Intrinsic and apparent short-time limits for fault healing: Theory, observations, and implications for velocity-dependent friction

Authors


Abstract

[1] The time-dependent healing of frictional strength, whose underlying mechanism may vary, is often logarithmic with time, after a certain time duration called the cutoff time. We theoretically show that the cutoff time depends on the initial strength at the beginning of quasi-stationary contact. This comes from the fact that the healing rate depends negatively and exponentially on the current strength. If healing starts with the minimum strength attained instantaneously upon the application of normal load, “intrinsic” cutoff time tcx, which reflects the reaction rate constant of the underlying physico-chemical process, will be observed. In general, the observed cutoff time is the sum of tcx and the effective contact time tini of asperities at the beginning of the healing. Hence, in slide-hold-slide (SHS) experiments, where tini is inversely proportional to the slip velocity Vprior in the sliding preceding the hold, two regimes are predicted. For high Vprior (tinitcx), the observed cutoff time is independent of Vprior, being ∼tcx. For low Vprior (tinitcx), the observed cutoff time is “apparent,” being ∼tini inversely proportional to Vprior. Both regimes have been identified in laboratory data. Incorporation of the present cutoff time theory into the rate and state evolution laws predicts that the velocity dependence of steady state will diminish at high velocities where effective contact time is ≪tcx. Furthermore, the two regimes for SHS tests are divided by this cutoff velocity. This also has been confirmed with laboratory data. Implications on the very long cutoff time observed for natural repeating earthquakes are discussed.

1. Introduction

[2] Time-dependent healing of frictional strength [e.g., Dieterich, 1972] is one of the key mechanisms that govern earthquake cycles [e.g., Beeler et al., 2001]. It is also a cornerstone of rate and state friction laws [e.g., Ruina, 1983], which have been successfully applied to explain a wide spectrum of earthquake phenomena [e.g., Scholz, 1998].

[3] When the frictional interface is in (quasi-) stationary contact, frictional strength often increases logarithmically with time t [e.g., Dokos, 1946; Dieterich, 1972].

display math

Here, Θ is the frictional strength normalized by the applied effective normal stress equation image. Since it is known that more shear stress (τ) is necessary to cause faster slip on the interface at the same state (strength), we need to define frictional strength Θequation image as the shear stress required to cause a slip of an arbitrarily chosen reference velocity V*, using the following constitutive equation:

display math

where a is an empirical constant called the coefficient of direct effect [e.g., Dieterich, 1979; Ruina, 1983]. Thus defined, Θ, taken distinct from the shear stress, is a natural extension of the classical frictional strength as a threshold for slip to occur [Nakatani, 2001]. Since this defined strength is determined by the internal physical states of the frictional interface, Θ can be regarded as a state variable in rate and state friction laws [Ruina, 1983]. In the present paper, Θ will be referred to as “strength” or “state” interchangeably.

[4] The logarithmic healing law (1) has two parameters. The first parameter b is the magnitude of strengthening per an e-fold increase of contact time (Figure 1a). Although b is often called the “healing rate,” we avoid this wording. As seen from the plot of Θ against linear time (Figure 1b), the real healing rate equation image is always decreasing with time. We reserve the term healing rate for equation image, not b.

Figure 1.

Form of function Θ(t) = Θ0 + b ln (t/tc + 1) plotted against (a) log time and (b) linear time. Note that the vertical scale is different between Figures 1a and 1b.

[5] The present paper focuses on the other parameter tc. This parameter is necessary to avoid the negative divergence of logarithm as t → 0 and is often called the “cutoff time” because the increase of Θ per an e-fold increase of time is much less than b for 0 ≤ ttc (Figure 1a). Again, however, “cutoff” is a misleading word because healing is more active at a smaller t even for ttc, as seen from Figure 1b. It is not that the process of healing is cut off for ttc. Rather, it is just that 0 ≤ ttc is too short a time interval to see a significant increase of Θ. Nonetheless, we keep the familiar terminology of cutoff time to refer to tc.

[6] The cutoff time tc is important not only for ttc but also for ttc. For ttc, ΔΘ(t) ≡ Θ(t) − Θ(t = 0) has a dependency of −bln(tc) on the cutoff time. So, without knowing the value of tc, one cannot tell the amount of strength recovery even for ttc. In many laboratory healing tests at room temperature [e.g., Dieterich, 1972], tc seems to be order 0.1–1 s, while tc up to half a day (5 × 104 s) has been registered in laboratory tests under hydrothermal conditions [Nakatani and Scholz, 2004a]. A cutoff time as long as 100 days (9 × 106 s) has been observed for healing of natural faults [Marone et al., 1995]. Assuming a typical value of b = 0.01 from laboratory experiments, a difference of tc by 8 orders of magnitude, for example, results in a difference of ΔΘ by 0.18 in terms of frictional coefficient. Predicting the amount of frictional healing ΔΘ in terms of frictional coefficient is important. For example, we can estimate the effective normal stress equation image of earthquake faults, an enigmatic problem [e.g., Wang et al., 1995; Scholz, 2000], by equation image = Δτ/ΔΘ, where Δτ is the interseismic strength recovery in terms of stress, which can be independently estimated from seismological and geodetic observations.

[7] From an experimental point of view, we should note that the −bln(tc) dependence of ΔΘ for ttc means that the value of tc can be determined from the data for ttc only. This is good news because conducting healing experiments for very short t is often technically difficult.

[8] Furthermore, recent process-based theories of logarithmic healing [e.g., Brechet and Estrin, 1994; Nakatani and Scholz, 2004b] suggest that tc is inversely proportional to the reaction rate constant of the underlying process. For example, experiments by Nakatani and Scholz [2004a, 2004b] have found an Arrhenius-type temperature dependence of 1/tc. On the other hand, there is a laboratory data set [Marone, 1998a] suggesting that tc depends on the slip velocity employed in the experiment. This implies that tc is not necessarily an intrinsic constant of the healing process.

[9] In this paper, we will theoretically show in section 2 that the observed tc is the sum of (1) the process rate-related characteristic time that appears in the original Brechet and Estrin (BE) model and (2) the effective contact time of asperities at the beginning of (quasi-) stationary contact. The latter factor was not considered in the BE model, but it will be shown that this is a natural consequence of general physical systems that result in logarithmic growth, including the BE model. Further, thus developed cutoff time theory is incorporated into the existing friction laws. This leads to a quantitative prediction of the high-velocity cutoff for the velocity dependence of strength in steady state sliding [e.g., Okubo and Dieterich, 1986], which is important in fault dynamics [e.g., Ruina, 1983]. In section 3, we will show that these theoretical predictions are consistent with the two experimental data sets mentioned above.

2. General Theory of the Cutoff Time of Logarithmic Growth

[10] Following the classic adhesive theory of friction [Bowden and Tabor, 1964], in which the macroscopic frictional strength is assumed to be in direct proportion to the real contact area, healing has been thought to result from the time-dependent increase of real contact area [e.g., Dieterich, 1972; Scholz and Engelder, 1976; Dieterich and Kilgore, 1994]. Brechet and Estrin [1994] followed this and showed that the specific form of logarithmic growth can be derived from the squashing of the asperity contacts with a strain rate depending exponentially on the driving stress, which decreases as the contact area grows. Although we think this is reasonable, we below develop a theory of cutoff time of logarithmic healing at a more general level detached from an actual physical mechanism. BE model is one of the possible concrete physical system that falls in the category of the systems discussed below. Specific correspondence will be shown in section 4.1.

2.1. Dependence of Cutoff Time on the Initial Condition

[11] Consider a time-evolving system y(t) whose growth rate decreases exponentially with its own level at the moment. Such a system can be generally described by

display math

where λ is a constant characterizing the exponential dependency and C is another system constant. For brevity, we adopt a following convention throughout the paper: z0 denotes the value of quantity z at t = 0, and equation image0equation image(t = 0). Δz is the difference of quantity z from z0. Δequation image denotes the time derivative of Δz.

