Origin of slow earthquake statistics in low-friction soft granular shear

Domain: earth_science
Published: 2025-12-01
Editorial outcome: major_revision

Abstract
Slow earthquakes differ from regular earthquakes in their slower moment release and size distribution dominated by smaller events. However, the physical origin of these slow earthquake statistics remains controversial. In this work, we experimentally demonstrate that their characteristics emerge from low-friction soft granular shear. To model slow-earthquake fault materials under hydrothermal conditions, we use a low-friction soft hydrogel particle layer floating on lubricating fluid and conduct stick-slip experiments. The observed slip events follow the same laws of both moment release rate and size distribution as with slow earthquakes, contrasting with frictional rigid granular shear. Slip size is determined by the competing effects of shear localization and pressure enhancement with decreasing porosity. These findings indicate that low friction and particle softness in sheared granular systems with sparse contact structures cause slow earthquake statistics, which may be driven by pore fluid dynamics and shear localization within hazardous fault zones.

Introduction
Slow earthquakes are distinctively slower slip phenomena than regular earthquakes of equivalent size 1,2. Between slow and regular earthquakes, the statistics of seismic moment release rate 2 and moment size distribution 3 often exhibit different characteristics. While the moment rate of regular earthquakes follows the cubic moment-duration scaling M0 ∝ T3 with seismic moment M0 and duration T (refs. 4,5), that of slow earthquakes seems to have a constant upper bound as M0 ∝ T (ref. 2). Moreover, while regular earthquakes exhibit a size distribution following a power law 6,7, slow earthquakes frequently obey exponential 3,8,9 or power-law distributions with a higher exponent 10.

To explain the long-lasting dynamics of slow earthquakes, numerous observations suggest the crucial role of viscous fluid around faults 11–13. Some numerical studies propose the effects of the diffusional dynamics of fault size evolution 14 and heterogeneous stress distribution 15. Furthermore, in the past decade, many gouge friction experiments have successfully reproduced unstable sliding with slow slip and/or rupture velocities 16–23.

However, the multiple unique statistical laws of slow earthquakes and their origin have scarcely been investigated through experiments. This is due to two main difficulties: reproducing the observed statistics within finite system sizes, and directly observing materials’ internal microstructures during gouge friction experiments. Despite such difficulties, a stick-slip experiment during antigorite serpentinite gouge dehydration successfully demonstrates the roughly linear scaling between stress drop and duration, corresponding to M0 ∝ T (ref. 17). Owing to the lack of experimental data simultaneously reproducing the multiple statistical properties, the mechanism responsible for the slow earthquake statistics remains controversial. Experimental elucidation of their enigmatic origin would clarify the relationship between slow and regular large earthquakes 2,24, which control the energy balance of plate tectonics 13.

To relate the macroscopic statistics of slow earthquakes to the microscopic dynamics of fault gouge 23,25–28, a combined approach of stick-slip experiments 29 and in situ fabric observations using soft matter has several advantages. Fault gouge materials generating slow earthquakes are presumed to be rich in low-modulus hydrous clay minerals 30, high-pressure pore fluids 12, and/or viscous phases associated with brittle-ductile transition zones 31. These factors make a fault low-frictional and soft. Therefore, the mixture of low-friction soft granular material and viscous liquid demonstrates behavior analogous to that of gouge 32–34. Experiments with soft materials at room temperature and atmospheric pressure allow us to arrest ruptures within the finite system size 21,35 and track each particle. In such a granular system, porosity not only represents the ratio of fluid or ductile phase, but also controls pressure, fault strength, fluid migration, and slip statistics. Accordingly, this serves as a useful framework for understanding the mechanism of slow earthquake faulting.

In this work, we experimentally investigated the statistics of low-friction soft granular slip events and the effect of porosity on them. We conducted quasi-two-dimensional (2D) rotary shear experiments on a monodisperse hydrogel granular layer with low friction and low elastic modulus, which floats on a viscous lubricating liquid surface as shown in Fig. 1a (see Methods for details). The experimental system with a constant volume enables the systematic variation of the bulk porosity ϕ and direct tracking of particle motion (Fig. 1b), both of which intrinsically govern the behavior of fluid-retaining faults. We captured hundreds of stick- and slow-slip events under shear, which become more pronounced with decreasing porosity (Fig. 1c, d). Slip events in this low-friction soft granular system simultaneously follow two statistical characteristics of slow earthquakes: the moment-duration scaling and size distribution 2,3, as shown in Fig. 2. Based on the in situ particle observation, shear localization at lower porosity (Fig. 3) explains these porosity-dependent statistics of slip moment (Fig. 4) as well as the depth distribution of earthquakes across the spectrum of their types (Fig. 5).

Results
Moment statistics of slip events
The fluid-lubricated soft granular layer under shear exhibits torque fluctuations corresponding to stick-slip events (Fig. 1c, Supplementary Fig. 1). In this lubricated system, the stick-slip behavior involves both partially decoupled stable shear during the stick phase and unstable slip with torque drops, accommodating total displacement similar to that in slow earthquakes (Fig. 1d). This transitional behavior between creep and stick-slip is typically observed in lubricated granular systems 32,33. At a porosity of ϕ ≳ 0.24, creep motion virtually dominates all deformation, whereas at ϕ = 0.18, the unstable slip events account for 32% of the total displacement. A decrease in porosity to approximately 0.20 causes a transition from creep to stick-slip behavior (Fig. 1c, d). The slip rate exceeds the steady shear rate, shown by the dashed black line in the inset of Fig. 1d. The frequency of slip events roughly increases with the porosity decrease (Supplementary Fig. 2, Supplementary Table 1). The decrease in porosity also causes an increase in both the mean torque strength and its drop amplitude (Fig. 1c, Supplementary Fig. 3a, b), which are defined in the inset of Fig. 1c. The maximum torque at the minimum porosity remains sufficiently low, resulting in no particle failure. The porosity dependence of the torque drop amplitude remains consistent across system sizes ranging from 100 to 300 mm in diameter (Supplementary Fig. 3b).

We statistically analyzed the moment M0 and its rate M_0 of the slip events, calculated from the torque drop amplitude, drop duration, and observed slip area (see Methods). Consistent with the variation in the torque drop amplitude (Supplementary Fig. 4a, b), a decrease in porosity from 0.41 to 0.18 progressively increases both the amount and rate of moment release (Fig. 2).

Statistically, the moment M0 follows an exponential size distribution, regardless of porosity (Fig. 2a). This is in contrast with regular earthquakes 6,7 and dry, confined granular materials 36–38, following a power-law size distribution, i.e., a scale-invariant process. Rather, our result is consistent with slow earthquakes following an exponential size distribution or a power-law size distribution with a higher exponent, both of which are difficult to statistically distinguish 3,10. In addition, a power-law size distribution truncated with an exponential function seem to be exhibited at a slower shear rate (Supplementary Fig. 5a). The characteristic size (slope) of the exponential size distribution decreases with increasing porosity ϕ, consistent with previous experimental studies 39. These exponential size distributions represent a scale-limited process, characterized by the area or amount of slip in slow earthquakes 3.

Our experiments also revealed an early linear moment-duration scaling of M0 ∝ T, indicating a constant moment rate M_0 (Fig. 2b). A slower shear rate results in approaching a linear moment-duration scaling (Supplementary Fig. 5b). An increase in timescale (slow shearing) consistently induces the linear scaling between size and duration in other granular experiments 37 and seismic observations 2, although the underlying mechanism remains unclear. Crucially, our laboratory experiments simultaneously reproduce the exponential size distribution and linear moment-duration scaling, key features of slow earthquakes.

In situ granular motions
To investigate the representative particle movements contributing to the observed moment statistics, we captured in situ particle arrangements as binary images every 0.1 s (Fig. 1b). Fig. 3a–c shows the comparison of the time-lapse images for ϕ = 0.23, 0.20, and 0.18, mapping the standard deviation of temporally varying binary values at each pixel over approximately 1100 s and 660 degrees of rotation (see Methods). These images reveal three types of regions formed at any porosity: the central shear band with active motion (continuous black region), the surrounding vibration region with particle drift shorter than the particle diameter, and the outer fixed region with no noticeable motion. The particles principally slip on the boundary plane within the granular layer between the shear band and the surrounding vibration region. When dry, angular quartz sand grains are sheared in a setup similar to ours, they also exhibit the central shear band and intermittent radial slip propagation into the surrounding region 27. This implies that the granular fabric we observed is qualitatively consistent, regardless of lubrication and particle shape. Following the structure of granular fault 40, the outer fixed region and the others correspond to the damage zone and the fault core, respectively; the shear band corresponds to the principal slip zone of a natural fault 41.

The shear band becomes more localized at lower porosities (Fig. 3a–c). We obtained a linear relationship between the shear band thickness w and ϕ (Fig. 4a). As ϕ decreased from 0.41 to 0.18, w decreased from ≳ 9d to 2d with the particle diameter d. The shear band thickness of less than ten particle diameters, an approximate typical value for dry frictional conditions 27,42, is associated with the correlation length of strain under low elastic modulus 43 and low friction. This shear localization is also observed in lubricated granular particles with an increase of pore-fluid viscosity 32,33. Notably, the released moment exhibits larger values under conditions of lower porosity despite the shear localization (Fig. 4b). Meanwhile, the porosity within the localized shear band remained constant at 0.3, which is higher (richer in fluid content) than the bulk ϕ, except under the highest ϕ value in run #79 (Supplementary Fig. 3e). Note that we only varied the bulk porosity ϕ; its spatial distribution spontaneously evolves and stabilizes under steady state.

Additionally, the surrounding vibration region exhibits radial anisotropy that persists throughout the entire run time (Fig. 3a–c). This implies the presence of fixed force-chain structures 44, which supports the shear band without their significant rearrangement during slip events. Furthermore, the outer fixed region implies that the mechanical correlation length is sufficiently short relative to the entire system size. This ensures that slip and rupture propagation are confined within the shear band, not extending throughout the entire system. The outer fixed region can apply low normal stress (radial pressure) to the shear band that has higher porosity, constraining its thickness w(ϕ).

We further examined the distribution and evolution of slip planes associated with each event. Figure 3d illustrates the typical types of particle motion with instantaneous displacement for different events at ϕ = 0.20. The black areas indicate particle differential motion over 2 s. Quantifying the number of these moving particles during each event reveal that slip events predominantly occurred on slip planes of similar size within the shear band for each porosity, independent of the moment M0 (see Methods; Supplementary Fig. 6). This justifies defining a representative constant value of slip plane area S(w(ϕ)) for each porosity. Figure 3e shows the temporal evolution of slip motions over 17.2 s, corresponding to the torque drops. The blue arrows indicate the rupture front, which is identified as the boundary between areas of particle mobility and the quiescent surroundings. Its propagation timescale is approximately 1 s. This rupture time is relatively shorter than the overall slip event duration T in this lubricated system. Thus, the low-friction and soft granular motion is primarily characterized by shear localization with decreasing porosity and slip on a plane of constant area for each porosity, regardless of the moment. These unique characteristics of the granular dynamics can explain the moment statistics.

Possible mechanisms for the statistics
The low-friction soft granular system exhibits both the exponential size distribution and linear moment-duration scaling (Fig. 2). These two statistical laws remain consistent across various experimental conditions (Supplementary Table 1), including polydispersity (Supplementary Fig. 7), liquid viscosity (Supplementary Fig. 8), and system size (Supplementary Fig. 9). This universality ensures that the two statistical laws can be explained by a simple model, as follows.

First, the exponential size distribution (Fig. 2a) should emerge from the strength of the granular layer, which is generally supported by force chains 44 and induced to slip by their buckling collapse 45. The stress of both force chains and consisting particles follows an exponential distribution 44,46. In contrast to our system, frictional particles under dry conditions and higher normal stress follow power-law size distributions with exponential truncation, regardless of system dimension 36–38. In the frictional systems under normal stress, the force chains form networks that can cascade into critical slip avalanches. However, lubrication and/or low normal stress (radial pressure) in this study inhibit the formation of force chain network, thereby restricting slip size to the scale of individual particles and suppressing moment release during slip events (Fig. 5a–c). When we substituted hydrogel particles with more rigid glass beads with frictional surfaces, we observed a superposition of exponential and power-law size distributions under lubrication (Supplementary Fig. 10a). This distribution is also observed under a lower shear rate condition (Supplementary Fig. 5a), which implies the distinction of individual events owing to complete relaxation and quasi-static loading. These results demonstrate that low friction, as well as low elastic modulus, low normal stress, and slower relaxation than shear rate, plays a crucial role in limiting the mechanically correlated length and preventing cascade collapse events. Consequently, granular systems with these properties manifest the exponential size distribution that reflect the heterogeneous stress distribution of individual force chains (Fig. 5a, b).

The characteristic scale of the size distribution can be determined by the balance between the enhanced force chain strength and localized slip area with normal stress. To explain the relationship between the mean moment and porosity, we assume that slips occur repeatedly on the outer boundary plane of the cylindrical shear band, as shown in the inset of Fig. 4a. Although each particle slips within the buckling force chains, we consider that most of the strain energy is dissipated by the slips on the shear-band boundary plane, resulting from collective particle rearrangements within the force chains. The shear band has a thickness w and a diameter δ of the interparticle contact patch. The seismic moment is defined as M0 = μSD, where μ is the shear modulus, S is the slip area, and D is the slip displacement 47. This moment resulting from the torque drop with an amplitude ΔΓ and associated slip with a plane area S = 2π(R + w)δ can be calculated as,

M0 = δ/C ΔΓ = 2πδ2/(C(R+w)) (R+w)Δσ, (1)

with the shear stress drop Δσ, a geometrical constant C, and the rotating cylinder radius R (see Methods). In this equation, the mechanical force drop ΔΓ/(R+w) is converted into the moment on the slip plane with the characteristic width δ. Equation (1) adequately explains the porosity-dependent characteristic moment 〈M0〉, with the fitted scaling 〈Δσ〉 ∝ ϕ−n (n = 5.0) and the fitted linear relationship w(ϕ), assuming constant δ (solid line in Fig. 4b). 〈〉 denotes the mean value.

This indicates that despite shear localization, the increases in normal stress and resultant shear stress predominantly contribute to the moment. This is consistent with the interpretation that the force chain strength governs the moment statistics 40. Granular materials typically exhibit a rigidity phase transition and normal stress enhancement with decreasing porosity 48, which is also observed in this study (Supplementary Fig. 3c). Thus, the size distribution is governed by the exponentially distributed stress drop Δσ and its corresponding slip displacement D, which vary with porosity (see Methods; Supplementary Fig. 4b).

