Green–Ampt approximations

The solution to the Green and Ampt infiltration equation is expressible in terms of the Lambert W−1 function. Approximations for Green and Ampt infiltration are thus derivable from approximations for the W−1 function and vice versa. An infinite family of asymptotic expansions to W−1 is presented. Although these expansions do not converge near the branch point of the W function (corresponds to Green–Ampt infiltration with immediate ponding), a method is presented for approximating W−1 that is exact at the branch point and asymptotically, with interpolation between these limits. Some existing and several new simple and compact yet robust approximations applicable to Green–Ampt infiltration and flux are presented, the most accurate of which has a maximum relative error of 5 × 10−5%. This error is orders of magnitude lower than any existing analytical approximations.