shear zone broadening controlled by thermal pressurization and poroelastic effects during model...

Journal of Geophysical Research: Solid Earth

Shear zone broadening controlled by thermal pressurizationand poroelastic effects during model earthquakes

J. Chen1 and A. W. Rempel1

1Department of Geological Sciences, University of Oregon, Eugene, Oregon, USA

Abstract As a result of the comminution that takes place over numerous earthquake cycles, maturefaults are characterized by thick layers of pulverized gouge with finite porosity that is saturated with waterat seismogenic depths. The heat generated during earthquakes raises the gouge temperature and thermalexpansion of the pore fluid, and surrounding solids produce elevated pore pressures that cause faultstrength to decrease in the process known as thermal pressurization. Building upon this framework, wedescribe a model that imposes a plane-strain configuration and shows that the stress variations causedby porothermoelasticity promote the Mohr-Coulomb failure of previously undeformed regions. Except inspecial cases where the friction is rate strengthening, we find that the frictional strength must varythroughout the post-failure region, which we identify in our model with the shear zone. We introduce astrain rate function that describes the overall influence of distributed slip on energy dissipation and faultstrength as the shear zone thickness expands. Using typical fault parameters at 7 km depth, the shear zonereaches several millimeters of thickness after 1 s sliding at an overall rate of 1 m/s. The expansion of theshear zone limits the temperature rise to several hundred degrees Celsius, and the average fault strengthfalls to about a tenth of the static frictional strength.

1. Introduction

Mature faults are characterized by thick layers of pulverized gouge that result from the comminution thattakes place over numerous earthquake cycles. There is abundant evidence, however, that during a singleearthquake the active shear zone typically occupies only a small fraction of the total gouge thickness. Boththe physical interactions that determine earthquake behavior and their geologic signatures recorded in faultrocks depend on shear zone width. For example, the rate with which coseismic heating is able to cause thepore fluid pressure to rise and lower fault strength during the process of thermal pressurization depends onthe width of the zone that accommodates shear [Sibson, 1973; Lachenbruch, 1980; Rice, 2006; Rempel andRice, 2006; Brantut et al., 2010; Ujiie and Tsutsumi, 2010; Garagash and Germanovich, 2012]. Similarly, shearwidth exerts a strong control on the maximum temperature rise, and hence the potential for melting faultconstituents and producing the records of past earthquake activity known as pseudotachylytes [Cowan, 1999;Otsuki et al., 2003; Spray, 2010]. In addition, the high-speed frictional behavior referred to as “flash weakening,”along with any other velocity-dependent mechanism for changing the friction coefficient itself, shouldgenerally be expected to depend on the total thickness over which shear strain is distributed during the life-time of contacts [Brown and Fialko, 2012; Chen and Rempel, 2014; Di Toro et al., 2004; Goldsby and Tullis, 2011;Han et al., 2011; Kohli et al., 2011; Rempel, 2006; Rempel and Weaver, 2008; Rice, 2006]. The process by whichstatic regions begin to deform, granules accelerate, and the shear width evolves demands further examinationin light of its importance for fault zone behavior.

Tremendous progress has been made in developing numerical models to track the evolution of fault strength,particularly, as it responds to the effects of thermal pressurization and flash heating. Most treatments setthe shear width as a fixed parameter (sometimes zero) without attempting to treat the physical interactionsthat control it. Intuition suggests that mechanisms causing increased weakening at higher slip rates shouldfavor thinner shear zones, whereas rate strengthening is expected to result in thicker shear zones that couldallow the relative velocities between shear zone particles to remain lower. This tendency is consistent withthe results of a recent detailed analysis that shows how stabilization by frictional rate strengthening anddilatancy competes with the localizing influence of thermal pressurization in model predictions of shear zonewidth [Rice et al., 2014; Platt et al., 2014]. When dilatancy is neglected and rate-weakening or rate-independent

RESEARCH ARTICLE10.1002/2014JB011641

Key Points:• Porothermoelasticity causes

mechanical instabilityduring earthquake

• Unstable region identified as shearzone expands

• Broader shear zone limits thetemperature and reduces the strength

Correspondence to:J. Chen,[email protected]

Citation:Chen, J., and A. W. Rempel (2015),Shear zone broadening controlledby thermal pressurization andporoelastic effects during modelearthquakes, J. Geophys. Res. SolidEarth, 120, doi:10.1002/2014JB011641.

Received 30 SEP 2014

Accepted 4 JUN 2015

Accepted article online 9 JUN 2015

©2015. American Geophysical Union.All Rights Reserved.

CHEN AND REMPEL SHEAR ZONE BROADENING 1

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Journal of Geophysical Research: Solid Earth 10.1002/2014JB011641

friction combine with thermal pressurization to control the shear zone strength according to the Terzaghieffective stress rule [Terzaghi, 1936], the only configuration of a continuously deforming shear zone that isstable to infinitesimal perturbations involves slip on a mathematical plane. Nevertheless, as discussed fur-ther below [see also Rice, 2006, Appendix A], the evolving stress state produced by porothermoelastic effectsleads to the development of a broadening zone in which increases in pore pressure and temperature causethe shear stress to exceed the local frictional strength. This suggests an apparent contradiction in that thezone poised for shear failure (i.e., bounded by the locations where shear stress matches frictional strength)becomes progressively wider even as the width predicted from a continuum treatment of the momentum bal-ance for the shear zone itself localizes dramatically. Here we present an alternative formulation that attemptsto address this discrepancy by relaxing the assumption that the entire shear zone deforms continuously.Instead, we consider the net effect of multiple transient localized slip events within a broadening shear zonewhose boundaries are defined by balancing the gouge frictional strength and the shear stress accordingto the Mohr-Coulomb criterion. Because our continuum analysis does not constrain precisely how strain isaccommodated within this growing frictionally unstable zone, we prescribe candidate uniform and Gaussiandistributions that satisfy the overall kinematic constraint and examine the effect of these choices on fault zoneevolution. Our results highlight the need to account for the discrete nature of gouge particle interactionswithin future simulations that are capable of generating rigorous predictions for the distribution of strain asslip progresses.

2. Background

Before delving into the analysis in section 3 and discussing the implications of our model results in section 4,we first provide a brief summary of pertinent observations from the field and the laboratory, as well as someimportant insights into the controls on the instantaneous shear zone width that emerge from force balanceconsiderations.

2.1. Field ObservationsThere is abundant field evidence to suggest that the relatively thick (i.e., meter-scale) gouge layers that char-acterize mature faults are in fact composed of complex internal structures, including numerous fine-scalefeatures that appear to record deformation that occurred during different seismic events. For example, obser-vations made along the San Gabriel and Punchbowl faults in Southern California show that decimeter-scalelayers of ultracataclasite are hosted within foliated cataclasite that extends to several meters in thickness[Chester et al., 1993; Chester and Chester, 1998; Chester et al., 2004]. Detailed analysis of ultracataclasite samplesfrom the San Gabriel fault suggests that extreme coseismic shear occurred within still narrower zones that areless than 5 mm thick. Further work along the Punchbowl fault by Chester et al. [2003] revealed that, within a∼1 mm zone of comminuted gouge, there is an internal structure to indicate that much of the shearing actu-ally took place in a layer of approximately 100–300μm thickness. Lockner et al. [2000] and Otsuki et al. [2003]studied the Nojima fault in Japan which ruptured in the 1995 Kobe earthquake and found millimeter-scalelayers of very fine gouge (with particle diameters <30 μm), alongside several millimeter-scale bands of pseu-dotachylyte. The microscopic structure of the 1999 Chi-Chi earthquake fault zone shows an approximately3 mm thick primary shear zone, characterized by a foliated gouge displaying an alternation of clay-rich andclast-rich layers [Boullier et al., 2009].

In the narrow shear zones described above, the large strains implied by typical seismic slips should have beenaccompanied by elevated pore pressures as a result of thermal pressurization. Indeed, Noda and Shimamoto[2005] measured the permeability of fault rocks from the active Hanaore fault near Kyoto, Japan, and con-cluded that the extremely low values (<10−17 m2), especially in the clayey foliated fault gouge and a finer zonewhich they referred to as “black clayey gouge,” enabled thermal pressurization to reduce the heat dissipationenough to prevent coseismic melting. However, as pointed out by Rice [2006], coseismic damage can dis-rupt pore-scale microstructure so that both heat and fluid escape from sufficiently thin shear zones becomeimportant in modifying the strength evolution during earthquakes. Temperature and pore pressure changescan activate other mechanisms, such as those described in section 3 below, which might offset the localizingeffects of thermal pressurization and lead to shear zone broadening.

