4.3 Hydrograph Characteristics, Curve Numbers, and Runoff Estimation Equations

Anatomy of a Storm Runoff Hydrograph

A hydrograph is a continuous graph plotting volumetric stormwater discharge rate ($Q$, typically in cubic feet per second, $\text{cfs}$, or cubic meters per second, $\text{m}^3/\text{s}$) on the vertical ($y$) axis against elapsed time ($t$, in minutes or hours) on the horizontal ($x$) axis at a specific watershed outlet or channel cross section.

Five Structural Elements of a Storm Hydrograph

1. Baseflow: The background dry-weather discharge sustained by groundwater seepage before surface runoff begins and after storm runoff recedes.
2. Rising Limb (Concentration Curve): The ascending portion of the hydrograph once rainfall exceeds initial abstractions ($I_a$). Its slope is governed by storm intensity and how rapidly the sub-basin concentrates overland flow.
3. Peak Discharge ($Q_p$) and Time to Peak ($T_p$): The crest of the hydrograph representing the maximum instantaneous flow rate ($\text{cfs}$). Time to Peak ($T_p$) is the elapsed time from the start of excess rainfall to the occurrence of $Q_p$. Conveyance channels, culverts, check dams, and basin spillways must be designed and inspected to pass $Q_p$ without overtopping or scouring.
4. Recession (Falling) Limb (Depletion Curve): The gradually descending curve after $Q_p$ as water stored on slopes, in depressions, and in shallow channel storage drains out of the watershed. The recession limb is typically flatter and longer than the rising limb.
5. Direct Runoff Volume: Because multiplying flow rate ($\text{ft}^3/\text{s}$) by time ($\text{s}$) equals volume ($\text{ft}^3$), the area under the hydrograph curve (above the baseflow separation line) equals the total volume of direct storm runoff (expressed in $\text{ft}^3$ or $\text{acre-feet}$). Temporary sediment basins and permanent detention ponds are sized using this volume.

How Construction and Land Development Alter the Hydrograph

When a forested or meadow watershed is cleared, stripped of topsoil, compacted by grading equipment, or covered with impervious pavement, its hydrograph undergoes five predictable transformations:

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1. The Rational Method for Peak Discharge ($Q = C \times i \times A$)

The Rational Method is the most widely used empirical equation for estimating the peak runoff rate ($Q_p$) on small catchments (typically limited to drainage areas $\le 200\text{ acres}$, and most accurate for uniform construction sub-basins $\le 20\text{ acres}$). The Rational Method calculates only the instantaneous peak flow rate ($\text{cfs}$) to size swales, ditches, inlets, and pipes; it does not directly compute total runoff volume or a full hydrograph.

$$Q = C \times i \times A$$

• $Q$ = Peak discharge rate in cubic feet per second ($\text{cfs}$; in U.S. customary units, $1\text{ acre-inch/hour} = 1.008\text{ cfs}$, so the commonly used Q = CiA approximation neglects a conversion factor of 1.008; exact unit conversion uses 1.008CiA).
• $C$ = Dimensionless runoff coefficient ($0 < C < 1.0$) representing the fraction of rainfall rate that becomes peak surface runoff:
  • Undisturbed woodland or forest: $C = 0.05\text{ to }0.20$
  • Established turf/meadow on sandy or loamy soil: $C = 0.10\text{ to }0.25$
  • Bare, newly graded construction site (compacted silt or clay): $C = 0.50\text{ to }0.75$
  • Impervious asphalt, concrete pavement, or roofs: $C = 0.85\text{ to }0.95$
  • If a sub-basin contains multiple surface types, calculate an area-weighted composite runoff coefficient: $C_w = \frac{\sum (C_j A_j)}{\sum A_j}$.
• $i$ = Average rainfall intensity in inches per hour ($\text{in/hr}$), read from the local IDF curve for the required return period ($T$) at a storm duration equal to the watershed's Time of Concentration ($T_c$).
• $A$ = Contributing watershed drainage area in acres ($\text{ac}$).

