Advances in experiments and numerical simulations on the effects of stress perturbations on fault slip

Yunmin Hung ,Shengli M ,Xiohui Li ,Ye Sho

a State Key Laboratory of Earthquake Dynamics,Institute of Geology,China Earthquake Administration,Beijing,100029,China

b Guangdong Earthquake Agency,Guangzhou,510070,China

Keywords:Coulomb stress change Rate-and state-dependent friction law Stress perturbation parameters Tectonic stress Acoustic emission

ABSTRACT Laboratory experiments and numerical simulations on rock friction perturbations,an important means for understanding the mechanism and influencing factors of stress-triggered earthquakes,are of great significance for studying earthquake mechanisms and earthquake hazard analysis.We reviews the experiments and numerical simulations on the effects of stress perturbations on fault slip,and the results show that stress perturbations can change fault stress and trigger earthquakes.The Coulomb failure criterion can shed light on some questions about stress-triggering earthquakes but cannot explain the time dependence of earthquake triggering nor be used to investigate the effect of heterogeneous stress perturbations.The amplitude and period are important factors affecting the correlation between stress perturbation and fault instability.The effect of the perturbation period on fault instability is still controversial,and the effect of the high-frequency perturbation on earthquakes may be underestimated.Normal and shear stress perturbation can trigger fault instability,but their effects on fault slip differ.It is necessary to distinguish whether the stress perturbation is dominated by shear or normal stress change when it triggers fault instability.Fault tectonic stress plays a decisive effect on the mode of fault instability and earthquake magnitude.Acoustic emission activity can reflect the changes in fault stress and the progression of fault nucleation,and identify the meta-instability stage and precursor of fault instability,providing a reference for earthquake prediction.

1.Introduction

Natural fault zones are continuously subject to variations in normal and shear stress,ranging from slowly and steadily changing tectonic stress caused by plate movement and other tectonic processes to the transient and periodic stresses caused by solid tides,surrounding strong earthquakes,reservoir water storage,and industrial water injection.These stresses act on faults to cause fault stress adjustment,change seismicity characteristics on faults,and lead to the advance or delay of strong earthquakes,which is the so-called“stress triggering”(e.g.,Harris,1998;Kilb et al.,2002;Stein et al.,1997).More and more studies have shown that,large earthquakes can alter the static stress field surrounding the rupture and change the seismicity of adjacent faults(King et al.,1994;Nalbant et al.,1998;Stein et al.,1997;Toda et al.,2008).For example,Stein et al.(1997) calculated the Coulomb stress changes from tem earthquakes withM≥ 6.7 that occurred on the North Anatolian fault from 1939 to 1992,and Nalbant et al.(1998) calculated the Coulomb stress changes from 29 earthquakes that occurred in the region of north Aegean Sea and northwest Turkey since 1912,the results indicate that the probability of damage earthquakes in Izmit area increases.Subsequently,anMW7.6 earthquake occurred in Izmit in 1999,confirming the valves of the studies.Toda et al.(2008) calculated the coseismic Coulomb stress change of the WenchuanMS8.0 earthquake in 2008 and found that the stress perturbation caused by the earthquake significantly increased the probability ofM≥ 6 andM≥ 7 earthquakes in the region from Kangding to Dawu and Maqin to Rangtag in the next ten years.Liu et al.(2022) calculated the coseismic Coulomb stress change of the MadoiMS7.4 earthquake in Qinghai and found that the risk of strong earthquakes on the Gadain South Margin fault,Madoi-Gadain fault,and Tibet Dagou-Changmahe fault would significantly increase in a short period(within about ten years),with the potential forMS≥6.0 earthquakes.On the other hand,large earthquakes also can trigger earthquake activity in the distance by sending transient and potentially destabilizing seismic waves (Brodsky et al.,2000;West et al.,2005;Yao et al.,2021).For example,West et al.(2005)found that the seismic waves of the SumatraMW9.0 earthquake in 2004 triggered 14 earthquake swarms at Mount Wrangell,Alaska.The seismic waves of the KaikuraMW7.8 earthquake in 2016 also triggered microearthquakes and tremors in New Zealand(Peng et al.,2018;Yao et al.,2021).

