Identifying a new paradigm for assessing irrigation system performance

Abstract

There are many definitions of irrigation system efficiency that are applied over a range of scales. Many traditional definitions considered only the water diverted as the water volume of concern. Considering also the water consumed in defining effective irrigation efficiency is a shift from the classical definition of system efficiency. In this paper, equations are derived for calculating the following system performance measures: the irrigation consumptive use coefficient, irrigation system efficiency, irrigation water and soil salinities, relative yield, and productivity of consumed, diverted and beneficially used water. The expressions are based on quite general assumptions and are valid for systems with a single water source and layouts composed of (or simplified to) irrigation units arranged in a row. The aim of these expressions is to illustrate how system performance is affected by the reuse of water which depends on the system’s hydraulic connections and the irrigation unit performance. Illustrations of the model are provided for systems in series and in parallel. Testing and refinement by removing some of the general assumptions underlying the model will be needed to develop practical applications that can be more confidently applied for comparison and improvement of irrigation systems.

Appendix

Irrigation consumptive use coefficient

The ICUC is the fraction of the irrigation water (I) destined for consumptive use (CU) (Burt et al. 1997). For the irrigation unit (denoted by subscript u):

$$ {\text{ICUC}}_{u} = \frac{{{\text{CU}}_{u} }}{{I_{u} }} $$

(16)

For simplicity, in the successive derivations it will be assumed that I _u , CU _u and therefore ICUC _u are constant for all units in the parallel system and that ICUC _u is constant in the series system.

For a system composed of n (1, 2, …) units (number of units indicated by the first subscript) arranged in series (series denoted by the second subscript, s):

$$ \begin{gathered} {\text{ICUC}}_{{n ,s}} = \frac{{\sum_{{i = 1}}^{n} {{\text{CU}}_{i} } }}{{I_{1} }} = \frac{{I_{1} {\text{ICUC}}_{u} + I_{1} (1 - {\text{ICUC}}_{u} ){\text{ICUC}}_{u} + \cdots + I_{1} (1 - {\text{ICUC}}_{u} )^{{n - 1}} {\text{ICUC}}_{u} }}{{I_{1} }} \hfill \\ = {\text{ICUC}}_{u} \frac{{1 - \left( {1 - {\text{ICUC}}_{u} } \right)^{n} }}{{1 - \left( {1 - {\text{ICUC}}_{u} } \right)}} = 1 - \left( {1 - {\text{ICUC}}_{u} } \right)^{n} \hfill \\ \end{gathered} $$

(17)

When the units are in parallel (parallel denoted by the second subscript, p), if n = 1:

$$ {\text{ICUC}}_{{1,p}} = {\text{ICUC}}_{u} $$

(18)

If n = 2:

$$ \begin{gathered} {\text{ICUC}}_{{2,p}} = \frac{{2{\text{CU}}_{u} }}{{I_{u} + \left[ {I_{u} - \left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )I_{u} } \right]}} = \frac{{2I_{u} {\text{ICUC}}_{u} }}{{2I_{u} - \left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )I_{u} }} \hfill \\ = \frac{{2{\text{ICUC}}_{u} }}{{2 - 1\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )}} \hfill \\ \end{gathered} $$

(19)

For n units:

$$ {\text{ICUC}}_{n}^{p} = \frac{{n{\text{ICUC}}_{u} }}{{n - (n - 1)\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )}} $$

(20)

In the parallel system, the unit irrigation water results from adding water from the main channel to the return flow coming from the upstream irrigation unit (fraction ρ ₂), till obtaining an amount of water I _u .

Irrigation efficiency

The IE is the fraction of the irrigation water that is beneficially used (IB) (Burt et al. 1997). For the irrigation unit (irrigation unit denoted by subscript u):

$$ {\text{IE}}_{u} = \frac{{{\text{IB}}_{u} }}{{I_{u} }} $$

(21)

For a system composed of n units in series:

$$ \begin{gathered} {\text{IE}}_{{n ,s}} = \frac{{h\sum_{{i = 1}}^{n} {{\text{CU}}_{i} } + I_{1} (1 - {\text{ICUC}}_{u} )^{n} f}}{{{\text{I}}_{1} }} = \frac{{hI_{1} \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)^{n} } \right] + I_{1} (1 - {\text{ICUC}}_{u} )^{n} f}}{{{\text{I}}_{1} }} \hfill \\ = h\left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)^{n} } \right] + f\left( {1 - {\text{ICUC}}_{u} } \right)^{n} \hfill \\ \end{gathered} $$

