Corresponding author: M. Hilpert, Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. (email@example.com)
 In a recent study by Hsu and Hilpert (2011), the Green-Ampt (GA) model for water infiltration was modified to account for a dynamic capillary pressure that depends on the wetting front velocity. In that study, the only transient flow, to which the modified GA approach was compared, was capillary rise. In this paper, transient downward infiltration experiments were performed using three different ponding depths. We demonstrated that infiltration can be effectively modeled using the modified GA approach. The parameters of the dynamic capillary pressure relationship were fitted and independent of the ponding depth. Furthermore, the experimental data could not be described accurately at early times with the classical GA approach. This was particularly evident when plotting the product between the Darcy velocity and the wetting front position against the wetting front position. In these plots, the initial unphysical infinite rate of infiltration, that occurs in the classical GA model, appears to be a primary element of inaccuracy. It was furthermore verified that the limited ability of the classical GA approach to describe the experimental data is not related to inertial forces arising during the initial phase of infiltration.
 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].
 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.  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  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.
 For downward infiltration, the modified GA model takes the form of the following ordinary differential equation (ODE) [Hsu and Hilpert, 2011]:
where S0 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, H0 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.
 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.
 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 , 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  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.
 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.
 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 : 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.
 A total of nine downward infiltration experiments were performed in sand columns. The experiments were carried out using a 60 cm long glass column with an inside diameter of about 2.5 cm. The cross-sectional area was determined to be 5.5 cm2. The columns were packed with 40–100 mesh silica sand (Fisher Scientific), which was first washed and dried to get rid of small particles and impurities. In order to obtain a uniform deposition of the sand grains, the columns were packed using a device formed by a funnel attached to a 120 cm long Tygon plastic tube, 1 cm in diameter. The plastic tubing was lowered down to the bottom of the column and sand was poured into the funnel at a constant rate. The packing device was kept at a constant distance of about 3 cm from the rising sand level during the packing operation. This way, the falling velocity of the sand grains remained constant, and discontinuities in the packing were avoided. A Marriotte bottle, acting as a constant head reservoir, was attached to the top of the column by means of Tygon tubing and a threaded connection with a stopcock that was screwed into the column. Figure 2 illustrates the experimental setup.
 Triplicate infiltration experiments were performed for three different ponding depths H0: 10, 20 and 40 cm. The variation in mass of the water reservoir was monitored over time with a balance (Ohaus SP2001) connected to a PC. The balance recorded the changing mass of the reservoir in 1 s time steps. Before each experiment, the column was repacked with oven-dried sand in order to facilitate establishing a zero initial water content ( ). Porosity was, therefore, a random variable and determined by first measuring the grain density of the sand using water pycnometry and the bulk density of each of the columns.
 Experiments were video recorded with a camera connected to a PC. Since a measuring tape was attached down the entire length of the column, we could determine l as a function of time t from the video recording. Therefore, we could also calculate the front velocity . The column was long enough such that the wetting front assumed its steady state velocity vss once it reached the bottom of the column. From the videos, we determined the travel time of the front along the bottom 5 cm of the column and then calculated vss. In order to get a better image resolution and therefore a better front position accuracy, the video recording was focused only on the bottom 20 cm of the column.
3. Experimental Results and Data Analysis
 The mass of water infiltrating the column, m, was calculated from the mass loss in the water reservoir. This loss was then converted into the cumulative infiltration
For each time step, the Darcy velocity was calculated through a finite difference expression for the cumulative infiltration, . From F, we could also infer the wetting front position
The mobile water content was estimated from the steady state Darcy velocity qss and vss via where qss was determined by averaging the q values that the front assumed while moving through the bottom 5 cm of the column.
