# Runoff curve number

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.

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:

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

where

$Q$ is runoff ([L]; in)
$P$ is rainfall ([L]; in)
$S$ is the potential maximum soil moisture retention after runoff begins ([L]; in)
$I_a$ 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 $I_a = 0.2S$, although more recent research has found that $I_a = 0.05S$ may be a more appropriate and accurate relationship.[2]

The runoff curve number, $CN$, is then related

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

$CN$ 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.

## Curve Number 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 0.3 in./h).
• 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 course textures. These soils have a moderate rate of water transmission (final infiltration rate of 0.15 to 0.30 in./h).
• 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 to 0.15 in./h).
• 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./h).

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 the CN adjustment section.

### Runoff curve numbers

#### Fully developed urban areas (vegetation established)

 Cover description Curve numbers for hydrologic soil group A B C D Open space (lawns, parks, golf courses, cemeteries, etc.) Poor condition (grass cover <50%) 68 79 86 89 Fair condition (grass cover 50 to 75%) 49 69 79 84 Good condition (grass cover >75%) 39 61 74 80 Impervious areas: Paved parking lots, roofs, driveways, etc. (excluding right of way) 98 98 98 98 Streets and roads: Paved; curbs and storm sewers (excluding right-of-way) 98 98 98 98 Paved; open ditches (including right-of-way) 83 89 92 93 Gravel (including right of way) 76 85 89 91 Dirt (including right-of-way) 72 82 87 89 Western desert urban areas: Natural desert landscaping (pervious area only) 63 77 85 88 Artificial desert landscaping (impervious weed barrier, desert shrub with 1- to 2-inch sand or gravel mulch and basin borders) 96 96 96 96 Urban districts: Commercial and business (85% imp.) 89 92 94 95 Industrial (72% imp.) 81 88 91 93 Residential districts by average lot size: 1/8 acre or less (town houses) (65% imp.)/small> 77 85 90 92 1/4 acre (38% imp.) 61 75 83 87 1/3 acre (30% imp.) 57 72 81 86 1/2 acre (25% imp.) 54 70 80 85 1 acre (20% imp.) 51 68 79 84 2 acres (12% imp.) 46 65 77 82

#### Developing urban areas

 Cover description Curve numbers for hydrologic soil group A B C D Newly graded areas (pervious areas only, no vegetation) 77 86 91 94

#### Cultivated agricultural lands

 Cover description Curve numbers for hydrologic soil group Cover type Treatment[A] Hydrologic condition A B C D Fallow Bare soil -- 77 86 91 94 Crop residue cover (CR) Poor 76 85 90 93 Good 74 83 88 90 Row crops Straight row (SR) Poor 72 81 88 91 Good 67 78 85 89 SR + CR Poor 71 80 87 90 Good 64 75 82 85 Contoured (C) Poor 70 79 84 88 Good 65 75 82 86 C + CR Poor 69 78 83 87 Good 64 74 81 85 Contoured & terraced (C&T) Poor 66 74 80 82 Good 62 71 78 81 C&T + R Poor 65 73 79 81 Good 61 70 77 80 Small grain SR Poor 65 76 84 88 Good 63 75 83 87 SR + CR Poor 64 75 83 86 Good 60 72 80 84 C Poor 63 74 82 85 Good 61 73 81 84 C + CR Poor 62 73 81 84 Good 60 72 80 83 C&T Poor 61 72 79 82 Good 59 70 78 81 C&T + R Poor 60 71 78 81 Good 58 69 77 80 Close-seeded or broadcast legumes or rotation meadow SR Poor 66 77 85 89 Good 58 72 81 85 C Poor 64 75 83 85 Good 55 69 78 83 C&T Poor 63 73 80 83 Good 51 67 76 80
 A Crop residue cover applies only if residue is on at least 5% of the surface throughout the year.