Second, the linear moment-duration scaling, M0 ∝ T (Fig. 2b), indicates a violation of self-similarity (fractal) holding in regular earthquakes, where M0 ∝ T3 (refs. 4,5). In regular earthquakes, self-similarity emerges from the relationship between slip area S and displacement D, where larger slip areas require greater displacements as D ∝ S. In contrast, our low-friction soft granular system exhibits the opposite trend, where D increases as S(w(ϕ)) and ϕ decrease (Supplementary Fig. 4c). We observed that the temporally-averaged slip velocity of the cylinder V ( ∝ D/T) remains constant regardless of porosity (Supplementary Figs. 3d, 11), resulting in the linear moment-duration scaling M0 ∝ μSVT ∝ T with constant S(w(ϕ)) for each porosity (Fig. 4a). Furthermore, when we investigated the effect of elastic modulus on the moment-duration scaling using glass beads, we observed a nonlinear scaling of M0 ∝ T2, especially in the high stress regime (Supplementary Fig. 10b). This suggests that the particle modulus or stress level is related to the constancy of slip area and velocity, which realizes the linear moment-duration scaling. Rigid particles, such as glass beads and fault gouge grains in the seismogenic zone, can preserve the fractal structures of force-chain networks at the percolation threshold 49 (Fig. 5c) or of rough interparticle patches (Fig. 5f), resulting in the cubic moment-duration scaling M0 ∝ T3. In contrast, low-modulus particles and/or low normal stress cause the monotonous structures of force chains (Fig. 5a, b) or interparticle patches (Fig. 5e), violating the self-similar scaling.

Note that, however, the effects of lubrication and system dimension cannot be ruled out. In particular, the self-similar moment-duration scaling M0 ∝ T3 is not realized in a 2D system due to anisotropic rupture propagation. The fault plane with a finite width, corresponding to the width δ in our granular layer, exhibits a strong dependence of the transition in the moment-duration scaling on stress heterogeneity 15. Any of these cases potentially lead to a loss of fractality in the distributions of displacement and patch area of slip by the force chain collapse (Supplementary Fig. 4c, 6), resulting in the linear moment-duration scaling M0 ∝ T (Fig. 2b).

The constant moment rate M_0 (= M0/T) in the linear moment-duration scaling can be expressed in terms of the system’s torsional stiffness K (Nm deg−1) using,

ΔΓ = 360KV, T = D/V = 1/c ω = 360V, (2)

with the tangential slip displacement D = VT, circumference c = 2πR, and angular velocity ω (deg s−1) of the cylinder (see Methods). The numerator of this equation represents the elastic shear deformation of the system, whereas the denominator applies a correction for the steady rotation of the motor during each slip. Using Equations (1) and (2), this model effectively explains the porosity-dependent mean moment rate 〈M_0〉, with the fitted shear band thickness w(ϕ) and the constant slip velocity V, assuming constant δ (solid line in Fig. 4c). V in Equation (2) is constant in this study (Supplementary Figs. 3d, 11), consistent with the constant torque drop rate ΔΓ/T and exponentially distributed slip displacement D (Supplementary Fig. 4b, c). As the shear band thickness has a minimum value of w ≃ d in this study, the moment rate should approach a maximum limit (dashed line in Fig. 4c). In natural fault systems, the principal slip zone thickness for each event typically ranges from 1 mm to 1 cm in the fault core 41. This characteristic length scale of shear localization possibly constrains both the moment scale and also the maximum moment rate observed in slow earthquakes 2. Our quantitative models for both the exponential size distribution and the linear moment-duration scaling provide a self-consistent framework for slow earthquake statistics, interpreted as the suppression of the force-chain network percolation and fractal patch formation.

Methods
We conducted rotary shear experiments on a floating granular monolayer, showing stick-slip behaviors. We measured torque and recorded the particle arrangement in situ under various porosities and other conditions (Supplementary Table 1). The temporal evolutions of mechanical torque and visual data were statistically quantified.

Experimental setup
The experimental setup is shown in Fig. 1a. We sheared a layer of particles floating on a heavy liquid surface in a cylindrical container (radius Rc) at room temperature and atmospheric pressure by inserting a rotating cylinder (radius R) into the center of the layer. The floating particles comprise an approximately one-particle-thick monolayer, enabling the tracking of all particles with a camera. This quasi-two-dimensional granular system removes the effects of basal and interparticle frictions, leading to enhanced particle dispersion. This lubricated granular layer is appropriate to simulate low effective normal stress owing to high pore pressure, widely suggested as a cause of regular and slow earthquakes 12,22. We used a cylindrical Couette system in two configurations, one with 2Rc = 300 mm and 2R = 18.84 mm, and the other with 2Rc = 100 mm and 2R = 18.72 mm. The ratio of the channel gap (Rc − R) to the particle diameter (d = 4.4 mm) was 32 and 21, respectively (Fig. 1b). The same 16 particles used as the experimental sample were glued in a single row around the side surface of the cylinder to roughen it, except for runs #27–31 with 17 particles. In the case of a bidisperse mixture described below (run #77), hydrogel particles (d = 4.4 mm) were used for this purpose.

We used three types of particles as a fault gouge analogue: opaque spherical polymer hydrogel particles (Leaf Corporation, Bio Beads, PEG) and spherical glass beads of two sizes. Hydrogel particles have a diameter of d = 4.4 mm with a standard deviation of 0.1 mm, with lower friction (coefficient of < 10−1) and lower modulus (∼ kPa) 66. The two types of glass beads have d = 4.1 or 2 mm, respectively, with higher friction (coefficient of ≳ 0.2) 67 and higher modulus (∼ GPa). To align particle centroids within the same plane on the liquid, we used a single type of monodisperse particle in each experimental run. To investigate the effect of particle arrangement, a bidisperse mixture of the hydrogels and 2 mm glass beads was also used. For preparation, we immersed the hydrogel particles in the heavy liquid solution for more than one week, as the hydrogel particles change volume owing to osmotic pressure depending on the concentration of the solution. The osmotic effect made the particles shrink by 2–3% in diameter 5 days after starting the permeation of the water-immersed particles.

To float the particles, we used two types of liquid. A transparent solution of sodium polytungstate (Na6(H2W12O40), TC-Tungsten Compounds GmbH, purity of > 99%) was prepared with a density and a viscosity of approximately 2.8 g cm−3 and 2 × 10−2 Pa s, or slightly above, respectively. Exceptionally, in runs #56 and #57, the density was approximately 2.6 g cm−3 and the relevant viscosity was slightly lower. To investigate the effect of viscosity, another solution was also prepared by adding a thickener of carboxymethyl cellulose (As One Co., Ltd., CMF-150) with a density of 2.8 g cm−3 and a viscosity of 1 Pa s. The densities made the particles remain suspended just below the liquid surface level, as schematically shown in Fig. 1a. To prepare the solution, sodium polytungstate powder was dissolved in deionized water by stirring, where its concentration determines both density and viscosity 68.

In all experiments, the depth of this liquid solution was approximately 10 mm, greater than twice the particle diameter. This enabled the particles to move under an adjacent particle without any stiffening by the resistance from the base of the container. However, we did not observe any significant vertical motion of hydrogel particles, potentially due to their low elastic modulus.

To rotate the cylinder and measure the torque, we used a B-type viscometer (BROOKFIELD, LVDV-II+Pro, maximum torque of 6.73 × 10−5 N m and rotation speed of 6 × 10−2 – 1.2 × 103 deg s−1). The motor of the viscometer was set to rotate unidirectionally at a constant rate ω, which was not directly measured. The cylinder and the motor were connected through a torsion spring with a rotational stiffness of K = 9.75 × 10−7 N m deg−1.

To record and track the motion of all particles in situ, the cylindrical container has a transparent base plane. We used a USB vision camera (OMRON SENTECH, STC-MCCM401U3V, effective pixels of 2048 × 2048, maximum frame rate of 89 fps) with a machine vision lens (RICOH OPTOWL, FL-BC1220-9M). For technical convenience, we used a mirror inclined at 45 deg and set the camera horizontally, with the optical axis aligned to the system center (Fig. 1a). When recording, we placed two panel light sources on both sides of the apparatus to enhance the contrast (Fig. 1b).

In the experimental procedure, we first filled the container with the liquid, added particles, manually stirred the mixture thoroughly, and subsequently inserted the cylinder of the viscometer. After adjusting the height of the glued particles to match that of the particle layer, we started rotating the cylinder while logging torque and recording images.

Experimental conditions
The shear experiments were performed in 23 runs, as summarized in Supplementary Table 1. We investigated the dependence of mechanical behaviors on porosity (0.18 ≲ ϕ < 0.41 and the corresponding number of hydrogel particles approximately 2700–3900), particle type (d = 4.4 mm hydrogel particles, 4.1 mm glass beads, and 2 mm glass beads), liquid viscosity (2 × 10−2 and 1 Pa s), container diameter (100 and 300 mm), and rotation angular velocity (ω = 0.06–60 deg s−1). For investigating the effect of the particle number, we only varied the total number within a container. The range of porosity was chosen to ensure that the granular layer can maintain a single-layer thickness and also exhibit any detectable torque resistance. To investigate the effect of particle arrangement, we further manually arranged an ordered polycrystalline structure with the intention in runs #67 and #70, where the size of each crystalline cluster was on the order of 10d (Fig. 3c). The particle arrangement in runs #85–87 with the rigid glass beads was also an ordered structure. We also prepared a random structure using a mixture of manually dispersed 3781 hydrogel particles (d = 4.4 mm) and approximately 3000 glass beads (d = 2 mm) in run #77.

The results were obtained at a constant rotation rate of 0.60 deg s−1, unless otherwise specified in each figure or dataset. This rotation rate corresponds to a strain rate of 10−2 s−1 at the surface of the rotating cylinder. The rotation rate of 0.60 deg s−1 is considered sufficiently slow to neglect the fluid inertial effect with a particle Reynolds number of Rep ≃ 0.1. This value was calculated assuming a viscosity of 2 × 10−2 Pa s, a velocity of 0.1 mm s−1, a fluid density of 2.8 g cm−3, and a particle diameter of 4 mm, respectively.

We estimated the porosity of the granular layer using the mean particle diameter d and total number counted using the recorded image (Fig. 1b). This value represents the bulk porosity and is used as the porosity value in this study. The standard deviation of porosity is 0.008, estimated from that of the hydrogel particle diameter (0.1 mm). We also evaluated whether the estimated porosity was consistent with the visually measured value. In particular, in the case of run #73 using approximately 4000 particles, many particles were out of the monolayer plane. The porosity was estimated to be between 0.15 (assuming all particles are confined within the monolayer) and 0.18 (in run #67 with fewer particles). We provisionally plotted these data in Fig. 4 and Supplementary Fig. 3 at ϕ = 0.15. Confining all particles within a single layer proved difficult at lower porosities (ϕ < 0.3), leading to some particles being located out of the plane.

Measurements
We measured the torque and recorded the images immediately after starting the rotational shear for hundreds to tens of thousands of seconds, depending on the runs. The torque was measured at a sampling rate of approximately 2 Hz. With the viscometer, the nominal lower limit of the effective measurement range was 6.73 × 10−7 N m (1% of the maximum). However, in practice, both the detection limit and resolution of the system were as low as 7 × 10−8 N m (0.1% of the maximum). We recorded the visual images with an exposure time of 0.1 s at a sampling rate of 10 fps. In our experimental setup, a unit pixel corresponds to approximately 0.16 mm at the center of the images. For all images, we did not apply any correction for optical refraction, distortion, and misalignment, which did not qualitatively affect the results of this study. The distortion between the center and edge of the 300 mm container was 2% at maximum, and the shear bands we analyzed were localized to the central part of the images. In each run, torque measurement and image acquisition were initiated manually at the same time. The accuracy of temporal synchronization between the mechanical data and the visual images was approximately 0.1 s, evaluated by the recording test at 10 fps. Examples of the recorded dataset are shown in Supplementary Fig. 1.

Analysis
We defined the run time by excluding data at the beginning and final 100 s of each run, and quantified both the mechanical and visual data over the remaining run duration. These exceptional data possibly include unsteady transient flow and noise owing to vibrations and perturbations by the operation of the viscometer.

First, using the mechanical data, we quantitatively characterized the temporal fluctuation of torque Γ(t). Torque drop duration T and drop amplitude ΔΓ = − (Γ(t + T) − Γ(t)) > 0 with start time t are characterized for each event. The values ΔΓ = ΔΓi, T = Ti, t = ti for the i-th event are defined such that Δ_Γ(t) < 0 is always satisfied in the range of ti ≤ t < ti + Ti and Δ_Γ(t) ≥ 0 at t = ti − dt, ti + Ti with a sampling interval of dt ≃ 0.5 s (inset of Fig. 1c).

Although we did not set any threshold value for the definition of the magnitude of torque and its drop amplitude, the events with ΔΓ ≤ 7 × 10−8 N m (the practically measurable minimum value) were excluded from the analysis. We analyzed approximately 10–6500 torque drop events, depending on the run duration, as presented in Supplementary Table 1. Event counts are normalized to the duration of each experimental run.

We calculated the cumulative frequency N(ΔΓ), defined as the total number of torque drop events per second with amplitudes equal to or greater than ΔΓ. As the cumulative frequency distribution follows a nearly exponential function (Supplementary Fig. 4a for torque drop, Fig. 2a for moment), we fit the following function to the data,

N(ΔΓ) = N0 exp (−ΔΓ/G), (3)

where N0 and G are the fitting parameters. When an ideal exponential distribution is satisfied, G = 〈ΔΓ〉, where 〈〉 denotes the mean value. We obtained the mean torque drop amplitude as the value of fitting parameter G.

From the torque data, we also estimated the rotating cylinder displacement u(t) along the side surface of the cylinder. To obtain the displacement, we applied the correction for the apparatus stiffness to the torque data 32. Using the torsional stiffness of the viscometer K = 9.75 × 10−7 (N m deg−1), constant angular velocity ω (deg s−1) of the motor, and radius of the rotating cylinder R, tangential displacement at the side surface of the cylinder can be calculated as,

u(t) = (Kωt − Γ(t)) / (360K/2πR), (4)

u(t) can be considered the displacement imposed on the granular layer itself, deforming it elastically and plastically. The resolution of the displacement is 0.01 mm, estimated based on that of the torque measurement.

Second, using the recorded raw images of particle arrangements, we obtained three types of images using Fiji/ImageJ software 69: original binary, time-lapse, and differential images. We first obtained original binary images based on the contrast difference between the particle areas and the background. To obtain them, the original video was converted to 8-bit grayscale. Then, background subtraction was performed using the “Subtract Background” function (100-pixel radius) in Fiji/ImageJ, followed by automatic thresholding using the “Default” method. We used raw images recorded over a 1000 s interval taken more than 1000 s after the beginning of the experiments, which capture the steady arrangements of particles. As an exception, we used images captured immediately after the beginning of run #67 and 600 s after the beginning of runs #74 and #80 to obtain adequate amounts of recorded images. These exceptional data possibly include unsteady transient flow and noise owing to vibrations and perturbations by the operation of the viscometer. Using the original binary images, we measured the number of particles (Fig. 1b, Supplementary Table 1).