In addition to the stress changes caused by earthquakes,the Earth's tides can produce stress variation at 0.001–0.004 MPa (Vidale et al.,1998).Many studies suggest that tidal stress also can trigger seismicity,even strong earthquakes (Peng et al.,2021;Tanaka,2010,2012).For example,a robust statistical correlation exists between the tidal stress phase and the onset time of strong earthquakes in subduction zones such as the 2011 Tohoku-OkiMW9.1 earthquake(Tanaka,2012)and the 2004 SumatraMW9.0 earthquake(Tanaka,2010).Two years before the 2011 TengchongMW5.1 andMW5.0 earthquakes,statistically significant correlations were observed between the tidally induced stresses and earthquake times(Peng et al.,2021).

In recent years,with the rapid growth of industrial activities such as unconventional oil and gas production,geothermal operations,wastewater disposal,and geological sequestration of CO2,induced seismicity associated with fluid injection has attracted wide attention.Studies show that industrial water injection can affect the stress of surrounding faults and induce seismicity through increasing pore pressure and poroelastic stressing(Atkinson et al.,2020;Ellsworth,2013;Eyre et al.,2019).In the case of long-duration water injections,such as wastewater disposal,water injection mainly leads to pore fluid diffusion along the pre-existing faults,increasing the pore pressure hence reducing the effective normal stress and bringing the fault close to the failure(e.g.,Keranen et al.,2014;Lei et al.,2013;Lei et al.,2008).Instead,in case of short-duration and rapid water injections,such as shale gas hydraulic fracturing,the water injection also can cause poroelastic stressing in the surroundings,then induce seismicity on faults without direct hydraulic connection to the injection well (Atkinson et al.,2020;Lei et al.,2019;Segall and Lu,2015).

Laboratory rock experiments and numerical simulations can quantitatively analyze the effects of stress perturbations on fault slip and are essential for understanding stress-triggering earthquake mechanisms.Up to now,much insight into the mechanism and condition of stresstriggering earthquakes has come from previous laboratory experiments and numerical simulations (Beeler and Lockner,2003;Boettcher and Marone,2004;Huang et al.,2016;Lockner and Beeler,1999;Ma et al.,2011;Savage and Marone,2007).Summarizing the previous results and promoting the in-depth study is of great theoretical and practical significance for studying earthquake mechanisms and prediction.Since there have been many reviews on field observation,this paper focuses on experiments and numerical simulations,analyzes the advancements and main problems in stress-triggering earthquake studies,and then proposes future study topics and directions.

2.Theoretical framework for fault reactivation and frictional stability

2.1.Coulomb failure criterion

Rock shear failure is usually described with the Coulomb failure criterion.In this mechanical framework,the Coulomb stress (Coulomb failure stress,CFS)on the fracture plane can be expressed as(King et al.,1994):

where τ is the shear stress on the fracture plane;σ is the normal stress on the fracture plane,which is positive under compression and negative under tension;Pis pore pressure;σ-Pis effective normal stress;μ is friction coefficient,generally ranging from 0.6 to 0.8;C0is rock cohesion.Faults generally have macroscopic slip plane,with cohesion negligible,so the Coulomb stress can be simplified as:

Because the absolute stress of a fault at the hypocenter depth is difficult to measure directly,the Coulomb stress change becomes an important method for analyzing the effects of stress changes on nearby faults' seismicity (Harris,1998).According to Coulomb's law,the Coulomb stress change on the fault can be expressed as:

where Δτ and Δσ are homogeneous shear (positive along the fault slip direction)and normal(positive under compression)stress change on the fault,respectively;ΔPis pore pressure change;and Δσ-ΔPis effective normal stress change.An increase in shear stress and a decrease in effective normal stress lead to an increase in Coulomb stress(ΔCFS>0),promoting the fault closer to failure.On the contrary,a decrease in shear stress and an increase in effective normal stress lead to a decrease in Coulomb stress(ΔCFS<0),deferring the fault away from failure.

2.2.Rate-and state-dependent friction law

The Coulomb failure criterion can predict the stress conditions for fault instability,but it does not address the question of frictional stability and whether the slip will be seismic or aseismic upon reactivation.The rate-and state-dependent friction law,capturing the primary macroscopic mechanical behavior of frictional slip,is commonly employed to describe fault friction and the resulting slip behavior,such as earthquake nucleation,dynamic or quasi-dynamic earthquake rupture propagation,and aftershocks (Dieterich,1992;Lapusta and Rice,2003).In this formalism,the friction stress on the fault can be described as(Dieterich,1979;Marone,1998;Ruina,1983):

where τ is shear stress,σ is normal stress,Vis slip rate,θ is a state variable indicating the contact state of the fault surface or the internal structure of fault gouge zone between fault surfaces(Marone,1998);a,b,andDcare empirical constants and characteristic distance,describing the friction characteristics of the fault (Fig.1);V0,θ0and μ0are references,corresponding to the steady-state friction.The evolution of state variables θ with time can be described by the“Aging” law(Dieterich,1979):

Fig.1.Transient response of the friction strength to loading rate changes and its evolution to a steady state.

or the“Slip” law(Ruina,1983):

Because the“Aging”law is more consistent with experimental results,it has been widely used in rock experiments and numerical simulations.Because the formalism also holds in the case of variable normal stress,Linker and Dieterich (1992) introduced a new parameter α,and the“Aging” law is changed to

The values of friction parametersa,b,μ0,Dc,and α can be obtained from laboratory friction experiments.

