Soil water content ϴ is a function of soil matric potential Ψm, being expressed as follows:
ϴ = f (Ψm) (1)
This function, known as the water retention curve, is commonly expressed by empirical exponential functions.
Based on [22], Eq. (1) takes the form:
ϴv = A e− [(log (10|Ψm|)]^2 / B^2 (2)
In which A and B are soil parameters, specific for each curve;, ϴv is soil water volumetric content (%), and Ψm is expressed in the soil matric potential, expressed in kPa.
An inflexion point was detected on Eq. (2), by confirming two curve branches with different slopes, one of them - the first branch - on the (wet) drainage zone (|Ψm| ≤ 10 kPa), and the second branch on the (dry) retention zone (|Ψm| ≥ 10 kPa), being tested successfully through their high correlation coefficients and by their low deviations. This inflexion is due to a cavitation initiated by entrapped air bubbles or the liquid’s own vapor pressure [35], provoking a second branch curve that occurs near that point.
Recurring to decimal logarithms [8], Eq. (2) takes a linear form, as follows:
log ϴv = log A – {[log (10 |Ψm|)]2 B− 2} log e (3)
Eq. (3) is defined by two right-lined segments with different slopes, the first one, represents the wet drainage zone (|Ψm| ≤ 10 kPa) and the second one the dry drainage zone (|Ψm| ≥ 10 kPa) being tested successfully for a large range of soils [34, 36] for several mineral soil types. Notice that this approach is not useful to be applied to mineral soils, even with some organic matter content, once that the soil total potential values can reach very low values for those soils, such as 1,500 kPa (wilting point). The 2nd branch of Eq. (3) (drier zone) was applied to the linear relationship of soil total water potential function and relative yield on irrigated mineral soil, for |Ψm| ≥ 10 kPa [37]. On the other hand, in this study the 1st right-lined segment, representing the wetter zone, was applied. In mineral soils, the suction range is generally conducted at lower tensions, because substrates are more porous and usually have large diameter pores. It enables water to drain at lower tensions [25], with most greenhouse crops using water held at matric potential Ψm values between –-1 kPa and − 10 kPa [38]. Hence, for this approach, the wetter zone of the curve will be adopted (drainage zone for mineral soils) for a matric potential |Ψm| ≤ 10 kPa. Eq. (3) may be easily graphically represented by a straight line, in a log-normal scale, as follows:
y = mx + b (4)
where the log scale axis “y” (ordenates) defines ϴv (%), the normal decimal scale axis “x” (abcisse) defines [log (10 |Ψm|)]2. The specific parameter – log e B− 2 is the slope “m”, and log A is the ordinate origin b.
Eq. (4) is useful to automatically control substrate irrigation. If tensiometers are applied, according to Eqs. (2), (3), and (4), compost volumetric water content difference between ϴv1 and ϴv2 [% (m3 water. m− 3 soil] can be obtained through the equations [10]:
ϴv1 ϴv2 − 1 = e {[ln (10 |Ψm2│)]^2 − [ ln (10 |Ψm│)]^2} / B^2 (non linear form) (5)
and
log ϴv1 - log ϴv2 = B− 2 {[log (10 |Ψm2|)]2 – log [(10 |Ψm1|]2} (linear form) (6)
where Ψm1 and Ψm2 are soil matric potential (kPa) values measured by the tensiometer.
Eq. (6) may be easily graphically represented in a log-normal scale, where log scale (ordenates) defines ϴv, and the normal (decimal) scale defines {[(ln (10 |Ψm2|)]2 – [(ln (10 |Ψm1|)]2 }, being B− 2 the slope.
The general water balance equation for the root zone during a specific area and time period is given by:
I + P + CR = ETa + Dr + R + ΔS (7)
where I is the net irrigation, P is the natural precipitation, CR is the capillary rise from the groundwater table to the root zone, ETa is the actual crop evapotranspiration, Dr is the drainage below the root zone, R is the runoff, and ΔS is the change in water storage within the root zone; the units for all these parameters are m3. Being capillary rise CR and runoff R zero during this specific time area and period and being ΔS, P and Dr negligible due to high irrigation frequency; and being very low or no natural precipitation P, Eq. (8) is simplified to:
I = ETa (8)
If ϴv1 and ϴv2 are, respectively, the volumetric water content (%) stored in the compost, after and before irrigation, during a specific area and time period, the crop net average irrigation amount I (m3) is defined by:
I = 100 (ϴv1 - ϴv2) Vs (9)
In which Vs is the compost average volume (m3).
However, total gross irrigation water amount IR is higher than I, given its dependence on water application efficiency E (dimensionless from 0 to1) due to irrigation water losses during water application [39, 40, 41], and can be determined as follows:
IR = I E− 1 (10)