Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels
Abstract
[1] We model ruptures on faults that weaken in response to flash heating of microscopic asperity contacts (within a rate-and-state framework) and thermal pressurization of pore fluid. These are arguably the primary weakening mechanisms on mature faults at coseismic slip rates, at least prior to large slip accumulation. Ruptures on strongly rate-weakening faults take the form of slip pulses or cracks, depending on the background stress. Self-sustaining slip pulses exist within a narrow range of stresses: below this range, artificially nucleated ruptures arrest; above this range, ruptures are crack-like. Natural earthquakes will occur as slip pulses if faults operate at the minimum stress required for propagation. Using laboratory-based flash heating parameters, propagation is permitted when the ratio of shear to effective normal stress on the fault is 0.2–0.3; this is mildly influenced by reasonable choices of hydrothermal properties. The San Andreas and other major faults are thought to operate at such stress levels. While the overall stress level is quite small, the peak stress at the rupture front is consistent with static friction coefficients of 0.6–0.9. Growing slip pulses have stress drops of ∼3 MPa; slip and the length of the slip pulse increase linearly with propagation distance at ∼0.14 and ∼30 m/km, respectively. These values are consistent with seismic and geologic observations. In contrast, cracks on faults of the same rheology have stress drops exceeding 20 MPa, and slip at the hypocenter increases with distance at ∼1 m/km.
1. Introduction
1.1. State of Stress on Faults
[2] Understanding the stress levels at which faults operate remains an outstanding issue in earthquake mechanics. Laboratory friction experiments at slow slip rates, far less than coseismic rates of ∼1 m/s, universally reveal friction coefficients, f, between 0.6 and 0.9 for almost all rock types (with the exception of clays and other weak layered minerals), a result known as Byerlee's law [Byerlee, 1978]. Assuming that pore pressure, p, is close to hydrostatic within the seismogenic upper crust, shear stress levels of ∼100 MPa are required to bring faults into a critically stressed state. There are, however, a number of lines of evidence that indicate that major faults support and are capable of hosting self-sustaining ruptures when the overall (i.e., spatially averaged) ratio of shear stress, τ, to effective normal stress, , is only a fraction of that predicted by Byerlee's law. This does not preclude the existence of small regions of locally high τ/
, which are the likely sites of rupture nucleation.
[3] It was recognized some time ago that frictional sliding of faults at such stresses should produce a heat signature that ought to be observable in the form of temperature anomalies across fault traces. The lack of observed temperature anomalies along the San Andreas Fault (SAF) places an upper bound on the average shear stress doing work during large seismic slips of ∼10 MPa [Brune et al., 1969; Lachenbruch and Sass, 1980; Lachenbruch et al., 1995; Williams et al., 2004]. While some have argued that groundwater flow might destroy the thermal signature of faulting at the stress levels predicted by Byerlee's law [Scholz, 2000], models of topologically driven flow along the SAF have shown that, at least along certain portions of the fault, this is unlikely to be the case [Saffer et al., 2003].
[4] Another set of constraints on the stress state on and around faults comes from in situ stress measurements (e.g., hydraulic fracturing and wellbore breakouts). Particular emphasis has been placed on determining the stresses acting on the SAF. Stresses adjacent to the SAF near Parkfield are consistent with a crust that is critically stressed at f ∼ 0.6–0.9, but the stress field is oriented such that the ratio of shear stress to effective normal stress, τ/, resolved onto the SAF is quite low [Zoback et al., 1987]. Measurements at 2 km depth in the San Andreas Fault Observatory at Depth (SAFOD) pilot hole, located 1.8 km from the trace of the SAF, indicate that the maximum horizontal compressive stress is inclined at 69 ± 14° from the local strike of the SAF [Hickman and Zoback, 2004]. This is consistent with the orientation of the regional stress field in central and southern California, which is reported by Townend and Zoback [2004] to be 68 ± 7°. The high angle between maximum compression and the fault implies that τ/
resolved on the SAF is only 0.2–0.4 [Hickman and Zoback, 2004]. Similar results were found within the Cajon Pass borehole along the SAF [Zoback and Healy, 1992].
[5] The nearly fault-normal orientation of maximum horizontal compression explains the existence of folds and thrust faults that strike parallel to the SAF [Mount and Suppe, 1987]. Inversions for the orientation of the stress field from earthquake focal mechanisms are less conclusive, though the data appear to suggest angles of at least 40–60°, if not greater [Hardebeck and Hauksson, 2001; Townend and Zoback, 2001; Provost and Houston, 2001, 2003; Hardebeck and Michael, 2004]. Scholz [2006] and Townend [2006] provide recent reviews of the controversy surrounding the state of stress on the SAF that explore these issues in greater detail.
[6] A further difficulty with coseismic slip at f ∼ 0.6–0.9 and hydrostatic p is that the predicted temperature rise on faults from shear heating will be sufficiently severe as to induce melting [McKenzie and Brune, 1972]. The temperature rises would be less extreme if slip were accommodated across a broad shear zone [Cardwell et al., 1978], and this could possibly preclude melting. Yet recent fieldwork on exhumed faults like the Punchbowl fault [Chester and Chester, 1998; Chester et al., 2004] and the Median Tectonic Line fault [Wibberley and Shimamoto, 2003] suggests a high degree of shear localization. Extreme localization within gouge layers during laboratory friction experiments has also been observed [Beeler et al., 1996; Mizoguchi and Shimamoto, 2004]. Analysis of core retrieved from a drill hole across the Chelungpu fault, which hosted the 1999 Mw 7.6 Chi-Chi earthquake, suggests slip was accommodated within a clay zone only 50–300 μm thick [Heermance et al., 2003]. Sibson [2003] and Rockwell and Ben-Zion [2007] discuss further field evidence for localization. Rice [2006] calculates that if sliding occurs at constant τ (with f = 0.6 and hydrostatic p that remains unchanged with slip), fault widths must exceed about 35 mm to prevent the onset of melting at 7 km depth [see also Rempel and Rice, 2006]. The observational signature of melting is the presence of pseudotachylytes in the fault zone. While pseudotachylytes have been observed along natural faults, they are generally associated with deeper faulting within the seismogenic zone and are far less common than expected [Sibson, 1975], so they either do not generally occur (at least near the surface) or are not preserved.
1.2. Weakening Mechanisms
[7] The scarcity of pseudotachylytes, despite the apparent thinness of the slip zone, suggests that coseismic slip occurs at much lower stress levels than those predicted by Byerlee's law. The explanation we consider in this study is that fault strength is dramatically reduced at coseismic slip rates (and that restrengthening occurs rapidly upon the cessation of slip). We do not discuss or intend to dismiss other possible resolutions of the stress level and heat flow issues, by assuming that fault materials have anomalously low f at all slip rates, or that f is not low but that p is near-lithostatic over much of the fault.
[8] Furthermore, our focus is on major faults. The stress measurements by Townend and Zoback [2000] make it clear that τ/ on optimally oriented planes within the crust frequently reaches typical static friction levels (0.6–0.9). It follows that overall stresses on many faults, especially those that are not major plate boundary faults, do operate at τ/
∼ 0.6–0.9. Exactly why these faults do not rupture at lower stress levels like major faults remains unknown, but one might speculate that some of the assumptions we make in developing our model of major faults operating at low stress levels do not apply to less mature faults. For example, we appeal to a high degree of shear localization in developing our model of flash heating, which is the essential ingredient in permitting low stress ruptures. Immature faults consist of a large number of unconnected segments that ultimately link together as the fault develops [Segall and Pollard, 1983]. Likewise, measurements of fault surface topography by Sagy et al. [2007] suggest that faults become progressively smoother with increasing slip. The additional complexity of immature fault zones would likely increase the stress level necessary for ruptures to propagate and might help explain differences in stress levels at which immature and mature faults operate. However, a complete resolution of this issue is beyond the scope of this work.
[9] Recent high speed friction experiments show that f decreases roughly as the inverse of the sliding velocity, V, for V above ∼0.1 m/s [Tsutsumi and Shimamoto, 1997; Tullis and Goldsby, 2003a, 2003b; Prakash and Yuan, 2004; Hirose and Shimamoto, 2005; Beeler et al., 2008]. This is presumably due to flash heating of microscopic asperity contacts, as first proposed in the field of dry metal friction [Bowden and Thomas, 1954; Archard, 1958/1959; Ettles, 1986; Lim and Ashby, 1987; Lim et al., 1989; Molinari et al., 1999] and recently applied to rock friction at elevated slip rates [Rice, 1999, 2006; Beeler and Tullis, 2003; Beeler et al., 2008]. This model appeals to a microscopic view of friction, in which rough surfaces that are brought together make actual contact only at a set of microscopic asperities, the net area of which is a small fraction of the nominal contact area. These asperities come into existence and slide only a short distance before being replaced by a new set of contacts. The stresses supported by these asperities are far larger than the macroscopic applied stresses; hence, the local rate of heat production during sliding is quite large. If, during the lifetime of an asperity contact, the local temperature reaches the melting temperature (or some comparable temperature at which thermally activated defects become highly mobile), the contact will weaken. The macroscopic resistance to sliding is the sum over all currently existing contacts. The onset of strong weakening seen in experiments at ∼0.1 m/s is attributed to the slip rate reaching a critical value at which asperities just begin to weaken before sliding out of existence. At higher slip rates, the contacts spend a larger portion of their existence in the weakened state (such that at some instant, a larger fraction of currently existing contacts are in the weakened state).
