11.1 Hydrology and Stormwater Management

10.1 Hydrology and Stormwater Management

1. Introduction to Engineering Hydrology

Engineering hydrology deals with the occurrence, distribution, circulation, and properties of water on and beneath the Earth's surface. For the PE Civil exam, understanding hydrologic principles is essential for predicting runoff estimation, assessing precipitation characteristics, conducting hydrograph analysis, and designing stormwater management infrastructure. The primary engineering goal is to quantify peak flow rates and runoff volumes to design structures such as culverts, storm sewers, and detention basin systems that protect lives and property from flooding.

2. Precipitation and Rainfall Analysis

Precipitation is the driving hydrologic force. The design of stormwater infrastructure is based on a design storm, which is a hypothetical rainfall event defined by its return period, duration, and temporal distribution.

Return Period and Probability

The return period ($T$), or recurrence interval, is the average time (in years) between occurrences of a storm that equals or exceeds a specified magnitude. The annual exceedance probability ($P$) is the probability that such a storm will occur in any given year: $P = \frac{1}{T}$ For example, a 100-year storm has an annual exceedance probability of $P = 1/100 = 0.01$, or 1%.

Intensity-Duration-Frequency (IDF) Curves

An intensity-duration-frequency (IDF) curve graphically represents the relationship between:

1. Rainfall Intensity ($i$, in inches per hour): The rate of rainfall.
2. Duration ($D$ or $t$, in minutes or hours): The length of the rainfall event.
3. Frequency (Return Period, $T$): The recurrence interval.

As storm duration increases, the average rainfall intensity decreases. Conversely, for a fixed duration, rarer storms (larger return periods) exhibit higher rainfall intensities. In practice, engineers use IDF equations or lookup curves developed by government agencies (such as NOAA Atlas 14) for specific geographic coordinates.

Probable Maximum Precipitation (PMP)

The probable maximum precipitation (PMP) is the theoretically greatest depth of precipitation for a given duration that is physically possible over a particular geographical area at a certain time of year. PMP is used as the design basis for high-hazard hydraulic structures—such as spillways of major dams—where structural failure would lead to catastrophic loss of life and economic damage.

Time of Concentration ($t_c$)

The time of concentration is the travel time required for runoff to travel from the hydraulically most remote point of the watershed to the outlet. In peak runoff estimation, the critical rainfall duration is assumed to be equal to $t_c$ because that represents the time at which the entire watershed is contributing flow to the outlet.

The time of concentration is computed as the sum of travel times ($T_t$) for three distinct flow regimes: $t_c = T_{\text{sheet}} + T_{\text{shallow}} + T_{\text{channel}}$

1. Sheet Flow: Very shallow flow over plane surfaces at the headwaters of a watershed. It is typically limited to a maximum length of 100 feet. The travel time is modeled using the Manning's kinematic solution: $T_{\text{sheet}} = \frac{0.007(n \cdot L)^{0.8}}{(P_2)^{0.5} \cdot S^{0.4}}$ Where $n$ is the Manning's roughness coefficient for sheet flow, $L$ is flow length (ft), $P_2$ is the 2-year, 24-hour rainfall depth (in), and $S$ is the slope of the land (ft/ft).
2. Shallow Concentrated Flow: Flow that begins to accumulate in swales, shallow depressions, or gutters. The velocity ($V$) is estimated based on the slope and whether the surface is paved or unpaved:
   • Unpaved: $V = 16.1345 \cdot S^{0.5}$
   • Paved: $V = 20.3282 \cdot S^{0.5}$ The travel time is $T_{\text{shallow}} = L / V$.
3. Open Channel Flow: Flow in defined channels, storm sewer pipes, or streams. The velocity is calculated using Manning's equation: $V = \frac{1.486}{n} \cdot R^{2/3} \cdot S^{1/2}$ Where $R$ is the hydraulic radius (ft) and $S$ is the channel slope (ft/ft). The travel time is $T_{\text{channel}} = L / V$.

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3. Runoff Estimation Methods

The Rational Method

The Rational Method is suitable for estimating peak runoff rates for small, urbanized watersheds (typically less than 200 acres). The fundamental formula is: $Q = C \cdot I \cdot A$ Where:

• $Q$ = Peak runoff rate (cubic feet per second, cfs).
• $C$ = Dimensionless runoff coefficient representing the fraction of rainfall that becomes runoff. It depends on land use, soil type, and watershed slope.
• $I$ = Rainfall intensity (inches per hour, in/hr) corresponding to a duration equal to the time of concentration ($t_c$).
• $A$ = Drainage area (acres).

Dimensional Compatibility: Note that $1 \text{ acre} \cdot 1 \text{ inch/hour} = 43,560 \text{ ft}^2 \cdot (1/12 \text{ ft}) / 3,600 \text{ seconds} \approx 1.008 \text{ cfs}$. Because this conversion factor is so close to unity, it is neglected in standard US Customary practice.

When a watershed has multiple land uses, a weighted runoff coefficient ($C_w$) is calculated: $C_w = \frac{\sum (C_i \cdot A_i)}{\sum A_i}$

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The SCS Curve Number (CN) Method

The SCS Curve Number method (developed by the USDA Soil Conservation Service, now NRCS) is used for larger, mixed-use watersheds. It estimates both the total volume and depth of runoff.

