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Stability Analysis of Infiltration¹

ABSTRACT

A rigorous hydrodynamic stability analysis of the delta-function model of infiltration is developed. Raats recently proposed as the criterion for instability that wetting front velocity should increase with front penetration. This detailed analysis, on the other hand, shows that the fundamental criterion is that the water pressure gradient immediately behind the front should oppose the flow. The two criteria agree only in the special case where five significant parameters of the system are all constant and independent of depth, or of depth to the front. The Raats criterion fails in the various more general cases with one or more of these quantities variable.

Quantitative criteria are developed for various specific circumstances in which the flow may become unstable: penetration to a non-wetting stratum; compression of air ahead of the front; redistribution following infiltration; increase of conductivity with depth.

The analysis indicates a definite maximum wave number for unstable disturbances, M_crit, and a wave number for disturbances which are amplified most rapidly, 3^- M_crit. Flows in a laboratory soil column may therefore remain stable because the cross-sectional dimensions are too small. Observed dimensions of fingers in unstable infiltration flows are consistent with the predicted wave number for maximum amplification.

The study is readily extended to time-dependent flows of two immiscible fluids in heterogeneous media. It applies also to time-dependent flows in heterogeneous Hele-Shaw cells.

Care should be exercised in applying results for the deltafunction model to flows with gradual wetting fronts, especially when nonwetting strata are involved.

¹ Contribution from CSIRO, Australia.

² Chief, CSIRO Division of Environmental Mechanics, P. O. Box 821, Canberra City, A.C.T. 2601 Australia.

Received for publication February 20, 1975. Accepted for publication July 14, 1975.

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