Why are earthquake slip zones so narrow?

Transcription

1 Why are earthquake slip zones so narrow? John W. Rudnicki Northwestern University Evanston, IL (with Jim Rice, Harvard University) October 9, 9 University o Minnesota. Geomechanics Seminar

2 F. Chester, J. Evans and R. Biegel, J. Geoph. Res., 98 (B1), (1993) 3-1 m (Damage highly cracked rock. Zone with macro aults or ractures extends ~ 1x urther.) 1-1 m (Sometimes described as oliated gouge, or or some aults, simply as gouge.) 1s-1s mm (But principal ailure surace is much thinner, typically < 1-5 mm!)

3 J.Geophys. Res. (1993)

4 Chester, F. M., and J. S. Chester, Ultracataclasite structure and riction processes o the Punchbowl ault, San Andreas system, Caliornia, Tectonophysics, 95 (1-): 199-1,1998 Prominent slip surace (pss) is located in the center o the layer and identiied by the black arrows. (Exhumed rom -4 km depth. Total slip 44 km. Several km o slip in earthquakes on pss.)

5 From J. S. Chester and D. L. Goldsby, SCEC Ann. Rpt., 3 (also, Chester et al., EOS, Trans AGU, 3) Punchbowl Fault prominent slip surace mm-thick layer; crystal-lattice preerred orientation (evidence: uniorm bireringence, bright layer); contains distinct microscopic slip suraces, 1 3 μm thick.

6 Q: Could coupling o luid and thermal diusion with slip be a actor in setting the width o the slip zone?

7 Stability o spatially uniorm, adiabatic, undrained, shear: y γ h / σ= n τ constant h/ h/ x σ n

8 Governing equations or shearing velocity V(y, t), shear stress τ(y, t), pore pressure p(y, t), and temperature T(y, t): Equation o Motion τ y = (inertia irrelevant), σ = const. Energy Balance V o T q h T τ = ρ c +, qh = K y t y y Fluid Mass Conservation m q ρ k p =, q = t y η y Even i y directions accelerations are large say, 1 1 g, stress change is small over narrow region near ault where luid and diusion processes are important. n Heat generated due to inelastic shear (neglect elastic) Neglects small terms due to work o normal stress and pore pressure and energy transer due to low o luid [Small work terms, σ h pq / ρ, and energy outlow eq.] Heat lux, Fourier s law n Fluid mass change (per unit reerence volume) Darcy s law

9 Dependence o Fluid Mass (per unit reerence volume) m= ρ n ( n = pore volume per unit reerence volume, ρ = luid mass density) el pl m ρ n n = n + ρ + t t t t should be measured or calculated or el one dimensional strain. 1 n p T = β +Λ n n n t t t ( β, Λ are compressb i ility and thermal expansivity o pore volume) 1 ρ n n ( β, Λ are compressibility and thermal expansivity o pore 1 ρ ρ p T = β Λ t t t m p T n = β Λ + t t t t ( n) n ( n) β n β + β pl is an elastic storage coeicient luid) Λ ( λ λ )/ β + β ratio o pore pressure change to temperature change during undrained, elastic deormation

10 Governing Equations τ =, τ = τ( t) y τ ρ α y t y α = K ρ c th V T T = c th / is thermal diusivity 1 Λ + = α hy t t β t y pl p T n p ( ) α = k / η n β β is hydraulic diusivity hy + n

11 Constitutive Relation: Friction Eective normal compressive stress τ = ( σ p) = ( V / y), (...) > In calculations, use ( ) ( ) ( γ) = γ + aln γ / γ where a n.15 is typical o experimental observations About.6 but can be substantially reduced at high slip rates by lash heating o asperities, gel ormation, melting. Simple rate-strengthening riction model; approximately valid only in stable regions in which rupture cannot nucleate, but may propagate through (or in unstable regions that have shear-heated to a rictionally stable T range). (Fuller rate-state description, with localization limiter, must be used in regions o unstable, rate-weakening, riction.)

12 Sliding velocity Fuller rate/state description. Vθ (, ) V 1 V Sliding distance Canonical experiment: aln( V/ V1) bln( V/ V1) ( ss V ) 1 d c ( ss V ) Sliding distance ( ) ( ) ( V) = V + aln V / V + bln( θ / θ ) dθ θv = g( θ, V) = 1 slowness or aging law dt d Ruina (1983), Dieterich c

13 Constitutive Relation: Dilatancy 1 ( ) pl pl n = n ( γ) = C + εln 1 + γ / C 1 where C and C are constants ε γ ε γ = t C + γ t γ t pl n Form introduced by Segall and Rice (1995) but neglects state dependence included by them. Inerred rom experiments by Marone et al. (199) and Byerlee and Lockner (1994). Fit with experiments implies ε 1 4 Note that dilatancy reaches a constant value or ixed strain-rate. Should also include a critical state type term that depends on eective stress. In linearized analysis this term only modiies a poroelastic constant.

