Laboratory Hydrofractures as Analogs to Tectonic Tremors

The fracture of Earth materials occurs over a wide range of time and length scales. Physical conditions, particularly the stress field and Earth material properties, may condition rupture in a specific fracture regime. In nature, fast and slow fractures occur concurrently: tectonic tremor events are fast enough to emit seismic waves and frequently accompany slow earthquakes, which are too slow to emit seismic waves and are referred to as aseismic slip events. In this study, we generate simultaneous seismic and aseismic processes in a laboratory setting by driving a penny‐shaped crack in a transparent sample with pressurized fluid. We leverage synchronized high‐speed imaging and high‐frequency acoustic emission (AE) sensing to visualize and listen to the various sequences of propagation (breaks) and arrest (sticks) of a fracture undergoing stick‐break instabilities. Slow radial crack propagation is facilitated by fast tangential fractures. Fluid viscosity and pressure regulate the fracture dynamics of slow and fast events, and control the inter‐event time and the energy released during individual fast events. These AE signals share behaviors with observations of episodic tremors in Cascadia, United States; these include: (a) bursty or intermittent slow propagation, and (b) nearly linear scaling of radiated energy with area. Our laboratory experiments provide a plausible model of tectonic tremor as an indicative of hydraulic fracturing facilitating shear slip during slow earthquakes.

• A new hydrofracturing experiment simulates tectonic tremors and aseismic slips by visualizing and listening to stick-break instabilities • Both laboratory and tectonic tremors behave with bursty or intermittent slow propagation and radiate energy that scales with rupture area • Our experiments suggest tectonic tremors are manifestations of hydraulic fracturing that contributes to induced slow earthquakes
The coexistence of slow and fast earthquakes also appears in lab experiments (Ostapchuk et al., 2021;Wu & McLaskey, 2019).Lab-generated slow-slip events are observed acoustically as low-amplitude tremors, while fast-slip events have impulsive and energetic acoustic signals (Bolton et al., 2022;Hulbert et al., 2019).Tremors are observed during the acceleration of pre-slip before large earthquakes (Zhuo et al., 2018).Studies have found that slow earthquakes may promote or inhibit large, fast megathrust earthquakes in subduction zones (Kato et al., 2012;Rolandone et al., 2018;Ruiz et al., 2014).Investigating slow earthquakes may improve our understanding of the triggering mechanism and potential area of fast earthquakes (Gao & Wang, 2017).
The observed coexistence of slow slip and tectonic tremor has led seismologists to use tremor, or its constituent low-frequency earthquakes (LFEs), as markers of slow earthquakes.Their timing can help detect and locate slow-slip events (Frank, 2016;Frank et al., 2014;Mouchon et al., 2023;Wech & Creager, 2011), estimate their size (Frank & Brodsky, 2019;Passarelli et al., 2021), and infer the mechanical and stress state on the fault (Bürgmann, 2018;Hulbert et al., 2020;Nishikawa et al., 2019).Despite the ample observations of tectonic tremors, their source mechanisms and relation to slow slip remain uncertain.Studies of radiation patterns of stacked LFEs suggest that the LFEs that constitute tectonic tremor may coincide in space with shear slip on the megathrust, where fluid pressures are thought to be high (Ide et al., 2007;Shelly et al., 2007).Recently, Farge et al. (2021) and Shapiro et al. (2018) suggested an alternative model whereby tremors may be markers of hydrofracture unclamping a fault that moves in shear motion, generating some component of tensile motions.While the physical model of a single force for the seismic source may be unrealistic (Takei & Kumazawa, 1994), the concept of tensile motions, particularly those facilitating shear slip, corroborates an alternative model to LFE generating processes proposed previously (e.g., Muñoz-Montecinos & Behr, 2023 and references therein).The subduction of Earth materials generates free fluid from the dehydration of minerals during their phase transformation (Behr & Bürgmann, 2021;Kirkpatrick et al., 2021).Such fluid generation and storage released are thought to play a major role in the genesis of tremors and slow slip involving fault dilatancy and compaction (Audet & Bürgmann, 2014;Condit et al., 2020Condit et al., , 2022;;Nakajima & Uchida, 2018;Tarling et al., 2019).Fluid diffusion, whether it is constant or intermittent, may be prevalent in subduction zones where brittle fractures are expected (Cruz-Atienza et al., 2018;Farge et al., 2021;Obara et al., 2012;Saffer & Wallace, 2015;Shapiro et al., 2018).
Although slow earthquakes have been induced in dry experiments by regulating slip rates, there have been limited investigations on the hydrofracturing mechanism to generate tectonic tremors.This study examines whether stick-break instabilities of tensile cracks may contribute to the processes by which tremors are observed during slow slip events.We inject overpressured viscous fluid in an intact transparent rigid material and observe the nucleation and propagation of fractures that generate seismic signals similar to tectonic tremors.We then investigate the interaction between fluids and fractures to understand better the evolution of slow slip rupture source area and radiated seismic energy in a laboratory setting.Finally, we observe and discuss similarities between the laboratory and natural cases, particularly the near proportionality between cumulative radiated seismic energy and tremor area and the intermittent-burst behavior of stick-break instabilities.

