Water Resources and Environmental

A. Basic hydrology

Hydrology: Rational Method, Rainfall, Runoff, and Flood Control

Convert rainfall into peak runoff with the rational and NRCS methods, read IDF curves through time of concentration, and size storage with return-period thinking.

Hydrology turns weather into a design number: how much water arrives at the inlet, the culvert, or the detention pond, and how fast. On the FE Civil exam this subtopic is dominated by one deceptively simple equation, the rational formula

Q = CiA

, surrounded by the supporting ideas it quietly depends on — rainfall intensity from an IDF curve, the time of concentration that selects that intensity, and the return period that fixes the storm. Points are lost not in the algebra but in the unit conventions (

A

in acres,

i

in in/hr, answer in cfs) and in choosing the right intensity. Everything here is in the FE Reference Handbook's Civil Engineering chapter under Hydrology.

The hydrologic cycle and the runoff budget

Precipitation that reaches a watershed is partitioned: some infiltrates, some is intercepted and evaporates or transpires, and the remainder runs off. A mass balance over a surface-water system says precipitation plus inflow minus outflow plus groundwater contribution, less evaporation, transpiration, and infiltration, equals the change in storage. Runoff methods are really shortcuts that bypass this full budget by lumping all the losses into a single coefficient or curve number. Knowing that

C

and

CN

are loss surrogates tells you immediately why an impervious parking lot has

C \to 0.95

while a forest has

C \approx 0.1

.

P + Q_{in} - Q_{out} + Q_g - E_s - T_s - I = \Delta S_s

The rational method

The rational method estimates the peak discharge from a small, mostly impervious catchment as the product of a runoff coefficient

C

, the rainfall intensity

i

, and the drainage area

A

. The key conceptual move is that the design intensity is the one whose duration equals the time of concentration

t_c

— the time for runoff from the hydraulically most remote point to reach the outlet — because only then is the entire watershed contributing simultaneously. In US customary units the formula is dimensionally a near-identity:

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

, so with

A

in acres and

i

in in/hr the answer falls out directly in cfs.

Q = C\,i\,A

Composite C and IDF curves

Real catchments are a patchwork of surfaces, so you area-weight the coefficients:

C_{comp} = \sum C_j A_j / \sum A_j

. The intensity itself comes from an intensity-duration-frequency (IDF) curve for the chosen return period — you enter with the storm duration (set equal to

t_c

) and read off

i

. Shorter

t_c

means higher intensity, which is why urbanizing a watershed (paving, storm drains) raises peaks twice over: it increases

C

and shortens

t_c

. If you are handed an IDF equation such as

i = a/(t_c + b)^m

, use

t_c

in the same units the curve was fitted in.

C_{comp} = \dfrac{\sum_j C_j A_j}{\sum_j A_j}

The NRCS curve-number method

For larger watersheds where the rational method is unreliable, the NRCS (SCS) method estimates runoff depth

Q

(inches) from storm depth

P

(inches) through a maximum retention

S

set by the curve number.

CN

ranges from about 30 (deep sand, woods) to 98 (pavement), folding in soil group, land use, and antecedent moisture. The

0.2S

term is the initial abstraction — rain that must fall before any runoff begins — so no runoff occurs until

P > 0.2S

. Multiply the depth

Q

by the area to get a runoff volume.

Q = \dfrac{(P - 0.2S)^2}{P + 0.8S}, \qquad S = \dfrac{1000}{CN} - 10

Unit hydrographs and storage

Where the rational and NRCS methods give only a peak or a volume, the unit hydrograph gives the whole shape of the runoff response: it is the direct-runoff hydrograph produced by one unit (one inch, say) of excess rainfall applied uniformly over a set duration. Because runoff response is treated as linear, you scale a unit hydrograph by the actual excess depth and superpose lagged copies for multi-period storms. Detention and stormwater storage exploit the same hydrograph: a basin stores the difference between the inflow and the allowable outflow, so the required storage is the area between the inflow and outflow hydrographs — the routing step that shaves the peak.

