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

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jb004096, 2006 Intrinsic and apparent short-time limits for fault healing: Theory, observations, and implications for velocity-dependent friction Masao Nakatani 1 and Christopher H. Scholz 2 Received 9 October 2005; revised 7 August 2006; accepted 31 August 2006; published 30 December [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 t cx, 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 t cx and the effective contact time t ini of asperities at the beginning of the healing. Hence, in slide-holdslide (SHS) experiments, where t ini is inversely proportional to the slip velocity V prior in the sliding preceding the hold, two regimes are predicted. For high V prior (t ini t cx ), the observed cutoff time is independent of V prior, being t cx. For low V prior (t ini t cx ), the observed cutoff time is apparent, being t ini inversely proportional to V prior. 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 t cx. 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. Citation: Nakatani, M., and C. H. Scholz (2006), Intrinsic and apparent short-time limits for fault healing: Theory, observations, and implications for velocity-dependent friction, J. Geophys. Res., 111,, doi: /2005jb 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]. QðÞ¼ t Qðt ¼ 0Þþ b ln t þ 1 : ð1þ t c Here, Q is the frictional strength normalized by the applied effective normal stress s. Since it is known that more shear stress (t) is necessary to cause faster slip on the interface at 1 Earthquake Research Institute, University of Tokyo, Tokyo, Japan. 2 Lamont-Doherty Earth Observatory of Columbia University, Palisades, New York, USA. Copyright 2006 by the American Geophysical Union /06/2005JB004096$09.00 the same state (strength), we need to define frictional strength Qs as the shear stress required to cause a slip of an arbitrarily chosen reference velocity V *, using the following constitutive equation: t ¼ Qs þ as ln V=V * ; ð2þ where a is an empirical constant called the coefficient of direct effect [e.g., Dieterich, 1979; Ruina, 1983]. Thus defined, Q, 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, Q can be regarded as a state variable in rate and state friction laws [Ruina, 1983]. In the present paper, Q 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 Q against linear time (Figure 1b), the real healing rate Q _ is always decreasing with time. We reserve the term healing rate for Q, _ not b. [5] The present paper focuses on the other parameter t c. This parameter is necessary to avoid the negative diver- 1of19

2 Figure 1. Form of function Q(t) =Q 0 + b ln (t/t c + 1) plotted against (a) log time and (b) linear time. Note that the vertical scale is different between Figures 1a and 1b. gence of logarithm as t! 0 and is often called the cutoff time because the increase of Q per an e-fold increase of time is much less than b for 0 t t c (Figure 1a). Again, however, cutoff is a misleading word because healing is more active at a smaller t even for t t c, as seen from Figure 1b. It is not that the process of healing is cut off for t t c. Rather, it is just that 0 t t c is too short a time interval to see a significant increase of Q. Nonetheless, we keep the familiar terminology of cutoff time to refer to t c. [6] The cutoff time t c is important not only for t t c but also for t t c. For t t c, DQ(t) Q(t) Q(t = 0) has a dependency of bln(t c ) on the cutoff time. So, without knowing the value of t c, one cannot tell the amount of strength recovery even for t t c. In many laboratory healing tests at room temperature [e.g., Dieterich, 1972], t c seems to be order s, while t c up to half a day ( s) has been registered in laboratory tests under hydrothermal conditions [Nakatani and Scholz, 2004a]. A cutoff time as long as 100 days ( 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 t c by 8 orders of magnitude, for example, results in a difference of DQ by 0.18 in terms of frictional coefficient. Predicting the amount of frictional healing DQ in terms of frictional coefficient is important. For example, we can estimate the effective normal stress s of earthquake faults, an enigmatic problem [e.g., Wang et al., 1995; Scholz, 2000], by s = Dt/DQ, where Dt 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(t c ) dependence of DQ for t t c means that the value of t c can be determined from the data for t t c 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 t c 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/t c. On the other hand, there is a laboratory data set [Marone, 1998a] suggesting that t c depends on the slip velocity employed in the experiment. This implies that t c is not necessarily an intrinsic constant of the healing process. [9] In this paper, we will theoretically show in section 2 that the observed t c is the sum of (1) the process raterelated 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 2of19

