Stability analysis of the unsaturated water flow equation: 1. Mathematical derivation

Abstract

This paper provides a theoretical stability analysis of gradual wetting fronts based on perturbation analysis. A traveling wave solution of the one‐dimensional vertical flow Richards' equation is used as the basic flow on which three‐dimensional perturbations are introduced. By locally linearizing the diffusivity form of the three‐dimensional Richards' equation a linear partial differential equation is obtained which governs the perturbation variables. The stability of each point at the wetting front is considered in a local coordinate system. The analysis of this perturbation equation at these points of the wetting front provides not only the relationship between the finger sizes and the nonponding infiltration rates at the soil surface but also the traveling speeds of the fingers rooted from these points. Once a perturbation is introduced at some point on the wetting front, there are three possibilities for the development of the perturbation. (1) The perturbation will monotonically decline with time; in this case, no fingers will form and the system is stable. (2) The perturbation does not decline with time, but its downward velocity is less than that of the stable basic wetting front; thus the distribution layer will gradually cover the fingers and the system will become stable. (3) The perturbation will increase with time and have a downward velocity greater than that of the stable wetting front; in this case, the finger will persistently grow in front of the stable wetting front and the system will become unstable. This analysis can be applied to an unsaturated homogeneous soil profile with uniform initial water content for the prediction of instability and for the estimation of finger characteristics over a wide range of infiltration rates.