Workshop on Cracks, Fractures and Faults in the

Earth: Bridging the Gaps, Bristol, UK, 18-20 June 2008

Thermo-Hydro-Mechanics

of Earthquake Rupture

James R. Rice (Harvard)

Collaborators:

Eric M. Dunham (Harvard), Nadia Lapusta (Caltech),

Hiroyuki Noda (Kyoto), Alan W. Rempel (Oregon),

John W. Rudnicki (Northwestern), Elizabeth L. Templeton (Harvard),

Victor C. Tsai (Harvard), Robert C. Viesca (Harvard)

Related theme:

How the presence of fluids, and damage/granulation

within fault-bordering materials, affect earthquake

rupture dynamics (and other large-scale fracturing)

A quick primer on the structure of mature, highly slipped faults:

Extreme localization of seismic slip but much wider damage zones

J.Geophys. Res. (1993)

10s-100s mm(But principalfailure surfacecan be much thinner,typically < 1-5 mm!)

1-10 m(Granulated,fault gouge)

~10-100 m

(Damaged highly cracked rock.)

F. Chester, J. Evans and R. Biegel, J. Geoph. Res., 98 (B1), 771-786 (1993)

Generic structure, mature fault zones:

Chester, F. M., and J. S. Chester, Ultracataclasite structure and friction processes of the

Punchbowl fault, San Andreas system, California, Tectonophysics, 295 (1-2): 199-221,1998

Prominent slip surface (pss) is located in the center of the layer and identified by the black arrows.

(Exhumed from 2-4 km depth. Total slip 44 km. "Several km" of slip on the pss.)

From J. S. Chester and D. L. Goldsby,

SCEC Ann. Rpt., 2003(also, Chester et al.,

EOS, Trans AGU, 2003)

Punchbowl Faultprominent slip

surface

~1 mm-thick layer;preferred orientation(evidence: uniformbirefringence,bright layer);contains distinct microscopicslip surfaces, 100-300 μm thick

5 mm

Particle size distribution for ultracataclasite hosting the Punchbowl pss

[J. Chester et al., Nature, 2005]

• N(D) / A = number of particles per unit samplearea with 2D / 3 < diameter < 4D / 3.

N(D) / A c / D2 for ~10-30 nm < D < ~70 μm.

• D50(= size such that 50% by wt. is larger) ~ 1 μm.

What's the right picture :

A miniature version of beach sand particles?

Clumping of small cohesive - sized particles, maybe like

a fine - grained polycrystal with defect - ridden g.b.'s?

San Andreas Fault at Mecca Hills, near Palm Springs. Easily shoveled gouge removed to reveal a more coherent slip surface.

(Caltech geology group from ~ 1960s or 1970s. Picture provided by J. Brune to Y. Ben-Zion, 2003)

Damage patterns in gouge within 1 m of, possibly, theprincipal slip zone the Sierra-Madre thrust fault; it liftsthe San Gabriel Mountains north of LA; exposed instream cut near NASA/JPL (photo from C. Sammis):

viewSan Gabr. Mtn.

Sier.-Madr.

Fault

North

h 3 mmWibberley (2003)

Median TectonicLine Fault, Japan

k,

q =k

f( p f g)

= fluid flux relative to solid host

[Faulkner et al., Nature '06]

Damage zone surrounding

the Caleta Coloso strike-slip

fault, Atacama fault system

of northern Chile.

• Cuts through granodioritic

crystalline rocks

• ~500 m total slip

• Fault zone structures

formed at 4–10 km depth

• Agrees with typical

estimates of widths of

seismic “low velocity”

zones about active faults

If rupture on major faults (usually) takes place in a narrow fault core, why are there wide granulated and damaged zones?

Perspective 1, Large-scale geological view:

• Major faults start their lives as independent, disconnected and poorly aligned fault segments (e.g., Pollard & Segall). The active locus smooths with slip (e.g., fewer step-overs per unit length, Wesnousky et al.).

Broad zones affected by prior faulting give way to narrower channels of continuing activity.

• Major fault traces, while smooth in the direction of slip, do have fractal-distributed Irregularities (Power and Tullis).

The more they slip the broader a zone which must be activated to accommodate the misfits.

Perspective 2, Processes in individual events:

• The rupture front concentrates stress over a wide zone, the more so the more rapid the propagation speed vr.

