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Ubiquitous weakening of faults due to thermal pressurization

September 2015
Nature Geoscience

DOI:10.1038/NGEO2554

Authors:

Robert C. Viesca

Tufts University

Dmitry Garagash

Dalhousie University

Laboratory simulations of earthquakes show that at high slip rates, faults can weaken significantly, aiding rupture [1–3] . Various mechanisms, such as thermal pressurization and flash heating, have been proposed to cause this weakening during laboratory experiments [1,4–6], yet the processes that aid fault slip in nature remain unknown. Measurements of seismic radiation during an earthquake can be used to estimate the frictional work asso- ciated with fault weakening, known as an event’s fracture energy [7–9]. Here we compile new and existing [8,9] measurements of fracture energy for earthquakes globally that vary in size from borehole microseismicity to great earthquakes. We observe a distinct transition in how fracture energy scales with event size, which implies that faults weaken di�erently during small and large earthquakes, and earthquakes are not self-similar. We use an elastodynamic numerical model of earthquake rupture to explore possible mechanisms. We find that thermal pressurization of pore fluid by the rapid shear heating of fault gouge can account for the observed scaling of fracture energy in small and large earthquakes, over seven orders of fault slip magnitude. We conclude that thermal pressurization is a widespread and prominent process for fault weakening.

Cascade of weakening at the tip of a dynamic shear rupture. a, Schematic of a slip-pulse mode of rupture: re-locking follows fault rupture during rupture propagation. b, Evolution of the fault slip rate and strength at the rupture tip (for example, shaded region in a), with the parameter set in Supplementary Table 2. Initial weakening occurs through reduction of the friction coefficient by flash weakening. Subsequently, thermal pressurization of fault gouge pore fluid reduces the effective normal stress. Two endmembers are considered: negligible (cyan lines) and non-negligible (red lines) flux of heat and pore fluid. Broken lines follow the evolution under flash weakening of friction without thermal pressurization (dashed line) or thermal pressurization under a flash-weakened coefficient of friction (dotted and dash-dotted lines).

…

Fault fracture energy and gradual fault weakening with slip. a, Inferred dependence of fault shear strength on slip at the tip of a propagating rupture. Weakening here occurs by thermal pressurization under a flash-weakened or an otherwise (approximately) constant coefficient of friction. b, Implications of the strength–slip relation for the slip dependence of fault fracture energy. Increasing darkness of the greyscale lines corresponds to increasing efficiency of heat and pore fluid drainage. Dashed lines assume weakening under either undrained-adiabatic (cyan) or ‘drained’, slip-on-a-plane (red) conditions.

…

Variation of inferred event fracture energy with event slip. a, Inferred fracture energy (G) and slip for events within continental crust and along subduction interplate interfaces. b, For subduction events, comparison of fracture energy based on seismological observations alone (G′) with those incorporating pre-event shear stress estimates and assuming near-complete coseismic strength drop (Gmax). c, For large crustal events, comparison of G′ with G inferred from kinematic inversions of fault slip history. In a–c the curves are three predictions for ruptures whose fracture energy are dominated by thermal pressurization. Dashed curves represent scenarios of negligible (cyan) and significant (red) diffusion of pore fluid and heat. The black curve represents an intermediate scenario.

…

Figure S.1: Variation of stress drop with seismic moment for events of Figure 3. Figure S.2: Variation of apparent stress with seismic moment for events of Figure 3. Figure S.3: Variation of inferred fracture energy (scaled by square of event slip) with seismic moment for events of Figure 3. Figure S.4: Locally inferred fracture energy (Imperial Valley) compared with inferred average values for crustal events.

…

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LETTERS

PUBLISHED ONLINE: 5 OCTOBER 2015 | DOI: 10.1038/NGEO2554

Ubiquitous weakening of faults due to

thermal pressurization

Robert C. Viesca

1,2

and Dmitry I. Garagash

Laboratory simulations of earthquakes show that at high slip

rates,faults can weakensigniﬁcantly, aiding rupture

1–3

. Various

mechanisms, such as thermal pressurization and ﬂash heating,

have been proposed to cause this weakening during laboratory

experiments

1,4–6

, yet the processes that aid fault slip in nature

remain unknown. Measurements of seismic radiation during an

earthquake can be used to estimate the frictional work asso-

ciated with fault weakening, known as an event’s fracture en-

ergy

7–9

. Here we compile new and existing

8,9

measurements of

fracture energy for earthquakes globally that vary in size from

borehole microseismicity to great earthquakes. We observe a

distinct transition in how fracture energy scales with eventsize,

which implies that faults weaken dierently during small and

large earthquakes, and earthquakes are not self-similar. We

use an elastodynamic numerical model of earthquake rupture

to explore possible mechanisms. We ﬁnd that thermal pressur-

ization of pore ﬂuid by the rapidshearheating of fault gougecan

account for the observed scaling of fracture energy in small and

large earthquakes, over seven orders of fault slip magnitude.

We conclude that thermal pressurization is a widespread and

prominent process for fault weakening.

Throughout the seismic cycle, faults experience a wide range of

slip rates. Over this range, l aboratory observations have indicated

that the frictional strength of faults may respond in a drastically

diﬀerent manner. At low, interseismic slip rates, rock friction

experiments show minor strength variations

. In contrast, at faster

slip rates, such as those during an earthquake-generating fault

rupture, strong weakening is observed

2,3

. An inherent problem

in inferring whether the mechanisms proposed for such strong

weakening operate on faults is that the depths at which they occur

are largely inaccessible. However, occasional forays allow direct

measurements and sampling, providing further indications that

coseismic strength may be low

11,12

The interest in determining whether strong weakening does

occur at high fault slip rates is heightened by implications for

earthquake hazard and fault operation. The tsunami hazard of

the 2011 Tohoku event was increased by large slip in the shallow,

near-trench regions of the subduction interface

13,14

. Thes e regions,

along with creeping segments of other major faults, have long been

considered to be stable as a result of rate-strengthening behaviour

at low slip rates; however, this st ability is undermined by strong

weakening at fast slip rates, such that these regions may participate

in the catastrophic rupture of the fault

15,16

. Low coseismic strength

also implies t hat the shear stress on a fault may be much lower than

the static strength and still support rapid rupture. This aﬀects the

manner in which a fault ruptures, as low shear stress increases the

likelihood of a rupture occurring as a propagating pulse of slip

17,18

a phenomena long observed seismologically

Here we use seismological observations to help constrain the

possible weakening mechanisms operating on faults. We compile

estimates of source parameters and independent estimates of

background stress over a wide range of event sizes among crustal

and subduction interplate events. From the source information,

we calculate t he average fault fracture energy, defined as the

work done on the fault in excess of a residual level of friction.

