Approximate analytical solution and parameter estimation for one‐dimensional horizontal absorption based on the van Genuchten–Mualem model

The van Genuchten–Mualem (VGM) model is used to describe the absorption of water by soil. The model is highly nonlinear, so developing a simple analytical or approximate method to predict the movement of soil water via the VGM model then estimating the model parameters is necessary. We present a approximate functional extremum solution (FES) for the problem of one‐dimensional horizontal absorption (1DHA) of water in soil under constant saturated boundary conditions and uniform initial conditions based on the least action and the variational principles. Simple linear relationships between the infiltration characteristics (distribution of soil water content, cumulative infiltration, infiltration rate, wetting front distance, and infiltration time) and the VGM model parameters were obtained. We developed a simple approximate analytical method for estimating the VGM model parameters by testing 1DHA. Numerical data simulated by HYDRUS‐1D were compared with the analytical results and used to verify our method's feasibility. The results indicated that the FES could accurately predict the infiltration characteristics, with relative errors between the approximate analytical and numerical solutions of 0.8 to 8.5% for the distribution of soil water content, 0.1 to 0.8% for cumulative infiltration, 1.1 to 6.4% for infiltration rate, and 0.1 to 6.9% for wetting front distance. Estimates of the parameters indicated that errors ranged from 3.2 to 28.3% for α (inverse of the mean pore diameter) and 0.4 to 8.0% for n (pore size distribution index). The proposed methods can be used to predict 1DHA and to estimate α and n in the VGM model.