Field evidence has not placed firm constraints on the potential for coseismic shear zone changes between thenarrowest foliations and the much broader surrounding gouge zones. However, there are suggestions that thedeforming zone might migrate during an earthquake, resulting in a network of shear zones that link multiple



foliated structures [Ben-Zion and Sammis, 2003; Schrank et al., 2008]. Exposures exhumed from seismogenicdepths have revealed much about the structure of mature fault zones that have experienced numerous earth-quakes, but ambiguities remain concerning the role of such factors as depth, lithology, poroelastic properties,and slip velocity in controlling the width of the shear zone that actively deforms during a single event, letalone the evolution of that shear zone width over an earthquake’s duration. Nevertheless, the preponderanceof evidence appears to indicate that active shear typically takes place in a zone that is both much narrowerthan the total gouge width and much broader than the median diameter of gouge particles.

2.2. Laboratory ExperimentsA more finely resolved temporal window into shear zone behavior is provided by laboratory observations.Localized fabrics that include the development of amorphous materials have been produced by laboratoryexperiments conducted at very low sliding rates (several microns per second) over short distances (usually lessthan several tens of centimeters, [see, e.g., Friedman et al., 1974; Yund et al., 1990]. More recently, evidence forshear localization has also been documented using rotary shear devices that achieve slip rates approachingtypical coseismic speeds (of order 1 m/s), often starting with intact rock samples, but developing gouge layersas products of frictional wear [see, e.g., Tsutsumi and Shimamoto, 1997; Brantut et al., 2008; Kitajima et al., 2011;Smith et al., 2012, 2015].

Mair and Marone [2000] conducted experiments to investigate shear heating in granular layers. Their resultswere consistent with the expectation that rate strengthening is associated with distributed shear, while rateweakening is associated with localized deformation. These experiments were later complemented by 3-Dnumerical simulations of fault gouge evolution [Mair and Abe, 2008; Abe and Mair, 2009] that suggested thatstrain localization may have been promoted by grain size reduction and development of a structural fabric.Han et al. [2011] observed shear localized in a ∼2 μm thick zone in granular periclase samples. Other exper-iments [Kuwano and Hatano, 2011; Sone and Shimamoto, 2009] suggest that most strain during high-speedslip accumulates in narrow shear zones of several tens to hundreds of microns in width, which is roughly anorder of magnitude greater than the characteristic particle size [see also Morgenstern and Tchalenko, 1967;Muir Wood, 2002].

The confining stresses in experiments are typically much lower than those present at seismogenic depths.This suggests the possibility that inertia may be much more important in laboratory settings [Platt et al., 2014]and cause the shear zone width to evolve more slowly than it does along faults during earthquakes. Neverthe-less, over a broad range of reported conditions, observed shear zone thicknesses typically exceed a minimumwidth in the range of 10–20 particle diameters.

2.3. Force Balance ConsiderationsMost continuum treatments contain the implicit assumption that strain is distributed throughout the entireshear zone at each instant in time. Since inertial effects are expected to be small, the momentum balance con-dition implies that forces must balance so the dynamic frictional strength is spatially uniform at each instantwithin such continuously deforming shear zones. Although the model described below in section 3 relaxes theassumption that the entire shear zone deforms continuously, an analysis of the stability of uniform, undrained,adiabatic shear can still be used to gain some insight into the controls on the preferred thickness w of theactively shearing region at any particular instant and how it depends on thermal and hydraulic properties,including the volumetric heat capacity 𝜌C, and the diffusivities for temperature 𝛼th and pore pressure 𝛼hy (seeFigure 1). Applying the Terzaghi effective stress rule [Terzaghi, 1936], the strength 𝜏 is expressed as the prod-uct of an effective friction coefficient f with the effective stress 𝜎n − p. Following a common approximationto the energy balance, we assume that all of the work performed during shear at strain rate �� is converted toheat at rate 𝜏�� = f

(𝜎n − p

)�� . Treating the total normal stress𝜎n as constant, then with all other material prop-

erties fixed, a drop in shear thickness w increases the strain rate �� , thereby raising the temperature faster toenhance the pressurization of pore fluid and weaken the fault. This localizing influence of thermal pressuriza-tion can be counteracted by rate-strengthening frictional behavior. For rate-weakening or rate-independentfriction, this idealized treatment predicts that localization decreases the shear thickness to zero so that sliptakes place on a mathematical plane.

A modest extension of the linear stability analysis in previous works [Rice et al., 2005a; Platt et al., 2010; Brantutand Sulem, 2012; Platt et al., 2014; Rice et al., 2014], for shear in a finite porothermoelastic layer for the case



Figure 1. A schematic of the model domain, with the macroscopic strain rate illustrated using a Gaussian distribution.

with rate-strengthening and temperature-dependent friction predicts that the maximum thickness of a layerundergoing uniform shear that is stable to infinitesimal plane wave perturbations can be expressed as

wcr ≈4𝜋2𝜌C(𝛼hy + 𝛼th)

fΛ[2V

(𝜕f∕𝜕V

)+ f

]− 𝜏

(𝜕f∕𝜕T

) 𝜕f𝜕V, (1)

where Λ =(𝜆f − 𝜆n

)∕(𝛽f + 𝛽n), and 𝜆 and 𝛽 are the expansivities and compressibilities of the fluid phase

(subscript f ), and pore volume (subscript n). Equation (1) gives a useful estimate of the shear zone width wwith uniform strain rate �� = V∕w, when the effective friction coefficient f is assumed to depend only on thetemperature T and slip rate V . (Noting that only the overall kinematic slip rate V is well constrained in mostpublished frictional data, we assume here that this is correlated with the local slip rate along particle surfacesthat actually influence the value of f .) Increases in temperature T are accompanied by increases in the porepressure p (i.e., reduced effective stress) because the thermal expansivity of water 𝜆f is greater than typicalvalues of 𝜆n, which accounts for the changes in void space that accompany the thermal expansion of gougeminerals. The greater the expansion factor Λ, the stronger the localizing influence of thermal pressurizationand the smaller the predicted shear width wcr. By contrast, the greater the combined diffusivity 𝛼hy + 𝛼th, themore rapidly thermal and pore pressure anomalies expand outward to reduce the strength further from thecentral plane and facilitate larger shear widths.

As shear progresses and changes in fault zone conditions cause parameters such as the hydraulic diffusivity𝛼hy and the expansion factor Λ to evolve, equation (1) can provide some intuition for the expected responseof shear width. Figure 2 illustrates how wcr depends on the hydraulic diffusivity 𝛼hy, which is expected toincrease as effective stress drops and granules separate over the course of an earthquake. Each curve is cal-culated using the different values of 𝜕f∕𝜕V and Λ that are noted on the plot. The figure suggests that boththe decreases in Λ and increases in 𝛼hy that are expected to result from coseismic changes in fault properties[e.g., Rempel, 2006] should produce increases in wcr for any fixed value of V

(𝜕f∕𝜕V

). It should be empha-

sized, however, that the results of this linear stability analysis are only suggestive rather than predictive, sincethe formulation is in reference to a base case with negligible flow of heat and mass, and nonlinear effects areneglected. Nevertheless, Platt et al. [2014] demonstrated that the shear zone width predicted by a nonlineartreatment that accounts for heat and mass transport (though without accounting for the coseismic evolutionof Λ or 𝛼hy) gives very similar results. For the entire range of hydraulic diffusivities plotted, even with the mostdramatic frictional strengthening considered in Figure 2 and the low value of Λ = 0.1 MPa/K chosen to corre-spond with the expansion coefficient at elevated temperatures and pore pressures, the predicted shear widthreaches a maximum of several centimeters.

While the thicknesses predicted by this simple analysis are consistent with much of the field and laboratoryevidences summarized above [see also Rice et al., 2014], there are other potential mechanisms for control-ling shear zone width that can operate without requiring rate-strengthening friction. In the next section, weconsider the mechanical stability of the gouge and incorporate the porothermoelastic constraints on the evo-lution of shear stress to show that the Mohr-Coulomb criteria predicts that the shear strength is exceededin a broadening zone, referred to below as the Mohr-Coulomb zone, that can become much wider than theshear thickness predicted from the linear stability analysis leading to equation (1). This leads us to propose analternative model for shear zone behavior that is characterized by transient, localized slip that can occur overa finite shear width even when the friction is rate weakening or velocity neutral.