3 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 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 _y ¼ C exp y ; ð3aþ l where l is a constant characterizing the exponential dependency and C is another system constant. For brevity, we adopt a following convention throughout the paper: z 0 denotes the value of quantity z at t = 0, and _z 0 _z(t =0).Dz is the difference of quantity z from z 0. D_z denotes the time derivative of Dz. [12] We rewrite (3a) into the following form in terms of the increment Dy D_yðÞ¼_y t 0 exp Dy : ð3bþ l From the (3a) form, we see that _y 0 = C exp( y 0 /l) depends on the initial value y 0. The solution of (3) is t DyðÞ¼l t ln þ 1 : ð4þ l=_y 0 This is a logarithmic growth with a cutoff time t c = l/_y 0. Therefore we see that the logarithmic healing (1) results from the following evolution law, which is of (3) type, _Q ¼ P exp Q : ð5aþ b Here P and b are system constants determined by the specific physical mechanism of the healing, and are independent of Q. Rewriting (5a) into the following form represented as to the increment is convenient: D Q _ ðþ¼ t Q _ 0 exp DQ : ð5bþ b The cutoff time t c of the resultant logarithmic healing (1) is given by t c ¼ b= _ Q 0 : ð6aþ [13] The essence of our theory developed below is that the initial healing rate Q 0 and hence t c depend on the initial state Q 0, as seen from (5a). Explicitly showing this, (6a) can be rewritten as t c ¼ bp 1 exp Q 0 : ð6bþ b [14] The differential equation (5) must hold for any healing mechanism that results in logarithmic time dependence, and hence the dependence of t c on initial condition (6) must be true for logarithmic healing in general Meaning of the Observed Cutoff Time: Intrinsic or Apparent? [15] As shown above, the cutoff time t c observed in log time healing reflects the initial strength Q 0. 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, Q 0 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 Q x. Since the X state for a given normal load should be uniquely determined by the material and geometrical properties of the frictional interface, Q x 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 with t QðÞ¼ t Q x þ b ln þ 1 ; ð7þ t cx t cx ¼ b= _ Q x ; where _ Q x is the healing rate for the interface with Q = Q x. As seen from (5a), _ Qx is a system constant, and therefore so is t cx. As shown in section 4.1, the cutoff time that Brechet and Estrin [1994] have derived corresponds to t cx. We refer to t cx 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]. ð8þ 3of19

4 Figure 2. Theoretical prediction (14) with different values of t ini, the effective contact time at the beginning of stationary contact. In slide-hold-slide (SHS) tests, t ini can be controlled as t ini is inversely proportional to the slip velocity V prior in the preceding slide period. [16] Now we proceed to derive t c in more general situations, where Q 0 can be different from Q x, 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 t eff ¼ D=V: ð9aþ 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 V prior in the preceding slide. Hence the effective contact time at the beginning of the hold, which we refer to as t ini,is given by t ini ¼ D=V prior : ð9bþ In this case, surface state at t = 0 is stronger than Q x, being Q 0 ¼ Q x þ b ln t ini þ 1 : ð10þ t cx Therefore the initial healing rate is smaller than _ Q x. From (5) and (10), we obtain _Q 0 ¼ Q _ x exp b ln t ð ini=t cx þ 1Þ : ð11þ b Using (6), (8), and (11), the observed cutoff time t c is given by t c ¼ t ini þ t cx : ð12þ [17] In the above, we introduced the initial contact time t ini 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 t ini for any given initial state Q 0 (>Q x )by Q 0 Q x þ b ln t ini þ 1 : ð13þ t cx [18] To sum up, we have shown that the time evolution of the system (5) with a given initial state Q 0 will follow t QðÞ¼Q t 0 þ b ln þ 1 : ð14þ t cx þ t ini The appearance of the initial state t ini (Q 0 ) 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 t c = t ini + t cx. However, as we will demonstrate with actual data sets in section 3, we can tell which of t ini and t cx is the dominant part of the observed t c by comparing tests with different V prior and thus t ini (=D/V prior ). In Figure 2, DQ predicted by 4of19