[10] A second weakening mechanism involves fluids in fault zones. The majority of coseismic slip occurs below the water table, such that fault zones, to the extent that they are porous, should be fully saturated with pore fluid. As the fault slides, it heats both the rock matrix and pore fluid. Both materials expand, but the thermal expansion coefficient of water far exceeds that of rock. The stiffness of the rock matrix limits the expansion of the fluid, which consequently pressurizes. If the fault zone material is sufficiently impermeable, then this pressurized fluid is effectively trapped over the duration of coseismic slip. The increase in fluid pressure decreases effective stress and hence fault strength. This process, known as thermal pressurization, has received considerable theoretical attention [Sibson, 1973; Lachenbruch, 1980; Mase and Smith, 1985, 1987; Lee and Delaney, 1987; Andrews, 2002, 2005; Wibberley, 2002; Noda, 2004; Cocco and Bizzarri, 2004; Noda and Shimamoto, 2005; Sulem et al., 2005; Bizzarri and Cocco, 2006a, 2006b; Suzuki and Yamashita, 2006; Rice, 2006; Rempel and Rice, 2006].
[11] Even if flash heating and/or thermal pressurization weaken faults and reduce heat production, a sufficiently long duration of sliding may result in macroscopic melting of rock and formation of a molten layer along the sliding interface; the ultimate fate of a fault zone for large amounts of slip depends on thermal and hydraulic parameters (e.g., permeability of fault gouge), as well as the initial effective normal stress and sliding rate [Rempel and Rice, 2006]. The onset of melting and its effect on fault strength, as well as the characterization of naturally occurring pseudotachylytes, has been the subject of much work [Sibson, 1975; Spray, 1987, 1992, 1993, 1995; Tsutsumi and Shimamoto, 1997; Matsuzawa, 2004; Fialko and Khazan, 2004; Hirose and Shimamoto, 2005; Di Toro et al., 2006; Sirono et al., 2006; Nielsen et al., 2008].
[12] Several other processes have been suggested to influence fault strength during earthquakes. These include elastohydrodynamic lubrication (due to pressure gradients established by viscous shear of fault zone fluids or a melt layer trapped between the walls of the fault) [Brodsky and Kanamori, 2001] and silica gel formation [Goldsby and Tullis, 2002; Di Toro et al., 2004]. Silica gel formation is likely restricted to faults in silica-rich host rock (e.g., granite), and gel lubrication, like thermal decomposition weakening [Han et al., 2007] and formation of a macroscopic melt layer, would only take place after sufficiently large slip, if at all. Thus we assume that the most universal mechanisms, especially during the early stages of rupture growth, are likely to be thermal pressurization and flash heating, which are the two processes that we study on in this work.
1.3. Rupture Modes on Statically Strong, Dynamically Weak Faults
[13] In light of the above discussion, the task before us is to reconcile laboratory observations that indicate rock friction coefficients between 0.6 and 0.9, at slow slip speeds, with the multiple lines of evidence suggesting that average stress levels on major faults are much smaller than those predicted by Byerlee's law. The hypothesis that we explore in this work is that faults are strong throughout the interseismic period, but weaken dramatically (by the mechanisms discussed previously) when sliding coseismically, a process referred to as dynamic weakening. Lapusta and Rice [2003] have shown, in earthquake cycle simulations that simultaneously incorporate slow loading in the interseismic period and fully elastodynamic coseismic ruptures, how faults having a shear strength that weakens dramatically with slip rate can host ruptures when the average shear stress on the fault is quite low. The low shear strength during coseismic slip results in minimal heat production.
[14] Intimately related to the concept of strongly rate-weakening faults, operating at low overall stress levels, is that ruptures take the form of self-healing slip pulses. There is strong observational support that ruptures propagate as slip pulses; slip inversions of high frequency seismic signals indicate that risetimes are much shorter than event durations [Heaton, 1990]. Three mechanisms have been shown to produce slip pulses: a contrast in material properties across the fault [Andrews and Ben-Zion, 1997], arrest waves from fault edges or from heterogeneity along the fault [Day, 1982; Johnson, 1990; Beroza and Mikumo, 1996; Bizzarri et al., 2001], and velocity weakening constitutive laws on understressed faults [Cochard and Madariaga, 1994, 1996; Perrin et al., 1995; Beeler and Tullis, 1996; Zheng and Rice, 1998].
[15] Zheng and Rice [1998] studied rupture propagation using rate- and state-dependent friction laws featuring velocity-weakening steady state fault strengths. They showed that when fault strength weakens sufficiently rapidly with slip velocity, there exists a critical background shear stress level, τpulse, below which ruptures must take the form of slip pulses. For background stresses above τpulse, ruptures can also take the form of expanding cracks (in the sense that slip near the hypocenter does not cease until the arrival of arrest waves emitted when the rupture reaches the fault edges). Self-sustaining slip pulses were found to exist only when stresses are right around τpulse; at much lower stress levels, artificially nucleated ruptures quickly arrest. It follows that quasi-statically loaded faults are capable of hosting ruptures (in the sense that an initially small event, nucleated in a region of locally high τ/, can propagate indefinitely) just as stresses reach the minimum level necessary for self-sustaining rupture propagation. At this level, ruptures will take the form of slip pulses. Earthquake cycle simulations by Lapusta and Rice [2003] which incorporate a steady state friction coefficient that weakens dramatically at coseismic slip rates, support this claim. Lapusta and Rice further demonstrate that nucleation occurs in small regions where τ/
locally exceeds τpulse/
based on the average
on the fault, and reaches levels comparable to those predicted using laboratory values of static friction. In our work, we also focus on faults that dramatically weaken at coseismic slip rates, but we incorporate a far more sophisticated, physics-, field-, and laboratory-based description of the dynamic weakening processes (in particular, flash heating and thermal pressurization). The combined effect of these weakening mechanisms cannot be written in a form that permits direct application of the theory by Zheng and Rice [1998] and extending their theory to encompass a broader set of constitutive laws is an additional focus of our work.
1.4. Overview
[16] In this study, we present a theoretical model that addresses the issues discussed, and we numerically simulate spontaneous ruptures on faults that weaken by flash heating and thermal pressurization. We use a combination of laboratory and field data to constrain the relevant thermal and hydraulic properties that enter the model, and explore the phenomenology of ruptures initiated by an artificial nucleation mechanism (classifying them as arresting slip pulses, growing slip pulses, and growing cracks). Because we make no compromises in physical parameters, our numerical simulations require incredibly small grid spacings for accurate solutions. We consequently limit our simulations to two-dimensional (mode III) ruptures and focus on the early stages of rupture growth. The strong velocity-weakening behavior of flash heating promotes self-healing slip pulses on faults with sufficiently low values of initial τ/; growing pulses occur when τ/
∼ 0.2–0.3. These values are consistent with those inferred to act on the SAF [Hickman and Zoback, 2004] (and presumably other major faults). Thermal pressurization can alter the rupture mode; efficient pressurization (occurring on more impermeable faults) extends the crack-like rupture regime to lower values of τ/
than those predicted by flash heating alone, though the background stress level remains the dominant predictor of rupture mode. We also find that the width of the shear zone influences the rupture mode, and that there are major differences (primarily with regard to maximum temperatures and pressures achieved at the rupture front) between models in which shear is distributed over 50–200 μm and the idealized model of slip on a mathematical plane discussed by Rice [2006]; that such small but nonzero thicknesses might be important in limiting maximum temperatures was also suggested by Rice [2006] and Rempel and Rice [2006].
2. Model
[17] Since our focus is on rupture propagation with dynamic weakening mechanisms, we choose to study ruptures in a setting for which the elastic response is easily understood. Furthermore, our direct use of laboratory and field estimates of relevant parameters requires us to use very small grid spacings compared to the propagation distances of interest. Consequently, we study two-dimensional, mode III ruptures in a uniform linear elastic whole space. The medium has shear modulus μ and shear wave speed cs. The fault lies along the plane z = 0. Slip is in the y direction, the rupture expands outward along the x axis, and all fields are functions only of x and z (and time, t). Initial stress, pore pressure, and material properties (to be discussed in the subsequent sections) are chosen to be representative of conditions at midseismogenic depth (7 km) and, in all cases, properties are chosen for consistency with laboratory constraints. These are given in Table 1 and are defined precisely in the following sections.
Elastic Properties | ||
---|---|---|
Shear modulus | μ | 30 GPa |
Shear wave speed | cs | 3 km/s |
Hydrothermal Properties | ||
Specific heata | ρc | 2.7 MJ/m3K |
Thermal diffusivitya | αth | 0.7 mm2/s |
Hydraulic diffusivitya | αhy | 0.86–3.52 mm2/s |
Undrained Δp/ΔTa | Λ | 0.34–0.98 MPa/K |
Damage indexa | r | 0–1 |
Width of shear zone | 2w | 25–200 μm |
Friction (Flash Heating) | ||
Unweakened contact strength | τc | 3 GPa |
Weakening temperature | Tw | 900°C |
Asperity diameter | D | 5 μm |
Fully weakened friction coefficientb | fw | 0.13 |
Friction (Rate and State) | ||
Reference slip velocity | V0 | 1 μm/s |
Steady state friction coefficient at V0 | f0 | 0.7 |
State evolution distance | L | 20 μm |
Direct effect parameter | a | 0.016 |
Evolution effect parameter | b | 0.02 |
Initial Conditions | ||
Normal stress | σ | 196 MPa |
Initial pore pressure | p0 | 70 MPa |
Initial temperature | T0 | 210°C |
Resulting initial Vw | Vw0 | 0.170 m/s |
Initial state variable | Θ0 | 0.59211 |
Background shear stress | τb | 24–35 MPa |
Initial τ at hypocenter | τb + τper | 100–150 MPa |
Width of perturbation | Dper | 1–3 cm |
- a Data compiled by Rice [2006, Table 2] for ambient T and p at 7-km depth.
- b From curve fit to data up to V = 0.4 m/s from experiments on granite by Tullis and Goldsby [2003b].