Flood frequency and return period

A flood's return period

T

is the inverse of its annual exceedance probability

p

— a 1%-annual-chance flood is the 100-year flood, meaning

p = 0.01

in any given year, not one flood per century. The risk that a

T

-year event is equaled or exceeded at least once in

n

years follows from the binomial complement, and it is alarmingly high: a 50-year design has nearly a 50% chance of being exceeded during a 30-year project life. This is the number behind freeboard, spillway sizing, and floodplain regulation.

T = \dfrac{1}{p}, \qquad R = 1 - \left(1 - \dfrac{1}{T}\right)^{n}

Exam strategy

Default to the rational method for small urban catchments and the NRCS method for larger or rural ones. For

Q = CiA

, confirm

A

is in acres and

i

in in/hr, then read the answer in cfs without a conversion factor. Always set the storm duration equal to

t_c

before reading the IDF curve — using a shorter duration overestimates intensity, a longer one underestimates it. For composite watersheds, area-weight

C

first. When a problem mentions a 'return period' or 'recurrence interval,' translate immediately to

p = 1/T

and, if a project life is given, compute the risk with

1-(1-p)^n

.

Key equations

Rational formula

Q = C\,i\,A

Peak discharge $Q$ (cfs) from runoff coefficient $C$ (dimensionless, 0-1), intensity $i$ (in/hr), and area $A$ (acres). Storm duration set to $t_c$ .

Composite runoff coefficient

C_{comp} = \dfrac{\sum_j C_j A_j}{\sum_j A_j}

Area-weighted $C$ for a mixed-cover watershed; each subarea $A_j$ has coefficient $C_j$ .

NRCS (SCS) runoff depth

Q = \dfrac{(P - 0.2S)^2}{P + 0.8S}

Runoff depth $Q$ (in) from storm depth $P$ (in); valid only when $P > 0.2S$ . Below that no runoff occurs.

Maximum retention from curve number

S = \dfrac{1000}{CN} - 10

$S$ (in) is the potential maximum retention; $CN$ (30-98) encodes soil group, land use, and antecedent moisture.

Time of concentration (Kirpich — NOT in FE Handbook)

t_c = 0.0078\,\dfrac{L^{0.77}}{S^{0.385}}

The FE Reference Handbook gives NO $t_c$ formula, so the exam will supply whatever method/constant it wants you to use; this Kirpich form (empirical, $t_c$ in min, $L$ = flow length in ft, $S$ = slope in ft/ft) is shown only so the symbol $t_c$ is familiar. Do not assume handbook provenance — use whatever the problem provides.

Surface-water hydrologic budget

P + Q_{in} - Q_{out} + Q_g - E_s - T_s - I = \Delta S_s

Mass balance: precipitation and inflows minus outflows and losses equal change in surface storage.

Return period and annual probability

T = \dfrac{1}{p}

$T$ (years) is the reciprocal of the annual exceedance probability $p$ . The 100-yr flood has $p = 0.01$ .

Risk of exceedance in n years

R = 1 - \left(1 - \dfrac{1}{T}\right)^{n}

Probability a $T$ -year event is exceeded at least once during $n$ years of project life.

Pan evaporation

E_L = P_c\,E_p

Lake evaporation $E_L$ from pan evaporation $E_p$ and pan coefficient $P_c$ (0.3-0.85, commonly 0.7).

Runoff volume from depth

V = Q\,(\text{in}) \times A \div 12

Volume (acre-ft) = runoff depth (in) times area (acres) divided by 12 (in/ft).

Worked examples

Peak discharge with a composite C

Problem. A 20-acre site drains as 4 ac of roof/pavement ( $C = 0.90$ ), 10 ac of lawn ( $C = 0.35$ ), and 6 ac of woods ( $C = 0.20$ ). For the design return period the intensity at the time of concentration is $i = 2.8\,\text{in/hr}$ . Find the peak discharge.

