The time dependence of rock healing as a universal relaxation process, a tutorial

The material properties of earth materials often change after the material has been perturbed (slow dynamics). For example, the seismic velocity of subsurface materials changes after earthquakes, and granular materials compact after being shaken. Such relaxation processes are associated by observables that change logarithmically with time. Since the logarithm diverges for short and long times, the relaxation can, strictly speaking, not have a log-time dependence. We present a self-contained description of a relaxation function that consists of a superposition of decaying exponentials that has log-time behaviour for intermediate times, but converges to zero for long times, and is finite for t = 0. The relaxation function depends on two parameters, the minimum and maximum relaxation time. These parameters can, in principle, be extracted from the observed relaxation. As an example, we present a crude model of a fracture that is closing under an external stress. Although the fracture model violates some of the assumptions on which the relaxation function is based, it follows the relaxation function well. We provide qualitative arguments that the relaxation process, just like the Gutenberg-Richter law, is applicable to a wide range of systems and has universal properties.