Rational Method

The Rational Method is the most widely used hydrologic model for sizing storm drainage systems, gutter inlets, roadside ditches, and small culverts. It is designed to estimate the peak runoff rate from small watersheds.

Governing Equations

USCS Units

$$Q = C i A$$

Where:

  • $Q$ = Peak runoff rate (cubic feet per second, cfs)
  • $C$ = Runoff coefficient (dimensionless)
  • $i$ = Average rainfall intensity (inches per hour, in/hr) for a duration equal to the time of concentration ($t_c$)
  • $A$ = Drainage area (acres, ac)

Note: Technically, $1\text{ in/hr} \cdot 1\text{ acre} = 1.008\text{ cfs}$. Because this conversion is so close to $1.0$, the conversion factor is neglected.

SI (Metric) Units

$$Q = \frac{C i A}{360} \approx 0.00278 C i A$$

Where:

  • $Q$ = Peak runoff rate (cubic meters per second, $m^3/s$)
  • $C$ = Runoff coefficient (dimensionless)
  • $i$ = Average rainfall intensity (millimeters per hour, mm/hr)
  • $A$ = Drainage area (hectares, ha)

Conversion: $1\text{ hectare} = 10,000\text{ m}^2 \approx 2.471\text{ acres}$.

Runoff Coefficient Frequency Adjustment ($C_f$)

For severe, rare storm events (return period $T \ge 25\text{ years}$), infiltration capacity is reduced due to soil saturation. To account for this, a frequency adjustment factor ($C_f$) is applied to the runoff coefficient.

The adjusted peak flow equation becomes:

$$Q = C_f C i A$$

The NCEES Reference Handbook provides the following standard adjustment factors:

Return Period ($T$)Frequency Factor ($C_f$)
$\le 10$-year1.00
25-year1.10
50-year1.20
100-year1.25

Critical Constraint: The product of $C_f \cdot C$ must never exceed $1.0$ (i.e., you cannot have more runoff than rainfall). If $C_f \cdot C > 1.0$, use $C_{adj} = 1.0$.

Limitations of the Rational Method

The Rational Method is based on simplified assumptions and should only be applied within its limits:

  1. Watershed Size: It is designed for small watersheds. The general rule of thumb (and AASHTO guidance) is that it should only be used for areas under 200 acres (some agencies permit up to 300 acres).
  2. Uniform Rainfall: It assumes that rainfall intensity is uniform over the entire drainage area and is constant for the duration of the storm.
  3. Peak Flow Only: It only provides the peak flow rate, not a hydrograph of flow over time. It does not account for routing or temporary storage within the watershed.

Worked Example: Design Flow for a Roadway Culvert

A proposed roadway cross-culvert drains a watershed of 45 acres. The watershed consists of:

  • 15 acres of paved asphalt roadway and parking ($C = 0.90$)
  • 30 acres of steep-sloped suburban residential lawns on clayey soil ($C = 0.40$)

The culvert must be designed to convey the 50-year storm. An analysis of the local IDF curves indicates that for a duration equal to the watershed’s time of concentration ($t_c = 22\text{ minutes}$), the 50-year rainfall intensity is $5.8\text{ in/hr}$.

Calculate the design peak runoff flow rate ($Q$) for the culvert.

Solution:

  1. Calculate the Composite Runoff Coefficient ($C_{comp}$):

    $$C_{comp} = \frac{\sum (C_i \cdot A_i)}{A_{total}}$$

    $$A_{total} = 15 + 30 = 45\text{ acres}$$

    $$C_{comp} = \frac{(0.90 \cdot 15) + (0.40 \cdot 30)}{45} = \frac{13.5 + 12.0}{45} = \frac{25.5}{45} = 0.567$$

  2. Select and Apply the Frequency Adjustment Factor ($C_f$): For a 50-year storm, the frequency adjustment factor is $C_f = 1.20$.

    Verify if the adjusted coefficient exceeds 1.0:

    $$C_{adj} = C_f \cdot C_{comp} = 1.20 \cdot 0.567 = 0.680$$

    Since $0.680 \le 1.0$, this value is valid.

  3. Calculate Peak Flow rate ($Q$):

    $$Q = C_{adj} i A$$

    $$Q = 0.680 \cdot 5.8\text{ in/hr} \cdot 45\text{ acres}$$

    $$Q = 177.5\text{ cfs}$$

Key Exam Pitfalls

  • Ignoring the Frequency Factor limit: Remember to apply $C_f$ for 25, 50, or 100-year storm frequencies. But remember that $C_{adj} = C_f \cdot C$ cannot exceed 1.0. If $C = 0.85$ and $C_f = 1.25$ (100-year storm), the product is $1.06$. You must use $1.00$ as your runoff coefficient.
  • Metric Conversion Mistake: In metric, the equation is $Q = 0.00278 C i A$. Do not forget the $0.00278$ (or dividing by 360) coefficient.