Such inferences have shown a nonlinear scaling of fracture energy

with slip

. Our extensive data set allows a comparison with a

dynamic rupture model that can incorporate multiple mechanisms

of strong weakening. We show that the single, dominant weakening

mechanism (thermal pressurization) may account for the observed

fracture energy scaling over many orders of magnitude of event size.

Two mechanisms for dramatic fault weakening among the most

likely to be ubiquitous are: the reduction of the eﬀective normal

stress by t hermal pressurization of pore fluid due to shear heating

of fault gouge

4,5,20

, and the reduction of the friction coeﬃcient by

localized flash heating of asperity contacts in the gouge

. Other

candidate strong-weakening mechanisms on faults include melting

and thermal decomposition

1,3,5

, and their o ccurrence depends on the

specific mineralogical composition of the gouge and, additionally,

may require significant coseismic temperature rise. The latter may

not always be attainable, particularly in a pulse-like rupture, owing

to drastic coseismic reduction of the shear heating associated with

the thermal pressurization

We determine how these two thermal weakening mechanisms

control the slip dependence of fracture energy by coupling their

influence on frictional strength to the dynamic elastic response

of the surrounding medium. Incorporating these mechanisms in

elastodynamic simulations of crustal-scale fault rupture is a current

computational challenge. The diﬃculties stem from the disparity of

length scales between the rupture extent and distances of strength

evolution, anticipated to be several orders of magnitude apart. We

circumvent this computational complexity by reducing the problem

to its essential elements to find evolution of slip and strength that are

consistent with elasticity and both manners of weakening. Specifi-

cally, we focus on the tip region of the r upture, where most strength

loss is expected to occur, and model it as a steadily propagating semi-

infinite crack (Methods and Supplementary Methods).

Strength evolution behind the rupture front follows the order

in which the mechanisms operate. This order is determined by

the relative size of slip s cales that characterize weakening under

each mechanism. For flash heating, the relevant quantity is t he sub-

millimetre slip scale

, L, associated with evolution of the friction

coeﬃcient, implying the coeﬃcient may drop to a new, dynamically

reduced value over short distances behind the front of a dynamic

rupture. For thermal pressurization, there are two characteristic

slip scales and each is associated with one of two limiting regimes.

Department of Civil and Environmental Engineering, Tufts University, Medford, Massachusetts 02155, USA.

Department of Civil and Resource

Engineering, Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada.

e-mail: robert.viesca@tufts.edu; garagash@dal.ca

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LETTERS

NATURE GEOSCIENCE DOI: 10.1038/NGEO2554

0.1

SlippingLocked Re-locked

−4

−3

−1

−2

−5

Normalized slip rate (V/V

∗

)

Near-tip region

of width ∼

∗

Normalized shear stress ( /f

)

and friction coecient (f/f

)

τσ

Normalized distance behind rupture tip (x/

∗

)

Figure 1 | Cascade of weakening at the tip of a dynamic shear rupture.

a, Schematic of a slip-pulse mode of rupture: re-locking follows fault

rupture during rupture propagation. b, Evolution of the fault slip rate and

strength at the rupture tip (for example, shaded region in a), with the

parameter set in Supplementary Table 2. Initial weakening occurs through

reduction of the friction coecient by ﬂash weakening. Subsequently,

thermal pressurization of fault gouge pore ﬂuid reduces the eective

normal stress. Two endmembers are considered: negligible (cyan lines) and

non-negligible (red lines) ﬂux of heat and pore ﬂuid. Broken lines follow the

evolution under ﬂash weakening of friction without thermal pressurization

(dashed line) or thermal pressurization under a ﬂash-weakened coecient

of friction (dotted and dash-dotted lines).

The first regime characterizes early moments of rupture, in which

pressurization occurs under undrained, adiabatic conditions, with

the slip sc ale

ρch

f 3

where ρc is the heat capacity, h is the thickness of the fault

gouge, f is the prevailing friction coeﬃcient, and 3 is the thermal

pressurization coeﬃcient relating increments of undrained pore

fluid pressure increase to increments in temperature. The second

regime occurs after suﬃciently long time (slip) allows large-scale

diﬀusion to obscure the details of shearing of the finitely thick gouge

zone. In this regime, heating is eﬀectively reduced to that of slip on

a plane, and weakening occurs with a characteristic slip

5,22

∗



ρc

f 3



4α

∗

where α is a lumped hydrothermal diﬀusivity

, and V

∗

is a

characteristic elastodynamic slip rate

. Among the propert ies

0.01 0.1 1 10 100

0.0

0.2

0.4

0.6

0.8

1.0

0.1

1 10 100

0.01

0.1

Normalized slip ( /

)

Normalized fracture energy (G/f

w0c

)

Normalized shear stress ( /f

)

τσ

∼

2/3

∼

Normalized slip ( /

)

δδ

Figure 2 | Fault fracture energy and gradual fault weakening with slip.

a, Inferred dependence of fault shear strength on slip at the tip of a

propagating rupture. Weakening here occurs by thermal pressurization

under a ﬂash-weakened or an otherwise (approximately) constant

coecient of friction. b, Implications of the strength–slip relation for the

slip dependence of fault fracture energy. Increasing darkness of the

greyscale lines corresponds to increasing eciency of heat and pore ﬂuid

drainage. Dashed lines assume weakening under either

undrained-adiabatic (cyan) or ‘drained’, slip-on-a-plane (red) conditions.

determining δ

and L

∗

with the potential for l argest variation

are the gouge hydrothermal diﬀusivity and the width of the

actively shear ing gouge. However, for sub-millimetre to millimetres-

thick shear zones

23–25

and hydraulic diﬀusivities within an order

of magnitude or above 1 mm

−1

(ref. 26), a centimetre-to-

decimetre δ

and millimetre-to-metre L

∗

can be reasonably expected

(Supplementary Table 1).

