A thermodynamics-based constitutive formulation for localised compaction bands

Abstract

We present a constitutive formulation for two-phase porous media based on an explicit consideration of the transport phenomena linking solid stress-deformation relations with transport of fluidized matter via a mixture theory approach of the two phases. In this formulation localized deformation (e.g. emergence of compaction bands) appears as a mass movement synchronization of the two phases which leads to a percolation phenomenon of the fluidized phase. Conservation of mass, momentum, energy as well as 2nd law constraints are cast into an extended reaction-diffusion equation, describing the interaction of thermodynamic forces and fluxes between the two phases. Of special interest is the non-local interaction of solid and fluid pressures that can grow from small perturbations into a localised standing wave instability marking the onset of an interconnected network of the weak phase. We present an analytical analysis of the linear stability of the new constitutive approach leading to a clear identification of the conditions for the onset of such stationary (Turing) instabilities. We apply the theory to a compaction experiment of a highly porous (45% porosity) carbonate rock where fluidization of the solid mass appears as a reactive source term and a non-local diffusion term. We show that the complexity of the experimental system and the emergence of localized structures, their mechanisms of propagation, their interaction and associated uncertainties can be captured by the two coupled fluid-solid pressure reaction-diffusion equations. The new constitutive approach caters for a minimal description of the problem yet can describe the full plethora of possible instabilities identified in a parametric study. Transient waves are also expected but difficult to detect as they have a complex waveform structure and small amplitude. The classical constitutive approach with rate-effects is treated as a quasi-static end member of this approach.