## 1. Introduction

[2] Liquid infiltration into the subsurface is an important hydrological process that, for example, occurs during groundwater recharge or when liquid contaminants are spilled onto soil. Such flows are generally best described by Richards' equation, which can account for 3-D variably saturated flow. If computational time is an issue or if it is difficult to characterize the subsurface, the 1-D Green-Ampt (GA) model [*Green and Ampt*, 1911] is a viable alternative. In fact, it is used in the Hydrologic Modeling System (HEC-HMS) developed by the US Army Corps of Engineers to simulate precipitation-runoff processes in watersheds [*Feldman*, 2000]. The GA model is also used to arrive at probabilistic forecasts of infiltration in heterogeneous soils [*Bresler and Dagan*, 1983; *Wang and Tartakovsky*, 2011].

[3] The GA model assumes a sharp infiltration front with a constant matric potential which has been related to soil hydraulic parameters [*Neuman*, 1976]. *Weisbrod et al.* [2009] have shown that the early stage of infiltration was best described by a GA model that accounts for dynamic effects by letting the front pressure depend on a contact angle that is significantly larger than the equilibrium contact angle and that remains constant during infiltration. Column infiltration experiments have shown, however, that the pressure head at the front depends on the flow velocity [*Weitz et al.*, 1987; *Geiger and Durnford*, 2000; *Annaka and Hanayama*, 2005]. Based on these experiments and dimensional analysis, *Hsu and Hilpert* [2011] posited a general functional form for the suction head at the front that accounts for porosity, initial water content, and wettability. They then incorporated this expression into the GA model and showed that transient capillary rise can be better described by the modified GA approach than by the classical one. In this paper, we examine whether this is also the case for transient downward infiltration, which can be expected to possess sharper wetting fronts and a different dynamic capillary pressure relation. This study could be more relevant to practical applications of the GA model, as it is more frequently used to model downward infiltration than capillary rise.

[4] For downward infiltration, the modified GA model takes the form of the following ordinary differential equation (ODE) [*Hsu and Hilpert*, 2011]:

where *S*_{0} is the equilibrium suction head, is the mobile water content (i.e., the difference between the initial water content and the saturated water content ), *K* is the saturated hydraulic conductivity, *l* is the wetting front position, is the front velocity, *H*_{0} is the ponding depth, is the interfacial tension, is the water density, *D* is the grain size, *g* is the acceleration of gravity, is a nondimensional function of porosity and , and are model parameters specific to the porous medium that quantify capillary nonequilibrium, and is the dynamic viscosity of water. Note that is typically smaller than due to entrapped air.

[5] Unlike the classical GA approach, the infiltration model presented in equation (1) accounts for a dynamic capillary pressure which depends on the velocity of the wetting front . The modified GA approach is also meant to overcome deficiencies in the classical GA approach, which predicts an initial infinite wetting front velocity due to the assumption of a constant capillary pressure.

[6] In this study, we performed downward infiltration experiments in sand columns to test whether the modified GA approach can better describe downward infiltration than the classical one, particularly during the early stages of infiltration. Like in the capillary rise study by *Hsu and Hilpert* [2011], we fitted the parameters in the relationship for the nonequilibrium suction. Comparisons between experiments and the modeling were based on traditional wetting front position *l* versus time *t* plots. However, Hsu and Hilpert also created plots originally proposed by *Tabuchi* [1971] that show the product between Darcy velocity and *l* versus *l*. The versus *l* plots highlighted the limited ability of the traditional GA approach to predict the convergence of the front toward the equilibrium water table during upward infiltration.

[7] We hypothesized that versus *l* plots also reveal shortcomings of the classical GA approach for modeling downward infiltration. In the classical GA approach, for which in equation (1), scales linearly with *l*, and *q* gradually approaches a steady state Darcy velocity during the late stage of infiltration that equals the hydraulic conductivity *K*. Moreover, as implying that which is unphysical. The modified GA approach, however, resolves this inconsistency as a velocity-dependent capillary pressure avoids the initially infinite rate of infiltration. Figure 1 shows hypothetical versus *l* plots for both the classical and modified GA approach. Our experiments indeed show that the modified GA approach describes downward infiltration experiments better than the classical one.

[8] We also checked whether potential differences between our infiltration experiments and the classical GA approach can be attributed to high Reynolds numbers Re occurring during the initial phase of infiltration that lead to a violation of Darcy**'**s law. Therefore we also modeled our experiments by a GA approach that accounts for inertial forces by incorporation of the Forcheimer model [*Gill*, 1976]. We used a parameterization suggested by *Ward* [1964]: where is the permeability, and is an empirical parameter. This relation results in the following modified GA approach:

We concluded that inertial forces did not play a relevant role in this study.