We eﬃciently resolve the details of the compounded weakening

at the rupture tip (Fig. 1). That the evolution of the friction

coeﬃcient is associated with such a short slip scale L—relative

to plausible values of δ

and L

∗

—implies that flash heating, if

active, reduces the friction coeﬃcient before any substantial thermal

pressurization. The subsequent manner in which eﬀective stress is

reduced depends on the ratio L

∗

/δ

, which can vary from ∼0.0001

to ∼100. For values of L

∗

/δ

above unity, the majority of the increase

in gouge pore fluid pressure in large slip events occurs in a manner

consistent with slip-on-a-plane pressurization. In t his fashion, the

strength drop in large-slip, high-L

∗

/δ

events occurs in a less eﬃcient

manner than the exp onential slip weakening of undrained-adiabatic

shear in the small-slip, low-L

∗

/δ

events.

Our dynamic rupture solutions provide a theoretical slip

dependence of fracture energy, def ine d as

G(δ) =



τ (δ

) −τ(δ)



dδ

(1)

2 NATURE GEOSCIENCE | ADVANCE ONLINE PUBLICATION | www.nature.com/naturegeoscience

NATURE GEOSCIENCE DOI: 10.1038/NGEO2554

LETTERS

G'—empty symbols

max

—solid symbols

Chile '85

(aftershocks)

Subduction interplate

Pagai Is.

Peru '07

Tohoku

Sumatra'04

Nias

Maule

Aleutian

Chile'95

Sanriku

Miyagi

Mentawai

Peru Feb'96

Peru Nov'96

Benkulu

Tokachi'03

−1

G'—empty symbols

G—solid symbols

Big crustal

Denali

Landers

Hector Mine

Northridge

Kobe

Fukuoka

Tottori

Imp. Valley

Morgan Hill

Colﬁorito

Parkﬁeld

Subduction interplate

Chile '85 Sequence, ref. 37

Ref. 36

Ref. 35

La No Ko IV HM

Fu De MH Pa To

Colf

KTB, refs 84,85

Groß Schönebeck, ref. 81

W. Nagano, ref. 86;

W. Nagano, ref. 82

Cajon Pass, ref. 8

Long Valley, ref. 83

Korabid Aftershocks, ref. 77

Friuli, ref. 79

Au Sable Aftershocks, ref. 78

Northridge Aftershocks, ref. 87

Loma Prieta Aftershocks, ref. 80

Big crustal, refs 9,75 (G):

−5

−4

−3

−2

−1

−2

−1

Slip (m)

−1

Slip (m)

Fracture energy (J m

−2

)

Fracture energy (J m

−2

)

Fracture energy (J m

−2

)

Peru'01

Small crustal (G'):

max

= G' +

δτ

∼

2/3

(

Figure 3 | Variation of inferred event fracture energy with event slip. a, Inferred fracture energy (G) and slip for events within continental crust and along

subduction interplate interfaces. b, For subduction events, comparison of fracture energy based on seismological observations alone (G

) with those

incorporating pre-event shear stress estimates and assuming near-complete coseismic strength drop (G

max

). c, For large crustal events, comparison of G

with G inferred from kinematic inversions of fault slip history. In a–c the curves are three predictions for ruptures whose fracture energy are dominated by

thermal pressurization. Dashed curves represent scenarios of negligible (cyan) and signiﬁcant (red) diusion of pore ﬂuid and heat. The black curve

represents an intermediate scenario.

where τ is the fault shear and δ is the average distance the fault

slipped in an event. B ecause the slip scale L is comparatively small,

the initial weakening due to flash heating has a negligible con-

tribution to t he total fault fracture energy, which is consequently

controlled by thermal pressurization

. The slip dependence is then

determined solely by the ratio L

∗

/δ

(Fig. 2). However, for any value

of the parameter, there are two endmember scalings of the fracture

energy: for small slip, the early time undrained-adiabatic deforma-

tion results in fracture energy scaling as G ∝δ

, and for large slip,

where shear heating resembles slip on a plane, we find that G∝δ

2/3

This latter large-slip scaling is in contrast to the finite fracture

energy (ρch ¯σ /3) reached in the limit of large slip under undrained-

adiabatic conditions. Our dynamic analysis, which couples evo-

lution of the slip rate to that of the strength, supports previous

kinematic models for evolution of the coseismic strength

5,22

. In the

latter a constant slip rate of coseismic magnitude is proscribed,

resulting in behaviour qualitatively comparable to, but quantita-

tively diﬀerent from (with the exception of the small-slip limit)

our model.

The definition of the fracture energy, equation (1), presumes

that the dissipation of energy is dominated by work done on

the fault surface—yet, inelastic deformation of the adjoining rock

is expected to occur. Fault rupture models allowing for such

deformation

suggest that oﬀ-fault dissipation, although scal-

ing with event size, remains at most comparable to the on-fault

dissipation (Methods).

We estimate the average fault fracture energy G for a wide range

of event sizes using previously published s eismological inferences

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LETTERS

NATURE GEOSCIENCE DOI: 10.1038/NGEO2554

of source parameters (Methods and Supplementary Tables 3–5 and

Supplementary Figs 1–3), namely the total moment and seismic

energy released and a characteristic source radius. On the basis of

seismic source spectra, for each event, we estimate an intermediate

energetic quantity G

=(1τ/2 −τ

)δ (where 1τ is the stat ic stress

drop and τ

is the apparent stress). By conservation of energy

G=G

+(τ

−τ

)δ, w here τ

is the minimum coseismic strength,

and τ

is the final (after-event) fault stress

. For crack-like ruptures,

reasonably characterized by minimal dynamic over/undershoot

(τ

≈τ

), G ≈G

. For large pulse-like ruptures, consistent with the

hypothesis of thermal pressurization that the coseismic strength

drop at large slip is near-complete (τ

≈0), G ≈ G

max

+τ

δ,

the estimate of which requires inferences of the absolute fault

stress (Methods).

