The runoff curve number (also called a curve number or simply CN) is an empirical parameter used in hydrology for predicting direct runoff or infiltration from rainfall excess.^[1] The curve number method was developed by the USDA Natural Resources Conservation Service, which was formerly called the Soil Conservation Service or SCS — the number is still popularly known as a "SCS runoff curve number" in the literature. The runoff curve number was developed from an empirical analysis of runoff from small catchments and hillslope plots monitored by the USDA. It is widely used and is an efficient method for determining the approximate amount of direct runoff from a rainfall event in a particular area.

Definition

The runoff curve number is based on the area's hydrologic soil group, land use, treatment and hydrologic condition. References, such as from USDA^[1] indicate the runoff curve numbers for characteristic land cover descriptions and a hydrologic soil group.

The runoff equation is:

${\displaystyle Q={\begin{cases}0&{\text{for ))P\leq I_{a}\\{\frac {(P-I_{a})^{2))((P-I_{a))+S))&{\text{for ))P>I_{a}\end{cases))}$

where

${\displaystyle Q}$ is runoff ([L]; in)

${\displaystyle P}$ is rainfall ([L]; in)

${\displaystyle S}$ is the potential maximum soil moisture retention after runoff begins ([L]; in)

${\displaystyle I_{a))$ is the initial abstraction ([L]; in), or the amount of water before runoff, such as infiltration, or rainfall interception by vegetation; historically, it has generally been assumed that ${\displaystyle I_{a}=0.2S}$, although more recent research has found that ${\displaystyle I_{a}=0.05S}$ may be a more appropriate relationship in urbanized watersheds where the CN is updated to reflect developed conditions.^[2]

The runoff curve number, ${\displaystyle CN}$, is then related

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

${\displaystyle CN}$ has a range from 30 to 100; lower numbers indicate low runoff potential while larger numbers are for increasing runoff potential. The lower the curve number, the more permeable the soil is. As can be seen in the curve number equation, runoff cannot begin until the initial abstraction has been met. It is important to note that the curve number methodology is an event-based calculation, and should not be used for a single annual rainfall value, as this will incorrectly miss the effects of antecedent moisture and the necessity of an initial abstraction threshold.

Selection

The NRCS curve number is related to soil type, soil infiltration capability, land use, and the depth of the seasonal high water table. To account for different soils' ability to infiltrate, NRCS has divided soils into four hydrologic soil groups (HSGs). They are defined as follows.^[1]

• HSG Group A (low runoff potential): Soils with high infiltration rates even when thoroughly wetted. These consist chiefly of deep, well-drained sands and gravels. These soils have a high rate of water transmission (final infiltration rate greater than 0.30 in (7.6 mm) per hour).
• HSG Group B: Soils with moderate infiltration rates when thoroughly wetted. These consist chiefly of soils that are moderately deep to deep, moderately well drained to well drained with moderately fine to moderately coarse textures. These soils have a moderate rate of water transmission (final infiltration rate of 0.15–0.30 in (3.8–7.6 mm) per hour).
• HSG Group C: Soils with slow infiltration rates when thoroughly wetted. These consist chiefly of soils with a layer that impedes downward movement of water or soils with moderately fine to fine textures. These soils have a slow rate of water transmission (final infiltration rate 0.05–0.15 in (1.3–3.8 mm) per hour).
• HSG Group D (high runoff potential): Soils with very slow infiltration rates when thoroughly wetted. These consist chiefly of clay soils with a high swelling potential, soils with a permanent high water table, soils with a claypan or clay layer at or near the surface, and shallow soils over nearly impervious materials. These soils have a very slow rate of water transmission (final infiltration rate less than 0.05 in (1.3 mm) per hour).

Selection of a hydrologic soil group should be done based on measured infiltration rates, soil survey (such as the NRCS Web Soil Survey), or judgement from a qualified soil science or geotechnical professional. The table below presents curve numbers for antecedent soil moisture condition II (average moisture condition). To alter the curve number based on moisture condition or other parameters, see Adjustments.

Values

Adjustments

Runoff is affected by the soil moisture before a precipitation event, the antecedent moisture condition (AMC). A curve number, as calculated above, may also be termed AMC II or ${\displaystyle CN_{II))$, or average soil moisture. The other moisture conditions are dry, AMC I or ${\displaystyle CN_{I))$, and moist, AMC III or ${\displaystyle CN_{III))$. The curve number can be adjusted by factors to ${\displaystyle CN_{II))$, where ${\displaystyle CN_{I))$ factors are less than 1 (reduce ${\displaystyle CN}$ and potential runoff), while ${\displaystyle CN_{III))$ factor are greater than 1 (increase ${\displaystyle CN}$ and potential runoff). The AMC factors can be looked up in the reference table below. Find the CN value for AMC II and multiply it by the adjustment factor based on the actual AMC to determine the adjusted curve number.

Initial abstraction ratio adjustment

The relationship ${\displaystyle I_{a}=0.2S}$ was derived from the study of many small, experimental watersheds . Since the history and documentation of this relationship are relatively obscure, more recent analysis used model fitting methods to determine the ratio of ${\displaystyle I_{a))$ to ${\displaystyle S}$ with hundreds of rainfall-runoff data from numerous U.S. watersheds. In the model fitting done by Hawkins et al. (2002)^[2] found that the ratio of ${\displaystyle I_{a))$ to ${\displaystyle S}$ varies from storm to storm and watershed to watershed and that the assumption of ${\displaystyle I_{a}/S=0.20}$ is usually high. More than 90 percent of ${\displaystyle I_{a}/S}$ ratios were less than 0.2. Based on this study, use of ${\displaystyle I_{a}/S}$ ratios of 0.05 rather than the commonly used value of 0.20 would seem more appropriate. Thus, the CN runoff equation becomes:

${\displaystyle Q={\begin{cases}0&{\text{for ))P\leq 0.05S\\{\frac {(P-0.05S_{0.05})^{2)){P+0.95S_{0.05))}&{\text{for ))P>0.05S\end{cases))}$

In this equation, note that the values of ${\displaystyle S_{0.05))$ are not the same as the one used in estimating direct runoff with an ${\displaystyle I_{a}/S}$ ratio of 0.20, because 5 percent of the storage is assumed to be the initial abstraction, not 20 percent. The relationship between ${\displaystyle S_{0.05))$ and ${\displaystyle S_{0.20))$ was obtained from model fitting results, giving the relationship:

${\displaystyle S_{0.05}=1.33{S_{0.20))^{1.15))$

The user, then, must do the following to use the adjusted 0.05 initial abstraction ratio:

1. Use the traditional tables of curve numbers to select the value appropriate for your watershed.
2. Calculate ${\displaystyle S_{0.20))$ using the traditional equation:${\displaystyle S={\frac {1000}{CN))-10}$
3. Convert this S value to ${\displaystyle S_{0.05))$ using the relationship above.
4. Calculate the runoff depth using the CN runoff equation above (with 0.05 substituted for the initial abstraction ratio).