Variational decomposition of Richards equation based on the minimum action principle

The classical Richards infiltration control equation belongs to partial differential equation and is strongly nonlinear, that it is difficult to obtain analytical solution. In this study, taking infiltration time as the minimum action, the time functional of infiltration path was established based on Richards equation to transform the vertical infiltration problem of unsaturated soil considering gravity into functional extremum problem, which was solved through the constructed equivalent Euler-Lagrange equation. The calculation results reveal a functional relationship of the diffusion coefficient D (6) with the distance of the generalized wetting front. Three solutions can be found when the form of diffusion coefficient D (9) is known: the moisture content at the wetting front under the optimal path, the minimum moisture content at the distant wetting front and the maximum entropy distribution of soil moisture content. Meanwhile, the accuracy of solving Richards equation by Boltzmann transformation and linear transformation is tested by examples. In the solving process, there is no new variable introduced to simplify the Richards equation, and the structure of the original equation remains, so that the solution is universal and can be used as a supplement to the mechanical calculation of unsaturated soil.