At arbitrary velocityV,a steady state exists that is reached whendθ/dt=0,i.e.,θ=θss=Dc/V,where the subscriptssrefers to the steady state.In this case,we can obtain a velocity-dependent parameter（a-b）=dμss/dln（V）,which characterizes friction slip stability.When（a-b）>0,the friction increases with an increasing sliding velocity,called velocity strengthening,slip is accommodated by stable sliding.(Fig.1).On the contrary,when（a-b）<0,the friction decreases as the sliding velocity increases,which is called velocity weakening,the fault may experience dynamic instability,meeting the conditions for earthquake nucleation (Fig.1).

When combined with the elastic dislocation theory,rate-and statedependent friction law states that if the stiffness of the elastic medium(K) is smaller than the critical stiffness (Kc) of the fault,the fault friction instability can occur.Coupling the rate-and state-dependent friction law with the spring-slider model,the critical stiffness (Kc) controls the transition from stable to unstable slip can be expressed as (Dieterich,1979;Rice,1993;Ruina,1983):

Therefore,fault slip after fault reactivation can be divided into three domains:1)if（a-b）>0,the fault slip is always stable;2)if（a-b）<0 andKKc,the fault is conditionally stable,depending on the relationship between velocity-dependent parameter（a-b）and characteristic distance,which can be expressed asDc（b-a）/Dc.

3.Experiments and numerical simulations of stress-triggering earthquakes

3.1.Mechanisms and conditions of stress-triggering

Existing studies mainly study the mechanisms and conditions of stress triggering through periodic stress perturbation experiments and numerical simulations.For example,Perfettini and Schmittbuhl(2001)studied the influence of normal and shear stress perturbation on a creeping fault using a spring-slider model subjected to rate-and state-dependent friction law.The results suggest the fault has a critical periodTcr=at the critical stiffness,indicating that only the perturbation period close to the critical periodTcrcan induce nucleating faults instability.Beeler and Lockner (2003) tested the effect of stress oscillation on granite fault slip and observed a critical period related to fault failure too.When the perturbation period is greater than the critical period,the correlation between fault instability and stress perturbation depends on perturbation amplitude,frequency,and stress rate.On the contrary,when the perturbation period is less than the critical period,the fault failure is mainly affected by the perturbation amplitude,and a much higher stress perturbation amplitude is required to induce a detectable correlation with the perturbation stress.Experiments on stress perturbations also demonstrate the vital role of perturbation period on fault instability over recent years and suggest that high-frequency perturbation has a limited effect on earthquake triggering(Boettcher and Marone,2004;Savage and Marone,2007).However,a few experiments and numerical simulations suggest that the effect of the high-frequency perturbation on fault failure may be undervalued.For example,Perfettini et al.(2003b) suggest high-frequency perturbation having a higher triggering potential than low-frequency perturbation from numerical simulations.Recent experiments and numerical simulations on transient stress perturbations also report that the effect of short-period pulse on fault instability is greater than that of long-period pulse with the same stress perturbation amplitude,indicating the effect of high-frequency and short-duration transient stresses on earthquakes may be underestimated(van der Elst and Savage,2015).