2.1. Flash Heating of Microscopic Asperity Contacts




[22] The use of a nonzero friction coefficient, fw, at elevated slip rates in the model of Rice [1999] was introduced by Beeler and Tullis [2003], Tullis and Goldsby [2003a, 2003b], and Beeler et al. [2008], who found its inclusion necessary to better match their experimental data. We use fw = 0.13, which is obtained by fitting to their granite data.



[24] The fraction of unweakened asperity area decreases with increasing slip velocity, implying that the contribution to the direct effect from those unweakened asperities also decreases. There are currently no experiments to constrain the transient evolution of the friction coefficient at coseismic slip rates, but Rice et al. [2001] point out that under the assumption that the direct effect arises from thermally activated shearing processes at asperity contacts, a should be linearly proportional to the absolute temperature at the contacts. In this work we use a = 0.016, which is obtained by linear extrapolation with absolute temperature from a typical value of 0.01 at room temperature [Ruina, 1983]. We also tried simulations in which we permitted a to vary as a function of T, but found little difference with simulations employing a constant a.
[25] We use a state evolution distance, L, comparable to the asperity diameter, D, under the assumption that sliding a distance D refreshes all contacts. Specifically, we use L = 20 μm for most of our simulations, although this requires an extremely small grid size to achieve an appropriate level of numerical accuracy. The high velocity friction experiments of Prakash and Clifton [1992] and Prakash [1998] show that, following extremely abrupt normal stress steps, f evolves toward its steady state value over slip distances around ∼3 μm; however, to date, no experiments have investigated the transient evolution of f following velocity steps at coseismic speeds. Furthermore, theoretical work by Noda [2008] implies that the evolution distance for flash heating may be much smaller than D. Suppose we keep the background temperature, T, constant by cooling, and impose steps in V for a fault comprised of single-sized asperities. Since all the asperities are renewed after slipping by D, evolution to the new steady state is perfectly achieved at a slip of D; it follows that L must be a fraction of D. Such small values of L are very difficult to use in simulations of rupture propagation. In addition, the flash heating constitutive law has a very high maximum weakening rate, −∂fss(V, T)/∂(lnV), at V just above Vw.
[26] In the above discussion, we have implicitly assumed that flash heating occurs on a mathematically planar fault, while many lines of evidence suggest that seismic shear is distributed over a zone of remarkably small but nevertheless finite width. There have been several attempts to understand the microscopic picture of a rapidly sheared gouge layer by means of the distinct element method. Morgan and Boettcher [1999] observed development of shear planes where deformation is localized, and da Cruz et al. [2005] reported that only about 15% of all contacts are slipping at a given time. These works suggest that intergranular slip occurs at a small number of the contacts, and the local slip rate is about the same as the macroscopic slip rate of the fault. On the other hand, there are observations of thin but finitely thick shear zones in both exhumed faults [Chester and Chester, 1998; Chester and Goldsby, 2003; Chester et al., 2004; Heermance et al., 2003; Mizoguchi and Shimamoto, 2004] and specimens after frictional experiments at high slip rates [Beeler et al., 1996; Mizoguchi and Shimamoto, 2004]. These shear zones are recognized by shape-preferred orientations of platy minerals, which clearly indicate that the total shear strain is distributed within it. Many of these studies suggest that shear localizes to zones of width ∼100 μm; see Rice [2006] for further discussion. In this work, our flash heating model is based on slip on a mathematical plane and we use the value of T at z = 0 when evaluating fss(V, T); however, we distribute the heat source over a finite width shear zone when calculating T and p changes in our thermal pressurization calculation. This approach is appropriate if, at any instant in time, only a small fraction of the contacts are slipping, and if the location of such slipping contacts moves around within the shear zone. In this case, the texture of severe shear deformation is developed over the distributed shear zone. It is also required that the timescale over which the localized slip surface migrates through the shear zone is much smaller than relevant thermal and hydraulic diffusion times.
2.2. Thermal Pressurization of Pore Fluid
[27] Thermal pressurization of pore fluid suppresses the temperature rise within and near the shear zone by reducing the effective normal stress and thus shear strength. This mechanism has been implemented in rupture propagation calculations by several authors [Andrews, 2002; Noda, 2004; Cocco and Bizzarri, 2004; Andrews, 2005; Bizzarri and Cocco, 2006a, 2006b; Suzuki and Yamashita, 2006] but never in combination with flash heating or other strongly velocity-weakening friction laws.