#### Other agricultural lands

 Cover description Curve numbers for hydrologic soil group Cover type Hydrologic condition A B C D Pasture, grassland, or range—continuous forage for grazing.A Poor 68 79 86 89 Fair 49 69 79 84 Good 39 61 74 80 Meadow—continuous grass, protected from grazing and generally mowed for hay. --- 30 58 71 78 Brush—brush-weed-grass mixture with brush the major element.B Poor 48 67 77 83 Fair 35 56 70 77 Good 30C 48 65 73 Woods—grass combination (orchard or tree farm).D Poor 57 73 82 86 Fair 43 65 76 82 Good 32 58 72 79 Woods.E Poor 45 66 77 83 Fair 36 60 73 79 Good 30 55 70 77 Farmsteads—buildings, lanes, driveways, and surrounding lots. --- 59 74 82 86
 A Poor: <50% ground cover or heavily grazed with no mulch; Fair: 50-75% ground cover and not heavily grazed; Good: >75% ground cover and light or only occasionally grazed. B Poor: <50% ground cover; Fair: 50-75% ground cover; Good: >75% ground cover. C Actual curve number is less than 30; use CN = 30 for runoff computation. D CN's shown were computed for areas with 50% woods and 50% grass (pasture) cover. Other combinations of conditions may be computed from the CN's for woods and pasture. E Poor: Forest litter, small trees, and brush are destroyed by heavy grazing or regular burning; Fair: Woods are grazed but not burned, and some forest litter covers the soil; Good: Woods are protected from grazing, and litter and brush adequately cover the soil.

#### Arid and semiarid rangelands

 Cover description Curve numbers for hydrologic soil group Cover type Hydrologic conditionA AB B C D Herbaceuous—mixture of grass, weeds, and low-growing brush, with brush the minor element Poor 80 87 93 Fair 71 81 89 Good 62 74 85 Oak-aspen—mountain brush mixture of oak brush, aspen, mountain mahogany, bitter brush, maple, and other brush Poor 66 74 79 Fair 48 57 63 Good 30 41 48 Pinyon-juniper—pinyon, juniper, or both; grass understory Poor 75 85 89 Fair 58 73 80 Good 41 61 71 Sagebrush with grass understory Poor 67 80 85 Fair 51 63 70 Good 35 47 55 Desert shrub—major plants include saltbush, geasewood, creosotebush, blackbrush, bursage, palo verde, mesquite, and cactus. Poor 63 77 85 88 Fair 55 72 81 86 Good 49 68 79 84
 A Poor: <30% ground cover (litter, grass, and brush overstory); Fair: 30 to 70% ground cover; Good: >70% ground cover. B Curve numbers for group A have been developed only for desert shrub.

Runoff is affected by the soil moisture before a precipitation event, or the antecedent moisture condition (AMC). A curve number, as calculated above, may also be termed AMC II or $CN_{II}$, or average soil moisture. The other moisture conditions are dry, AMC I or $CN_{I}$, and moist, AMC III or $CN_{III}$. The curve number can be adjusted by factors to $CN_{II}$, where $CN_{I}$ factors are less than 1 (reduce $CN$ and potential runoff), while $CN_{III}$ factor are greater than 1 (increase $CN$ 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.

### Adjustments to Select Curve Number for Soil Moisture Conditions[3]

 Curve Number (AMC II) Factors to Convert Curve Number for AMC II to AMC I or III AMC I (dry) AMC III (wet) 10 0.40 2.22 20 0.45 1.85 30 0.50 1.67 40 0.55 1.50 50 0.62 1.40 60 0.67 1.30 70 0.73 1.21 80 0.79 1.14 90 0.87 1.07 100 1.00 1.00

The relationship $I_a = 0.2S$ 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 $I_a$ to $S$ 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 $I_a$ to $S$ varies from storm to storm and watershed to watershed and that the assumption of $I_a/S=0.20$ is usually high. More than 90 percent of $I_a/S$ ratios were less than 0.2. Based on this study, use of $I_a/S$ ratios of 0.05 rather than the commonly used value of 0.20 would seem more appropriate. Thus, the CN runoff equation becomes:

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

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

$S_{0.05}=1.33{S_{0.20}}^{1.15}$

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 $S_{0.20}$ using the traditional equation:
$S = \frac{1000}{CN} - 10$
1. Convert this S value to $S_{0.05}$ using the relationship above.
2. Calculate the runoff depth using the CN runoff equation above (with 0.05 substituted for the initial abstraction ratio).