A Reaction-Cross-Diffusion Formulation for the Evolution of Compaction Bands

We present a new thermodynamically consistent model for the time-dependent evolution of compaction bands in porous rocks. The model extends a closed-form analytical solution of compaction band formation for porous media into the time domain. The nucleation of compaction bands is predicted for a critical competition between the nonlinear reaction-diffusion processes of the power-law viscous creep of the matrix in competition with the rates of reaction-diffusion processes of the pore fluid. The width and spacing of compaction bands is regularized through dynamic renormalization of reaction-diffusion processes over a nonlocal zone which in turn governs the style of propagation of the compacting zone. The numerical models are tested against laboratory results for the evolution of compaction bands in sandstone. The results show that the model is able to accurately capture the formation and evolution of compaction bands controlled by a simple parameter space of self-diffusion of compaction of the global matrix $D_M$ and the cross-coupled feedback between solid pressure and mobility of the fluid in the reacting zone $d_H$. Accordingly, three different styles of compaction observed in nature can be reproduced: (a) Classical McKenzie solution with diffuse growth of compaction over the compacted domain; (b) Growth of a rhythmic pattern of compaction bands progressing into the far field (Turing pattern); (c) Growth on any perturbations decreasing their wavelength/thickness over time.