Fault map and slip,Sowers et al. (1994).

Pre-stress direction,Hardebeck and

Hauksson (2001).

Portion of 1992 Landers(Mohave Desert) earthquake:

• Mature faults can includecomplex transitions.

• Damaged border zone can beactivated during earthquakes.

slip direction

Quandary in seismology:

• Lab estimates of rock friction coefficient f usually high, f ~ 0.6-0.8.

Shear strength = f x ( n – p), where:

n = normal stress clamping the fault shut p = pore pressure in infiltrating fluid phase (groundwater)

• Fault slip zones are thin.

==> If those f prevail during seismic slip, we should find

• measurable heat outflow near major faults, and

• extensive melting along exhumed faults.

Neither effect is generally found.

Actually, major faults don’t support much shear stress!

[Hickman& Zoback,GRL 2004]

lower

ratio of

SAF––––––

( n )SAF

than forother

orientations

Regional max. hor. stress Hmax at steep angle with SAF

[Hickman & Zoback, GRL, 2004]

Stress state measured near the SAF resolves very little shear stress on it

[Noda et al., in prep.]

One line of explanation: Weak faults: = f x ( n – p)

• Fault core materials are different, have very low f.

• f isn’t low, but pore pressure p is high over much of the fault.

Another line: Statically strong but dynamically weak faults, e.g.,due to thermal weakening in rapid, large slip:

• Processes expected to be important from start of seismic slip:

- Flash heating of asperity contacts, reduces f in rapid slip.

- Thermal pressurization of pore fluid, reduces effective stress.

• Other processes that may set in at large enough slip or rise in T:

- Thermal decomposition, fluid product phase at high pressure.

- Gel(?) formation at large slip in wet silica-rich faults.

- Melting at large slip, if above set has not limited increase of T.

Dynamic rupture simulations, using lab-constrained poroelastic, thermal, and friction parameters

[combining flash heating and thermal pressurization of pore fluid]

[Noda, Dunham & Rice, work in progress 2006-2008]

First examples to follow: = (x,t), = (x,t);

slip and shear stress in zdirection (mode III; somesimilar mode II done too)

n

b

Elastodynamics

Relates history of slip (x,t) to history of stress (x,t) [recall, = f ( n p)] :

(x,t) = o(x,t)μ

2csV (x,t) + (x,t) where V (x,t) =

(x,t)

t ,

equals background stress b outside nucleation zone

(x,t) = linear functional of slip history (x ,t ) within wave cone of (x,t) :

Could use BIE, FD, FE, or Spectral formulation; Spectral (for slip between uniform half spaces) is :

(x,t)

(x,t)=

n

Dn (t)

n (t)exp(inkx)

n (t) =0

tCn (t t )Dn (t )dt

Need a constitutive relation between V (t) and (t) at each x to close the system.

We'll do that based on flash heating and thermal pressurization of pore fluid.

(x,t)

x

t

V = slip rate

asperity diameter

T = asperity temperature

Tf = average temperature of fault surfaces

Flash heating of microscopic frictional asperity contacts

contact shear stress

T, asperity temperature

Tf Tw , weakening temperature

c

"Weakening" slip rate:

Vw = th

D

Tw Tf

c c

2

f =fo , V Vw ,

fw + fo fw( )VwV

, V Vw ,

[Beeler and Tullis, 2003,

2006; Rice, 1999, 2006]

0.36 m/s

Recent lab set-up to study weakening at high slip rates (Tullis & Goldsby, 2003)

Instron testing frame, rotary shear of annular

ring up to 45 mm slip

[Tullis & Goldsby, SCECAnn. Mtg., 2003; EOS, 2003]

Results for Arkansasnovaculite (quartzite),granite, Tanco albite

(feldspar), gabbro, andcalcite.

For quartzite, granite, feldspar, gabbro, calcite :Vw (0.14, 0.14, 0.28, 0.11, 0.27) m/s

For quartzite, granite, gabbro : fo (0.64, 0.82, 0.88) , fw (0.12, 0.13, 0.15)

Flash weakening in rapid slip:

Bare rock on rock vs. fault gouge

Is the fault core a granular material flowing like

beach sand, or is it a fine grain polycrystal with

weak but cohesive grain boundaries?

