Basic Hydrology for the FE Civil Exam: Rainfall, Runoff and Watersheds

NCEES FE Civil spec: Water Resources and Environmental Engineering, A. Basic hydrology (e.g., infiltration, rainfall, runoff, watersheds). FE Reference Handbook: Civil Engineering chapter, Hydrology/Water Resources section.

What the FE Civil Spec Covers

Section A of Water Resources and Environmental Engineering is basic hydrology: infiltration, rainfall, runoff and watersheds. Expect one or two questions that ask you to turn rainfall into a runoff peak or a runoff volume. The two workhorse methods are the rational method for peak flow from small catchments and the NRCS (SCS) curve number method for runoff depth. You may also see watershed-average rainfall (arithmetic mean or Thiessen polygons), unit hydrographs and infiltration curves.

Key Concepts

Watershed: the area draining to one outlet. Its time of concentration $t_{c}$ is the travel time from the hydraulically most distant point to the outlet.
Runoff coefficient C: the fraction of rainfall that becomes peak runoff. Pavement and roofs are 0.7 to 0.95; lawns and woods are 0.05 to 0.35. For mixed land use, area-weight it.
Curve number CN: an index from 30 to 100 that combines soil group and land cover. Higher CN means more runoff.
Infiltration: water entering the soil. Capacity usually decays with time (Horton type curve), so cumulative infiltration is the area under the capacity curve.
Unit hydrograph: the outlet hydrograph from 1 unit (1 in or 1 cm) of excess rain over a set duration. Scale its ordinates by the excess depth and add lagged copies (superposition).

Core Formulas

Rational method, US units (Q in cfs, i in in/hr, A in acres):

Q = C i A

Composite runoff coefficient:

C_{co m p} = \frac{\sum C _{j} A _{j}}{\sum A _{j}}

NRCS curve number runoff depth (P, S and Q in inches), with initial abstraction 0.2S:

S = \frac{1000}{C N} - 10 Q = \frac{( P - 0.2 S ) ^{2}}{P + 0.8 S} (P > 0.2 S)

Thiessen (area-weighted) average rainfall:

\overset{ˉ}{P} = \frac{\sum P _{j} A _{j}}{\sum A _{j}}

Where It Is in the FE Reference Handbook

Search the Civil Engineering chapter, Hydrology/Water Resources section. The NRCS rainfall-runoff equations and the rational formula sit at the start of that section. Search terms that work: "NRCS", "Rational Formula", "curve number".

Worked Example

A 25 acre development has 15 acres of residential lots (C = 0.40) and 10 acres of pavement (C = 0.90). The design intensity for a duration equal to $t_{c}$ is 3.2 in/hr. (a) Find the peak runoff with the rational method. (b) For a 4.0 in storm on the same site with CN = 80, find the runoff depth and runoff volume.

Step 1, composite C:

C = \frac{0.40 ( 15 ) + 0.90 ( 10 )}{25} = \frac{6 + 9}{25} = 0.60

Step 2, peak flow:

Q = (0.60) (3.2 in/hr) (25 ac) = 48 cfs

Step 3, retention and runoff depth:

S = \frac{1000}{80} - 10 = 2.5 in, 0.2 S = 0.5 in

Q = \frac{( 4.0 - 0.5 ) ^{2}}{4.0 + 0.8 ( 2.5 )} = \frac{12.25}{6.0} = 2.04 in

Step 4, runoff volume:

V = \frac{2.04}{12} ft \times 25 ac = 4.25 ac-ft \approx 185, 000 ft^{3}

Answer: about 48 cfs peak, and 2.04 in of runoff (4.25 acre-feet) from the 4 in storm.

How FE Questions Are Framed

Hydrology questions are short but have several traps in the givens. A rational method question may give the area in hectares and intensity in mm/h, so decide early which form of the formula you are using. A curve number question may give a soil group and land cover instead of CN, in which case you look up CN in a table provided with the problem. Unit hydrograph questions usually give ordinates at even time steps and two or more blocks of excess rain: multiply each block depth by the ordinates, shift each result by one time step, and add the columns. Infiltration questions with an exponential capacity curve need an integral, which the calculator can do numerically.

Remember the difference between the two outputs. The rational method gives a peak flow rate and says nothing about volume. The NRCS method gives a runoff depth over the whole watershed, which becomes a volume when you multiply by area. If a later question asks you to size storage, you need a volume, not a peak.

Common Mistakes

Averaging runoff coefficients without weighting them by area.
Using an intensity for the wrong duration. The duration should equal the time of concentration.
Forgetting the 0.2S initial abstraction, or applying the CN equation when P is less than 0.2S (runoff is then zero).
Mixing SI and US forms of the rational formula. The 0.278 factor only applies with mm/h and square kilometers.
Reporting runoff depth in inches when the question asks for a volume. Multiply depth by watershed area.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

Using the rational method Q = 0.278CIA, estimate peak runoff for C = 0.60, rainfall intensity I = 50 mm/hr, and drainage area A = 2 km². See the worked solution or practice the Hydrology & Stormwater deck
A 10 ha catchment has a runoff coefficient of 0.60 and a design rainfall intensity of 50 mm/h. Using the rational method, the peak runoff is most nearly: See the worked solution or practice the Hydrology & Stormwater deck
A watershed has a composite NRCS curve number CN = 75. A design storm delivers P = 5.0 in of rainfall. Using the NRCS (SCS) rainfall-runoff relation with the standard initial abstraction of 0.2S, the direct runoff depth is most nearly: See the worked solution or practice the Hydrology & Stormwater deck
A watershed is divided into two Thiessen polygons that cover it completely with no overlap. Gauge 1 represents 2 ha and recorded 20 mm of rainfall; gauge 2 represents 3 ha and recorded 40 mm during the same storm. What is the average storm rainfall depth over the watershed? See the worked solution or practice the Hydrology & Stormwater deck
A 1-hour, 1-cm unit hydrograph has discharge ordinates of 0, 2, 5, 3 and 0 m³/s at t = 0, 1, 2, 3 and 4 h (zero afterward). Effective rainfall is 1.0 cm during hour 1 (0-1 h) and 0.5 cm during hour 2 (1-2 h). Ignore baseflow. What is the peak direct-runoff discharge? See the worked solution or practice the Hydrology & Stormwater deck
Under continuously ponded conditions with enough water supply, infiltration capacity is f(t) = 5 + 20e^(−t) mm/h, with t in hours. How much water infiltrates during the first hour? See the worked solution or practice the Hydrology & Stormwater deck

Related Calculators and Examples

Frequently Asked Questions

When should I use the rational method instead of the NRCS curve number method on the FE exam?

Use the rational method when the question asks for a peak discharge from a small watershed and gives a runoff coefficient, an intensity and an area. Use the NRCS curve number method when the question gives a curve number and a storm depth and asks for a runoff depth or runoff volume.

Does the rational formula need a unit conversion factor?

In US units, Q = CiA with i in inches per hour and A in acres gives cubic feet per second directly (the true factor is 1.008, taken as 1). In SI units with i in mm/h and A in km squared, Q = 0.278CiA gives cubic meters per second.

What rainfall intensity goes into the rational formula?

The intensity for a storm whose duration equals the time of concentration of the watershed, read from the intensity-duration-frequency curve for the design return period.

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