Figure 2. Predicted shear zone thickness wcr from the linear stability analysis, shown as a function of the hydraulicdiffusivity 𝛼hy and expansion factor Λ for different degrees of rate strengthening.

3. Dynamic Shearing Model

Linear stability analysis predicts that more pronounced rate-strengthening friction favors broader shear zonewidths wcr that lower strain rate for a given total slip rate. However, there are problems with this as a generalexplanation for the controls on shear zone width throughout the seismogenic zone. Earthquake nucle-ation requires rate weakening [e.g., Chester and Chester, 1998; Scholz, 1988; Dieterich, 1994; Garagash andGermanovich, 2012; Viesca and Rice, 2012], whereas the simple theory predicts that only planar slip is stablein this case. Many exhumed faults appear to have finite shear zone thicknesses that are more than an orderof magnitude larger than the median grain diameter; the stability analysis suggests that rate strengthen-ing, or some other mechanism must be invoked for significant coseismic shear in such zones to accumulate.It is certainly true that slip can nucleate in rate-weakening regions and migrate to rate-strengtheningregions. However, even when this does not occur, increases in fault temperature can cause an evolution torate-strengthening friction [e.g., Blanpied et al., 1995], and there are several other independent indications thata transition may take place from pronounced rate-weakening to rate-strengthening friction at coseismic sliprates. Such behavior emerges from treatments of progressive flash heating by the development and growthof thermally activated weak layers along contact surfaces [Rempel and Weaver, 2008; Chen and Rempel, 2014],particle dynamics models for shear at low effective stresses [Kuwano and Hatano, 2011], and observed changesin resistance at large slip distances as melting conditions are approached in high-velocity rock friction exper-iments [Spray, 1987; Tsutsumi and Shimamoto, 1997; Hirose and Shimamoto, 2005; Brown and Fialko, 2012].However, momentum balance considerations are difficult to satisfy within a finite region that experiences atransition from rate weakening to rate strengthening, as the following simple argument shows.

3.1. Momentum BalanceAs long as the Terzaghi effective stress rule holds, the momentum balance requires that, for regions whereinertia is negligible, the shear stress must satisfy

𝜕

𝜕y[f (𝜎n − p)] = 0. (2)

If we assume a constitutive relationship for friction where f depends only on the temperature T and thestrain rate �� and assume a differentiable ��(y), then with 𝜎n spatially uniform, the equation above can beexpanded to

𝜕f𝜕T

𝜕T𝜕y

+ 𝜕f𝜕��

𝜕��

𝜕y= f𝜎n − p

𝜕p𝜕y. (3)

Inconsistencies arise if we attempt to solve �� from equation (3) using a generic frictional constitutive law witha transition from rate weakening at low strain rates to rate strengthening at a threshold in �� [e.g., Kuwano andHatano, 2011; Chen and Rempel, 2014] (similar arguments hold when the rheological change coincides with



a temperature threshold). If we consider y = 0 as the midplane of shearing, then for y> 0 we expect for oursystem that 𝜕T∕𝜕y < 0, 𝜕p∕𝜕y < 0, and as long as the friction f is temperature weakening (or only weaklytemperature strengthening), this implies that

𝜕f𝜕��

𝜕��

𝜕y< 0.

If the shear zone is in a rate-strengthening regime, 𝜕f∕𝜕�� > 0 so we must have 𝜕��∕𝜕y < 0. This implies that�� decreases with |y| until reaching a distance yL, beyond which it is small enough that the friction is rateweakening. If deformation were allowed beyond |y|> yL, the momentum balance could only be satisfied with𝜕��∕𝜕y> 0, but any increase in �� would imply a return to rate strengthening once more; hence, the assumedrheology is consistent only with shear thicknesses bounded by yL. We deduce that, even if the frictional behav-ior transitions from rate weakening to rate strengthening beyond a threshold strain rate, strain can only beaccommodated within a finite region when the strain rate is fast enough to cause rate strengthening, with animplied discontinuous drop to zero strain rate at the shear zone boundary; this is compatible with the resultsof the linear stability analysis discussed in section 2.3 above, which implies that uniform, undrained, adiabaticdeformation can be stable only when it is accommodated within a width less than wcr. In the limit where fis rate neutral (and 𝜕f∕𝜕T < 0), yL shrinks to zero since there is no way to satisfy equation (3) in a zone offinite thickness with the expected direction of pore pressure and temperature gradients. Under typical seis-mogenic conditions, accounting for acceleration does not alter the behavior substantially since inertial effectsare generally quite small [Platt et al., 2014].

These arguments suggest that in cases where the fault constituents are not uniformly velocity strengthening,the thickness of the region that shears at any particular time must be extremely narrow, tending toward amathematical plane. The field and laboratory observations discussed in section 2 support the common exis-tence of very thin shear zones that nevertheless span finite widths that exceed typical particle diameters byan order of magnitude or more. It is reasonable to question how this behavior can be reconciled with themomentum balance constraint when the frictional rheology is rate weakening or rate neutral. To this end,we note that in the parameter regime of interest where inertia is unimportant, the momentum balance leadsto a constraint only on the instantaneous thickness of the shearing region. The potential exists for shear tobe distributed over a broader zone that hosts transient slip in highly localized patches with the shear stressthroughout given by the frictional strength according to the Terzaghi effective stress rule applied on the activesliding surface. Together, these observations motivate further investigation into controls on shear zone widththat are suggested by porothermoelastic effects.

3.2. Conservation of Fluid and HeatWhen the shear zone is deforming, frictional heat causes the pore fluid inside the gouge to expand, resultingin an outward fluid flow qf as illustrated in Figure 1. Ignoring dehydration reactions for simplicity, changes inthe fluid mass inside the shear zone must be compensated by outward flow, so that continuity requires

𝜕(𝜌f n)𝜕t

+𝜕qf

𝜕y= 0, qf = −

𝜌f k

𝜂f

𝜕p𝜕y, (4)

where 𝜌f is the density of pore fluid, n is the porosity, and the outward fluid flow qf follows Darcy’s law withpermeability k, viscosity 𝜂f , and pore pressure p(y, t). If we assume that n only changes elastically and furthersimplify the model by taking the fluid density 𝜌f and fluid diffusivity 𝛼hy = k∕[𝜂fn(𝛽f +𝛽n)] as spatially uniform,equation (4) can be written as

𝜕p𝜕t

− Λ𝜕T𝜕t

= 𝛼hy𝜕2p𝜕y2

, (5)

whereΛ = (𝜆f −𝜆n)∕(𝛽f +𝛽n). For clarity of presentation we treatΛ and 𝛼hy as constant throughout the modeldomain. However, we note that the effects of damage can cause the drained compressibility and permeabilityto increase within the shear zone [e.g., Rice, 2006]. We reserve further discussion of such complications untillater, after first presenting the remaining essential components of our model.

For simplicity, we assume uniform thermal properties in both the shear zone and the ambient gouge. Weconsider a shear zone with width w that starts to deform with strain rate �� and overall slip rate V at t = 0. Theconservation of energy requires that

𝜏�� = 𝜌C𝜕T𝜕t

+𝜕qh

𝜕y, qh = −K

𝜕T𝜕y, (6)



Table 1. Variation of Volume of the Porous Material and ItsConstituents for the Three Independent ComponentsIllustrated in Figure 3a

ΔV of Constituents

Problem Fluid Phase Solid Phase

(a) −nV0𝛽fΔp −(1 − n)V0𝛽sΔp

(b) −V0𝛽s(Δ𝜎c − Δp)(c) nV0𝜆fΔT (1 − n)V0𝜆sΔT

aThe three components are (a) hydrostatic loading of satu-rated material, (b) effective stress loading of drained material,and (c) thermal heating of saturated material.

where the heat flow qh depends on the thermalconductivity K , which we treat as spatially uni-form. Defining the thermal diffusivity as 𝛼th ≡K∕(𝜌C), and substituting it for 𝜏 using the Terza-ghi effective stress rule, the temperature evolu-tion within the shear zone is

𝜕T𝜕t

= 𝛼th𝜕2T𝜕y2

+ f ��𝜌C

(𝜎n − p), |y| ≤ w∕2, (7)

while outside the shear zone, since there is noheat source, the temperature evolution is

𝜕T𝜕t


, |y|>w∕2. (8)

For compatibility with the specified kinematic condition, we also require that the strain rate satisfy

∫w∕2

−w∕2�� dy = V. (9)

3.3. Poroelastic Effective Stress VariationWe can view shear zone broadening as a process of mobilization in which the deformation starts (i.e., theshear zone boundary propagates) when a mechanical threshold is reached. Macroscopically, the shear zoneundergoes large finite deformation, but along its boundary while the strain is still small and reversible, wehypothesize that the essential features of mobilization can be determined from a poroelastic analysis. Theonset of Mohr-Coulomb failure is the focus of this analysis. We next review how changes in pore pressure andtemperature can cause the effective stress in the direction of shear to evolve more quickly than the effectivestress normal to the shear plane, following the line of reasoning presented earlier by Rice [2006, Appendix A].This difference will be shown in the next subsection to lead to a criterion for potential changes in shearzone width.