(22)

If the units are in parallel and n = 1:

$$ {\text{IE}}_{{1,p}} = {\text{IE}}_{u} $$

(23)

For n = 2:

$$ {\text{IE}}_{{2,p}} = \frac{{I_{u} {\text{IE}}_{u} + {\text{I}}_{u} {\text{IE}}_{u} - \rho _{b} I_{u} (1 - {\text{ICUC}}_{u} )}}{{I_{u} + \left[ {I_{u} - \left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} ){\text{I}}_{u} } \right]}} = \frac{{2{\text{IE}}_{u} - 1\rho _{b} (1 - {\text{ICUC}}_{u} )}}{{2 - 1\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )}} $$

(24)

For n units in parallel:

$$ {\text{IE}}_{{n,p}} = \frac{{n{\text{IE}}_{u} - (n - 1)\rho _{b} (1 - {\text{ICUC}}_{u} )}}{{n - (n - 1)\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )}}. $$

(25)

Equations 24 and 25 contain the assumption that if water is considered beneficial once, even if not consumed, it cannot be considered beneficial again. Otherwise, IE could take values greater than unity.

Irrigation water salinity

We assumed a steady state regime and no internal input from soil salts, i.e. the salts leaving an irrigation unit are those that entered it.

In a series system, the irrigation water for a unit is the return flow from the upstream unit. Thus, the evapo-concentration of the irrigation water of unit j determines the salinity of the irrigation water for unit j + 1. Denoting irrigation salinity by CI, using subscript (1, 2,…, j,…, n) to indicate the order of the irrigation unit in the system layout and a second subscript, s, to denote arrangement in series, for a generic irrigation unit, j:

$$ \frac{{{\text{CI}}_{{j,s}} }}{{{\text{CI}}_{1} }} = \left( {\frac{1}{{1 - {\text{ICUC}}_{u} }}} \right)^{{j - 1}} . $$

(26)

For calculating the same ratio in a system in parallel (arrangement in parallel denoted by subscript p), first we need to calculate the channel flow (F). If we express I _u as a fraction (k) of the head flow (F ₁), then the flow in the channel upstream irrigation unit j is:

$$ \begin{gathered} F_{j} = F_{1} - (j - 1)I_{u} + (j - 1)(1 - {\text{ICUC}}_{u} )\rho _{1} I_{u} + (j - 2)(1 - {\text{ICUC}}_{u} )\rho _{2} I_{u} \hfill \\ = F_{1} \left[ {1 - \left( {j - 1} \right)k + \left( {j - 1} \right)\left( {1 - {\text{ICUC}}_{u} } \right)\rho _{1} k + \left( {j - 2} \right)\left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} k} \right]. \hfill \\ \end{gathered} $$

(27)

The salinity of the water in the channel (CC) at the entrance to irrigation unit j is:

$$ \begin{gathered} {\text{CC}}_{j} = \frac{{\left\{ {F_{{j - 1}} - \left[ {I_{u} - I_{u} \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]} \right\}{\text{CC}}_{{j - 1}} + I_{u} \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{1} \frac{{{\text{CI}}_{{j - 1}} }}{{1 - {\text{ICUC}}_{u} }}}}{{F_{j} }} \hfill \\ = \frac{1}{{F_{j} }}\left\{ {F_{{j - 1}} - F_{1} k\left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]} \right\}{\text{CC}}_{{j - 1}} + \frac{1}{{F_{j} }}\left\{ {F_{1} k\rho _{1} } \right\}{\text{CI}}_{{j - 1}} \hfill \\ \end{gathered} $$

(28)

where $CI_{{j - 1}}$ is the salinity of the irrigation water at unit j − 1. The salinity of the irrigation water at unit j is:

$$ \begin{gathered} {\text{CI}}_{j} = \frac{{\left[ {I_{u} - I_{u} \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]{\text{CC}}_{j} + I_{u} \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} \frac{{{\text{CI}}_{{j - 1}} }}{{1 - {\text{ICUC}}_{u} }}}}{{I_{u} }} \hfill \\ = \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]{\text{CC}}_{j} + \rho _{2} {\text{CI}}_{{j - 1}} \hfill \\ \end{gathered} $$

