Velocity- and slip-dependent weakening in simulated fault gouge: Implications for multimode fault slip

Abstract

In addition to the velocity dependence of friction, slip dependence may play a major role before and during earthquake slip in fault zones. We performed laboratory friction experiments on simulated fault gouges, measuring both the velocity and slip dependence of friction in velocity step tests. The pure velocity-dependent component of friction measured over short displacements shows both velocity strengthening and velocity weakening friction, depending on the amount of slip considered. However, we observe that increases in sliding velocity can induce slip weakening behavior which overwhelms the velocity dependence resulting in large overall weakening, especially at rates > 1 µm/s. On natural tectonic faults, this suggests that a velocity perturbation, such as coseismic rupture propagating onto a fault patch, could induce instability via large slip weakening. Therefore, a fault which is experiencing a transient slip or slow earthquakes may be more easily induced to slip coseismically if a dynamic rupture from large earthquake propagates onto the fault.

Key Points

• Slip weakening can be large compared to velocity weakening
• Slip weakening can be induced by velocity perturbations
• Slow slip faults may be more prone to instability by slip weakening slip

1 Introduction

Evaluation of the frictional behavior of fault gouges in order to predict a wide range of natural phenomena, including earthquakes, steady creep, afterslip, or transient slow slip events, depends critically on laboratory shearing experiments. One commonly used method is imposing near-instantaneous changes in sliding velocity and measuring the resulting change in the shear strength, τ:

(1)

where c is the cohesive strength, μ is the coefficient of friction, and σ_n' is the effective normal stress [Handin, 1969]. The cohesive strength of gouges is usually assumed to be negligible so that strength is controlled by μ, in this case known as the coefficient of sliding friction. In applying such tests to natural fault slip, a decrease in strength (friction) upon an increase in velocity (i.e., velocity weakening behavior) is required for slip instability that facilitates earthquakes, which exhibit a stress drop as the fault accelerates to peak slip velocity [e.g., Dieterich and Kilgore, 1996; Marone, 1998; Scholz, 2002]. Conversely, velocity strengthening behavior is favorable for steady creep or earthquake afterslip. Slow earthquakes and other types of discrete slip transients may be generated by nearly velocity-neutral behavior [Liu and Rice, 2005; Rubin, 2008] or a combination of velocity strengthening and weakening [Shibazaki and Iio, 2003].

In laboratory data, the change in friction resulting from a change in the imposed sliding velocity V can be described by an empirical law:

(2)

(3)

where a, b₁, and b₂ are unitless constants and D_c1, and D_c2 are critical slip distances, over which friction evolves over durations measured as θ₁ and θ₂, the state variables (Figure 1a) [Dieterich, 1979, 1981]. In most cases, only one state variable is necessary to describe the data; in this case the last term in equation 2 may be eliminated by setting b₂ and θ₂ to 0. In this formulation, known as rate- and state-dependent (RSF) friction laws, the velocity-dependence of friction is quantified when the coefficient of friction is assumed to be at steady state so that equations 2 and 3 reduce to

(4)

where b = b₁ + b₂ if two state variables are used. a − b > 0 describes velocity strengthening, and a − b < 0 velocity weakening behavior. In many cases a − b can be determined simply by measuring Δμ before and after the velocity step, after sufficient shear displacement such that steady state at the new velocity V is reached; i.e., a sufficient distance beyond D_c, which is typically on the order of tens of microns [e.g., Marone and Kilgore, 1993]. More sophisticated approaches employ an inverse modeling technique in combination with knowledge of the testing apparatus stiffness in order to extract the parameters a, b, and D_c [Reinen and Weeks, 1993; Blanpied et al., 1998]; we employ such techniques here.

Figure 1

(a) Schematic illustration of the experimental apparatus, (b) examples of friction data with (10–100 µm/s) and without a velocity step (10 µm/s and 100 µm/s), (c) example of experimental data 1 mm after the velocity step from 10 to 100 µm/s in Figure 1b overlain by an inverse model, and (d) example of experimental data 8 mm after the velocity step from 10 to 100 µm/s in Figure 1b overlain by an inverse model.