If the drained compressibility is 𝛽d and the solid grain compressibility is 𝛽s, for a control volume V0 with poros-ity n, variations in the reference confining stress Δ𝜎c = 𝜎c − 𝜎c,o, the pore pressure Δp = p − po and thetemperature ΔT = T − To produce a change in total volume that can be expressed as

ΔV∕V0 = 𝜀kk = −𝛽dΔ𝜎c + (𝛽d − 𝛽s)Δp + 𝜆sΔT . (10)

Here the first term on the right is the drained volume change due to the confining stress variation, the secondterm represents the effects of pressure changes within the pore fluid, and the last term accounts for thermalexpansion. When there is a deviation from the reference state, the coupled thermomechanical loading ofporous material can be decomposed into three independent problems. As noted in Table 1 and illustrated inFigure 3, the overall forcing is equivalent to (a) hydrostatic loading of saturated material, (b) loading of drainedmaterial by changes in effective stress, and (c) thermal loading of drained material. With these considerations,the solid volume variation is [Ghabezloo and Sulem, 2009]

ΔVs∕V0 = −(1 − n)𝛽sΔp − 𝛽s(Δ𝜎c − Δp) + (1 − n)𝜆sΔT . (11)

Figure 3. Thermomechanical loading of porous material can be decomposed into three independent components, asdescribed in the text.



The change in porosity n can be written as the difference between the total volume change and the volumechange of the solid phase alone, so that

Δn =ΔV − ΔVs

V0

= − (𝛽d − 𝛽s)Δ𝜎c + [𝛽d − (1 + n)𝛽s]Δp + 𝜆snΔT .(12)

At fixed confining stress 𝜎c, equation (12) implies that the elastic change in n is

dn = n(𝛽ndp + 𝜆ndT) (13)

where 𝛽n = (𝛽d − 𝛽s)∕n − 𝛽s.

The stress-strain relation can be written in terms of the deviatoric stress as

𝜎ij − 𝜎ij,o − (𝜎kk − 𝜎kk,o)𝛿ij∕3 = 2𝜇(𝜀ij − 𝜀kk𝛿ij∕3). (14)

where 𝜇 is the shear modulus. Following Cocco and Rice [2002] and Rice [2006], the shear zone behavioris expected to be well approximated as uniaxial strain in the y direction. Setting 𝜀kk = 𝜀yy and identifying𝜎n = −𝜎yy , equation (14) can be used to show that

− (𝜎n − 𝜎n,o) − (𝜎kk − 𝜎kk,o)∕3 = 4𝜇𝜀yy∕3, (15)

and 𝜎xx − 𝜎xx,o − (𝜎kk − 𝜎kk,o)∕3 = −2𝜇𝜀yy∕3, (16)

which can be combined to write𝜎xx − 𝜎xx,o = −(𝜎n − 𝜎n,o) − 2𝜇𝜀yy . (17)

We interpret the confining stress 𝜎c as −𝜎kk∕3 and use the expression for 𝜀kk in terms of 𝛽d and 𝛽s fromequation (10) to write

𝜎xx − 𝜎xx,o + (1 − 𝛽s∕𝛽d)Δp

= −𝜈d

1 − 𝜈d[Δ𝜎n − (1 − 𝛽s∕𝛽d)Δp] −

2(1 + 𝜈d)3(1 − 𝜈d)

𝜇𝜆sΔT .(18)

where the Poisson’s ratio under drained conditions 𝜈d follows

𝜈d =3 − 2𝜇𝛽d

6 + 2𝜇𝛽d. (19)

Setting Δ𝜎n = 𝜎n − 𝜎n,o = 0 and assuming an initial isotropic stress state with 𝜎xx,o ≈ −𝜎n,o, the effectivecompressive stress in the deformation plane is

𝜎xx = −𝜎xx − p = (𝜎n − p) +1 − 2𝜈d

1 − 𝜈d(1 − 𝛽s∕𝛽d)Δp +

2(1 + 𝜈d)3(1 − 𝜈d)

𝜇𝜆sΔT , (20)

Alongside the effective stress components 𝜎xx and 𝜎yy = −𝜎yy − p ≈ 𝜎n − p, the shear stress is 𝜏xy = f 𝜎yy .

A simple numerical illustration is useful for showing how changes in the effective stress components arise.Following Rice [2006] by taking 𝛽s = 1.6×10−11 Pa−1,𝜆s = 2.4×10−5 K−1, 𝛽d = 5.8×10−11 Pa−1, and𝜇 = 13 GPa,the effective stress in the deformation plane satisfies

− 𝜎xx − p ≈ (𝜎n − p) + 0.55Δp + 0.31ΔT MPa/K. (21)

Equation (21) clearly predicts that pore pressure increases Δp and temperature increases ΔT cause the effec-tive stress parallel to the shearing direction 𝜎xx ≡ −𝜎xx −p to undergo larger changes than the effective stress𝜎yy ≡ 𝜎n − p in the direction normal to shear. The Mohr-Coulomb criterion will next be used to examine howthe resulting changes to the stress state can determine whether a region is poised to become mobilized orrequired to stay substantially undeformed.



Figure 4. Mohr circle of the stress state showing the failure criterion in solid red. The failure envelope with frictionf passes through the origin as the cohesion is taken to be zero within the comminuted gouge. The blue dashed circleis a stress state that will not result in failure, the red circle is a stress state of failure onset, and the red dashed circleindicates where the Mohr circle would be if following the trend. Both effective principal stresses are decreasing withrising pore pressure but at different rates, ��11 decreases more slowly than ��33 due to the poroelastic effect. As a result,the Mohr circle becomes enlarged and moves close to the failure envelope.

3.4. Mohr-Coulomb CriterionThe stress state described above implies that the principal stresses can be written as

𝜎11 = 𝜎xx cos2 𝜓 + 2𝜏xy sin𝜓 cos𝜓 + 𝜎yy sin2 𝜓 (22)

𝜎33 = 𝜎xx sin2 𝜓 − 2𝜏xy sin𝜓 cos𝜓 + 𝜎yy cos2 𝜓, (23)

where the rotation angle 𝜓 is determined by setting 𝜏13 = 0, which gives tan(2𝜓) = 2𝜏xy∕(��xx − ��yy). Treat-ing the cohesion as zero within the highly comminuted gouge for simplicity, the Mohr-Coulomb criterionillustrated in Figure 4 is satisfied when

𝜎11 + 𝜎33

2sin 𝜃 =

|𝜎11 − 𝜎33|2

, (24)

where tan 𝜃 = f . Equation (21) is of the form 𝜎xx = 𝜎yy + AΔp + BΔT , which motivates the definition ofr ≡ 𝜎xx∕𝜎yy − 1 to gauge the evolution of the stress state as changes take place to the local pore pressureand temperature. Substituting equations (22) and (23) for the principal stresses into equation (24) and per-forming a sequence of manipulations, it can be shown that the Mohr-Coulomb criterion is satisfied at rc = 2f 2

so that tan (2𝜓) = f−1, where we associate the friction coefficient f = f0 with its static value prior to anyof the weakening that would commonly be associated with high-speed slip. Previously unmobilized gougeis incorporated within an expanding zone that is poised for shear failure once r reaches rc on its bound-ary and the Mohr circle intersects with the failure envelope (Figure 4). This boundary can be tracked as thelocation where

𝜏 = f (𝜎n − p)

=1 − 2𝜈d

1 − 𝜈d(1 − 𝛽s∕𝛽d)

Δp2f

+2(1 + 𝜈d)3(1 − 𝜈d)

𝜇𝜆sΔT2f.