Potential Maximum Retention ($S$)

The potential maximum retention ($S$, in inches) represents the storage capacity of the soil-cover complex after runoff begins. It is defined by the Curve Number ($CN$): $S = \frac{1000}{CN} - 10$ $CN$ ranges from 0 (100% infiltration, no runoff) to 100 (impervious surface, 100% runoff).

Hydrologic Soil Groups (HSGs)

Soils are classified into four groups based on their minimum infiltration rates:

• Group A (Low runoff potential): Deep, well-drained sands and gravels with high infiltration rates.
• Group B: Moderately deep, moderately well-drained soils with moderate infiltration rates (e.g., sandy loam).
• Group C: Soils with a layer that impedes downward movement of water, or clayey loams. Low infiltration rates.
• Group D (High runoff potential): Clay soils with very low infiltration rates and high swelling potential.

Runoff Depth Equations

The depth of direct runoff ($Q_d$, in inches) is given by: $Q_d = \frac{(P - I_a)^2}{P - I_a + S} \quad \text{for } P > I_a$ Where $P$ is the total rainfall depth (inches) and $I_a$ is the initial abstraction (inches). The standard empirical relationship assumes: $I_a = 0.2 \cdot S$ Substituting this yields the standard runoff equation: $Q_d = \frac{(P - 0.2S)^2}{P + 0.8S} \quad \text{for } P > 0.2S$ If $P \le 0.2S$, then $Q_d = 0$.

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4. Hydrograph Analysis

A hydrograph is a continuous plot of discharge versus time at a specific point along a channel. It represents the integrated response of a watershed to a precipitation event.

Hydrograph Components

1. Rising Limb: The portion of the curve where discharge increases due to accumulating runoff.
2. Peak Discharge ($Q_p$): The maximum flow rate recorded.
3. Recession Limb: The downward portion of the curve representing the release of stored water from the channel and soils.
4. Baseflow: The dry-weather flow of a stream, typically supplied by groundwater.
5. Direct Runoff: The flow resulting directly from storm precipitation (calculated by subtracting baseflow from the total hydrograph).

Unit Hydrograph (UH) Theory

A unit hydrograph represents the direct runoff hydrograph resulting from 1 inch of excess rainfall (direct runoff) generated uniformly over the watershed at a constant rate for a specified duration ($D$-hours). UH theory relies on two principles:

• Linearity (Proportionality): The ordinates of a direct runoff hydrograph are directly proportional to the runoff depth. If a $D$-hour UH has a peak of 100 cfs, then 2.5 inches of excess rainfall over $D$ hours will produce a peak of 250 cfs.
• Superposition (Additivity): The total runoff hydrograph from multiple consecutive periods of excess rainfall is the sum of the individual lagged runoff hydrographs.

S-Hydrograph Method

To convert a unit hydrograph of duration $D_1$ to a duration $D_2$:

1. Construct an S-hydrograph by lagging the $D_1$-hour unit hydrograph by $D_1$ intervals and summing the ordinates.
2. Lag the S-hydrograph by the target duration $D_2$.
3. Subtract the lagged S-hydrograph from the original S-hydrograph.
4. Multiply the resulting ordinates by the ratio $D_1 / D_2$ to obtain the new $D_2$-hour unit hydrograph.

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5. Stormwater Management and Routing

Urbanization increases the peak discharge and volume of runoff. Stormwater management systems, such as detention basin design projects, are utilized to mitigate these impacts by temporarily storing stormwater and releasing it at a rate that matches pre-development conditions.

Reservoir Routing (Modified Puls Method)

Routing tracks the shape and timing of a flood wave as it moves through a channel or reservoir. For reservoirs, the continuity equation is: $\frac{I_1 + I_2}{2} - \frac{O_1 + O_2}{2} = \frac{S_2 - S_1}{\Delta t}$ Where $I$ is inflow, $O$ is outflow, $S$ is storage, and $\Delta t$ is the routing time step. The Modified Puls method rearranges this to: $(I_1 + I_2) + \left(\frac{2S_1}{\Delta t} - O_1\right) = \frac{2S_2}{\Delta t} + O_2$ A routing table is constructed using a relationship curve of $O$ vs. $\frac{2S}{\Delta t} + O$. This enables the determination of outflow ($O_2$) and storage ($S_2$) at each successive time step. As water is routed through a basin, it experiences:

• Attenuation: The peak outflow ($O_p$) is lower than the peak inflow ($I_p$).
• Lag: The peak outflow occurs later in time than the peak inflow.

Outlet Structure Hydraulics

• Orifice Flow: Submerged circular or rectangular openings. $Q = C_d \cdot A \cdot \sqrt{2gH}$ Where $C_d$ is the discharge coefficient (typically $\approx 0.60$), $A$ is the orifice area ($\text{ft}^2$), $g$ is $32.2 \text{ ft/s}^2$, and $H$ is the head (ft) from the water surface to the centroid of the orifice.
• Weir Flow: Open-channel overflow.
  • Rectangular Weir: $Q = C_w \cdot L \cdot H^{1.5}$ Where $C_w$ is the weir coefficient, $L$ is the crest length (ft), and $H$ is the head (ft) above the crest.
  • V-Notch (Triangular) Weir: $Q = C_v \cdot \tan\left(\frac{\theta}{2}\right) \cdot H^{2.5}$ Where $\theta$ is the notch angle. This structure is highly sensitive to small head changes, making it ideal for controlling low flows.