Experimental Approach
The experimental apparatus is schematically shown in Figure 1a.More details, including a depiction of the experimental apparatus, are provided in Figures S1 and S2 in Supporting Information S1.To build a stiff and transparent sample with a notch included in the design to inject the fluid, we apply stereolithography (STL) 3D printing of polymethylmethacrylate (PMMA) using a FormLabs Form 3 printer.The sample consists of a transparent cylinder of 100 mm in diameter and 32 mm in height.We use a high-pressure pump (Teledyne Isco 65D) to inject the fluid into the PMMA sample at a constant flow rate of 0.3 mL/min.A pressure transducer measures the inline fluid pressure and indicates that the sample starts breaking at a pressure of about 30 MPa (Figure S3 in Supporting Information S1).At this instant, the compressed fluid expands suddenly and drives a fracture.The fluid is dyed with fluorescein, allowing us to visualize the fracture and the fluid independently.We use a high-speed camera (Vision Research Phantom TMX 6410) that records 500 × 500 pixel images (200 μm/pixel) at 100,000 frames per second (10 μs/image).With the images from the high-speed camera we create subtracted images, taking the difference of each pixel between each frame and the first one, for tracking cumulative fracture area.We also create differential images, taking the difference of each pixel between two consecutive images, for visualizing detailed tangential fracturing.
In addition, we use four Glaser-type broadband sensors (KRNBB-PC) to record acoustic signals associated with the fracturing process.Those acoustic-emission (AE) receivers have an exceptional frequency range of 20-1,000 kHz with nearly flat instrumental response in this frequency range (Mclaskey & Glaser, 2012).These four AE sensors are placed at each quadrant of the specimen and record acoustic signals at a frequency of 2 MHz.The experimental schematic is displayed in Figure 1a and detailed in Figures S1-S3 in Supporting Information S1.We perform two fracturing experiments varying the injected fluid viscosity to be μ = 1 cP and 800 cP in each.The complete fracture front and AE recordings are displayed in Figures S4 and S5 in Supporting Information S1.We also display the spectrograms of these AE signals in Figure S6 in Supporting Information S1.

AE Nucleation Location
The difference in first arrival times of the AE signals allows us to estimate the nucleation location of the AE sources.
Often, the AE signals interfere with each other (Figure S5 in Supporting Information S1), especially due to waves reverberating in the sample, so we do not use all of them.In the high-viscosity experiment, we pick the arrival times of acoustic waves from discernible stick-break events.We utilize the relative arrival time between receivers that are sufficient to locate the events, and subsequently increase the location accuracy using waveform cross-correlations.We pre-calculate all theoretical travel times using the compressional (P-) wavespeed found by performing calibration tests (Figure S7 in Supporting Information S1) and all possible source locations in a polar coordinate system (radial distance, r, from the center of the specimen and, θ, the azimuth angle taken from the East direction).We then perform a grid search as a global inversion to find the original location of the AE signal.The misfit function L is defined as: where i and j denote the indices of the receivers.TT is the theoretical travel-time difference between receivers i and j. tt is the observed arrival time difference between receivers i and j (origin time cancels).The neighboring receivers are paired as four groups for the inversion.We demonstrate the example process of inverting the location of the first fracture event from their four arrival picks in Figure S8 in Supporting Information S1.All fracture event locations are processed identically.

Estimating Radial and Transverse Fracture Area
By image processing, we track the contour of the slow radial fracture front to compute the radial fracture area over frame time.Since we are imaging at a constant frame rate of 100,000 frames per second, we can calculate the area of the crack surface every 10 μs.As observed in Figures S4 and S8 in Supporting Information S1, measurements include both the radius of the slowly expanding aseismic radial crack and the area of fractures that propagate tangentially to this front at seismic speeds.The latter manifest as illuminated pixels in sequential differential images and as AE signals.The details can be seen in Movies S1 and S2 and are described in the later results.
We focus on the propagation phases to avoid signal saturation at the beginning of the experiment and boundary effects as the radial crack reaches out of bound, in both low-viscosity and high-viscosity fracturing experiments.
There are some differences in area measurement for the two experiments.In the high-viscosity experiment, we calculate the transverse fracture area of discrete AE events, which lasts longer than the 10 μs measurement interval, as the differential area between the start and end times of the AE events.These individual fracture areas are then summed up over time and because there is no radial propagation in between AE events, this equals the cumulative radial fracture area.In the low-viscosity experiment, due to continuous and weak AE signals, we simply define the start and end times of a moving window of a fixed length (10 μs).An example of estimating the radial fracture area as a function of time is shown in Figure S9 in Supporting Information S1.The radial fracture area consists of multiple transverse fracture areas.