Two types of image processing were applied to the original binary images: time-lapse and differential images (Fig. 3). For the time-lapse images (Fig. 3a–c), we mapped the standard deviation for each pixel over the entire duration of 1000 s. Using the time-lapse images, we characterized the spatial distribution of shear bands (Fig. 3a–c) and measured their thickness (Fig. 4a). To define the shear band thickness w, we applied three iterations of erosion followed by three iterations of dilation to the time-lapse images, and then computed the radial distribution of pixel intensity (standard deviation). This processing allows us to extract regions of active particle motion. We empirically identified the shear band edge as the radial position slightly outside the active zone of particle motion with a normalized standard deviation of 250/255, considering the smearing out of the active zone.

For the differential images, we also calculated the absolute difference in brightness between binary snapshots, holding a 2-second gap every 0.1 s (10 fps). We chose the relatively long time gap (2 s) to capture adequate differences between frames as several pixels in each differential image. To reduce the random optical noise, we further applied a Gaussian blur with a standard deviation of 1.00 to the images, and performed a second thresholding in the range 70–255 using the Li method in Fiji/ImageJ. Using the differential images thus obtained, we quantitatively analyzed the motion area where particles have swept (Fig. 3d, e, Supplementary Fig. 1). In addition, to investigate the apparent slip plane size for each event, we also estimated the number of drifting particles p at each time. To calculate p, we used the ratio of the differential area over the entire region, a, to that per particle over the region with a radius of r = R + d inscribing the fixed particles on the cylinder, ac, resulting in p = a/ac. This value p can be used as an estimate of the equivalent slip plane area S. We statistically analyzed p averaged over each duration T for each event (Supplementary Fig. 6). Note that we did not apply static (ω = 0) noise subtraction to the differential images due to difficulties in quantifying the noise. Even with correction, however, p decreased by a constant, causing only a parallel shift in Supplementary Fig. 6 without any qualitative influence on the results.

Estimation of the moment
Moment M0 of slip associated with stress drop Δσ is represented as,

M0 = μSD = L S/C Δσ, (5)

where μ is the shear modulus, S is the slip area, D is the slip displacement, L is the slip patch width, Δσ is the stress drop, and C is the geometrical constant 47. The stress drop is represented as 47,

Δσ = Cμ D/L, (6)

Considering the cylindrical shape of a shear band with stress applied to its outer boundary plane, as shown in the inset of Fig. 4a, the shear stress drop Δσ(r) with radial distance r from the center of the rotation axis is related to the torque drop amplitude ΔΓ(r) as,

Δσ(r) = ΔΓ(r) / (2πr2δ), (7)

with the diameter δ of the interparticle contact area, and L = δ, S = 2πrδ under the assumption that the moment arm is mechanically homogeneous. The measured torque drop amplitude on the cylinder ΔΓ can be considered in equilibrium with the torque drop amplitude on the outer boundary plane of the shear band ΔΓ(r = R + w), with radius of the rotating cylinder R and shear band thickness w. Using Equations (5) and (7), we can obtain Equation (1). We assumed the following values: C = 7π/16 for the circular crack at each contact patch 47, and constant δ = 0.1 mm, which is expected to increase with decreasing porosity in practice.

Model fitting of the moment and moment rate
Moment M0 (ΔΓ) in Equation (1) is represented using torque drop amplitude ΔΓ as,

ΔΓ = (R+w) [2π(R+w)δ] Δσ, (8)

when considering the cylindrical shear band. The mean moment data 〈M0〉(ϕ) in Fig. 4b, obtained from the exponential fitting of the cumulative frequency distribution (Fig. 2a), are adequately explained when we use the w(ϕ) relationship and fit 〈Δσ〉 = Aϕ−n to the mean torque drop amplitude data 〈ΔΓ〉(ϕ), except for run #73 (solid curve in Supplementary Fig. 3b). The fitting results are A = 4.7 × 10−13 Pa and n = 5.0, shown as a solid curve in Fig. 4b.

Moment rate M_0 = M0/T is represented as,

M_0 = M0 / (D/V) = V, (9)

