Two-phase flow of water and air during aerated subsurface drip irrigation Ninghu Su a,b, * , David J. Midmore b a School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China b Primary Industries Research Centre, Central Queensland University, Rockhampton, Qld 4702, Australia Received 28 October 2004; revised 28 October 2004; accepted 17 February 2005 Abstract 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. q 2005 Elsevier B.V. All rights reserved. Keywords: Two-phase flow; Airflow; Water flow; Subsurface irrigation; Mathematical models 1. Introduction In this paper, we investigate aerated subsurface irrigation, i.e. the concurrent flow of water and air. We nominally name this method ‘oxygation’ as compared to ‘fertigation’ or ‘chemigation’ when fertilizers are added to the irrigation water supply. Oxygation is a typical two-phase flow problem. Irrigation is an age-old practice and has been continuously refined to meet different requirements. While beneficial to plants, irrigation, when performed improperly, may have unwanted side effects on the environment such as irrigation-induced salinity in soils, groundwater and surface water bodies. In addition to those side effects, water applied during irrigation may not necessarily assist plants to function properly and achieve increased yield as expected. During an irrigation event, with whatever delivery technique, the purging of the air in the soil by the infiltrating front of water will create a verifiable level of anaerobic condition in the irrigated zones (Sojka, 1992; Drew and Stolzy, 1996; Arteca, 1997; Rawler et al., 2002; Blokhina et al., 2003). Consequently, by reducing oxygen concentration in the soil as it is Journal of Hydrology 313 (2005) 158–165 www.elsevier.com/locate/jhydrol 0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2005.02.025 * Corresponding author. E-mail address: n.su@cqu.edu.au (N. Su).