Determining the velocity dependence of friction by calculating a − b in this manner is subject to a particular limitation which we address in this study: the coefficient of friction must be at steady state, or independent of displacement, both before and after the velocity step. In practice, however, a dependence of friction on slip is commonly observed. Typically, this appears to be superimposed on the velocity step and is therefore removed from the data [e.g., Blanpied et al., 1998]. However, such detrending practices are not always straightforward, for example, if the slip dependence of friction differs before and after the velocity step. A further complication is that due to displacement limitations in many testing apparatuses, the length of an individual velocity step is usually limited to < 1 mm, hindering evaluation of friction slip dependence over longer displacements. Slip dependence can have significant effects on frictional behavior, and laboratory measurements of slip weakening friction have been demonstrated to be sufficient for the generation of slow earthquakes [Ikari et al., 2013]. Here we evaluate the relative importance of slip- and velocity-dependent friction in laboratory velocity-stepping experiments and assess the impact of friction slip dependence on natural faults.

2 Experimental Methods

As an analogue fault gouge, we used a mixture of silt-sized quartz and commercially obtained clay-rich sediment (Grüne Tonerde, Argiletz Laboratories). X-ray diffraction (XRD) shows that the clay-rich sediment is composed of 62% phyllosilicates (primarily smectite, illite, and mixed layer clays) with other mineralogic constituents being quartz and calcite. The clay, quartz, and distilled water are mixed into a stiff paste in a 3:3:2 proportion by weight. Experiments were conducted in a single-direct shear apparatus following Ikari and Kopf [2011]. In this device, the sample is a cylindrical volume in which the lower half is displaced relative to the upper half, which enforces localized, planar shear deformation (Figure 1a). Upon application of the normal load, pore fluid is allowed to drain via porous metal frits in fluid communication with an open reservoir containing distilled water. Samples were sheared after the rate of height change under load became negligible so that excess pore pressure is assumed to be fully dissipated and that the applied normal stress is equal to the effective normal stress σ_n'. All tests were conducted at σ_n' = 2 MPa, with total shear displacements of up to ~16 mm.

In each test, we sheared the samples at a constant initial velocity V_o for 7 mm followed by a single stepwise increase in velocity to a final value V. The sample was then allowed to shear at V for at least 8 mm, in order to evaluate friction over longer displacements than typical friction experiments which employ < 1 mm of shear at a given velocity (Figure 1b). We conducted three types of velocity-stepping tests within the range 0.1–100 µm/s: (1) constant velocity increases (V/V_o = 10), (2) constant initial velocity (V_o = 0.1 µm/s) with variable V/V_o, and (3) constant final velocity (V = 100 µm/s) with variable V/V_o. We supplemented these experiments with constant velocity experiments (no velocity steps) conducted at 0.1, 1, 10, or 100 µm/s.

We evaluate the velocity dependence of friction using two methods. In the first method, we use an inverse modeling technique [Reinen and Weeks, 1993] to extract the parameters a, b, and D_c (Figure 1c). This method includes explicitly removing friction slip dependence and yields the conventionally measured a − b parameter frequently reported in laboratory friction studies. For each velocity-stepping experiment, the model was applied over a distance of ≤ 1 mm as was commonly employed in previous studies and also over 8 mm for comparison. In the second method, we measure a net change in friction (Δμ) between an initial value at V_o immediately preceding the velocity step, to a friction value at V after 1 and 8 mm of slip (Figure 1d). As this measurement is a net change in friction, it includes effects of both velocity and slip dependence. We then compare values of Δμ/ΔlnV measured using both methods.

We measure the slip dependence of friction η (mm⁻¹) using a linear fit to the data 1–2 mm immediately preceding the velocity step (at V_o) and 3–8 mm after the velocity step (at V) where

(5)

(Figure 1b) [Ikari et al., 2013]. For the measurement of slip dependence at V, the 3 mm distance from the velocity step was chosen as to be sufficiently greater than typical values of D_c, in order to avoid biasing of η as friction rapidly decays following the initial peak defined by a. For constant- velocity experiments, we measured η over the same range as for V in velocity-stepping experiments, between ~11 and 16 mm total displacement.

3 Results

We observe Δμ/ΔlnV in the range −0.027 to 0.004, with mostly instances of overall weakening following velocity steps (Figure 2). Model-extracted values of a − b, which represent the strictly velocity-dependent friction without the influence of slip dependence, range from −0.003 to 0.004 over 1 mm displacement and −0.013 to −0.002 over 8 mm displacement (Table 1). Values of Δμ/ΔlnV are consistently lower for net measurements (which include the effects of slip dependence) compared to modeled velocity step data. There is generally very little dependence of Δμ/ΔlnV on slip velocity (for constant tenfold increases) or the size of the velocity perturbation V/V_o, with some notable exceptions: for experiments in which V/V_o = 10, modeled data over 1 mm shows increasing Δμ/ΔlnV as a function of velocity, while the net strength measurements over 8 mm show a strongly decreasing trend. For experiments with constant V = 100 µm/s, net measurements of Δμ/ΔlnV over 8 mm increase (become less negative) with increasing velocity perturbation size V/V_o; however, we also note that in this case, Δμ/ΔlnV decreases as a function of increasing V_o.