[32] Rempel and Rice [2006] and Rice [2006] estimated hydrothermal properties based on a collection of experimental and theoretical works [Burnham et al., 1969; Keenan et al., 1978; Lachenbruch, 1980; Tödheide, 1972; Vosteen and Schellschmidt, 2003; Wibberley, 2002; Wibberley and Shimamoto, 2003] for intact and damaged elastic fault walls composed of ultracataclasite at 7 km depth, 196 MPa normal stress, an ambient pore pressure of 70 MPa, and an ambient temperature of 210°C [see Table 2 in Rice, 2006]. These values are ρc = 2.7 MPa/K, Λ = 0.98 (intact) and 0.32 (damaged) MPa/K, αth = 0.70 mm2/s, αhy = 0.86 (intact) and 3.52 (damaged) mm2/s, which yield αHT = 23.6 (intact) and 464.1 (damaged) mm2/s. The hydraulic properties of rock are from the Median Tectonic Line fault in southwest Japan [Wibberley, 2002; Wibberley and Shimamoto, 2003]. The properties are summarized in Table 1.
[33] There is considerable uncertainty in choosing the appropriate value of permeability. Wibberley and Shimamoto [2003] measured permeability using nitrogen gas as the pore fluid, and Faulkner and Rutter [2000, 2003] showed for that for nonswelling clays the water permeability can be an order of magnitude smaller than the gas permeability. Permeability in and around fault zones can also be quite anisotropic with respect to the direction of flow relative to foliation of the gouge. Faulkner and Rutter [1998] found variations over three orders of magnitude with direction relative to the fault; the lowest permeability is typically that perpendicular to the fault, which is the relevant direction for our models. Additionally, studies of clayey gouge from the Neodani [Tsutsumi et al., 2004] and Hanaore [Noda and Shimamoto, 2005] faults in southwest Japan, and the Chelungpu fault in Taiwan [Tanikawa and Shimamoto, 2009] yielded much higher permeabilities (10−19 to 10−16 m2 at around 100 MPa effective pressure) than those measured by Wibberley and Shimamoto [2003] with the same methodology. The clayey gouge in these works typically contains a swelling clay for which the difference in gas and water permeability can be much greater than that estimated by Faulkner and Rutter [2000, 2003].


[35] Previous studies [Noda, 2004; Cocco and Bizzarri, 2004; Andrews, 2005; Bizzarri and Cocco, 2006a, 2006b; Suzuki and Yamashita, 2006] typically used shear zones that are a few centimeters thick. However, observational studies of exhumed faults [Chester and Chester, 1998; Chester et al., 2004; Heermance et al., 2003; Mizoguchi and Shimamoto, 2004] and specimens after rotary shear experiments [Beeler et al., 1996; Mizoguchi and Shimamoto, 2004] suggest that shear deformation is extremely concentrated in zones that are at most ∼200 μm in width (and there is no evidence that rules out the possibility that at any instant, shear is localized in an even narrower zone). In this study we use at most 2w = 200 μm. In our numerical simulations, we integrated equations (11) and (12) using a conventional second-order finite difference method (in space) on a one-dimensional grid perpendicular to the fault plane at every node on the fault. Time stepping was performed using an adaptive Runge-Kutta method [Dormand, 1996]. Details of the numerical method are given in the appendices.





2.3. Predicted Background Stress Levels Separating Pulses and Cracks (Constant T and p)
[37] As discussed before, friction laws having strong velocity-weakening behavior (like flash heating) can cause ruptures to take the form of self-healing slip pulses, provided that the background shear stress level is below a critical level termed τpulse by Zheng and Rice [1998]. To begin, we consider faults with a steady state strength that depends only on V. In the context of our model, we do this by holding the background temperature, T, and pore pressure, p, fixed at their initial values, such that the steady state strength is τss(V) = 0fss(V, T0). In the following, we recall the results of Zheng and Rice [1998].








[41] For a given material, friction law, and effective normal stress level, the only free parameter is the background stress level, τb. The maximum value of τb for which (27) holds is defined as τpulse [Zheng and Rice, 1998]. At stresses below τpulse, ruptures, if they exist, must assume the form of slip pulses. Laboratory experiments have confirmed that ruptures on understressed faults do take the form of slip pulses [Lykotrafitis et al., 2006].
[42] Figure 1 shows the steady state friction coefficient for our flash heating model as a function of V using the initial values of T and p (and hence ), as well as the radiation-damping line that defines τpulse. Note that τpulse/
0 depends on initial effective normal stress,
0. For
0 = 126 MPa, τpulse/
0 is very small (0.2475) due to the extreme velocity weakening behavior at elevated slip rates. Note also that τpulse/
0 increases with decreasing
0 since the slope of a radiation-damping line increases when normalized by
0. It is interesting that Hickman and Zoback [2004] have noted a decrease in the value of τ/
resolved onto planes parallel to the SAF with increasing depth in the SAFOD pilot hole.



[43] For extremely low values of 0 (around a few MPa), there exists a radiation-damping line which fits tangentially to fss(V, T0) in the low velocity (V < Vw) regime and has an intercept at V = 0 at a value of τ/
0 comparable to typical static friction coefficients. We do not explore the importance of this in the present study. Rather, instead of assuming that the value of pore pressure is sufficiently high to decrease the effective normal stress to these extremely low values, we consider faults that are stressed at levels far below the levels associated with steady sliding at slow slip rates.




[45] To assess whether or not crack-like solutions can exist, we must determine appropriate values of q to use in equation (30). For our model, the variables that comprise q are T and p on the fault. The transition to a slip pulse rupture mode occurs first in the hypocentral region, so it is necessary to estimate characteristic values of T and p there. To do so, we must first discuss our nucleation procedure.
2.4. Nucleation Procedure
[46] The extreme velocity weakening associated with flash heating makes τpulse very low (see Figure 1), and we focus on faults with background shear stresses, τb, near τpulse. To nucleate ruptures on faults with overall (i.e., average) stress levels that are much smaller than those associated with static friction coefficients of 0.6–0.8, either the frictional constitutive parameters or the initial stress level must be quite different on some portion of the fault. Under such circumstances, ruptures that are nucleated in one of these regions can propagate into low-stressed regions [Perrin et al., 1995; Zheng and Rice, 1998; Bizzarri et al., 2001]. Another possibility is that rupture could nucleate on a fault having a different orientation than the main fault (e.g., a branch on which the ratio of τ/ is close to the static friction level) and subsequently jump onto the main fault plane on which τ is close to τpulse. In the laboratory setting, ruptures can be nucleated on understressed faults by an explosion on the simulated fault; this lowers
locally to initiate slip [Lykotrafitis et al., 2006].
[47] In this work, we employ a similar, artificial nucleation procedure by applying a sudden perturbation in shear stress at the center of the fault. We caution that this is not likely to be a realistic description of the nucleation of earthquakes on natural faults. At t = 0−, we set τ0(x, 0−) = τb. We also set Θ(x, 0) = 0.59211, which is determined so that the friction coefficient at the rupture front is comparable to a typical static friction coefficient of 0.85. Our estimate of the peak friction coefficient at the rupture front is obtained by assuming that state evolution is negligible and that τ responds only by the direct effect.
[48] At t = 0, we apply a Gaussian-shaped stress perturbation, the amplitude and standard deviation of which are τper and Dper, respectively. We use Dper = 2.5 cm and a value of τper that satisfies τ0(0, 0+) = τb + τper = 107.458 MPa unless otherwise noted (see Table 1). This amplitude is chosen so that V(0, 0+) is 1 m/s. Weakening due to flash heating occurs immediately in the nucleation zone since V > Vw there, which drives the rupture out at a speed near the shear wave speed. We would like to concentrate on the mathematical criterion between crack-like and self-healing pulse solutions, and implementing a more realistic nucleation process is beyond the scope of this work. Because of limitations in numerical resources and calculation time, it is difficult to simulate faults that are even tens of meters long, so we typically simulate ruptures propagating just over 2 m in each direction (though we do extend a few select cases out to 16 m in each direction). While these short distances are insufficient for fully examining the thermal pressurization process, the rupture mode (crack or pulse) is established over these distances with the parameters selected above.
2.5. Predicted Background Stress Levels Separating Pulses and Cracks (Variable T and p)
[49] Now that we have described our nucleation procedure, we can return to our task of estimating characteristic values of T and p, which we need to evaluate equation (30). T and p change in response to thermal pressurization driven by shear heating. Our approach is to estimate V and τ (and hence the shear heating rate) within the nucleation zone and then calculate ΔT and Δp, the changes in T and p that result from this amount of shear heating over the duration of the nucleation process. A precise description of how we estimate these changes is given in Appendix B.
[50] For each set of simulation parameters, we calculate ΔT and Δp and use these values to shift the steady state strength curve as shown in Figure 3. We define τdynapulse as the value of τpulse for these dynamic conditions; since ΔT and Δp are functions of such parameters as the shear zone width, background shear stress, and hydrothermal properties, then τdynapulse also depends on these parameters. There are no crack-like solutions for τb < τdynapulse, so the equality τb = τdynapulse delimits the actual boundary between crack-like and pulse-like solutions.


