Explicit Formula for the Three-Parameter Infiltration Equation

The two-parameter (i.e., sorptivity and maximum hydraulic conductivity) Green-Ampt (GA) and Talsma-Parlange (TP) infiltration equations bracket the range of possible responses for cumulative water inflow into initially dry or uniformly moist soil profiles. We show that these implicit equations can be represented in explicit form (infiltration as a function of time) using the Lambert W function. Of these formulas, TP has been shown to be closest to measured data. The three-parameter equation was introduced to interpolate between these limits, with α as the interpolating parameter. The GA case corresponds to α = 0 whereas TP results for α = 1. Besides the limiting cases, another exact explicit solution is available for α = 1/2. Guided by the behaviour of the Lambert W function, an analytical iteration scheme is presented for solution of the three-parameter equation. However, by constructing an accurate explicit approximation (maximum relative error less than 0.05%), we show that iteration is not needed in most circumstances. An exception is the one-dimensional intercept, for which one iteration of the analytical approximation might be needed.