Fall AGU 1998 - Dilatant Stabilization of Earthquake Rupture

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Dilatant Stabilization of Subduction Earthquake Rupture into the Shallow Thrust Interface

¹ Department of Earth and Planetary Sciences and Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

² Now at Bullard Laboratories, University of Cambridge, Cambridge, CB3 0EZ, UK

Abstract

Dilatancy in a fluid-infiltrated fault zone may fully or partially stabilize frictional failure. In the rate/state friction context, when a steadily sliding fault is fully undrained, the effective stress has to exceed

in order for unstable slip to be able to nucleate under sufficiently reduced stiffness [Segall and Rice, JGR, 1995]. Here f is the friction coefficient, assumed to decrease in steady state at rate b - a with logarithm of slip-velocity,

is the rate of increase of inelastic porosity with log slip-velocity, and b is a compressibility parameter. Such stabilization can occur when the characteristic time for equilibration of fault pore pressure with that of the surroundings, T_p, is long compared to the rupture time scale; T_pcontrols the amount that dilatancy reduces the pore pressure, and thus increases the effective clamping stress, to mitigate against frictional weakening.

We address dilatancy here as a factor controlling rupture in the shallow, certainly fluid-infiltrated, portion of a subduction fault zone. This is done using a simple 2D plane-strain model in which slip varies with down-dip distance and time. The governing equations, solved quasi-dynamically, incorporate the temperature (and hence depth) dependence of b - a, represent inelastic porosity changes as above, and treat equilibration of pore pressure between the fault and its surroundings by a lumped reservoir model with characteristic diffusion time T_p. We present results for T_p= 10^-8 yr and 10^-1 yr, in which cases the fault responds as if were, respectively, fully drained and undrained on the dynamic rupture propagation time-scale. There are corresponding nucleation sizes and , the latter existing only at sufficiently great depths that the effective stress exceeds , 30 MPa in our simulations. Both cases exhibit periodic large events with characteristics that are representative for subduction zones, and ruptures nucleate at similar depths in the two cases. However, slip propagating up-dip extends all the way to the trench for the drained fault, but the rupture front slows and comes to a halt at shallow depths in the undrained case.

Dilatant effects like those modeled may explain the typically aseismic response of the shallow thrust zone, and could be a primary factor controlling the magnitude of tsunami generation, since coupling of slip to wave generation is strongest for slip extending to near the trench.

Background / Motivation

Tsunami waves generated by "great" subduction earthquakes.

Newman and Okal [1997]: ratio of energy released to total seismic moment of large subduction earthquakes was order of magnitude lower for those producing tsunamis.

Geist and Dmowska [1996, 1998]: tsunami waveform sensitive to slip distribution - significantly different results than from simple uniform slip models

Bilek and Lay [1998] in survey of various subduction zones found trend of increasing source duration with shallower nucleation depth.

- Also pointed out that for Japan trench, Tokachi '92 and Sanriku '94 main events showed evidence of slow initiation phases along up-dip region and shallow post-seismic "slow" slip measured days - 1 year after event.

Rudnicki and Chen [1998], Segall and Rice [1995]: Dilatancy interacts with pore fluids and can stabilize rupture. Effect should be greatest in shallow (low ) part of fault zone.

Figure 1: (a) 2-D Subduction Zone model: Dip-slip thrust fault of length W, at an angle

in an elastic half space. Along fault (0 < x < W) slip varies with depth and time, rate- and state-dependent friction laws apply and properties are depth variable. Further down-dip (W < x <

) slip is imposed at a uniform rate, V_plate = 60 mm/yr.

(b) Fault zone showing notation - sense of mass flux and shear and normal stress.

Conservation of Fluid Mass:

q_mass = fluid mass flux per unit area

= porosity

= fluid density

Variations in density as a function of pore pressure:

= isothermal fluid compressibility (e.g. 5x10^-4 MPa^-1)

Following Segall and Rice [1995], distinguish between elastic and plastic pore deformation and write the change in porosity as the sum of an elastic and plastic component [and ignore thermal effects]:

= inelastic dilatancy rate,

= elastic pore compressibility (e.g. 5x10^-3 MPa^-1) at fixed .

Integrating over fault width d, combining constants into term C_m, and defining :

Fluid Transport:

From Darcy's law, and introducing lumped parameter approach:

= effective permeability

= dynamic viscosity

p_amb= ambient fluid pressure

Combine expressions for fluid mass and transport to give

Relation for Pressure:

= drainage time for fluid re-equilibration with p_ambfollowing approach of Rudnicki and Chen [1988] and Segall and Rice [1995].

Representation of Inelastic Dilatancy:

Observations based on Marone et al. [1990] for dilatancy in fault gouge:

f evolves similarly (when detrended) to the contact lifetime state variable in the rate- and state-dependent friction law

Assume then that , and so at steady state slip:

= dilatancy coefficient (e.g. 10^-4)

Therefore the form of is:

and

for the particular (Dieterich-Ruina) form of the friction law above.