[51] In order for a crack-like solution to exist, it is necessary for equation (24) to be satisfied. At the beginning of rupture propagation, all calculations show crack-like behavior in the sense that the center of the ruptured area is slipping, which is consistent with the locally high τ0 within the nucleation region. However, at later times when the rupture has propagated further, the average value of τ0 becomes closer to τb and, if τb is too small to satisfy equation (27) or equation (30), the rupture can no longer be crack-like.
3. Rupture Mode Phase Diagrams
3.1. Neglecting Changes of T and p
[52] Before proceeding to more complicated cases, we first consider ruptures on faults where T and p are constrained to their initial values, so that strength is solely a function of V and the analysis of Zheng and Rice [1998] applies directly. Note that T is the macroscopic temperature on the fault, not that at the asperity contacts; hence, neglecting changes in T does not mean that we are neglecting flash heating. The upper inset of Figure 4a shows the rupture mode as a function of background stress, indicating that, as expected, there are no crack-like solutions at stresses below τpulse = 0.2475 0. Ruptures take the form of growing slip pulses in a rather narrow region of τb around τpulse. For the cases with τb ≥ 0.2778
0, the transition to a slip pulse rupture mode does not occur during the duration of the calculation. A detailed analysis of the transition to a slip pulse rupture mode is presented in Appendix C.

3.2. Sensitivity to Width of Shear Zone
[53] Figure 4a shows a phase diagram of the rupture propagation mode in a (τb, 2w) plane for r = 0.8. It should be emphasized that we obtain crack-like solutions for stress levels below τpulse, which is the theoretical threshold below which crack-like solutions are impossible in the absence of temperature and pore pressure changes [Zheng and Rice, 1998]. As the background shear stress decreases, ruptures change from expanding cracks (Figure 2a) to growing pulses (Figure 2b). For even smaller values of τb, ruptures are arrested (Figure 2c). There are clear boundaries between these solution types. It is also notable that the range of background shear stress levels admitting crack-like solutions is wider for smaller 2w, implying that thermal pressurization of pore fluid (which is more effective for narrower shear zones) extends the crack-like solution regime. As 2w increases, the boundary between crack-like and pulse-like solutions becomes less dependent on 2w, but the effect of T and p changes are still remarkable in an absolute sense, compared with the case in which they are neglected.
3.3. Sensitivity to Hydraulic Properties
[54] Figure 4b shows a phase diagram of the rupture mode as a function of τb and the damage index r (or, equivalently, αHT) with 2w = 100 μm. We again obtain crack-like solutions below τpulse, and the phase diagram shows that changes in T and p make crack-like solutions more likely; the range of background shear stress levels permitting crack-like solutions is wider for smaller r or αHT. The effect of varying hydraulic properties (within what we feel to be a realistic range) is almost as large as the effect of varying the width of the shear zone.
[55] Note that the cases in which T and p changes are neglected cannot be plotted on this section of parameter space. When the change in p is neglected with Λ = 0, T increases very rapidly and easily reaches Tw. Our constitutive model is not appropriate at such high temperatures.
3.4. Sensitivity to Nucleation
[56] The criterion for determining the rupture mode when T and p changes are permitted (i.e., our estimation of τdynapulse) depends on the nucleation parameters. Here we investigate the two nucleation parameters separately. All calculations are performed with r = 0.8 and 2w = 100 μm. Figures 5a and 5b show phase diagrams in the (τb, Dper) plane with fixed τb + τper = 107.4579 MPa, and in the (τb, τb + τper) plane with fixed Dper = 2.5 cm, respectively. The rupture mode indeed depends on the size of perturbation, with large perturbations (either in length and/or amplitude) favoring crack-like ruptures. Note that while our estimation of τdynapulse gives the right trend in the crack-pulse phase boundary, it does not precisely trace the boundary. This is presumably due to several overly simplifying approximations used to estimate the changes in p and T during the nucleation process; these are discussed in more detail in Appendix B.

4. Observational Constraints
[57] In the previous sections, we have established that ruptures on faults governed by flash heating and thermal pressurization can take the form of either slip pulses or cracks, and we have determined the conditions that give rise to each particular rupture mode. Since the crack-to-pulse transition occurs rapidly in our simulations, it was only necessary to propagate ruptures out to a few meters. However, much longer propagation distances are required to ascertain certain rupture characteristics that can be compared to observations. In particular, we are interested in addressing issues raised at the start of this manuscript regarding heat flow and the stress state around major faults. Additionally, our model makes predictions regarding observables like stress drop and the scaling of slip with rupture length. To probe these questions, we have conducted two long simulations, one for a crack-like rupture and the other for a growing slip pulse, in which propagation is permitted out to 16 m in each direction; the parameters are marked in Figure 4 by “Figures 6, 7, 9”. The 16 m distance is the limit of our current computational resources, and our ruptures correspond to magnitude ∼1 earthquakes (to the extent that propagation distance in our two-dimensional models can be mapped onto radius for a rupture in three dimensions). Source properties for earthquakes of this size have been studied using borehole seismograms [e.g., Abercrombie, 1995], and Yamada et al. [2005] have performed finite fault slip inversions of several mining-induced earthquakes with magnitudes between 0.8 and 1.4.
4.1. Weakening Process and Implied Slip-Weakening Relationship
[58] Our model incorporates several weakening mechanisms, and we wish to discuss here how these mechanisms act in concert during the rupture process to alter fault strength. Figure 6 shows the reduction of fault strength as a function of slip at x = 8 m for our two long simulations, and Figure 7 shows the history of τ, V, T, and p at this same location.