[12] We rewrite (3a) into the following form in terms of the increment Δy

display math

From the (3a) form, we see that equation image0 = C exp(−y0/λ) depends on the initial value y0. The solution of (3) is

display math

This is a logarithmic growth with a cutoff time tc = λ/equation image0. Therefore we see that the logarithmic healing (1) results from the following evolution law, which is of (3) type,

display math

Here P and b are system constants determined by the specific physical mechanism of the healing, and are independent of Θ. Rewriting (5a) into the following form represented as to the increment is convenient:

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The cutoff time tc of the resultant logarithmic healing (1) is given by

display math

[13] The essence of our theory developed below is that the initial healing rate equation image0 and hence tc depend on the initial state Θ0, as seen from (5a). Explicitly showing this, (6a) can be rewritten as

display math

[14] The differential equation (5) must hold for any healing mechanism that results in logarithmic time dependence, and hence the dependence of tc on initial condition (6) must be true for logarithmic healing in general.

2.2. Meaning of the Observed Cutoff Time: Intrinsic or Apparent?

[15] As shown above, the cutoff time tc observed in log time healing reflects the initial strength Θ0. In situations where periods of slip and stationary contact alternates, such as in laboratory slide-hold-slide (SHS) tests and recurring earthquakes on the same patch of a fault, Θ0 can take different values depending on the history of the slip in the preceding slide (coseismic) period [e.g., Dieterich, 1978; Ruina, 1983]. On the other hand, an unambiguous choice for reference strength would be the minimum value achieved instantaneously upon the application of the normal load. We refer to this state as the X state and denote it with Θx. Since the X state for a given normal load should be uniquely determined by the material and geometrical properties of the frictional interface, Θx is a system constant. For simplicity, we focus on evolution under a constant normal stress. If we start measuring the healing from this state, it should be

display math

with

display math

where equation imagex is the healing rate for the interface with Θ = Θx. As seen from (5a), equation imagex is a system constant, and therefore so is tcx. As shown in section 4.1, the cutoff time that Brechet and Estrin [1994] have derived corresponds to tcx. We refer to tcx as the “intrinsic cutoff time,” as it can be related to the reaction rate constant of the underlying physico-chemical process [Brechet and Estrin, 1994; Nakatani and Scholz, 2004b].

[16] Now we proceed to derive tc in more general situations, where Θ0 can be different from Θx, as mentioned earlier. The effective contact time concept [e.g., Dieterich, 1978] assumes that the steady state at a given slip velocity V is equivalent to the state achieved in stationary contact for an average age of asperity contacts

display math

Here D is a characteristic length scale for the asperity population. In case of SHS tests, the state at the beginning of hold must be the state set by the steady state sliding at Vprior in the preceding slide. Hence the effective contact time at the beginning of the hold, which we refer to as tini, is given by

display math

In this case, surface state at t = 0 is stronger than Θx, being

display math

Therefore the initial healing rate is smaller than equation imagex. From (5) and (10), we obtain

display math

Using (6), (8), and (11), the observed cutoff time tc is given by

display math

[17] In the above, we introduced the initial contact time tini using the average contact time in steady state sliding. This, however, was merely to use a familiar example. In general, we may define “equivalent initial contact time” tini for any given initial state Θ0 (>Θx) by

display math

[18] To sum up, we have shown that the time evolution of the system (5) with a given initial state Θ0 will follow

display math

The appearance of the initial state tini0) in the observed cutoff time is a direct result of the governing differential equation (5) and hence is inevitable for any logarithmic growth.

[19] Usually, we can observe only the sum tc = tini + tcx. However, as we will demonstrate with actual data sets in section 3, we can tell which of tini and tcx is the dominant part of the observed tc by comparing tests with different Vprior and thus tini (=D/Vprior). In Figure 2, ΔΘ predicted by (14) is plotted for different values of tini. For low Vprior such that tinitcx, (14) is approximated by

display math
display math

Hence the healing curve shifts to the right as Vprior is decreased because tini is inversely proportional to Vprior. For high Vprior such that tinitcx, (14) is approximated by

display math

Hence the healing curve is virtually fixed, independent of Vprior.

Figure 2.

Theoretical prediction (14) with different values of tini, the effective contact time at the beginning of stationary contact. In slide-hold-slide (SHS) tests, tini can be controlled as tini is inversely proportional to the slip velocity Vprior in the preceding slide period.

[20] Conversely, if the observed tc is found to be independent of the Vprior employed, we can tell that the observed tc is the intrinsic cutoff time tcx. At the same time, we can tell the effective contact time for the employed Vprior is much less than tcx. In contrast, if the observed tc is found to be inversely proportional to Vprior, we can tell that the observed tc is “apparent,” being the effective contact time for the Vprior. In this case, the value of the length dimension constant D in (9) is determined to be Vprior/tc. At the same time, tcx is known to be much less than the observed tc. Of course, there also exists an intermediate regime of Vprior where tini is of the similar order of magnitude as tcx. In this case, the observed tc will increase as Vprior decreases, but the dependence is weaker than the inverse proportionality.

2.3. State Evolution Laws and the Cutoff Time

[21] The logarithmic healing effect of friction has been represented in the existing state evolution laws for friction. However, the cutoff time, either intrinsic or apparent, was not considered when those laws were constructed. Here we examine the existing evolution laws in light of the presently proposed cutoff time theory and make the necessary modifications to incorporate it. Modified laws will be used in the data interpretation in section 3.

[22] Currently, two different evolution laws called the “slip law” and the “slowness law” are widely used, each of which can only reproduce limited aspects of the observations. We start with the steady state where the both evolution laws agree and then will discuss issues specific to each evolution law.

2.3.1. Velocity Dependence of Steady State and Its High-Velocity Cutoff

[23] When slip at a constant velocity is maintained for sufficiently long times, it is believed that the state of the frictional interface reaches a steady value for that velocity [e.g., Ruina, 1983]. In the traditional evolution laws, the steady state is given by

display math

where Θ* is the value of steady state for a reference velocity V*. This is in common to the slip and slowness laws. The negative velocity dependence is explained [e.g., Marone, 1998b] by the effective contact time concept (9a). Equation (17) predicts that Θss(V) decreases with an increase of log velocity, without limit. This obviously corresponds to the logarithmic healing without the minimum bound associated with the X state discussed in section 2.2.

[24] The modified version can be obtained by simply inserting the effective contact time (9a) into the healing equation (7), where the intrinsic cutoff time resulting from the X state has been incorporated.

display math

The newly incorporated constants D and tcx are observable in SHS tests employing appropriate Vprior conditions, as shown in section 2.2.

[25] Contrasting with (17), this modified version predicts the negative dependence of steady state on log velocity diminishes at high velocities as the effective contact time becomes less than the intrinsic cutoff time and hence has little effects on the state. This may be better seen by rewriting (18) as

display math

where

display math

Existence of such a high-velocity cutoff has been suggested earlier [e.g., Okubo and Dieterich, 1986].

[26] The cutoff velocity Vcx can be directly observed in velocity step (V step) tests [Blanpied et al., 1987; Weeks, 1993; Nakatani and Scholz, 2004a], where velocity dependence of the steady state can be measured as the magnitude of so-called “evolution effect” [e.g., Marone, 1998b]. As seen from (20), the present theory explains the Vcx as the velocity at which teff becomes equal to tcx. This leads to an interesting prediction on the relation between the V step tests and SHS tests. Using (20), we can rewrite (9b) into

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The cutoff time (12) observed in SHS tests is therefore expressed as

display math

Thus the high- and low-velocity regimes for healing tests are divided by the cutoff velocity Vcx for the velocity dependence of steady state; If SHS tests are done with VpriorVcx, apparent cutoff time inversely proportional to Vprior is observed as described by (15). If VpriorVcx, intrinsic cutoff time tcx independent of Vprior is observed as described by (16). Equation (22) describes the behavior for any Vprior including the intermediate regime. In the data analysis in section 3, consistency between SHS tests and the V step tests will be discussed.