14 Spatially uniorm solution (Lachenbruch, 198): V( y, t) = V ( y) = γ y ( γ = uniorm shearing rate), pyt (, ) = p( t), τ( yt, ) = τ ( t) = ( γ )[ σ p( t)], o n σ p () t = [ σ p ()]exp( H γ t) [ call this σ ()] t H n n o 1 where o o is a characteristic strain or decay o strength o H = ( γ ) Λ/ ρ c ( γ ) / [1.7(1 + β / β )], o o n T( y, t) = T ( t), ρ cdt ( t)/ dt = ( γ ) σ ( t) γ o o Is this solution stable (I.e., to spatial non-uniormities)? ANS: NO! For constant riction actor, deormation would localize to a plane. What are the eects o rate-strengthening riction and dilatancy? o

15 Linearized perturbation about time-dependent spatially uniorm solution: (irst neglect dilatancy) V( y, t) = γ y+ V ( y, t), p( y, t) = p ( t) + p ( y, t), o 1 1 T( yt, ) = T( t) + T( yt, ), = ( γ ) + ( γ ) V( yt, )/ y 1 o o 1 Linearized Equations: V1 ( γo) p1+ ( γo) σ( t) = y y V1 V1 o T1 T 1 ( γo) σ( t) ( γo) p1γo + ( γo) σ( t) γo = ρ c αth, y y t y p T p t t y Λ = αhy Boundary conditions: No heat (adiabatic) or mass (undrained low): T / y =, p / y = at y = ± h/ 1 1 Nature o solution with spatial dependence exp(πiy/λ): σ (t) V 1 (y,t) y σ (t) exp( Hγ o t) V 1 (y,t) y, p 1 (y,t), T 1 (y,t) exp(st)exp(πiy / λ) exp[(s+ Hγ o )t]exp(πiy / λ)

16 s = s(λ) satisies: 1 Instability condition : (s) ( ) ( ) ( ) 4πα 4 ch παhy γ γ zh γ = s s where z = = = λ λ γ γ a V y R > λ > λ = π cr α ( αth + αhy ) th + α hy ( z+ ) H γ αα th hy V Near λ = λcr, Im( s) zh, oscillatory h (implies unloading i perturbations become large enough but not included here.) Typically o order -6, in results o low shear-rate experiments.

17 Parameter Values γ α th = = = o 1 / s, ρ c.7 MPa/ K.65 mm / s (range.5 to.7 mm / s) λ = β η = 3 o 4 1 / K,.7 / GPa, 1 Pa-s λ λ Λ + β β o n 1.6 / (1 n / ) MPa/ K ( γ ) =.6,.4,. z = 4, 7, 13 appropriate or water at 8 km depth assuming hydrostatic conditions (1 MPa/km) and a typical thermal gradient ( 5 K/km Low end hydromechanical parameters : n =.4, β / β = 1 α hy H =.35 n = Λ = 1.8 mm /s,.8 MPa/C, ( γ ) based on laboratory measurments o ault gouge material (Wibberley [] and Wibberley and Shimamoto [3] and Lockner et al. [] High end hydromechanical parameters: n =.6, β / β = 4 α hy 4.8 mm /s,.3 MPa/C, H =.1 ( γ ) n = Λ = modiied to estimate eects o increased damaged caused by elevated stresses near the edge o the propagating rupture.

18 Numerical Values o Critical Wavelength ( γ ) =.6,.4,. Low end Parameters λ cr.36 mm,.534(.441) mm, 1.3(.63) mm High end Parameters λ cr = =.849 mm, 1.58(1.4) mm,.43(1.47) mm Highest value typical or rock, but riction at rapid rates o shearing could be reduced by lash heating at asperities, gel ormation, etc. Possible " sel - consistent" estimate o shear layer thickness h at large shear: γ = V / h, V = 1 m/s, h λ α + α th hy h 4π ( z+ ) HV Low end Parameters h = 1.9 μm, 6.5(19.4) μm, 16(38.9) μm High end Parameters h = 7 μm, 158(18) μm, 59(16) μm cr would not expect to see layer thicknesses larger than the critical wavelength since these would localize. order 1 to several 1 μm

19 1 Critical Wavelength, λ crit (mm) Homogeneous shear possible at small h ( λ crit > h ) h = λ crit High End, =. High End, =.6 Low End, =.6 Low End, =. 1E-3 1E Layer width, h (mm)