In Fig. 3a we compile our frac ture energy estimates and estimates

made previously

8,9

. Estimates of G are available for the largest events:

for those occurring on a subduction interplate interface, we assume

G =G

max

and make use of published inferences of fault stress; for

those occurring within the crust, we rely on available estimates of G

from kinematic inversions

, independent of our estimates of G

from

source spectra. In both cases, we find that G

largely underestimates

the fracture energy (Fig. 3b,c), which may indicate that the strong-

weakening slip-pulse scenario represents rupture behaviour of large

subduction and crustal events

. However, for the smallest events

(borehole seismicity and crustal aftershock sequences, Fig. 3a),

studies of the stress field and dynamic rupture models imply that

rupture may frequently be crack-like, in which case we may take

G ≈ G

. Inferences of the stress field in regions surrounding the

KTB and Cajon Pass boreholes indicate a critically stressed crust,

in which high values of the shear-to-normal-stress ratio on opti-

mally oriented faults

29,30

favour crack-like rupture propagation

17,18

Alternatively, these small events may be limited by a fault extent

such that a rupture nucleated in a crack-like mode reaches fault

boundaries and arrests before a transition to a pulse-like mode

ever occurs.

The estimates of fracture energy are in general agreement with

the prediction of the dynamic rupture tip solution with thermal

pressurization over more than six decades of event slip. For small

event sizes, we observe that the scaling of fracture energy with slip is

similar to δ

, consistent with linear slip weakening under undrained-

adiabatic pressurization for event slip smaller than δ

. At large event

sizes, the apparent quadratic scaling of the fracture energy breaks

down and is replaced by a sublinear scaling characteristic of slip-on-

a-plane (or ‘drained’) pressurization. Remarkably, the t he oretically

predicted evolution of fracture energy with slip (Fig. 3, curves) is

made using a single value of δ

(0.1 m) and a single value of the

nominal fault strength f ¯σ

(110 MPa). The former may imply similar

coseismic shear zone thicknesses on diﬀerent faults; the latter may

indicate either approximately constant values of f and ¯σ

in the crust,

or, alternatively, a variable high or low (flash-weakened) value of

friction, inversely related to a variable eﬀective normal stress level

from event to event.

Although multiple weakening mechanisms may operate on

faults, the results suggest that thermal pressurization may be the

dominant contributor to fault fracture energy. Furthermore, that

fracture energy seems to follow a scaling attributable to thermal

pressurization, considering a large number of events occurring on a

wide variety of fault zones, also suggests the potential prevalence of

this mechanism. Understanding how faults lose their strength dur-

ing the rapid slip of an earthquake-generating rupture is important,

for instance, to constrain the minimum level of stress a fault requires

to rupture catastrophically.

Methods

Methods and any associated references are available in the online

version of the paper.

Received 13 January 2015; accepted 3 September 2015;

published online 5 October 2015

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27. Brantut, N. & Rice, J. R. How pore fluid pressurization influences

crack tip processes during dynamic rupture. Geophys. Res. L ett. 38,

L24314 (2011).

28. Andrews, D. J. Rupture dynamics with energy loss outside the slip zone.

J. Geophys. Res. 110, B01307 (2005).

4 NATURE GEOSCIENCE | ADVANCE ONLINE PUBLICATION | www.nature.com/naturegeoscience

NATURE GEOSCIENCE DOI: 10.1038/NGEO2554

LETTERS

29. Zoback, M. D. & Healy, J. H. In situ stress measurements to 3.5 km depth in

the Cajon Pass scientific research borehole: Implications for the mechanics of

crustal faulting. J. Geophys. Res. 97, 5039–5057 (1992).

30. Zoback, M. D. & Harjes, H.-P. Injection-induced earthquakes and crustal

stress at 9 km depth at the KTB deep dr illing site, Germany. J. Geophys. Res.

102, 18477–18491 (1997).

Acknowledgements

This study was supported by the Natural Science and Engineering R esearch Council

of Canada (Discovery grant 371606), the National Science Foundation (grant

EAR-1344993), and the Southern California Earthquake Center (SCEC; contribution

No. 6004). SCEC is funded by NSF Cooperative Agreement EAR-1033462 and

USGS Cooperative Agreement G12AC20038.

Author contributions

Both authors contributed to developing the main ideas, interpreting the results and

producing the manuscript.

Additional information

Supplementary information is available in the online version of the paper. Reprints and

permissions information is available online at www.nature.com/reprints.

Correspondence and requests for materials should be addressed to R.C.V. or D.I.G.

Competing ﬁnancial interests

The authors declare no competing financial interests.

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LETTERS

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Methods

The dependence of fracture energy with event slip is theoretically estimated by

examining the dependence of strength w ith slip at the tip of a propagating rupture.

The region near the tip is modelled as a steadily propagating semi-infinite crack

rupturing a planar fault dividing two linearly elastic half-spaces. The fault strength

is determined by an evolving friction coeﬃcient and eﬀective normal stress. The

evolution of the former is slip-rate- and state-dependent, with strong weakening at

high slip rates

2,18

. The evolution of fault pore fluid pressure alters the eﬀective

normal stress and depends on the history of shear heating and hydrothermal

diﬀusion

4,5,18

. We formulate a sharp-tip rupture approximation that requires a

regularization of the frictional e volution to allow for a slip velocity that is

vanishingly small at the rupture front. The resulting coupled nonlinear integral

equations are solved by amending a pseudospectral boundary integral equation

method. Laboratory measurements of frictional and hydrothermal properties of

fault material indicate that evolution of pore pressure and friction occur over

disparate fault length and slip scales. Our method of solution allows the resolution

of the compounded weakening with large-scale separation. When calcu lating the

fracture energy, the details of frict ional evolution are reasonably neglected, and the

model reduces to one with a single parameter that captures the relative eﬃciency of

hydrothermal diﬀusion

. We compile estimates of fracture energ y for crustal and

subduction interplate ruptures using published seismological data and, where

available, estimates of pre-event shear stress for large subduction events. Primarily,

we infer key source properties (seismic moment, radiated energy, and characteristic

event radius) on the basis of the recorded source spectra, which we use in turn to

estimate the average event f racture energy and slip

Seismological estimates of fracture energy. In this work we compiled

seismological estimates of the fracture energy of a large number of events (from

borehole seismicity to great subduction megathrust eart hquakes) which are based

on two distinct approaches. The first approach

9,31

relies on the finite-fault

kinematic inversion for the slip-history, and corresponding calculation of the

elastic stress change over the fault to calculate G directly via its definition

G(δ) =

[τ (δ

) −τ (δ)]dδ

(2)