2.3.2. Slip Law

[27] This type of evolution law assumes that evolution of state occurs with slip displacement. Specifically, it says that the state evolves exponentially with slip over a characteristic evolution distance of Dc, toward the steady state value for the given velocity [e.g., Ruina, 1983].

display math

Although slip law does not represent true time-dependent healing [e.g., Beeler et al., 1994; Nakatani and Mochizuki, 1996; Beeler and Tullis, 1997], the logarithmic healing effect is partially included as the negative dependence of steady state on slip velocity discussed already, which was traditionally represented by (17). As we have replaced it with (18) to incorporate our cutoff time theory, the modified slip law is given as

display math

[28] The parameter D only affects the effective contact time observed as the apparent cutoff time in SHS tests, not affecting the slip-dependent transient, which is solely described by Dc. Hence (24) can describe these two independent observations at the same time. Necessity of the two separate length dimension parameters in principle has become apparent only now as our cutoff time theory has shown that the effective contact time is an observable quantity, and so is D.

[29] Note that the relevance of (24) to time-dependent healing is limited to the setup of its initial condition through the steady state (18) in the preceding slide. It cannot be used to predict the healing itself, as the traditional slip law cannot.

2.3.3. Slowness Law

[30] The other widely used evolution law is slowness law, which allows true time-dependent healing [e.g., Beeler et al., 1994; Nakatani and Mochizuki, 1996; Beeler and Tullis, 1997]. The standard slowness law is

display math

Here the first and second terms represent the time-dependent healing and slip weakening, respectively [e.g., Beeler and Tullis, 1997; Nakatani, 2001]. Steady state is realized by the balance of the two terms. By setting equation image to zero, (25) gives

display math

which is exactly the same as (17).

[31] The same length dimension parameter L appears in the both terms of (25), but its meaning is clearly different. The meaning of L in the slip-weakening term

display math

is easily seen. It specifies the rate of the slip weakening to be −b/L per unit slip [Nakatani, 2001]. This role of L is akin to the role of Dc in the slip law in that both of them specify the rapidness of slip-dependent change.

[32] The healing term of (25), i.e.,

display math

has the form of (5), i.e., the general governing equation to produce logarithmic growth. Therefore (28) does predict the apparent cutoff time depending on the initial state, though this feature of the traditional slowness law had not been recognized until Nakatani [2001, Appendix A2.1] pointed out that a straightforward integration of (28) with the initial condition given by (26) leads to logarithmic healing with a cutoff time of tc = L/Vprior. As seen from (5b) and (6a), the preexponential factor of (28) constrains that tc for healing starting with Θ* (i.e., steady state at V*) is L/V*. Hence the parameter L in (28) is the constant relating the slip velocity and the effective contact time, playing the exact same role as D in (9).

[33] Thus the slowness law (25) cannot correctly describe the observations of slip-dependent evolution distance and the effective contact time at the same time. One may think that this problem can be solved by assigning separate parameters for the two terms as

display math

Nakatani [2001, Appendix A2.1] casually proposed this. However, as shown in Appendix A, this actually does not work.

[34] Now, we modify the slowness law to reflect the present cutoff time theory including the intrinsic cutoff time. The healing term should have the form of (5) to produce logarithmic growth. Furthermore, with a constraint that the healing starting with Θ0 = Θx must have a cutoff time of tcx, the healing term is fixed as

display math

[35] The time-dependent evolution with this equation from a general initial state Θ0 has been already given by (14), where both the apparent cutoff time and the intrinsic cutoff time are predicted. For the steady state to be (18) under the existence of the healing term (30), we need to modify the slip-weakening term (27) to

display math

[36] Hence the complete modified evolution law becomes

display math

The modified slowness law (32), differing from (25), has the length dimension constant in only the second term. We have introduced this constant D from the steady state equation (18), where D is the constant relating slip velocity and the effective contact time (9a). However, D also plays the other role of governing the slip weakening, as seen from (31) which describes linear slip weakening at a rate of −b/D per unit slip for D/Vtcx. Therefore (32) cannot correctly describe the observed effective contact time and the slip-dependent transient at the same time, the same problem as in the traditional slowness law (25). However, besides this problem, it is already known that the linear slip-dependent evolution predicted by the slowness law does not agree with laboratory observations [e.g., Marone, 1998b; Nakatani, 2001; Kato and Tullis, 2001], whatever value is chosen for its length dimension constant. So, we limit the use of (32) to the phenomena not affected much by slip-dependent evolution, such as healing in quasi-stationary contact. Accordingly, D will be determined so that the observed effective contact time is correctly reproduced.

3. Interpretation of Laboratory Data

[37] As shown in section 2.2, we can tell if the observed cutoff time is apparent or intrinsic by examining its dependence on Vprior. In this section, we will demonstrate this point by examining two laboratory experiments. In each experiment, SHS tests were done for two different Vprior. As we intend a stringent test of the theory, we determine healing parameters from the data for one Vprior and see how well the theory with these parameter values predicts the data for the other Vprior. In addition, we also show how the behavior of the cutoff time in each experiment is consistent with the constraints on Vcx from the corresponding V step tests.

[38] Before showing the data, we below explain what exactly SHS tests measure, in order to see how the data should be compared with the theory. Figure 3 schematically illustrates a typical SHS test, where the loading ram is held stationary for a “hold” period of an arbitrary duration t. Each hold period is preceded by a slide period in which steady state sliding at a given slip velocity Vprior is achieved. Strengthening achieved during the hold period is then measured by imposing the slide at a given slip velocity Vreload. Although the slip velocity at the interface can differ from the load point velocity held constant at Vreload due to the elastic deformation of the loading system subject to the changing shear load, slip velocity is guaranteed to coincide with Vreload at the peak shear stress because the shear load does not change at this moment. Hence we know that the strength at this moment as Θpeak = τpeak/equation imagea ln(Vreload/V*). On the other hand, the strength at the beginning of the hold (t = 0) is known from the steady state stress as Θini = τss(Vprior)/equation imagea ln(Vprior/V*). If we choose Vreload equal to Vreload and look at the difference of Vpeak from τss, as is usually practiced in SHS tests, including the experiments examined in this paper, the direct effect terms in Θpeak and Θini cancel out, leading to

display math

Therefore the increase of strength (Θpeak − Θini) can be directly measured as (τpeakτss)/equation image, not requiring the correction for the direct effect. This is especially valuable in the present analysis, where the amount of strengthening, not only its time dependency, is used.

Figure 3.

Stress (solid curve in the upper axis), strength (dashed curve in the upper axis), and slip (lower axis) in a type of slide-hold-slide test, where the loading ram is held stationary during the hold period for a given duration (t). Dotted curve is the theoretically expected healing curve for an ideal hold period without slip.

[39] In the following, we compare the directly observable quantity (Θpeak − Θini) with the theoretical healing (14)(16), which results from the healing term (30) of the modified slowness law (32). Strictly saying, this is not correct for two reasons. First, Θ can change somewhat during reloading (from Θend to Θpeak in Figure 3), mainly because significant preslip occurs as the shear stress is raised. Second, because of slow slip continuing during hold, even Θend can be less than the ideal healing (dotted curve in Figure 3), where slip-dependent evolution is not accounted for. However, these effects of slip do not seem to affect our conclusions as will be shown by the modeling (details in Appendix B) using the full evolution laws (24) and (32). We proceed in this way to show that the observed behaviors of cutoff time result entirely from the nature of time-dependent healing.

3.1. Interpretation of Marone's [1998a] Healing Tests

[40] We first examine the data from Marone's [1998a] healing tests on a simulated gouge layer under a constant normal stress of 25 MPa. An initially 2.1 mm thick layer of nominally dry quartz gouge (initial particle size 50–150μm) was sheared within rough (200 μm RMS) granite surfaces. Experiments were done at room temperature. Two series of SHS tests (Figure 3) were done, one series with Vprior = Vrelaod = 10 μm/s and the other with Vprior = Vrelaod = 1 μm/s. Results for the both series are shown in Figure 4, where (τpeakτini)/equation image is plotted. Since Vprior = Vrelaod in the both series, this is equal to Θpeak − Θini as explained earlier.

