Approximate water vertical infiltration extremum path in unsaturated soils

Abstract

According to the principle of stationary action, water vertically infiltrates unsaturated soils along an infiltration-duration extremum path with linear deviation. The solution of the Euler–Lagrange equation with appropriate boundary conditions yields a linear relationship between the moisture diffusion function and the ratio of the spatial location to the wetting front distance. By integrating the law of mass conservation at the boundary, the temporal evolution of the wetting front is implicitly derived. A linear term is introduced to compensate for the deviation in calculating the position of the wetting front owing to the Taylor series approximation. Based on the representation of the diffusion function in relation to the boundary diffusivity, the spatial location and the position of the wetting front, a direct way to determine hydraulic conductivity is presented. Unlike existing methods, the proposed approach involves fewer hypotheses and is more theoretically robust. Moreover, the material parameters in the proposed approach have a clear physical meaning and can be explicitly determined. Using the Brooks–Corey hydraulic function, the evolution and distribution characteristics of the water content are explicitly determined, while the advancing wetting front is implicitly represented. The results of the proposed solution for different types of soil are in agreement with these numerical and analytical solutions.