where D is the tangential slip displacement and V = D/T is the mean tangential slip velocity, both at the cylinder surface. As D is calculated from displacement u(Γ) in Equation (4), we obtain Equation (2), where ΔΓ/T is trivially expressed with a single variable V. The analytical curve of Equation (9) using Equation (1) and (2) is shown in Fig. 4c as a solid line, with a constant mean value of V = 0.13 mm s−1 (Supplementary Figs. 3d, 11) and w(ϕ) relationship (Fig. 4a).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript titled “Origin of slow earthquake statistics in low-friction soft granular shear” by Yuto Sasaki and Hiroaki\nKatsuragi offers compelling experimental evidence for the role of low friction between viscoelastic particles in reproducing\ntwo key features of slow earthquakes: exponential frequency-moment distribution and linear scaling between moment and\nduration. Thanks to an interesting variation of the classical Couette cell, the authors are able to demonstrate how the two\naforementioned micro- to meso-scale properties (particle viscoelasticity and interparticle friction) of their experiment can be\ntuned (by using hydrogel particles or glass beads) to alter the shape of the frequency distribution of torque drops and the\nscaling between torque drop and duration. This is an interesting demonstration of the spectrum of frictional slip behaviors\nand how the latter are controlled by system properties. As such, in my view, the insights into the physics governing slow\nslip will be of interest to the slow-earthquake community.\nThe authors use a variation of a well-established methodology (see-through Couette cell, with particle tracking) to produce\nresults that support their main argument sufficiently. To my understanding there is no obvious flaw in their methods and\ndiscussion of their results. The manuscript contains sufficient information to allow replication of the work, at least in principle.\nThe data supporting the manuscript are accessible and I confirm that they can be accessed and viewed (plotted with\nMatlab).\nTo better couple the manuscript with previous, laboratory-based studies looking into the physics of (slow) earthquakes, I\nwould recommend that the authors compare their findings with those of Leeman et al. (2016). For example, can a\ncomparison be made between the role of effective stiffness of a frictional system comprising of hard particles, in the context\nof rate- and state-dependent friction, and the effect of lubrication in a system with soft particles? This could come as an\nextension to the comment in line 254 to 257. Another study that I think could be discussed is on the lubricating effect of a\nfluid on a granular medium, by Reber et al. (2014). A brief discussion of the effect of fluid viscosity and packing fraction on\nthe macroscopic strength of the material could be illuminating. Also, the summarizing Fig. 3 from Reber et al. (2014) could\nbe discussed in the context of Fig. 5 from this manuscript. Finally, Hayman et al. (2011) reported an evolution in the\nfrequency distribution of slip from power law to exponential with increasing packing fraction, in a quasi-2D system of hard\nparticles. The authors could consider updating their discussion in, for example, line 157 to 159 and line 200 to 206,\ntaking into account, e.g., Fig. 3 and Fig. 6 from Hayman et al. (2011).\nFinally, some questions, comments and corrections:\nLine 9: by “limited” do the authors mean in terms of maximum magnitude?\nLine 17: The authors could consider a bridging sentence that would make the transition from earthquakes to hydrogel\nparticle layers smoother.\nLine 47: Perhaps it would be simpler to say “… the difference between slow and regular, large earthquakes which\ncontrol…”? For sure “underground” seems unnecessary.\nLine 54: Change “stick” to “stick-“ as in stick-slip.\nLine 64, 65: I am not sure what a precursory slip movement looks like in extended data Fig. 1. Perhaps some indicative\nguiding lines or a blow-up of one example could help.\nLine 67, 77/Fig. 1D: consider adding an inset (e.g., showing a step or gentle undulation) to better demonstrate what is\ndescribed in the text.\nLine 76: consider adding a statement such as “therefore particles do not fail”, as a clarification\nFig. 2: Perhaps “logarithmic scale” is more appropriate than “logarithmic coordinates”?\nLine 115: Add reference to Fig. 4B (larger moment at lower porosity)?\nLine 129: do the authors refer to the confining pressure applied to the shear band from the particles in the outer region?\nLine 138: Should it not be 17.2 s instead of 15.2 based on Fig. 3E?\nLine 198: Does “pressure condition” refer to the confining pressure?\nLine 224: Should “Fig. 5E” instead point to “Fig. 2B”?\nLine 252: Consider “As *with* slow earthquakes”?\nLine 281: Change “plughing” to “ploughing”, if that what is indeed meant.\nLine 289, 290: Consider simplifying to \"… potentially inducing devastating regular earthquakes, such as… \".\nLine 326: Change “polytundstate” to “polytungstate”.\nLine 388: Change “lightning” to “lighting”.\nLine 503 & 505: If I understood correctly, both references to “Fig. 2B” should be pointing to “Extended data Fig. 2B” (i.e., not\nthe one in the main text).\nLine 603, reference #21: Change “nankai” to Nankai.\nLine 670, reference # 47: issue with printing of special characters.\np. 31, extended data Fig. 6A, B: change “Nm” to “N m” in the x-axis caption for consistency.\np. 32, extended data Fig. 8B: change “Nm” to “N m” in the x-axis caption for consistency.\nLast but not least, I would like to thank the authors for a thought-provoking manuscript that I read with great interest!\nReferences\n1. Leeman, J. R., Saffer, D. M., Scuderi, M. M. & Marone, C. Laboratory observations of slow earthquakes and the spectrum\nof tectonic fault slip modes. Nat Commun 7, 11104 (2016).\n2. Hayman, N. W., Ducloué, L., Foco, K. L. & Daniels, K. E. Granular Controls on Periodicity of Stick-Slip Events: Kinematics\nand Force-Chains in an Experimental Fault. Pure Appl. Geophys. 168, 2239–2257 (2011).\n3. Reber, J. E., Hayman, N. W. & Lavier, L. L. Stick-slip and creep behavior in lubricated granular material: Insights into the\nbrittle-ductile transition. Geophys. Res. Lett. 41, 3471–3477 (2014).", "reviewer_2": "Connecting the deformation of a wet granular system to slow-slip events is intriguing and potentially of wide interest.\nThe experiments are extensive, and the statistical analysis is intriguing.\nOne of the shortcomings of the current manuscript is a better motivation why these experiments are relevant to investigating\nslow slip. The setup and materials are obviously quite different and the connection to recorded slow-slip event locations is\nnot obvious. While the statistics is pointing towards that these experiments indeed captures some behavior, it would be\nbeneficial if there would also be other supporting arguments for the choice of experimental setup and material. In addition,\nexperiments have already been conducted with grains embedded in a fluid where the slip was monitored. While these\nexperiments did not include any statistics it might be fair to cite them non the less (e.g., Reber et al., 2014, GRL). I wonder if\nthe paper would profit from some restructuring of the text. The reader has to wait until the very end to get a better idea on\nhow the experiments potentially relate to slow-slip scenarios. Could you strengthen the relevance at the beginning more?\nHow the porosity in the experiments is related to slow-slip systems is not apparent in the present manuscript. The paper\nwould benefit from a clearer elaboration on this point. What would drive a similar change in porosity in subduction zones?\nSome details of the experimental setup that are quite important (when you use a uniform grain size versus a bimodal grain\nsize) are only introduced in line 148. This makes it hard to appreciate this detail in the results presented previously.\nExperiment setup: I assume you are keeping the volume of the experimental ‘box’ the same and vary the number of beads to\nachieve different porosities. Is this correct? If so, how do you control for changes in the local porosity? I assume that the\nbeads can move around in the fluid and that the porosity will not be equal in the entire experimental setup. Does this matter?\nSome elaboration on this point would be helpful. What would be possible scenarios where you would expect that the\nporosity is changing similarly to what the experiments are simulating?", "reviewer_3": "This study elucidates one of micromechanical origins of the scaling laws ubiquitous in slow earthquakes. There are many\nstudies on the scaling laws of regular earthquakes, but those for slow earthquakes are limited. I thus appreciate this study as\none of early attempts. An important finding of this paper is that soil mechanical effects could be an explanation for\ncomprehensive slow earthquake scaling laws. It is in contrast with many previous studies that had focused on the frictional\nproperties of real contact areas, rather than granular rearrangements inducing gouge contact reconfigurations. Therefore, I\nrecommend publishing this paper as a proposal of an alternative view on the observed laws of earthquakes.\nHowever, given the great novelty of their proposal, my serious concern is that their contribution to this field is not given\nacross to potential readers in the current format. I am inclined to offer a major revision then.\nMy concern is mostly about writing, but the lack of references is the most harmful deficit for this work. Introduction rewriting\nand result comparisons to previous studies are necessary. These comparisons would largely modify the current structure of\nthis paper, so I am not willing to discuss the details.\nOn introduction-----------------------------\nInsufficient references on laboratory studies. At least I know several studies already conducted, for example:\n0. Laboratory slow earthquakes: Kaproth & Maronem 2013; Leeman Marone Saffer, 2018\n1. Scaling laws of laboratory slow earthquakes: Nielsen et al. 2016 (http://doi.org/10.1002/2015GL067490), Wu & McLaskey\n2019\n2. Gouge effects on the frictional properties: Nakatani 1998; Karner Marone 2001; Chambon Schmittbuhl Corfdir 2002\n3. Computationally simulated gouge (I am not familiar with this): e.g. 10.1002/essoar.10504966.1\nReferring to the literature 0&1, the previous efforts from laboratories must be outlined. Referring to the literature 2 (&3), the\nauthors should emphasize the significance of gouge effects for frictional properties. I am seriously afraid that this study is\noverlooked by other experimental research groups without such an effort to connect this study to the literature. As a favorable\nreviewer of this study, I should offer a thorough restructuring of the introduction section.\nAlso, I know that there are already other laboratory-based explanations on the slow earthquake scaling laws (cf. the\nliterature 1 above), so I consider this study needs to be read as an alternative view of the observed scaling laws. This point\nwas less written than I had expected. I would like the authors to rewrite it.\nNote: here I use granular, gouge, and soil-mechanical effects in a similar meaning. I think the relation between granularity\nand gouge presence are worth mentioning in the paper, as many readers related to earthquake mechanics are unfamiliar\nwith granular physics. If such jumbling the two is inaccurate to read this paper, additional mentioning is even necessary.\nOn results-----------\nComparisons to the established literature are insufficient in this article. Was this study able to explain some scaling laws\nunexplained in previous studies? Overviewing Nielsen et al. 2016, Wu & McLaskey 2019 mentioned earlier, scalings\nthemselves seem to be explained in their analysis already. Again, I admit the worth of this paper in proposing another origin\nof slow-earthquake-like scalings, but even this point is not clear in the current format lacking the references of previous\nlaboratory efforts. In-situ measurements and some other approaches seem advanced but I am not sure. In summary, their\nsection \"New implications to slow earthquakes\" must be thoroughly rewritten by referring to the previous laboratory efforts.\nPerhaps the same applies to other secitions.\nSeveral questions on the result limitations and details----------\nQuasi-two-dimensionality:\nThe scaling laws could depend on the dimensions of the experimental set up. Is the dimensional limitation of this study\nunrelated to the results they obtained? The same question applies to monodispersity and sphericity of the system\ninvestigated in this study. These points are not well discussed.\nScaling laws vs porosity:\nIn their results Fig. 2A, the porosity determines the prefactor of the frequency scaling law. Please quantify it (probably\nanother power law?). Is the variety of the frequency prefactor quantitatively consistent with observations? I found qualitative\nconsistency mentioned in their section \"New implications to slow earthquakes.\"\nFluid viscosity, monodispersity, sphericity:\nI appreciate their comprehensive confirmation of the result validity, but the link for Extended Data Table 1 and Fig. 5\njumped to Table 1 and Fig. 5 in the paper. The extended data seem not available yet. It may not be the fault of the authors.\nI end my comment on this article. This research is obviously backed up with hard work of the authors in laboratories. So I\nam sure that the results of this paper are worth publishing. At the same time, in reading this article, I felt this paper was written by\nan early career researcher from a different field of research. It may be fruitful to get advice from laboratory research groups\noriented to earthquake mechanics."}, "author_response": "Response letter on the rivised manuscript titled “Origin of\nslow earthquake statistics in low-friction soft granular shear”\nDear Editor and Reviewers,\nWe sincerely appreciate the editor and all three reviewers for your invaluable\ncontributions, as well as your careful, insightful, and positive comments. All\nreviewers consistently emphasized the importance of clearly positioning our\nstudy within the existing literature, recognizing the significance of our findings.\nIn response, we have revised the manuscript with particular attention to the\nIntroduction and Discussion. Additionally, your thoughtful feedback led us to\nextensively revise the manuscript, and the overall content including manuscript\ntexts and figures has been significantly improved and enriched. As part of the\noverall revision, we have added three new figures: Supplementary Figures 2, 7,\nand 8. Within the Nature Communications format requirements and word limits,\nwe have incorporated all suggested and additional revisions.\nBelow, we provide a point-by-point response to each comment. All received com-\nments are highlighted in brown, our responses in black, and excerpts from the\nrevised manuscript in blue along with the line numbers. For your reference, we\nalso attach a file that shows the differences between the previous version and the\nrevised manuscript. In this file, deletions are indicated in red, and modification-\ns/additions/relocations are highlighted in blue. Please note that line numbers and\nreference numbers differ between the revised manuscript and the difference file.\nTherefore, the line numbers cited in this response letter do not correspond to\nthose in the revised manuscript, but to the difference file. The reference list in\nthis letter matches that of the revised manuscript.\nWe sincerely hope our revisions meet the expectations of the editor and review-\ners, and we look forward to your favorable consideration of our manuscript.\npositive considerations. We sincerely appre-\n[1] The manuscript titled “Origin of slow\nciate your effort to access our data and even\nearthquake statistics in low-friction soft\nverify the graph plotting. According to this\ngranular shear” by Yuto Sasaki and Hiroaki\nrevision, we updated the open-source data\nKatsuragi offers compelling experimental\nin the Zenodo repository and its reference\nevidence for the role of low friction between\n[70] in the manuscript.\nviscoelastic particles in reproducing two\nkey features of slow earthquakes: exponen-\n[3] To better couple the manuscript with pre-\ntial frequency-moment distribution and lin-\nvious, laboratory-based studies looking into\near scaling between moment and duration.\nthe physics of (slow) earthquakes, I would\nThanks to an interesting variation of the\nrecommend that the authors compare their\nclassical Couette cell, the authors are able\nfindings with those of Leeman et al. (2016).\nto demonstrate how the two aforementioned\nFor example, can a comparison be made\nmicro- to meso-scale properties (particle\nbetween the role of effective stiffness of a\nviscoelasticity and interparticle friction) of\nfrictional system comprising of hard par-\ntheir experiment can be tuned (by using\nticles, in the context of rate- and state-\nhydrogel particles or glass beads) to alter the\ndependent friction, and the effect of lubri-\nshape of the frequency distribution of torque\ncation in a system with soft particles? This\ndrops and the scaling between torque drop\ncould come as an extension to the comment\nand duration. This is an interesting demon-\nin line 254 to 257.\nstration of the spectrum of frictional slip\nReply) Thank you for your comments\nbehaviors and how the latter are controlled\nand suggestions. We fully agree that exam-\nby system properties. As such, in my view,\nining the effective stiffness in the references\nthe insights into the physics governing slow\n(Leeman et al., 2016 [18]; [20]) is of great\nslip will be of interest to the slow-earthquake\nimportance. Reviewer #3 also raised this\ncommunity.\npoint, and we kindly ask you to refer to\nReply) Thank you for your encouraging\nissue [46.2] for a comprehensive discussion\ncomments.\non this point.\n[2] The authors use a variation of a well-\n[4] Another study that I think could be\nestablished methodology (see-through Cou-\ndiscussed is on the lubricating effect of a\nette cell, with particle tracking) to produce\nfluid on a granular medium, by Reber et al.\nresults that support their main argument suf-\n(2014). A brief discussion of the effect of\nficiently. To my understanding there is no\nfluid viscosity and packing fraction on the\nobvious flaw in their methods and discussion\nmacroscopic strength of the material could be\nof their results. The manuscript contains\nilluminating. Also, the summarizing Fig. 3\nsufficient information to allow replication of\nfrom Reber et al. (2014) could be discussed\ninthecontextofFig.5fromthismanuscript.