A quasi‐solution of Richards' Equation for the downward infiltration of water into soil

Abstract

A new equation is presented that expresses depth Z of soil water content θ at time t as an explicit function of θ and t and holds for all t ≥ 0. The new Z ( θ , t ) includes the classical J. R. Philip (1957 b , c ) solution for asymptotically large times. At moderate times, the new Z ( θ , t ) is matched to the classical Philip solution expressed as ϕ t ½ + χ t + ψ t ⅔ + ω; t 2 + …, where ϕ, χ, ϕ, ω,… are different functions of θ alone. The matching process enables the characterizing functions (or parameters) within Z ( θ , t ) to be evaluated in straightforward fashion from the moderate‐time Philip functions (ϕ, χ, ψ, ω,…). When evaluated for either Philip's data or the Green and Ampt step‐function solution, the new Z ( θ , t ) in each case describes the water content profiles with excellent accuracy for all times. Also, for all t ≥ 0, the new Z ( θ , t ) can be directly integrated to provide a new equation for cumulative quantity of water infiltrated or flux, both as explicit functions of time.