Estimating the AE Energy
To calculate the radiated energy of a single AE event, we need to correct the acoustic signals for path effects as the amplitude decays due to geometrical spreading and attenuation with distance.Because surface waves dominate the amplitudes of the wavefield that propagate in the traction-free sample, we use the square root of the distance as the geometrical spreading term.We model attenuation as an exponential decay with an attenuation parameter for the material absorption and scattering (Ono, 2018).Therefore, the following equation is utilized to correct the signal amplitude loss: where A 0 is raw AE signal in voltage, x is the distance between transverse fracture front and receiver, which is the receiver-or azimuth-specific (e.g., the radius is measured in the northwest direction to correct the signals recorded by the receiver located in the same corner).As suggested by Ono (2018), the attenuation coefficient α = 21.5 for PMMA using a peak acoustic frequency of 75 kHz.The spectrograms of recorded acoustic signals demonstrate the peaks of the dominant acoustic power around 40 kHz, as shown in Figure S6 in Supporting Information S1.We then perform a sensitivity analysis over a range of attenuation coefficients between 0.01 and 100.
Here we choose α = 20 to correct for attenuation.We report that an attenuation coefficient of less than 50 does not impact our main results.We show the results for these correction experiments in Figure S10 in Supporting Information S1.
After the signal correction (Equation 2), we calculate the total AE radiated energy for four individual receivers using: where E is the radiated energy calculated from voltage (AE) data, which is shown to be proportional to the kinetic energy in a drop-ball test, and therefore proportional to the elastic strain energy (Figures S11 and S12 in Supporting Information S1).We acknowledge that A is proportional to displacements and in voltage units, but our empirical calibration demonstrates that it is also a valid measure of radiated energy.Other studies have also approximated the AE radiated energy using voltage data to approximate the released elastic strain energy (Ghaffari et al., 2021;Grosse et al., 2021).A similar processing for tectonic tremor signals, such as attenuation correction and integration of velocity squared, is performed to calibrate and calculate signal energy (Ide et al., 2008;Wech, 2021;Yabe & Ide, 2014).
In Equation 3, t0 and t1 are selected for each experiment for the energy integration.For the high-viscosity fluid experiment, we manually pick t0 and t1 to be the window for individual discernible AE events after high-pass filtering signals .The time window varies from the start of the fracture AE event until the start of the next event.For the low-viscosity fluid experiment, we use an interval of 10 μs.We then compare the cumulative AE energy with the evolving fracture surface area.

Stick-Break Instabilities
We conduct two experiments with varying fracturing fluids: the first with water (low viscosity, μ = 1 cP) and the second with a mixture of glycerol and water (high viscosity, μ = 800 cP).As studied by Cochard et al. (2023), a stick-break instability occurs in all experiments.The fracture propagates slowly in the radial dimension overall (thousands of times smaller than the Rayleigh wave speed), but the slow radial fracture is accommodated by seismic transverse fractures, which propagate rapidly (close to Rayleigh wave speed).The break amplitude (i.e., when the radial fracture front advances and transverse fracturing occurs) and the stick time (i.e., time between breaks) increase as the fluid viscosity increases, which were also shown by Cochard et al. (2023).Here, we focus on a single small but fast seismic fracture.
In the high fluid viscosity experiment, the local nucleation of the fracture occurs mainly in the northeastern quadrant of the sample.Following the initial nucleation, the fracture expands transversely very rapidly, as shown by red patches in Figure 1b.A single transverse fracture takes approximately 90 μs to revolve around the entire perimeter of the fracture.Given the fracture length, we estimate the fracture speed of ∼1,000 m/s, very close to the experimentally measured Rayleigh-wave velocity in the 3D printed material (see glass-capillary break calibration in Figure S7 in Supporting Information S1).The AE signals of a single event recorded by the four sensors at each quadrant of the sample allow us to confirm the dynamics of these events.The first AE signal arrives at each sensor with a delay based on the location of its initiation point (Figure 1c).Thus, we can triangulate the signals to locate the position at which the elastic waves originated from with high accuracy (see Methods and Figure S8 in Supporting Information S1 for further details).We find that the position of the source of the signals coincides with the exact location of the fracture nucleation point identified visually, as shown in Figure 1b.We conclude that the radiated elastic energy is released in the form of elastic waves due to the fast anti-plane propagation of a tensile crack.Furthermore, we reveal that the whole course of the transverse fracture propagation radiates seismic signals since fortunately, we capture the surface wave associated with another fracture event in the later propagation stage (Figure S8 in Supporting Information S1).
To provide a basis for comparison, we conduct a water-induced fracturing experiment.In contrast with the high viscosity fluid experiment, we observe numerous individual events occurring with smaller amplitudes but much more frequently.The very small time interval between the nucleation of stick-break events in this case causes strong interference in their signals, which resemble seismic tremors or swarms.Despite the difference in frequency and amplitude, every event follows the same growth cycle from nucleation to arrest, as shown by tracking each individual fracture event in Figure S9 in Supporting Information S1.