(29)

If ρ ₁ = 0, then the ratio $\frac{{{\text{CI}}_{{j,p}} }}{{{\text{CI}}_{1} }}$ becomes independent of CI₁ and F ₁:

$$ \frac{{{\text{CI}}_{1} }}{{{\text{CI}}_{1} }} = 1. $$

(30)

$$ \frac{{{\text{CI}}_{{ 2,p}} }}{{{\text{CI}}_{1} }} = \frac{1}{{{\text{CI}}_{1} }}\left[ {(1 - {\text{ICUC}}_{u} )\rho _{2} \frac{{{\text{CI}}_{1} }}{{(1 - {\text{ICUC}}_{u} )}} + \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]{\text{CI}}_{1} } \right] = \rho _{2} + \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]. $$

(31)

$$ \begin{gathered} \frac{{{\text{CI}}_{{3,p}} }}{{{\text{CI}}_{1} }} = \frac{1}{{{\text{CI}}_{1} }}\left[ {(1 - {\text{ICUC}}_{u} )\rho _{2} \frac{{{\text{CI}}_{2} }}{{(1 - {\text{ICUC}}_{u} )}} + \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]{\text{CI}}_{1} } \right] \hfill \\ = \frac{1}{{{\text{CI}}_{1} }}\left[ {\rho _{2} {\text{CI}}_{1} \left\{ {\rho _{2} + \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]} \right\} + \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]{\text{CI}}_{1} } \right] \hfill \\ = \rho _{2}^{2} + \rho _{2} \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right] + \left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right] = \rho _{2}^{2} + (\rho _{2} + 1)\left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]. \hfill \\ \end{gathered} $$

(32)

$$ \frac{{{\text{CI}}_{{j,p}} }}{{{\text{CI}}_{1} }} = \rho _{2}^{{j - 1}} + \left( {\rho _{2}^{{j - 2}} + \rho _{2}^{{j - 3}} + \cdots + \rho _{2} + 1} \right)\left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right] = \rho _{2}^{{j - 1}} + \left( {\frac{{1 - \rho _{2}^{{j - 1}} }}{{1 - \rho _{2} }}} \right)\left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]. $$

(33)

Soil salinity

The assumptions for calculating soil salinity (CS) are: (1) the salinity of the soil solution is the mean of the irrigation and drainage (CD) water salinities, (2) drainage occurs when soil water content is greater than field capacity and (3) soil water content at field capacity is half the soil water content at saturation.

Then, for a system in series:

$$ \frac{{{\text{CS}}_{{j,s}} }}{{{\text{CI}}_{1} }} = \frac{1}{2}\frac{1}{2}\left[ {\left( {\frac{1}{{1 - {\text{ICUC}}_{u} }}} \right)^{{j - 1}} + \left( {\frac{1}{{1 - {\text{ICUC}}_{u} }}} \right)^{j} } \right] = \frac{{2 - {\text{ICUC}}_{u} }}{{4\left( {1 - {\text{ICUC}}_{u} } \right)^{j} }}. $$

(34)

If the system is in parallel, first, we define the ‘leaching fraction’ (LF), assuming a steady state regime, as:

$$ {\text{LF}} = \frac{{g(1 - {\text{ICUC}}_{u} )}}{{{\text{ICUC}}_{u} + g(1 - {\text{ICUC}}_{u} )}} = \frac{{{\text{CI}}_{{j,p}} }}{{{\text{CD}}_{{j,p}} }} $$

(35)

where g is the fraction of non-consumed water that goes to percolation. Therefore,

$$ \frac{{{\text{CS}}_{{j,p}} }}{{{\text{CI}}_{{j,p}} }} = \frac{1}{{{\text{CI}}_{{j,p}} }}\frac{1}{2}\left( {\frac{{{\text{CI}}_{{j,p}} + {\raise0.7ex\hbox{${{\text{CI}}_{{j,p}} }$} \!\mathord{\left/ {\vphantom {{{\text{CI}}_{{j,p}} } {{\text{LF}}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{\text{LF}}}$}}}}{2}} \right) = \frac{{{\text{LF}} + 1}}{{4{\text{LF}}}} = \frac{{2g(1 - {\text{ICUC}}_{u} ) + {\text{ICUC}}_{u} }}{{4{\text{g}}(1 - {\text{ICUC}}_{u} )}} = \frac{1}{2} + \frac{{{\text{ICUC}}_{u} }}{{4g\left( {1 - {\text{ICUC}}_{u} } \right)}}. $$