[59] The rapid rise of stress from τb to a peak stress of τ ∼ 100 MPa (τ/ ∼ 0.85) is associated with the direct effect of rate-and-state friction; during this time, state evolution is negligible. As state begins to evolve, f plummets toward fw since V is already well above Vw. By δ ∼ 0.1 mm, this evolution process is complete and f has essentially reached its steady state value. Thermal pressurization is responsible for subsequent weakening. Initially the response is effectively that of undrained, adiabatic deformation; diffusion only becomes significant when the hydraulic or thermal diffusion lengths exceed the width of the shear zone (i.e.,
> w). This occurs after about 0.2 ms (for α = 3 mm2/s and w = 50 μm); the corresponding value of slip at this time for the ruptures shown in Figure 7 is about 1 mm. The total duration of slip for the slip pulse is about 0.15 ms at this point, so diffusion is only just becoming significant when the trailing edge of the slip pulse passes this point. This is illustrated in panels (c) and (d) of Figure 7, which show both the actual histories of ΔT and Δp in the simulations and the histories that would have resulted in the absence of thermal and hydraulic diffusion for the same shear heating histories (the latter are shown as dashed lines marked “undrained, adiabatic”).
[60] The peak slip velocity is very high (∼300 m/s) at the rupture tip, but values of V above 100 m/s only persist for about 1 μs. These high slip velocities imply fault-parallel strains of order 0.1 (estimated as (V/2)/vr by assuming steady state propagation at a rupture speed vr close to cs). Such high strains are incompatible with the assumption of a linear elastic rheology; inelastic deformation would presumably result and limit slip velocities to more reasonable values. The large stresses at the rupture front would also damage the fault walls, enhancing the ability of high pressure fluids to escape the shear zone. We approximate this by using, in some of our simulations, hydraulic properties consistent with damage-induced increases in permeability and matrix compressibility.
[61] While the strength drop associated with these ruptures is large (∼90 MPa), the static stress drop (i.e., the difference between τb and the final stress after the event) for the slip pulse is about 3 MPa at this point. This is due to the combination of a low background stress level and the increase of stress that occurs when the fault ceases to slide. The latter is caused by an immediate return of f to its unweakened value in the absence of flash heating and a more gradual decrease of p as high pressure fluids diffuse away from the shear zone. A stress drop of 3 MPa is quite consistent with seismic data. Typical values are around 1–10 MPa (with much variation [e.g., Kanamori and Anderson, 1975]). Abercrombie [1995] and Ide and Beroza [1997] have shown that there is no systematic variation of average static stress drop over ten orders of magnitude in seismic moment.
[62] The fracture energy associated with the ruptures shown in Figure 6 is about 0.2 kJ/m2. This value is quite comparable to seismic inferences of fracture energy for earthquakes with slip of ∼1 mm [Abercrombie and Rice, 2005, Figure 8] although the form of the stress versus slip curve (rising at the end for the slip pulse) violates some of the approximations used in that study (e.g., that τ during sliding was approximately equal to the final stress after the event).
[63] The fact that stress returns nearly to τb after the passage of a slip pulse suggests that only a small amount of additional loading is necessary for the fault to host another rupture (as opposed to what would happen if stress drop were complete). Paleoseismic investigations by Weldon et al. [2004] along the SAF at Wrightwood reveal that periods of high accumulated strain are followed by either a single large earthquake or a series of relatively large (with meters of surface displacement) and frequent events separated by intervals of about 60 years. These observations suggest that after a modest static stress drop caused by a rupture, the fault is still relatively close to having the ability to rupture again.
[64] The spatial pattern of static stress drop is also interesting. Close to the nucleation region, there is little drop in stress; in fact, stress increases in the immediate vicinity of the hypocenter. This increase is characteristic of self-similar slip pulses (which have a logarithmic singularity in stress at the hypocenter), as discussed by Nielsen and Madariaga [2003]; and suggests that, in the absence of other sources of heterogeneity, the hypocentral region would be the most likely spot for subsequent events to nucleate.
4.2. Statically Strong, Dynamically Weak Faults
[65] As discussed in the introduction, dynamic weakening mechanisms provide one possible explanation for the absence of measurable heat flow and the low stress levels inferred to be acting on major faults. The strong rate dependence of friction predicts that the background stress level on faults should be around τpulse, far lower than the prediction of Byerlee's law. Nonetheless, the peak stress at the rupture tip exceeds 100 MPa and values of τ/ there are ∼0.85, within the range of laboratory measurements of static friction (the latter being determined by our choice of the initial state Θ0). Once nucleated, ruptures are driven by the stress concentration at their tips.

[67] We can also interpret the stress state around major faults like the SAF in light of our model. Townend and Zoback [2004] report that the regional direction of maximum horizontal compression, SHmax, in central and southern California is 68 ± 7° as measured from the local strike of the SAF. Assuming that τ/ within the crust is bounded by typical static friction levels (0.6–0.9), as is observed at many locations [Townend and Zoback, 2000], we can constrain the maximum value of τ/
on planes parallel to the SAF (Figure 8). For an angle of 68° between the fault and SHmax and a static friction coefficient of 0.8, τ/
< 0.30 on the SAF. Direct measurement of stresses in the SAFOD pilot hall [Hickman and Zoback, 2004] reveal that τ/
on the SAF decreases with depth from 0.6 at about 1.2 km depth to 0.24 at the bottom of the pilot hole at 2.1 km depth. Furthermore, a number of geodynamic studies find values for the long-term average stress supported by major faults in the range of τ/
≈ 0.2–0.3 [Bird and Kong, 1994; Humphreys and Coblentz, 2007]. These low values of τ/
are exactly what our model predicts, without any alteration of hydrothermal and frictional properties from their laboratory-measured values.


4.3. Scaling of Slip and Slip Pulse Length With Propagation Distance
[68] Over the modeled propagation distance of 16 m, our ruptures appear to be growing in an approximately self-similar manner. In particular, the total slip locked in by the slip pulse increases nearly linearly with propagation distance, as shown in Figure 9a. Figure 9b shows the slip history of the crack-like rupture, with that of the slip pulse overlaid for comparison. Slip at the center of the crack-like rupture increases at about 1 mm/m rupture length. When a crack-like rupture terminates, stopping phases propagate inward to lock the fault; during this time, additional slip occurs, such that the final displacement at the center could be nearly twice as large as the instantaneous value shown here. Then the average final slip in the area ruptured in a crack-like event will be on the order of 1 mm/m rupture length. On the other hand, for a growing slip pulse, the final slip grows with distance from the center at about 0.14 mm/m (Figure 9a). Assuming this M-shaped linear slip profile persists as the rupture grows, the average slip per rupture length is 0.07 mm/m, which is in good agreement with measurements of 0.01–0.1 mm/m for large earthquakes [Scholz, 1982; Scholz et al., 1986; Manighetti et al., 2005].

[69] Another indication of self-similar growth is that, after an initial transient related to the nucleation process and the transition to a slip pulse rupture mode, the length of the slip pulse increases linearly with time. By assuming a rupture speed of cs (as is seen in our simulations), we can convert time into propagation distance; see Figure 10, which shows the length of the slip pulse in Figure 9a as a function of propagation distance. We used the condition V > Vth for threshold velocities ranging from Vth = 0.001–0.1 m/s to define the extent of the actively slipping region. Regardless of the value of Vth, the length of the slip pulse grows at a rate of about 30 m/km, which is in acceptable agreement with, though about three times smaller than, lengths based on risetimes inferred from slip inversions of large events compiled by Heaton [1990], though those measurements are limited in their resolution by the finite bandwidth of seismic data. Multiple time window slip inversions of magnitude ∼1 mining-induced earthquakes, which have source dimensions of 10–20 m, suggest risetimes around 3–4 ms (much larger than risetimes in our simulations; see Figure 7a), though these values are likely upper bounds since the width of the time windows used was 1.5 ms [Yamada et al., 2005].

