Cross-diffusion Waves in Hydro-Poro-Mechanics

Abstract

We propose a new class of wave-phenomena in multiphase solids (and granular media) triggered by Hydro-Poro-Mechanical coupling and cross-diffusion feedbacks of porous materials. We define cross-diffusion as the phenomenon when a generalized thermodynamic force induces a generalized thermodynamic flux of another kind. Addition of cross-diffusion relaxes the adiabatic constraints on the reaction part of the system and corrects the mathematical ill-posedness. We identify the important aspect of cross-diffusion terms and present a linear stability analysis of the governing partial differential equations (PDE’s). Multiple transient wave instabilities are found as solutions of the coupled PDE’s. In the long-wavelength limit (long-time scale) these waves feed into solitary waves that are standing wave patterns frozen into the porous medium at various scales. We revisit earlier work showing that the wavenumber of the standing wave is entirely defined by the ratio of the mechanical over the fluid (self-diffusion) coefficients of the coupled reaction-cross-diffusion equations. Diffusion coefficients are hence identified as material parameters controlling the criterion for nucleation of waves and the signature of both transient cross- and stationary self-diffusion waves. We show examples of self- and cross-diffusion waves in nature and laboratory experiments as stationary and time-lapse diffusional waves. Our approach offers a simple mathematical framework for analysis of coupled hydro-mechanical porous medium, providing a new fundamental perspective for analyses of the initiation of macroscopic instabilities and transient precursors in many disciplines.