Derivation of a two-term infiltration equation from the Green–Ampt model

For a uniform and infinitely deep soil at constant initial volumetric soil water content θn, the soil surface is subjected instantaneously to a ponded-water head that increases with the square root of time t1/2 after the initial ponding of water. An integration procedure is used in a rigorous derivation of the one-dimensional Green–Ampt flux equation. The resulting ordinary differential equation is solved exactly to yield the two-term infiltration equation wherein the coefficient of t is the sated hydraulic conductivity, but the ponded-head t1/2 function must be the same as the one found earlier by a different but exact mathematical derivation. The present findings reveal and describe this previously unknown kinship between the Green–Ampt infiltration model and the two-term infiltration equation, and underscore in still another way the serious limitations of the two-term equation.