5 (14) is plotted for different values of t ini. For low V prior such that t ini t cx, (14) is approximated by t DQðÞffib t ln þ 1 t ini ð15aþ t ¼ b ln þ 1 : ð15bþ D=V prior Hence the healing curve shifts to the right as V prior is decreased because t ini is inversely proportional to V prior. For high V prior such that t ini t cx, (14) is approximated by DQðÞffib t ln t t cx þ 1 : ð16þ Hence the healing curve is virtually fixed, independent of V prior. [20] Conversely, if the observed t c is found to be independent of the V prior employed, we can tell that the observed t c is the intrinsic cutoff time t cx. At the same time, we can tell the effective contact time for the employed V prior is much less than t cx. In contrast, if the observed t c is found to be inversely proportional to V prior, we can tell that the observed t c is apparent, being the effective contact time for the V prior. In this case, the value of the length dimension constant D in (9) is determined to be V prior /t c. At the same time, t cx is known to be much less than the observed t c.of course, there also exists an intermediate regime of V prior where t ini is of the similar order of magnitude as t cx.inthis case, the observed t c will increase as V prior decreases, but the dependence is weaker than the inverse proportionality 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 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 Q ss ðvþ ¼ Q * þ b ln V * =V ; ð17þ where Q * 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 Q ss (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. Q ss ðv Þ ¼ Q x þ b ln D=V þ 1 : ð18þ t cx The newly incorporated constants D and t cx are observable in SHS tests employing appropriate V prior 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 Q ss ðv Þ ¼ Q x þ b ln V cx V þ 1 ; ð19þ where V cx D=t cx : ð20þ Existence of such a high-velocity cutoff has been suggested earlier [e.g., Okubo and Dieterich, 1986]. [26] The cutoff velocity V cx 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 V cx as the velocity at which t eff becomes equal to t cx. This leads to an interesting prediction on the relation between the V step tests and SHS tests. Using (20), we can rewrite (9b) into V cx t ini ¼ t cx : V prior ð21þ The cutoff time (12) observed in SHS tests is therefore expressed as t c ¼ t cx 1 þ V cx =V prior : ð22þ Thus the high- and low-velocity regimes for healing tests are divided by the cutoff velocity V cx for the velocity dependence of steady state; If SHS tests are done with V prior V cx, apparent cutoff time inversely proportional to V prior is observed as described by (15). If V prior V cx, intrinsic cutoff time t cx independent of V prior is observed as described by (16). Equation (22) describes the behavior for any V prior including the intermediate regime. In the data analysis in section 3, consistency between SHS tests and the V step tests will be discussed 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 characteris- 5of19

6 tic evolution distance of D c, toward the steady state value for the given velocity [e.g., Ruina, 1983]. _Q ¼ V fq Q ss ðv Þg: ð23þ D c 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 _Q ¼ V Q Q x þ b ln D=V þ 1 D c t cx : ð24þ [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 D c. 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 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 _Q ¼ b exp Q * Q b L=V * b L V: ð25þ Here the first and second terms represent the timedependent 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 _ Q to zero, (25) gives Q ss ðvþ ¼ Q * þ b ln V * =V ; ð26þ 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 _Q ¼ b L V ð27þ 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 D c in the slip law in that both of them specify the rapidness of slip-dependent change. [32] The healing term of (25), i.e., _Q ¼ b exp Q * Q ; ð28þ L=V * b 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 t c = L/V prior. As seen from (5b) and (6a), the preexponential factor of (28) constrains that t c for healing starting with Q * (i.e., steady state at V * )isl/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 _Q ¼ b L 0 exp Q * Q b =V * b L V: ð29þ 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 Q 0 = Q x must have a cutoff time of t cx, the healing term is fixed as _Q ¼ b exp Q x Q : ð30þ t cx b [35] The time-dependent evolution with this equation from a general initial state Q 0 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 b _Q ¼ : D=V þ t cx ð31þ [36] Hence the complete modified evolution law becomes _Q ¼ b exp Q x Q t cx b b : D=V þ t cx ð32þ 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 6of19

