Stability analysis of water movement in unsaturated porous materials: 2. Numerical studies

Abstract

In an earlier paper a detailed theory was presented in which the principles of hydrodynamic stability analysis were used to develop a linear perturbation equation for vertical water movement with nonsharp fronts. In the present study the analysis is applied to the stability of several soil water systems some of which are potentially unstable. The first stage in the analysis requires the generation of water content and pressure head profiles using a computer‐based numerical solution for the flow equation for unsaturated porous materials. These profiles permit the calculation of certain differential terms which appear in the specification of the coefficients required in the stability analysis. Results for the growth rate of imposed disturbances of wavelengths of 1, 5, and 25 cm together with the appropriate eigenvectors are presented for infiltration into a homogeneous profile, redistribution following infiltration, infiltration into a scale heterogeneous medium, and infiltration into a fine‐over‐coarse stratified profile. Although instability was expected for the redistribution, heterogeneous, and layered systems, the results, whilst showing a noticeable trend towards instability, did not predict the occurrence of such a condition. The reason for this is discussed, in particular the evidence which suggests that small amounts of initial water can have a significant effect on the damping‐out of instabilities that otherwise would occur under dry initial conditions. Reference is made to a projected further paper documenting the effect of initial water content on wetting front instability and to the need for additional numerical studies at very low initial water contents.

An overview of instability and fingering during immiscible fluid flow in porous and fractured media

“…If any eigenvalue is positive, it may cause instability under the assumed conditions. Numerical results were given by Diment and Watson (1983) for several cases including redistribution following infiltration, infiltration into a heterogeneous medium in which hydraulic conductivity increases with depth, and infiltration into a fine-over-coarse two-layered system. In no case was the actual flow problem solved; only the eigenvalues of R were determined.…”

Criterion for instability of steady-state unsaturated flows

“…Diment et al (1982) formulated the eigenvalue problem for stability analysis of the unsaturated flow problem, similar to the formulation made here. However in a subsequent numerical solution to that problem, Diment and Watson (1983) did not predict any condition for linear instability and found all the flows they examined to be stable against small perturbations.…”

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

“…It accounts for gravity, capillarity, and the fact that the permeability to water is reduced because the porous medium is only partially saturated with water. The inability of the Richards equation to explain fingered flow is well documented [35,33,51,78,36,38,37,79,61].…”