The other method

relies on estimating an energetic quantity G

(see (3) below)

from the source spectra only—that is, from the s eismically inferred moment M

radiated energy E

, and e quivalent source radius r (inferred for the conner

frequency, for example, by way of a circular rupture model

32,33

). As we show below,

can be used to approximate the fracture energy G for expanding crack-like

ruptures, whereas a measure we call G

max

(see (5) below) additionally requires an

independent estimate of t he fault stress (either b efore or after the event) and is a

suitable approximation of the frac ture energy of large, pulse-like ruptures.

and G

max

approximations of the fracture energy. Potential energy release in an

earthquake can be written as

1W /A=(τ

+τ

)δ/2 =E

/A+G +F

where E

/A= radiated energy, G = fracture energy and F =τ

δ = frictional heat

per unit slipped fault surface. The energetic quantity

=(τ

−τ

)δ/2 −τ

δ (3)

can be estimated from source spectra by using the following expressions for the

stress drop τ

−τ

=7M

/16r

, the apparent stress τ

/πr

δ (A=πr

) and the

average slip δ =M

/µπr

. Fracture energy can then be expressed as

G=G

(

−τ

)

δ (4)

Modelling of crack-like events with arbitrary slip show smal l dynamic stress

overshoot

(that is, minimum coseismic stress τ

is only slightly larger than the

final static stress τ

), potentially allowing it to be neglected in the expression (4) for

the fracture energy. That is,

crack-like events,τ

≈τ

: G≈G

On the other hand, we expect near-complete coseismic strength loss, τ

≈0, for

large slip events on faults with strong dynamic weakening, as well as a finite final

static stress τ

for pulse-like events (for which stress is expected to partially recover

in the wake of the pulse from its minimum, coseismic value τ

). Thus,

large pulse-like events,τ

τ

: G≈G

max

+τ

δ (5)

Fracture energy estimates of big crustal and subduction events. The relevant

source data and corresponding estimates of fracture energy G (for large crustal

events

9,34

), as well as our estimates of energetic quantities G

(for large

subduction

35–51

and crustal events

8,9,52–72

) and G

max

(for great subduction events for

which independent estimates of t he fault prestress, τ

∼15 MPa, are available

73,74

)

are shown in Supplementary Tables 3 and 4 for selected earthquakes.

The uncertainty in G-estimates for large events based on finite-fault kinematic

inversions may stem from non-uniqueness of finite source inversion for a given

event, as well as from diﬀerent s ets of assumptions involved w ith modelling

coseismic stress changes b ased on a g iven fault slip kinematics. For example,

Tinti et al.

9,75

estimate the average fracture energy over fault area with significant

slip for the Landers event to be 51 MJ m

−2

and 42 MJ m

−2

for the two respective

kinematic slip models

52,76

. A diﬀerent study

of the same two kinematic models for

Landers yielded 90 MJ m

−2

and 32 MJ m

−2

, respectively. This is a fairly

representative cas e, which suggests that the uncertainties of G-estimates can b e

roughly constrained to within a factor of two.

Fracture energy estimates of borehole seismicity and crustal aftershock sequences.

We use source parameters from a number of recent studies

8,77–87

of aftershock

sequences and borehole-recorded microseismicity (Supplementar y Table 5) to

infer values of the energetic quantity G

and average event slip. In doing so, we

recalculate the source parameters from the reported individual spectral

measurements using a consistent set of assumptions. For all but t wo event

sequences (Chile 1985 and Northridge 1994 aftershock sequences), we use sp ectral

relations

32,33

between the corner frequencies (f

, f

) and the source radius r of a

circular rupture to estimate

r =geometric mean





(6)

where k

and k

are tabulated functions

of rupture velocity and azimuth θ (the ray

take-oﬀ angle with the fault normal), and the

are the corresponding average

values over the representative azimuth range for an assumed representative value of

the rupture velocity. (When one corner frequency is reported but not the

other—that is, either f

or f

—we modify (6) accordingly). We assume v

=0.9c

(ref. 8), and select average values of k as either

=0.30,

=0.39 (average over 0 ≤θ ≤90

◦

) (7)

=0.21,

=0.32 (average over 45

◦

≤θ ≤90

◦

) (8)

depending on the assumed azimuthal coverage of events. Specifically, we have

assumed the entire azimuthal coverage, (7), for all borehole-recorded events and

the locally recorded Au Sable Forks 2002 aftershock sequence; and the upper-half

of that range, (8), for all other crustal aftershock sequences that we analysed. The

latter choice can be understood on an example of strike-slip (vertical) fault with

receiving stations located on the earth surface at or near the fault (with the distance

to t he fault not exceeding a typical event depth). When both corner frequencies are

reported, we apply a filter 0.75 ≤f

≤2.25 to remove the events with

unrealistically small/large values of the frequency ratio (this is motivated by

elastodynamic solutions

with v

=0.9c

, which suggest the corner frequency ratio

values bet ween 1 and 1.5, depending on the azimuth). These removed events

normally account for a small percentage of the total number of events in a sequence

(for example, 3% of KTB borehole events, 6% of Loma Prieta aftershocks) with the

exception of the Western Nagano events analysed in ref. 86, of which about 40%

had unrealistic corner frequency ratios. The static stress drop, average event slip,

and energetic quantity G

are estimated from source values M

, r, and E

, as

explained earlier

In the case of Northridge 1994 aftershock sequence, reported signal durations

signal

(ref. 87) allow an estimation of the equivalent circular rupture radius

r =v

signal



1+max



sinθ,



−1

(9)

where, as before, we assume v

=0.9c

, and average

sinθ

=0.9 over the azimuthal

range 45

◦

≤θ ≤90

◦

. The latter estimate of

sinθ

is justified for the Northridge

aftershocks, for which we can calculate

sinθ

≈0.92 using the stations’ locations

and hypocentre/fault-plane solutions

. We note that our values of the source radii

for Northridge aftershocks are on average a half of the values originally reported

This discrepancy can be tracked to the assumption that a pulse duration t

pulse

(used

in previous source radius estimates

) is half of the signal duration t

signal

. On the

basis of the preceding estimates

(equations (9) and (11) therein), we can obtain



pulse



≈0.265



signal



for the studied Northridge aftershocks

, which suggests that

the pulse durations and corresponding source radii were possibly overestimated in

their study. We also note that the original range of stress-drop values is 2–66 MPa

and that our values range from 3 to 100 MPa.

