Evapotranspiration and water balance from your own farm data
How ETo comes out of the weather station through the Penman-Monteith equation (FAO-56), how the crop coefficient turns ETo into crop water use, and a step-by-step numerical example that ends in cubic meters per week.
Evapotranspiration is the water that leaves the field for the air: part evaporates from the soil, part transpires through the plant. It is the crop’s actual water use, and it is the number irrigation scheduling has to replace. A farm with its own weather station can calculate it every day, with the climate of its own fields.
This article walks through the calculation. It is not hard, but it has several steps, and it is worth understanding each one so you can trust the result the platform delivers.
The three numbers in the calculation
- ETo, reference evapotranspiration: how much water a well-watered 12 cm grass would lose in that climate. It depends only on the weather, not on the crop.
- Kc, crop coefficient: how much your crop uses relative to the grass, at each growth stage. It comes from a table (FAO-56, Table 12) and can be adjusted locally.
- ETc = Kc × ETo, the crop’s water use, in millimeters per day.
The FAO equation
The standard method is the Penman-Monteith equation in its FAO form (Paper 56, equation 6):
ETo = [0.408 Δ (Rn − G) + γ (900 / (T + 273)) u2 (es − ea)] / [Δ + γ (1 + 0.34 u2)]
Where ETo is in mm/day; Rn is net radiation at the surface (MJ/m²/day); G is soil heat flux, which is taken as zero in the daily calculation; T is mean air temperature at 2 m (°C); u2 is wind speed at 2 m (m/s); es is saturation vapor pressure and ea is actual vapor pressure (kPa); Δ is the slope of the vapor pressure curve (kPa/°C); γ is the psychrometric constant (kPa/°C).
Each term comes from a station sensor: temperature and humidity give es, ea and Δ; the anemometer gives u2; the pyranometer gives the solar radiation that becomes Rn; the station’s elevation gives γ.
Worked example
A September day on a farm at 600 m elevation in the Brazilian Cerrado (northwest Minas Gerais), with the station data:
- Maximum temperature 32 °C, minimum 18 °C (mean 25 °C)
- Maximum relative humidity 80%, minimum 35%
- Wind at 2 m: 2.0 m/s
- Solar radiation: 22 MJ/m²/day
Vapor pressure. Saturation vapor pressure is es(T) = 0.6108 × exp(17.27 T / (T + 237.3)). At 32 °C it is 4.75 kPa; at 18 °C, 2.06 kPa. The mean is es = 3.41 kPa. Actual vapor pressure uses the humidity: ea = [2.06 × 0.80 + 4.75 × 0.35] / 2 = 1.66 kPa. The deficit es − ea = 1.75 kPa.
Slope of the curve. Δ = 4098 × es(25 °C) / (25 + 237.3)² = 0.189 kPa/°C.
Psychrometric constant. At 600 m the atmospheric pressure is P = 101.3 × ((293 − 0.0065 × 600) / 293)^5.26 = 94.4 kPa, and γ = 0.000665 × P = 0.0628 kPa/°C.
Net radiation. The shortwave part is Rns = (1 − 0.23) × 22 = 16.9 MJ/m²/day. The longwave loss, calculated with the formula in chapter 3 of FAO-56 from the temperatures, ea and the ratio of measured to clear-sky radiation, comes to about 4.9. So Rn = 16.9 − 4.9 = 12.1 MJ/m²/day.
Putting the equation together. Numerator: 0.408 × 0.189 × 12.1 = 0.93; plus 0.0628 × (900 / 298) × 2.0 × 1.75 = 0.66; total 1.59. Denominator: 0.189 + 0.0628 × (1 + 0.34 × 2.0) = 0.294.
ETo = 1.59 / 0.294 = 5.4 mm/day.
That is a typical value for the end of the dry season in the Brazilian Cerrado: strong sun, dry air, moderate wind.
From ETo to irrigation depth
Take corn at grain fill, with Kc = 1.20 (FAO-56, Table 12, mid-season value). The day’s water use is ETc = 1.20 × 5.4 = 6.5 mm/day.
Over a week without rain, the crop used 7 × 6.5 = 45.5 mm. If it rained, effective rainfall is subtracted. That is the water balance: crop use minus rainfall, day by day, against what the soil still holds.
The depth the pivot has to apply is larger than the crop use, because part of the water is lost to evaporation from the spray and to non-uniform application. With an application efficiency of 85%, a usual value for center pivots that the equipment manufacturer should confirm, the gross depth is 45.5 / 0.85 = 53.5 mm.
From depth to pumped volume
One millimeter over one hectare is 10 m³. On a 100 ha pivot, 53.5 mm per week is 53,500 m³ per week, or about 7,640 m³ per day. If the pump runs 20 hours a day, that is 382 m³/h.
What the platform does on its own
- Reads the station every few minutes and closes each day with Tmax, Tmin, humidity, wind and radiation.
- Calculates ETo with the equation above, using the station’s elevation.
- Applies the Kc for the crop stage you registered and subtracts the measured rainfall.
- Shows the depth to replace and, with the hour meter and the flow meter, the depth that was actually applied.
What to do now
- Check whether your station measures solar radiation. Without it, ETo is estimated from temperature (the Hargreaves method), with a larger error.
- Register the crop and the dates of each growth stage, so Kc follows the cycle.
- Confirm the application efficiency with the pivot manufacturer; it changes the gross depth.
If you want to know how much water your intake needs to supply, talk to one of our engineers.
Sources: FAO Irrigation and Drainage Paper 56, Crop Evapotranspiration (Allen, Pereira, Raes and Smith, 1998), equations 6, 7, 8, 11, 13, 38 to 40 and Table 12.