(25)

In Figure 5 we illustrate how the shear resistance 𝜏 = f (𝜎n − p) and the driving force from poroelastic effectsdetermine the mobilization or Mohr-Coulomb boundary. Thermal pressurization increases the pore pressure,and hydraulic diffusion controls its decrease toward ambient levels far from the shear zone, leading to corre-sponding decreases to the effective stress and hence the frictional strength. At the same time, pressure andtemperature increases enhance the driving force for mobilization described by the right side of equation (25)as the symmetry plane is approached. Inside the Mohr-Coulomb boundary, the driving force is greater thanthe frictional strength that opposes shear, so the granules throughout this region are poised for mobilization.



Figure 5. A schematic view of the balances that determine the location of the Mohr-Coulomb boundary. The solidcurves represent the driving force resulting from poroelastic effects when the fault is damaged (red) and undamaged(black); this is defined by the right side of equation (25) as a linear combination of pore pressure and temperaturechanges, which decay with distance from the symmetry plane. The blue dashed line is the resisting frictional strength,which increases as the pore pressure drops with distance from the symmetry plane. The intersection described byequation (25) is the boundary inside which Mohr-Coulomb failure will develop and material is poised to shear. Alsoshown is the overall shear strength 𝜏 that is defined by an average over time so that it lies between the minimumstrength along the symmetry plane and the higher values along the Mohr-Coulomb boundary.

3.5. Distributed ShearWe have shown that the Mohr-Coulomb boundary defines the region with the potential for failure. However,with velocity-weakening or rate neutral friction, the shear stress 𝜏 cannot be equated with the spatially varyingfrictional strength f (y, t)[𝜎n −p(y, t)]while simultaneously satisfying the momentum balance throughout thisregion. We note that Figure 5 suggests that the effective stress and hence the frictional strength might benearly constant within a much thinner zone where the pore pressure gradient is negligible. However, betweenthis thin zone and the Mohr-Coulomb boundary, there is a significant gradient in frictional strength.

Since all potential contact surfaces inside the Mohr-Coulomb boundary are post-failure, in principle, slip couldoccur anywhere within this region. To avoid violating local momentum balance constraints during sliding withthe shear stress equal to the frictional strength, we suggest that shearing might be considered to take placesequentially along numerous local shear zones that are thin, possibly planar, and transient. If each local shearzone is sufficiently thin, the pore pressure variations in the region that slides during any particular instantare negligible so that fault-parallel frictional forces are balanced. No continuum modeling approach can cap-ture all of the detailed mechanical interactions that take place at the granular scale. However, we can definea strain-rate function that is designed to approximate the overall shearing behavior at the scale over whicha continuum modeling approach is appropriate. Evidence that finite shear zones along natural faults do notdeform uniformly throughout a particular slip event comes, for example, from detailed observations of thenarrow bands gently inclined to the strike of the main fault, known as Riedel shears [Logan and Teufel, 1986].Laboratory observations of the force chains that arise in a shear zone full of granules [e.g., Daniels and Hayman,2008] also exhibit notable transient local behavior, including detectable variations in tangential stress. Weexpect finite microstructures such as these to prohibit localization to the idealized mathematical plane pre-dicted from linear-stability considerations in a continuum treatment. Moreover, we suggest that observedand inferred force fluctuations on a granular scale can cause the locus of instantaneous shear to move withina broader zone to produce a macroscopic shear width. Indeed, when the Terzaghi effective stress rule holds,the shear stress at any instant is expected to be set by the frictional strength along the actively sliding sur-face, while granular contacts are stressed to that same level elsewhere, but remain static (assisted in weakerregions by the geometrical frustration that causes jamming).

We propose that distributed shear can be viewed as the superposition of numerous transient shearsthroughout the post-failure region defined by the Mohr-Coulomb boundary and referred to below as the



Figure 6. Two candidate strain rate distributions. Uniform shearing is shown in red and a Gaussian distribution is shownin black. For this latter case, the shear core in which most strain is accommodated corresponds with the location wherey = 𝜎, whereas the Mohr-Coulomb boundary is at y = w∕2 = 𝜎L.

Mohr-Coulomb zone. In a continuum treatment, the cumulative effect of this deformation can be approxi-mated by defining a macroscopic strain rate distribution. We examine two candidate distributions that areillustrated in Figure 6: (1) uniform shearing, in which transient shears are equally likely throughout the growingpost-failure region; and (2) Gaussian shearing, in which transient shears occur preferentially in places wherethe effective stress is lowest. For the first case, we approximate a uniform strain rate distribution by setting

��(y,w) = V2w

− V2w

erf[

k𝛾

(2|y|

w− 1

)], (26)

which approaches a step function with large k𝛾 ∼ 100; here the shear zone width w is set by tracking theMohr-Coulomb boundary at y = w∕2. For the numerical illustrations that follow, we assume that the frictioninside this boundary is weakened by flash processes [e.g., Rice, 2006] so that f = fw , while outside the boundarywhere the friction is not weakened we assign f = f0. For the second case, we use a Gaussian distributioncentered on the symmetry plane

��(y,w) ∼ ��0 exp(−

y2

2𝜎2

).

Here we set the standard deviation to𝜎 = w∕(2L)where the parameter L is chosen large enough that the strainrate is negligible outside the Mohr-Coulomb boundary, and �� is chosen to satisfy the kinematic constraintfrom equation (9) so that

��(y,w) =√

2𝜋

LV

werf(L∕√

2)exp

(−

2L2y2

w2

), (27)

where the term erf(L∕√

2) ≈ 1 when L ≥ 2.5.

Outside the Mohr-Coulomb boundary the assigned strain rate distributions ensure that �� ≈ 0, so the frictionalheating term in equation (7) is negligible. For the Gaussian strain rate distribution it is useful to refer to the“shear core” as the region in which most strain is accumulated, which we define here for convenience by|y| < 𝜎. It is possible to define the shear core using other criteria, but we show below that this simple criterionsuffices to ensure that lateral variations in the frictional strength are relatively small inside this zone whereshear is concentrated.

Both equations (26) and (27) satisfy the integral kinematic constraint from equation (9). We have shown thatspatial variations in pore pressure and temperature produce inhomogeneities in the local strength f (𝜎n−p). Toaccount for the net effect of these variations and the distribution of strain within the Mohr-Coulomb bound-ary, it is useful to define an overall coseismic shear strength along the fault as the time-averaged strengthresulting from many transient slips at different locations within the failure region, so that

𝜏 = 1V ∫

∞

−∞f (𝜎n − p)�� dy. (28)

For our prescribed uniform and Gaussian strain rate distributions, �� ≈ 0 for |y|>w∕2, so the overallshear strength 𝜏 corresponds with the equivalent uniform strength that would generate the same amount of



Table 2. Nominal Parameter Values Chosen to Correspond WithTypical Conditions at 7 km Depth, Following Rice [2006] andRempel and Weaver [2008]

Parameters Value

V (m/s) 1

𝜌 (g/cm3) 2.65

C (J/ (kg ∘C)) 732

𝛼th (mm2/s) 2.2

𝛼hy (mm2/s) 44

p0 (MPa) 70

T0 (∘C) 210

𝜎n (MPa) 196

Λ (MPa/∘C) 0.92

𝛽d (Pa−1) 5.8 × 10−11

𝛽s (Pa−1) 1.6 × 10−11

𝜇 (GPa) 13

𝜆s (∘C−1) 2.4 × 10−5

frictional heat, which is the fault strength thatwould be inferred from long-term observationsof the heat anomaly [e.g., Lachenbruch, 1980;Lachenbruch and Sass, 1980].

3.6. Effects of DamageThe essential equations that describe ourmodel have already been presented. However,in choosing representative parameter valuesit is important to note that damage can playa central role in modifying local conditionsduring earthquakes. When a mature fault isslipping, the ultracataclastic core where slipis concentrated can have altered poroelasticproperties compared with the material outsidethe damaged zone [Evans et al., 1997; Wibberleyand Shimamoto, 2003]. New flow passages canform to increase both the permeability andcompressibility as a result of (a) fault dilatancyduring coseismic slip [Marone et al., 1991], (b)breakage of water-swelling clays and alteration

of veins formed during the interseismic period [Chester et al., 1993], (c) porosity increases due to elevatedpore pressure [Seront et al., 1998; Lockner et al., 2000; Wibberley and Shimamoto, 2003; Mizoguchi et al., 2008],and (d) local fracturing and activation of the damage zone along and near the shear zone walls as a resultof large stress fluctuations in the vicinity of the propagating rupture tip [Rice, 1980; Poliakov et al., 2002; Riceet al., 2005b; Faulkner and Armitage, 2013; Candela et al., 2014].