The experiments and numerical simulations above show that the correlation between fault stick-slip and stress perturbation is both amplitude-and frequency-dependent.However,some studies show that fault stick-slip is mainly affected by perturbation amplitude,and the perturbation period plays little effect (Gao and Shi,2022;Huang et al.,2009;Lockner and Beeler,1999).For example,the laboratory work of Lockner and Beeler(1999)shows that the degree of correlation between the timing of simulated earthquakes (stick-slip events) and the imposed periodic loading is most sensitive to the perturbation amplitude,with weaker dependence on perturbation period and loading rate.Moreover,the correlation between the two increases with perturbation amplitude:when the perturbation amplitude is less than 0.05 MPa,the correlation is negligible;when the perturbation amplitude ranges from 0.05 to 0.1 MPa,the correlation turns from weak to strong;when the perturbation amplitude is more significant than 0.1 MPa,the correlation is obvious.In addition to the correlation between stress perturbation and the timing of simulated earthquakes,periodic stress perturbation experiments and numerical simulations also suggest that perturbation amplitude can influence the stress drop,time interval,and the clock advance(Δt) as the fault is triggered.For example,the periodic stress perturbation experiments conducted on double-direct shear configuration show that the stress drop and time interval of stick-slip events tends to be scattered as the perturbation amplitude increases (Huang et al.,2009,2016).A numerical simulation based on the rate-and state-dependent friction law indicated that fault instability time caused by sinusoid periodic stress perturbation is always more advanced than tectonic stress.The clock advance increases with the perturbation amplitude.The perturbation frequency has little effect on the instability time(Gao and Shi,2022).

Another line of studies focused on the time dependence of stress perturbation triggering earthquakes.For example,Gomberg et al.(1997)studied dynamic stress triggering using the spring-slider model with the rate-and state-dependent friction law and found that the clock advance caused by transient stress perturbation depends nonlinearly on when in the earthquake cycle the transient load is applied.The time of instability depends nonlinearly on the transient loading rate.Higher-frequency and/or longer-duration seismic waves increase the amount of clock advance.To understand the physics of dynamic triggering and the influence of dynamic stressing on earthquake recurrence,Johnson et al.(2008)performed double-direct shear experiments on granular media by applying acoustic perturbation and found that the frequency delayed of small-magnitude events is related to the change of fault contact caused by acoustic perturbation.Ferdowsi et al.(2015) observed the delayed triggering of slip in the granular gouge using the discrete element method and indicated that the dynamic stress perturbation influences the network of weak contacts in the granular gouge layer,reducing the slip contact ratio,shear modulus,and friction strength of the gouge,thus causing a clock advance.With the discrete element method,Blank et al.(2021)investigated the effect of the dynamic perturbation on heterogeneous prestressed faults and found that the perturbation amplitude,the state of the asperity,and the deformation of the surrounding material jointly control the delay time between perturbation and triggering event.They propose that the aseismic slip in a weak fault section triggered by the perturbation might account for the delayed triggering.More recently,Dong et al.(2022) reported the time delay in the near-field dynamic triggering experiment.The results show that the time delay is strongly related to a “disturbed” rupture nucleation process,and the interaction between alteration of fault contact state combined with perturbed nucleation controlled the physical process of dynamic near-field triggering.

3.2.Difference between normal and shear stress perturbation

According to the Coulomb failure criterion,the Coulomb stress change is the comprehensive effect of normal and shear stress change.Some studies suggest that normal and shear stress perturbation have the same effect in inducing fault instability(Perfettini et al.,2003a,2003b),so calculating the Coulomb stress change caused by stress perturbation and analyzing its effect on fault stress has become an excellent tool for stress-triggering (Harris,1998).However,other experimental studies showed that the effect of normal and shear stress perturbation on fault slip is different.For example,the work of Cui et al.(2006)shows that the shear stress perturbation only has a specific effect on stick-slip time,while the normal stress perturbation with the same amplitude completely breaks the fault stick-slip behaviors,making the stick-slip curve irregular,changing the stress drop and time interval of the stick-slip events.With a double-direct shear configuration,Huang et al.(2016)applied the stress perturbation with similar amplitude in shear and normal stress,then compared their effects on fault slip behaviors.The results show that the normal stress perturbation makes stress drops scattered as the amplitude increases more obviously than the shear stress perturbation,suggesting that the normal stress perturbation has a more significant influence on fault slip than the shear stress perturbation when they have the same amplitude (Fig.2).It is thought that the mechanisms of these two perturbations are different: the shear stress perturbation only alters the driving force of the fault,and the normal stress perturbation changes the contact state of asperity on fault.

