A two-dimensional water balance model for micro-irrigated hedgerow tree crops

Abstract

Distribution of water and energy is non-uniform in widely spaced, micro-irrigated, hedgerow crops. For accurate water use predictions, this two-dimensional variation in the energy and water balance must be adequately accounted for. To this end, a user-friendly, two-dimensional, mechanistic soil water balance model (SWB-2D), has been developed. Energy is partitioned at the surface depending on solar orientation, row direction and canopy size, shape and leaf area density. Water is assumed to be distributed uniformly at the surface in the case of rainfall, whilst micro-irrigation only wets a limited portion of the field. Crop water uptake is calculated as a function of evaporative demand, soil water potential and root density. Evaporation is also calculated as being either limited by available energy or by water supply. Water is redistributed in the soil in two dimensions with a finite difference solution to the Richards’ equation. A field trial was set up to test the 2-D soil water balance model in a citrus orchard at Syferkuil (Pietersburg, South Africa). Model predictions generally compared well to actual soil water content measured with time domain reflectometry probes. Scenario modelling and analyses were carried out by varying some input parameters (row orientation, canopy width, wetted diameter and fraction of roots in the wetted volume of soil) and observing variations in the output of the soil water balance. The model holds potential for improving irrigation scheduling and efficiency through increased understanding and accuracy in estimating soil water reserves, since it accounts for the differing conditions in the under-tree irrigated strip and inter-row rainfed areas.

Appendix

Finite difference flux equations and derivatives

In the finite difference flux equations, positive values represent fluxes into the control volume:

$$ {\text{UR}}x = \frac{{{\left( {\Phi {\text{lr}}_{{i - 1,j}} - \Phi {\text{ll}}_{{i - 1,j}} } \right)}\;{\left( {Z_{i} - Z_{{i - 1}} } \right)}}} {{2{\left( {X_{{j + 1}} - X_{j} } \right)}}} $$

(A1)

$$ {\text{LR}}x = \frac{{{\left( {\Phi {\text{ur}}_{{i,j}} - \Phi {\text{ul}}_{{i,j}} } \right)}\;{\left( {Z_{{i + 1}} - Z_{i} } \right)}}} {{2{\left( {X_{{j + 1}} - X_{j} } \right)}}} $$

(A2)

$$ {\text{UL}}x = \frac{{{\left( {\Phi {\text{ll}}_{{i - 1,j - 1}} - \Phi {\text{lr}}_{{i - 1,j - 1}} } \right)}{\left( {Z_{i} - Z_{{i - 1}} } \right)}}} {{2{\left( {X_{j} - X_{{j - 1}} } \right)}}} $$

(A3)

$$ {\text{LL}}x = \frac{{{\left( {\Phi {\text{ul}}_{{i,j - 1}} - \Phi {\text{ur}}_{{i,j - 1}} } \right)}\;{\left( {Z_{{i + 1}} - Z_{i} } \right)}}} {{2{\left( {X_{j} - X_{{j - 1}} } \right)}}} $$

(A4)

$$ {\text{UR}}z = \frac{{{\left( {\Phi {\text{ul}}_{{i - 1,j}} - \Phi {\text{ll}}_{{i - 1,j}} } \right)}\;{\left( {X_{{j + 1}} - X_{j} } \right)}}} {{2{\left( {Z_{i} - Z_{{i - 1}} } \right)}}}\; + \;\frac{{g\begin{array}{*{20}c} {{}} \\ \end{array} K{\text{ul}}_{{i - 1,j}} \;{\left( {X_{{j + 1}} - X_{j} } \right)}}} {2} $$

(A5)

$$ {\text{LR}}z = \frac{{{\left( {\Phi {\text{ll}}_{{i,j}} - \Phi {\text{ul}}_{{i,j}} } \right)}\;{\left( {X_{{j + 1}} - X_{j} } \right)}}} {{2{\left( {Z_{{i + 1}} - Z_{i} } \right)}}}\begin{array}{*{20}c} {{}} \\ \end{array} - \begin{array}{*{20}c} {{}} \\ \end{array} \frac{{g\begin{array}{*{20}c} {{}} \\ \end{array} K{\text{ul}}_{{i,j}} \;{\left( {X_{{j + 1}} - X_{j} } \right)}}} {2} $$