(36)

Finally, we can relate soil salinity to the irrigation water salinity at the entrance of the system by multiplying numerator and denominator by CI₁ and by making use of the unit’s irrigation water salinity relative to the initial irrigation water salinity derived in the previous section:

$$ \frac{{{\text{CS}}_{{j,p}} }}{{{\text{CI}}_{1} }} = \frac{{{\text{CI}}_{{j,p}} }}{{{\text{CI}}_{1} }}\left[ {\frac{1}{2} + \frac{{{\text{ICUC}}_{u} }}{{4g\left( {1 - {\text{ICUC}}_{u} } \right)}}} \right] = \left[ {\rho _{2}^{{j - 1}} + \left( {\frac{{1 - \rho _{2}^{{j - 1}} }}{{1 - \rho _{2} }}} \right)\left[ {1 - \left( {1 - {\text{ICUC}}_{u} } \right)\rho _{2} } \right]} \right]\left[ {\frac{1}{2} + \frac{{{\text{ICUC}}_{u} }}{{4g\left( {1 - {\text{ICUC}}_{u} } \right)}}} \right]. $$

(37)

Relative yield

Maas and Hoffman (1977) proposed a model to estimate relative yield (RY, %) of crops limited by soil salinity. For an irrigation unit j:

$$ {\text{RY}}_{j} = 100,\,{\text{if CS}}_{j} \le {\text{CS}}_{{{\text{th}}}} $$

(38)

$$ {\text{RY}}_{j} = \max \left[ {100 - B_{{\text{o}}} ({\text{CS}}_{j} - {\text{CS}}_{{{\text{th}}}} ) ; 0} \right],\,{\text{if CS}}_{j} > {\text{CS}}_{{{\text{th}}}} $$

(39)

where CS_th is the threshold of soil salinity above which yield is reduced and B _o is the decrease in relative yield per unit of increase in soil salinity above the threshold. CS _j may be calculated from Eq. 34 or 37 if CI₁ is known.

Irrigation water productivity

Here we consider three definitions of irrigation water productivity: production per unit of water diverted (WP2), production per unit of irrigation water consumed (WP3) and production per unit of irrigation water beneficially used (WP4):

$$ {\text{WP}}2_{{n,s}} = \frac{{Y_{{\max }} \sum_{{j = 1}}^{n} {(1 - {\text{ICUC}}_{u} )^{{j - 1}} {\text{RY}}_{j} } }}{{100I_{u} }} $$

(40)

$$ {\text{WP}}2_{{n,p}} = \frac{{Y_{{\max }} \sum_{{j = 1}}^{n} {{\text{RY}}_{j} } }}{{100I_{u} \left[ {n - \left( {n - 1} \right)\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )} \right]}} $$

(41)

$$ {\text{WP}}3_{{n,s}} = \frac{{Y_{{\max }} \sum_{{j = 1}}^{n} {(1 - {\text{ICUC}}_{u} )^{{j - 1}} {\text{RY}}_{j} } }}{{100I_{u} {\text{ICUC}}_{n} }} $$

(42)

$$ {\text{WP}}3_{{n,p}} = \frac{{Y_{{\max }} \sum_{{j = 1}}^{n} {{\text{RY}}_{j} } }}{{100I_{u} \left[ {n - \left( {n - 1} \right)\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )} \right]{\text{ICUC}}_{n} }} $$

(43)

$$ {\text{WP}}4_{{n,s}} = \frac{{Y_{{\max }} \sum_{{j = 1}}^{n} {(1 - {\text{ICUC}}_{u} )^{{j - 1}} {\text{RY}}_{j} } }}{{100I_{u} {\text{IE}}_{n} }} $$

(44)

$$ {\text{WP}}4_{{n,p}} = \frac{{Y_{{\max }} \sum_{{j = 1}}^{n} {{\text{RY}}_{j} } }}{{100I_{u} \left[ {n - \left( {n - 1} \right)\left( {1 - {\text{ICUC}}_{u} } \right)(\rho _{1} + \rho _{2} )} \right]{\text{IE}}_{n} }} $$

(45)

where Y _max is the maximum yield achievable in the environment under consideration.