Consistency Between Observed Fracture Discontinuities and AE Signals
To further analyze the causal relationship between stick-break events and AE signals, we compare and associate transient AE signals with the radial fracture radii in the subtracted images and intensity variations from differential images (Figure 2).We plot the kymographs, time representations of radial fracture radii taken in the northeast direction, for both fracturing cases with different fluid viscosities.In these representations, we highlight the fracture front with a dashed red line and the fluid front with a solid cyan line.The episodic displacement and stop pattern over the radial fracturing process is clearly observed on the top panel of Figures 2a and 2b.In the high-viscosity fracturing fluid experiment, although the fracture speed is fast (≥180 m/s) for a single transverse event, the average radial propagation speed is slow (∼4 m/s), resulting from periods of pauses (i.e., stick events).
The velocity of the fluid expansion is close to the average radial fracture propagation speed.The kymographs at other radial directions reveal similar fracture propagations as the contour of the fracture front is approximately symmetrical (Figure 1b and Movie S1).Besides visualizing the optical images, four AE sensors record the acoustic signals emitted by the fracture nucleation and propagation.We can identify at least three clear events with recognizable associations with fracture discontinuities and peak intensity rates.The AE wave packet may include a single event or multiple events.As we observe, several smaller events that follow the third event occur in short periods, which results in strong interference between the signals.The density of concentrated signals thus suggests the frequent occurrence of transverse fractures.This gives rise to a faster average fracture propagation than earlier propagation with longer stops.
In contrast, for the low fluid viscosity experiment the kymography indicates a rather continuous slow fracture propagation relative to the first observation.The fluid front is very close to the fracture front, making them hardly differentiable.The pressure fluctuations during the fluid expansion induce frequent and small fractures.The average fluid expansion equals the average slow radial fracture velocity (∼2 m/s) and is close to the one estimated in the high viscous case.The intensity-rate curve with multiple peaks discloses a fast-paced fracture propagation.Accordingly, we observe continuous acoustic signals filled with concurrent events with signals that overlap with each other, which thus makes individual fracture events difficult to distinguish, as shown in Figure 2b.Unlike the high-viscosity experiment (Figure 2a), uncovering the origin of these signals is a greater challenge.The acoustic signals from either low-viscosity or high-viscosity experiments display very similar waveforms to tectonic tremors consisting of either clear or unclear discrete seismic events, as often observed in Japan and Cascadia.

Scaling Relationship Between Fracture Area and AE Energy
We investigate the relationship between fracture area and AE energy and thus the fracture size can be characterized with AE energy.In the case of a fracturing fluid of 800 cP, there are multiple successive and very discernible tensile fracture events.By measuring the start and end times of each of these clear fracture events, we can calculate the fracture area by subtraction of images between the end and the beginning of the event and measuring the area (pixels) highlighted with high image intensity.We correct the effects of geometrical spreading and attenuation of the acoustic signals corresponding to each event and calculate their radiated energy as described above.We plot the cumulative AE energy against the cumulative fracture area and highlight a nearly linear relation for the first three events in Figure 3a.To better support the interpretation of a near-linear relation, we split the acoustic signals arbitrarily after the third fracture event into two parts.After calculating the fracture surface area changes and AE energy based on the split time, we consider them as two individual points.The larger variance at later times is also observed compared to the earlier times because of the uneven distribution of the events with respect to the different sensors.Averaging the energy over four sensors can eliminate the influence of azimuthal variations to a large extent.We confirm that the near-linear relation between cumulative radiated energy and cumulative fracture area remains valid using sensor-averaged measurements.
In the water (low viscosity) fracturing experiment, we cannot calculate fracture area and AE energy for individual fracture events, which are inseparable in time (Figure 2b).Instead, we calculate fracture area and AE energy for each frame interval (10 μs).The acoustic signals are corrected for geometrical spreading and attenuation using the transverse fracture front as location, calculate the distances between the transverse fracture fronts and the sensors, and calculate their respective energy, similarly to the previous case.We show the cumulative energy against the cumulative area in Figure 3b again and observe the near-linear relationship between the two again.There is an exception for the NW sensor, which we interpret as a deviation from the circularity of the rupture front (e.g., Movie S2).Overall, the consistent observations in both high-and low-viscosity cases suggest that the fracture energy increases with the fracture area almost linearly, regardless of the fluid viscosity.This relation is further theoretically explained in Text S1 in Supporting Information S1.

Comparison With Tectonic Tremors
We observe two types of fracturing behaviors in our laboratory experiments: (a) the slow average radial fracture velocity (2-4 m/s) actually occurs as intermittent short-duration rapid radial advances (i.e., breaks) separated by long nearly stationary periods (i.e., stick events); (b) the fast advances nucleate at the radial fracture front and propagate transversely much faster (∼1,000 m/s), at the Rayleigh-wave speed of the material.The radial fracture advancement is driven by the stress concentration applied at the crack tip.As shown in Cochard et al. (2023), the increase in the fluid viscosity leads to larger stick-break events, though does not alter much the overall radial fracture speed.In such a tensile-mode fracture, the stick-break instability results from extended fracture propagation because the pressure front is heterogeneous (i.e., is not strictly uniform in Figure 1) despite the fluid being injected at a uniform rate, and seems ubiquitous to all-natural fractures (Hull, 1999).Although in nature the slow slip often plausibly is too small to be measured, tectonic tremor is thought to consist of LFEs and may be broadband (Ide, 2019;Kaneko et al., 2018;Masuda et al., 2020), suggestive of a universal slow earthquake model (Ide, 2008(Ide, , 2010b)).The tremors generated in our laboratory experiments also exhibit broadband characteristics (Figure S6 in Supporting Information S1).

