A qualitative analysis of the dynamics of a sheared and pressurized layer of saturated soil

Abstract

A layer of a saturated binary mixture of soil and water, both of which are true density preserving, is considered. This layer is subjected from above to normal and shear tractions and an inflow of water, and from below to drainage of water and abrasion of till from the rock bed. Sliding processes along the top and bottom interfaces, as well as deformational creep of the sediment and water constituents within the layer generate heat, but here the purely mechanical problem is analysed. We study the steady–state plane flow with vanishing abrasion and balanced inflow and drainage of water. The differential equations governing the horizontal creep flows of the sediment and water decouple from the equations describing the vertical profiles of the vertical water velocity and the solid volume fraction. A stiff second–order ordinary differential equation is shown to describe the distribution of the latter; it genuinely depends on the inflow of water, the fluid viscosity and the thermodynamic pressure, a variable not present in classical formulations and introduced by Svendsen and Hutter in 1995. The singular nature of this equation is resolved by methods of matched asymptotic expansions. It is shown that in conformity with thermodynamics in the sediment layer, two regions of more dense and less dense solid fraction arise, one of which is a boundary layer. This boundary layer is formed where the fluid enters the sediment layer. Moreover, by using the thickness of this boundary layer as the internal length parameter, we find several constitutive equations for the thermodynamic pressure which adequately describe this creeping flow problem. Viewed as a model for the saturated till layer below ice sheets, the analysis shows that the question of whether soft basal sliding may develop catastrophically is a question of thermodynamics rather than dynamics.