\nthe work, at least in principle. The data sup-\nporting the manuscript are accessible and I\nReply) Thank you very much for point-\nconfirm that they can be accessed and viewed\ning us to the relevant literature. We have\n(plotted with Matlab).\nrecognized this as an important point, and\nReviewer #2 has also mentioned this in\n2\nissue [33].Asyoucorrectlypointedout,the system of hard particles. The authors could\nincrease in macroscopic strength observed in consider updating their discussion in, for\nour study with increasing packing fraction example, line 157 to 159 and line 200 to 206,\ncorresponds to the findings and equation (1) taking into account, e.g., Fig. 3 and Fig. 6\nin the reference (Reber et al., 2014 [33]), from Hayman et al. (2011).\nwhere an increase in fluid viscosity leads Reply) We appreciate your insightful\nto an increase in macroscopic strength (i.e., comment. As you correctly pointed out, the\neffective viscosity). In accordance with the reference (Hayman et al., 2011 [61]) also\neffective viscosity increase, the transition reports a power-law distribution with a high\nfrom creep to stick-slip occurred in the ref- exponent (greater than 5), or an exponen-\nerence [33] and in ours. Furthermore, the tial distribution, under high packing frac-\nobserved localization of the shear band in tion conditions. We fully recognize that this\nthe presence of fluid, compared to the typ- studyisoneofthefewvaluablereportsshow-\nical values under dry conditions, is also in ingsuchexponential-likestatistics.However,\nagreement with the results reported in fig. 3 while reference [61] also uses a 2D exper-\nfrom the reference [33] and related studies. imental setup, their results contrast with\nWe have reflected on these points in the ours: as the packing fraction increases, the\nrevised manuscript as follows. In addition, event size becomes smaller, and the power-\nas mentioned by Reviewer #2, we have also law exponent increases (or the distribution\ncited the reference in the Introduction to becomes exponential). We believe this dis-\nstrengthenthevalueofourexperimentalsys- crepancystemsfromdifferencesintheexper-\ntem.Thefollowingexcerptsshowtherevised imentalsetups.Althoughbothourstudyand\nparts we made, highlighted in blue.\nreference [61] use a constant-volume condi-\nLines 91–93) “This transitional behav- tion, their experiment applies simple shear\nior between creep and stick-slip is typically through top and bottom boundaries and,\nobserved in the lubricated granular system due to practical constraints, also employs\n[32, 33].” sidewalls. As shear deformation proceeds,\nLines 155–157) “This shear localiza- the aspect ratio of the system decreases, and\ntion with porosity decrease is also observed the hard particles become increasingly con-\nin lubricated granular particles with an strained by the confining geometry. In other\nincrease of pore-fluid viscosity [32, 33].” words, despite the nominally constant vol-\nLines 366–368) “Our results suggest ume, the effective free volume available for\nthat with much fluid or ductile matrix at particle motion decreases with strain. Such\nhigher porosity, shear localizes within the a setup causes the entire system to behave\nprincipal slip zone [33], which is mechani- more elastically, and, as a result, the stress\ncally supported by force chains comprising does not reach a steady state, a behavior\nthe rest of the fault core (Fig. 5a).” distinct from what is typically observed in\nRegarding another excerpt, we kindly shear experiments such as rotary shear [39]\nask you to also refer to) issue [32] or gouge friction tests. This effect becomes\nparticularly pronounced under high pack-\n[5] Finally, Hayman et al. (2011) reported ing fractions, likely leading to the observed\nan evolution in the frequency distribution suppression of large events in reference [61]\nof slip from power law to exponential with (see their figs. 3a, 3b, 6a, and 6b). There-\nincreasing packing fraction, in a quasi-2D fore, while we value the findings of the\n3\nreference [61], a direct comparison with our Reply) Thank you for your comments.\nrotary shear experiments is not straightfor- Asyoucorrectlynoted,by“limited”werefer\nward. Although fig. 3c in the reference [61] to the relatively low maximum event mag-\nshowsanincreaseinrigidityandeventdura- nitude and the higher relative frequency of\ntion with packing fraction, which may be smaller events compared to larger ones (a\nconsistent with our findings, these results steeper frequency–magnitude distribution).\nlikely reflect not only intrinsic properties of Tomakethispointclearer,wehavemodified\nthe granular system but also effects spe- the wording as follows.\ncific to the simple shear setup. Accordingly, Lines 7–8) “Slow earthquakes differ\nin the final paragraph of the manuscript, from regular earthquakes in their slower\nwe mention this geometrical problem as an moment release and size distribution domi-\nimportant topic for future work, along with nated by smaller events.”\nabriefreferencetothediscussionabove[61].\nWe compare our results to other granular [7] Line 17: The authors could consider a\nexperiments of rotary shear [39] and simple bridging sentence that would make the tran-\nshearwithoutaspect-ratioconstraints[37]in sition from earthquakes to hydrogel particle\nthe revised manuscript, as follows.\nlayers smoother.\nLines 119–120) “The characteristic size Reply)Thankyouforthesuggestion.We\n(slope) of the exponential size distribution have now added a statement regarding the\ndecreases with increasing porosity ϕ, con- motivation for using hydrogel particles, as\nsistent with previous experimental studies follows.\n[39].” Lines 17–20) “To model slow-\nLines 125–129) “A slower shear rate earthquake fault materials under\nresults in approaching a linear moment- hydrothermal conditions, we use a low-\nduration scaling (Supplementary Fig. 5b). friction soft hydrogel particle layer floating\nAnincreaseintimescale(slowshearing)con- on lubricating fluid and conduct stick-slip\nsistently induces the linear scaling between experiments.”\nsize and duration in other granular exper-\niments [37] and seismic observations [2], [8] Line 47: Perhaps it would be simpler\nalthough the underlying mechanism remains to say “... the difference between slow and\nunclear.” regular,largeearthquakeswhichcontrol...”?\nLines 395–397)“Asanexampleofunre- For sure “underground” seems unnecessary.\nsolved inconsistency, a simple shear setup Reply) Thank you for your comments.\nwithevolvingaspectratioalsoexhibitsexpo- We have revised the wording for improved\nnential size distributions, while its porosity clarity, as follows.\ndependence is opposed to ours [61].” Lines 55–58) “Experimental elucida-\nRegarding another excerpt, we kindly tion of their enigmatic origin would clarify\nask you to also refer to) issue [47.2] the relationship between slow and regular\nlarge earthquakes [2, 24], which control the\n[6] Finally, some questions, comments and energy balance of plate tectonics [13].”\ncorrections: Line 9: by “limited” do the\nauthors mean in terms of maximum magni- [9] Line 54: Change “stick” to “stick-“ as in\ntude? stick-slip.\nWe have revised the wording accordingly as\nfollows.\nLines 77–79) “We captured hundreds of\nstick-andslow-slipeventsunderquasi-static\nshear, which become more pronounced with\ndecreasing porosity (Fig. 1c, d)”\n[10] Line 64, 65: I am not sure what a pre-\ncursory slip movement looks like in extended\ndata Fig. 1. Perhaps some indicative guiding\nlines or a blow-up of one example could help.\nReply) Thank you for your comments.\nIn response to your comments, we carefully\nre-analyzed the data and found that events\nwith and without precursory slip are indeed\nmixed. We are sincerely grateful for your\ninsightful feedback. Since preslip does not\nalways occur, we have removed the corre-\nsponding description from the manuscript.\nAs you suggested, we have added a magni-\nfied view in Supplementary Fig. 1 to better\nshow the relationship between torque and\ndisplacement. We have also included guide\nlines to indicate the onset times of the\ntorque-dropeventsasdefinedinouranalysis.\nLines 87–89) “The fluid-lubricated soft\ngranular layer under shear exhibits torque\nfluctuations corresponding to stick–slip\nevents (Fig. 1c, Supplementary Fig. 1).”\nCaption for Supplementary Fig. 1)\n“Correlation between the torque and\nparticle motion. The data were obtained\nin run #83 (ϕ = 0.20). (a) Torque data.\nThe vertical dashed lines indicate the onset\nof each event defined by torque drop. (b)\nDifferential area (particle motion area) per\nsecond, normalized to the whole area. The\nvertical dashed lines indicate the onset of\neach event defined by torque drop. (c) Nor-\nmalized cross correlation coefficient between\nthe torque and differential area ratio per\nsecond, calculated after subtracting the\n5\nmean from each time series (t = 1,890–\n2,387 s). The correlation is plotted as a\nfunction of the lag time of torque relative to\nthe differential area ratio (particle motion).\nThe peak correlation at slightly negative\nlag time implies synchronization where the\npeak of accelerated particle motion follows\nthe corresponding torque drop onset.”\n[11] Line 67, 77/Fig. 1D: consider adding\nan inset (e.g., showing a step or gentle\nundulation) to better demonstrate what is\ndescribed in the text.\nReply) Thank you for your comment. As\nyou suggested, we have added an inset in\nFig. 1d for clarity.\nLines 95–97) “A decrease in porosity\nto approximately 0.20 causes a transition\nfromcreeptostick-slipbehavior(Fig.1c,d).\nThe slip rate exceeds the steady shear rate,\nshown by the dashed black line in the inset\nof Fig. 1d.”\nCaption for Fig. 1d) “(d) Temporal\nevolution of tangential displacement at\nthe cylinder surface, with color coding\ncorresponding to panel (c). Inset shows\nan enlarged view of the boxed area. The\ndashed line represents the steady shear rate\nof 0.099 mms−1.”\n[12] Line 76: consider adding a statement\nsuch as “therefore particles do not fail”,\nas a clarification\nReply) Thank you for your suggestion.\nWe have revised the expression as follows.\nLines 101-102) “The maximum torque\nat the minimum porosity is still substan-\ntially low, resulting in no particle failure.”\n[13] Fig. 2: Perhaps “logarithmic scale” is\nmore appropriate than “logarithmic coordi-\nnates”?\nReply)Thankyouforyourcomment.We\nhave corrected the proper wording as follows\n6\n(andwehaveappliedthesamecorrectionsto corresponding to the torque drops.”\nthe corresponding parts of the other figures\nas well).\n[17] Line 198: Does “pressure condition”\nCaption for Fig. 2) “Statistical prop-\nrefer to the confining pressure?\nerties of the released moment. (a)\nReply) Thank you for your comment. As\nCumulative frequency distribution of the\nyoupointedout,werefertothenormalstress\nmoment(persecond).Thelineartrendsona\nor the confining pressure in 2D correspond-\nsemi-log scale demonstrate exponential dis-\ningtothevariousporosities,andthiswasthe\ntributions, with fitting curves shown as solid\nintention behind the original wording. How-\nlines. Inset shows the same data in a log-log\never, since the system maintains a constant\nplot. The colors and symbols correspond\nvolume and the pressure is not a trivial con-\nto those in (b). (b) Relationship between\nstant value, we decided to avoid confusion\nmoment and duration on a log-log scale for\nby mentioning only the control parameter ϕ\nporosities ranging from ϕ = 0.41 to 0.18.”\nand have removed references to pressure, as\nfollows.\n[14] Line 115: Add reference to Fig. 4B\nLines 237–239) “Thus, the size dis-\n(larger moment at lower porosity)?\ntribution is governed by the exponentially\nReply)Thankyouforyourcomment.We\ndistributed stress drop ∆σ and its corre-\nhave added the reference as follows.\nsponding slip displacement D, which vary\nLines 157–158) “Notably, the released\nwith porosity (see Methods; Supplementary\nmoment exhibits larger values under con-\nFig. 4).”\nditions of lower porosity despite the shear\nlocalization (Fig. 4b).”\n[18] Line 224: Should “Fig. 5E” instead\npoint to “Fig. 2B”?\n[15] Line 129: do the authors refer to the\nReply) Thank you for your comment.\nconfining pressure applied to the shear band\nWe have revised the figure references.\nfrom the particles in the outer region?\nReply) Thank you for your comment.\n[19] Line 252: Consider “As *with* slow\nYes, we refer to the radial normal stress. To\nearthquakes”?\nimprove clarity, we have revised the expres-\nReply) Thank you for your comment.\nsion as follows.\nAlthough the corresponding expression no\nLines 170–172) “The outer fixed region\nlonger appears in the main text due to\ncan apply low normal stress (radial pres-\nthe overall revision of the manuscript, we\nsure)totheshearbandwithhigherporosity,\nhave retained a similar expression in the\nconstraining its thickness w(ϕ).”\nAbstract.\n[16] Line138:Shoulditnotbe17.2sinstead\n[20] Line 281: Change “plughing” to\nof 15.2 based on Fig. 3E?\n“ploughing”, if that what is indeed meant.\nReply) We sincerely appreciate your\nReply) Thank you for your comment.\nfeedback and careful review. We have cor-\nThe corresponding part has been removed\nrected the numerical value.\nas a result of the overall revision of the\nLines 179–180) “Fig. 3e shows the tem-\nmanuscript.\nporal evolution of slip motions over 17.2 s,\n7\n[21] Line 289, 290: Consider simplifying to Reply) Thank you for your comment.\n”... potentially inducing devastating regular We have corrected all the unit formatting\nearthquakes, such as... ”.\nin the manuscript, including Supplementary\nReply) Thank you for your comment.\nFig. 10a, b as you mentioned (please note\nWe have revised the wording for improved\nthat the figure numbers have been updated\nclarity as per your suggestion.\ndue to the addition of new figures).\n[22] Line 326: Change “polytundstate” to\n[28] p. 32, extended data Fig. 8B: change\n“polytungstate”.\n“Nm” to “N m” in the x-axis caption for\nReply) Thank you for your comment.\nconsistency.\nWe have corrected this word.\nReply) Thank you for your comment.\nWe have corrected all the unit formatting\n[23] Line 388: Change “lightning” to “light-\nin the manuscript, including Supplementary\ning”.\nFig. 11b as you mentioned (please note that\nReply) Thank you for your comment.\nthe figure numbers have been updated due\nThe corresponding part has been removed\nto the addition of new figures).\nas a result of the overall revision of the\nmanuscript.\n[29] Last but not least, I would like to\nthank the authors for a thought-provoking\n[24] Line 503 & 505: If I understood cor-\nmanuscript that I read with great interest!\nrectly, both references to “Fig. 2B” should be\nReply) We again sincerely appreciate\npointing to “Extended data Fig. 2B” (i.e.,\nyour interest in our work, as well as your\nnot the one in the main text).\nencouraging and thoughtful comments and\nReply) Thank you for your feedback. We\ninsightful suggestions.\nhave corrected all the figure references and\ntheir hyperlinks, including the mentioned\n[30] References 1. Leeman, J. R., Saffer,\npart (please note that the figure numbers\nD. M., Scuderi, M. M. & Marone, C. Labo-\nhave been updated due to the addition of\nratory observations of slow earthquakes and\nnew figures).\nthespectrumoftectonicfaultslipmodes.Nat\nCommun 7, 11104 (2016). 2. Hayman, N.\n[25] Line 603, reference #21: Change\nW., Duclou´e, L., Foco, K. L. & Daniels,\n“nankai” to Nankai.\nK. E. Granular Controls on Periodicity of\nReply) Thank you for your comment.\nStick-Slip Events: Kinematics and Force-\nWe have corrected the reference list.\nChainsinanExperimentalFault.PureAppl.\nGeophys. 168, 2239–2257 (2011). 3. Reber,\n[26] Line 670, reference # 47: issue with\nJ. E., Hayman, N. W. & Lavier, L. L.\nprinting of special characters.\nStick-slip and creep behavior in lubricated\nReply) Thank you for your comment.\ngranular material: Insights into the brittle-\nWe have corrected the reference list.\nductile transition. Geophys. Res. Lett. 41,\n3471–3477 (2014).\n[27] p. 31, extended data Fig. 6A, B: change\nReply) Thank you for providing the\n“Nm” to “N m” in the x-axis caption for\nreference information. We now include all\nconsistency.\n8\nthese papers [18, 33, 61] and relevant dis-\ncussion in the revised manuscript.\ntioned the necessity and value of our exper-\n[31] Connecting the deformation of a\nimental setup and material choices as sum-\nwet granular system to slow-slip events is\nmarized above. We have revised as follows.\nintriguing and potentially of wide interest.\nIn addition, we have revised the related part\nThe experiments are extensive, and the sta-\nintheabstractasmentionedbyReviewer#1\ntistical analysis is intriguing.\n(issue [7]).\nReply) Thank you for your supportive\nLines 59–70)“Torelatethemacroscopic\ncomments.\nstatistics of slow earthquakes to the micro-\nscopic dynamics of fault gouge [23, 25–28], a\n[32] One of the shortcomings of the cur-\ncombined approach of stick-slip experiments\nrent manuscript is a better motivation why\n[29]andinsitufabricobservationsusingsoft\nthese experiments are relevant to investi-\nmatter has several advantages. Fault gouge\ngating slow slip. The setup and materials\nmaterials generating slow earthquakes are\nare obviously quite different and the connec-\npresumed to be rich in low-modulus hydrous\ntion to recorded slow-slip event locations is\nclay minerals [30], high-pressure pore fluids\nnot obvious. While the statistics is pointing\n[12], and/or viscous phases associated with\ntowards that these experiments indeed cap-\nbrittle-ductile transition zones [31]. These\ntures some behavior, it would be beneficial\nfactors make a fault low-frictional and soft.\nif there would also be other supporting argu-\nTherefore, the mixture of low-friction soft\nments for the choice of experimental setup\ngranular material and viscous liquid demon-\nand material.\nstrates analogical behavior similar to the\nReply) Thank you very much for your\ngouge materials [32–34]. Experiments with\ncritical and valuable comments. As you\nsoft materials at room temperature and\npointed out, our experimental setup and\natmospheric pressure allow us to arrest rup-\nconditions differ significantly from the natu-\ntures within the finite system size [21, 35]\nralfaultgougeshearunderhightemperature\nand track each particle. In such a gran-\nand high pressure. However, regarding the\nular system, porosity not only represents\nexperimental materials, hydrogels are par-\nthe ratio of fluid or ductile phase, but\nticularly suitable for reproducing the low-\nalso controls pressure, fault strength, fluid\nmodulus and lubricated hydrous fault clays,\nmigration, and slip statistics. Accordingly,\nwhich are metamorphosed under hydrother-\nthis serves as a useful framework for under-\nmal conditions. Using low-modulus materi-\nstanding the mechanism of slow earthquake\nals is also effective to arrest ruptures within\nfaulting.”