Plausible Mechanisms for Tremor Generation
The role of fluid migration in the generation of tectonic tremor has been proposed in multiple studies.First, where tremor occurs fluids likely come from the dehydration of minerals as they transform with increasing temperatures and pressure (van Keken et al., 2011).Their pore-fluid pressure is inferred to be high, if not close to lithostatic (Audet & Bürgmann, 2014;Condit & French, 2022).Cruz-Atienza et al. (2018) suggested that the rapid tremor migration events that occur backward relative to the slow, aseismic rupture front are induced by the slow propagation (∼10 m/s) of a wave of high pore pressure, which then reduces the effective normal stress and allows slip.Such pore pressure diffusion is evoked in the more general case of slow-slip and tectonic tremor by Farge et al. (2021).Tremor fronts migration described as diffusive has been observed and inferred to reflect fluid processes in numerous other studies (Amoruso & Crescentini, 2009;Ando et al., 2012;Farge et al., 2021;Ide, 2008Ide, , 2010a;;Ide & Maury, 2018;Nakata et al., 2011;Obara et al., 2012;Sagae et al., 2023).
The rupture mechanisms for LFEs that compose tremors are observed to be consistent with shear slip along the megathrust (Baratin et al., 2018;Ide et al., 2007;Imanishi et al., 2016;Royer & Bostock, 2014;Shelly et al., 2007) that can be activated by overpressurized pore fluid pressure (Cruz-Atienza et al., 2018;Tarling et al., 2019).Recent investigations by Shapiro et al. (2018) and Farge et al. (2021) argued similar measurements of seismic radiation pattern can be generated by a dipole or single-force mechanism, owing to insufficient station coverage.While shear deformations are evident in geological observations, the associated tensile veins generated at the depth of tectonic tremors also suggest that tensile fracture plays an important role in accommodating the deformation in the context of high fluid pressure (Behr & Bürgmann, 2021;Condit & French, 2022;Schmidt & Platt, 2022;Sibson, 2017).Here, we propose that tectonic tremor may not solely be shear dislocation alongside shear slow slip but could also facilitate shear slip through hydraulic fracturing that promotes the propagation of pore fluids and overall shear deformation (Figure 4).We discuss two observations that coincide in the laboratory and nature.

Stick-Break Instabilities as a Model for Intermittent Tectonic Tremor
Our laboratory experiments have demonstrated that the slow propagation of hydrofractures proceeds in a stick-break fashion, with inter-event times and fracture sizes driven by pressurized fluids that increase with fluid viscosity (Cochard et al., 2023).In nature, tectonic tremors also happen in bursts and exhibit temporal and spatial clustering (Frank et al., 2018;Husker et al., 2012;Jolivet & Frank, 2020).We present tremor observations during an example slow slip event in Cascadia in Figure 5.Despite a general trend of tremors migrating southward in that specific event, we also observe intermittent occurrence of tremor, punctuated by pauses with no radiation interpreted as temporary stalling of tremor-initiating slow slip (Frank & Brodsky, 2019;Frank et al., 2018;Jolivet & Frank, 2020;Mouchon et al., 2023;Rousset et al., 2019), and large variability in centroid locations of the tremors occurring in 2-hr intervals.Aspects of these observations mirror our lab-based findings, albeit with greater complexity, and may suggest the potential reopening of sealed or healed fault valves (e.g., backward ruptures).Unfortunately, our lab experiment is unconfined and does not include healing/sealing mechanisms, therefore, we do not observe all these behaviors in our laboratory experiments.Moreover, the intermittent and bursty characteristics of natural tremors are noticeable when considering the radiated energy of the tremors.Comparing this with the AE energies observed in Figure 2 reveals that the stick-break instabilities could be an effective proxy model for simulating tectonic tremor-like events; we further corroborate this proposition in the next section.

