Explicit expressions for Green‐Ampt (delta function diffusivity) infiltration rate and cumulative storage

The sharp wetting front model of infiltration (Green and Ampt, 1911; Philip, 1954) yields through simple integration an exact solution relating the infiltration rate (i), cumulative infiltration (I), and time (t). The relation, however, is implicit for i or I; i.e., it is of the form t = A[I − B ln (1 + I/B)]. Numerical iteration is required to find the infiltration rate, and furthermore, analytic manipulations are limited using this traditional formulation of the Green‐Ampt infiltration. In this note we present an accurate expression for the infiltration rate in the form of a rapidly converging series in the variable τ = t/(t + χ). Truncating the series at four terms yields a useful expression for i(t). The proposed four‐term expression gives less than 2% error at all times and is readily integrated to yield the cumulative infiltration I(t). In conjunction with the exact expression for time (t) given i or I, the proposed expression is useful in infiltration/runoff calculations that necessitate the time compression approximation (TCA).