And if the latter, does it respond like bare rock

on rock?

How about gouge? For the same gabbro: Tullis & Goldsby [2003], freshly machined, no initial gouge. Tsutsumi & Shimamoto [1997], studied after ~20 m slow slip at n = 1.5 MPa, so much gouge was likely to be present.

(Tullis & Goldsby, 2003) (Beeler, Tullis & Goldsby, 2007)

No initial gouge: Vw ~ 1.1 m/s Presumed abundant gouge: Vw ~ 2.3 m/s

Flash heating (in simulations, Noda, Dunham & Rice, 2007-08)

Rate- and state-dependent friction law (in form of

“slip law” [Dietrich, 1979; Ruina, 1983])

fss =fo , V Vw ,

fw + fo fw( )VwV

, V Vw , with Vw = th

D

Tw T

c c

2

Flash heating at microscopic contacts, model for fss (steady state friction)

[Beeler and Tullis, 2003, 2006; Rice, 1999, 2006]

df

dt=a

V

dV

dt

V

Lf fss( ) (with L = 5-10 μm, a = 0.015, fo = 0.82, fw = 0.13)

Effective stress law:

= e f = ( n p z=0 ) f

pore pressure on slip surface

given, fixed compressive normal stress

friction coefficient

Governing equations, 1-space-dimension shearing field, constant normal stress n:

m =mass of pore fluid

unit (reference) vol. of porous medium= f n

Thermalpressurizationof pore fluid

(Habib 67, 75, Sibson 73, Anderson 80,Lachenbruch 80, Voight & Faust 82,

Mase & Smith 85, 87, Lee & Delaney 87, Vardoulakis 02, Andrews 04,

many more in recent yrs.)

• Fluid mass conservation :

m

t+

q fy

=0 , q f =f k

f

p

y

p

t=

T

t

1 npl

t+

1

f y f hyp

y ,

hy = k / f ;

0.3-1.0 (MPa/ºC), n( f + n ) 5-30 10 11/Pa;

f , n = fluid compressibility, pore space expansivity.

[Rice, JGR, 2006]

• Energy equation :

= cT

t+

qh

y , qh = K

T

y ,

= u / y 0 , = f ( n p) when > 0

(effective stress law, grains in frictional contact):

f ( n p) = cT

t yc th

T

y ;

c 2.7 MPa/ºC ; th =K

c0.7 mm2 /s.

temperature

porepressure

velocity

heatflux

fluidmass flux

Thermal pressurization

Effective stress law:

= e f = ( n p z=0 ) f

2

2

z

T

t

Tth

=

t

T

z

p

t

phy +=

2

2

Vfz

Tc e

z

th2

1

0

=

=

0

0

=

=zz

p

Thermal pressurization (T.P.), infinitesimally thin slipping zone

(later, we will see finite thickness cases)

B.C.

pore pressure on slip surface

Conservation of energy

Conservation of fluid mass

given, fixed compressive normal stress

friction coefficient

c 2

hy + th( )2 60 mm2 /s (low end) L* 4 mm, if V = 1 m/s and f = 0.25.

450 mm2 /s (high end) L* 30 mm, if V = 1 m/s and f = 0.25.

Evaluations for 7 km depth, a typical centroidal depth of crustal rupture zone;

n overburden = 196 MPa, po = pamb = hydrostatic = 70 MPa, Tamb 210 ºC:

Assuming relatively intact gouge, MTL lab-measuredproperties [Wibberley & Shimamoto, JSG, 2003].

Accounting arbitrarily for damage at the rupture front andduring subsequent shear, permeability kdamaged = 5-10 kintact,

drained skeletal compressibility ddamaged =1.5-2 d

intact.

[Rice, JGR, 2006;Rempel & Rice, JGR, 2006]

per = 30MPa

Dper = 5cm

pulse

[Noda, Dunham

& Rice, in prep.]

per = 30MPa

Dper = 5cm

pulse

[Noda, Dunham

& Rice, in prep.]

per = 30MPa

Dper = 5cm

pulse

[Noda, Dunham

& Rice, in prep.]

per = 30MPa

Dper = 5cm

pulse

[Noda, Dunham

& Rice, in prep.]

per = 30MPa

Dper = 5cm

pulse

~ 100 μs

p

[Noda, Dunham & Rice, in prep.] Longest simulation to date: ~30 m of rupture length

A growing pulse-like solution which propagates for ~ 30 m.