Scaling Relation Between Tectonic Tremor Source Area and Radiated Seismic Energy
Since the tremors usually present a low signal-noise ratio and lack resolvable low-frequency signals, the seismic moment (i.e., the total size and slip) is difficult to measure.Radiated seismic energy has been utilized as an alternative measure of size to define the tremor magnitude (Maeda & Obara, 2009;Wech, 2021).Our scaling relation between radiated energy and fracture area provides the first experimental evidence supporting use of radiated seismic energy as a measure of tremor source area or size.We use an enhanced catalog of Wech ( 2021) with 1,056 tremor swarms in the Cascadia region from 2017 to 2023, each assumed to be driven by a slow slip event.We utilize the 500 swarms with the greatest number of tremors for measuring their tremor areas and energies.We measured the tremor area and energy by employing a grid-based approach that discretizes the entire Cascadia transition zone into many grid cells with a grid size of 7.5 × 7.5 km 2 .For each swarm (slow slip event), in a similar manner to summing pixels illuminated after fracturing in the laboratory, we sum up the areas of grid cells that include tremors and the corresponding tremor energies within 2-hr intervals over the whole course of the event.In Figure 6, we display the relations between cumulative tremor area and energy for all 500 swarms.Similar results can be found when using data from Ulberg (2018) serves as an independent verification (Figure S14 in Supporting Information S1).We notice that the northern tremor swarms are systematically louder than the southern ones, which may be controlled by the geographic variations in fault strength.
The area-energy relation demonstrates that the daily radiated seismic energy varies quasi-linearly with the tremor source area, as shown in Figure 6.The seismic signals recorded during these swarms show very similar behavior to the AE signals recorded during the stick-break instability experiments.In addition, the observed spread of energy-area scaling coefficients may indicate the variations in apparent stress states on the subducted slab among events.The variability could be attributed further to heterogeneities in material properties and stress loading of the subducted slab (Kano et al., 2018;Kotowski & Behr, 2019).In the laboratory experiment, we also observe a slight asymmetry, which shows preferred nucleation and propagation direction due to the heterogeneity of the loading from the fluid pressure on the fracture tip.Therefore, the pre-existing stress condition and heterogeneity of material properties accentuate the complexity of rupturing process in the field scenarios.The tensile-shear model is also described by Sibson (2017).Evidence of these crack-induced tremors can be seen in many exhumed rock samples (Behr & Bürgmann, 2021;Condit & French, 2022).Alternative models also include jamming of granular media (Sammis & Bostock, 2021;Sibson, 2017).
Since the range of individual event sizes was not large enough, we only focus on the relation between cumulative values for energy and area for the comparison.Regardless, relating radiated energy and fracture area may be interpreted in several ways.Ide (2008Ide ( , 2010b) ) proposed a Brownian model for shear events for tectonic tremor that predicts a random distribution of single moment rates functions and a proportionality, or near proportionality between tremor area and radiated energy.A second consideration might be to think about seismic moment from   Wech, 2021).The color of the circles indicates the centroid latitude of each swarm, with northern to southern events shaded from red to blue, and highlighting the more energetic tremors in the north (Wech, 2021).
(a) The black line highlights the example event in Figure 5a.Swarms systematically radiate greater energies from north to south.(b) The slopes of the curves in (a), measured as the linear regression coefficients, are displayed versus the radiated energy.We find a median regression coefficient, n, as the power for E ∼ A^n to be 1.16 for the 500 swarms studied, which lies between the laboratory values of n = 1.19 and n = 1.07 for the high-and low-viscosity experiments, respectively.The power exponent is more variable for the small swarm relative to larger ones.We refer to Figure S15 in Supporting Information S1 for more details.
these tensile fractures.Eaton et al. (2014) concluded that the seismic moment is proportional to fluid pressure and cubed source radius.Our finding that radiated energy scales nearly, or quasi-linearly with area suggests that the event slip is nearly constant with tremor area.Slip-opening invariance is quite plausible given the aspect ratios of veins (Fisher & Brantley, 2014).The detailed analysis on the energy-area relationship is referred to Text S1 in Supporting Information S1.