Assessing oﬀ-fault contributions to fr acture energy. Dynamic rupture models

indicate that stress concentrations at the tip of a dynamically propagating rupture

may be suﬃcient to cause significant oﬀ-fault inelastic deformation

28,90

, which may

be reflected in the observed damage zones of surrounding faults

91,92

. This

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LETTERS

coseismic, oﬀ-fault inelastic deformation is part of the total fracture energy of an

event. Constraints on the magnitude of that contribution (relative to on-fault

processes, such as those studied here) may be deduced from dynamic rupture

models in which inelastic deformation is allowed to occur

28,93

. In these models, the

extent of inelastic deformation is proportional to the rupture propagation

distance, which implies a linear scaling of the oﬀ-fault dissipation with event

slip for crack-like and unsteady (g rowing) slip-pulse rupture modes. We seek an

order-of-magnitude constraint on the oﬀ-fault contribution to the fracture energy

by determining the prefactor of that linear scaling. To do so we use detailed

reports on the extent and intensity of oﬀ-fault deformation in dynamic rupture

models

, along with a relation between rupture size and mean slip. We find that

the oﬀ-fault contribution to the fracture energy G

oﬀ

≈(0.1–10) MJ m

−2

(δ/1 m),

where the lower limit roughly corresponds to the case of low fault shear stress

inferred on major plate-bounding faults

. This estimate is approximately an

order of magnitude below the fracture energy estimates from seismological

observations for larger events (Fig. 3, δ & 0.1 m). Although we caution that this is a

preliminary estimate, this suggests that on-fault contributions to fracture energy

dominate when fault strength weakens slowly with slip, as for weakening by

thermal pressurization.

‘Homogeneous’ fault model predictions for ruptures with heterogeneous slip.

Kinematic slip inversions of large events often show complex/spatially

heterogeneous slip. It is therefore a relevant question as to how much can we

learn about the physics of earthquake slip by contrasting fault-averages of

heterogeneous fields inferred from observations to predictions for the s ame

field based on ‘homogeneous’ fault models. Systematic studies of such

comparisons, involving predictive solutions for heterogeneous faults, are needed

to answer this question conclusively, and are outside of the scope of this

contribution. Yet, we can suggest two types of observations that may lend some

credence to the application of homogeneous fault model predictions to their

heterogeneous prototypes.

It has been observed that slip and fracture energy inferred from

finite-fault kinematic inversions, although heterogeneous, can strongly correlate

9,96

This suggests that the fault-area-averaged fracture energy inferred from an

event’s kinematic inversion

observ

can be meaningfully compared to the

theoretical predictions from a homogeneous fault model (that is, fault with

homogeneous properties and prestress) if the scale of the heterogeneity is larger

than the characteristic dimension of a dynamically slipping patch (making

pulse-like ruptures more viable candidates than crack-like ones); and the slip

magnitude versus fault area distribution is known from the inversion. For

example, assuming a linear distribution of area with slip in the range between 0 and

, one can readily evaluate the fault-averaged prediction



theory

(δ)



to be

contrasted to values inferred from observations—that is, to

observ

. Such a

comparison is shown on Fig. 3, after introducing an additional simplification by

using G

theory

(

) in place of



theory

(δ)



. The maximum error associated with this

last simplification is between +33% and −5% for a power-law G

theory

versus δ

relation, whose exponent may vary from 2 to 2/3, respectively, with increasing slip,

as predicted by the thermal pressurization model. This latter error is

inconsequential in the log–log scale comparisons over many orders of magnitude

of slip and G (Fig. 3).

Another observation pertains to the question how well do the ‘local’ G

versus slip data inferred f rom a kinematic inversion agree with the fault-averaged

data. We provide an illustration of such an agreement in Supplementary Fig. 1,

where we plot the local fracture energy versus slip data (cyan stars) from selected

locations in the finite-fault model of the Imperial Valley event

together with the

fault-average values for other large crustal earthquakes (as in Fig. 3c). Evidently, the

local fault data scales in approximately the same way as the averaged data from

diﬀerent events (including the fault-average value for the Imperial Valley

event itself).

Modelling of the tip of a dynamic fault rupture. Our fault rupture model reduces

to coupled nonlinear integral equations relating the evolution of f riction and the

development of fault pore fluid pressures to the elastodynamics of a propagating

rupture. Below we summarize the governing equations and highlight key

assumptions that enable eﬃcient, approximate solutions for the slip- and

slip-rate-weakening behaviour behind the rupture front. Knowledge of this

behaviour enables a calcul ation of the fracture energy (including its slip

dependence) as it depends on the manner of dynamic weakening.

The fault shear strength is determined by

τ =f (σ −p) (10)

where f is the frict ion coeﬃcient, σ is the fault-normal stress, and p is the fault

pore pressure.

Numerical solutions

indicate that the classical friction coeﬃcient dependence

on the rate and ‘state’ of frictional contacts

10,97–99

amended for high-velocity thermal

weakening (fl ash heating) results in a marked slip acceleration and corresponding

friction/stress build up to some peak value (f

) immediately on arrival of the

rupture front, followed by weakening towards the high-velocity (‘flash-heated’),

steady-state friction value (f

f

). This allows one to significantly simplify the

modelling by approximating the rupture front as a sharp crack with a p eak level of

friction f

established at the tip, whereas the decline in the friction coeﬃcient

immediately behind the sharp front is dominated by the evolution of the state

variable over the characteristic slip scale L, which on integration yields an

exponential decay with slip

27,100

f −f

≈(f

−f

) exp(−δ/L) (11)

To solve for simultaneous evolution of the fault pore pressure, we consider the

frictional heating of the fault and simultaneous heat flux in the fault-normal

direction. Assuming that t he fault slip velocity is accommodated by a Gaussian

distribution of the strain-rate ˙γ over a thin layer of gouge of nominal width h,

˙γ =(V /h) exp[−π(y/h)

], and g iven a history of shear stress and slip velocity at a

point along the fault, the current pore pressure at the gouge centre at that point can

be given by

p(t) =p

ρch

τ (t

)V (t



t −t

;



(12)

where t

/(4α) is a characteristic diﬀusion timescale, and K is a convolution

kernel whos e form depends on the ratio of hydraulic-to-thermal diﬀusivities

/α

(refs 5,21,101).

