Approximate solution of a one-dimensional soil water infiltration equation based on the Brooks-Corey model

- The Principle of Least Action and the Variational Principle were used to analyze soil water content distribution.
- The variational principle was combined with the Taylor expansion to solve the extreme value problem.
- A new method was proposed to predict the distribution of soil water content based on the Brooks-Corey Model.

The solution of a one-dimensional unsaturated soil water infiltration equation can provide theoretical support when simulating soil water distribution. To solve the equation for one-dimensional vertical infiltration of soil moisture in unsaturated soil with initially uniform moisture content distribution, a new method is proposed based on the Principle of Least Action and the Variational Principle. We used second-, third- and fourth-order Taylor series expansions to obtain an approximate analytical solution for a one-dimensional, constant water head, vertical infiltration equation. The results were compared with a numerical solution obtained using the HYDRUS-1D software. Our results showed that the soil water content profile, accumulated infiltration and infiltration rate can be calculated by the new method, and the soil water content calculated by the second-, third- and fourth-order approximate analytical solutions was closer to the numerical solution in the higher water content areas. The relative error of the soil water content for clay loam, silt, loam and sandy loam ranged from 0.7187% to 3.0848%, 0.9667% to 2.2847% 0.6310% to 2.2943%, 1.7504% to 6.8823%, respectively. In four soils modeled numerically, the fourth-order results were the best, with the smallest relative errors and the best coefficients of determination. The longer the wetting front distance, the smaller the calculated errors. Thus, the fourth-order approximate analytical solution can be used to simulate the long-term vertical infiltration process. If the results do not require the highest error precision, the second-order approximate analytical solution was simple in form and also sufficient to be used to describe the water content of the vertical infiltration process. Our results show that the approximate analytical solution proposed provides an effective approach to predict soil water content in the vertical infiltration process.