Conclusions
The intermittency and migration of tectonic tremors are frequently observed in subduction zones (Farge et al., 2021;Kao et al., 2006;Obara et al., 2010;Wech & Bartlow, 2014).Tremors are often hypothesized as occurring passively as shear events initiated by slow slip, from seismogeodetic observations (Amoruso & Crescentini, 2009;Ando et al., 2012;Frank et al., 2018;Ide, 2008).However, recent geological investigations (Condit & French, 2022;Mindaleva et al., 2020;Schmidt & Platt, 2022;Ujiie et al., 2018) and numerical simulations (Cruz-Atienza et al., 2018;Farge et al., 2021) reveal that fluid propagation can potentially generate seismic behaviors similar to tectonic tremor via a combined mechanism of shear and tensile deformations.While we focus only on the tensile mechanisms due to our experimental limitations, our results suggest previous studies of tremor intermittency that attribute it exclusively to sporadic fault locking and stalled tremor-initiating slow shear slip (Frank & Brodsky, 2019;Frank et al., 2018;Jolivet & Frank, 2020;Mouchon et al., 2023;Rousset et al., 2019) should also consider a role for variable pore pressures that may cause tensile fracturing and sealing.Additionally, in the model we propose in Figure 4, the cracking and sealing of fractures actually modify the potential for shear slip and thus tremor cannot be considered strictly as a result of the overall driving slow slip event, but instead as a facilitator of shear slip.
We perform a laboratory experiment in ambient pressure and temperature conditions of a pure slow tensile hydrofracturing event driven by a radially spreading fluid pressure front, accompanied by fast fracture propagating transversely to the front.Unprecedented high spatial and temporal resolution of the fracture dynamics reveal a stick-break instability (Cochard et al., 2023), whereby AEs from the fast break events behave like seismic tectonic tremor.The fluid decompression controls the slow radial fracture propagation pattern (Figure S13 in Supporting Information S1).The observations of decelerated radial propagations, which are due to decreases in transient fluid pressure, explain that the migration of tremors slows down, as observed in Figure 5b.This temporal evolution has been observed in numerous other studies (Amoruso & Crescentini, 2009;Ando et al., 2012;Farge et al., 2021;Ide, 2008Ide, , 2010a;;Ide & Maury, 2018;Nakata et al., 2011;Obara et al., 2010Obara et al., , 2012;;Sagae et al., 2023), describing it as diffusive and thus in some cases attributing it to pore pressure migration.The laboratory stick-break stabilities also are similar to the intermittencies of tremor rupture observed in Cascadia.Our study further suggests a near-linear scaling of cumulative energy with active breaking area for both natural observation of tectonic tremor in Cascadia and the laboratory experiments.The idealized geometry of our experiment likely only represents elements of a rather complex system of veins and existing cracks.
The rate of fast transverse fracture and size vary with fluid viscosity.While the overall behavior of the slower rupture is not affected much by fluid viscosity, we observe pauses and breaks with various durations during the slow radial fracture propagation (Figure 2 and Figure S4 in Supporting Information S1).We observe that in the high-viscosity fluid experiments, the slow radial fracture propagates with large break amplitudes followed by long pauses.In contrast, in the low-viscosity fluid, the experiment proceeds with small break amplitudes and short pauses.The tremor intermittency might be an indication of fluid viscosity, as in our experiment.Despite these differences, the overall propagation of the slow radial fracture (and fluid diffusion) occurs at a similar rate and coincides with fluid propagation.Simple models accounting for the different fracture styles and fluid viscosity and with an intact, simple internal stress structure showed that the fluid viscosity impacted the stress loading conditions (i.e., the lag between the fluid and the fracture front; Cochard et al., 2023).In nature, the viscosity of geofluids may vary tremendously (Audétat & Keppler, 2004), as well as pre-existing permeability (Kato et al., 2004).These factors may control the migration of fluid pressure and therefore tremor and slow slip propagation patterns, which deserves further exploration.
Our experimental results suggest a plausible model whereby fluid propagation generates seismic events with characteristics that are comparable to field observations.The transverse breaks propagate at much higher speeds (≥180 m/s) than the overall propagation speed (2-4 m/s), and are accompanied by fast fractures that propagate transversely to the radial rupture front and radiate AEs.This highlights the importance of imaging with a high temporal resolution to observe the fracture dynamics in detail.Gombert and Hawthorne (2023) recently observed short-time tremor bursts that may migrate with a faster speed (3-25 m/s) than long-time tremor events.We suspect that even shorter bursts that comprise those individual bursts might reach higher velocities, but that temporal resolution limits the observations.Furthermore, local nucleation observed at some locations along the fracture front, followed by transverse expansion of the fracture, suggests that the transient fluid pressure may rapidly vary differential stress and cause complex deformations (Condit & French, 2022;Schmidt & Platt, 2022).
To conclude, our experimental approach permits measurement of a fracturing transparent sample in a nearly circular fracture geometry with unprecedented high spatiotemporal resolution, providing novel insights into the generation and characterization of tectonic tremors and slow slip.Further work should include differential stress conditions to evaluate the contribution of shear stress on the propagation of these fractures.There remain alternative models to tremor generations at depth.One of them describes slow earthquakes and tremor as deformation of shear zones within granular media, where particle jamming from compression and expansion builds force chains (Sammis & Bostock, 2021;Vo et al., 2020) that may explain magnitude-frequency distributions of LFEs.While our study suggests a role for tensile cracking (i.e., hydrofracture) in tremor genesis, it is rather likely that the reality is a combination of various mechanisms.On rare occasions have slow slip earthquakes not been accompanied by tremor (Hirose et al., 2023), which suggests that hydrofracture mechanisms for tectonic tremor are not required for slow slip and additional physics ought to be understood.and Bob Graham for helpful discussions on AE sensor calibrations.We are grateful to the editor (Thorsten Becker) and the three anonymous reviewers who provided a lively discussion during the reviewing process and greatly improved the robustness and quality of our arguments.We also thank James MacArthur for making us one amplifier for AE signals.
We are further grateful to Carl Ulberg and Ken Creager for providing their tectonic tremors and helpful discussions on the tremor data processing.This research was supported by China University of Petroleum (Beijing) through Harvard-CUPB joint laboratory on Petroleum Science and the Harvard MRSEC (NSF, DMR-2011754).Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

Figure 1 .
Figure 1.Experimental apparatus, subtracted images, and acoustic signals for the high-viscous fluid experiment (μ = 800 cP).(a) Schematic of the experimental apparatus with a high-speed camera and four acoustic emission (AE) sensors placed at each sample quadrant (SW, NW, NE, SE).The filled triangles represent the AE sensor locations.The dashed red and solid cyan lines are the fracture and fluid fronts, respectively.The cyan cone is the injected fluid.See additional experimental details in Figures S1 and S2 in Supporting Information S1.(b) Three subtracted images from the first image after T = 2.4 ms of the first burst show nucleation, propagation, and arrest of a single AE event.The red patch and arrow represent the transverse fracture area and direction, respectively.(c) AE signals associated with the fracture event in (b), the black arrows denote the picks of the first arrivals that we use for locating the AE source, which is marked as the orange star in (b).