Figure 4.

Results of SHS tests by Marone [1998a]. Data values were read from Figure 1b of Marone [1998a]. The ideal healing curve (15) fits the 10 μm/s series data with b = 0.0039 and D = 3.2 μm (blue dotted curve). The red dotted curve is the prediction of (15) for the 1 μm/s series with the same parameter values, which results in a 10 times larger tc.

[41] Both the 10 and 1 μm/s series show time-dependent healing with a similar log linear slope. All phenomenology in this experiment is typical of the healing mechanism due to solid-state contact deformation [e.g., Scholz and Engelder, 1976; Dieterich and Conrad, 1984]. A major finding here is the clear separation of the two trends from the tests employing the different slip velocities, as Marone [1998a, p. 69] pointed out that “a ten times increase in loading rate has about the same effect on (τpeakτini)/equation image as a ten times increase in hold time.”

[42] With our cutoff time theory, we interpret this phenomenon as the reciprocal dependence of the (apparent) cutoff time on Vprior(15). From the 10 μm/s series data, we obtain b = 0.039 and tc = 0.32 s (the blue dotted curve). This constrains that D = 3.2 μm (=10 μm/s × 0.32 s). The red dotted curve shows healing with a ten times larger tc, as predicted for Vprior = 1 μm/s by (15) with the same parameter values. The 1 μm/s series data roughly agrees with this. Hence we can conclude that the cutoff times observed in this experiment are apparent, being the effective contact time set in the slide preceding each hold period. Since such tc (=tini) ∝ Vprior−1 scaling is limited to tinitcx, intrinsic cutoff time is constrained to be ≪0.32 s, the smaller of the observed cutoff times.

[43] This latter conclusion of tcxD/Vprior is consistent with the fact that evolution effect was observed for this range of velocity in the V step tests done on the same condition [Marone, 1998a], which suggests VpriorVcx as discussed in section 2.3.1.

[44] At a closer look of Figure 4, we notice that the majority of the 1 μm/s series data is shifted from the 10 μm/s series trend by more than one decade, which is the maximum expected from the cutoff time theory. There might be something more that contributes to the observed shift. We note, however, that the data points for the 1 μm/s series appear to be divided into two subgroups. The upper subgroup agrees with our theoretical prediction very well.

[45] Finally, we evaluate the effects of slip-dependent evolution we neglected in the above analysis. First, we evaluate the change of Θ during reloading (from Θend to Θpeak in Figure 3). Since this change is predominated by the strength loss by slip weakening [e.g., Nakatani, 2001], we used the modified slip law (24). For the slip-dependent evolution distance Dc, we used 7.1 μm, consistent with V step tests on the same system [Marone, 1998a]. Details of this correction procedure are given in Appendix B1.1. Figure 5 shows the Θend value (open symbols) inferred from each observed Θpeak (dots). The corrected value (Θend − Θini) is generally greater than the observed (Θpeak − Θini) by ∼20%. However, the separation between the trends for different velocities is still consistent with tcVprior−1. The blue and red dotted curves are the predictions by (15) for the two velocities, with b = 0.0042, D = 1.9 μm, as suggested by the Θend for the 10 μm/s series. Second, we evaluate the effect of slow slip during the hold period, which can make the Θend smaller than the ideal healing (dotted curves) (15), where only the healing term (30) is considered. The red and blue solid curves are the predictions of the full slowness law (32), with the same parameter values (details in Appendix B1.2). Because of the slip-weakening term, they are slightly below the ideal healing curves, especially in the early part of the hold period, where slip velocity is still significant. However, the difference diminishes as the hold time extends. This would be due to the negative dependence of the healing rate on the current state (5), a general feature of logarithmic growth. After the slip has become very slow (because of the relaxed shear stress and the increased strength [Nakatani, 2001]), Θ grows faster than the ideal healing curve at the same hold time because the current state is less until it catches up. Therefore we expect that it is generally safe to neglect the effect of slow slip in the hold period as long as we look at the result for sufficiently large hold times.

Figure 5.

Value of Θend (open symbols) recovered from Θpeak (dots) values of Marone's [1998a] experiments (details in Appendix B1). The blue dotted curve is the ideal healing (15) fitting the 10 μm/s series Θend with b = 0.0042 and D = 1.9 μm. The red dotted curve is the prediction of (15) for the 1 μm/s series with the same parameter values, which results in a 10 times larger tc. These predictions by (15) only reflect the healing term (30) of the slowness law. They do not differ very much from the predictions by the full slowness law (32), which are indicated with the blue and red solid curves for the 10 μm/s series and the 1 μm/s series, respectively.

3.2. Interpretation of Hydrothermal Healing Tests

[46] Figure 6 shows the data from hydrothermal healing of a simulated quartz gouge layer at 200°C. The effective normal stress was 100 MPa. The pore space was filled with water at 10 MPa. An initially 3 mm thick layer of crushed quartz (particle size <63 μm, including many fines <1 μm) was sheared within stainless-steel surfaces with grooves (0.4 mm depth) perpendicular to the sliding direction. Two series of SHS tests are shown in Figure 6, one series with Vprior = Vreload = 13 μm/s, the other series with Vprior = Vreload = 1.3 μm/s. The former series is from the experiment shown in Figure 4a of Nakatani and Scholz [2004a]. They proved, by showing that the healing does not occur if the pore fluid is H2O in vapor phase, that the solution transfer through hot interstitial water is the underlying mechanism of the healing observed in this experiment. The log time healing showed b = 0.013, a value significantly larger than the range of b observed for the solid-state healing mechanism. The latter series is from a new experiment, which exactly mimics the former experiment except that a 10 times slower slip velocity was employed. This latter experiment showed a time-dependent healing with a similarly large b. In addition, both experiments showed slip-dependent erasure of healing over a very long Dc of several hundreds of microns (Table 1), very different from 1 to 10 μm associated with solid-state healing mechanism [e.g., Marone, 1998b]. Hence the healing mechanism in the latter series is thought to be of the solution transfer type, the same as that for the former series.

Figure 6.

Results of SHS tests, where the healing was due to solution transfer mechanisms through hot (200°C) interstitial water. The data (blue triangles) for the 13 μm/s series are from the experiment shown in Figure 4a of Nakatani and Scholz [2004a]. The 1.3 μm/s series data (black circles) are from a new experiment exactly mimicking it except the slip velocity. The 13 μm/s series data show a logarithmic healing with b = 0.013 and tc = 1200 s (the blue dotted curve). If tcx were much less than tini, a 10 times larger tc would be expected from (15) for the 1.3 μm/s series (red doted curve), which does not agree with the data. With a constraint of Vcx = 1 μm/s suggested by V step tests (Appendix C), tc for the 1.3 μm/s series is theoretically predicted to be 1.64 times (black dotted curve) of that for the 13 μm/s series, which agrees with the data. The one dot chain curve and two dots chain curve are predictions for the 1.3 μm/s series with Vcx = 2 μm/s and Vcx = 0.5 μm/s, respectively.

Table 1. Hydrothermal Healing Data
Number of HoldsPosition, mmHold Time, s(τendτss)/equation image(τpeakτss)/equation imageDc, μm
Tests With Vprior = Vreload = 13 μm/s (Figure 4a ofNakatani and Scholz [2004a])
13.282800−0.110.058950
24.8a3600−0.090.012660
36.3255600−0.090.068300a
48.010800−0.080.036170a
59.475600−0.090.05160
 
Tests With Vprior= Vreload= 1.3 μm/s (Additional Tests Done for This Paper)
13.082800−0.120.059780
24.63600−0.100.014740
36.4255600−0.090.064610
48.410800−0.090.029435
510.175600−0.090.040170

[47] The hold periods of these experiments were realized by quickly decreasing the shear stress to a prescribed level (τhold) less than the dynamic friction and holding the shear stress at that level for a given duration of time t (Figure 7), rather than by holding the displacement of loading ram stationary (Figure 3), but the difference is not important here. The earlier discussion about the cancellation of direct effect leading to (33) holds to these experiments as well, which justifies the comparison of healing amount between tests employing different slip velocities.

