The time validity of Philip's two-term infiltration equation: An elusive theoretical quantity?

The two-term infiltration equation  is commonly used to determine the sorptivity, , and product, , of the dimensionless multiple and saturated soil hydraulic conductivity from cumulative vertical infiltration measurements (L) at times (T). This reduced form of the quasi-analytical power series solution of Richardson's equation of Philip enjoys a solid physical underpinning but at the expense of a limited time validity. Using simulated infiltration data, Jaiswal et al. have shown this time validity to equal about 2.5 cm of cumulative infiltration. The goals of this work are twofold. First, we investigate the extent to which cumulative infiltration measurements larger than 2.5 cm bias the estimates of and . Second, we investigate the impact of epistemic errors on the inferred time validities and parameters. Partial infiltration curves up to 2.5 cm of cumulative vertical infiltration improve substantially the agreement between actual and least squares estimates of and . But this only holds if the data generating infiltration process follows Richardson's equation and experimental conditions satisfy assumptions of soil homogeneity and a uniform initial water content. Otherwise, autocorrelated cumulative infiltration residuals will bias the least squares estimates of and . Our findings reiterate and reinvigorate earlier conclusions of Haverkamp et al. and show that epistemic errors deteriorate the physical significance of the coefficients of infiltration functions. As a result, the parameters of infiltration functions cannot simply be used in storm water and vadose zone flow models to forecast runoff and recharge at field and landscape scales unless these predictions are accompanied by realistic uncertainty bounds. We conclude that the time validity of Philip's two-term equation is an elusive theoretical quantity with arbitrary physical meaning.