We describe the method used by Fowler and Krantz ( 1993 ) which reduces the Miller model of frost heave to two linear ordinary differential equations. Numerical solutions are presented and qualitative comparison is made with a numerical solution to the full system of equations given by O'Neill and Miller ( 1985 ).

Summary A 3‐D theory of groundwater flow and consolidation that allows for compressibility of the soil or sediment matrix is that due to Biot (1941). However, this theory is inapplicable to virgin consolidation tests, where the deformation of the soil along the critical state surface is accomplished by elastic–plastic flow. In fact, Biot’s model cannot even be applied to overconsolidated soils where the deformation is elastic, as it does not conserve the mass of the solid. In this paper, we provide a modified version of the model, and show how it can be extended to allow for elastic–plastic deformation and large‐amplitude deformations in elastic–plastic compaction. A particular example of the potential application of this theory is to the groundwater flow induced by the compaction of sediments by an overriding ice sheet, and we show how the commonly used Dupuit approximation can be employed to simplify the theory.