3.3.Effect of tectonic stress

Besides stress perturbation parameters and directions,previous experiments and numerical simulations show that tectonic stress is also an important factor affecting the effect of stress perturbation (Dieterich et al.,2015;Gischig,2015;Gomberg et al.,1997,1998;Huang et al.,2009,2016,2021),and high tectonic stress faults are more susceptible to stress perturbation(Huang et al.,2009).For example,the dynamic stress triggering studies based on the rate-and state-dependent friction law and the spring-slider model show that the clock advance caused by stress perturbation depends nonlinearly on the time when in the earthquake cycle the perturbation is applied(Gomberg et al.,1997,1998).Moreover,the relationship between clock advance and the start time of perturbation is different in static and dynamic stress perturbation: the clock advance Δtcaused by static stress perturbation applied late in the cycle is smaller than that in the early part.In contrast,the dynamic stress perturbation applied late in the cycle produces a more significant clock advance Δt(Gomberg et al.,1998).The numerical simulation of Perfettini et al.(2003a) shows that when the fault stress is less than 90%,the clock advance caused by static stress perturbation is approximately constant,but the clock advance is approximately equal to the remaining time of the earthquake cycle as the fault stress is more than 90%,showing the instantaneous triggering.Apart from the clock advance,fault tectonic stress can influence fault instability mode and strength.For example,previous periodic stress perturbation experiments on the double-direct samples showed that the faults with higher normal stress are more sensitive to the stress perturbation(Huang et al.,2009,2016),and the mean of stress drop increases linearly with normal stress(Huang et al.,2016).Recently,the numerical simulation of local stress perturbation shows that the perturbation stress applied at higher criticality stressed fault could induce the fault fracture exceeding the perturbation range(Fig.3a),and the magnitude of the inducing earthquake is controlled by fault size.While the same perturbation is applied on low-critically stressed faults,it only induces a self-arrested rupture around the perturbation zone(Fig.3b),and the magnitude of the inducing earthquake is affected by the perturbation zone(Huang et al.,2021).The numerical simulation on 1D fault quasi-dynamic rupture propagation under homogeneous shear stress (Gischig,2015) and 2D fault rupture subject to the rate-and state-dependent friction law under heterogeneous shear stress(Dieterich et al.,2015) get similar results,indicating that fault tectonic stress controls the fault rupture mode and the magnitude of an induced earthquake.

Fig.2.(a) Stress curve and (b) stick-slip stress drop distribution for stress perturbation applied on normal and shear stress at 5 MPa average normal stress(Huang et al.,2016).In shear stress perturbation,the periodic displacement perturbations are applied in the shear direction(see Fig.2a),and the equivalent stress perturbation amplitude is 0.02,0.03,0.06 and 0.1 MPa.In normal stress perturbation,the periodic stress perturbations with amplitude of 0.02,0.033,0.05,0.1,and 0.2 MP are applied in the normal direction of the fault.The period of displacement and stress perturbation is 10 s.

Fig.3.Distribution of perturbation-induced fault slip velocity under different tectonic stresses (ts′) (Huang et al.,2021).(a) When ts′=0.9,the fault slip caused by perturbation directly extends to the fault boundary;(b) when ts′=0.7,the perturbation only induces local fault rupture (black frame),confined by the perturbation boundary,and the overall failure is observed later as the tectonic stress increases.The dotted lines indicate the perturbation boundary.

3.4.Effect of Local and heterogeneous stress perturbation

The stress perturbation caused by static stress variation due to earthquakes and the dynamic stress due to seismic waves and solid tides generally is applied to the whole fault or fault section,and the stress is homogeneous on the fault.However,in industrial water injections such as shale gas hydraulic fracturing,pressure diffusion away from an induced fracture is strongly inhibited by the low permeability of shales.The stress perturbation of water injection on surrounding faults is inherently local and heterogeneous.Previous experiments and numerical simulations have shown that the perturbation stress leading to fault reactivation can significantly exceed that predicted by the Coulomb failure criterion when the stress is localized and heterogeneous (Huang et al.,2021;Ji et al.,2020;Passel‵egue et al.,2018),suggesting that the Coulomb failure criterion may not be applicable for injection-induced earthquakes.For example,laboratory experiments of fault reactivation due to fluid injection(Passel‵egue et al.,2018)suggest that high-rate fluid injection causes significant fluid pressure heterogeneities,and the fluid pressure inducing fault reactivation far exceeds the predicted pressure.In the work of Ji et al.(2020),the peak fluid pressure that induces fault instability also exceeds the predicted pressure and the gap between the two increases with the degree of fluid pressure heterogeneity.Recently,Huang et al.(2021) conducted several numerical simulations on the localized stress perturbation by applying heterogeneous perturbations with different widths (1–15 km) to a critically stressed fault (ts′=0.9).The results show that the perturbation stress(ΔCFSp)exceeds the stress predicted by the Coulomb failure criterion,inversely proportional to the size of the perturbation zone but proportional to stressing rate.The size and spatial distribution of perturbation strongly impacted rupture timing,location,size,etc.(Fig.4).