Elastic Expression for Shear Stress - constant :

Shear stress due to down-dip slip d on fault:

K = Stiffness matrix - relates slip at one point on fault to the resulting shear stress at another,

= "seismic radiation damping" term [Rice, 1993] .

(physically: instantaneously transmits stress changes along fault which would result after wave propagation from full elastodynamic analysis) Shear stress as a function of friction coefficient f and effective normal stress

This system of equations is solved at each step to give V, q, and p.

Depth Variation of Pressure:

Critical Stiffness and Nucleation Size, k_crit and :

Discretized model - to properly model continuum:

= size above which cells would be capable of failing independently of one another - insufficiently "stiff"

= nucleation size - minimum length of fault to slip simultaneously for instability to develop

Critical Stiffness:

For single degree of freedom system following friction law:

Critical Cell / Nucleation Size:

Critical stiffness related to cell size from dislocation theory:

Under Drained Conditions (T_p << slip time)

Under Undrained Conditions (T_p >> slip time)

For drained case,

> 0 along whole fault - potentially unstable everywhere

However, for undrained case, in order for > 0,

and so fault can be stabilized at low effective normal stress (shallow depths).
Example: for typical parameters,

= 10^-4, f = 0.6,

= 5x10^-4 MPa^-1,

= 0.004

> 30 MPa For most of the fault

= 100 MPa so criteria satisfied, but at shallow depths

< 30 MPa

fault stabilized.

Plots that illustrate this effect for:

Slip

Figure 2: (a) Slip (m) versus distance down-dip (km). Lines plotted every 5 years. Parameters for "reference model" simulation (Appendix), T_p = 10^-8 yr (~ 0.3 s) corresponding to a fully drained fault.

T_p

^-1

Figure 3: Slip versus distance down-dip as in Figure 2. Solid lines plotted every 5 years and dashed lines plotted for constant increments of moment release; (a) for T_p = 10^-8 yr, and (b) for T_p = 10^-1 yr.

Accumulated Slip

Figure 4: 3D plot of cumulative slip (m) versus distance down-dip (km) and time (s) for a single main event; (a) for T_p = 10^-8 yr, and (b) for T_p = 10^-1 yr.

Porosity

Figure 5: Porosity versus distance down-dip (km) at x = 14 km along fault. (a) for T_p = 10^-8 yr, and (b) for T_p = 10^-1 yr.

Excess Pore Pressure

Figure 6: Excess pore pressure

(bars) versus distance down-dip (km) at x = 14 km along fault. (a) for T_p = 10^-8 yr, and (b) for T_p = 10^-1 yr.

Conclusions

Cyclic behavior of large earthquakes with representative values of slip and velocity results from our 2D models of subduction earthquake rupture incorporating rate- and state-dependent friction.

Introducing dilatancy allows fault dilation during shearing to reduce pore pressure p and so increase effective normal stress , potentially stabilizing fault.

Degree of stabilization depends explicitly on:

T_p

drained

undrained

- controls extent of difference in behavior under drained or undrained conditions.

In undrained case, for unstable slip .

Tsunami potential appears to correlate strongly with rupture into shallow portion of thrust interface - such behavior shown here to depend directly on degree of dilatant stabilization.

References:

Bilek, S. L., and T. Lay, Variation of interplate fault zone properties with depth in the Japan subduction zone, Science, 281, 1175-1178, 1998.

Geist, E., and R. Dmowska, Mechanics of dip-slip faulting related to the generation of local tsunamis, EOS Trans. Amer. Geophys. Union, 77, no. 46 (Fall Meeting Supplement), p. F510, 1996.

Newman A. V. and E. A. Okal, Teleseismic estimates of seismic source energy: Towards real-time identification of tsunami earthquakes, EOS Trans. Amer. Geophys. Union, 78, no. 17 (Spring Meeting Supplement), p. S215, 1997

Marone, C., C. B. Raleigh, and C. H. Scholz, Frictional behavior and constitutive modeling of simulated fault gouge, J. Geophys. Res., 95, pp. 7007-7025, 1990.

Rice, J. R., Spatio-temporal complexity of slip on a fault; J. Geophys. Res., 98, 9885-9907, 1993.

Rudnicki, J. W., and C. H. Chen, Stabilization of rapid frictional slip on a weakening fault by dilatant hardening, J. Geophys. Res., 93, 4745-4757, 1988.

Segall, P., and J. R. Rice, Dilatancy, compaction, and slip instability of a fluid infiltrated fault, J. Geophys. Res., 100, 22155-22171, 1995.

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Page established by Mark Taylor, Bullard Laboratories, Dept. of Earth Sciences, University of Cambridge, April 15 1999.

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