\nthe finite-size experimental system [21, 35].\nThe density and viscosity of the heavy liq-\n[33] In addition, experiments have already\nuid can be adjusted by concentration, which\nbeen conducted with grains embedded in a\nopens the possibility of future experimen-\nfluid where the slip was monitored. While\ntal studies simulating slow earthquakes in\nthese experiments did not include any statis-\nthe brittle-ductile transition zone, using this\ntics it might be fair to cite them non the less\nconvenient solution as a starting point. In\n(e.g., Reber et al., 2014, GRL).\nReply) Thank you for your comments.\nWe have come to recognize the significance\nof this point, and Reviewer #1 also raised\n10\na similar point in issue [4]. We have cited\nthe significance of fault-gouge porosity men-\nrelevant studies on deformation experiments\ntioned in the Introduction (issue [32]), we\nof wet granular systems (Reber et al., 2014,\nhave discussed the porosity dependency in\nGRL [33]; [32]). We kindly ask you to refer\nthe context of the subduction zone system,\nto issues [32, 4] for the Introduction and\nas follows.\nthe Results/Discussion, respectively.\nLines 360–363) “Applying our granular\nmodel to fault gouge, the porosity depen-\n[34] I wonder if the paper would profit from\ndence of moment statistics can qualitatively\nsome restructuring of the text. The reader\nexplain the depth distribution of both slow\nhas to wait until the very end to get a\nand regular earthquakes along the subduct-\nbetter idea on how the experiments poten-\ning plate interface [59].”\ntially relate to slow-slip scenarios. Could\nLines 379–392) “Furthermore, the\nyou strengthen the relevance at the beginning\ndependence of moment on porosity should\nmore?\nbe observed when local porosity changes\nReply) We sincerely appreciate the\naccording to the pressure of migrating pore\nimportant feedback and insightful sug-\nfluid. Tectonic tremor events, a type of slow\ngestions. Accordingly, we have explicitly\nearthquake, exhibit a temporal change in\nclarified the motivation behind connecting\nthe mean size and occurrence rate, both\nour experiments to slow earthquakes in\nof which tend to be positively correlated\nthe Introduction. This point is included in\n[60]. This correlation is consistent with our\nissue [32], and therefore we kindly ask you\nresults (Fig. 2a, 4b, Supplementary Fig. 2),\nto refer to it regarding the excerpts.\nwhile the seismological study claims that\nthe size and rate decrease with pore fluid\n[35] How the porosity in the experiments is\npressure reduction.”\nrelated to slow-slip systems is not apparent\nLines 401–405) “Spatiotemporal obser-\nin the present manuscript. The paper\nvations of slow earthquake statistics could\nwould benefit from a clearer elaboration on\nhelp monitor fluid-filled pores [60], fault\nthis point. What would drive a similar change\nstrengthorstress[24],andshearlocalization\nin porosity in subduction zones?\npotentially inducing devastating regular\nReply) We appreciate your critical feed-\nearthquakes such as the 2024 M 7.5 Noto\nback.WerevisedtheIntroductiontoempha-\nearthquake [65].”\nsize the importance of porosity, which not\nonly represents the depth (pressure) but\n[36] Some details of the experimental setup\nalso controls the gouge strength and pore\nthat are quite important (when you use a\nfluid dynamics in faults. Furthermore, in\nuniform grain size versus a bimodal grain\nthe Discussion, we pointed out that poros-\nsize) are only introduced in line 148. This\nity in slow earthquake faults temporally and\nmakes it hard to appreciate this detail in the\nspatially varies, controlled by the migra-\nresults presented previously.\ntion of high-pressure pore fluids, which is one\nReply) Thank you for your comment.\nof the driving factors of slow earthquakes.\nWe have added the details of the sample\nWe also cited a seismological observational\npreparation in Methods and Supplementary\nstudy implying this phenomenon [60] and\nTable 1, as follows.\ndiscussed its implications. In addition to\nLines 428–442) “We used three types of\n11\nparticles as a fault gouge analogue: opaque\nrandom structure using a mixture of manu-\nspherical polymer hydrogel particles (Leaf\nally dispersed 3781 hydrogel particles (d =\nCorporation,BioBeads,PEG)andspherical\n4.4mm)andapproximately3000glassbeads\nglass beads of two sizes. Hydrogel particles\n(d = 2 mm) in run #77.”\nhave a diameter of d = 4.4 mm with a stan-\nSupplementary Table 1) (please refer).\ndarddeviationof0.1mm,withlowerfriction\n(coefficient of < 10−1) and lower modulus\n[37] Experiment setup: I assume you are\n(∼ kPa) [66]. The two types of glass beads\nkeeping the volume of the experimental ‘box’\nhave d = 4.1 or 2 mm, respectively, with\nthe same and vary the number of beads to\nhigher friction (coefficient of ≳ 0.2 [67]) and\nachieve different porosities. Is this correct?\nhigher modulus (∼ GPa). To align particle\nReply) Thank you for your comment.\ncentroids within the same plane on the liq-\nIt is exactly correct. We have revised the\nuid, we used a single type of monodisperse\ncorresponding part for clarity, as follows.\nparticle in each experimental run. To inves-\nLines 75–77) “The experimental sys-\ntigate the effect of particle arrangement,\ntem with a constant volume enables the\na bidisperse mixture of the hydrogels and\nsystematic variation of the bulk porosity\n2 mm glass beads was also used.”\nϕ and direct tracking of particle motion\nLines 495–510) “The shear experiments\n(Fig. 1b), both of which intrinsically govern\nwere performed in 23 runs, as summarized\nthe behavior of fluid-retaining faults.”\nin Supplementary Table 1. We investigated\nthe dependence of mechanical behaviors on\n[38] If so, how do you control for changes\nporosity (0.18 ≲ ϕ < 0.41 and the cor-\nin the local porosity? I assume that the\nresponding number of hydrogel particles in\nbeads can move around in the fluid and that\napproximately 2,700–3,900), particle type\nthe porosity will not be equal in the entire\n(d = 4.4 mm hydrogel particles, 4.1 mm\nexperimental setup. Does this matter? Some\nglass beads, and 2 mm glass beads), liquid\nelaboration on this point would be helpful.\nviscosity (2 × 10−2 and 1 Pas), container\nReply) Thank you for your thoughtful\ndiameter (100 and 300 mm), and rotation\ncomment.Inthisexperiment,wesimplyvary\nangular velocity (ω = 0.06–60 degs−1). For\nthe number of particles dispersed within a\ninvestigating the effect of the particle num-\nfixedsystemvolume,andtheporosityisspa-\nber,weonlyvariedthetotalnumberwithina\ntially distributed. The shear band in the\ncontainer. The range of porosity was chosen\ncentral region consistently shows a porosity\nto ensure that the granular layer can main-\nof around 0.3, regardless of the bulk poros-\ntain single-layer thickness and also exhibit\nity. This value is relatively high compared to\nany detectable torque resistance. To inves-\nthe bulk, and its high porosity corresponds\ntigate the effect of particle arrangement, we\ntotheregionwhereshearlocalizationoccurs,\nfurther manually arranged an ordered poly-\nsince particles are able to move around in\ncrystalline structure with the intention in\nthe fluid, as you mention. This porosity dis-\nruns #67 and #70, where the size of each\ntribution emerges spontaneously during the\ncrystalline cluster was on the order of 10d\nexperiment and is related to the unresolved\n(Fig. 3c). The particle arrangement in runs\nproblem of shear band self-organization.\n#85–87 with the rigid glass beads was also\nTherefore, in this experiment, we are\nan ordered structure. We also prepared a\nnot controlling local porosity but only the\n12\nbulk porosity. Although the local poros-\nstatistics to reflect the pore fluid dynam-\nity within the shear band remains nearly\nics within faults. We have revised the\nconstant under different bulk porosity con-\nmanuscript accordingly, and kindly ask you\nditions, the shear band width varies. This\nto refer to issues [32, 35] regarding the\nsuggests that the change in event statistics\nexcerpts.\nis due to the difference in shear band width,\nnot the porosity within the shear band.\nInvestigating the influence of local porosity\non statistical properties is difficult with-\nout first understanding why high-porosity\nshear bands form spontaneously. This is\na fundamentally important problem to be\naddressed in future work. We have revised\nthe manuscript to clarify these points as\nfollows.\nLines 158–164) “Meanwhile, the poros-\nity within the localized shear band remained\nconstant at 0.3, which is higher (richer in\nfluid content) than the bulk ϕ, except under\nthe highest ϕ value in run #79 (Supplemen-\ntary Fig. 3e). Note that we only varied the\nbulkporosityϕ;itsspatialdistributionspon-\ntaneouslyevolvesandstabilizesundersteady\nstate.”\nLines 518–520) “We estimated the\nporosity of the granular layer using the\nmean particle diameter d and total number\ncounted using the recorded image (Fig. 1b).\nThis value represents the bulk porosity and\nis used as the porosity value in this study.”\nWe also kindly ask you to refer to the\nsecond excerpt of) issue [36].\n[39] What would be possible scenarios where\nyou would expect that the porosity is chang-\ning similarly to what the experiments are\nsimulating?\nReply) Thank you for your comments.\nAs the reference [60] observed, porosity in\nslow earthquake faults could temporally\nand spatially vary, controlled by the migra-\ntion of high-pressure pore fluids. Therefore,\nwe expect the porosity-dependent moment\nInsufficient references on laboratory stud-\n[40] This study elucidates one of the\nies. At least I know several studies already\nmicromechanical origins of the scaling laws\nconducted, for example: 0. Laboratory slow\nubiquitous in slow earthquakes. There are\nearthquakes:Kaproth&Maronem2013;Lee-\nmany studies on the scaling laws of regular\nman Marone Saffer, 2018\nearthquakes, but those for slow earthquakes\nReply) Thank you for your comments.\nare limited. I thus appreciate this study as\nWe have cited previous experimental stud-\none of early attempts. An important find-\nies on slow earthquakes to acknowledge\ning of this paper is that soil mechanical\ntheir important contributions (Kaproth &\nMarone, 2013 [16]; Leeman, Marone, & Saf-\neffects could be an explanation for\nfer[20]),aswellasotherexperimentalefforts\ncomprehensive slow earthquake scaling laws.\n[18, 19, 22]. Reviewer #1 has also raised\nIt is in contrast with many previous studies\nthe relevant reference (Leeman et al., 2016\nthat had focused on the frictional properties of\n[18]) in issue [3]. For the related revision\nreal contact areas, rather than granular rear-\nin both the Introduction and Discussion, we\nrangements inducing gouge contact reconfig-\nkindly ask you to refer to issue [46].\nurations. Therefore, I recommend publishing\nthis paper as a proposal of an alternative\n[43] 1. Scaling laws of laboratory\nview on the observed laws of earthquakes.\nslow earthquakes: Nielsen et al. 2016\nReply) We are truly grateful for your\n(http://doi.org/10.1002/2015GL067490),\nthoughtful review and favorable recommen-\nWu & McLaskey 2019\ndation.\nReply) Thank you for your comment.\n[41] However, given the great novelty of\nWe have also cited the reference (Wu &\nMclaskey, 2019 [21]) in the main text,\ntheir proposal, my serious concern is that\nacknowledging one of the carefully designed\ntheir contribution to this field is not given\nexperimental efforts to reproduce slow\nacross to potential readers in the current for-\nearthquakes. In Addition, we have cited\nmat. I am inclined to offer a major revision\nthe suggested reference (Nielsen et al., 2016\nthen. My concern is mostly about writing,\n[63]) and review paper including compre-\nbut the lack of references is the most harmful\nhensive comparisons between experimental\ndeficit for this work. Introduction rewriting\nand observational data [64]. For the related\nand result comparisons to previous stud-\nrevision in both the Introduction and Dis-\nies are necessary. These comparisons would\ncussion, we kindly ask you to refer to issue\nlargely modify the current structure of this\n[46].\npaper, so I am not willing to discuss the\ndetails.\n[44] 2. Gouge effects on the frictional prop-\nReply) We appreciate your critical and\nerties: Nakatani 1998; Karner Marone 2001;\ninsightful comments. In response, we have\nChambon Schmittbuhl Corfdir 2002\nrevised the Introduction, Results, and Dis-\nReply) We sincerely appreciate your\ncussion to improve their clarity and have\nsuggestion of relevant literature. We have\nincorporated a comparison between our\nadded these references (Nakatani, 1998 [25];\nresults and previous studies, as detailed\nKarner & Marone, 2001 [26]; Chambon,\nbelow.\nrelated revision in both the Introduction and eration as well [16, 17]. The reduction of\nDiscussion, we kindly ask you to refer to issue normalstressandtheincreaseinthenormal-\n[47]. izedstiffnessalsosuppressthestressedpatch\ngrowth. This results in the slow rupture\n[45] 3. Computationally simulated gouge (I faulting process, that is, slow earthquakes\nam not familiar with this): e.g. 10.1002/es- [21], which is consistent with experiments\nsoar.10504966.1 under high pore fluid pressure [22].\nReply) Thank you for your critical and However, these references address nei-\nwide-ranging suggestions. We have cited the ther the moment–duration scaling nor size\ncorresponding peer-reviewed publication distribution, the scope of our study, which\nwith the discrete element method (DEM) in arewidelyrecognizedasoneofthestatistical\nthe main text (Casas, Mollom, & Daouadji, relationships in slow earthquake observa-\n2022, Pageoph [62]), while the suggested tions. One exception is a stick-slip exper-\narticle is a preprint (Casas, Mollom, & iment using dehydrating antigorite gouge\nDaouadji, 2020, ESSOAr). For the related in the reference [17], where the moment-\nrevision in both the Introduction and Dis- duration scaling of slow earthquakes is dis-\ncussion, we kindly ask you to refer to issue cussed. Given that this paper is most closely\n[47]. related to the main theme of our study, we\nhave decided to include an additional men-\n[46] Referring to the literature 0&1, the tion of it as a main focus of this study\nprevious efforts from laboratories must be in the Introduction. In accordance with\noutlined.\nthese previous studies, we have revised the\nReply) Thank you for your comment.\nmanuscript as follows.\nWe have acknowledged the previous experi-\nLines 45–55) “Furthermore, in the past\nmental efforts and added their contributions decade, many gouge friction experiments\nboth in the Introduction and Discussion as have successfully reproduced unstable slid-\nfollows.\ning with slow slip and/or rupture velocities\n[16–23]. However, the multiple unique\n[46.1] In the Introduction, we have pro- statistical laws of slow earthquakes and\nvided an overview of previous experimen- their origin have scarcely been investigated\ntal studies, including those suggested. Most through experiments, which is the focus of\nlaboratories experimentally reproduce slow this study. This is due to two main diffi-\nearthquakes by reducing the normal stress culties: reproducing the observed statistics\n(i.e., increasing the stiffness normalized with within finite system sizes, and directly\nthe critical stiffness k/k ) on the fault plane observing materials’ internal microstruc-\nc\n[16–23]. The normal stress reduction effec- tures during gouge friction experiments.\ntively strengthens the normalized stiffness Despite such difficulties, a stick-slip exper-\nof system, approaching the transitional con- iment during antigorite serpentinite gouge\ndition from unstable to stable slip. This dehydration successfully demonstrates the\ntransitional slip velocity is slower and the roughly linear scaling between stress drop\nslip duration is longer than highly unstable and duration, corresponding to M ∝ T\n0\nfrictional slip [18, 20]. The velocity depen- [17]. Owing to the lack of experimen-\ndence of friction in the normalized system tal data simultaneously reproducing the\n15\nmultiple statistical properties, the mecha- suchaninterpretationrequiresapropereval-\nnism responsible for the distinct statistical uation of the stiffnesses k′, k . Nevertheless,\nc\nproperties of slow earthquakes remains con- we have added a statement suggesting that\ntroversial.” theconditionsfortheobservedslowslipmay\nbe close to the stability boundary between\n[46.2] In the Results and Discussion, stable and unstable slip, as follows.\nwe have added the comparisons with related Lines 340–359) “The mechanism of\npapers. In the following, we discuss the slow earthquakes proposed by gouge friction\ncomparisons unrelated to any scaling rela- experiments is based on frictional insta-\ntionships [19, 21, 22], for which we kindly bility, which appears to differ from our\nask you to refer to issue [51]. As men- scenario despite many consistencies. How-\ntionedbyReviewer#1inissue [3],wefully ever, the low normal stress condition in our\nagree that examining the effective stiffness experimental system is consistent with the\nk′ = k/σ′ and critical stiffness k in slow slow-slip gouge friction experiments charac-\nn c\nslip phenomena [18, 20] is of great impor- terized by high system stiffness normalized\ntance. We addressed the possibility that the to low critical stiffness [16–18, 20, 21]. When\nobserved statistics of slow earthquakes may the explanation that transitional slip near\nemerge because the system stiffness k is the boundary between stable and unstable\nclose to the critical stiffness k . However, frictional sliding corresponds to slow slip\nc\nin our 2D lubricated Couette system, even phenomena [18, 20, 55] is accepted, the\nmeasuring the radial normal stress σ′ is stiffness condition in this study possibly lies\nn\na challenging task due to technical issues.