The Fluid Dynamics of Solid Mechanical Shear Zones

Abstract

Shear zones in outcrops and core drillings on active faults commonly reveal two scales of localization, with centimeter to tens of meters thick deformation zones embedding much narrower zones of mm-scale to cm-scale. The narrow zones are often attributed to some form of fast instability such as earthquakes or slow slip events. Surprisingly, the double localisation phenomenon seem to be independent of the mode of failure, as it is observed in brittle cataclastic fault zones as well as ductile mylonitic shear zones. In both, a very thin layer of chemically altered, ultra fine grained ultracataclasite or ultramylonite is noted. We present an extension to the classical solid mechanical theory where both length scales emerge as part of the same evolutionary process of shearing the host rock. We highlight the important role of any type of solid-fluid phase transitions that govern the second degree localisation process in the core of the shear zone. In both brittle and ductile shear zones, chemistry stops the localisation process caused by a multiphysics feedback loop leading to an unstable slip. The microstructural evolutionary processes govern the time-scale of the transition between slow background shear and fast, intermittent instabilities in the fault zone core. The fast cataclastic fragmentation processes are limiting the rates of forming the ultracataclasites in the brittle domain, while the slow dynamic recrystallisation prolongs the transition to ultramylonites into a slow slip instability in the ductile realm.

Appendix: Poro-Chemical Model

At high temperatures the solid $AB$ breaks down, producing excess $B$ fluid, and increasing the fluid pore pressure through a general fluid-release reaction of the form:

$$\begin{aligned} \nu _1 AB_{\rm s} \rightleftharpoons \nu _2 A_{\rm s} + \nu _3 B_{\rm f} \end{aligned}.$$

(20)

We assume the following relations for the partial molar reaction rates for the species,

$$\begin{aligned} r_{AB}&= - \left[ \frac{\rho _{AB}}{M_{AB}} (1 - \phi )(1 - s) \right] ^{\nu _1} k_{\rm F} \hbox { exp}(-Q_{\rm F}/RT) \nonumber \\ r_A&= \left[ \frac{\rho _{A}}{M_A} (1 - \phi ) s \right] ^{\nu _2} k_{\rm R} \hbox { exp}(-Q_{\rm R}/RT) \nonumber \\ r_B&= \left[ \Delta \phi _{\rm chem} \frac{\rho _{B}}{M_B} \right] ^{\nu _3} k_{\rm R} \hbox { exp}(-Q_{\rm R}/RT). \end{aligned}$$

(21)

From the stoichiometry of the considered reaction, Eq. (20), it should hold that:

$$\begin{aligned} -\frac{r_{AB}}{\nu _1} = \frac{r_A}{\nu _2} = \frac{r_B}{\nu _3}. \end{aligned}$$

(22)

From Eqs. (21–22), and for $\nu _1=\nu _2=\nu _3=1$ we derive the poro-chemical model

$$\begin{aligned} \Delta \phi _{\rm chem}&= A_{\phi } \frac{1-\phi _0}{1 + \frac{\rho _{B}}{\rho _{A}} \frac{M_{A}}{M_{B}} \frac{1}{s}}, \nonumber \\ s&= \frac{\omega _{\rm rel}}{1+\omega _{rel}}, \quad \hbox{and}\nonumber \\ r_{\rm rel}&= \frac{\rho _{AB}}{\rho _{A}} \frac{M_{A}}{M_{AB}} K_{\rm c} \hbox { exp} \left( \frac{\Delta h}{RT} \right). \end{aligned}$$

(23)

In Eq. (23), $K_{\rm c} = k_{\rm F} / k_{\rm R}$ is the ratio of the pre-exponential factors of the Arrhenius reaction rates and $\Delta h= Q_{\rm R} - Q_{\rm F}$ the difference of the forward and reverse activation energies. The parameter $A_{\phi }$ is a coefficient that determines the amount of the interconnected pore-volume (porosity) created due to the reaction. We assume that all the fluid generated contributes to the interconnected pore volume, and, thus, set $A_{\phi } = 1$.

Following these considerations, the rates of the forward ($\omega _{\rm F}$) and reverse ($\omega _{\rm R}$) first order reactions can be calculated to be

$$\begin{aligned} r_{\rm F} = r_{AB}&= \frac{\rho _{AB}}{M_{AB}} (1 - \phi )(1 - s) k_{\rm F} \hbox {e}^{-Q_{\rm F}/RT}\end{aligned}.$$

(24)

$$\begin{aligned} r_{\rm R}&= r_{A}r_{B} = \frac{\rho _{A} \rho _{B}}{M_A M_B} (1 - \phi ) s \Delta \phi _{\rm chem} k_{\rm R} \hbox {e}^{-Q_{\rm R}/RT}. \end{aligned}$$

(25)

Note that, for simplicity, we have assumed in Eq. (21) that the two products are produced with the same pre-exponential factor and activation energies. If this is not the case, the above model should be modified accordingly. The net reaction rate would then be $r = r_{\rm F} - r_{\rm R} \frac{M_{AB}}{\rho _{AB}}$ (the reverse reaction rate was normalized with the reference concentration $\frac{\rho _{AB}}{M_{AB}}$ for dimensional purposes), which however would be essentially irreversible ($r_{\rm F} \gg r_{\rm R}$) in the case $K_{\rm c} = k_{\rm F} / k_{\rm R} \gg 1$.