Abstract

Biot’s equation is solved for a three-layer system in which the centre granular layer is sandwiched between two virtually undeformable, but slightly permeable slabs, and then sheared. The intention is to use the outcome of the mathematical model to make plausible statements about the internal stress, deformation and flow state in partially molten rock. Such statements may be quite sensitive to the choice of the modelling parameters. We begin by setting out the mathematical details, and then present a sensitivity analysis. The loading consists of three elements: a shear stress, monotonically applied on the granular centre layer; a linearly increasing fluid pressure at the bottom of the bottom layer and independently a linearly increasing fluid pressure at the top of the top layer. A simplified specialisation is introduced in which the middle layer is much more permeable than the top and bottom layers. The evolution of the excess pore pressure in the layer is determined for long times. There are two distinct cases. The first is the one in which the externally applied fluid pressures are significant, the second is the limit in which these pressure increases are insignificant. In the latter case the long-term behaviour of the fluid pressure in the three layers is entirely determined by the applied shear stress rate and a stable steady state appears, in other words, the leading term in the long-time expansion settles down to a time independent value. When there is an external fluid pressure increase no steady state settles and the leading term in the long-time expansion is proportional to the time t. The instability that is created in this way needs examination as the hydraulic failure that is associated with it will take place at the position where the pressure increase is greatest. The largest pressure in the middle layer occurs at the position y = 0 for positive pressure increases at the bottom. This is then the point where failure may first be expected in the granular mass for slow loading. Progressively then the layer becomes wider as shear continues and the pressure becomes more and more negative. Now, there comes a point when the granular layer itself crushes further which causes the permeability to decrease as the grain size becomes smaller. The latter effect will decrease the magnitude of the pressure increase and a steady state is reached when the growth in the layer and the reduction in permeability due to further crushing even one another out.