\nnear the stability boundary. As the criti-\nIt would also require measurements of fric- cal stiffness reflects the characteristics of\ntional parameters of (a − b) and D under rate- and state-dependent friction [56, 57],\nc\nvariousslipvelocityconditionsusinganother the persistence of slow slip velocities has\nhigh-stiffness apparatus. For these reasons, been explained within this framework\na direct evaluation of k′ and k within the [16, 17, 55, 58]. Accordingly, the bulk\nc\nframework of the rate-and-state friction law frictional properties and relevant porosity\nis currently not straightforward. We recog- evolution of our lubricated soft granular\nnize that elucidating our lubricated system system warrant further investigations in\nin the framework of the rate-and-state fric- this context.”\ntion remains a highly important issue to be\naddressed in future studies.\nWe also note that the characteristics of\nslow slip are less prominent in the case of\nhard glass beads (e.g., the frequency dis-\ntribution of event size deviates from an\nexponential distribution), which may sug-\ngest a decrease in the normalized stiffness\nk/k ,thoughthisremainsspeculative.Alter-\nc\nnatively, the presence of lubrication may\nincrease the normalized stiffness k/k and\nc\nenable the emergence of slow slip events as\nobserved in our study. However, as above,\n16\n[47] Referring to the literature 2 (&3), the behavior. This is consistent with both our\nauthors should emphasize the significance of interpretationandtheresultsfromthegouge\ngouge effects for frictional properties.\nexperimentsabove.Thisstudymainlypoints\nReply) Thank you for your critical feed- out that when cohesive forces exist between\nback. We deeply appreciate your suggestion, particles, Riedel shear bands, commonly\nwhich has strengthened our discussion and observed in gouge friction experiments, can\ninterpretation. In accordance with your form. In our results as well, cohesive forces\ncomment, we acknowledge the significance may be present due to surface tension, and\noftheforcechaindevelopmentandbuckling, the loosely packed shear band and the radi-\ni.e., gouge effects of dilation and com- ally aligned surrounding vibration region\npaction. We have mentioned the importance (Fig. 3a–c) might be related to Y-shear\nof the granularity of gouge material both in and R’-shear types of Riedel shear bands,\nthe Introduction and Discussion as follows. respectively[40,62].Sincethisremainsspec-\nulative and given the word limit, we have\n[47.1] In the Introduction, we have omitted this discussion but briefly men-\nrevised the manuscript to highlight the tioned this topic in the manuscript. Key\nimportance of the microscopic behavior openquestiontobeexploredishowcohesion\nof particles and force chains within gouge affects the fabric and the loss of fractal-\nmaterials in governing macroscopic physical ity in the contact surface, which we have\nproperties such as frictional characteristics. discussed as a possible cause of the lin-\nTherefore, we have revised the manuscript ear moment–duration scaling. Additionally,\nand kindly ask you to refer to issue [32] other numerical simulations of interacting\nregarding the excerpt.\nmany-body systems suggest that friction\nmodifies the size distribution (Clancy et al.,\n[47.2] In the Results and Discussion, we 2006, Burridge-knopoff model: Exploration\nhave added some comparisons with gouge of dynamic phases, PRE; Saitoh, 2021, The\nfriction experiments, focusing on the effects role of friction in statistics and scaling laws\nof force chains and dilatancy. These gouge of avalanches, EPJE). Although the results\neffects, as you suggested, strongly support of these two studies are important, it is not\nour interpretation that force chain dynam- straightforward to directly compare them\nicscontrolstheslipstatisticswithingranular with ours, and we could not include these\nmaterials. The experimental studies [23, 25– two in the manuscript withinthe word limit.\n28] describe how gouge particles undergo We have revised the manuscript to better\ndilation and compaction through the forma- reflect these previous efforts, as follows.\ntion and buckling of force chains, leading Lines 143–146) “When dry, angular\nto shear loading and slip. These results quartz sand grains are sheared in a setup\nare highly consistent with our experimental similar to ours, they also exhibit the cen-\nresults and strengthen our discussion. The tral shear band and intermittent radial slip\nreference [27], in particular, reports results propagationintothesurroundingregion[27].\nconsistent with ours by experiments using This implies that the granular fabric we\nirregularly shaped, dry quartz sand grains observed is qualitatively consistent, regard-\nin a similar experimental setup. This study less of lubrication and particle shape.”\nsuggests that stick-slip granular dynam- Lines 153–155) “The shear band thick-\nics and relevant fabric formation, as we ness of less than ten particle diameters,\nobserved,occurregardlessoflubricationcon- which is an approximate typical value for\n17\ndry frictional conditions [27, 42], is associ- slow earthquake scaling laws (cf. the litera-\nated with the shorter correlation length of ture 1 above), so I consider this study needs\nstrainunderlowelasticmodulus[43]andlow to be read as an alternative view of the\nfriction.” observed scaling laws. This point was less\nLines 334–339) “Moreover, a critical written than I had expected. I would like the\nrole of force chain dynamics and its dilatan- authors to rewrite it.\ncy/compaction effect has also been recog- Reply) Thank you for your comments.\nnized by gouge friction experiments [23, 25, In the Introduction, we have revised the\n26,28].Thehigherporositywithinourlocal- manuscript to clarify our position as an\nized shear band also implies the dilatancy alternative explanation of slow earthquakes,\neffect. These studies support our interpreta- compared to the existing experimental stud-\ntion that force chain development and buck- ies. Exactly as you pointed out, our study\nling collapse [45] correspond to the loading takes a fundamentally different perspective\nand stress drop phases, respectively.” from previous fault gouge friction experi-\nLines 393–395) “Building upon our ments, in terms of experimental setup and\nfindings, this simple experimental system observation target. We aim to capture the\ncan facilitate further promising investiga- essential statistical nature of slow earth-\ntions into the effects of pressure in 3D, quakes using a physically simpler system,\nsystem geometry [40, 61], granular fabric which is analogous to a fluid-retaining\n[23, 28], and rheology [34, 62].” gouge or a brittle-ductile mixture. We agree\nthat this uniqueness is important. Among\n[48] I am seriously afraid that this study the scaling relationships identified in pre-\nis overlooked by other experimental research vious gouge friction experiments that have\ngroups without such an effort to connect successfully reproduced slow earthquakes\nthis study to the literature. As a favor- [17, 19, 21, 23], the one most relevant to our\nable reviewer of this study, I should offer study is the relationship between stress drop\na thorough restructuring of the introduction andeventduration,experimentallyimplying\nsection.\nthe seismological scaling of M ∝ T [17]. In\ninsightful and important feedback. In our study as an alternative view are clearly\nresponse, we have revised the manuscript mentioned. Thus, we kindly ask you to refer\nto better position our study within the to the excerpts in issues [32, 46.1].\ncontext of prior research and to highlight\ncomparisons with previous results. In par- [50] Note: here I use granular, gouge, and\nticular, the Introduction has been revised to soil-mechanical effects in a similar mean-\nmore clearly delineate the differences from ing. I think the relation between granularity\nearlier studies and to define the scope of the and gouge presence are worth mentioning\npresent work. We kindly ask you to refer to in the paper, as many readers related to\nissues [46, 47, 49, 51]. earthquake mechanics are unfamiliar\nwith granular physics. If such jumbling the\n[49] Also, I know that there are already two is inaccurate to read this paper, additional\nother laboratory-based explanations on the mentioning is even necessary.\nReply) Thank you for your comments.\nWe completely agree with your suggestion,\n18\nand we recognize that fault gouge can be granular materials within a viscous-fluid-\nphysically treated as a granular material. In dominated matrix, leading to some novel\nline with your suggestion, we have clarified insights into brittle-ductile behavior statis-\nthe two corresponding relationships in the tics in this study.\nrevised manuscript. Since this is related to Nevertheless, there are five relationships\nthe necessity and value of our experimental already observed in lab slow earthquakes:\nsetup, we kindly ask you to refer to issue (1) stress drop vs duration, (2) stress drop\n[32] regarding the excerpt.\nvs maximum slip velocity, (3) stress drop vs\nrupture velocity, (4) moment rate spectrum,\n[51] On results———– Comparisons to the and (5) fracture energy vs slip displacement.\nestablished literature are insufficient in this (1)Anegativecorrelationbetweenstress\narticle. Was this study able to explain some drop and duration has been commonly\nscaling laws unexplained in previous stud- observed in gouge friction experiments\nies? Overviewing Nielsen et al. 2016, Wu [19, 21, 23], while ours and antigorite-\n& McLaskey 2019 mentioned earlier, scal- dehydration-assisted slow slips [17] exhibit\nings themselves seem to be explained in their the positive scaling. The cause of this incon-\nanalysis already. Again, I admit the worth sistency remains unclear. However, one pos-\nof this paper in proposing another origin of sibleexplanationisthepresenceofabundant\nslow-earthquake-like scalings, but even this fluid in ours and dehydrating gouge, and\npoint is not clear in the current format thispointhasbeenaddedtothemanuscript.\nlacking the references of previous laboratory The fact that our results are consistent with\nefforts.\nthose under dehydration process as slow\nReply) Thank you for your critical com- earthquake faults, is particularly thought-\nments. We completely agree on the impor- provoking.\ntance of comparing with previous stud- (2) The relationship between stress drop\nies, especially regarding some scaling laws. and maximum slip velocity is different\nBesides the scaling relationship captured between slow and regular lab earthquakes\nin our study, roughly five scaling relation- [19, 21]. However, depending on experi-\nshipshavebeenidentifiedinpreviousstudies ments, lab slow earthquakes show qualita-\nusing gouge friction experiments that repro- tively various deviations from the propor-\nduce slow earthquakes, as described below. tional scaling in regular earthquakes, com-\nOur results, however, provide another scal- ing from the elastic theory (Brune, 1970,\ning relationship obtained from a different Tectonic stress and the spectra of seismic\nperspective along with geophysical observ- shear waves from earthquakes, JGR). Fur-\nables, distinct from any of the five scal- thermore, related geophysical observational\nings. Importantly, our experimental results dataareverysparse,warrantingfuturework\nsimultaneously realize multiple seismolog- in both fields.\nical statistical laws, the size distribution (3) The relationship between stress drop\nand moment-duration scaling, which have and rupture velocity is a positive correlation\nnot yet been reported in gouge friction [21, 22]. As in other gouge experiments, our\nexperiments but have been reported in seis- system is not enough to capture the rupture\nmological observations. Our experimental growth, which we leave for future work.\nsystem fundamentally differs from previous\ngougefrictionexperimentsinthatitinvolves\n19\n(4) The moment rate spectrum (moment is obtained with dehydrating antigorite ser-\nrate vs frequency), different between reg- pentinite [17]. This result is similar to ours\nular and slow earthquakes, has been cap- (Supplementary Fig. 4b). The same study\ntured by both geophysical observations ([1]; also reports a constant slip velocity, smaller\nHawthorne et al., 2018, Observing and mod- stressdrop,andlongerdurationunderdehy-\nelingthespectrumofaslowslipevent,JGR) dration, which are observed in both our sys-\nand by stick-slip experiments [21]. Actually, tem and another viscously lubricated granu-\nthe statistical differences that we observed lar shear [32]. These consistently correspond\nshould fundamentally result in this spec- to slow earthquake characteristics and the\ntral difference (Hawthorne et al., 2018). As linear moment-duration scaling of M ∝ T\n0\nwe did not measure acoustic emissions, this in the presence of fluid. In contrast, most\nspectral difference is another future work.\ngouge friction experiments under conditions\n(5) Finally, another seismological scaling ranging from dry to 100% relative humid-\nof fracture energy dissipation as a function ity show the negative correlation between\nof slip displacement is obtained for regu- stress drop and duration across slow to fast\nlar earthquakes, which can be consistently slip events [19, 21, 23]. Wetting conditions\ncompared between laboratory experiments of the liquid volume confined within pores\nand geophysical observations [63, 64]. Slow might play a role in this difference. In addi-\nearthquakes are expected to exhibit another tion,notethatourexperimentsonlyproduce\ncharacteristicfractureenergyscalinglawdue slow earthquake equivalents in terms of the\ntoviscoelasticdissipationwithfluidandduc- statistics.”\ntile phases. However, to our knowledge, no Lines 397–399) “Our study lays the\nobservational or experimental data of the groundwork for addressing the scaling\nscalinglawforfractureenergyinthecontext between stress drop and slip velocity [19, 21,\nof slow earthquakes are currently available, 22] and the fracture energy scaling of slow\nevenintherelevantreferences[63,64].Thus, earthquakes [63, 64].”\nwe have mentioned this issue briefly in the Additional comparisons and discussions\nfinal paragraph as an important, ongoing beyond these scalings are also provided\nissue that should be addressed in future in) issue [46.2, 47.2]\nstudies, both experimentally and observa-\ntionally.\n[52] In-situ measurements and some other\nIn summary, our study investigates the approaches seem advanced but I am not sure.\nmultiple statistical laws in the lab, funda- Insummary,theirsection”Newimplications\nmentally important to geophysical observa- to slow earthquakes” must be thoroughly\ntional results and other scaling laws. In the rewritten by referring to the previous labo-\nDiscussion, we have refined and limited the ratory efforts. Perhaps the same applies to\nscope of our discussion to maintain focus other secitions.\nafter extensive deliberation within the word Reply) We sincerely appreciate your\nlimitation, and mentioned the scaling (1), insightful comments. As far as we have\n(2), and (5), as follows.\ninvestigated, in-situ tracking of gouge par-\nLines308–319)“Ingougefrictionexper- ticle motion during friction experiments to\niments for slow slip phenomena, the lin- probe the microscopic mechanisms remains\near scaling of stress drop and resultant scarce even today. Most studies approach\nmoment, roughly proportional to duration, the underlying mechanisms through macro-\n20\nscopic mechanical measurements and 3D setups. As for moment-duration scal-\nmicrostructural observations of gouge mate- ing,therearefewpriorexperimentalstudies,\nrials recovered after the whole experiments. making it difficult to discuss in detail, but\nWe have thoroughly revised the Discussion we cannot rule out the influence of dimen-\nsection and the manuscript as a whole by sionality. We recognize it as an issue that\nincorporating comparisons with previous requires further investigation. In actual sub-\nstudies to place our findings in the appro- duction zones, the depth range of seismicity\npriate context.\nis limited, which constrains the propaga-\ntion of fault rupture along the dip direction.\n[53] Several questions on the result In this sense, the quasi-2D shear config-\nlimitations and details———- Quasi-two- uration in our experiments is also physi-\ndimensionality: The scaling laws could callymeaningfulinthatrupturepropagation\ndepend on the dimensions of the experi- is restricted in the dip direction. Indeed,\nmental set up. Is the dimensional limitation there are numerical studies reporting that\nof this study unrelated to the results they the moment-duration scalings can be repro-\nobtained? The same question applies to duced on finite-width faults [15]. We have\nmonodispersity and sphericity of the system revised the manuscript as follows.\ninvestigated in this study. These points are Lines 197–217) “First, the exponential\nnot well discussed.\nsize distribution (Fig. 2a) should emerge\nReply) Thank you for your comments.\nfrom the strength of the granular layer,\n(1) We consider that dimensionality can which is generally supported by force chains\naffect the results. (2) In contrast, we believe [44]andinducedtoslipbytheirbucklingcol-\nthat particle size dispersion does not sig- lapse[45].Thestressofbothforcechainsand\nnificantly affect the results qualitatively. (3) consisting particles follows an exponential\nAs for particle shape, it may influence the distribution [44, 46]. In contrast to our sys-\nresults, but it is currently difficult to eval- tem,frictionalparticlesunderdryconditions\nuate, and we regard it as a topic for future and higher normal stress follow power-law\ninvestigation. Below, we explain each point size distributions with exponential trunca-\nin turn.\ntion, regardless of system dimension [36–38].\n(1) Regarding dimensionality, while it Inthefrictionalsystemsundernormalstress,\nthe force chains form networks that can\nmay not directly influence the size distri-\ncascade into critical slip avalanches. How-\nbution, it may affect the moment-duration ever, lubrication and/or low normal stress\nscaling. For the size distribution, power- (radial pressure) in this study inhibit the\nlaw behavior has been reported even in 2D formation of force chain network, thereby\nexperiments [36], suggesting that confine- restricting slip size to the scale of indi-\nment pressure (i.e., normal stress in the vidual particles and suppressing moment\nradial direction in our system) is likely a release during slip events (Fig. 5a–c). When\nmore critical parameter than dimensional- we substituted hydrogel particles with more\nity. Indeed, since many studies reporting rigid glass beads with frictional surfaces, we\npower-law distributions are based on 3D observed a superposition of exponential and\nexperiments, power-law behavior may more power-law size distributions under lubrica-\neasily emerge under confining pressure in tion(SupplementaryFig.10a).Theseresults\n21\ndemonstrate that low friction, as well as the self-similarity of faulting, which can be\nlow elastic modulus and low normal stress, applied to fault gouge materials.”