Approximate analytical solutions for one-dimensional soil water movement equation with arbitrary initial conditions

Simple analytical solutions of one-dimensional horizontal and vertical soil water movement are of great importance for the soil hydraulic parameters estimation and precision irrigation. We present the approximate analytical solutions for the problem of horizontal absorption and vertical infiltration in soil under arbitrary initial conditions and constant boundary conditions based on the principle of least action and the variational principle. The algebraic functions of soil water content (SWC) profile, cumulative infiltration, infiltration rate and infiltration time about the wetting front distance/depth (WFD) and the parameters of Brooks-Corey model are obtained. Compared with the numerical results simulated by HYDRUS-1D, the algebraic results calculated by approximate analytical solutions have high precision to predict the horizontal and vertical soil water infiltration based on the known WFD. Contrary to the linear relationship in horizontal absorption with uniform initial conditions, the relationships between cumulative infiltration and WFD, infiltration rate and reciprocal of WFD, square of WFD and infiltration time are nonlinear for the arbitrary heterogeneous initial conditions. However, the initial conditions have little effect on the relationship between infiltration rate and WFD when the initial SWCs are lower than 0.6 θs. Based on the nonlinear relationships of infiltration rate and WFD in vertical infiltration, the WFD and time of the stable infiltration are obtained.