 From the experimental data, we created the versus l plot shown in Figure 3. Due to the finite digital resolution of the balance (0.1 g), the recorded mass did not change during each time step. Therefore, we coarsened the F(t) data by eliminating data points during which mass did not change. The coarsened time steps increased with l. Relatively small time steps of 2 s were used at the beginning of the experiments. The plot clearly shows that larger ponding depths result in larger Darcy velocities. As expected, the early time Darcy velocity decreased considerably, eventually reaching steady state (where the slope of the versus l curves becomes uniform). Linear regression using the values between 50 to 60 cm yielded acceptable R-Squared values for all nine experiments (between 0.82 and 0.98). The slope of the straight lines were taken as an indicative value of the hydraulic conductivity K in each sand column. The average hydraulic conductivity in the nine infiltration experiments was found to be cm s−1. The density of the sand grains was g cm−3. The porosity in the nine columns was .
 In order to model the wetting front position l and the front velocity as a function of time t, equation (1) was solved numerically. For the classical GA approach, however, we could use the analytical solutions developed by Green and Ampt. Our aim was to investigate whether the experimental data could be accurately described by the modified GA approach, which incorporates a dynamic capillary pressure term, rather than by the simpler classical GA approach. Known parameters used in solving equation (1) were H0 and the grain size D which varied between 0.15 and 0.40 mm and was taken as the arithmetic mean grain size, mm. For the fluid properties, we assumed Pa s, kg m−3 and N m−1. The mobile water content was determined from the steady state Darcy and front velocities. We fixed in order to reduce the number of fit parameters. According to Hsu and Hilpert , a good model fit requires a 1. In their analysis of infiltration experiments performed by Geiger and Durnford , the best fit for was found to be 0.3, which we also used for this study.
 For each of the nine column experiments, unknown parameters were inversely determined from the measurements using a MATLAB implementation of an implicit filtering method [Kelley, 2011]. Implicit filtering uses a projected quasi-Newton algorithm to solve bound constrained optimization problems. Unknown parameters in solving equation (1) were , K, S0, and the functional form of . As in the Hsu and Hilpert  analysis, some unknown parameters were combined to obtain . The final fit parameters were fneq, S0, and K. The objective function used in the implicit filtering accounts for the difference between experimental and computational versus l data. The l versus t data were also taken into account in order to be able to accurately model the temporal behavior of the infiltration experiments.
Table 1 summarizes the results of the inverse modeling for both the classical and modified GA equations. Results obtained for the same ponding depth H0 were condensed in that we present the average and standard deviation for each fit parameter. Among the modified GA parameters, S0 did not show any specific trend with respect to changes in H0. Furthermore, fitted ranges of S0 for the three different sets of experiments overlap between 60 and 62 cm. The fneq parameter also showed good consistency among all experiments, in that, the uncertainty intervals for the three H0 values overlap with one another. Moreover, all fneq values fell in the range defined by Hsu and Hilpert  in their analysis of the Geiger and Durnford  infiltration experiments. The K values proved very consistent and all fell in the K range from the linear regression analysis presented in section 3. For the classical GA model, we fitted K and S0 (see Table 1). The S0 values seem to be underestimated, whereas the K values are consistent with the linear regression analysis.
Table 1. Porosity , and Fit Parameters for the Classical and Modified GA Model According to Equation (1)a
Modified GA Fit Parameters
Classical GA Fit Parameters
K [cm s−1]
K [cm s−1]
Parameters were averaged for the triplicate downward infiltration experiments carried out at the same H0.
0.365 ± 0.007
56 ± 11
652.94 ± 51.40
0.0285 ± 0.0034
11 ± 13
0.0252 ± 0.0065
0.361 ± 0.006
72 ± 12
740.09 ± 167.01
0.0289 ± 0.0037
5 ± 8
0.0331 ± 0.0095
0.365 ± 0.076
45 ± 17
435.45 ± 180.76
0.0309 ± 0.0080
23 ± 31
0.0247 ± 0.0075
 We show model fits for only one experiment which is indicative of all nine experiments. Figure 4a shows the model predictions of versus l along with experimental data for both the modified and the classical GA approach for a ponding depth 20 cm. By accounting for a dynamic capillary pressure, the modified GA model is fully capable of accurately describing the nonlinearity manifested at the beginning of the infiltration experiment. However, noticeable discrepancies appear under the classical GA approach due to the assumption of a constant pressure head present at the front throughout the whole infiltration event. Furthermore, in assuming an unphysical infinite wetting front velocity ( ) at the beginning of infiltration when l = 0, the classical GA approach fails in modeling the initial conditions of the experiment.