Figure 7.

Stress (solid curve in the upper axis), strength (dashed curve in the upper axis), and slip (lower axis) in a type of SHS test, where the hold period is realized by quickly decreasing the shear stress to a prescribed level (τhold) less than the dynamic friction and then holding the shear stress at that level for a given duration (t). Dotted curve is the theoretically expected healing curve for an ideal hold period without slip.

[48] Contrasting to the experiment examined in section 3.1, Figure 6 does not show a clear separation between the trends for the two velocities. The blue dotted curve indicates a logarithmic healing curve with b = 0.013 and tc = 1200 s, as suggested by the 13 μm/s series data. It is obvious that the 1.3 μm/s series data are close to this curve, rather than showing an inversely proportional dependency of the cutoff time on Vprior indicated by the red dotted curve. According to our cutoff time theory, such a cutoff time independent of Vprior is expected to be observed for initial conditions tinitcx, as described by (16). In this case, the observed cutoff time should largely come from the intrinsic cutoff time tcx.

[49] As discussed in section 2.3.1, the above condition of tinitcx is equivalent to VpriorVcx, where Vcx is the cutoff velocity for the evolution effect observed in V step tests. We now check this point, using the data from V step tests done on the same system at the same condition [Nakatani and Scholz, 2004a]. The data suggest Vcx ∼ 1 μm/s (see Appendix C), so, the condition VpriorVcx is actually not met at least for the slower Vprior of 1.3 μm/s, questioning our interpretation above following (16). Hence we now reinterpret the SHS data, using the nonapproximated equation (22), which can handle any Vprior including the intermediate range comparable with Vcx. For the 13 μm/s series, (22) with Vcx = 1 μm/s predicts tc = (1 + 1 μm/s/13 μm/s) tcx = 1.077 tcx. So, this series is safely regarded as a high-velocity case, where the process rate-related intrinsic cutoff time is observed. Consistently, an Arrhenius-type temperature dependence has been found for the cutoff time observed in tests employing Vprior = 13 μm/s [Nakatani and Scholz, 2004a, 2004b]. On the other hand, a significant effect of tini is expected for the 1.3 μm/s series as (22) predicts tc = (1 + 1 μm/s/1.3 μm/s) tcx = 1.77 tcx. Corresponding healing curve is shown by the black dotted curve in Figure 6. Thus trends for the two velocities are expected to be separated by a 1.64 (= 1.77/1.077) times difference in tc. This is different from our earlier interpretation of tc independent of Vprior, however, the expected separation is still minor and consistent with the data. Therefore we conclude that the SHS data are consistent with V step results suggesting Vcx = 1 μm/s in that the separation between the trends for Vprior = 13 μm/s and 1.3 μm/s is not great, definitely smaller than a decade difference in tc (red dotted curve) expected for VpriorVcx. This interpretation holds for the possible range of Vcx between 0.5 and 2 μm/s suggested from the V step tests (Appendix C). The shaded area in Figure 6 corresponds to the range of tc for Vprior = 1.3 μm/s predicted by (22) with this range of Vcx, which all agrees with the data within the data scatter but clearly differs from the red dotted curve expected for VpriorVcx.

[50] Now that we have confirmed the consistency between the SHS and V step tests, we can combine them to determine the values of fundamental parameters (tcx and D) of healing. Combining the Vcx = 1 μm/s from the V step tests with the SHS results of tc = 1200 s for Vprior = 13 μm/s, we obtain tcx = 1114 s, using (22). From tcx = 1114 s and Vcx = 1 μm/s, we obtain D = 1114 μm, using (20). This determination of D is based on that the effective contact time at steady state sliding at Vcx is tcx. So, though less direct, D is still determined from a measurement of effective contact time, as in the determination of D for the VpriorVcx case in section 3.1. The blue and black dotted curves in Figure 6 correspond to the predictions for the 13 and 1.3 μm/s series, respectively, by (14) (or, equivalently, (30)) with these parameter values. If we take the upper bound of Vcx = 2 μm/s, we obtain tcx = 1040 s, and D = 2080 μm. The one dot chain curve in Figure 6 corresponds to the prediction for the 1.3 μm/s series with those parameter values. Note that prediction for the 13 μm/s series remains the same (blue dotted curve) because the parameters are determined to fit this series. If we take the lower bound of Vcx = 0.5 μm/s, we obtain tcx = 1156 s, and D = 578 μm. The two dots chain curve in Figure 6 corresponds to the prediction for the 1.3 μm/s series with those parameter values. Note that the estimation of tcx is not much affected by the value of Vcx in this range because the Vprior = 13 μm/s is greater than Vcx by a large margin so that tctcx holds.

[51] Finally, we evaluate the effects of slip-dependent evolution. First, we evaluate the change of Θ during reloading (from Θend to Θpeak in Figure 7), as detailed in Appendix B2.1. The result is shown in Figure 8, where Θend value (open symbols) inferred from each observed Θpeak (dots) is plotted. Thanks to the gentle slip weakening (i.e., large Dc) associated with this healing, the inferred Θend is not so different from Θpeak. Hence all the conclusions made earlier on the basis of Θpeak remain intact. Of course, slightly different parameter values are suggested, but the difference is minor. Second, the effect of slip-dependent evolution during hold is negligible; The predictions of the full evolution law (32) (blue and black solid curves, Appendix B2.2) and the predictions of the healing term (30) only (blue and black dotted curves) are almost the same. In Figure 8, we only plotted the results with Vcx = 1 μm/s for clarity. The effect of Vcx was almost identical to that shown in Figure 6.

Figure 8.

Value of Θend (open symbols) recovered from Θpeak (dots) values of hydrothermal healing experiments (Figure 6). Details are in Appendix B2. The blue dotted curve is the ideal healing (14), which fits the 13 μm/s series Θend with b = 0.0137, tcx = 980 s, and D = 980 μm. The black dotted curve is the prediction for the 1.3 μm/s series with the same parameter values. These predictions of (14) only reflect the healing term (30) of the slowness law, but they virtually coincide with the predictions of the full slowness law (32), which are indicated with the blue and black solid curves for the 13 μm/s series and the 1.3 μm/s series, respectively.

4. Discussion

4.1. Relation With the Contact Growth Mechanism

[52] In section 2, we started from a phenomenological differential equation (5) to derive the cutoff time theory, without referring to the physics that produces (5), i.e., a negative and exponential dependence of healing rate on the current strength. We did so to keep the generality of the theory, but intuitive understanding was difficult. In this section, we interpret the workings of the theory in terms of physics, following the Brechet and Estrin (BE) [1994] model, which produces (5) by invoking the negative dependence of contact normal stress on the real contact area, which, in this model, is regarded as the physical entity of the state variable. As discussed by Nakatani and Scholz [2004b], the BE model can be applied to many different healing mechanisms as long as the contact deformation rate has an exponential dependence on the driving contact normal stress. Such deformation mechanisms include dislocation glide [e.g., Frost and Ashby, 1982], stress corrosion [e.g., Scholz, 1972], and pressure solution [e.g., Rutter, 1976]. Equation (5) or similar can be realized by other ways as well. In Hickman and Evans's [1992] experiment on a halite lens contact, contact growth rate was controlled not by contact stress, but by the curvature of the contact edge, which changes systematically with the contact size. In this case, our cutoff theory would apply, at least qualitatively, because the contact growth rate is controlled by contact size, though the dependency may not be exponential and a qualitatively different behavior may result. On the other hand, if the contact growth is controlled by some external factors such as the chemical saturation of pore fluid [e.g., Olsen et al., 1998], our theory will not apply.

