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

Abstract

S U M M A R YThe 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.

Creep Deformation in Vaca Muerta Shale From Nanoindentation to Triaxial Experiments

“…A similar idea has been used to model the log time relaxation characteristic of the so‐called slow dynamics observed in many materials including rocks (Gibbs et al, ; TenCate et al, ; Zaitsev et al, , ). To illustrate the concept, Snieder et al () showed that summing individual relaxation functions with exponential Maxwell kinetics, f i ( t )=exp(‐ t / τ i )/ τ i , where τ i is the relaxation time, gives an aggregate relaxation function characterized by a long log( t ) behavior at intermediate times. One important feature of this model is that the duration of the log( t ) segment can be arbitrarily extended by increasing the range of relaxation times, that is, [ τ min , τ max ].…”

“…One important feature of this model is that the duration of the log( t ) segment can be arbitrarily extended by increasing the range of relaxation times, that is, [ τ min , τ max ]. In the same paper, Snieder et al () presented another illustrative model with no variations in relaxation time but which, nevertheless, produced approximate log( t ) kinetics. This model considers the time‐dependent closure of a rough fracture contained in a linear viscoelastic Kelvin material (dashpot and spring in parallel ).…”

“…As time evolves, new contacts are formed and stresses are transferred to the new contacts, causing fracture closure to decelerate. Snieder et al () observed approximate log time fracture closure at intermediate times when the distribution of asperity heights (i.e., the distribution of times of contact) was sufficiently broad.…”

“…Several previous studies suggest that t c in various shales is between 0.4 and 1.2 days. Using this range, the creep moduli observed in Vaca Muerta yield a long‐term viscosity term of 0.6 to 2 × 10 7 GPa.s. Finally, motivated by the study of Snieder et al (), we devised a conceptual model containing viscoelastic elements that remain inactive until certain activation strains (represented as displacements in the model) are reached. When there are many different elements with widely distributed activation strains, the model produces time‐dependent global deformation approximately proportional to log( t ), and the response eventually becomes asymptotically linear in time when all the elements are activated.…”

Complex Near‐Surface Rheology Inferred From the Response of Greater Tokyo to Strong Ground Motions

“…Following the strong ground motions, a relaxation process may follow as the grains rearrange themselves and/or cracks close as observed in rock samples during laboratory experiments (e.g., Johnson & Jia, ; Ten Cate & Shankland, ; TenCate et al, ) and in the shallow subsurface after earthquakes (Nakata & Snieder, ; Rubinstein & Beroza, ; Sawazaki et al, ; Schaff & Beroza, ). This relaxation process can be modeled through viscoelastic rheology as the medium progressively recovers to its preevent state (e.g., Hamiel et al, ; Lebedev & Ostrovsky, ; Lieou et al, ; Lyakhovsky, Ben‐Zion, & Agnon, ; Snieder et al, ). For extreme levels of strain, a pure plastic rheology better describes the material response, whereby a permanent and not recoverable deformation occurs.…”