We consider the steady, dynamic propagation of a sharply tipped mode-II or

mode-III crack (in-plane or anti-plane rupture) with rupture velocity v

. We define

x as the distance behind the rupture tip such that t =x/v

is the time since passage

of the rupture front, and the slip velocity is given by

V =v

dδ

(13)

where δ is the relative fault displacement. In view of (13), the shear stress

distribution along the fault, ahead and behind the rupture tip, is given by

102

τ (x)=

¯µ

2πv

∞

V (s)

s−x

ds (14)

where ¯µ is a shear modulus that depends on both the mode of rupture as

well as v

Solutions to (10)–(14) are found by first discretizing the equations spatially

along two sets of singular integral quadrature points

103

with quantities defined on

the one set related to those on the other by barycentric interpolation

104

. The

resulting nonlinear equations for the discretized slip rate, friction and eﬀective

normal stress are solved via Newton–Raphson iteration, with details given in a

subsection to follow. The solution yields a relation between shear strength and slip,

τ (δ) (as in Fig. 2), which allows an estimate of the slip dependence of the fracture

energy G via (2).

Endmembers of the fracture energy of thermal pressurization. When the

slip-weakening scales for t he flash heating (L) and thermal pressurization

(δ

or L

∗

) separate—that is, Lmax(δ

∗

)—a cascade of weakening is

observed in our numerical solutions, which show that the early flash heating

weakening acts to set up the nominal level of friction (f ≈f

) on a

dynamically slipping fault, although its contribution to the event fracture

energy is negligible compared to that of the ther mal pressurization ac ting

over larger slip and spatial scales. Although the numerical solution allows

one to recover the complete dependence of the fracture energy on slip, it is

instructive to provide the analytical endmembers of this dependence for

relatively small and large values of slip, when the dynamic response of the

sheared fault gouge can b e described as essentially undrained and

drained, respectively.

The fault gouge response to the frictional heating and pressurization is

undrained (drained) when the slip time t is much smaller (larger) than the

hydrothermal diﬀusion timescale t

defined in the above. Elastodynamics (14) and

the characteristic scale of the strength reduction by the thermal pressurization

(τ

∗

¯σ

) suggest the characteristic slip velocity V

∗

=(f

¯σ

/ ¯µ)v

near the rupture

front. Then, the fault response is expected to be undrained (drained) when the

dynamically accrued slip is much smaller (larger) than the characteristic diﬀusion

slip δ

∗

, which interestingly is not an independent slip scale (that is, it can

be expressed in terms of the two previously introduced TP slip-weakening

scales, δ

=δ

∗

Undrained fracture energy. When δ δ

(or t t

) and f ≈f

, the constitutive

equations (10) with (12) and t =x/v

reduce to the slip-law

, τ =f

¯σ

exp(−δ/δ

and the corresponding f racture energy dep endence on slip is found from (2),

G=f

¯σ



1−(1 +δ/δ

) exp(−δ/δ

)



. The small-slip limit, δ δ

, corresponds to

the quadratic slip dependence

undrained, small slip: G ≈f

¯σ

/2δ

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Drained fracture energy. In the large, drained slip limit δ δ

(or t t

), (10)

with (12) and t =x/v

reduce to the slip-on-a-plane constitutive equation

τ (x)−f

¯σ

=−

ρc

√

τ (s)V (s)ds

√

4πα(x −s)

(15)

We look for the large-slip, far-field (large distance x from the rupture front)

asymptotic solution of (14) and (15) among the candidate monomial solutions

105

(14), V /V

∗

=Bx

and τ /τ

∗

=−(B/2)cot(πλ)x

, where the values of the exponent

and the prefactor, λ =−1/4 and B =

√

2(x

∗

/π)

1/4

, with x

∗

=( ¯µ/τ

∗

, are recovered

by substitution into (15) and balancing the dominant terms in the limit x →∞.

Integrating for slip and relating it to the strength, we can recover the corresponding,

drained large-slip law τ /τ

∗

≈(2/3π)

1/3

(δ/L

∗

)

−1/3

. The corresponding expression for

the fracture energy slip dependence follows by integrating (2),

drained, large slip: G≈(12π)

−1/3

¯σ

∗1/3

2/3

Numerical method of solution for slip rate near the rupture tip. When the total

slip incurred in dynamic rupture is much larger than the characteristic

slip-weakening distance, most of the strength reduction is accommodated in the

near-tip region of the rupture. This region can be modelled as a semi-infinite shear

crack steadily propagating at the instantaneous rupture velocity v

(ref. 21). The

corresponding elastodynamic equation in the coordinate system x moving with the

crack tip (x =0) is given by

τ (x)=τ

∞

¯µ

2π

∞

dδ

s−x

(16)

where ¯µ =µ

√

1−(v

)

is the apparent shear modulus for a mode-III crack, c

the shear wave speed of the surrounding medium, µ is the shear modulus, and τ

∞

is the far-field driving stress. Requiring no stress intensity at the tip and a constant

stress drop far b ehind determines the asymptotic behaviour of the slip gradient

towards the crack tip and far from it

dδ



x→0

∼

√

dδ



x→∞

∼

√

(17)

We use t he following coordinate transform to map x ∈[0, ∞) to z ∈[−1, 1]

x =

1+z

1−z

and arrive at

τ [x(z)]=τ

∞

¯µ

2π

−1

(1−z)(1−u)

dδ

u−z

which takes the familiar form of the Hilbert transform when expressed as

g(z) =

¯µ

2π

−1

h(u)

u−z

du (18)

with

h(z) =(1−z)

dδ

g(z) =2

τ [x(z)]−τ

∞

1−z

(19)

Reflecting the asymptotics of (17), h(z) behaves as

h(z →−1)∼

√

1+z h(z →1) ∼

√

1−z

With this anticipated we may define a function φ (z) such that

h(z) =

1+z

1−z

φ(z) (20)