Solution. Area-weight the coefficient:

C_{comp} = \dfrac{0.90(4) + 0.35(10) + 0.20(6)}{20} = \dfrac{3.6 + 3.5 + 1.2}{20} = \dfrac{8.3}{20} = 0.415

. Apply the rational formula:

Q = C_{comp}\,i\,A = (0.415)(2.8)(20) = 23.2\,\text{cfs}

. Sanity check:

A

in acres and

i

in in/hr give cfs directly (the

1.008

factor is

\approx 1

). A fully paved 20-ac site at this intensity would give

0.95(2.8)(20) = 53\,\text{cfs}

, so

23.2\,\text{cfs}

for a mixed site is reasonable. Final:

Q \approx 23.2\,\text{cfs}

.

NRCS runoff depth and volume

Problem. A 40-acre watershed has a curve number $CN = 78$ . A storm drops $P = 4.5\,\text{in}$ . Find the runoff depth and the runoff volume in acre-feet.

Solution. Retention:

S = \dfrac{1000}{78} - 10 = 12.82 - 10 = 2.82\,\text{in}

. Initial abstraction

0.2S = 0.564\,\text{in} < P

, so runoff occurs. Depth:

Q = \dfrac{(4.5 - 0.564)^2}{4.5 + 0.8(2.82)} = \dfrac{(3.936)^2}{4.5 + 2.256} = \dfrac{15.49}{6.756} = 2.29\,\text{in}

. Volume:

V = 2.29\,\text{in} \times 40\,\text{ac} / 12 = 7.64\,\text{acre-ft}

. Sanity check: runoff depth (

2.29

in) is about half the rainfall (

4.5

in), consistent with a moderately pervious

CN = 78

. Final:

Q = 2.29\,\text{in}

,

V \approx 7.64\,\text{acre-ft}

.

Return period and project risk

Problem. A temporary cofferdam will stand for 30 years and is designed for the 50-year flood. What is the annual exceedance probability of the design flood, and what is the probability it is exceeded at least once during the 30-year life?

Solution. Annual probability:

p = 1/T = 1/50 = 0.02 = 2\%

. Risk over the project life:

R = 1 - (1 - p)^{n} = 1 - (1 - 0.02)^{30} = 1 - (0.98)^{30}

.

(0.98)^{30} = 0.545

, so

R = 1 - 0.545 = 0.455 = 45.5\%

. Sanity check: nearly a coin-flip chance of exceedance over 30 years despite a '50-year' label — the classic counterintuitive result that the return period is not a guarantee of spacing. Final:

p = 2.00\%

,

R \approx 45.5\%

.

Common pitfalls

•Treating the '100-year flood' as one flood per century. It is a 1% annual-chance event; back-to-back 100-year floods are entirely possible.
•Using the wrong rainfall duration. The design intensity in $Q = CiA$ is read at a duration equal to the time of concentration — using a shorter duration inflates $i$ and the peak.
•Unit slips in the rational formula. $A$ must be in acres and $i$ in in/hr to get cfs; mixing in square miles or mm/hr without converting wrecks the answer.
•Forgetting the initial abstraction in the NRCS method. No runoff occurs until $P > 0.2S$ ; plugging small storms into the formula can give a negative numerator.
•Averaging runoff coefficients by count instead of area-weighting. Always weight $C$ by each subarea before applying the rational formula.
•Confusing runoff depth (inches over the watershed) with volume. Multiply depth by area and divide by 12 to get acre-feet.
•Applying the rational method to large or complex watersheds. It is intended for small (typically < 200 ac), drainage-controlled areas; use NRCS or a unit hydrograph for larger basins.

References

NCEES FE Reference Handbook — Civil Engineering: Hydrology
NCEES FE Reference Handbook — Civil Engineering: NRCS/SCS Rainfall-Runoff
USDA NRCS, National Engineering Handbook Part 630 (Hydrology) — Curve-number method background

B. Basic hydraulics

Open-Channel Flow and the Manning Equation

Size channels with Manning's equation, find normal and critical depth, and use the Froude number and specific energy to classify subcritical versus supercritical flow.