(a) Distribution of slip showing a linear increase with x by 0.14 mm/m.

(b) Distribution of V and – b at t = 5250 μs. The peak slip rate V is extremely

high (~ 300 m/s). Stress change – b is positive near the center, transiently

large and negative near the rupture front, but much smaller on “healed” parts

of the fault.

low init

nhigh init

n

[Lykotrafitis, Rosakis & Ravichandran, Sci. 2006; Lu, Lapusta & Rosakis, PNAS 2007]

Summary, possible thermo-hydro-mechanical weakening

processes in fault zones during earthquake rupture:

• Flash heating of microscopic asperity contacts: -Reduces f in rapid slip; [ = f x ( n – p) ] -Promotes self-healing rupture mode.

• Thermal pressurization of pore fluid by shear heating: -Reduces n – p. [ = f x ( n – p) ]

Consequences:

• Major faults can operate at low overall stress levels; fits wellbore and regional stress constraints.

• There is very little heat generation; no heat-flow paradox.

• Rupture will typically be in a self-healing (vs. crack-like) mode, fitting seismic constraints.

Material Contrast Across Fault

Far-field elastic dissimilarity(highly studied)

Near-fault permeability& poro-elastic dissimilarity(ignored until recently)

Across-fault dissimilarity of poroelastic response

[Rudnicki & Rice, JGR 2006; Dunham & Rice, JGR in press 2008]

Pore pressure change under undrained conditions :

pu = B kk

3= BKu kk (B ~ 0.3-0.8 for gouge)

(mm to cm scale)

Simplified rupture studies, using slip-weakening failure model

n p

[Dunham & Rice, JGR, in press 2008]

Case with opposing preferred propagation directions basedon far-field elastic versus near-fault poroelastic dissimilarity

Questions on concepts for

thermo-hydro-mechanical analyses:

Is the fluid phase interconnected and appropriate

for application of porous media concepts

(permeability, k), and of poroelasticity (Biot

theory, , B, etc.) and poroplasticity (effective

stress, – p) concepts?

Does it become interconnected by material yielding

in passage of concentrated strain field at the rupture

tip?

Rupture studies, using Mohr-Coulomb type elastic-plasticmodel off the fault, and slip-weakening failure model on fault

General Stress States :

1

3 kk

=1

2sijsij , sij = ij

1

3 ij kk

d ij = deij + d

pij

de ij =1

2Gdsij +

1

9K ijd kk

d p ij =sij2

d p +1

3 ijdp

=sij2

+ij

3

skl2

+μ kl

3

d kl

h

Hardening hd p = d + μd

Dilatancy d p = d p

1 1

Special case: Drucker - Prager Model :

μ independent of

yield: = b μ

h =db

d pdμ

d p

b

Off-fault elastic-plastic constitutive model

=1

2sijsij ,

sij = ij1

3 kk ijDP and MC yield criteria equivalent when

int =max + min

2= kk

3 i.e., when sint = 0( )

(we assume such condition for ij0 )

Closeness to failure (CF ) of initial stress state

m =max + min

2

(CF, S and are interdependent)

n and in plane of max and min

Effect of pre-stress

angle on

distribution of

off-fault yielding

(for fixed value,

S = 1, of seismic

stress ratio)

(no cohesion, c = 0,

no hardening, h = 0)

[Templeton & Rice, JGR in press, 2008]

Views ofpart of theright halfof modeleddomain;field issymmetricrelative tocoordinateorigin,located atcenter ofleft side ofeach panel

We’ve taken cohesion c = 0; how much

cohesion is needed to eliminate plastic flow?