3.5.Stress perturbation and acoustic emission

Fig.4.(a) Diagram of stress distribution,(b) simulated hypocenters in fault instability,and (c) the perturbation stress (ΔCFSp and ΔCFSa) required to induce fault reactivation in different widths (1 km–15 km) (Huang et al.,2021).The stress perturbations are applied at ts′=0.9,and the stress rate is 2.7 MPa/day.P1 to P5 represent the stress distribution of different perturbations.ΔCFSa is the average perturbation stress,which is the average of perturbation stress in the entire perturbation zone.

Acoustic emission (AE) is generated by the rapid release of energy from rock rupture and is the most similar signal to earthquakes in the laboratory.It is repeatedly used to study the nucleation of faulting and has shed some light on earthquake precursors.For example,Ma et al.(1996) studied the temporal-spatial distribution of strain,fault displacement,and acoustic emission events in fault deformation and rupture with different geometric structures under biaxial loading,analyzing the characteristics of specific instability events.Ma et al.(2004) discussed the effect of the heterogeneity of rock samples on the temporal-spatial distribution of acoustic emission and found that pre-existing microcracks increased the acoustic emission rate rapidly before fracture nucleation,and theb-value fluctuated on a decreasing background.The macrostructures controlled the AE spatial distribution.The experiments on 5°bend faults also show that acoustic emission events migrated along faults during fault sliding (Guo et al.,2011;Yun et al.,2011),and high-energy acoustic emission events mainly occur in fault instability (Yun et al.,2011).Recently,the fault stick-slip experiments at different loading rates and lateral pressures also show that acoustic emission events increased abruptly at fault instability,accompanied by high-level acoustic emission events (Zhao et al.,2022).

In addition to fault slip nucleation,acoustic emission is also used to investigate the effect of stress perturbation on earthquake activity.For example,in the early study,Byerlee and Lockner (1977) analyzed the effect of fluid injection on rock rupture propagation and found that acoustic emission events migrate as a cluster along the sample,following the migration of the waterfront closely under constant stress.Masuda et al.(1990) performed a similar injection experiment on dry granite.They found that the acoustic emission gradually migrated from the water injection point to the surrounding area and stated the positive feedback between acoustic emission (AE) activity and pore pressure diffusion.The work of Stanchits et al.(2011) also confirmed the close relationship between pore pressure diffusion and acoustic emission activity.Other studies focused on the effect of stress perturbation on acoustic emission during fault friction instability.For example,Ma et al.(2011) found that the stress perturbation could advance the occurrence time of acoustic emission event before the stick-slip,and this effect increased with average normal stress and perturbation amplitude.The shear stress perturbation (increase) had more influence on acoustic emission than normal stress perturbation (decrease),and the faster stress rate was beneficial to the occurrence of acoustic emission events.

4.Main knowledge

4.1. Applicability of stress triggered earthquake models

A large number of observations show that static and dynamic stress changes caused by strong earthquakes,solid tides,etc.,can induce seismicity both on surrounding and remote faults (e.g.,Brodsky,2006;Harris,1998;Hill,2010;Peng et al.,2021;Peng et al.,2018;Stein et al.,1997;Tanaka,2010,2012;Toda et al.,2008).Previous studies mainly investigate the effect of strong earthquakes and other perturbations on triggering earthquakes by calculating Coulomb stress changes.For example,many studies quickly calculated the coseismic Coulomb stress changes caused by the WenchuanMS8.0 earthquake and then analyzed its influence on regional seismicity (e.g.,Parsons et al.,2008;Toda et al.,2008).The stress perturbation experiments also mainly focused on the stress threshold for perturbation triggering fault instability through the Coulomb stress changes (e.g.,Huang et al.,2009;Huang et al.,2016;Lockner and Beeler,1999).However,the experiments and simulations on water injection-induced earthquakes show that the peak pressure for inducing fault activation exceeds the stress predicted by the Coulomb failure criterion (Huang et al.,2021;Ji et al.,2020;Passel‵egue et al.,2018) and the gap between the two increases with the degree of fluid pressure heterogeneity (Ji et al.,2020).These studies suggest that the knowledge from the Coulomb failure criterion may not be suitable for local and heterogeneous stress perturbation.

