Two-phase flow of water and air during aerated subsurface drip irrigation

Here, we investigate the two-phase flow problem as in aerated subsurface irrigation. We extend McWhorter's one-dimensional equation for the concurrent flow of air and water (CEFAW) to three dimensions, and present explicit solutions subject to linear functions for the two-phase unsaturated hydraulic conductivity, diffusivity and fractional flow function. We present both steady- and unsteady-state solutions to the CEFAW corresponding to two types of constant continuous sources (a line source and a point source), which are relevant to aerated subsurface irrigation using emitters. The two-dimensional solution appears as the modified Bessel function of the second kind of order zero while the three-dimensional one as a complex exponential function. Graphic illustrations of the mathematical solutions indicate asymmetrical distributions of water content around the supply source due to gravity at the steady state established at large time, which differ in patterns and magnitudes for the two types of supply sources.