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

Core Ideas Least action and variational principles were used to simulate unsaturated vertical infiltration. A functional extremum explicit solution to Richards' equation was obtained. A new method describes nonlinear relationships between cumulative infiltration, infiltration rate and wetting‐front depth. The method could be used to accurately predict soil water content and estimate soil hydraulic parameters. 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 ( h d ), and the wetting front depth ( z f ) 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 z f , with the relative error ( R e ) of < 4.1% and the coefficient of determination ( R 2 ) > 0.986. The MFES was also used to describe the nonlinear relationship between the infiltration rate ( i ) and the reciprocal of z f for soils with h d ≥10 cm, with R e < 6.0% and R 2 > 0.8453. A simple method for estimating soil saturated hydraulic conductivity ( K s ) was proposed from the nonlinear relationship between i and z f , and the sensitivities of parameters in the Brooks–Corey model were also analyzed.