Figure 2

Results of velocity-stepping experiments with (a) constant V/V_o = 10, (b) variable V/V_o with V_o = 0.1 m/s, and (c) variable V/V_o with V = 100 µm/s.

Table 1. Model-Extracted Velocity-Dependent Friction Parametersa

Experiment	Displacement Range (mm)	Vo (μm/s)	V (μm/s)	a	b1	Dc1 (μm)	b2	Dc2 (μm)	a − b	SD a	SD b1	SD Dc1 (μm)	SD b2	SD Dc2 (μm)
B201	1	0.1	1	0.0063	0.0056	19.5	0.0014	206.2	−0.0008	0.00022	0.00024	1.35	0.00012	42.2
	8			0.0062	0.0057	20.1	0.0076	539.2	−0.0071	0.00037	0.00038	1.96	0.00009	8.8
B217	1	1	10	0.0081	0.0069	63.3			0.0013	0.00010	0.00010	1.07
	8			0.0082	0.0054	49.1	0.0049	1010.9	−0.0022	0.00016	0.00017	1.78	0.00004	12.3
B209	1	10	100	0.0047	0.0021	83.6			0.0026	0.00028	0.00029	22.61
	8			0.0045	0.0024	110.7	0.0049	1759.3	−0.0028	0.00027	0.00028	24.53	0.00013	114.7
B198	1	0.1	10	0.0070	0.0043	15.6	0.0061	90.0	−0.0033	0.00015	0.00018	0.87	0.00012	1.8
	8			0.0063	0.0078	38.9	0.0110	933.7	−0.0126	0.00009	0.00009	0.62	0.00002	3.3
B178	1	1	100	0.0059	0.0028	129.6			0.0031	0.00010	0.00014	12.32
	8			0.0059	0.0026	157.3	0.0048	896.6	−0.0016	0.00007	0.00009	7.75	0.00007	14.7
B210	1	0.1	100	0.0069	0.0010	13.9	0.0023	65.6	0.0037	0.00052	0.00078	22.30	0.00632	19.2
	8			0.0065	0.0040	74.8	0.0073	944.7	−0.0048	0.00007	0.00008	2.02	0.00005	10.1

• ^a SD = standard deviation.

For all velocity-step experiments, the critical slip distance D_c was extracted by inverse modeling. In some cases two state variables provided a better data fit; in these cases we report D_c2, which is always greater than D_c1. Modeled data over 1 mm yielded D_c values < 100 µm in most cases, consistent with previous laboratory measurements [e.g., Dieterich, 1981; Marone and Kilgore, 1993; Ikari et al., 2009] (Figure 3). However, when 8 mm of data is modeled, D_c is generally ~ 1 mm and can be up to ~1.8 mm. No significant trends are observed as a function of slip velocity or velocity perturbation size.

Figure 3

Critical slip distance D_c as a function of velocity step size V/V_o.

Slip-dependent friction rates range from η = −0.007 to 0.014 mm⁻¹ (Figure 4). Most η values measured before velocity steps (at V_o) are positive, while all measurements after velocity steps (at V) are negative. The largest difference in η before and after a velocity step tends to occur when both V_o and V are large. Post-velocity step η values are also lower than measurements of η measured at the same velocity and displacement in constant velocity experiments, which cluster near zero (η = −0.001 to 0.0005 for 1, 10, and 100 µm/s). Constant-velocity test η values increase as a function of slip velocity while postvelocity step values decrease so that the difference increases with velocity for rates > 1 µm/s.

Figure 4

Slip dependence of friction rate η in velocity-stepping experiments with (a) constant V/V_o = 10, (b) variable V/V_o with V_o = 0.1 m/s, and (c) variable V/V_o with V = 100 µm/s. (d) Comparison between η measured after velocity steps (at V) (blue circles) and η measured in constant velocity experiments at the same displacement (black triangles), showing the effect of a velocity perturbation on friction slip dependence.