7 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. 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/V t cx. Therefore (32) cannot correctly describe the observed effective contact time and the slipdependent 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 V prior. In this section, we will demonstrate this point by examining two laboratory experiments. In each experiment, SHS tests were done for two different V prior.as we intend a stringent test of the theory, we determine healing parameters from the data for one V prior and see how well the theory with these parameter values predicts the data for the other V prior. In addition, we also show how the behavior of the cutoff time in each experiment is consistent with the constraints on V cx 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 V prior is achieved. Strengthening achieved during the hold period is then measured by imposing the slide at a given slip velocity V reload. Although the slip velocity at the interface can differ from the load point velocity held constant at V reload due to the elastic deformation of the loading system subject to the changing shear load, slip velocity is guaranteed to coincide with V reload 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 Q peak = t peak /s a ln(v reload /V * ). On the other hand, the strength at the beginning of the hold (t = 0) is known from the steady state stress as Q ini = t ss (V prior )/s a ln(v prior /V * ). If we choose V reload equal to V reload and look at the difference of V peak from t ss,asis usually practiced in SHS tests, including the experiments examined in this paper, the direct effect terms in Q peak and Q ini cancel out, leading to Q peak Q ini ¼ t peak t ss =s: ð33þ Therefore the increase of strength (Q peak Q ini ) can be directly measured as (t peak t ss )/s, not requiring the 7of19

8 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 mm/s series data with b = and D =3.2mm (blue dotted curve). The red dotted curve is the prediction of (15) for the 1 mm/s series with the same parameter values, which results in a 10 times larger t c. 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. [39] In the following, we compare the directly observable quantity (Q peak Q ini ) 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, Q can change somewhat during reloading (from Q end to Q peak in Figure 3), mainly because significant preslip occurs as the shear stress is raised. Second, because of slow slip continuing during hold, even Q end 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 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 mm) was sheared within rough (200 mm RMS) granite surfaces. Experiments were done at room temperature. Two series of SHS tests (Figure 3) were done, one series with V prior = V relaod =10mm/s and the other with V prior = V relaod =1mm/s. Results for the both series are shown in Figure 4, where (t peak t ini )/s is plotted. Since V prior = V relaod in the both series, this is equal to Q peak Q ini as explained earlier. [41] Both the 10 and 1 mm/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 (t peak t ini )/s 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 V prior (15). From the 10 mm/s series data, we obtain b = and t c = 0.32 s (the blue dotted curve). This constrains that D =3.2mm (=10 mm/s 0.32 s). The red dotted curve shows healing with a ten times larger t c,as predicted for V prior = 1 mm/s by (15) with the same parameter values. The 1 mm/s series data roughly agrees with this. Hence we can conclude that the cutoff times 8of19