[53] The BE model analyzes the squashing of contacting asperities by the applied constant normal load. A simplified geometry has been adopted, where the real contact area of the interface consists of many column-like asperities of the same dimension. The material of the asperities is assumed to follow a Peierls-type constitutive law:

display math

Here, equation image is the strain rate of the asperity creep normal to the interface, Γ is a preexponential factor, and S is a constant denoting the magnitude of characteristic stress change that results in an e-fold change of strain rate. These two parameters are material properties, which may depend on temperature. The contact normal stress σ is inversely proportional to the real contact area A and hence is a variable. Compression is taken to be positive in strain and stress.

[54] The model treats the situation where AAx, where Ax is the real contact area achieved instantaneously upon the application of normal load, i.e., the X state. The contact normal stress at the X state, σx, is a system constant in the order of penetration hardness [e.g., Dieterich and Kilgore, 1994; Berthoud et al., 1999]. For a given normal load N, Ax = N/σx and hence Ax is a system constant, too. For AAx, the contact normal stress is given by

display math

From the constancy of asperity volume, we obtain

display math

From (34)(36), we obtain an evolution equation for the real contact area, i.e., the physical entity of the logarithmic healing:

display math

This is a differential equation of (3a) type. Therefore the predicted behavior of the contact area growth will follow the general theory of logarithmic growth developed in section 2. Assuming the adhesion theory of friction [Bowden and Tabor, 1964], that is, Θ ∝ A, (37) can be easily translated into the evolution law of frictional strength

display math

Comparing (38) with (5a), we can confirm that the two constants in (5a) are given as functions of the system constants only, as assumed in section 2.

[55] Now, we show how the section 2 theory works in the BE model. Equations (37) and (38) are exactly equivalent to the BE model, though expressed in a different form. Brechet and Estrin [1994] solved this system only for a special initial condition of Θ0 = Θx and hence the parameters of the resultant logarithmic growth were perfectly fixed by the system constants. However, in reality, the initial state can be different from the X state and hence the cutoff time depends on the initial condition as shown in section 2. This is easier to see if we rewrite (37) and (38) into (3b) form. They are

display math

with

display math
display math

with

display math

respectively.

[56] The parameters of the resultant logarithmic healing are thus given by

display math
display math

The parameter b depends on the system constants only, but the cutoff time tc depends on initial state Θ0 as well, through the dependence of initial healing rate on Θ0(42). Paraphrased in more concrete terms, it goes like this. A more healed interface has a larger initial contact area A0 and hence a lower contact stress, resulting in a smaller healing rate at t = 0 (see (40)). A smaller initial healing rate will result in a longer cutoff time (44), which is a general feature of logarithmic healing (6a). Quantitatively, an increase of Θ0 by b causes an e-fold increase of tc (see (42)(44)). Since this is an effect of the initial state through its effect on initial contact stress σ0, the effective contact time at the beginning of hold, tini, has little effect on tc as long as tinitcx because Θ0 remains nearly constant at ∼Θx, resulting in σ0σx and hence tctcx. (More precisely saying, 0 > σ0σx ≫ −S.) On the other hand, if tinitcx, Θ0 increases by b per an e-fold increase of tini, causing a decrease of σ0 by S, which in turn results in an e-fold decrease of equation image0 and an e-fold increase of tc.

4.2. Parameter Values and the Underlying Physical Processes of Friction

[57] The cutoff time observed in the two experiments examined in section 3 scaled with Vprior quite differently. Although the healing mechanisms are known to be different between these two experiments, we do not think this is the direct reason. Rather, we have made an interpretation through a theory common to different healing mechanisms (Figure 2); the cutoff time was inversely proportional to Vprior in one experiment (section 3.1) because Vprior was ≪Vcx, while the cutoff time was little affected by Vprior in the other experiment (section 3.2) because Vprior was ≫Vcx. The difference of healing mechanism is relevant only through the difference of Vcx, which is the ratio of more fundamental parameters D and tcx. In the below, we discuss implications of the values of these parameters constrained in section 3.

[58] The value of the intrinsic cutoff time tcx was very different for the two mechanisms; about one thousand seconds for hydrothermal healing by Nakatani and Scholz [2004a] and ≪1 s for the solid-state healing by Marone [1998a]. The difference would be much more pronounced if the tcx for the hydrothermal experiment, which is known to increase with decreasing temperature following Arrhenius relationship, is extrapolated to room temperature for fair comparison with the latter room T experiment. Such a huge difference in tcx is of no surprise because the rate-limiting physico-chemical process is different; Solution transfer for hydrothermal healing [Nakatani and Scholz, 2004a] and stress corrosion [e.g., Dieterich and Conrad, 1984] for solid-state healing.

[59] In section 3, we have determined the value of parameter D for each mechanism. We determined it from the data reflecting the effective contact time for steady state at a known slip velocity as defined by (9a), not referring to slip-dependent transient behavior. So, as discussed in section 2.3, there is no a priori reason that the parameter D has to coincide with the other length dimension parameter Dc, which is determined from the rapidness of the slip-dependent transient. Nonetheless, the values of D and Dc in each experiment fell in a similar range. This is even more impressive when we consider that the values are very different between the two experiments; several micrometers for the solid-state healing and several hundreds to one thousand micrometers for solution transfer healing. Though we have only two examples so far, these results may suggest that the same physical structure controls the slip-dependent transient and the average contact time associated with the same healing mechanism. Further investigation of D and Dc should be warranted to understand how the inhomogeneous structures in the fault zone work in different frictional behaviors. In addition, we note that establishing the relation of D and Dc, even empirically, would have practical merit. Both of these parameters are necessary to predict the behaviors of natural faults such as a seismic cycle, but observations on natural faults are often only enough to determine one of them. If we know the relation between them, we can infer the other parameter. Retrospectively, it could be said that rate and state friction modeling so far has assumed D = Dc, without justification.

[60] Between the two experiments, both D and Dc differed by 2 orders of magnitude. This cannot be readily attributed to different geometrical settings. The gouge layer thickness was similar. The gouge was definitely finer in the hydrothermal healing test, which showed much larger D and Dc. Hence the difference suggests that the two healing mechanisms may work on different hierarchies of structures in the fault zone; Solid-state healing occurs at individual contacts, whereas development of structures at scales far beyond the individual contacts is involved in solution transfer healing. This was pointed out by Nakatani and Scholz [2004a] from the observation of Dc only. Now, the additional observation of D supports this.

4.3. Implication of the Cutoff Time Observed in Healing of Natural Faults

[61] Marone et al. [1995] has found that stress drop of repeating earthquakes on the Calaveras Fault increases with the logarithm of recurrence interval, with a cutoff time of about 100 days. This is much longer than the cutoff time observed in laboratory experiments. Marone [1998a] argued that this be explained by his laboratory finding of a slip velocity effect on healing, which we have discussed in section 3.1. The direct basis of his argument is that the rate and state friction law could reproduce both the laboratory and natural observations of cutoff time, by assuming appropriate coseismic slip velocity. He presumed that the cutoff time in both cases results from the suppression of healing by slip weakening caused by afterslip. However, as we discuss below, the cutoff time in the two cases actually emerged from clearly different reasons.

[62] The laboratory observation is that the cutoff time decreases with the preceding coseismic slip velocity Vprior. This cutoff time cannot result from the slip weakening due to afterslip, which would result in a cutoff time increasing with Vprior, as Marone [1998a] presumed. Instead, the laboratory observation of tcVprior−1 scaling suggests that the observed cutoff time is the effective contact time set in the preceding coseismic slip (section 3.1). This type of cutoff time is expected as a general nature of logarithmic growth (section 2.2). Further, we have figured out that the slowness-type rate and state law, which Marone [1998a] used, predicts the same scaling for the same reason (section 2.3.3).

