Scaling of Viscous Shear Zones with Depth Dependent Viscosity and Power Law Stress-strain Rate Dependence

One of the unresolved questions concerning fault deformation is the degree and cause of localisation of shear at depth beneath a fault. Geologic observations of exhumed shear zones indicate that whilst the motion is no longer planar, it can still be localised near the down-dip extension of the fault; however, the degree of localisation is uncertain. We employ simple analytic and numerical models to investigate the structural form of distributed shear beneath a strike-slip fault, and the relative importance of the physical mechanisms that have the potential to localise a shear zone. As we are concerned with long-term structure the model is time-averaged across the earthquake cycle, consisting of an idealised strike-slip fault within a rigid lid over a viscous layer.

For a depth dependent viscosity, η = η₀ exp (−z/z₀), we find a shear zone develops with a half-width δ_w ~ √z₀ for small z₀, where lengths are non-dimensionalised by the layer thickness (d km). Including a non-linear stress-strain rate relation (ε ̇ ∝ σⁿ) scales δ_w by 1/√n, comparable to deformation length scales in thin viscous sheet calculations. We find that the primary control on δ_w is the depth dependence of viscosity arising from the increase in temperature with depth. As this relationship is exponential, scaling relations give a half-width that scales approximately as

$$\delta_w\approx T(z=1/2){\sqrt\frac{Rd}{nQ\beta}} km,$$

with T (K), gas constant R (J/mol K), activation energy Q (J/mol), and geotherm β (K/km). Figure illustrates shear zones for a dry olivine composition. For n = 1 the shear zone half-width is δ_w = 4 km, which reduces to δ_w = 2.3 km when n = 3; other parameter choices consistent with laboratory-derived rheological properties give δ_wfrom 2-6 km.

The inclusion of shear-stress heating only reduces δ_w by an additional 5-25%, depending on the initial width of the shear zone; in the case of dry olivine with n = 3 we get δ_w = 1.8 km. This reduction in width occurs over a thermal diffusion timescale of ~5Ma; hence the full effect will only occur if the fault location is stationary relative to the viscous layer. Whilst the width of the shear zone may not decrease significantly, local temperature increases range from 50-300◦C with a viscosity reduction of up to 5 orders of magnitude and a concomitant reduction in driving stresses.