Steady Flows in Unsaturated Soils Are Stable

The stability of steady, vertically upward and downward flow of water in a homogeneous layer of soil is analyzed. Three equivalent dimensionless forms of the Richards equation are introduced, namely the pressure head, saturation, and matric flux potential forms. To illustrate general results and derive special results, use is made of several representative classes of soils. For all classes of soils with a Lipschitz continuous relationship between the hydraulic conductivity and the matric flux potential, steady flows are shown to be unique. In addition, linear stability of these steady flows is proved. To this end, use is made of the energy method, in which one considers (weighted) L ²-norms of the perturbations of the steady flows. This gives a general restriction of the dependence of the hydraulic conductivity upon the matric flux potential, yielding linear stability and exponential decay with time of a specific weighted L ²-norm. It is shown that for other norms the ultimate decay towards the steady-solution is preceded by transient growth. An extension of the Richards equation to take into account dynamic memory effects is also considered. It is shown that the stability condition for the standard Richards equation implies linear stability of the steady solution of the extended model.