Figure 4b shows similar findings for the l versus t plots. The classical GA model fails to predict early infiltration behavior. It predicts an initially infinite slope because the predicted initial Darcy velocity is infinite too. Since the classical GA approach overpredicts measured l-values during the early stage of infiltration, the best model fit underpredicts the experimental l data for large infiltration times. Conversely, the modified GA approach describes the experimental data well in the early as well as in the late stage of infiltration.
 In all infiltration experiments, we observed Re 3 where the highest Re values occurred at the beginning of infiltration. Due to the finite temporal resolution of our F measurements, it is possible that during the first second of infiltration higher Re values occurred and that Darcy's law was violated. We therefore modeled the data by equation (2), the GA approach that accounts for inertial forces. As shown in Figure 5, this model can be fitted quite well to the experimental versus l data and, contrary to the classical GA approach, conforms to as . However, a good fit is only possible if cF is considered to be a fit parameter which then assumes an unrealistically high value of (other fit parameters: cm s−1, cm). If Ward's value for cF is used, the round shoulder for small l values almost vanishes and results in poor model fit (K and S0 values from Table 1 for the classical GA approach). Therefore, modeling our experimental data requires accounting for a dynamic capillary pressure rather than for inertial forces.
5. Discussion and Summary
 Downward infiltration experiments in sand columns were performed for three different ponding depths. The experimental data were presented in form of the traditional l versus t plots. The versus l plots originally proposed by Tabuchi  were also created in order to highlight deficiencies of the modeling approaches. All experiments were modeled by using both the classical and modified GA approach. The fit parameters for the modified GA approach showed good consistency across experiments, indicating an equilibrium suction head of about 60 cm for the media used in this study. The fneq parameters fell in the range between 255 and 907 (Table 1) and were also in close agreement with the Hsu and Hilpert  study. Fitted K values showed close agreement with the linear regression analysis performed using the versus l experimental results. We did not use the exponent in the dynamic capillary pressure relation as a fit parameter, because then the error bounds for all fit parameters would have been unreasonably large. In the future, it would be good to estimate the parameters for the nonequilibrium capillary pressure for a given porous medium from independent infiltration experiments that are performed at different flow rates which are held constant during the course of an experiment (like the Geiger and Durnford experiments). Future work should also examine infiltration into prewetted columns as an initial water content significantly affects the capillary pressure at an infiltration front [DiCarlo, 2007]. One could then determine the values of the more fundamental parameter and the function that make up the lumped parameter fneq.
 The modified GA approach in equation (1) describes all experiments performed in this study accurately. On the other hand, the classical GA approach cannot catch the early stage of the experiments when the front velocity drops steeply. Fitted S0 values for the classical GA approach appear to be underestimated, likely because the optimization cannot accurately determine S0 at early infiltration times, when the modeled wetting front velocity is unphysically infinite. K values were, however, in line with the linear regression analysis. By assuming that the Darcy-scale dynamic capillary pressure depends on the velocity of the wetting front, the modified GA approach brings a substantial improvement over the classical GA approach.
 There was no evidence that inertial forces contributed to the shortcomings of the classical GA approach. Measured Re numbers were less than 3, which would suggest that inertial effects were negligible in our experiments. However, since we could not exclude the possibility of higher Re numbers occurring during the very initial phases of infiltration, we incorporated Forcheimer's equation into the GA model. We showed that inertial forces cannot quantitatively explain the measured versus l plots, even though they remove the unphysical positive value of for