=

Case for which plasticity suppresses (or long delays) transition to supershear speed

Case for which plasticity delays, but still allows, transition to supershear speed

Material Contrast Across Fault

1 / 2 = 1.2

CS1 / CS2 = 1.6

CP1 / CP2 = 1.6

Fast

Slow

Material Contrast:(well-posed regime)

• Normal stress along fault coupled to slip

• For elastic case, preferential rupture direction due to normal stress variation

along fault (in dir. of slip on slow side -- i.e., to the left here)

• Problem studied by Weertman, Adams, Ranjith, Rice, Cochard, Andrews

and Ben-Zion, Harris and Day, and many others

[Templeton & Rice, EOS, F 2006]

fast

slowLocation of Plastic Strain

no materialcontrast

[Templeton

& Rice,

EOS, F06]

identical

identical

elastically preferred direction

Rupture Propagation , Low fast

slow

max

max

=14°S = 1.0

Elastic Elastic-Plastic

Reversal in preferredpropagation direction

PreferredDirection

[Templeton

& Rice,

EOS, F06]

Initial stress state

Effect of Fluid Saturation on Yielding during Undrained Response

[Viesca, Templeton& Rice, to JGR 2008]

Yield locus for (total) stress changes under undrained conditions,

slope = (1 – B) tan

[B = Skempton coef.]

Short time scale On fault, p = 0 here. Yield phrased in termsundrained off-fault, (A special case; more of effective stress,p = B kk / 3 general: Rudnicki & ij = ij + p ij .

for elastic response. Rice, JGR, 2006.)

Poro - elastic - plastic constitutive response

Elastic strain increment :

de ij =dsij2G

+ij

3K

d kk

3+ dp ;

= Biot pore pressure factor [ = 1K

Ks in simple cases] .

Here sij = ij1

3 ij kk .

Plastic strain increment :

d p ij =sij2

+ij

3

skl2

+μ kl

3

d kl + kldp

h ;

Terzaghi effective stress often appropriate

[specific cases identified (Rice, ASCE - EMD, 1977)] .

Here =1

2sijsij .

Increment of fluid mass content :

dm

f=K

d kk

3+dp

B+ d p kk ;

B = Skempton factor [ =1 K / Ks

1 (1 + n)K / Ks + nK / K f

, in smp. cs.]

d pn = d p kk (specific cases: [Rice, ASCE - EMD, 1977] ).

Form invariance, drained and undrained

[Rudnicki, 1984]

Drained, dp = 0 :

d ij =dsij2G

+ijd kk

9K

+sij2

+ij

3

skl2

+μ kl

3

d kl

h .

Undrained, dm = 0 :

d ij =dsij2G

+ijd kk

9Ku

+sij2

+

ˆij

3

skl2

+μ kl

3

d kl

h ,

where:

Ku =K

1 B= undrained bulk modulus ,

h = h + μ KB / ,

μ = (1 B)μ , ˆ = (1 B) .

[Viesca et al., to JGR, 2008]

Effect of

Skempton

coefficient B;

shallow pre-

stress angle

= 14º.

( = 0,

h = 0,

c = 0. )

Effect of

Skempton

coefficient B;

steep pre-

stress angle

= 56º.

( = 0,

h = 0,

c = 0. )

[Viesca et al., to JGR, 2008]

Effect of

dilatancy

(induces

suction in

pore fluid);

= 14º

and 56º.

( h = 0,

c = 0. )

[Viesca et al., to JGR, 2008]

d p kk (= dpn) = d p

• Location of plastic straining depends on angle, , of mostcompressive principal pre-stress:

– For < 10°, plastic strain occurs only on the compressionalside of the fault

– For > 45°, plastic strain occurs only on the extensionalside of the fault

• Extent of plastic straining depends on measures of proximinty ofthe pre-stress to yield levels on the fault (low S) and off the fault(high high CF). S and CF are interdependent for fixed .

• Plasticity can prevent/delay the transition to supershearpropagation speed in some cases.

• Off-fault plastic straining can reverse the elastically predictedpreferred direction of rupture propagation for bimaterialruptures

– For < 25°, preferred propagation direction may be reversed

– For > 45°, preferred propagation direction is unchanged

• Pore fluids, with undrained p = – B kk/3 on seismic timescale, increase extent of plasticity on compressional side,decrease on extensional side.

• Dilatancy, with pore fluids, induces suction and decreasesextent of plasticity relative to non-dilatant case.

Summary of some effects of off-fault elastic-plastic response

Why should shear be so extremely localized?

Governing equations, 1-space-dimension shearing field, constant normal stress n:

m =mass of pore fluid

unit (reference) vol. of porous medium= f n

Thermalpressurizationof pore fluid

(Habib 67, 75, Sibson 73, Anderson 80,Lachenbruch 80, Voight & Faust 82,

Mase & Smith 85, 87, Lee & Delaney 87, Vardoulakis 02, Andrews 04,

many more in recent yrs.)