4 Discussion and Implications

4.1 Significance of Slip-Dependent Friction Trends

The results of our velocity-stepping experiments indicate that changes in longer-term slip dependence have a dominant effect on friction. Comparison between model-extracted and net-measured Δμ/ΔlnV values show that considering only the velocity dependence of friction does not capture the overall amount of weakening induced by the velocity step. Furthermore, comparison between short-displacement (1 mm) and long-displacement (8 mm) velocity step evaluations show that velocity strengthening can only be interpreted when short displacements are considered, whereas evaluation over longer distances indicates strictly weakening. Therefore, evaluating velocity steps over distances shorter than 1 mm may lead to underestimation of total eventual weakening. In some cases, a gouge may be interpreted to exhibit “velocity strengthening behavior” when net weakening should actually be expected. This is particularly important because this may cause an expectation of stable slip; however, frictional slip instability due to slip weakening friction is actually possible [Ikari et al., 2013].

We observe that slip weakening induced by the velocity perturbations predominantly controls frictional behavior. Postvelocity step slip-dependent frictional rates at 100 µm/s indicate narrow range from η = −0.007 to −0.006, as well as those at 10 µm/s for which η = −0.006 to −0.005 for all experiments (Figure 4). This suggests that the slip-dependence of friction is independent of velocity history but dependent on slip velocity itself. However, we observe that the friction slip dependence after a velocity step is strongly reduced compared to slip dependence at the same velocity with no step, suggesting that the velocity perturbation induces slip weakening. (Figure 4d).

The origin of the induced slip weakening is uncertain. One possibility is that slip weakening behavior represents the gouge seeking a steady state friction value that is normally lower when no perturbation has occurred. The slip-dependent frictional rates observed after velocity steps correlate with the differences in friction during constant velocity experiments (Figure S1 in the supporting information). The friction coefficients from the constant velocity experiments vary from 0.55 at 0.1 µm/s to 0.28 at 100 µm/s (Figure S1a). The slip occurrence of slip weakening friction after velocity steps seems to correlate with the difference in friction measured during constant velocity experiments (Figure S1b). However, we point out that this difference in steady state friction does not necessarily explain the rate of weakening; i.e., the friction may drop rapidly for a small difference or slowly for a large difference, and vice versa.

It is similarly difficult to relate friction slip dependence to physical changes in the shearing material. In terms of rate-dependent friction, the parameters a, b₁, and D_c1 are thought to be related to the evolution of grain-scale contact asperities and/or porosity in response to the velocity step [e.g., Dieterich, 1981; Sleep, 1997; Blanpied et al., 1998]. However, the mechanisms associated with b₂ and D_c2 are still unknown and mainly employed to provide a better fit to experimental data [e.g., Blanpied and Tullis, 1986; Blanpied et al., 1998]. Comparison between velocity step data modeled over 1 mm and over 8 mm shows that a, b₁, and D_c1 are largely unaffected by the displacement considered in the analysis (Table 1). However, a large difference is seen in b₂ and D_c2, which are sometimes not necessary to describe the data over 1 mm, and results in a significant decrease in a − b. This suggests that if grain-scale asperity contact mechanics describe a, b₁, and D_c1, then a completely different mechanism is associated with b₂ and D_c2 which must operate on a millimeter scale.

Effects of porosity evolution, dilation and compaction, and shear localization [Marone and Kilgore, 1993; Sleep, 1997; Niemeijer and Spiers, 2007] may play a role but would be limited by the geometry of our apparatus that enforces localized shear. Furthermore, dilation/compaction trends, as derived from records of normal displacement as a function of shear displacement from our experiments, do not show any systematic behavior related to friction slip dependence or velocity perturbations. Of the six experiments with a velocity step, only two showed any kind of signal in the normal displacement that clearly correlates with the step; these exhibited a small change from slightly compactive to slightly dilatant behavior. Both of these experiments showed prominent slip weakening after the velocity step, and since dilatancy is expected to be associated with lower pore pressure and therefore strain hardening, this suggests that pore pressure fluctuations and dilatancy trends cannot explain the frictional behavior. We speculate that surface roughness evolution could be the effect captured [e.g., Power et al., 1988].

