

.

Key equations

Rational formula

Q = C\,i\,A

$Q$ = peak discharge (cfs); $C$ = runoff coefficient; $i$ = intensity at duration $t_c$ (in/hr); $A$ = area (acres). $1\,\text{ac-in/hr}\approx1\,\text{cfs}$ .

Composite runoff coefficient

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

Area-weighted $C$ for a mixed-cover catchment.

Time of concentration

t_c = T_{ti} + t_{sc} + t_{ch}

Sum of sheet, shallow-concentrated, and channel travel times along the longest hydraulic path.

Sheet-flow travel time

T_{ti} = \dfrac{K_u}{i^{0.4}}\left(\dfrac{nL}{\sqrt{S}}\right)^{0.6}

Kinematic-wave overland time (min), handbook §6.5.4.2 form — exponent $0.6$ on $nL/\sqrt{S}$ , $0.4$ on $i$ ; $K_u=0.933$ USCS, $6.92$ SI; $n$ = overland roughness, $L\le300\,\text{ft}$ , $S$ = slope, $i$ = intensity.

Shallow concentrated velocity

V = k\sqrt{S}

$k\approx20.3$ (paved), $16.1$ (unpaved), $V$ in ft/s, $S$ in ft/ft; then $t_{sc}=L/(60V)$ in minutes.

Channel travel time

t_{ch} = \dfrac{L}{60\,V}, \quad V = \dfrac{1.486}{n}R^{2/3}S^{1/2}

Manning velocity for the channel/pipe segment (USCS); $t_{ch}$ in minutes with $L$ in ft, $V$ in ft/s.

IDF intensity

i = \dfrac{c\,T^{m}}{(t_c + f)^{e}}

Design intensity (in/hr); enter at duration $=t_c$ (minutes) and the design return period $T$ (yr).

Worked examples

Composite C, time of concentration, and peak flow

Problem. An 8-acre catchment is 5 ac pavement ( $C=0.90$ ) and 3 ac lawn ( $C=0.25$ ). The flow path is: 120 ft of sheet flow over short grass ( $n=0.15$ , $S=0.015$ ); 400 ft of unpaved shallow concentrated flow at $S=0.02$ ; and 1000 ft of channel at $V=3.5\,\text{ft/s}$ . The 10-year IDF is $i = 94/(t_c+10)^{0.79}$ (in/hr, $t_c$ in min). Find the 10-year peak discharge.

Solution. Composite coefficient:

C_w = \dfrac{0.90(5)+0.25(3)}{8} = \dfrac{4.5+0.75}{8} = 0.656

. Shallow concentrated:

V = 16.1\sqrt{0.02} = 2.28\,\text{ft/s}

;

t_{sc} = 400/(60\cdot2.28) = 2.93\,\text{min}

. Channel:

t_{ch} = 1000/(60\cdot3.5) = 4.76\,\text{min}

. Sheet flow (handbook §6.5.4.2 form; iterate, first assume

i\approx7

):

T_{ti} = \dfrac{0.933}{7^{0.4}}\left(\dfrac{0.15\cdot120}{\sqrt{0.015}}\right)^{0.6} = \dfrac{0.933}{2.18}\,(146.9)^{0.6} = 0.428(19.97) = 8.5\,\text{min}

. Then

t_c = 8.5 + 2.93 + 4.76 = 16.2\,\text{min}

. IDF at 16.2 min:

i = 94/(26.2)^{0.79} = 94/13.2 = 7.1\,\text{in/hr}

, consistent with the assumed value, so no further iteration. Peak:

Q = C_w\,i\,A = 0.656(7.1)(8) = 37.3\,\text{cfs}

. Sanity check: a fully paved 8-ac site at this intensity would give

0.95(7.1)(8)=54\,\text{cfs}

, so

37\,\text{cfs}

for a 5/8-paved site is reasonable. Final:

Q \approx 37\,\text{cfs}

.

Q = 0.656\,(4.6)\,(8) \approx 24.1\ \text{cfs}

Two sub-areas with different times of concentration

Problem. A storm drain serves area 1 (6 ac, $C=0.80$ , $t_c=10\,\text{min}$ ) and area 2 (10 ac, $C=0.40$ , $t_c=25\,\text{min}$ ) draining to a common outlet. The IDF gives $i=120/(t_c+15)$ (in/hr, min). Determine the design peak at the outlet.

Solution. The design

t_c

is the longest path to the outlet,

25\,\text{min}

, so all sub-areas contribute at that duration:

i = 120/(25+15) = 120/40 = 3.0\,\text{in/hr}

. Composite

C

:

C_w = \dfrac{0.80(6)+0.40(10)}{16} = \dfrac{4.8+4.0}{16} = 0.55

. Peak:

Q = C_w\,i\,A = 0.55(3.0)(16) = 26.4\,\text{cfs}

. Check the partial-area case: using only area 1 at its shorter

t_c=10\,\text{min}

gives

i=120/25=4.8\,\text{in/hr}

and

Q_1=0.80(4.8)(6)=23.0\,\text{cfs}

— less than 26.4, so the full-area, longer-

t_c

case governs here. Final:

Q \approx 26.4\,\text{cfs}

.

Q = 0.55\,(3.0)\,(16) = 26.4\ \text{cfs}

Common pitfalls

•Using a storm duration other than $t_c$ to pick the intensity. The rational method requires duration $=t_c$ so the whole area contributes at peak.
•Simple-averaging the runoff coefficient instead of area-weighting it; a small paved fraction can dominate a large pervious area.
•Ignoring the iteration: sheet-flow $T_{ti}$ depends on $i$ , which depends on $t_c$ — assume, compute, re-read until consistent.
•Unit slips: $A$ must be in acres and $i$ in in/hr for the answer to come out in cfs; feeding $t_c$ to the IDF in hours when the fit expects minutes is a frequent error.
•Applying the rational method to a watershed larger than ~200 ac (or asking it for a runoff volume or hydrograph) — switch to NRCS or a unit hydrograph.
•Forgetting the partial-area check: a smaller, more-impervious sub-area at its shorter $t_c$ can produce a higher peak than the whole catchment at the longer $t_c$ .
•Capping sheet flow at the wrong length; the kinematic form applies only to the first ~300 ft, after which flow concentrates.

References

NCEES PE Civil Reference Handbook — §6.5.2.1 Rational Formula Method
NCEES PE Civil Reference Handbook — §6.5.4 Time of Concentration
FHWA HEC-22, Urban Drainage Design Manual — runoff coefficients and travel-time segments

F. Rainfall and stream gauging stations

Rainfall and Stream Gauging Stations

Areal rainfall by arithmetic, Thiessen, and isohyetal methods; stream discharge by the velocity-area mid-section method; rating curves; and double-mass consistency checks.

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G. Depletions

Depletions: Evaporation, Infiltration and Diversions

The hydrologic budget, evaporation/evapotranspiration estimates, infiltration by the Horton and phi-index methods, deep percolation losses, and diversions in a water balance.

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H. Stormwater management and treatment

Detention/Retention Sizing, Routing and BMPs

Detention versus retention, storage-indication (modified Puls) reservoir routing, peak-shaving to a release limit, and water-quality BMPs.

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