Worked Calculation: Pre-Clearing vs. Active Construction Grading

Consider a $12\text{-acre}$ sub-basin discharging into a perimeter diversion swale designed for the $10\text{-year}$ storm:

• Pre-Construction (Wooded Meadow): $C = 0.15$ and $T_c = 30\text{ minutes}$. Reading the local $10\text{-year}$ IDF curve at a $30\text{-minute}$ duration gives $i = 3.2\text{ in/hr}$:

$$Q_{\text{pre}} = 0.15 \times 3.2\text{ in/hr} \times 12\text{ ac} = 5.76\text{ cfs}$$

• Active Mass Grading (Bare Compacted Clay Loam): Clearing and scraper traffic raise the runoff coefficient to $C = 0.65$ and cut the Time of Concentration to $T_c = 12\text{ minutes}$. Entering the $10\text{-year}$ IDF curve at the shorter $12\text{-minute}$ duration yields a higher design intensity of $i = 5.0\text{ in/hr}$:

$$Q_{\text{construction}} = 0.65 \times 5.0\text{ in/hr} \times 12\text{ ac} = 39.0\text{ cfs}$$

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2. The NRCS (SCS) Curve Number Method (TR-55)

Developed by the USDA Natural Resources Conservation Service in Technical Release 55 (TR-55), the Curve Number ($CN$) Method calculates direct surface runoff depth ($Q$, in inches) and total runoff volume from a $24\text{-hour}$ design storm for watersheds up to $2,000\text{ acres}$:

$$Q = \frac{(P - I_a)^2}{(P - I_a) + S}$$

where:

• $Q$ = Direct surface runoff depth over the watershed ($\text{inches}$).
• $P$ = Total storm rainfall depth ($\text{inches}$).
• $S$ = Potential maximum retention after runoff begins ($\text{inches}$), calculated from the dimensionless Runoff Curve Number ($CN$, ranging from $30\text{ to }100$) as:

$$S = \frac{1000}{CN} - 10$$

• $I_a$ = Initial abstraction ($\text{inches}$)—all rainfall losses that occur before surface runoff begins, including canopy interception, surface depression storage, evaporation, and initial infiltration prior to ponding. Using the standard NRCS empirical approximation $I_a = 0.2S$, the runoff equation becomes:

$$Q = \frac{(P - 0.2S)^2}{P + 0.8S} \quad \text{(valid only when } P > 0.2S\text{; if } P \le 0.2S\text{, then } Q = 0\text{)}$$

A higher Curve Number ($CN$) reflects a less permeable or more disturbed surface (such as newly graded bare construction soil, where $CN = 77$ for HSG A, $86$ for HSG B, $91$ for HSG C, and $94$ for HSG D, or impervious pavement where $CN = 98$). As $CN$ increases toward $100$, potential retention $S$ shrinks toward $0$, initial abstraction $I_a$ approaches $0$, and runoff depth $Q$ approaches total rainfall $P$.

Worked Calculation: NRCS Curve Number Runoff Depth and Basin Volume

A $10\text{-acre}$ newly graded, bare-earth construction site on Group B soil ($CN = 86$) experiences a $2\text{-year, 24-hour}$ storm of $P = 3.50\text{ inches}$.

1. Calculate Potential Maximum Retention ($S$):

$$S = \frac{1000}{86} - 10 = 11.63 - 10 = 1.63\text{ inches}$$

1. Verify Initial Abstraction ($I_a$):

$$I_a = 0.2 \times S = 0.2 \times 1.63 = 0.326\text{ inches}$$

Because $P = 3.50\text{ in} > 0.326\text{ in}$, direct surface runoff occurs.

1. Calculate Direct Runoff Depth ($Q$):

$$Q = \frac{(3.50 - 0.326)^2}{3.50 + 0.8(1.63)} = \frac{(3.174)^2}{3.50 + 1.304} = \frac{10.074}{4.804} \approx 2.10\text{ inches}$$

1. Convert Runoff Depth ($Q$) to Total Runoff Volume ($V_r$) in Cubic Feet:

$$V_r = 2.10\text{ in} \times \left(\frac{1\text{ ft}}{12\text{ in}}\right) \times 10\text{ ac} \times 43,560\text{ ft}^2/\text{ac} = 0.175\text{ ft} \times 435,600\text{ ft}^2 = 76,230\text{ ft}^3 \text{ (or } 1.75\text{ acre-feet)}$$

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3. Open-Channel Hydraulics: Continuity and Manning's Equations

Once runoff enters a diversion swale, roadside ditch, or storm pipe, CESSWI inspectors evaluate channel capacity and erosive velocity using two fundamental hydraulic equations:

The Continuity Equation

$$Q = V \times A$$

where $Q$ is volumetric flow rate ($\text{cfs}$ or $\text{ft}^3/\text{s}$), $V$ is average cross-sectional velocity ($\text{ft/s}$), and $A$ is the wetted cross-sectional flow area ($\text{ft}^2$). For any given flow rate $Q$, constricting the cross-sectional area ($A$) forces velocity ($V$) to increase proportionally.

