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#### Garden with Insight v1.0 Help: Erosion - Water Erosion

The EPIC component for water-induced erosion simulates erosion caused by rainfall and runoff and by irrigation (sprinkler and furrow). To simulate rainfall/runoff erosion, EPIC contains six equations -- the USLE (Wischmeier and Smith, 1978), the Onstad-Foster modification of the USLE (Onstad and Foster, 1975), the MUSLE (Williams, 1975), two recently developed variations of MUSLE (MUST and MUSS), and a MUSLE structure that accepts input coefficients (MUSI). Only one of the equations (MUSI) interacts with other EPIC components. The six equations are identical except for their energy components. The USLE depends strictly upon rainfall as an indicator of erosive energy. The MUSLE and its variations use only runoff variables to simulate erosion and sediment yield. Runoff variables increase the prediction accuracy, eliminate the need for a delivery ratio (used in the USLE to estimate sediment yield), and enable the equation to give single storm estimates of sediment yields. The USLE give only annual estimates. The Onstad- Foster equation contains a combination of the USLE and MUSLE energy factors.

Water Erosion From Rainfall

The water erosion model uses an equation of the form [Equation 118] and [Equation 119], where Y is the sediment yield in t/ha, K is the soil erodibility factor, CE is the crop management factor, PE is the erosion control practice factor, LS is the slope length and steepness factor, ROKF is the coarse fragment factor, Q is the runoff volume in mm, q(p) is the peak runoff rate in mm/hr, A is the watershed area in ha, MUST is a new equation theoretically developed from sediment concentration databases, MUSS is a new equation developed by fitting small watershed data (no channel erosion), and MUSI allows user input of four coefficients (b(y1)..b(y4)). The PE value is determined initially by considering the conservation practices to be applied. The runoff model supplies estimates of Q and q(p).

Equation 118

Y = chi * K * CE * PE * LS * ROKF
Code:
same
Variables:
Y = WaterErosion_tPha
chi = energyComponent
K = soilErodibilityFactor
CE = cropManagementFactor
PE = erosionControlPracticeFactor
LS = slopeLengthAndSteepnessFactor
ROKF = coarseFragmentFactor

Now all the energy equations. We will start with the MUSLE and its derivatives then move to the USLE and Onstad-Foster.
MUSLE

Equation 119

chi = 1.586 * power(Q * q(p), 0.65) * power(A, 0.12)
Code:
chi = 1.586 * power(Q * q(p), 0.65) * power(A + 1.0, 0.12)
Variables:
chi = WaterErosionEnergyComponentByMUSLE
Q = runoffVolume_mm
q(p) = peakRunoffRate_mmPhr
A = watershedArea_ha
MUST

Equation 119

chi = 2.5 * sqrt(Q * q(p))
Code:
same
Variables:
chi = WaterErosionEnergyComponentByMUST
Q = runoffVolume_mm
q(p) = peakRunoffRate_mmPhr
MUSS

Equation 119

chi = 0.79 * power(Q * q(p), 0.65) * power(A, 0.009)
Code:
chi = 0.79 * power(Q * q(p), 0.65) * power(A + 1.0, 0.009)
Variables:
chi = WaterErosionEnergyComponentByMUSS
Q = runoffVolume_mm
q(p) = peakRunoffRate_mmPhr
A = watershedArea_ha
MUSI

Equation 119

chi = by(1) * power(Q, by(2)) * power(q(p), by(3)) * power(A, by(4))
Code:
chi = by(1) * power(Q, by(2)) * power(q(p), by(3)) * power(A + 1.0, by(4))
Variables:
chi = WaterErosionEnergyComponentByMUSS
Q = runoffVolume_mm
q(p) = peakRunoffRate_mmPhr
A = watershedArea_ha
by(1)(2)(3)(4) = userCoeffsForMUSIPtr

For the energy component for USLE and Onstad-Foster, we have to move forward to equations 125-136. This is going to get a bit mucky...

To estimate the daily rainfall energy in the absence of time-distributed rainfall, it is assumed that the rainfall rate is exponentially distributed

r(t) = r(p) * exp(-t / kappa) (Equation 124)

where r is the rainfall rate at time t in mm/hr, r(p) is the peak rainfall rate in mm/hr and kappa is the decay constant in hrs. Equation 124 contains no assumption about the sequence of rainfall rates (time distribution).

The USLE energy equation in metric units is

RE = delta-r * (12.1 + 8.9 log (delta-r / delta-t)) (Equation 125)

where RE is the rainfall energy for water erosion equations and delta-r is a rainfall amount in mm during a time interval delta-t in hours. The energy equation can be expressed analytically as

RE = 12.1 * 0~infinity r dt + 8.9 0~infinity log r dt (Equation 126)

Substituting equation 124 into equation 126 and integrating gives the equation for estimating daily rainfal energy:

RE = R * (12.1 + 8.9 * (log r(p) - 0.434)) (Equation 127)

where R is the daily rainfall amount in mm. The rainfall energy factor, EI, is obtained by multiplying equation 127 by the maximum 0.5-hr rainfall intensity (r(0.5)) and converting to the proper units: [Equation 128].