• Fluid mass conservation :

m

t+

q fy

=0 , q f =f k

f

p

y

p

t=

T

t

1 npl

t+

1

f y f hyp

y ,

hy = k / f ;

0.3-1.0 (MPa/ºC), n( f + n ) 5-30 10 11/Pa;

f , n = fluid compressibility, pore space expansivity.

[Rice, JGR, 2006]

• Energy equation :

= cT

t+

qh

y , qh = K

T

y ,

= u / y 0 , = f ( n p) when > 0

(effective stress law, grains in frictional contact):

f ( n p) = cT

t yc th

T

y ;

c 2.7 MPa/ºC ; th =K

c0.7 mm2 /s.

temperature

porepressure

velocity

heatflux

fluidmass flux

m = f n , n =volume of pore-fluid

unit (reference) volume of porous medium

m = f n + f n, n = nel + npl

f / f = f p f T , nel / n = n p + nT

m / f = (p T ) + n pl ,

where storage factor = = n( f + n ) , and dp

dT elastic,undrained

= =f n

f + n .

Thus m

t+

q f

y= 0 , q f =

f k

f

p

y

p

t=

T

t

1 n pl

t+

1

f y f hyp

y , diffusivity hy =

k

f .

Some calculations, giving form of fluid mass conservation equation:

inelastic dilatancy rate

The spatially uniform solution: Adiabatic, undrained, uniform shear, at constant shear rate

u(y,t) = uo(y) = oy ( o = uniform shearing rate),

p(y,t) = po(t) , T (y,t) = To(t).

• (y,t) = o(t) = f n po(t)( ) (we assume f = constant in large shear with = constant)

• o(t) o = cdTo(t)

dt

• dpo(t)

dt=

dTo(t)

dt

1 dnpl

dt (we assume

dnpl

dt= 0 in large shear with = constant)

Solution (Lachenbruch [1980]):

n po(t) = n po(0)( )expf

c ot [call this o(t)]

o(t) = f o(t) (weakening strain c

f10-20 , f 0.4)

u(y,t) = oy

Is the spatially homogeneous solution stable?

Answer is no. Shear should localize to a mathematical plane if

f = constant and npl / t = 0 .

Aim in our work [Rice, Rudnicki and Tsai]: Add (separately at present) two stabilizing features. Does shear then localize? If so, what thickness?

Features are:

(1) Rate-strengthening friction coefficient, .Approximately valid only in stable regions in which rupture cannotnucleate, but may propagate through, or in unstable regions which haveshear-heated to a frictionally stable T range.

(2) Dilatancy, increasing with rate of sustained shear [NOT DISCUSSED HERE],

For simplicity, treatments ignore rate and state transients inresponse to changes in shearing rate, and use "steady state" responsein defining

npl = npl ( ) with dnpl ( ) / d > 0.

f ( ) and npl ( ).

f = f ( ) with df ( ) / d > 0

Study of rate-strengthening of friction coefficient f (now neglecting dilatancy)

Linearized perturbation about time-dependent spatially uniform solution:

u(y,t) = oy + u1(y,t) ,

(y,t)= u(y,t) / y = o + 1(y,t) ,

p(y,t) = po(t) + p1(y,t) ,

T (y,t) = To(t) + T1(y,t) ,

f = f ( o ) + f ( o ) 1(y,t)

rate strengthening, f ( o ) > 0

[Rice & Rudnicki,in preparation]

• y

f ( o )p1 + f ( o ) 1 o(t)( ) = 0

• f ( o ) o(t) 1 f ( o )p1 o + f ( o ) 1 o(t) o = cT1

t th

2T1

y2

• p1

t

T1

t= hy

2p1

y2

• y= 0 , =f ( n p) , n = const.

• f ( n p)u

y= c

T

t yc th

T

y

• p

t

T

t=

1

f y f hyp

y

Study of rate-strengthening of friction coefficient f (now neglecting dilatancy)

Linearized perturbation about time-dependent spatially uniform solution:

u(y,t) = oy + u1(y,t) ,

(y,t)= u(y,t) / y = o + 1(y,t) ,

p(y,t) = po(t) + p1(y,t) ,

T (y,t) = To(t) + T1(y,t) ,

f = f ( o ) + f ( o ) 1(y,t)

rate strengthening, f ( o ) > 0

Nature of solution with spatial dependence exp(2 iy/ ) (wavelength ):

p1(y,t) , T1(y,t) exp(st)exp(2 iy / ) ;

1(y,t) exp s +f

c o t exp(2 iy / )

s = s( ) satisfies:

Typically z of order 20-40,in results oflow shear-rateexperiments at high T.