Within a damaged shear zone the drained compressibility is expected to increase and this affects the value of𝛽

dmgn =

(𝛽

dmgd − 𝛽s

)∕n − 𝛽s, whereas we identify 𝜆dmg

n ≈ 𝜆s. We expect the damaged compressibility 𝛽dmgd to

be significantly higher than the compressibility of the surrounding bulk framework 𝛽d [Wibberley, 2002]. Whena contrast in these parameters takes place across the shear zone boundary,Λ and 𝛼hy must also change. How-ever, numerical tests using our nominal conditions demonstrate that expected contrasts in these parametershave a limited impact on the pore pressure evolution described by equation (5). For simplicity, we reservefurther exploration of the potential effects produced by abrupt changes in Λ and 𝛼hy for future work.

In the damaged region, the effective compressive stress takes the same form as equation (20), but 𝛽d and𝜈d are replaced by 𝛽dmg

d and 𝜈dmgd , defined as above with equation (19). Here we follow Rice [2006] and use

a simplified damage treatment where 𝛽d is replaced by 𝛽dmgd , and we keep all other parameters the same as

those used for equation (21). The value of 𝛽dmgd is assumed greater than 𝛽d , but it must be smaller than 3∕(2𝜇)

to avoid producing a negative Poisson’s ratio. If we use 𝛽dmgd = 1.5𝛽d , the fault-parallel effective stress in the

damaged zone is

− 𝜎dmgxx − p ≈ (𝜎n − p) + 0.74Δp + 0.25ΔTMPa/K. (29)

Comparison with equation (21) reveals that there is an enhanced contribution from pore pressure changes,while temperature changes are less important than when intact gouge parameters are employed. The anal-ysis suggests that damage might produce appreciable changes in constitutive behavior and influence thepropagation of the Mohr-Coulomb boundary. This possibility is explored briefly in section 4 below.

4. Results

Using the nominal, undamaged parameters listed in Table 2, we modeled the temperature and pore pres-sure evolution during coseismic shear, as described by equations (5), (7), and (8), where the Mohr-Coulombboundary is constrained by equation (25). Details of the computational method are provided in Appendix A.Except where specified otherwise, for the illustrations that follow, we initiate fault motion at constant slidingrate V = 1 m/s and impose an initial shear zone thickness of w0 = 0.2 mm, with a uniform weakened fric-tion of fw = 0.2, while f0 = 0.6 along the Mohr-Coulomb boundary. We start with a base case in which any



Figure 7. Evolution of shear zone width with different initial values. As long as w0 is much smaller than the final width,the curves collapse upon each other after an initial transient.

plane within the post-failure region has an equal likelihood of failure. Consequently, the strain rate inside theshear zone is assumed uniform, as approximated by equation (26) with w∕2 assigned to correspond with theMohr-Coulomb boundary.

First, we test the robustness of the model to changes in initial conditions. Although we specify an arbitraryinitial shear zone thickness, Figure 7 illustrates that different initial thickness values only affect the behaviorat early times, and the predicted shear zone thickness at later times is insensitive to w0. At intermediate timesthe thickness is nearly proportional to

√t as a result of the diffusive propagation of changes in pore pres-

sure and temperature. At later times, the thickness evolves more slowly than√

t as pressure and temperatureconditions near the symmetry plane approach steady values.

The temperature and the pore pressure evolution are shown by the snapshots in Figures 8a and 8b. Intersec-tions with the thick black line mark the location and the temperature and pore pressure at the Mohr-Coulomb(and shear zone) boundary at the corresponding times (0.1 s intervals, corresponding with 0.1 m slip incre-ments). Following a brief initial transient, the temperature and pore pressure along the Mohr-Coulombboundary tend to nearly steady values. As the shear zone broadens, heat is dissipated over larger gougevolumes and this has the effect of restricting the maximum rises to the temperature and pore pressure. Overthe 1 m of slip that is represented by the final profiles shown, the pore pressure approaches a significant frac-tion of lithostatic pressure and the rate of frictional heat generation is appreciably reduced as a result of thelowered effective stress. In the end, only a modest temperature rise of less than 350∘C is predicted along themidplane of the shear zone.

In Figures 8c and 8d we show predictions for a case with the same parameter choices, but a different strain ratedistribution that reflects the assumption that the weakest failure planes within the post-failure zone accom-modate a larger proportion of the overall deformation. We model this by choosing a Gaussian strain ratedistribution, given by equation (27), that is focused within a shear core determined by L=5 to be one fifth thewidth w of the post-failure zone. It is clear that along the 1𝜎 boundary, within which strain is concentrated, thepore pressure and, consequently, the frictional strength, is nearly constant. Moreover, because the effectivezone of concentrated shear is much more localized than the entire Mohr-Coulomb width, the temperaturerise is higher than that predicted for the uniform strain rate case.

From equation (25), the friction coefficient plays a central role in controlling the growth of the post-failureMohr-Coulomb zone. Higher friction generates more frictional heat inside the shear zone and leads to largerincreases in pore pressure and temperature. This results in a broader Mohr-Coulomb zone, as demonstrated inFigure 9. For the case with Gaussian shear and L = 5, the Mohr-Coulomb zone thickness is far less sensitive andonly increases slightly with fw . When the strain is more concentrated (higher L), the increase in pore pressureacts to reduce the strength and alleviate the influence of higher fw . As a result, the Mohr-Coulomb thicknessevolves more slowly and stays much thinner than in the case without a shear core (i.e., uniform strain).



Figure 8. Modeled evolution of (a,c) temperature and (b,d) normalized pore pressure evolution during 1 s. Figures 8a and 8b use uniform strain rate distributionapproximated by equation (26), while Figures 8c and 8d use Gaussian shearing with L = 5. The pore pressure perturbation propagates faster than thetemperature perturbation, since 𝛼hy is greater than 𝛼th, as noted in Table 2.

As the shear zone expands, frictional heat is dissipated over a broader region. This causes the maximum tem-perature to be lower than in cases where the shear zone thickness is fixed at its initial level, as illustrated inFigure 10a. When the shear zone thickness is kept at 0.2 mm, the maximum temperature rise is about 760∘C,which is similar with the maximum temperature rise 750∘C for slip on a mathematical plane predicted by Rice[2006], but with a Gaussian strain rate distribution and shear core defined using the nominal value of L = 5,the maximum temperature rise is only about 520∘C. When the strain rate is uniform the temperature risebarely reaches 350∘C. More localized shear modeled by increasing the value of L and thereby reducing thethickness of the shear core causes the maximum temperature rise to increase. From Figure 10b, predictionsfor the evolution of average strength using different strain rate distributions are quite similar. This implies thatthe sensitivity of the maximum temperature to the strain rate distribution mainly results from differences inthe gouge volume over which heat is dissipated.

The hydraulic diffusivity 𝛼hy and thermal diffusivity 𝛼th also have a significant influence on the model behavior.Figures 11a and 11b show the evolution of shear zone thickness, maximum temperature rise, and averageshear strength for uniform shearing. Figures 11c and 11d illustrate the case of Gaussian shear with L = 5. Foreach set of calculations, the thermal diffusivity 𝛼th was kept constant, but we used the different values for theratio 𝛼hy∕𝛼th that are noted in the legend. With higher hydraulic diffusivity the pore pressure perturbationexpands outward more rapidly. This ultimately tends to result in a correspondingly broader shear zone, butthe evolution can be complicated. For example, if 𝛼hy is too high, the pore pressure is slow to build initiallyand the Mohr-Coulomb zone grows comparatively gradually at first. Figures 11a and 11c demonstrate that



Figure 9. Shear zone thickness at 1 s with different friction value with uniform shearing and shear core with L = 5. Asthe friction increases, it becomes easier for the Mohr-Coulomb boundary to expand. It must be noted that with theshear core, the thicknesses of the Mohr-Coulomb zone and the core are much smaller than the cases without the core.

the dependence of w on 𝛼hy can be complicated and nonmonotonic, with higher 𝛼hy favoring more rapidlyevolving shear zone thicknesses at later times, but an intermediate regime in which changes in w with 𝛼hy

can be of either sign. The dependence of temperature and strength on 𝛼hy is more straightforward. Becausehigher 𝛼hy improves shear zone drainage, the fault strength is able to remain comparatively high (the solidcurves in Figures 11b and 11d) and, as a consequence, the maximum temperature rise is increased.

