11.1 Rainfall Hydrology and Rational Method

11.1 Rainfall Hydrology and Rational Method

Introduction to Hydrologic Principles

Hydrology in civil engineering focuses on the distribution, movement, and management of water on the Earth's surface. For the PE Civil Transportation exam, hydrologic analysis is fundamental for designing highway drainage structures, culverts, storm sewers, and roadside ditches. The design of these systems starts with predicting the quantity of runoff generated by a given rainfall event.

The hydrologic cycle describes the continuous movement of water. When precipitation falls on a watershed, it is distributed into several pathways:

• Infiltration: Water that penetrates the soil surface.
• Evapotranspiration: Water returned to the atmosphere through evaporation and plant transpiration.
• Depression Storage: Water trapped in puddles and surface depressions.
• Surface Runoff: Water that flows over the land surface once infiltration and storage capacities are exceeded.

In transportation drainage design, the primary objective is to manage surface runoff to prevent roadway flooding and hydroplaning.

The Rational Method

The Rational Method is the most widely used empirical formula for estimating peak runoff rates from small, urbanized watersheds. It is based on the assumption that a steady, uniform rainfall intensity over the entire drainage area will eventually produce a steady peak runoff rate.

Limitations of the Rational Method

The Rational Method should only be applied under the following conditions:

1. The drainage area is small, typically less than 200 acres (some agencies limit its use to 100 acres).
2. The rainfall intensity is uniform over the entire watershed.
3. The rainfall duration is at least equal to the time of concentration of the watershed.
4. The peak runoff occurs when the entire watershed is contributing to the outlet.

The Rational Equation

The peak runoff rate is calculated using the Rational Equation: $Q = C \cdot i \cdot A$

Where:

• $Q$ = Peak runoff rate (cubic feet per second, cfs)
• $C$ = Runoff coefficient (dimensionless, representing the fraction of rainfall that becomes runoff)
• $i$ = Rainfall intensity (inches per hour, in/hr)
• $A$ = Drainage area (acres)

Note on Units: One acre-inch per hour is equivalent to 1.008 cfs. Because this value is extremely close to 1.0, the conversion factor is omitted in US Customary units, and the equation is written directly as $Q = C i A$. In SI units, the formula is modified to: $Q = 0.00278 \cdot C \cdot i \cdot A$ where $Q$ is in $m^3/s$, $i$ is in mm/hr, and $A$ is in hectares.

Runoff Coefficients ($C$)

The runoff coefficient ($C$) reflects the watershed's characteristics, including soil type, land use, slope, and surface imperviousness. Soils with high infiltration rates (Group A) have lower $C$ values, while impervious surfaces (asphalt, concrete) have high $C$ values.

Composite Runoff Coefficient

If a watershed contains multiple land uses, a composite runoff coefficient ($C_{composite}$) must be calculated as the area-weighted average: $C_{composite} = \frac{\sum (C_j \cdot A_j)}{\sum A_j} = \frac{C_1 A_1 + C_2 A_2 + \dots + C_n A_n}{A_{total}}$

Time of Concentration ($t_c$)

The time of concentration ($t_c$) is defined as the time required for water to travel from the hydraulically most remote point in the watershed to the outlet. Sizing drainage structures requires setting the design rainfall duration equal to $t_c$, because this represents the condition under which the entire watershed contributes runoff simultaneously, maximizing the peak flow.

$t_c$ is calculated as the sum of travel times for three distinct flow regimes: $t_c = t_{sheet} + t_{shallow} + t_{channel}$

1. Sheet Flow ($t_{sheet}$)

Sheet flow occurs in the uppermost reaches of a watershed where water flows as a thin film over the land surface (usually limited to a maximum length of 100 feet, or 300 feet under older guidelines). The travel time is commonly calculated using the Manning's Kinematic Wave equation: $t_{sheet} = \frac{0.007(n \cdot L)^{0.8}}{(P_2)^{0.5} \cdot S^{0.4}}$ Where:

• $t_{sheet}$ = Travel time (hours)
• $n$ = Manning's roughness coefficient for sheet flow
• $L$ = Flow length (ft)
• $P_2$ = 2-year, 24-hour rainfall depth (in)
• $S$ = Slope of the land (ft/ft)

2. Shallow Concentrated Flow ($t_{shallow}$)

After a short distance (usually 100-300 feet), sheet flow begins to accumulate in small rills or depressions, forming shallow concentrated flow. The velocity ($V$) of shallow concentrated flow is calculated based on whether the surface is paved or unpaved:

• Unpaved: $V = 16.1345 \cdot S^{0.5}$ (fps)
• Paved: $V = 20.3282 \cdot S^{0.5}$ (fps) Travel time is then computed as $t_{shallow} = L / (3600 \cdot V)$ in hours.

3. Open-Channel or Pipe Flow ($t_{channel}$)

Once flow enters a defined channel, gutter, or storm sewer pipe, the velocity is determined using Manning's equation (discussed in detail in Section 11.2). The travel time is calculated as: $t_{channel} = \frac{L}{60 \cdot V}$ where $t_{channel}$ is in minutes, $L$ is channel length (ft), and $V$ is velocity (fps).

Minimum $t_c$ Rule: Many municipal drainage manuals specify a minimum time of concentration (typically 5 minutes for highly urbanized areas and 10 to 15 minutes for suburban/rural areas) to prevent unrealistically high design intensities.

Water-Quality Volume (WQV)

In addition to managing peak flows, modern transportation design requires treating stormwater runoff to remove pollutants. The Water-Quality Volume (WQV) represents the volume of runoff generated by a small, frequent storm (typically the 90th percentile storm, or a depth of 1.0 inch) that must be captured and treated.

WQV is calculated in acre-feet using the following formula: $WQV = \frac{P \cdot R_v \cdot A}{12}$ Where:

• $WQV$ = Water-Quality Volume (acre-feet)
• $P$ = Design rainfall depth (inches, typically 1.0 in)
• $R_v$ = Volumetric runoff coefficient (dimensionless)
• $A$ = Drainage area (acres)
• $12$ = Conversion factor (inches to feet)

The volumetric runoff coefficient ($R_v$) represents the fraction of rainfall converted to runoff for small storms and is estimated using Schueler’s formula: $R_v = 0.05 + 0.009 \cdot I$ where $I$ is the percent imperviousness of the drainage area (expressed as a percentage, e.g., use 75 for $75%$, not 0.75).