[70] The rate at which the slip pulse length increases with propagation distance can be related to the relative speed of the rupture front, vr, and the healing front, vh. The rate of 30 m/km = 0.03 is equal to, at least for large times, (vrt − vht)/(vrt) = 1 − vh/vr, implying that vh = 0.97 vr. The self-similar slip pulse solutions of Nielsen and Madariaga [2003] and Broberg [1999, p. 418] have the following property: Given a value of G, the energy supplied to the crack tip per unit area advance of the crack, relative to G0, the value of G for a quasi-statically expanding crack of the same length, there is a unique relationship between vr and vh. The very low values of G in our simulations (a consequence of small L) require that vr be close to the limit speed cs. However, in this limit, small variations in vr at fixed G/G0 are associated with large variations in vh, making it difficult to constrain expected values of vh. Exactly what determines the precise value of vr (and hence vh) in our simulations remains unclear. In particular, we caution that the measured rate of increase of slip and pulse length with propagation distance may be sensitive to the artificial nucleation process we employ. This issue requires further study.
5. Summary
[71] The thermal weakening mechanisms of flash heating of microscopic asperities and thermal pressurization of pore fluid have been implemented into a spectral boundary integral equation code for elastodynamic rupture propagation. Flash heating is encapsulated in a framework of rate-and-state friction that includes the direct effect and evolution toward steady state conditions over a slip distance L. We use values of L comparable to asperity dimensions (∼20 μm), as suggested by laboratory experiments, and a range of hydrothermal properties consistent with laboratory measurements of fault gouge and reasonable assumptions about the degree of damage induced by the passage of high stress concentrations near rupture fronts.
[72] For faults with a purely velocity-weakening steady state strength, Zheng and Rice [1998] have shown the existence of a critical stress level, τpulse, below which crack-like ruptures cannot exist. The value of τpulse/ predicted by the observed velocity dependence of flash heating is about 0.2–0.3. We have generalized the theory of Zheng and Rice [1998] to accommodate weakening mechanisms that involve a set of variables (like T and p in our model) that do not achieve velocity-dependent steady states over the duration of a rupture. When temperature and pore pressure are permitted to evolve in response to shear heating, theoretical considerations and a series of calculations reveal that additional weakening due to thermal pressurization enables ruptures to propagate in the crack-like rupture mode at background stresses slightly below τpulse. However, while thermal pressurization mildly increases the range of crack-like solutions in parameter space, it does not prevent the occurrence of self-healing slip pulses. While there is only minor sensitivity of the crack-pulse phase boundary to the thermal pressurization parameters, the efficiency of thermal pressurization is likely to be important in determining the average stress doing work within ruptures at larger amounts of slip than those we have been able to model in this work. Furthermore, we have found that temperature and pressure conditions within the shear zone are quite sensitive to its width, even for widths below 100 μm.
[73] Our models suggest that faults will host ruptures at the minimum stress level required for a rupture, once nucleated, to propagate in a self-sustaining manner. This critical level is around τpulse/ ∼ 0.2–0.3, which is consistent with the stress state inferred to be acting on major faults like the SAF. At this critical stress level, ruptures take the form of slip pulses. Such slip pulses have reasonable static stress drops of ∼3 MPa and expand in a nearly self-similar manner, such that slip increases with propagation distance at a rate of ∼0.14 mm/m. The average stress doing work during slip, τheat, is below 15 MPa, and this quantity is expected to decrease with increasing propagation distance since thermal pressurization will have more time to reduce fault strength.
[74] We have chosen to use parameter values taken directly from laboratory experiments, which constrains us to model two-dimensional ruptures out to only 16 m in each direction from the hypocenter, extrapolating our results to larger source dimensions. An outstanding question we have not addressed is how different the ground motion generated by our ruptures might be from those using more common parameter values. For example, most rupture models (like those in the Southern California Earthquake Center code validation tests [Harris and Archuleta, 2004; Harris et al., 2009]) feature strength drops that are only ∼10 MPa larger than static stress drops; in our simulations, the strength drop is around ∼90 MPa.
Acknowledgments
[106] The largest simulations were performed on Harvard's BlueGene/L; we thank Aaron Culich, Robert Parrot, and Joyanta Sircar of the Harvard SEAS/IT group and Kirk Jordan of IBM for technical support. We also thank associate editor Dan Faulkner and two anonymous reviewers for their insightful suggestions. The study was supported at Harvard by NSF-EAR award 0510193 and by the Southern California Earthquake Center as funded by Cooperative Agreements NSF EAR-0106924 and USGS 02HQAG0008 (SCEC contribution number 1228).
Appendix A:: Justification of Steady State Strength Assumption





Appendix B:: Characteristic Changes in T and p During Nucleation
[77] In this appendix, we estimate characteristic changes in T and p during the nucleation process. We seek only an estimate here and will consequently employ several approximations. Our approach is to estimate V and τ (and hence the shear heating rate) within the nucleation zone and then calculate the changes in T and p that result from this amount of shear heating over the duration of the nucleation process.
[78] Using the Gaussian shape for our stress perturbation leads to a complicated evolution of V and τ within the nucleation zone. To simplify things, we consider instead an equivalent boxcar-shaped perturbation of amplitude τp and half-width Dp. These parameters are related to τper and Dper by setting Dp = 3 Dper (so that 99.7% of the Gaussian-shaped perturbation lies within the boxcar-shaped perturbation) and preserving the net force imparted by the perturbation. It follows that τp = 0.42 τper. The characteristic duration of the nucleation process is taken to be the time it takes a shear wave to cross the boxcar perturbation, i.e., 2 Dp/cs = 6 Dper/cs. For Dper = 2.5 cm, this is 50 μs.
[79] We further assume that when the perturbation is applied at t = 0, f drops instantaneously to fw. This is justified if the resulting V far exceeds Vw and the state evolution time is small compared to the duration of the nucleation process. Assuming small changes in p, the fault strength is then τ = fw0. For a spatially uniform stress field, the elastodynamic response is the radiation-damping response; for an applied load of τb + τp within the boxcar perturbation, that suggests τ ≈ τb + τp − (μV)/(2cs). Setting stress equal to strength and eliminating τp in favor of τper, we find V ≈ 2 cs (τb + 0.42 τper − fw
0)/μ. Typical values of V are ∼10 m/s, so V ≫ Vw. At this V, slipping a distance of L = 20 μm takes only ∼2 μs, which is far less than the characteristic duration of nucleation.





Appendix C:: Details of the Crack-Pulse Transition
C1. Transition Behavior Neglecting T and p Changes


[83] Figure C1a shows the evolution of (V, f) at x = 0.1 m for the pulse-like rupture indicated in Figure 4a as “Figure C1a.” Figure C1b presents a magnified view of the transition to the slip pulse rupture mode. fss(V, T0) and a radiation-damping line which fits tangentially to fss(V, T0) are also plotted. [1] to [4] in these figures mark the following sequence of events. As the rupture front approaches, the fault is loaded with almost no state evolution because V is very small [1]. At the rupture front, f achieves its maximum value. Behind the rupture front, state evolution and unloading (corresponding to a decrease in ϕ) lower f slightly below fss(V, T0) [2]. As the fault decelerates, f follows a path nearly parallel to fss(V, T0) [3]. When V reaches Vpulse, defined as the slip rate where fss(V, T0) and a radiation-damping line intersect tangentially (Figure 1), f departs from a path parallel to fss(V, T0) and moves onto a trajectory that approximately follows the radiation-damping line that is just tangent to fss(V, T0) [4]. This transition is similar to one reported by Cochard and Madariaga [1994] although V dropped to zero abruptly in their case due to the use of a purely rate-dependent friction law.



[84] Figure C1c shows slip rate histories at x = 0.1 m for different values of τb/0. The horizontal line represents Vpulse, where the transition to a pulse-like solution occurs. With increasing τb/
0, the transition time increases monotonically. Note that in the case with τb/
0 = 0.2778, the transition does not occur during the duration of the calculation but V keeps decreasing. Here the question arises, is this solution incorrectly identified as crack-like simply because the calculation is too short? Figure C1d is a plot of the inverse of the crack-pulse transition time as a function of τb/
0, which reveals that the transition time appears to diverge at a finite value of τb/
0 around 0.28. Therefore we conclude that there are indeed crack-like solutions that will never become pulses, no matter how far they propagate.
C2. Transition Behavior Accounting for T and p Changes
[85] When T and p are permitted to change, certain properties of the crack-like solutions also change. Figure C2a shows the evolution of slip rate for the crack-like rupture shown in Figure 2a (r = 0.8, 2w = 25 μm, τb/0 = 0.2143). Deceleration of fault motion takes place behind the rupture front followed later by acceleration. We take this acceleration as an indication of the transition to a crack-like solution. The calculations with T and p changes were carried out at least until we recognized the transition to a crack-like solution or growing/arresting slip pulse solution. Figure C2b shows a plot of friction coefficient versus slip rate at x = 0.1 m. The trajectory traces in the direction of decreasing V just below fss(V, T) behind the rupture tip although fss(V, T) depends on T. However, at a certain point, the trajectory reverses and moves in the direction of higher velocities and lower friction coefficients. In this case, V, T, and p keep increasing (Figure C2c), and it is likely that this point will eventually achieve either a complete stress drop or the onset of melting; its fate depends on hydraulic properties and the width of the shear zone [Rempel and Rice, 2006]. Figure C2d shows the history of f and fss(V, T) at x = 0.1 m, indicating that fss(V, T) becomes larger than f at the transition to a crack-like solution and, after that, f traces just above fss(V, T).