(A6)

$$ {\text{UL}}z = \frac{{{\left( {\Phi {\text{ur}}_{{i - 1,j - 1}} - \Phi {\text{lr}}_{{i - 1,j - 1}} } \right)}\;{\left( {X_{j} - X_{{j - 1}} } \right)}}} {{2{\left( {Z_{i} - Z_{{i - 1}} } \right)}}}\begin{array}{*{20}c} {{}} \\ \end{array} + \begin{array}{*{20}c} {{}} \\ \end{array} \frac{{g\begin{array}{*{20}c} {{}} \\ \end{array} K{\text{ur}}_{{i - 1,j - 1}} \;{\left( {X_{j} - X_{{j - 1}} } \right)}}} {2} $$

(A7)

$$ {\text{LL}}z = \frac{{{\left( {\Phi {\text{lr}}_{{i,j - 1}} - \Phi {\text{ur}}_{{i,j - 1}} } \right)}\;{\left( {X_{j} - X_{{j - 1}} } \right)}}} {{2{\left( {Z_{{i + 1}} - Z_{i} } \right)}}}\begin{array}{*{20}c} {{}} \\ \end{array} - \begin{array}{*{20}c} {{}} \\ \end{array} \frac{{g\begin{array}{*{20}c} {{}} \\ \end{array} K{\text{ur}}_{{i,j - 1}} {\left( {X_{j} - X_{{j - 1}} } \right)}}} {2} $$

(A8)

Note that vertical fluxes include gravitational components.

The derivatives of the flux equations and storage term are:

$$ \frac{{\partial {\text{UR}}x}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{ll}}_{{i - 1,j}} {\left( {Z_{i} - Z_{{i - 1}} } \right)}}} {{2{\left( {X_{{j + 1}} - X_{j} } \right)}}} $$

(A9)

$$ \frac{{\partial {\text{LR}}x}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{ul}}_{{i,j}} {\left( {Z_{{i + 1}} - Z_{i} } \right)}}} {{2{\left( {X_{{j + 1}} - X_{j} } \right)}}} $$

(A10)

$$ \frac{{\partial {\text{UL}}x}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{lr}}_{{i - 1,j - 1}} {\left( {Z_{i} - Z_{{i - 1}} } \right)}}} {{2{\left( {X_{j} - X_{{j - 1}} } \right)}}} $$

(A11)

$$ \frac{{\partial {\text{LL}}x}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{ur}}_{{i,j - 1}} {\left( {Z_{{i + 1}} - Z_{i} } \right)}}} {{2{\left( {X_{j} - X_{{j - 1}} } \right)}}} $$

(A12)

$$ \frac{{\partial {\text{UR}}z}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{ll}}_{{i - 1,j}} {\left( {X_{{j + 1}} - X_{j} } \right)}}} {{2{\left( {Z_{i} - Z_{{i - 1}} } \right)}}} $$

(A13)

$$ \frac{{\partial {\text{LR}}z}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{ul}}_{{i,j}} {\left( {X_{{j + 1}} - X_{j} } \right)}}} {{2{\left( {Z_{{i + 1}} - Z_{i} } \right)}}} - \frac{{g{\left( {X_{{j + 1}} - X_{j} } \right)}}} {2}\begin{array}{*{20}c} {{}} \\ \end{array} \frac{{\partial K{\text{ul}}_{{i,j}} }} {{\partial \psi _{{i,j}} }} $$

(A14)

$$ \frac{{\partial {\text{UL}}z}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{lr}}_{{i - 1,j - 1}} {\left( {X_{j} - X_{{j - 1}} } \right)}}} {{2{\left( {Z_{i} - Z_{{i - 1}} } \right)}}} $$

(A15)

$$ \frac{{\partial {\text{LL}}z}} {{\partial \psi _{{i,j}} }} = \frac{{ - K{\text{ur}}_{{i,j - 1}} {\left( {X_{j} - X_{{j - 1}} } \right)}}} {{2{\left( {Z_{{i + 1}} - Z_{i} } \right)}}} - \frac{{g{\left( {X_{j} - X_{{j - 1}} } \right)}}} {2}\begin{array}{*{20}c} {{}} \\ \end{array} \frac{{\partial K{\text{ur}}_{{i,j - 1}} }} {{\partial \psi _{{i,j}} }} $$

(A16)

$$ \frac{{\partial S}} {{\partial \psi _{{i,j}} }} = \rho _{w} \frac{{\partial \theta ^{{t + \Delta t}}_{{i,j}} }} {{\partial \psi _{{i,j}} }}{\left( {\frac{{{\left( {X_{{j + 1}} - X_{{j - 1}} } \right)}{\left( {Z_{{i + 1}} - Z_{{i - 1}} } \right)}}} {{4\Delta t}}} \right)} $$

(A17)