20 Linearized perturbation about time-dependent spatially uniorm solution: (now neglect rate-strengthening) V( y, t) = γ y+ V ( y, t), p( y, t) = p ( t) + p ( y, t), o 1 1 T( yt, ) = T( t) + T( yt, ), = ( γ ) + ( γ ) V( yt, )/ y 1 o o 1 Linearized Equations: y ( p ) 1 = o T1 T 1 σ() t γ1 p1 = ρ c αth, t y p T ε γ p t t t y Λ + = αhy βγ Boundary conditions: No heat (adiabatic) or mass (undrained low): T / y =, p / y = at y = ± h/ 1 1 Nature o solution with spatial dependence exp(πiy/λ): γ, p ( y, t), T( y, t) exp( st)exp( πiy/ λ) 1 1 1

21 Resulting Equations p () t = 1 Λ T() t + ( εβ / γ ) γ () t = ( εβ / γ ) C 1 1 d γ 1 γ () tˆ B 1 { exp( tˆ) Θ } = CBΘ dtˆ 1 1 No perturbation in pore pressure! C = Λ ( γ / εβ ) T() + γ () 1 1 a constant o the order o the magnitude o initial perturbations where tˆ = H γ t ( p ) B = σ β / ε - Θ = n BH γλ / 4πα Solution: th tˆ 1 γ ( tˆ) = exp B 1 ( 1 exp( tˆ) tˆ/ Θ γ () + BCΘ exp B 1 ( 1 exp( ξ) ξ / Θ) dξ tˆ 1 γ () t = γ ( r) dr 1 1 H I Θ > 1, coeicient is positive at small times, tˆ, implying rapid growth, but becomes negative as tˆ, implying exponential decay.

22 1 6 B = 4, Θ = e B = 13, Θ = sqrt (e) All results or C = γ 1 (Hγ t)/γ 1 () unperturbed solution Approximate result is the maximum strain-rate multiplied by the time at which it occurs t ˆ = ln Θ Hγ t Hγ 1 ( t )/γ 1 () accumulated strain approximation B = 4, Θ = e =.716 Hγ 1 ( t )/γ 1 () 1 1 accumulated strain approximation B = 13, Θ = sqrt(e) = Hγ t Hγ t

23 16 Θ crit 1 8 Threshold 1 4 Threshold B

24 Θ>Θ crit or instability 3 4 (based on an accumulated strain o 1 or 1 ) λ > λ = π crit 4 3 For B = 4, Θ =.5 (.15) or an accumulated strain 1 (1 ) crit Using this and low end parameters λ crit =.3, Θ α crit th BH γ.368,.5 mm or ( γ ) =.6,.4,. For B = 13, Θ = 1.6 or an accumulated strain 1 crit Using this and high end parameters λ crit =.11,..58,.365 mm or ( γ ) =.6,.4,. (About 4 to 1 smaller than or rate-strengthening) For "sel - consistent" estimate o layer thickness: γ = V / h, V = 1 m/s and setting λ = h low end parameters: 9., 13.5, 7.1 μm high end parameters: 4.41, 6.66, 13.3 μm crit 4

25 1 Critical Wavelength, λ crit (mm) Homogeneous shear possible at small h ( λ crit > h ) h = λ crit Viscous, High Viscous, Low Dilatant, High Dilatant, Low 1E-3 1E Layer width, h (mm)

26 γ (t) / γ γ (t) / γ No dilatancy n = 13 n = s = γ t n = s = γ t γ (t) / γ t γ (t) / γ t No dilatancy n = 13 n = s = γ t s = γ t Combined eects o rate-strengthening and dilatancy. High end parameters with B = 13, h = 1mm. Let column is strain-rate; right is strain No dilatancy Wavelength just less than critical (n = 13) based on rate-strengthening only. Wavelength just greater than critical (n = 1) based on rate-strengthening only. Is stabilized by addition o dilatancy. γ (t) / γ n = 11 γ (t) / γ t For a slightly larger wavelength (n = 11), strain-rate appears to be going to zero, but strains are becoming large ~ s = γ t s = γ t I wavelength is increased urther (n = 1) strains 1 6

27 Implications: Even with velocity strengthening and dilatancy must expect large shear strain to be conined to a thin zone, less than diusion penetration distances o heat and luid in moderate and larger events. Justiies use o model based on slip on a plane. Observed 1-5 mm deormed zone thickness in gouge may be a precursor thickness (i.e., λcr based on an initial, broad h) not the thickness o the large shear zone.

28 Limitations (in process o overcoming!?): Linearized perturbation analysis (mainly o two eects separately) Need ull nonlinear numerical analysis (Victor Tsai did some preliminary calculations) More elaborate description o dilatancy. Include variation o luid properties with pressure and temperature changes.

29 Thanks! Questions?