Equation 128

EI = R * (12.1 + 8.9 * (log(r(p)) - 0.434)) * r(0.5) / 1000
Code:
r(0.5) = 2.0 * R * alpha(0.5)
they do not mention how to get r(0.5) in publication
also add lower bound at 0
Variables:
EI = RainfallEnergyFactorForUSLE
R = rainfallWithoutSnowmeltForDay_mm
r(p) = peakRainfallRate_mmPhr
r(0.5) = maxRainfallIntensityInHalfHour_mmPhr

Equation 128 USLE part

chi = EI
Code:
chi = EI * 1.292
Variables:
chi = WaterErosionEnergyComponentByUSLE
EI = rainfallEnergyFactorForUSLE

To compute values for r(p), equation 124 is integrated to give

R = r(p) * kappa (Equation 129)

and

R(t) = R * (1 - exp(-t / kappa)) (Equation 130)

The value of R(0.5) can be estimated by using alpha(0.5) as mentioned in the Hydrology section of this section:

R(0.5) = alpha(0.5) * R (Equation 131)

To determine the value of r(p), equations 131 and 129 are substituted into equation 130 to give [Equation 132].

Equation 132

r(p) = -2 * R * ln(1.0 - alpha(0.5))
Code:
same
Variables:
r(p) = PeakRainfallRate_mmPhr
R = rainfallForDay_mm
alpha(0.5) = propTotalRainFallsInFirstHalfHour_frn

Since rainfall rates vary seasonally, alpha(0.5) is evaluated for each month by using Weather Service information (U.S. Department of Commerce, 1979). The frequency with which the maximum 0.5-hr rainfall amount occurs is estimated by using the Hazen plotting position equation (Hazen, 1930) [Equation 133] where F is the frequency with which the largest of a total of tau events occurs. The total number of events for each month is the product of the number of years of record and the average number of rainfall events for the month.

Equation 133

F = 1 / (2 * tau)
Code:
same
Variables:
F = FrequencyOfRainfallEvents_Pyr
tau = yearsRecordMaxHalfHourRain

To estimate the mean value of alpha(0.5), it is necessary to estimate the mean value of R(0.5). The value of R(0.5) can be computed easily if the maximum 0.5-hr rainfall amounts are assumed to be exponentially distributed. From the exponential distribution, the expression for the mean 0.5-hr rainfall amount is

R(0.5) = R(0.5F)/-log(F) (Equation 134)

where R(0.5) is the mean maximum 0.5-hr rainfall amount, R(0.5F) is the maximum 0.5-hr rainfall amount for frequency F, and subscript k refers to the month.

The mean alpha(0.5) is computed with the equation [Equation 135] where R is the mean amount of rainfall for each event (averagy monthly rainfall/average number of days of rainfall) and subscript k refers to the month. Daily values of alpha(0.5) are generated from a triangular distribution.

Equation 135

alpha(0.5) = R(0.5) / R
Code:
alpha(0.5) = -R(0.5) / log((F / numWetDaysForMonth) * R)
Variables:
F = frequencyOfRainfallEvents
alpha(0.5) = MeanPropTotalRainFallsInFirstHalfHourForMonth_frn
R(0.5) = meanMaxHalfHourRainfallAmountForMonth_mm
R = meanRainfallPerEventForMonth_mm = meanTotalRainfallForMonth_mm / numWetDaysForMonth

The lower limit of alpha(0.5), determined by a uniform rainfall, is to 0.5/24 or 0.0208. The upper limit of alpha(0.5) is set by considering a large rainfall event. In a large event, it is highly unlikely that all the rainfall occurs in 0.5 hr (alpha(0.5) = 1.0). The upper limit of alpha(0.5) can be estimated by substituting a high value for r(p) (250 mm/hr is generally near the upper limit of rainfall intensity) into equation 130 [Equation 136] where alpha(0.5u) is the upper limit of alpha(0.5). The peak of the alpha(0.5) triangular distribution is alpha(0.5) from equation 135. So the lower limit is 0.0208, the upper limit is the aResult of equation 136, and the peak is the aResult of equation 135.

Equation 136

alpha(0.5u) = 1.0 - exp(-125 / R)
where R is mean daily rainfall for the month
Code:
alpha(0.5u) = 1.0 - exp(-125 / (R + 5))
where R is rainfall for the day
Variables:
alpha(0.5u) = MaxPropTotalRainFallsInFirstHalfHourForMonth_frn
R = meanRainfallPerEventForMonth_mm = meanTotalRainfallForMonth_mm / numWetDaysForMonth
R = rainfallWithoutSnowmeltForDay_mm

For the USLE method, X is the same as EI (equation 118), but for the Onstad-Foster method, we need a function for that part of equation 118.