Manning's Open-Channel Flow Equation

$$V = \frac{1.486}{n} R^{2/3} S^{1/2} \quad \text{and} \quad Q = \frac{1.486}{n} A R^{2/3} S^{1/2} \quad \text{(U.S. Customary Units)}$$

• $V$ = Average flow velocity ($\text{ft/s}$). (In SI metric units, the constant $1.486$ is replaced by $1.0$: $V = \frac{1}{n} R^{2/3} S^{1/2}$ in $\text{m/s}$.)
• $n$ = Manning's roughness coefficient, which quantifies boundary friction resistance:
  • Smooth HDPE/PVC pipe or troweled concrete: $n \approx 0.011\text{ to }0.013$
  • Bare, newly excavated earth channel: $n \approx 0.018\text{ to }0.025$
  • Angular rock riprap lining: $n \approx 0.035\text{ to }0.045$
  • Dense turf grass or Turf Reinforcement Mat (TRM): $n \approx 0.040\text{ to }0.150$
• $R$ = Hydraulic radius ($\text{ft}$), defined as the wetted cross-sectional area ($A$, in $\text{ft}^2$) divided by the wetted perimeter ($P_w$, in $\text{ft}$)—the length of the channel bottom and side slopes in direct contact with water (excluding the top free water surface exposed to air):

$$R = \frac{A}{P_w}$$

• $S$ = Longitudinal channel bed slope ($\text{ft/ft}$, vertical drop divided by horizontal run).

Applying Manning's Equation During Field Inspections

Every erosion control measure installed in a conveyance channel manipulates one of the variables in Manning's equation to keep flow velocity ($V$) below the soil's permissible non-erodible velocity (typically $1.5\text{ to }2.5\text{ ft/s}$ for bare sandy/silty soils, $3.5\text{ to }5.0\text{ ft/s}$ for established sod, and $6.0\text{ to }12.0+\text{ ft/s}$ for heavy rock riprap or TRMs):

• Increasing Roughness ($n \uparrow \implies V \downarrow$): Lining a bare earth ditch ($n = 0.020$) with angular riprap or a turf reinforcement mat ($n = 0.040$) doubles $n$ in the denominator, cutting computed velocity in half if hydraulic radius and slope are held fixed; at fixed discharge, water depth and hydraulic radius change and a new hydraulic solution is needed and protecting the subgrade from scour.
• Reducing Hydraulic Radius ($R \downarrow \implies V \downarrow$): Reshaping a deep, narrow V-shaped ditch into a wide, shallow parabolic or trapezoidal swale increases the wetted perimeter ($P_w$) relative to area ($A$), lowering $R$ and slowing velocity.
• Flattening Effective Slope ($S \downarrow \implies V \downarrow$): Installing check dams (spaced so the toe of the upstream check dam sits at the same elevation as the crest of the next check dam downstream) flattens the energy grade slope ($S$), creating low-velocity ponded reaches that trap coarse sediment.

Limits of runoff methods

The modified Rational Method extends the peak-rate concept using simplified inflow hydrographs for detention screening; storage is obtained from the difference between inflow and allowable outflow over time, not from a peak cfs value alone. A unit hydrograph represents the watershed response to one unit depth of excess rainfall over a specified duration; scaled, time-shifted responses can be added under its linear assumptions. NRCS soil-cover/curve-number methods estimate runoff depth, and a separate routing or hydrograph method is needed for the peak and detention design. Designers must select the applicable regional rainfall distribution: the legacy Type I/IA/II/III map is instructional background, not a replacement for updated regional NRCS/NOAA rainfall data. Inspectors read the method and inputs stated in the approved calculations. NRCS hydrologic technical resources were checked October 13, 2026.