Criterion for instability of steady-state unsaturated flows

The stability of steady-state solutions to the unsaturated flow equation is examined. Conditions under which infinitesimal disturbances are amplified are determined by linear stability analysis. Uniform suction head profiles are shown to be linearly stable to three-dimensional disturbances. The stability of nonuniform suction head profiles to planar (gc ₁ ? χ ₂) disturbances is examined. When the steady-state suction head solution (Ψ) increases with depth, χ₃, (dΨ/dχ₃ > 0), a condition for the amplification of infinitesimal planar disturbances is identified as $$\frac{{d^2 K(\Psi )}}{{d\Psi ^2 }} > \frac{{\left( {\frac{{dK(\Psi )}}{{d\Psi }}} \right)^2 }}{{K(\Psi )}},$$ , where K(Ψ) is the hydraulic conductivity versus suction head characteristic of the porous medium. The same condition applies when dΨ/dχ₃ < -1. Therefore when the rate of change of the slope of the K - Ψ characteristic curves is larger than the squared slope divided by K, even small disturbances can be amplified exponentially. The smallest wavelength of unstable planar perturbations is shown to be inversely related to the coarseness of the soil. Conditions under which the instability criterion is met are delineated for some commonly employed K - Ψ curves.