Substituting this expression in (18) we find that the integral takes a form

approximable by a Gauss–Chebyshev quadrature of the third kind

−1

1+u

1−u

F(u)du ≈

j=1

F(u

) (21)

with u

=cos[π(2j−1)/(2n +1)] and w

=2π(1+t

)/(2n+1). When F(u) is the

singular kernel 1/(u −z), the approximation continues to hold provided z is

evaluated at points z

=cos[2πi/(2n+1)] for i =1, 2,. .. , n (ref. 103). This reduces

the evaluation of (16) to that of

g(z

) =

¯µ

2π

j=1

φ(u

)

−z

(22)

To evaluate the left-hand side of (22), we calc ulate f and ¯σ at points z

and note

that for ruptures including undrained thermal pressurization, τ

∞

=0. Considering

friction, we may integrate a slip-law formulation for friction evolution (refs 18,98),

neglecting the direct eﬀect as discussed and making use of the Galilean

transformation to the crack-tip coordinate x =v

f (x) −f

=−

dδ



f −f

(V )



We transform the coordinates via (17), replacing dδ/dz with its representation by

φ(z), and include a Heaviside step function H(z) to extend the domain of

integration to one consistent with the quadrature (21)

f [x(z)]−f

=−

−1

H(z −u)φ(u)

1+u

1−u

f −f

(V )

1−u

such that

f [x(z

)]−f

=−

j=1

φ(u

)

f [x(u

)]−f

(V )

1−u

(23)

where S

=1 if u

< z

, and 0 otherwise, and f

(V ) may be evaluated by recognizing

that V =v

dδ/dx with dδ/dx =φ[z(x)]

√

1−z

/2.

We may find an expression for the eﬀective stress σ −p via (12) in a similar

manner. Following the shift to the crack-tip coordinate,

¯σ (x) − ¯σ

=−

ρch

τ (s)

dδ



x −s

;



ds (24)

Subsequently, the coordinate transformation from x and s to z and u, similar use of

the Heaviside function, and the quadrature approximation of the integral, enable

the approximation of ¯σ (x) as

¯σ [x(z

)]− ¯σ

=−

ρch

j=1

φ(u

f [x(u

)]¯σ [x(u

)]

1−u

(25)

where K



(x(z

) −x(u

))/(v

)



In (23) and (25), f and ¯σ are evaluated at two sets of points at x(z

) and x(u

To reduce the set of solution variables, we interpolate the quantity p (that is, f or ¯σ )

from x(u

) to x(z

) by

p[x(z

)]=

j=1

p[x(u

)] (26)

We make convenient use of the points u

lying at the zeros of a Chebyshev

polynomial and use the barycentric form of Lagrange interpolation

104

, for which



−u



m=1

−u

and, as u

are the zeros of a Chebyshev polynomial of the third kind, the weights are

=(−1)

cos

sinθ

=π

2j−1

2n+1

Using the shorthand notation w here indices i and j stand for the evaluation at

the z

and u

nodes, respectively, (for example, f

=f [x(z

)], f

=f [x(u

)], and,

according to (26), f

j=1

), and defining m

=1/(1−u

), we arrive at the

system of 3n equations

−

j=1



−f

)



¯σ

= ¯σ

−

ρch

j=1

¯σ

(27)

¯σ

1−z

¯µ

2π

j=1

−z

for 3n unknowns in the order of f

, ¯σ

, and φ

. On evaluating the Jacobian of (27), its

solution is amenable to the Newton–Raphson iteration method.

The quadrature (22) can also be used in the solution to the problem of the

exponentially slip-weakening semi-infinite crack. Such a manner of slip weakening

occurs, for example, for the case of undrained-adiabatic thermal pressurization

For this case, when evaluating g (z), we use

τ −τ

∞

=f ¯σ

exp(−δ/δ

∗

)

in which slip is evaluated at points z by applying the coordinate transform to the

integral δ(x) =

(dδ/dx

)dx

to ar rive at

δ[x(z)]=

−1

H(z −u)

1+u

1−u

φ(u)

1−u

du (28)

NATURE GEOSCIENCE | www.nature.com/naturegeoscience

NATURE GEOSCIENCE DOI: 10.1038/NGEO2554

LETTERS

which is approximated at z

nodes as

with terms as previously defined.

Amendment to account for non-classical asymptotic behaviour. The above

method may be amended to account for an exponent of the far-field slip gradient

other than one-half, that is,

dδ



x→0

∼

√

dδ



x→∞

=Ax



−

< λ < 0



where the latter asymptote is assumed to be known explicitly. For example, in a

preceding section concerning endmember cases, we discuss a case where the

weakening in the far field occ urs by slip-on-a-plane thermal pressurization, for

which λ =−1/4 and A =[(4/π)(f ¯σ

/ ¯µ)

∗

]

1/4

. We use the behaviour of h(z)

(equation (19)) in the far field, h(z) ∼(1 −z)

−1−λ

, to amend the definition of the

function φ(z) (equation (20)) by introducing an additional term (φ

asy

(z)) that

reflects the asymptotics

h(z) =

1+z

1−z

(

asy

(z) +1φ(z)

)

, φ

asy

(z) ≡A



1−z



1/2+λ

Here 1φ(z) is f inite and continuously diﬀerentiable at the interval’s ends allowing

one, as previously, to use Gauss–Chebyshev quadrature methods to evaluate

related integrals.

Denote the contributions from the φ

asy

-term in h(z) to the elastic shear stress

(18) with (19), the eﬀective stress (24), and slip (28) as

asy

[x(z)]=

¯µ

2π

1−z

−1

1+u

1−u

asy

(u)

u−z

¯σ

asy

[x(z)]=−

ρch

−1

1+u

1−u

asy

(u)τ

asy

[x(u)]

1−u



x(z) −x(u)



asy

[x(z)]=

2π

−1

1+u

1−u

asy

(u)

1−u

We can write the amended set (27) of 3n algebraic equations as

−

j=1

(φ

asy

+1φ

)



−f

)



¯σ

− ¯σ

asy

= ¯σ

−

ρch

j=1



(φ

asy

+1φ

¯σ

−(φτ )

asy



2(f

¯σ

−τ

asy

)

1−z

¯µ

2π

j=1

1φ

−z

in 3n unknowns chosen here in the order of f

, 1 ¯σ

Ubiquitous weakening of faults due to thermal pressurization

Abstract and Figures