Garden with Insight v1.0 Help: Hydrology - Surface Runoff - Runoff Volume

Surface runoff is predicted for daily rainfall by using the SCS curve number equation (U.S. Department of Agriculture, Soil Conservation Service 1972) [Equation1] where Q is daily runoff in mm, R is daily rainfall in mm, and s is a retention parameter. The retention parameter, s, varies (a) among watersheds because soils, land use, management, and slope all vary, and (b) with time because of changes in soil water content.

Equation 1:

if R > 0.2 * s
Q = sqr(R - 0.2 * s) / (R + 0.8 * s)
else
Q = 0.0
Code:
same, modifications are made for impact of dikes
Variables:
Q = RunoffVolume_mm
R = rainfallForDay_mm
s = retentionParameter_mm

The parameter s is related to curve number (CN) by the SCS equation (U.S. Dept. of Ag., Soil Conservation Service 1972) [Equation 2]. The constant, 254, in equation 2 gives s in mm. Thus, R and Q are also expressed in mm.

Equation 2:

S = 254 * (100 / CN - 1)
Code:
same
Variables:
CN = curveNumber

CN2- the curve number for moisture condition 2, or average curve number -- can be obtained easily for any area by using the SCS hydrology handbook (U.S. Dept. Ag., SCS 1972). The handbook tables consider soils, land use, and management. Asuuming the handbook CN2 value is appropriate for a 5% slope, we developed the following equation for adjusting that value for other slopes. [Equation 3] where CN2s is the handbook CN2 value adjusted for slope, CN3 is the curve number for moisture condition 3 (wet), and S is the average slope of the watershed. Values of CN1, the curve number for moisture condition 1 (dry), and CN3 corresponding to CN2 are also tabulated in the handbook. For computing purposes, CN1 and CN3 were related to CN2 with the equations [Equation 4] [Equation 5].

Equation 3:

CN(2s) = 1/3 * (CN(3) - CN(2)) * (1 - 2 * exp(-13.86 * S)) + CN(2)
Code:
same
Variables:
CN(2s) = CurveNumberAvgForSlope
CN(2) = curveNumberAvg
CN(3) = curveNumberWet
S = slopeSteepness_mPm

Equation 4:

CN(1) = CN(2) - 20 * (100 - CN(2)) / (100 - CN(2) + exp(2.533 - 0.0636 * (100 - CN(2))))
Code:
same except for added upper bound at 0.4 * CN(2)
Variables:
CN(1) = CurveNumberDry
CN(2) = curveNumberAvg

Equation 5:

CN(3) = CN(2) * exp(0.00673 * (100 - CN(2)))
Code:
same
Variables:
CN(3) = CurveNumberWet
CN(2) = curveNumberAvg

Fluctuations in soil water content cause the retention parameter to change according to the equation [Equation 6] where s1 is the value of s associated with CN1, FFC is the fraction of field capacity, and w1 and w2 are shape parameters.

Equation 6:

s = s1 * (1 - FFC / (FFC + exp(w1 - w2 * FFC)))
Code:
if FFC > 0
s = s1 * (1 - FFC * 100 / (FFC * 100 + exp(w1 - w2 * FFC * 100)))
else
s = s1 * (1 - FFC)
Variables:
s1 = retentionParameter_mm
FFC = fractionOfFieldCapacity
w1 = shapeParamOne
w2 = shapeParamTwo

FFC is computed with the equation [Equation 7] where SW is the soil water content in the root zone, WP is the wilting point water content (1,500 kPa for many soils) and FC is the field capacity water content (33 kPa for many soils).

Equation 7:

FFC = (SW - WP) / (FC - WP)
Code:
same
Variables:
FFC = FractionOfFieldCapacity
SW = waterContent_mm
WP = wiltingPoint_mm
FC = fieldCapacity_mm

Values for w1 and w2 are obtained from a simultaneous solution of equation 6 according to the assumptions that s=s2 when FFC=0.6 and s=s3 when (SW-FC)/(PO-FC) = 0.5 [Equation 8] [Equation 9] where s3 is the CN3 retention parameter and the porosity-field capacity ratio POFC is computed with the equation [Equation 10] where PO is the porosity of the soil layer l. Equations 8 and 9 assure that CN1 corresponds with the wilting point and that the curve number cannot exceed 100.