The effects of the alterations to the poroelastic properties produced by damage of the magnitude illustratedby equation (29) cause relatively small perturbations to w in comparison with the effects of changes in 𝛼hy. InFigures 11a and 11c, the Mohr-Coulomb boundaries for a damaged shear zone, shown by the dashed curves,track the undamaged (solid) curves with only small deviations. The temperature and strength evolutionare nearly unchanged from those for the undamaged cases, so for clarity we omit these results fromFigures 11b and 11d. This is not to suggest that damage is unimportant, however. It should be emphasizedthat in addition to changes in poroelastic parameters, increases in hydraulic diffusivity are expected to be akey outcome of damage; Figure 11 demonstrates the importance of the role played by 𝛼hy.

5. Discussion5.1. Shear Zone BehaviorUsing nominal parameter values based upon conditions typical of a seismogenic depth of 7 km, our resultsshow that the post-failure Mohr-Coulomb zone, in which changes in pore pressure and temperature causethe frictional strength to be exceeded by the fault parallel stress, can grow to tens of millimeters in thicknessafter 1 s and 1 m of slip. Depending on how shear is distributed within this zone, the maximum coseismictemperature rise can be less than half that predicted when shear is confined to a constant nominal thickness of0.2 mm. There is field evidence along faults to support the existence of shear zones that span at least this rangeof widths. Indeed, for the Gaussian strain rate distributions we model where shear is concentrated within justa fraction of the entire Mohr-Coulomb zone width w, we anticipate that the inferred shear thickness wouldbe more comparable to the width of the shear core w∕L.

The calculations described above demonstrate that even as the post-failure zone grows, it remains quite thin.However, it is worth emphasizing that, in contrast to the predictions of a simple linear stability analysis, ifshear is able to be distributed throughout the post failure zone, then the overall shear zone thickness is alwaysfinite, even when the frictional behavior is rate weakening or rate neutral. The origin of this difference is rootedin differing assumptions for how strain is distributed through time rather than only through space. Withoutrate-strengthening friction or some other delocalizing mechanism, the variations in pore pressure that driveoutward fluid flow prevent lateral forces from being balanced when shear takes place across a finite thick-ness, throughout which the Terzaghi effective stress principle applies. Our model circumvents this problem by



Figure 10. (a) The temperature rise on the central plane and (b) the average strength for varying shear zone thicknessand fixed shear zone thickness. With a fixed shear zone, the temperature rise will reach about 800∘C, but if the shearzone is expanding with a shear core, the maximum temperature is lowered to about 650∘C. If the strain rate is uniformand the whole Mohr-Coulomb zone is shearing, the temperature rise is even lower. With different degrees of shearconcentration (controlled by different L values), the temperature rise can vary between the case of uniform shearing andfixed shear zone width. The average strength evolution with different degrees of shear concentration are almostindistinguishable, but it is closer to the strength evolution with fixed shear zone width.

postulating that strain is distributed through a finite zone as the superposition of numerous transient shearsthat are not coincident in time but combine to yield an effective strain rate distribution that accommodatesthe overall rate of seismic slip. The suggestion is that particle contacts inside the Mohr-Coulomb boundarycan be temporarily immobilized and that during these static periods the stress is not constrained by theirfailure strength.

The actual strain rate distribution is not constrained within the continuum framework we consider; thetwo cases discussed above are introduced as convenient illustrations to probe the model behavior. Granu-lar mechanics experiments have shown evidence for finite force fluctuations in space and time at contactsbetween particles within shear zones [e.g., Daniels and Hayman, 2008]. We can probe the average ampli-tude of the analogous stress variations in our model to see how this aspect of the predicted behavior isaffected by the simple strain rate distributions we apply. Figure 12 shows how changing the strain rate distri-bution by changing the value of L influences changes in the overall shear zone width w and fault strength. AsL increases so that the strain rate is more localized where strength is lowest, Figure 12a shows that the over-all Mohr-Coulomb width increases slightly more gradually and tends toward a single curve that is relatively



Figure 11. (a) and (c) The curves showing the shear zone thickness increases when the ratio of 𝛼hy∕𝛼th changes from 20 to 100, but as the ratio grows higher, thepore pressure perturbation is not as large for the Mohr-Coulomb criterion to be satisfied, which slows down the thickness growth in the early stage. (b) and (d)The higher 𝛼hy keeps the shear zone more drained, and as long as the heat advection is small (see Appendix B for detailed discussion), the maximum temperatureis higher and the average shear strength is higher. The effect of damage is illustrated as the dashed curves in Figures 11a and 11c, which has only minor influenceon the shear zone evolution.

insensitive to further changes in L. Figure 12b compares the difference between the minimum strength 𝜏|0

(i.e., along the symmetry plane) and the averaged strength 𝜏 obtained from equation (28); also shown arethe predictions for uniform shear, which can be viewed as truncated Gaussian shearing with L = 0. As Lbecomes larger, the difference 𝜏−𝜏|0 becomes smaller, and the strength distribution in the zone where shearis concentrated becomes more uniform. Theoretically, when L→∞, the Gaussian shearing case becomes pla-nar shearing, with 𝜏 = 𝜏|0. In each case shown, after an initial transient the perturbations in shear strengthtend toward nearly steady and relatively small fractional values even as the average fault strength contin-ues to gradually decrease, as shown, for example, in Figure 10b. In fact, the curve for Gaussian shearing withL = 2.5 actually exhibits a decrease in strength perturbation after about 0.6 s; for L = 5 a similar decrease isevident after about 0.7 s. Future modeling work that is designed to explicitly account for the discrete natureof particle-particle interactions should enable predictive models that can test the validity and limitations ofthe strain rate distributions adopted here.

Mature fault zones often display complex structures with evidence for extremely localized principal slip zonesthat are interpreted to have developed over numerous earthquakes, bordered by granular zones with folia-tions and Riedel shears that are inclined to the principal slip direction [e.g., Chester and Logan, 1987]. Afterprincipal slip zones are formed, the proportion of slip that is still nevertheless accommodated in the surround-ing granular gouge is often not well constrained. However, records of the measured force fluctuations over thecourse of discrete granular shear experiments suggest that the distributions of these fluctuations can remainremarkably similar over slip distances that are large in comparison to the lateral granular width [Daniels andHayman, 2008]. Similarly, following a brief initial transient, the fluctuations in strength shown in Figure 12



Figure 12. (a) The influence of L values on shear zone width and(b) the ratio of minimum strength to average fault strength varies withthe strain rate distribution. With smaller L, the strain is less localized,so the difference between average strength and minimum strengthis smaller, but still greater than the difference for uniform shearing.Larger L corresponds to more localized deformation, and the differenceis smaller.

demonstrate relatively constant ampli-tudes of stress fluctuations over the 1 sduration of a model earthquake, duringwhich the overall shear zone width canbroaden substantially. Moreover, sincethese are relative stress fluctuations andthe average strength 𝜏 decreases mono-tonically (see Figure 10), in order toachieve a constant absolute amplitude ofstress fluctuation in our model, we wouldneed to allow L to decrease through timeso that the shear core occupies an increas-ing portion of the Mohr-Coulomb zone asshear progresses. For larger earthquakes,this suggests that force fluctuations canbe limited to modest levels even whenthe strain rate distribution is uniform.

5.2. Modest Temperature RiseWhen the shear zone broadens with time,the maximum temperature rise is smallerthan would be expected if the thick-ness were assumed fixed at its initialvalue. It is possible that shear zone broad-ening might dissipate sufficient heat tobe partially responsible for preventingmacroscopic melting and explaining theapparent scarcity of pseudotachylytesalong mature faults. In Figure 10 we cansee that when the ambient temperatureis 210∘C (a typical ambient value for afault at 7 km depth) with uniform shear,the maximum temperature rise is around350∘C. The low temperature keeps theshear zone well below the melting tem-perature for common fault constituents(i.e., ∼1000∘C). Moreover, because thecoseismic temperature rise can some-times be inferred or at least boundedfrom petrological analyses and labora-tory experiments [e.g., Kuo et al., 2011;

den Hartog et al., 2012; Rowe et al., 2012], predictions of shear zone thickness and observations alongwell-characterized faults hold potential as a future test of model behavior and a further constraint on thestrain rate distribution.