[63] On the other hand, the long cutoff time of ∼100 days predicted by Marone's [1998a] simulation for natural earthquake conditions cannot be the effective contact time. For the conditions of D = 5 mm and Vprior = 1 m/s adopted there, the effective contact time is only 5 ms. Since the evolution law (25) he used does not include tcx, this is the only possibility for the cutoff time emerging from the healing term. Therefore the long cutoff time predicted in that simulation must come from the other effect, the slip-weakening term due to the large afterslip following fast coseismic slip as Marone presumed.

[64] In conclusion, the scaling of cutoff time with slip velocity observed in Marone's [1998a] experiment has nothing to do with the long cutoff time observed in the natural earthquake sequences. In addition, we note that afterslip, the evaluation of which requires more rigorous treatment of dynamic seismic cycle [Beeler et al., 2001], is not the only possibility to explain the long cutoff time. Other more straightforward explanations, such as a stronger healing mechanism emerging after a large intrinsic cutoff time tcx, as suggested by Nakatani and Scholz [2004a], are possible as well.

5. Conclusions

[65] Logarithmic growth results from a system whose growth rate depends negatively and exponentially on its current level. Cutoff time, after which time the log linear growth becomes conspicuous, inevitably appears in such a system and is inversely proportional to the initial healing rate, which in turn depends on the initial strength. Depending on the prior slip history, initial strength can vary, with its minimum value being the strength attained instantaneously upon the application of normal load. This state (X state) is a system constant and so is the cutoff time for healing starting with X state. This intrinsic cutoff time tcx can reflect the reaction rate constant of the underlying physico-chemical process [e.g., Nakatani and Scholz, 2004b]. For healing starting with a given initial state, the observed cutoff time is the sum of tcx and the effective contact time tini for the initial state, which is defined as the contact time necessary to attain the given state by healing from the X state. This theory has been developed in section 2, detached from physics realizing the healing. In section 4.1, we have shown correspondence with a representative physical model, where healing is modeled as the growth of real contact area driven by contact normal stress [Brechet and Estrin, 1994].

[66] In SHS tests, the steady state sliding at Vprior preceding the hold period sets tini = D/Vprior, where D is a length dimension constant. It is theoretically predicted that (1) for a high Vprior such that tinitcx, intrinsic cutoff time tcx independent of Vprior will be observed, and (2) for a low Vprior such that tinitcx, apparent cutoff time inversely proportional to Vprior will be observed, which is actually the effective contact time in the preceding slide. We have examined the scaling of the cutoff time with Vprior observed in two experiments. One experiment [Marone, 1998a], where a solid-state healing mechanism occurred, seems to be a low Vprior case. The other experiment, where healing by hydrothermal solution transfer mechanism occurred, seems to be a high (to intermediate) Vprior case. Our interpretation is that the different scaling reflects the different relations of Vprior with the values of parameters D and tcx intrinsic to each healing mechanism, rather than the difference in mechanism itself. Although the both regimes should be observed for a single healing mechanism if the employed Vprior range is wide enough, such data are not presently available.

[67] Further, our cutoff time theory predicts that the velocity dependence of steady state has a high-velocity cutoff Vcx = D/tcx, at which effective contact time is equal to tcx. Therefore the low and high Vprior regimes for SHS tests are divided by Vcx. Constraints on Vcx obtained from V step tests done in the two experiments mentioned above were consistent with this.

[68] As our cutoff time theory has shown that the effective contact time is observable, we could determine the length dimension parameter D relating the effective contact time and slip velocity, independently of the other length dimension parameter Dc describing the rapidness of slip-dependent evolution. The values of D and Dc were similar in each of the two experiments analyzed, while the values differed by 2 orders of magnitude between the two experiments. This suggests that slip-dependent evolution and effective contact time associated with the same healing mechanism are controlled by the same physical structure in the fault zone, while different healing mechanisms concern different structures.

[69] In order to incorporate the presently developed cutoff time theory, we have proposed some modifications on the existing evolution laws, both for slip law and slowness law. The modified laws refer to the X state by incorporating the intrinsic cutoff time tcx. Also, the value of effective contact time in steady state has been explicitly represented as D/V, whereas traditional laws just mention its inverse proportionality to V. Slip-dependent evolution is described with a separate parameter Dc in the modified slip law. We could not do this for the slowness law; both the slip dependence and effective contact time are described by the single parameter D, the same problem as in the traditional slowness law. In addition, we have found out that the traditional slowness law does predict the apparent cutoff time.

[70] Marone [1998a] argued that his laboratory finding of velocity dependence of the cutoff time can explain the long cutoff time for the healing of natural earthquakes [Marone et al., 1995] because his simulation with the slowness law predicts both. However, we have realized that the cutoff time in the laboratory case (and its simulation) is the effective contact time in the preceding slide, whereas the very long cutoff time in his simulation of the natural earthquakes resulted from the slip weakening due to the long-lasting afterslip following very rapid coseismic slip. The cutoff time due to the former reason decreases with the coseismic slip velocity and cannot explain the long cutoff time for the natural earthquakes.

Appendix A:: Apparent Cutoff Time Predicted by the Slowness Law Employing Two Separate Length Dimension Constants

[71] Nakatani [2001, Appendix] proposed (29) so that the observed apparent cutoff time can be described with an additional length dimension constant L′ in the healing term, independently of the slip-dependent evolution described with L in the slip-weakening term. As seen from (5b) and (6a), the first term of (29) predicts that the cutoff time is L′/V* for the healing that starts from Θ = Θ*, which sounds promising. However, Θss(V) from (29) is

display math

[72] So Θ* is now the steady state for (L/L′)V*, not for V*. The steady state for V* is Θ* + bln(L/L′), instead. Integration of the first term (i.e., healing term) of (29) with the initial condition Θ0 = Θ* + bln(L/L′) gives a logarithmic healing with a cutoff time of L/V*, where L′ all but cancels out. It is straightforward to further show that, for general Vprior, the cutoff time becomes L/Vprior, which is dictated by the L in the slip-weakening term (i.e., second term of (29)), not reflecting L′.

Appendix B:: Data Modeling Using Full Evolution Laws Including Slip-Dependent Effect

B1. Experiments of Marone [1998a]

B1.1. Modeling of Reloading Period

[73] We begin with inferring Θend by modeling the change of Θ in reloading (from Θend to Θpeak in Figure 3). We use the modified slip law (24) because the evolution in reloading is mainly slip-dependent. We simulate a reloading period with the initial condition of a given Θend and calculate the resulting Θpeak. By doing this with many different values of Θend, we construct the relation of Θend and Θpeak (the thick black curve in Figure B1). Using this curve, Θend is recovered for each observed Θpeak. The result is shown in Figure 5.

Figure B1.

Value of Θend predicted for reloading from a given Θend. Results for Vreload = 10 μm/s are shown. The modified slip law (24) was used in these simulations. The value of τend was given for the three simulations shown by the thin colored lines. For the simulation shown by the thick black line, which corresponds to the type of SHS tests shown in Figure 3, τend was calculated through the simulation of the hold period using the constitutive law (2) coupled with the modified slowness law (32). Parameter values used were as follows: a = 0.0024, b = 0.0039, tcx = 1 × 10−6 s, D = 3.2 μm, Dc = 7.1 μm, Vprior = Vreload = 10 μm/s, k = 9 × 10−4 (frictional coefficient/μm).

[74] An additional initial condition τend, the shear stress at the end of the hold period (Figure 3), must be specified in simulating reloading. In this type of SHS tests, τend decreases with hold time because of the slow slip continuing on the interface while the load point is held stationary. In constructing the Θend − Θpeak relation for this type of test (the thick black curve in Figure B1), we used τend obtained from the simulation of the hold period using the modified slowness law (32). Although we are not sure of how well we could predict τend, the effect of τend (colored curves in Figure B1) on the predicted Θend − Θpeak relation is negligible except for very small Θend, which lies outside the range of data we analyze. The curves shown in Figure B1 are for Vreload = 10 μm/s. Curves for Vreload = 1 μm/s were very similar (not shown).