Figure 2 .
Figure 2. Kymograph, pixel intensity variation rate, and acoustic emission (AE) signals.Top panels: kymograph of the subtracted images taken in a NE direction of the specimens.The dashed red and solid cyan lines represent the fracture and fluid fronts in the upper panels, respectively.Middle panels: time series of the image intensity rate, which is measured as the averaged pixels of the differential images.Lower panels: acoustic signals recorded by the four AE sensors.The amplitude unit "a.u." represents an "arbitrary unit" for their normalization to the peak amplitude.The back lines are the radiated energy rates of acoustic signals.(a) For a high fluid viscosity (diluted glycol, μ = 800 cP) experiment and (b) for a low viscosity fluid (water, μ = 0 cP) experiment.The full-length measurements are displayed in Figures S4 and S5 in Supporting Information S1.Note: the time series is trimmed to focus on the hydrofracture propagation instead of its beginnings, which suffer from signal saturation, and endings, which are affected by sample edges effects.

Figure 3 .
Figure 3. Laboratory relationship between cumulative fractured area and cumulative radiated energy.The colored dots represent the measurements of sensors at different azimuths.The black dots are the mean estimate of four sensor-specific values.The cumulative radiated energy is proportional to the fracture area for individual fracture events.The gray line is the best-fit linear regression using the averaged measurements (black dots).(a) High fluid viscosity experiment: the first three data points come from three clear events denoted as pink stars shown in the schematic diagram.They are recognized as the first three events in the kymograph panel in Figure 2a.The late two measurements represent the signals after the third event, which are indistinguishable and divided into two groups arbitrarily.(b) Low fluid viscosity experiment: the acoustic emission (AE) cumulative energy is calculated directly on the continuous vibrations.

Figure 4 .
Figure 4. Illustration of tremor generation mechanisms in an overpressurized fault zone in subduction.Two primary mechanisms may influence tremor dynamics.The first, the single-force mechanism (Farge et al., 2021; Shapiro et al., 2018), proposes that tremors occur during transient fluid pressures as barriers unclog and fluid flows, which are indicated by blue arrows in the lower left panel.The second mechanism proposed in this study suggests a mixed mode of shear and tensile fracture resulting from increased fluid pressure.The black arrows represent seismic source body forces in the lower right panel.The tensile-shear model is also described bySibson (2017).Evidence of these crack-induced tremors can be seen in many exhumed rock samples(Behr & Bürgmann, 2021;Condit & French, 2022).Alternative models also include jamming of granular media(Sammis & Bostock, 2021;Sibson, 2017).

Figure 5 .
Figure 5. Example Tremor Swarm in Cascadia.(a) Individual tremor locations during one slow slip event that started 10 March 2019, color-coded by occurrence time since origin.(b) Bursty tremors.The upper panel shows the southward propagation of centroid location along the latitude over time since its origin.Each dot denotes the relative distance in kilometers of centroid latitude of tremors within 2-hr intervals relative to the original location.The error bar indicates the latitude range (in kilometers) of each interval.The blue curve represents the averaged or smooth propagation over time.The lower panel shows the curves of the number and summed tremor energy in 2-hr intervals binned in the same way as in the upper panel.The yellow patches highlight the quiescence periods, in which no tremor is observed.We find the episodicity of tremors similar to laboratory observations in Figure 2.

Figure 6 .
Figure6.Field relationship between cumulative tremor area and cumulative radiated energy.The relationship between cumulative tremor area and cumulative radiated energy for 500 inferred tremor swarms that took place in the Cascadia region from 2017 to 2023 (adapted fromWech, 2021).The color of the circles indicates the centroid latitude of each swarm, with northern to southern events shaded from red to blue, and highlighting the more energetic tremors in the north(Wech, 2021).(a) The black line highlights the example event in Figure5a.Swarms systematically radiate greater energies from north to south.(b) The slopes of the curves in (a), measured as the linear regression coefficients, are displayed versus the radiated energy.We find a median regression coefficient, n, as the power for E ∼ A^n to be 1.16 for the 500 swarms studied, which lies between the laboratory values of n = 1.19 and n = 1.07 for the high-and low-viscosity experiments, respectively.The power exponent is more variable for the small swarm relative to larger ones.We refer to FigureS15in Supporting Information S1 for more details.