## 1. Introduction

[2] The drainage of fluids from homogeneous porous media has been studied for a long time. Studies may be classified into two groups: the saturated flow approach and the saturated-unsaturated flow approach.

[3] The first group focuses on water flow below the water table only. *Youngs* [1960] derived an approximate equation describing the cumulative outflow based on the capillary tube analogy and a constant drainable porosity. *Arbhabhirama and Ahmed* [1973] considered the effect of pore size distribution in the capillary tube model and derived approximate solutions for nonsteady column drainage. On the basis of a concept of instantaneous and complete drainage at the water table, *Ligon et al.* [1962] derived equations to predict the drawdown of the surface of saturation in the column and the rate of outflow. *Kroszynski* [1975] presented a more detailed list of such kind of work.

[4] The second group regards the water both above and below the water table as a continuum and uses the Richards' equation [*Richards*, 1931] to describe the flow process. *Fujioka and Kitamura* [1964] assumed a constant moisture diffusivity to linearize the Richards' equation and derived an approximate solution for hydraulic head distribution in the column. *Kroszynski* [1975] derived an approximate analytical solution for predicting the pressure head and the drawdown of the surface of saturation. Many researchers [e.g., *Philip*, 1960; *Sander et al.*, 1988; *Ross and Parlange*, 1994; *Parlange et al.*, 1997; *Hogarth and Parlange*, 2000; *Menziani et al.*, 2007] derived analytical solutions to the Richards' equation to investigate water flow in an unsaturated zone. *Triadis and Broadbridge* [2010] systematically summarized analytical solutions of the Richards' equation.

[5] Numerical solutions for the drainage of vertical soil columns were given by a number of researchers, such as *Watson* [1967], *Hornberger and Remson* [1970], and *Kroszynski* [1975]. A lot of numerical algorithms have been derived to solve the Richards' equation for saturated-unsaturated flow [e.g., *Celia et al.*, 1990; *Kirkland et al.*, 1992; *Clement et al.*, 1994; *Rathfelder and Abriola*, 1994; *Babajimopoulos*, 2000; *Kavetski et al.*, 2002; *Casulli and Zanolli*, 2010]. A detailed list of these mathematical models was given by *Clement et al.* [1994]. A full list of numerical methods to solve the Richards' equation is given by *Kosugi* [2008], *Casulli and Zanolli* [2010], and *Juncu et al.* [2010].

[6] The drainage of fluids from porous media is traditionally described by the single-phase water flow approach, and the air phase is ignored. However, there are some situations where the role of the air phase cannot be neglected. *Bouwer and Rice* [1978] concluded that delayed release of pore water from a pumped, unconfined aquifer can be caused by restricted air movement in the vadose zone due to layers of high water content. *Vachaud et al.* [1973] observed that the pressure in the stratified unsaturated zone is significantly different from the external atmospheric pressure and concluded that the impact of air must be taken into account in determining the soil water suction. They suggested that the flow equations must be written in terms of two-phase immiscible fluid flow. Under these circumstances, the single-phase water flow theory is inadequate and the two-phase immiscible fluid flow approach must be used to interpret the drainage process. *Jiao and Guo* [2009] conducted a theoretical study of pumping-induced airflow in an unconfined aquifer capped with a low-permeability layer and concluded that if airflow caused by the low-permeability cap was ignored, the errors in estimated drawdown and then aquifer parameters could be significant.

[7] Multiphase flow in porous media has been studied for more than 60 years. There has been a lot of work on two-phase flow in porous media since the 1970s in both soil water hydrology and oil reservoir engineering. A full list of such work is impossible herein. *Morel-Seytoux* [1973] presented an excellent development of the various forms of the multiphase flow governing equations. *Parker* [1989] and *Muccino et al.* [1998] presented comprehensive reviews on multiphase flow in porous media. The various two-phase flow models were reviewed by *Vauclin* [1989]. Solutions for simultaneous flow of air and water in the vadose zone are mainly on infiltration into the vadose zone with air effects [e.g., *Touma et al.*, 1984; *Morel-Seytoux and Billica*, 1985; *Touma and Vauclin*, 1986; *Weir and Kissling*, 1992; *Wang et al.*, 1997; *Cueto-Felgueroso and Juanes*, 2008]. *Weeks* [2002] and *Guo et al.* [2008] investigated the effects of airflow induced by rain infiltration on water level changes in a well in an unconfined aquifer. *Jiao and Li* [2004] and *Guo and Jiao* [2008] developed numerical solutions for air-water two-phase flow caused by periodic water level fluctuations. *Li and Jiao* [2005] derived an analytical solution for airflow in the unsaturated zone induced by fluctuating water table.

[8] The numerical simulation of multiphase flow has attracted more attention than experimental work [*Kueper and Frind*, 1991]. A number of numerical simulators have been developed to simulate multiphase flow [e.g., *Faust*, 1985; *Kuppusamy et al.*, 1987; *Faust et al.*, 1989; *Kueper and Frind*, 1991; *Celia and Binning*, 1992; *Class et al.*, 2002; *Amaziane et al.*, 2010]. Well-documented software is also available, such as TOUGH2 [*Pruess et al.*, 1999] and NAPL [*Guarnaccia et al.*, 1997]. *Miller et al.* [1998] reviewed the modeling of multiphase flow and transport in heterogeneous porous media. *Niessner and Hassanizadeh* [2008] developed a new numerical model for two-phase flow in porous media including fluid-fluid interfacial area. *Hoteit and Firoozabadi* [2008] considered capillary heterogeneity in numerical modeling of two-phase flow in heterogeneous porous media. On the basis of the immiscible two-phase flow of water and oil in saturated heterogeneous soil columns, *Aggelopoulos and Tsakiroglou* [2009] developed a multiple flow path model for immiscible displacement in heterogeneous soil columns.

[9] The governing equations of two-phase flow are strongly nonlinear, so analytical solutions incorporating fully the effects of gravity and capillarity in transient multiphase flow through heterogeneous porous media are not tractable [*Kueper and Frind*, 1991]. *Buckley and Leverett* [1942] first derived a classical solution for one-dimensional, horizontal two-phase flow without capillary forces. *McWhorter and Sunada* [1990] derived solutions for the horizontal, transient flow of two viscous, incompressible fluids, in which the effect of capillarity was fully incorporated. *Fokas and Yortsos* [1982] derived an exact solution for one-dimensional two-phase flow in a semi-infinite, horizontal reservoir for a constant flux boundary condition. *Rogers et al.* [1983] extended the solution of *Fokas and Yortsos* [1982] to take gravitational effects into consideration. *Sander et al.* [1993] derived an exact solution to the one-dimensional diffusion-convection equation for two-phase flow in porous media with Dirichlet boundary conditions. *Sander et al.* [2005] derived exact solutions to radially symmetric two-phase flow into an infinite medium under a constant flux boundary condition. More recently, on the basis of the work of *McWhorter and Sunada* [1990], *Fučík et al.* [2007] derived a semianalytical solution for one-dimensional two-phase flow through a homogeneous porous medium.

[10] The objective of this study is to investigate experimentally and theoretically the effects of air phase on water flow during the drainage of a vertical sand column. Cumulative outflow and vacuum in the vadose zone of the column were measured. An attempt is made to derive solutions for one-dimensional, vertical air-water two-phase flow problem and use the solutions to interpret the experimental data.