Functional Extremum Solution and Parameter Estimation for One-Dimensional Vertical Infiltration using the Brooks–Corey Model

A simple analytical solution of one-dimensional vertical infiltration of soil water is of great importance for the estimation of soil hydraulic properties and for precision irrigation. We use a new approximate analytical method based on the principle of least action and the variational principle for simulating the vertical infiltration of water in unsaturated soil for a constant-saturation upper boundary and for estimating infiltration parameters. The method proposes the functional extremum problem of one-dimensional vertical infiltration of water in unsaturated soil using the Brooks–Corey model. A functional extremum solution (FES) to Richards’ equation was obtained using the Euler–Lagrange equation and the integral mean-value theorem. A modified functional extremum solution (MFES) was also proposed empirically to improve precision based on the FES of horizontal absorption and numerical data simulated by Hydrus-1D. The relationships between a parameter derived by the value theorem of the integral (α), the air-entry suction (hd), and the wetting front depth (zf) were analyzed by comparing MFES with the numerical distribution of soil water content. The MFES was used to describe the nonlinear relationship between cumulative infiltration and zf, with the relative error (Rₑ) of 0.986. The MFES was also used to describe the nonlinear relationship between the infiltration rate (i) and the reciprocal of zf for soils with hd ≥10 cm, with Rₑ 0.8453. A simple method for estimating soil saturated hydraulic conductivity (Kₛ) was proposed from the nonlinear relationship between i and zf, and the sensitivities of parameters in the Brooks–Corey model were also analyzed.