It is worth commenting on the observations of in recent Japan Trench Fast Drilling Project (JFAST) studies inthe context of our model results. Samples obtained from the Tohoku-Oki megathrust at ∼800 meters belowseafloor contain a high abundance of smectite [Ujiie et al., 2013]. When the temperature is beyond ∼200∘C,smectite dehydrates and eventually transforms to illite [Pollastro, 1993; Schleicher et al., 2014]. However, thetemperature during the large inferred slip at this location (∼50 m) calculated using a simple thermal modelwas reported at around 1500∘C [Fulton et al., 2013]. This is problematic, since smectite should be unstablewhen such high temperatures are achieved. Kinetic barriers may impede dehydration over short coseismictimescales and provide a partial explanation for the retention of smectite. Alternatively, if the shear zoneexpanded during this earthquake, heat may have dissipated over a sufficiently large volume of gouge to allowthe temperature to remain relatively cool.



5.3. Slip-History-Related Property AlterationThe calculations shown here treat fault material properties as spatially uniform and temporally indepen-dent. Even in our crude attempts to approximate the effects of damage, we also treat the damaged materialproperties as spatially uniform and temporally independent. It is reasonable, however, to expect that rup-ture propagation, comminution, and finite strain during earthquakes might produce significant changes tohydraulic properties near the shear core. We showed that the ratio of 𝛼hy∕𝛼th has a strong control on the max-imum temperature and the shear zone thickness, while in fact this ratio likely varies across the shear zone,and may evolve as well over the duration of slip. Moreover, in addition to the broken fabrics caused by com-minution, chemical reactions are also expected to occur in the shear zone [e.g., Brantut et al., 2010; Brantut andSulem, 2012]. In parts of the San Andreas fault, for example, talc may be generated from the breakdown of ser-pentinite coseismically [Moore and Rymer, 2007], as suggested by laboratory experiments [Rutter and Brodie,1988; Hirose and Bystricky, 2007]. The modeling framework we provide here is amenable to extensions thatincorporate the evolution of permeability and the effects of chemical reactions, to provide a more completeunderstanding of fault behavior during earthquakes.

5.4. Anisotropy Effects Within the Shear ZoneOur formulation treats the fault zone materials as isotropic, both for thermal properties and hydraulic proper-ties. However, even for monomineralic rocks, the thermal diffusivity can be greatly affected by the orientationof the grains, and aligned fractures and foliations, along with the distribution of veins, can cause significantanisotropy in hydraulic diffusivity. We have shown that poroelastic effects cause differences between theeffective compressive stresses in directions parallel and perpendicular to the slip direction. This differentialstress can also be caused by other mechanisms. For example, clays develop fabrics that are observed in nat-ural faults [e.g., Solum and van der Pluijm, 2009] and nonclay synthetic gouges also develop foliations duringshear that induce permeability variations and may cause deviations in frictional strength [Zhang and Tullis,1998; Zhang et al., 1999; Okazaki et al., 2013]. Anisotropy may augment or retard the rotation of the principalstress axes that ultimately leads to the boundary of mobilization that we identify as the Mohr-Coulomb zone.Further, model extensions to account for such effects await more firm constraints that might be provided bydetailed observations of the inner structure and physical properties in natural fault zones.

6. Conclusion

Field and borehole observations along faults and measurements during and after experiments on granularsystems provide clear evidence that slip is accommodated over narrow, but finite, widths that greatly exceedthe average particle diameter. By examining the effects of porothermoelasticity constrained by plane strain,our model yields predictions for the evolution of strength, temperature, and shear width that are consistentwith the available constraints. We have demonstrated that thermal pressurization causes effective stress vari-ations to produce a post-failure zone that is weaker than the local frictional strength, and the boundary alongwhich the shear stress and strength are equal propagates into the undeformed surrounding gouge. Impor-tantly, this progressive widening of the post-failure zone occurs whether the frictional constitutive behavior israte weakening or rate strengthening. We postulate that shear is accommodated throughout the post-failurezone and that slip along numerous transient surfaces can be captured in our continuum model by defining aneffective strain rate distribution. Model predictions for fault strength are not strongly affected by the details ofthe particular strain rate distribution that is assigned. Predictions for the temperature rise along the symmetryplane are more sensitive, with higher temperatures expected as shear is increasingly concentrated where thefrictional strength is lowest. Our model is intentionally focused on an idealized depiction of fault slip, and weacknowledge that other neglected mechanical and chemical processes that occur during earthquakes maybe important. Efforts to refine the choice of appropriate strain rate distribution and fault constitutive prop-erties would benefit from further comparison against experimental data and improved constraints on faultzone temperature rise.

Appendix A: Numerical Method

To simplify the calculation, the pore pressure p and temperature T are first normalized to

T = ΛT − T0

𝜎n − p0, and p =

p − p0

𝜎n − p0,



so that the normalized governing equations are

𝜕p𝜕t

= 𝛼hy𝜕2p𝜕y2

+ 𝜕T𝜕t,

𝜕T𝜕t


+ Λf ��𝜌C

(1 − p).(A1)

And the strain rate distribution satisfies �� ||||y|>w∕2 → 0. The initial conditions are

T = 0, p = 0,

and the boundary conditions are

𝜕T𝜕y

||||y=0 = 0,𝜕p𝜕y

||||y=0= 0, T|y→∞ = 0, and p|y→∞ = 0.

From these equations, we further use a scaling transform 𝜉 = 2y∕w and 𝜏 = t to fix the moving boundary,then apply central difference to the transformed space. The transformed equations are

𝜕p𝜕𝜏

=4𝛼hy

w2

𝜕2p𝜕𝜉2

+ 𝜕T𝜕𝜏

+ 𝜉Γ(𝜕p𝜕𝜉

− 𝜕T𝜕𝜉

),

𝜕T𝜕𝜏

=4𝛼th

w2

𝜕2T𝜕𝜉2

+ 𝜉Γ𝜕T𝜕𝜉

+ Λf ��𝜌C

(1 − p),(A2)

where Γ = (1∕w)dw∕d𝜏 is the relative broadening rate of the shear zone.

The equations are discretized using centered differences for space, and we solve for the discretized temper-ature and pressure through time using the ode toolbox in Matlab. This enables the value of r to be trackedto impose the Mohr-Coulomb criterion r = rc on the shear zone boundary from equation (25) and determineits evolving width w. The strain rate �� is assumed to follow the Gaussian distribution in equation (27) or theuniform distribution approximated by equation (26).

Appendix B: Significance of Advection

One hidden assumption for the energy conservation equation (7) to hold is that the advection of heat is neg-ligible compared to the conduction. This is true when the permeability of the shear zone is low, but whetheradvection can still be ignored when we put the ratio of 𝛼hy∕𝛼th to 500 requires a more careful examination.The heat advection rate in the fluid is

qa ∼ qfCfΔT ∼ −𝜌fkCfΔT

𝜂f

ΔpΔy

= −𝜌f𝛼hyn𝛽(𝜎n − p0)CfΔTΔy

,

while the heat conduction rate isqh ∼ −K

ΔTΔy

= −𝜌sCs𝛼thΔTΔy

,

where the subscript s denotes the properties of the shear zone. Let qa∕qh ≪ 1, then we have

𝛼hy∕𝛼th ≪𝜌sCs

n𝜌fCf𝛽(𝜎n − p0),

and if we take the porosity n as 5% − 10%, the ratio is at least 1650, which means that for our model, as longas 𝛼hy ≪ 1650 𝛼th, the advection of heat is not important.

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AcknowledgmentsPortions of this work were supportedby the Southern California EarthquakeCenter through SCEC grant 12122 andby the National Science Foundationthrough grant EAR-1114380. Themanuscript was substantially improvedin response to the thoughtful reviews,comments, and criticisms of John D.Platt, Dmitri Garagash, two anonymousreviewers, and Associate EditorAlexandre Schubnel.


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shear zone broadening controlled by thermal pressurization and poroelastic effects during model...

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