Equations 8, 9:

w1 = ln(60 / (1 - s2/s1) - 60) + 60 * w2
w2 = (ln(60 / (1 - s2/s1) - 60) - ln(POFC / (1 - s3/s1) - POFC)) / (POFC - 60)
Code:
w1 = 1 - s2/smx + 60
w2 = 1 - s3/smx + POFC
x1 = leftside(w1) or 60
w1 = rightside(w1) or 1 - s2/smx
x2 = leftside(w2) or POFC
w2 = rightside(w2) or 1 - s3/s2
w1 = ln(x1/w1 - x1) + x1 * w2
= ln(60 / (1 - s2/smx) - 60) + 60 * w2
= same except for s2/s1 is replaced by s2/smx
w2 = (ln(x1/w1 - x1) - ln(x2/w2 - x2)) / (x2 - x1)
= (ln(60 / (1 - s2/smx) - 60) - ln(POFC / (1 - s3/smx) - POFC)) / (POFC - 60)
= same except for s2/s1 is replaced by s2/smx and s3/s1 by s3/smx
(if you assume s1 is the same as smx here, they are the same)
( Note: 60 was changed to 0.6 and POFC changed from 0-100 to 0-1 later )
Variables:
w1 = curveNumberCoeffsPtr[0]
w2 = curveNumberCoeffsPtr[1]
s2 = retentionParamAvg_mm
s3 = retentionParamWet_mm
smx = retentionParamFinal_mm
POFC = porosityToFieldCapacityRatio

Equation 10:

POFC = 100 + 50 * (sum(1-M)(PO - FC) / sum(1-M)(FC - WP))
Code:
POFC = 100 + 50 * ((sum(1-M)(PO - WP) / sum(1-m)(FC - WP)) - 1.0) + 0.5
(later amended to POFC = 1.0 + 0.5 * ((sum(1-M)(PO - WP) / sum(1-m)(FC - WP)) - 1.0) + 0.005 to make 0-1 instead of 0-100 )
Variables:
POFC = PorosityToFieldCapacityRatio
PO = porosity_mm
FC = fieldCapacity_mm
WP = wiltingPoint_mm

The FFC value obtained in equation 7 represents soil water uniformly distributed through the top 1.0 m of soil. Runoff estimates can be improved if the depth distribution of soil water is known. For example, water distributed near the soil surface results in more runoff than the same volume of water uniformly distributed through the top meter of soil. Also, a soil surface associated with a uniform distribution of soil water results in more runoff than a soil surface that is dry. Since EPIC estimates the water content of each soil layer daily, the depth distribution is available. The effect of depth distribution on runoff is expressed in the depth weighting class function [Equation 11] where FFC* is the depth weighted FFC value for use in equation 6, Z is the depth in m to the bottom of soil layer l, and M is the number of soil layers. Equation 11 performs two functions: (1) it reduces the influence of lower layers because FFCl is divided by Zl and (2) it gives proper weight to thick layers relative to thin layers because FFC is multiplied by the layer thickness.

Equation 11:

FFC* = sum(1-M)(FFC * (Z(l) - Z(l-1)) / Z(l)) / sum(1-M)((Z(l) - Z(l-1))/Z(l))
Z(l) <= 1.0 m
Code:
same
Variables:
FFC* = DepthWeightedFractionOfFieldCapacity
FFC = fractionOfFieldCapacity_frn
Z = depth_m
M = numLayers

There is also a provision for estimating runoff from frozen soil. If the temperature in the second soil layer is less than 0 degrees C, the retention parameter is reduced using the equation [Equation 12], where sf is the retention parameter for frozen ground. Equation 12 increased runoff for frozen soils but allows significant infiltration when soils are dry.

Equation 12:

s(f) = 0.1 s
Code:
Variables:
s(f) = ModifiedRetentionParamForFrozenGround
s = retentionParam

The final step in estimating the runoff volume is an attempt to account for uncertainty. The retention parameter or curve number estimate is based on land use, management, hydrologic soil group, land slope, and soil water content and distribution and is adjusted for frozen soil. However, many complex natural processes and artificial diversions that affect runoff are not+ accounted for in the model. Thus, the final curve number estimate is generated from a triangular distribution to account for this uncertain variation. The mean of the triangle is the best estimate of the curve number based on using equations 2, 3, 6, 7, 11, and 12. The extremes as +-5 curve numbers from the mean. The generated curve number is substituted into equation 2 to estimate runoff with equation 1.