Dynamic disk simulations[Chevoir et al., 05]:z > 1000 foreffective stress> 10 MPa

zf

c os = s +4 2

th2

s +4 2

hy2

where z =f ( o )

o f ( o )=

f

a b

0.5

0.02= 25

[Rice & Rudnicki,in preparation]

• y

f ( o )p1 + f ( o ) 1 o(t)( ) = 0

• f ( o ) o(t) 1 f ( o )p1 o + f ( o ) 1 o(t) o = cT1

t th

2T1

y2

• p1

t

T1

t= hy

2p1

y2

Condition for linear instability of flow profile :

Re s( )( ) +f

c o > 0 > cr 2c

f

( th + hy )

(z + 2) o z =

f ( o )

o f ( o )>> 1

Comment : Near = cr , Im s( )( ) zth hy

th + hy

f

c o (oscillatory; unloading?)

fast; compare to o(t) expf

c ot

Shear of a rate - strengthening layer of thickness h at net "slip" rate V ( o = V / h) :

> cr 2c

f

( th + hy )h

(z + 2)V

drawn for:

z = 40

V = 1 m/s

th = 1 mm2/s

High end

parameters

Low end

parameters

Homogeneous

shear not possible

Homogeneous

shear is possible

Linear Stability Analysis [Rice & Rudnicki, in preparation]

Shear of a rate-strengthening layer of thickness h at net "slip" rate V ( o = V / h).

Perturbation solution with spatial dependence exp(2 iy / ) (wavelength ):

Stable if < cr , Unstable if > cr .

cr = h

h = 4 2 c

f

( th + hy )

(z + 2)V

z =f ( )

df ( ) / d

Implications:• Even with velocity strengthening [with large z (= ), e.g., of order z > 10],we must expect large shear strain to be confined to a thin zone.

• Suggests validity of use of dynamic fault model based on slip on a plane.

• Observed 1-5 mm deformed zone thickness in gouge may be a precursor thickness(i.e., cr based on an initial, broad h) not the thickness of the large shear zone.

Possible self - consistent estimate of shear layer thickness h at large shear :

Set o =V

h, cr h h

4 2( th + hy ) c

(z + 2) f V

Results (using z = 20, V = 1 m/s, th = 0.7 mm2 /s, c = 2.7 MPa/ºC):

Low end ( = 0.70 MPa/ºC, hy = 1.5 mm2 /s): h = 8.4 μm / f = 21 μm if f = 0.4.

Based on lab properties of intact Median Tectonic Line gouge (Wibberley and Shimamoto,

2003) at effective confining stress = 125 MPa and T = 200ºC -- corresponding to ~7 km depth.

High end ( = 0.34 MPa/ºC, hy = 3.5 mm2 /s): h = 59 μm / f = 148 μm if f = 0.4.

Accounts for increased permeability k and drained compressibility d due to fresh damage

of fault gouge introduced at the rupture front just before shear (kdmg= 5-10k, d

dmg= 1.5-2 d )

f / [ df / d ]

Nonlinear numerical analysis: Typical dynamics[V. Tsai, studies in progress, 2005-] T , p , and ( n – p) parameter dependence:

Blanpied et al. [1998], Keenan et al. [1978]; Wibberley & Shimamoto [2003]

50=

( )log

Some basic questions about fault core and damage zone:

1. Is the fault core a granular material flowing like beach

sand? Or is it a fine-grain polycrystal with weak but

cohesive grain boundaries?

2. Is the fluid phase interconnected and appropriate for

application of concepts of porous media (Darcy flow,

permeability, k)? And of poroelastic (Biot theory, , B,

etc.) and poroplastic (effective stress, – p) concepts?

3. Is Mohr-Coulomb plasticity adequate? What is cohesion

c? How does it degrade with shear? What experiments

would help? Should granular deformation models with

dilatancy, like for critical state soil mechanics, be used?