[86] Figure C3a shows details of the growing pulse solution in Figure 4b (r = 0.8, 2w = 50 μm, τb/0 = 0.2143). This case is very similar to the case without T and p changes; behind the rupture front, f traces just below fss(V, T) before changing its track to a radiation-damping line (Figure C3b). Figure C3c shows the history of V, T, and p at x = 0.1 m, showing how diffusion of heat and fluid away from the shear zone reduce T and p after fault motion ceases; this is accompanied by an increase in fault strength. In pulse-like solutions, f is always below fss(V, T) after the rupture front passes, and the transition to a pulse-like solution takes place before one to a crack-like solution (Figure C3d). Following the discussion in the previous sections, we discuss this in terms of Vpulse.

C3. Improved Condition for Nonexistence of Crack-like Ruptures
[87] We can also use the singular crack model discussed in Appendix A to obtain an improved estimate of the critical stress level below which crack-like ruptures cannot exist. Our strategy is to estimate the elastodynamic relation between τ and V from the singular crack solution, and ask whether or not there is a simultaneous solution to this equation and the steady state strength expression (equation 25).




Appendix D:: Numerical Implementation
[89] In this appendix, we describe the implementation of thermal pressurization and flash heating (in the framework of rate-and-state friction) in our spectral boundary integral equation code. We first discretize the fault using a uniform grid spacing Δx (in the boundary integral equation method, only the fault is discretized). Diffusion of heat and pore fluid is calculated using a finite difference method by setting, at each point along the fault, a one-dimensional grid perpendicular to the fault with uniform spacing Δz. This is illustrated in Figure D1a.










[92] The initial conditions δ(x, 0) and {θi(x, 0)} are given. Note that neither τ(x, 0) nor V(x, 0) can be specified independently of the given initial conditions; instead, they are calculated by solving the constraint equation (D3). Combining equations (D1)–(D4) yields a set of coupled functional differential equations for δ(x, t) and {θi(x, t)}.
[93] An extensive literature exists on the numerical solution of functional differential equations [e.g., Baker, 1996, 2000; Brunner, 2004]. Solution methods are similar to those for ordinary differential equations (ODE), with the additional necessity of choosing an appropriate quadrature rule to evaluate the functional term Fn(t). With the exception of certain low order quadrature methods, this requires a continuous approximation of Dn(t) for any past time t. The main classes of ODE solvers, such as one-step (e.g., Runge-Kutta [Enright et al., 1994]) and multistep methods, can be brought to bear on functional differential equations [Tavernini, 1971; Cryer and Tavernini, 1972]. The particular approach we use in this study is a second-order method similar to Heun's method [Cryer and Tavernini, 1972] and the method proposed by Lapusta et al. [2000] within the context of the spectral boundary integral equation elastodynamic framework. The method can be thought of as a second-order predictor-corrector method, for which higher-order generalizations in the form of multistep methods have been studied by Jackeiwicz [1986]. The method falls within the class of continuous Runge-Kutta methods [Baker, 1996; Bellen and Zennaro, 2003] which permit evaluation of Dn(t′) (and {θi(x, t′)}, though this is not necessary) for any t′ in [0, t] (which permits the use of high-order quadrature rules to evaluate the functional term). The second-order method we use provides second-order accurate values of Dn(t′) via a quadratic interpolation polynomial, as discussed by Bellen and Zennaro [2003, p. 122] and Maset et al. [2005]. For those wishing to develop higher order one-step methods, Maset et al. [2005] has introduced a family of explicit continuous Runge-Kutta methods up to fourth order. For neutral functional differential equations (i.e., ones in which the time derivative of the field appears within the functional term), an approximation to the rates at past times is also needed and an iterative solution for the rates at the current time step is required [Enright and Hu, 1997]. This case arises for our equations if the functional in (D4) is integrated by parts to extract the static elastic response. The remaining convolution integral involves the spatial Fourier transform of V(x,t) at the current and previous times [Perrin et al., 1995; Geubelle and Rice, 1995]. Jackeiwicz [1986] cautions that the approximation of the rates must be at least one order greater than the approximation of the fields; otherwise, the method suffers from order reduction.


[95] The specific method to advance from the current time t to t + Δt follows. Assume that the values of δ(x, t) and {θi(x, t)} are known from the previous time step, as well as the functional term ϕ(x, t) and the slip history δ(x, t′) for t′ in [0, t].
[96] 1. Set the slip rate at the beginning of the time step by setting stress equal to strength and solving, for V(x, t), the nonlinear equation (D3).

[98] 3. Evaluate the stress transfer functional ϕ(x, t + Δt) using the procedure discussed below.

[100] 5. Iterate by repeating steps 3 and 4 as many times as desired to obtain increasingly more accurate values of δ(x, t + Δt), {θi (x, t + Δt)}, and ϕ(x, t + Δt). A single iteration, as taken in Heun's method, is used in this work.





Appendix E:: Convergence Analyses

[103] h* is obtained from a linear stability analysis about states of steady sliding. Under the highly nonlinear conditions at a rupture front, the length scale that must be resolved for an accurate solution is that characterizing the strength drop, from approximately f00 to fw
0. This length scale is ∼μL/(f0 − fw)
0 times a monotonically decreasing function of vr. Contraction of the process zone with increasing vr makes this problem even more numerically demanding than implied by equation (E1).
[104] We first conducted a set of convergence tests in which we refined either Δx or Δz while holding the other grid spacing fixed. With these tests we were able to determine the relative contribution to the total error coming from the spatial discretization of the elastodynamic equation and from the diffusion equations. To maximize computational efficiency, it is desirable that the relative error contributions be approximately equal. In other words, nothing is gained by using an incredibly fine grid spacing for the diffusion equations without simultaneously using a fine grid spacing to discretize the fault. We found that the error contributions from the two spatial discretizations were approximately equal for Δx/Δz = 100.
[105] We then performed a convergence test in which we refined both Δx and Δz simultaneously, with a fixed ratio of Δx/Δz = 100, for the case with 2w = 100 μm, r = 0.8, and τb/0 = 0.2381. Figure E1a shows the L∞ errors (the maximum, over the entire fault, of the absolute value of the error) in δ, V, τ, T, and p at t = 84 μs, with respect to the case with Δx = 3.125 × 10−2 mm and Δz = 3.125 × 10−4 mm, showing second-order convergence. Note that the L∞ errors are almost always determined by a single grid point at the rupture front. The magnitude of the numerical error is not negligibly small at the rupture front; for Δx = 0.5 and 1 mm, the L∞ error in V is on the order of 1 m/s and the peak value of V is around 15.6 m/s (making a 6% relative error). However, the main target of this work is to determine the rupture mode (crack or pulse), and the distinguishing features of these rupture modes are first seen in the central region of the rupture (Figures C2a and C3a). Figure E1b shows the error in the fields at the central pair of grid points, showing that the numerical error in V at the center of the rupture is very small, and the solution converges with second-order accuracy; with Δx = 1 mm, the error in V is below 1 mm/s. In the calculations presented in the main part of this paper, we use Δx = 0.5 mm and Δz = 5 μm in the cases with 2w = 100 μm, such that the expected numerical errors are on the order of 0.01 μm in δ, 0.1 mm/s in V, 0.1 kPa in τ, 10−3 K in T, and 1 kPa in p at the center of the rupture. In simulations with different values of 2w, we change Δz to maintain a fixed number of grid points within the shear zone by setting Δz = 0.1w.

