Junior Applied Mathematics Seminar

As temperatures are increasing, so is the presence of meltwater lakes sitting on the surface of the Greenland Ice Sheet. Such lakes have the possibility of draining through cracks in the ice to the bedrock. Observed discharge rates have found that these lakes can drain at three times the flow rate of Niagara Falls. Current models of subglacial drainage systems are unable to cope with such a large and sudden volume of water. This motivates the idea of a 'subglacial blister' which propagates and slowly dissipates underneath the ice sheet. We present a basic hydrofracture model for understanding this process, before carrying out a number of extensions to observe the effects of turbulence, topography, leak-off and finite ice thickness.

The flow of viscous fluid through a soft porous medium exerts drag on the matrix and induces non-uniform deformation. This behaviour can become increasingly complicated when the medium has a complex rheology, such that deformations exhibit elastic (reversible) and plastic (irreversible) behaviour, or when the rheology has a viscous component, making the response of the medium rate dependent. This is perhaps particularly the case when compaction is repeated over many cycles, or when additional forces (e.g. gravity or an external load) act simultaneously with flow to compact the medium, as in many industrial and geophysical applications. Here, we explore the interaction of viscous effects with elastic and plastic media from a theoretical standpoint, focussing on unidirectional compaction. We initially consider how the medium responds to the reversal of flow forcing when some of its initial deformation is non-recoverable. More generally, we explore how spatial variations in stress arising from fluid flow interact with the stress history of the sample when some element of its rheology is plastic and rate-dependent, and characterise the response of the medium depending on the nature of its constitutive laws for effective stress and permeability.