9 Figure 5. Value of Q end (open symbols) recovered from Q peak (dots) values of Marone s [1998a] experiments (details in Appendix B1). The blue dotted curve is the ideal healing (15) fitting the 10 mm/s series Q end with b = and D =1.9mm. The red dotted curve is the prediction of (15) for the 1 mm/s series with the same parameter values, which results in a 10 times larger t c. 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 mm/s series and the 1 mm/s series, respectively. observed in this experiment are apparent, being the effective contact time set in the slide preceding each hold period. 1 Since such t c (=t ini ) / V prior scaling is limited to t ini t cx, intrinsic cutoff time is constrained to be 0.32 s, the smaller of the observed cutoff times. [43] This latter conclusion of t cx D/V prior 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 V prior V cx as discussed in section [44] At a closer look of Figure 4, we notice that the majority of the 1 mm/s series data is shifted from the 10 mm/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 mm/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 Q during reloading (from Q end to Q peak 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 D c, we used 7.1 mm, 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 Q end value (open symbols) inferred from each observed Q peak (dots). The corrected value (Q end Q ini ) is generally greater than the observed (Q peak Q ini ) by 20%. However, the separation between the trends for different velocities is still consistent with t c / V 1 prior. The blue and red dotted curves are the predictions by (15) for the two velocities, with b = , D =1.9mm, as suggested by the Q end for the 10 mm/s series. Second, we evaluate the effect of slow slip during the hold period, which can make the Q end 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 9of19

10 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 mm/s series are from the experiment shown in Figure 4a of Nakatani and Scholz [2004a]. The 1.3 mm/s series data (black circles) are from a new experiment exactly mimicking it except the slip velocity. The 13 mm/s series data show a logarithmic healing with b = and t c = 1200 s (the blue dotted curve). If t cx were much less than t ini, a 10 times larger t c would be expected from (15) for the 1.3 mm/s series (red doted curve), which does not agree with the data. With a constraint of V cx =1mm/s suggested by V step tests (Appendix C), t c for the 1.3 mm/s series is theoretically predicted to be 1.64 times (black dotted curve) of that for the 13 mm/s series, which agrees with the data. The one dot chain curve and two dots chain curve are predictions for the 1.3 mm/s series with V cx =2mm/s and V cx =0.5mm/s, respectively. 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]), Q 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 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 mm, including many fines <1 mm) 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 V prior = V reload =13mm/s, the other series with V prior = V reload =1.3mm/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 H 2 O 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 D c of several hundreds of microns (Table 1), very different from 1 to 10 mm 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. 10 of 19

11 Table 1. Hydrothermal Healing Data Number of Holds Position, mm Hold Time, s (t end t ss )/s (t peak t ss )/s D c, mm Tests With V prior = V reload = 13 mm/s (Figure 4a of Nakatani and Scholz [2004a]) a a a Tests With V prior =V reload =1.3mm/s (Additional Tests Done for This Paper) a Somewhat greater than the values plotted in Figure 9a of Nakatani and Scholz [2004a], which were mistaken. [47] The hold periods of these experiments were realized by quickly decreasing the shear stress to a prescribed level (t hold ) 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. [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 = and t c = 1200 s, as suggested by the 13 mm/s series data. It is obvious that the 1.3 mm/s series data are close to this curve, rather than showing an inversely proportional dependency of the cutoff time on V prior indicated by the red dotted curve. According to our cutoff time theory, such a cutoff time independent of V prior is expected to be observed for initial conditions t ini t cx, as described by (16). In this case, the observed cutoff time should largely come from the intrinsic cutoff time t cx. [49] As discussed in section 2.3.1, the above condition of t ini t cx is equivalent to V prior V cx, where V cx 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 V cx 1 mm/s (see Appendix C), so, the condition V prior V cx is actually not met at least for the slower V prior of 1.3 mm/s, questioning our interpretation above following (16). Hence we now reinterpret the SHS data, using the nonapproximated equation (22), which can handle any V prior including the intermediate range comparable with V cx. For the 13 mm/s series, (22) with V cx =1mm/s predicts t c =(1+1mm/s/13 mm/s) t cx = t cx. So, this series is safely regarded as a high-velocity 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 (t hold ) 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. 11 of 19