Equation 118

chi = 0.646 * EI + 0.45 * power(Q * q(p), 0.33)
Code:
same
Variables:
EI = rainfallEnergyFactorForUSLE
Q = runoffVolume_mm
q(p) = peakRunoffRate_mmPhr

Now to fill in the other parts of the water erosion equation, we go back to equations 120-123 and stick equation 137 on the end.

The value of LS is calculated with the equation (Wischmeier and Smith, 1978) [Equation 120] where S is the land surface slope in m/m, lamda is the slope length in m, and xi is a parameter dependent upon slope.

Equation 120

LS = pow(lambda / 22.1, xi) * (65.41 * sqr(S) + 4.56 * S + 0.065)
Code:
same
Variables:
LS = SlopeLengthAndSteepnessFactor
lambda = slopeLength_m
xi = slopeLengthFactorParam
S = slopeSteepness_mPm

The value of xi varies with slope and is estimated with the equation [Equation 121].

Equation 121

xi = 0.3 * S / (S + exp(-1.47 -61.09 * S)) + 0.2
Code:
same
Variables:
xi = SlopeLengthFactorParam
S = slopeSteepness_mPm

The crop management factor is evaluated for all days when runoff occurs by using the equation [Equation 122] where CE(mn,j) is the minimum value of the crop management factor for crop j and CV is the soil cover (above ground biomass plus residue) in t/ha.

Equation 122

CE = exp[(ln(0.8) - ln(CE(mn,j))) * exp(-1.15 * CV) + ln(CE(mn,j))
Code:
same (ln(0.8) is -0.2231)
Variables:
CE = CropManagementFactor
CE(mn,j) = patchMeanMinCropManagementFactor
CV = patchTotalAboveGroundBiomassAndResidue_tPha

The soil erodibility factor, K, is evaluated for the top soil layer at the start of each year of simulation with the equation [Equation 123] where SAN, SIL, CLA, and C are the sand, silt, clay, and organic carbon contents of the soil in percent and SN1 = 1 - SAN / 100. Equation 123 allows K to vary from about 0.1 to 0.5. The first term gives low K values for soils with high coarse-sand contents and high values for soil with little sand. The fine sand content is estimated as the product of sand and silt divided by 100. The expression for coarse sand in the first term is simply the difference between sand and the estimated fine sand. The second term reduces K for soils that have high clay to silt ratios. The third term reduces K for soils with high organic carbon contents. The fourth term reduces K further for soils with extremely high sand contents (SAN>70%).

Equation 123

K = [0.2 + 0.3 * exp(-0.0256 * SAN * (1 - SIL/100))
* pow(SIL / (CLA + SIL), 0.3)
* [1 - 0.25 * C / (C + exp(3.72 - 2.95 * C))
* [1 - 0.7 * (1-SAN/100) / ((1-SAN/100) + exp(-5.51 + 22.9 * (1-SAN/100)))
Code:
same
Variables:
K = soilErodibilityFactor
SAN = soilSandContent_pct
SIL = siltContent_pct
CLA = clayContent_pct
C = organicC_pct
SN1 = nonSandContent_frn

The coarse fragment factor is estimated with the equation (Simanton et al. 1984) [Equation 137] where ROK is the percent of coarse fragments in the surface soil layer.

Equation 137

ROKF = exp(-0.03 * ROK)
Code:
same
Variables:
ROKF = CoarseFragmentFactor
ROK = rockContentSurfaceSoilLayer_pct

Water Erosion From Irrigation

Erosion caused by applying irrigation water in furrows is estimated with MUST [Equation 138] where CE, the crop management factor, has a constant value of 0.5. The volume of runoff is estimated as the product of the irrigation volume applied and the irrigation runoff ratio. The peak runoff rate is estimated for each furrow by using Mannings equation and assuming that the flow depth is 0.75 of the ridge height and that the furrow is triangular. If irrigation water is applied to land without furrows, the peak runoff rate is estimated by dividing the runoff volume by the duration (24 hrs).

Equation 138

Y = 2.5 * power(Q * q(p), 0.5) * K * CE * PE * LS
Code:
Y = 11.8 * 1000 * power(Q * q(p), 0.56) * K * CE * PE * LS / distance_m
Variables:
Y = WaterErosionFromIrrigation_tPha
Q = furrowVolume_mm
q(p) = peakRunoffRateForFurrow_m3Psec
K = soilErodibilityFactor
CE = kCropManagementFactor = 0.5
PE = erosionControlPracticeFactor
LS = slopeLengthAndSteepnessFactor

 Home ... News ... Products ... Download ... Order ... Support ... Consulting ... Company Updated: May 4, 1998. Questions/comments on site to webmaster@kurtz-fernhout.com. Copyright © 1998 Paul D. Fernhout & Cynthia F. Kurtz.