\nplays a crucial role in limiting the mechan- (2) Regarding particle size distribu-\nically correlated length and preventing cas- tion, we performed experiments using both\ncade collapse events. Consequently, granular monodisperse and bidisperse particles and\nsystems with these properties manifest the confirmed that the results did not change\nmoment statistics that reflect the heteroge- qualitatively. The particles used in our\nneous stress distribution of individual force experiments have a size variation of about\nchains (Fig. 5a, b).” 2%, which may prevent them from forming\nLines 261–272) “In contrast, low- a perfect crystalline arrangement. Since we\nmodulus particles and/or low normal stress have revised this point, including the effects\ncause the monotonous structures of force of viscosity, monodispersity, and sphericity,\nchains (Fig. 5a, b) or interparticle patches we kindly ask you to refer to issue [55],\n(Fig. 5e), violating the self-similar scaling. regarding the excerpts.\nNote that, however, the effects of lubrica- (3) Finally, regarding particle shape,\ntion and system dimension cannot be ruled particle shape and its surface roughness\nout. In particular, the self-similar moment- should affect the force chain development,\ndurationscalingM ∝ T3 isnotrealizedina gouge fabric [28], and also event statistics,\n0\n2D system due to anisotropic rupture prop- which also warrants further investigation.\nagation. The fault plane with a finite width, At the very least, similar fabrics and par-\ncorresponding to the width δ in our gran- ticle motions to those in our study have\nular layer, exhibits a strong dependence of been observed in experiments using sand\nthe transition in the moment-duration scal- grains [27], suggesting that our results are\ningonstressheterogeneity[15].Anyofthese consistent in that regard. However, how the\ncases potentially lead to a loss of fractal- statistical laws may change remains nontriv-\nity in the distributions of displacement and ial. As we have discussed in the manuscript,\npatch area of slip by the force chain col- the fractal nature of interparticle contacts\nlapse(SupplementaryFig.4c,6),resultingin could influence statistical properties, and\nthe linear moment-duration scaling M ∝ T further systematic investigation is needed.\n0\n(Fig. 2b).” Moreover, since gouge materials in subduc-\nLines 299–307) “Our results of slow tion zones are typically platy and aligned, it\nearthquake statistics indicate that lubri- may be necessary to investigate the statis-\ncatedsoftparticlesunderlowradialpressure tics using such an anisotropic setup as well.\nand with finite fault width exhibit local Since this point is related to the gouge\nforce-chain collapse and monotonous struc- effects mentioned above, we kindly ask you\nturesofinterparticlepatches.Inthiscontext, to refer to the first and last excerpts of\nparticles comprising force chains can be con- issue [47.2].\nsidered as gouge grains or even as brittle\nblocks across multiple scales [50, 51]. Low [54] Scaling laws vs porosity: In their results\nradial pressure and finite fault width cor- Fig. 2A, the porosity determines the pref-\nrespond to low effective normal stress due actor of the frequency scaling law. Please\nto pore fluid pressure and depth limitation quantify it (probably another power law?).\nof the seismogenic zone, respectively. These Is the variety of the frequency prefactor\nfactors mitigate the cascading processes in quantitatively consistent with observations?\nin their section ”New implications to slow events roughly increases with the porosity\nearthquakes.” decrease(SupplementaryFig.2,Supplemen-\nReply) Thank you for your thought- tary Table 1).”\nful comment. We quantified the relationship Supplementary Fig. 2) (please refer).\nbetween the prefactor N (event frequency)\n0\nand porosity ϕ. Although some runs exhibit We also kindly ask you to refer to the\na trend that appears to follow a power- discussion in the second excerpt of) issue\nlaw size distribution, as you mentioned, it [35].\nis not always the case, and we therefore [55] Fluid viscosity, monodispersity,\nanalyze and fit the data using an expo- sphericity: I appreciate their comprehensive\nnential function. As shown in the newly confirmation of the result validity, but the\nadded Supplementary Fig. 2, the prefactor link for Extended Data Table 1 and Fig. 5\ngenerally decreases with increasing poros- jumped to Table 1 and Fig. 5 in the paper.\nity, indicating that higher porosity leads to The extended data seem not available yet. It\na lower event frequency, which is consis- may not be the fault of the authors.\ntent with intuitive expectations. Observa- Reply) Thank you for your comments.\ntional data also show a positive correlation We confirmed and corrected all the links\nbetween the average event size (the inverse throughout the manuscript. We also com-\nof the exponential slope) and the overall pleted the information in Supplementary\nevent occurrence frequency (the exponential Table 1 and added the relevant figures (Sup-\nprefactor) [60], which is consistent with our plementary Fig. 7 and 8) as follows.\nexperimental findings. However, seismologi- Lines 189–196) “The low-friction\ncal data are interpreted such that, as fluid soft granular system exhibits both the\npressure decreases over time due to fluid exponential size distribution and linear\ndrainage, both the average event size and moment-duration scaling (Fig. 2). These\nthe event rate decrease [60]. If we interpret two statistical laws remain consistent across\nthe decrease in fluid pressure as a proxy various experimental conditions (Supple-\nfor decreasing porosity, then the porosity mentary Table 1), including polydispersity\ndependence observed in the field would be (Supplementary Fig. 7), liquid viscosity\nopposite to our experimental results. Since (Supplementary Fig. 8), and system size\nporepressureisinfluencedbyvariousfactors (Supplementary Fig. 9). This universality\nsuch as gouge permeability, elastic proper- ensures that the two statistical laws can be\nties, and fluid pressure or mobility. This explained by a simple model, as follows.”\nimpliesthatadirectcorrespondencebetween\nexperimentsandobservationsisnotstraight- [56] I end my comment on this article. This\nforward. Furthermore, natural faults are research is obviously backed up with hard\nsubjected to confining pressure, resulting in work of the authors in laboratories. So I am\nvery low porosity conditions. This highlights sure that the results of this paper are worth\ntheneedforfutureexperimentsunderhigher publishing. At the same time, in reading this\nnormal stress to investigate the effects of article, I felt this paper was written by an\nlow-porosityconditionsmorethoroughly.We early career researcher from a different field\nhave revised the manuscript as follows.\nof research. It may be fruitful to get advice\n23\nfrom laboratory research groups oriented to\nearthquake mechanics.\nReply) We are deeply grateful for your\ninsightful and helpful comments and sug-\ngestions. We sincerely apologize for any lack\nof clarity in our original writing, but your\nfeedback has been invaluable in improving\nthe manuscript, and we believe it is now\nmore accessible to a broader readership.\nIn addition, we have had the opportunity\nto engage in discussions with researchers\nworking on the experimental mechanics of\nearthquakes at both domestic and interna-\ntional conferences, including the European\nGeosciences Union General Assembly 2025\nand the Japan Geoscience Union meeting\n2025. Insights from those discussions have\nalso been incorporated into the revised\nmanuscript. We hope the revised version\nmeets your expectations and would greatly\nappreciate your consideration.\nK. Deep low-frequency tremors as a\n[1] Ide, S., Beroza, G. C., Shelly, D. R.\nproxy for slip monitoring at plate inter-\n& Uchide, T. A scaling law for slow\nface. Geophysical Research Letters 35,\nearthquakes. Nature 447,76–79(2007).\nL13304 (2008).\n[2] Ide,S.&Beroza,G.C. 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Source\ndata for “Origin of slow earthquake\nstatistics in low-friction soft gran-\nular shear” Version v2. Zenodo\nhttps://doi.org/10.5281/zenodo.16624257\n(2025).\nof slow earthquake statistics in low-friction soft granular\nshear”\nDear Editor and Reviewers,\nWe sincerely appreciate your invaluable feedback, which has enhanced the sig-\nnificance and value of this paper. Accordingly, we have revised the manuscript,\nincorporating all the suggested points, editorial requests, and some minor correc-\ntions. In addition, we have also made some revisions to improve readability after\ncarefully rereading the manuscript. Below, we provide a point-by-point response\nto each comment. All received comments are highlighted in brown, our responses\nin black, and excerpts from the revised manuscript in blue along with the line\nnumbers. For your reference, we also attach a file that shows the di↵erences\nbetween the previous version and the revised manuscript. In this file, deletions\nare indicated in red, and modifications/additions/relocations are highlighted in\nblue. Please note that line numbers di↵er between the revised manuscript and\nthe di↵erence file. Therefore, the line numbers cited in this response letter do\nnot correspond to those in the revised manuscript, but to the difference file. The\nreference numbers in this letter correspond to those in the revised manuscript.\nWe sincerely look forward to your favorable consideration of our revision.\n33].”\n[1] Having studied the revised manuscript as\nLines 317–319) “Spatiotemporal obser-\nwell as the response letter, I believe that the\nvations of slow earthquake statistics could\ncommentsImadeontheoriginallysubmitted\nhelp monitor fluid-filled fault zones [60],\nmanuscript have been adequately addressed.\nfault strength or stress [24], and shear\nThe authors have made a considerable e↵ort\nlocalization that could lead to devastat-\nto link their findings to published litera-\ning regular earthquakes such as the 2024\nture, following recommendations from all\nM 7.5 Noto earthquake [65].”\nw\nthree reviewers. Compared to the originally\nsubmitted version, the revised manuscript\n[3] Finally, I noticed that the data reposi-\no↵ers stronger motivation for the study and\ntory has been updated. I believe there is an\nimproved discussion.\nissue with the time columns in the updated\nReply) Thank you for recognizing the\nrawdata.csv. It seems to me that the round-\nrevised points and providing supportive\ning of the time values is incorrect, leading\ncomments.\nto several blocks of e.g., torque values having\nthe same time value. Therefore, I would ask\n[2] At the discretion of the authors and the\nthe authors to look into this matter.\nassociate editor, I recommend that the text\nReply) We really appreciate your criti-\nbe simplified when possible or adjusted for\ncal feedback. We have checked all data files,\nclarity. For example: - in line 52 of the\nincluding the mentioned time values, for\nrevised manuscript, I would change “demon-\nrounding errors (to only three significant\nstrates analogical behavior similar to the\nfigures). All CSV files have been regener-\ngouge materials” to “demonstrates behavior\nated from the original data files. We have\nanalogous to that of gouge”. - in lines 77\nuploaded them and updated the citation in\n& 78, I would change the wording to “is\nthe manuscript as [70]. We have confirmed\ntypically observed in lubricated granular sys-\nthat the measured and calculated values in\ntems”. - the last sentence of the discussion\nthem retain adequate precision.\nfeels long-winded. Consider a di↵erent final\nclause: “and shear localization that could\nlead to devastating...”. Additionally, I would\nreplace“pores”inline310with“faultzones”\nor something of similar spatial scale.\nReply)Thankyouforyourcomments.In\naccordance with them, we have revised the\nmanuscript.Thefollowingexcerptsshowthe\nrevised parts we made, highlighted in blue.\nLines 51–53) “Therefore, the mixture of\nlow-friction soft granular material and vis-\ncousliquiddemonstratesbehavioranalogous\nto that of gouge [32–34].”\nLines 76–78) “This transitional behav-\nior between creep and stick-slip is typically\n[4] The authors addressed my earlier com-\nments in a satisfactory manner.\nReply) Thank you for your comments.\n[5] Below are just a couple of small issues I\ncame across while reading the new version.\nL 36: Delete: , which is the focus of this\nstudy. It disrupts the flow of the paragraph.\nL 85: Confusing sentence. Substantially low?\nWhy substantially? L253: I’m not sure I\nunderstand the point you want to make with\nrespect to the experiments only capturing\nslow slip events in terms of statistics. This\nseems obvious, as this is the tool you used\nto investigate the problem. L255: Somewhat\ncumbersome first sentence to the paragraph.\nReply) Thank you for your feedback. We\nhave revised the manuscript accordingly as\nfollows.\nLines 35–36) “However, the multiple\nunique statistical laws of slow earthquakes\nand their origin have scarcely been investi-\ngated through experiments.”\nLines 85–86) “The maximum torque at\nthe minimum porosity remains su ciently\nlow, resulting in no particle failure.”\nLines255–260)“Incontrast,mostgouge\nfriction experiments under conditions rang-\ning from dry to 100% relative humidity and\nhighnormalstressshowthenegativecorrela-\ntionbetweenstressdropanddurationacross\nslow to fast slip events [19, 21, 23]. The sat-\nuration state of the liquid volume confined\nwithin pores, as well as normal stress, might\nplay a role in this di↵erence.”\nLines 261–262) “We could not find\nany previous work on the size distribution\nin slow slip gouge friction experiments, as\nmost studies have focused on slip velocity.”\ntise, I would like to leave the comments on\n[6] The authors did a great job of revision.\nthem to the other two experts. I encourage\nThey have responded to the questions and\nthe publication of this paper in the current\ncomments from the three reviewers through\nformat.\na thorough refinement of the paper.\nReply) We are truly grateful for your\nReply) We appreciate your supportive\nthoughtfulreviewandencouragingfeedback.\ncomments.\n[9] Just one unfamiliar wording: wetting\n[7] Still, the relation between a thin shear\nconditions -> wet conditions?\nzone in a precut rock and a self-organized\nReply) Thank you for your comment.\nshear zone in a fat granular band is unclear,\nWe have corrected the wording and kindly\nbut it is obviously future work.\nask you to refer to the third excerpt of\nReply) Thank you for your comment.\nissue [5].\nWe completely agree with you. Quantita-\ntive comparison between fault rock gouge\n[10] A minor confusion (of mine): The\nand soft granular media needs some scal-\nauthors describe their lab ruptures as col-\ning using dimensionless numbers, such as\nlapses of force chains in a bulk (e.g. L213,\nthe inertial number [43]. At least, based on\nL237), but they mention that the main\nour experimental results, porosity reduction\nrupture zones are basically near the side\nlocalizes the shear band thickness, and fault\nwall (above eq.1). The former sounds the\nrock brittle gouge under high normal stress\nobserved rupture is a bulk property, whereas\nis expected to have a “thin” shear zone as\nthe latter sounds the observed rupture is\nshown in Fig. 5c. In response to this com-\nan interfacial property, contributed from the\nment, we have revised the manuscript as\nlab machine as well as from their analogue.\nfollows.\nWhich is true? The possibility of the lat-\nLines 312–315) “Our study lays the\nter makes me consider that their laboratory\ngroundwork for addressing the scaling\nobservations might not derive from a soft\nbetween stress drop and slip velocity\nmaterial alone but from the interplay of the\n[19, 21, 22], the fracture energy scaling of\nsoft and sti↵ materials.\nslow earthquakes [63, 64], and the modeling\nReply) We appreciate your critical feed-\nof seismic moment as well as shear band\nback. We have recognized no wall slip on\nthickness [51] through nondimensionaliza-\nthe side wall of the machine rotary cylinder,\ntion.”\nwhich is also roughened with fixed particles.\nIn this paper, we interpret that interparticle\n[8] Regarding my comments on the previous\ndisplacements within each force chain in the\nversion of the manuscript, the current revi-\nbulk are infinitesimal, and the force-chain\nsion addresses the lack of comparison with\nbuckling explicitly and collectively causes\nprevious studies and seismological observa-\ntions, and the paper sectioning has been\nthe “slip on the boundary plane within the\nimproved accordingly. I appreciate that sev-\ngranular layer between the shear band and\neralquestionsIonceaskedabouttheirresults\nsurrounding vibration region.” In this gran-\nhave also been answered. Because the details\nular system, a shear velocity profile expo-\nnentially decaysintheradial direction, lead-\ning to clear discontinuous slip on the outer\n4\nlateral surface of the shear band. There-\nfore, the observed slip statistics derive from\nthe self-organized interplay between the soft\nshear band (principal slip zone) and the\nouter sti↵ granular zone (outer fault core).\nTo avoid misreading, we have replaced the\nword“outersidewall”with“outerboundary\nplane” of the shear band. In addition, as we\nhavediscussedinthemanuscriptinresponse\nto your previous comments, the system sti↵-\nness plays a critical role in slip stability.\nIn accordance with these points, we have\nrevised the manuscript as follows.\nLines 118–119) “The particles princi-\npally slip on the boundary plane within the\ngranular layer between the shear band and\nsurrounding vibration region.”\nLines between 183 & 184) “To explain\nthe relationship between the mean moment\nand porosity, we assume that slips occur\nrepeatedly on the outer boundary plane of\nthe cylindrical shear band, as shown in the\ninset of Fig. 5a. Although each particle slips\nwithinthebucklingforcechains,weconsider\nthat most of the strain energy is dissipated\nby the slips on the shear-band boundary\nplane, resulting from collective particle\nrearrangements within the force chains.”\n5"}]

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