4.2 Implications for Fault Slip Behavior

On natural faults, the slip rates at which we conducted our experiments (0.1–100 µm/s) are most relevant for transient slip events, such as slow earthquakes [e.g., Ide et al., 2007]. For example, very low-frequency (VLF) earthquakes which occur within the shallow accretionary prism in the Nankai Trough have a slip velocity of 50 µm/s to 2 mm/s [Ito and Obara, 2006]. Recently, Ikari et al. [2013] demonstrated with natural samples from fault zones in the Nankai wedge that slip-dependent fault weakening rates observed at laboratory slip velocities (0.03–100 µm/s) is sufficient for unstable or quasi-stable slip that may result in slow earthquakes, because slip weakening represents a stress drop per displacement. If the stress drops rapidly enough per increment of displacement, it would satisfy a critical stiffness criterion for unstable slip analogous to that developed for velocity weakening friction by Scholz [1998].

Another notable location that experiences slow earthquakes is the Japan Trench subduction zone [Fukao and Kanjo, 1980; Kawasaki et al., 2001]. In the Tohoku region, two slow slip events have been recently observed: one in 2008 and one in 2011, beginning 40 days prior to the 2011 M_w = 9 Tohoku-Oki earthquake [Ito et al., 2013]. One reason that these slow events are noteworthy is because they overlap with the Tohoku earthquake rupture area, which generated extraordinarily large near-trench coseismic slip of several tens of meters [Fujiwara et al., 2011; Ito et al., 2011; Kodaira et al., 2012]. This indicates that the shallow plate boundary at the Japan Trench can fail as both slow and normal earthquakes. The 2011 slow slip event was probably ongoing at the time of the Tohoku earthquake and induced a M_w = 7 interplate earthquake before slipping ceased. Afterslip of the M_w = 7 interplate earthquake was also occurring at the downdip end of the plate boundary fault during the arrival of the main shock [Ohta et al., 2012]. This observation suggests that slip velocities were elevated over a large portion of the plate interface before the arrival of coseismic slip in the same area. The average slip velocity in the 2011 slow slip event was ~0.1–1 µm/s [Ito et al., 2013]; maximum rates thus were probably > 1 µm/s. Our data suggest that at these rates, large slip weakening might be expected to be induced as a response to a positive velocity perturbation. In the Japan Trench example, this perturbation would be the propagation of rupture from an earthquake which nucleated downdip.

If the Tohoku fault zone behaves in a similar manner to our analogue gouge, the large coseismic slip at the Japan Trench during the 2011 Tohoku earthquake may have been assisted by an ongoing slow slip event which caused slightly elevated slip rates at which large, induced slip weakening is possible. However, natural samples from the Tohoku plate boundary are composed of 60–80% smectite [Kameda et al., 2015], which is not similar to our experimental gouge; therefore, additional investigation is necessary to verify if the Tohoku samples exhibit similar behavior to our observations here. However, the analogue gouges used in this study are more relevant to fault zones with a relatively high (>50%) component of nonclay minerals, for example, the Nankai Trough offshore Cape Muroto, Japan [Shipboard Scientific Party, 2001]. Because the Nankai Trough is known for very low frequency earthquakes [e.g., Ito and Obara, 2006], this region may be a candidate to experience induced slip weakening and perhaps enhanced frictional instability in the event of coincident slow earthquakes and coseismic rupture propagation.

5 Conclusions

We conducted velocity-stepping friction experiments using a clay-quartz mixture and evaluated the effect of slip velocity and velocity perturbation size on the velocity and slip dependence of friction. We observe that slip-dependent friction trends, primarily slip weakening, are induced by the velocity steps which have a dominant effect on frictional strength evolution. Evaluating friction over short distances (≤1 mm) after the velocity step, or using inverse modeling that removes friction slip-dependence, can severely underestimate the amount of overall weakening that appears over longer displacements. In some cases, an increase in sliding velocity can induce a slip-hardening fault to become slip weakening. The slip-dependent friction rate is independent of velocity history but dependent on slip velocity itself; moreover, large slip weakening induced by velocity steps is expected at slip rates > 1 µm/s. These results also suggest that slip weakening should be expected for a velocity perturbation that occurs at higher velocities, even if the perturbation is smaller. On natural faults with an elevated slip rate, for example, a velocity perturbation would therefore be expected to result in strongly slip weakening behavior and thus may cause the fault to tend toward unstable potentially coseismic slip.

Acknowledgments

This work was supported by the Japan Society for the Promotion of Science KAKENHI (grant 26257206) to Y.I. and by the Deutsche Forschungsgemeinschaft grant IK107/1-1 to M.I. We thank Achim Kopf for his helpful discussions and assistance in the laboratory. We also thank Andrew Newman, Harold Tobin, and an anonymous reviewer for their constructive comments.