[75] Parameter values necessary for the above simulation were determined as follows. At this stage, we allow the use of different values for some healing-related parameters (b and D) for tests with different velocities because the purpose now is to recover the loss of Θ during each reloading as accurately as possible. For machine stiffness, we used k = 9 × 10−4 (frictional coefficient/μm) [Marone, 1998a]. Among the friction parameters, the evolution distance Dc has the largest effect; a smaller Dc results in a greater difference of Θpeak from Θend. We have confirmed that the evolution in the V step test on the same system is reproduced with the slip law with Dc = 7.1 μm, the value of L that Marone found from the same test with the slowness law. Another length parameter D has a minor effect through its effect on Θss(V), toward which the slip weakening occurs. We used D = 3.2 μm, as suggested by the tc = 0.32 s for the Θpeak data of the 10 μm/s series (Figure 4). For the 1 μm/s series, we used D = 8.7 μm, as suggested by the tc = 8.7 s for the 1 μm/s series. For the initial estimate of b, we took 0.0039 for the 10 μm/s series and 0.0043 for the 1 μm/s series, as suggested by the log linear slope of the Θpeak data of each series. For a, we took a value of 0.0024 for the both series. This value was determined by subtracting (ba) = 0.0017 directly observable in V step tests from the average of b values suggested by the Θpeak trend at each Vprior. Ideally, the parameter values estimated from Θend, instead of from Θpeak, should be used in the modeling. We have confirmed that the result is only slightly modified by redoing the modeling with the parameter values suggested by the inferred Θend trends shown in Figure 5.

B1.2. Modeling of Hold Period

[76] Next, we model the hold period to see how well the modified slowness law (32) explains the Θend values for the both series. This must be done with the same parameter values for the both series. From the Θend trend for 10 μm/s series, we obtain b = 0.0042 and tc = 0.19 s. As this is a D/Vpriortcx case (section 3.1), we obtain D = 0.19 s × 10 μm/s = 1.9 μm. We take tcx to be 1 × 10−6 s, the value of which does not affect the prediction as long as tcxD/Vprior. Predictions of Θend by the full slowness law (32) are shown by the solid curves in Figure 5. In using (32), slip velocity was calculated by (2), where we used a = 0.0029. This value was determined by subtracting (ba) = 0.0017 from the average of b values from the Θend trends at each Vprior.

[77] In the above, the parameters b = 0.0042 and D = 1.9 μm were determined by fitting the 10 μm/s series data only, for we intended a stringent test how well the parameter determined from the tests at one velocity can predict the results at the other velocity. Of course, from a viewpoint of getting the best information from the whole data set, b and D values should be determined to fit both series equally well. In this case, we obtain b = 0.0046 and D = 3.7 μm.

B2. Hydrothermal Healing Experiments

B2.1. Modeling of Reloading Period

[78] We follow the same logical steps as in section B1. Initially, we recover Θend by simulating the reloading period with (24). The auxiliary initial condition τend is known in this case because it is a controlled parameter (Table 1) in this type of SHS tests (Figure 7). The machine stiffness has been measured for the apparatus used, k = 8 × 10−4 (frictional coefficient/μm). The parameter Dc has been obtained from the slip-weakening curve following each hold (Table 1). Generally, tests done later in a run resulted in smaller Dc values [Nakatani and Scholz, 2004a]. Other parameter values were determined as follows. From the Θpeak data, we estimate b = 0.013 and tc = 1200 s for the 13 μm/s series and b = 0.012 and tc = 1400 s for the 1.3 μm/s series. As done in section 3.2, combining these tc values with another constraint of Vcx ∼ 1 μm/s from the V step tests (Appendix C), we obtain tcx = 1114 s and D = Vcxtcx = 1114 μm for the 13 μm/s series. For the 1.3 μm/s series, tcx = 790 s and D = 790 μm is obtained in the same way. As mentioned in B1.1, at this stage where we infer the loss of Θ in reloading for each test, we allow the use of different values for some parameters for the two series. For a, we use a = 0.0087 throughout, a representative value from the direct effect observed in V step tests [Nakatani and Scholz, 2004a]. Since Dc in those tests were large, being >200 μm, a can be determined in this way fairly reliably [e.g., Nakatani, 2001].

[79] Thus recovered Θend is plotted in Figure 8 with open symbols. The magnitude of correction varies for each data point, reflecting the difference in Dc. However, even the largest correction is not larger than the corrections for Marone's [1998a] data (Figure 5) because Dc was large, with a minimum of ∼60 μm.

B2.2. Modeling of Hold Period

[80] Next, we model the hold period with the modified slowness law (32). From the Θend trend for the 13 μm/s series, we obtain b = 0.0137 and tc = 1057 s. With a constraint of Vcx = 1 μm/s, we obtain tcx = 980 s and D = 980 μm. Predictions of Θend by the full slowness law (32) are shown by the solid curves in Figure 8. If we take the upper bound of Vcx = 2 μm/s, tcx = 916 s and D = 1832 μm. For the lower bound of Vcx = 0.5 μm/s, tcx = 1018 s and D = 509 μm.

Appendix C:: Determination of Vcx Associated With Hydrothermal Healing of Nakatani and Scholz [2004a]

[81] Velocity dependence of steady state can be measured in a V step test as the magnitude of evolution effect, which follows an instantaneous stress jump upon velocity step [e.g., Marone, 1998b]. Its magnitude is the difference of Θss(V) between the velocities before and after the step. From (19), we see that the magnitude for a tenfold velocity step between Vm/equation image and equation imageVm is

display math

Here we use Vm, the geometric average of the velocities before and after the step, as the representative velocity of a test. The function (C1) stays at ∼ln 10 × b for VmVcx, while it is ∼0 for VmVcx.

[82] Figure C1 shows the results of tenfold V step tests done on the same system at the same condition [Nakatani and Scholz, 2004a, Figures 12b–12d] as the SHS tests examined in section 3.2. Tests were done at three different velocity ranges Vm = 0.41, 4.1, and 41 μm/s, which corresponds to the steps between 0.13–1.3, 1.3–13, and 13–130 μm/s, respectively. By fitting the data by (C1) with b fixed at 0.13, the value suggested from the SHS tests, we estimate that Vcx is about 1 μm/s, definitely constrained between 0.5 and 2 μm/s.

Figure C1.

Velocity dependence of the magnitude of the evolution effect observed in the velocity stepping tests done at the same hydrothermal condition as that for the tests shown in Figure 6. Data are read from Figures 12b–12d of Nakatani and Scholz [2004a], where a tenfold velocity step was imposed. The velocity range of each test is represented by Vm, the geometrical average of the velocities before and after the step. Plotted together are the theoretical predictions by (C1) with different values of Vcx. The value of b = 0.013 was taken from the log linear slope of the healing tests on the same system (Figure 6). Note that the data from the fastest velocity range (steps between 13 and 130 μm/s, Vm = 41 μm/s) were not used to constrain Vcx value because the observed evolution effect did not seem to reflect the hydrothermal healing mechanism.

[83] Here, we did not consider the data from the velocity step tests in the highest velocity range (Vm = 41 μm/s), which lies above the trend expected from (C1). This is because we believe that the evolution effect observed in this velocity range, which showed much shorter Dc than those observed in the velocity step tests at the slower velocity ranges and also in the slip weakening of the healing tests [Nakatani and Scholz, 2004a], are not related to the hydrothermal healing mechanism we are analyzing. We rather interpret that the velocity step results at Vm = 41 μm/s constrains that ΔΘvstep(Vm = 41 μm/s) associated with the hydrothermal healing mechanism was ∼0, consistent with the prediction of (C1) with the parameter values determined above.

Acknowledgments

[84] This article was greatly improved by constructive reviews by N. M. Beeler and T.-f. Wong. Discussions with J. Rice, N. Kato, and S. Yoshida were helpful. Work partly supported by JSPS grants 15340143 and 18253003.

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