12 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 V prior =13mm/s [Nakatani and Scholz, 2004a, 2004b]. On the other hand, a significant effect of t ini is expected for the 1.3 mm/s series as (22) predicts t c = (1 + 1 mm/s/1.3 mm/s) t cx = 1.77 t cx. 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 t c. This is different from our earlier interpretation of t c independent of V prior, 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 V cx =1mm/s in that the separation between the trends for V prior =13mm/s and 1.3 mm/s is not great, definitely smaller than a decade difference in t c (red dotted curve) expected for V prior V cx. This interpretation holds for the possible range of V cx between 0.5 and 2 mm/s suggested from the V step tests (Appendix C). The shaded area in Figure 6 corresponds to the range of t c for V prior =1.3mm/s predicted by (22) with this range of V cx, which all agrees with the data within the data scatter but clearly differs from the red dotted curve expected for V prior V cx. [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 (t cx and D) of healing. Combining the V cx =1mm/s from the V step tests with the SHS results of t c = 1200 s for V prior =13mm/s, we obtain t cx = 1114 s, using (22). From t cx = 1114 s and V cx = 1 mm/s, we obtain D =1114mm, using (20). This determination of D is based on that the effective contact time at steady state sliding at V cx is t cx. So, though less direct, D is still determined from a measurement of effective contact time, as in the determination of D for the V prior V cx case in section 3.1. The blue and black dotted curves in Figure 6 correspond to the predictions for the 13 and 1.3 mm/s series, respectively, by (14) (or, equivalently, (30)) with these parameter values. If we take the upper bound of V cx = 2 mm/s, we obtain t cx = 1040 s, and D = 2080 mm. The one dot chain curve in Figure 6 corresponds to the prediction for the 1.3 mm/s series with those parameter values. Note that prediction for the 13 mm/s series remains the same (blue dotted curve) because the parameters are determined to fit this series. If we take the lower bound of V cx =0.5mm/s, we obtain t cx = 1156 s, and D = 578 mm. The two dots chain curve in Figure 6 corresponds to the prediction for the 1.3 mm/s series with those parameter values. Note that the estimation of t cx is not much affected by the value of V cx in this range because the V prior =13mm/s is greater than V cx by a large margin so that t c t cx holds. [51] Finally, we evaluate the effects of slip-dependent evolution. First, we evaluate the change of Q during reloading (from Q end to Q peak in Figure 7), as detailed in Appendix B2.1. The result is shown in Figure 8, where Q end value (open symbols) inferred from each observed Q peak (dots) is plotted. Thanks to the gentle slip weakening (i.e., large D c ) associated with this healing, the inferred Q end is not so different from Q peak. Hence all the conclusions made earlier on the basis of Q peak 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 V cx = 1 mm/s for clarity. The effect of V cx was almost identical to that shown in Figure 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: _e ¼ G exp s : ð34þ S Here, _e is the strain rate of the asperity creep normal to the interface, G 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 s 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 A ffi A x, where A x is the real contact area achieved instantaneously upon the application of normal load, i.e., the X state. The contact 12 of 19

13 Figure 8. Value of Q end (open symbols) recovered from Q peak (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 mm/s series Q end with b = , t cx = 980 s, and D = 980 mm. The black dotted curve is the prediction for the 1.3 mm/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 mm/s series and the 1.3 mm/s series, respectively. normal stress at the X state, s x, 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, A x = N/s x and hence A x is a system constant, too. For A ffi A x, the contact normal stress is given by sðaþ ffi s x 1 A A x A x : ð35þ From the constancy of asperity volume, we obtain _e ¼ _ A A x : ð36þ From (34) (36), we obtain an evolution equation for the real contact area, i.e., the physical entity of the logarithmic healing: _A ffi A x G exp 2s x exp A=A x : ð37þ S S=s x 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, Q / A, (37) can be easily translated into the evolution law of frictional strength _Q ffi Q x G exp 2s x exp Q=Q x : ð38þ S S=s x 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 Q 0 = Q x and hence the parameters of the resultant logarithmic growth were perfectly fixed by the 13 of 19