According to the Coulomb failure criterion,faulting occurs when the shear stress acting on the fault exceeds its frictional strength,implying nearly instantaneous triggering.Numerous observations have demonstrated the instantaneous triggering,meaning that the inducing earthquake mainly occurs when the seismic waves pass (e.g.,Brodsky et al.,2000;Peng et al.,2009;West et al.,2005).However,in many cases,the onset of the triggering earthquake may be several hours to days later than the passage of the seismic waves,or the triggering seismicity may last for a longer time than the seismic waves passing,manifesting as a delayed triggering(Nissen et al.,2016;Peng et al.,2011).The time dependence in dynamic stress triggering is also observed in laboratory experiments and numerical simulations (e.g.,Dong et al.,2022;Johnson et al.,2008;Perfettini et al.,2003b;van der Elst and Savage,2015).These results conflict with the Coulomb failure criterion,indicating that the Coulomb stress changes cannot explain the delayed triggering,either.To explain the mechanisms of delayed triggering,several ideas have been proposed based on observations,laboratory experiments,and numerical simulations (e.g.,Blank et al.,2021;Freed,2005;Johnson et al.,2008;Peng et al.,2011).The rationality and applicability of these hypotheses need support from more earthquake cases and verification from more laboratory experiments and numerical simulations.

In addition,the Coulomb failure criterion cannot tackle the question of frictional stability,that is,whether the slip will be seismic or aseismic upon reactivation.According to the rate-and state-dependent friction law (Dieterich,1979;Marone,1998;Ruina,1983),the fault frictional stability is affected by factors such as rock physical properties,local elastic stiffness,and fault frictional properties (Kolawole et al.,2019;Rice and Ruina,1983),and the velocity weakening is a necessary condition for earthquake nucleation.To understand the mechanisms and conditions of stress-triggering earthquakes,it is necessary to carry out rock experiments and numerical simulations with the rate-and state-dependent friction law to discuss the velocity dependence of different faults and the effects of stress perturbation on the velocity dependence.

4.2.Factors influencing stress-triggering

Many studies have shown that the triggering of stick-slip events is both amplitude-and frequency-dependent (Beeler and Lockner,2003;Boettcher and Marone,2004;Savage and Marone,2007).The experiments and numerical simulations with different conditions have gotten the consensus insights on the correlation between stress perturbation amplitude and fault instability.The correlation between perturbation and fault instability increases with perturbation amplitude (e.g.,Beeler and Lockner,2003;Gao and Shi,2022;Huang et al.,2009;Huang et al.,2016;Lockner and Beeler,1999).Theoretical analyses and experiments on periodic stress perturbation suggest that the fault has a critical period controlling its response mode to the perturbation.When the period of stress perturbation is longer than the critical period,the correlation between fault instability and perturbation depends on the perturbation amplitude and frequency as well as the stress rate.When the period of the perturbation stress is less than the critical period,the fault failure is mainly affected by perturbation amplitude,and a much higher stress perturbation amplitude is required to induce a detectable correlation with the perturbation stress (Beeler and Lockner,2003;Boettcher and Marone,2004;Savage and Marone,2007).These results suggest that high-frequency stress perturbation slightly influences fault instability.However,the research on transient perturbation shows that short-period pulses have a more significant effect on fault instability than long-period pulses with the same stress perturbation amplitude,suggesting the effect of high-frequency transient stress on earthquakes may be underestimated(van der Elst and Savage,2015).In addition,some experimental results show that stick-slip is mainly affected by perturbation amplitude and has little to do with the perturbation period(Huang et al.,2009;Lockner and Beeler,1999).Therefore,the relationship between perturbation period and fault instability,especially the effect of the high-frequency perturbation on fault instability,still needs further studies on experiments and numerical simulations.

Earthquakes may alter surrounding faults' normal and shear stress,triggering seismicity.Some studies report that normal and shear stress perturbation has the same effect in triggering fault instability(Perfettini et al.,2003a,2003b),so calculating the Coulomb stress change on the fault has become an important means to evaluate the fault instability.However,other experiments show that the mechanism of normal and shear stress perturbation differs significantly,and normal stress perturbations play a more significant effect on fault instability than shear stress perturbation (Huang et al.,2016).When the two perturbations trigger faulting,faults show significant differences in slip curves (Cui et al.,2006)and acoustic emission(Ma et al.,2011).Therefore,it is necessary to analyze the effects of perturbation magnitude and to distinguish whether the shear or normal stress changes are dominant when analyzing the effect of stress perturbations on fault slip by Coulomb stress changes.

Apart from perturbation parameters such as amplitude and period,tectonic stress can also influence the effect of stress perturbation on fault instability,and high-stress faults are more susceptible to stress perturbation (Huang et al.,2009).Previous experiments and numerical simulations show that tectonic stress can affect clock advance Δt(Gomberg et al.,1998;Perfettini et al.,2003a),fault rupture mode and strength(Dieterich et al.,2015;Gischig,2015;Huang et al.,2016,2021),and high-stress faults are more prone to large earthquakes.Therefore,when investigating the effect of stress perturbation on fault slip,it is necessary to analyze the influence of perturbation parameters such as amplitude and period on stress perturbation,consider the role of tectonic stress,and pay more attention to the high tectonic stress faults.

4.3.Significance of acoustic emission to earthquake prediction

The spatial-temporal variation of the minor earthquake is of great value to earthquake prediction.Laboratory studies show that acoustic emission (AE) can reflect the stress of faults and shows the precursor to fault instability,such as increasing the occurring rate (e.g.,Guo et al.,2011;Yun et al.,2011;Zhao et al.,2022).Therefore,investigating the acoustic emission during fault friction instability can shed light on earthquake prediction from Coulomb stress changes and earthquake activity.

Experiments and numerical simulations on stress perturbation show that acoustic emission can also reflect the process of stress perturbation and its effect on fracture propagation and fault nucleation.For example,the fluid injection experiments show that acoustic emission activity can reflect fluid diffusion and fracture propagation (Byerlee and Lockner,1977;Masuda et al.,1990;Stanchits et al.,2011).The stress perturbation experiments show that perturbation direction(normal and shear stress),stressing rate,and tectonic stress can influence the temporal distribution of acoustic emissions (Ma et al.,2011).Limited by the experimental technology and equipment,the relationship between stress perturbation and acoustic emission spatial distribution has not been involved in the study.The temporal-spatial variation of acoustic emission,especially the change close to the instability,is essential for identifying meta-instability.Measuring and accurately locating the acoustic emission events during fault stick-slip and then identifying the precursor of fault instability from the temporal-spatial changes and statistical characteristics of acoustic emission will provide more references for the study of earthquake mechanisms and prediction.

5.Perspective

5.1.Study on triggering mechanism

Stress perturbation caused by earthquakes and other sources triggering subsequent earthquakes has been demonstrated by many observations and supported by several experiments and numerical simulations.Existing studies have extensively discussed the conditions and influence factors of stress triggering,getting significant results and promoting the understanding of stress-triggering mechanisms.However,stress triggering,exceptionally dynamic triggering,is a complex mechanical process.There are still some debates about the effect of high-frequency transient stress on triggering earthquakes,which needs to be deeply and extensively studied from multiple perspectives and aspects,such as field observation,rock friction experiments,and numerical simulation.

5.2.Study on the effect of heterogeneous stress perturbation

In recent years,with the rapid development of unconventional energy exploitation,water injection-induced earthquakes have increased rapidly worldwide,becoming the focus of seismic study.Affected by the formation permeability and others,the stress variation caused by water injection is inherently local and heterogeneous.Recent experiments and numerical simulations suggest that the knowledge from homogeneous stress perturbation,such as the Coulomb failure criterion,may not be suitable for injection-induced earthquakes.To understand the mechanisms of injection-induced earthquakes and reduce/avoid earthquake disasters,it is necessary to strengthen the study on the effect of local and heterogeneous stress perturbation.

5.3.Study on acoustic emission of stress perturbation

The acoustic emission during fault friction can reflect the stress and nucleation of the fault,providing a reference for predicting strong earthquakes by earthquake activity.Studying the temporal-spatial evolution of acoustic emission caused by stress perturbation helps understand the mechanical process of stress perturbation,identify the metainstability of fault slip,and capture precursors of fault failure,providing a reference for earthquake prediction.Past laboratory studies of stress perturbation have focused on the temporal change of the acoustic emission (e.g.,Ma et al.,2011),but the relationship between perturbation and spatial distribution of acoustic emission is still limited.To understand fault nucleation caused by stress perturbations and capture the precursor of fault instability,it is necessary to study the effect of stress perturbation on acoustic emission deeply.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Author agreement and acknowledgment

The authors thank Deputy Editor-in-Chief Baoshan Wang and two anonymous reviewers for their constructive and valuable comments.and the corresponding author would like to declare on behalf of my coauthors that none of the material in the paper has been published or is under consideration for publication elsewhere.All the authors listed have approved the manuscript that is enclosed.This work is supported by the National Natural Science Foundation of China(U1839211)and the Spark Program of Earthquake Science and Technology (XH20044),and the State Key Laboratory of Earthquake Dynamics(No.LED2018B06).